From a7f39918ee500bbf3b3fbd0d0c7808b2053b7bdf Mon Sep 17 00:00:00 2001 From: Richard Kreckel Date: Thu, 4 May 2000 19:32:30 +0000 Subject: [PATCH] - Added lots of @cindex to produce an index. - Minor updates / fixed typos. --- doc/cln.dvi | Bin 157376 -> 257596 bytes doc/cln.html | 420 +++- doc/cln.info | 489 +++- doc/cln.ps | 5519 ++++++++++++++++++++++++++++------------------ doc/cln.tex | 358 ++- doc/cln.texi | 358 ++- doc/cln_1.html | 13 +- doc/cln_10.html | 18 +- doc/cln_11.html | 20 +- doc/cln_12.html | 17 +- doc/cln_13.html | 2 +- doc/cln_2.html | 16 +- doc/cln_3.html | 39 +- doc/cln_4.html | 211 +- doc/cln_5.html | 14 +- doc/cln_6.html | 2 +- doc/cln_7.html | 34 +- doc/cln_8.html | 18 +- doc/cln_9.html | 38 +- doc/cln_toc.html | 4 +- 20 files changed, 5252 insertions(+), 2338 deletions(-) diff --git a/doc/cln.dvi b/doc/cln.dvi index a1e064dcae4275401a1a67e6e5617eaffe63bd2c..086cf54ac849ccff0928188b6bc52f3035b1a310 100644 GIT binary patch literal 257596 zcmeEvdw|?kb^nr?O@ahO1ge5S5`^7k-+r3^*r*E{qp1}UwJK`WYEk>6t&d8rI-{|bs@STawe?ZvB z`~C4t>#{rZ{oZ@-x#xY(J@;1&PI>*^pFRKb7W|9+@Vtw5CvUtcS1OK`iWl|u^z?M~ ztnTVrwZ5;nukTJgaS9(h-oK~uZ*uyHmX@h|7regzTa))bwfwLCW%9{a{Pm;T?`=8z zQ#ZFP!3YcEnW6FCzQ@`Dgx$BXMO6&0O75P#& z=Ufv@4W*M)(~BQ@(;IA;LpqC=NkFBR(`JQaAdQlreECBGI?ET zD4i;dCKHtf*`E&m8Q^gINa!dp_`v!n`@j3+4dedVi^bla`i4=#(@WwQV5}>i%NDwF z`H@XdEtX9roI3h!Oq$IVQ}HAb$ARU|vA1li zyKp%AfD_9nF;udU%pZ6Qz)U!~tP@M8&8%Hh_l3ZZ?D_8y_%d}E{7ep$B~nW;AXmbb zH!h)+O(gRM{A?ms(kUWbCMRnHqp>k;4uwST z`f{{Z8H&fMToj4_Zvao*YD5b`omxDWRYG=31uUD!hyUFa;moKg zRDOU!xqW0LbpX2#%-qCLYL^nTR(fTUu`Iq622K?nimW8o3w?&EVnaju zu~Cw{IP;Ry7{AUOTLKz0>m+FX#wK7W72tY{b~0OZStE5nN)Aj57t*s`FihqXZ=+9 z%=DLQO<8ekZiG4`G;>-%b?Q*foHUo`PssZkCTkovL~^^!PoJBl{)i``KZSxZL-eEH z@~tg7E@YS$y%Xm{4oAeGm;$Z@a$IQxIZE>*K<=^h zND_VnHV~r~CPZPz#sxXQPt4~^4p|(`EpjBx0UJq%8}Tzhg=asvyOxt>) zIL?4W=>Yt|%oQ2u?Rh4d$zdtFIoN>W`Y>Pa%D~1XkST#t#V!ri0=0`Q9P>afr81dh zf}*v9rin9}jEzzKax>$!OZ+Cr9Q~Kpz4J9O`oboLUsYcNkUIS!RicIS3CGv_3>zT{e?k5=)AV1?ZvKt|;XHsK@ z8nnfc;d(5esX-hP9D>gsLRkp{)_5d+!jr=|(8)F5oR z_&wm5f^%&Q3;-1#b21Pb^fSgFD@9acWn;fMIGTz(qkJ6uP8Lor0k;tv;BGipfHf;B z6~+g4HK4a)j6-X~sE4(k?KI#e;AIj%>cPNl;DC{wi1G$-w+Nf(D+}*!RsV^*)-YWg z_>&?4llIbOhPMpO;X*-?;V=Poz^Ac?Gj zvJn@DB4?dyaIxSgKc_uSBAUpI73q^=e<&%sRQh>!8f!JKolsaF6Gt&5Qv3js7K5`7szOgu9 z6wkm|M}~t;E0zuH6Y2|+5cJ1TDG#kpH=FyNfqtMDWeKTaLxo?D3|of;?iQ$let*Fk z&CzJz53(krROV6;Rh1@zb_fG;_J9jmqMkDrnNysn<@C=rk(7PL83U!J)q}nJl-R=` z><`PxANQ%lv^6_i>x_ZltpZ~V&@{k6%lt3cIWC0Vugx@}Sy?BPF zFaIaMXZZN_cn5DSO0*3;x%<0!d~|n_}Q zrb>cC0OOIO&_20twSOM)HVLnFWce_ZPYyEdpl7ZPCu2p(Uscd_6xNp)?|ZrNW;@EQ z|FjdtXq{ZwG?Xt8vcfXpbS5|)RFC^LsTH!hMVmSPoe@Hz|nIIKBsoz!U>1&}1?!B`d>L&9pMw8@f6KAL`fusu|dP?T>YGDS2E$v^Onl7z)aCK54tmzqigO02CuoU^M)J2WV6o7 zXX|ufee;$VmOo3;!&spDp)qW)h!;rLVy>D?tBFNvG*;|56#0OtbdU%T`WLsv^PqcB zJ;gS+Hu4v6l_W|}EOu0NpDCF*bFG6ufIKXUIGf1jFD=MdPEC+bnnrA*(N}oUCmY zIj%7=8afkV*#dH7qmi;naGo9|%X%7tTLGgJiWqMavni_93$I)`Sw8=VH`FgKw|?f& zySg#haEcb+)Zrz!!wbb%@(l?K2$R1ncv1*0K8h$jNUSy}0bL+iDaC4v4(FyiAFN%< z$;sm)1mz=86?BRvUSnO}R3EX=1M*}H5%OVg40Q!b?4)Z-mS}I0%U6gIQh}%?A;e*q z05oSmBtdTx2B{j*t}Ml%sy8g4!u3JrAs-1p6nycWzg<_~h6N47>8%yI4z&)x3}862 zNN;+5%Mv?NOTzk9VF)a143>y&U{R70t6*G|;OYT=G~PsN!}bHV50dZl`$aOr3n`zv z;)h&y>p3^ejI{7&3!EA3TziglNrWACj10P zGV&3e)7d$iOplSAc3~l$R^PNUn9!WK`C>dmJr!dqss+LIP4&U;yLNBhy0vR0nT4c} zgY1^jaX&I?9P&!!hJT|^uI^hrHLM}NfpX-AuUVr3X2Ng_;0@UlLJs&im3P6hJ|-Uy zXl?|8pt0a+5}z8u<7xux#5p($!xY=bh@Io%w}uGyi{AJ$WbEn#14zrQ`^uBGUE*!V zk>C*#AizQ4NazJ||HJ2+=eKO#eDyB#&X#m+M4yHgOl!|U%v(t2FQv6mX=sfq{+7 zM&j{h&VWF(aardM{C?w&_;lq;`OWvak2rnO&^L_J`7+pjYr6XU<2jup%bYz}b-8Wi zs2!Rw`5wOUMt2%{+&_~~$Qo#MnzdcM!D)uGxlYdrnq^LXGcafP-Qxq09Ele?`?}Wk zc3sxvhojHw3=cNL;^#4f9~{qWyonU=L(4aY|_M= z$bv9J5pAqQUcU$V^mMx2m@Q5r2H{QvT$MMJPNU+34E|d8pjB?rG6^-gFf(E@BN9qX z#_}+c^Po=|zY^JT1*WCigozuu=>cQw8KV;Gsw2M5IAM)K!-_Vb0a_%80}(U88KOpT zxVStI#2P^R>{kb&-tb{z6tmZ4dT~1%rnIKwO)%~d7&`z#ZeSF%+AIgA43wNvFS9|0 z_}9hfc1|+n)l=LmA&AmA?D?JlC7xT2j$Fa%UF~eUq8#nHp2|~&tH?vt2g^TVR!!9CY0B|J!p+qiwDRgTh0jsEUM;{PlrMa+o_3B+6etR4ClsdPa zJMgHeMKP<9=^@|%&V=d-AjzG+zl5sfO-H6M5JrFWo!KB<+cnN0wa4V+p!2T%U9qn6 z;`hBoK8E8Da?U#tz?3AbyRfF#^$!h}qd$6+=O<%vpdEmQ6$T}XX1f9lhJkeTt}aI` ztAi;7Fw!YjiA=a_Rcy#E#gU{JYXIn8_^fx#yqLfmW`LY*xr<@fj`G5HALWK5k?sL4 za9@CHEK_y}>yT9xV5lWZ=_G>Bg=~Q4@s7>$#RC&cjC3?dj zB$C511VlRyMgQO$da*R@)Kj)$GP-zhU5en5$GiQh53z&AeT&|31FTHEsc_gg)$!^; 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z&ZsiJw`0*wAX4`o#(Ngm5mC3UdIRD(fRN|Q*l(;wFodnU!@MRFqerGW~Y zXF=UF`Fr~QS@8I&Gt!#IuB(wcgd4aXUj6D-MPGz}*4 zWC2*CYl{Z#;2G%7TyN|aTIWG{wI2=>U+3KN&`ut1tFg(-RW$Y+rQb8VnUwna)n zlB>8F#4CFnCXKxZuTa_%5oI59=K1zNi8MvII$@De>qst ze;iC)KL8P;s~@fr#J=N-X{GrSo^;s&-Ne3b)wu}SeZ=X0h@upS!q4m}PD(umIV4Pr z8B?%Sl7>K}Wdl@2-FYadfzw3zD7;3ZQBagwa$SNV3aT{`c?Bpvxho^$$vh z8xTR0Z7V zajIA`a0-% - + CLN, a Class Library for Numbers @@ -243,8 +243,9 @@ The kernel of CLN has been written in assembly language for some CPUs (i386, m68k, sparc, mips, arm).
  • -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. + +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3.
  • It uses Karatsuba multiplication, which is significantly faster @@ -253,9 +254,9 @@ for large numbers than the standard multiplication algorithm. For very large numbers (more than 12000 decimal digits), it uses Schönhage-Strassen -multiplication, which is an asymptotically -optimal multiplication algorithm, for multiplication, division and -radix conversion. + +multiplication, which is an asymptotically optimal multiplication +algorithm, for multiplication, division and radix conversion.

    @@ -292,7 +293,7 @@ This section describes how to install the CLN package on your system. To build CLN, you need a C++ compiler. Actually, you need GNU g++ 2.7.0 or newer. On HPPA, you need GNU g++ 2.8.0 or newer. -I recommend GNU egcs 1.1 or newer. +I recommend GNU g++ 2.95 or newer.

    @@ -315,6 +316,8 @@ of static and global variables, a feature which I could implement for GNU g++ only. +

    +

    2.1.2 Make utility

    @@ -323,6 +326,8 @@ implement for GNU g++ only. To build CLN, you also need to have GNU make installed. +

    +

    2.1.3 Sed utility

    @@ -446,11 +451,6 @@ With full `-O2', g++ miscompiles the division routines --enable-shared to work, you need egcs-1.1.2 or newer. -

    -On MIPS (SGI Irix 6), pass option --without-gmp to configure. gmp does -not work when compiled in `n32' binary format on Irix. - -

    By default, only a static library is built. You can build CLN as a shared library too, by calling configure with the option `--enable-shared'. @@ -467,6 +467,9 @@ library.

    2.3 Installing the library

    +

    + +

    As with any autoconfiguring GNU software, installation is as easy as this: @@ -552,6 +555,8 @@ Rational number Floating-point number

    + + The base class cl_number is an abstract base class. It is not useful to declare a variable of this type except if you want to completely disable compile-time type checking and use run-time type @@ -559,17 +564,24 @@ checking instead.

    + + + The class cl_N comprises real and complex numbers. There is no special class for complex numbers since complex numbers with imaginary part 0 are automatically converted to real numbers.

    + The class cl_R comprises real numbers of different kinds. It is an abstract class.

    + + + The class cl_RA comprises exact real numbers: rational numbers, including integers. There is no special class for non-integral rational numbers since rational numbers with denominator 1 are automatically converted @@ -577,6 +589,7 @@ to integers.

    + The class cl_F implements floating-point approximations to real numbers. It is an abstract class. @@ -584,6 +597,9 @@ It is an abstract class.

    3.1 Exact numbers

    +

    + +

    Some numbers are represented as exact numbers: there is no loss of information @@ -625,6 +641,9 @@ is completely transparent.

    3.2 Floating-point numbers

    +

    + +

    Not all real numbers can be represented exactly. (There is an easy mathematical @@ -635,6 +654,7 @@ CLN implements ordinary floating-point numbers, with mantissa and exponent.

    + The elementary operations (+, -, *, /, ...) only return approximate results. For example, the value of the expression (cl_F) 0.3 + (cl_F) 0.4 prints as `0.70000005', not as @@ -665,6 +685,7 @@ Floating point numbers come in four flavors:

    • + Short floats, type cl_SF. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 17 mantissa bits (including the "hidden" bit). @@ -672,6 +693,7 @@ They don't consume heap allocation.
    • + Single floats, type cl_FF. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 24 mantissa bits (including the "hidden" bit). @@ -680,6 +702,7 @@ This corresponds closely to the C/C++ type `float'.
    • + Double floats, type cl_DF. They have 1 sign bit, 11 exponent bits (including the exponent's sign), and 53 mantissa bits (including the "hidden" bit). @@ -688,6 +711,7 @@ This corresponds closely to the C/C++ type `double'.
    • + Long floats, type cl_LF. They have 1 sign bit, 32 exponent bits (including the exponent's sign), and n mantissa bits (including the "hidden" bit), where n >= 64. @@ -708,6 +732,7 @@ with larger exponent range.

      + As a user of CLN, you can forget about the differences between the four floating-point types and just declare all your floating-point variables as being of type cl_F. This has the advantage that @@ -723,6 +748,9 @@ the floating point contagion rule happened to change in the future.)

      3.3 Complex numbers

      +

      + +

      Complex numbers, as implemented by the class cl_N, have a real @@ -739,6 +767,9 @@ through application of sqrt or transcendental functions.

      3.4 Conversions

      +

      + +

      Conversions from any class to any its superclasses ("base classes" in @@ -796,6 +827,7 @@ Conversions from `const char *' are provided for the classes cl_R, cl_N. The easiest way to specify a value which is outside of the range of the C++ built-in types is therefore to specify it as a string, like this: + 

          cl_I order_of_rubiks_cube_group = "43252003274489856000";
@@ -815,12 +847,16 @@ the functions
 
 
      int cl_I_to_int (const cl_I& x)
 
      
+
 
      unsigned int cl_I_to_uint (const cl_I& x)
 
      
+
 
      long cl_I_to_long (const cl_I& x)
 
      
+
 
      unsigned long cl_I_to_ulong (const cl_I& x)
 
      
+
 Returns x as element of the C type ctype. If x is not
 representable in the range of ctype, a runtime error occurs.
 
@@ -837,8 +873,10 @@ the functions
 
 
      float cl_float_approx (const type& x)
 
      
+
 
      double cl_double_approx (const type& x)
 
      
+
 Returns an approximation of x of C type ctype.
 If abs(x) is too close to 0 (underflow), 0 is returned.
 If abs(x) is too large (overflow), an IEEE infinity is returned.
@@ -849,8 +887,10 @@ Conversions from any class to any of its subclasses ("derived classes" in
 C++ terminology) are not provided. Instead, you can assert and check
 that a value belongs to a certain subclass, and return it as element of that
 class, using the `As' and `The' macros.
+
 As(type)(value) checks that value belongs to
 type and returns it as such.
+
 The(type)(value) assumes that value belongs to
 type and returns it as such. It is your responsibility to ensure
 that this assumption is valid.
@@ -967,10 +1007,12 @@ defines the following operations:
 
 
      type operator + (const type&, const type&)
 
      
+
 Addition.
 
 
      type operator - (const type&, const type&)
 
      
+
 Subtraction.
 
 
      type operator - (const type&)
@@ -979,18 +1021,22 @@ Returns the negative of the argument.
 
 
      type plus1 (const type& x)
 
      
+
 Returns x + 1.
 
 
      type minus1 (const type& x)
 
      
+
 Returns x - 1.
 
 
      type operator * (const type&, const type&)
 
      
+
 Multiplication.
 
 
      type square (const type& x)
 
      
+
 Returns x * x.
 
 
@@ -1004,10 +1050,12 @@ defines the following operations:
 
 
      type operator / (const type&, const type&)
 
      
+
 Division.
 
 
      type recip (const type&)
 
      
+
 Returns the reciprocal of the argument.
 
 
@@ -1023,6 +1071,7 @@ Instead, cl_I defines an "exact quotient" function:
 
 
      cl_I exquo (const cl_I& x, const cl_I& y)
 
      
+
 Checks that y divides x, and returns the quotient x/y.
 
 
@@ -1034,12 +1083,14 @@ The following exponentiation functions are defined:
 
 
      cl_I expt_pos (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_RA expt_pos (const cl_RA& x, const cl_I& y)
 
      
 y must be > 0. Returns x^y.
 
 
      cl_RA expt (const cl_RA& x, const cl_I& y)
 
      
+
 
      cl_R expt (const cl_R& x, const cl_I& y)
 
      
 
      cl_N expt (const cl_N& x, const cl_I& y)
@@ -1057,6 +1108,7 @@ defines the following operation:
 
 
      type abs (const type& x)
 
      
+
 Returns the absolute value of x.
 This is x if x >= 0, and -x if x <= 0.
 
@@ -1082,6 +1134,7 @@ defines the following operation:
 
 
      type signum (const type& x)
 
      
+
 Returns the sign of x, in the same number format as x.
 This is defined as x / abs(x) if x is non-zero, and
 x if x is zero. If x is real, the value is either
@@ -1100,10 +1153,12 @@ Each of the classes cl_RA, cl_I defines the following
 
 
      cl_I numerator (const type& x)
 
      
+
 Returns the numerator of x.
 
 
      cl_I denominator (const type& x)
 
      
+
 Returns the denominator of x.
 
 
@@ -1124,6 +1179,7 @@ The class cl_N defines the following operation:
 
 
      cl_N complex (const cl_R& a, const cl_R& b)
 
      
+
 Returns the complex number a+bi, that is, the complex number with
 real part a and imaginary part b.
 
@@ -1136,14 +1192,17 @@ Each of the classes cl_N, cl_R defines the following o
 
 
      cl_R realpart (const type& x)
 
      
+
 Returns the real part of x.
 
 
      cl_R imagpart (const type& x)
 
      
+
 Returns the imaginary part of x.
 
 
      type conjugate (const type& x)
 
      
+
 Returns the complex conjugate of x.
 
 
@@ -1164,6 +1223,9 @@ We have the relations
 
 
 
      4.5 Comparisons
      
+
      
+
+
 
 
      
 Each of the classes cl_N, cl_R, cl_RA, cl_I,
@@ -1175,18 +1237,22 @@ defines the following operations:
 
 
      bool operator == (const type&, const type&)
 
      
+
 
      bool operator != (const type&, const type&)
 
      
+
 Comparison, as in C and C++.
 
 
      uint32 cl_equal_hashcode (const type&)
 
      
+
 Returns a 32-bit hash code that is the same for any two numbers which are
 the same according to ==. This hash code depends on the number's value,
 not its type or precision.
 
 
      cl_boolean zerop (const type& x)
 
      
+
 Compare against zero: x == 0
 
 
@@ -1200,33 +1266,42 @@ defines the following operations:
 
 
      cl_signean cl_compare (const type& x, const type& y)
 
      
+
 Compares x and y. Returns +1 if x>y,
 -1 if x<y, 0 if x=y.
 
 
      bool operator <= (const type&, const type&)
 
      
+
 
      bool operator < (const type&, const type&)
 
      
+
 
      bool operator >= (const type&, const type&)
 
      
+
 
      bool operator > (const type&, const type&)
 
      
+
 Comparison, as in C and C++.
 
 
      cl_boolean minusp (const type& x)
 
      
+
 Compare against zero: x < 0
 
 
      cl_boolean plusp (const type& x)
 
      
+
 Compare against zero: x > 0
 
 
      type max (const type& x, const type& y)
 
      
+
 Return the maximum of x and y.
 
 
      type min (const type& x, const type& y)
 
      
+
 Return the minimum of x and y.
 
 
@@ -1242,6 +1317,9 @@ there is no floating point number whose value is exactly 1/3.
 
 
 
      4.6 Rounding functions
      
+
      
+
+
 
 
      
 When a real number is to be converted to an integer, there is no "best"
@@ -1311,15 +1389,19 @@ defines the following operations:
 
 
      cl_I floor1 (const type& x)
 
      
+
 Returns floor(x).
 
      cl_I ceiling1 (const type& x)
 
      
+
 Returns ceiling(x).
 
      cl_I truncate1 (const type& x)
 
      
+
 Returns truncate(x).
 
      cl_I round1 (const type& x)
 
      
+
 Returns round(x).
 
 
@@ -1408,12 +1490,16 @@ defines the following operations:
 
      
 
      type_div_t floor2 (const type& x, const type& y)
 
      
+
 
      type_div_t ceiling2 (const type& x, const type& y)
 
      
+
 
      type_div_t truncate2 (const type& x, const type& y)
 
      
+
 
      type_div_t round2 (const type& x, const type& y)
 
      
+
 
 
 
      
@@ -1432,12 +1518,16 @@ defines the following operations:
 
 
      type ffloor (const type& x)
 
      
+
 
      type fceiling (const type& x)
 
      
+
 
      type ftruncate (const type& x)
 
      
+
 
      type fround (const type& x)
 
      
+
 
 
 
      
@@ -1477,12 +1567,16 @@ defines the following operations:
 
      
 
      type_fdiv_t ffloor2 (const type& x)
 
      
+
 
      type_fdiv_t fceiling2 (const type& x)
 
      
+
 
      type_fdiv_t ftruncate2 (const type& x)
 
      
+
 
      type_fdiv_t fround2 (const type& x)
 
      
+
 
 
      
 and similarly for class cl_R, but with quotient type cl_F.
@@ -1537,8 +1631,10 @@ The classes cl_R, cl_I define the following operations
 
 
      type mod (const type& x, const type& y)
 
      
+
 
      type rem (const type& x, const type& y)
 
      
+
 
 
 
@@ -1555,6 +1651,7 @@ defines the following operation:
 
 
      type sqrt (const type& x)
 
      
+
 x must be >= 0. This function returns the square root of x,
 normalized to be >= 0. If x is the square of a rational number,
 sqrt(x) will be a rational number, else it will return a
@@ -1569,6 +1666,7 @@ The classes cl_RA, cl_I define the following operation
 
 
      cl_boolean sqrtp (const type& x, type* root)
 
      
+
 This tests whether x is a perfect square. If so, it returns true
 and the exact square root in *root, else it returns false.
 
@@ -1581,6 +1679,7 @@ Furthermore, for integers, similarly:
 
 
      cl_boolean isqrt (const type& x, type* root)
 
      
+
 x should be >= 0. This function sets *root to
 floor(sqrt(x)) and returns the same value as sqrtp:
 the boolean value (expt(*root,2) == x).
@@ -1595,6 +1694,7 @@ define the following operation:
 
 
      cl_boolean rootp (const type& x, const cl_I& n, type* root)
 
      
+
 x must be >= 0. n must be > 0.
 This tests whether x is an nth power of a rational number.
 If so, it returns true and the exact root in *root, else it returns
@@ -1610,6 +1710,7 @@ for class cl_N:
 
 
      cl_N sqrt (const cl_N& z)
 
      
+
 Returns the square root of z, as defined by the formula
 sqrt(z) = exp(log(z)/2). Conversion to a floating-point type
 or to a complex number are done if necessary. The range of the result is the
@@ -1622,6 +1723,9 @@ The result is an exact number only if z is an exact number.
 
 
 
      4.8 Transcendental functions
      
+
      
+
+
 
 
      
 The transcendental functions return an exact result if the argument
@@ -1638,6 +1742,7 @@ For example, cos(0) = 1 returns the rational number 1.
 
 
      cl_R exp (const cl_R& x)
 
      
+
 
      cl_N exp (const cl_N& x)
 
      
 Returns the exponential function of x. This is e^x where
@@ -1646,10 +1751,12 @@ is the entire complex plane excluding 0.
 
 
      cl_R ln (const cl_R& x)
 
      
+
 x must be > 0. Returns the (natural) logarithm of x.
 
 
      cl_N log (const cl_N& x)
 
      
+
 Returns the (natural) logarithm of x. If x is real and positive,
 this is ln(x). In general, log(x) = log(abs(x)) + i*phase(x).
 The range of the result is the strip in the complex plane
@@ -1657,6 +1764,7 @@ The range of the result is the strip in the complex plane
 
 
      cl_R phase (const cl_N& x)
 
      
+
 Returns the angle part of x in its polar representation as a
 complex number. That is, phase(x) = atan(realpart(x),imagpart(x)).
 This is also the imaginary part of log(x).
@@ -1678,6 +1786,7 @@ Returns the logarithm of a with respect to base b.
 
 
      cl_N expt (const cl_N& x, const cl_N& y)
 
      
+
 Exponentiation: Returns x^y = exp(y*log(x)).
 
 
@@ -1689,6 +1798,7 @@ The constant e = exp(1) = 2.71828... is returned by the following functions:
 
 
      cl_F cl_exp1 (cl_float_format_t f)
 
      
+
 Returns e as a float of format f.
 
 
      cl_F cl_exp1 (const cl_F& y)
@@ -1708,6 +1818,7 @@ Returns e as a float of format cl_default_float_format.
 
 
      cl_R sin (const cl_R& x)
 
      
+
 Returns sin(x). The range of the result is the interval
 -1 <= sin(x) <= 1.
 
@@ -1717,6 +1828,7 @@ Returns sin(z). The range of the result is the entire complex plane
 
 
      cl_R cos (const cl_R& x)
 
      
+
 Returns cos(x). The range of the result is the interval
 -1 <= cos(x) <= 1.
 
@@ -1726,25 +1838,31 @@ Returns cos(z). The range of the result is the entire complex plane
 
 
      struct cl_cos_sin_t { cl_R cos; cl_R sin; };
 
      
+
 
      cl_cos_sin_t cl_cos_sin (const cl_R& x)
 
      
 Returns both sin(x) and cos(x). This is more efficient than
+
 computing them separately. The relation cos^2 + sin^2 = 1 will
 hold only approximately.
 
 
      cl_R tan (const cl_R& x)
 
      
+
 
      cl_N tan (const cl_N& x)
 
      
 Returns tan(x) = sin(x)/cos(x).
 
 
      cl_N cis (const cl_R& x)
 
      
+
 
      cl_N cis (const cl_N& x)
 
      
 Returns exp(i*x). The name `cis' means "cos + i sin", because
 e^(i*x) = cos(x) + i*sin(x).
 
+
+
 
      cl_N asin (const cl_N& z)
 
      
 Returns arcsin(z). This is defined as
@@ -1757,6 +1875,7 @@ with realpart = pi/2 and imagpart > 0.
 
 
      cl_N acos (const cl_N& z)
 
      
+
 Returns arccos(z). This is defined as
 arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i
 and satisfies arccos(-z) = pi - arccos(z).
@@ -1765,6 +1884,8 @@ The range of the result is the strip in the complex domain
 with realpart = 0 and imagpart < 0 and the numbers
 with realpart = pi and imagpart > 0.
 
+
+
 
      cl_R atan (const cl_R& x, const cl_R& y)
 
      
 Returns the angle of the polar representation of the complex number
@@ -1793,13 +1914,16 @@ with realpart = pi/2 and imagpart <= 0.
 
 
 
      
-The constant pi = 3.14... is returned by the following functions:
+
+
+Archimedes' constant pi = 3.14... is returned by the following functions:
 
 
 
      
 
 
      cl_F cl_pi (cl_float_format_t f)
 
      
+
 Returns pi as a float of format f.
 
 
      cl_F cl_pi (const cl_F& y)
@@ -1819,6 +1943,7 @@ Returns pi as a float of format cl_default_float_format.
 
 
      cl_R sinh (const cl_R& x)
 
      
+
 Returns sinh(x).
 
 
      cl_N sinh (const cl_N& z)
@@ -1827,6 +1952,7 @@ Returns sinh(z). The range of the result is the entire complex plan
 
 
      cl_R cosh (const cl_R& x)
 
      
+
 Returns cosh(x). The range of the result is the interval
 cosh(x) >= 1.
 
@@ -1836,20 +1962,24 @@ Returns cosh(z). The range of the result is the entire complex plan
 
 
      struct cl_cosh_sinh_t { cl_R cosh; cl_R sinh; };
 
      
+
 
      cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x)
 
      
+
 Returns both sinh(x) and cosh(x). This is more efficient than
 computing them separately. The relation cosh^2 - sinh^2 = 1 will
 hold only approximately.
 
 
      cl_R tanh (const cl_R& x)
 
      
+
 
      cl_N tanh (const cl_N& x)
 
      
 Returns tanh(x) = sinh(x)/cosh(x).
 
 
      cl_N asinh (const cl_N& z)
 
      
+
 Returns arsinh(z). This is defined as
 arsinh(z) = log(z+sqrt(1+z^2)) and satisfies
 arsinh(-z) = -arsinh(z).
@@ -1860,6 +1990,7 @@ with imagpart = pi/2 and realpart < 0.
 
 
      cl_N acosh (const cl_N& z)
 
      
+
 Returns arcosh(z). This is defined as
 arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2)).
 The range of the result is the half-strip in the complex domain
@@ -1868,6 +1999,7 @@ excluding the numbers with realpart = 0 and -pi < imagpar
 
 
      cl_N atanh (const cl_N& z)
 
      
+
 Returns artanh(z). This is defined as
 artanh(z) = (log(1+z)-log(1-z)) / 2 and satisfies
 artanh(-z) = -artanh(z). The range of the result is
@@ -1880,6 +2012,9 @@ with imagpart = pi/2 and realpart >= 0.
 
 
 
      4.8.4 Euler gamma
      
+
      
+
+
 
 
      
 Euler's constant C = 0.577... is returned by the following functions:
@@ -1889,6 +2024,7 @@ Euler's constant C = 0.577... is returned by the following functions:
 
 
      cl_F cl_eulerconst (cl_float_format_t f)
 
      
+
 Returns Euler's constant as a float of format f.
 
 
      cl_F cl_eulerconst (const cl_F& y)
@@ -1902,12 +2038,14 @@ Returns Euler's constant as a float of format cl_default_float_format
 Catalan's constant G = 0.915... is returned by the following functions:
+
 
 
 
      
 
 
      cl_F cl_catalanconst (cl_float_format_t f)
 
      
+
 Returns Catalan's constant as a float of format f.
 
 
      cl_F cl_catalanconst (const cl_F& y)
@@ -1922,6 +2060,9 @@ Returns Catalan's constant as a float of format cl_default_float_format4.8.5 Riemann zeta
+
      
+
+
 
 
      
 Riemann's zeta function at an integral point s>1 is returned by the
@@ -1932,6 +2073,7 @@ following functions:
 
 
      cl_F cl_zeta (int s, cl_float_format_t f)
 
      
+
 Returns Riemann's zeta function at s as a float of format f.
 
 
      cl_F cl_zeta (int s, const cl_F& y)
@@ -1971,54 +2113,69 @@ on each of the bit positions in parallel.
 
 
      cl_I lognot (const cl_I& x)
 
      
+
 
      cl_I operator ~ (const cl_I& x)
 
      
+
 Logical not, like ~x in C. This is the same as -1-x.
 
 
      cl_I logand (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_I operator & (const cl_I& x, const cl_I& y)
 
      
+
 Logical and, like x & y in C.
 
 
      cl_I logior (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_I operator | (const cl_I& x, const cl_I& y)
 
      
+
 Logical (inclusive) or, like x | y in C.
 
 
      cl_I logxor (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_I operator ^ (const cl_I& x, const cl_I& y)
 
      
+
 Exclusive or, like x ^ y in C.
 
 
      cl_I logeqv (const cl_I& x, const cl_I& y)
 
      
+
 Bitwise equivalence, like ~(x ^ y) in C.
 
 
      cl_I lognand (const cl_I& x, const cl_I& y)
 
      
+
 Bitwise not and, like ~(x & y) in C.
 
 
      cl_I lognor (const cl_I& x, const cl_I& y)
 
      
+
 Bitwise not or, like ~(x | y) in C.
 
 
      cl_I logandc1 (const cl_I& x, const cl_I& y)
 
      
+
 Logical and, complementing the first argument, like ~x & y in C.
 
 
      cl_I logandc2 (const cl_I& x, const cl_I& y)
 
      
+
 Logical and, complementing the second argument, like x & ~y in C.
 
 
      cl_I logorc1 (const cl_I& x, const cl_I& y)
 
      
+
 Logical or, complementing the first argument, like ~x | y in C.
 
 
      cl_I logorc2 (const cl_I& x, const cl_I& y)
 
      
+
 Logical or, complementing the second argument, like x | ~y in C.
 
      
 
@@ -2028,6 +2185,7 @@ These operations are all available though the function
 
 
      cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
 
      
+
 
      
 
      
 where op must have one of the 16 values (each one stands for a function
@@ -2036,6 +2194,21 @@ which combines two bits into one bit): boole_clr, boole_setboole_and, boole_ior, boole_xor, boole_eqv,
 boole_nand, boole_nor, boole_andc1, boole_andc2,
 boole_orc1, boole_orc2.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
 
      
@@ -2046,16 +2219,19 @@ Other functions that view integers as bit strings:
 
 
      cl_boolean logtest (const cl_I& x, const cl_I& y)
 
      
+
 Returns true if some bit is set in both x and y, i.e. if
 logand(x,y) != 0.
 
 
      cl_boolean logbitp (const cl_I& n, const cl_I& x)
 
      
+
 Returns true if the nth bit (from the right) of x is set.
 Bit 0 is the least significant bit.
 
 
      uintL logcount (const cl_I& x)
 
      
+
 Returns the number of one bits in x, if x >= 0, or
 the number of zero bits in x, if x < 0.
 
@@ -2069,6 +2245,7 @@ struct cl_byte { uintL size; uintL position; };

      + represents the bit interval containing the bits position...position+size-1 of an integer. The constructor cl_byte(size,position) constructs a cl_byte. @@ -2078,16 +2255,19 @@ The constructor cl_byte(size,position) constructs a cl_bytecl_I ldb (const cl_I& n, const cl_byte& b)

      + extracts the bits of n described by the bit interval b and returns them as a nonnegative integer with b.size bits.
      cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
      + Returns true if some bit described by the bit interval b is set in n.
      cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
      + Returns n, with the bits described by the bit interval b replaced by newbyte. Only the lowest b.size bits of newbyte are relevant. @@ -2102,11 +2282,13 @@ functions are their counterparts without shifting:
      cl_I mask_field (const cl_I& n, const cl_byte& b)
      + returns an integer with the bits described by the bit interval b copied from the corresponding bits in n, the other bits zero.
      cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
      + returns an integer where the bits described by the bit interval b come from newbyte and the other bits come from n. @@ -2137,33 +2319,39 @@ for common arithmetic operations:
      cl_boolean oddp (const cl_I& x)
      + Returns true if the least significant bit of x is 1. Equivalent to mod(x,2) != 0.
      cl_boolean evenp (const cl_I& x)
      + Returns true if the least significant bit of x is 0. Equivalent to mod(x,2) == 0.
      cl_I operator << (const cl_I& x, const cl_I& n)
      + Shifts x by n bits to the left. n should be >=0. Equivalent to x * expt(2,n).
      cl_I operator >> (const cl_I& x, const cl_I& n)
      + Shifts x by n bits to the right. n should be >=0. Bits shifted out to the right are thrown away. Equivalent to floor(x / expt(2,n)).
      cl_I ash (const cl_I& x, const cl_I& y)
      + Shifts x by y bits to the left (if y>=0) or by -y bits to the right (if y<=0). In other words, this returns floor(x * expt(2,y)).
      uintL integer_length (const cl_I& x)
      + Returns the number of bits (excluding the sign bit) needed to represent x in two's complement notation. This is the smallest n >= 0 such that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that @@ -2171,12 +2359,14 @@ in two's complement notation. This is the smallest n >= 0 such that
      uintL ord2 (const cl_I& x)
      + x must be non-zero. This function returns the number of 0 bits at the right of x in two's complement notation. This is the largest n >= 0 such that 2^n divides x.
      uintL power2p (const cl_I& x)
      + x must be > 0. This function checks whether x is a power of 2. If x = 2^(n-1), it returns n. Else it returns 0. (See also the function logp.) @@ -2190,6 +2380,7 @@ If x = 2^(n-1), it returns n. Else it returns 0.
      uint32 gcd (uint32 a, uint32 b)
      +
      cl_I gcd (const cl_I& a, const cl_I& b)
      This function returns the greatest common divisor of a and b, @@ -2197,6 +2388,7 @@ normalized to be >= 0.
      cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
      + This function ("extended gcd") returns the greatest common divisor g of a and b and at the same time the representation of g as an integral linear combination of a and b: @@ -2208,11 +2400,13 @@ value, in the following sense: If a and b are non-zero
      cl_I lcm (const cl_I& a, const cl_I& b)
      + This function returns the least common multiple of a and b, normalized to be >= 0.
      cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
      +
      cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
      a must be > 0. b must be >0 and != 1. If log(a,b) is @@ -2228,17 +2422,20 @@ it returns false.
      cl_I factorial (uintL n)
      + n must be a small integer >= 0. This function returns the factorial n! = 1*2*...*n.
      cl_I doublefactorial (uintL n)
      + n must be a small integer >= 0. This function returns the doublefactorial n!! = 1*3*...*n or n!! = 2*4*...*n, respectively.
      cl_I binomial (uintL n, uintL k)
      + n and k must be small integers >= 0. This function returns the binomial coefficient (n choose k) = n! / k! (n-k)! @@ -2265,6 +2462,7 @@ defines the following operations.
      type scale_float (const type& x, sintL delta)
      +
      type scale_float (const type& x, const cl_I& delta)
      Returns x*2^delta. This is more efficient than an explicit multiplication @@ -2280,27 +2478,32 @@ representation of floating-point numbers.
      sintL float_exponent (const type& x)
      + Returns the exponent e of x. For x = 0.0, this is 0. For x non-zero, this is the unique integer with 2^(e-1) <= abs(x) < 2^e.
      sintL float_radix (const type& x)
      + Returns the base of the floating-point representation. This is always 2.
      type float_sign (const type& x)
      + Returns the sign s of x as a float. The value is 1 for x >= 0, -1 for x < 0.
      uintL float_digits (const type& x)
      + Returns the number of mantissa bits in the floating-point representation of x, including the hidden bit. The value only depends on the type of x, not on its value.
      uintL float_precision (const type& x)
      + Returns the number of significant mantissa bits in the floating-point representation of x. Since denormalized numbers are not supported, this is the same as float_digits(x) if x is non-zero, and @@ -2309,6 +2512,11 @@ this is the same as float_digits(x) if x is non-zero,

      The complete internal representation of a float is encoded in the type + + + + + cl_decoded_float (or cl_decoded_sfloat, cl_decoded_ffloat, cl_decoded_dfloat, cl_decoded_lfloat, respectively), defined by @@ -2326,6 +2534,7 @@ and returned by the function

      cl_decoded_typefloat decode_float (const type& x)
      + For x non-zero, this returns (-1)^s, e, m with x = (-1)^s * 2^e * m and 0.5 <= m < 1.0. For x = 0, it returns (-1)^s=1, e=0, m=0. @@ -2336,7 +2545,7 @@ it returns (-1)^s=1, e=0, m=0. A complete decoding in terms of integers is provided as type 
      -struct cl_idecoded_float {
+struct cl_idecoded_float {
         cl_I mantissa; cl_I exponent; cl_I sign;
 };
      @@ -2349,6 +2558,7 @@ by the following function:
      cl_idecoded_float integer_decode_float (const type& x)
      + For x non-zero, this returns (-1)^s, e, m with x = (-1)^s * 2^e * m and m an integer with float_digits(x) bits. For x = 0, it returns (-1)^s=1, e=0, m=0. @@ -2364,6 +2574,7 @@ Some other function, implemented only for class cl_F:
      cl_F float_sign (const cl_F& x, const cl_F& y)
      + This returns a floating point number whose precision and absolute value is that of y and whose sign is that of x. If x is zero, it is treated as positive. Same for y. @@ -2372,6 +2583,9 @@ zero, it is treated as positive. Same for y.

      4.11 Conversion functions

      +

      + + @@ -2385,6 +2599,7 @@ The type cl_float_format_t describes a floating-point format.

      cl_float_format_t cl_float_format (uintL n)
      + Returns the smallest float format which guarantees at least n decimal digits in the mantissa (after the decimal point). @@ -2394,6 +2609,7 @@ Returns the floating point format of x.
      cl_float_format_t cl_default_float_format
      + Global variable: the default float format used when converting rational numbers to floats. @@ -2409,6 +2625,7 @@ defines the following operations:
      cl_F cl_float (const type&x, cl_float_format_t f)
      + Returns x as a float of format f.
      cl_F cl_float (const type&x, const cl_F& y)
      @@ -2431,28 +2648,34 @@ Every floating-point format has some characteristic numbers:
      cl_F most_positive_float (cl_float_format_t f)
      + Returns the largest (most positive) floating point number in float format f.
      cl_F most_negative_float (cl_float_format_t f)
      + Returns the smallest (most negative) floating point number in float format f.
      cl_F least_positive_float (cl_float_format_t f)
      + Returns the least positive floating point number (i.e. > 0 but closest to 0) in float format f.
      cl_F least_negative_float (cl_float_format_t f)
      + Returns the least negative floating point number (i.e. < 0 but closest to 0) in float format f.
      cl_F float_epsilon (cl_float_format_t f)
      + Returns the smallest floating point number e > 0 such that 1+e != 1.
      cl_F float_negative_epsilon (cl_float_format_t f)
      + Returns the smallest floating point number e > 0 such that 1-e != 1. @@ -2469,6 +2692,7 @@ defines the following operation:
      cl_RA rational (const type& x)
      + Returns the value of x as an exact number. If x is already an exact number, this is x. If x is a floating-point number, the value is a rational number whose denominator is a power of 2. @@ -2483,6 +2707,7 @@ the function
      cl_RA rationalize (const cl_R& x)
      + If x is a floating-point number, it actually represents an interval of real numbers, and this function returns the rational number with smallest denominator (and smallest numerator, in magnitude) @@ -2524,6 +2749,7 @@ a complicated but deterministic way.

      The global variable + 

       cl_random_state cl_default_random_state
@@ -2540,12 +2766,14 @@ below are called without cl_random_state argument.
 
      
 
      uint32 random32 ()
 
      
+
 Returns a random unsigned 32-bit number. All bits are equally random.
 
 
      cl_I random_I (cl_random_state& randomstate, const cl_I& n)
 
      
 
      cl_I random_I (const cl_I& n)
 
      
+
 n must be an integer > 0. This function returns a random integer x
 in the range 0 <= x < n.
 
@@ -2553,6 +2781,7 @@ in the range 0 <= x < n.
 
      
 
      cl_F random_F (const cl_F& n)
 
      
+
 n must be a float > 0. This function returns a random floating-point
 number of the same format as n in the range 0 <= x < n.
 
@@ -2560,6 +2789,7 @@ number of the same format as n in the range 0 <= x <
 
      
 
      cl_R random_R (const cl_R& n)
 
      
+
 Behaves like random_I if n is an integer and like random_F
 if n is a float.
 
@@ -2567,6 +2797,9 @@ if n is a float.
 
 
 
      4.13 Obfuscating operators
      
+
      
+
+
 
 
      
 The modifying C/C++ operators +=, -=, *=, /=,
@@ -2581,6 +2814,7 @@ to get happy, then add

      + to the beginning of your source files, before the inclusion of any CLN include files. This flag will enable the following operators: @@ -2594,12 +2828,16 @@ For the classes cl_N, cl_R, cl_RA,

      type& operator += (type&, const type&)
      +
      type& operator -= (type&, const type&)
      +
      type& operator *= (type&, const type&)
      +
      type& operator /= (type&, const type&)
      +

      @@ -2616,14 +2854,19 @@ For the class cl_I:

      type& operator &= (type&, const type&)
      +
      type& operator |= (type&, const type&)
      +
      type& operator ^= (type&, const type&)
      +
      type& operator <<= (type&, const type&)
      +
      type& operator >>= (type&, const type&)
      +

      @@ -2635,6 +2878,7 @@ For the classes cl_N, cl_R, cl_RA,

      type& operator ++ (type& x)
      + The prefix operator ++x.
      void operator ++ (type& x, int) @@ -2643,6 +2887,7 @@ The postfix operator x++.
      type& operator -- (type& x)
      + The prefix operator --x.
      void operator -- (type& x, int) @@ -2659,10 +2904,16 @@ efficient.

      5. Input/Output

      +

      + +

      5.1 Internal and printed representation

      +

      + +

      All computations deal with the internal representations of the numbers. @@ -2676,8 +2927,10 @@ Several external representations may denote the same number, for example,

      Converting an internal to an external representation is called "printing", + converting an external to an internal representation is called "reading". -In CLN, is it always true that conversion of an internal to an external + +In CLN, it is always true that conversion of an internal to an external representation and then back to an internal representation will yield the same internal representation. Symbolically: read(print(x)) == x. This is called "print-read consistency". @@ -3058,7 +3311,7 @@ using this variable name. Default is "x".

      The global variable cl_default_print_flags contains the default values, -used by the function fprint, +used by the function fprint. @@ -3163,10 +3416,16 @@ Tests whether the given number is an element of the number ring R.

      7. Modular integers

      +

      + +

      7.1 Modular integer rings

      +

      + +

      CLN implements modular integers, i.e. integers modulo a fixed integer N. @@ -3212,9 +3471,11 @@ Modular integer rings are constructed using the function

      cl_modint_ring cl_find_modint_ring (const cl_I& N)
      + This function returns the modular ring `Z/NZ'. It takes care of finding out about special cases of N, like powers of two and odd numbers for which Montgomery multiplication will be a win, + and precomputes any necessary auxiliary data for computing modulo N. There is a cache table of rings, indexed by N (or, more precisely, by abs(N)). This ensures that the precomputation costs are reduced @@ -3229,8 +3490,10 @@ Modular integer rings can be compared for equality:
      bool operator== (const cl_modint_ring&, const cl_modint_ring&)
      +
      bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
      + These compare two modular integer rings for equality. Two different calls to cl_find_modint_ring with the same argument necessarily return the same ring because it is memoized in the cache table. @@ -3248,22 +3511,27 @@ Given a modular integer ring R, the following members can be used.
      cl_I R->modulus
      + This is the ring's modulus, normalized to be nonnegative: abs(N).
      cl_MI R->zero()
      + This returns 0 mod N.
      cl_MI R->one()
      + This returns 1 mod N.
      cl_MI R->canonhom (const cl_I& x)
      + This returns x mod N.
      cl_I R->retract (const cl_MI& x)
      + This is a partial inverse function to R->canonhom. It returns the standard representative (>=0, <N) of x. @@ -3271,6 +3539,7 @@ standard representative (>=0, <N) of x
      cl_MI R->random()
      + This returns a random integer modulo N. @@ -3282,15 +3551,18 @@ The following operations are defined on modular integers.
      cl_modint_ring x.ring ()
      + Returns the ring to which the modular integer x belongs.
      cl_MI operator+ (const cl_MI&, const cl_MI&)
      + Returns the sum of two modular integers. One of the arguments may also be a plain integer.
      cl_MI operator- (const cl_MI&, const cl_MI&)
      + Returns the difference of two modular integers. One of the arguments may also be a plain integer. @@ -3300,50 +3572,61 @@ Returns the negative of a modular integer.
      cl_MI operator* (const cl_MI&, const cl_MI&)
      + Returns the product of two modular integers. One of the arguments may also be a plain integer.
      cl_MI square (const cl_MI&)
      + Returns the square of a modular integer.
      cl_MI recip (const cl_MI& x)
      + Returns the reciprocal x^-1 of a modular integer x. x must be coprime to the modulus, otherwise an error message is issued.
      cl_MI div (const cl_MI& x, const cl_MI& y)
      + Returns the quotient x*y^-1 of two modular integers x, y. y must be coprime to the modulus, otherwise an error message is issued.
      cl_MI expt_pos (const cl_MI& x, const cl_I& y)
      + y must be > 0. Returns x^y.
      cl_MI expt (const cl_MI& x, const cl_I& y)
      + Returns x^y. If y is negative, x must be coprime to the modulus, else an error message is issued.
      cl_MI operator<< (const cl_MI& x, const cl_I& y)
      + Returns x*2^y.
      cl_MI operator>> (const cl_MI& x, const cl_I& y)
      + Returns x*2^-y. When y is positive, the modulus must be odd, or an error message is issued.
      bool operator== (const cl_MI&, const cl_MI&)
      +
      bool operator!= (const cl_MI&, const cl_MI&)
      + Compares two modular integers, belonging to the same modular integer ring, for equality.
      cl_boolean zerop (const cl_MI& x)
      + Returns true if x is 0 mod N. @@ -3356,8 +3639,10 @@ input/output).
      void fprint (cl_ostream stream, const cl_MI& x)
      +
      cl_ostream operator<< (cl_ostream stream, const cl_MI& x)
      + Prints the modular integer x on the stream. The output may depend on the global printer settings in the variable cl_default_print_flags. @@ -3365,6 +3650,9 @@ on the global printer settings in the variable cl_default_print_flags8. Symbolic data types +

      + +

      CLN implements two symbolic (non-numeric) data types: strings and symbols. @@ -3373,6 +3661,9 @@ CLN implements two symbolic (non-numeric) data types: strings and symbols.

      8.1 Strings

      +

      + +

      The class @@ -3397,6 +3688,7 @@ Strings are constructed through the following constructors:

      cl_string (const char * s)
      + Returns an immutable copy of the (zero-terminated) C string s.
      cl_string (const char * ptr, unsigned long len) @@ -3417,17 +3709,21 @@ Assignment from cl_string and const char *.
      s.length()
      +
      strlen(s)
      + Returns the length of the string s.
      s[i]
      + Returns the ith character of the string s. i must be in the range 0 <= i < s.length().
      bool equal (const cl_string& s1, const cl_string& s2)
      + Compares two strings for equality. One of the arguments may also be a plain const char *. @@ -3435,6 +3731,9 @@ plain const char *.

      8.2 Symbols

      +

      + +

      Symbols are uniquified strings: all symbols with the same name are shared. @@ -3453,6 +3752,7 @@ Symbols are constructed through the following constructor:

      cl_symbol (const cl_string& s)
      + Looks up or creates a new symbol with a given name. @@ -3469,12 +3769,17 @@ Conversion to cl_string: Returns the string which names the symbol
      bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
      + Compares two symbols for equality. This is very fast.

      9. Univariate polynomials

      +

      + + + @@ -3586,6 +3891,7 @@ return the same polynomial ring.

      cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R)
      +
      cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
      cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R) @@ -3620,27 +3926,33 @@ Given a univariate polynomial ring R, the following members can be
      cl_ring R->basering()
      + This returns the base ring, as passed to `cl_find_univpoly_ring'.
      cl_UP R->zero()
      + This returns 0 in R, a polynomial of degree -1.
      cl_UP R->one()
      + This returns 1 in R, a polynomial of degree <= 0.
      cl_UP R->canonhom (const cl_I& x)
      + This returns x in R, a polynomial of degree <= 0.
      cl_UP R->monomial (const cl_ring_element& x, uintL e)
      + This returns a sparse polynomial: x * X^e, where X is the indeterminate.
      cl_UP R->create (sintL degree)
      + Creates a new polynomial with a given degree. The zero polynomial has degree -1. After creating the polynomial, you should put in the coefficients, using the set_coeff member function, and then call the finalize @@ -3655,12 +3967,14 @@ The following are the only destructive operations on univariate polynomials.
      void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
      + This changes the coefficient of X^index in x to be y. After changing a polynomial and before applying any "normal" operation on it, you should call its finalize member function.
      void finalize (cl_UP& x)
      + This function marks the endpoint of destructive modifications of a polynomial. It normalizes the internal representation so that subsequent computations have less overhead. Doing normal computations on unnormalized polynomials may @@ -3675,14 +3989,17 @@ The following operations are defined on univariate polynomials.
      cl_univpoly_ring x.ring ()
      + Returns the ring to which the univariate polynomial x belongs.
      cl_UP operator+ (const cl_UP&, const cl_UP&)
      + Returns the sum of two univariate polynomials.
      cl_UP operator- (const cl_UP&, const cl_UP&)
      + Returns the difference of two univariate polynomials.
      cl_UP operator- (const cl_UP&) @@ -3691,44 +4008,54 @@ Returns the negative of a univariate polynomial.
      cl_UP operator* (const cl_UP&, const cl_UP&)
      + Returns the product of two univariate polynomials. One of the arguments may also be a plain integer or an element of the base ring.
      cl_UP square (const cl_UP&)
      + Returns the square of a univariate polynomial.
      cl_UP expt_pos (const cl_UP& x, const cl_I& y)
      + y must be > 0. Returns x^y.
      bool operator== (const cl_UP&, const cl_UP&)
      +
      bool operator!= (const cl_UP&, const cl_UP&)
      + Compares two univariate polynomials, belonging to the same univariate polynomial ring, for equality.
      cl_boolean zerop (const cl_UP& x)
      + Returns true if x is 0 in R.
      sintL degree (const cl_UP& x)
      + Returns the degree of the polynomial. The zero polynomial has degree -1.
      cl_ring_element coeff (const cl_UP& x, uintL index)
      + Returns the coefficient of X^index in the polynomial x.
      cl_ring_element x (const cl_ring_element& y)
      + Evaluation: If x is a polynomial and y belongs to the base ring, then `x(y)' returns the value of the substitution of y into x.
      cl_UP deriv (const cl_UP& x)
      + Returns the derivative of the polynomial x with respect to the indeterminate X. @@ -3742,8 +4069,10 @@ input/output).
      void fprint (cl_ostream stream, const cl_UP& x)
      +
      cl_ostream operator<< (cl_ostream stream, const cl_UP& x)
      + Prints the univariate polynomial x on the stream. The output may depend on the global printer settings in the variable cl_default_print_flags. @@ -3761,18 +4090,26 @@ The following functions return special polynomials.
      cl_UP_I cl_tschebychev (sintL n)
      + + Returns the n-th Tchebychev polynomial (n >= 0).
      cl_UP_I cl_hermite (sintL n)
      + + Returns the n-th Hermite polynomial (n >= 0).
      cl_UP_RA cl_legendre (sintL n)
      + + Returns the n-th Legendre polynomial (n >= 0).
      cl_UP_I cl_laguerre (sintL n)
      + + Returns the n-th Laguerre polynomial (n >= 0). @@ -3789,6 +4126,9 @@ of these polynomials from their definition can be found in the

      10.1 Why C++ ?

      +

      + +

      Using C++ as an implementation language provides @@ -3802,6 +4142,7 @@ Efficiency: It compiles to machine code.

    • + Portability: It runs on all platforms supporting a C++ compiler. Because of the availability of GNU C++, this includes all currently used 32-bit and 64-bit platforms, independently of the quality of the vendor's C++ compiler. @@ -3811,7 +4152,7 @@ of the availability of GNU C++, this includes all currently used 32-bit and Type safety: The C++ compilers knows about the number types and complains if, for example, you try to assign a float to an integer variable. However, a drawback is that C++ doesn't know about generic types, hence a restriction -like that operation+ (const cl_MI&, const cl_MI&) requires that both +like that operator+ (const cl_MI&, const cl_MI&) requires that both arguments belong to the same modular ring cannot be expressed as a compile-time information. @@ -3847,6 +4188,8 @@ Object sharing: An operation like x+0 returns x withou it.
    • + + Garbage collection: A reference counting mechanism makes sure that any number object's storage is freed immediately when the last reference to the object is gone. @@ -3879,8 +4222,8 @@ The kernel of CLN has been written in assembly language for some CPUs (i386, m68k, sparc, mips, arm).
    • -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3.
    • For large numbers, CLN uses, instead of the standard O(N^2) @@ -3891,6 +4234,7 @@ algorithm. For very large numbers (more than 12000 decimal digits), CLN uses Schönhage-Strassen + multiplication, which is an asymptotically optimal multiplication algorithm.
    • @@ -3902,6 +4246,9 @@ of division and radix conversion.

      10.4 Garbage collection

      +

      + +

      All the number classes are reference count classes: They only contain a pointer @@ -3937,6 +4284,9 @@ environment variables, or directly substitute the appropriate values.

      11.1 Compiler options

      +

      + +

      Until you have installed CLN in a public place, the following options are @@ -3973,6 +4323,10 @@ linking a CLN application it is sufficient to give the flag -lcln.

      11.2 Include files

      +

      + + +

      Here is a summary of the include files and their contents. @@ -4168,6 +4522,7 @@ Includes all of the above.

      A function which computes the nth Fibonacci number can be written as follows. + @@ -4244,9 +4599,17 @@ automatically reclaimed (garbage collected). Only the result survives and gets passed to the caller. +

      +The file fibonacci.cc in the subdirectory examples +contains this implementation together with an even faster algorithm. + +

      11.4 Debugging support

      +

      + +

      When debugging a CLN application with GNU gdb, two facilities are @@ -4279,6 +4642,7 @@ CLN offers a function cl_print, callable from the debugger, for printing number objects. In order to get this function, you have to define the macro `CL_DEBUG' and then include all the header files for which you want cl_print debugging support. For example: + 

       #define CL_DEBUG
@@ -4305,6 +4669,7 @@ only with number objects and similar. Therefore CLN offers a member function
 debug_print() on all CLN types. The same macro `CL_DEBUG'
 is needed for this member function to be implemented. Under gdb,
 you call it like this:
+
 
 
       (gdb) print s
@@ -4325,6 +4690,9 @@ Unfortunately, this feature does not seem to work under all circumstances.
 
 
 
      12. Customizing
      
+
      
+
+
 
 
 
@@ -4343,12 +4711,16 @@ void cl_abort (void);
 
      
 
 
      
+
 This function must not return control to its caller.
 
