CLN is a library for computations with all kinds of numbers. It has a rich set of number classes:
The subtypes of the complex numbers among these are exactly the
types of numbers known to the Common Lisp language. Therefore
CLN can be used for Common Lisp implementations, giving
`CLN' another meaning: it becomes an abbreviation of
"Common Lisp Numbers".
The CLN package implements
+, -, *, /, sqrt,
comparisons, ...),
and, or, not, ...),
CLN is a C++ library. Using C++ as an implementation language provides
+, -, *, =,
==, ... operators as in C or C++.
CLN is memory efficient:
CLN is speed efficient:
i386, m68k, sparc, mips, arm).
CLN aims at being easily integrated into larger software packages:
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 g++ 2.95 or newer.
The following C++ features are used: classes, member functions, overloading of functions and operators, constructors and destructors, inline, const, multiple inheritance, templates.
The following C++ features are not used:
new, delete, virtual inheritance,
exceptions.
CLN relies on semi-automatic ordering of initializations of static and global variables, a feature which I could implement for GNU g++ only.
To build CLN, you also need to have GNU make installed.
To build CLN on HP-UX, you also need to have GNU sed installed.
This is because the libtool script, which creates the CLN library, relies
on sed, and the vendor's sed utility on these systems is too
limited.
As with any autoconfiguring GNU software, installation is as easy as this:
$ ./configure $ make $ make check
If on your system, `make' is not GNU make, you have to use
`gmake' instead of `make' above.
The configure command checks out some features of your system and
C++ compiler and builds the Makefiles. The make command
builds the library. This step may take 4 hours on an average workstation.
The make check runs some test to check that no important subroutine
has been miscompiled.
The configure command accepts options. To get a summary of them, try
$ ./configure --help
Some of the options are explained in detail in the `INSTALL.generic' file.
You can specify the C compiler, the C++ compiler and their options through
the following environment variables when running configure:
CC
CFLAGS
CXX
CXXFLAGS
Examples:
$ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure $ CC="gcc -V 2.7.2" CFLAGS="-O -g" \ CXX="g++ -V 2.7.2" CXXFLAGS="-O -g" ./configure $ CC="gcc -V 2.8.1" CFLAGS="-O -fno-exceptions" \ CXX="g++ -V 2.8.1" CXXFLAGS="-O -fno-exceptions" ./configure $ CC="gcc -V egcs-2.91.60" CFLAGS="-O2 -fno-exceptions" \ CXX="g++ -V egcs-2.91.60" CFLAGS="-O2 -fno-exceptions" ./configure
Note that for these environment variables to take effect, you have to set
them (assuming a Bourne-compatible shell) on the same line as the
configure command. If you made the settings in earlier shell
commands, you have to export the environment variables before
calling configure. In a csh shell, you have to use the
`setenv' command for setting each of the environment variables.
On Linux, g++ needs 15 MB to compile the tests. So you should better
have 17 MB swap space and 1 MB room in $TMPDIR.
If you use g++ version 2.7.x, don't add `-O2' to the CXXFLAGS,
because `g++ -O' generates better code for CLN than `g++ -O2'.
If you use g++ version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or
gcc-2.95.x, I recommend adding `-fno-exceptions' to the CXXFLAGS.
This will likely generate better code.
If you use g++ version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc,
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.
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 configure with the options
`--enable-shared --disable-static'.
If you use g++ version egcs-2.91.x (egcs-1.1) on Sparc, you cannot
use `--enable-shared' because g++ would miscompile parts of the
library.
Starting with version 1.0.4, CLN may be configured to make use of a
preinstalled gmp library. Please make sure that you have at
least gmp version 3.0 installed since earlier versions are
unsupported and likely not to work. Enabling this feature by calling
configure with the option `--with-gmp' is known to be quite
a boost for CLN's performance.
If you have installed the gmp library and its header file in
some place where your compiler cannot find it by default, you must help
configure by setting CPPFLAGS and LDFLAGS. Here is
an example:
$ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \ CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
As with any autoconfiguring GNU software, installation is as easy as this:
$ make install
The `make install' command installs the library and the include files
into public places (`/usr/local/lib/' and `/usr/local/include/',
if you haven't specified a --prefix option to configure).
This step may require superuser privileges.
If you have already built the library and wish to install it, but didn't
specify --prefix=... at configure time, just re-run
configure, giving it the same options as the first time, plus
the --prefix=... option.
You can remove system-dependent files generated by make through
$ make clean
You can remove all files generated by make, thus reverting to a
virgin distribution of CLN, through
$ make distclean
CLN implements the following class hierarchy:
Number
cl_number
<cl_number.h>
|
|
Real or complex number
cl_N
<cl_complex.h>
|
|
Real number
cl_R
<cl_real.h>
|
+-------------------+-------------------+
| |
Rational number Floating-point number
cl_RA cl_F
<cl_rational.h> <cl_float.h>
| |
| +-------------+-------------+-------------+
Integer | | | |
cl_I Short-Float Single-Float Double-Float Long-Float
<cl_integer.h> cl_SF cl_FF cl_DF cl_LF
<cl_sfloat.h> <cl_ffloat.h> <cl_dfloat.h> <cl_lfloat.h>
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
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
to integers.
The class cl_F implements floating-point approximations to real numbers.
It is an abstract class.
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
representation. On exact numbers, the elementary operations (+,
-, *, /, comparisons, ...) compute the completely
correct result.
In CLN, the exact numbers are:
Rational numbers are always normalized to the form
numerator/denominator where the numerator and denominator
are coprime integers and the denominator is positive. If the resulting
denominator is 1, the rational number is converted to an integer.
Small integers (typically in the range -2^30...2^30-1,
for 32-bit machines) are especially efficient, because they consume no heap
allocation. Otherwise the distinction between these immediate integers
(called "fixnums") and heap allocated integers (called "bignums")
is completely transparent.
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 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.
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
`0.7'. Rounding errors like this one are inevitable when computing
with floating-point numbers.
Nevertheless, CLN rounds the floating-point results of the operations +,
-, *, /, sqrt according to the "round-to-even"
rule: It first computes the exact mathematical result and then returns the
floating-point number which is nearest to this. If two floating-point numbers
are equally distant from the ideal result, the one with a 0 in its least
significant mantissa bit is chosen.
Similarly, testing floating point numbers for equality `x == y'
is gambling with random errors. Better check for `abs(x - y) < epsilon'
for some well-chosen epsilon.
Floating point numbers come in four flavors:
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.
cl_FF.
They have 1 sign bit, 8 exponent bits (including the exponent's sign),
and 24 mantissa bits (including the "hidden" bit).
In CLN, they are represented as IEEE single-precision floating point numbers.
This corresponds closely to the C/C++ type `float'.
cl_DF.
They have 1 sign bit, 11 exponent bits (including the exponent's sign),
and 53 mantissa bits (including the "hidden" bit).
