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<HTML>
<HEAD>
<!-- Created by texi2html 1.56k from cln.texi on 28 August 2000 -->
<TITLE>CLN, a Class Library for Numbers - 1. Introduction</TITLE>
</HEAD>
<BODY>
Go to the first, previous, <A HREF="cln_2.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
<P><HR><P>
<H1><A NAME="SEC1" HREF="cln_toc.html#TOC1">1. Introduction</A></H1>
<P>
CLN is a library for computations with all kinds of numbers.
It has a rich set of number classes:
<UL>
<LI>
Integers (with unlimited precision),
<LI>
Rational numbers,
<LI>
Floating-point numbers:
<UL>
<LI>
Short float,
<LI>
Single float,
<LI>
Double float,
<LI>
Long float (with unlimited precision),
</UL>
<LI>
Complex numbers,
<LI>
Modular integers (integers modulo a fixed integer),
<LI>
Univariate polynomials.
</UL>
<P>
The subtypes of the complex numbers among these are exactly the
types of numbers known to the Common Lisp language. Therefore
<CODE>CLN</CODE> can be used for Common Lisp implementations, giving
<SAMP>`CLN'</SAMP> another meaning: it becomes an abbreviation of
"Common Lisp Numbers".
<P>
The CLN package implements
<UL>
<LI>
Elementary functions (<CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, <CODE>sqrt</CODE>,
comparisons, ...),
<LI>
Logical functions (logical <CODE>and</CODE>, <CODE>or</CODE>, <CODE>not</CODE>, ...),
<LI>
Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
functions and their inverse functions).
</UL>
<P>
CLN is a C++ library. Using C++ as an implementation language provides
<UL>
<LI>
efficiency: it compiles to machine code,
<LI>
type safety: the C++ compiler knows about the number types and complains
if, for example, you try to assign a float to an integer variable.
<LI>
algebraic syntax: You can use the <CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>, <CODE>=</CODE>,
<CODE>==</CODE>, ... operators as in C or C++.
</UL>
<P>
CLN is memory efficient:
<UL>
<LI>
Small integers and short floats are immediate, not heap allocated.
<LI>
Heap-allocated memory is reclaimed through an automatic, non-interruptive
garbage collection.
</UL>
<P>
CLN is speed efficient:
<UL>
<LI>
The kernel of CLN has been written in assembly language for some CPUs
(<CODE>i386</CODE>, <CODE>m68k</CODE>, <CODE>sparc</CODE>, <CODE>mips</CODE>, <CODE>arm</CODE>).
<LI>
<A NAME="IDX1"></A>
On all CPUs, CLN may be configured to use the superefficient low-level
routines from GNU GMP version 3.
<LI>
It uses Karatsuba multiplication, which is significantly faster
for large numbers than the standard multiplication algorithm.
<LI>
For very large numbers (more than 12000 decimal digits), it uses
Schönhage-Strassen
<A NAME="IDX2"></A>
multiplication, which is an asymptotically optimal multiplication
algorithm, for multiplication, division and radix conversion.
</UL>
<P>
CLN aims at being easily integrated into larger software packages:
<UL>
<LI>
The garbage collection imposes no burden on the main application.
<LI>
The library provides hooks for memory allocation and exceptions.
<LI>
<A NAME="IDX3"></A>
All non-macro identifiers are hidden in namespace <CODE>cln</CODE> in
order to avoid name clashes.
</UL>
<P><HR><P>
Go to the first, previous, <A HREF="cln_2.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
</BODY>
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<HTML>
<HEAD>
<!-- Created by texi2html 1.56k from cln.texi on 28 August 2000 -->
<TITLE>CLN, a Class Library for Numbers - 10. Internals</TITLE>
</HEAD>
<BODY>
Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_9.html">previous</A>, <A HREF="cln_11.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
<P><HR><P>
<H1><A NAME="SEC59" HREF="cln_toc.html#TOC59">10. Internals</A></H1>
<H2><A NAME="SEC60" HREF="cln_toc.html#TOC60">10.1 Why C++ ?</A></H2>
<P>
<A NAME="IDX327"></A>
<P>
Using C++ as an implementation language provides
<UL>
<LI>
Efficiency: It compiles to machine code.
<LI>
<A NAME="IDX328"></A>
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.
<LI>
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>operator+ (const cl_MI&#38;, const cl_MI&#38;)</CODE> requires that both
arguments belong to the same modular ring cannot be expressed as a compile-time
information.
<LI>
Algebraic syntax: The elementary operations <CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>,
<CODE>=</CODE>, <CODE>==</CODE>, ... can be used in infix notation, which is more
convenient than Lisp notation <SAMP>`(+ x y)'</SAMP> or C notation <SAMP>`add(x,y,&#38;z)'</SAMP>.
</UL>
<P>
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.
<H2><A NAME="SEC61" HREF="cln_toc.html#TOC61">10.2 Memory efficiency</A></H2>
<P>
In order to save memory allocations, CLN implements:
<UL>
<LI>
Object sharing: An operation like <CODE>x+0</CODE> returns <CODE>x</CODE> without copying
it.
<LI>
<A NAME="IDX329"></A>
<A NAME="IDX330"></A>
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.
<LI>
Small integers are represented as immediate values instead of pointers
to heap allocated storage. This means that integers <CODE>&#62; -2^29</CODE>,
<CODE>&#60; 2^29</CODE> don't consume heap memory, unless they were explicitly allocated
on the heap.
</UL>
<H2><A NAME="SEC62" HREF="cln_toc.html#TOC62">10.3 Speed efficiency</A></H2>
<P>
Speed efficiency is obtained by the combination of the following tricks
and algorithms:
<UL>
<LI>
Small integers, being represented as immediate values, don't require
memory access, just a couple of instructions for each elementary operation.
<LI>
The kernel of CLN has been written in assembly language for some CPUs
(<CODE>i386</CODE>, <CODE>m68k</CODE>, <CODE>sparc</CODE>, <CODE>mips</CODE>, <CODE>arm</CODE>).
<LI>
On all CPUs, CLN may be configured to use the superefficient low-level
routines from GNU GMP version 3.
<LI>
For large numbers, CLN uses, instead of the standard <CODE>O(N^2)</CODE>
algorithm, the Karatsuba multiplication, which is an
<CODE>O(N^1.6)</CODE>
algorithm.
<LI>
For very large numbers (more than 12000 decimal digits), CLN uses
Schönhage-Strassen
<A NAME="IDX331"></A>
multiplication, which is an asymptotically optimal multiplication
algorithm.
<LI>
These fast multiplication algorithms also give improvements in the speed
of division and radix conversion.
</UL>
<H2><A NAME="SEC63" HREF="cln_toc.html#TOC63">10.4 Garbage collection</A></H2>
<P>
<A NAME="IDX332"></A>
<P>
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.
<P>
Memory occupied by number objects are automatically reclaimed as soon as
their reference count drops to zero.
<P>
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.
<P><HR><P>
Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_9.html">previous</A>, <A HREF="cln_11.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
</BODY>
</HTML>

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<HTML>
<HEAD>
<!-- Created by texi2html 1.56k from cln.texi on 28 August 2000 -->
<TITLE>CLN, a Class Library for Numbers - 11. Using the library</TITLE>
</HEAD>
<BODY>
Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_10.html">previous</A>, <A HREF="cln_12.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
<P><HR><P>
<H1><A NAME="SEC64" HREF="cln_toc.html#TOC64">11. Using the library</A></H1>
<P>
For the following discussion, we will assume that you have installed
the CLN source in <CODE>$CLN_DIR</CODE> and built it in <CODE>$CLN_TARGETDIR</CODE>.
For example, for me it's <CODE>CLN_DIR="$HOME/cln"</CODE> and
<CODE>CLN_TARGETDIR="$HOME/cln/linuxelf"</CODE>. You might define these as
environment variables, or directly substitute the appropriate values.
<H2><A NAME="SEC65" HREF="cln_toc.html#TOC65">11.1 Compiler options</A></H2>
<P>
<A NAME="IDX333"></A>
<P>
Until you have installed CLN in a public place, the following options are
needed:
<P>
When you compile CLN application code, add the flags
<PRE>
-I$CLN_DIR/include -I$CLN_TARGETDIR/include
</PRE>
<P>
to the C++ compiler's command line (<CODE>make</CODE> variable CFLAGS or CXXFLAGS).
When you link CLN application code to form an executable, add the flags
<PRE>
$CLN_TARGETDIR/src/libcln.a
</PRE>
<P>
to the C/C++ compiler's command line (<CODE>make</CODE> variable LIBS).
<P>
If you did a <CODE>make install</CODE>, the include files are installed in a
public directory (normally <CODE>/usr/local/include</CODE>), hence you don't
need special flags for compiling. The library has been installed to a
public directory as well (normally <CODE>/usr/local/lib</CODE>), hence when
linking a CLN application it is sufficient to give the flag <CODE>-lcln</CODE>.
<H2><A NAME="SEC66" HREF="cln_toc.html#TOC66">11.2 Compatibility to old CLN versions</A></H2>
<P>
<A NAME="IDX334"></A>
<A NAME="IDX335"></A>
<P>
As of CLN version 1.1 all non-macro identifiers were hidden in namespace
<CODE>cln</CODE> in order to avoid potential name clashes with other C++
libraries. If you have an old application, you will have to manually
port it to the new scheme. The following principles will help during
the transition:
<UL>
<LI>
All headers are now in a separate subdirectory. Instead of including
<CODE>cl_</CODE><VAR>something</VAR><CODE>.h</CODE>, include
<CODE>cln/</CODE><VAR>something</VAR><CODE>.h</CODE> now.
<LI>
All public identifiers (typenames and functions) have lost their
<CODE>cl_</CODE> prefix. Exceptions are all the typenames of number types,
(cl_N, cl_I, cl_MI, ...), rings, symbolic types (cl_string,
cl_symbol) and polynomials (cl_UP_<VAR>type</VAR>). (This is because their
names would not be mnemonic enough once the namespace <CODE>cln</CODE> is
imported. Even in a namespace we favor <CODE>cl_N</CODE> over <CODE>N</CODE>.)
<LI>
All public <EM>functions</EM> that had by a <CODE>cl_</CODE> in their name still
carry that <CODE>cl_</CODE> if it is intrinsic part of a typename (as in
<CODE>cl_I_to_int ()</CODE>).
</UL>
<P>
When developing other libraries, please keep in mind not to import the
namespace <CODE>cln</CODE> in one of your public header files by saying
<CODE>using namespace cln;</CODE>. This would propagate to other applications
and can cause name clashes there.
<H2><A NAME="SEC67" HREF="cln_toc.html#TOC67">11.3 Include files</A></H2>
<P>
<A NAME="IDX336"></A>
<A NAME="IDX337"></A>
<P>
Here is a summary of the include files and their contents.
<DL COMPACT>
<DT><CODE>&#60;cln/object.h&#62;</CODE>
<DD>
General definitions, reference counting, garbage collection.
<DT><CODE>&#60;cln/number.h&#62;</CODE>
<DD>
The class cl_number.
<DT><CODE>&#60;cln/complex.h&#62;</CODE>
<DD>
Functions for class cl_N, the complex numbers.
<DT><CODE>&#60;cln/real.h&#62;</CODE>
<DD>
Functions for class cl_R, the real numbers.
<DT><CODE>&#60;cln/float.h&#62;</CODE>
<DD>
Functions for class cl_F, the floats.
<DT><CODE>&#60;cln/sfloat.h&#62;</CODE>
<DD>
Functions for class cl_SF, the short-floats.
<DT><CODE>&#60;cln/ffloat.h&#62;</CODE>
<DD>
Functions for class cl_FF, the single-floats.
<DT><CODE>&#60;cln/dfloat.h&#62;</CODE>
<DD>
Functions for class cl_DF, the double-floats.
<DT><CODE>&#60;cln/lfloat.h&#62;</CODE>
<DD>
Functions for class cl_LF, the long-floats.