 
 
 
 
      12.2 Floating-point underflow
      
+
      
+
+
 
 
      
 Floating point underflow denotes the situation when a floating-point number
@@ -4362,9 +4734,8 @@ cl_boolean cl_inhibit_floating_point_underflow
 
 
      
 to cl_true, the error will be inhibited, and a floating-point zero
-will be generated instead.
-The default value of cl_inhibit_floating_point_underflow is
-cl_false.
+will be generated instead.  The default value of 
+cl_inhibit_floating_point_underflow is cl_false.
 
 
 
@@ -4374,6 +4745,7 @@ The default value of cl_inhibit_floating_point_underflow is
 
      
 The output of the function fprint may be customized by changing the
 value of the global variable cl_default_print_flags.
+
 
 
 
@@ -4397,6 +4769,8 @@ void (*cl_free_hook) (void* ptr)      = ...;

      + + The cl_malloc_hook function must not return a NULL pointer. @@ -4416,7 +4790,7 @@ Jump to:

      -This document was generated on 14 January 2000 using +This document was generated on 4 May 2000 using texi2html 1.56k. diff --git a/doc/cln.info b/doc/cln.info index 2b72212..2adcbc2 100644 --- a/doc/cln.info +++ b/doc/cln.info @@ -215,8 +215,8 @@ CLN is speed efficient: * The kernel of CLN has been written in assembly language for some CPUs (`i386', `m68k', `sparc', `mips', `arm'). - * On all CPUs, CLN uses the superefficient low-level routines from - GNU GMP version 2. + * On all CPUs, CLN may be configured to use the superefficient + low-level routines from GNU GMP version 3. * It uses Karatsuba multiplication, which is significantly faster for large numbers than the standard multiplication algorithm. @@ -267,7 +267,7 @@ C++ compiler To build CLN, you need a C++ compiler. Actually, you need GNU `g++ 2.7.0' or newer. On HPPA, you need GNU `g++ 2.8.0' or newer. I -recommend GNU `egcs 1.1' or newer. +recommend GNU `g++ 2.95' or newer. The following C++ features are used: classes, member functions, overloading of functions and operators, constructors and destructors, @@ -375,9 +375,6 @@ add either `-O' or `-O2 -fno-schedule-insns' to the CXXFLAGS. With full `-O2', `g++' miscompiles the division routines. Also, for -enable-shared to work, you need egcs-1.1.2 or newer. -On MIPS (SGI Irix 6), pass option `--without-gmp' to configure. gmp does -not work when compiled in `n32' binary format on Irix. - By default, only a static library is built. You can build CLN as a shared library too, by calling `configure' with the option `--enable-shared'. To get it built as a shared library only, call @@ -1310,7 +1307,8 @@ Trigonometric functions numbers with `realpart = -pi/2' and `imagpart >= 0' and the numbers with `realpart = pi/2' and `imagpart <= 0'. -The constant pi = 3.14... is returned by the following functions: +Archimedes' constant pi = 3.14... is returned by the following +functions: `cl_F cl_pi (cl_float_format_t f)' Returns pi as a float of format `f'. @@ -1943,7 +1941,7 @@ number, for example, "20.0" and "20.000". Converting an internal to an external representation is called "printing", converting an external to an internal representation is -called "reading". In CLN, is it always true that conversion of an +called "reading". In CLN, it is always true that conversion of an internal to an external representation and then back to an internal representation will yield the same internal representation. Symbolically: `read(print(x)) == x'. This is called "print-read @@ -2186,7 +2184,7 @@ The structure type `cl_print_flags' contains the following fields: printed using this variable name. Default is `"x"'. The global variable `cl_default_print_flags' contains the default -values, used by the function `fprint', +values, used by the function `fprint'.  File: cln.info, Node: Rings, Next: Modular integers, Prev: Input/Output, Up: Top @@ -2755,7 +2753,7 @@ Using C++ as an implementation language provides * Type safety: The C++ compilers knows about the number types and complains if, for example, you try to assign a float to an integer variable. However, a drawback is that C++ doesn't know about - generic types, hence a restriction like that `operation+ (const + generic types, hence a restriction like that `operator+ (const cl_MI&, const cl_MI&)' requires that both arguments belong to the same modular ring cannot be expressed as a compile-time information. @@ -2808,8 +2806,8 @@ and algorithms: * The kernel of CLN has been written in assembly language for some CPUs (`i386', `m68k', `sparc', `mips', `arm'). - * On all CPUs, CLN uses the superefficient low-level routines from - GNU GMP version 2. + * On all CPUs, CLN may be configured to use the superefficient + low-level routines from GNU GMP version 3. * For large numbers, CLN uses, instead of the standard `O(N^2)' algorithm, the Karatsuba multiplication, which is an `O(N^1.6)' @@ -3138,6 +3136,9 @@ When the function returns, all the local variables in the function are automatically reclaimed (garbage collected). Only the result survives and gets passed to the caller. +The file `fibonacci.cc' in the subdirectory `examples' contains this +implementation together with an even faster algorithm. +  File: cln.info, Node: Debugging support, Prev: An Example, Up: Using the library @@ -3275,81 +3276,401 @@ Index * Menu: +* abs (): Elementary functions. +* abstract class: Ordinary number types. +* acos (): Trigonometric functions. +* acosh (): Hyperbolic functions. +* advocacy: Why C++ ?. +* Archimedes' constant: Trigonometric functions. +* As() (): Conversions. +* ash (): Logical functions. +* asin: Trigonometric functions. +* asin (): Trigonometric functions. +* asinh (): Hyperbolic functions. +* atan: Trigonometric functions. +* atan (): Trigonometric functions. +* atanh (): Hyperbolic functions. +* basering (): Functions on univariate polynomials. +* binomial (): Combinatorial functions. +* boole (): Logical functions. +* boole_1: Logical functions. +* boole_2: Logical functions. +* boole_and: Logical functions. +* boole_andc1: Logical functions. +* boole_andc2: Logical functions. +* boole_c1: Logical functions. +* boole_c2: Logical functions. +* boole_clr: Logical functions. +* boole_eqv: Logical functions. +* boole_nand: Logical functions. +* boole_nor: Logical functions. +* boole_orc1: Logical functions. +* boole_orc2: Logical functions. +* boole_set: Logical functions. +* boole_xor: Logical functions. +* canonhom () <1>: Functions on univariate polynomials. +* canonhom (): Functions on modular integers. +* Catalan's constant: Euler gamma. +* ceiling1 (): Rounding functions. +* ceiling2 (): Rounding functions. +* cis (): Trigonometric functions. +* cl_abort (): Error handling. +* cl_byte: Logical functions. +* cl_catalanconst (): Euler gamma. +* cl_compare (): Comparisons. +* cl_cos_sin (): Trigonometric functions. +* cl_cos_sin_t: Trigonometric functions. +* cl_cosh_sinh (): Hyperbolic functions. +* cl_cosh_sinh_t: Hyperbolic functions. +* CL_DEBUG: Debugging support. +* cl_decoded_dfloat: Functions on floating-point numbers. +* cl_decoded_ffloat: Functions on floating-point numbers. +* cl_decoded_float: Functions on floating-point numbers. +* cl_decoded_lfloat: Functions on floating-point numbers. +* cl_decoded_sfloat: Functions on floating-point numbers. +* cl_default_float_format: Conversion to floating-point numbers. +* cl_default_print_flags: Customizing I/O. +* cl_default_random_state: Random number generators. +* cl_DF: Floating-point numbers. +* cl_double_approx (): Conversions. +* cl_equal_hashcode (): Comparisons. +* cl_eulerconst (): Euler gamma. +* cl_F <1>: Floating-point numbers. +* cl_F: Ordinary number types. +* cl_FF: Floating-point numbers. +* cl_find_modint_ring (): Modular integer rings. +* cl_find_univpoly_ring (): Univariate polynomial rings. +* cl_float: Conversion to floating-point numbers. +* cl_float_approx (): Conversions. +* cl_float_format (): Conversion to floating-point numbers. +* cl_free_hook (): Customizing the memory allocator. +* cl_hermite (): Special polynomials. +* cl_I_to_int (): Conversions. +* cl_I_to_long (): Conversions. +* cl_I_to_uint (): Conversions. +* cl_I_to_ulong (): Conversions. +* cl_idecoded_float: Functions on floating-point numbers. +* cl_laguerre (): Special polynomials. +* cl_legendre (): Special polynomials. +* cl_LF: Floating-point numbers. +* cl_malloc_hook (): Customizing the memory allocator. +* cl_N: Ordinary number types. +* cl_number: Ordinary number types. +* cl_pi: Trigonometric functions. +* cl_R: Ordinary number types. +* cl_RA: Ordinary number types. +* cl_SF: Floating-point numbers. +* cl_string (): Strings. +* cl_symbol (): Symbols. +* cl_tschebychev (): Special polynomials. +* cl_zeta (): Riemann zeta. +* coeff (): Functions on univariate polynomials. +* comparison: Comparisons. +* compiler options: Compiler options. +* complex (): Elementary complex functions. +* complex number <1>: Complex numbers. +* complex number: Ordinary number types. +* conjugate (): Elementary complex functions. +* conversion <1>: Conversion functions. +* conversion: Conversions. +* cos (): Trigonometric functions. +* cosh (): Hyperbolic functions. +* create (): Functions on univariate polynomials. +* customizing: Customizing. +* debug_print (): Debugging support. +* debugging: Debugging support. +* decode_float (): Functions on floating-point numbers. +* degree (): Functions on univariate polynomials. +* denominator (): Elementary rational functions. +* deposit_field (): Logical functions. +* deriv (): Functions on univariate polynomials. +* div (): Functions on modular integers. +* doublefactorial (): Combinatorial functions. +* dpb (): Logical functions. +* equal () <1>: Symbols. +* equal (): Strings. +* etract (): Functions on modular integers. +* Euler's constant: Euler gamma. +* evenp (): Logical functions. +* exact number: Exact numbers. +* exp (): Exponential and logarithmic functions. +* exp1 (): Exponential and logarithmic functions. +* expt () <1>: Functions on modular integers. +* expt () <2>: Exponential and logarithmic functions. +* expt (): Elementary functions. +* expt_pos () <1>: Functions on univariate polynomials. +* expt_pos () <2>: Functions on modular integers. +* expt_pos (): Elementary functions. +* exquo (): Elementary functions. +* factorial (): Combinatorial functions. +* fceiling (): Rounding functions. +* fceiling2 (): Rounding functions. +* ffloor (): Rounding functions. +* ffloor2 (): Rounding functions. +* Fibonacci number: An Example. +* finalize (): Functions on univariate polynomials. +* float_digits (): Functions on floating-point numbers. +* float_epsilon (): Conversion to floating-point numbers. +* float_exponent (): Functions on floating-point numbers. +* float_negative_epsilon (): Conversion to floating-point numbers. +* float_precision (): Functions on floating-point numbers. +* float_radix (): Functions on floating-point numbers. +* float_sign (): Functions on floating-point numbers. +* floating-point number: Floating-point numbers. +* floor1 (): Rounding functions. +* floor2 (): Rounding functions. +* fprint () <1>: Functions on univariate polynomials. +* fprint (): Functions on modular integers. +* fround (): Rounding functions. +* fround2 (): Rounding functions. +* ftruncate (): Rounding functions. +* ftruncate2 (): Rounding functions. +* garbage collection <1>: Garbage collection. +* garbage collection: Memory efficiency. +* gcd (): Number theoretic functions. +* GMP: Introduction. +* header files: Include files. +* Hermite polynomial: Special polynomials. +* imagpart (): Elementary complex functions. +* include files: Include files. +* Input/Output: Input/Output. +* installation: Installing the library. +* integer: Ordinary number types. +* integer_decode_float (): Functions on floating-point numbers. +* integer_length (): Logical functions. +* isqrt (): Roots. +* Laguerre polynomial: Special polynomials. +* lcm (): Number theoretic functions. +* ldb (): Logical functions. +* ldb_test (): Logical functions. +* least_negative_float (): Conversion to floating-point numbers. +* least_positive_float (): Conversion to floating-point numbers. +* Legende polynomial: Special polynomials. +* length (): Strings. +* ln (): Exponential and logarithmic functions. +* log (): Exponential and logarithmic functions. +* logand (): Logical functions. +* logandc1 (): Logical functions. +* logandc2 (): Logical functions. +* logbitp (): Logical functions. +* logcount (): Logical functions. +* logeqv (): Logical functions. +* logior (): Logical functions. +* lognand (): Logical functions. +* lognor (): Logical functions. +* lognot (): Logical functions. +* logorc1 (): Logical functions. +* logorc2 (): Logical functions. +* logp (): Number theoretic functions. +* logtest (): Logical functions. +* logxor (): Logical functions. +* make: C++ compiler. +* mask_field (): Logical functions. +* max (): Comparisons. +* min (): Comparisons. +* minus1 (): Elementary functions. +* minusp (): Comparisons. +* mod (): Rounding functions. +* modifying operators: Obfuscating operators. +* modular integer: Modular integers. +* modulus: Functions on modular integers. +* monomial (): Functions on univariate polynomials. +* Montgomery multiplication: Modular integer rings. +* most_negative_float (): Conversion to floating-point numbers. +* most_positive_float (): Conversion to floating-point numbers. +* numerator (): Elementary rational functions. +* oddp (): Logical functions. +* one () <1>: Functions on univariate polynomials. +* one (): Functions on modular integers. +* operator != () <1>: Functions on univariate polynomials. +* operator != () <2>: Functions on modular integers. +* operator != () <3>: Modular integer rings. +* operator != (): Comparisons. +* operator & (): Logical functions. +* operator &= (): Obfuscating operators. +* operator () (): Functions on univariate polynomials. +* operator * () <1>: Functions on univariate polynomials. +* operator * () <2>: Functions on modular integers. +* operator * (): Elementary functions. +* operator *= (): Obfuscating operators. +* operator + () <1>: Functions on univariate polynomials. +* operator + () <2>: Functions on modular integers. +* operator + (): Elementary functions. +* operator ++ (): Obfuscating operators. +* operator += (): Obfuscating operators. +* operator - () <1>: Functions on univariate polynomials. +* operator - () <2>: Functions on modular integers. +* operator - (): Elementary functions. +* operator -- (): Obfuscating operators. +* operator -= (): Obfuscating operators. +* operator / (): Elementary functions. +* operator /= (): Obfuscating operators. +* operator < (): Comparisons. +* operator << () <1>: Functions on univariate polynomials. +* operator << () <2>: Functions on modular integers. +* operator << (): Logical functions. +* operator <<= (): Obfuscating operators. +* operator <= (): Comparisons. +* operator == () <1>: Functions on univariate polynomials. +* operator == () <2>: Functions on modular integers. +* operator == () <3>: Modular integer rings. +* operator == (): Comparisons. +* operator > (): Comparisons. +* operator >= (): Comparisons. +* operator >> () <1>: Functions on modular integers. +* operator >> (): Logical functions. +* operator >>= (): Obfuscating operators. +* operator [] (): Strings. +* operator ^ (): Logical functions. +* operator ^= (): Obfuscating operators. +* operator | (): Logical functions. +* operator |= (): Obfuscating operators. +* operator ~ (): Logical functions. +* ord2 (): Logical functions. +* phase (): Exponential and logarithmic functions. +* pi: Trigonometric functions. +* plus1 (): Elementary functions. +* plusp (): Comparisons. +* polynomial: Univariate polynomials. +* portability: Why C++ ?. +* power2p (): Logical functions. +* printing: Internal and printed representation. +* random (): Functions on modular integers. +* random32 (): Random number generators. +* random_F (): Random number generators. +* random_I (): Random number generators. +* random_R (): Random number generators. +* rational (): Conversion to rational numbers. +* rational number: Ordinary number types. +* rationalize (): Conversion to rational numbers. +* reading: Internal and printed representation. +* real number: Ordinary number types. +* realpart (): Elementary complex functions. +* recip () <1>: Functions on modular integers. +* recip (): Elementary functions. +* reference counting: Memory efficiency. +* rem (): Rounding functions. +* representation: Internal and printed representation. +* Riemann's zeta: Riemann zeta. +* ring: Modular integer rings. +* ring (): Functions on univariate polynomials. +* ring(): Functions on modular integers. +* rootp (): Roots. +* round1 (): Rounding functions. +* round2 (): Rounding functions. +* rounding: Rounding functions. +* rounding error: Floating-point numbers. +* Rubik's cube: Conversions. +* scale_float (): Functions on floating-point numbers. +* Schönhage-Strassen: Speed efficiency. +* Schönhage-Strassen multiplication: Introduction. +* sed: Make utility. +* set_coeff (): Functions on univariate polynomials. +* signum (): Elementary functions. +* sin (): Trigonometric functions. +* sinh (): Hyperbolic functions. +* sqrt (): Roots. +* sqrtp (): Roots. +* square () <1>: Functions on univariate polynomials. +* square () <2>: Functions on modular integers. +* square (): Elementary functions. +* string: Strings. +* strlen (): Strings. +* symbol: Symbols. +* symbolic type: Symbolic data types. +* tan (): Trigonometric functions. +* tanh (): Hyperbolic functions. +* The() (): Conversions. +* transcendental functions: Transcendental functions. +* truncate1 (): Rounding functions. +* truncate2 (): Rounding functions. +* Tschebychev polynomial: Special polynomials. +* underflow: Floating-point underflow. +* univariate polynomial: Univariate polynomials. +* WANT_OBFUSCATING_OPERATORS: Obfuscating operators. +* xgcd (): Number theoretic functions. +* zero () <1>: Functions on univariate polynomials. +* zero (): Functions on modular integers. +* zerop () <1>: Functions on univariate polynomials. +* zerop () <2>: Functions on modular integers. +* zerop (): Comparisons. + +  Tag Table: Node: Top957 Node: Introduction3124 -Node: Installation5626 -Node: Prerequisites5920 -Node: C++ compiler6118 -Node: Make utility6833 -Node: Sed utility7019 -Node: Building the library7339 -Node: Installing the library10650 -Node: Cleaning up11373 -Node: Ordinary number types11698 -Node: Exact numbers14045 -Node: Floating-point numbers15210 -Node: Complex numbers18789 -Node: Conversions19286 -Node: Functions on numbers22752 -Node: Constructing numbers23455 -Node: Constructing integers23827 -Node: Constructing rational numbers24117 -Node: Constructing floating-point numbers24592 -Node: Constructing complex numbers25712 -Node: Elementary functions26076 -Node: Elementary rational functions28543 -Node: Elementary complex functions29115 -Node: Comparisons29943 -Node: Rounding functions31842 -Node: Roots37619 -Node: Transcendental functions39500 -Node: Exponential and logarithmic functions40056 -Node: Trigonometric functions42073 -Node: Hyperbolic functions45416 -Node: Euler gamma47489 -Node: Riemann zeta48405 -Node: Functions on integers48961 -Node: Logical functions49249 -Node: Number theoretic functions55202 -Node: Combinatorial functions56569 -Node: Functions on floating-point numbers57247 -Node: Conversion functions60478 -Node: Conversion to floating-point numbers60758 -Node: Conversion to rational numbers62981 -Node: Random number generators64035 -Node: Obfuscating operators65709 -Node: Input/Output67439 -Node: Internal and printed representation67649 -Node: Input functions70191 -Node: Output functions74742 -Node: Rings78478 -Node: Modular integers80402 -Node: Modular integer rings80602 -Node: Functions on modular integers82692 -Node: Symbolic data types85702 -Node: Strings85965 -Node: Symbols87030 -Node: Univariate polynomials87932 -Node: Univariate polynomial rings88190 -Node: Functions on univariate polynomials93144 -Node: Special polynomials96925 -Node: Internals97645 -Node: Why C++ ?97859 -Node: Memory efficiency99360 -Node: Speed efficiency100058 -Node: Garbage collection101122 -Node: Using the library101949 -Node: Compiler options102483 -Node: Include files103401 -Node: An Example107042 -Node: Debugging support110067 -Node: Customizing112417 -Node: Error handling112645 -Node: Floating-point underflow113219 -Node: Customizing I/O113858 -Node: Customizing the memory allocator114151 -Node: Index115108 +Node: Installation5646 +Node: Prerequisites5940 +Node: C++ compiler6138 +Node: Make utility6853 +Node: Sed utility7039 +Node: Building the library7359 +Node: Installing the library10541 +Node: Cleaning up11264 +Node: Ordinary number types11589 +Node: Exact numbers13936 +Node: Floating-point numbers15101 +Node: Complex numbers18680 +Node: Conversions19177 +Node: Functions on numbers22643 +Node: Constructing numbers23346 +Node: Constructing integers23718 +Node: Constructing rational numbers24008 +Node: Constructing floating-point numbers24483 +Node: Constructing complex numbers25603 +Node: Elementary functions25967 +Node: Elementary rational functions28434 +Node: Elementary complex functions29006 +Node: Comparisons29834 +Node: Rounding functions31733 +Node: Roots37510 +Node: Transcendental functions39391 +Node: Exponential and logarithmic functions39947 +Node: Trigonometric functions41964 +Node: Hyperbolic functions45315 +Node: Euler gamma47388 +Node: Riemann zeta48304 +Node: Functions on integers48860 +Node: Logical functions49148 +Node: Number theoretic functions55101 +Node: Combinatorial functions56468 +Node: Functions on floating-point numbers57146 +Node: Conversion functions60377 +Node: Conversion to floating-point numbers60657 +Node: Conversion to rational numbers62880 +Node: Random number generators63934 +Node: Obfuscating operators65608 +Node: Input/Output67338 +Node: Internal and printed representation67548 +Node: Input functions70090 +Node: Output functions74641 +Node: Rings78377 +Node: Modular integers80301 +Node: Modular integer rings80501 +Node: Functions on modular integers82591 +Node: Symbolic data types85601 +Node: Strings85864 +Node: Symbols86929 +Node: Univariate polynomials87831 +Node: Univariate polynomial rings88089 +Node: Functions on univariate polynomials93043 +Node: Special polynomials96824 +Node: Internals97544 +Node: Why C++ ?97758 +Node: Memory efficiency99258 +Node: Speed efficiency99956 +Node: Garbage collection101040 +Node: Using the library101867 +Node: Compiler options102401 +Node: Include files103319 +Node: An Example106960 +Node: Debugging support110110 +Node: Customizing112460 +Node: Error handling112688 +Node: Floating-point underflow113262 +Node: Customizing I/O113901 +Node: Customizing the memory allocator114194 +Node: Index115151  End Tag Table diff --git a/doc/cln.ps b/doc/cln.ps index 2b21c11..1241843 100644 --- 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Fp(&)30 b(x\))480 4749 y Fr(Returns)f +Fp(x)h(-)h(1)p Fr(.)0 4920 y Fk(t)m(yp)s(e)36 b Fp(operator)28 +b(*)i(\(const)e Fk(t)m(yp)s(e)5 b Fp(&,)31 b(const)d +Fk(t)m(yp)s(e)5 b Fp(&\))480 5045 y Fr(Multiplication.)0 +5215 y Fk(t)m(yp)s(e)36 b Fp(square)28 b(\(const)h Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 5340 y Fr(Returns)f Fp(x)h(*)h(x)p +Fr(.)p eop %%Page: 14 16 -14 15 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +14 15 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 b(on)f(n)m(um)m(b)s(ers)2463 b(14)0 366 y(Eac)m(h)39 -b(of)g(the)g(classes)h Fn(cl_N)p Fp(,)g Fn(cl_R)p Fp(,)f -Fn(cl_RA)p Fp(,)h Fn(cl_F)p Fp(,)g Fn(cl_SF)p Fp(,)f -Fn(cl_FF)p Fp(,)h Fn(cl_DF)p Fp(,)g Fn(cl_LF)d Fp(de\014nes)h(the)g -(follo)m(wing)0 491 y(op)s(erations:)0 796 y Fi(t)m(yp)s(e)e -Fn(operator)28 b(/)i(\(const)e Fi(t)m(yp)s(e)5 b Fn(&,)31 -b(const)d Fi(t)m(yp)s(e)5 b Fn(&\))480 921 y Fp(Division.)0 -1101 y Fi(t)m(yp)s(e)36 b Fn(recip)29 b(\(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))480 1225 y Fp(Returns)29 b(the)i(recipro)s(cal)g(of)g(the)f -(argumen)m(t.)0 1530 y(The)d(class)g Fn(cl_I)f Fp(do)s(esn't)h -(de\014ne)g(a)g(`)p Fn(/)p Fp(')g(op)s(eration)h(b)s(ecause)f(in)g(the) -g(C/C)p Fn(++)f Fp(language)j(this)e(op)s(erator,)h(applied)0 +b(of)g(the)g(classes)h Fp(cl_N)p Fr(,)g Fp(cl_R)p Fr(,)f +Fp(cl_RA)p Fr(,)h Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)f +Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)g Fp(cl_LF)d Fr(de\014nes)h(the)g +(follo)m(wing)0 491 y(op)s(erations:)0 796 y Fk(t)m(yp)s(e)e +Fp(operator)28 b(/)i(\(const)e Fk(t)m(yp)s(e)5 b Fp(&,)31 +b(const)d Fk(t)m(yp)s(e)5 b Fp(&\))480 921 y Fr(Division.)0 +1101 y Fk(t)m(yp)s(e)36 b Fp(recip)29 b(\(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))480 1225 y Fr(Returns)29 b(the)i(recipro)s(cal)g(of)g(the)f +(argumen)m(t.)0 1530 y(The)d(class)g Fp(cl_I)f Fr(do)s(esn't)h +(de\014ne)g(a)g(`)p Fp(/)p Fr(')g(op)s(eration)h(b)s(ecause)f(in)g(the) +g(C/C)p Fp(++)f Fr(language)j(this)e(op)s(erator,)h(applied)0 1655 y(to)j(in)m(tegral)h(t)m(yp)s(es,)f(denotes)f(the)h(`)p -Fn(floor)p Fp(')e(or)h(`)p Fn(truncate)p Fp(')f(op)s(eration)i(\(whic)m +Fp(floor)p Fr(')e(or)h(`)p Fp(truncate)p Fr(')f(op)s(eration)i(\(whic)m (h)f(one)h(of)f(these,)h(is)g(implemen)m(ta-)0 1780 y(tion)j(dep)s (enden)m(t\).)48 b(\(See)34 b(Section)f(4.6)i([Rounding)d(functions],)i -(page)g(17\))g(Instead,)g Fn(cl_I)e Fp(de\014nes)g(an)h(\\exact)0 -1904 y(quotien)m(t")f(function:)0 2209 y Fn(cl_I)d(exquo)g(\(const)g -(cl_I&)g(x,)h(const)e(cl_I&)h(y\))480 2334 y Fp(Chec)m(ks)i(that)g -Fn(y)f Fp(divides)g Fn(x)p Fp(,)g(and)g(returns)f(the)h(quotien)m(t)i -Fn(x)p Fp(/)p Fn(y)p Fp(.)0 2638 y(The)e(follo)m(wing)i(exp)s(onen)m -(tiation)f(functions)f(are)h(de\014ned:)0 2943 y Fn(cl_I)e(expt_pos)f +(page)g(17\))g(Instead,)g Fp(cl_I)e Fr(de\014nes)g(an)h(\\exact)0 +1904 y(quotien)m(t")f(function:)0 2209 y Fp(cl_I)d(exquo)g(\(const)g +(cl_I&)g(x,)h(const)e(cl_I&)h(y\))480 2334 y Fr(Chec)m(ks)i(that)g +Fp(y)f Fr(divides)g Fp(x)p Fr(,)g(and)g(returns)f(the)h(quotien)m(t)i +Fp(x)p Fr(/)p Fp(y)p Fr(.)0 2638 y(The)e(follo)m(wing)i(exp)s(onen)m +(tiation)f(functions)f(are)h(de\014ned:)0 2943 y Fp(cl_I)e(expt_pos)f (\(const)h(cl_I&)g(x,)h(const)f(cl_I&)g(y\))0 3068 y(cl_RA)g(expt_pos)f (\(const)h(cl_RA&)f(x,)i(const)f(cl_I&)g(y\))480 3193 -y(y)h Fp(m)m(ust)f(b)s(e)h Fn(>)g Fp(0.)41 b(Returns)30 -b Fn(x^y)p Fp(.)0 3373 y Fn(cl_RA)f(expt)g(\(const)g(cl_RA&)f(x,)i +y(y)h Fr(m)m(ust)f(b)s(e)h Fp(>)g Fr(0.)41 b(Returns)30 +b Fp(x^y)p Fr(.)0 3373 y Fp(cl_RA)f(expt)g(\(const)g(cl_RA&)f(x,)i (const)f(cl_I&)g(y\))0 3497 y(cl_R)g(expt)g(\(const)g(cl_R&)g(x,)h (const)f(cl_I&)g(y\))0 3622 y(cl_N)g(expt)g(\(const)g(cl_N&)g(x,)h -(const)f(cl_I&)g(y\))480 3747 y Fp(Returns)g Fn(x^y)p -Fp(.)0 4052 y(Eac)m(h)39 b(of)g(the)g(classes)h Fn(cl_R)p -Fp(,)g Fn(cl_RA)p Fp(,)f Fn(cl_I)p Fp(,)h Fn(cl_F)p Fp(,)g -Fn(cl_SF)p Fp(,)f Fn(cl_FF)p Fp(,)h Fn(cl_DF)p Fp(,)g -Fn(cl_LF)d Fp(de\014nes)h(the)g(follo)m(wing)0 4176 y(op)s(eration:)0 -4481 y Fi(t)m(yp)s(e)e Fn(abs)29 b(\(const)g Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 4606 y Fp(Returns)f(the)i(absolute)g(v)-5 -b(alue)31 b(of)f Fn(x)p Fp(.)41 b(This)29 b(is)i Fn(x)f -Fp(if)g Fn(x)g(>=)g(0)p Fp(,)g(and)g Fn(-x)g Fp(if)g -Fn(x)g(<=)g(0)p Fp(.)0 4911 y(The)g(class)h Fn(cl_N)e -Fp(implemen)m(ts)g(this)h(as)h(follo)m(ws:)0 5215 y Fn(cl_R)e(abs)h -(\(const)e(cl_N)i(x\))480 5340 y Fp(Returns)f(the)i(absolute)g(v)-5 -b(alue)31 b(of)f Fn(x)p Fp(.)p eop +(const)f(cl_I&)g(y\))480 3747 y Fr(Returns)g Fp(x^y)p +Fr(.)0 4052 y(Eac)m(h)39 b(of)g(the)g(classes)h Fp(cl_R)p +Fr(,)g Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)g +Fp(cl_SF)p Fr(,)f Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)g +Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)0 4176 y(op)s(eration:)0 +4481 y Fk(t)m(yp)s(e)e Fp(abs)29 b(\(const)g Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 4606 y Fr(Returns)f(the)i(absolute)g(v)-5 +b(alue)31 b(of)f Fp(x)p Fr(.)41 b(This)29 b(is)i Fp(x)f +Fr(if)g Fp(x)g(>=)g(0)p Fr(,)g(and)g Fp(-x)g Fr(if)g +Fp(x)g(<=)g(0)p Fr(.)0 4911 y(The)g(class)h Fp(cl_N)e +Fr(implemen)m(ts)g(this)h(as)h(follo)m(ws:)0 5215 y Fp(cl_R)e(abs)h +(\(const)e(cl_N)i(x\))480 5340 y Fr(Returns)f(the)i(absolute)g(v)-5 +b(alue)31 b(of)f Fp(x)p Fr(.)p eop %%Page: 15 17 -15 16 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +15 16 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 b(on)f(n)m(um)m(b)s(ers)2463 b(15)0 366 y(Eac)m(h)22 -b(of)g(the)g(classes)h Fn(cl_N)p Fp(,)g Fn(cl_R)p Fp(,)f -Fn(cl_RA)p Fp(,)g Fn(cl_I)p Fp(,)h Fn(cl_F)p Fp(,)g Fn(cl_SF)p -Fp(,)f Fn(cl_FF)p Fp(,)g Fn(cl_DF)p Fp(,)g Fn(cl_LF)f -Fp(de\014nes)g(the)g(follo)m(wing)0 491 y(op)s(eration:)0 -798 y Fi(t)m(yp)s(e)36 b Fn(signum)28 b(\(const)h Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 922 y Fp(Returns)g(the)i(sign)f(of)g -Fn(x)p Fp(,)h(in)f(the)g(same)g(n)m(um)m(b)s(er)e(format)h(as)i -Fn(x)p Fp(.)42 b(This)31 b(is)g(de\014ned)f(as)i Fn(x)e(/)g(abs\(x\)) -480 1047 y Fp(if)g Fn(x)g Fp(is)h(non-zero,)g(and)f Fn(x)g -Fp(if)g Fn(x)g Fp(is)h(zero.)41 b(If)30 b Fn(x)g Fp(is)g(real,)i(the)e +b(of)g(the)g(classes)h Fp(cl_N)p Fr(,)g Fp(cl_R)p Fr(,)f +Fp(cl_RA)p Fr(,)g Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)g Fp(cl_SF)p +Fr(,)f Fp(cl_FF)p Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f +Fr(de\014nes)g(the)g(follo)m(wing)0 491 y(op)s(eration:)0 +798 y Fk(t)m(yp)s(e)36 b Fp(signum)28 b(\(const)h Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 922 y Fr(Returns)g(the)i(sign)f(of)g +Fp(x)p Fr(,)h(in)f(the)g(same)g(n)m(um)m(b)s(er)e(format)h(as)i +Fp(x)p Fr(.)42 b(This)31 b(is)g(de\014ned)f(as)i Fp(x)e(/)g(abs\(x\)) +480 1047 y Fr(if)g Fp(x)g Fr(is)h(non-zero,)g(and)f Fp(x)g +Fr(if)g Fp(x)g Fr(is)h(zero.)41 b(If)30 b Fp(x)g Fr(is)g(real,)i(the)e (v)-5 b(alue)31 b(is)g(either)f(0)h(or)f(1)h(or)g(-1.)0 -1531 y Fq(4.3)68 b(Elemen)l(tary)32 b(rational)g(functions)0 -1808 y Fp(Eac)m(h)f(of)g(the)f(classes)i Fn(cl_RA)p Fp(,)d -Fn(cl_I)g Fp(de\014nes)g(the)i(follo)m(wing)h(op)s(erations:)0 -2115 y Fn(cl_I)d(numerator)f(\(const)h Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 2240 y Fp(Returns)f(the)i(n)m(umerator)e(of)i -Fn(x)p Fp(.)0 2422 y Fn(cl_I)e(denominator)f(\(const)g -Fi(t)m(yp)s(e)5 b Fn(&)31 b(x\))480 2546 y Fp(Returns)e(the)i -(denominator)f(of)g Fn(x)p Fp(.)0 2853 y(The)c(n)m(umerator)h(and)f +1531 y Fs(4.3)68 b(Elemen)l(tary)32 b(rational)g(functions)0 +1808 y Fr(Eac)m(h)f(of)g(the)f(classes)i Fp(cl_RA)p Fr(,)d +Fp(cl_I)g Fr(de\014nes)g(the)i(follo)m(wing)h(op)s(erations:)0 +2115 y Fp(cl_I)d(numerator)f(\(const)h Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 2240 y Fr(Returns)f(the)i(n)m(umerator)e(of)i +Fp(x)p Fr(.)0 2422 y Fp(cl_I)e(denominator)f(\(const)g +Fk(t)m(yp)s(e)5 b Fp(&)31 b(x\))480 2546 y Fr(Returns)e(the)i +(denominator)f(of)g Fp(x)p Fr(.)0 2853 y(The)c(n)m(umerator)h(and)f (denominator)g(of)h(a)h(rational)g(n)m(um)m(b)s(er)c(are)k(normalized)e (in)h(suc)m(h)g(a)g(w)m(a)m(y)h(that)g(they)f(ha)m(v)m(e)0 2978 y(no)j(factor)i(in)e(common)e(and)i(the)h(denominator)e(is)i(p)s -(ositiv)m(e.)0 3462 y Fq(4.4)68 b(Elemen)l(tary)32 b(complex)f -(functions)0 3739 y Fp(The)f(class)h Fn(cl_N)e Fp(de\014nes)h(the)g -(follo)m(wing)i(op)s(eration:)0 4046 y Fn(cl_N)d(complex)g(\(const)f -(cl_R&)h(a,)h(const)f(cl_R&)g(b\))480 4171 y Fp(Returns)i(the)h -(complex)f(n)m(um)m(b)s(er)f Fn(a+bi)p Fp(,)h(that)i(is,)f(the)g -(complex)g(n)m(um)m(b)s(er)d(with)j(real)g(part)g Fn(a)g -Fp(and)480 4295 y(imaginary)e(part)g Fn(b)p Fp(.)0 4602 -y(Eac)m(h)h(of)g(the)f(classes)i Fn(cl_N)p Fp(,)d Fn(cl_R)g -Fp(de\014nes)h(the)g(follo)m(wing)i(op)s(erations:)0 -4909 y Fn(cl_R)d(realpart)f(\(const)h Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 5033 y Fp(Returns)f(the)i(real)g(part)f(of)h -Fn(x)p Fp(.)0 5215 y Fn(cl_R)e(imagpart)f(\(const)h Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 5340 y Fp(Returns)f(the)i(imaginary)f(part)g(of)h -Fn(x)p Fp(.)p eop +(ositiv)m(e.)0 3462 y Fs(4.4)68 b(Elemen)l(tary)32 b(complex)f +(functions)0 3739 y Fr(The)f(class)h Fp(cl_N)e Fr(de\014nes)h(the)g +(follo)m(wing)i(op)s(eration:)0 4046 y Fp(cl_N)d(complex)g(\(const)f +(cl_R&)h(a,)h(const)f(cl_R&)g(b\))480 4171 y Fr(Returns)i(the)h +(complex)f(n)m(um)m(b)s(er)f Fp(a+bi)p Fr(,)h(that)i(is,)f(the)g +(complex)g(n)m(um)m(b)s(er)d(with)j(real)g(part)g Fp(a)g +Fr(and)480 4295 y(imaginary)e(part)g Fp(b)p Fr(.)0 4602 +y(Eac)m(h)h(of)g(the)f(classes)i Fp(cl_N)p Fr(,)d Fp(cl_R)g +Fr(de\014nes)h(the)g(follo)m(wing)i(op)s(erations:)0 +4909 y Fp(cl_R)d(realpart)f(\(const)h Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 5033 y Fr(Returns)f(the)i(real)g(part)f(of)h +Fp(x)p Fr(.)0 5215 y Fp(cl_R)e(imagpart)f(\(const)h Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 5340 y Fr(Returns)f(the)i(imaginary)f(part)g(of)h +Fp(x)p Fr(.)p eop %%Page: 16 18 -16 17 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(16)0 366 y Fi(t)m(yp)s(e)36 -b Fn(conjugate)27 b(\(const)i Fi(t)m(yp)s(e)5 b Fn(&)31 -b(x\))480 491 y Fp(Returns)e(the)i(complex)f(conjugate)i(of)e -Fn(x)p Fp(.)0 795 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(relations)180 -1071 y Fn(x)f(=)g(complex\(realpart\(x\),)25 b(imagpart\(x\)\))180 +16 17 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(16)0 366 y Fk(t)m(yp)s(e)36 +b Fp(conjugate)27 b(\(const)i Fk(t)m(yp)s(e)5 b Fp(&)31 +b(x\))480 491 y Fr(Returns)e(the)i(complex)f(conjugate)i(of)e +Fp(x)p Fr(.)0 795 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(relations)180 +1071 y Fp(x)f(=)g(complex\(realpart\(x\),)25 b(imagpart\(x\)\))180 1223 y(conjugate\(x\))i(=)j(complex\(realpart\(x\),)25 -b(-imagpart\(x\)\))0 1693 y Fq(4.5)68 b(Comparisons)0 -1969 y Fp(Eac)m(h)22 b(of)g(the)g(classes)h Fn(cl_N)p -Fp(,)g Fn(cl_R)p Fp(,)f Fn(cl_RA)p Fp(,)g Fn(cl_I)p Fp(,)h -Fn(cl_F)p Fp(,)g Fn(cl_SF)p Fp(,)f Fn(cl_FF)p Fp(,)g -Fn(cl_DF)p Fp(,)g Fn(cl_LF)f Fp(de\014nes)g(the)g(follo)m(wing)0 -2094 y(op)s(erations:)0 2397 y Fn(bool)29 b(operator)f(==)i(\(const)f -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 2522 y(bool)29 b(operator)f(!=)i(\(const)f -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))480 2646 y Fp(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p -Fn(++)p Fp(.)0 2825 y Fn(uint32)g(cl_equal_hashcode)c(\(const)k -Fi(t)m(yp)s(e)5 b Fn(&\))480 2950 y Fp(Returns)31 b(a)h(32-bit)i(hash)d +b(-imagpart\(x\)\))0 1693 y Fs(4.5)68 b(Comparisons)0 +1969 y Fr(Eac)m(h)22 b(of)g(the)g(classes)h Fp(cl_N)p +Fr(,)g Fp(cl_R)p Fr(,)f Fp(cl_RA)p Fr(,)g Fp(cl_I)p Fr(,)h +Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)f Fp(cl_FF)p Fr(,)g +Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)g(follo)m(wing)0 +2094 y(op)s(erations:)0 2397 y Fp(bool)29 b(operator)f(==)i(\(const)f +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 2522 y(bool)29 b(operator)f(!=)i(\(const)f +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))480 2646 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p +Fp(++)p Fr(.)0 2825 y Fp(uint32)g(cl_equal_hashcode)c(\(const)k +Fk(t)m(yp)s(e)5 b Fp(&\))480 2950 y Fr(Returns)31 b(a)h(32-bit)i(hash)d (co)s(de)h(that)h(is)f(the)g(same)f(for)h(an)m(y)g(t)m(w)m(o)i(n)m(um)m (b)s(ers)29 b(whic)m(h)j(are)g(the)h(same)480 3075 y(according)e(to)g -Fn(==)p Fp(.)40 b(This)29 b(hash)g(co)s(de)i(dep)s(ends)d(on)i(the)g(n) +Fp(==)p Fr(.)40 b(This)29 b(hash)g(co)s(de)i(dep)s(ends)d(on)i(the)g(n) m(um)m(b)s(er's)e(v)-5 b(alue,)31 b(not)f(its)g(t)m(yp)s(e)g(or)h -(preci-)480 3199 y(sion.)0 3378 y Fn(cl_boolean)d(zerop)h(\(const)f -Fi(t)m(yp)s(e)5 b Fn(&)31 b(x\))480 3503 y Fp(Compare)e(against)j -(zero:)41 b Fn(x)30 b(==)g(0)0 3806 y Fp(Eac)m(h)39 b(of)g(the)g -(classes)h Fn(cl_R)p Fp(,)g Fn(cl_RA)p Fp(,)f Fn(cl_I)p -Fp(,)h Fn(cl_F)p Fp(,)g Fn(cl_SF)p Fp(,)f Fn(cl_FF)p -Fp(,)h Fn(cl_DF)p Fp(,)g Fn(cl_LF)d Fp(de\014nes)h(the)g(follo)m(wing)0 -3931 y(op)s(erations:)0 4235 y Fn(cl_signean)28 b(cl_compare)f(\(const) -i Fi(t)m(yp)s(e)5 b Fn(&)30 b(x,)g(const)f Fi(t)m(yp)s(e)5 -b Fn(&)30 b(y\))480 4359 y Fp(Compares)f Fn(x)h Fp(and)g -Fn(y)p Fp(.)40 b(Returns)30 b Fn(+)p Fp(1)g(if)g Fn(x>y)p -Fp(,)g(-1)h(if)g Fn(x=)i(\(const)f -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 4912 y(bool)29 b(operator)f(>)i(\(const)f -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))480 5036 y Fp(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p -Fn(++)p Fp(.)0 5215 y Fn(cl_boolean)f(minusp)g(\(const)h -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x\))480 5340 y Fp(Compare)f(against)j -(zero:)41 b Fn(x)30 b(<)g(0)p eop +(preci-)480 3199 y(sion.)0 3378 y Fp(cl_boolean)d(zerop)h(\(const)f +Fk(t)m(yp)s(e)5 b Fp(&)31 b(x\))480 3503 y Fr(Compare)e(against)j +(zero:)41 b Fp(x)30 b(==)g(0)0 3806 y Fr(Eac)m(h)39 b(of)g(the)g +(classes)h Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)f Fp(cl_I)p +Fr(,)h Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)f Fp(cl_FF)p +Fr(,)h Fp(cl_DF)p Fr(,)g Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)0 +3931 y(op)s(erations:)0 4235 y Fp(cl_signean)28 b(cl_compare)f(\(const) +i Fk(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fk(t)m(yp)s(e)5 +b Fp(&)30 b(y\))480 4359 y Fr(Compares)f Fp(x)h Fr(and)g +Fp(y)p Fr(.)40 b(Returns)30 b Fp(+)p Fr(1)g(if)g Fp(x>y)p +Fr(,)g(-1)h(if)g Fp(x=)i(\(const)f +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 4912 y(bool)29 b(operator)f(>)i(\(const)f +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))480 5036 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p +Fp(++)p Fr(.)0 5215 y Fp(cl_boolean)f(minusp)g(\(const)h +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x\))480 5340 y Fr(Compare)f(against)j +(zero:)41 b Fp(x)30 b(<)g(0)p eop %%Page: 17 19 -17 18 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(17)0 366 y Fn(cl_boolean)28 -b(plusp)h(\(const)f Fi(t)m(yp)s(e)5 b Fn(&)31 b(x\))480 -491 y Fp(Compare)e(against)j(zero:)41 b Fn(x)30 b(>)g(0)0 -656 y Fi(t)m(yp)s(e)36 b Fn(max)29 b(\(const)g Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x,)g(const)f Fi(t)m(yp)s(e)5 b Fn(&)30 b(y\))480 -780 y Fp(Return)g(the)g(maxim)m(um)d(of)k Fn(x)f Fp(and)g -Fn(y)p Fp(.)0 945 y Fi(t)m(yp)s(e)36 b Fn(min)29 b(\(const)g -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x,)g(const)f Fi(t)m(yp)s(e)5 -b Fn(&)30 b(y\))480 1069 y Fp(Return)g(the)g(minim)m(um)d(of)j -Fn(x)g Fp(and)g Fn(y)p Fp(.)0 1363 y(When)h(a)h(\015oating)g(p)s(oin)m +17 18 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(17)0 366 y Fp(cl_boolean)28 +b(plusp)h(\(const)f Fk(t)m(yp)s(e)5 b Fp(&)31 b(x\))480 +491 y Fr(Compare)e(against)j(zero:)41 b Fp(x)30 b(>)g(0)0 +656 y Fk(t)m(yp)s(e)36 b Fp(max)29 b(\(const)g Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x,)g(const)f Fk(t)m(yp)s(e)5 b Fp(&)30 b(y\))480 +780 y Fr(Return)g(the)g(maxim)m(um)d(of)k Fp(x)f Fr(and)g +Fp(y)p Fr(.)0 945 y Fk(t)m(yp)s(e)36 b Fp(min)29 b(\(const)g +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fk(t)m(yp)s(e)5 +b Fp(&)30 b(y\))480 1069 y Fr(Return)g(the)g(minim)m(um)d(of)j +Fp(x)g Fr(and)g Fp(y)p Fr(.)0 1363 y(When)h(a)h(\015oating)g(p)s(oin)m (t)f(n)m(um)m(b)s(er)e(and)h(a)i(rational)g(n)m(um)m(b)s(er)d(are)j (compared,)f(the)g(\015oat)h(is)f(\014rst)f(con)m(v)m(erted)j(to)0 1488 y(a)e(rational)h(n)m(um)m(b)s(er)c(using)i(the)h(function)f -Fn(rational)p Fp(.)39 b(Since)31 b(a)g(\015oating)g(p)s(oin)m(t)g(n)m +Fp(rational)p Fr(.)39 b(Since)31 b(a)g(\015oating)g(p)s(oin)m(t)g(n)m (um)m(b)s(er)d(actually)k(represen)m(ts)0 1612 y(an)h(in)m(terv)-5 b(al)34 b(of)g(real)g(n)m(um)m(b)s(ers,)d(the)j(result)f(migh)m(t)g(b)s -(e)f(surprising.)48 b(F)-8 b(or)34 b(example,)g Fn +(e)f(surprising.)48 b(F)-8 b(or)34 b(example,)g Fp (\(cl_F\)\(cl_R\)"1/3")26 b(==)0 1737 y(\(cl_R\)"1/3")h -Fp(returns)i(false)i(b)s(ecause)g(there)f(is)h(no)f(\015oating)h(p)s +Fr(returns)i(false)i(b)s(ecause)g(there)f(is)h(no)f(\015oating)h(p)s (oin)m(t)g(n)m(um)m(b)s(er)d(whose)i(v)-5 b(alue)31 b(is)f(exactly)i -Fn(1/3)p Fp(.)0 2166 y Fq(4.6)68 b(Rounding)30 b(functions)0 -2440 y Fp(When)37 b(a)g(real)h(n)m(um)m(b)s(er)d(is)i(to)h(b)s(e)e(con) +Fp(1/3)p Fr(.)0 2166 y Fs(4.6)68 b(Rounding)30 b(functions)0 +2440 y Fr(When)37 b(a)g(real)h(n)m(um)m(b)s(er)d(is)i(to)h(b)s(e)e(con) m(v)m(erted)j(to)e(an)g(in)m(teger,)k(there)c(is)g(no)g(\\b)s(est")g (rounding.)60 b(The)36 b(desired)0 2564 y(rounding)g(function)g(dep)s (ends)f(on)i(the)g(application.)61 b(The)36 b(Common)e(Lisp)j(and)f (ISO)g(Lisp)g(standards)g(o\013er)0 2689 y(four)30 b(rounding)f -(functions:)0 2983 y Fn(floor\(x\))96 b Fp(This)30 b(is)g(the)h -(largest)g(in)m(teger)h Fn(<)p Fp(=)p Fn(x)p Fp(.)0 3148 -y Fn(ceiling\(x\))480 3272 y Fp(This)e(is)g(the)h(smallest)f(in)m -(teger)i Fn(>)p Fp(=)p Fn(x)p Fp(.)0 3437 y Fn(truncate\(x\))480 -3561 y Fp(Among)e(the)g(in)m(tegers)i(b)s(et)m(w)m(een)f(0)g(and)e -Fn(x)h Fp(\(inclusiv)m(e\))i(the)f(one)g(nearest)f(to)i -Fn(x)p Fp(.)0 3726 y Fn(round\(x\))96 b Fp(The)29 b(in)m(teger)h -(nearest)f(to)h Fn(x)p Fp(.)40 b(If)29 b Fn(x)f Fp(is)h(exactly)i +(functions:)0 2983 y Fp(floor\(x\))96 b Fr(This)30 b(is)g(the)h +(largest)g(in)m(teger)h Fp(<)p Fr(=)p Fp(x)p Fr(.)0 3148 +y Fp(ceiling\(x\))480 3272 y Fr(This)e(is)g(the)h(smallest)f(in)m +(teger)i Fp(>)p Fr(=)p Fp(x)p Fr(.)0 3437 y Fp(truncate\(x\))480 +3561 y Fr(Among)e(the)g(in)m(tegers)i(b)s(et)m(w)m(een)f(0)g(and)e +Fp(x)h Fr(\(inclusiv)m(e\))i(the)f(one)g(nearest)f(to)i +Fp(x)p Fr(.)0 3726 y Fp(round\(x\))96 b Fr(The)29 b(in)m(teger)h +(nearest)f(to)h Fp(x)p Fr(.)40 b(If)29 b Fp(x)f Fr(is)h(exactly)i (halfw)m(a)m(y)f(b)s(et)m(w)m(een)g(t)m(w)m(o)g(in)m(tegers,)h(c)m(ho)s (ose)f(the)f(ev)m(en)480 3850 y(one.)0 4144 y(These)h(functions)g(ha)m (v)m(e)i(di\013eren)m(t)e(adv)-5 b(an)m(tages:)0 4418 -y Fn(floor)38 b Fp(and)g Fn(ceiling)g Fp(are)h(translation)h(in)m(v)-5 -b(arian)m(t:)60 b Fn(floor\(x+n\))28 b(=)i(floor\(x\))e(+)i(n)39 -b Fp(and)f Fn(ceiling\(x+n\))27 b(=)0 4543 y(ceiling\(x\))h(+)i(n)g -Fp(for)g(ev)m(ery)h Fn(x)f Fp(and)g(ev)m(ery)h(in)m(teger)h -Fn(n)p Fp(.)0 4817 y(On)46 b(the)h(other)g(hand,)j Fn(truncate)44 -b Fp(and)i Fn(round)f Fp(are)i(symmetric:)72 b Fn(truncate\(-x\))27 -b(=)j(-truncate\(x\))44 b Fp(and)0 4941 y Fn(round\(-x\))28 -b(=)i(-round\(x\))p Fp(,)38 b(and)g(furthermore)e Fn(round)h -Fp(is)h(un)m(biased:)57 b(on)38 b(the)h(\\a)m(v)m(erage",)44 +y Fp(floor)38 b Fr(and)g Fp(ceiling)g Fr(are)h(translation)h(in)m(v)-5 +b(arian)m(t:)60 b Fp(floor\(x+n\))28 b(=)i(floor\(x\))e(+)i(n)39 +b Fr(and)f Fp(ceiling\(x+n\))27 b(=)0 4543 y(ceiling\(x\))h(+)i(n)g +Fr(for)g(ev)m(ery)h Fp(x)f Fr(and)g(ev)m(ery)h(in)m(teger)h +Fp(n)p Fr(.)0 4817 y(On)46 b(the)h(other)g(hand,)j Fp(truncate)44 +b Fr(and)i Fp(round)f Fr(are)i(symmetric:)72 b Fp(truncate\(-x\))27 +b(=)j(-truncate\(x\))44 b Fr(and)0 4941 y Fp(round\(-x\))28 +b(=)i(-round\(x\))p Fr(,)38 b(and)g(furthermore)e Fp(round)h +Fr(is)h(un)m(biased:)57 b(on)38 b(the)h(\\a)m(v)m(erage",)44 b(it)39 b(rounds)d(do)m(wn)0 5066 y(exactly)c(as)f(often)g(as)f(it)h (rounds)e(up.)0 5340 y(The)h(functions)g(are)h(related)g(lik)m(e)h (this:)p eop %%Page: 18 20 -18 19 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(18)180 366 y Fn(ceiling\(m/n\))27 +18 19 bop 0 -116 a 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Fi(t)m(yp)s(e)5 -b Fn(&)31 b(x,)e Fi(t)m(yp)s(e)5 b Fn(*)31 b(root\))480 -3408 y(x)f Fp(should)g(b)s(e)g Fn(>)p Fp(=)g(0.)42 b(This)30 -b(function)h(sets)g Fn(*root)e Fp(to)i Fn(floor\(sqrt\(x\)\))c -Fp(and)j(returns)g(the)h(same)480 3533 y(v)-5 b(alue)31 -b(as)f Fn(sqrtp)p Fp(:)40 b(the)30 b(b)s(o)s(olean)h(v)-5 -b(alue)31 b Fn(\(expt\(*root,2\))26 b(==)k(x\))p Fp(.)0 -3829 y(F)-8 b(or)31 b Fn(n)p Fp(th)f(ro)s(ots,)h(the)f(classes)i -Fn(cl_RA)p Fp(,)d Fn(cl_I)g Fp(de\014ne)h(the)g(follo)m(wing)i(op)s -(eration:)0 4125 y Fn(cl_boolean)c(rootp)h(\(const)f -Fi(t)m(yp)s(e)5 b Fn(&)31 b(x,)e(const)g(cl_I&)g(n,)h -Fi(t)m(yp)s(e)5 b Fn(*)30 b(root\))480 4250 y(x)36 b -Fp(m)m(ust)f(b)s(e)g Fn(>)p Fp(=)g(0.)58 b Fn(n)36 b -Fp(m)m(ust)f(b)s(e)g Fn(>)h Fp(0.)58 b(This)35 b(tests)i(whether)e -Fn(x)h Fp(is)g(an)g Fn(n)p Fp(th)f(p)s(o)m(w)m(er)h(of)g(a)h(rational) +3284 y Fp(cl_boolean)c(isqrt)h(\(const)f Fk(t)m(yp)s(e)5 +b Fp(&)31 b(x,)e Fk(t)m(yp)s(e)5 b Fp(*)31 b(root\))480 +3408 y(x)f Fr(should)g(b)s(e)g Fp(>)p Fr(=)g(0.)42 b(This)30 +b(function)h(sets)g Fp(*root)e Fr(to)i Fp(floor\(sqrt\(x\)\))c +Fr(and)j(returns)g(the)h(same)480 3533 y(v)-5 b(alue)31 +b(as)f Fp(sqrtp)p Fr(:)40 b(the)30 b(b)s(o)s(olean)h(v)-5 +b(alue)31 b Fp(\(expt\(*root,2\))26 b(==)k(x\))p Fr(.)0 +3829 y(F)-8 b(or)31 b Fp(n)p Fr(th)f(ro)s(ots,)h(the)f(classes)i +Fp(cl_RA)p Fr(,)d Fp(cl_I)g Fr(de\014ne)h(the)g(follo)m(wing)i(op)s +(eration:)0 4125 y Fp(cl_boolean)c(rootp)h(\(const)f +Fk(t)m(yp)s(e)5 b Fp(&)31 b(x,)e(const)g(cl_I&)g(n,)h +Fk(t)m(yp)s(e)5 b Fp(*)30 b(root\))480 4250 y(x)36 b +Fr(m)m(ust)f(b)s(e)g Fp(>)p Fr(=)g(0.)58 b Fp(n)36 b +Fr(m)m(ust)f(b)s(e)g Fp(>)h Fr(0.)58 b(This)35 b(tests)i(whether)e +Fp(x)h Fr(is)g(an)g Fp(n)p Fr(th)f(p)s(o)m(w)m(er)h(of)g(a)h(rational) 480 4374 y(n)m(um)m(b)s(er.)h(If)30 b(so,)h(it)g(returns)e(true)h(and)g -(the)h(exact)h(ro)s(ot)e(in)g Fn(*root)p Fp(,)g(else)h(it)g(returns)e +(the)h(exact)h(ro)s(ot)e(in)g Fp(*root)p Fr(,)g(else)h(it)g(returns)e (false.)0 4670 y(The)h(only)g(square)g(ro)s(ot)h(function)f(whic)m(h)g (accepts)i(negativ)m(e)h(n)m(um)m(b)s(ers)28 b(is)i(the)h(one)f(for)g -(class)h Fn(cl_N)p Fp(:)0 4966 y Fn(cl_N)e(sqrt)g(\(const)g(cl_N&)g -(z\))480 5091 y Fp(Returns)38 b(the)i(square)f(ro)s(ot)h(of)f -Fn(z)p Fp(,)j(as)d(de\014ned)g(b)m(y)g(the)g(form)m(ula)g -Fn(sqrt\(z\))28 b(=)i(exp\(log\(z\)/2\))p Fp(.)480 5215 +(class)h Fp(cl_N)p Fr(:)0 4966 y Fp(cl_N)e(sqrt)g(\(const)g(cl_N&)g +(z\))480 5091 y Fr(Returns)38 b(the)i(square)f(ro)s(ot)h(of)f +Fp(z)p Fr(,)j(as)d(de\014ned)g(b)m(y)g(the)g(form)m(ula)g +Fp(sqrt\(z\))28 b(=)i(exp\(log\(z\)/2\))p Fr(.)480 5215 y(Con)m(v)m(ersion)44 b(to)g(a)f(\015oating-p)s(oin)m(t)i(t)m(yp)s(e)e (or)h(to)g(a)f(complex)g(n)m(um)m(b)s(er)e(are)j(done)f(if)g(necessary) -8 b(.)480 5340 y(The)31 b(range)h(of)g(the)g(result)g(is)g(the)g(righ) -m(t)g(half)g(plane)g Fn(realpart\(sqrt\(z\)\))25 b(>=)30 -b(0)h Fp(including)h(the)p eop +m(t)g(half)g(plane)g Fp(realpart\(sqrt\(z\)\))25 b(>=)30 +b(0)h Fr(including)h(the)p eop %%Page: 22 24 -22 23 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +22 23 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 b(on)f(n)m(um)m(b)s(ers)2463 b(22)480 366 y(p)s(ositiv)m(e)35 b(imaginary)e(axis)h(and)g(0,)h(but)e(excluding)h(the)g(negativ)m(e)i (imaginary)d(axis.)52 b(The)33 b(result)480 491 y(is)d(an)h(exact)h(n)m -(um)m(b)s(er)c(only)i(if)g Fn(z)g Fp(is)h(an)f(exact)i(n)m(um)m(b)s -(er.)0 930 y Fq(4.8)68 b(T)-11 b(ranscenden)l(tal)31 -b(functions)0 1204 y Fp(The)c(transcenden)m(tal)h(functions)e(return)h +(um)m(b)s(er)c(only)i(if)g Fp(z)g Fr(is)h(an)f(exact)i(n)m(um)m(b)s +(er.)0 930 y Fs(4.8)68 b(T)-11 b(ranscenden)l(tal)31 +b(functions)0 