In CLN, they are represented as IEEE double-precision floating point numbers.
This corresponds closely to the C/C++ type `double'.
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.
The precision of a long float is unlimited, but once created, a long float
has a fixed precision. (No "lazy recomputation".)
Of course, computations with long floats are more expensive than those with smaller floating-point formats.
CLN does not implement features like NaNs, denormalized numbers and 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.
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
when you change the precision of some computation (say, from cl_DF
to cl_LF), you don't have to change the code, only the precision
of the initial values. Also, many transcendental functions have been
declared as returning a cl_F when the argument is a cl_F,
but such declarations are missing for the types cl_SF, cl_FF,
cl_DF, cl_LF. (Such declarations would be wrong if
the floating point contagion rule happened to change in the future.)
Complex numbers, as implemented by the class cl_N, have a real
part and an imaginary part, both real numbers. A complex number whose
imaginary part is the exact number 0 is automatically converted
to a real number.
Complex numbers can arise from real numbers alone, for example
through application of sqrt or transcendental functions.
Conversions from any class to any its superclasses ("base classes" in C++ terminology) is done automatically.
Conversions from the C built-in types `long' and `unsigned long'
are provided for the classes cl_I, cl_RA, cl_R,
cl_N and cl_number.
Conversions from the C built-in types `int' and `unsigned int'
are provided for the classes cl_I, cl_RA, cl_R,
cl_N and cl_number. However, these conversions emphasize
efficiency. Their range is therefore limited:
In a declaration like `cl_I x = 10;' the C++ compiler is able to
do the conversion of 10 from `int' to `cl_I' at compile time
already. On the other hand, code like `cl_I x = 1000000000;' is
in error.
So, if you want to be sure that an `int' whose magnitude is not guaranteed
to be < 2^29 is correctly converted to a `cl_I', first convert it to a
`long'. Similarly, if a large `unsigned int' is to be converted to a
`cl_I', first convert it to an `unsigned long'.
Conversions from the C built-in type `float' are provided for the classes
cl_FF, cl_F, cl_R, cl_N and cl_number.
Conversions from the C built-in type `double' are provided for the classes
cl_DF, cl_F, cl_R, cl_N and cl_number.
Conversions from `const char *' are provided for the classes
cl_I, cl_RA,
cl_SF, cl_FF, cl_DF, cl_LF, cl_F,
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";
Note that this conversion is done at runtime, not at compile-time.
Conversions from cl_I to the C built-in types `int',
`unsigned int', `long', `unsigned long' are provided through
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)
x as element of the C type ctype. If x is not
representable in the range of ctype, a runtime error occurs.
Conversions from the classes cl_I, cl_RA,
cl_SF, cl_FF, cl_DF, cl_LF, cl_F and
cl_R
to the C built-in types `float' and `double' are provided through
the functions
float cl_float_approx (const type& x)
double cl_double_approx (const type& x)
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.
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.
Example:
cl_I x = ...;
if (!(x >= 0)) abort();
cl_I ten_x = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
// In general, it would be a rational number.
Each of the number classes declares its mathematical operations in the
corresponding include file. For example, if your code operates with
objects of type cl_I, it should #include <cl_integer.h>.
Here is how to create number objects "from nothing".
cl_I objects are most easily constructed from C integers and from
strings. See section 3.4 Conversions.
cl_RA objects can be constructed from strings. The syntax
for rational numbers is described in section 5.1 Internal and printed representation.
Another standard way to produce a rational number is through application
of `operator /' or `recip' on integers.
cl_F objects with low precision are most easily constructed from
C `float' and `double'. See section 3.4 Conversions.
To construct a cl_F with high precision, you can use the conversion
from `const char *', but you have to specify the desired precision
within the string. (See section 5.1 Internal and printed representation.)
Example:
cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
will set `e' to the given value, with a precision of 40 decimal digits.
The programmatic way to construct a cl_F with high precision is
through the cl_float conversion function, see
section 4.11.1 Conversion to floating-point numbers. For example, to compute
e to 40 decimal places, first construct 1.0 to 40 decimal places
and then apply the exponential function:
cl_float_format_t precision = cl_float_format(40); cl_F e = exp(cl_float(1,precision));
Non-real cl_N objects are normally constructed through the function
cl_N complex (const cl_R& realpart, const cl_R& imagpart)
See section 4.4 Elementary complex functions.
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
type operator + (const type&, const type&)
type operator - (const type&, const type&)
type operator - (const type&)
type plus1 (const type& x)
x + 1.
type minus1 (const type& x)
x - 1.
type operator * (const type&, const type&)
type square (const type& x)
x * x.
Each of the classes cl_N, cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
type operator / (const type&, const type&)
type recip (const type&)
The class cl_I doesn't define a `/' operation because
in the C/C++ language this operator, applied to integral types,
denotes the `floor' or `truncate' operation (which one of these,
is implementation dependent). (See section 4.6 Rounding functions.)
Instead, cl_I defines an "exact quotient" function:
cl_I exquo (const cl_I& x, const cl_I& y)
y divides x, and returns the quotient x/y.
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)
x^y.
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operation:
type abs (const type& x)
x.
This is x if x >= 0, and -x if x <= 0.
The class cl_N implements this as follows:
cl_R abs (const cl_N x)
x.
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operation:
type signum (const type& x)
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
0 or 1 or -1.
Each of the classes cl_RA, cl_I defines the following operations:
cl_I numerator (const type& x)
x.
cl_I denominator (const type& x)
x.
The numerator and denominator of a rational number are normalized in such a way that they have no factor in common and the denominator is positive.
The class cl_N defines the following operation:
cl_N complex (const cl_R& a, const cl_R& b)
a+bi, that is, the complex number with
real part a and imaginary part b.
Each of the classes cl_N, cl_R defines the following operations:
cl_R realpart (const type& x)
x.
cl_R imagpart (const type& x)
x.
type conjugate (const type& x)
x.
We have the relations
x = complex(realpart(x), imagpart(x))
conjugate(x) = complex(realpart(x), -imagpart(x))
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
bool operator == (const type&, const type&)
bool operator != (const type&, const type&)
uint32 cl_equal_hashcode (const type&)
==. This hash code depends on the number's value,
not its type or precision.
cl_boolean zerop (const type& x)
x == 0
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
cl_signean cl_compare (const type& x, const type& y)
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&)
cl_boolean minusp (const type& x)
x < 0
cl_boolean plusp (const type& x)
x > 0
type max (const type& x, const type& y)
x and y.
type min (const type& x, const type& y)
x and y.
When a floating point number and a rational number are compared, the float
is first converted to a rational number using the function rational.
Since a floating point number actually represents an interval of real numbers,
the result might be surprising.
For example, (cl_F)(cl_R)"1/3" == (cl_R)"1/3" returns false because
there is no floating point number whose value is exactly 1/3.