<DT><CODE>&#60;cln/rational.h&#62;</CODE>
<DD>
Functions for class cl_RA, the rational numbers.
<DT><CODE>&#60;cln/integer.h&#62;</CODE>
<DD>
Functions for class cl_I, the integers.
<DT><CODE>&#60;cln/io.h&#62;</CODE>
<DD>
Input/Output.
<DT><CODE>&#60;cln/complex_io.h&#62;</CODE>
<DD>
Input/Output for class cl_N, the complex numbers.
<DT><CODE>&#60;cln/real_io.h&#62;</CODE>
<DD>
Input/Output for class cl_R, the real numbers.
<DT><CODE>&#60;cln/float_io.h&#62;</CODE>
<DD>
Input/Output for class cl_F, the floats.
<DT><CODE>&#60;cln/sfloat_io.h&#62;</CODE>
<DD>
Input/Output for class cl_SF, the short-floats.
<DT><CODE>&#60;cln/ffloat_io.h&#62;</CODE>
<DD>
Input/Output for class cl_FF, the single-floats.
<DT><CODE>&#60;cln/dfloat_io.h&#62;</CODE>
<DD>
Input/Output for class cl_DF, the double-floats.
<DT><CODE>&#60;cln/lfloat_io.h&#62;</CODE>
<DD>
Input/Output for class cl_LF, the long-floats.
<DT><CODE>&#60;cln/rational_io.h&#62;</CODE>
<DD>
Input/Output for class cl_RA, the rational numbers.
<DT><CODE>&#60;cln/integer_io.h&#62;</CODE>
<DD>
Input/Output for class cl_I, the integers.
<DT><CODE>&#60;cln/input.h&#62;</CODE>
<DD>
Flags for customizing input operations.
<DT><CODE>&#60;cln/output.h&#62;</CODE>
<DD>
Flags for customizing output operations.
<DT><CODE>&#60;cln/malloc.h&#62;</CODE>
<DD>
<CODE>malloc_hook</CODE>, <CODE>free_hook</CODE>.
<DT><CODE>&#60;cln/abort.h&#62;</CODE>
<DD>
<CODE>cl_abort</CODE>.
<DT><CODE>&#60;cln/condition.h&#62;</CODE>
<DD>
Conditions/exceptions.
<DT><CODE>&#60;cln/string.h&#62;</CODE>
<DD>
Strings.
<DT><CODE>&#60;cln/symbol.h&#62;</CODE>
<DD>
Symbols.
<DT><CODE>&#60;cln/proplist.h&#62;</CODE>
<DD>
Property lists.
<DT><CODE>&#60;cln/ring.h&#62;</CODE>
<DD>
General rings.
<DT><CODE>&#60;cln/null_ring.h&#62;</CODE>
<DD>
The null ring.
<DT><CODE>&#60;cln/complex_ring.h&#62;</CODE>
<DD>
The ring of complex numbers.
<DT><CODE>&#60;cln/real_ring.h&#62;</CODE>
<DD>
The ring of real numbers.
<DT><CODE>&#60;cln/rational_ring.h&#62;</CODE>
<DD>
The ring of rational numbers.
<DT><CODE>&#60;cln/integer_ring.h&#62;</CODE>
<DD>
The ring of integers.
<DT><CODE>&#60;cln/numtheory.h&#62;</CODE>
<DD>
Number threory functions.
<DT><CODE>&#60;cln/modinteger.h&#62;</CODE>
<DD>
Modular integers.
<DT><CODE>&#60;cln/V.h&#62;</CODE>
<DD>
Vectors.
<DT><CODE>&#60;cln/GV.h&#62;</CODE>
<DD>
General vectors.
<DT><CODE>&#60;cln/GV_number.h&#62;</CODE>
<DD>
General vectors over cl_number.
<DT><CODE>&#60;cln/GV_complex.h&#62;</CODE>
<DD>
General vectors over cl_N.
<DT><CODE>&#60;cln/GV_real.h&#62;</CODE>
<DD>
General vectors over cl_R.
<DT><CODE>&#60;cln/GV_rational.h&#62;</CODE>
<DD>
General vectors over cl_RA.
<DT><CODE>&#60;cln/GV_integer.h&#62;</CODE>
<DD>
General vectors over cl_I.
<DT><CODE>&#60;cln/GV_modinteger.h&#62;</CODE>
<DD>
General vectors of modular integers.
<DT><CODE>&#60;cln/SV.h&#62;</CODE>
<DD>
Simple vectors.
<DT><CODE>&#60;cln/SV_number.h&#62;</CODE>
<DD>
Simple vectors over cl_number.
<DT><CODE>&#60;cln/SV_complex.h&#62;</CODE>
<DD>
Simple vectors over cl_N.
<DT><CODE>&#60;cln/SV_real.h&#62;</CODE>
<DD>
Simple vectors over cl_R.
<DT><CODE>&#60;cln/SV_rational.h&#62;</CODE>
<DD>
Simple vectors over cl_RA.
<DT><CODE>&#60;cln/SV_integer.h&#62;</CODE>
<DD>
Simple vectors over cl_I.
<DT><CODE>&#60;cln/SV_ringelt.h&#62;</CODE>
<DD>
Simple vectors of general ring elements.
<DT><CODE>&#60;cln/univpoly.h&#62;</CODE>
<DD>
Univariate polynomials.
<DT><CODE>&#60;cln/univpoly_integer.h&#62;</CODE>
<DD>
Univariate polynomials over the integers.
<DT><CODE>&#60;cln/univpoly_rational.h&#62;</CODE>
<DD>
Univariate polynomials over the rational numbers.
<DT><CODE>&#60;cln/univpoly_real.h&#62;</CODE>
<DD>
Univariate polynomials over the real numbers.
<DT><CODE>&#60;cln/univpoly_complex.h&#62;</CODE>
<DD>
Univariate polynomials over the complex numbers.
<DT><CODE>&#60;cln/univpoly_modint.h&#62;</CODE>
<DD>
Univariate polynomials over modular integer rings.
<DT><CODE>&#60;cln/timing.h&#62;</CODE>
<DD>
Timing facilities.
<DT><CODE>&#60;cln/cln.h&#62;</CODE>
<DD>
Includes all of the above.
</DL>
<H2><A NAME="SEC68" HREF="cln_toc.html#TOC68">11.4 An Example</A></H2>
<P>
A function which computes the nth Fibonacci number can be written as follows.
<A NAME="IDX338"></A>
<PRE>
#include &#60;cln/integer.h&#62;
#include &#60;cln/real.h&#62;
using namespace cln;
// Returns F_n, computed as the nearest integer to
// ((1+sqrt(5))/2)^n/sqrt(5). Assume n&#62;=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 );
}
</PRE>
<P>
Let's explain what is going on in detail.
<P>
The include file <CODE>&#60;cln/integer.h&#62;</CODE> is necessary because the type
<CODE>cl_I</CODE> is used in the function, and the include file <CODE>&#60;cln/real.h&#62;</CODE>
is needed for the type <CODE>cl_R</CODE> and the floating point number functions.
The order of the include files does not matter. In order not to write out
<CODE>cln::</CODE><VAR>foo</VAR> we can safely import the whole namespace <CODE>cln</CODE>.
<P>
Then comes the function declaration. The argument is an <CODE>int</CODE>, the
result an integer. The return type is defined as <SAMP>`const cl_I'</SAMP>, not
simply <SAMP>`cl_I'</SAMP>, because that allows the compiler to detect typos like
<SAMP>`fibonacci(n) = 100'</SAMP>. It would be possible to declare the return
type as <CODE>const cl_R</CODE> (real number) or even <CODE>const cl_N</CODE> (complex
number). We use the most specialized possible return type because functions
which call <SAMP>`fibonacci'</SAMP> 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.
<P>
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.
<P>
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.
<P>
The type <CODE>cl_R</CODE> for sqrt5 and, in the following line, phi is the only
possible choice. You cannot write <CODE>cl_F</CODE> because the C++ compiler can
only infer that <CODE>cl_float(5,prec)</CODE> is a real number. You cannot write
<CODE>cl_N</CODE> because a <SAMP>`round1'</SAMP> does not exist for general complex
numbers.
<P>
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.
<P>
The file <CODE>fibonacci.cc</CODE> in the subdirectory <CODE>examples</CODE>
contains this implementation together with an even faster algorithm.
<H2><A NAME="SEC69" HREF="cln_toc.html#TOC69">11.5 Debugging support</A></H2>
<P>
<A NAME="IDX339"></A>
<P>
When debugging a CLN application with GNU <CODE>gdb</CODE>, two facilities are
available from the library:
<UL>
<LI>The library does type checks, range checks, consistency checks at
many places. When one of these fails, the function <CODE>cl_abort()</CODE> is
called. Its default implementation is to perform an <CODE>exit(1)</CODE>, so
you won't have a core dump. But for debugging, it is best to set a
breakpoint at this function:
<PRE>
(gdb) break cl_abort
</PRE>
When this breakpoint is hit, look at the stack's backtrace:
<PRE>
(gdb) where
</PRE>
<LI>The debugger's normal <CODE>print</CODE> command doesn't know about
CLN's types and therefore prints mostly useless hexadecimal addresses.
CLN offers a function <CODE>cl_print</CODE>, callable from the debugger,
for printing number objects. In order to get this function, you have
to define the macro <SAMP>`CL_DEBUG'</SAMP> and then include all the header files
for which you want <CODE>cl_print</CODE> debugging support. For example:
<A NAME="IDX340"></A>
<PRE>
#define CL_DEBUG
#include &#60;cln/string.h&#62;
</PRE>
Now, if you have in your program a variable <CODE>cl_string s</CODE>, and
inspect it under <CODE>gdb</CODE>, the output may look like this:
<PRE>
(gdb) print s
$7 = {&#60;cl_gcpointer&#62; = { = {pointer = 0x8055b60, heappointer = 0x8055b60,
word = 134568800}}, }
(gdb) call cl_print(s)
(cl_string) ""
$8 = 134568800
</PRE>
Note that the output of <CODE>cl_print</CODE> 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
<CODE>debug_print()</CODE> on all CLN types. The same macro <SAMP>`CL_DEBUG'</SAMP>
is needed for this member function to be implemented. Under <CODE>gdb</CODE>,
you call it like this:
<A NAME="IDX341"></A>
<PRE>
(gdb) print s
$7 = {&#60;cl_gcpointer&#62; = { = {pointer = 0x8055b60, heappointer = 0x8055b60,
word = 134568800}}, }
(gdb) call s.debug_print()
(cl_string) ""
(gdb) define cprint
&#62;call ($1).debug_print()
&#62;end
(gdb) cprint s
(cl_string) ""
</PRE>
Unfortunately, this feature does not seem to work under all circumstances.
</UL>
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<H1><A NAME="SEC70" HREF="cln_toc.html#TOC70">12. Customizing</A></H1>
<P>
<A NAME="IDX342"></A>
<H2><A NAME="SEC71" HREF="cln_toc.html#TOC71">12.1 Error handling</A></H2>
<P>
When a fatal error occurs, an error message is output to the standard error
output stream, and the function <CODE>cl_abort</CODE> 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 <CODE>cl_abort</CODE>
yourself, with the prototype
<PRE>
#include &#60;cln/abort.h&#62;
void cl_abort (void);
</PRE>
<P>
<A NAME="IDX343"></A>
This function must not return control to its caller.
<H2><A NAME="SEC72" HREF="cln_toc.html#TOC72">12.2 Floating-point underflow</A></H2>
<P>
<A NAME="IDX344"></A>
<P>
Floating point underflow denotes the situation when a floating-point number
is to be created which is so close to <CODE>0</CODE> that its exponent is too
low to be represented internally. By default, this causes a fatal error.