1204 y Fr(The)c(transcenden)m(tal)h(functions)e(return)h (an)g(exact)h(result)f(if)g(the)h(argumen)m(t)e(is)i(exact)g(and)f(the) g(result)g(is)g(exact)0 1329 y(as)k(w)m(ell.)44 b(Otherwise)31 b(they)g(m)m(ust)f(return)g(inexact)i(n)m(um)m(b)s(ers)d(ev)m(en)j(if)f (the)g(argumen)m(t)g(is)g(exact.)44 b(F)-8 b(or)32 b(example,)0 -1453 y Fn(cos\(0\))d(=)h(1)g Fp(returns)f(the)h(rational)i(n)m(um)m(b)s -(er)c Fn(1)p Fp(.)0 1860 y Fk(4.8.1)63 b(Exp)s(onen)m(tial)30 -b(and)g(logarithmic)g(functions)0 2156 y Fn(cl_R)f(exp)h(\(const)e +1453 y Fp(cos\(0\))d(=)h(1)g Fr(returns)f(the)h(rational)i(n)m(um)m(b)s +(er)c Fp(1)p Fr(.)0 1860 y Fm(4.8.1)63 b(Exp)s(onen)m(tial)30 +b(and)g(logarithmic)g(functions)0 2156 y Fp(cl_R)f(exp)h(\(const)e (cl_R&)h(x\))0 2281 y(cl_N)g(exp)h(\(const)e(cl_N&)h(x\))480 -2405 y Fp(Returns)j(the)h(exp)s(onen)m(tial)h(function)f(of)g -Fn(x)p Fp(.)48 b(This)32 b(is)h Fn(e^x)f Fp(where)g Fn(e)h -Fp(is)g(the)g(base)g(of)g(the)g(natural)480 2530 y(logarithms.)41 +2405 y Fr(Returns)j(the)h(exp)s(onen)m(tial)h(function)f(of)g +Fp(x)p Fr(.)48 b(This)32 b(is)h Fp(e^x)f Fr(where)g Fp(e)h +Fr(is)g(the)g(base)g(of)g(the)g(natural)480 2530 y(logarithms.)41 b(The)29 b(range)i(of)g(the)f(result)h(is)f(the)h(en)m(tire)g(complex)f -(plane)g(excluding)h(0.)0 2700 y Fn(cl_R)e(ln)h(\(const)f(cl_R&)g(x\)) -480 2824 y(x)h Fp(m)m(ust)f(b)s(e)h Fn(>)g Fp(0.)41 b(Returns)30 -b(the)g(\(natural\))h(logarithm)g(of)f(x.)0 2994 y Fn(cl_N)f(log)h -(\(const)e(cl_N&)h(x\))480 3119 y Fp(Returns)43 b(the)i(\(natural\))f -(logarithm)h(of)f(x.)82 b(If)44 b Fn(x)f Fp(is)i(real)f(and)g(p)s -(ositiv)m(e,)49 b(this)44 b(is)g Fn(ln\(x\))p Fp(.)81 -b(In)480 3243 y(general,)34 b Fn(log\(x\))29 b(=)h(log\(abs\(x\)\))d(+) -j(i*phase\(x\))p Fp(.)45 b(The)32 b(range)g(of)h(the)g(result)f(is)h +(plane)g(excluding)h(0.)0 2700 y Fp(cl_R)e(ln)h(\(const)f(cl_R&)g(x\)) +480 2824 y(x)h Fr(m)m(ust)f(b)s(e)h Fp(>)g Fr(0.)41 b(Returns)30 +b(the)g(\(natural\))h(logarithm)g(of)f(x.)0 2994 y Fp(cl_N)f(log)h +(\(const)e(cl_N&)h(x\))480 3119 y Fr(Returns)43 b(the)i(\(natural\))f +(logarithm)h(of)f(x.)82 b(If)44 b Fp(x)f Fr(is)i(real)f(and)g(p)s +(ositiv)m(e,)49 b(this)44 b(is)g Fp(ln\(x\))p Fr(.)81 +b(In)480 3243 y(general,)34 b Fp(log\(x\))29 b(=)h(log\(abs\(x\)\))d(+) +j(i*phase\(x\))p Fr(.)45 b(The)32 b(range)g(of)h(the)g(result)f(is)h (the)f(strip)g(in)480 3368 y(the)f(complex)f(plane)g -Fn(-pi)g(<)g(imagpart\(log\(x\)\))25 b(<=)30 b(pi)p Fp(.)0 -3538 y Fn(cl_R)f(phase)g(\(const)g(cl_N&)g(x\))480 3662 -y Fp(Returns)38 b(the)h(angle)g(part)g(of)g Fn(x)f Fp(in)g(its)h(p)s +Fp(-pi)g(<)g(imagpart\(log\(x\)\))25 b(<=)30 b(pi)p Fr(.)0 +3538 y Fp(cl_R)f(phase)g(\(const)g(cl_N&)g(x\))480 3662 +y Fr(Returns)38 b(the)h(angle)g(part)g(of)g Fp(x)f Fr(in)g(its)h(p)s (olar)g(represen)m(tation)g(as)g(a)g(complex)g(n)m(um)m(b)s(er.)63 -b(That)480 3787 y(is,)34 b Fn(phase\(x\))28 b(=)i -(atan\(realpart\(x\),imagpart)o(\(x\)\))o Fp(.)43 b(This)33 +b(That)480 3787 y(is,)34 b Fp(phase\(x\))28 b(=)i +(atan\(realpart\(x\),imagpart)o(\(x\)\))o Fr(.)43 b(This)33 b(is)g(also)h(the)f(imaginary)g(part)g(of)480 3911 y -Fn(log\(x\))p Fp(.)41 b(The)31 b(range)g(of)g(the)h(result)f(is)g(the)g -(in)m(terv)-5 b(al)32 b Fn(-pi)e(<)g(phase\(x\))e(<=)i(pi)p -Fp(.)42 b(The)31 b(result)g(will)480 4036 y(b)s(e)f(an)g(exact)i(n)m -(um)m(b)s(er)c(only)i(if)h Fn(zerop\(x\))d Fp(or)i(if)h -Fn(x)f Fp(is)g(real)h(and)f(p)s(ositiv)m(e.)0 4206 y -Fn(cl_R)f(log)h(\(const)e(cl_R&)h(a,)h(const)f(cl_R&)g(b\))480 -4330 y(a)j Fp(and)g Fn(b)g Fp(m)m(ust)g(b)s(e)g Fn(>)g -Fp(0.)48 b(Returns)31 b(the)i(logarithm)g(of)f Fn(a)g -Fp(with)h(resp)s(ect)f(to)i(base)e Fn(b)p Fp(.)47 b Fn(log\(a,b\))28 -b(=)480 4455 y(ln\(a\)/ln\(b\))p Fp(.)38 b(The)30 b(result)g(can)g(b)s -(e)g(exact)i(only)f(if)f Fn(a)g(=)g(1)g Fp(or)g(if)h -Fn(a)f Fp(and)g Fn(b)g Fp(are)g(b)s(oth)g(rational.)0 -4624 y Fn(cl_N)f(log)h(\(const)e(cl_N&)h(a,)h(const)f(cl_N&)g(b\))480 -4749 y Fp(Returns)g(the)i(logarithm)f(of)h Fn(a)f Fp(with)g(resp)s(ect) -g(to)i(base)e Fn(b)p Fp(.)40 b Fn(log\(a,b\))29 b(=)h -(log\(a\)/log\(b\))p Fp(.)0 4919 y Fn(cl_N)f(expt)g(\(const)g(cl_N&)g -(x,)h(const)f(cl_N&)g(y\))480 5043 y Fp(Exp)s(onen)m(tiation:)42 -b(Returns)29 b Fn(x^y)g(=)h(exp\(y*log\(x\)\))p Fp(.)0 +Fp(log\(x\))p Fr(.)41 b(The)31 b(range)g(of)g(the)h(result)f(is)g(the)g +(in)m(terv)-5 b(al)32 b Fp(-pi)e(<)g(phase\(x\))e(<=)i(pi)p +Fr(.)42 b(The)31 b(result)g(will)480 4036 y(b)s(e)f(an)g(exact)i(n)m +(um)m(b)s(er)c(only)i(if)h Fp(zerop\(x\))d Fr(or)i(if)h +Fp(x)f Fr(is)g(real)h(and)f(p)s(ositiv)m(e.)0 4206 y +Fp(cl_R)f(log)h(\(const)e(cl_R&)h(a,)h(const)f(cl_R&)g(b\))480 +4330 y(a)j Fr(and)g Fp(b)g Fr(m)m(ust)g(b)s(e)g Fp(>)g +Fr(0.)48 b(Returns)31 b(the)i(logarithm)g(of)f Fp(a)g +Fr(with)h(resp)s(ect)f(to)i(base)e Fp(b)p Fr(.)47 b Fp(log\(a,b\))28 +b(=)480 4455 y(ln\(a\)/ln\(b\))p Fr(.)38 b(The)30 b(result)g(can)g(b)s +(e)g(exact)i(only)f(if)f Fp(a)g(=)g(1)g Fr(or)g(if)h +Fp(a)f Fr(and)g Fp(b)g Fr(are)g(b)s(oth)g(rational.)0 +4624 y Fp(cl_N)f(log)h(\(const)e(cl_N&)h(a,)h(const)f(cl_N&)g(b\))480 +4749 y Fr(Returns)g(the)i(logarithm)f(of)h Fp(a)f Fr(with)g(resp)s(ect) +g(to)i(base)e Fp(b)p Fr(.)40 b Fp(log\(a,b\))29 b(=)h +(log\(a\)/log\(b\))p Fr(.)0 4919 y Fp(cl_N)f(expt)g(\(const)g(cl_N&)g +(x,)h(const)f(cl_N&)g(y\))480 5043 y Fr(Exp)s(onen)m(tiation:)42 +b(Returns)29 b Fp(x^y)g(=)h(exp\(y*log\(x\)\))p Fr(.)0 5340 y(The)g(constan)m(t)i(e)e(=)g(exp\(1\))i(=)e(2.71828)p -Fl(:)15 b(:)g(:)34 b Fp(is)c(returned)g(b)m(y)g(the)g(follo)m(wing)i +Fn(:)15 b(:)g(:)34 b Fr(is)c(returned)g(b)m(y)g(the)g(follo)m(wing)i (functions:)p eop %%Page: 23 25 -23 24 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(23)0 366 y Fn(cl_F)29 -b(cl_exp1)g(\(cl_float_format_t)c(f\))480 491 y Fp(Returns)k(e)i(as)g -(a)g(\015oat)g(of)f(format)g Fn(f)p Fp(.)0 657 y Fn(cl_F)f(cl_exp1)g -(\(const)f(cl_F&)h(y\))480 782 y Fp(Returns)g(e)i(in)f(the)h(\015oat)g -(format)f(of)g Fn(y)p Fp(.)0 948 y Fn(cl_F)f(cl_exp1)g(\(void\))480 -1073 y Fp(Returns)g(e)i(as)g(a)g(\015oat)g(of)f(format)g -Fn(cl_default_float_format)p Fp(.)0 1472 y Fk(4.8.2)63 -b(T)-10 b(rigonometric)30 b(functions)0 1767 y Fn(cl_R)f(sin)h(\(const) -e(cl_R&)h(x\))480 1892 y Fp(Returns)g Fn(sin\(x\))p Fp(.)39 +23 24 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(23)0 366 y Fp(cl_F)29 +b(cl_exp1)g(\(cl_float_format_t)c(f\))480 491 y Fr(Returns)k(e)i(as)g +(a)g(\015oat)g(of)f(format)g Fp(f)p Fr(.)0 657 y Fp(cl_F)f(cl_exp1)g +(\(const)f(cl_F&)h(y\))480 782 y Fr(Returns)g(e)i(in)f(the)h(\015oat)g +(format)f(of)g Fp(y)p Fr(.)0 948 y Fp(cl_F)f(cl_exp1)g(\(void\))480 +1073 y Fr(Returns)g(e)i(as)g(a)g(\015oat)g(of)f(format)g +Fp(cl_default_float_format)p Fr(.)0 1472 y Fm(4.8.2)63 +b(T)-10 b(rigonometric)30 b(functions)0 1767 y Fp(cl_R)f(sin)h(\(const) +e(cl_R&)h(x\))480 1892 y Fr(Returns)g Fp(sin\(x\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f(in)m(terv)-5 -b(al)32 b Fn(-1)e(<=)f(sin\(x\))g(<=)h(1)p Fp(.)0 2058 -y Fn(cl_N)f(sin)h(\(const)e(cl_N&)h(z\))480 2183 y Fp(Returns)g -Fn(sin\(z\))p Fp(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f -(en)m(tire)i(complex)e(plane.)0 2349 y Fn(cl_R)f(cos)h(\(const)e(cl_R&) -h(x\))480 2474 y Fp(Returns)g Fn(cos\(x\))p Fp(.)39 b(The)30 +b(al)32 b Fp(-1)e(<=)f(sin\(x\))g(<=)h(1)p Fr(.)0 2058 +y Fp(cl_N)f(sin)h(\(const)e(cl_N&)h(z\))480 2183 y Fr(Returns)g +Fp(sin\(z\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f +(en)m(tire)i(complex)e(plane.)0 2349 y Fp(cl_R)f(cos)h(\(const)e(cl_R&) +h(x\))480 2474 y Fr(Returns)g Fp(cos\(x\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f(in)m(terv)-5 -b(al)32 b Fn(-1)e(<=)f(cos\(x\))g(<=)h(1)p Fp(.)0 2640 -y Fn(cl_N)f(cos)h(\(const)e(cl_N&)h(x\))480 2764 y Fp(Returns)g -Fn(cos\(z\))p Fp(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f -(en)m(tire)i(complex)e(plane.)0 2931 y Fn(struct)f(cl_cos_sin_t)e({)j +b(al)32 b Fp(-1)e(<=)f(cos\(x\))g(<=)h(1)p Fr(.)0 2640 +y Fp(cl_N)f(cos)h(\(const)e(cl_N&)h(x\))480 2764 y Fr(Returns)g +Fp(cos\(z\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f +(en)m(tire)i(complex)e(plane.)0 2931 y Fp(struct)f(cl_cos_sin_t)e({)j (cl_R)f(cos;)g(cl_R)g(sin;)h(};)0 3055 y(cl_cos_sin_t)d(cl_cos_sin)h -(\(const)g(cl_R&)h(x\))480 3180 y Fp(Returns)k(b)s(oth)h -Fn(sin\(x\))f Fp(and)g Fn(cos\(x\))p Fp(.)51 b(This)34 +(\(const)g(cl_R&)h(x\))480 3180 y Fr(Returns)k(b)s(oth)h +Fp(sin\(x\))f Fr(and)g Fp(cos\(x\))p Fr(.)51 b(This)34 b(is)g(more)g(e\016cien)m(t)i(than)e(computing)f(them)g(sepa-)480 -3304 y(rately)-8 b(.)42 b(The)30 b(relation)i Fn(cos^2)d(+)h(sin^2)e(=) -j(1)f Fp(will)g(hold)g(only)h(appro)m(ximately)-8 b(.)0 -3471 y Fn(cl_R)29 b(tan)h(\(const)e(cl_R&)h(x\))0 3595 -y(cl_N)g(tan)h(\(const)e(cl_N&)h(x\))480 3720 y Fp(Returns)g -Fn(tan\(x\))g(=)h(sin\(x\)/cos\(x\))p Fp(.)0 3886 y Fn(cl_N)f(cis)h +3304 y(rately)-8 b(.)42 b(The)30 b(relation)i Fp(cos^2)d(+)h(sin^2)e(=) +j(1)f Fr(will)g(hold)g(only)h(appro)m(ximately)-8 b(.)0 +3471 y Fp(cl_R)29 b(tan)h(\(const)e(cl_R&)h(x\))0 3595 +y(cl_N)g(tan)h(\(const)e(cl_N&)h(x\))480 3720 y Fr(Returns)g +Fp(tan\(x\))g(=)h(sin\(x\)/cos\(x\))p Fr(.)0 3886 y Fp(cl_N)f(cis)h (\(const)e(cl_R&)h(x\))0 4011 y(cl_N)g(cis)h(\(const)e(cl_N&)h(x\))480 -4135 y Fp(Returns)h Fn(exp\(i*x\))p Fp(.)41 b(The)30 -b(name)g(`)p Fn(cis)p Fp(')h(means)f(\\cos)i Fn(+)f Fp(i)g(sin",)g(b)s -(ecause)g Fn(e^\(i*x\))e(=)h(cos\(x\))e(+)480 4260 y(i*sin\(x\))p -Fp(.)0 4426 y Fn(cl_N)h(asin)g(\(const)g(cl_N&)g(z\))480 -4551 y Fp(Returns)24 b Fn(arcsin\(z\))p Fp(.)37 b(This)24 -b(is)h(de\014ned)f(as)h Fn(arcsin\(z\))j(=)i(log\(iz+sqrt\(1-z^2\)\)/i) -20 b Fp(and)k(sat-)480 4675 y(is\014es)g Fn(arcsin\(-z\))j(=)j -(-arcsin\(z\))p Fp(.)36 b(The)23 b(range)i(of)f(the)g(result)f(is)h +4135 y Fr(Returns)h Fp(exp\(i*x\))p Fr(.)41 b(The)30 +b(name)g(`)p Fp(cis)p Fr(')h(means)f(\\cos)i Fp(+)f Fr(i)g(sin",)g(b)s +(ecause)g Fp(e^\(i*x\))e(=)h(cos\(x\))e(+)480 4260 y(i*sin\(x\))p +Fr(.)0 4426 y Fp(cl_N)h(asin)g(\(const)g(cl_N&)g(z\))480 +4551 y Fr(Returns)24 b Fp(arcsin\(z\))p Fr(.)37 b(This)24 +b(is)h(de\014ned)f(as)h Fp(arcsin\(z\))j(=)i(log\(iz+sqrt\(1-z^2\)\)/i) +20 b Fr(and)k(sat-)480 4675 y(is\014es)g Fp(arcsin\(-z\))j(=)j +(-arcsin\(z\))p Fr(.)36 b(The)23 b(range)i(of)f(the)g(result)f(is)h (the)g(strip)g(in)f(the)h(complex)g(do-)480 4800 y(main)g -Fn(-pi/2)29 b(<=)h(realpart\(arcsin\(z\)\))25 b(<=)30 -b(pi/2)p Fp(,)25 b(excluding)g(the)h(n)m(um)m(b)s(ers)c(with)j -Fn(realpart)480 4925 y(=)30 b(-pi/2)c Fp(and)g Fn(imagpart)i(<)i(0)d -Fp(and)f(the)h(n)m(um)m(b)s(ers)e(with)i Fn(realpart)h(=)i(pi/2)25 -b Fp(and)i Fn(imagpart)h(>)i(0)p Fp(.)0 5091 y Fn(cl_N)f(acos)g -(\(const)g(cl_N&)g(z\))480 5215 y Fp(Returns)19 b Fn(arccos\(z\))p -Fp(.)35 b(This)20 b(is)g(de\014ned)f(as)i Fn(arccos\(z\))27 +Fp(-pi/2)29 b(<=)h(realpart\(arcsin\(z\)\))25 b(<=)30 +b(pi/2)p Fr(,)25 b(excluding)g(the)h(n)m(um)m(b)s(ers)c(with)j +Fp(realpart)480 4925 y(=)30 b(-pi/2)c Fr(and)g Fp(imagpart)i(<)i(0)d +Fr(and)f(the)h(n)m(um)m(b)s(ers)e(with)i Fp(realpart)h(=)i(pi/2)25 +b Fr(and)i Fp(imagpart)h(>)i(0)p Fr(.)0 5091 y Fp(cl_N)f(acos)g +(\(const)g(cl_N&)g(z\))480 5215 y Fr(Returns)19 b Fp(arccos\(z\))p +Fr(.)35 b(This)20 b(is)g(de\014ned)f(as)i Fp(arccos\(z\))27 b(=)j(pi/2)g(-)g(arcsin\(z\))d(=)k(log\(z+i*sqrt\(1-)480 -5340 y(z^2\)\)/i)36 b Fp(and)i(satis\014es)g Fn(arccos\(-z\))28 -b(=)i(pi)g(-)g(arccos\(z\))p Fp(.)61 b(The)38 b(range)h(of)f(the)g +5340 y(z^2\)\)/i)36 b Fr(and)i(satis\014es)g Fp(arccos\(-z\))28 +b(=)i(pi)g(-)g(arccos\(z\))p Fr(.)61 b(The)38 b(range)h(of)f(the)g (result)h(is)f(the)p eop %%Page: 24 26 -24 25 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +24 25 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 b(on)f(n)m(um)m(b)s(ers)2463 b(24)480 366 y(strip)40 -b(in)f(the)i(complex)e(domain)g Fn(0)30 b(<=)g(realpart\(arcsin\(z\)\)) -25 b(<=)30 b(pi)p Fp(,)42 b(excluding)e(the)g(n)m(um-)480 -491 y(b)s(ers)33 b(with)g Fn(realpart)28 b(=)i(0)k Fp(and)f -Fn(imagpart)28 b(<)i(0)k Fp(and)f(the)h(n)m(um)m(b)s(ers)e(with)h -Fn(realpart)28 b(=)i(pi)k Fp(and)480 616 y Fn(imagpart)28 -b(>)i(0)p Fp(.)0 785 y Fn(cl_R)f(atan)g(\(const)g(cl_R&)g(x,)h(const)f -(cl_R&)g(y\))480 910 y Fp(Returns)37 b(the)i(angle)g(of)g(the)f(p)s +b(in)f(the)i(complex)e(domain)g Fp(0)30 b(<=)g(realpart\(arcsin\(z\)\)) +25 b(<=)30 b(pi)p Fr(,)42 b(excluding)e(the)g(n)m(um-)480 +491 y(b)s(ers)33 b(with)g Fp(realpart)28 b(=)i(0)k Fr(and)f +Fp(imagpart)28 b(<)i(0)k Fr(and)f(the)h(n)m(um)m(b)s(ers)e(with)h +Fp(realpart)28 b(=)i(pi)k Fr(and)480 616 y Fp(imagpart)28 +b(>)i(0)p Fr(.)0 785 y Fp(cl_R)f(atan)g(\(const)g(cl_R&)g(x,)h(const)f +(cl_R&)g(y\))480 910 y Fr(Returns)37 b(the)i(angle)g(of)g(the)f(p)s (olar)g(represen)m(tation)h(of)g(the)f(complex)g(n)m(um)m(b)s(er)e -Fn(x+iy)p Fp(.)63 b(This)38 b(is)480 1034 y Fn(atan\(y/x\))27 -b Fp(if)i Fn(x>0)p Fp(.)40 b(The)28 b(range)i(of)f(the)h(result)f(is)h -(the)f(in)m(terv)-5 b(al)31 b Fn(-pi)e(<)h(atan\(x,y\))e(<=)i(pi)p -Fp(.)39 b(The)480 1159 y(result)j(will)g(b)s(e)f(an)g(exact)i(n)m(um)m -(b)s(er)d(only)h(if)h Fn(x)30 b(>)g(0)41 b Fp(and)h Fn(y)f -Fp(is)h(the)f(exact)j Fn(0)p Fp(.)74 b(W)-10 b(ARNING:)43 +Fp(x+iy)p Fr(.)63 b(This)38 b(is)480 1034 y Fp(atan\(y/x\))27 +b Fr(if)i Fp(x>0)p Fr(.)40 b(The)28 b(range)i(of)f(the)h(result)f(is)h +(the)f(in)m(terv)-5 b(al)31 b Fp(-pi)e(<)h(atan\(x,y\))e(<=)i(pi)p +Fr(.)39 b(The)480 1159 y(result)j(will)g(b)s(e)f(an)g(exact)i(n)m(um)m +(b)s(er)d(only)h(if)h Fp(x)30 b(>)g(0)41 b Fr(and)h Fp(y)f +Fr(is)h(the)f(exact)j Fp(0)p Fr(.)74 b(W)-10 b(ARNING:)43 b(In)480 1283 y(Common)26 b(Lisp,)i(this)g(function)g(is)g(called)h(as) -g Fn(\(atan)g(y)h(x\))p Fp(,)e(with)g(rev)m(ersed)g(order)g(of)g -(argumen)m(ts.)0 1453 y Fn(cl_R)h(atan)g(\(const)g(cl_R&)g(x\))480 -1578 y Fp(Returns)36 b Fn(arctan\(x\))p Fp(.)59 b(This)37 -b(is)g(the)g(same)g(as)g Fn(atan\(1,x\))p Fp(.)59 b(The)37 +g Fp(\(atan)g(y)h(x\))p Fr(,)e(with)g(rev)m(ersed)g(order)g(of)g +(argumen)m(ts.)0 1453 y Fp(cl_R)h(atan)g(\(const)g(cl_R&)g(x\))480 +1578 y Fr(Returns)36 b Fp(arctan\(x\))p Fr(.)59 b(This)37 +b(is)g(the)g(same)g(as)g Fp(atan\(1,x\))p Fr(.)59 b(The)37 b(range)g(of)h(the)f(result)g(is)h(the)480 1702 y(in)m(terv)-5 -b(al)34 b Fn(-pi/2)29 b(<)h(atan\(x\))e(<)i(pi/2)p Fp(.)47 +b(al)34 b Fp(-pi/2)29 b(<)h(atan\(x\))e(<)i(pi/2)p Fr(.)47 b(The)32 b(result)g(will)h(b)s(e)f(an)h(exact)h(n)m(um)m(b)s(er)c(only) -j(if)g Fn(x)f Fp(is)h(the)480 1827 y(exact)f Fn(0)p Fp(.)0 -1996 y Fn(cl_N)d(atan)g(\(const)g(cl_N&)g(z\))480 2121 -y Fp(Returns)43 b Fn(arctan\(z\))p Fp(.)80 b(This)44 -b(is)g(de\014ned)g(as)g Fn(arctan\(z\))28 b(=)i +j(if)g Fp(x)f Fr(is)h(the)480 1827 y(exact)f Fp(0)p Fr(.)0 +1996 y Fp(cl_N)d(atan)g(\(const)g(cl_N&)g(z\))480 2121 +y Fr(Returns)43 b Fp(arctan\(z\))p Fr(.)80 b(This)44 +b(is)g(de\014ned)g(as)g Fp(arctan\(z\))28 b(=)i (\(log\(1+iz\)-log\(1-iz\)\))25 b(/)30 b(2i)480 2245 -y Fp(and)23 b(satis\014es)h Fn(arctan\(-z\))k(=)i(-arctan\(z\))p -Fp(.)35 b(The)23 b(range)h(of)g(the)g(result)f(is)h(the)g(strip)f(in)g -(the)h(com-)480 2370 y(plex)45 b(domain)f Fn(-pi/2)29 -b(<=)h(realpart\(arctan\(z\)\))25 b(<=)30 b(pi/2)p Fp(,)47 +y Fr(and)23 b(satis\014es)h Fp(arctan\(-z\))k(=)i(-arctan\(z\))p +Fr(.)35 b(The)23 b(range)h(of)g(the)g(result)f(is)h(the)g(strip)f(in)g +(the)h(com-)480 2370 y(plex)45 b(domain)f Fp(-pi/2)29 +b(<=)h(realpart\(arctan\(z\)\))25 b(<=)30 b(pi/2)p Fr(,)47 b(excluding)e(the)h(n)m(um)m(b)s(ers)c(with)480 2494 -y Fn(realpart)28 b(=)i(-pi/2)42 b Fp(and)h Fn(imagpart)28 -b(>=)i(0)42 b Fp(and)h(the)h(n)m(um)m(b)s(ers)c(with)j -Fn(realpart)28 b(=)i(pi/2)43 b Fp(and)480 2619 y Fn(imagpart)28 -b(<=)i(0)p Fp(.)0 2915 y(The)g(constan)m(t)i(pi)e(=)g(3.14)p -Fl(:)15 b(:)g(:)33 b Fp(is)d(returned)f(b)m(y)h(the)h(follo)m(wing)h -(functions:)0 3212 y Fn(cl_F)d(cl_pi)g(\(cl_float_format_t)d(f\))480 -3336 y Fp(Returns)j(pi)i(as)f(a)h(\015oat)g(of)f(format)g -Fn(f)p Fp(.)0 3506 y Fn(cl_F)f(cl_pi)g(\(const)g(cl_F&)g(y\))480 -3631 y Fp(Returns)g(pi)i(in)f(the)g(\015oat)h(format)f(of)g -Fn(y)p Fp(.)0 3800 y Fn(cl_F)f(cl_pi)g(\(void\))480 3925 -y Fp(Returns)g(pi)i(as)f(a)h(\015oat)g(of)f(format)g -Fn(cl_default_float_format)p Fp(.)0 4331 y Fk(4.8.3)63 -b(Hyp)s(erb)s(olic)31 b(functions)0 4627 y Fn(cl_R)e(sinh)g(\(const)g -(cl_R&)g(x\))480 4752 y Fp(Returns)g Fn(sinh\(x\))p Fp(.)0 -4921 y Fn(cl_N)g(sinh)g(\(const)g(cl_N&)g(z\))480 5046 -y Fp(Returns)g Fn(sinh\(z\))p Fp(.)39 b(The)30 b(range)h(of)f(the)h +y Fp(realpart)28 b(=)i(-pi/2)42 b Fr(and)h Fp(imagpart)28 +b(>=)i(0)42 b Fr(and)h(the)h(n)m(um)m(b)s(ers)c(with)j +Fp(realpart)28 b(=)i(pi/2)43 b Fr(and)480 2619 y Fp(imagpart)28 +b(<=)i(0)p Fr(.)0 2915 y(Arc)m(himedes')g(constan)m(t)i(pi)e(=)g(3.14)p +Fn(:)15 b(:)g(:)33 b Fr(is)d(returned)f(b)m(y)h(the)h(follo)m(wing)h +(functions:)0 3212 y Fp(cl_F)d(cl_pi)g(\(cl_float_format_t)d(f\))480 +3336 y Fr(Returns)j(pi)i(as)f(a)h(\015oat)g(of)f(format)g +Fp(f)p Fr(.)0 3506 y Fp(cl_F)f(cl_pi)g(\(const)g(cl_F&)g(y\))480 +3631 y Fr(Returns)g(pi)i(in)f(the)g(\015oat)h(format)f(of)g +Fp(y)p Fr(.)0 3800 y Fp(cl_F)f(cl_pi)g(\(void\))480 3925 +y Fr(Returns)g(pi)i(as)f(a)h(\015oat)g(of)f(format)g +Fp(cl_default_float_format)p Fr(.)0 4331 y Fm(4.8.3)63 +b(Hyp)s(erb)s(olic)31 b(functions)0 4627 y Fp(cl_R)e(sinh)g(\(const)g +(cl_R&)g(x\))480 4752 y Fr(Returns)g Fp(sinh\(x\))p Fr(.)0 +4921 y Fp(cl_N)g(sinh)g(\(const)g(cl_N&)g(z\))480 5046 +y Fr(Returns)g Fp(sinh\(z\))p Fr(.)39 b(The)30 b(range)h(of)f(the)h (result)f(is)h(the)f(en)m(tire)h(complex)g(plane.)0 5215 -y Fn(cl_R)e(cosh)g(\(const)g(cl_R&)g(x\))480 5340 y Fp(Returns)g -Fn(cosh\(x\))p Fp(.)39 b(The)30 b(range)h(of)f(the)h(result)f(is)h(the) -f(in)m(terv)-5 b(al)32 b Fn(cosh\(x\))c(>=)i(1)p Fp(.)p +y Fp(cl_R)e(cosh)g(\(const)g(cl_R&)g(x\))480 5340 y Fr(Returns)g +Fp(cosh\(x\))p Fr(.)39 b(The)30 b(range)h(of)f(the)h(result)f(is)h(the) +f(in)m(terv)-5 b(al)32 b Fp(cosh\(x\))c(>=)i(1)p Fr(.)p eop %%Page: 25 27 -25 26 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(25)0 366 y Fn(cl_N)29 -b(cosh)g(\(const)g(cl_N&)g(z\))480 491 y Fp(Returns)g -Fn(cosh\(z\))p Fp(.)39 b(The)30 b(range)h(of)f(the)h(result)f(is)h(the) -f(en)m(tire)h(complex)g(plane.)0 667 y Fn(struct)e(cl_cosh_sinh_t)d({)k +25 26 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(25)0 366 y Fp(cl_N)29 +b(cosh)g(\(const)g(cl_N&)g(z\))480 491 y Fr(Returns)g +Fp(cosh\(z\))p Fr(.)39 b(The)30 b(range)h(of)f(the)h(result)f(is)h(the) +f(en)m(tire)h(complex)g(plane.)0 667 y Fp(struct)e(cl_cosh_sinh_t)d({)k (cl_R)f(cosh;)g(cl_R)h(sinh;)e(};)0 792 y(cl_cosh_sinh_t)e -(cl_cosh_sinh)i(\(const)g(cl_R&)h(x\))480 916 y Fp(Returns)44 -b(b)s(oth)h Fn(sinh\(x\))e Fp(and)i Fn(cosh\(x\))p Fp(.)83 +(cl_cosh_sinh)i(\(const)g(cl_R&)h(x\))480 916 y Fr(Returns)44 +b(b)s(oth)h Fp(sinh\(x\))e Fr(and)i Fp(cosh\(x\))p Fr(.)83 b(This)44 b(is)i(more)e(e\016cien)m(t)j(than)e(computing)f(them)480 -1041 y(separately)-8 b(.)42 b(The)30 b(relation)i Fn(cosh^2)c(-)j -(sinh^2)d(=)i(1)g Fp(will)h(hold)f(only)g(appro)m(ximately)-8 -b(.)0 1217 y Fn(cl_R)29 b(tanh)g(\(const)g(cl_R&)g(x\))0 +1041 y(separately)-8 b(.)42 b(The)30 b(relation)i Fp(cosh^2)c(-)j +(sinh^2)d(=)i(1)g Fr(will)h(hold)f(only)g(appro)m(ximately)-8 +b(.)0 1217 y Fp(cl_R)29 b(tanh)g(\(const)g(cl_R&)g(x\))0 1342 y(cl_N)g(tanh)g(\(const)g(cl_N&)g(x\))480 1466 y -Fp(Returns)g Fn(tanh\(x\))g(=)h(sinh\(x\)/cosh\(x\))p -Fp(.)0 1643 y Fn(cl_N)f(asinh)g(\(const)g(cl_N&)g(z\))480 -1767 y Fp(Returns)22 b Fn(arsinh\(z\))p Fp(.)36 b(This)22 -b(is)i(de\014ned)d(as)j Fn(arsinh\(z\))k(=)i(log\(z+sqrt\(1+z^2\)\))18 -b Fp(and)k(satis\014es)480 1892 y Fn(arsinh\(-z\))28 -b(=)i(-arsinh\(z\))p Fp(.)37 b(The)27 b(range)h(of)g(the)g(result)g(is) +Fr(Returns)g Fp(tanh\(x\))g(=)h(sinh\(x\)/cosh\(x\))p +Fr(.)0 1643 y Fp(cl_N)f(asinh)g(\(const)g(cl_N&)g(z\))480 +1767 y Fr(Returns)22 b Fp(arsinh\(z\))p Fr(.)36 b(This)22 +b(is)i(de\014ned)d(as)j Fp(arsinh\(z\))k(=)i(log\(z+sqrt\(1+z^2\)\))18 +b Fr(and)k(satis\014es)480 1892 y Fp(arsinh\(-z\))28 +b(=)i(-arsinh\(z\))p Fr(.)37 b(The)27 b(range)h(of)g(the)g(result)g(is) g(the)g(strip)f(in)h(the)g(complex)f(domain)480 2016 -y Fn(-pi/2)i(<=)h(imagpart\(arsinh\(z\)\))25 b(<=)30 -b(pi/2)p Fp(,)39 b(excluding)f(the)h(n)m(um)m(b)s(ers)c(with)j -Fn(imagpart)28 b(=)i(-)480 2141 y(pi/2)f Fp(and)h Fn(realpart)e(>)i(0)g -Fp(and)g(the)g(n)m(um)m(b)s(ers)e(with)i Fn(imagpart)f(=)h(pi/2)f -Fp(and)h Fn(realpart)e(<)i(0)p Fp(.)0 2317 y Fn(cl_N)f(acosh)g(\(const) -g(cl_N&)g(z\))480 2442 y Fp(Returns)d Fn(arcosh\(z\))p -Fp(.)37 b(This)26 b(is)h(de\014ned)f(as)h Fn(arcosh\(z\))h(=)i +y Fp(-pi/2)i(<=)h(imagpart\(arsinh\(z\)\))25 b(<=)30 +b(pi/2)p Fr(,)39 b(excluding)f(the)h(n)m(um)m(b)s(ers)c(with)j +Fp(imagpart)28 b(=)i(-)480 2141 y(pi/2)f Fr(and)h Fp(realpart)e(>)i(0)g +Fr(and)g(the)g(n)m(um)m(b)s(ers)e(with)i Fp(imagpart)f(=)h(pi/2)f +Fr(and)h Fp(realpart)e(<)i(0)p Fr(.)0 2317 y Fp(cl_N)f(acosh)g(\(const) +g(cl_N&)g(z\))480 2442 y Fr(Returns)d Fp(arcosh\(z\))p +Fr(.)37 b(This)26 b(is)h(de\014ned)f(as)h Fp(arcosh\(z\))h(=)i (2*log\(sqrt\(\(z+1\)/2\)+sqr)o(t\(\(z)o(-)480 2566 y(1\)/2\)\))p -Fp(.)102 b(The)51 b(range)h(of)g(the)g(result)f(is)g(the)h(half-strip)f -(in)h(the)f(complex)g(domain)g Fn(-pi)29 b(<)480 2691 +Fr(.)102 b(The)51 b(range)h(of)g(the)g(result)f(is)g(the)h(half-strip)f +(in)h(the)f(complex)g(domain)g Fp(-pi)29 b(<)480 2691 y(imagpart\(arcosh\(z\)\))c(<=)30 b(pi,)f(realpart\(arcosh\(z\)\))c(>=) -30 b(0)p Fp(,)76 b(excluding)67 b(the)f(n)m(um)m(b)s(ers)480 -2815 y(with)30 b Fn(realpart)e(=)i(0)g Fp(and)g Fn(-pi)g(<)g(imagpart)e -(<)i(0)p Fp(.)0 2991 y Fn(cl_N)f(atanh)g(\(const)g(cl_N&)g(z\))480 -3116 y Fp(Returns)38 b Fn(artanh\(z\))p Fp(.)64 b(This)39 -b(is)g(de\014ned)f(as)h Fn(artanh\(z\))28 b(=)i -(\(log\(1+z\)-log\(1-z\)\))25 b(/)30 b(2)39 b Fp(and)480 -3241 y(satis\014es)f Fn(artanh\(-z\))27 b(=)j(-artanh\(z\))p -Fp(.)58 b(The)37 b(range)g(of)g(the)g(result)g(is)g(the)h(strip)e(in)h -(the)g(com-)480 3365 y(plex)45 b(domain)f Fn(-pi/2)29 -b(<=)h(imagpart\(artanh\(z\)\))25 b(<=)30 b(pi/2)p Fp(,)47 +30 b(0)p Fr(,)76 b(excluding)67 b(the)f(n)m(um)m(b)s(ers)480 +2815 y(with)30 b Fp(realpart)e(=)i(0)g Fr(and)g Fp(-pi)g(<)g(imagpart)e +(<)i(0)p Fr(.)0 2991 y Fp(cl_N)f(atanh)g(\(const)g(cl_N&)g(z\))480 +3116 y Fr(Returns)38 b Fp(artanh\(z\))p Fr(.)64 b(This)39 +b(is)g(de\014ned)f(as)h Fp(artanh\(z\))28 b(=)i +(\(log\(1+z\)-log\(1-z\)\))25 b(/)30 b(2)39 b Fr(and)480 +3241 y(satis\014es)f Fp(artanh\(-z\))27 b(=)j(-artanh\(z\))p +Fr(.)58 b(The)37 b(range)g(of)g(the)g(result)g(is)g(the)h(strip)e(in)h +(the)g(com-)480 3365 y(plex)45 b(domain)f Fp(-pi/2)29 +b(<=)h(imagpart\(artanh\(z\)\))25 b(<=)30 b(pi/2)p Fr(,)47 b(excluding)e(the)h(n)m(um)m(b)s(ers)c(with)480 3490 -y Fn(imagpart)28 b(=)i(-pi/2)42 b Fp(and)h Fn(realpart)28 -b(<=)i(0)42 b Fp(and)h(the)h(n)m(um)m(b)s(ers)c(with)j -Fn(imagpart)28 b(=)i(pi/2)43 b Fp(and)480 3614 y Fn(realpart)28 -b(>=)i(0)p Fp(.)0 4038 y Fk(4.8.4)63 b(Euler)30 b(gamma)0 -4313 y Fp(Euler's)g(constan)m(t)i(C)e(=)g(0.577)p Fl(:)15 -b(:)g(:)33 b 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-(the)g(\015oat)h(format)f(of)h Fn(y)p Fp(.)0 1294 y Fn(cl_F)e -(cl_catalanconst)d(\(void\))480 1419 y Fp(Returns)j(Catalan's)j -(constan)m(t)g(as)e(a)h(\015oat)g(of)g(format)e Fn -(cl_default_float_format)p Fp(.)0 1881 y Fk(4.8.5)63 -b(Riemann)29 b(zeta)0 2160 y Fp(Riemann's)g(zeta)j(function)e(at)h(an)g -(in)m(tegral)h(p)s(oin)m(t)e Fn(s>1)f Fp(is)i(returned)e(b)m(y)h(the)h -(follo)m(wing)h(functions:)0 2470 y Fn(cl_F)d(cl_zeta)g(\(int)g(s,)h -(cl_float_format_t)25 b(f\))480 2594 y Fp(Returns)k(Riemann's)h(zeta)i -(function)e(at)h Fn(s)f Fp(as)g(a)h(\015oat)g(of)g(format)e -Fn(f)p Fp(.)0 2779 y Fn(cl_F)g(cl_zeta)g(\(int)g(s,)h(const)f(cl_F&)f -(y\))480 2904 y Fp(Returns)h(Riemann's)h(zeta)i(function)e(at)h -Fn(s)f Fp(in)g(the)g(\015oat)h(format)f(of)h Fn(y)p Fp(.)0 -3088 y Fn(cl_F)e(cl_zeta)g(\(int)g(s\))480 3213 y Fp(Returns)e -(Riemann's)g(zeta)j(function)d(at)i Fn(s)f Fp(as)g(a)h(\015oat)f(of)h -(format)e Fn(cl_default_float_format)p Fp(.)0 3708 y -Fq(4.9)68 b(F)-11 b(unctions)30 b(on)g(in)l(tegers)0 -4170 y Fk(4.9.1)63 b(Logical)30 b(functions)0 4450 y -Fp(In)m(tegers,)e(when)c(view)m(ed)i(as)g(in)f(t)m(w)m(o's)i(complemen) +b(constan)m(t)f(G)g(=)f(0.915)p Fn(:)15 b(:)g(:)33 b +Fr(is)e(returned)e(b)m(y)h(the)h(follo)m(wing)h(functions:)0 +676 y Fp(cl_F)d(cl_catalanconst)d(\(cl_float_format_t)g(f\))480 +800 y Fr(Returns)j(Catalan's)j(constan)m(t)g(as)e(a)h(\015oat)g(of)g +(format)e Fp(f)p Fr(.)0 985 y Fp(cl_F)g(cl_catalanconst)d(\(const)j +(cl_F&)g(y\))480 1110 y Fr(Returns)g(Catalan's)j(constan)m(t)g(in)e +(the)g(\015oat)h(format)f(of)h Fp(y)p Fr(.)0 1294 y Fp(cl_F)e +(cl_catalanconst)d(\(void\))480 1419 y Fr(Returns)j(Catalan's)j +(constan)m(t)g(as)e(a)h(\015oat)g(of)g(format)e Fp +(cl_default_float_format)p Fr(.)0 1881 y Fm(4.8.5)63 +b(Riemann)29 b(zeta)0 2160 y Fr(Riemann's)g(zeta)j(function)e(at)h(an)g +(in)m(tegral)h(p)s(oin)m(t)e Fp(s>1)f Fr(is)i(returned)e(b)m(y)h(the)h +(follo)m(wing)h(functions:)0 2470 y Fp(cl_F)d(cl_zeta)g(\(int)g(s,)h +(cl_float_format_t)25 b(f\))480 2594 y Fr(Returns)k(Riemann's)h(zeta)i +(function)e(at)h Fp(s)f Fr(as)g(a)h(\015oat)g(of)g(format)e +Fp(f)p Fr(.)0 2779 y Fp(cl_F)g(cl_zeta)g(\(int)g(s,)h(const)f(cl_F&)f +(y\))480 2904 y Fr(Returns)h(Riemann's)h(zeta)i(function)e(at)h +Fp(s)f Fr(in)g(the)g(\015oat)h(format)f(of)h Fp(y)p Fr(.)0 +3088 y Fp(cl_F)e(cl_zeta)g(\(int)g(s\))480 3213 y Fr(Returns)e +(Riemann's)g(zeta)j(function)d(at)i Fp(s)f Fr(as)g(a)h(\015oat)f(of)h +(format)e Fp(cl_default_float_format)p Fr(.)0 3708 y +Fs(4.9)68 b(F)-11 b(unctions)30 b(on)g(in)l(tegers)0 +4170 y Fm(4.9.1)63 b(Logical)30 b(functions)0 4450 y +Fr(In)m(tegers,)e(when)c(view)m(ed)i(as)g(in)f(t)m(w)m(o's)i(complemen) m(t)e(notation,)j(can)e(b)s(e)f(though)m(t)h(as)f(in\014nite)h(bit)f (strings)g(where)0 4574 y(the)31 b(bits')f(v)-5 b(alues)31 b(ev)m(en)m(tually)h(are)f(constan)m(t.)42 b(F)-8 b(or)31 -b(example,)431 4832 y Fn(17)47 b(=)h(......00010001)431 -4936 y(-6)f(=)h(......11111010)0 5215 y Fp(The)30 b(logical)i(op)s +b(example,)431 4832 y Fp(17)47 b(=)h(......00010001)431 +4936 y(-6)f(=)h(......11111010)0 5215 y Fr(The)30 b(logical)i(op)s (erations)f(view)f(in)m(tegers)h(as)g(suc)m(h)f(bit)g(strings)g(and)f (op)s(erate)i(on)f(eac)m(h)h(of)g(the)f(bit)g(p)s(ositions)g(in)0 5340 y(parallel.)p eop %%Page: 27 29 -27 28 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(27)0 366 y Fn(cl_I)29 +27 28 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(27)0 366 y Fp(cl_I)29 b(lognot)g(\(const)g(cl_I&)f(x\))0 491 y(cl_I)h(operator)f(~)i(\(const) -f(cl_I&)g(x\))480 616 y Fp(Logical)j(not,)f(lik)m(e)h -Fn(~x)e Fp(in)g(C.)g(This)g(is)g(the)h(same)e(as)i Fn(-1-x)p -Fp(.)0 776 y Fn(cl_I)e(logand)g(\(const)g(cl_I&)f(x,)i(const)f(cl_I&)g +f(cl_I&)g(x\))480 616 y Fr(Logical)j(not,)f(lik)m(e)h +Fp(~x)e Fr(in)g(C.)g(This)g(is)g(the)h(same)e(as)i Fp(-1-x)p +Fr(.)0 776 y Fp(cl_I)e(logand)g(\(const)g(cl_I&)f(x,)i(const)f(cl_I&)g (y\))0 900 y(cl_I)g(operator)f(&)i(\(const)f(cl_I&)g(x,)h(const)f -(cl_I&)g(y\))480 1025 y Fp(Logical)j(and,)e(lik)m(e)i -Fn(x)e(&)g(y)g Fp(in)g(C.)0 1185 y Fn(cl_I)f(logior)g(\(const)g(cl_I&)f +(cl_I&)g(y\))480 1025 y Fr(Logical)j(and,)e(lik)m(e)i +Fp(x)e(&)g(y)g Fr(in)g(C.)0 1185 y Fp(cl_I)f(logior)g(\(const)g(cl_I&)f (x,)i(const)f(cl_I&)g(y\))0 1309 y(cl_I)g(operator)f(|)i(\(const)f -(cl_I&)g(x,)h(const)f(cl_I&)g(y\))480 1434 y Fp(Logical)j(\(inclusiv)m -(e\))g(or,)f(lik)m(e)h Fn(x)e(|)g(y)g Fp(in)g(C.)0 1594 -y Fn(cl_I)f(logxor)g(\(const)g(cl_I&)f(x,)i(const)f(cl_I&)g(y\))0 +(cl_I&)g(x,)h(const)f(cl_I&)g(y\))480 1434 y Fr(Logical)j(\(inclusiv)m +(e\))g(or,)f(lik)m(e)h Fp(x)e(|)g(y)g Fr(in)g(C.)0 1594 +y Fp(cl_I)f(logxor)g(\(const)g(cl_I&)f(x,)i(const)f(cl_I&)g(y\))0 1718 y(cl_I)g(operator)f(^)i(\(const)f(cl_I&)g(x,)h(const)f(cl_I&)g -(y\))480 1843 y Fp(Exclusiv)m(e)i(or,)g(lik)m(e)h Fn(x)e(^)g(y)g -Fp(in)g(C.)0 2003 y Fn(cl_I)f(logeqv)g(\(const)g(cl_I&)f(x,)i(const)f -(cl_I&)g(y\))480 2127 y Fp(Bit)m(wise)j(equiv)-5 b(alence,)32 -b(lik)m(e)g Fn(~\(x)d(^)h(y\))g Fp(in)g(C.)0 2288 y Fn(cl_I)f(lognand)g +(y\))480 1843 y Fr(Exclusiv)m(e)i(or,)g(lik)m(e)h Fp(x)e(^)g(y)g +Fr(in)g(C.)0 2003 y Fp(cl_I)f(logeqv)g(\(const)g(cl_I&)f(x,)i(const)f +(cl_I&)g(y\))480 2127 y Fr(Bit)m(wise)j(equiv)-5 b(alence,)32 +b(lik)m(e)g Fp(~\(x)d(^)h(y\))g Fr(in)g(C.)0 2288 y Fp(cl_I)f(lognand)g (\(const)f(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 2412 -y Fp(Bit)m(wise)j(not)f(and,)f(lik)m(e)h Fn(~\(x)f(&)g(y\))g -Fp(in)g(C.)0 2572 y Fn(cl_I)f(lognor)g(\(const)g(cl_I&)f(x,)i(const)f -(cl_I&)g(y\))480 2697 y Fp(Bit)m(wise)j(not)f(or,)f(lik)m(e)i -Fn(~\(x)d(|)h(y\))g Fp(in)g(C.)0 2857 y Fn(cl_I)f(logandc1)f(\(const)h -(cl_I&)g(x,)h(const)f(cl_I&)g(y\))480 2981 y Fp(Logical)j(and,)e +y Fr(Bit)m(wise)j(not)f(and,)f(lik)m(e)h Fp(~\(x)f(&)g(y\))g +Fr(in)g(C.)0 2572 y Fp(cl_I)f(lognor)g(\(const)g(cl_I&)f(x,)i(const)f +(cl_I&)g(y\))480 2697 y Fr(Bit)m(wise)j(not)f(or,)f(lik)m(e)i +Fp(~\(x)d(|)h(y\))g Fr(in)g(C.)0 2857 y Fp(cl_I)f(logandc1)f(\(const)h +(cl_I&)g(x,)h(const)f(cl_I&)g(y\))480 2981 y Fr(Logical)j(and,)e (complemen)m(ting)g(the)h(\014rst)e(argumen)m(t,)h(lik)m(e)i -Fn(~x)e(&)g(y)g Fp(in)g(C.)0 3141 y Fn(cl_I)f(logandc2)f(\(const)h -(cl_I&)g(x,)h(const)f(cl_I&)g(y\))480 3266 y Fp(Logical)j(and,)e +Fp(~x)e(&)g(y)g Fr(in)g(C.)0 3141 y Fp(cl_I)f(logandc2)f(\(const)h +(cl_I&)g(x,)h(const)f(cl_I&)g(y\))480 3266 y Fr(Logical)j(and,)e (complemen)m(ting)g(the)h(second)f(argumen)m(t,)g(lik)m(e)i -Fn(x)e(&)g(~y)g Fp(in)g(C.)0 3426 y Fn(cl_I)f(logorc1)g(\(const)f -(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 3550 y Fp(Logical)j(or,)f +Fp(x)e(&)g(~y)g Fr(in)g(C.)0 3426 y Fp(cl_I)f(logorc1)g(\(const)f +(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 3550 y Fr(Logical)j(or,)f (complemen)m(ting)f(the)g(\014rst)g(argumen)m(t,)g(lik)m(e)i -Fn(~x)d(|)i(y)f Fp(in)g(C.)0 3710 y Fn(cl_I)f(logorc2)g(\(const)f -(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 3835 y Fp(Logical)j(or,)f +Fp(~x)d(|)i(y)f Fr(in)g(C.)0 3710 y Fp(cl_I)f(logorc2)g(\(const)f +(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 3835 y Fr(Logical)j(or,)f (complemen)m(ting)f(the)g(second)h(argumen)m(t,)f(lik)m(e)i -Fn(x)e(|)g(~y)f Fp(in)h(C.)0 4127 y(These)g(op)s(erations)h(are)g(all)g +Fp(x)e(|)g(~y)f Fr(in)h(C.)0 4127 y(These)g(op)s(erations)h(are)g(all)g (a)m(v)-5 b(ailable)33 b(though)d(the)g(function)0 4418 -y Fn(cl_I)f(boole)g(\(cl_boole)f(op,)h(const)g(cl_I&)g(x,)h(const)f -(cl_I&)g(y\))0 4692 y Fp(where)i Fn(op)g Fp(m)m(ust)g(ha)m(v)m(e)i(one) +y Fp(cl_I)f(boole)g(\(cl_boole)f(op,)h(const)g(cl_I&)g(x,)h(const)f +(cl_I&)g(y\))0 4692 y Fr(where)i Fp(op)g Fr(m)m(ust)g(ha)m(v)m(e)i(one) f(of)g(the)g(16)h(v)-5 b(alues)32 b(\(eac)m(h)h(one)f(stands)g(for)f(a) i(function)e(whic)m(h)h(com)m(bines)f(t)m(w)m(o)i(bits)0 -4817 y(in)m(to)24 b(one)f(bit\):)38 b Fn(boole_clr)p -Fp(,)22 b Fn(boole_set)p Fp(,)h Fn(boole_1)p Fp(,)f Fn(boole_2)p -Fp(,)h Fn(boole_c1)p Fp(,)g Fn(boole_c2)p Fp(,)f Fn(boole_and)p -Fp(,)h Fn(boole_)0 4941 y(ior)p Fp(,)37 b Fn(boole_xor)p -Fp(,)e Fn(boole_eqv)p Fp(,)g Fn(boole_nand)p Fp(,)g Fn(boole_nor)p -Fp(,)g Fn(boole_andc1)p Fp(,)g Fn(boole_andc2)p Fp(,)f -Fn(boole_orc1)p Fp(,)0 5066 y Fn(boole_orc2)p Fp(.)0 +4817 y(in)m(to)24 b(one)f(bit\):)38 b Fp(boole_clr)p +Fr(,)22 b Fp(boole_set)p Fr(,)h Fp(boole_1)p Fr(,)f Fp(boole_2)p +Fr(,)h Fp(boole_c1)p Fr(,)g Fp(boole_c2)p Fr(,)f Fp(boole_and)p +Fr(,)h Fp(boole_)0 4941 y(ior)p Fr(,)37 b Fp(boole_xor)p +Fr(,)e Fp(boole_eqv)p Fr(,)g Fp(boole_nand)p Fr(,)g Fp(boole_nor)p +Fr(,)g Fp(boole_andc1)p Fr(,)g Fp(boole_andc2)p Fr(,)f +Fp(boole_orc1)p Fr(,)0 5066 y Fp(boole_orc2)p Fr(.)0 5340 y(Other)c(functions)g(that)h(view)f(in)m(tegers)i(as)f(bit)f (strings:)p eop %%Page: 28 30 -28 29 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(28)0 366 y Fn(cl_boolean)28 +28 29 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(28)0 366 y Fp(cl_boolean)28 b(logtest)g(\(const)h(cl_I&)g(x,)g(const)g(cl_I&)g(y\))480 -491 y Fp(Returns)g(true)i(if)f(some)g(bit)g(is)h(set)f(in)h(b)s(oth)e -Fn(x)h Fp(and)g Fn(y)p Fp(,)g(i.e.)42 b(if)30 b Fn(logand\(x,y\))d(!=)j -(0)p Fp(.)0 671 y Fn(cl_boolean)e(logbitp)g(\(const)h(cl_I&)g(n,)g -(const)g(cl_I&)g(x\))480 795 y Fp(Returns)f(true)g(if)g(the)h -Fn(n)p Fp(th)f(bit)g(\(from)g(the)g(righ)m(t\))h(of)g -Fn(x)f Fp(is)h(set.)40 b(Bit)30 b(0)e(is)h(the)g(least)g(signi\014can)m -(t)h(bit.)0 975 y Fn(uintL)f(logcount)f(\(const)h(cl_I&)g(x\))480 -1100 y Fp(Returns)d(the)i(n)m(um)m(b)s(er)c(of)k(one)f(bits)g(in)g -Fn(x)p Fp(,)h(if)f Fn(x)g(>)p Fp(=)f(0,)j(or)e(the)g(n)m(um)m(b)s(er)e -(of)i(zero)h(bits)f(in)g Fn(x)p Fp(,)h(if)f Fn(x)f(<)h -Fp(0.)0 1404 y(The)j(follo)m(wing)i(functions)e(op)s(erate)h(on)f(in)m +491 y Fr(Returns)g(true)i(if)f(some)g(bit)g(is)h(set)f(in)h(b)s(oth)e +Fp(x)h Fr(and)g Fp(y)p Fr(,)g(i.e.)42 b(if)30 b Fp(logand\(x,y\))d(!=)j +(0)p Fr(.)0 671 y Fp(cl_boolean)e(logbitp)g(\(const)h(cl_I&)g(n,)g +(const)g(cl_I&)g(x\))480 795 y Fr(Returns)f(true)g(if)g(the)h +Fp(n)p Fr(th)f(bit)g(\(from)g(the)g(righ)m(t\))h(of)g +Fp(x)f Fr(is)h(set.)40 b(Bit)30 b(0)e(is)h(the)g(least)g(signi\014can)m +(t)h(bit.)0 975 y Fp(uintL)f(logcount)f(\(const)h(cl_I&)g(x\))480 +1100 y Fr(Returns)d(the)i(n)m(um)m(b)s(er)c(of)k(one)f(bits)g(in)g +Fp(x)p Fr(,)h(if)f Fp(x)g(>)p Fr(=)f(0,)j(or)e(the)g(n)m(um)m(b)s(er)e +(of)i(zero)h(bits)f(in)g Fp(x)p Fr(,)h(if)f Fp(x)f(<)h +Fr(0.)0 1404 y(The)j(follo)m(wing)i(functions)e(op)s(erate)h(on)f(in)m (terv)-5 b(als)31 b(of)g(bits)f(in)g(in)m(tegers.)42 -b(The)30 b(t)m(yp)s(e)240 1660 y Fn(struct)46 b(cl_byte)g({)h(uintL)g -(size;)f(uintL)g(position;)g(};)0 1937 y Fp(represen)m(ts)39 +b(The)30 b(t)m(yp)s(e)240 1660 y Fp(struct)46 b(cl_byte)g({)h(uintL)g +(size;)f(uintL)g(position;)g(};)0 1937 y Fr(represen)m(ts)39 b(the)g(bit)g(in)m(terv)-5 b(al)41 b(con)m(taining)f(the)f(bits)g -Fn(position)p Fl(:)15 b(:)g(:)o Fn(position+size-1)35 -b Fp(of)k(an)g(in)m(teger.)68 b(The)0 2061 y(constructor)31 -b Fn(cl_byte\(size,position\))24 b Fp(constructs)31 b(a)f -Fn(cl_byte)p Fp(.)0 2366 y Fn(cl_I)f(ldb)h(\(const)e(cl_I&)h(n,)h -(const)f(cl_byte&)f(b\))480 2490 y Fp(extracts)g(the)f(bits)f(of)h -Fn(n)f Fp(describ)s(ed)g(b)m(y)g(the)h(bit)g(in)m(terv)-5 -b(al)27 b Fn(b)g Fp(and)f(returns)f(them)g(as)i(a)g(nonnegativ)m(e)480 -2615 y(in)m(teger)32 b(with)e Fn(b.size)e Fp(bits.)0 -2794 y Fn(cl_boolean)g(ldb_test)g(\(const)g(cl_I&)h(n,)h(const)f -(cl_byte&)f(b\))480 2919 y Fp(Returns)h(true)i(if)f(some)g(bit)g +Fp(position)p Fn(:)15 b(:)g(:)o Fp(position+size-1)35 +b Fr(of)k(an)g(in)m(teger.)68 b(The)0 2061 y(constructor)31 +b Fp(cl_byte\(size,position\))24 b Fr(constructs)31 b(a)f +Fp(cl_byte)p Fr(.)0 2366 y Fp(cl_I)f(ldb)h(\(const)e(cl_I&)h(n,)h +(const)f(cl_byte&)f(b\))480 2490 y Fr(extracts)g(the)f(bits)f(of)h +Fp(n)f Fr(describ)s(ed)g(b)m(y)g(the)h(bit)g(in)m(terv)-5 +b(al)27 b Fp(b)g Fr(and)f(returns)f(them)g(as)i(a)g(nonnegativ)m(e)480 +2615 y(in)m(teger)32 b(with)e Fp(b.size)e Fr(bits.)0 +2794 y Fp(cl_boolean)g(ldb_test)g(\(const)g(cl_I&)h(n,)h(const)f +(cl_byte&)f(b\))480 2919 y Fr(Returns)h(true)i(if)f(some)g(bit)g (describ)s(ed)f(b)m(y)i(the)f(bit)h(in)m(terv)-5 b(al)31 -b Fn(b)f Fp(is)h(set)g(in)f Fn(n)p Fp(.)0 3099 y Fn(cl_I)f(dpb)h +b Fp(b)f Fr(is)h(set)g(in)f Fp(n)p Fr(.)0 3099 y Fp(cl_I)f(dpb)h (\(const)e(cl_I&)h(newbyte,)f(const)h(cl_I&)g(n,)h(const)f(cl_byte&)f -(b\))480 3223 y Fp(Returns)f Fn(n)p Fp(,)h(with)f(the)h(bits)g(describ) -s(ed)e(b)m(y)i(the)f(bit)h(in)m(terv)-5 b(al)29 b Fn(b)e -Fp(replaced)h(b)m(y)g Fn(newbyte)p Fp(.)38 b(Only)27 -b(the)480 3348 y(lo)m(w)m(est)32 b Fn(b.size)d Fp(bits)h(of)h -Fn(newbyte)d Fp(are)j(relev)-5 b(an)m(t.)0 3652 y(The)28 -b(functions)h Fn(ldb)f Fp(and)g Fn(dpb)g Fp(implicitly)h(shift.)40 +(b\))480 3223 y Fr(Returns)f Fp(n)p Fr(,)h(with)f(the)h(bits)g(describ) +s(ed)e(b)m(y)i(the)f(bit)h(in)m(terv)-5 b(al)29 b Fp(b)e +Fr(replaced)h(b)m(y)g Fp(newbyte)p Fr(.)38 b(Only)27 +b(the)480 3348 y(lo)m(w)m(est)32 b Fp(b.size)d Fr(bits)h(of)h +Fp(newbyte)d Fr(are)j(relev)-5 b(an)m(t.)0 3652 y(The)28 +b(functions)h Fp(ldb)f Fr(and)g Fp(dpb)g Fr(implicitly)h(shift.)40 b(The)29 b(follo)m(wing)h(functions)e(are)i(their)f(coun)m(terparts)g -(without)0 3777 y(shifting:)0 4081 y Fn(cl_I)g(mask_field)f(\(const)g -(cl_I&)h(n,)h(const)f(cl_byte&)f(b\))480 4206 y Fp(returns)j(an)g(in)m +(without)0 3777 y(shifting:)0 4081 y Fp(cl_I)g(mask_field)f(\(const)g +(cl_I&)h(n,)h(const)f(cl_byte&)f(b\))480 4206 y Fr(returns)j(an)g(in)m (teger)j(with)d(the)h(bits)g(describ)s(ed)f(b)m(y)h(the)g(bit)g(in)m -(terv)-5 b(al)33 b Fn(b)e Fp(copied)i(from)d(the)i(corre-)480 -4330 y(sp)s(onding)d(bits)h(in)g Fn(n)p Fp(,)g(the)h(other)g(bits)f -(zero.)0 4510 y Fn(cl_I)f(deposit_field)e(\(const)i(cl_I&)g(newbyte,)f +(terv)-5 b(al)33 b Fp(b)e Fr(copied)i(from)d(the)i(corre-)480 +4330 y(sp)s(onding)d(bits)h(in)g Fp(n)p Fr(,)g(the)h(other)g(bits)f +(zero.)0 4510 y Fp(cl_I)f(deposit_field)e(\(const)i(cl_I&)g(newbyte,)f (const)h(cl_I&)g(n,)g(const)g(cl_byte&)f(b\))480 4634 -y Fp(returns)34 b(an)h(in)m(teger)i(where)d(the)i(bits)f(describ)s(ed)f -(b)m(y)h(the)h(bit)f(in)m(terv)-5 b(al)36 b Fn(b)f Fp(come)g(from)f -Fn(newbyte)480 4759 y Fp(and)c(the)g(other)h(bits)f(come)h(from)d -Fn(n)p Fp(.)0 5063 y(The)i(follo)m(wing)i(relations)f(hold:)180 -5340 y Fn(ldb)e(\(n,)h(b\))g(=)g(mask_field\(n,)d(b\))i(>>)h -(b.position)p Fp(,)p eop +y Fr(returns)34 b(an)h(in)m(teger)i(where)d(the)i(bits)f(describ)s(ed)f +(b)m(y)h(the)h(bit)f(in)m(terv)-5 b(al)36 b Fp(b)f Fr(come)g(from)f +Fp(newbyte)480 4759 y Fr(and)c(the)g(other)h(bits)f(come)h(from)d +Fp(n)p Fr(.)0 5063 y(The)i(follo)m(wing)i(relations)f(hold:)180 +5340 y Fp(ldb)e(\(n,)h(b\))g(=)g(mask_field\(n,)d(b\))i(>>)h +(b.position)p Fr(,)p eop %%Page: 29 31 -29 30 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(29)180 366 y Fn(dpb)29 +29 30 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(29)180 366 y Fp(dpb)29 b(\(newbyte,)f(n,)i(b\))g(=)g(deposit_field)d(\(newbyte)h(<<)h -(b.position,)f(n,)i(b\))p Fp(,)180 522 y Fn(deposit_field\(newbyte,n,)o +(b.position,)f(n,)i(b\))p Fr(,)180 522 y Fp(deposit_field\(newbyte,n,)o (b\))24 b(=)30 b(n)g(^)g(mask_field\(n,b\))c(^)31 b -(mask_field\(new_byte,b\))p Fp(.)0 832 y(The)36 b(follo)m(wing)i(op)s +(mask_field\(new_byte,b\))p Fr(.)0 832 y(The)36 b(follo)m(wing)i(op)s (erations)f(on)g(in)m(tegers)h(as)f(bit)f(strings)h(are)g(e\016cien)m (t)h(shortcuts)e(for)h(common)e(arithmetic)0 957 y(op)s(erations:)0 -1267 y Fn(cl_boolean)28 b(oddp)h(\(const)f(cl_I&)h(x\))480 -1392 y Fp(Returns)g(true)i(if)f(the)h(least)g(signi\014can)m(t)g(bit)g -(of)f Fn(x)g Fp(is)h(1.)41 b(Equiv)-5 b(alen)m(t)31 b(to)g -Fn(mod\(x,2\))d(!=)i(0)p Fp(.)0 1578 y Fn(cl_boolean)e(evenp)h(\(const) -f(cl_I&)h(x\))480 1702 y Fp(Returns)g(true)i(if)f(the)h(least)g -(signi\014can)m(t)g(bit)g(of)f Fn(x)g Fp(is)h(0.)41 b(Equiv)-5 -b(alen)m(t)31 b(to)g Fn(mod\(x,2\))d(==)i(0)p Fp(.)0 -1888 y Fn(cl_I)f(operator)f(<<)i(\(const)f(cl_I&)g(x,)g(const)g(cl_I&)g -(n\))480 2013 y Fp(Shifts)h Fn(x)g Fp(b)m(y)g Fn(n)g -Fp(bits)g(to)h(the)g(left.)41 b Fn(n)30 b Fp(should)g(b)s(e)f -Fn(>)p Fp(=0.)41 b(Equiv)-5 b(alen)m(t)31 b(to)g Fn(x)f(*)g -(expt\(2,n\))p Fp(.)0 2199 y Fn(cl_I)f(operator)f(>>)i(\(const)f(cl_I&) -g(x,)g(const)g(cl_I&)g(n\))480 2323 y Fp(Shifts)24 b -Fn(x)g Fp(b)m(y)g Fn(n)g Fp(bits)g(to)h(the)g(righ)m(t.)39 -b Fn(n)24 b Fp(should)f(b)s(e)h Fn(>)p Fp(=0.)39 b(Bits)25 +1267 y Fp(cl_boolean)28 b(oddp)h(\(const)f(cl_I&)h(x\))480 +1392 y Fr(Returns)g(true)i(if)f(the)h(least)g(signi\014can)m(t)g(bit)g +(of)f Fp(x)g Fr(is)h(1.)41 b(Equiv)-5 b(alen)m(t)31 b(to)g +Fp(mod\(x,2\))d(!=)i(0)p Fr(.)0 1578 y Fp(cl_boolean)e(evenp)h(\(const) +f(cl_I&)h(x\))480 1702 y Fr(Returns)g(true)i(if)f(the)h(least)g +(signi\014can)m(t)g(bit)g(of)f Fp(x)g Fr(is)h(0.)41 b(Equiv)-5 +b(alen)m(t)31 b(to)g Fp(mod\(x,2\))d(==)i(0)p Fr(.)0 +1888 y Fp(cl_I)f(operator)f(<<)i(\(const)f(cl_I&)g(x,)g(const)g(cl_I&)g +(n\))480 2013 y Fr(Shifts)h Fp(x)g Fr(b)m(y)g Fp(n)g +Fr(bits)g(to)h(the)g(left.)41 b Fp(n)30 b Fr(should)g(b)s(e)f +Fp(>)p Fr(=0.)41 b(Equiv)-5 b(alen)m(t)31 b(to)g Fp(x)f(*)g +(expt\(2,n\))p Fr(.)0 2199 y Fp(cl_I)f(operator)f(>>)i(\(const)f(cl_I&) +g(x,)g(const)g(cl_I&)g(n\))480 2323 y Fr(Shifts)24 b +Fp(x)g Fr(b)m(y)g Fp(n)g Fr(bits)g(to)h(the)g(righ)m(t.)39 +b Fp(n)24 b Fr(should)f(b)s(e)h Fp(>)p Fr(=0.)39 b(Bits)25 b(shifted)f(out)g(to)h(the)g(righ)m(t)g(are)f(thro)m(wn)480 2448 y(a)m(w)m(a)m(y)-8 b(.)43 b(Equiv)-5 b(alen)m(t)31 -b(to)g Fn(floor\(x)e(/)h(expt\(2,n\)\))p Fp(.)0 2634 -y Fn(cl_I)f(ash)h(\(const)e(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 -2759 y Fp(Shifts)34 b Fn(x)h Fp(b)m(y)g Fn(y)f Fp(bits)h(to)h(the)f -(left)h(\(if)f Fn(y>)p Fp(=0\))g(or)g(b)m(y)g Fn(-y)f -Fp(bits)h(to)h(the)f(righ)m(t)h(\(if)f Fn(y<)p Fp(=0\).)54 +b(to)g Fp(floor\(x)e(/)h(expt\(2,n\)\))p Fr(.)0 2634 +y Fp(cl_I)f(ash)h(\(const)e(cl_I&)h(x,)h(const)f(cl_I&)g(y\))480 +2759 y Fr(Shifts)34 b Fp(x)h Fr(b)m(y)g Fp(y)f Fr(bits)h(to)h(the)f +(left)h(\(if)f Fp(y>)p Fr(=0\))g(or)g(b)m(y)g Fp(-y)f +Fr(bits)h(to)h(the)f(righ)m(t)h(\(if)f Fp(y<)p Fr(=0\).)54 b(In)35 b(other)480 2883 y(w)m(ords,)30 b(this)g(returns)g -Fn(floor\(x)e(*)i(expt\(2,y\)\))p Fp(.)0 3069 y Fn(uintL)f -(integer_length)d(\(const)j(cl_I&)g(x\))480 3194 y Fp(Returns)40 +Fp(floor\(x)e(*)i(expt\(2,y\)\))p Fr(.)0 3069 y Fp(uintL)f +(integer_length)d(\(const)j(cl_I&)g(x\))480 3194 y Fr(Returns)40 b(the)h(n)m(um)m(b)s(er)e(of)j(bits)f(\(excluding)g(the)h(sign)f(bit\)) -g(needed)g(to)h(represen)m(t)f Fn(x)g Fp(in)f(t)m(w)m(o's)480 +g(needed)g(to)h(represen)m(t)f Fp(x)g Fr(in)f(t)m(w)m(o's)480 3318 y(complemen)m(t)28 b(notation.)41 b(This)27 b(is)i(the)f(smallest) -h(n)f Fn(>)p Fp(=)f(0)i(suc)m(h)f(that)h(-2)p Fn(^)p -Fp(n)f Fn(<)p Fp(=)g(x)g Fn(<)g Fp(2)p Fn(^)p Fp(n.)40 -b(If)28 b(x)g Fn(>)g Fp(0,)480 3443 y(this)i(is)h(the)f(unique)g(n)g -Fn(>)g Fp(0)g(suc)m(h)g(that)h(2)p Fn(^)p Fp(\(n-1\))h -Fn(<)p Fp(=)e(x)g Fn(<)g Fp(2)p Fn(^)p Fp(n.)0 3629 y -Fn(uintL)f(ord2)g(\(const)g(cl_I&)g(x\))480 3753 y(x)36 -b Fp(m)m(ust)g(b)s(e)g(non-zero.)61 b(This)36 b(function)g(returns)g +h(n)f Fp(>)p Fr(=)f(0)i(suc)m(h)f(that)h(-2)p Fp(^)p +Fr(n)f Fp(<)p Fr(=)g(x)g Fp(<)g Fr(2)p Fp(^)p Fr(n.)40 +b(If)28 b(x)g Fp(>)g Fr(0,)480 3443 y(this)i(is)h(the)f(unique)g(n)g +Fp(>)g Fr(0)g(suc)m(h)g(that)h(2)p Fp(^)p Fr(\(n-1\))h +Fp(<)p Fr(=)e(x)g Fp(<)g Fr(2)p Fp(^)p Fr(n.)0 3629 y +Fp(uintL)f(ord2)g(\(const)g(cl_I&)g(x\))480 3753 y(x)36 +b Fr(m)m(ust)g(b)s(e)g(non-zero.)61 b(This)36 b(function)g(returns)g (the)h(n)m(um)m(b)s(er)d(of)j(0)g(bits)g(at)h(the)f(righ)m(t)g(of)g -Fn(x)f Fp(in)480 3878 y(t)m(w)m(o's)c(complemen)m(t)d(notation.)42 -b(This)30 b(is)g(the)h(largest)h(n)d Fn(>)p Fp(=)h(0)h(suc)m(h)f(that)h -(2)p Fn(^)p Fp(n)f(divides)g Fn(x)p Fp(.)0 4064 y Fn(uintL)f(power2p)f -(\(const)h(cl_I&)g(x\))480 4188 y(x)24 b Fp(m)m(ust)g(b)s(e)g -Fn(>)g Fp(0.)40 b(This)23 b(function)i(c)m(hec)m(ks)h(whether)e -Fn(x)g Fp(is)h(a)g(p)s(o)m(w)m(er)g(of)g(2.)39 b(If)24 -b Fn(x)g Fp(=)h(2)p Fn(^)p Fp(\(n-1\),)i(it)e(returns)480 +Fp(x)f Fr(in)480 3878 y(t)m(w)m(o's)c(complemen)m(t)d(notation.)42 +b(This)30 b(is)g(the)h(largest)h(n)d Fp(>)p Fr(=)h(0)h(suc)m(h)f(that)h +(2)p Fp(^)p Fr(n)f(divides)g Fp(x)p Fr(.)0 4064 y Fp(uintL)f(power2p)f +(\(const)h(cl_I&)g(x\))480 4188 y(x)24 b Fr(m)m(ust)g(b)s(e)g +Fp(>)g Fr(0.)40 b(This)23 b(function)i(c)m(hec)m(ks)h(whether)e +Fp(x)g Fr(is)h(a)g(p)s(o)m(w)m(er)g(of)g(2.)39 b(If)24 +b Fp(x)g Fr(=)h(2)p Fp(^)p Fr(\(n-1\),)i(it)e(returns)480 4313 y(n.)40 b(Else)31 b(it)g(returns)e(0.)41 b(\(See)31 -b(also)g(the)g(function)f Fn(logp)p Fp(.\))0 4780 y Fk(4.9.2)63 -b(Num)m(b)s(er)30 b(theoretic)g(functions)0 5091 y Fn(uint32)f(gcd)g +b(also)g(the)g(function)f Fp(logp)p Fr(.\))0 4780 y Fm(4.9.2)63 +b(Num)m(b)s(er)30 b(theoretic)g(functions)0 5091 y Fp(uint32)f(gcd)g (\(uint32)g(a,)g(uint32)g(b\))0 5215 y(cl_I)g(gcd)h(\(const)e(cl_I&)h -(a,)h(const)f(cl_I&)g(b\))480 5340 y Fp(This)h(function)g(returns)f -(the)h(greatest)j(common)28 b(divisor)i(of)h Fn(a)f Fp(and)g -Fn(b)p Fp(,)g(normalized)g(to)h(b)s(e)f Fn(>)p Fp(=)f(0.)p +(a,)h(const)f(cl_I&)g(b\))480 5340 y Fr(This)h(function)g(returns)f +(the)h(greatest)j(common)28 b(divisor)i(of)h Fp(a)f Fr(and)g +Fp(b)p Fr(,)g(normalized)g(to)h(b)s(e)f Fp(>)p Fr(=)f(0.)p eop %%Page: 30 32 -30 31 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2463 b(30)0 366 y Fn(cl_I)29 +30 31 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +b(on)f(n)m(um)m(b)s(ers)2463 b(30)0 366 y Fp(cl_I)29 b(xgcd)g(\(const)g(cl_I&)g(a,)h(const)f(cl_I&)g(b,)g(cl_I*)g(u,)h -(cl_I*)f(v\))480 491 y Fp(This)h(function)h(\(\\extended)g(gcd"\))h -(returns)e(the)h(greatest)h(common)d(divisor)i Fn(g)f -Fp(of)h Fn(a)g Fp(and)f Fn(b)h Fp(and)480 616 y(at)39 +(cl_I*)f(v\))480 491 y Fr(This)h(function)h(\(\\extended)g(gcd"\))h +(returns)e(the)h(greatest)h(common)d(divisor)i Fp(g)f +Fr(of)h Fp(a)g Fr(and)f Fp(b)h Fr(and)480 616 y(at)39 b(the)f(same)g(time)g(the)g(represen)m(tation)i(of)e -Fn(g)g Fp(as)g(an)g(in)m(tegral)i(linear)f(com)m(bination)f(of)h -Fn(a)f Fp(and)480 740 y Fn(b)p Fp(:)53 b Fn(u)36 b Fp(and)g -Fn(v)g Fp(with)h Fn(u*a+v*b)28 b(=)i(g)p Fp(,)38 b Fn(g)e(>)p -Fp(=)g(0.)60 b Fn(u)36 b Fp(and)g Fn(v)h Fp(will)g(b)s(e)f(normalized)g +Fp(g)g Fr(as)g(an)g(in)m(tegral)i(linear)f(com)m(bination)f(of)h +Fp(a)f Fr(and)480 740 y Fp(b)p Fr(:)53 b Fp(u)36 b Fr(and)g +Fp(v)g Fr(with)h Fp(u*a+v*b)28 b(=)i(g)p Fr(,)38 b Fp(g)e(>)p +Fr(=)g(0.)60 b Fp(u)36 b Fr(and)g Fp(v)h Fr(will)g(b)s(e)f(normalized)g (to)h(b)s(e)f(of)h(smallest)480 865 y(p)s(ossible)g(absolute)h(v)-5 b(alue,)40 b(in)e(the)f(follo)m(wing)i(sense:)55 b(If)37 -b Fn(a)h Fp(and)e Fn(b)i Fp(are)g(non-zero,)i(and)d Fn(abs\(a\))480 -989 y(!=)30 b(abs\(b\))p Fp(,)35 b Fn(u)g Fp(and)f Fn(v)h -Fp(will)h(satisfy)g(the)f(inequalities)i Fn(abs\(u\))29 -b(<=)g(abs\(b\)/\(2*g\))p Fp(,)34 b Fn(abs\(v\))29 b(<=)480 -1114 y(abs\(a\)/\(2*g\))p Fp(.)0 1275 y Fn(cl_I)g(lcm)h(\(const)e -(cl_I&)h(a,)h(const)f(cl_I&)g(b\))480 1400 y Fp(This)h(function)g +b Fp(a)h Fr(and)e Fp(b)i Fr(are)g(non-zero,)i(and)d Fp(abs\(a\))480 +989 y(!=)30 b(abs\(b\))p Fr(,)35 b Fp(u)g Fr(and)f Fp(v)h +Fr(will)h(satisfy)g(the)f(inequalities)i Fp(abs\(u\))29 +b(<=)g(abs\(b\)/\(2*g\))p Fr(,)34 b Fp(abs\(v\))29 b(<=)480 +1114 y(abs\(a\)/\(2*g\))p Fr(.)0 1275 y Fp(cl_I)g(lcm)h(\(const)e +(cl_I&)h(a,)h(const)f(cl_I&)g(b\))480 1400 y Fr(This)h(function)g (returns)f(the)h(least)i(common)d(m)m(ultiple)h(of)g -Fn(a)g Fp(and)g Fn(b)p Fp(,)g(normalized)g(to)h(b)s(e)f -Fn(>)p Fp(=)g(0.)0 1562 y Fn(cl_boolean)e(logp)h(\(const)f(cl_I&)h(a,)h +Fp(a)g Fr(and)g Fp(b)p Fr(,)g(normalized)g(to)h(b)s(e)f +Fp(>)p Fr(=)g(0.)0 1562 y Fp(cl_boolean)e(logp)h(\(const)f(cl_I&)h(a,)h (const)f(cl_I&)g(b,)h(cl_RA*)e(l\))0 1686 y(cl_boolean)g(logp)h (\(const)f(cl_RA&)h(a,)h(const)f(cl_RA&)f(b,)i(cl_RA*)f(l\))480 -1811 y(a)35 b Fp(m)m(ust)g(b)s(e)g Fn(>)h Fp(0.)57 b -Fn(b)36 b Fp(m)m(ust)e(b)s(e)h Fn(>)p Fp(0)h(and)f(!