When a real number is to be converted to an integer, there is no "best" rounding. The desired rounding function depends on the application. The Common Lisp and ISO Lisp standards offer four rounding functions:
floor(x)
x.
ceiling(x)
x.
truncate(x)
x (inclusive) the one nearest to x.
round(x)
x. If x is exactly halfway between two
integers, choose the even one.
These functions have different advantages:
floor and ceiling are translation invariant:
floor(x+n) = floor(x) + n and ceiling(x+n) = ceiling(x) + n
for every x and every integer n.
On the other hand, truncate and round are symmetric:
truncate(-x) = -truncate(x) and round(-x) = -round(x),
and furthermore round is unbiased: on the "average", it rounds
down exactly as often as it rounds up.
The functions are related like this:
ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1
for rational numbers m/n (m, n integers, n>0), and
truncate(x) = sign(x) * floor(abs(x))
Each of the classes cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
cl_I floor1 (const type& x)
floor(x).
cl_I ceiling1 (const type& x)
ceiling(x).
cl_I truncate1 (const type& x)
truncate(x).
cl_I round1 (const type& x)
round(x).
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
cl_I floor1 (const type& x, const type& y)
floor(x/y).
cl_I ceiling1 (const type& x, const type& y)
ceiling(x/y).
cl_I truncate1 (const type& x, const type& y)
truncate(x/y).
cl_I round1 (const type& x, const type& y)
round(x/y).
These functions are called `floor1', ... here instead of `floor', ..., because on some systems, system dependent include files define `floor' and `ceiling' as macros.
In many cases, one needs both the quotient and the remainder of a division. It is more efficient to compute both at the same time than to perform two divisions, one for quotient and the next one for the remainder. The following functions therefore return a structure containing both the quotient and the remainder. The suffix `2' indicates the number of "return values". The remainder is defined as follows:
quotient = floor(x),
remainder = x - quotient,
quotient = floor(x,y),
remainder = x - quotient*y,
and similarly for the other three operations.
Each of the classes cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
struct type_div_t { cl_I quotient; type remainder; };
type_div_t floor2 (const type& x)
type_div_t ceiling2 (const type& x)
type_div_t truncate2 (const type& x)
type_div_t round2 (const type& x)
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
struct type_div_t { cl_I quotient; type remainder; };
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)
Sometimes, one wants the quotient as a floating-point number (of the same format as the argument, if the argument is a float) instead of as an integer. The prefix `f' indicates this.
Each of the classes
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
type ffloor (const type& x)
type fceiling (const type& x)
type ftruncate (const type& x)
type fround (const type& x)
and similarly for class cl_R, but with return type cl_F.
The class cl_R defines the following operations:
cl_F ffloor (const type& x, const type& y)
cl_F fceiling (const type& x, const type& y)
cl_F ftruncate (const type& x, const type& y)
cl_F fround (const type& x, const type& y)
These functions also exist in versions which return both the quotient and the remainder. The suffix `2' indicates this.
Each of the classes
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
struct type_fdiv_t { type quotient; type remainder; };
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.
The class cl_R defines the following operations:
struct type_fdiv_t { cl_F quotient; cl_R remainder; };
type_fdiv_t ffloor2 (const type& x, const type& y)
type_fdiv_t fceiling2 (const type& x, const type& y)
type_fdiv_t ftruncate2 (const type& x, const type& y)
type_fdiv_t fround2 (const type& x, const type& y)
Other applications need only the remainder of a division. The remainder of `floor' and `ffloor' is called `mod' (abbreviation of "modulo"). The remainder `truncate' and `ftruncate' is called `rem' (abbreviation of "remainder").
mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y
rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y
If x and y are both >= 0, mod(x,y) = rem(x,y) >= 0.
In general, mod(x,y) has the sign of y or is zero,
and rem(x,y) has the sign of x or is zero.
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)
Each of the classes cl_R,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
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
floating-point approximation.
The classes cl_RA, cl_I define the following operation:
cl_boolean sqrtp (const type& x, type* root)
x is a perfect square. If so, it returns true
and the exact square root in *root, else it returns false.
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).
For nth roots, the classes cl_RA, cl_I
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
false.
The only square root function which accepts negative numbers is the one
for class cl_N:
cl_N sqrt (const cl_N& z)
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
right half plane realpart(sqrt(z)) >= 0
including the positive imaginary axis and 0, but excluding
the negative imaginary axis.
The result is an exact number only if z is an exact number.
The transcendental functions return an exact result if the argument
is exact and the result is exact as well. Otherwise they must return
inexact numbers even if the argument is exact.
For example, cos(0) = 1 returns the rational number 1.
cl_R exp (const cl_R& x)
cl_N exp (const cl_N& x)
x. This is e^x where
e is the base of the natural logarithms. The range of the result
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)
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
-pi < imagpart(log(x)) <= pi.
cl_R phase (const cl_N& x)
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).
The range of the result is the interval -pi < phase(x) <= pi.
The result will be an exact number only if zerop(x) or
if x is real and positive.
cl_R log (const cl_R& a, const cl_R& b)
a and b must be > 0. Returns the logarithm of a with
respect to base b. log(a,b) = ln(a)/ln(b).
The result can be exact only if a = 1 or if a and b
are both rational.
cl_N log (const cl_N& a, const cl_N& b)
a with respect to base b.
log(a,b) = log(a)/log(b).
cl_N expt (const cl_N& x, const cl_N& y)
x^y = exp(y*log(x)).
The constant e = exp(1) = 2.71828... is returned by the following functions:
cl_F cl_exp1 (cl_float_format_t f)
f.
cl_F cl_exp1 (const cl_F& y)
y.
cl_F cl_exp1 (void)
cl_default_float_format.
cl_R sin (const cl_R& x)
sin(x). The range of the result is the interval
-1 <= sin(x) <= 1.
cl_N sin (const cl_N& z)
sin(z). The range of the result is the entire complex plane.
cl_R cos (const cl_R& x)
cos(x). The range of the result is the interval
-1 <= cos(x) <= 1.
cl_N cos (const cl_N& x)
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)
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)
tan(x) = sin(x)/cos(x).
cl_N cis (const cl_R& x)
cl_N cis (const cl_N& x)
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)
arcsin(z). This is defined as
arcsin(z) = log(iz+sqrt(1-z^2))/i and satisfies
arcsin(-z) = -arcsin(z).
The range of the result is the strip in the complex domain
-pi/2 <= realpart(arcsin(z)) <= pi/2, excluding the numbers
with realpart = -pi/2 and imagpart < 0 and the numbers
with realpart = pi/2 and imagpart > 0.
cl_N acos (const cl_N& z)
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).
The range of the result is the strip in the complex domain
0 <= realpart(arcsin(z)) <= pi, excluding the numbers
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)
x+iy. This is atan(y/x) if x>0. The range of
the result is the interval -pi < atan(x,y) <= pi. The result will
be an exact number only if x > 0 and y is the exact 0.