If you set the global variable
<PRE>
cl_boolean cl_inhibit_floating_point_underflow
</PRE>
<P>
to <CODE>cl_true</CODE>, the error will be inhibited, and a floating-point zero
will be generated instead. The default value of
<CODE>cl_inhibit_floating_point_underflow</CODE> is <CODE>cl_false</CODE>.
<H2><A NAME="SEC73" HREF="cln_toc.html#TOC73">12.3 Customizing I/O</A></H2>
<P>
The output of the function <CODE>fprint</CODE> may be customized by changing the
value of the global variable <CODE>default_print_flags</CODE>.
<A NAME="IDX345"></A>
<H2><A NAME="SEC74" HREF="cln_toc.html#TOC74">12.4 Customizing the memory allocator</A></H2>
<P>
Every memory allocation of CLN is done through the function pointer
<CODE>malloc_hook</CODE>. Freeing of this memory is done through the function
pointer <CODE>free_hook</CODE>. The default versions of these functions,
provided in the library, call <CODE>malloc</CODE> and <CODE>free</CODE> and check
the <CODE>malloc</CODE> result against <CODE>NULL</CODE>.
If you want to provide another memory allocator, you need to define
the variables <CODE>malloc_hook</CODE> and <CODE>free_hook</CODE> yourself,
like this:
<PRE>
#include &#60;cln/malloc.h&#62;
namespace cln {
void* (*malloc_hook) (size_t size) = ...;
void (*free_hook) (void* ptr) = ...;
}
</PRE>
<P>
<A NAME="IDX346"></A>
<A NAME="IDX347"></A>
The <CODE>cl_malloc_hook</CODE> function must not return a <CODE>NULL</CODE> pointer.
<P>
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.
<P><HR><P>
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<TITLE>CLN, a Class Library for Numbers - Index</TITLE>
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<P><HR><P>
<H1><A NAME="SEC75" HREF="cln_toc.html#TOC75">Index</A></H1>
<P>
Jump to:
<P>
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<TITLE>CLN, a Class Library for Numbers - 2. Installation</TITLE>
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<P><HR><P>
<H1><A NAME="SEC2" HREF="cln_toc.html#TOC2">2. Installation</A></H1>
<P>
This section describes how to install the CLN package on your system.
<H2><A NAME="SEC3" HREF="cln_toc.html#TOC3">2.1 Prerequisites</A></H2>
<H3><A NAME="SEC4" HREF="cln_toc.html#TOC4">2.1.1 C++ compiler</A></H3>
<P>
To build CLN, you need a C++ compiler.
Actually, you need GNU <CODE>g++ 2.90</CODE> or newer, the EGCS compilers will
do.
I recommend GNU <CODE>g++ 2.95</CODE> or newer.
<P>
The following C++ features are used:
classes, member functions, overloading of functions and operators,
constructors and destructors, inline, const, multiple inheritance,
templates and namespaces.
<P>
The following C++ features are not used:
<CODE>new</CODE>, <CODE>delete</CODE>, virtual inheritance, exceptions.
<P>
CLN relies on semi-automatic ordering of initializations
of static and global variables, a feature which I could
implement for GNU g++ only.
<H3><A NAME="SEC5" HREF="cln_toc.html#TOC5">2.1.2 Make utility</A></H3>
<P>
<A NAME="IDX4"></A>
<P>
To build CLN, you also need to have GNU <CODE>make</CODE> installed.
<H3><A NAME="SEC6" HREF="cln_toc.html#TOC6">2.1.3 Sed utility</A></H3>
<P>
<A NAME="IDX5"></A>
<P>
To build CLN on HP-UX, you also need to have GNU <CODE>sed</CODE> installed.
This is because the libtool script, which creates the CLN library, relies
on <CODE>sed</CODE>, and the vendor's <CODE>sed</CODE> utility on these systems is too
limited.
<H2><A NAME="SEC7" HREF="cln_toc.html#TOC7">2.2 Building the library</A></H2>
<P>
As with any autoconfiguring GNU software, installation is as easy as this:
<PRE>
$ ./configure
$ make
$ make check
</PRE>
<P>
If on your system, <SAMP>`make'</SAMP> is not GNU <CODE>make</CODE>, you have to use
<SAMP>`gmake'</SAMP> instead of <SAMP>`make'</SAMP> above.
<P>
The <CODE>configure</CODE> command checks out some features of your system and
C++ compiler and builds the <CODE>Makefile</CODE>s. The <CODE>make</CODE> command
builds the library. This step may take 4 hours on an average workstation.
The <CODE>make check</CODE> runs some test to check that no important subroutine
has been miscompiled.
<P>
The <CODE>configure</CODE> command accepts options. To get a summary of them, try
<PRE>
$ ./configure --help
</PRE>
<P>
Some of the options are explained in detail in the <SAMP>`INSTALL.generic'</SAMP> file.
<P>
You can specify the C compiler, the C++ compiler and their options through
the following environment variables when running <CODE>configure</CODE>:
<DL COMPACT>
<DT><CODE>CC</CODE>
<DD>
Specifies the C compiler.
<DT><CODE>CFLAGS</CODE>
<DD>
Flags to be given to the C compiler when compiling programs (not when linking).
<DT><CODE>CXX</CODE>
<DD>
Specifies the C++ compiler.
<DT><CODE>CXXFLAGS</CODE>
<DD>
Flags to be given to the C++ compiler when compiling programs (not when linking).
</DL>
<P>
Examples:
<PRE>
$ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
$ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
$ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
</PRE>
<P>
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
<CODE>configure</CODE> command. If you made the settings in earlier shell
commands, you have to <CODE>export</CODE> the environment variables before
calling <CODE>configure</CODE>. In a <CODE>csh</CODE> shell, you have to use the
<SAMP>`setenv'</SAMP> command for setting each of the environment variables.
<P>
Currently CLN works only with the GNU <CODE>g++</CODE> compiler, and only in
optimizing mode. So you should specify at least <CODE>-O</CODE> in the CXXFLAGS,
or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use <CODE>-O</CODE>.)
<P>
If you use <CODE>g++</CODE> version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or
gcc-2.95.x, I recommend adding <SAMP>`-fno-exceptions'</SAMP> to the CXXFLAGS.
This will likely generate better code.
<P>
If you use <CODE>g++</CODE> version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc,
add either <SAMP>`-O'</SAMP>, <SAMP>`-O1'</SAMP> or <SAMP>`-O2 -fno-schedule-insns'</SAMP> to the
CXXFLAGS. With full <SAMP>`-O2'</SAMP>, <CODE>g++</CODE> miscompiles the division routines.
Also, if you have <CODE>g++</CODE> version egcs-1.1.1 or older on Sparc, you must
specify <SAMP>`--disable-shared'</SAMP> because <CODE>g++</CODE> would miscompile parts of
the library.
<P>
By default, both a shared and a static library are built. You can build
CLN as a static (or shared) library only, by calling <CODE>configure</CODE> with
the option <SAMP>`--disable-shared'</SAMP> (or <SAMP>`--disable-static'</SAMP>). While
shared libraries are usually more convenient to use, they may not work
on all architectures. Try disabling them if you run into linker
problems. Also, they are generally somewhat slower than static
libraries so runtime-critical applications should be linked statically.
<H3><A NAME="SEC8" HREF="cln_toc.html#TOC8">2.2.1 Using the GNU MP Library</A></H3>
<P>
<A NAME="IDX6"></A>
<P>
Starting with version 1.1, CLN may be configured to make use of a
preinstalled <CODE>gmp</CODE> library. Please make sure that you have at
least <CODE>gmp</CODE> version 3.0 installed since earlier versions are
unsupported and likely not to work. Enabling this feature by calling
<CODE>configure</CODE> with the option <SAMP>`--with-gmp'</SAMP> is known to be quite
a boost for CLN's performance.
<P>
If you have installed the <CODE>gmp</CODE> library and its header file in
some place where your compiler cannot find it by default, you must help
<CODE>configure</CODE> by setting <CODE>CPPFLAGS</CODE> and <CODE>LDFLAGS</CODE>. Here is
an example:
<PRE>
$ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
</PRE>
<H2><A NAME="SEC9" HREF="cln_toc.html#TOC9">2.3 Installing the library</A></H2>
<P>
<A NAME="IDX7"></A>
<P>
As with any autoconfiguring GNU software, installation is as easy as this:
<PRE>
$ make install
</PRE>
<P>
The <SAMP>`make install'</SAMP> command installs the library and the include files
into public places (<TT>`/usr/local/lib/'</TT> and <TT>`/usr/local/include/'</TT>,
if you haven't specified a <CODE>--prefix</CODE> option to <CODE>configure</CODE>).
This step may require superuser privileges.
<P>
If you have already built the library and wish to install it, but didn't
specify <CODE>--prefix=...</CODE> at configure time, just re-run
<CODE>configure</CODE>, giving it the same options as the first time, plus
the <CODE>--prefix=...</CODE> option.
<H2><A NAME="SEC10" HREF="cln_toc.html#TOC10">2.4 Cleaning up</A></H2>
<P>
You can remove system-dependent files generated by <CODE>make</CODE> through
<PRE>
$ make clean
</PRE>
<P>
You can remove all files generated by <CODE>make</CODE>, thus reverting to a
virgin distribution of CLN, through
<PRE>
$ make distclean
</PRE>
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<H1><A NAME="SEC11" HREF="cln_toc.html#TOC11">3. Ordinary number types</A></H1>
<P>
CLN implements the following class hierarchy:
<PRE>
Number
cl_number
&#60;cln/number.h&#62;
|
|
Real or complex number
cl_N
&#60;cln/complex.h&#62;
|
|
Real number
cl_R
&#60;cln/real.h&#62;
|
+-------------------+-------------------+
| |
Rational number Floating-point number
cl_RA cl_F
&#60;cln/rational.h&#62; &#60;cln/float.h&#62;
| |
| +--------------+--------------+--------------+
Integer | | | |
cl_I Short-Float Single-Float Double-Float Long-Float
&#60;cln/integer.h&#62; cl_SF cl_FF cl_DF cl_LF
&#60;cln/sfloat.h&#62; &#60;cln/ffloat.h&#62; &#60;cln/dfloat.h&#62; &#60;cln/lfloat.h&#62;
</PRE>
<P>
<A NAME="IDX8"></A>
<A NAME="IDX9"></A>
The base class <CODE>cl_number</CODE> 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.
<P>
<A NAME="IDX10"></A>
<A NAME="IDX11"></A>
<A NAME="IDX12"></A>
The class <CODE>cl_N</CODE> comprises real and complex numbers. There is
no special class for complex numbers since complex numbers with imaginary
part <CODE>0</CODE> are automatically converted to real numbers.
<P>
<A NAME="IDX13"></A>
The class <CODE>cl_R</CODE> comprises real numbers of different kinds. It is an
abstract class.
<P>
<A NAME="IDX14"></A>
<A NAME="IDX15"></A>
<A NAME="IDX16"></A>
The class <CODE>cl_RA</CODE> 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</CODE> are automatically converted
to integers.
<P>
<A NAME="IDX17"></A>
The class <CODE>cl_F</CODE> implements floating-point approximations to real numbers.
It is an abstract class.
<H2><A NAME="SEC12" HREF="cln_toc.html#TOC12">3.1 Exact numbers</A></H2>
<P>
<A NAME="IDX18"></A>
<P>
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 (<CODE>+</CODE>,
<CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, comparisons, ...) compute the completely
correct result.
<P>
In CLN, the exact numbers are:
<UL>
<LI>
rational numbers (including integers),
<LI>
complex numbers whose real and imaginary parts are both rational numbers.
</UL>
<P>
Rational numbers are always normalized to the form
<CODE><VAR>numerator</VAR>/<VAR>denominator</VAR></CODE> where the numerator and denominator
are coprime integers and the denominator is positive. If the resulting
denominator is <CODE>1</CODE>, the rational number is converted to an integer.
<P>
Small integers (typically in the range <CODE>-2^30</CODE>...<CODE>2^30-1</CODE>,
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.