=)h(1.)57 +1811 y(a)35 b Fr(m)m(ust)g(b)s(e)g Fp(>)h Fr(0.)57 b +Fp(b)36 b Fr(m)m(ust)e(b)s(e)h Fp(>)p Fr(0)h(and)f(!=)h(1.)57 b(If)36 b(log\(a,b\))i(is)d(rational)i(n)m(um)m(b)s(er,)f(this)f (function)480 1935 y(returns)29 b(true)h(and)g(sets)h(*l)g(=)f (log\(a,b\),)j(else)e(it)g(returns)e(false.)0 2325 y -Fk(4.9.3)63 b(Com)m(binatorial)29 b(functions)0 2618 -y Fn(cl_I)g(factorial)f(\(uintL)h(n\))480 2742 y(n)h -Fp(m)m(ust)f(b)s(e)h(a)h(small)f(in)m(teger)h Fn(>)p -Fp(=)f(0.)41 b(This)30 b(function)g(returns)f(the)i(factorial)h -Fn(n)p Fp(!)40 b(=)30 b Fn(1*2*)p Fl(:)15 b(:)g(:)p Fn(*n)p -Fp(.)0 2904 y Fn(cl_I)29 b(doublefactorial)d(\(uintL)j(n\))480 -3028 y(n)44 b Fp(m)m(ust)g(b)s(e)g(a)h(small)f(in)m(teger)i -Fn(>)p Fp(=)e(0.)84 b(This)43 b(function)i(returns)e(the)i -(doublefactorial)h Fn(n)p 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Fr(,)g Fp(cl_FF)p +Fr(,)g Fp(cl_DF)p Fr(,)h Fp(cl_LF)e Fr(de\014nes)i(the)h(follo)m(wing)g +(op)s(erations.)0 4966 y Fk(t)m(yp)s(e)36 b Fp(scale_float)27 +b(\(const)i Fk(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(sintL)f(delta\))0 +5091 y Fk(t)m(yp)s(e)36 b Fp(scale_float)27 b(\(const)i +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f(cl_I&)g(delta\))480 +5215 y Fr(Returns)35 b Fp(x*2^delta)p Fr(.)57 b(This)35 b(is)h(more)g(e\016cien)m(t)i(than)e(an)g(explicit)i(m)m(ultiplication) -f(b)s(ecause)f(it)480 5340 y(copies)31 b Fn(x)f Fp(and)g(mo)s(di\014es) +f(b)s(ecause)f(it)480 5340 y(copies)31 b Fp(x)f Fr(and)g(mo)s(di\014es) f(the)h(exp)s(onen)m(t.)p eop %%Page: 31 33 -31 32 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +31 32 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 b(on)f(n)m(um)m(b)s(ers)2463 b(31)0 366 y(The)31 b(follo)m(wing)j (functions)d(pro)m(vide)h(an)g(abstract)h(in)m(terface)g(to)g(the)f (underlying)f(represen)m(tation)i(of)f(\015oating-)0 -491 y(p)s(oin)m(t)e(n)m(um)m(b)s(ers.)0 792 y Fn(sintL)f -(float_exponent)d(\(const)j Fi(t)m(yp)s(e)5 b Fn(&)30 -b(x\))480 916 y Fp(Returns)g(the)h(exp)s(onen)m(t)g Fn(e)f -Fp(of)h Fn(x)p Fp(.)41 b(F)-8 b(or)32 b Fn(x)e(=)g(0.0)p -Fp(,)g(this)h(is)g(0.)42 b(F)-8 b(or)31 b Fn(x)g Fp(non-zero,)g(this)g -(is)g(the)g(unique)480 1041 y(in)m(teger)h(with)e Fn(2^\(e-1\))e(<=)i -(abs\(x\))f(<)h(2^e)p Fp(.)0 1217 y Fn(sintL)f(float_radix)e(\(const)i -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x\))480 1341 y Fp(Returns)f(the)i(base)g +491 y(p)s(oin)m(t)e(n)m(um)m(b)s(ers.)0 792 y Fp(sintL)f +(float_exponent)d(\(const)j Fk(t)m(yp)s(e)5 b Fp(&)30 +b(x\))480 916 y Fr(Returns)g(the)h(exp)s(onen)m(t)g Fp(e)f +Fr(of)h Fp(x)p Fr(.)41 b(F)-8 b(or)32 b Fp(x)e(=)g(0.0)p +Fr(,)g(this)h(is)g(0.)42 b(F)-8 b(or)31 b Fp(x)g Fr(non-zero,)g(this)g +(is)g(the)g(unique)480 1041 y(in)m(teger)h(with)e Fp(2^\(e-1\))e(<=)i +(abs\(x\))f(<)h(2^e)p Fr(.)0 1217 y Fp(sintL)f(float_radix)e(\(const)i +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x\))480 1341 y Fr(Returns)f(the)i(base)g (of)f(the)h(\015oating-p)s(oin)m(t)g(represen)m(tation.)42 -b(This)30 b(is)g(alw)m(a)m(ys)i Fn(2)p Fp(.)0 1518 y -Fi(t)m(yp)s(e)k Fn(float_sign)27 b(\(const)i Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 1642 y Fp(Returns)f(the)i(sign)f -Fn(s)g Fp(of)h Fn(x)f Fp(as)h(a)f(\015oat.)42 b(The)30 -b(v)-5 b(alue)31 b(is)f(1)h(for)f Fn(x)g(>)p Fp(=)g(0,)h(-1)g(for)f -Fn(x)g(<)g Fp(0.)0 1818 y Fn(uintL)f(float_digits)e(\(const)i -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x\))480 1943 y Fp(Returns)24 +b(This)30 b(is)g(alw)m(a)m(ys)i Fp(2)p Fr(.)0 1518 y +Fk(t)m(yp)s(e)k Fp(float_sign)27 b(\(const)i Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 1642 y Fr(Returns)f(the)i(sign)f +Fp(s)g Fr(of)h Fp(x)f Fr(as)h(a)f(\015oat.)42 b(The)30 +b(v)-5 b(alue)31 b(is)f(1)h(for)f Fp(x)g(>)p Fr(=)g(0,)h(-1)g(for)f +Fp(x)g(<)g Fr(0.)0 1818 y Fp(uintL)f(float_digits)e(\(const)i +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x\))480 1943 y Fr(Returns)24 b(the)h(n)m(um)m(b)s(er)e(of)i(man)m(tissa)g(bits)f(in)h(the)g -(\015oating-p)s(oin)m(t)h(represen)m(tation)g(of)f Fn(x)p -Fp(,)h(including)480 2067 y(the)31 b(hidden)e(bit.)40 +(\015oating-p)s(oin)m(t)h(represen)m(tation)g(of)f Fp(x)p +Fr(,)h(including)480 2067 y(the)31 b(hidden)e(bit.)40 b(The)30 b(v)-5 b(alue)31 b(only)g(dep)s(ends)d(on)i(the)h(t)m(yp)s(e)f -(of)h Fn(x)p Fp(,)f(not)h(on)f(its)h(v)-5 b(alue.)0 2243 -y Fn(uintL)29 b(float_precision)d(\(const)j Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x\))480 2368 y Fp(Returns)21 b(the)h(n)m(um)m(b)s(er)d(of)j +(of)h Fp(x)p Fr(,)f(not)h(on)f(its)h(v)-5 b(alue.)0 2243 +y Fp(uintL)29 b(float_precision)d(\(const)j Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x\))480 2368 y Fr(Returns)21 b(the)h(n)m(um)m(b)s(er)d(of)j (signi\014can)m(t)h(man)m(tissa)f(bits)g(in)f(the)h(\015oating-p)s(oin) -m(t)h(represen)m(tation)g(of)f Fn(x)p Fp(.)480 2493 y(Since)32 +m(t)h(represen)m(tation)g(of)f Fp(x)p Fr(.)480 2493 y(Since)32 b(denormalized)f(n)m(um)m(b)s(ers)e(are)j(not)f(supp)s(orted,)g(this)g -(is)h(the)f(same)g(as)h 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Fp(F)-8 b(or)38 b Fn(x)f Fp(non-zero,)j -(this)d(returns)f Fn(\(-1\)^s)p Fp(,)h Fn(e)p Fp(,)i -Fn(m)e Fp(with)g Fn(x)30 b(=)g(\(-1\)^s)f(*)h(2^e)f(*)h(m)37 -b Fp(and)g Fn(0.5)29 b(<=)h(m)g(<)480 4453 y(1.0)p Fp(.)63 -b(F)-8 b(or)39 b Fn(x)f Fp(=)g(0,)i(it)f(returns)e Fn(\(-1\)^s)p -Fp(=1,)i Fn(e)p Fp(=0,)h Fn(m)p Fp(=0.)64 b Fn(e)37 b -Fp(is)i(the)f(same)f(as)i(returned)e(b)m(y)h(the)480 -4578 y(function)30 b Fn(float_exponent)p Fp(.)0 4878 +Fp(cl_decoded_float)d Fr(\(or)k Fp(cl_)0 3042 y(decoded_sfloat)p +Fr(,)39 b Fp(cl_decoded_ffloat)p Fr(,)f Fp(cl_decoded_dfloat)p +Fr(,)h Fp(cl_decoded_lfloat)p Fr(,)f(resp)s(ectiv)m(ely\),)0 +3167 y(de\014ned)29 b(b)m(y)240 3421 y Fp(struct)46 b(cl_decoded_)p +Fk(t)m(yp)s(e)5 b Fp(float)44 b({)622 3525 y Fk(t)m(yp)s(e)53 +b Fp(mantissa;)45 b(cl_I)i(exponent;)e Fk(t)m(yp)s(e)53 +b Fp(sign;)240 3628 y(};)0 3903 y Fr(and)30 b(returned)f(b)m(y)h(the)h +(function)0 4204 y Fp(cl_decoded_)p Fk(t)m(yp)s(e)5 b +Fp(float)26 b(decode_float)h(\(const)i 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Fn(cl_R)f(random_R)f(\(cl_random_state&)e(randomstate,)h +Fp(n)f Fr(in)g(the)h(range)f Fp(0)h(<=)e(x)h(<)g(n)p +Fr(.)0 932 y Fp(cl_R)f(random_R)f(\(cl_random_state&)e(randomstate,)h (const)i(cl_R&)g(n\))0 1057 y(cl_R)g(random_R)f(\(const)h(cl_R&)g(n\)) -480 1181 y Fp(Beha)m(v)m(es)j(lik)m(e)g Fn(random_I)c -Fp(if)i Fn(n)g Fp(is)h(an)f(in)m(teger)i(and)e(lik)m(e)h -Fn(random_F)d Fp(if)j Fn(n)f Fp(is)g(a)h(\015oat.)0 1709 -y Fq(4.13)68 b(Obfuscating)31 b(op)t(erators)0 1991 y -Fp(The)i(mo)s(difying)f(C/C)p Fn(++)h Fp(op)s(erators)h -Fn(+=)p Fp(,)g Fn(-=)p Fp(,)g Fn(*=)p Fp(,)h Fn(/=)p -Fp(,)f Fn(&=)p Fp(,)g Fn(|=)p Fp(,)g Fn(^=)p Fp(,)g Fn(<<=)p -Fp(,)g Fn(>>=)f Fp(are)h(not)g(a)m(v)-5 b(ailable)36 +480 1181 y Fr(Beha)m(v)m(es)j(lik)m(e)g Fp(random_I)c +Fr(if)i Fp(n)g Fr(is)h(an)f(in)m(teger)i(and)e(lik)m(e)h +Fp(random_F)d Fr(if)j Fp(n)f Fr(is)g(a)h(\015oat.)0 1709 +y Fs(4.13)68 b(Obfuscating)31 b(op)t(erators)0 1991 y +Fr(The)i(mo)s(difying)f(C/C)p Fp(++)h Fr(op)s(erators)h +Fp(+=)p Fr(,)g Fp(-=)p Fr(,)g Fp(*=)p Fr(,)h Fp(/=)p +Fr(,)f Fp(&=)p Fr(,)g Fp(|=)p Fr(,)g Fp(^=)p Fr(,)g Fp(<<=)p +Fr(,)g Fp(>>=)f Fr(are)h(not)g(a)m(v)-5 b(ailable)36 b(b)m(y)e(default)0 2116 y(b)s(ecause)41 b(their)g(use)g(tends)g(to)g (mak)m(e)g(programs)f(unreadable.)72 b(It)41 b(is)h(trivial)g(to)g(get) g(a)m(w)m(a)m(y)h(without)e(them.)0 2240 y(Ho)m(w)m(ev)m(er,)33 b(if)d(y)m(ou)h(feel)g(that)g(y)m(ou)g(absolutely)g(need)f(these)h(op)s (erators)f(to)h(get)h(happ)m(y)-8 b(,)30 b(then)g(add)240 -2502 y Fn(#define)46 b(WANT_OBFUSCATING_OPERATO)o(RS)0 -2785 y Fp(to)34 b(the)g(b)s(eginning)f(of)h(y)m(our)f(source)h +2502 y Fp(#define)46 b(WANT_OBFUSCATING_OPERATO)o(RS)0 +2785 y Fr(to)34 b(the)g(b)s(eginning)f(of)h(y)m(our)f(source)h (\014les,)h(b)s(efore)e(the)h(inclusion)f(of)h(an)m(y)g(CLN)f(include)g (\014les.)51 b(This)33 b(\015ag)h(will)0 2910 y(enable)d(the)f(follo)m (wing)i(op)s(erators:)0 3192 y(F)-8 b(or)31 b(the)g(classes)g -Fn(cl_N)p Fp(,)f Fn(cl_R)p Fp(,)f Fn(cl_RA)p Fp(,)g Fn(cl_F)p -Fp(,)h Fn(cl_SF)p Fp(,)f Fn(cl_FF)p Fp(,)g Fn(cl_DF)p -Fp(,)g Fn(cl_LF)p Fp(:)0 3509 y Fi(t)m(yp)s(e)5 b Fn(&)30 -b(operator)e(+=)i(\()p Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f -Fi(t)m(yp)s(e)5 b Fn(&\))0 3633 y Fi(t)m(yp)s(e)g Fn(&)30 -b(operator)e(-=)i(\()p Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f -Fi(t)m(yp)s(e)5 b Fn(&\))0 3758 y Fi(t)m(yp)s(e)g Fn(&)30 -b(operator)e(*=)i(\()p Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f -Fi(t)m(yp)s(e)5 b Fn(&\))0 3883 y Fi(t)m(yp)s(e)g Fn(&)30 -b(operator)e(/=)i(\()p Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f -Fi(t)m(yp)s(e)5 b Fn(&\))0 4165 y Fp(F)-8 b(or)31 b(the)g(class)g -Fn(cl_I)p Fp(:)0 4482 y Fi(t)m(yp)s(e)5 b Fn(&)30 b(operator)e(+=)i(\() -p Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 4606 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(-=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 4731 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(*=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 4855 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(&=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 4980 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(|=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 5104 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(^=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 5229 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(<<=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))0 5353 y Fi(t)m(yp)s(e)g Fn(&)30 b(operator)e(>>=)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&,)30 b(const)f Fi(t)m(yp)s(e)5 -b Fn(&\))p eop +Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)f Fp(cl_RA)p Fr(,)g Fp(cl_F)p +Fr(,)h Fp(cl_SF)p Fr(,)f Fp(cl_FF)p Fr(,)g Fp(cl_DF)p +Fr(,)g Fp(cl_LF)p Fr(:)0 3509 y Fk(t)m(yp)s(e)5 b Fp(&)30 +b(operator)e(+=)i(\()p Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f +Fk(t)m(yp)s(e)5 b Fp(&\))0 3633 y Fk(t)m(yp)s(e)g Fp(&)30 +b(operator)e(-=)i(\()p Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f +Fk(t)m(yp)s(e)5 b Fp(&\))0 3758 y Fk(t)m(yp)s(e)g Fp(&)30 +b(operator)e(*=)i(\()p Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f +Fk(t)m(yp)s(e)5 b Fp(&\))0 3883 y Fk(t)m(yp)s(e)g Fp(&)30 +b(operator)e(/=)i(\()p Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f +Fk(t)m(yp)s(e)5 b Fp(&\))0 4165 y Fr(F)-8 b(or)31 b(the)g(class)g +Fp(cl_I)p Fr(:)0 4482 y Fk(t)m(yp)s(e)5 b Fp(&)30 b(operator)e(+=)i(\() +p Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 4606 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(-=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 4731 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(*=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 4855 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(&=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 4980 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(|=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 5104 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(^=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 5229 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(<<=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))0 5353 y Fk(t)m(yp)s(e)g Fp(&)30 b(operator)e(>>=)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fk(t)m(yp)s(e)5 +b Fp(&\))p eop %%Page: 36 38 -36 37 bop 0 -116 a Fp(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 +36 37 bop 0 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 b(on)f(n)m(um)m(b)s(ers)2463 b(36)0 366 y(F)-8 b(or)31 -b(the)g(classes)g Fn(cl_N)p Fp(,)f Fn(cl_R)p Fp(,)f Fn(cl_RA)p -Fp(,)g Fn(cl_I)p Fp(,)h Fn(cl_F)p Fp(,)f Fn(cl_SF)p Fp(,)g -Fn(cl_FF)p Fp(,)h Fn(cl_DF)p Fp(,)f Fn(cl_LF)p Fp(:)0 -665 y Fi(t)m(yp)s(e)5 b Fn(&)30 b(operator)e(++)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x\))480 790 y Fp(The)g(pre\014x)f(op)s -(erator)i Fn(++x)p Fp(.)0 964 y Fn(void)e(operator)f(++)i(\()p -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x,)g(int\))480 1089 y Fp(The)g(p)s -(ost\014x)f(op)s(erator)i Fn(x++)p Fp(.)0 1263 y Fi(t)m(yp)s(e)5 -b Fn(&)30 b(operator)e(--)i(\()p Fi(t)m(yp)s(e)5 b Fn(&)30 -b(x\))480 1388 y Fp(The)g(pre\014x)f(op)s(erator)i Fn(--x)p -Fp(.)0 1562 y Fn(void)e(operator)f(--)i(\()p Fi(t)m(yp)s(e)5 -b Fn(&)30 b(x,)g(int\))480 1687 y Fp(The)g(p)s(ost\014x)f(op)s(erator)i -Fn(x--)p Fp(.)0 1985 y(Note)j(that)e(b)m(y)g(using)g(these)h +b(the)g(classes)g Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)f Fp(cl_RA)p +Fr(,)g Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)g +Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)p Fr(:)0 +665 y Fk(t)m(yp)s(e)5 b Fp(&)30 b(operator)e(++)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x\))480 790 y Fr(The)g(pre\014x)f(op)s +(erator)i Fp(++x)p Fr(.)0 964 y Fp(void)e(operator)f(++)i(\()p +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(int\))480 1089 y Fr(The)g(p)s +(ost\014x)f(op)s(erator)i Fp(x++)p Fr(.)0 1263 y Fk(t)m(yp)s(e)5 +b Fp(&)30 b(operator)e(--)i(\()p Fk(t)m(yp)s(e)5 b Fp(&)30 +b(x\))480 1388 y Fr(The)g(pre\014x)f(op)s(erator)i Fp(--x)p +Fr(.)0 1562 y Fp(void)e(operator)f(--)i(\()p Fk(t)m(yp)s(e)5 +b Fp(&)30 b(x,)g(int\))480 1687 y Fr(The)g(p)s(ost\014x)f(op)s(erator)i +Fp(x--)p Fr(.)0 1985 y(Note)j(that)e(b)m(y)g(using)g(these)h (obfuscating)g(op)s(erators,)g(y)m(ou)f(w)m(ouldn't)g(gain)h -(e\016ciency:)46 b(In)31 b(CLN)h(`)p Fn(x)e(+=)g(y;)p -Fp(')i(is)0 2110 y(exactly)g(the)f(same)e(as)i(`)p Fn(x)f(=)g(x+y;)p -Fp(',)g(not)h(more)e(e\016cien)m(t.)p eop +(e\016ciency:)46 b(In)31 b(CLN)h(`)p Fp(x)e(+=)g(y;)p +Fr(')i(is)0 2110 y(exactly)g(the)f(same)e(as)i(`)p Fp(x)f(=)g(x+y;)p +Fr(',)g(not)h(more)e(e\016cien)m(t.)p eop %%Page: 37 39 -37 38 bop 0 -116 a Fp(Chapter)30 b(5:)41 b(Input/Output)2784 -b(37)0 366 y Fm(5)80 b(Input/Output)0 933 y Fq(5.1)68 +37 38 bop 0 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2784 +b(37)0 366 y Fo(5)80 b(Input/Output)0 933 y Fs(5.1)68 b(In)l(ternal)32 b(and)e(prin)l(ted)g(represen)l(tation)0 -1209 y Fp(All)h(computations)f(deal)h(with)f(the)h(in)m(ternal)g +1209 y Fr(All)h(computations)f(deal)h(with)f(the)h(in)m(ternal)g (represen)m(tations)g(of)g(the)f(n)m(um)m(b)s(ers.)0 1484 y(Ev)m(ery)i(n)m(um)m(b)s(er)d(has)j(an)f(external)i(represen)m (tation)g(as)f(a)g(sequence)g(of)g(ASCI)s(I)e(c)m(haracters.)46 b(Sev)m(eral)33 b(external)0 1609 y(represen)m(tations)e(ma)m(y)f (denote)h(the)g(same)f(n)m(um)m(b)s(er,)e(for)i(example,)g -Fn(")p Fp(20.0)p Fn(")i Fp(and)d Fn(")p Fp(20.000)p Fn(")p -Fp(.)0 1884 y(Con)m(v)m(erting)38 b(an)g(in)m(ternal)g(to)g(an)f +Fp(")p Fr(20.0)p Fp(")i Fr(and)d Fp(")p Fr(20.000)p Fp(")p +Fr(.)0 1884 y(Con)m(v)m(erting)38 b(an)g(in)m(ternal)g(to)g(an)f (external)i(represen)m(tation)f(is)g(called)g(\\prin)m(ting",)j(con)m (v)m(erting)e(an)e(external)0 2009 y(to)43 b(an)e(in)m(ternal)i (represen)m(tation)g(is)f(called)h(\\reading".)76 b(In)41 -b(CLN,)h(is)g(it)g(alw)m(a)m(ys)i(true)d(that)i(con)m(v)m(ersion)g(of)0 +b(CLN,)h(it)g(is)g(alw)m(a)m(ys)i(true)d(that)i(con)m(v)m(ersion)g(of)0 2133 y(an)d(in)m(ternal)g(to)g(an)g(external)g(represen)m(tation)h(and) e(then)g(bac)m(k)i(to)f(an)g(in)m(ternal)g(represen)m(tation)h(will)f (yield)0 2258 y(the)34 b(same)e(in)m(ternal)j(represen)m(tation.)50 -b(Sym)m(b)s(olically:)d Fn(read\(print\(x\)\))27 b(==)i(x)p -Fp(.)50 b(This)33 b(is)g(called)i(\\prin)m(t-read)0 2382 +b(Sym)m(b)s(olically:)d Fp(read\(print\(x\)\))27 b(==)i(x)p +Fr(.)50 b(This)33 b(is)g(called)i(\\prin)m(t-read)0 2382 y(consistency".)0 2658 y(Di\013eren)m(t)d(t)m(yp)s(es)e(of)h(n)m(um)m (b)s(ers)d(ha)m(v)m(e)j(di\013eren)m(t)g(external)g(represen)m(tations) h(\(case)f(is)g(insigni\014can)m(t\):)0 2959 y(In)m(tegers)167 -b(External)30 b(represen)m(tation:)41 b Fi(sign)p Fn({)p -Fi(digit)r Fn(}+)p Fp(.)g(The)29 b(reader)g(also)i(accepts)g(the)e -(Common)e(Lisp)i(syn-)480 3084 y(taxes)39 b Fi(sign)p -Fn({)p Fi(digit)r Fn(}+.)f Fp(with)h(a)f(trailing)i(dot)e(for)h -(decimal)f(in)m(tegers)i(and)d(the)i Fn(#)p Fi(n)p Fn(R)p -Fp(,)g Fn(#b)p Fp(,)i Fn(#o)p Fp(,)f Fn(#x)480 3208 y -Fp(pre\014xes.)0 3386 y(Rational)32 b(n)m(um)m(b)s(ers)480 -3510 y(External)37 b(represen)m(tation:)56 b Fi(sign)p -Fn({)p Fi(digit)r Fn(}+/{)p Fi(digit)r Fn(}+)p Fp(.)k(The)37 -b Fn(#)p Fi(n)p Fn(R)p Fp(,)h Fn(#b)p Fp(,)g Fn(#o)p -Fp(,)h Fn(#x)d Fp(pre\014xes)g(are)i(al-)480 3635 y(lo)m(w)m(ed)32 +b(External)30 b(represen)m(tation:)41 b Fk(sign)p Fp({)p +Fk(digit)r Fp(}+)p Fr(.)g(The)29 b(reader)g(also)i(accepts)g(the)e +(Common)e(Lisp)i(syn-)480 3084 y(taxes)39 b Fk(sign)p +Fp({)p Fk(digit)r Fp(}+.)f Fr(with)h(a)f(trailing)i(dot)e(for)h +(decimal)f(in)m(tegers)i(and)d(the)i Fp(#)p Fk(n)p Fp(R)p +Fr(,)g Fp(#b)p Fr(,)i Fp(#o)p Fr(,)f Fp(#x)480 3208 y +Fr(pre\014xes.)0 3386 y(Rational)32 b(n)m(um)m(b)s(ers)480 +3510 y(External)37 b(represen)m(tation:)56 b Fk(sign)p +Fp({)p Fk(digit)r Fp(}+/{)p Fk(digit)r Fp(}+)p Fr(.)k(The)37 +b Fp(#)p Fk(n)p Fp(R)p Fr(,)h Fp(#b)p Fr(,)g Fp(#o)p +Fr(,)h Fp(#x)d Fr(pre\014xes)g(are)i(al-)480 3635 y(lo)m(w)m(ed)32 b(here)e(as)g(w)m(ell.)0 3812 y(Floating-p)s(oin)m(t)j(n)m(um)m(b)s (ers)480 3937 y(External)48 b(represen)m(tation:)76 b -Fi(sign)p Fn({)p Fi(digit)r Fn(})p Fp(*)p Fi(exp)s(onen)m(t)50 -b Fp(or)d Fi(sign)p Fn({)p Fi(digit)r Fn(})p Fp(*)p Fn(.{)p -Fi(digit)r Fn(})p Fp(*)p Fi(exp)s(onen)m(t)k Fp(or)480 -4061 y Fi(sign)p Fn({)p Fi(digit)r Fn(})p Fp(*)p Fn(.{)p -Fi(digit)r Fn(}+)p Fp(.)40 b(A)26 b(precision)g(sp)s(eci\014er)f(of)h -(the)h(form)p 2684 4061 28 4 v 57 w Fi(prec)k Fp(ma)m(y)26 +Fk(sign)p Fp({)p Fk(digit)r Fp(})p Fr(*)p Fk(exp)s(onen)m(t)50 +b Fr(or)d Fk(sign)p Fp({)p Fk(digit)r Fp(})p Fr(*)p Fp(.{)p +Fk(digit)r Fp(})p Fr(*)p Fk(exp)s(onen)m(t)k Fr(or)480 +4061 y Fk(sign)p Fp({)p Fk(digit)r Fp(})p Fr(*)p Fp(.{)p +Fk(digit)r Fp(}+)p Fr(.)40 b(A)26 b(precision)g(sp)s(eci\014er)f(of)h +(the)h(form)p 2684 4061 28 4 v 57 w Fk(prec)k Fr(ma)m(y)26 b(b)s(e)f(app)s(ended.)37 b(There)480 4186 y(m)m(ust)h(b)s(e)h(at)i (least)g(one)e(digit)i(in)e(the)h(non-exp)s(onen)m(t)f(part.)69 b(The)39 b(exp)s(onen)m(t)g(has)h(the)f(syn)m(tax)480 -4310 y Fi(expmark)m(er)d(expsign)30 b Fn({)p Fi(digit)r -Fn(}+)p Fp(.)41 b(The)30 b(exp)s(onen)m(t)g(mark)m(er)g(is)660 -4461 y(`)p Fn(s)p Fp(')h(for)f(short-\015oats,)660 4612 -y(`)p Fn(f)p Fp(')h(for)f(single-\015oats,)660 4763 y(`)p -Fn(d)p Fp(')h(for)f(double-\015oats,)660 4914 y(`)p Fn(L)p -Fp(')h(for)f(long-\015oats,)480 5091 y(or)25 b(`)p Fn(e)p -Fp(',)i(whic)m(h)e(denotes)h(a)g(default)f(\015oat)h(format.)39 +4310 y Fk(expmark)m(er)d(expsign)30 b Fp({)p Fk(digit)r +Fp(}+)p Fr(.)41 b(The)30 b(exp)s(onen)m(t)g(mark)m(er)g(is)660 +4461 y(`)p Fp(s)p Fr(')h(for)f(short-\015oats,)660 4612 +y(`)p Fp(f)p Fr(')h(for)f(single-\015oats,)660 4763 y(`)p +Fp(d)p Fr(')h(for)f(double-\015oats,)660 4914 y(`)p Fp(L)p +Fr(')h(for)f(long-\015oats,)480 5091 y(or)25 b(`)p Fp(e)p +Fr(',)i(whic)m(h)e(denotes)h(a)g(default)f(\015oat)h(format.)39 b(The)25 b(precision)g(sp)s(ecifying)g(su\016x)g(has)g(the)g(syn-)480 -5215 y(tax)p 638 5215 V 57 w Fi(prec)30 b Fp(where)23 -b Fi(prec)30 b Fp(denotes)24 b(the)g(n)m(um)m(b)s(er)e(of)i(v)-5 +5215 y(tax)p 638 5215 V 57 w Fk(prec)30 b Fr(where)23 +b Fk(prec)30 b Fr(denotes)24 b(the)g(n)m(um)m(b)s(er)e(of)i(v)-5 b(alid)25 b(man)m(tissa)f(digits)g(\(in)g(decimal,)i(excluding)480 5340 y(leading)31 b(zero)s(es\),)h(cf.)41 b(also)31 b(function)f(`)p -Fn(cl_float_format)p Fp('.)p eop +Fp(cl_float_format)p Fr('.)p eop %%Page: 38 40 -38 39 bop 0 -116 a Fp(Chapter)30 b(5:)41 b(Input/Output)2784 +38 39 bop 0 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2784 b(38)0 366 y(Complex)29 b(n)m(um)m(b)s(ers)480 491 y(External)i (represen)m(tation:)660 646 y(In)h(algebraic)i(notation:)46 -b Fi(realpart)r Fn(+)p Fi(imagpart)r Fn(i)p Fp(.)g(Of)32 -b(course,)h(if)g Fi(imagpart)h Fp(is)e(negativ)m(e,)k(its)660 +b Fk(realpart)r Fp(+)p Fk(imagpart)r Fp(i)p Fr(.)g(Of)32 +b(course,)h(if)g Fk(imagpart)h Fr(is)e(negativ)m(e,)k(its)660 771 y(prin)m(ted)23 b(represen)m(tation)h(b)s(egins)f(with)g(a)h(`)p -Fn(-)p Fp(',)h(and)e(the)h(`)p Fn(+)p Fp(')f(b)s(et)m(w)m(een)h -Fi(realpart)i Fp(and)d Fi(imagpart)660 895 y Fp(ma)m(y)37 +Fp(-)p Fr(',)h(and)e(the)h(`)p Fp(+)p Fr(')f(b)s(et)m(w)m(een)h +Fk(realpart)i Fr(and)d Fk(imagpart)660 895 y Fr(ma)m(y)37 b(b)s(e)h(omitted.)63 b(Note)39 b(that)g(this)e(notation)j(cannot)e(b)s -(e)f(used)g(when)g(the)h Fi(imagpart)i Fp(is)660 1020 +(e)f(used)g(when)g(the)h Fk(imagpart)i Fr(is)660 1020 y(rational)c(and)f(the)g(rational)h(n)m(um)m(b)s(er's)d(base)i(is)g -Fn(>)p Fp(18,)i(b)s(ecause)e(the)g(`)p Fn(i)p Fp(')g(is)g(then)g(read)f +Fp(>)p Fr(18,)i(b)s(ecause)e(the)g(`)p Fp(i)p Fr(')g(is)g(then)g(read)f (as)i(a)660 1144 y(digit.)660 1299 y(In)30 b(Common)e(Lisp)h(notation:) -42 b Fn(#C\()p Fi(realpart)32 b(imagpart)r Fn(\))p Fp(.)0 -1799 y Fq(5.2)68 b(Input)30 b(functions)0 2078 y Fp(Including)24 -b Fn()e Fp(de\014nes)i(a)h(t)m(yp)s(e)g Fn(cl_istream)p -Fp(,)e(whic)m(h)h(is)h(the)g(t)m(yp)s(e)g(of)g(the)g(\014rst)f(argumen) +42 b Fp(#C\()p Fk(realpart)32 b(imagpart)r Fp(\))p Fr(.)0 +1799 y Fs(5.2)68 b(Input)30 b(functions)0 2078 y Fr(Including)24 +b Fp()e Fr(de\014nes)i(a)h(t)m(yp)s(e)g Fp(cl_istream)p +Fr(,)e(whic)m(h)h(is)h(the)g(t)m(yp)s(e)g(of)g(the)g(\014rst)f(argumen) m(t)g(to)h(all)h(input)0 2203 y(functions.)38 b(Unless)24 b(y)m(ou)g(build)f(and)g(use)g(CLN)g(with)h(the)g(macro)f(CL)p 2375 2203 28 4 v 32 w(IO)p 2511 2203 V 33 w(STDIO)f(b)s(eing)i -(de\014ned,)g Fn(cl_istream)0 2327 y Fp(is)30 b(the)h(same)f(as)g -Fn(istream&)p Fp(.)0 2607 y(The)g(v)-5 b(ariable)180 -2887 y Fn(cl_istream)28 b(cl_stdin)0 3197 y Fp(con)m(tains)k(the)e +(de\014ned,)g Fp(cl_istream)0 2327 y Fr(is)30 b(the)h(same)f(as)g +Fp(istream&)p Fr(.)0 2607 y(The)g(v)-5 b(ariable)180 +2887 y Fp(cl_istream)28 b(cl_stdin)0 3197 y Fr(con)m(tains)k(the)e (standard)g(input)f(stream.)0 3476 y(These)h(are)h(the)f(simple)g -(input)f(functions:)0 3787 y Fn(int)g(freadchar)f(\(cl_istream)g -(stream\))480 3911 y Fp(Reads)33 b(a)h(c)m(haracter)h(from)c -Fn(stream)p Fp(.)48 b(Returns)32 b Fn(cl_EOF)f Fp(\(not)j(a)g(`)p -Fn(char)p Fp('!\))48 b(if)33 b(the)h(end)e(of)i(stream)480 +(input)f(functions:)0 3787 y Fp(int)g(freadchar)f(\(cl_istream)g +(stream\))480 3911 y Fr(Reads)33 b(a)h(c)m(haracter)h(from)c +Fp(stream)p Fr(.)48 b(Returns)32 b Fp(cl_EOF)f Fr(\(not)j(a)g(`)p +Fp(char)p Fr('!\))48 b(if)33 b(the)h(end)e(of)i(stream)480 4036 y(w)m(as)d(encoun)m(tered)g(or)f(an)g(error)g(o)s(ccurred.)0 -4221 y Fn(int)f(funreadchar)f(\(cl_istream)f(stream,)h(int)i(c\))480 -4346 y Fp(Puts)37 b(bac)m(k)i Fn(c)f Fp(on)m(to)h Fn(stream)p -Fp(.)61 b Fn(c)38 b Fp(m)m(ust)e(b)s(e)i(the)g(result)g(of)g(the)g -(last)g Fn(freadchar)e Fp(op)s(eration)i(on)480 4471 -y Fn(stream)p Fp(.)0 4781 y(Eac)m(h)d(of)g(the)g(classes)g -Fn(cl_N)p Fp(,)g Fn(cl_R)p Fp(,)g Fn(cl_RA)p Fp(,)f Fn(cl_I)p -Fp(,)h Fn(cl_F)p Fp(,)f Fn(cl_SF)p Fp(,)h Fn(cl_FF)p -Fp(,)f Fn(cl_DF)p Fp(,)h Fn(cl_LF)e Fp(de\014nes,)i(in)f -Fn()p Fp(,)29 -b(the)i(follo)m(wing)h(input)d(function:)0 5215 y Fn(cl_istream)f -(operator>>)f(\(cl_istream)g(stream,)i Fi(t)m(yp)s(e)5 -b Fn(&)30 b(result\))480 5340 y Fp(Reads)g(a)h(n)m(um)m(b)s(er)d(from)h -Fn(stream)g Fp(and)h(stores)g(it)h(in)f(the)h Fn(result)p -Fp(.)p eop +4221 y Fp(int)f(funreadchar)f(\(cl_istream)f(stream,)h(int)i(c\))480 +4346 y Fr(Puts)37 b(bac)m(k)i Fp(c)f Fr(on)m(to)h Fp(stream)p +Fr(.)61 b Fp(c)38 b Fr(m)m(ust)e(b)s(e)i(the)g(result)g(of)g(the)g +(last)g Fp(freadchar)e Fr(op)s(eration)i(on)480 4471 +y Fp(stream)p Fr(.)0 4781 y(Eac)m(h)d(of)g(the)g(classes)g +Fp(cl_N)p Fr(,)g Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)f Fp(cl_I)p +Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p +Fr(,)f Fp(cl_DF)p Fr(,)h Fp(cl_LF)e Fr(de\014nes,)i(in)f +Fp()p Fr(,)29 +b(the)i(follo)m(wing)h(input)d(function:)0 5215 y Fp(cl_istream)f +(operator>>)f(\(cl_istream)g(stream,)i Fk(t)m(yp)s(e)5 +b Fp(&)30 b(result\))480 5340 y Fr(Reads)g(a)h(n)m(um)m(b)s(er)d(from)h +Fp(stream)g Fr(and)h(stores)g(it)h(in)f(the)h Fp(result)p +Fr(.)p eop %%Page: 39 41 -39 40 bop 0 -116 a Fp(Chapter)30 b(5:)41 b(Input/Output)2784 +39 40 bop 0 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2784 b(39)0 366 y(The)30 b(most)g(\015exible)g(input)g(functions,)g -(de\014ned)f(in)h Fn()p -Fp(,)28 b(are)j(the)f(follo)m(wing:)0 659 y Fn(cl_N)f(read_complex)e +(de\014ned)f(in)h Fp()p +Fr(,)28 b(are)j(the)f(follo)m(wing:)0 659 y Fp(cl_N)f(read_complex)e (\(cl_istream)h(stream,)g(const)h(cl_read_flags&)d(flags\))0 783 y(cl_R)j(read_real)f(\(cl_istream)f(stream,)i(const)g (cl_read_flags&)d(flags\))0 908 y(cl_F)j(read_float)f(\(cl_istream)f (stream,)h(const)h(cl_read_flags&)e(flags\))0 1033 y(cl_RA)i (read_rational)e(\(cl_istream)g(stream,)h(const)h(cl_read_flags&)e (flags\))0 1157 y(cl_I)i(read_integer)e(\(cl_istream)h(stream,)g(const) -h(cl_read_flags&)d(flags\))480 1282 y Fp(Reads)d(a)g(n)m(um)m(b)s(er)e -(from)h Fn(stream)p Fp(.)36 b(The)23 b Fn(flags)e Fp(are)i(parameters)g +h(cl_read_flags&)d(flags\))480 1282 y Fr(Reads)d(a)g(n)m(um)m(b)s(er)e +(from)h Fp(stream)p Fr(.)36 b(The)23 b Fp(flags)e Fr(are)i(parameters)g (whic)m(h)f(a\013ect)j(the)e(input)f(syn)m(tax.)480 1406 y(Whitespace)32 b(b)s(efore)e(the)g(n)m(um)m(b)s(er)e(is)j(silen)m(tly) -h(skipp)s(ed.)0 1568 y Fn(cl_N)d(read_complex)e(\(const)i +h(skipp)s(ed.)0 1568 y Fp(cl_N)d(read_complex)e(\(const)i (cl_read_flags&)d(flags,)j(const)g(char)g(*)h(string,)f(const)g(char)g (*)0 1692 y(string_limit,)e(const)i(char)g(*)h(*)g(end_of_parse\))0 1817 y(cl_R)f(read_real)f(\(const)h(cl_read_flags&)d(flags,)j(const)g @@ -3328,119 +3602,119 @@ h(skipp)s(ed.)0 1568 y Fn(cl_N)d(read_complex)e(\(const)i y(string_limit,)e(const)i(char)g(*)h(*)g(end_of_parse\))0 2564 y(cl_I)f(read_integer)e(\(const)i(cl_read_flags&)d(flags,)j(const) g(char)g(*)h(string,)f(const)g(char)g(*)0 2688 y(string_limit,)e(const) -i(char)g(*)h(*)g(end_of_parse\))480 2813 y Fp(Reads)g(a)h(n)m(um)m(b)s +i(char)g(*)h(*)g(end_of_parse\))480 2813 y Fr(Reads)g(a)h(n)m(um)m(b)s (er)d(from)h(a)i(string)f(in)g(memory)-8 b(.)39 b(The)30 -b Fn(flags)f Fp(are)i(parameters)f(whic)m(h)g(a\013ect)i(the)480 +b Fp(flags)f Fr(are)i(parameters)f(whic)m(h)g(a\013ect)i(the)480 2938 y(input)26 b(syn)m(tax.)40 b(The)27 b(string)g(starts)g(at)h -Fn(string)d Fp(and)h(ends)g(at)i Fn(string_limit)c Fp(\(exclusiv)m(e)29 -b(limit\).)480 3062 y Fn(string_limit)24 b Fp(ma)m(y)j(also)h(b)s(e)e -Fn(NULL)p Fp(,)h(denoting)h(the)f(en)m(tire)h(string,)g(i.e.)40 -b(equiv)-5 b(alen)m(t)29 b(to)f Fn(string_)480 3187 y(limit)h(=)h -(string)f(+)h(strlen\(string\))p Fp(.)57 b(If)37 b Fn(end_of_parse)d -Fp(is)j Fn(NULL)p Fp(,)h(the)g(string)f(in)g(memory)480 +Fp(string)d Fr(and)h(ends)g(at)i Fp(string_limit)c Fr(\(exclusiv)m(e)29 +b(limit\).)480 3062 y Fp(string_limit)24 b Fr(ma)m(y)j(also)h(b)s(e)e +Fp(NULL)p Fr(,)h(denoting)h(the)f(en)m(tire)h(string,)g(i.e.)40 +b(equiv)-5 b(alen)m(t)29 b(to)f Fp(string_)480 3187 y(limit)h(=)h +(string)f(+)h(strlen\(string\))p Fr(.)57 b(If)37 b Fp(end_of_parse)d +Fr(is)j Fp(NULL)p Fr(,)h(the)g(string)f(in)g(memory)480 3311 y(m)m(ust)26 b(con)m(tain)i(exactly)h(one)f(n)m(um)m(b)s(er)d(and) h(nothing)h(more,)g(else)h(a)g(fatal)g(error)f(will)g(b)s(e)g -(signalled.)480 3436 y(If)33 b Fn(end_of_parse)c Fp(is)k(not)h -Fn(NULL)p Fp(,)f Fn(*end_of_parse)c Fp(will)k(b)s(e)g(assigned)g(a)h(p) +(signalled.)480 3436 y(If)33 b Fp(end_of_parse)c Fr(is)k(not)h +Fp(NULL)p Fr(,)f Fp(*end_of_parse)c Fr(will)k(b)s(e)g(assigned)g(a)h(p) s(oin)m(ter)f(past)g(the)g(last)480 3560 y(parsed)h(c)m(haracter)i -(\(i.e.)55 b Fn(string_limit)31 b Fp(if)k(nothing)f(came)h(after)g(the) +(\(i.e.)55 b Fp(string_limit)31 b Fr(if)k(nothing)f(came)h(after)g(the) g(n)m(um)m(b)s(er\).)51 b(Whitespace)480 3685 y(is)30 -b(not)h(allo)m(w)m(ed.)0 3977 y(The)f(structure)g Fn(cl_read_flags)d -Fp(con)m(tains)k(the)g(follo)m(wing)g(\014elds:)0 4270 -y Fn(cl_read_syntax_t)26 b(syntax)480 4394 y Fp(The)c(p)s(ossible)f +b(not)h(allo)m(w)m(ed.)0 3977 y(The)f(structure)g Fp(cl_read_flags)d +Fr(con)m(tains)k(the)g(follo)m(wing)g(\014elds:)0 4270 +y Fp(cl_read_syntax_t)26 b(syntax)480 4394 y Fr(The)c(p)s(ossible)f (results)h(of)h(the)f(read)g(op)s(eration.)38 b(P)m(ossible)23 -b(v)-5 b(alues)23 b(are)f Fn(syntax_number)p Fp(,)f Fn(syntax_)480 -4519 y(real)p Fp(,)37 b Fn(syntax_rational)p Fp(,)e Fn(syntax_integer)p -Fp(,)g Fn(syntax_float)p Fp(,)g Fn(syntax_sfloat)p Fp(,)g -Fn(syntax_)480 4643 y(ffloat)p Fp(,)29 b Fn(syntax_dfloat)p -Fp(,)e Fn(syntax_lfloat)p Fp(.)0 4805 y Fn(cl_read_lsyntax_t)f(lsyntax) -480 4929 y Fp(Sp)s(eci\014es)e(the)i(language-dep)s(enden)m(t)g(syn)m +b(v)-5 b(alues)23 b(are)f Fp(syntax_number)p Fr(,)f Fp(syntax_)480 +4519 y(real)p Fr(,)37 b Fp(syntax_rational)p Fr(,)e Fp(syntax_integer)p +Fr(,)g Fp(syntax_float)p Fr(,)g Fp(syntax_sfloat)p Fr(,)g +Fp(syntax_)480 4643 y(ffloat)p Fr(,)29 b Fp(syntax_dfloat)p +Fr(,)e Fp(syntax_lfloat)p Fr(.)0 4805 y Fp(cl_read_lsyntax_t)f(lsyntax) +480 4929 y Fr(Sp)s(eci\014es)e(the)i(language-dep)s(enden)m(t)g(syn)m (tax)f(v)-5 b(arian)m(t)26 b(for)f(the)g(read)g(op)s(eration.)40 b(P)m(ossible)26 b(v)-5 b(alues)480 5054 y(are)480 5215 -y Fn(lsyntax_standard)960 5340 y Fp(accept)32 b(standard)d(algebraic)j +y Fp(lsyntax_standard)960 5340 y Fr(accept)32 b(standard)d(algebraic)j (notation)g(only)-8 b(,)31 b(no)g(complex)f(n)m(um)m(b)s(ers,)p eop %%Page: 40 42 -40 41 bop 0 -116 a Fp(Chapter)30 b(5:)41 b(Input/Output)2784 -b(40)480 366 y Fn(lsyntax_algebraic)960 491 y Fp(accept)32 -b(the)e(algebraic)i(notation)g Fi(x)6 b Fn(+)p Fi(y)i -Fn(i)30 b Fp(for)g(complex)g(n)m(um)m(b)s(ers,)480 679 -y Fn(lsyntax_commonlisp)960 804 y Fp(accept)44 b(the)f -Fn(#b)p Fp(,)i Fn(#o)p Fp(,)g Fn(#x)d Fp(syn)m(taxes)h(for)g(binary)-8 +40 41 bop 0 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2784 +b(40)480 366 y Fp(lsyntax_algebraic)960 491 y Fr(accept)32 +b(the)e(algebraic)i(notation)g Fk(x)6 b Fp(+)p Fk(y)i +Fp(i)30 b Fr(for)g(complex)g(n)m(um)m(b)s(ers,)480 679 +y Fp(lsyntax_commonlisp)960 804 y Fr(accept)44 b(the)f +Fp(#b)p Fr(,)i Fp(#o)p Fr(,)g Fp(#x)d Fr(syn)m(taxes)h(for)g(binary)-8 b(,)45 b(o)s(ctal,)i(hexadecimal)c(n)m(um)m(b)s(ers,)960 -928 y Fn(#)p Fi(base)5 b Fn(R)42 b Fp(for)g(rational)h(n)m(um)m(b)s -(ers)d(in)h(a)i(giv)m(en)g(base,)i Fn(#c\()p Fi(realpart)33 -b(imagpart)r Fn(\))41 b Fp(for)960 1053 y(complex)30 -b(n)m(um)m(b)s(ers,)480 1241 y Fn(lsyntax_all)960 1365 -y Fp(accept)i(all)f(of)g(these)f(extensions.)0 1553 y -Fn(unsigned)e(int)i(rational_base)480 1678 y Fp(The)g(base)g(in)g(whic) +928 y Fp(#)p Fk(base)5 b Fp(R)42 b Fr(for)g(rational)h(n)m(um)m(b)s +(ers)d(in)h(a)i(giv)m(en)g(base,)i Fp(#c\()p Fk(realpart)33 +b(imagpart)r Fp(\))41 b Fr(for)960 1053 y(complex)30 +b(n)m(um)m(b)s(ers,)480 1241 y Fp(lsyntax_all)960 1365 +y Fr(accept)i(all)f(of)g(these)f(extensions.)0 1553 y +Fp(unsigned)e(int)i(rational_base)480 1678 y Fr(The)g(base)g(in)g(whic) m(h)h(rational)g(n)m(um)m(b)s(ers)d(are)j(read.)0 1866 -y Fn(cl_float_format_t)26 b(float_flags.default_flo)o(at_f)o(orm)o(at) -480 1990 y Fp(The)k(\015oat)h(format)f(used)f(when)h(reading)g -(\015oats)h(with)f(exp)s(onen)m(t)g(mark)m(er)g(`)p Fn(e)p -Fp('.)0 2178 y Fn(cl_float_format_t)c(float_flags.default_lfl)o(oat_)o -(for)o(mat)480 2303 y Fp(The)k(\015oat)h(format)f(used)f(when)h +y Fp(cl_float_format_t)26 b(float_flags.default_flo)o(at_f)o(orm)o(at) +480 1990 y Fr(The)k(\015oat)h(format)f(used)f(when)h(reading)g +(\015oats)h(with)f(exp)s(onen)m(t)g(mark)m(er)g(`)p Fp(e)p +Fr('.)0 2178 y Fp(cl_float_format_t)c(float_flags.default_lfl)o(oat_)o +(for)o(mat)480 2303 y Fr(The)k(\015oat)h(format)f(used)f(when)h (reading)g(\015oats)h(with)f(exp)s(onen)m(t)g(mark)m(er)g(`)p -Fn(l)p Fp('.)0 2491 y Fn(cl_boolean)e(float_flags.mantissa_de)o(pen)o -(dent)o(_flo)o(at_)o(form)o(at)480 2615 y Fp(When)k(this)g(\015ag)g(is) +Fp(l)p Fr('.)0 2491 y Fp(cl_boolean)e(float_flags.mantissa_de)o(pen)o +(dent)o(_flo)o(at_)o(form)o(at)480 2615 y Fr(When)k(this)g(\015ag)g(is) g(true,)g(\015oats)h(sp)s(eci\014ed)e(with)g(more)g(digits)i(than)f (corresp)s(onding)f(to)h(the)g(ex-)480 2740 y(p)s(onen)m(t)25 b(mark)m(er)f(they)i(con)m(tain,)h(but)e(without)p 2123 -2740 28 4 v 58 w Fi(nnn)f Fp(su\016x,)h(will)h(get)g(a)g(precision)f +2740 28 4 v 58 w Fk(nnn)f Fr(su\016x,)h(will)h(get)g(a)g(precision)f (corresp)s(onding)480 2864 y(to)31 b(their)g(n)m(um)m(b)s(er)d(of)i -(signi\014can)m(t)h(digits.)0 3374 y Fq(5.3)68 b(Output)30 -b(functions)0 3655 y Fp(Including)40 b Fn()d -Fp(de\014nes)j(a)h(t)m(yp)s(e)f Fn(cl_ostream)p Fp(,)h(whic)m(h)f(is)g +(signi\014can)m(t)h(digits.)0 3374 y Fs(5.3)68 b(Output)30 +b(functions)0 3655 y Fr(Including)40 b Fp()d +Fr(de\014nes)j(a)h(t)m(yp)s(e)f Fp(cl_ostream)p Fr(,)h(whic)m(h)f(is)g (the)h(t)m(yp)s(e)f(of)h(the)f(\014rst)g(argumen)m(t)g(to)h(all)0 3780 y(output)e(functions.)66 b(Unless)39 b(y)m(ou)g(build)f(and)g(use) h(CLN)f(with)h(the)g(macro)g(CL)p 2847 3780 V 32 w(IO)p 2983 3780 V 32 w(STDIO)f(b)s(eing)h(de\014ned,)0 3904 -y Fn(cl_ostream)28 b Fp(is)i(the)h(same)e(as)i Fn(ostream&)p -Fp(.)0 4185 y(The)f(v)-5 b(ariable)180 4466 y Fn(cl_ostream)28 -b(cl_stdout)0 4778 y Fp(con)m(tains)k(the)e(standard)g(output)g -(stream.)0 5059 y(The)g(v)-5 b(ariable)180 5340 y Fn(cl_ostream)28 +y Fp(cl_ostream)28 b Fr(is)i(the)h(same)e(as)i Fp(ostream&)p +Fr(.)0 4185 y(The)f(v)-5 b(ariable)180 4466 y Fp(cl_ostream)28 +b(cl_stdout)0 4778 y Fr(con)m(tains)k(the)e(standard)g(output)g +(stream.)0 5059 y(The)g(v)-5 b(ariable)180 5340 y Fp(cl_ostream)28 b(cl_stderr)p eop %%Page: 41 43 -41 42 bop 0 -116 a Fp(Chapter)30 b(5:)41 b(Input/Output)2784 +41 42 bop 0 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2784 b(41)0 366 y(con)m(tains)32 b(the)e(standard)g(error)g(output)g (stream.)0 643 y(These)g(are)h(the)f(simple)g(output)g(functions:)0 -948 y Fn(void)f(fprintchar)f(\(cl_ostream)f(stream,)h(char)i(c\))480 -1072 y Fp(Prin)m(ts)g(the)h(c)m(haracter)h Fn(x)e Fp(literally)i(on)e -(the)h Fn(stream)p Fp(.)0 1252 y Fn(void)e(fprint)g(\(cl_ostream)e -(stream,)i(const)f(char)i(*)g(string\))480 1376 y Fp(Prin)m(ts)g(the)h -Fn(string)d Fp(literally)33 b(on)d(the)g Fn(stream)p -Fp(.)0 1556 y Fn(void)f(fprintdecimal)e(\(cl_ostream)g(stream,)i(int)g +948 y Fp(void)f(fprintchar)f(\(cl_ostream)f(stream,)h(char)i(c\))480 +1072 y Fr(Prin)m(ts)g(the)h(c)m(haracter)h Fp(x)e Fr(literally)i(on)e +(the)h Fp(stream)p Fr(.)0 1252 y Fp(void)e(fprint)g(\(cl_ostream)e +(stream,)i(const)f(char)i(*)g(string\))480 1376 y Fr(Prin)m(ts)g(the)h +Fp(string)d Fr(literally)33 b(on)d(the)g Fp(stream)p +Fr(.)0 1556 y Fp(void)f(fprintdecimal)e(\(cl_ostream)g(stream,)i(int)g (x\))0 1681 y(void)g(fprintdecimal)e(\(cl_ostream)g(stream,)i(const)g -(cl_I&)g(x\))480 1805 y Fp(Prin)m(ts)h(the)h(in)m(teger)h -Fn(x)e Fp(in)g(decimal)g(on)g(the)h Fn(stream)p Fp(.)0 -1985 y Fn(void)e(fprintbinary)e(\(cl_ostream)h(stream,)g(const)h(cl_I&) -g(x\))480 2110 y Fp(Prin)m(ts)h(the)h(in)m(teger)h Fn(x)e -Fp(in)g(binary)f(\(base)i(2,)g(without)g(pre\014x\))e(on)h(the)h -Fn(stream)p Fp(.)0 2289 y Fn(void)e(fprintoctal)f(\(cl_ostream)f -(stream,)h(const)h(cl_I&)g(x\))480 2414 y Fp(Prin)m(ts)h(the)h(in)m -(teger)h Fn(x)e Fp(in)g(o)s(ctal)h(\(base)g(8,)g(without)g(pre\014x\))e -(on)i(the)f Fn(stream)p Fp(.)0 2594 y Fn(void)f(fprinthexadecimal)d +(cl_I&)g(x\))480 1805 y Fr(Prin)m(ts)h(the)h(in)m(teger)h +Fp(x)e Fr(in)g(decimal)g(on)g(the)h Fp(stream)p Fr(.)0 +1985 y Fp(void)e(fprintbinary)e(\(cl_ostream)h(stream,)g(const)h(cl_I&) +g(x\))480 2110 y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e +Fr(in)g(binary)f(\(base)i(2,)g(without)g(pre\014x\))e(on)h(the)h +Fp(stream)p Fr(.)0 2289 y Fp(void)e(fprintoctal)f(\(cl_ostream)f +(stream,)h(const)h(cl_I&)g(x\))480 2414 y Fr(Prin)m(ts)h(the)h(in)m +(teger)h Fp(x)e Fr(in)g(o)s(ctal)h(\(base)g(8,)g(without)g(pre\014x\))e +(on)i(the)f Fp(stream)p Fr(.)0 2594 y Fp(void)f(fprinthexadecimal)d (\(cl_ostream)h(stream,)i(const)f(cl_I&)h(x\))480 2718 -y Fp(Prin)m(ts)h(the)h(in)m(teger)h Fn(x)e Fp(in)g(hexadecimal)g -(\(base)h(16,)h(without)e(pre\014x\))g(on)g(the)h Fn(stream)p -Fp(.)0 3023 y(Eac)m(h)k(of)g(the)g(classes)g Fn(cl_N)p -Fp(,)g Fn(cl_R)p Fp(,)g Fn(cl_RA)p Fp(,)f Fn(cl_I)p Fp(,)h -Fn(cl_F)p Fp(,)f Fn(cl_SF)p Fp(,)h Fn(cl_FF)p Fp(,)f -Fn(cl_DF)p Fp(,)h Fn(cl_LF)e Fp(de\014nes,)i(in)f Fn()p Fp(,)29 b(the)i(follo)m(wing)h -(output)e(functions:)0 3451 y Fn(void)f(fprint)g(\(cl_ostream)e -(stream,)i(const)f Fi(t)m(yp)s(e)5 b Fn(&)31 b(x\))0 +y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e Fr(in)g(hexadecimal)g +(\(base)h(16,)h(without)e(pre\014x\))g(on)g(the)h Fp(stream)p +Fr(.)0 3023 y(Eac)m(h)k(of)g(the)g(classes)g Fp(cl_N)p +Fr(,)g Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h +Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p Fr(,)f +Fp(cl_DF)p Fr(,)h Fp(cl_LF)e Fr(de\014nes,)i(in)f Fp()p Fr(,)29 b(the)i(follo)m(wing)h +(output)e(functions:)0 3451 y Fp(void)f(fprint)g(\(cl_ostream)e +(stream,)i(const)f Fk(t)m(yp)s(e)5 b Fp(&)31 b(x\))0 3576 y(cl_ostream)d(operator<<)f(\(cl_ostream)g(stream,)i(const)g -Fi(t)m(yp)s(e)5 b Fn(&)30 b(x\))480 3700 y Fp(Prin)m(ts)38 -b(the)h(n)m(um)m(b)s(er)d Fn(x)i Fp(on)g(the)h Fn(stream)p -Fp(.)63 b(The)38 b(output)g(ma)m(y)g(dep)s(end)f(on)h(the)h(global)g +Fk(t)m(yp)s(e)5 b Fp(&)30 b(x\))480 3700 y Fr(Prin)m(ts)38 +b(the)h(n)m(um)m(b)s(er)d Fp(x)i Fr(on)g(the)h Fp(stream)p +Fr(.)63 b(The)38 b(output)g(ma)m(y)g(dep)s(end)f(on)h(the)h(global)g (prin)m(ter)480 3825 y(settings)k(in)e(the)h(v)-5 b(ariable)43 -b Fn(cl_default_print_flags)p Fp(.)69 b(The)42 b Fn(ostream)e -Fp(\015ags)i(and)f(settings)480 3950 y(\(\015ags,)31 +b Fp(cl_default_print_flags)p Fr(.)69 b(The)42 b Fp(ostream)e +Fr(\015ags)i(and)f(settings)480 3950 y(\(\015ags,)31 b(width)f(and)g(lo)s(cale\))i(are)f(ignored.)0 4254 y(The)f(most)g -(\015exible)g(output)g(function,)g(de\014ned)g(in)g Fn()p Fp(,)28 b(are)j(the)f(follo)m(wing:)240 -4510 y Fn(void)47 b(print_complex)92 b(\(cl_ostream)44 +(\015exible)g(output)g(function,)g(de\014ned)g(in)g Fp()p Fr(,)28 b(are)j(the)f(follo)m(wing:)240 +4510 y Fp(void)47 b(print_complex)92 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1242 4614 y(const)j(cl_N&)f(z\);)240 4717 y(void)h(print_real)236 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1242 @@ -3452,51 +3726,51 @@ b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1242 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)p eop %%Page: 42 44 -42 43 bop 0 -116 a Fp(Chapter)30 b(5:)41 b(Input/Output)2784 -b(42)1242 366 y Fn(const)47 b(cl_I&)f(z\);)0 640 y Fp(Prin)m(ts)30 -b(the)h(n)m(um)m(b)s(er)d Fn(x)i Fp(on)g(the)h Fn(stream)p -Fp(.)39 b(The)30 b Fn(flags)f Fp(are)h(parameters)g(whic)m(h)g +42 43 bop 0 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2784 +b(42)1242 366 y Fp(const)47 b(cl_I&)f(z\);)0 640 y Fr(Prin)m(ts)30 +b(the)h(n)m(um)m(b)s(er)d Fp(x)i Fr(on)g(the)h Fp(stream)p +Fr(.)39 b(The)30 b Fp(flags)f Fr(are)h(parameters)g(whic)m(h)g (a\013ect)i(the)f(output.)0 914 y(The)f(structure)g(t)m(yp)s(e)g -Fn(cl_print_flags)d Fp(con)m(tains)k(the)g(follo)m(wing)h(\014elds:)0 -1213 y Fn(unsigned)c(int)i(rational_base)480 1338 y Fp(The)g(base)g(in) +Fp(cl_print_flags)d Fr(con)m(tains)k(the)g(follo)m(wing)h(\014elds:)0 +1213 y Fp(unsigned)c(int)i(rational_base)480 1338 y Fr(The)g(base)g(in) g(whic)m(h)h(rational)g(n)m(um)m(b)s(ers)d(are)j(prin)m(ted.)40 -b(Default)32 b(is)e Fn(10)p Fp(.)0 1512 y Fn(cl_boolean)e -(rational_readably)480 1637 y Fp(If)i(this)h(\015ag)g(is)g(true,)g +b(Default)32 b(is)e Fp(10)p Fr(.)0 1512 y Fp(cl_boolean)e +(rational_readably)480 1637 y Fr(If)i(this)h(\015ag)g(is)g(true,)g (rational)h(n)m(um)m(b)s(ers)c(are)j(prin)m(ted)f(with)h(radix)f(sp)s (eci\014ers)g(in)g(Common)f(Lisp)480 1761 y(syn)m(tax)i(\()p -Fn(#)p Fi(n)p Fn(R)f Fp(or)g Fn(#b)g Fp(or)g Fn(#o)g -Fp(or)g Fn(#x)g Fp(pre\014xes,)g(trailing)h(dot\).)41 -b(Default)32 b(is)e(false.)0 1936 y Fn(cl_boolean)e(float_readably)480 -2060 y Fp(If)i(this)g(\015ag)g(is)g(true,)h(t)m(yp)s(e)f(sp)s(eci\014c) +Fp(#)p Fk(n)p Fp(R)f Fr(or)g Fp(#b)g Fr(or)g Fp(#o)g +Fr(or)g Fp(#x)g Fr(pre\014xes,)g(trailing)h(dot\).)41 +b(Default)32 b(is)e(false.)0 1936 y Fp(cl_boolean)e(float_readably)480 +2060 y Fr(If)i(this)g(\015ag)g(is)g(true,)h(t)m(yp)s(e)f(sp)s(eci\014c) g(exp)s(onen)m(t)g(mark)m(ers)f(ha)m(v)m(e)j(precedence)e(o)m(v)m(er)i -('E'.)f(Default)g(is)480 2185 y(false.)0 2359 y Fn(cl_float_format_t)26 -b(default_float_format)480 2484 y Fp(Floating)35 b(p)s(oin)m(t)e(n)m +('E'.)f(Default)g(is)480 2185 y(false.)0 2359 y Fp(cl_float_format_t)26 +b(default_float_format)480 2484 y Fr(Floating)35 b(p)s(oin)m(t)e(n)m (um)m(b)s(ers)d(of)j(this)g(format)g(will)g(b)s(e)f(prin)m(ted)h(using) f(the)h('E')g(exp)s(onen)m(t)g(mark)m(er.)480 2608 y(Default)e(is)g -Fn(cl_float_format_ffloat)p Fp(.)0 2782 y Fn(cl_boolean)d -(complex_readably)480 2907 y Fp(If)37 b(this)f(\015ag)i(is)f(true,)h 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b(the)f(default)g(v) -5 b(alues,)41 b(used)d(b)m(y)g(the)h(function)0 3878 -y Fn(fprint)p Fp(,)p eop +y Fp(fprint)p Fr(.)p eop %%Page: 43 45 -43 44 bop 0 -116 a Fp(Chapter)30 b(6:)41 b(Rings)3120 -b(43)0 366 y Fm(6)80 b(Rings)0 754 y Fp(CLN)30 b(has)g(a)h(class)g(of)g -(abstract)g(rings.)1433 1010 y Fn(Ring)1338 1113 y(cl_ring)1290 -1217 y()0 1494 y Fp(Rings)f(can)h(b)s(e)f(compared)f(for)h -(equalit)m(y:)0 1798 y Fn(bool)f(operator==)f(\(const)g(cl_ring&,)g +43 44 bop 0 -116 a Fr(Chapter)30 b(6:)41 b(Rings)3120 +b(43)0 366 y Fo(6)80 b(Rings)0 754 y Fr(CLN)30 b(has)g(a)h(class)g(of)g +(abstract)g(rings.)1433 1010 y Fp(Ring)1338 1113 y(cl_ring)1290 +1217 y()0 1494 y Fr(Rings)f(can)h(b)s(e)f(compared)f(for)h +(equalit)m(y:)0 1798 y Fp(bool)f(operator==)f(\(const)g(cl_ring&,)g (const)h(cl_ring&\))0 1923 y(bool)g(operator!=)f(\(const)g(cl_ring&,)g -(const)h(cl_ring&\))480 2047 y Fp(These)h(compare)g(t)m(w)m(o)i(rings)e +(const)h(cl_ring&\))480 2047 y 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Fp(Returns)k(the)h(di\013erence)h(of)f(t)m(w)m(o)i(mo)s(dular)c +5091 y Fp(cl_MI)c(operator-)f(\(const)g(cl_MI&,)h(const)g(cl_MI&\))480 +5215 y Fr(Returns)k(the)h(di\013erence)h(of)f(t)m(w)m(o)i(mo)s(dular)c (in)m(tegers.)53 b(One)33 b(of)i(the)f(argumen)m(ts)f(ma)m(y)h(also)h (b)s(e)f(a)480 5340 y(plain)c(in)m(teger.)p eop %%Page: 47 49 -47 48 bop 0 -116 a Fp(Chapter)30 b(7:)41 b(Mo)s(dular)30 -b(in)m(tegers)2670 b(47)0 366 y Fn(cl_MI)29 b(operator-)f(\(const)g -(cl_MI&\))480 491 y Fp(Returns)h(the)i(negativ)m(e)i(of)d(a)h(mo)s -(dular)d(in)m(teger.)0 663 y Fn(cl_MI)h(operator*)f(\(const)g(cl_MI&,)h -(const)g(cl_MI&\))480 787 y Fp(Returns)37 b(the)i(pro)s(duct)e(of)h(t)m +47 48 bop 0 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30 +b(in)m(tegers)2670 b(47)0 366 y Fp(cl_MI)29 b(operator-)f(\(const)g +(cl_MI&\))480 491 y Fr(Returns)h(the)i(negativ)m(e)i(of)d(a)h(mo)s +(dular)d(in)m(teger.)0 663 y Fp(cl_MI)h(operator*)f(\(const)g(cl_MI&,)h +(const)g(cl_MI&\))480 787 y Fr(Returns)37 b(the)i(pro)s(duct)e(of)h(t)m (w)m(o)i(mo)s(dular)c(in)m(tegers.)65 b(One)38 b(of)g(the)h(argumen)m (ts)e(ma)m(y)h(also)h(b)s(e)f(a)480 912 y(plain)30 b(in)m(teger.)0 -1084 y Fn(cl_MI)f(square)g(\(const)f(cl_MI&\))480 1208 -y Fp(Returns)h(the)i(square)f(of)h(a)f(mo)s(dular)f(in)m(teger.)0 -1380 y Fn(cl_MI)g(recip)g(\(const)g(cl_MI&)f(x\))480 -1505 y Fp(Returns)c(the)i(recipro)s(cal)f Fn(x^-1)f Fp(of)i(a)f(mo)s -(dular)e(in)m(teger)k Fn(x)p Fp(.)38 b Fn(x)25 b Fp(m)m(ust)f(b)s(e)h +1084 y Fp(cl_MI)f(square)g(\(const)f(cl_MI&\))480 1208 +y Fr(Returns)h(the)i(square)f(of)h(a)f(mo)s(dular)f(in)m(teger.)0 +1380 y Fp(cl_MI)g(recip)g(\(const)g(cl_MI&)f(x\))480 +1505 y Fr(Returns)c(the)i(recipro)s(cal)f Fp(x^-1)f Fr(of)i(a)f(mo)s +(dular)e(in)m(teger)k Fp(x)p Fr(.)38 b Fp(x)25 b Fr(m)m(ust)f(b)s(e)h (coprime)f(to)i(the)f(mo)s(dulus,)480 1629 y(otherwise)31 -b(an)f(error)g(message)h(is)f(issued.)0 1801 y Fn(cl_MI)f(div)g +b(an)f(error)g(message)h(is)f(issued.)0 1801 y Fp(cl_MI)f(div)g (\(const)g(cl_MI&)g(x,)h(const)e(cl_MI&)h(y\))480 1926 -y Fp(Returns)j(the)h(quotien)m(t)h Fn(x*y^-1)e Fp(of)h(t)m(w)m(o)h(mo)s -(dular)d(in)m(tegers)j Fn(x)p Fp(,)g Fn(y)p Fp(.)48 b -Fn(y)33 b Fp(m)m(ust)e(b)s(e)i(coprime)f(to)i(the)480 +y Fr(Returns)j(the)h(quotien)m(t)h Fp(x*y^-1)e Fr(of)h(t)m(w)m(o)h(mo)s +(dular)d(in)m(tegers)j Fp(x)p Fr(,)g Fp(y)p Fr(.)48 b +Fp(y)33 b Fr(m)m(ust)e(b)s(e)i(coprime)f(to)i(the)480 2050 y(mo)s(dulus,)28 b(otherwise)j(an)f(error)g(message)g(is)h -(issued.)0 2222 y Fn(cl_MI)e(expt_pos)f(\(const)h(cl_MI&)f(x,)i(const)f -(cl_I&)g(y\))480 2347 y(y)h Fp(m)m(ust)f(b)s(e)h Fn(>)g -Fp(0.)41 b(Returns)30 b Fn(x^y)p Fp(.)0 2519 y Fn(cl_MI)f(expt)g +(issued.)0 2222 y Fp(cl_MI)e(expt_pos)f(\(const)h(cl_MI&)f(x,)i(const)f +(cl_I&)g(y\))480 2347 y(y)h Fr(m)m(ust)f(b)s(e)h Fp(>)g +Fr(0.)41 b(Returns)30 b Fp(x^y)p Fr(.)0 2519 y Fp(cl_MI)f(expt)g (\(const)g(cl_MI&)f(x,)i(const)f(cl_I&)g(y\))480 2643 -y Fp(Returns)d Fn(x^y)p Fp(.)39 b(If)26 b Fn(y)h Fp(is)g(negativ)m(e,)j -Fn(x)d Fp(m)m(ust)f(b)s(e)h(coprime)f(to)i(the)f(mo)s(dulus,)e(else)j +y Fr(Returns)d Fp(x^y)p Fr(.)39 b(If)26 b Fp(y)h Fr(is)g(negativ)m(e,)j +Fp(x)d Fr(m)m(ust)f(b)s(e)h(coprime)f(to)i(the)f(mo)s(dulus,)e(else)j (an)f(error)g(message)480 2768 y(is)j(issued.)0 2940 -y Fn(cl_MI)f(operator<<)f(\(const)g(cl_MI&)h(x,)h(const)f(cl_I&)f(y\)) -480 3064 y Fp(Returns)h Fn(x*2^y)p Fp(.)0 3236 y Fn(cl_MI)g(operator>>) +y Fp(cl_MI)f(operator<<)f(\(const)g(cl_MI&)h(x,)h(const)f(cl_I&)f(y\)) +480 3064 y Fr(Returns)h Fp(x*2^y)p Fr(.)0 3236 y Fp(cl_MI)g(operator>>) f(\(const)g(cl_MI&)h(x,)h(const)f(cl_I&)f(y\))480 3361 -y Fp(Returns)g Fn(x*2^-y)p Fp(.)38 b(When)29 b Fn(y)f -Fp(is)h(p)s(ositiv)m(e,)i(the)e(mo)s(dulus)d(m)m(ust)i(b)s(e)g(o)s(dd,) +y Fr(Returns)g Fp(x*2^-y)p Fr(.)38 b(When)29 b Fp(y)f +Fr(is)h(p)s(ositiv)m(e,)i(the)e(mo)s(dulus)d(m)m(ust)i(b)s(e)g(o)s(dd,) g(or)h(an)g(error)f(message)h(is)480 3485 y(issued.)0 -3657 y Fn(bool)g(operator==)f(\(const)g(cl_MI&,)h(const)g(cl_MI&\))0 +3657 y Fp(bool)g(operator==)f(\(const)g(cl_MI&,)h(const)g(cl_MI&\))0 3782 y(bool)g(operator!