WARNING: In Common Lisp, this function is called as (atan y x),
with reversed order of arguments.
cl_R atan (const cl_R& x)
arctan(x). This is the same as atan(1,x). The range
of the result is the interval -pi/2 < atan(x) < pi/2. The result
will be an exact number only if x is the exact 0.
cl_N atan (const cl_N& z)
arctan(z). This is defined as
arctan(z) = (log(1+iz)-log(1-iz)) / 2i and satisfies
arctan(-z) = -arctan(z). The range of the result is
the strip in the complex domain
-pi/2 <= realpart(arctan(z)) <= pi/2, excluding the numbers
with realpart = -pi/2 and imagpart >= 0 and the numbers
with realpart = pi/2 and imagpart <= 0.
Archimedes' constant pi = 3.14... is returned by the following functions:
cl_F cl_pi (cl_float_format_t f)
f.
cl_F cl_pi (const cl_F& y)
y.
cl_F cl_pi (void)
cl_default_float_format.
cl_R sinh (const cl_R& x)
sinh(x).
cl_N sinh (const cl_N& z)
sinh(z). The range of the result is the entire complex plane.
cl_R cosh (const cl_R& x)
cosh(x). The range of the result is the interval
cosh(x) >= 1.
cl_N cosh (const cl_N& z)
cosh(z). The range of the result is the entire complex plane.
struct cl_cosh_sinh_t { cl_R cosh; cl_R sinh; };
cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x)
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)
tanh(x) = sinh(x)/cosh(x).
cl_N asinh (const cl_N& z)
arsinh(z). This is defined as
arsinh(z) = log(z+sqrt(1+z^2)) and satisfies
arsinh(-z) = -arsinh(z).
The range of the result is the strip in the complex domain
-pi/2 <= imagpart(arsinh(z)) <= pi/2, excluding the numbers
with imagpart = -pi/2 and realpart > 0 and the numbers
with imagpart = pi/2 and realpart < 0.
cl_N acosh (const cl_N& z)
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
-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0,
excluding the numbers with realpart = 0 and -pi < imagpart < 0.
cl_N atanh (const cl_N& z)
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
the strip in the complex domain
-pi/2 <= imagpart(artanh(z)) <= pi/2, excluding the numbers
with imagpart = -pi/2 and realpart <= 0 and the numbers
with imagpart = pi/2 and realpart >= 0.
Euler's constant C = 0.577... is returned by the following functions:
cl_F cl_eulerconst (cl_float_format_t f)
f.
cl_F cl_eulerconst (const cl_F& y)
y.
cl_F cl_eulerconst (void)
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)
f.
cl_F cl_catalanconst (const cl_F& y)
y.
cl_F cl_catalanconst (void)
cl_default_float_format.
Riemann's zeta function at an integral point s>1 is returned by the
following functions:
cl_F cl_zeta (int s, cl_float_format_t f)
s as a float of format f.
cl_F cl_zeta (int s, const cl_F& y)
s in the float format of y.
cl_F cl_zeta (int s)
s as a float of format
cl_default_float_format.
Integers, when viewed as in two's complement notation, can be thought as infinite bit strings where the bits' values eventually are constant. For example,
17 = ......00010001
-6 = ......11111010
The logical operations view integers as such bit strings and operate on each of the bit positions in parallel.
cl_I lognot (const cl_I& x)
cl_I operator ~ (const cl_I& x)
~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)
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)
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)
x ^ y in C.
cl_I logeqv (const cl_I& x, const cl_I& y)
~(x ^ y) in C.
cl_I lognand (const cl_I& x, const cl_I& y)
~(x & y) in C.
cl_I lognor (const cl_I& x, const cl_I& y)
~(x | y) in C.
cl_I logandc1 (const cl_I& x, const cl_I& y)
~x & y in C.
cl_I logandc2 (const cl_I& x, const cl_I& y)
x & ~y in C.
cl_I logorc1 (const cl_I& x, const cl_I& y)
~x | y in C.
cl_I logorc2 (const cl_I& x, const cl_I& y)
x | ~y in C.
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
which combines two bits into one bit): boole_clr, boole_set,
boole_1, boole_2, boole_c1, boole_c2,
boole_and, boole_ior, boole_xor, boole_eqv,
boole_nand, boole_nor, boole_andc1, boole_andc2,
boole_orc1, boole_orc2.
Other functions that view integers as bit strings:
cl_boolean logtest (const cl_I& x, const cl_I& y)
x and y, i.e. if
logand(x,y) != 0.
cl_boolean logbitp (const cl_I& n, const cl_I& x)
nth bit (from the right) of x is set.
Bit 0 is the least significant bit.
uintL logcount (const cl_I& x)
x, if x >= 0, or
the number of zero bits in x, if x < 0.
The following functions operate on intervals of bits in integers. The type
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.
cl_I ldb (const cl_I& n, const cl_byte& b)
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)
b is set in
n.
cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
n, with the bits described by the bit interval b
replaced by newbyte. Only the lowest b.size bits of
newbyte are relevant.
The functions ldb and dpb implicitly shift. The following
functions are their counterparts without shifting:
cl_I mask_field (const cl_I& n, const cl_byte& b)
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)
b
come from newbyte and the other bits come from n.
The following relations hold:
ldb (n, b) = mask_field(n, b) >> b.position,
dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b),
deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b).
The following operations on integers as bit strings are efficient shortcuts for common arithmetic operations:
cl_boolean oddp (const cl_I& x)
x is 1. Equivalent to
mod(x,2) != 0.
cl_boolean evenp (const cl_I& x)
x is 0. Equivalent to
mod(x,2) == 0.
cl_I operator << (const cl_I& x, const cl_I& n)
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)
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)
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)
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.
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.)
uint32 gcd (uint32 a, uint32 b)
cl_I gcd (const cl_I& a, const cl_I& b)
a and b,
normalized to be >= 0.
cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
g of
a and b and at the same time the representation of g
as an integral linear combination of a and b:
u and v with u*a+v*b = g, g >= 0.
u and v will be normalized to be of smallest possible absolute
value, in the following sense: If a and b are non-zero, and
abs(a) != abs(b), u and v will satisfy the inequalities
abs(u) <= abs(b)/(2*g), abs(v) <= abs(a)/(2*g).
cl_I lcm (const cl_I& a, const cl_I& b)
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
rational number, this function returns true and sets *l = log(a,b), else
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)!
for 0 <= k <= n, 0 else.
Recall that a floating-point number consists of a sign s, an
exponent e and a mantissa m. The value of the number is
(-1)^s * 2^e * m.
Each of the classes
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations.
type scale_float (const type& x, sintL delta)
type scale_float (const type& x, const cl_I& delta)
x*2^delta. This is more efficient than an explicit multiplication
because it copies x and modifies the exponent.