<H2><A NAME="SEC13" HREF="cln_toc.html#TOC13">3.2 Floating-point numbers</A></H2>
<P>
<A NAME="IDX19"></A>
<P>
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.
<P>
<A NAME="IDX20"></A>
The elementary operations (<CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, ...)
only return approximate results. For example, the value of the expression
<CODE>(cl_F) 0.3 + (cl_F) 0.4</CODE> prints as <SAMP>`0.70000005'</SAMP>, not as
<SAMP>`0.7'</SAMP>. Rounding errors like this one are inevitable when computing
with floating-point numbers.
<P>
Nevertheless, CLN rounds the floating-point results of the operations <CODE>+</CODE>,
<CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, <CODE>sqrt</CODE> 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 <CODE>0</CODE> in its least
significant mantissa bit is chosen.
<P>
Similarly, testing floating point numbers for equality <SAMP>`x == y'</SAMP>
is gambling with random errors. Better check for <SAMP>`abs(x - y) &#60; epsilon'</SAMP>
for some well-chosen <CODE>epsilon</CODE>.
<P>
Floating point numbers come in four flavors:
<UL>
<LI>
<A NAME="IDX21"></A>
Short floats, type <CODE>cl_SF</CODE>.
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.
<LI>
<A NAME="IDX22"></A>
Single floats, type <CODE>cl_FF</CODE>.
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 <SAMP>`float'</SAMP>.
<LI>
<A NAME="IDX23"></A>
Double floats, type <CODE>cl_DF</CODE>.
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 <SAMP>`double'</SAMP>.
<LI>
<A NAME="IDX24"></A>
Long floats, type <CODE>cl_LF</CODE>.
They have 1 sign bit, 32 exponent bits (including the exponent's sign),
and n mantissa bits (including the "hidden" bit), where n &#62;= 64.
The precision of a long float is unlimited, but once created, a long float
has a fixed precision. (No "lazy recomputation".)
</UL>
<P>
Of course, computations with long floats are more expensive than those
with smaller floating-point formats.
<P>
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.
<P>
<A NAME="IDX25"></A>
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</CODE>. This has the advantage that
when you change the precision of some computation (say, from <CODE>cl_DF</CODE>
to <CODE>cl_LF</CODE>), 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 <CODE>cl_F</CODE> when the argument is a <CODE>cl_F</CODE>,
but such declarations are missing for the types <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>,
<CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>. (Such declarations would be wrong if
the floating point contagion rule happened to change in the future.)
<H2><A NAME="SEC14" HREF="cln_toc.html#TOC14">3.3 Complex numbers</A></H2>
<P>
<A NAME="IDX26"></A>
<P>
Complex numbers, as implemented by the class <CODE>cl_N</CODE>, have a real
part and an imaginary part, both real numbers. A complex number whose
imaginary part is the exact number <CODE>0</CODE> is automatically converted
to a real number.
<P>
Complex numbers can arise from real numbers alone, for example
through application of <CODE>sqrt</CODE> or transcendental functions.
<H2><A NAME="SEC15" HREF="cln_toc.html#TOC15">3.4 Conversions</A></H2>
<P>
<A NAME="IDX27"></A>
<P>
Conversions from any class to any its superclasses ("base classes" in
C++ terminology) is done automatically.
<P>
Conversions from the C built-in types <SAMP>`long'</SAMP> and <SAMP>`unsigned long'</SAMP>
are provided for the classes <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_R</CODE>,
<CODE>cl_N</CODE> and <CODE>cl_number</CODE>.
<P>
Conversions from the C built-in types <SAMP>`int'</SAMP> and <SAMP>`unsigned int'</SAMP>
are provided for the classes <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_R</CODE>,
<CODE>cl_N</CODE> and <CODE>cl_number</CODE>. However, these conversions emphasize
efficiency. Their range is therefore limited:
<UL>
<LI>
The conversion from <SAMP>`int'</SAMP> works only if the argument is &#60; 2^29 and &#62; -2^29.
<LI>
The conversion from <SAMP>`unsigned int'</SAMP> works only if the argument is &#60; 2^29.
</UL>
<P>
In a declaration like <SAMP>`cl_I x = 10;'</SAMP> the C++ compiler is able to
do the conversion of <CODE>10</CODE> from <SAMP>`int'</SAMP> to <SAMP>`cl_I'</SAMP> at compile time
already. On the other hand, code like <SAMP>`cl_I x = 1000000000;'</SAMP> is
in error.
So, if you want to be sure that an <SAMP>`int'</SAMP> whose magnitude is not guaranteed
to be &#60; 2^29 is correctly converted to a <SAMP>`cl_I'</SAMP>, first convert it to a
<SAMP>`long'</SAMP>. Similarly, if a large <SAMP>`unsigned int'</SAMP> is to be converted to a
<SAMP>`cl_I'</SAMP>, first convert it to an <SAMP>`unsigned long'</SAMP>.
<P>
Conversions from the C built-in type <SAMP>`float'</SAMP> are provided for the classes
<CODE>cl_FF</CODE>, <CODE>cl_F</CODE>, <CODE>cl_R</CODE>, <CODE>cl_N</CODE> and <CODE>cl_number</CODE>.
<P>
Conversions from the C built-in type <SAMP>`double'</SAMP> are provided for the classes
<CODE>cl_DF</CODE>, <CODE>cl_F</CODE>, <CODE>cl_R</CODE>, <CODE>cl_N</CODE> and <CODE>cl_number</CODE>.
<P>
Conversions from <SAMP>`const char *'</SAMP> are provided for the classes
<CODE>cl_I</CODE>, <CODE>cl_RA</CODE>,
<CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>, <CODE>cl_F</CODE>,
<CODE>cl_R</CODE>, <CODE>cl_N</CODE>.
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:
<A NAME="IDX28"></A>
<PRE>
cl_I order_of_rubiks_cube_group = "43252003274489856000";
</PRE>
<P>
Note that this conversion is done at runtime, not at compile-time.
<P>
Conversions from <CODE>cl_I</CODE> to the C built-in types <SAMP>`int'</SAMP>,
<SAMP>`unsigned int'</SAMP>, <SAMP>`long'</SAMP>, <SAMP>`unsigned long'</SAMP> are provided through
the functions
<DL COMPACT>
<DT><CODE>int cl_I_to_int (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX29"></A>
<DT><CODE>unsigned int cl_I_to_uint (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX30"></A>
<DT><CODE>long cl_I_to_long (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX31"></A>
<DT><CODE>unsigned long cl_I_to_ulong (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX32"></A>
Returns <CODE>x</CODE> as element of the C type <VAR>ctype</VAR>. If <CODE>x</CODE> is not
representable in the range of <VAR>ctype</VAR>, a runtime error occurs.
</DL>
<P>
Conversions from the classes <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>,
<CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>, <CODE>cl_F</CODE> and
<CODE>cl_R</CODE>
to the C built-in types <SAMP>`float'</SAMP> and <SAMP>`double'</SAMP> are provided through
the functions
<DL COMPACT>
<DT><CODE>float float_approx (const <VAR>type</VAR>&#38; x)</CODE>
<DD>
<A NAME="IDX33"></A>
<DT><CODE>double double_approx (const <VAR>type</VAR>&#38; x)</CODE>
<DD>
<A NAME="IDX34"></A>
Returns an approximation of <CODE>x</CODE> of C type <VAR>ctype</VAR>.
If <CODE>abs(x)</CODE> is too close to 0 (underflow), 0 is returned.
If <CODE>abs(x)</CODE> is too large (overflow), an IEEE infinity is returned.
</DL>
<P>
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'</SAMP> and <SAMP>`The'</SAMP> macros.
<A NAME="IDX35"></A>
<CODE>As(<VAR>type</VAR>)(<VAR>value</VAR>)</CODE> checks that <VAR>value</VAR> belongs to
<VAR>type</VAR> and returns it as such.
<A NAME="IDX36"></A>
<CODE>The(<VAR>type</VAR>)(<VAR>value</VAR>)</CODE> assumes that <VAR>value</VAR> belongs to
<VAR>type</VAR> and returns it as such. It is your responsibility to ensure
that this assumption is valid.
Example:
<PRE>
cl_I x = ...;
if (!(x &#62;= 0)) abort();
cl_I ten_x = The(cl_I)(expt(10,x)); // If x &#62;= 0, 10^x is an integer.
// In general, it would be a rational number.
</PRE>
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<TITLE>CLN, a Class Library for Numbers - 5. Input/Output</TITLE>
</HEAD>
<BODY>
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<P><HR><P>
<H1><A NAME="SEC44" HREF="cln_toc.html#TOC44">5. Input/Output</A></H1>
<P>
<A NAME="IDX237"></A>
<H2><A NAME="SEC45" HREF="cln_toc.html#TOC45">5.1 Internal and printed representation</A></H2>
<P>
<A NAME="IDX238"></A>
<P>
All computations deal with the internal representations of the numbers.
<P>
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".
<P>
Converting an internal to an external representation is called "printing",
<A NAME="IDX239"></A>
converting an external to an internal representation is called "reading".
<A NAME="IDX240"></A>
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</CODE>.
This is called "print-read consistency".
<P>
Different types of numbers have different external representations (case
is insignificant):
<DL COMPACT>
<DT>Integers
<DD>
External representation: <VAR>sign</VAR>{<VAR>digit</VAR>}+. The reader also accepts the
Common Lisp syntaxes <VAR>sign</VAR>{<VAR>digit</VAR>}+<CODE>.</CODE> with a trailing dot
for decimal integers
and the <CODE>#<VAR>n</VAR>R</CODE>, <CODE>#b</CODE>, <CODE>#o</CODE>, <CODE>#x</CODE> prefixes.
<DT>Rational numbers
<DD>
External representation: <VAR>sign</VAR>{<VAR>digit</VAR>}+<CODE>/</CODE>{<VAR>digit</VAR>}+.
The <CODE>#<VAR>n</VAR>R</CODE>, <CODE>#b</CODE>, <CODE>#o</CODE>, <CODE>#x</CODE> prefixes are allowed
here as well.
<DT>Floating-point numbers
<DD>
External representation: <VAR>sign</VAR>{<VAR>digit</VAR>}*<VAR>exponent</VAR> or
<VAR>sign</VAR>{<VAR>digit</VAR>}*<CODE>.</CODE>{<VAR>digit</VAR>}*<VAR>exponent</VAR> or
<VAR>sign</VAR>{<VAR>digit</VAR>}*<CODE>.</CODE>{<VAR>digit</VAR>}+. A precision specifier
of the form _<VAR>prec</VAR> may be appended. There must be at least
one digit in the non-exponent part. The exponent has the syntax
<VAR>expmarker</VAR> <VAR>expsign</VAR> {<VAR>digit</VAR>}+.
The exponent marker is
<UL>
<LI>
<SAMP>`s'</SAMP> for short-floats,
<LI>
<SAMP>`f'</SAMP> for single-floats,
<LI>
<SAMP>`d'</SAMP> for double-floats,
<LI>
<SAMP>`L'</SAMP> for long-floats,
</UL>
or <SAMP>`e'</SAMP>, which denotes a default float format. The precision specifying
suffix has the syntax _<VAR>prec</VAR> where <VAR>prec</VAR> denotes the number of
valid mantissa digits (in decimal, excluding leading zeroes), cf. also
function <SAMP>`cl_float_format'</SAMP>.
<DT>Complex numbers
<DD>
External representation:
<UL>
<LI>
In algebraic notation: <CODE><VAR>realpart</VAR>+<VAR>imagpart</VAR>i</CODE>. Of course,
if <VAR>imagpart</VAR> is negative, its printed representation begins with
a <SAMP>`-'</SAMP>, and the <SAMP>`+'</SAMP> between <VAR>realpart</VAR> and <VAR>imagpart</VAR>
may be omitted. Note that this notation cannot be used when the <VAR>imagpart</VAR>
is rational and the rational number's base is &#62;18, because the <SAMP>`i'</SAMP>
is then read as a digit.