=)f(\(const)g(cl_MI&,)h(const)g(cl_MI&\))480 -3906 y Fp(Compares)24 b(t)m(w)m(o)i(mo)s(dular)d(in)m(tegers,)28 +3906 y Fr(Compares)24 b(t)m(w)m(o)i(mo)s(dular)d(in)m(tegers,)28 b(b)s(elonging)e(to)f(the)h(same)e(mo)s(dular)f(in)m(teger)k(ring,)f -(for)f(equal-)480 4031 y(it)m(y)-8 b(.)0 4203 y Fn(cl_boolean)28 -b(zerop)h(\(const)f(cl_MI&)h(x\))480 4327 y Fp(Returns)g(true)i(if)f -Fn(x)g Fp(is)g Fn(0)g(mod)g(N)p Fp(.)0 4625 y(The)g(follo)m(wing)i +(for)f(equal-)480 4031 y(it)m(y)-8 b(.)0 4203 y Fp(cl_boolean)28 +b(zerop)h(\(const)f(cl_MI&)h(x\))480 4327 y Fr(Returns)g(true)i(if)f +Fp(x)g Fr(is)g Fp(0)g(mod)g(N)p Fr(.)0 4625 y(The)g(follo)m(wing)i (output)e(functions)g(are)g(de\014ned)g(\(see)h(also)g(the)g(c)m -(hapter)g(on)f(input/output\).)0 4923 y Fn(void)f(fprint)g +(hapter)g(on)f(input/output\).)0 4923 y Fp(void)f(fprint)g (\(cl_ostream)e(stream,)i(const)f(cl_MI&)h(x\))0 5047 y(cl_ostream)f(operator<<)f(\(cl_ostream)g(stream,)i(const)g(cl_MI&)f -(x\))480 5172 y Fp(Prin)m(ts)36 b(the)h(mo)s(dular)e(in)m(teger)j -Fn(x)e Fp(on)g(the)h Fn(stream)p Fp(.)57 b(The)36 b(output)h(ma)m(y)f +(x\))480 5172 y Fr(Prin)m(ts)36 b(the)h(mo)s(dular)e(in)m(teger)j +Fp(x)e Fr(on)g(the)h Fp(stream)p Fr(.)57 b(The)36 b(output)h(ma)m(y)f (dep)s(end)e(on)j(the)g(global)480 5296 y(prin)m(ter)30 -b(settings)h(in)f(the)h(v)-5 b(ariable)31 b Fn(cl_default_print_flags)p -Fp(.)p eop +b(settings)h(in)f(the)h(v)-5 b(ariable)31 b Fp(cl_default_print_flags)p +Fr(.)p eop %%Page: 48 50 -48 49 bop 0 -116 a Fp(Chapter)30 b(8:)41 b(Sym)m(b)s(olic)29 -b(data)i(t)m(yp)s(es)2536 b(48)0 366 y Fm(8)80 b(Sym)l(b)t(olic)31 -b(data)f(t)l(yp)t(es)0 784 y Fp(CLN)g(implemen)m(ts)f(t)m(w)m(o)j(sym)m +48 49 bop 0 -116 a Fr(Chapter)30 b(8:)41 b(Sym)m(b)s(olic)29 +b(data)i(t)m(yp)s(es)2536 b(48)0 366 y Fo(8)80 b(Sym)l(b)t(olic)31 +b(data)f(t)l(yp)t(es)0 784 y Fr(CLN)g(implemen)m(ts)f(t)m(w)m(o)j(sym)m (b)s(olic)d(\(non-n)m(umeric\))h(data)h(t)m(yp)s(es:)41 -b(strings)30 b(and)g(sym)m(b)s(ols.)0 1310 y Fq(8.1)68 -b(Strings)0 1593 y Fp(The)30 b(class)1290 1855 y Fn(String)1242 -1959 y(cl_string)1195 2062 y()0 2345 y Fp(implemen)m(ts)f +b(strings)30 b(and)g(sym)m(b)s(ols.)0 1310 y Fs(8.1)68 +b(Strings)0 1593 y Fr(The)30 b(class)1290 1855 y Fp(String)1242 +1959 y(cl_string)1195 2062 y()0 2345 y Fr(implemen)m(ts)f (imm)m(utable)g(strings.)0 2628 y(Strings)h(are)h(constructed)f (through)g(the)h(follo)m(wing)g(constructors:)0 2944 -y Fn(cl_string)d(\(const)h(char)g(*)h(s\))480 3069 y -Fp(Returns)f(an)i(imm)m(utable)e(cop)m(y)i(of)f(the)h -(\(zero-terminated\))h(C)e(string)g Fn(s)p Fp(.)0 3260 -y Fn(cl_string)e(\(const)h(char)g(*)h(ptr,)f(unsigned)f(long)h(len\)) -480 3385 y Fp(Returns)34 b(an)h(imm)m(utable)f(cop)m(y)h(of)g(the)h -Fn(len)e Fp(c)m(haracters)i(at)g Fn(ptr[0])p Fp(,)f Fl(:)15 -b(:)g(:)q Fp(,)36 b Fn(ptr[len-1])p Fp(.)52 b(NUL)480 +y Fp(cl_string)d(\(const)h(char)g(*)h(s\))480 3069 y +Fr(Returns)f(an)i(imm)m(utable)e(cop)m(y)i(of)f(the)h +(\(zero-terminated\))h(C)e(string)g Fp(s)p Fr(.)0 3260 +y Fp(cl_string)e(\(const)h(char)g(*)h(ptr,)f(unsigned)f(long)h(len\)) +480 3385 y Fr(Returns)34 b(an)h(imm)m(utable)f(cop)m(y)h(of)g(the)h +Fp(len)e Fr(c)m(haracters)i(at)g Fp(ptr[0])p Fr(,)f Fn(:)15 +b(:)g(:)q Fr(,)36 b Fp(ptr[len-1])p Fr(.)52 b(NUL)480 3509 y(c)m(haracters)32 b(are)f(allo)m(w)m(ed.)0 3826 y(The)f(follo)m(wing)i(functions)e(are)g(a)m(v)-5 b(ailable)33 -b(on)d(strings:)0 4142 y Fn(operator)e(=)480 4267 y Fp(Assignmen)m(t)i -(from)f Fn(cl_string)f Fp(and)h Fn(const)g(char)h(*)p -Fp(.)0 4458 y Fn(s.length\(\))0 4583 y(strlen\(s\))480 -4707 y Fp(Returns)f(the)i(length)g(of)f(the)h(string)f -Fn(s)p Fp(.)0 4899 y Fn(s[i])288 b Fp(Returns)24 b(the)h -Fn(i)p Fp(th)f(c)m(haracter)j(of)e(the)f(string)h Fn(s)p -Fp(.)39 b Fn(i)24 b Fp(m)m(ust)g(b)s(e)g(in)h(the)f(range)i -Fn(0)k(<=)f(i)h(<)g(s.length\(\))p Fp(.)0 5091 y Fn(bool)f(equal)g +b(on)d(strings:)0 4142 y Fp(operator)e(=)480 4267 y Fr(Assignmen)m(t)i +(from)f Fp(cl_string)f Fr(and)h Fp(const)g(char)h(*)p +Fr(.)0 4458 y Fp(s.length\(\))0 4583 y(strlen\(s\))480 +4707 y Fr(Returns)f(the)i(length)g(of)f(the)h(string)f +Fp(s)p Fr(.)0 4899 y Fp(s[i])288 b Fr(Returns)24 b(the)h +Fp(i)p Fr(th)f(c)m(haracter)j(of)e(the)f(string)h Fp(s)p +Fr(.)39 b Fp(i)24 b Fr(m)m(ust)g(b)s(e)g(in)h(the)f(range)i +Fp(0)k(<=)f(i)h(<)g(s.length\(\))p Fr(.)0 5091 y Fp(bool)f(equal)g (\(const)g(cl_string&)e(s1,)j(const)f(cl_string&)e(s2\))480 -5215 y Fp(Compares)34 b(t)m(w)m(o)i(strings)f(for)g(equalit)m(y)-8 +5215 y Fr(Compares)34 b(t)m(w)m(o)i(strings)f(for)g(equalit)m(y)-8 b(.)57 b(One)35 b(of)g(the)h(argumen)m(ts)e(ma)m(y)h(also)h(b)s(e)f(a)g -(plain)g Fn(const)480 5340 y(char)29 b(*)p Fp(.)p eop +(plain)g Fp(const)480 5340 y(char)29 b(*)p Fr(.)p eop %%Page: 49 51 -49 50 bop 0 -116 a Fp(Chapter)30 b(8:)41 b(Sym)m(b)s(olic)29 -b(data)i(t)m(yp)s(es)2536 b(49)0 366 y Fq(8.2)68 b(Sym)l(b)t(ols)0 -640 y Fp(Sym)m(b)s(ols)40 b(are)j(uniqui\014ed)d(strings:)64 +49 50 bop 0 -116 a Fr(Chapter)30 b(8:)41 b(Sym)m(b)s(olic)29 +b(data)i(t)m(yp)s(es)2536 b(49)0 366 y Fs(8.2)68 b(Sym)l(b)t(ols)0 +640 y Fr(Sym)m(b)s(ols)40 b(are)j(uniqui\014ed)d(strings:)64 b(all)43 b(sym)m(b)s(ols)e(with)h(the)g(same)g(name)f(are)i(shared.)75 b(This)41 b(means)g(that)0 765 y(comparison)34 b(of)i(t)m(w)m(o)g(sym)m (b)s(ols)e(is)h(fast)h(\(e\013ectiv)m(ely)i(just)d(a)g(p)s(oin)m(ter)g @@ -3706,39 +3980,39 @@ b(strings)f(m)m(ust)f(in)g(the)h(w)m(orst)h(case)g(w)m(alk)f(b)s(oth)f b(are)h(used,)g(for)g(example,)0 1014 y(as)31 b(tags)g(for)f(prop)s (erties,)g(as)h(names)e(of)i(v)-5 b(ariables)31 b(in)f(p)s(olynomial)f (rings,)i(etc.)0 1288 y(Sym)m(b)s(ols)e(are)h(constructed)h(through)f -(the)g(follo)m(wing)i(constructor:)0 1587 y Fn(cl_symbol)c(\(const)h -(cl_string&)e(s\))480 1711 y Fp(Lo)s(oks)j(up)g(or)g(creates)i(a)f(new) +(the)g(follo)m(wing)i(constructor:)0 1587 y 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b(The)28 b(base)g(ring)g(and)0 1554 y(the)h(indeterminate)f(are)g(explicitly)i(part)f(of)f(ev)m(ery)h (p)s(olynomial.)40 b(CLN)28 b(do)s(esn't)g(allo)m(w)i(y)m(ou)e(to)i (\(acciden)m(tally\))0 1679 y(mix)h(elemen)m(ts)h(of)f(di\013eren)m(t)h -(p)s(olynomial)f(rings,)h(e.g.)46 b Fn(\(a^2+1\))28 b(*)i(\(b^3-1\))g -Fp(will)i(result)f(in)h(a)g(run)m(time)e(error.)0 1803 +(p)s(olynomial)f(rings,)h(e.g.)46 b Fp(\(a^2+1\))28 b(*)i(\(b^3-1\))g +Fr(will)i(result)f(in)h(a)g(run)m(time)e(error.)0 1803 y(\(Ideally)h(this)g(should)e(return)g(a)i(m)m(ultiv)-5 b(ariate)31 b(p)s(olynomial,)f(but)g(they)h(are)f(not)h(y)m(et)g (implemen)m(ted)e(in)h(CLN.\))0 2077 y(The)g(classes)h(of)g(univ)-5 b(ariate)31 b(p)s(olynomial)f(rings)f(are)1529 2330 y -Fn(Ring)1433 2434 y(cl_ring)1385 2538 y()1576 +Fp(Ring)1433 2434 y(cl_ring)1385 2538 y()1576 2642 y(|)1576 2746 y(|)1051 2849 y(Univariate)45 b(polynomial)g(ring) 1290 2953 y(cl_univpoly_ring)1290 3057 y()1576 3161 y(|)765 3264 y(+----------------+------)o(----)o(----)o(---)o(--+) @@ -3757,11 +4031,11 @@ y(Rational)45 b(polynomial)g(ring)142 b(|)288 4821 y y(+----------------+)765 5236 y(|)288 5340 y(Integer)46 b(polynomial)f(ring)p eop %%Page: 51 53 -51 52 bop 0 -116 a Fp(Chapter)30 b(9:)41 b(Univ)-5 b(ariate)32 -b(p)s(olynomials)2423 b(51)288 366 y Fn(cl_univpoly_integer_rin)o(g)335 -470 y()0 784 y Fp(and)30 b(the)g(corresp)s +51 52 bop 0 -116 a Fr(Chapter)30 b(9:)41 b(Univ)-5 b(ariate)32 +b(p)s(olynomials)2423 b(51)288 366 y Fp(cl_univpoly_integer_rin)o(g)335 +470 y()0 784 y Fr(and)30 b(the)g(corresp)s (onding)g(classes)h(of)g(univ)-5 b(ariate)31 b(p)s(olynomials)e(are) -1147 1077 y Fn(Univariate)45 b(polynomial)1481 1181 y(cl_UP)1290 +1147 1077 y Fp(Univariate)45 b(polynomial)1481 1181 y(cl_UP)1290 1285 y()1576 1389 y(|)765 1492 y (+----------------+------)o(----)o(----)o(---)o(--+)765 1596 y(|)763 b(|)907 b(|)383 1700 y(Complex)46 b(polynomial)331 @@ -3776,27 +4050,27 @@ b(polynomial)283 b(|)622 3049 y(cl_UP_RA)570 b(|)335 3153 y()89 b(|)1576 3257 y(|)765 3360 y(+----------------+)765 3464 y(|)383 3568 y(Integer)46 b(polynomial)622 3672 y(cl_UP_I)335 3776 y()0 -4089 y Fp(Univ)-5 b(ariate)32 b(p)s(olynomial)d(rings)h(are)h -(constructed)g(using)f(the)g(functions)0 4468 y Fn(cl_univpoly_ring)c +4089 y Fr(Univ)-5 b(ariate)32 b(p)s(olynomial)d(rings)h(are)h +(constructed)g(using)f(the)g(functions)0 4468 y Fp(cl_univpoly_ring)c (cl_find_univpoly_ring)e(\(const)29 b(cl_ring&)f(R\))0 4593 y(cl_univpoly_ring)e(cl_find_univpoly_ring)e(\(const)29 b(cl_ring&)f(R,)i(const)f(cl_symbol&)f(varname\))480 -4717 y Fp(This)36 b(function)h(returns)f(the)i(p)s(olynomial)e(ring)h -(`)p Fn(R[X])p Fp(',)h(unnamed)d(or)i(named.)60 b Fn(R)37 -b Fp(ma)m(y)g(b)s(e)f(an)480 4842 y(arbitrary)g(ring.)56 +4717 y Fr(This)36 b(function)h(returns)f(the)i(p)s(olynomial)e(ring)h +(`)p Fp(R[X])p Fr(',)h(unnamed)d(or)i(named.)60 b Fp(R)37 +b Fr(ma)m(y)g(b)s(e)f(an)480 4842 y(arbitrary)g(ring.)56 b(This)35 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Fp(cl_univpoly_complex_ring)24 b(cl_find_univpoly_ring)g(\(const)29 b(cl_complex_ring&)d(R\))0 491 y(cl_univpoly_complex_ring)e(cl_find_univpoly_ring)g(\(const)29 b(cl_complex_ring&)d(R,)k(const)0 616 y(cl_symbol&)e(varname\))0 @@ -3814,48 +4088,48 @@ b(cl_integer_ring&)d(R\))0 1612 y(cl_univpoly_integer_ring)e (cl_find_univpoly_ring)h(\(const)j(cl_modint_ring&)e(R\))0 1985 y(cl_univpoly_modint_ring)e(cl_find_univpoly_ring)h(\(const)j (cl_modint_ring&)e(R,)k(const)0 2110 y(cl_symbol&)e(varname\))480 -2234 y Fp(These)c(functions)f(are)i(equiv)-5 b(alen)m(t)26 -b(to)e(the)h(general)g Fn(cl_find_univpoly_ring)p Fp(,)20 +2234 y Fr(These)c(functions)f(are)i(equiv)-5 b(alen)m(t)26 +b(to)e(the)h(general)g Fp(cl_find_univpoly_ring)p Fr(,)20 b(only)k(the)g(return)480 2359 y(t)m(yp)s(e)31 b(is)f(more)f(sp)s (eci\014c,)i(according)g(to)g(the)g(base)f(ring's)h(t)m(yp)s(e.)0 -2845 y Fq(9.2)68 b(F)-11 b(unctions)30 b(on)g(univ)-7 -b(ariate)31 b(p)t(olynomials)0 3123 y Fp(Giv)m(en)g(a)g(univ)-5 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Fa(.)d(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)48 b Fp(56)299 +(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)48 b Fr(56)299 4379 y(10.4)92 b(Garbage)32 b(collection)e Fa(.)15 b(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)57 b Fp(56)0 4628 y Fq(11)135 b(Using)46 b(the)f(library)26 +g(.)g(.)57 b Fr(56)0 4628 y Fs(11)135 b(Using)46 b(the)f(library)26 b Fb(.)20 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f (.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)71 -b Fq(57)299 4784 y Fp(11.1)92 b(Compiler)30 b(options)12 +b Fs(57)299 4784 y Fr(11.1)92 b(Compiler)30 b(options)12 b Fa(.)j(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)41 b Fp(57)299 +(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)41 b Fr(57)299 4909 y(11.2)92 b(Include)30 b(\014les)10 b Fa(.)15 b(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)39 b Fp(57)299 5033 +g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)39 b Fr(57)299 5033 y(11.3)92 b(An)30 b(Example)19 b Fa(.)c(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)49 b Fp(61)299 5158 y(11.4)92 b(Debugging)32 +(.)g(.)g(.)g(.)g(.)49 b Fr(61)299 5158 y(11.4)92 b(Debugging)32 b(supp)s(ort)17 b Fa(.)c(.)i(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 -b Fp(62)p eop -%%Page: -3 71 --3 70 bop 3824 -116 a Fp(iii)0 83 y Fq(12)135 b(Customizing)40 +b Fr(62)p eop +%%Page: -3 74 +-3 73 bop 3824 -116 a Fr(iii)0 83 y Fs(12)135 b(Customizing)40 b Fb(.)20 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f (.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)84 -b Fq(64)299 239 y Fp(12.1)92 b(Error)30 b(handling)17 +b Fs(64)299 239 y Fr(12.1)92 b(Error)30 b(handling)17 b Fa(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 -b Fp(64)299 363 y(12.2)92 b(Floating-p)s(oin)m(t)33 b(under\015o)m(w)11 +b Fr(64)299 363 y(12.2)92 b(Floating-p)s(oin)m(t)33 b(under\015o)m(w)11 b Fa(.)j(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)40 b Fp(64)299 488 y(12.3)92 b(Customizing)30 +(.)g(.)g(.)g(.)40 b Fr(64)299 488 y(12.3)92 b(Customizing)30 b(I/O)16 b Fa(.)f(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)45 -b Fp(64)299 612 y(12.4)92 b(Customizing)30 b(the)h(memory)d(allo)s +b Fr(64)299 612 y(12.4)92 b(Customizing)30 b(the)h(memory)d(allo)s (cator)k Fa(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)59 -b Fp(65)0 861 y Fq(Index)31 b Fb(.)19 b(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h +b Fr(65)0 861 y Fs(Index)31 b Fb(.)19 b(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h (.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.) g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)76 -b Fq(66)p eop +b Fs(66)p eop %%Trailer end userdict /end-hook known{end-hook}if diff --git a/doc/cln.tex b/doc/cln.tex index b83dfa8..21f263e 100644 --- a/doc/cln.tex +++ b/doc/cln.tex @@ -191,8 +191,9 @@ CLN is speed efficient: The kernel of CLN has been written in assembly language for some CPUs (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}). @item -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. +@cindex GMP +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3. @item It uses Karatsuba multiplication, which is significantly faster for large numbers than the standard multiplication algorithm. @@ -200,13 +201,14 @@ for large numbers than the standard multiplication algorithm. For very large numbers (more than 12000 decimal digits), it uses @iftex Sch{@"o}nhage-Strassen +@cindex Sch{@"o}nhage-Strassen multiplication @end iftex @ifinfo Schönhage-Strassen +@cindex Schönhage-Strassen multiplication @end ifinfo -multiplication, which is an asymptotically -optimal multiplication algorithm, for multiplication, division and -radix conversion. +multiplication, which is an asymptotically optimal multiplication +algorithm, for multiplication, division and radix conversion. @end itemize @noindent @@ -232,7 +234,7 @@ This section describes how to install the CLN package on your system. To build CLN, you need a C++ compiler. Actually, you need GNU @code{g++ 2.7.0} or newer. On HPPA, you need GNU @code{g++ 2.8.0} or newer. -I recommend GNU @code{egcs 1.1} or newer. +I recommend GNU @code{g++ 2.95} or newer. The following C++ features are used: classes, member functions, @@ -281,10 +283,12 @@ initializations will not work. @end itemize @end ignore +@cindex @code{make} @subsection Make utility To build CLN, you also need to have GNU @code{make} installed. +@cindex @code{sed} @subsection Sed utility To build CLN on HP-UX, you also need to have GNU @code{sed} installed. @@ -397,9 +401,6 @@ add either @samp{-O} or @samp{-O2 -fno-schedule-insns} to the CXXFLAGS. With full @samp{-O2}, @code{g++} miscompiles the division routines. Also, for --enable-shared to work, you need egcs-1.1.2 or newer. -On MIPS (SGI Irix 6), pass option @code{--without-gmp} to configure. gmp does -not work when compiled in @samp{n32} binary format on Irix. - By default, only a static library is built. You can build CLN as a shared library too, by calling @code{configure} with the option @samp{--enable-shared}. To get it built as a shared library only, call @code{configure} with the options @@ -411,6 +412,7 @@ library. @section Installing the library +@cindex installation As with any autoconfiguring GNU software, installation is as easy as this: @@ -477,28 +479,39 @@ Rational number Floating-point number @end example +@cindex @code{cl_number} +@cindex abstract class The base class @code{cl_number} is an abstract base class. It is not useful to declare a variable of this type except if you want to completely disable compile-time type checking and use run-time type checking instead. +@cindex @code{cl_N} +@cindex real number +@cindex complex number The class @code{cl_N} comprises real and complex numbers. There is no special class for complex numbers since complex numbers with imaginary part @code{0} are automatically converted to real numbers. +@cindex @code{cl_R} The class @code{cl_R} comprises real numbers of different kinds. It is an abstract class. +@cindex @code{cl_RA} +@cindex rational number +@cindex integer The class @code{cl_RA} comprises exact real numbers: rational numbers, including integers. There is no special class for non-integral rational numbers since rational numbers with denominator @code{1} are automatically converted to integers. +@cindex @code{cl_F} The class @code{cl_F} implements floating-point approximations to real numbers. It is an abstract class. @section Exact numbers +@cindex exact number Some numbers are represented as exact numbers: there is no loss of information when such a number is converted from its mathematical value to its internal @@ -528,6 +541,7 @@ is completely transparent. @section Floating-point numbers +@cindex floating-point number Not all real numbers can be represented exactly. (There is an easy mathematical proof for this: Only a countable set of numbers can be stored exactly in @@ -535,6 +549,7 @@ a computer, even if one assumes that it has unlimited storage. But there are uncountably many real numbers.) So some approximation is needed. CLN implements ordinary floating-point numbers, with mantissa and exponent. +@cindex rounding error The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{}) only return approximate results. For example, the value of the expression @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as @@ -556,12 +571,14 @@ Floating point numbers come in four flavors: @itemize @bullet @item +@cindex @code{cl_SF} Short floats, type @code{cl_SF}. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 17 mantissa bits (including the ``hidden'' bit). They don't consume heap allocation. @item +@cindex @code{cl_FF} Single floats, type @code{cl_FF}. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 24 mantissa bits (including the ``hidden'' bit). @@ -569,6 +586,7 @@ In CLN, they are represented as IEEE single-precision floating point numbers. This corresponds closely to the C/C++ type @samp{float}. @item +@cindex @code{cl_DF} Double floats, type @code{cl_DF}. They have 1 sign bit, 11 exponent bits (including the exponent's sign), and 53 mantissa bits (including the ``hidden'' bit). @@ -576,6 +594,7 @@ In CLN, they are represented as IEEE double-precision floating point numbers. This corresponds closely to the C/C++ type @samp{double}. @item +@cindex @code{cl_LF} Long floats, type @code{cl_LF}. They have 1 sign bit, 32 exponent bits (including the exponent's sign), and n mantissa bits (including the ``hidden'' bit), where n >= 64. @@ -591,6 +610,7 @@ gradual underflow. If the exponent range of some floating-point type is too limited for your application, choose another floating-point type with larger exponent range. +@cindex @code{cl_F} As a user of CLN, you can forget about the differences between the four floating-point types and just declare all your floating-point variables as being of type @code{cl_F}. This has the advantage that @@ -604,6 +624,7 @@ the floating point contagion rule happened to change in the future.) @section Complex numbers +@cindex complex number Complex numbers, as implemented by the class @code{cl_N}, have a real part and an imaginary part, both real numbers. A complex number whose @@ -615,6 +636,7 @@ through application of @code{sqrt} or transcendental functions. @section Conversions +@cindex conversion Conversions from any class to any its superclasses (``base classes'' in C++ terminology) is done automatically. @@ -656,6 +678,7 @@ Conversions from @samp{const char *} are provided for the classes @code{cl_R}, @code{cl_N}. The easiest way to specify a value which is outside of the range of the C++ built-in types is therefore to specify it as a string, like this: +@cindex Rubik's cube @example cl_I order_of_rubiks_cube_group = "43252003274489856000"; @end example @@ -667,9 +690,13 @@ the functions @table @code @item int cl_I_to_int (const cl_I& x) +@cindex @code{cl_I_to_int ()} @itemx unsigned int cl_I_to_uint (const cl_I& x) +@cindex @code{cl_I_to_uint ()} @itemx long cl_I_to_long (const cl_I& x) +@cindex @code{cl_I_to_long ()} @itemx unsigned long cl_I_to_ulong (const cl_I& x) +@cindex @code{cl_I_to_ulong ()} Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not representable in the range of @var{ctype}, a runtime error occurs. @end table @@ -682,7 +709,9 @@ the functions @table @code @item float cl_float_approx (const @var{type}& x) +@cindex @code{cl_float_approx ()} @itemx double cl_double_approx (const @var{type}& x) +@cindex @code{cl_double_approx ()} Returns an approximation of @code{x} of C type @var{ctype}. If @code{abs(x)} is too close to 0 (underflow), 0 is returned. If @code{abs(x)} is too large (overflow), an IEEE infinity is returned. @@ -692,8 +721,10 @@ Conversions from any class to any of its subclasses (``derived classes'' in C++ terminology) are not provided. Instead, you can assert and check that a value belongs to a certain subclass, and return it as element of that class, using the @samp{As} and @samp{The} macros. +@cindex @code{As() ()} @code{As(@var{type})(@var{value})} checks that @var{value} belongs to @var{type} and returns it as such. +@cindex @code{The() ()} @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to @var{type} and returns it as such. It is your responsibility to ensure that this assumption is valid. @@ -777,24 +808,30 @@ defines the following operations: @table @code @item @var{type} operator + (const @var{type}&, const @var{type}&) +@cindex @code{operator + ()} Addition. @item @var{type} operator - (const @var{type}&, const @var{type}&) +@cindex @code{operator - ()} Subtraction. @item @var{type} operator - (const @var{type}&) Returns the negative of the argument. @item @var{type} plus1 (const @var{type}& x) +@cindex @code{plus1 ()} Returns @code{x + 1}. @item @var{type} minus1 (const @var{type}& x) +@cindex @code{minus1 ()} Returns @code{x - 1}. @item @var{type} operator * (const @var{type}&, const @var{type}&) +@cindex @code{operator * ()} Multiplication. @item @var{type} square (const @var{type}& x) +@cindex @code{square ()} Returns @code{x * x}. @end table @@ -804,9 +841,11 @@ defines the following operations: @table @code @item @var{type} operator / (const @var{type}&, const @var{type}&) +@cindex @code{operator / ()} Division. @item @var{type} recip (const @var{type}&) +@cindex @code{recip ()} Returns the reciprocal of the argument. @end table @@ -818,6 +857,7 @@ Instead, @code{cl_I} defines an ``exact quotient'' function: @table @code @item cl_I exquo (const cl_I& x, const cl_I& y) +@cindex @code{exquo ()} Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}. @end table @@ -825,10 +865,12 @@ The following exponentiation functions are defined: @table @code @item cl_I expt_pos (const cl_I& x, const cl_I& y) +@cindex @code{expt_pos ()} @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y) @code{y} must be > 0. Returns @code{x^y}. @item cl_RA expt (const cl_RA& x, const cl_I& y) +@cindex @code{expt ()} @itemx cl_R expt (const cl_R& x, const cl_I& y) @itemx cl_N expt (const cl_N& x, const cl_I& y) Returns @code{x^y}. @@ -840,6 +882,7 @@ defines the following operation: @table @code @item @var{type} abs (const @var{type}& x) +@cindex @code{abs ()} Returns the absolute value of @code{x}. This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}. @end table @@ -857,6 +900,7 @@ defines the following operation: @table @code @item @var{type} signum (const @var{type}& x) +@cindex @code{signum ()} Returns the sign of @code{x}, in the same number format as @code{x}. This is defined as @code{x / abs(x)} if @code{x} is non-zero, and @code{x} if @code{x} is zero. If @code{x} is real, the value is either @@ -870,9 +914,11 @@ Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations: @table @code @item cl_I numerator (const @var{type}& x) +@cindex @code{numerator ()} Returns the numerator of @code{x}. @item cl_I denominator (const @var{type}& x) +@cindex @code{denominator ()} Returns the denominator of @code{x}. @end table @@ -886,6 +932,7 @@ The class @code{cl_N} defines the following operation: @table @code @item cl_N complex (const cl_R& a, const cl_R& b) +@cindex @code{complex ()} Returns the complex number @code{a+bi}, that is, the complex number with real part @code{a} and imaginary part @code{b}. @end table @@ -894,12 +941,15 @@ Each of the classes @code{cl_N}, @code{cl_R} defines the following operations: @table @code @item cl_R realpart (const @var{type}& x) +@cindex @code{realpart ()} Returns the real part of @code{x}. @item cl_R imagpart (const @var{type}& x) +@cindex @code{imagpart ()} Returns the imaginary part of @code{x}. @item @var{type} conjugate (const @var{type}& x) +@cindex @code{conjugate ()} Returns the complex conjugate of @code{x}. @end table @@ -914,6 +964,7 @@ We have the relations @section Comparisons +@cindex comparison Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I}, @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF} @@ -921,15 +972,19 @@ defines the following operations: @table @code @item bool operator == (const @var{type}&, const @var{type}&) +@cindex @code{operator == ()} @itemx bool operator != (const @var{type}&, const @var{type}&) +@cindex @code{operator != ()} Comparison, as in C and C++. @item uint32 cl_equal_hashcode (const @var{type}&) +@cindex @code{cl_equal_hashcode ()} Returns a 32-bit hash code that is the same for any two numbers which are the same according to @code{==}. This hash code depends on the number's value, not its type or precision. @item cl_boolean zerop (const @var{type}& x) +@cindex @code{zerop ()} Compare against zero: @code{x == 0} @end table @@ -939,25 +994,34 @@ defines the following operations: @table @code @item cl_signean cl_compare (const @var{type}& x, const @var{type}& y) +@cindex @code{cl_compare ()} Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y}, -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}. @item bool operator <= (const @var{type}&, const @var{type}&) +@cindex @code{operator <= ()} @itemx bool operator < (const @var{type}&, const @var{type}&) +@cindex @code{operator < ()} @itemx bool operator >= (const @var{type}&, const @var{type}&) +@cindex @code{operator >= ()} @itemx bool operator > (const @var{type}&, const @var{type}&) +@cindex @code{operator > ()} Comparison, as in C and C++. @item cl_boolean minusp (const @var{type}& x) +@cindex @code{minusp ()} Compare against zero: @code{x < 0} @item cl_boolean plusp (const @var{type}& x) +@cindex @code{plusp ()} Compare against zero: @code{x > 0} @item @var{type} max (const @var{type}& x, const @var{type}& y) +@cindex @code{max ()} Return the maximum of @code{x} and @code{y}. @item @var{type} min (const @var{type}& x, const @var{type}& y) +@cindex @code{min ()} Return the minimum of @code{x} and @code{y}. @end table @@ -970,6 +1034,7 @@ there is no floating point number whose value is exactly @code{1/3}. @section Rounding functions +@cindex rounding When a real number is to be converted to an integer, there is no ``best'' rounding. The desired rounding function depends on the application. @@ -1017,12 +1082,16 @@ defines the following operations: @table @code @item cl_I floor1 (const @var{type}& x) +@cindex @code{floor1 ()} Returns @code{floor(x)}. @item cl_I ceiling1 (const @var{type}& x) +@cindex @code{ceiling1 ()} Returns @code{ceiling(x)}. @item cl_I truncate1 (const @var{type}& x) +@cindex @code{truncate1 ()} Returns @code{truncate(x)}. @item cl_I round1 (const @var{type}& x) +@cindex @code{round1 ()} Returns @code{round(x)}. @end table @@ -1082,9 +1151,13 @@ defines the following operations: @table @code @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @}; @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y) +@cindex @code{floor2 ()} @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y) +@cindex @code{ceiling2 ()} @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y) +@cindex @code{truncate2 ()} @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y) +@cindex @code{round2 ()} @end table Sometimes, one wants the quotient as a floating-point number (of the @@ -1097,9 +1170,13 @@ defines the following operations: @table @code @item @var{type} ffloor (const @var{type}& x) +@cindex @code{ffloor ()} @itemx @var{type} fceiling (const @var{type}& x) +@cindex @code{fceiling ()} @itemx @var{type} ftruncate (const @var{type}& x) +@cindex @code{ftruncate ()} @itemx @var{type} fround (const @var{type}& x) +@cindex @code{fround ()} @end table and similarly for class @code{cl_R}, but with return type @code{cl_F}. @@ -1123,9 +1200,13 @@ defines the following operations: @table @code @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @}; @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x) +@cindex @code{ffloor2 ()} @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x) +@cindex @code{fceiling2 ()} @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x) +@cindex @code{ftruncate2 ()} @itemx @var{type}_fdiv_t fround2 (const @var{type}& x) +@cindex @code{fround2 ()} @end table and similarly for class @code{cl_R}, but with quotient type @code{cl_F}. @@ -1159,7 +1240,9 @@ The classes @code{cl_R}, @code{cl_I} define the following operations: @table @code @item @var{type} mod (const @var{type}& x, const @var{type}& y) +@cindex @code{mod ()} @itemx @var{type} rem (const @var{type}& x, const @var{type}& y) +@cindex @code{rem ()} @end table @@ -1171,6 +1254,7 @@ defines the following operation: @table @code @item @var{type} sqrt (const @var{type}& x) +@cindex @code{sqrt ()} @code{x} must be >= 0. This function returns the square root of @code{x}, normalized to be >= 0. If @code{x} is the square of a rational number, @code{sqrt(x)} will be a rational number, else it will return a @@ -1181,6 +1265,7 @@ The classes @code{cl_RA}, @code{cl_I} define the following operation: @table @code @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root) +@cindex @code{sqrtp ()} This tests whether @code{x} is a perfect square. If so, it returns true and the exact square root in @code{*root}, else it returns false. @end table @@ -1189,6 +1274,7 @@ Furthermore, for integers, similarly: @table @code @item cl_boolean isqrt (const @var{type}& x, @var{type}* root) +@cindex @code{isqrt ()} @code{x} should be >= 0. This function sets @code{*root} to @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}: the boolean value @code{(expt(*root,2) == x)}. @@ -1199,6 +1285,7 @@ define the following operation: @table @code @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root) +@cindex @code{rootp ()} @code{x} must be >= 0. @code{n} must be > 0. This tests whether @code{x} is an @code{n}th power of a rational number. If so, it returns true and the exact root in @code{*root}, else it returns @@ -1210,6 +1297,7 @@ for class @code{cl_N}: @table @code @item cl_N sqrt (const cl_N& z) +@cindex @code{sqrt ()} Returns the square root of @code{z}, as defined by the formula @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type or to a complex number are done if necessary. The range of the result is the @@ -1221,7 +1309,7 @@ The result is an exact number only if @code{z} is an exact number. @section Transcendental functions - +@cindex transcendental functions The transcendental functions return an exact result if the argument is exact and the result is exact as well. Otherwise they must return @@ -1233,21 +1321,25 @@ For example, @code{cos(0) = 1} returns the rational number @code{1}. @table @code @item cl_R exp (const cl_R& x) +@cindex @code{exp ()} @itemx cl_N exp (const cl_N& x) Returns the exponential function of @code{x}. This is @code{e^x} where @code{e} is the base of the natural logarithms. The range of the result is the entire complex plane excluding 0. @item cl_R ln (const cl_R& x) +@cindex @code{ln ()} @code{x} must be > 0. Returns the (natural) logarithm of x. @item cl_N log (const cl_N& x) +@cindex @code{log ()} Returns the (natural) logarithm of x. If @code{x} is real and positive, this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}. The range of the result is the strip in the complex plane @code{-pi < imagpart(log(x)) <= pi}. @item cl_R phase (const cl_N& x) +@cindex @code{phase ()} Returns the angle part of @code{x} in its polar representation as a complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}. This is also the imaginary part of @code{log(x)}. @@ -1266,6 +1358,7 @@ Returns the logarithm of @code{a} with respect to base @code{b}. @code{log(a,b) = log(a)/log(b)}. @item cl_N expt (const cl_N& x, const cl_N& y) +@cindex @code{expt ()} Exponentiation: Returns @code{x^y = exp(y*log(x))}. @end table @@ -1273,6 +1366,7 @@ The constant e = exp(1) = 2.71828@dots{} is returned by the following functions: @table @code @item cl_F cl_exp1 (cl_float_format_t f) +@cindex @code{exp1 ()} Returns e as a float of format @code{f}. @item cl_F cl_exp1 (const cl_F& y) @@ -1287,6 +1381,7 @@ Returns e as a float of format @code{cl_default_float_format}. @table @code @item cl_R sin (const cl_R& x) +@cindex @code{sin ()} Returns @code{sin(x)}. The range of the result is the interval @code{-1 <= sin(x) <= 1}. @@ -1294,6 +1389,7 @@ Returns @code{sin(x)}. The range of the result is the interval Returns @code{sin(z)}. The range of the result is the entire complex plane. @item cl_R cos (const cl_R& x) +@cindex @code{cos ()} Returns @code{cos(x)}. The range of the result is the interval @code{-1 <= cos(x) <= 1}. @@ -1301,20 +1397,26 @@ Returns @code{cos(x)}. The range of the result is the interval Returns @code{cos(z)}. The range of the result is the entire complex plane. @item struct cl_cos_sin_t @{ cl_R cos; cl_R sin; @}; +@cindex @code{cl_cos_sin_t} @itemx cl_cos_sin_t cl_cos_sin (const cl_R& x) Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than +@cindex @code{cl_cos_sin ()} computing them separately. The relation @code{cos^2 + sin^2 = 1} will hold only approximately. @item cl_R tan (const cl_R& x) +@cindex @code{tan ()} @itemx cl_N tan (const cl_N& x) Returns @code{tan(x) = sin(x)/cos(x)}. @item cl_N cis (const cl_R& x) +@cindex @code{cis ()} @itemx cl_N cis (const cl_N& x) Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because @code{e^(i*x) = cos(x) + i*sin(x)}. +@cindex @code{asin} +@cindex @code{asin ()} @item cl_N asin (const cl_N& z) Returns @code{arcsin(z)}. This is defined as @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies @@ -1329,6 +1431,7 @@ results for arsinh. @end ignore @item cl_N acos (const cl_N& z) +@cindex @code{acos ()} Returns @code{arccos(z)}. This is defined as @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i} @ignore @@ -1344,6 +1447,8 @@ with @code{realpart = pi} and @code{imagpart > 0}. Proof: This follows from the results about arcsin. @end ignore +@cindex @code{atan} +@cindex @code{atan ()} @item cl_R atan (const cl_R& x, const cl_R& y) Returns the angle of the polar representation of the complex number @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of @@ -1371,10 +1476,13 @@ Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function. @end table -The constant pi = 3.14@dots{} is returned by the following functions: +@cindex pi +@cindex Archimedes' constant +Archimedes' constant pi = 3.14@dots{} is returned by the following functions: @table @code @item cl_F cl_pi (cl_float_format_t f) +@cindex @code{cl_pi} Returns pi as a float of format @code{f}. @item cl_F cl_pi (const cl_F& y) @@ -1389,12 +1497,14 @@ Returns pi as a float of format @code{cl_default_float_format}. @table @code @item cl_R sinh (const cl_R& x) +@cindex @code{sinh ()} Returns @code{sinh(x)}. @item cl_N sinh (const cl_N& z) Returns @code{sinh(z)}. The range of the result is the entire complex plane. @item cl_R cosh (const cl_R& x) +@cindex @code{cosh ()} Returns @code{cosh(x)}. The range of the result is the interval @code{cosh(x) >= 1}. @@ -1402,16 +1512,20 @@ Returns @code{cosh(x)}. The range of the result is the interval Returns @code{cosh(z)}. The range of the result is the entire complex plane. @item struct cl_cosh_sinh_t @{ cl_R cosh; cl_R sinh; @}; +@cindex @code{cl_cosh_sinh_t} @itemx cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x) +@cindex @code{cl_cosh_sinh ()} Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will hold only approximately. @item cl_R tanh (const cl_R& x) +@cindex @code{tanh ()} @itemx cl_N tanh (const cl_N& x) Returns @code{tanh(x) = sinh(x)/cosh(x)}. @item cl_N asinh (const cl_N& z) +@cindex @code{asinh ()} Returns @code{arsinh(z)}. This is defined as @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies @code{arsinh(-z) = -arsinh(z)}. @@ -1443,6 +1557,7 @@ log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2))) @end ignore @item cl_N acosh (const cl_N& z) +@cindex @code{acosh ()} Returns @code{arcosh(z)}. This is defined as @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}. The range of the result is the half-strip in the complex domain @@ -1488,6 +1603,7 @@ Otherwise, -z is in Range(sqrt). @end ignore @item cl_N atanh (const cl_N& z) +@cindex @code{atanh ()} Returns @code{artanh(z)}. This is defined as @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies @code{artanh(-z) = -artanh(z)}. The range of the result is @@ -1519,11 +1635,13 @@ Proof: Write z = x+iy. Examine @subsection Euler gamma +@cindex Euler's constant Euler's constant C = 0.577@dots{} is returned by the following functions: @table @code @item cl_F cl_eulerconst (cl_float_format_t f) +@cindex @code{cl_eulerconst ()} Returns Euler's constant as a float of format @code{f}. @item cl_F cl_eulerconst (const cl_F& y) @@ -1534,9 +1652,11 @@ Returns Euler's constant as a float of format @code{cl_default_float_format}. @end table Catalan's constant G = 0.915@dots{} is returned by the following functions: +@cindex Catalan's constant @table @code @item cl_F cl_catalanconst (cl_float_format_t f) +@cindex @code{cl_catalanconst ()} Returns Catalan's constant as a float of format @code{f}. @item cl_F cl_catalanconst (const cl_F& y) @@ -1548,12 +1668,14 @@ Returns Catalan's constant as a float of format @code{cl_default_float_format}. @subsection Riemann zeta +@cindex Riemann's zeta Riemann's zeta function at an integral point @code{s>1} is returned by the following functions: @table @code @item cl_F cl_zeta (int s, cl_float_format_t f) +@cindex @code{cl_zeta ()} Returns Riemann's zeta function at @code{s} as a float of format @code{f}. @item cl_F cl_zeta (int s, const cl_F& y) @@ -1582,46 +1704,62 @@ on each of the bit positions in parallel. @table @code @item cl_I lognot (const cl_I& x) +@cindex @code{lognot ()} @itemx cl_I operator ~ (const cl_I& x) +@cindex @code{operator ~ ()} Logical not, like @code{~x} in C. This is the same as @code{-1-x}. @item cl_I logand (const cl_I& x, const cl_I& y) +@cindex @code{logand ()} @itemx cl_I operator & (const cl_I& x, const cl_I& y) +@cindex @code{operator & ()} Logical and, like @code{x & y} in C. @item cl_I logior (const cl_I& x, const cl_I& y) +@cindex @code{logior ()} @itemx cl_I operator | (const cl_I& x, const cl_I& y) +@cindex @code{operator | ()} Logical (inclusive) or, like @code{x | y} in C. @item cl_I logxor (const cl_I& x, const cl_I& y) +@cindex @code{logxor ()} @itemx cl_I operator ^ (const cl_I& x, const cl_I& y) +@cindex @code{operator ^ ()} Exclusive or, like @code{x ^ y} in C. @item cl_I logeqv (const cl_I& x, const cl_I& y) +@cindex @code{logeqv ()} Bitwise equivalence, like @code{~(x ^ y)} in C. @item cl_I lognand (const cl_I& x, const cl_I& y) +@cindex @code{lognand ()} Bitwise not and, like @code{~(x & y)} in C. @item cl_I lognor (const cl_I& x, const cl_I& y) +@cindex @code{lognor ()} Bitwise not or, like @code{~(x | y)} in C. @item cl_I logandc1 (const cl_I& x, const cl_I& y) +@cindex @code{logandc1 ()} Logical and, complementing the first argument, like @code{~x & y} in C. @item cl_I logandc2 (const cl_I& x, const cl_I& y) +@cindex @code{logandc2 ()} Logical and, complementing the second argument, like @code{x & ~y} in C. @item cl_I logorc1 (const cl_I& x, const cl_I& y) +@cindex @code{logorc1 ()} Logical or, complementing the first argument, like @code{~x | y} in C. @item cl_I logorc2 (const cl_I& x, const cl_I& y) +@cindex @code{logorc2 ()} Logical or, complementing the second argument, like @code{x | ~y} in C. @end table These operations are all available though the function @table @code @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y) +@cindex @code{boole ()} @end table where @code{op} must have one of the 16 values (each one stands for a function which combines two bits into one bit): @code{boole_clr}, @code{boole_set}, @@ -1629,19 +1767,38 @@ which combines two bits into one bit): @code{boole_clr}, @code{boole_set}, @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv}, @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2}, @code{boole_orc1}, @code{boole_orc2}. +@cindex @code{boole_clr} +@cindex @code{boole_set} +@cindex @code{boole_1} +@cindex @code{boole_2} +@cindex @code{boole_c1} +@cindex @code{boole_c2} +@cindex @code{boole_and} +@cindex @code{boole_xor} +@cindex @code{boole_eqv} +@cindex @code{boole_nand} +@cindex @code{boole_nor} +@cindex @code{boole_andc1} +@cindex @code{boole_andc2} +@cindex @code{boole_orc1} +@cindex @code{boole_orc2} + Other functions that view integers as bit strings: @table @code @item cl_boolean logtest (const cl_I& x, const cl_I& y) +@cindex @code{logtest ()} Returns true if some bit is set in both @code{x} and @code{y}, i.e. if @code{logand(x,y) != 0}. @item cl_boolean logbitp (const cl_I& n, const cl_I& x) +@cindex @code{logbitp ()} Returns true if the @code{n}th bit (from the right) of @code{x} is set. Bit 0 is the least significant bit. @item uintL logcount (const cl_I& x) +@cindex @code{logcount ()} Returns the number of one bits in @code{x}, if @code{x} >= 0, or the number of zero bits in @code{x}, if @code{x} < 0. @end table @@ -1651,20 +1808,24 @@ The type @example struct cl_byte @{ uintL size; uintL position; @}; @end example +@cindex @code{cl_byte} represents the bit interval containing the bits @code{position}@dots{}@code{position+size-1} of an integer. The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}. @table @code @item cl_I ldb (const cl_I& n, const cl_byte& b) +@cindex @code{ldb ()} extracts the bits of @code{n} described by the bit interval @code{b} and returns them as a nonnegative integer with @code{b.size} bits. @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b) +@cindex @code{ldb_test ()} Returns true if some bit described by the bit interval @code{b} is set in @code{n}. @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b) +@cindex @code{dpb ()} Returns @code{n}, with the bits described by the bit interval @code{b} replaced by @code{newbyte}. Only the lowest @code{b.size} bits of @code{newbyte} are relevant. @@ -1675,10 +1836,12 @@ functions are their counterparts without shifting: @table @code @item cl_I mask_field (const cl_I& n, const cl_byte& b) +@cindex @code{mask_field ()} returns an integer with the bits described by the bit interval @code{b} copied from the corresponding bits in @code{n}, the other bits zero. @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b) +@cindex @code{deposit_field ()} returns an integer where the bits described by the bit interval @code{b} come from @code{newbyte} and the other bits come from @code{n}. @end table @@ -1699,39 +1862,47 @@ for common arithmetic operations: @table @code @item cl_boolean oddp (const cl_I& x) +@cindex @code{oddp ()} Returns true if the least significant bit of @code{x} is 1. Equivalent to @code{mod(x,2) != 0}. @item cl_boolean evenp (const cl_I& x) +@cindex @code{evenp ()} Returns true if the least significant bit of @code{x} is 0. Equivalent to @code{mod(x,2) == 0}. @item cl_I operator << (const cl_I& x, const cl_I& n) +@cindex @code{operator << ()} Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0. Equivalent to @code{x * expt(2,n)}. @item cl_I operator >> (const cl_I& x, const cl_I& n) +@cindex @code{operator >> ()} Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0. Bits shifted out to the right are thrown away. Equivalent to @code{floor(x / expt(2,n))}. @item cl_I ash (const cl_I& x, const cl_I& y) +@cindex @code{ash ()} Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or by @code{-y} bits to the right (if @code{y}<=0). In other words, this returns @code{floor(x * expt(2,y))}. @item uintL integer_length (const cl_I& x) +@cindex @code{integer_length ()} Returns the number of bits (excluding the sign bit) needed to represent @code{x} in two's complement notation. This is the smallest n >= 0 such that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that 2^(n-1) <= x < 2^n. @item uintL ord2 (const cl_I& x) +@cindex @code{ord2 ()} @code{x} must be non-zero. This function returns the number of 0 bits at the right of @code{x} in two's complement notation. This is the largest n >= 0 such that 2^n divides @code{x}. @item uintL power2p (const cl_I& x) +@cindex @code{power2p ()} @code{x} must be > 0. This function checks whether @code{x} is a power of 2. If @code{x} = 2^(n-1), it returns n. Else it returns 0. (See also the function @code{logp}.) @@ -1742,11 +1913,13 @@ If @code{x} = 2^(n-1), it returns n. Else it returns 0. @table @code @item uint32 gcd (uint32 a, uint32 b) +@cindex @code{gcd ()} @itemx cl_I gcd (const cl_I& a, const cl_I& b) This function returns the greatest common divisor of @code{a} and @code{b}, normalized to be >= 0. @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v) +@cindex @code{xgcd ()} This function (``extended gcd'') returns the greatest common divisor @code{g} of @code{a} and @code{b} and at the same time the representation of @code{g} as an integral linear combination of @code{a} and @code{b}: @@ -1757,10 +1930,12 @@ value, in the following sense: If @code{a} and @code{b} are non-zero, and @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}. @item cl_I lcm (const cl_I& a, const cl_I& b) +@cindex @code{lcm ()} This function returns the least common multiple of @code{a} and @code{b}, normalized to be >= 0. @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l) +@cindex @code{logp ()} @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l) @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is rational number, this function returns true and sets *l = log(a,b), else @@ -1772,15 +1947,18 @@ it returns false. @table @code @item cl_I factorial (uintL n) +@cindex @code{factorial ()} @code{n} must be a small integer >= 0. This function returns the factorial @code{n}! = @code{1*2*@dots{}*n}. @item cl_I doublefactorial (uintL n) +@cindex @code{doublefactorial ()} @code{n} must be a small integer >= 0. This function returns the doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or @code{n}!! = @code{2*4*@dots{}*n}, respectively. @item cl_I binomial (uintL n, uintL k) +@cindex @code{binomial ()} @code{n} and @code{k} must be small integers >= 0. This function returns the binomial coefficient @tex @@ -1805,6 +1983,7 @@ defines the following operations. @table @code @item @var{type} scale_float (const @var{type}& x, sintL delta) +@cindex @code{scale_float ()} @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta) Returns @code{x*2^delta}. This is more efficient than an explicit multiplication because it copies @code{x} and modifies the exponent. @@ -1815,23 +1994,28 @@ representation of floating-point numbers. @table @code @item sintL float_exponent (const @var{type}& x) +@cindex @code{float_exponent ()} Returns the exponent @code{e} of @code{x}. For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique integer with @code{2^(e-1) <= abs(x) < 2^e}. @item sintL float_radix (const @var{type}& x) +@cindex @code{float_radix ()} Returns the base of the floating-point