The following functions provide an abstract interface to the underlying representation of floating-point numbers.
sintL float_exponent (const type& x)
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)
2.
type float_sign (const type& x)
s of x as a float. The value is 1 for
x >= 0, -1 for x < 0.
uintL float_digits (const type& x)
x, including the hidden bit. The value only depends on the type
of x, not on its value.
uintL float_precision (const type& x)
x. Since denormalized numbers are not supported,
this is the same as float_digits(x) if x is non-zero, and
0 if x = 0.
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
struct cl_decoded_typefloat {
type mantissa; cl_I exponent; type sign;
};
and returned by the function
cl_decoded_typefloat decode_float (const type& x)
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.
e is the same as returned by the function float_exponent.
A complete decoding in terms of integers is provided as type
struct cl_idecoded_float {
cl_I mantissa; cl_I exponent; cl_I sign;
};
by the following function:
cl_idecoded_float integer_decode_float (const type& x)
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.
WARNING: The exponent e is not the same as the one returned by
the functions decode_float and float_exponent.
Some other function, implemented only for class cl_F:
cl_F float_sign (const cl_F& x, const cl_F& y)
y and whose sign is that of x. If x is
zero, it is treated as positive. Same for y.
The type cl_float_format_t describes a floating-point format.
cl_float_format_t cl_float_format (uintL n)
n
decimal digits in the mantissa (after the decimal point).
cl_float_format_t cl_float_format (const cl_F& x)
x.
cl_float_format_t cl_default_float_format
To convert a real number to a float, each of the types
cl_R, cl_F, cl_I, cl_RA,
int, unsigned int, float, double
defines the following operations:
cl_F cl_float (const type&x, cl_float_format_t f)
x as a float of format f.
cl_F cl_float (const type&x, const cl_F& y)
x in the float format of y.
cl_F cl_float (const type&x)
x as a float of format cl_default_float_format if
it is an exact number, or x itself if it is already a float.
Of course, converting a number to a float can lose precision.
Every floating-point format has some characteristic numbers:
cl_F most_positive_float (cl_float_format_t f)
f.
cl_F most_negative_float (cl_float_format_t f)
f.
cl_F least_positive_float (cl_float_format_t f)
f.
cl_F least_negative_float (cl_float_format_t f)
f.
cl_F float_epsilon (cl_float_format_t f)
1+e != 1.
cl_F float_negative_epsilon (cl_float_format_t f)
1-e != 1.
Each of the classes cl_R, cl_RA, cl_F
defines the following operation:
cl_RA rational (const type& x)
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.
In order to convert back, say, (cl_F)(cl_R)"1/3" to 1/3, there is
the function
cl_RA rationalize (const cl_R& x)
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)
which lies in this interval.
If x is already an exact number, this function returns x.
If x is any float, one has
cl_float(rational(x),x) = x
cl_float(rationalize(x),x) = x
A random generator is a machine which produces (pseudo-)random numbers.
The include file <cl_random.h> defines a class cl_random_state
which contains the state of a random generator. If you make a copy
of the random number generator, the original one and the copy will produce
the same sequence of random numbers.
The following functions return (pseudo-)random numbers in different formats. Calling one of these modifies the state of the random number generator in a complicated but deterministic way.
cl_random_state cl_default_random_state
contains a default random number generator. It is used when the functions
below are called without cl_random_state argument.
uint32 random32 (cl_random_state& randomstate)
uint32 random32 ()
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.
cl_F random_F (cl_random_state& randomstate, const cl_F& 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.
cl_R random_R (cl_random_state& randomstate, const cl_R& n)
cl_R random_R (const cl_R& n)
random_I if n is an integer and like random_F
if n is a float.
The modifying C/C++ operators +=, -=, *=, /=,
&=, |=, ^=, <<=, >>=
are not available by default because their
use tends to make programs unreadable. It is trivial to get away without
them. However, if you feel that you absolutely need these operators
to get happy, then add
#define 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:
For the classes cl_N, cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF:
type& operator += (type&, const type&)
type& operator -= (type&, const type&)
type& operator *= (type&, const type&)
type& operator /= (type&, const type&)
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&)
type& operator ^= (type&, const type&)
type& operator <<= (type&, const type&)
type& operator >>= (type&, const type&)
For the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF:
type& operator ++ (type& x)
++x.
void operator ++ (type& x, int)
x++.
type& operator -- (type& x)
--x.
void operator -- (type& x, int)
x--.
Note that by using these obfuscating operators, you wouldn't gain efficiency: In CLN `x += y;' is exactly the same as `x = x+y;', not more efficient.
All computations deal with the internal representations of the numbers.
Every number has an external representation as a sequence of ASCII characters. 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",
converting an external to an internal representation is 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 consistency".
Different types of numbers have different external representations (case is insignificant):
. with a trailing dot
for decimal integers
and the #nR, #b, #o, #x prefixes.
/{digit}+.
The #nR, #b, #o, #x prefixes are allowed
here as well.
.{digit}*exponent or
sign{digit}*.{digit}+. A precision specifier
of the form _prec may be appended. There must be at least
one digit in the non-exponent part. The exponent has the syntax
expmarker expsign {digit}+.
The exponent marker is
realpart+imagparti. Of course,
if imagpart is negative, its printed representation begins with
a `-', and the `+' between realpart and imagpart
may be omitted. Note that this notation cannot be used when the imagpart
is rational and the rational number's base is >18, because the `i'
is then read as a digit.
#C(realpart imagpart).
Including <cl_io.h> defines a type cl_istream, which is
the type of the first argument to all input functions. Unless you build
and use CLN with the macro CL_IO_STDIO being defined, cl_istream
is the same as istream&.
The variable
cl_istream cl_stdin
contains the standard input stream.
These are the simple input functions:
int freadchar (cl_istream stream)
stream. Returns cl_EOF (not a `char'!)
if the end of stream was encountered or an error occurred.
int funreadchar (cl_istream stream, int c)
c onto stream. c must be the result of the
last freadchar operation on stream.
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines, in <cl_type_io.h>, the following input function:
cl_istream operator>> (cl_istream stream, type& result)
stream and stores it in the result.
The most flexible input functions, defined in <cl_type_io.h>,
are the following:
cl_N read_complex (cl_istream stream, const cl_read_flags& flags)
cl_R read_real (cl_istream stream, const cl_read_flags& flags)
cl_F read_float (cl_istream stream, const cl_read_flags& flags)
cl_RA read_rational (cl_istream stream, const cl_read_flags& flags)
cl_I read_integer (cl_istream stream, const cl_read_flags& flags)
stream. The flags are parameters which
affect the input syntax. Whitespace before the number is silently skipped.
cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
flags are parameters which
affect the input syntax. The string starts at string and ends at
string_limit (exclusive limit). string_limit may also be
NULL, denoting the entire string, i.e. equivalent to
string_limit = string + strlen(string). If end_of_parse is
NULL, the string in memory must contain exactly one number and nothing
more, else a fatal error will be signalled. If end_of_parse
is not NULL, *end_of_parse will be assigned a pointer past
the last parsed character (i.e. string_limit if nothing came after
the number). Whitespace is not allowed.