<LI>
In Common Lisp notation: <CODE>#C(<VAR>realpart</VAR> <VAR>imagpart</VAR>)</CODE>.
</UL>
</DL>
<H2><A NAME="SEC46" HREF="cln_toc.html#TOC46">5.2 Input functions</A></H2>
<P>
Including <CODE>&#60;cln/io.h&#62;</CODE> defines a type <CODE>cl_istream</CODE>, which is
the type of the first argument to all input functions. <CODE>cl_istream</CODE>
is the same as <CODE>std::istream&#38;</CODE>.
<P>
The variable
<UL>
<LI>
<CODE>cl_istream stdin</CODE>
</UL>
<P>
contains the standard input stream.
<P>
These are the simple input functions:
<DL COMPACT>
<DT><CODE>int freadchar (cl_istream stream)</CODE>
<DD>
Reads a character from <CODE>stream</CODE>. Returns <CODE>cl_EOF</CODE> (not a <SAMP>`char'</SAMP>!)
if the end of stream was encountered or an error occurred.
<DT><CODE>int funreadchar (cl_istream stream, int c)</CODE>
<DD>
Puts back <CODE>c</CODE> onto <CODE>stream</CODE>. <CODE>c</CODE> must be the result of the
last <CODE>freadchar</CODE> operation on <CODE>stream</CODE>.
</DL>
<P>
Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
defines, in <CODE>&#60;cln/<VAR>type</VAR>_io.h&#62;</CODE>, the following input function:
<DL COMPACT>
<DT><CODE>cl_istream operator&#62;&#62; (cl_istream stream, <VAR>type</VAR>&#38; result)</CODE>
<DD>
Reads a number from <CODE>stream</CODE> and stores it in the <CODE>result</CODE>.
</DL>
<P>
The most flexible input functions, defined in <CODE>&#60;cln/<VAR>type</VAR>_io.h&#62;</CODE>,
are the following:
<DL COMPACT>
<DT><CODE>cl_N read_complex (cl_istream stream, const cl_read_flags&#38; flags)</CODE>
<DD>
<DT><CODE>cl_R read_real (cl_istream stream, const cl_read_flags&#38; flags)</CODE>
<DD>
<DT><CODE>cl_F read_float (cl_istream stream, const cl_read_flags&#38; flags)</CODE>
<DD>
<DT><CODE>cl_RA read_rational (cl_istream stream, const cl_read_flags&#38; flags)</CODE>
<DD>
<DT><CODE>cl_I read_integer (cl_istream stream, const cl_read_flags&#38; flags)</CODE>
<DD>
Reads a number from <CODE>stream</CODE>. The <CODE>flags</CODE> are parameters which
affect the input syntax. Whitespace before the number is silently skipped.
<DT><CODE>cl_N read_complex (const cl_read_flags&#38; flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
<DD>
<DT><CODE>cl_R read_real (const cl_read_flags&#38; flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
<DD>
<DT><CODE>cl_F read_float (const cl_read_flags&#38; flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
<DD>
<DT><CODE>cl_RA read_rational (const cl_read_flags&#38; flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
<DD>
<DT><CODE>cl_I read_integer (const cl_read_flags&#38; flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
<DD>
Reads a number from a string in memory. The <CODE>flags</CODE> are parameters which
affect the input syntax. The string starts at <CODE>string</CODE> and ends at
<CODE>string_limit</CODE> (exclusive limit). <CODE>string_limit</CODE> may also be
<CODE>NULL</CODE>, denoting the entire string, i.e. equivalent to
<CODE>string_limit = string + strlen(string)</CODE>. If <CODE>end_of_parse</CODE> is
<CODE>NULL</CODE>, the string in memory must contain exactly one number and nothing
more, else a fatal error will be signalled. If <CODE>end_of_parse</CODE>
is not <CODE>NULL</CODE>, <CODE>*end_of_parse</CODE> will be assigned a pointer past
the last parsed character (i.e. <CODE>string_limit</CODE> if nothing came after
the number). Whitespace is not allowed.
</DL>
<P>
The structure <CODE>cl_read_flags</CODE> contains the following fields:
<DL COMPACT>
<DT><CODE>cl_read_syntax_t syntax</CODE>
<DD>
The possible results of the read operation. Possible values are
<CODE>syntax_number</CODE>, <CODE>syntax_real</CODE>, <CODE>syntax_rational</CODE>,
<CODE>syntax_integer</CODE>, <CODE>syntax_float</CODE>, <CODE>syntax_sfloat</CODE>,
<CODE>syntax_ffloat</CODE>, <CODE>syntax_dfloat</CODE>, <CODE>syntax_lfloat</CODE>.
<DT><CODE>cl_read_lsyntax_t lsyntax</CODE>
<DD>
Specifies the language-dependent syntax variant for the read operation.
Possible values are
<DL COMPACT>
<DT><CODE>lsyntax_standard</CODE>
<DD>
accept standard algebraic notation only, no complex numbers,
<DT><CODE>lsyntax_algebraic</CODE>
<DD>
accept the algebraic notation <CODE><VAR>x</VAR>+<VAR>y</VAR>i</CODE> for complex numbers,
<DT><CODE>lsyntax_commonlisp</CODE>
<DD>
accept the <CODE>#b</CODE>, <CODE>#o</CODE>, <CODE>#x</CODE> syntaxes for binary, octal,
hexadecimal numbers,
<CODE>#<VAR>base</VAR>R</CODE> for rational numbers in a given base,
<CODE>#c(<VAR>realpart</VAR> <VAR>imagpart</VAR>)</CODE> for complex numbers,
<DT><CODE>lsyntax_all</CODE>
<DD>
accept all of these extensions.
</DL>
<DT><CODE>unsigned int rational_base</CODE>
<DD>
The base in which rational numbers are read.
<DT><CODE>cl_float_format_t float_flags.default_float_format</CODE>
<DD>
The float format used when reading floats with exponent marker <SAMP>`e'</SAMP>.
<DT><CODE>cl_float_format_t float_flags.default_lfloat_format</CODE>
<DD>
The float format used when reading floats with exponent marker <SAMP>`l'</SAMP>.
<DT><CODE>cl_boolean float_flags.mantissa_dependent_float_format</CODE>
<DD>
When this flag is true, floats specified with more digits than corresponding
to the exponent marker they contain, but without <VAR>_nnn</VAR> suffix, will get a
precision corresponding to their number of significant digits.
</DL>
<H2><A NAME="SEC47" HREF="cln_toc.html#TOC47">5.3 Output functions</A></H2>
<P>
Including <CODE>&#60;cln/io.h&#62;</CODE> defines a type <CODE>cl_ostream</CODE>, which is
the type of the first argument to all output functions. <CODE>cl_ostream</CODE>
is the same as <CODE>std::ostream&#38;</CODE>.
<P>
The variable
<UL>
<LI>
<CODE>cl_ostream stdout</CODE>
</UL>
<P>
contains the standard output stream.
<P>
The variable
<UL>
<LI>
<CODE>cl_ostream stderr</CODE>
</UL>
<P>
contains the standard error output stream.
<P>
These are the simple output functions:
<DL COMPACT>
<DT><CODE>void fprintchar (cl_ostream stream, char c)</CODE>
<DD>
Prints the character <CODE>x</CODE> literally on the <CODE>stream</CODE>.
<DT><CODE>void fprint (cl_ostream stream, const char * string)</CODE>
<DD>
Prints the <CODE>string</CODE> literally on the <CODE>stream</CODE>.
<DT><CODE>void fprintdecimal (cl_ostream stream, int x)</CODE>
<DD>
<DT><CODE>void fprintdecimal (cl_ostream stream, const cl_I&#38; x)</CODE>
<DD>
Prints the integer <CODE>x</CODE> in decimal on the <CODE>stream</CODE>.
<DT><CODE>void fprintbinary (cl_ostream stream, const cl_I&#38; x)</CODE>
<DD>
Prints the integer <CODE>x</CODE> in binary (base 2, without prefix)
on the <CODE>stream</CODE>.
<DT><CODE>void fprintoctal (cl_ostream stream, const cl_I&#38; x)</CODE>
<DD>
Prints the integer <CODE>x</CODE> in octal (base 8, without prefix)
on the <CODE>stream</CODE>.
<DT><CODE>void fprinthexadecimal (cl_ostream stream, const cl_I&#38; x)</CODE>
<DD>
Prints the integer <CODE>x</CODE> in hexadecimal (base 16, without prefix)
on the <CODE>stream</CODE>.
</DL>
<P>
Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
defines, in <CODE>&#60;cln/<VAR>type</VAR>_io.h&#62;</CODE>, the following output functions:
<DL COMPACT>
<DT><CODE>void fprint (cl_ostream stream, const <VAR>type</VAR>&#38; x)</CODE>
<DD>
<DT><CODE>cl_ostream operator&#60;&#60; (cl_ostream stream, const <VAR>type</VAR>&#38; x)</CODE>
<DD>
Prints the number <CODE>x</CODE> on the <CODE>stream</CODE>. The output may depend
on the global printer settings in the variable <CODE>default_print_flags</CODE>.
The <CODE>ostream</CODE> flags and settings (flags, width and locale) are
ignored.
</DL>
<P>
The most flexible output function, defined in <CODE>&#60;cln/<VAR>type</VAR>_io.h&#62;</CODE>,
are the following:
<PRE>
void print_complex (cl_ostream stream, const cl_print_flags&#38; flags,
const cl_N&#38; z);
void print_real (cl_ostream stream, const cl_print_flags&#38; flags,
const cl_R&#38; z);
void print_float (cl_ostream stream, const cl_print_flags&#38; flags,
const cl_F&#38; z);
void print_rational (cl_ostream stream, const cl_print_flags&#38; flags,
const cl_RA&#38; z);
void print_integer (cl_ostream stream, const cl_print_flags&#38; flags,
const cl_I&#38; z);
</PRE>
<P>
Prints the number <CODE>x</CODE> on the <CODE>stream</CODE>. The <CODE>flags</CODE> are
parameters which affect the output.
<P>
The structure type <CODE>cl_print_flags</CODE> contains the following fields:
<DL COMPACT>
<DT><CODE>unsigned int rational_base</CODE>
<DD>
The base in which rational numbers are printed. Default is <CODE>10</CODE>.
<DT><CODE>cl_boolean rational_readably</CODE>
<DD>
If this flag is true, rational numbers are printed with radix specifiers in
Common Lisp syntax (<CODE>#<VAR>n</VAR>R</CODE> or <CODE>#b</CODE> or <CODE>#o</CODE> or <CODE>#x</CODE>
prefixes, trailing dot). Default is false.
<DT><CODE>cl_boolean float_readably</CODE>
<DD>
If this flag is true, type specific exponent markers have precedence over 'E'.
Default is false.
<DT><CODE>cl_float_format_t default_float_format</CODE>
<DD>
Floating point numbers of this format will be printed using the 'E' exponent
marker. Default is <CODE>cl_float_format_ffloat</CODE>.
<DT><CODE>cl_boolean complex_readably</CODE>
<DD>
If this flag is true, complex numbers will be printed using the Common Lisp
syntax <CODE>#C(<VAR>realpart</VAR> <VAR>imagpart</VAR>)</CODE>. Default is false.
<DT><CODE>cl_string univpoly_varname</CODE>
<DD>
Univariate polynomials with no explicit indeterminate name will be printed
using this variable name. Default is <CODE>"x"</CODE>.
</DL>
<P>
The global variable <CODE>default_print_flags</CODE> contains the default values,
used by the function <CODE>fprint</CODE>.
<P><HR><P>
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<TITLE>CLN, a Class Library for Numbers - 6. Rings</TITLE>
</HEAD>
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<H1><A NAME="SEC48" HREF="cln_toc.html#TOC48">6. Rings</A></H1>
<P>
CLN has a class of abstract rings.
