1 \input texinfo @c -*-texinfo-*-
4 @settitle CLN, a Class Library for Numbers
5 @c @setchapternewpage off
10 @c I hate putting "@noindent" in front of every paragraph.
16 * CLN: (cln). Class Library for Numbers (C++).
21 @c Don't need the other types of indices.
32 This file documents @sc{cln}, a Class Library for Numbers.
34 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
35 Richard B. Kreckel, @code{<kreckel@@ginac.de>}.
37 Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004.
38 Copyright (C) Richard B. Kreckel 2000, 2001, 2002, 2003, 2004.
40 Permission is granted to make and distribute verbatim copies of
41 this manual provided the copyright notice and this permission notice
42 are preserved on all copies.
45 Permission is granted to process this file through TeX and print the
46 results, provided the printed document carries copying permission
47 notice identical to this one except for the removal of this paragraph
48 (this paragraph not being relevant to the printed manual).
51 Permission is granted to copy and distribute modified versions of this
52 manual under the conditions for verbatim copying, provided that the entire
53 resulting derived work is distributed under the terms of a permission
54 notice identical to this one.
56 Permission is granted to copy and distribute translations of this manual
57 into another language, under the above conditions for modified versions,
58 except that this permission notice may be stated in a translation approved
64 @c prevent ugly black rectangles on overfull hbox lines:
67 @title CLN, a Class Library for Numbers
69 @author by Bruno Haible
71 @vskip 0pt plus 1filll
72 Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004.
74 Copyright @copyright{} Richard Kreckel 2000, 2001, 2002, 2003, 2004.
77 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
78 Richard Kreckel, @code{<kreckel@@ginac.de>}.
80 Permission is granted to make and distribute verbatim copies of
81 this manual provided the copyright notice and this permission notice
82 are preserved on all copies.
84 Permission is granted to copy and distribute modified versions of this
85 manual under the conditions for verbatim copying, provided that the entire
86 resulting derived work is distributed under the terms of a permission
87 notice identical to this one.
89 Permission is granted to copy and distribute translations of this manual
90 into another language, under the above conditions for modified versions,
91 except that this permission notice may be stated in a translation approved
102 @node Top, Introduction, (dir), (dir)
105 @c * Introduction:: Introduction
109 @node Introduction, Top, Top, Top
110 @comment node-name, next, previous, up
111 @chapter Introduction
114 CLN is a library for computations with all kinds of numbers.
115 It has a rich set of number classes:
119 Integers (with unlimited precision),
125 Floating-point numbers:
135 Long float (with unlimited precision),
142 Modular integers (integers modulo a fixed integer),
145 Univariate polynomials.
149 The subtypes of the complex numbers among these are exactly the
150 types of numbers known to the Common Lisp language. Therefore
151 @code{CLN} can be used for Common Lisp implementations, giving
152 @samp{CLN} another meaning: it becomes an abbreviation of
153 ``Common Lisp Numbers''.
156 The CLN package implements
160 Elementary functions (@code{+}, @code{-}, @code{*}, @code{/}, @code{sqrt},
161 comparisons, @dots{}),
164 Logical functions (logical @code{and}, @code{or}, @code{not}, @dots{}),
167 Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
168 functions and their inverse functions).
172 CLN is a C++ library. Using C++ as an implementation language provides
176 efficiency: it compiles to machine code,
178 type safety: the C++ compiler knows about the number types and complains
179 if, for example, you try to assign a float to an integer variable.
181 algebraic syntax: You can use the @code{+}, @code{-}, @code{*}, @code{=},
182 @code{==}, @dots{} operators as in C or C++.
186 CLN is memory efficient:
190 Small integers and short floats are immediate, not heap allocated.
192 Heap-allocated memory is reclaimed through an automatic, non-interruptive
197 CLN is speed efficient:
201 The kernel of CLN has been written in assembly language for some CPUs
202 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
205 On all CPUs, CLN may be configured to use the superefficient low-level
206 routines from GNU GMP version 3.
208 It uses Karatsuba multiplication, which is significantly faster
209 for large numbers than the standard multiplication algorithm.
211 For very large numbers (more than 12000 decimal digits), it uses
213 Sch{@"o}nhage-Strassen
214 @cindex Sch{@"o}nhage-Strassen multiplication
218 @cindex Schnhage-Strassen multiplication
220 multiplication, which is an asymptotically optimal multiplication
221 algorithm, for multiplication, division and radix conversion.