The structure cl_read_flags contains the following fields:
cl_read_syntax_t syntax
syntax_number, syntax_real, syntax_rational,
syntax_integer, syntax_float, syntax_sfloat,
syntax_ffloat, syntax_dfloat, syntax_lfloat.
cl_read_lsyntax_t lsyntax
lsyntax_standard
lsyntax_algebraic
x+yi for complex numbers,
lsyntax_commonlisp
#b, #o, #x syntaxes for binary, octal,
hexadecimal numbers,
#baseR for rational numbers in a given base,
#c(realpart imagpart) for complex numbers,
lsyntax_all
unsigned int rational_base
cl_float_format_t float_flags.default_float_format
cl_float_format_t float_flags.default_lfloat_format
cl_boolean float_flags.mantissa_dependent_float_format
Including <cl_io.h> defines a type cl_ostream, which is
the type of the first argument to all output functions. Unless you build
and use CLN with the macro CL_IO_STDIO being defined, cl_ostream
is the same as ostream&.
The variable
cl_ostream cl_stdout
contains the standard output stream.
The variable
cl_ostream cl_stderr
contains the standard error output stream.
These are the simple output functions:
void fprintchar (cl_ostream stream, char c)
x literally on the stream.
void fprint (cl_ostream stream, const char * string)
string literally on the stream.
void fprintdecimal (cl_ostream stream, int x)
void fprintdecimal (cl_ostream stream, const cl_I& x)
x in decimal on the stream.
void fprintbinary (cl_ostream stream, const cl_I& x)
x in binary (base 2, without prefix)
on the stream.
void fprintoctal (cl_ostream stream, const cl_I& x)
x in octal (base 8, without prefix)
on the stream.
void fprinthexadecimal (cl_ostream stream, const cl_I& x)
x in hexadecimal (base 16, without prefix)
on the stream.
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines, in <cl_type_io.h>, the following output functions:
void fprint (cl_ostream stream, const type& x)
cl_ostream operator<< (cl_ostream stream, const type& x)
x on the stream. The output may depend
on the global printer settings in the variable cl_default_print_flags.
The ostream flags and settings (flags, width and locale) are
ignored.
The most flexible output function, defined in <cl_type_io.h>,
are the following:
void print_complex (cl_ostream stream, const cl_print_flags& flags,
const cl_N& z);
void print_real (cl_ostream stream, const cl_print_flags& flags,
const cl_R& z);
void print_float (cl_ostream stream, const cl_print_flags& flags,
const cl_F& z);
void print_rational (cl_ostream stream, const cl_print_flags& flags,
const cl_RA& z);
void print_integer (cl_ostream stream, const cl_print_flags& flags,
const cl_I& z);
Prints the number x on the stream. The flags are
parameters which affect the output.
The structure type cl_print_flags contains the following fields:
unsigned int rational_base
10.
cl_boolean rational_readably
#nR or #b or #o or #x
prefixes, trailing dot). Default is false.
cl_boolean float_readably
cl_float_format_t default_float_format
cl_float_format_ffloat.
cl_boolean complex_readably
#C(realpart imagpart). Default is false.
cl_string univpoly_varname
"x".
The global variable cl_default_print_flags contains the default values,
used by the function fprint.
CLN has a class of abstract rings.
Ring
cl_ring
<cl_ring.h>
Rings can be compared for equality:
bool operator== (const cl_ring&, const cl_ring&)
bool operator!= (const cl_ring&, const cl_ring&)
Given a ring R, the following members can be used.
void R->fprint (cl_ostream stream, const cl_ring_element& x)
cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
cl_ring_element R->zero ()
cl_boolean R->zerop (const cl_ring_element& x)
cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
cl_ring_element R->uminus (const cl_ring_element& x)
cl_ring_element R->one ()
cl_ring_element R->canonhom (const cl_I& x)
cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
cl_ring_element R->square (const cl_ring_element& x)
cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
The following rings are built-in.
cl_null_ring cl_0_ring
cl_complex_ring cl_C_ring
cl_N.
cl_real_ring cl_R_ring
cl_R.
cl_rational_ring cl_RA_ring
cl_RA.
cl_integer_ring cl_I_ring
cl_I.
Type tests can be performed for any of cl_C_ring, cl_R_ring,
cl_RA_ring, cl_I_ring:
cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
CLN implements modular integers, i.e. integers modulo a fixed integer N.
The modulus is explicitly part of every modular integer. CLN doesn't
allow you to (accidentally) mix elements of different modular rings,
e.g. (3 mod 4) + (2 mod 5) will result in a runtime error.
(Ideally one would imagine a generic data type cl_MI(N), but C++
doesn't have generic types. So one has to live with runtime checks.)
The class of modular integer rings is
Ring
cl_ring
<cl_ring.h>
|
|
Modular integer ring
cl_modint_ring
<cl_modinteger.h>
and the class of all modular integers (elements of modular integer rings) is
Modular integer
cl_MI
<cl_modinteger.h>
Modular integer rings are constructed using the function
cl_modint_ring cl_find_modint_ring (const cl_I& N)
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
to a minimum.
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&)
cl_find_modint_ring with the same argument necessarily return the
same ring because it is memoized in the cache table.
Given a modular integer ring R, the following members can be used.
cl_I R->modulus
abs(N).
cl_MI R->zero()
0 mod N.
cl_MI R->one()
1 mod N.
cl_MI R->canonhom (const cl_I& x)
x mod N.
cl_I R->retract (const cl_MI& x)
R->canonhom. It returns the
standard representative (>=0, <N) of x.
cl_MI R->random(cl_random_state& randomstate)
cl_MI R->random()
N.
The following operations are defined on modular integers.
cl_modint_ring x.ring ()
x belongs.
cl_MI operator+ (const cl_MI&, const cl_MI&)
cl_MI operator- (const cl_MI&, const cl_MI&)
cl_MI operator- (const cl_MI&)
cl_MI operator* (const cl_MI&, const cl_MI&)
cl_MI square (const cl_MI&)
cl_MI recip (const cl_MI& x)
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)
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)
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)
x*2^y.
cl_MI operator>> (const cl_MI& x, const cl_I& y)
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&)
cl_boolean zerop (const cl_MI& x)
x is 0 mod N.
The following output functions are defined (see also the chapter on input/output).
void fprint (cl_ostream stream, const cl_MI& x)
cl_ostream operator<< (cl_ostream stream, const cl_MI& x)
x on the stream. The output may depend
on the global printer settings in the variable cl_default_print_flags.
CLN implements two symbolic (non-numeric) data types: strings and symbols.
The class
String
cl_string
<cl_string.h>
implements immutable strings.