<PRE>
Ring
cl_ring
&#60;cln/ring.h&#62;
</PRE>
<P>
Rings can be compared for equality:
<DL COMPACT>
<DT><CODE>bool operator== (const cl_ring&#38;, const cl_ring&#38;)</CODE>
<DD>
<DT><CODE>bool operator!= (const cl_ring&#38;, const cl_ring&#38;)</CODE>
<DD>
These compare two rings for equality.
</DL>
<P>
Given a ring <CODE>R</CODE>, the following members can be used.
<DL COMPACT>
<DT><CODE>void R-&#62;fprint (cl_ostream stream, const cl_ring_element&#38; x)</CODE>
<DD>
<A NAME="IDX241"></A>
<DT><CODE>cl_boolean R-&#62;equal (const cl_ring_element&#38; x, const cl_ring_element&#38; y)</CODE>
<DD>
<A NAME="IDX242"></A>
<DT><CODE>cl_ring_element R-&#62;zero ()</CODE>
<DD>
<A NAME="IDX243"></A>
<DT><CODE>cl_boolean R-&#62;zerop (const cl_ring_element&#38; x)</CODE>
<DD>
<A NAME="IDX244"></A>
<DT><CODE>cl_ring_element R-&#62;plus (const cl_ring_element&#38; x, const cl_ring_element&#38; y)</CODE>
<DD>
<A NAME="IDX245"></A>
<DT><CODE>cl_ring_element R-&#62;minus (const cl_ring_element&#38; x, const cl_ring_element&#38; y)</CODE>
<DD>
<A NAME="IDX246"></A>
<DT><CODE>cl_ring_element R-&#62;uminus (const cl_ring_element&#38; x)</CODE>
<DD>
<A NAME="IDX247"></A>
<DT><CODE>cl_ring_element R-&#62;one ()</CODE>
<DD>
<A NAME="IDX248"></A>
<DT><CODE>cl_ring_element R-&#62;canonhom (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX249"></A>
<DT><CODE>cl_ring_element R-&#62;mul (const cl_ring_element&#38; x, const cl_ring_element&#38; y)</CODE>
<DD>
<A NAME="IDX250"></A>
<DT><CODE>cl_ring_element R-&#62;square (const cl_ring_element&#38; x)</CODE>
<DD>
<A NAME="IDX251"></A>
<DT><CODE>cl_ring_element R-&#62;expt_pos (const cl_ring_element&#38; x, const cl_I&#38; y)</CODE>
<DD>
<A NAME="IDX252"></A>
</DL>
<P>
The following rings are built-in.
<DL COMPACT>
<DT><CODE>cl_null_ring cl_0_ring</CODE>
<DD>
The null ring, containing only zero.
<DT><CODE>cl_complex_ring cl_C_ring</CODE>
<DD>
The ring of complex numbers. This corresponds to the type <CODE>cl_N</CODE>.
<DT><CODE>cl_real_ring cl_R_ring</CODE>
<DD>
The ring of real numbers. This corresponds to the type <CODE>cl_R</CODE>.
<DT><CODE>cl_rational_ring cl_RA_ring</CODE>
<DD>
The ring of rational numbers. This corresponds to the type <CODE>cl_RA</CODE>.
<DT><CODE>cl_integer_ring cl_I_ring</CODE>
<DD>
The ring of integers. This corresponds to the type <CODE>cl_I</CODE>.
</DL>
<P>
Type tests can be performed for any of <CODE>cl_C_ring</CODE>, <CODE>cl_R_ring</CODE>,
<CODE>cl_RA_ring</CODE>, <CODE>cl_I_ring</CODE>:
<DL COMPACT>
<DT><CODE>cl_boolean instanceof (const cl_number&#38; x, const cl_number_ring&#38; R)</CODE>
<DD>
<A NAME="IDX253"></A>
Tests whether the given number is an element of the number ring R.
</DL>
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<H1><A NAME="SEC49" HREF="cln_toc.html#TOC49">7. Modular integers</A></H1>
<P>
<A NAME="IDX254"></A>
<H2><A NAME="SEC50" HREF="cln_toc.html#TOC50">7.1 Modular integer rings</A></H2>
<P>
<A NAME="IDX255"></A>
<P>
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. <CODE>(3 mod 4) + (2 mod 5)</CODE> will result in a runtime error.
(Ideally one would imagine a generic data type <CODE>cl_MI(N)</CODE>, but C++
doesn't have generic types. So one has to live with runtime checks.)
<P>
The class of modular integer rings is
<PRE>
Ring
cl_ring
&#60;cln/ring.h&#62;
|
|
Modular integer ring
cl_modint_ring
&#60;cln/modinteger.h&#62;
</PRE>
<P>
<A NAME="IDX256"></A>
<P>
and the class of all modular integers (elements of modular integer rings) is
<PRE>
Modular integer
cl_MI
&#60;cln/modinteger.h&#62;
</PRE>
<P>
Modular integer rings are constructed using the function
<DL COMPACT>
<DT><CODE>cl_modint_ring find_modint_ring (const cl_I&#38; N)</CODE>
<DD>
<A NAME="IDX257"></A>
This function returns the modular ring <SAMP>`Z/NZ'</SAMP>. It takes care
of finding out about special cases of <CODE>N</CODE>, like powers of two
and odd numbers for which Montgomery multiplication will be a win,
<A NAME="IDX258"></A>
and precomputes any necessary auxiliary data for computing modulo <CODE>N</CODE>.
There is a cache table of rings, indexed by <CODE>N</CODE> (or, more precisely,
by <CODE>abs(N)</CODE>). This ensures that the precomputation costs are reduced
to a minimum.
</DL>
<P>
Modular integer rings can be compared for equality:
<DL COMPACT>
<DT><CODE>bool operator== (const cl_modint_ring&#38;, const cl_modint_ring&#38;)</CODE>
<DD>
<A NAME="IDX259"></A>
<DT><CODE>bool operator!= (const cl_modint_ring&#38;, const cl_modint_ring&#38;)</CODE>
<DD>
<A NAME="IDX260"></A>
These compare two modular integer rings for equality. Two different calls
to <CODE>find_modint_ring</CODE> with the same argument necessarily return the
same ring because it is memoized in the cache table.
</DL>
<H2><A NAME="SEC51" HREF="cln_toc.html#TOC51">7.2 Functions on modular integers</A></H2>
<P>
Given a modular integer ring <CODE>R</CODE>, the following members can be used.
<DL COMPACT>
<DT><CODE>cl_I R-&#62;modulus</CODE>
<DD>
<A NAME="IDX261"></A>
This is the ring's modulus, normalized to be nonnegative: <CODE>abs(N)</CODE>.
<DT><CODE>cl_MI R-&#62;zero()</CODE>
<DD>
<A NAME="IDX262"></A>
This returns <CODE>0 mod N</CODE>.
<DT><CODE>cl_MI R-&#62;one()</CODE>
<DD>
<A NAME="IDX263"></A>
This returns <CODE>1 mod N</CODE>.
<DT><CODE>cl_MI R-&#62;canonhom (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX264"></A>
This returns <CODE>x mod N</CODE>.
<DT><CODE>cl_I R-&#62;retract (const cl_MI&#38; x)</CODE>
<DD>
<A NAME="IDX265"></A>
This is a partial inverse function to <CODE>R-&#62;canonhom</CODE>. It returns the
standard representative (<CODE>&#62;=0</CODE>, <CODE>&#60;N</CODE>) of <CODE>x</CODE>.
<DT><CODE>cl_MI R-&#62;random(random_state&#38; randomstate)</CODE>
<DD>
<DT><CODE>cl_MI R-&#62;random()</CODE>
<DD>
<A NAME="IDX266"></A>
This returns a random integer modulo <CODE>N</CODE>.
</DL>
<P>
The following operations are defined on modular integers.
<DL COMPACT>
<DT><CODE>cl_modint_ring x.ring ()</CODE>
<DD>
<A NAME="IDX267"></A>
Returns the ring to which the modular integer <CODE>x</CODE> belongs.
<DT><CODE>cl_MI operator+ (const cl_MI&#38;, const cl_MI&#38;)</CODE>
<DD>
<A NAME="IDX268"></A>
Returns the sum of two modular integers. One of the arguments may also
be a plain integer.
<DT><CODE>cl_MI operator- (const cl_MI&#38;, const cl_MI&#38;)</CODE>
<DD>
<A NAME="IDX269"></A>
Returns the difference of two modular integers. One of the arguments may also
be a plain integer.
<DT><CODE>cl_MI operator- (const cl_MI&#38;)</CODE>
<DD>
Returns the negative of a modular integer.
<DT><CODE>cl_MI operator* (const cl_MI&#38;, const cl_MI&#38;)</CODE>
<DD>
<A NAME="IDX270"></A>
Returns the product of two modular integers. One of the arguments may also
be a plain integer.
<DT><CODE>cl_MI square (const cl_MI&#38;)</CODE>
<DD>
<A NAME="IDX271"></A>
Returns the square of a modular integer.
<DT><CODE>cl_MI recip (const cl_MI&#38; x)</CODE>
<DD>
<A NAME="IDX272"></A>
Returns the reciprocal <CODE>x^-1</CODE> of a modular integer <CODE>x</CODE>. <CODE>x</CODE>
must be coprime to the modulus, otherwise an error message is issued.
<DT><CODE>cl_MI div (const cl_MI&#38; x, const cl_MI&#38; y)</CODE>
<DD>
<A NAME="IDX273"></A>
Returns the quotient <CODE>x*y^-1</CODE> of two modular integers <CODE>x</CODE>, <CODE>y</CODE>.
<CODE>y</CODE> must be coprime to the modulus, otherwise an error message is issued.
<DT><CODE>cl_MI expt_pos (const cl_MI&#38; x, const cl_I&#38; y)</CODE>
<DD>
<A NAME="IDX274"></A>
<CODE>y</CODE> must be &#62; 0. Returns <CODE>x^y</CODE>.
<DT><CODE>cl_MI expt (const cl_MI&#38; x, const cl_I&#38; y)</CODE>
<DD>
<A NAME="IDX275"></A>
Returns <CODE>x^y</CODE>. If <CODE>y</CODE> is negative, <CODE>x</CODE> must be coprime to the
modulus, else an error message is issued.
<DT><CODE>cl_MI operator&#60;&#60; (const cl_MI&#38; x, const cl_I&#38; y)</CODE>
<DD>
<A NAME="IDX276"></A>
Returns <CODE>x*2^y</CODE>.
<DT><CODE>cl_MI operator&#62;&#62; (const cl_MI&#38; x, const cl_I&#38; y)</CODE>
<DD>
<A NAME="IDX277"></A>
Returns <CODE>x*2^-y</CODE>. When <CODE>y</CODE> is positive, the modulus must be odd,
or an error message is issued.
<DT><CODE>bool operator== (const cl_MI&#38;, const cl_MI&#38;)</CODE>
<DD>
<A NAME="IDX278"></A>
<DT><CODE>bool operator!= (const cl_MI&#38;, const cl_MI&#38;)</CODE>
<DD>
<A NAME="IDX279"></A>
Compares two modular integers, belonging to the same modular integer ring,
for equality.
<DT><CODE>cl_boolean zerop (const cl_MI&#38; x)</CODE>
<DD>
<A NAME="IDX280"></A>
Returns true if <CODE>x</CODE> is <CODE>0 mod N</CODE>.
</DL>
<P>
The following output functions are defined (see also the chapter on
input/output).
<DL COMPACT>
<DT><CODE>void fprint (cl_ostream stream, const cl_MI&#38; x)</CODE>
<DD>
<A NAME="IDX281"></A>
<DT><CODE>cl_ostream operator&#60;&#60; (cl_ostream stream, const cl_MI&#38; x)</CODE>
<DD>
<A NAME="IDX282"></A>
Prints the modular integer <CODE>x</CODE> on the <CODE>stream</CODE>. The output may depend
on the global printer settings in the variable <CODE>default_print_flags</CODE>.