225 CLN aims at being easily integrated into larger software packages:
229 The garbage collection imposes no burden on the main application.
231 The library provides hooks for memory allocation and exceptions.
234 All non-macro identifiers are hidden in namespace @code{cln} in
235 order to avoid name clashes.
239 @chapter Installation
241 This section describes how to install the CLN package on your system.
244 @section Prerequisites
246 @subsection C++ compiler
248 To build CLN, you need a C++ compiler.
249 Actually, you need GNU @code{g++ 2.95} or newer.
251 The following C++ features are used:
252 classes, member functions, overloading of functions and operators,
253 constructors and destructors, inline, const, multiple inheritance,
254 templates and namespaces.
256 The following C++ features are not used:
257 @code{new}, @code{delete}, virtual inheritance, exceptions.
259 CLN relies on semi-automatic ordering of initializations
260 of static and global variables, a feature which I could
261 implement for GNU g++ only.
264 @comment cl_modules.h requires g++
265 Therefore nearly any C++ compiler will do.
267 The following C++ compilers are known to compile CLN:
270 GNU @code{g++ 2.7.0}, @code{g++ 2.7.2}
275 The following C++ compilers are known to be unusable for CLN:
278 On SunOS 4, @code{CC 2.1}, because it doesn't grok @code{//} comments
279 in lines containing @code{#if} or @code{#elif} preprocessor commands.
281 On AIX 3.2.5, @code{xlC}, because it doesn't grok the template syntax
282 in @code{cl_SV.h} and @code{cl_GV.h}, because it forces most class types
283 to have default constructors, and because it probably miscompiles the
284 integer multiplication routines.
286 On AIX 4.1.4.0, @code{xlC}, because when optimizing, it sometimes converts
287 @code{short}s to @code{int}s by zero-extend.
291 On HPPA, GNU @code{g++ 2.7.x}, because the semi-automatic ordering of
292 initializations will not work.
296 @subsection Make utility
299 To build CLN, you also need to have GNU @code{make} installed.
301 Only GNU @code{make} 3.77 is unusable for CLN; other versions work fine.
303 @subsection Sed utility
306 To build CLN on HP-UX, you also need to have GNU @code{sed} installed.
307 This is because the libtool script, which creates the CLN library, relies
308 on @code{sed}, and the vendor's @code{sed} utility on these systems is too
312 @section Building the library
314 As with any autoconfiguring GNU software, installation is as easy as this:
322 If on your system, @samp{make} is not GNU @code{make}, you have to use
323 @samp{gmake} instead of @samp{make} above.
325 The @code{configure} command checks out some features of your system and
326 C++ compiler and builds the @code{Makefile}s. The @code{make} command
327 builds the library. This step may take about an hour on an average workstation.
328 The @code{make check} runs some test to check that no important subroutine
329 has been miscompiled.
331 The @code{configure} command accepts options. To get a summary of them, try
337 Some of the options are explained in detail in the @samp{INSTALL.generic} file.
339 You can specify the C compiler, the C++ compiler and their options through
340 the following environment variables when running @code{configure}:
344 Specifies the C compiler.
347 Flags to be given to the C compiler when compiling programs (not when linking).
350 Specifies the C++ compiler.
353 Flags to be given to the C++ compiler when compiling programs (not when linking).
359 $ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
360 $ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
361 CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
362 $ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
363 CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
364 $ CC="gcc -V 3.0.4" CFLAGS="-O2 -finline-limit=1000 -fno-exceptions" \
365 CXX="g++ -V 3.0.4" CFLAGS="-O2 -finline-limit=1000 -fno-exceptions" \
369 @comment cl_modules.h requires g++
370 You should not mix GNU and non-GNU compilers. So, if @code{CXX} is a non-GNU
371 compiler, @code{CC} should be set to a non-GNU compiler as well. Examples:
374 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O" ./configure
375 $ CC="gcc -V 2.7.0" CFLAGS="-g" CXX="g++ -V 2.7.0" CXXFLAGS="-g" ./configure
378 On SGI Irix 5, if you wish not to use @code{g++}:
381 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O -Olimit 16000" ./configure
384 On SGI Irix 6, if you wish not to use @code{g++}:
387 $ CC="cc -32" CFLAGS="-O" CXX="CC -32" CXXFLAGS="-O -Olimit 34000" \
388 ./configure --without-gmp
389 $ CC="cc -n32" CFLAGS="-O" CXX="CC -n32" CXXFLAGS="-O \
390 -OPT:const_copy_limit=32400 -OPT:global_limit=32400 -OPT:fprop_limit=4000" \
391 ./configure --without-gmp
395 Note that for these environment variables to take effect, you have to set
396 them (assuming a Bourne-compatible shell) on the same line as the
397 @code{configure} command. If you made the settings in earlier shell
398 commands, you have to @code{export} the environment variables before
399 calling @code{configure}. In a @code{csh} shell, you have to use the
400 @samp{setenv} command for setting each of the environment variables.