Strings are constructed through the following constructors:
cl_string (const char * s)
s.
cl_string (const char * ptr, unsigned long len)
len characters at
ptr[0], ..., ptr[len-1]. NUL characters are allowed.
The following functions are available on strings:
operator =
cl_string and const char *.
s.length()
strlen(s)
s.
s[i]
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)
const char *.
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 comparison), whereas comparison of two strings must in the worst case walk both strings until their end. Symbols are used, for example, as tags for properties, as names of variables in polynomial rings, etc.
Symbols are constructed through the following constructor:
cl_symbol (const cl_string& s)
The following operations are available on symbols:
cl_string (const cl_symbol& sym)
cl_string: Returns the string which names the symbol
sym.
bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
CLN implements univariate polynomials (polynomials in one variable) over an
arbitrary ring. The indeterminate variable may be either unnamed (and will be
printed according to cl_default_print_flags.univpoly_varname, which
defaults to `x') or carry a given name. The base ring and the
indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
(accidentally) mix elements of different polynomial rings, e.g.
(a^2+1) * (b^3-1) will result in a runtime error. (Ideally this should
return a multivariate polynomial, but they are not yet implemented in CLN.)
The classes of univariate polynomial rings are
Ring
cl_ring
<cl_ring.h>
|
|
Univariate polynomial ring
cl_univpoly_ring
<cl_univpoly.h>
|
+----------------+-------------------+
| | |
Complex polynomial ring | Modular integer polynomial ring
cl_univpoly_complex_ring | cl_univpoly_modint_ring
<cl_univpoly_complex.h> | <cl_univpoly_modint.h>
|
+----------------+
| |
Real polynomial ring |
cl_univpoly_real_ring |
<cl_univpoly_real.h> |
|
+----------------+
| |
Rational polynomial ring |
cl_univpoly_rational_ring |
<cl_univpoly_rational.h> |
|
+----------------+
|
Integer polynomial ring
cl_univpoly_integer_ring
<cl_univpoly_integer.h>
and the corresponding classes of univariate polynomials are
Univariate polynomial
cl_UP
<cl_univpoly.h>
|
+----------------+-------------------+
| | |
Complex polynomial | Modular integer polynomial
cl_UP_N | cl_UP_MI
<cl_univpoly_complex.h> | <cl_univpoly_modint.h>
|
+----------------+
| |
Real polynomial |
cl_UP_R |
<cl_univpoly_real.h> |
|
+----------------+
| |
Rational polynomial |
cl_UP_RA |
<cl_univpoly_rational.h> |
|
+----------------+
|
Integer polynomial
cl_UP_I
<cl_univpoly_integer.h>
Univariate polynomial rings are constructed using the functions
cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R)
cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
R may be an arbitrary ring. This function takes care of finding out
about special cases of R, such as the rings of complex numbers,
real numbers, rational numbers, integers, or modular integer rings.
There is a cache table of rings, indexed by R and varname.
This ensures that two calls of this function with the same arguments will
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)
cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R)
cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R)
cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R)
cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
cl_find_univpoly_ring,
only the return type is more specific, according to the base ring's type.
Given a univariate polynomial ring R, the following members can be used.
cl_ring R->basering()
cl_UP R->zero()
0 in R, a polynomial of degree -1.
cl_UP R->one()
1 in R, a polynomial of degree <= 0.
cl_UP R->canonhom (const cl_I& x)
x in R, a polynomial of degree <= 0.
cl_UP R->monomial (const cl_ring_element& x, uintL e)
x * X^e, where X is the
indeterminate.
cl_UP R->create (sintL degree)
-1. After creating the polynomial, you should put in the coefficients,
using the set_coeff member function, and then call the finalize
member function.
The following are the only destructive operations on univariate polynomials.
void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
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)
The following operations are defined on univariate polynomials.
cl_univpoly_ring x.ring ()
x belongs.
cl_UP operator+ (const cl_UP&, const cl_UP&)
cl_UP operator- (const cl_UP&, const cl_UP&)
cl_UP operator- (const cl_UP&)
cl_UP operator* (const cl_UP&, const cl_UP&)
cl_UP square (const cl_UP&)
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&)
cl_boolean zerop (const cl_UP& x)
x is 0 in R.
sintL degree (const cl_UP& x)
-1.
cl_ring_element coeff (const cl_UP& x, uintL index)
X^index in the polynomial x.
cl_ring_element x (const cl_ring_element& y)
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)
x with respect to the
indeterminate X.
The following output functions are defined (see also the chapter on input/output).
void fprint (cl_ostream stream, const cl_UP& x)
cl_ostream operator<< (cl_ostream stream, const cl_UP& x)
x on the stream. The output may
depend on the global printer settings in the variable
cl_default_print_flags.
The following functions return special polynomials.
cl_UP_I cl_tschebychev (sintL n)
cl_UP_I cl_hermite (sintL n)
cl_UP_RA cl_legendre (sintL n)
cl_UP_I cl_laguerre (sintL n)
Information how to derive the differential equation satisfied by each
of these polynomials from their definition can be found in the
doc/polynomial/ directory.
Using C++ as an implementation language provides
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.
+, -, *,
=, ==, ... can be used in infix notation, which is more
convenient than Lisp notation `(+ x y)' or C notation `add(x,y,&z)'.
With these language features, there is no need for two separate languages, one for the implementation of the library and one in which the library's users can program. This means that a prototype implementation of an algorithm can be integrated into the library immediately after it has been tested and debugged. No need to rewrite it in a low-level language after having prototyped in a high-level language.
In order to save memory allocations, CLN implements:
x+0 returns x without copying
it.
> -2^29,
< 2^29 don't consume heap memory, unless they were explicitly allocated
on the heap.
Speed efficiency is obtained by the combination of the following tricks and algorithms:
i386, m68k, sparc, mips, arm).
O(N^2)
algorithm, the Karatsuba multiplication, which is an
O(N^1.6)
algorithm.
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 number objects, only the objects' reference count are manipulated.
Memory occupied by number objects are automatically reclaimed as soon as their reference count drops to zero.
For number rings, another strategy is implemented: There is a cache of, for example, the modular integer rings. A modular integer ring is destroyed only if its reference count dropped to zero and the cache is about to be resized. The effect of this strategy is that recently used rings remain cached, whereas undue memory consumption through cached rings is avoided.
For the following discussion, we will assume that you have installed
the CLN source in $CLN_DIR and built it in $CLN_TARGETDIR.
For example, for me it's CLN_DIR="$HOME/cln" and
CLN_TARGETDIR="$HOME/cln/linuxelf". You might define these as
environment variables, or directly substitute the appropriate values.
Until you have installed CLN in a public place, the following options are needed:
When you compile CLN application code, add the flags
-I$CLN_DIR/include -I$CLN_TARGETDIR/include
to the C++ compiler's command line (make variable CFLAGS or CXXFLAGS).