</DL>
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<TITLE>CLN, a Class Library for Numbers - 8. Symbolic data types</TITLE>
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<H1><A NAME="SEC52" HREF="cln_toc.html#TOC52">8. Symbolic data types</A></H1>
<P>
<A NAME="IDX283"></A>
<P>
CLN implements two symbolic (non-numeric) data types: strings and symbols.
<H2><A NAME="SEC53" HREF="cln_toc.html#TOC53">8.1 Strings</A></H2>
<P>
<A NAME="IDX284"></A>
<A NAME="IDX285"></A>
<P>
The class
<PRE>
String
cl_string
&#60;cln/string.h&#62;
</PRE>
<P>
implements immutable strings.
<P>
Strings are constructed through the following constructors:
<DL COMPACT>
<DT><CODE>cl_string (const char * s)</CODE>
<DD>
Returns an immutable copy of the (zero-terminated) C string <CODE>s</CODE>.
<DT><CODE>cl_string (const char * ptr, unsigned long len)</CODE>
<DD>
Returns an immutable copy of the <CODE>len</CODE> characters at
<CODE>ptr[0]</CODE>, ..., <CODE>ptr[len-1]</CODE>. NUL characters are allowed.
</DL>
<P>
The following functions are available on strings:
<DL COMPACT>
<DT><CODE>operator =</CODE>
<DD>
Assignment from <CODE>cl_string</CODE> and <CODE>const char *</CODE>.
<DT><CODE>s.length()</CODE>
<DD>
<A NAME="IDX286"></A>
<DT><CODE>strlen(s)</CODE>
<DD>
<A NAME="IDX287"></A>
Returns the length of the string <CODE>s</CODE>.
<DT><CODE>s[i]</CODE>
<DD>
<A NAME="IDX288"></A>
Returns the <CODE>i</CODE>th character of the string <CODE>s</CODE>.
<CODE>i</CODE> must be in the range <CODE>0 &#60;= i &#60; s.length()</CODE>.
<DT><CODE>bool equal (const cl_string&#38; s1, const cl_string&#38; s2)</CODE>
<DD>
<A NAME="IDX289"></A>
Compares two strings for equality. One of the arguments may also be a
plain <CODE>const char *</CODE>.
</DL>
<H2><A NAME="SEC54" HREF="cln_toc.html#TOC54">8.2 Symbols</A></H2>
<P>
<A NAME="IDX290"></A>
<A NAME="IDX291"></A>
<P>
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.
<P>
Symbols are constructed through the following constructor:
<DL COMPACT>
<DT><CODE>cl_symbol (const cl_string&#38; s)</CODE>
<DD>
Looks up or creates a new symbol with a given name.
</DL>
<P>
The following operations are available on symbols:
<DL COMPACT>
<DT><CODE>cl_string (const cl_symbol&#38; sym)</CODE>
<DD>
Conversion to <CODE>cl_string</CODE>: Returns the string which names the symbol
<CODE>sym</CODE>.
<DT><CODE>bool equal (const cl_symbol&#38; sym1, const cl_symbol&#38; sym2)</CODE>
<DD>
<A NAME="IDX292"></A>
Compares two symbols for equality. This is very fast.
</DL>
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<TITLE>CLN, a Class Library for Numbers - 9. Univariate polynomials</TITLE>
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<H1><A NAME="SEC55" HREF="cln_toc.html#TOC55">9. Univariate polynomials</A></H1>
<P>
<A NAME="IDX293"></A>
<A NAME="IDX294"></A>
<H2><A NAME="SEC56" HREF="cln_toc.html#TOC56">9.1 Univariate polynomial rings</A></H2>
<P>
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 <CODE>default_print_flags.univpoly_varname</CODE>, which
defaults to <SAMP>`x'</SAMP>) 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.
<CODE>(a^2+1) * (b^3-1)</CODE> will result in a runtime error. (Ideally this should
return a multivariate polynomial, but they are not yet implemented in CLN.)
<P>
The classes of univariate polynomial rings are
<PRE>
Ring
cl_ring
&#60;cln/ring.h&#62;
|
|
Univariate polynomial ring
cl_univpoly_ring
&#60;cln/univpoly.h&#62;
|
+----------------+-------------------+
| | |
Complex polynomial ring | Modular integer polynomial ring
cl_univpoly_complex_ring | cl_univpoly_modint_ring
&#60;cln/univpoly_complex.h&#62; | &#60;cln/univpoly_modint.h&#62;
|
+----------------+
| |
Real polynomial ring |
cl_univpoly_real_ring |
&#60;cln/univpoly_real.h&#62; |
|
+----------------+
| |
Rational polynomial ring |
cl_univpoly_rational_ring |
&#60;cln/univpoly_rational.h&#62; |
|
+----------------+
|
Integer polynomial ring
cl_univpoly_integer_ring
&#60;cln/univpoly_integer.h&#62;
</PRE>
<P>
and the corresponding classes of univariate polynomials are
<PRE>
Univariate polynomial
cl_UP
&#60;cln/univpoly.h&#62;
|
+----------------+-------------------+
| | |
Complex polynomial | Modular integer polynomial
cl_UP_N | cl_UP_MI
&#60;cln/univpoly_complex.h&#62; | &#60;cln/univpoly_modint.h&#62;
|
+----------------+
| |
Real polynomial |
cl_UP_R |
&#60;cln/univpoly_real.h&#62; |
|
+----------------+
| |
Rational polynomial |
cl_UP_RA |
&#60;cln/univpoly_rational.h&#62; |
|
+----------------+
|
Integer polynomial
cl_UP_I
&#60;cln/univpoly_integer.h&#62;
</PRE>
<P>
Univariate polynomial rings are constructed using the functions
<DL COMPACT>
<DT><CODE>cl_univpoly_ring find_univpoly_ring (const cl_ring&#38; R)</CODE>
<DD>
<DT><CODE>cl_univpoly_ring find_univpoly_ring (const cl_ring&#38; R, const cl_symbol&#38; varname)</CODE>
<DD>
This function returns the polynomial ring <SAMP>`R[X]'</SAMP>, unnamed or named.
<CODE>R</CODE> may be an arbitrary ring. This function takes care of finding out
about special cases of <CODE>R</CODE>, 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 <CODE>R</CODE> and <CODE>varname</CODE>.
This ensures that two calls of this function with the same arguments will
return the same polynomial ring.
<DT><CODE>cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring&#38; R)</CODE>
<DD>
<A NAME="IDX295"></A>
<DT><CODE>cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring&#38; R, const cl_symbol&#38; varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring&#38; R)</CODE>
<DD>
<DT><CODE>cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring&#38; R, const cl_symbol&#38; varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring&#38; R)</CODE>
<DD>
<DT><CODE>cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring&#38; R, const cl_symbol&#38; varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring&#38; R)</CODE>
<DD>
<DT><CODE>cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring&#38; R, const cl_symbol&#38; varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring&#38; R)</CODE>
<DD>
<DT><CODE>cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring&#38; R, const cl_symbol&#38; varname)</CODE>
<DD>
These functions are equivalent to the general <CODE>find_univpoly_ring</CODE>,
only the return type is more specific, according to the base ring's type.
</DL>
<H2><A NAME="SEC57" HREF="cln_toc.html#TOC57">9.2 Functions on univariate polynomials</A></H2>
<P>
Given a univariate polynomial ring <CODE>R</CODE>, the following members can be used.
<DL COMPACT>
<DT><CODE>cl_ring R-&#62;basering()</CODE>
<DD>
<A NAME="IDX296"></A>
This returns the base ring, as passed to <SAMP>`find_univpoly_ring'</SAMP>.
<DT><CODE>cl_UP R-&#62;zero()</CODE>
<DD>
<A NAME="IDX297"></A>
This returns <CODE>0 in R</CODE>, a polynomial of degree -1.
<DT><CODE>cl_UP R-&#62;one()</CODE>
<DD>
<A NAME="IDX298"></A>
This returns <CODE>1 in R</CODE>, a polynomial of degree &#60;= 0.
<DT><CODE>cl_UP R-&#62;canonhom (const cl_I&#38; x)</CODE>
<DD>
<A NAME="IDX299"></A>
This returns <CODE>x in R</CODE>, a polynomial of degree &#60;= 0.
<DT><CODE>cl_UP R-&#62;monomial (const cl_ring_element&#38; x, uintL e)</CODE>
<DD>
<A NAME="IDX300"></A>
This returns a sparse polynomial: <CODE>x * X^e</CODE>, where <CODE>X</CODE> is the
indeterminate.
<DT><CODE>cl_UP R-&#62;create (sintL degree)</CODE>
<DD>
<A NAME="IDX301"></A>
Creates a new polynomial with a given degree. The zero polynomial has degree
<CODE>-1</CODE>. After creating the polynomial, you should put in the coefficients,
using the <CODE>set_coeff</CODE> member function, and then call the <CODE>finalize</CODE>
member function.
</DL>
<P>
The following are the only destructive operations on univariate polynomials.
<DL COMPACT>
<DT><CODE>void set_coeff (cl_UP&#38; x, uintL index, const cl_ring_element&#38; y)</CODE>
<DD>
<A NAME="IDX302"></A>
This changes the coefficient of <CODE>X^index</CODE> in <CODE>x</CODE> to be <CODE>y</CODE>.
After changing a polynomial and before applying any "normal" operation on it,
you should call its <CODE>finalize</CODE> member function.
<DT><CODE>void finalize (cl_UP&#38; x)</CODE>
<DD>
<A NAME="IDX303"></A>
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
produce wrong results or crash the program.
</DL>
<P>
The following operations are defined on univariate polynomials.
<DL COMPACT>
<DT><CODE>cl_univpoly_ring x.ring ()</CODE>
<DD>
<A NAME="IDX304"></A>
Returns the ring to which the univariate polynomial <CODE>x</CODE> belongs.
<DT><CODE>cl_UP operator+ (const cl_UP&#38;, const cl_UP&#38;)</CODE>
<DD>
<A NAME="IDX305"></A>
Returns the sum of two univariate polynomials.
<DT><CODE>cl_UP operator- (const cl_UP&#38;, const cl_UP&#38;)</CODE>
<DD>
<A NAME="IDX306"></A>
Returns the difference of two univariate polynomials.
<DT><CODE>cl_UP operator- (const cl_UP&#38;)</CODE>
<DD>
Returns the negative of a univariate polynomial.
<DT><CODE>cl_UP operator* (const cl_UP&#38;, const cl_UP&#38;)</CODE>
<DD>
<A NAME="IDX307"></A>
Returns the product of two univariate polynomials. One of the arguments may
also be a plain integer or an element of the base ring.
<DT><CODE>cl_UP square (const cl_UP&#38;)</CODE>
<DD>
<A NAME="IDX308"></A>
Returns the square of a univariate polynomial.
<DT><CODE>cl_UP expt_pos (const cl_UP&#38; x, const cl_I&#38; y)</CODE>
<DD>
<A NAME="IDX309"></A>
<CODE>y</CODE> must be &#62; 0. Returns <CODE>x^y</CODE>.
<DT><CODE>bool operator== (const cl_UP&#38;, const cl_UP&#38;)</CODE>
<DD>
<A NAME="IDX310"></A>
<DT><CODE>bool operator!= (const cl_UP&#38;, const cl_UP&#38;)</CODE>
<DD>
<A NAME="IDX311"></A>
Compares two univariate polynomials, belonging to the same univariate
polynomial ring, for equality.
<DT><CODE>cl_boolean zerop (const cl_UP&#38; x)</CODE>
<DD>
<A NAME="IDX312"></A>
Returns true if <CODE>x</CODE> is <CODE>0 in R</CODE>.