402 Currently CLN works only with the GNU @code{g++} compiler, and only in
403 optimizing mode. So you should specify at least @code{-O} in the CXXFLAGS,
404 or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use @code{-O}.)
406 If you use @code{g++} 3.0.x or 3.1, I recommend adding
407 @samp{-finline-limit=1000} to the CXXFLAGS. This is essential for good code.
409 If you use @code{g++} gcc-2.95.x or gcc-3.x , I recommend adding
410 @samp{-fno-exceptions} to the CXXFLAGS. This will likely generate better code.
412 If you use @code{g++} from gcc-3.0.4 or older on Sparc, add either
413 @samp{-O}, @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the
414 CXXFLAGS. With full @samp{-O2}, @code{g++} miscompiles the division
415 routines. If you use @code{g++} older than 2.95.3 on Sparc you should
416 also specify @samp{--disable-shared} because of bad code produced in the
417 shared library. Also, do not use gcc-3.0 on Sparc for compiling CLN, it
420 If you use @code{g++} on OSF/1 or Tru64 using gcc-2.95.x, you should
421 specify @samp{--disable-shared} because of linker problems with
422 duplicate symbols in shared libraries. If you use @code{g++} from
423 gcc-3.0.n, with n larger than 1, you should @emph{not} add
424 @samp{-fno-exceptions} to the CXXFLAGS, since that will generate wrong
425 code (gcc-3.1 is okay again, as is gcc-3.0).
427 Also, please do not compile CLN with @code{g++} using the @code{-O3}
428 optimization level. This leads to inferior code quality.
430 If you use @code{g++} from gcc-3.1, it will need 235 MB of virtual memory.
431 You might need some swap space if your machine doesn't have 512 MB of RAM.
433 By default, both a shared and a static library are built. You can build
434 CLN as a static (or shared) library only, by calling @code{configure} with
435 the option @samp{--disable-shared} (or @samp{--disable-static}). While
436 shared libraries are usually more convenient to use, they may not work
437 on all architectures. Try disabling them if you run into linker
438 problems. Also, they are generally somewhat slower than static
439 libraries so runtime-critical applications should be linked statically.
441 If you use @code{g++} from gcc-3.1 with option @samp{-g}, you will need
442 some disk space: 335 MB for building as both a shared and a static library,
443 or 130 MB when building as a shared library only.
446 @subsection Using the GNU MP Library
449 Starting with version 1.1, CLN may be configured to make use of a
450 preinstalled @code{gmp} library. Please make sure that you have at
451 least @code{gmp} version 3.0 installed since earlier versions are
452 unsupported and likely not to work. Enabling this feature by calling
453 @code{configure} with the option @samp{--with-gmp} is known to be quite
454 a boost for CLN's performance.
456 If you have installed the @code{gmp} library and its header file in
457 some place where your compiler cannot find it by default, you must help
458 @code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
462 $ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
463 CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
467 @section Installing the library
470 As with any autoconfiguring GNU software, installation is as easy as this:
476 The @samp{make install} command installs the library and the include files
477 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
478 if you haven't specified a @code{--prefix} option to @code{configure}).
479 This step may require superuser privileges.
481 If you have already built the library and wish to install it, but didn't
482 specify @code{--prefix=@dots{}} at configure time, just re-run
483 @code{configure}, giving it the same options as the first time, plus
484 the @code{--prefix=@dots{}} option.
489 You can remove system-dependent files generated by @code{make} through
495 You can remove all files generated by @code{make}, thus reverting to a
496 virgin distribution of CLN, through
503 @chapter Ordinary number types
505 CLN implements the following class hierarchy:
513 Real or complex number
522 +-------------------+-------------------+
524 Rational number Floating-point number
526 <cln/rational.h> <cln/float.h>
528 | +--------------+--------------+--------------+
530 cl_I Short-Float Single-Float Double-Float Long-Float
531 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
532 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
535 @cindex @code{cl_number}
536 @cindex abstract class
537 The base class @code{cl_number} is an abstract base class.
538 It is not useful to declare a variable of this type except if you want
539 to completely disable compile-time type checking and use run-time type
544 @cindex complex number
545 The class @code{cl_N} comprises real and complex numbers. There is
546 no special class for complex numbers since complex numbers with imaginary
547 part @code{0} are automatically converted to real numbers.