representation. This is always @code{2}. @item @var{type} float_sign (const @var{type}& x) +@cindex @code{float_sign ()} Returns the sign @code{s} of @code{x} as a float. The value is 1 for @code{x} >= 0, -1 for @code{x} < 0. @item uintL float_digits (const @var{type}& x) +@cindex @code{float_digits ()} Returns the number of mantissa bits in the floating-point representation of @code{x}, including the hidden bit. The value only depends on the type of @code{x}, not on its value. @item uintL float_precision (const @var{type}& x) +@cindex @code{float_precision ()} Returns the number of significant mantissa bits in the floating-point representation of @code{x}. Since denormalized numbers are not supported, this is the same as @code{float_digits(x)} if @code{x} is non-zero, and @@ -1839,6 +2023,11 @@ this is the same as @code{float_digits(x)} if @code{x} is non-zero, and @end table The complete internal representation of a float is encoded in the type +@cindex @code{cl_decoded_float} +@cindex @code{cl_decoded_sfloat} +@cindex @code{cl_decoded_ffloat} +@cindex @code{cl_decoded_dfloat} +@cindex @code{cl_decoded_lfloat} @code{cl_decoded_float} (or @code{cl_decoded_sfloat}, @code{cl_decoded_ffloat}, @code{cl_decoded_dfloat}, @code{cl_decoded_lfloat}, respectively), defined by @example @@ -1851,6 +2040,7 @@ and returned by the function @table @code @item cl_decoded_@var{type}float decode_float (const @var{type}& x) +@cindex @code{decode_float ()} For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0. @@ -1859,6 +2049,7 @@ it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0. A complete decoding in terms of integers is provided as type @example +@cindex @code{cl_idecoded_float} struct cl_idecoded_float @{ cl_I mantissa; cl_I exponent; cl_I sign; @}; @@ -1867,6 +2058,7 @@ by the following function: @table @code @item cl_idecoded_float integer_decode_float (const @var{type}& x) +@cindex @code{integer_decode_float ()} For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)} bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0. @@ -1878,6 +2070,7 @@ Some other function, implemented only for class @code{cl_F}: @table @code @item cl_F float_sign (const cl_F& x, const cl_F& y) +@cindex @code{float_sign ()} This returns a floating point number whose precision and absolute value is that of @code{y} and whose sign is that of @code{x}. If @code{x} is zero, it is treated as positive. Same for @code{y}. @@ -1885,6 +2078,7 @@ zero, it is treated as positive. Same for @code{y}. @section Conversion functions +@cindex conversion @subsection Conversion to floating-point numbers @@ -1892,6 +2086,7 @@ The type @code{cl_float_format_t} describes a floating-point format. @table @code @item cl_float_format_t cl_float_format (uintL n) +@cindex @code{cl_float_format ()} Returns the smallest float format which guarantees at least @code{n} decimal digits in the mantissa (after the decimal point). @@ -1899,6 +2094,7 @@ decimal digits in the mantissa (after the decimal point). Returns the floating point format of @code{x}. @item cl_float_format_t cl_default_float_format +@cindex @code{cl_default_float_format} Global variable: the default float format used when converting rational numbers to floats. @end table @@ -1910,6 +2106,7 @@ defines the following operations: @table @code @item cl_F cl_float (const @var{type}&x, cl_float_format_t f) +@cindex @code{cl_float} Returns @code{x} as a float of format @code{f}. @item cl_F cl_float (const @var{type}&x, const cl_F& y) Returns @code{x} in the float format of @code{y}. @@ -1924,23 +2121,29 @@ Every floating-point format has some characteristic numbers: @table @code @item cl_F most_positive_float (cl_float_format_t f) +@cindex @code{most_positive_float ()} Returns the largest (most positive) floating point number in float format @code{f}. @item cl_F most_negative_float (cl_float_format_t f) +@cindex @code{most_negative_float ()} Returns the smallest (most negative) floating point number in float format @code{f}. @item cl_F least_positive_float (cl_float_format_t f) +@cindex @code{least_positive_float ()} Returns the least positive floating point number (i.e. > 0 but closest to 0) in float format @code{f}. @item cl_F least_negative_float (cl_float_format_t f) +@cindex @code{least_negative_float ()} Returns the least negative floating point number (i.e. < 0 but closest to 0) in float format @code{f}. @item cl_F float_epsilon (cl_float_format_t f) +@cindex @code{float_epsilon ()} Returns the smallest floating point number e > 0 such that @code{1+e != 1}. @item cl_F float_negative_epsilon (cl_float_format_t f) +@cindex @code{float_negative_epsilon ()} Returns the smallest floating point number e > 0 such that @code{1-e != 1}. @end table @@ -1952,6 +2155,7 @@ defines the following operation: @table @code @item cl_RA rational (const @var{type}& x) +@cindex @code{rational ()} Returns the value of @code{x} as an exact number. If @code{x} is already an exact number, this is @code{x}. If @code{x} is a floating-point number, the value is a rational number whose denominator is a power of 2. @@ -1962,6 +2166,7 @@ the function @table @code @item cl_RA rationalize (const cl_R& x) +@cindex @code{rationalize ()} If @code{x} is a floating-point number, it actually represents an interval of real numbers, and this function returns the rational number with smallest denominator (and smallest numerator, in magnitude) @@ -1993,6 +2198,7 @@ Calling one of these modifies the state of the random number generator in a complicated but deterministic way. The global variable +@cindex @code{cl_default_random_state} @example cl_random_state cl_default_random_state @end example @@ -2002,26 +2208,31 @@ below are called without @code{cl_random_state} argument. @table @code @item uint32 random32 (cl_random_state& randomstate) @itemx uint32 random32 () +@cindex @code{random32 ()} Returns a random unsigned 32-bit number. All bits are equally random. @item cl_I random_I (cl_random_state& randomstate, const cl_I& n) @itemx cl_I random_I (const cl_I& n) +@cindex @code{random_I ()} @code{n} must be an integer > 0. This function returns a random integer @code{x} in the range @code{0 <= x < n}. @item cl_F random_F (cl_random_state& randomstate, const cl_F& n) @itemx cl_F random_F (const cl_F& n) +@cindex @code{random_F ()} @code{n} must be a float > 0. This function returns a random floating-point number of the same format as @code{n} in the range @code{0 <= x < n}. @item cl_R random_R (cl_random_state& randomstate, const cl_R& n) @itemx cl_R random_R (const cl_R& n) +@cindex @code{random_R ()} Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F} if @code{n} is a float. @end table @section Obfuscating operators +@cindex modifying operators The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=}, @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=} @@ -2032,6 +2243,7 @@ to get happy, then add @example #define WANT_OBFUSCATING_OPERATORS @end example +@cindex @code{WANT_OBFUSCATING_OPERATORS} to the beginning of your source files, before the inclusion of any CLN include files. This flag will enable the following operators: @@ -2040,9 +2252,13 @@ For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @table @code @item @var{type}& operator += (@var{type}&, const @var{type}&) +@cindex @code{operator += ()} @itemx @var{type}& operator -= (@var{type}&, const @var{type}&) +@cindex @code{operator -= ()} @itemx @var{type}& operator *= (@var{type}&, const @var{type}&) +@cindex @code{operator *= ()} @itemx @var{type}& operator /= (@var{type}&, const @var{type}&) +@cindex @code{operator /= ()} @end table For the class @code{cl_I}: @@ -2052,10 +2268,15 @@ For the class @code{cl_I}: @itemx @var{type}& operator -= (@var{type}&, const @var{type}&) @itemx @var{type}& operator *= (@var{type}&, const @var{type}&) @itemx @var{type}& operator &= (@var{type}&, const @var{type}&) +@cindex @code{operator &= ()} @itemx @var{type}& operator |= (@var{type}&, const @var{type}&) +@cindex @code{operator |= ()} @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&) +@cindex @code{operator ^= ()} @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&) +@cindex @code{operator <<= ()} @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&) +@cindex @code{operator >>= ()} @end table For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I}, @@ -2063,12 +2284,14 @@ For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I}, @table @code @item @var{type}& operator ++ (@var{type}& x) +@cindex @code{operator ++ ()} The prefix operator @code{++x}. @item void operator ++ (@var{type}& x, int) The postfix operator @code{x++}. @item @var{type}& operator -- (@var{type}& x) +@cindex @code{operator -- ()} The prefix operator @code{--x}. @item void operator -- (@var{type}& x, int) @@ -2081,8 +2304,10 @@ efficient. @chapter Input/Output +@cindex Input/Output @section Internal and printed representation +@cindex representation All computations deal with the internal representations of the numbers. @@ -2091,8 +2316,10 @@ Several external representations may denote the same number, for example, "20.0" and "20.000". Converting an internal to an external representation is called ``printing'', +@cindex printing converting an external to an internal representation is called ``reading''. -In CLN, is it always true that conversion of an internal to an external +@cindex reading +In CLN, it is always true that conversion of an internal to an external representation and then back to an internal representation will yield the same internal representation. Symbolically: @code{read(print(x)) == x}. This is called ``print-read consistency''. @@ -2366,7 +2593,7 @@ using this variable name. Default is @code{"x"}. @end table The global variable @code{cl_default_print_flags} contains the default values, -used by the function @code{fprint}, +used by the function @code{fprint}. @chapter Rings @@ -2433,8 +2660,10 @@ Tests whether the given number is an element of the number ring R. @chapter Modular integers +@cindex modular integer @section Modular integer rings +@cindex ring CLN implements modular integers, i.e. integers modulo a fixed integer N. The modulus is explicitly part of every modular integer. CLN doesn't @@ -2468,9 +2697,11 @@ Modular integer rings are constructed using the function @table @code @item cl_modint_ring cl_find_modint_ring (const cl_I& N) +@cindex @code{cl_find_modint_ring ()} This function returns the modular ring @samp{Z/NZ}. It takes care of finding out about special cases of @code{N}, like powers of two and odd numbers for which Montgomery multiplication will be a win, +@cindex Montgomery multiplication and precomputes any necessary auxiliary data for computing modulo @code{N}. There is a cache table of rings, indexed by @code{N} (or, more precisely, by @code{abs(N)}). This ensures that the precomputation costs are reduced @@ -2481,7 +2712,9 @@ Modular integer rings can be compared for equality: @table @code @item bool operator== (const cl_modint_ring&, const cl_modint_ring&) +@cindex @code{operator == ()} @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&) +@cindex @code{operator != ()} These compare two modular integer rings for equality. Two different calls to @code{cl_find_modint_ring} with the same argument necessarily return the same ring because it is memoized in the cache table. @@ -2493,23 +2726,29 @@ Given a modular integer ring @code{R}, the following members can be used. @table @code @item cl_I R->modulus +@cindex @code{modulus} This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}. @item cl_MI R->zero() +@cindex @code{zero ()} This returns @code{0 mod N}. @item cl_MI R->one() +@cindex @code{one ()} This returns @code{1 mod N}. @item cl_MI R->canonhom (const cl_I& x) +@cindex @code{canonhom ()} This returns @code{x mod N}. @item cl_I R->retract (const cl_MI& x) +@cindex @code{etract ()} This is a partial inverse function to @code{R->canonhom}. It returns the standard representative (@code{>=0}, @code{random(cl_random_state& randomstate) @itemx cl_MI R->random() +@cindex @code{random ()} This returns a random integer modulo @code{N}. @end table @@ -2517,13 +2756,16 @@ The following operations are defined on modular integers. @table @code @item cl_modint_ring x.ring () +@cindex @code{ring()} Returns the ring to which the modular integer @code{x} belongs. @item cl_MI operator+ (const cl_MI&, const cl_MI&) +@cindex @code{operator + ()} Returns the sum of two modular integers. One of the arguments may also be a plain integer. @item cl_MI operator- (const cl_MI&, const cl_MI&) +@cindex @code{operator - ()} Returns the difference of two modular integers. One of the arguments may also be a plain integer. @@ -2531,40 +2773,51 @@ a plain integer. Returns the negative of a modular integer. @item cl_MI operator* (const cl_MI&, const cl_MI&) +@cindex @code{operator * ()} Returns the product of two modular integers. One of the arguments may also be a plain integer. @item cl_MI square (const cl_MI&) +@cindex @code{square ()} Returns the square of a modular integer. @item cl_MI recip (const cl_MI& x) +@cindex @code{recip ()} Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x} must be coprime to the modulus, otherwise an error message is issued. @item cl_MI div (const cl_MI& x, const cl_MI& y) +@cindex @code{div ()} Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}. @code{y} must be coprime to the modulus, otherwise an error message is issued. @item cl_MI expt_pos (const cl_MI& x, const cl_I& y) +@cindex @code{expt_pos ()} @code{y} must be > 0. Returns @code{x^y}. @item cl_MI expt (const cl_MI& x, const cl_I& y) +@cindex @code{expt ()} Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the modulus, else an error message is issued. @item cl_MI operator<< (const cl_MI& x, const cl_I& y) +@cindex @code{operator << ()} Returns @code{x*2^y}. @item cl_MI operator>> (const cl_MI& x, const cl_I& y) +@cindex @code{operator >> ()} Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd, or an error message is issued. @item bool operator== (const cl_MI&, const cl_MI&) +@cindex @code{operator == ()} @itemx bool operator!= (const cl_MI&, const cl_MI&) +@cindex @code{operator != ()} Compares two modular integers, belonging to the same modular integer ring, for equality. @item cl_boolean zerop (const cl_MI& x) +@cindex @code{zerop ()} Returns true if @code{x} is @code{0 mod N}. @end table @@ -2573,17 +2826,21 @@ input/output). @table @code @item void fprint (cl_ostream stream, const cl_MI& x) +@cindex @code{fprint ()} @itemx cl_ostream operator<< (cl_ostream stream, const cl_MI& x) +@cindex @code{operator << ()} Prints the modular integer @code{x} on the @code{stream}. The output may depend on the global printer settings in the variable @code{cl_default_print_flags}. @end table @chapter Symbolic data types +@cindex symbolic type CLN implements two symbolic (non-numeric) data types: strings and symbols. @section Strings +@cindex string The class @@ -2599,6 +2856,7 @@ Strings are constructed through the following constructors: @table @code @item cl_string (const char * s) +@cindex @code{cl_string ()} Returns an immutable copy of the (zero-terminated) C string @code{s}. @item cl_string (const char * ptr, unsigned long len) @@ -2613,19 +2871,24 @@ The following functions are available on strings: Assignment from @code{cl_string} and @code{const char *}. @item s.length() +@cindex @code{length ()} @itemx strlen(s) +@cindex @code{strlen ()} Returns the length of the string @code{s}. @item s[i] +@cindex @code{operator [] ()} Returns the @code{i}th character of the string @code{s}. @code{i} must be in the range @code{0 <= i < s.length()}. @item bool equal (const cl_string& s1, const cl_string& s2) +@cindex @code{equal ()} Compares two strings for equality. One of the arguments may also be a plain @code{const char *}. @end table @section Symbols +@cindex symbol Symbols are uniquified strings: all symbols with the same name are shared. This means that comparison of two symbols is fast (effectively just a pointer @@ -2638,6 +2901,7 @@ Symbols are constructed through the following constructor: @table @code @item cl_symbol (const cl_string& s) +@cindex @code{cl_symbol ()} Looks up or creates a new symbol with a given name. @end table @@ -2649,11 +2913,14 @@ Conversion to @code{cl_string}: Returns the string which names the symbol @code{sym}. @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2) +@cindex @code{equal ()} Compares two symbols for equality. This is very fast. @end table @chapter Univariate polynomials +@cindex polynomial +@cindex univariate polynomial @section Univariate polynomial rings @@ -2749,6 +3016,7 @@ This ensures that two calls of this function with the same arguments will return the same polynomial ring. @item cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R) +@cindex @code{cl_find_univpoly_ring ()} @itemx cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname) @itemx cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R) @itemx cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname) @@ -2768,22 +3036,28 @@ Given a univariate polynomial ring @code{R}, the following members can be used. @table @code @item cl_ring R->basering() +@cindex @code{basering ()} This returns the base ring, as passed to @samp{cl_find_univpoly_ring}. @item cl_UP R->zero() +@cindex @code{zero ()} This returns @code{0 in R}, a polynomial of degree -1. @item cl_UP R->one() +@cindex @code{one ()} This returns @code{1 in R}, a polynomial of degree <= 0. @item cl_UP R->canonhom (const cl_I& x) +@cindex @code{canonhom ()} This returns @code{x in R}, a polynomial of degree <= 0. @item cl_UP R->monomial (const cl_ring_element& x, uintL e) +@cindex @code{monomial ()} This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the indeterminate. @item cl_UP R->create (sintL degree) +@cindex @code{create ()} Creates a new polynomial with a given degree. The zero polynomial has degree @code{-1}. After creating the polynomial, you should put in the coefficients, using the @code{set_coeff} member function, and then call the @code{finalize} @@ -2794,11 +3068,13 @@ The following are the only destructive operations on univariate polynomials. @table @code @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y) +@cindex @code{set_coeff ()} This changes the coefficient of @code{X^index} in @code{x} to be @code{y}. After changing a polynomial and before applying any "normal" operation on it, you should call its @code{finalize} member function. @item void finalize (cl_UP& x) +@cindex @code{finalize ()} This function marks the endpoint of destructive modifications of a polynomial. It normalizes the internal representation so that subsequent computations have less overhead. Doing normal computations on unnormalized polynomials may @@ -2809,47 +3085,60 @@ The following operations are defined on univariate polynomials. @table @code @item cl_univpoly_ring x.ring () +@cindex @code{ring ()} Returns the ring to which the univariate polynomial @code{x} belongs. @item cl_UP operator+ (const cl_UP&, const cl_UP&) +@cindex @code{operator + ()} Returns the sum of two univariate polynomials. @item cl_UP operator- (const cl_UP&, const cl_UP&) +@cindex @code{operator - ()} Returns the difference of two univariate polynomials. @item cl_UP operator- (const cl_UP&) Returns the negative of a univariate polynomial. @item cl_UP operator* (const cl_UP&, const cl_UP&) +@cindex @code{operator * ()} Returns the product of two univariate polynomials. One of the arguments may also be a plain integer or an element of the base ring. @item cl_UP square (const cl_UP&) +@cindex @code{square ()} Returns the square of a univariate polynomial. @item cl_UP expt_pos (const cl_UP& x, const cl_I& y) +@cindex @code{expt_pos ()} @code{y} must be > 0. Returns @code{x^y}. @item bool operator== (const cl_UP&, const cl_UP&) +@cindex @code{operator == ()} @itemx bool operator!= (const cl_UP&, const cl_UP&) +@cindex @code{operator != ()} Compares two univariate polynomials, belonging to the same univariate polynomial ring, for equality. @item cl_boolean zerop (const cl_UP& x) +@cindex @code{zerop ()} Returns true if @code{x} is @code{0 in R}. @item sintL degree (const cl_UP& x) +@cindex @code{degree ()} Returns the degree of the polynomial. The zero polynomial has degree @code{-1}. @item cl_ring_element coeff (const cl_UP& x, uintL index) +@cindex @code{coeff ()} Returns the coefficient of @code{X^index} in the polynomial @code{x}. @item cl_ring_element x (const cl_ring_element& y) +@cindex @code{operator () ()} Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring, then @samp{x(y)} returns the value of the substitution of @code{y} into @code{x}. @item cl_UP deriv (const cl_UP& x) +@cindex @code{deriv ()} Returns the derivative of the polynomial @code{x} with respect to the indeterminate @code{X}. @end table @@ -2859,7 +3148,9 @@ input/output). @table @code @item void fprint (cl_ostream stream, const cl_UP& x) +@cindex @code{fprint ()} @itemx cl_ostream operator<< (cl_ostream stream, const cl_UP& x) +@cindex @code{operator << ()} Prints the univariate polynomial @code{x} on the @code{stream}. The output may depend on the global printer settings in the variable @code{cl_default_print_flags}. @@ -2871,15 +3162,23 @@ The following functions return special polynomials. @table @code @item cl_UP_I cl_tschebychev (sintL n) +@cindex @code{cl_tschebychev ()} +@cindex Tschebychev polynomial Returns the n-th Tchebychev polynomial (n >= 0). @item cl_UP_I cl_hermite (sintL n) +@cindex @code{cl_hermite ()} +@cindex Hermite polynomial Returns the n-th Hermite polynomial (n >= 0). @item cl_UP_RA cl_legendre (sintL n) +@cindex @code{cl_legendre ()} +@cindex Legende polynomial Returns the n-th Legendre polynomial (n >= 0). @item cl_UP_I cl_laguerre (sintL n) +@cindex @code{cl_laguerre ()} +@cindex Laguerre polynomial Returns the n-th Laguerre polynomial (n >= 0). @end table @@ -2891,6 +3190,7 @@ of these polynomials from their definition can be found in the @chapter Internals @section Why C++ ? +@cindex advocacy Using C++ as an implementation language provides @@ -2899,6 +3199,7 @@ Using C++ as an implementation language provides Efficiency: It compiles to machine code. @item +@cindex portability Portability: It runs on all platforms supporting a C++ compiler. Because of the availability of GNU C++, this includes all currently used 32-bit and 64-bit platforms, independently of the quality of the vendor's C++ compiler. @@ -2907,7 +3208,7 @@ of the availability of GNU C++, this includes all currently used 32-bit and Type safety: The C++ compilers knows about the number types and complains if, for example, you try to assign a float to an integer variable. However, a drawback is that C++ doesn't know about generic types, hence a restriction -like that @code{operation+ (const cl_MI&, const cl_MI&)} requires that both +like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both arguments belong to the same modular ring cannot be expressed as a compile-time information. @@ -2934,6 +3235,8 @@ In order to save memory allocations, CLN implements: Object sharing: An operation like @code{x+0} returns @code{x} without copying it. @item +@cindex garbage collection +@cindex reference counting Garbage collection: A reference counting mechanism makes sure that any number object's storage is freed immediately when the last reference to the object is gone. @@ -2958,8 +3261,8 @@ memory access, just a couple of instructions for each elementary operation. The kernel of CLN has been written in assembly language for some CPUs (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}). @item -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3. @item For large numbers, CLN uses, instead of the standard @code{O(N^2)} algorithm, the Karatsuba multiplication, which is an @@ -2976,9 +3279,11 @@ algorithm. For very large numbers (more than 12000 decimal digits), CLN uses @iftex Sch{@"o}nhage-Strassen +@cindex Sch{@"o}nhage-Strassen @end iftex @ifinfo Schönhage-Strassen +@cindex Schönhage-Strassen @end ifinfo multiplication, which is an asymptotically optimal multiplication algorithm. @@ -2989,6 +3294,7 @@ of division and radix conversion. @section Garbage collection +@cindex garbage collection All the number classes are reference count classes: They only contain a pointer to an object in the heap. Upon construction, assignment and destruction of @@ -3014,6 +3320,7 @@ environment variables, or directly substitute the appropriate values. @section Compiler options +@cindex compiler options Until you have installed CLN in a public place, the following options are needed: @@ -3037,6 +3344,8 @@ linking a CLN application it is sufficient to give the flag @code{-lcln}. @section Include files +@cindex include files +@cindex header files Here is a summary of the include files and their contents. @@ -3167,6 +3476,7 @@ Includes all of the above. @section An Example A function which computes the nth Fibonacci number can be written as follows. +@cindex Fibonacci number @example #include @@ -3227,8 +3537,11 @@ When the function returns, all the local variables in the function are automatically reclaimed (garbage collected). Only the result survives and gets passed to the caller. +The file @code{fibonacci.cc} in the subdirectory @code{examples} +contains this implementation together with an even faster algorithm. @section Debugging support +@cindex debugging When debugging a CLN application with GNU @code{gdb}, two facilities are available from the library: @@ -3253,6 +3566,7 @@ CLN offers a function @code{cl_print}, callable from the debugger, for printing number objects. In order to get this function, you have to define the macro @samp{CL_DEBUG} and then include all the header files for which you want @code{cl_print} debugging support. For example: +@cindex @code{CL_DEBUG} @example #define CL_DEBUG #include @@ -3275,6 +3589,7 @@ only with number objects and similar. Therefore CLN offers a member function @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG} is needed for this member function to be implemented. Under @code{gdb}, you call it like this: +@cindex @code{debug_print ()} @example (gdb) print s $7 = @{ = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60, @@ -3292,6 +3607,7 @@ Unfortunately, this feature does not seem to work under all circumstances. @chapter Customizing +@cindex customizing @section Error handling @@ -3304,10 +3620,12 @@ yourself, with the prototype #include void cl_abort (void); @end example +@cindex @code{cl_abort ()} This function must not return control to its caller. @section Floating-point underflow +@cindex underflow Floating point underflow denotes the situation when a floating-point number is to be created which is so close to @code{0} that its exponent is too @@ -3317,15 +3635,15 @@ If you set the global variable cl_boolean cl_inhibit_floating_point_underflow @end example to @code{cl_true}, the error will be inhibited, and a floating-point zero -will be generated instead. -The default value of @code{cl_inhibit_floating_point_underflow} is -@code{cl_false}. +will be generated instead. The default value of +@code{cl_inhibit_floating_point_underflow} is @code{cl_false}. @section Customizing I/O The output of the function @code{fprint} may be customized by changing the value of the global variable @code{cl_default_print_flags}. +@cindex @code{cl_default_print_flags} @section Customizing the memory allocator @@ -3343,6 +3661,8 @@ like this: void* (*cl_malloc_hook) (size_t size) = @dots{}; void (*cl_free_hook) (void* ptr) = @dots{}; @end example +@cindex @code{cl_malloc_hook ()} +@cindex @code{cl_free_hook ()} The @code{cl_malloc_hook} function must not return a @code{NULL} pointer. It is not possible to change the memory allocator at runtime, because diff --git a/doc/cln.texi b/doc/cln.texi index 5f89fc2..2ddb3b9 100644 --- a/doc/cln.texi +++ b/doc/cln.texi @@ -313,8 +313,9 @@ CLN is speed efficient: The kernel of CLN has been written in assembly language for some CPUs (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}). @item -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. +@cindex GMP +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3. @item It uses Karatsuba multiplication, which is significantly faster for large numbers than the standard multiplication algorithm. @@ -322,13 +323,14 @@ for large numbers than the standard multiplication algorithm. For very large numbers (more than 12000 decimal digits), it uses @iftex Sch{@"o}nhage-Strassen +@cindex Sch{@"o}nhage-Strassen multiplication @end iftex @ifinfo Schönhage-Strassen +@cindex Schönhage-Strassen multiplication @end ifinfo -multiplication, which is an asymptotically -optimal multiplication algorithm, for multiplication, division and -radix conversion. +multiplication, which is an asymptotically optimal multiplication +algorithm, for multiplication, division and radix conversion. @end itemize @noindent @@ -370,7 +372,7 @@ This section describes how to install the CLN package on your system. To build CLN, you need a C++ compiler. Actually, you need GNU @code{g++ 2.7.0} or newer. On HPPA, you need GNU @code{g++ 2.8.0} or newer. -I recommend GNU @code{egcs 1.1} or newer. +I recommend GNU @code{g++ 2.95} or newer. The following C++ features are used: classes, member functions, @@ -419,11 +421,13 @@ initializations will not work. @end itemize @end ignore +@cindex @code{make} @node Make utility, Sed utility, C++ compiler, Prerequisites @subsection Make utility To build CLN, you also need to have GNU @code{make} installed. +@cindex @code{sed} @node Sed utility, , Make utility, Prerequisites @subsection Sed utility @@ -538,9 +542,6 @@ add either @samp{-O} or @samp{-O2 -fno-schedule-insns} to the CXXFLAGS. With full @samp{-O2}, @code{g++} miscompiles the division routines. Also, for --enable-shared to work, you need egcs-1.1.2 or newer. -On MIPS (SGI Irix 6), pass option @code{--without-gmp} to configure. gmp does -not work when compiled in @samp{n32} binary format on Irix. - By default, only a static library is built. You can build CLN as a shared library too, by calling @code{configure} with the option @samp{--enable-shared}. To get it built as a shared library only, call @code{configure} with the options @@ -553,6 +554,7 @@ library. @node Installing the library, Cleaning up, Building the library, Installation @section Installing the library +@cindex installation As with any autoconfiguring GNU software, installation is as easy as this: @@ -621,23 +623,33 @@ Rational number Floating-point number @end example +@cindex @code{cl_number} +@cindex abstract class The base class @code{cl_number} is an abstract base class. It is not useful to declare a variable of this type except if you want to completely disable compile-time type checking and use run-time type checking instead. +@cindex @code{cl_N} +@cindex real number +@cindex complex number The class @code{cl_N} comprises real and complex numbers. There is no special class for complex numbers since complex numbers with imaginary part @code{0} are automatically converted to real numbers. +@cindex @code{cl_R} The class @code{cl_R} comprises real numbers of different kinds. It is an abstract class. +@cindex @code{cl_RA} +@cindex rational number +@cindex integer The class @code{cl_RA} comprises exact real numbers: rational numbers, including integers. There is no special class for non-integral rational numbers since rational numbers with denominator @code{1} are automatically converted to integers. +@cindex @code{cl_F} The class @code{cl_F} implements floating-point approximations to real numbers. It is an abstract class. @@ -651,6 +663,7 @@ It is an abstract class. @node Exact numbers, Floating-point numbers, Ordinary number types, Ordinary number types @section Exact numbers +@cindex exact number Some numbers are represented as exact numbers: there is no loss of information when such a number is converted from its mathematical value to its internal @@ -681,6 +694,7 @@ is completely transparent. @node Floating-point numbers, Complex numbers, Exact numbers, Ordinary number types @section Floating-point numbers +@cindex floating-point number Not all real numbers can be represented exactly. (There is an easy mathematical proof for this: Only a countable set of numbers can be stored exactly in @@ -688,6 +702,7 @@ a computer, even if one assumes that it has unlimited storage. But there are uncountably many real numbers.) So some approximation is needed. CLN implements ordinary floating-point numbers, with mantissa and exponent. +@cindex rounding error The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{}) only return approximate results. For example, the value of the expression @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as @@ -709,12 +724,14 @@ Floating point numbers come in four flavors: @itemize @bullet @item +@cindex @code{cl_SF} Short floats, type @code{cl_SF}. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 17 mantissa bits (including the ``hidden'' bit). They don't consume heap allocation. @item +@cindex @code{cl_FF} Single floats, type @code{cl_FF}. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 24 mantissa bits (including the ``hidden'' bit). @@ -722,6 +739,7 @@ In CLN, they are represented as IEEE single-precision floating point numbers. This corresponds closely to the C/C++ type @samp{float}. @item +@cindex @code{cl_DF} Double floats, type @code{cl_DF}. They have 1 sign bit, 11 exponent bits (including the exponent's sign), and 53 mantissa bits (including the ``hidden'' bit). @@ -729,6 +747,7 @@ In CLN, they are represented as IEEE double-precision floating point numbers. This corresponds closely to the C/C++ type @samp{double}. @item +@cindex @code{cl_LF} Long floats, type @code{cl_LF}. They have 1 sign bit, 32 exponent bits (including the exponent's sign), and n mantissa bits (including the ``hidden'' bit), where n >= 64. @@ -744,6 +763,7 @@ gradual underflow. If the exponent range of some floating-point type is too limited for your application, choose another floating-point type with larger exponent range. +@cindex @code{cl_F} As a user of CLN, you can forget about the differences between the four floating-point types and just declare all your floating-point variables as being of type @code{cl_F}. This has the advantage that @@ -758,6 +778,7 @@ the floating point contagion rule happened to change in the future.) @node Complex numbers, Conversions, Floating-point numbers, Ordinary number types @section Complex numbers +@cindex complex number Complex numbers, as implemented by the class @code{cl_N}, have a real part and an imaginary part, both real numbers. A complex number whose @@ -770,6 +791,7 @@ through application of @code{sqrt} or transcendental functions. @node Conversions, , Complex numbers, Ordinary number types @section Conversions +@cindex conversion Conversions from any class to any its superclasses (``base classes'' in C++ terminology) is done automatically. @@ -811,6 +833,7 @@ Conversions from @samp{const char *} are provided for the classes @code{cl_R}, @code{cl_N}. The easiest way to specify a value which is outside of the range of the C++ built-in types is therefore to specify it as a string, like this: +@cindex Rubik's cube @example cl_I order_of_rubiks_cube_group = "43252003274489856000"; @end example @@ -822,9 +845,13 @@ the functions @table @code @item int cl_I_to_int (const cl_I& x) +@cindex @code{cl_I_to_int ()} @itemx unsigned int cl_I_to_uint (const cl_I& x) +@cindex @code{cl_I_to_uint ()} @itemx long cl_I_to_long (const cl_I& x) +@cindex @code{cl_I_to_long ()} @itemx unsigned long cl_I_to_ulong (const cl_I& x) +@cindex @code{cl_I_to_ulong ()} Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not representable in the range of @var{ctype}, a runtime error occurs. @end table @@ -837,7 +864,9 @@ the functions @table @code @item float cl_float_approx (const @var{type}& x) +@cindex @code{cl_float_approx ()} @itemx double cl_double_approx (const @var{type}& x) +@cindex @code{cl_double_approx ()} Returns an approximation of @code{x} of C type @var{ctype}. If @code{abs(x)} is too close to 0 (underflow), 0 is returned. If @code{abs(x)} is too large (overflow), an IEEE infinity is returned. @@ -847,8 +876,10 @@ Conversions from any class to any of its subclasses (``derived classes'' in C++ terminology) are not provided. Instead, you can assert and check that a value belongs to a certain subclass, and return it as element of that class, using the @samp{As} and @samp{The} macros. +@cindex @code{As() ()} @code{As(@var{type})(@var{value})} checks that @var{value} belongs to @var{type} and returns it as such. +@cindex @code{The() ()} @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to @var{type} and returns it as such. It is your responsibility to ensure that this assumption is valid. @@ -962,24 +993,30 @@ defines the following operations: @table @code @item @var{type} operator + (const @var{type}&, const @var{type}&) +@cindex @code{operator + ()} Addition. @item @var{type} operator - (const @var{type}&, const @var{type}&) +@cindex @code{operator - ()} Subtraction. @item @var{type} operator - (const @var{type}&) Returns the negative of the argument. @item @var{type} plus1 (const @var{type}& x) +@cindex @code{plus1 ()} Returns @code{x + 1}. @item @var{type} minus1 (const @var{type}& x) +@cindex @code{minus1 ()} Returns @code{x - 1}. @item @var{type} operator * (const @var{type}&, const @var{type}&) +@cindex @code{operator * ()} Multiplication. @item @var{type} square (const @var{type}& x) +@cindex @code{square ()} Returns @code{x * x}. @end table @@ -989,9 +1026,11 @@ defines the following operations: @table @code @item @var{type} operator / (const @var{type}&, const @var{type}&) +@cindex @code{operator / ()} Division. @item @var{type} recip (const @var{type}&) +@cindex @code{recip ()} Returns the reciprocal of the argument. @end table @@ -1003,6 +1042,7 @@ Instead, @code{cl_I} defines an ``exact quotient'' function: @table @code @item cl_I exquo (const cl_I& x, const cl_I& y) +@cindex @code{exquo ()} Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}. @end table @@ -1010,10 +1050,12 @@ The following exponentiation functions are defined: @table @code @item cl_I expt_pos (const cl_I& x, const cl_I& y) +@cindex @code{expt_pos ()} @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y) @code{y} must be > 0. Returns @code{x^y}. @item cl_RA expt (const cl_RA& x, const cl_I& y) +@cindex @code{expt ()} @itemx cl_R expt (const cl_R& x, const cl_I& y) @itemx cl_N expt (const cl_N& x, const cl_I& y) Returns @code{x^y}. @@ -1025,6 +1067,7 @@ defines the following operation: @table @code @item @var{type} abs (const @var{type}& x) +@cindex @code{abs ()} Returns the absolute value of @code{x}. This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}. @end table @@ -1042,6 +1085,7 @@ defines the following operation: @table @code @item @var{type} signum (const @var{type}& x) +@cindex @code{signum ()} Returns the sign of @code{x}, in the same number format as @code{x}. This is defined as @code{x / abs(x)} if @code{x} is non-zero, and @code{x} if @code{x} is zero. If @code{x} is real, the value is either @@ -1056,9 +1100,11 @@ Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations: @table @code @item cl_I numerator (const @var{type}& x) +@cindex @code{numerator ()} Returns the numerator of @code{x}. @item cl_I denominator (const @var{type}& x) +@cindex @code{denominator ()} Returns the denominator of @code{x}. @end table @@ -1073,6 +1119,7 @@ The class @code{cl_N} defines the following operation: @table @code @item cl_N complex (const cl_R& a, const cl_R& b) +@cindex @code{complex ()} Returns the complex number @code{a+bi}, that is, the complex number with real part @code{a} and imaginary part @code{b}. @end table @@ -1081,12 +1128,15 @@ Each of the classes @code{cl_N}, @code{cl_R} defines the following operations: @table @code @item cl_R realpart (const @var{type}& x) +@cindex @code{realpart ()} Returns the real part of @code{x}. @item cl_R imagpart (const @var{type}& x) +@cindex @code{imagpart ()} Returns the imaginary part of @code{x}. @item @var{type} conjugate (const @var{type}& x) +@cindex @code{conjugate ()} Returns the complex conjugate of @code{x}. @end table @@ -1102,6 +1152,7 @@ We have the relations @node Comparisons, Rounding functions, Elementary complex functions, Functions on numbers @section Comparisons +@cindex comparison Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I}, @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF} @@ -1109,15 +1160,19 @@ defines the following operations: @table @code @item bool operator == (const @var{type}&, const @var{type}&) +@cindex @code{operator == ()} @itemx bool operator != (const @var{type}&, const @var{type}&) +@cindex @code{operator != ()} Comparison, as in C and C++. @item uint32 cl_equal_hashcode (const @var{type}&) +@cindex @code{cl_equal_hashcode ()} Returns a 32-bit hash code that is the same for any two numbers which are the same according to @code{==}. This hash code depends on the number's value, not its type or precision. @item cl_boolean zerop (const @var{type}& x) +@cindex @code{zerop ()} Compare against zero: @code{x == 0} @end table @@ -1127,25 +1182,34 @@ defines the following operations: @table @code @item cl_signean cl_compare (const @var{type}& x, const @var{type}& y) +@cindex @code{cl_compare ()} Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y}, -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}. @item bool operator <= (const @var{type}&, const @var{type}&) +@cindex @code{operator <= ()} @itemx bool operator < (const @var{type}&, const @var{type}&) +@cindex @code{operator < ()} @itemx bool operator >= (const @var{type}&, const @var{type}&) +@cindex @code{operator >= ()} @itemx bool operator > (const @var{type}&, const @var{type}&) +@cindex @code{operator > ()} Comparison, as in C and C++. @item cl_boolean minusp (const @var{type}& x) +@cindex @code{minusp ()} Compare against zero: @code{x < 0} @item cl_boolean plusp (const @var{type}& x) +@cindex @code{plusp ()} Compare against zero: @code{x > 0} @item @var{type} max (const @var{type}& x, const @var{type}& y) +@cindex @code{max ()} Return the maximum of @code{x} and @code{y}. @item @var{type} min (const @var{type}& x, const @var{type}& y) +@cindex @code{min ()} Return the minimum of @code{x} and @code{y}. @end table @@ -1159,6 +1223,7 @@ there is no floating point number whose value is exactly @code{1/3}. @node Rounding functions, Roots, Comparisons, Functions on numbers @section Rounding functions +@cindex rounding When a real number is to be converted to an integer, there is no ``best'' rounding. The desired rounding function depends on the application. @@ -1206,12 +1271,16 @@ defines the following operations: @table @code @item cl_I floor1 (const @var{type}& x) +@cindex @code{floor1 ()} Returns @code{floor(x)}. @item cl_I ceiling1 (const @var{type}& x) +@cindex @code{ceiling1 ()} Returns @code{ceiling(x)}. @item cl_I truncate1 (const @var{type}& x) +@cindex @code{truncate1 ()} Returns @code{truncate(x)}. @item cl_I round1 (const @var{type}& x) +@cindex @code{round1 ()} Returns @code{round(x)}. @end table @@ -1271,9 +1340,13 @@ defines the following operations: @table @code @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @}; @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y) +@cindex @code{floor2 ()} @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y) +@cindex @code{ceiling2 ()} @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y) +@cindex @code{truncate2 ()} @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y) +@cindex @code{round2 ()} @end table Sometimes, one wants the quotient as a floating-point number (of the @@ -1286,9 +1359,13 @@ defines the following operations: @table @code @item @var{type} ffloor (const @var{type}& x) +@cindex @code{ffloor ()} @itemx @var{type} fceiling (const @var{type}& x) +@cindex @code{fceiling ()} @itemx @var{type} ftruncate (const @var{type}& x) +@cindex @code{ftruncate ()} @itemx @var{type} fround (const @var{type}& x) +@cindex @code{fround ()} @end table and similarly for class @code{cl_R}, but with return type @code{cl_F}. @@ -1312,9 +1389,13 @@ defines the following operations: @table @code @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @}; @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x) +@cindex @code{ffloor2 ()} @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x) +@cindex @code{fceiling2 ()} @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x) +@cindex @code{ftruncate2 ()} @itemx @var{type}_fdiv_t fround2 (const @var{type}& x) +@cindex @code{fround2 ()} @end table and similarly for class @code{cl_R}, but with quotient type @code{cl_F}. @@ -1348,7 +1429,9 @@ The classes @code{cl_R}, @code{cl_I} define the following operations: @table @code @item @var{type} mod (const @var{type}& x, const @var{type}& y) +@cindex @code{mod ()} @itemx @var{type} rem (const @var{type}& x, const @var{type}& y) +@cindex @code{rem ()} @end table @@ -1361,6 +1444,7 @@ defines the following operation: @table @code @item @var{type} sqrt (const @var{type}& x) +@cindex @code{sqrt ()} @code{x} must be >= 0. This function returns the square root of @code{x}, normalized to be >= 0. If @code{x} is the square of a rational number, @code{sqrt(x)} will be a rational number, else it will return a @@ -1371,6 +1455,7 @@ The classes @code{cl_RA}, @code{cl_I} define the following operation: @table @code @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root) +@cindex @code{sqrtp ()} This tests whether @code{x} is a perfect square. If so, it returns true and the exact square root in @code{*root}, else it returns false. @end table @@ -1379,6 +1464,7 @@ Furthermore, for integers, similarly: @table @code @item cl_boolean isqrt (const @var{type}& x, @var{type}* root) +@cindex @code{isqrt ()} @code{x} should be >= 0. This function sets @code{*root} to @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}: the boolean value @code{(expt(*root,2) == x)}. @@ -1389,6 +1475,7 @@ define the following operation: @table @code @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root) +@cindex @code{rootp ()} @code{x} must be >= 0. @code{n} must be > 0. This tests whether @code{x} is an @code{n}th power of a rational number. If so, it returns true and the exact root in @code{*root}, else it returns @@ -1400,6 +1487,7 @@ for class @code{cl_N}: @table @code @item cl_N sqrt (const cl_N& z) +@cindex @code{sqrt ()} Returns the square root of @code{z}, as defined by the formula @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type or to a complex number are done if necessary. The range of the result is the @@ -1412,7 +1500,7 @@ The result is an