<DT><CODE>sintL degree (const cl_UP&#38; x)</CODE>
<DD>
<A NAME="IDX313"></A>
Returns the degree of the polynomial. The zero polynomial has degree <CODE>-1</CODE>.
<DT><CODE>cl_ring_element coeff (const cl_UP&#38; x, uintL index)</CODE>
<DD>
<A NAME="IDX314"></A>
Returns the coefficient of <CODE>X^index</CODE> in the polynomial <CODE>x</CODE>.
<DT><CODE>cl_ring_element x (const cl_ring_element&#38; y)</CODE>
<DD>
<A NAME="IDX315"></A>
Evaluation: If <CODE>x</CODE> is a polynomial and <CODE>y</CODE> belongs to the base ring,
then <SAMP>`x(y)'</SAMP> returns the value of the substitution of <CODE>y</CODE> into
<CODE>x</CODE>.
<DT><CODE>cl_UP deriv (const cl_UP&#38; x)</CODE>
<DD>
<A NAME="IDX316"></A>
Returns the derivative of the polynomial <CODE>x</CODE> with respect to the
indeterminate <CODE>X</CODE>.
</DL>
<P>
The following output functions are defined (see also the chapter on
input/output).
<DL COMPACT>
<DT><CODE>void fprint (cl_ostream stream, const cl_UP&#38; x)</CODE>
<DD>
<A NAME="IDX317"></A>
<DT><CODE>cl_ostream operator&#60;&#60; (cl_ostream stream, const cl_UP&#38; x)</CODE>
<DD>
<A NAME="IDX318"></A>
Prints the univariate polynomial <CODE>x</CODE> on the <CODE>stream</CODE>. The output may
depend on the global printer settings in the variable
<CODE>default_print_flags</CODE>.
</DL>
<H2><A NAME="SEC58" HREF="cln_toc.html#TOC58">9.3 Special polynomials</A></H2>
<P>
The following functions return special polynomials.
<DL COMPACT>
<DT><CODE>cl_UP_I tschebychev (sintL n)</CODE>
<DD>
<A NAME="IDX319"></A>
<A NAME="IDX320"></A>
Returns the n-th Chebyshev polynomial (n &#62;= 0).
<DT><CODE>cl_UP_I hermite (sintL n)</CODE>
<DD>
<A NAME="IDX321"></A>
<A NAME="IDX322"></A>
Returns the n-th Hermite polynomial (n &#62;= 0).
<DT><CODE>cl_UP_RA legendre (sintL n)</CODE>
<DD>
<A NAME="IDX323"></A>
<A NAME="IDX324"></A>
Returns the n-th Legendre polynomial (n &#62;= 0).
<DT><CODE>cl_UP_I laguerre (sintL n)</CODE>
<DD>
<A NAME="IDX325"></A>
<A NAME="IDX326"></A>
Returns the n-th Laguerre polynomial (n &#62;= 0).
</DL>
<P>
Information how to derive the differential equation satisfied by each
of these polynomials from their definition can be found in the
<CODE>doc/polynomial/</CODE> directory.
<P><HR><P>
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<TITLE>CLN, a Class Library for Numbers - Table of Contents</TITLE>
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<H1>CLN, a Class Library for Numbers</H1>
<ADDRESS>by Bruno Haible</ADDRESS>
<P>
<P><HR><P>
<UL>
<LI><A NAME="TOC1" HREF="cln_1.html#SEC1">1. Introduction</A>
<LI><A NAME="TOC2" HREF="cln_2.html#SEC2">2. Installation</A>
<UL>
<LI><A NAME="TOC3" HREF="cln_2.html#SEC3">2.1 Prerequisites</A>
<UL>
<LI><A NAME="TOC4" HREF="cln_2.html#SEC4">2.1.1 C++ compiler</A>
<LI><A NAME="TOC5" HREF="cln_2.html#SEC5">2.1.2 Make utility</A>
<LI><A NAME="TOC6" HREF="cln_2.html#SEC6">2.1.3 Sed utility</A>
</UL>
<LI><A NAME="TOC7" HREF="cln_2.html#SEC7">2.2 Building the library</A>
<UL>
<LI><A NAME="TOC8" HREF="cln_2.html#SEC8">2.2.1 Using the GNU MP Library</A>
</UL>
<LI><A NAME="TOC9" HREF="cln_2.html#SEC9">2.3 Installing the library</A>
<LI><A NAME="TOC10" HREF="cln_2.html#SEC10">2.4 Cleaning up</A>
</UL>
<LI><A NAME="TOC11" HREF="cln_3.html#SEC11">3. Ordinary number types</A>
<UL>
<LI><A NAME="TOC12" HREF="cln_3.html#SEC12">3.1 Exact numbers</A>
<LI><A NAME="TOC13" HREF="cln_3.html#SEC13">3.2 Floating-point numbers</A>
<LI><A NAME="TOC14" HREF="cln_3.html#SEC14">3.3 Complex numbers</A>
<LI><A NAME="TOC15" HREF="cln_3.html#SEC15">3.4 Conversions</A>
</UL>
<LI><A NAME="TOC16" HREF="cln_4.html#SEC16">4. Functions on numbers</A>
<UL>
<LI><A NAME="TOC17" HREF="cln_4.html#SEC17">4.1 Constructing numbers</A>
<UL>
<LI><A NAME="TOC18" HREF="cln_4.html#SEC18">4.1.1 Constructing integers</A>
<LI><A NAME="TOC19" HREF="cln_4.html#SEC19">4.1.2 Constructing rational numbers</A>
<LI><A NAME="TOC20" HREF="cln_4.html#SEC20">4.1.3 Constructing floating-point numbers</A>
<LI><A NAME="TOC21" HREF="cln_4.html#SEC21">4.1.4 Constructing complex numbers</A>
</UL>
<LI><A NAME="TOC22" HREF="cln_4.html#SEC22">4.2 Elementary functions</A>
<LI><A NAME="TOC23" HREF="cln_4.html#SEC23">4.3 Elementary rational functions</A>
<LI><A NAME="TOC24" HREF="cln_4.html#SEC24">4.4 Elementary complex functions</A>
<LI><A NAME="TOC25" HREF="cln_4.html#SEC25">4.5 Comparisons</A>
<LI><A NAME="TOC26" HREF="cln_4.html#SEC26">4.6 Rounding functions</A>
<LI><A NAME="TOC27" HREF="cln_4.html#SEC27">4.7 Roots</A>
<LI><A NAME="TOC28" HREF="cln_4.html#SEC28">4.8 Transcendental functions</A>
<UL>
<LI><A NAME="TOC29" HREF="cln_4.html#SEC29">4.8.1 Exponential and logarithmic functions</A>
<LI><A NAME="TOC30" HREF="cln_4.html#SEC30">4.8.2 Trigonometric functions</A>
<LI><A NAME="TOC31" HREF="cln_4.html#SEC31">4.8.3 Hyperbolic functions</A>
<LI><A NAME="TOC32" HREF="cln_4.html#SEC32">4.8.4 Euler gamma</A>
<LI><A NAME="TOC33" HREF="cln_4.html#SEC33">4.8.5 Riemann zeta</A>
</UL>
<LI><A NAME="TOC34" HREF="cln_4.html#SEC34">4.9 Functions on integers</A>
<UL>
<LI><A NAME="TOC35" HREF="cln_4.html#SEC35">4.9.1 Logical functions</A>
<LI><A NAME="TOC36" HREF="cln_4.html#SEC36">4.9.2 Number theoretic functions</A>
<LI><A NAME="TOC37" HREF="cln_4.html#SEC37">4.9.3 Combinatorial functions</A>
</UL>
<LI><A NAME="TOC38" HREF="cln_4.html#SEC38">4.10 Functions on floating-point numbers</A>
<LI><A NAME="TOC39" HREF="cln_4.html#SEC39">4.11 Conversion functions</A>
<UL>
<LI><A NAME="TOC40" HREF="cln_4.html#SEC40">4.11.1 Conversion to floating-point numbers</A>
<LI><A NAME="TOC41" HREF="cln_4.html#SEC41">4.11.2 Conversion to rational numbers</A>
</UL>
<LI><A NAME="TOC42" HREF="cln_4.html#SEC42">4.12 Random number generators</A>
<LI><A NAME="TOC43" HREF="cln_4.html#SEC43">4.13 Obfuscating operators</A>
</UL>
<LI><A NAME="TOC44" HREF="cln_5.html#SEC44">5. Input/Output</A>
<UL>
<LI><A NAME="TOC45" HREF="cln_5.html#SEC45">5.1 Internal and printed representation</A>
<LI><A NAME="TOC46" HREF="cln_5.html#SEC46">5.2 Input functions</A>
<LI><A NAME="TOC47" HREF="cln_5.html#SEC47">5.3 Output functions</A>
</UL>
<LI><A NAME="TOC48" HREF="cln_6.html#SEC48">6. Rings</A>
<LI><A NAME="TOC49" HREF="cln_7.html#SEC49">7. Modular integers</A>
<UL>
<LI><A NAME="TOC50" HREF="cln_7.html#SEC50">7.1 Modular integer rings</A>
<LI><A NAME="TOC51" HREF="cln_7.html#SEC51">7.2 Functions on modular integers</A>
</UL>
<LI><A NAME="TOC52" HREF="cln_8.html#SEC52">8. Symbolic data types</A>
<UL>
<LI><A NAME="TOC53" HREF="cln_8.html#SEC53">8.1 Strings</A>
<LI><A NAME="TOC54" HREF="cln_8.html#SEC54">8.2 Symbols</A>
</UL>
<LI><A NAME="TOC55" HREF="cln_9.html#SEC55">9. Univariate polynomials</A>
<UL>
<LI><A NAME="TOC56" HREF="cln_9.html#SEC56">9.1 Univariate polynomial rings</A>
<LI><A NAME="TOC57" HREF="cln_9.html#SEC57">9.2 Functions on univariate polynomials</A>
<LI><A NAME="TOC58" HREF="cln_9.html#SEC58">9.3 Special polynomials</A>
</UL>
<LI><A NAME="TOC59" HREF="cln_10.html#SEC59">10. Internals</A>
<UL>
<LI><A NAME="TOC60" HREF="cln_10.html#SEC60">10.1 Why C++ ?</A>
<LI><A NAME="TOC61" HREF="cln_10.html#SEC61">10.2 Memory efficiency</A>
<LI><A NAME="TOC62" HREF="cln_10.html#SEC62">10.3 Speed efficiency</A>
<LI><A NAME="TOC63" HREF="cln_10.html#SEC63">10.4 Garbage collection</A>
</UL>
<LI><A NAME="TOC64" HREF="cln_11.html#SEC64">11. Using the library</A>
<UL>
<LI><A NAME="TOC65" HREF="cln_11.html#SEC65">11.1 Compiler options</A>
<LI><A NAME="TOC66" HREF="cln_11.html#SEC66">11.2 Compatibility to old CLN versions</A>
<LI><A NAME="TOC67" HREF="cln_11.html#SEC67">11.3 Include files</A>
<LI><A NAME="TOC68" HREF="cln_11.html#SEC68">11.4 An Example</A>
<LI><A NAME="TOC69" HREF="cln_11.html#SEC69">11.5 Debugging support</A>
</UL>
<LI><A NAME="TOC70" HREF="cln_12.html#SEC70">12. Customizing</A>
<UL>
<LI><A NAME="TOC71" HREF="cln_12.html#SEC71">12.1 Error handling</A>
<LI><A NAME="TOC72" HREF="cln_12.html#SEC72">12.2 Floating-point underflow</A>
<LI><A NAME="TOC73" HREF="cln_12.html#SEC73">12.3 Customizing I/O</A>
<LI><A NAME="TOC74" HREF="cln_12.html#SEC74">12.4 Customizing the memory allocator</A>
</UL>
<LI><A NAME="TOC75" HREF="cln_13.html#SEC75">Index</A>
</UL>
<P><HR><P>
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