550 The class @code{cl_R} comprises real numbers of different kinds. It is an
554 @cindex rational number
556 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
557 integers. There is no special class for non-integral rational numbers
558 since rational numbers with denominator @code{1} are automatically converted
562 The class @code{cl_F} implements floating-point approximations to real numbers.
563 It is an abstract class.
566 @section Exact numbers
569 Some numbers are represented as exact numbers: there is no loss of information
570 when such a number is converted from its mathematical value to its internal
571 representation. On exact numbers, the elementary operations (@code{+},
572 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
575 In CLN, the exact numbers are:
579 rational numbers (including integers),
581 complex numbers whose real and imaginary parts are both rational numbers.
584 Rational numbers are always normalized to the form
585 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
586 are coprime integers and the denominator is positive. If the resulting
587 denominator is @code{1}, the rational number is converted to an integer.
589 @cindex immediate numbers
590 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
591 for 32-bit machines) are especially efficient, because they consume no heap
592 allocation. Otherwise the distinction between these immediate integers
593 (called ``fixnums'') and heap allocated integers (called ``bignums'')
594 is completely transparent.
597 @section Floating-point numbers
598 @cindex floating-point number
600 Not all real numbers can be represented exactly. (There is an easy mathematical
601 proof for this: Only a countable set of numbers can be stored exactly in
602 a computer, even if one assumes that it has unlimited storage. But there
603 are uncountably many real numbers.) So some approximation is needed.
604 CLN implements ordinary floating-point numbers, with mantissa and exponent.
606 @cindex rounding error
607 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
608 only return approximate results. For example, the value of the expression
609 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
610 @samp{0.7}. Rounding errors like this one are inevitable when computing
611 with floating-point numbers.
613 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
614 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
615 rule: It first computes the exact mathematical result and then returns the
616 floating-point number which is nearest to this. If two floating-point numbers
617 are equally distant from the ideal result, the one with a @code{0} in its least
618 significant mantissa bit is chosen.
620 Similarly, testing floating point numbers for equality @samp{x == y}
621 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
622 for some well-chosen @code{epsilon}.
624 Floating point numbers come in four flavors:
629 Short floats, type @code{cl_SF}.
630 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
631 and 17 mantissa bits (including the ``hidden'' bit).
632 They don't consume heap allocation.
636 Single floats, type @code{cl_FF}.
637 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
638 and 24 mantissa bits (including the ``hidden'' bit).
639 In CLN, they are represented as IEEE single-precision floating point numbers.
640 This corresponds closely to the C/C++ type @samp{float}.
644 Double floats, type @code{cl_DF}.
645 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
646 and 53 mantissa bits (including the ``hidden'' bit).
647 In CLN, they are represented as IEEE double-precision floating point numbers.
648 This corresponds closely to the C/C++ type @samp{double}.
652 Long floats, type @code{cl_LF}.
653 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
654 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
655 The precision of a long float is unlimited, but once created, a long float
656 has a fixed precision. (No ``lazy recomputation''.)
659 Of course, computations with long floats are more expensive than those
660 with smaller floating-point formats.
662 CLN does not implement features like NaNs, denormalized numbers and
663 gradual underflow. If the exponent range of some floating-point type
664 is too limited for your application, choose another floating-point type
665 with larger exponent range.
668 As a user of CLN, you can forget about the differences between the
669 four floating-point types and just declare all your floating-point
670 variables as being of type @code{cl_F}. This has the advantage that
671 when you change the precision of some computation (say, from @code{cl_DF}
672 to @code{cl_LF}), you don't have to change the code, only the precision
673 of the initial values. Also, many transcendental functions have been
674 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
675 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
676 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
677 the floating point contagion rule happened to change in the future.)
680 @section Complex numbers
681 @cindex complex number
683 Complex numbers, as implemented by the class @code{cl_N}, have a real
684 part and an imaginary part, both real numbers. A complex number whose
685 imaginary part is the exact number @code{0} is automatically converted
688 Complex numbers can arise from real numbers alone, for example
689 through application of @code{sqrt} or transcendental functions.