exact number only if @code{z} is an exact number. @node Transcendental functions, Functions on integers, Roots, Functions on numbers @section Transcendental functions - +@cindex transcendental functions The transcendental functions return an exact result if the argument is exact and the result is exact as well. Otherwise they must return @@ -1433,21 +1521,25 @@ For example, @code{cos(0) = 1} returns the rational number @code{1}. @table @code @item cl_R exp (const cl_R& x) +@cindex @code{exp ()} @itemx cl_N exp (const cl_N& x) Returns the exponential function of @code{x}. This is @code{e^x} where @code{e} is the base of the natural logarithms. The range of the result is the entire complex plane excluding 0. @item cl_R ln (const cl_R& x) +@cindex @code{ln ()} @code{x} must be > 0. Returns the (natural) logarithm of x. @item cl_N log (const cl_N& x) +@cindex @code{log ()} Returns the (natural) logarithm of x. If @code{x} is real and positive, this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}. The range of the result is the strip in the complex plane @code{-pi < imagpart(log(x)) <= pi}. @item cl_R phase (const cl_N& x) +@cindex @code{phase ()} Returns the angle part of @code{x} in its polar representation as a complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}. This is also the imaginary part of @code{log(x)}. @@ -1466,6 +1558,7 @@ Returns the logarithm of @code{a} with respect to base @code{b}. @code{log(a,b) = log(a)/log(b)}. @item cl_N expt (const cl_N& x, const cl_N& y) +@cindex @code{expt ()} Exponentiation: Returns @code{x^y = exp(y*log(x))}. @end table @@ -1473,6 +1566,7 @@ The constant e = exp(1) = 2.71828@dots{} is returned by the following functions: @table @code @item cl_F cl_exp1 (cl_float_format_t f) +@cindex @code{exp1 ()} Returns e as a float of format @code{f}. @item cl_F cl_exp1 (const cl_F& y) @@ -1488,6 +1582,7 @@ Returns e as a float of format @code{cl_default_float_format}. @table @code @item cl_R sin (const cl_R& x) +@cindex @code{sin ()} Returns @code{sin(x)}. The range of the result is the interval @code{-1 <= sin(x) <= 1}. @@ -1495,6 +1590,7 @@ Returns @code{sin(x)}. The range of the result is the interval Returns @code{sin(z)}. The range of the result is the entire complex plane. @item cl_R cos (const cl_R& x) +@cindex @code{cos ()} Returns @code{cos(x)}. The range of the result is the interval @code{-1 <= cos(x) <= 1}. @@ -1502,20 +1598,26 @@ Returns @code{cos(x)}. The range of the result is the interval Returns @code{cos(z)}. The range of the result is the entire complex plane. @item struct cl_cos_sin_t @{ cl_R cos; cl_R sin; @}; +@cindex @code{cl_cos_sin_t} @itemx cl_cos_sin_t cl_cos_sin (const cl_R& x) Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than +@cindex @code{cl_cos_sin ()} computing them separately. The relation @code{cos^2 + sin^2 = 1} will hold only approximately. @item cl_R tan (const cl_R& x) +@cindex @code{tan ()} @itemx cl_N tan (const cl_N& x) Returns @code{tan(x) = sin(x)/cos(x)}. @item cl_N cis (const cl_R& x) +@cindex @code{cis ()} @itemx cl_N cis (const cl_N& x) Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because @code{e^(i*x) = cos(x) + i*sin(x)}. +@cindex @code{asin} +@cindex @code{asin ()} @item cl_N asin (const cl_N& z) Returns @code{arcsin(z)}. This is defined as @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies @@ -1530,6 +1632,7 @@ results for arsinh. @end ignore @item cl_N acos (const cl_N& z) +@cindex @code{acos ()} Returns @code{arccos(z)}. This is defined as @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i} @ignore @@ -1545,6 +1648,8 @@ with @code{realpart = pi} and @code{imagpart > 0}. Proof: This follows from the results about arcsin. @end ignore +@cindex @code{atan} +@cindex @code{atan ()} @item cl_R atan (const cl_R& x, const cl_R& y) Returns the angle of the polar representation of the complex number @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of @@ -1572,10 +1677,13 @@ Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function. @end table -The constant pi = 3.14@dots{} is returned by the following functions: +@cindex pi +@cindex Archimedes' constant +Archimedes' constant pi = 3.14@dots{} is returned by the following functions: @table @code @item cl_F cl_pi (cl_float_format_t f) +@cindex @code{cl_pi} Returns pi as a float of format @code{f}. @item cl_F cl_pi (const cl_F& y) @@ -1591,12 +1699,14 @@ Returns pi as a float of format @code{cl_default_float_format}. @table @code @item cl_R sinh (const cl_R& x) +@cindex @code{sinh ()} Returns @code{sinh(x)}. @item cl_N sinh (const cl_N& z) Returns @code{sinh(z)}. The range of the result is the entire complex plane. @item cl_R cosh (const cl_R& x) +@cindex @code{cosh ()} Returns @code{cosh(x)}. The range of the result is the interval @code{cosh(x) >= 1}. @@ -1604,16 +1714,20 @@ Returns @code{cosh(x)}. The range of the result is the interval Returns @code{cosh(z)}. The range of the result is the entire complex plane. @item struct cl_cosh_sinh_t @{ cl_R cosh; cl_R sinh; @}; +@cindex @code{cl_cosh_sinh_t} @itemx cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x) +@cindex @code{cl_cosh_sinh ()} Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will hold only approximately. @item cl_R tanh (const cl_R& x) +@cindex @code{tanh ()} @itemx cl_N tanh (const cl_N& x) Returns @code{tanh(x) = sinh(x)/cosh(x)}. @item cl_N asinh (const cl_N& z) +@cindex @code{asinh ()} Returns @code{arsinh(z)}. This is defined as @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies @code{arsinh(-z) = -arsinh(z)}. @@ -1645,6 +1759,7 @@ log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2))) @end ignore @item cl_N acosh (const cl_N& z) +@cindex @code{acosh ()} Returns @code{arcosh(z)}. This is defined as @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}. The range of the result is the half-strip in the complex domain @@ -1690,6 +1805,7 @@ Otherwise, -z is in Range(sqrt). @end ignore @item cl_N atanh (const cl_N& z) +@cindex @code{atanh ()} Returns @code{artanh(z)}. This is defined as @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies @code{artanh(-z) = -artanh(z)}. The range of the result is @@ -1722,11 +1838,13 @@ Proof: Write z = x+iy. Examine @node Euler gamma, Riemann zeta, Hyperbolic functions, Transcendental functions @subsection Euler gamma +@cindex Euler's constant Euler's constant C = 0.577@dots{} is returned by the following functions: @table @code @item cl_F cl_eulerconst (cl_float_format_t f) +@cindex @code{cl_eulerconst ()} Returns Euler's constant as a float of format @code{f}. @item cl_F cl_eulerconst (const cl_F& y) @@ -1737,9 +1855,11 @@ Returns Euler's constant as a float of format @code{cl_default_float_format}. @end table Catalan's constant G = 0.915@dots{} is returned by the following functions: +@cindex Catalan's constant @table @code @item cl_F cl_catalanconst (cl_float_format_t f) +@cindex @code{cl_catalanconst ()} Returns Catalan's constant as a float of format @code{f}. @item cl_F cl_catalanconst (const cl_F& y) @@ -1752,12 +1872,14 @@ Returns Catalan's constant as a float of format @code{cl_default_float_format}. @node Riemann zeta, , Euler gamma, Transcendental functions @subsection Riemann zeta +@cindex Riemann's zeta Riemann's zeta function at an integral point @code{s>1} is returned by the following functions: @table @code @item cl_F cl_zeta (int s, cl_float_format_t f) +@cindex @code{cl_zeta ()} Returns Riemann's zeta function at @code{s} as a float of format @code{f}. @item cl_F cl_zeta (int s, const cl_F& y) @@ -1794,46 +1916,62 @@ on each of the bit positions in parallel. @table @code @item cl_I lognot (const cl_I& x) +@cindex @code{lognot ()} @itemx cl_I operator ~ (const cl_I& x) +@cindex @code{operator ~ ()} Logical not, like @code{~x} in C. This is the same as @code{-1-x}. @item cl_I logand (const cl_I& x, const cl_I& y) +@cindex @code{logand ()} @itemx cl_I operator & (const cl_I& x, const cl_I& y) +@cindex @code{operator & ()} Logical and, like @code{x & y} in C. @item cl_I logior (const cl_I& x, const cl_I& y) +@cindex @code{logior ()} @itemx cl_I operator | (const cl_I& x, const cl_I& y) +@cindex @code{operator | ()} Logical (inclusive) or, like @code{x | y} in C. @item cl_I logxor (const cl_I& x, const cl_I& y) +@cindex @code{logxor ()} @itemx cl_I operator ^ (const cl_I& x, const cl_I& y) +@cindex @code{operator ^ ()} Exclusive or, like @code{x ^ y} in C. @item cl_I logeqv (const cl_I& x, const cl_I& y) +@cindex @code{logeqv ()} Bitwise equivalence, like @code{~(x ^ y)} in C. @item cl_I lognand (const cl_I& x, const cl_I& y) +@cindex @code{lognand ()} Bitwise not and, like @code{~(x & y)} in C. @item cl_I lognor (const cl_I& x, const cl_I& y) +@cindex @code{lognor ()} Bitwise not or, like @code{~(x | y)} in C. @item cl_I logandc1 (const cl_I& x, const cl_I& y) +@cindex @code{logandc1 ()} Logical and, complementing the first argument, like @code{~x & y} in C. @item cl_I logandc2 (const cl_I& x, const cl_I& y) +@cindex @code{logandc2 ()} Logical and, complementing the second argument, like @code{x & ~y} in C. @item cl_I logorc1 (const cl_I& x, const cl_I& y) +@cindex @code{logorc1 ()} Logical or, complementing the first argument, like @code{~x | y} in C. @item cl_I logorc2 (const cl_I& x, const cl_I& y) +@cindex @code{logorc2 ()} Logical or, complementing the second argument, like @code{x | ~y} in C. @end table These operations are all available though the function @table @code @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y) +@cindex @code{boole ()} @end table where @code{op} must have one of the 16 values (each one stands for a function which combines two bits into one bit): @code{boole_clr}, @code{boole_set}, @@ -1841,19 +1979,38 @@ which combines two bits into one bit): @code{boole_clr}, @code{boole_set}, @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv}, @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2}, @code{boole_orc1}, @code{boole_orc2}. +@cindex @code{boole_clr} +@cindex @code{boole_set} +@cindex @code{boole_1} +@cindex @code{boole_2} +@cindex @code{boole_c1} +@cindex @code{boole_c2} +@cindex @code{boole_and} +@cindex @code{boole_xor} +@cindex @code{boole_eqv} +@cindex @code{boole_nand} +@cindex @code{boole_nor} +@cindex @code{boole_andc1} +@cindex @code{boole_andc2} +@cindex @code{boole_orc1} +@cindex @code{boole_orc2} + Other functions that view integers as bit strings: @table @code @item cl_boolean logtest (const cl_I& x, const cl_I& y) +@cindex @code{logtest ()} Returns true if some bit is set in both @code{x} and @code{y}, i.e. if @code{logand(x,y) != 0}. @item cl_boolean logbitp (const cl_I& n, const cl_I& x) +@cindex @code{logbitp ()} Returns true if the @code{n}th bit (from the right) of @code{x} is set. Bit 0 is the least significant bit. @item uintL logcount (const cl_I& x) +@cindex @code{logcount ()} Returns the number of one bits in @code{x}, if @code{x} >= 0, or the number of zero bits in @code{x}, if @code{x} < 0. @end table @@ -1863,20 +2020,24 @@ The type @example struct cl_byte @{ uintL size; uintL position; @}; @end example +@cindex @code{cl_byte} represents the bit interval containing the bits @code{position}@dots{}@code{position+size-1} of an integer. The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}. @table @code @item cl_I ldb (const cl_I& n, const cl_byte& b) +@cindex @code{ldb ()} extracts the bits of @code{n} described by the bit interval @code{b} and returns them as a nonnegative integer with @code{b.size} bits. @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b) +@cindex @code{ldb_test ()} Returns true if some bit described by the bit interval @code{b} is set in @code{n}. @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b) +@cindex @code{dpb ()} Returns @code{n}, with the bits described by the bit interval @code{b} replaced by @code{newbyte}. Only the lowest @code{b.size} bits of @code{newbyte} are relevant. @@ -1887,10 +2048,12 @@ functions are their counterparts without shifting: @table @code @item cl_I mask_field (const cl_I& n, const cl_byte& b) +@cindex @code{mask_field ()} returns an integer with the bits described by the bit interval @code{b} copied from the corresponding bits in @code{n}, the other bits zero. @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b) +@cindex @code{deposit_field ()} returns an integer where the bits described by the bit interval @code{b} come from @code{newbyte} and the other bits come from @code{n}. @end table @@ -1911,39 +2074,47 @@ for common arithmetic operations: @table @code @item cl_boolean oddp (const cl_I& x) +@cindex @code{oddp ()} Returns true if the least significant bit of @code{x} is 1. Equivalent to @code{mod(x,2) != 0}. @item cl_boolean evenp (const cl_I& x) +@cindex @code{evenp ()} Returns true if the least significant bit of @code{x} is 0. Equivalent to @code{mod(x,2) == 0}. @item cl_I operator << (const cl_I& x, const cl_I& n) +@cindex @code{operator << ()} Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0. Equivalent to @code{x * expt(2,n)}. @item cl_I operator >> (const cl_I& x, const cl_I& n) +@cindex @code{operator >> ()} Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0. Bits shifted out to the right are thrown away. Equivalent to @code{floor(x / expt(2,n))}. @item cl_I ash (const cl_I& x, const cl_I& y) +@cindex @code{ash ()} Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or by @code{-y} bits to the right (if @code{y}<=0). In other words, this returns @code{floor(x * expt(2,y))}. @item uintL integer_length (const cl_I& x) +@cindex @code{integer_length ()} Returns the number of bits (excluding the sign bit) needed to represent @code{x} in two's complement notation. This is the smallest n >= 0 such that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that 2^(n-1) <= x < 2^n. @item uintL ord2 (const cl_I& x) +@cindex @code{ord2 ()} @code{x} must be non-zero. This function returns the number of 0 bits at the right of @code{x} in two's complement notation. This is the largest n >= 0 such that 2^n divides @code{x}. @item uintL power2p (const cl_I& x) +@cindex @code{power2p ()} @code{x} must be > 0. This function checks whether @code{x} is a power of 2. If @code{x} = 2^(n-1), it returns n. Else it returns 0. (See also the function @code{logp}.) @@ -1955,11 +2126,13 @@ If @code{x} = 2^(n-1), it returns n. Else it returns 0. @table @code @item uint32 gcd (uint32 a, uint32 b) +@cindex @code{gcd ()} @itemx cl_I gcd (const cl_I& a, const cl_I& b) This function returns the greatest common divisor of @code{a} and @code{b}, normalized to be >= 0. @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v) +@cindex @code{xgcd ()} This function (``extended gcd'') returns the greatest common divisor @code{g} of @code{a} and @code{b} and at the same time the representation of @code{g} as an integral linear combination of @code{a} and @code{b}: @@ -1970,10 +2143,12 @@ value, in the following sense: If @code{a} and @code{b} are non-zero, and @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}. @item cl_I lcm (const cl_I& a, const cl_I& b) +@cindex @code{lcm ()} This function returns the least common multiple of @code{a} and @code{b}, normalized to be >= 0. @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l) +@cindex @code{logp ()} @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l) @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is rational number, this function returns true and sets *l = log(a,b), else @@ -1986,15 +2161,18 @@ it returns false. @table @code @item cl_I factorial (uintL n) +@cindex @code{factorial ()} @code{n} must be a small integer >= 0. This function returns the factorial @code{n}! = @code{1*2*@dots{}*n}. @item cl_I doublefactorial (uintL n) +@cindex @code{doublefactorial ()} @code{n} must be a small integer >= 0. This function returns the doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or @code{n}!! = @code{2*4*@dots{}*n}, respectively. @item cl_I binomial (uintL n, uintL k) +@cindex @code{binomial ()} @code{n} and @code{k} must be small integers >= 0. This function returns the binomial coefficient @tex @@ -2020,6 +2198,7 @@ defines the following operations. @table @code @item @var{type} scale_float (const @var{type}& x, sintL delta) +@cindex @code{scale_float ()} @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta) Returns @code{x*2^delta}. This is more efficient than an explicit multiplication because it copies @code{x} and modifies the exponent. @@ -2030,23 +2209,28 @@ representation of floating-point numbers. @table @code @item sintL float_exponent (const @var{type}& x) +@cindex @code{float_exponent ()} Returns the exponent @code{e} of @code{x}. For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique integer with @code{2^(e-1) <= abs(x) < 2^e}. @item sintL float_radix (const @var{type}& x) +@cindex @code{float_radix ()} Returns the base of the floating-point representation. This is always @code{2}. @item @var{type} float_sign (const @var{type}& x) +@cindex @code{float_sign ()} Returns the sign @code{s} of @code{x} as a float. The value is 1 for @code{x} >= 0, -1 for @code{x} < 0. @item uintL float_digits (const @var{type}& x) +@cindex @code{float_digits ()} Returns the number of mantissa bits in the floating-point representation of @code{x}, including the hidden bit. The value only depends on the type of @code{x}, not on its value. @item uintL float_precision (const @var{type}& x) +@cindex @code{float_precision ()} Returns the number of significant mantissa bits in the floating-point representation of @code{x}. Since denormalized numbers are not supported, this is the same as @code{float_digits(x)} if @code{x} is non-zero, and @@ -2054,6 +2238,11 @@ this is the same as @code{float_digits(x)} if @code{x} is non-zero, and @end table The complete internal representation of a float is encoded in the type +@cindex @code{cl_decoded_float} +@cindex @code{cl_decoded_sfloat} +@cindex @code{cl_decoded_ffloat} +@cindex @code{cl_decoded_dfloat} +@cindex @code{cl_decoded_lfloat} @code{cl_decoded_float} (or @code{cl_decoded_sfloat}, @code{cl_decoded_ffloat}, @code{cl_decoded_dfloat}, @code{cl_decoded_lfloat}, respectively), defined by @example @@ -2066,6 +2255,7 @@ and returned by the function @table @code @item cl_decoded_@var{type}float decode_float (const @var{type}& x) +@cindex @code{decode_float ()} For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0. @@ -2074,6 +2264,7 @@ it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0. A complete decoding in terms of integers is provided as type @example +@cindex @code{cl_idecoded_float} struct cl_idecoded_float @{ cl_I mantissa; cl_I exponent; cl_I sign; @}; @@ -2082,6 +2273,7 @@ by the following function: @table @code @item cl_idecoded_float integer_decode_float (const @var{type}& x) +@cindex @code{integer_decode_float ()} For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)} bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0. @@ -2093,6 +2285,7 @@ Some other function, implemented only for class @code{cl_F}: @table @code @item cl_F float_sign (const cl_F& x, const cl_F& y) +@cindex @code{float_sign ()} This returns a floating point number whose precision and absolute value is that of @code{y} and whose sign is that of @code{x}. If @code{x} is zero, it is treated as positive. Same for @code{y}. @@ -2101,6 +2294,7 @@ zero, it is treated as positive. Same for @code{y}. @node Conversion functions, Random number generators, Functions on floating-point numbers, Functions on numbers @section Conversion functions +@cindex conversion @menu * Conversion to floating-point numbers:: @@ -2114,6 +2308,7 @@ The type @code{cl_float_format_t} describes a floating-point format. @table @code @item cl_float_format_t cl_float_format (uintL n) +@cindex @code{cl_float_format ()} Returns the smallest float format which guarantees at least @code{n} decimal digits in the mantissa (after the decimal point). @@ -2121,6 +2316,7 @@ decimal digits in the mantissa (after the decimal point). Returns the floating point format of @code{x}. @item cl_float_format_t cl_default_float_format +@cindex @code{cl_default_float_format} Global variable: the default float format used when converting rational numbers to floats. @end table @@ -2132,6 +2328,7 @@ defines the following operations: @table @code @item cl_F cl_float (const @var{type}&x, cl_float_format_t f) +@cindex @code{cl_float} Returns @code{x} as a float of format @code{f}. @item cl_F cl_float (const @var{type}&x, const cl_F& y) Returns @code{x} in the float format of @code{y}. @@ -2146,23 +2343,29 @@ Every floating-point format has some characteristic numbers: @table @code @item cl_F most_positive_float (cl_float_format_t f) +@cindex @code{most_positive_float ()} Returns the largest (most positive) floating point number in float format @code{f}. @item cl_F most_negative_float (cl_float_format_t f) +@cindex @code{most_negative_float ()} Returns the smallest (most negative) floating point number in float format @code{f}. @item cl_F least_positive_float (cl_float_format_t f) +@cindex @code{least_positive_float ()} Returns the least positive floating point number (i.e. > 0 but closest to 0) in float format @code{f}. @item cl_F least_negative_float (cl_float_format_t f) +@cindex @code{least_negative_float ()} Returns the least negative floating point number (i.e. < 0 but closest to 0) in float format @code{f}. @item cl_F float_epsilon (cl_float_format_t f) +@cindex @code{float_epsilon ()} Returns the smallest floating point number e > 0 such that @code{1+e != 1}. @item cl_F float_negative_epsilon (cl_float_format_t f) +@cindex @code{float_negative_epsilon ()} Returns the smallest floating point number e > 0 such that @code{1-e != 1}. @end table @@ -2175,6 +2378,7 @@ defines the following operation: @table @code @item cl_RA rational (const @var{type}& x) +@cindex @code{rational ()} Returns the value of @code{x} as an exact number. If @code{x} is already an exact number, this is @code{x}. If @code{x} is a floating-point number, the value is a rational number whose denominator is a power of 2. @@ -2185,6 +2389,7 @@ the function @table @code @item cl_RA rationalize (const cl_R& x) +@cindex @code{rationalize ()} If @code{x} is a floating-point number, it actually represents an interval of real numbers, and this function returns the rational number with smallest denominator (and smallest numerator, in magnitude) @@ -2217,6 +2422,7 @@ Calling one of these modifies the state of the random number generator in a complicated but deterministic way. The global variable +@cindex @code{cl_default_random_state} @example cl_random_state cl_default_random_state @end example @@ -2226,20 +2432,24 @@ below are called without @code{cl_random_state} argument. @table @code @item uint32 random32 (cl_random_state& randomstate) @itemx uint32 random32 () +@cindex @code{random32 ()} Returns a random unsigned 32-bit number. All bits are equally random. @item cl_I random_I (cl_random_state& randomstate, const cl_I& n) @itemx cl_I random_I (const cl_I& n) +@cindex @code{random_I ()} @code{n} must be an integer > 0. This function returns a random integer @code{x} in the range @code{0 <= x < n}. @item cl_F random_F (cl_random_state& randomstate, const cl_F& n) @itemx cl_F random_F (const cl_F& n) +@cindex @code{random_F ()} @code{n} must be a float > 0. This function returns a random floating-point number of the same format as @code{n} in the range @code{0 <= x < n}. @item cl_R random_R (cl_random_state& randomstate, const cl_R& n) @itemx cl_R random_R (const cl_R& n) +@cindex @code{random_R ()} Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F} if @code{n} is a float. @end table @@ -2247,6 +2457,7 @@ if @code{n} is a float. @node Obfuscating operators, , Random number generators, Functions on numbers @section Obfuscating operators +@cindex modifying operators The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=}, @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=} @@ -2257,6 +2468,7 @@ to get happy, then add @example #define WANT_OBFUSCATING_OPERATORS @end example +@cindex @code{WANT_OBFUSCATING_OPERATORS} to the beginning of your source files, before the inclusion of any CLN include files. This flag will enable the following operators: @@ -2265,9 +2477,13 @@ For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @table @code @item @var{type}& operator += (@var{type}&, const @var{type}&) +@cindex @code{operator += ()} @itemx @var{type}& operator -= (@var{type}&, const @var{type}&) +@cindex @code{operator -= ()} @itemx @var{type}& operator *= (@var{type}&, const @var{type}&) +@cindex @code{operator *= ()} @itemx @var{type}& operator /= (@var{type}&, const @var{type}&) +@cindex @code{operator /= ()} @end table For the class @code{cl_I}: @@ -2277,10 +2493,15 @@ For the class @code{cl_I}: @itemx @var{type}& operator -= (@var{type}&, const @var{type}&) @itemx @var{type}& operator *= (@var{type}&, const @var{type}&) @itemx @var{type}& operator &= (@var{type}&, const @var{type}&) +@cindex @code{operator &= ()} @itemx @var{type}& operator |= (@var{type}&, const @var{type}&) +@cindex @code{operator |= ()} @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&) +@cindex @code{operator ^= ()} @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&) +@cindex @code{operator <<= ()} @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&) +@cindex @code{operator >>= ()} @end table For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I}, @@ -2288,12 +2509,14 @@ For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I}, @table @code @item @var{type}& operator ++ (@var{type}& x) +@cindex @code{operator ++ ()} The prefix operator @code{++x}. @item void operator ++ (@var{type}& x, int) The postfix operator @code{x++}. @item @var{type}& operator -- (@var{type}& x) +@cindex @code{operator -- ()} The prefix operator @code{--x}. @item void operator -- (@var{type}& x, int) @@ -2307,6 +2530,7 @@ efficient. @node Input/Output, Rings, Functions on numbers, Top @chapter Input/Output +@cindex Input/Output @menu * Internal and printed representation:: @@ -2316,6 +2540,7 @@ efficient. @node Internal and printed representation, Input functions, Input/Output, Input/Output @section Internal and printed representation +@cindex representation All computations deal with the internal representations of the numbers. @@ -2324,8 +2549,10 @@ Several external representations may denote the same number, for example, "20.0" and "20.000". Converting an internal to an external representation is called ``printing'', +@cindex printing converting an external to an internal representation is called ``reading''. -In CLN, is it always true that conversion of an internal to an external +@cindex reading +In CLN, it is always true that conversion of an internal to an external representation and then back to an internal representation will yield the same internal representation. Symbolically: @code{read(print(x)) == x}. This is called ``print-read consistency''. @@ -2601,7 +2828,7 @@ using this variable name. Default is @code{"x"}. @end table The global variable @code{cl_default_print_flags} contains the default values, -used by the function @code{fprint}, +used by the function @code{fprint}. @node Rings, Modular integers, Input/Output, Top @@ -2670,6 +2897,7 @@ Tests whether the given number is an element of the number ring R. @node Modular integers, Symbolic data types, Rings, Top @chapter Modular integers +@cindex modular integer @menu * Modular integer rings:: @@ -2678,6 +2906,7 @@ Tests whether the given number is an element of the number ring R. @node Modular integer rings, Functions on modular integers, Modular integers, Modular integers @section Modular integer rings +@cindex ring CLN implements modular integers, i.e. integers modulo a fixed integer N. The modulus is explicitly part of every modular integer. CLN doesn't @@ -2711,9 +2940,11 @@ Modular integer rings are constructed using the function @table @code @item cl_modint_ring cl_find_modint_ring (const cl_I& N) +@cindex @code{cl_find_modint_ring ()} This function returns the modular ring @samp{Z/NZ}. It takes care of finding out about special cases of @code{N}, like powers of two and odd numbers for which Montgomery multiplication will be a win, +@cindex Montgomery multiplication and precomputes any necessary auxiliary data for computing modulo @code{N}. There is a cache table of rings, indexed by @code{N} (or, more precisely, by @code{abs(N)}). This ensures that the precomputation costs are reduced @@ -2724,7 +2955,9 @@ Modular integer rings can be compared for equality: @table @code @item bool operator== (const cl_modint_ring&, const cl_modint_ring&) +@cindex @code{operator == ()} @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&) +@cindex @code{operator != ()} These compare two modular integer rings for equality. Two different calls to @code{cl_find_modint_ring} with the same argument necessarily return the same ring because it is memoized in the cache table. @@ -2737,23 +2970,29 @@ Given a modular integer ring @code{R}, the following members can be used. @table @code @item cl_I R->modulus +@cindex @code{modulus} This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}. @item cl_MI R->zero() +@cindex @code{zero ()} This returns @code{0 mod N}. @item cl_MI R->one() +@cindex @code{one ()} This returns @code{1 mod N}. @item cl_MI R->canonhom (const cl_I& x) +@cindex @code{canonhom ()} This returns @code{x mod N}. @item cl_I R->retract (const cl_MI& x) +@cindex @code{etract ()} This is a partial inverse function to @code{R->canonhom}. It returns the standard representative (@code{>=0}, @code{random(cl_random_state& randomstate) @itemx cl_MI R->random() +@cindex @code{random ()} This returns a random integer modulo @code{N}. @end table @@ -2761,13 +3000,16 @@ The following operations are defined on modular integers. @table @code @item cl_modint_ring x.ring () +@cindex @code{ring()} Returns the ring to which the modular integer @code{x} belongs. @item cl_MI operator+ (const cl_MI&, const cl_MI&) +@cindex @code{operator + ()} Returns the sum of two modular integers. One of the arguments may also be a plain integer. @item cl_MI operator- (const cl_MI&, const cl_MI&) +@cindex @code{operator - ()} Returns the difference of two modular integers. One of the arguments may also be a plain integer. @@ -2775,40 +3017,51 @@ a plain integer. Returns the negative of a modular integer. @item cl_MI operator* (const cl_MI&, const cl_MI&) +@cindex @code{operator * ()} Returns the product of two modular integers. One of the arguments may also be a plain integer. @item cl_MI square (const cl_MI&) +@cindex @code{square ()} Returns the square of a modular integer. @item cl_MI recip (const cl_MI& x) +@cindex @code{recip ()} Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x} must be coprime to the modulus, otherwise an error message is issued. @item cl_MI div (const cl_MI& x, const cl_MI& y) +@cindex @code{div ()} Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}. @code{y} must be coprime to the modulus, otherwise an error message is issued. @item cl_MI expt_pos (const cl_MI& x, const cl_I& y) +@cindex @code{expt_pos ()} @code{y} must be > 0. Returns @code{x^y}. @item cl_MI expt (const cl_MI& x, const cl_I& y) +@cindex @code{expt ()} Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the modulus, else an error message is issued. @item cl_MI operator<< (const cl_MI& x, const cl_I& y) +@cindex @code{operator << ()} Returns @code{x*2^y}. @item cl_MI operator>> (const cl_MI& x, const cl_I& y) +@cindex @code{operator >> ()} Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd, or an error message is issued. @item bool operator== (const cl_MI&, const cl_MI&) +@cindex @code{operator == ()} @itemx bool operator!= (const cl_MI&, const cl_MI&) +@cindex @code{operator != ()} Compares two modular integers, belonging to the same modular integer ring, for equality. @item cl_boolean zerop (const cl_MI& x) +@cindex @code{zerop ()} Returns true if @code{x} is @code{0 mod N}. @end table @@ -2817,7 +3070,9 @@ input/output). @table @code @item void fprint (cl_ostream stream, const cl_MI& x) +@cindex @code{fprint ()} @itemx cl_ostream operator<< (cl_ostream stream, const cl_MI& x) +@cindex @code{operator << ()} Prints the modular integer @code{x} on the @code{stream}. The output may depend on the global printer settings in the variable @code{cl_default_print_flags}. @end table @@ -2825,6 +3080,7 @@ on the global printer settings in the variable @code{cl_default_print_flags}. @node Symbolic data types, Univariate polynomials, Modular integers, Top @chapter Symbolic data types +@cindex symbolic type CLN implements two symbolic (non-numeric) data types: strings and symbols. @@ -2835,6 +3091,7 @@ CLN implements two symbolic (non-numeric) data types: strings and symbols. @node Strings, Symbols, Symbolic data types, Symbolic data types @section Strings +@cindex string The class @@ -2850,6 +3107,7 @@ Strings are constructed through the following constructors: @table @code @item cl_string (const char * s) +@cindex @code{cl_string ()} Returns an immutable copy of the (zero-terminated) C string @code{s}. @item cl_string (const char * ptr, unsigned long len) @@ -2864,20 +3122,25 @@ The following functions are available on strings: Assignment from @code{cl_string} and @code{const char *}. @item s.length() +@cindex @code{length ()} @itemx strlen(s) +@cindex @code{strlen ()} Returns the length of the string @code{s}. @item s[i] +@cindex @code{operator [] ()} Returns the @code{i}th character of the string @code{s}. @code{i} must be in the range @code{0 <= i < s.length()}. @item bool equal (const cl_string& s1, const cl_string& s2) +@cindex @code{equal ()} Compares two strings for equality. One of the arguments may also be a plain @code{const char *}. @end table @node Symbols, , Strings, Symbolic data types @section Symbols +@cindex symbol Symbols are uniquified strings: all symbols with the same name are shared. This means that comparison of two symbols is fast (effectively just a pointer @@ -2890,6 +3153,7 @@ Symbols are constructed through the following constructor: @table @code @item cl_symbol (const cl_string& s) +@cindex @code{cl_symbol ()} Looks up or creates a new symbol with a given name. @end table @@ -2901,12 +3165,15 @@ Conversion to @code{cl_string}: Returns the string which names the symbol @code{sym}. @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2) +@cindex @code{equal ()} Compares two symbols for equality. This is very fast. @end table @node Univariate polynomials, Internals, Symbolic data types, Top @chapter Univariate polynomials +@cindex polynomial +@cindex univariate polynomial @menu * Univariate polynomial rings:: @@ -3009,6 +3276,7 @@ This ensures that two calls of this function with the same arguments will return the same polynomial ring. @item cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R) +@cindex @code{cl_find_univpoly_ring ()} @itemx cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname) @itemx cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R) @itemx cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname) @@ -3029,22 +3297,28 @@ Given a univariate polynomial ring @code{R}, the following members can be used. @table @code @item cl_ring R->basering() +@cindex @code{basering ()} This returns the base ring, as passed to @samp{cl_find_univpoly_ring}. @item cl_UP R->zero() +@cindex @code{zero ()} This returns @code{0 in R}, a polynomial of degree -1. @item cl_UP R->one() +@cindex @code{one ()} This returns @code{1 in R}, a polynomial of degree <= 0. @item cl_UP R->canonhom (const cl_I& x) +@cindex @code{canonhom ()} This returns @code{x in R}, a polynomial of degree <= 0. @item cl_UP R->monomial (const cl_ring_element& x, uintL e) +@cindex @code{monomial ()} This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the indeterminate. @item cl_UP R->create (sintL degree) +@cindex @code{create ()} Creates a new polynomial with a given degree. The zero polynomial has degree @code{-1}. After creating the polynomial, you should put in the coefficients, using the @code{set_coeff} member function, and then call the @code{finalize} @@ -3055,11 +3329,13 @@ The following are the only destructive operations on univariate polynomials. @table @code @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y) +@cindex @code{set_coeff ()} This changes the coefficient of @code{X^index} in @code{x} to be @code{y}. After changing a polynomial and before applying any "normal" operation on it, you should call its @code{finalize} member function. @item void finalize (cl_UP& x) +@cindex @code{finalize ()} This function marks the endpoint of destructive modifications of a polynomial. It normalizes the internal representation so that subsequent computations have less overhead. Doing normal computations on unnormalized polynomials may @@ -3070,47 +3346,60 @@ The following operations are defined on univariate polynomials. @table @code @item cl_univpoly_ring x.ring () +@cindex @code{ring ()} Returns the ring to which the univariate polynomial @code{x} belongs. @item cl_UP operator+ (const cl_UP&, const cl_UP&) +@cindex @code{operator + ()} Returns the sum of two univariate polynomials. @item cl_UP operator- (const cl_UP&, const cl_UP&) +@cindex @code{operator - ()} Returns the difference of two univariate polynomials. @item cl_UP operator- (const cl_UP&) Returns the negative of a univariate polynomial. @item cl_UP operator* (const cl_UP&, const cl_UP&) +@cindex @code{operator * ()} Returns the product of two univariate polynomials. One of the arguments may also be a plain integer or an element of the base ring. @item cl_UP square (const cl_UP&) +@cindex @code{square ()} Returns the square of a univariate polynomial. @item cl_UP expt_pos (const cl_UP& x, const cl_I& y) +@cindex @code{expt_pos ()} @code{y} must be > 0. Returns @code{x^y}. @item bool operator== (const cl_UP&, const cl_UP&) +@cindex @code{operator == ()} @itemx bool operator!= (const cl_UP&, const cl_UP&) +@cindex @code{operator != ()} Compares two univariate polynomials, belonging to the same univariate polynomial ring, for equality. @item cl_boolean zerop (const cl_UP& x) +@cindex @code{zerop ()} Returns true if @code{x} is @code{0 in R}. @item sintL degree (const cl_UP& x) +@cindex @code{degree ()} Returns the degree of the polynomial. The zero polynomial has degree @code{-1}. @item cl_ring_element coeff (const cl_UP& x, uintL index) +@cindex @code{coeff ()} Returns the coefficient of @code{X^index} in the polynomial @code{x}. @item cl_ring_element x (const cl_ring_element& y) +@cindex @code{operator () ()} Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring, then @samp{x(y)} returns the value of the substitution of @code{y} into @code{x}. @item cl_UP deriv (const cl_UP& x) +@cindex @code{deriv ()} Returns the derivative of the polynomial @code{x} with respect to the indeterminate @code{X}. @end table @@ -3120,7 +3409,9 @@ input/output). @table @code @item void fprint (cl_ostream stream, const cl_UP& x) +@cindex @code{fprint ()} @itemx cl_ostream operator<< (cl_ostream stream, const cl_UP& x) +@cindex @code{operator << ()} Prints the univariate polynomial @code{x} on the @code{stream}. The output may depend on the global printer settings in the variable @code{cl_default_print_flags}. @@ -3133,15 +3424,23 @@ The following functions return special polynomials. @table @code @item cl_UP_I cl_tschebychev (sintL n) +@cindex @code{cl_tschebychev ()} +@cindex Tschebychev polynomial Returns the n-th Tchebychev polynomial (n >= 0). @item cl_UP_I cl_hermite (sintL n) +@cindex @code{cl_hermite ()} +@cindex Hermite polynomial Returns the n-th Hermite polynomial (n >= 0). @item cl_UP_RA cl_legendre (sintL n) +@cindex @code{cl_legendre ()} +@cindex Legende polynomial Returns the n-th Legendre polynomial (n >= 0). @item cl_UP_I cl_laguerre (sintL n) +@cindex @code{cl_laguerre ()} +@cindex Laguerre polynomial Returns the n-th Laguerre polynomial (n >= 0). @end table @@ -3162,6 +3461,7 @@ of these polynomials from their definition can be found in the @node Why C++ ?, Memory efficiency, Internals, Internals @section Why C++ ? +@cindex advocacy Using C++ as an implementation language provides @@ -3170,6 +3470,7 @@ Using C++ as an implementation language provides Efficiency: It compiles to machine code. @item +@cindex portability Portability: It runs on all platforms supporting a C++ compiler. Because of the availability of GNU C++, this includes all currently used 32-bit and 64-bit platforms, independently of the quality of the vendor's C++ compiler. @@ -3178,7 +3479,7 @@ of the availability of GNU C++, this includes all currently used 32-bit and Type safety: The C++ compilers knows about the number types and complains if, for example, you try to assign a float to an integer variable. However, a drawback is that C++ doesn't know about generic types, hence a restriction -like that @code{operation+ (const cl_MI&, const cl_MI&)} requires that both +like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both arguments belong to the same modular ring cannot be expressed as a compile-time information. @@ -3206,6 +3507,8 @@ In order to save memory allocations, CLN implements: Object sharing: An operation like @code{x+0} returns @code{x} without copying it. @item +@cindex garbage collection +@cindex reference counting Garbage collection: A reference counting mechanism makes sure that any number object's storage is freed immediately when the last reference to the object is gone. @@ -3231,8 +3534,8 @@ memory access, just a couple of instructions for each elementary operation. The kernel of CLN has been written in assembly language for some CPUs (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}). @item -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3. @item For large numbers, CLN uses, instead of the standard @code{O(N^2)} algorithm, the Karatsuba multiplication, which is an @@ -3249,9 +3552,11 @@ algorithm. For very large numbers (more than 12000 decimal digits), CLN uses @iftex Sch{@"o}nhage-Strassen +@cindex Sch{@"o}nhage-Strassen @end iftex @ifinfo Schönhage-Strassen +@cindex Schönhage-Strassen @end ifinfo multiplication, which is an asymptotically optimal multiplication algorithm. @@ -3263,6 +3568,7 @@ of division and radix conversion. @node Garbage collection, , Speed efficiency, Internals @section Garbage collection +@cindex garbage collection All the number classes are reference count classes: They only contain a pointer to an object in the heap. Upon construction, assignment and destruction of @@ -3297,6 +3603,7 @@ environment variables, or directly substitute the appropriate values. @node Compiler options, Include files, Using the library, Using the library @section Compiler options +@cindex compiler options Until you have installed CLN in a public place, the following options are needed: @@ -3321,6 +3628,8 @@ linking a CLN application it is sufficient to give the flag @code{-lcln}. @node Include files, An Example, Compiler options, Using the library @section Include files +@cindex include files +@cindex header files Here is a summary of the include files and their contents. @@ -3452,6 +3761,7 @@ Includes all of the above. @section An Example A function which computes the nth Fibonacci number can be written as follows. +@cindex Fibonacci number @example #include @@ -3512,9 +3822,12 @@ When the function returns, all the local variables in the function are automatically reclaimed (garbage collected). Only the result survives and gets passed to the caller. +The file @code{fibonacci.cc} in the subdirectory @code{examples} +contains this implementation together with an even faster algorithm. @node Debugging support, , An Example, Using the library @section Debugging support +@cindex debugging When debugging a CLN application with GNU @code{gdb}, two facilities are available from the library: @@ -3539,6 +3852,7 @@ CLN offers a function @code{cl_print}, callable from the debugger, for printing number objects. In order to get this function, you have to define the macro @samp{CL_DEBUG} and then include all the header files for which you want @code{cl_print} debugging support. For example: +@cindex @code{CL_DEBUG} @example #define CL_DEBUG #include @@ -3561,6 +3875,7 @@ only with number objects and similar. Therefore CLN offers a member function @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG} is needed for this member function to be implemented. Under @code{gdb}, you call it like this: +@cindex @code{debug_print ()} @example (gdb) print s $7 = @{ = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60, @@ -3579,6 +3894,7 @@ Unfortunately, this feature does not seem to work under all circumstances. @node Customizing, Index, Using the library, Top @chapter Customizing +@cindex customizing @menu * Error handling:: @@ -3599,11 +3915,13 @@ yourself, with the prototype #include void cl_abort (void); @end example +@cindex @code{cl_abort ()} This function must not return control to its caller. @node Floating-point underflow, Customizing I/O, Error handling, Customizing @section Floating-point underflow +@cindex underflow Floating point underflow denotes the situation when a floating-point number is to be created which is so close to @code{0} that its exponent is too @@ -3613,9 +3931,8 @@ If you set the global variable cl_boolean cl_inhibit_floating_point_underflow @end example to @code{cl_true}, the error will be inhibited, and a floating-point zero -will be generated instead. -The default value of @code{cl_inhibit_floating_point_underflow} is -@code{cl_false}. +will be generated instead. The default value of +@code{cl_inhibit_floating_point_underflow} is @code{cl_false}. @node Customizing I/O, Customizing the memory allocator, Floating-point underflow, Customizing @@ -3623,6 +3940,7 @@ The default value of @code{cl_inhibit_floating_point_underflow} is The output of the function @code{fprint} may be customized by changing the value of the global variable @code{cl_default_print_flags}. +@cindex @code{cl_default_print_flags} @node Customizing the memory allocator, , Customizing I/O, Customizing @@ -3641,6 +3959,8 @@ like this: void* (*cl_malloc_hook) (size_t size) = @dots{}; void (*cl_free_hook) (void* ptr) = @dots{}; @end example +@cindex @code{cl_malloc_hook ()} +@cindex @code{cl_free_hook ()} The @code{cl_malloc_hook} function must not return a @code{NULL} pointer. It is not possible to change the memory allocator at runtime, because diff --git a/doc/cln_1.html b/doc/cln_1.html index 6a1cb07..4f72621 100644 --- a/doc/cln_1.html +++ b/doc/cln_1.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 1. Introduction @@ -134,8 +134,9 @@ The kernel of CLN has been written in assembly language for some CPUs (i386, m68k, sparc, mips, arm).