When you link CLN application code to form an executable, add the flags
$CLN_TARGETDIR/src/libcln.a
to the C/C++ compiler's command line (make variable LIBS).
If you did a make install, the include files are installed in a
public directory (normally /usr/local/include), hence you don't
need special flags for compiling. The library has been installed to a
public directory as well (normally /usr/local/lib), hence when
linking a CLN application it is sufficient to give the flag -lcln.
Here is a summary of the include files and their contents.
<cl_object.h>
<cl_number.h>
<cl_complex.h>
<cl_real.h>
<cl_float.h>
<cl_sfloat.h>
<cl_ffloat.h>
<cl_dfloat.h>
<cl_lfloat.h>
<cl_rational.h>
<cl_integer.h>
<cl_io.h>
<cl_complex_io.h>
<cl_real_io.h>
<cl_float_io.h>
<cl_sfloat_io.h>
<cl_ffloat_io.h>
<cl_dfloat_io.h>
<cl_lfloat_io.h>
<cl_rational_io.h>
<cl_integer_io.h>
<cl_input.h>
<cl_output.h>
<cl_malloc.h>
cl_malloc_hook, cl_free_hook.
<cl_abort.h>
cl_abort.
<cl_condition.h>
<cl_string.h>
<cl_symbol.h>
<cl_proplist.h>
<cl_ring.h>
<cl_null_ring.h>
<cl_complex_ring.h>
<cl_real_ring.h>
<cl_rational_ring.h>
<cl_integer_ring.h>
<cl_numtheory.h>
<cl_modinteger.h>
<cl_V.h>
<cl_GV.h>
<cl_GV_number.h>
<cl_GV_complex.h>
<cl_GV_real.h>
<cl_GV_rational.h>
<cl_GV_integer.h>
<cl_GV_modinteger.h>
<cl_SV.h>
<cl_SV_number.h>
<cl_SV_complex.h>
<cl_SV_real.h>
<cl_SV_rational.h>
<cl_SV_integer.h>
<cl_SV_ringelt.h>
<cl_univpoly.h>
<cl_univpoly_integer.h>
<cl_univpoly_rational.h>
<cl_univpoly_real.h>
<cl_univpoly_complex.h>
<cl_univpoly_modint.h>
<cl_timing.h>
<cln.h>
A function which computes the nth Fibonacci number can be written as follows.
#include <cl_integer.h>
#include <cl_real.h>
// Returns F_n, computed as the nearest integer to
// ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
const cl_I fibonacci (int n)
{
// Need a precision of ((1+sqrt(5))/2)^-n.
cl_float_format_t prec = cl_float_format((int)(0.208987641*n+5));
cl_R sqrt5 = sqrt(cl_float(5,prec));
cl_R phi = (1+sqrt5)/2;
return round1( expt(phi,n)/sqrt5 );
}
Let's explain what is going on in detail.
The include file <cl_integer.h> is necessary because the type
cl_I is used in the function, and the include file <cl_real.h>
is needed for the type cl_R and the floating point number functions.
The order of the include files does not matter.
Then comes the function declaration. The argument is an int, the
result an integer. The return type is defined as `const cl_I', not
simply `cl_I', because that allows the compiler to detect typos like
`fibonacci(n) = 100'. It would be possible to declare the return
type as const cl_R (real number) or even const cl_N (complex
number). We use the most specialized possible return type because functions
which call `fibonacci' will be able to profit from the compiler's type
analysis: Adding two integers is slightly more efficient than adding the
same objects declared as complex numbers, because it needs less type
dispatch. Also, when linking to CLN as a non-shared library, this minimizes
the size of the resulting executable program.
The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest integer. In order to get a correct result, the absolute error should be less than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)). To this end, the first line computes a floating point precision for sqrt(5) and phi.
Then sqrt(5) is computed by first converting the integer 5 to a floating point number and than taking the square root. The converse, first taking the square root of 5, and then converting to the desired precision, would not work in CLN: The square root would be computed to a default precision (normally single-float precision), and the following conversion could not help about the lacking accuracy. This is because CLN is not a symbolic computer algebra system and does not represent sqrt(5) in a non-numeric way.
The type cl_R for sqrt5 and, in the following line, phi is the only
possible choice. You cannot write cl_F because the C++ compiler can
only infer that cl_float(5,prec) is a real number. You cannot write
cl_N because a `round1' does not exist for general complex
numbers.
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.
When debugging a CLN application with GNU gdb, two facilities are
available from the library:
cl_abort() is
called. Its default implementation is to perform an exit(1), so
you won't have a core dump. But for debugging, it is best to set a
breakpoint at this function:
(gdb) break cl_abortWhen this breakpoint is hit, look at the stack's backtrace:
(gdb) where
print command doesn't know about
CLN's types and therefore prints mostly useless hexadecimal addresses.
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 #include <cl_string.h>Now, if you have in your program a variable
cl_string s, and
inspect it under gdb, the output may look like this:
(gdb) print s
$7 = {<cl_gcpointer> = { = {pointer = 0x8055b60, heappointer = 0x8055b60,
word = 134568800}}, }
(gdb) call cl_print(s)
(cl_string) ""
$8 = 134568800
Note that the output of cl_print goes to the program's error output,
not to gdb's standard output.
Note, however, that the above facility does not work with all CLN types,
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
$7 = {<cl_gcpointer> = { = {pointer = 0x8055b60, heappointer = 0x8055b60,
word = 134568800}}, }
(gdb) call s.debug_print()
(cl_string) ""
(gdb) define cprint
>call ($1).debug_print()
>end
(gdb) cprint s
(cl_string) ""
Unfortunately, this feature does not seem to work under all circumstances.
When a fatal error occurs, an error message is output to the standard error
output stream, and the function cl_abort is called. The default
version of this function (provided in the library) terminates the application.
To catch such a fatal error, you need to define the function cl_abort
yourself, with the prototype
#include <cl_abort.h> void cl_abort (void);
This function must not return control to its caller.
Floating point underflow denotes the situation when a floating-point number
is to be created which is so close to 0 that its exponent is too
low to be represented internally. By default, this causes a fatal error.
If you set the global variable
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.
The output of the function fprint may be customized by changing the
value of the global variable cl_default_print_flags.
Every memory allocation of CLN is done through the function pointer
cl_malloc_hook. Freeing of this memory is done through the function
pointer cl_free_hook. The default versions of these functions,
provided in the library, call malloc and free and check
the malloc result against NULL.
If you want to provide another memory allocator, you need to define
the variables cl_malloc_hook and cl_free_hook yourself,
like this:
#include <cl_malloc.h> void* (*cl_malloc_hook) (size_t size) = ...; void (*cl_free_hook) (void* ptr) = ...;
The cl_malloc_hook function must not return a NULL pointer.
It is not possible to change the memory allocator at runtime, because it is already called at program startup by the constructors of some global variables.
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