    • -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. + +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3.
    • It uses Karatsuba multiplication, which is significantly faster @@ -144,9 +145,9 @@ for large numbers than the standard multiplication algorithm. For very large numbers (more than 12000 decimal digits), it uses Schönhage-Strassen -multiplication, which is an asymptotically -optimal multiplication algorithm, for multiplication, division and -radix conversion. + +multiplication, which is an asymptotically optimal multiplication +algorithm, for multiplication, division and radix conversion.

    diff --git a/doc/cln_10.html b/doc/cln_10.html index 059d504..d858b76 100644 --- a/doc/cln_10.html +++ b/doc/cln_10.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 10. Internals @@ -14,6 +14,9 @@ Go to the first, previous, 10.1 Why C++ ? +

    + +

    Using C++ as an implementation language provides @@ -27,6 +30,7 @@ Efficiency: It compiles to machine code.

  • + Portability: It runs on all platforms supporting a C++ compiler. Because of the availability of GNU C++, this includes all currently used 32-bit and 64-bit platforms, independently of the quality of the vendor's C++ compiler. @@ -36,7 +40,7 @@ of the availability of GNU C++, this includes all currently used 32-bit and Type safety: The C++ compilers knows about the number types and complains if, for example, you try to assign a float to an integer variable. However, a drawback is that C++ doesn't know about generic types, hence a restriction -like that operation+ (const cl_MI&, const cl_MI&) requires that both +like that operator+ (const cl_MI&, const cl_MI&) requires that both arguments belong to the same modular ring cannot be expressed as a compile-time information. @@ -72,6 +76,8 @@ Object sharing: An operation like x+0 returns x withou it.
  • + + Garbage collection: A reference counting mechanism makes sure that any number object's storage is freed immediately when the last reference to the object is gone. @@ -104,8 +110,8 @@ The kernel of CLN has been written in assembly language for some CPUs (i386, m68k, sparc, mips, arm).
  • -On all CPUs, CLN uses the superefficient low-level routines from GNU -GMP version 2. +On all CPUs, CLN may be configured to use the superefficient low-level +routines from GNU GMP version 3.
  • For large numbers, CLN uses, instead of the standard O(N^2) @@ -116,6 +122,7 @@ algorithm. For very large numbers (more than 12000 decimal digits), CLN uses Schönhage-Strassen + multiplication, which is an asymptotically optimal multiplication algorithm.
  • @@ -127,6 +134,9 @@ of division and radix conversion.

    10.4 Garbage collection

    +

    + +

    All the number classes are reference count classes: They only contain a pointer diff --git a/doc/cln_11.html b/doc/cln_11.html index 61ab64b..411c047 100644 --- a/doc/cln_11.html +++ b/doc/cln_11.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 11. Using the library @@ -22,6 +22,9 @@ environment variables, or directly substitute the appropriate values.

    11.1 Compiler options

    +

    + +

    Until you have installed CLN in a public place, the following options are @@ -58,6 +61,10 @@ linking a CLN application it is sufficient to give the flag -lcln.

    11.2 Include files

    +

    + + +

    Here is a summary of the include files and their contents. @@ -253,6 +260,7 @@ Includes all of the above.

    A function which computes the nth Fibonacci number can be written as follows. + @@ -329,9 +337,17 @@ automatically reclaimed (garbage collected). Only the result survives and gets passed to the caller. +

    +The file fibonacci.cc in the subdirectory examples +contains this implementation together with an even faster algorithm. + +

    11.4 Debugging support

    +

    + +

    When debugging a CLN application with GNU gdb, two facilities are @@ -364,6 +380,7 @@ CLN offers a function cl_print, callable from the debugger, for printing number objects. In order to get this function, you have to define the macro `CL_DEBUG' and then include all the header files for which you want cl_print debugging support. For example: + 

     #define CL_DEBUG
@@ -390,6 +407,7 @@ only with number objects and similar. Therefore CLN offers a member function
 debug_print() on all CLN types. The same macro `CL_DEBUG'
 is needed for this member function to be implemented. Under gdb,
 you call it like this:
+
 
 
     (gdb) print s
diff --git a/doc/cln_12.html b/doc/cln_12.html
index 0070b35..9399348 100644
--- a/doc/cln_12.html
+++ b/doc/cln_12.html
@@ -1,6 +1,6 @@
 
 
-
+
 
 CLN, a Class Library for Numbers - 12. Customizing
 
@@ -10,6 +10,9 @@ Go to the first, previous, 12. Customizing
+
    
+
+
 
 
 
@@ -28,12 +31,16 @@ void cl_abort (void);
 
    
 
 
    
+
 This function must not return control to its caller.
 
 
 
 
 
    12.2 Floating-point underflow
    
+
    
+
+
 
 
    
 Floating point underflow denotes the situation when a floating-point number
@@ -47,9 +54,8 @@ cl_boolean cl_inhibit_floating_point_underflow
 
 
    
 to cl_true, the error will be inhibited, and a floating-point zero
-will be generated instead.
-The default value of cl_inhibit_floating_point_underflow is
-cl_false.
+will be generated instead.  The default value of 
+cl_inhibit_floating_point_underflow is cl_false.
 
 
 
@@ -59,6 +65,7 @@ The default value of cl_inhibit_floating_point_underflow is
 
    
 The output of the function fprint may be customized by changing the
 value of the global variable cl_default_print_flags.
+
 
 
 
@@ -82,6 +89,8 @@ void (*cl_free_hook) (void* ptr)      = ...;

    + + The cl_malloc_hook function must not return a NULL pointer. diff --git a/doc/cln_13.html b/doc/cln_13.html index 7186cd1..1240d1e 100644 --- a/doc/cln_13.html +++ b/doc/cln_13.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - Index diff --git a/doc/cln_2.html b/doc/cln_2.html index 2a92f22..d9f2136 100644 --- a/doc/cln_2.html +++ b/doc/cln_2.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 2. Installation @@ -27,7 +27,7 @@ This section describes how to install the CLN package on your system. To build CLN, you need a C++ compiler. Actually, you need GNU g++ 2.7.0 or newer. On HPPA, you need GNU g++ 2.8.0 or newer. -I recommend GNU egcs 1.1 or newer. +I recommend GNU g++ 2.95 or newer.

    @@ -50,6 +50,8 @@ of static and global variables, a feature which I could implement for GNU g++ only. +

    +

    2.1.2 Make utility

    @@ -58,6 +60,8 @@ implement for GNU g++ only. To build CLN, you also need to have GNU make installed. +

    +

    2.1.3 Sed utility

    @@ -181,11 +185,6 @@ With full `-O2', g++ miscompiles the division routines --enable-shared to work, you need egcs-1.1.2 or newer. -

    -On MIPS (SGI Irix 6), pass option --without-gmp to configure. gmp does -not work when compiled in `n32' binary format on Irix. - -

    By default, only a static library is built. You can build CLN as a shared library too, by calling configure with the option `--enable-shared'. @@ -202,6 +201,9 @@ library.

    2.3 Installing the library

    +

    + +

    As with any autoconfiguring GNU software, installation is as easy as this: diff --git a/doc/cln_3.html b/doc/cln_3.html index 9db9d05..ed3eae4 100644 --- a/doc/cln_3.html +++ b/doc/cln_3.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 3. Ordinary number types @@ -45,6 +45,8 @@ Rational number Floating-point number

    + + The base class cl_number is an abstract base class. It is not useful to declare a variable of this type except if you want to completely disable compile-time type checking and use run-time type @@ -52,17 +54,24 @@ checking instead.

    + + + The class cl_N comprises real and complex numbers. There is no special class for complex numbers since complex numbers with imaginary part 0 are automatically converted to real numbers.

    + The class cl_R comprises real numbers of different kinds. It is an abstract class.

    + + + The class cl_RA comprises exact real numbers: rational numbers, including integers. There is no special class for non-integral rational numbers since rational numbers with denominator 1 are automatically converted @@ -70,6 +79,7 @@ to integers.

    + The class cl_F implements floating-point approximations to real numbers. It is an abstract class. @@ -77,6 +87,9 @@ It is an abstract class.

    3.1 Exact numbers

    +

    + +

    Some numbers are represented as exact numbers: there is no loss of information @@ -118,6 +131,9 @@ is completely transparent.

    3.2 Floating-point numbers

    +

    + +

    Not all real numbers can be represented exactly. (There is an easy mathematical @@ -128,6 +144,7 @@ CLN implements ordinary floating-point numbers, with mantissa and exponent.

    + The elementary operations (+, -, *, /, ...) only return approximate results. For example, the value of the expression (cl_F) 0.3 + (cl_F) 0.4 prints as `0.70000005', not as @@ -158,6 +175,7 @@ Floating point numbers come in four flavors:

    • + Short floats, type cl_SF. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 17 mantissa bits (including the "hidden" bit). @@ -165,6 +183,7 @@ They don't consume heap allocation.
    • + Single floats, type cl_FF. They have 1 sign bit, 8 exponent bits (including the exponent's sign), and 24 mantissa bits (including the "hidden" bit). @@ -173,6 +192,7 @@ This corresponds closely to the C/C++ type `float'.
    • + Double floats, type cl_DF. They have 1 sign bit, 11 exponent bits (including the exponent's sign), and 53 mantissa bits (including the "hidden" bit). @@ -181,6 +201,7 @@ This corresponds closely to the C/C++ type `double'.
    • + Long floats, type cl_LF. They have 1 sign bit, 32 exponent bits (including the exponent's sign), and n mantissa bits (including the "hidden" bit), where n >= 64. @@ -201,6 +222,7 @@ with larger exponent range.

      + As a user of CLN, you can forget about the differences between the four floating-point types and just declare all your floating-point variables as being of type cl_F. This has the advantage that @@ -216,6 +238,9 @@ the floating point contagion rule happened to change in the future.)

      3.3 Complex numbers

      +

      + +

      Complex numbers, as implemented by the class cl_N, have a real @@ -232,6 +257,9 @@ through application of sqrt or transcendental functions.

      3.4 Conversions

      +

      + +

      Conversions from any class to any its superclasses ("base classes" in @@ -289,6 +317,7 @@ Conversions from `const char *' are provided for the classes cl_R, cl_N. The easiest way to specify a value which is outside of the range of the C++ built-in types is therefore to specify it as a string, like this: + 

          cl_I order_of_rubiks_cube_group = "43252003274489856000";
@@ -308,12 +337,16 @@ the functions
 
 
      int cl_I_to_int (const cl_I& x)
 
      
+
 
      unsigned int cl_I_to_uint (const cl_I& x)
 
      
+
 
      long cl_I_to_long (const cl_I& x)
 
      
+
 
      unsigned long cl_I_to_ulong (const cl_I& x)
 
      
+
 Returns x as element of the C type ctype. If x is not
 representable in the range of ctype, a runtime error occurs.
 
@@ -330,8 +363,10 @@ the functions
 
 
      float cl_float_approx (const type& x)
 
      
+
 
      double cl_double_approx (const type& x)
 
      
+
 Returns an approximation of x of C type ctype.
 If abs(x) is too close to 0 (underflow), 0 is returned.
 If abs(x) is too large (overflow), an IEEE infinity is returned.
@@ -342,8 +377,10 @@ Conversions from any class to any of its subclasses ("derived classes" in
 C++ terminology) are not provided. Instead, you can assert and check
 that a value belongs to a certain subclass, and return it as element of that
 class, using the `As' and `The' macros.
+
 As(type)(value) checks that value belongs to
 type and returns it as such.
+
 The(type)(value) assumes that value belongs to
 type and returns it as such. It is your responsibility to ensure
 that this assumption is valid.
diff --git a/doc/cln_4.html b/doc/cln_4.html
index f9ac86e..4bf113f 100644
--- a/doc/cln_4.html
+++ b/doc/cln_4.html
@@ -1,6 +1,6 @@
 
 
-
+
 
 CLN, a Class Library for Numbers - 4. Functions on numbers
 
@@ -109,10 +109,12 @@ defines the following operations:
 
 
      type operator + (const type&, const type&)
 
      
+
 Addition.
 
 
      type operator - (const type&, const type&)
 
      
+
 Subtraction.
 
 
      type operator - (const type&)
@@ -121,18 +123,22 @@ Returns the negative of the argument.
 
 
      type plus1 (const type& x)
 
      
+
 Returns x + 1.
 
 
      type minus1 (const type& x)
 
      
+
 Returns x - 1.
 
 
      type operator * (const type&, const type&)
 
      
+
 Multiplication.
 
 
      type square (const type& x)
 
      
+
 Returns x * x.
 
 
@@ -146,10 +152,12 @@ defines the following operations:
 
 
      type operator / (const type&, const type&)
 
      
+
 Division.
 
 
      type recip (const type&)
 
      
+
 Returns the reciprocal of the argument.
 
 
@@ -165,6 +173,7 @@ Instead, cl_I defines an "exact quotient" function:
 
 
      cl_I exquo (const cl_I& x, const cl_I& y)
 
      
+
 Checks that y divides x, and returns the quotient x/y.
 
 
@@ -176,12 +185,14 @@ The following exponentiation functions are defined:
 
 
      cl_I expt_pos (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_RA expt_pos (const cl_RA& x, const cl_I& y)
 
      
 y must be > 0. Returns x^y.
 
 
      cl_RA expt (const cl_RA& x, const cl_I& y)
 
      
+
 
      cl_R expt (const cl_R& x, const cl_I& y)
 
      
 
      cl_N expt (const cl_N& x, const cl_I& y)
@@ -199,6 +210,7 @@ defines the following operation:
 
 
      type abs (const type& x)
 
      
+
 Returns the absolute value of x.
 This is x if x >= 0, and -x if x <= 0.
 
@@ -224,6 +236,7 @@ defines the following operation:
 
 
      type signum (const type& x)
 
      
+
 Returns the sign of x, in the same number format as x.
 This is defined as x / abs(x) if x is non-zero, and
 x if x is zero. If x is real, the value is either
@@ -242,10 +255,12 @@ Each of the classes cl_RA, cl_I defines the following
 
 
      cl_I numerator (const type& x)
 
      
+
 Returns the numerator of x.
 
 
      cl_I denominator (const type& x)
 
      
+
 Returns the denominator of x.
 
 
@@ -266,6 +281,7 @@ The class cl_N defines the following operation:
 
 
      cl_N complex (const cl_R& a, const cl_R& b)
 
      
+
 Returns the complex number a+bi, that is, the complex number with
 real part a and imaginary part b.
 
@@ -278,14 +294,17 @@ Each of the classes cl_N, cl_R defines the following o
 
 
      cl_R realpart (const type& x)
 
      
+
 Returns the real part of x.
 
 
      cl_R imagpart (const type& x)
 
      
+
 Returns the imaginary part of x.
 
 
      type conjugate (const type& x)
 
      
+
 Returns the complex conjugate of x.
 
 
@@ -306,6 +325,9 @@ We have the relations
 
 
 
      4.5 Comparisons
      
+
      
+
+
 
 
      
 Each of the classes cl_N, cl_R, cl_RA, cl_I,
@@ -317,18 +339,22 @@ defines the following operations:
 
 
      bool operator == (const type&, const type&)
 
      
+
 
      bool operator != (const type&, const type&)
 
      
+
 Comparison, as in C and C++.
 
 
      uint32 cl_equal_hashcode (const type&)
 
      
+
 Returns a 32-bit hash code that is the same for any two numbers which are
 the same according to ==. This hash code depends on the number's value,
 not its type or precision.
 
 
      cl_boolean zerop (const type& x)
 
      
+
 Compare against zero: x == 0
 
 
@@ -342,33 +368,42 @@ defines the following operations:
 
 
      cl_signean cl_compare (const type& x, const type& y)
 
      
+
 Compares x and y. Returns +1 if x>y,
 -1 if x<y, 0 if x=y.
 
 
      bool operator <= (const type&, const type&)
 
      
+
 
      bool operator < (const type&, const type&)
 
      
+
 
      bool operator >= (const type&, const type&)
 
      
+
 
      bool operator > (const type&, const type&)
 
      
+
 Comparison, as in C and C++.
 
 
      cl_boolean minusp (const type& x)
 
      
+
 Compare against zero: x < 0
 
 
      cl_boolean plusp (const type& x)
 
      
+
 Compare against zero: x > 0
 
 
      type max (const type& x, const type& y)
 
      
+
 Return the maximum of x and y.
 
 
      type min (const type& x, const type& y)
 
      
+
 Return the minimum of x and y.
 
 
@@ -384,6 +419,9 @@ there is no floating point number whose value is exactly 1/3.
 
 
 
      4.6 Rounding functions
      
+
      
+
+
 
 
      
 When a real number is to be converted to an integer, there is no "best"
@@ -453,15 +491,19 @@ defines the following operations:
 
 
      cl_I floor1 (const type& x)
 
      
+
 Returns floor(x).
 
      cl_I ceiling1 (const type& x)
 
      
+
 Returns ceiling(x).
 
      cl_I truncate1 (const type& x)
 
      
+
 Returns truncate(x).
 
      cl_I round1 (const type& x)
 
      
+
 Returns round(x).
 
 
@@ -550,12 +592,16 @@ defines the following operations:
 
      
 
      type_div_t floor2 (const type& x, const type& y)
 
      
+
 
      type_div_t ceiling2 (const type& x, const type& y)
 
      
+
 
      type_div_t truncate2 (const type& x, const type& y)
 
      
+
 
      type_div_t round2 (const type& x, const type& y)
 
      
+
 
 
 
      
@@ -574,12 +620,16 @@ defines the following operations:
 
 
      type ffloor (const type& x)
 
      
+
 
      type fceiling (const type& x)
 
      
+
 
      type ftruncate (const type& x)
 
      
+
 
      type fround (const type& x)
 
      
+
 
 
 
      
@@ -619,12 +669,16 @@ defines the following operations:
 
      
 
      type_fdiv_t ffloor2 (const type& x)
 
      
+
 
      type_fdiv_t fceiling2 (const type& x)
 
      
+
 
      type_fdiv_t ftruncate2 (const type& x)
 
      
+
 
      type_fdiv_t fround2 (const type& x)
 
      
+
 
 
      
 and similarly for class cl_R, but with quotient type cl_F.
@@ -679,8 +733,10 @@ The classes cl_R, cl_I define the following operations
 
 
      type mod (const type& x, const type& y)
 
      
+
 
      type rem (const type& x, const type& y)
 
      
+
 
 
 
@@ -697,6 +753,7 @@ defines the following operation:
 
 
      type sqrt (const type& x)
 
      
+
 x must be >= 0. This function returns the square root of x,
 normalized to be >= 0. If x is the square of a rational number,
 sqrt(x) will be a rational number, else it will return a
@@ -711,6 +768,7 @@ The classes cl_RA, cl_I define the following operation
 
 
      cl_boolean sqrtp (const type& x, type* root)
 
      
+
 This tests whether x is a perfect square. If so, it returns true
 and the exact square root in *root, else it returns false.
 
@@ -723,6 +781,7 @@ Furthermore, for integers, similarly:
 
 
      cl_boolean isqrt (const type& x, type* root)
 
      
+
 x should be >= 0. This function sets *root to
 floor(sqrt(x)) and returns the same value as sqrtp:
 the boolean value (expt(*root,2) == x).
@@ -737,6 +796,7 @@ define the following operation:
 
 
      cl_boolean rootp (const type& x, const cl_I& n, type* root)
 
      
+
 x must be >= 0. n must be > 0.
 This tests whether x is an nth power of a rational number.
 If so, it returns true and the exact root in *root, else it returns
@@ -752,6 +812,7 @@ for class cl_N:
 
 
      cl_N sqrt (const cl_N& z)
 
      
+
 Returns the square root of z, as defined by the formula
 sqrt(z) = exp(log(z)/2). Conversion to a floating-point type
 or to a complex number are done if necessary. The range of the result is the
@@ -764,6 +825,9 @@ The result is an exact number only if z is an exact number.
 
 
 
      4.8 Transcendental functions
      
+
      
+
+
 
 
      
 The transcendental functions return an exact result if the argument
@@ -780,6 +844,7 @@ For example, cos(0) = 1 returns the rational number 1.
 
 
      cl_R exp (const cl_R& x)
 
      
+
 
      cl_N exp (const cl_N& x)
 
      
 Returns the exponential function of x. This is e^x where
@@ -788,10 +853,12 @@ is the entire complex plane excluding 0.
 
 
      cl_R ln (const cl_R& x)
 
      
+
 x must be > 0. Returns the (natural) logarithm of x.
 
 
      cl_N log (const cl_N& x)
 
      
+
 Returns the (natural) logarithm of x. If x is real and positive,
 this is ln(x). In general, log(x) = log(abs(x)) + i*phase(x).
 The range of the result is the strip in the complex plane
@@ -799,6 +866,7 @@ The range of the result is the strip in the complex plane
 
 
      cl_R phase (const cl_N& x)
 
      
+
 Returns the angle part of x in its polar representation as a
 complex number. That is, phase(x) = atan(realpart(x),imagpart(x)).
 This is also the imaginary part of log(x).
@@ -820,6 +888,7 @@ Returns the logarithm of a with respect to base b.
 
 
      cl_N expt (const cl_N& x, const cl_N& y)
 
      
+
 Exponentiation: Returns x^y = exp(y*log(x)).
 
 
@@ -831,6 +900,7 @@ The constant e = exp(1) = 2.71828... is returned by the following functions:
 
 
      cl_F cl_exp1 (cl_float_format_t f)
 
      
+
 Returns e as a float of format f.
 
 
      cl_F cl_exp1 (const cl_F& y)
@@ -850,6 +920,7 @@ Returns e as a float of format cl_default_float_format.
 
 
      cl_R sin (const cl_R& x)
 
      
+
 Returns sin(x). The range of the result is the interval
 -1 <= sin(x) <= 1.
 
@@ -859,6 +930,7 @@ Returns sin(z). The range of the result is the entire complex plane
 
 
      cl_R cos (const cl_R& x)
 
      
+
 Returns cos(x). The range of the result is the interval
 -1 <= cos(x) <= 1.
 
@@ -868,25 +940,31 @@ Returns cos(z). The range of the result is the entire complex plane
 
 
      struct cl_cos_sin_t { cl_R cos; cl_R sin; };
 
      
+
 
      cl_cos_sin_t cl_cos_sin (const cl_R& x)
 
      
 Returns both sin(x) and cos(x). This is more efficient than
+
 computing them separately. The relation cos^2 + sin^2 = 1 will
 hold only approximately.
 
 
      cl_R tan (const cl_R& x)
 
      
+
 
      cl_N tan (const cl_N& x)
 
      
 Returns tan(x) = sin(x)/cos(x).
 
 
      cl_N cis (const cl_R& x)
 
      
+
 
      cl_N cis (const cl_N& x)
 
      
 Returns exp(i*x). The name `cis' means "cos + i sin", because
 e^(i*x) = cos(x) + i*sin(x).
 
+
+
 
      cl_N asin (const cl_N& z)
 
      
 Returns arcsin(z). This is defined as
@@ -899,6 +977,7 @@ with realpart = pi/2 and imagpart > 0.
 
 
      cl_N acos (const cl_N& z)
 
      
+
 Returns arccos(z). This is defined as
 arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i
 and satisfies arccos(-z) = pi - arccos(z).
@@ -907,6 +986,8 @@ The range of the result is the strip in the complex domain
 with realpart = 0 and imagpart < 0 and the numbers
 with realpart = pi and imagpart > 0.
 
+
+
 
      cl_R atan (const cl_R& x, const cl_R& y)
 
      
 Returns the angle of the polar representation of the complex number
@@ -935,13 +1016,16 @@ with realpart = pi/2 and imagpart <= 0.
 
 
 
      
-The constant pi = 3.14... is returned by the following functions:
+
+
+Archimedes' constant pi = 3.14... is returned by the following functions:
 
 
 
      
 
 
      cl_F cl_pi (cl_float_format_t f)
 
      
+
 Returns pi as a float of format f.
 
 
      cl_F cl_pi (const cl_F& y)
@@ -961,6 +1045,7 @@ Returns pi as a float of format cl_default_float_format.
 
 
      cl_R sinh (const cl_R& x)
 
      
+
 Returns sinh(x).
 
 
      cl_N sinh (const cl_N& z)
@@ -969,6 +1054,7 @@ Returns sinh(z). The range of the result is the entire complex plan
 
 
      cl_R cosh (const cl_R& x)
 
      
+
 Returns cosh(x). The range of the result is the interval
 cosh(x) >= 1.
 
@@ -978,20 +1064,24 @@ Returns cosh(z). The range of the result is the entire complex plan
 
 
      struct cl_cosh_sinh_t { cl_R cosh; cl_R sinh; };
 
      
+
 
      cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x)
 
      
+
 Returns both sinh(x) and cosh(x). This is more efficient than
 computing them separately. The relation cosh^2 - sinh^2 = 1 will
 hold only approximately.
 
 
      cl_R tanh (const cl_R& x)
 
      
+
 
      cl_N tanh (const cl_N& x)
 
      
 Returns tanh(x) = sinh(x)/cosh(x).
 
 
      cl_N asinh (const cl_N& z)
 
      
+
 Returns arsinh(z). This is defined as
 arsinh(z) = log(z+sqrt(1+z^2)) and satisfies
 arsinh(-z) = -arsinh(z).
@@ -1002,6 +1092,7 @@ with imagpart = pi/2 and realpart < 0.
 
 
      cl_N acosh (const cl_N& z)
 
      
+
 Returns arcosh(z). This is defined as
 arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2)).
 The range of the result is the half-strip in the complex domain
@@ -1010,6 +1101,7 @@ excluding the numbers with realpart = 0 and -pi < imagpar
 
 
      cl_N atanh (const cl_N& z)
 
      
+
 Returns artanh(z). This is defined as
 artanh(z) = (log(1+z)-log(1-z)) / 2 and satisfies
 artanh(-z) = -artanh(z). The range of the result is
@@ -1022,6 +1114,9 @@ with imagpart = pi/2 and realpart >= 0.
 
 
 
      4.8.4 Euler gamma
      
+
      
+
+
 
 
      
 Euler's constant C = 0.577... is returned by the following functions:
@@ -1031,6 +1126,7 @@ Euler's constant C = 0.577... is returned by the following functions:
 
 
      cl_F cl_eulerconst (cl_float_format_t f)
 
      
+
 Returns Euler's constant as a float of format f.
 
 
      cl_F cl_eulerconst (const cl_F& y)
@@ -1044,12 +1140,14 @@ Returns Euler's constant as a float of format cl_default_float_format
 Catalan's constant G = 0.915... is returned by the following functions:
+
 
 
 
      
 
 
      cl_F cl_catalanconst (cl_float_format_t f)
 
      
+
 Returns Catalan's constant as a float of format f.
 
 
      cl_F cl_catalanconst (const cl_F& y)
@@ -1064,6 +1162,9 @@ Returns Catalan's constant as a float of format cl_default_float_format4.8.5 Riemann zeta
+
      
+
+
 
 
      
 Riemann's zeta function at an integral point s>1 is returned by the
@@ -1074,6 +1175,7 @@ following functions:
 
 
      cl_F cl_zeta (int s, cl_float_format_t f)
 
      
+
 Returns Riemann's zeta function at s as a float of format f.
 
 
      cl_F cl_zeta (int s, const cl_F& y)
@@ -1113,54 +1215,69 @@ on each of the bit positions in parallel.
 
 
      cl_I lognot (const cl_I& x)
 
      
+
 
      cl_I operator ~ (const cl_I& x)
 
      
+
 Logical not, like ~x in C. This is the same as -1-x.
 
 
      cl_I logand (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_I operator & (const cl_I& x, const cl_I& y)
 
      
+
 Logical and, like x & y in C.
 
 
      cl_I logior (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_I operator | (const cl_I& x, const cl_I& y)
 
      
+
 Logical (inclusive) or, like x | y in C.
 
 
      cl_I logxor (const cl_I& x, const cl_I& y)
 
      
+
 
      cl_I operator ^ (const cl_I& x, const cl_I& y)
 
      
+
 Exclusive or, like x ^ y in C.
 
 
      cl_I logeqv (const cl_I& x, const cl_I& y)
 
      
+
 Bitwise equivalence, like ~(x ^ y) in C.
 
 
      cl_I lognand (const cl_I& x, const cl_I& y)
 
      
+
 Bitwise not and, like ~(x & y) in C.
 
 
      cl_I lognor (const cl_I& x, const cl_I& y)
 
      
+
 Bitwise not or, like ~(x | y) in C.
 
 
      cl_I logandc1 (const cl_I& x, const cl_I& y)
 
      
+
 Logical and, complementing the first argument, like ~x & y in C.
 
 
      cl_I logandc2 (const cl_I& x, const cl_I& y)
 
      
+
 Logical and, complementing the second argument, like x & ~y in C.
 
 
      cl_I logorc1 (const cl_I& x, const cl_I& y)
 
      
+
 Logical or, complementing the first argument, like ~x | y in C.
 
 
      cl_I logorc2 (const cl_I& x, const cl_I& y)
 
      
+
 Logical or, complementing the second argument, like x | ~y in C.
 
      
 
@@ -1170,6 +1287,7 @@ These operations are all available though the function
 
 
      cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
 
      
+
 
      
 
      
 where op must have one of the 16 values (each one stands for a function
@@ -1178,6 +1296,21 @@ which combines two bits into one bit): boole_clr, boole_setboole_and, boole_ior, boole_xor, boole_eqv,
 boole_nand, boole_nor, boole_andc1, boole_andc2,
 boole_orc1, boole_orc2.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
 
      
@@ -1188,16 +1321,19 @@ Other functions that view integers as bit strings:
 
 
      cl_boolean logtest (const cl_I& x, const cl_I& y)
 
      
+
 Returns true if some bit is set in both x and y, i.e. if
 logand(x,y) != 0.
 
 
      cl_boolean logbitp (const cl_I& n, const cl_I& x)
 
      
+
 Returns true if the nth bit (from the right) of x is set.
 Bit 0 is the least significant bit.
 
 
      uintL logcount (const cl_I& x)
 
      
+
 Returns the number of one bits in x, if x >= 0, or
 the number of zero bits in x, if x < 0.
 
@@ -1211,6 +1347,7 @@ struct cl_byte { uintL size; uintL position; };

      + represents the bit interval containing the bits position...position+size-1 of an integer. The constructor cl_byte(size,position) constructs a cl_byte. @@ -1220,16 +1357,19 @@ The constructor cl_byte(size,position) constructs a cl_bytecl_I ldb (const cl_I& n, const cl_byte& b)

      + extracts the bits of n described by the bit interval b and returns them as a nonnegative integer with b.size bits.
      cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
      + Returns true if some bit described by the bit interval b is set in n.
      cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
      + Returns n, with the bits described by the bit interval b replaced by newbyte. Only the lowest b.size bits of newbyte are relevant. @@ -1244,11 +1384,13 @@ functions are their counterparts without shifting:
      cl_I mask_field (const cl_I& n, const cl_byte& b)
      + returns an integer with the bits described by the bit interval b copied from the corresponding bits in n, the other bits zero.
      cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
      + returns an integer where the bits described by the bit interval b come from newbyte and the other bits come from n. @@ -1279,33 +1421,39 @@ for common arithmetic operations:
      cl_boolean oddp (const cl_I& x)
      + Returns true if the least significant bit of x is 1. Equivalent to mod(x,2) != 0.
      cl_boolean evenp (const cl_I& x)
      + Returns true if the least significant bit of x is 0. Equivalent to mod(x,2) == 0.
      cl_I operator << (const cl_I& x, const cl_I& n)
      + Shifts x by n bits to the left. n should be >=0. Equivalent to x * expt(2,n).
      cl_I operator >> (const cl_I& x, const cl_I& n)
      + Shifts x by n bits to the right. n should be >=0. Bits shifted out to the right are thrown away. Equivalent to floor(x / expt(2,n)).
      cl_I ash (const cl_I& x, const cl_I& y)
      + Shifts x by y bits to the left (if y>=0) or by -y bits to the right (if y<=0). In other words, this returns floor(x * expt(2,y)).
      uintL integer_length (const cl_I& x)
      + Returns the number of bits (excluding the sign bit) needed to represent x in two's complement notation. This is the smallest n >= 0 such that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that @@ -1313,12 +1461,14 @@ in two's complement notation. This is the smallest n >= 0 such that
      uintL ord2 (const cl_I& x)
      + x must be non-zero. This function returns the number of 0 bits at the right of x in two's complement notation. This is the largest n >= 0 such that 2^n divides x.
      uintL power2p (const cl_I& x)
      + x must be > 0. This function checks whether x is a power of 2. If x = 2^(n-1), it returns n. Else it returns 0. (See also the function logp.) @@ -1332,6 +1482,7 @@ If x = 2^(n-1), it returns n. Else it returns 0.
      uint32 gcd (uint32 a, uint32 b)
      +
      cl_I gcd (const cl_I& a, const cl_I& b)
      This function returns the greatest common divisor of a and b, @@ -1339,6 +1490,7 @@ normalized to be >= 0.
      cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
      + This function ("extended gcd") returns the greatest common divisor g of a and b and at the same time the representation of g as an integral linear combination of a and b: @@ -1350,11 +1502,13 @@ value, in the following sense: If a and b are non-zero
      cl_I lcm (const cl_I& a, const cl_I& b)
      + This function returns the least common multiple of a and b, normalized to be >= 0.
      cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
      +
      cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
      a must be > 0. b must be >0 and != 1. If log(a,b) is @@ -1370,17 +1524,20 @@ it returns false.
      cl_I factorial (uintL n)
      + n must be a small integer >= 0. This function returns the factorial n! = 1*2*...*n.
      cl_I doublefactorial (uintL n)
      + n must be a small integer >= 0. This function returns the doublefactorial n!! = 1*3*...*n or n!! = 2*4*...*n, respectively.
      cl_I binomial (uintL n, uintL k)
      + n and k must be small integers >= 0. This function returns the binomial coefficient (n choose k) = n! / k! (n-k)! @@ -1407,6 +1564,7 @@ defines the following operations.
      type scale_float (const type& x, sintL delta)
      +
      type scale_float (const type& x, const cl_I& delta)
      Returns x*2^delta. This is more efficient than an explicit multiplication @@ -1422,27 +1580,32 @@ representation of floating-point numbers.
      sintL float_exponent (const type& x)
      + Returns the exponent e of x. For x = 0.0, this is 0. For x non-zero, this is the unique integer with 2^(e-1) <= abs(x) < 2^e.
      sintL float_radix (const type& x)
      + Returns the base of the floating-point representation. This is always 2.
      type float_sign (const type& x)
      + Returns the sign s of x as a float. The value is 1 for x >= 0, -1 for x < 0.
      uintL float_digits (const type& x)
      + Returns the number of mantissa bits in the floating-point representation of x, including the hidden bit. The value only depends on the type of x, not on its value.
      uintL float_precision (const type& x)
      + Returns the number of significant mantissa bits in the floating-point representation of x. Since denormalized numbers are not supported, this is the same as float_digits(x) if x is non-zero, and @@ -1451,6 +1614,11 @@ this is the same as float_digits(x) if x is non-zero,

      The complete internal representation of a float is encoded in the type + + + + + cl_decoded_float (or cl_decoded_sfloat, cl_decoded_ffloat, cl_decoded_dfloat, cl_decoded_lfloat, respectively), defined by @@ -1468,6 +1636,7 @@ and returned by the function

      cl_decoded_typefloat decode_float (const type& x)
      + For x non-zero, this returns (-1)^s, e, m with x = (-1)^s * 2^e * m and 0.5 <= m < 1.0. For x = 0, it returns (-1)^s=1, e=0, m=0. @@ -1478,7 +1647,7 @@ it returns (-1)^s=1, e=0, m=0. A complete decoding in terms of integers is provided as type 
      -struct cl_idecoded_float {
+struct cl_idecoded_float {
         cl_I mantissa; cl_I exponent; cl_I sign;
 };
      @@ -1491,6 +1660,7 @@ by the following function:
      cl_idecoded_float integer_decode_float (const type& x)
      + For x non-zero, this returns (-1)^s, e, m with x = (-1)^s * 2^e * m and m an integer with float_digits(x) bits. For x = 0, it returns (-1)^s=1, e=0, m=0. @@ -1506,6 +1676,7 @@ Some other function, implemented only for class cl_F:
      cl_F float_sign (const cl_F& x, const cl_F& y)
      + This returns a floating point number whose precision and absolute value is that of y and whose sign is that of x. If x is zero, it is treated as positive. Same for y. @@ -1514,6 +1685,9 @@ zero, it is treated as positive. Same for y.

      4.11 Conversion functions

      +

      + + @@ -1527,6 +1701,7 @@ The type cl_float_format_t describes a floating-point format.

      cl_float_format_t cl_float_format (uintL n)
      + Returns the smallest float format which guarantees at least n decimal digits in the mantissa (after the decimal point). @@ -1536,6 +1711,7 @@ Returns the floating point format of x.
      cl_float_format_t cl_default_float_format
      + Global variable: the default float format used when converting rational numbers to floats. @@ -1551,6 +1727,7 @@ defines the following operations:
      cl_F cl_float (const type&x, cl_float_format_t f)
      + Returns x as a float of format f.
      cl_F cl_float (const type&x, const cl_F& y)
      @@ -1573,28 +1750,34 @@ Every floating-point format has some characteristic numbers:
      cl_F most_positive_float (cl_float_format_t f)
      + Returns the largest (most positive) floating point number in float format f.
      cl_F most_negative_float (cl_float_format_t f)
      + Returns the smallest (most negative) floating point number in float format f.
      cl_F least_positive_float (cl_float_format_t f)
      + Returns the least positive floating point number (i.e. > 0 but closest to 0) in float format f.
      cl_F least_negative_float (cl_float_format_t f)
      + Returns the least negative floating point number (i.e. < 0 but closest to 0) in float format f.
      cl_F float_epsilon (cl_float_format_t f)
      + Returns the smallest floating point number e > 0 such that 1+e != 1.
      cl_F float_negative_epsilon (cl_float_format_t f)
      + Returns the smallest floating point number e > 0 such that 1-e != 1. @@ -1611,6 +1794,7 @@ defines the following operation:
      cl_RA rational (const type& x)
      + Returns the value of x as an exact number. If x is already an exact number, this is x. If x is a floating-point number, the value is a rational number whose denominator is a power of 2. @@ -1625,6 +1809,7 @@ the function
      cl_RA rationalize (const cl_R& x)
      + If x is a floating-point number, it actually represents an interval of real numbers, and this function returns the rational number with smallest denominator (and smallest numerator, in magnitude) @@ -1666,6 +1851,7 @@ a complicated but deterministic way.

      The global variable + 

       cl_random_state cl_default_random_state
@@ -1682,12 +1868,14 @@ below are called without cl_random_state argument.
 
      
 
      uint32 random32 ()
 
      
+
 Returns a random unsigned 32-bit number. All bits are equally random.
 
 
      cl_I random_I (cl_random_state& randomstate, const cl_I& n)
 
      
 
      cl_I random_I (const cl_I& n)
 
      
+
 n must be an integer > 0. This function returns a random integer x
 in the range 0 <= x < n.
 
@@ -1695,6 +1883,7 @@ in the range 0 <= x < n.
 
      
 
      cl_F random_F (const cl_F& n)
 
      
+
 n must be a float > 0. This function returns a random floating-point
 number of the same format as n in the range 0 <= x < n.
 
@@ -1702,6 +1891,7 @@ number of the same format as n in the range 0 <= x <
 
      
 
      cl_R random_R (const cl_R& n)
 
      
+
 Behaves like random_I if n is an integer and like random_F
 if n is a float.
 
@@ -1709,6 +1899,9 @@ if n is a float.
 
 
 
      4.13 Obfuscating operators
      
+
      
+
+
 
 
      
 The modifying C/C++ operators +=, -=, *=, /=,
@@ -1723,6 +1916,7 @@ to get happy, then add

      + to the beginning of your source files, before the inclusion of any CLN include files. This flag will enable the following operators: @@ -1736,12 +1930,16 @@ For the classes cl_N, cl_R, cl_RA,

      type& operator += (type&, const type&)
      +
      type& operator -= (type&, const type&)
      +
      type& operator *= (type&, const type&)
      +
      type& operator /= (type&, const type&)
      +

      @@ -1758,14 +1956,19 @@ For the class cl_I:

      type& operator &= (type&, const type&)
      +
      type& operator |= (type&, const type&)
      +
      type& operator ^= (type&, const type&)
      +
      type& operator <<= (type&, const type&)
      +
      type& operator >>= (type&, const type&)
      +

      @@ -1777,6 +1980,7 @@ For the classes cl_N, cl_R, cl_RA,

      type& operator ++ (type& x)
      + The prefix operator ++x.
      void operator ++ (type& x, int) @@ -1785,6 +1989,7 @@ The postfix operator x++.
      type& operator -- (type& x)
      + The prefix operator --x.
      void operator -- (type& x, int) diff --git a/doc/cln_5.html b/doc/cln_5.html index 4df2aed..7f83961 100644 --- a/doc/cln_5.html +++ b/doc/cln_5.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 5. Input/Output @@ -10,10 +10,16 @@ Go to the first, previous, 5. Input/Output +

      + +

      5.1 Internal and printed representation

      +

      + +

      All computations deal with the internal representations of the numbers. @@ -27,8 +33,10 @@ Several external representations may denote the same number, for example,

      Converting an internal to an external representation is called "printing", + converting an external to an internal representation is called "reading". -In CLN, is it always true that conversion of an internal to an external + +In CLN, it is always true that conversion of an internal to an external representation and then back to an internal representation will yield the same internal representation. Symbolically: read(print(x)) == x. This is called "print-read consistency". @@ -409,7 +417,7 @@ using this variable name. Default is "x".

      The global variable cl_default_print_flags contains the default values, -used by the function fprint, +used by the function fprint.

      diff --git a/doc/cln_6.html b/doc/cln_6.html index 4d9413f..4ccf7aa 100644 --- a/doc/cln_6.html +++ b/doc/cln_6.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 6. Rings diff --git a/doc/cln_7.html b/doc/cln_7.html index e065b2b..007ae5a 100644 --- a/doc/cln_7.html +++ b/doc/cln_7.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 7. Modular integers @@ -10,10 +10,16 @@ Go to the first, previous, 7. Modular integers +

      + +

      7.1 Modular integer rings

      +

      + +

      CLN implements modular integers, i.e. integers modulo a fixed integer N. @@ -59,9 +65,11 @@ Modular integer rings are constructed using the function

      cl_modint_ring cl_find_modint_ring (const cl_I& N)
      + This function returns the modular ring `Z/NZ'. It takes care of finding out about special cases of N, like powers of two and odd numbers for which Montgomery multiplication will be a win, + and precomputes any necessary auxiliary data for computing modulo N. There is a cache table of rings, indexed by N (or, more precisely, by abs(N)). This ensures that the precomputation costs are reduced @@ -76,8 +84,10 @@ Modular integer rings can be compared for equality:
      bool operator== (const cl_modint_ring&, const cl_modint_ring&)
      +
      bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
      + These compare two modular integer rings for equality. Two different calls to cl_find_modint_ring with the same argument necessarily return the same ring because it is memoized in the cache table. @@ -95,22 +105,27 @@ Given a modular integer ring R, the following members can be used.
      cl_I R->modulus
      + This is the ring's modulus, normalized to be nonnegative: abs(N).
      cl_MI R->zero()
      + This returns 0 mod N.
      cl_MI R->one()
      + This returns 1 mod N.
      cl_MI R->canonhom (const cl_I& x)
      + This returns x mod N.
      cl_I R->retract (const cl_MI& x)
      + This is a partial inverse function to R->canonhom. It returns the standard representative (>=0, <N) of x. @@ -118,6 +133,7 @@ standard representative (>=0, <N) of x
      cl_MI R->random()
      + This returns a random integer modulo N. @@ -129,15 +145,18 @@ The following operations are defined on modular integers.
      cl_modint_ring x.ring ()
      + Returns the ring to which the modular integer x belongs.
      cl_MI operator+ (const cl_MI&, const cl_MI&)
      + Returns the sum of two modular integers. One of the arguments may also be a plain integer.
      cl_MI operator- (const cl_MI&, const cl_MI&)
      + Returns the difference of two modular integers. One of the arguments may also be a plain integer. @@ -147,50 +166,61 @@ Returns the negative of a modular integer.
      cl_MI operator* (const cl_MI&, const cl_MI&)
      + Returns the product of two modular integers. One of the arguments may also be a plain integer.
      cl_MI square (const cl_MI&)
      + Returns the square of a modular integer.
      cl_MI recip (const cl_MI& x)
      + Returns the reciprocal x^-1 of a modular integer x. x must be coprime to the modulus, otherwise an error message is issued.
      cl_MI div (const cl_MI& x, const cl_MI& y)
      + Returns the quotient x*y^-1 of two modular integers x, y. y must be coprime to the modulus, otherwise an error message is issued.
      cl_MI expt_pos (const cl_MI& x, const cl_I& y)
      + y must be > 0. Returns x^y.
      cl_MI expt (const cl_MI& x, const cl_I& y)
      + Returns x^y. If y is negative, x must be coprime to the modulus, else an error message is issued.
      cl_MI operator<< (const cl_MI& x, const cl_I& y)
      + Returns x*2^y.
      cl_MI operator>> (const cl_MI& x, const cl_I& y)
      + Returns x*2^-y. When y is positive, the modulus must be odd, or an error message is issued.
      bool operator== (const cl_MI&, const cl_MI&)
      +
      bool operator!= (const cl_MI&, const cl_MI&)
      + Compares two modular integers, belonging to the same modular integer ring, for equality.
      cl_boolean zerop (const cl_MI& x)
      + Returns true if x is 0 mod N. @@ -203,8 +233,10 @@ input/output).
      void fprint (cl_ostream stream, const cl_MI& x)
      +
      cl_ostream operator<< (cl_ostream stream, const cl_MI& x)
      + Prints the modular integer x on the stream. The output may depend on the global printer settings in the variable cl_default_print_flags. diff --git a/doc/cln_8.html b/doc/cln_8.html index a0381f6..3dd2018 100644 --- a/doc/cln_8.html +++ b/doc/cln_8.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 8. Symbolic data types @@ -10,6 +10,9 @@ Go to the first, previous, 8. Symbolic data types +

      + +

      CLN implements two symbolic (non-numeric) data types: strings and symbols. @@ -18,6 +21,9 @@ CLN implements two symbolic (non-numeric) data types: strings and symbols.

      8.1 Strings

      +

      + +

      The class @@ -42,6 +48,7 @@ Strings are constructed through the following constructors:

      cl_string (const char * s)
      + Returns an immutable copy of the (zero-terminated) C string s.
      cl_string (const char * ptr, unsigned long len) @@ -62,17 +69,21 @@ Assignment from cl_string and const char *.
      s.length()
      +
      strlen(s)
      + Returns the length of the string s.
      s[i]
      + Returns the ith character of the string s. i must be in the range 0 <= i < s.length().
      bool equal (const cl_string& s1, const cl_string& s2)
      + Compares two strings for equality. One of the arguments may also be a plain const char *. @@ -80,6 +91,9 @@ plain const char *.

      8.2 Symbols

      +

      + +

      Symbols are uniquified strings: all symbols with the same name are shared. @@ -98,6 +112,7 @@ Symbols are constructed through the following constructor:

      cl_symbol (const cl_string& s)
      + Looks up or creates a new symbol with a given name. @@ -114,6 +129,7 @@ Conversion to cl_string: Returns the string which names the symbol
      bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
      + Compares two symbols for equality. This is very fast. diff --git a/doc/cln_9.html b/doc/cln_9.html index be510a2..08c8bfe 100644 --- a/doc/cln_9.html +++ b/doc/cln_9.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - 9. Univariate polynomials @@ -10,6 +10,10 @@ Go to the first, previous, 9. Univariate polynomials +

      + + + @@ -121,6 +125,7 @@ return the same polynomial ring.

      cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R)
      +
      cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
      cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R) @@ -155,27 +160,33 @@ Given a univariate polynomial ring R, the following members can be
      cl_ring R->basering()
      + This returns the base ring, as passed to `cl_find_univpoly_ring'.
      cl_UP R->zero()
      + This returns 0 in R, a polynomial of degree -1.
      cl_UP R->one()
      + This returns 1 in R, a polynomial of degree <= 0.
      cl_UP R->canonhom (const cl_I& x)
      + This returns x in R, a polynomial of degree <= 0.
      cl_UP R->monomial (const cl_ring_element& x, uintL e)
      + This returns a sparse polynomial: x * X^e, where X is the indeterminate.
      cl_UP R->create (sintL degree)
      + Creates a new polynomial with a given degree. The zero polynomial has degree -1. After creating the polynomial, you should put in the coefficients, using the set_coeff member function, and then call the finalize @@ -190,12 +201,14 @@ The following are the only destructive operations on univariate polynomials.
      void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
      + This changes the coefficient of X^index in x to be y. After changing a polynomial and before applying any "normal" operation on it, you should call its finalize member function.
      void finalize (cl_UP& x)
      + This function marks the endpoint of destructive modifications of a polynomial. It normalizes the internal representation so that subsequent computations have less overhead. Doing normal computations on unnormalized polynomials may @@ -210,14 +223,17 @@ The following operations are defined on univariate polynomials.
      cl_univpoly_ring x.ring ()
      + Returns the ring to which the univariate polynomial x belongs.
      cl_UP operator+ (const cl_UP&, const cl_UP&)
      + Returns the sum of two univariate polynomials.
      cl_UP operator- (const cl_UP&, const cl_UP&)
      + Returns the difference of two univariate polynomials.
      cl_UP operator- (const cl_UP&) @@ -226,44 +242,54 @@ Returns the negative of a univariate polynomial.
      cl_UP operator* (const cl_UP&, const cl_UP&)
      + Returns the product of two univariate polynomials. One of the arguments may also be a plain integer or an element of the base ring.
      cl_UP square (const cl_UP&)
      + Returns the square of a univariate polynomial.
      cl_UP expt_pos (const cl_UP& x, const cl_I& y)
      + y must be > 0. Returns x^y.
      bool operator== (const cl_UP&, const cl_UP&)
      +
      bool operator!= (const cl_UP&, const cl_UP&)
      + Compares two univariate polynomials, belonging to the same univariate polynomial ring, for equality.
      cl_boolean zerop (const cl_UP& x)
      + Returns true if x is 0 in R.
      sintL degree (const cl_UP& x)
      + Returns the degree of the polynomial. The zero polynomial has degree -1.
      cl_ring_element coeff (const cl_UP& x, uintL index)
      + Returns the coefficient of X^index in the polynomial x.
      cl_ring_element x (const cl_ring_element& y)
      + Evaluation: If x is a polynomial and y belongs to the base ring, then `x(y)' returns the value of the substitution of y into x.
      cl_UP deriv (const cl_UP& x)
      + Returns the derivative of the polynomial x with respect to the indeterminate X. @@ -277,8 +303,10 @@ input/output).
      void fprint (cl_ostream stream, const cl_UP& x)
      +
      cl_ostream operator<< (cl_ostream stream, const cl_UP& x)
      + Prints the univariate polynomial x on the stream. The output may depend on the global printer settings in the variable cl_default_print_flags. @@ -296,18 +324,26 @@ The following functions return special polynomials.
      cl_UP_I cl_tschebychev (sintL n)
      + + Returns the n-th Tchebychev polynomial (n >= 0).
      cl_UP_I cl_hermite (sintL n)
      + + Returns the n-th Hermite polynomial (n >= 0).
      cl_UP_RA cl_legendre (sintL n)
      + + Returns the n-th Legendre polynomial (n >= 0).
      cl_UP_I cl_laguerre (sintL n)
      + + Returns the n-th Laguerre polynomial (n >= 0). diff --git a/doc/cln_toc.html b/doc/cln_toc.html index ecf7ce9..5dd7415 100644 --- a/doc/cln_toc.html +++ b/doc/cln_toc.html @@ -1,6 +1,6 @@ - + CLN, a Class Library for Numbers - Table of Contents @@ -115,7 +115,7 @@
    • Index

    -This document was generated on 14 January 2000 using +This document was generated on 4 May 2000 using texi2html 1.56k.