695 Conversions from any class to any its superclasses (``base classes'' in
696 C++ terminology) is done automatically.
698 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
699 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
700 @code{cl_N} and @code{cl_number}.
702 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
703 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
704 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
705 efficiency. Their range is therefore limited:
709 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
711 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
714 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
715 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
716 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
718 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
719 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
720 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
721 @samp{cl_I}, first convert it to an @samp{unsigned long}.
723 Conversions from the C built-in type @samp{float} are provided for the classes
724 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
726 Conversions from the C built-in type @samp{double} are provided for the classes
727 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
729 Conversions from @samp{const char *} are provided for the classes
730 @code{cl_I}, @code{cl_RA},
731 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
732 @code{cl_R}, @code{cl_N}.
733 The easiest way to specify a value which is outside of the range of the
734 C++ built-in types is therefore to specify it as a string, like this:
737 cl_I order_of_rubiks_cube_group = "43252003274489856000";
739 Note that this conversion is done at runtime, not at compile-time.
741 Conversions from @code{cl_I} to the C built-in types @samp{int},
742 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
746 @item int cl_I_to_int (const cl_I& x)
747 @cindex @code{cl_I_to_int ()}
748 @itemx unsigned int cl_I_to_uint (const cl_I& x)
749 @cindex @code{cl_I_to_uint ()}
750 @itemx long cl_I_to_long (const cl_I& x)
751 @cindex @code{cl_I_to_long ()}
752 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
753 @cindex @code{cl_I_to_ulong ()}
754 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
755 representable in the range of @var{ctype}, a runtime error occurs.
758 Conversions from the classes @code{cl_I}, @code{cl_RA},
759 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
761 to the C built-in types @samp{float} and @samp{double} are provided through
765 @item float float_approx (const @var{type}& x)
766 @cindex @code{float_approx ()}
767 @itemx double double_approx (const @var{type}& x)
768 @cindex @code{double_approx ()}
769 Returns an approximation of @code{x} of C type @var{ctype}.
770 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
771 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
774 Conversions from any class to any of its subclasses (``derived classes'' in
775 C++ terminology) are not provided. Instead, you can assert and check
776 that a value belongs to a certain subclass, and return it as element of that
777 class, using the @samp{As} and @samp{The} macros.
778 @cindex @code{As()()}
779 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
780 @var{type} and returns it as such.
781 @cindex @code{The()()}
782 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
783 @var{type} and returns it as such. It is your responsibility to ensure
784 that this assumption is valid. Since macros and namespaces don't go
785 together well, there is an equivalent to @samp{The}: the template
793 if (!(x >= 0)) abort();
794 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
795 // In general, it would be a rational number.
796 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
801 @chapter Functions on numbers
803 Each of the number classes declares its mathematical operations in the
804 corresponding include file. For example, if your code operates with
805 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
808 @section Constructing numbers
810 Here is how to create number objects ``from nothing''.
813 @subsection Constructing integers
815 @code{cl_I} objects are most easily constructed from C integers and from
816 strings. See @ref{Conversions}.
819 @subsection Constructing rational numbers
821 @code{cl_RA} objects can be constructed from strings. The syntax
822 for rational numbers is described in @ref{Internal and printed representation}.
823 Another standard way to produce a rational number is through application
824 of @samp{operator /} or @samp{recip} on integers.
827 @subsection Constructing floating-point numbers
829 @code{cl_F} objects with low precision are most easily constructed from
830 C @samp{float} and @samp{double}. See @ref{Conversions}.
832 To construct a @code{cl_F} with high precision, you can use the conversion
833 from @samp{const char *}, but you have to specify the desired precision
834 within the string. (See @ref{Internal and printed representation}.)
837 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
839 will set @samp{e} to the given value, with a precision of 40 decimal digits.
841 The programmatic way to construct a @code{cl_F} with high precision is
842 through the @code{cl_float} conversion function, see
843 @ref{Conversion to floating-point numbers}. For example, to compute
844 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
845 and then apply the exponential function:
847 float_format_t precision = float_format(40);
848 cl_F e = exp(cl_float(1,precision));
852 @subsection Constructing complex numbers
854 Non-real @code{cl_N} objects are normally constructed through the function
856 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
858 See @ref{Elementary complex functions}.
861 @section Elementary functions
863 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
864 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
865 defines the following operations:
868 @item @var{type} operator + (const @var{type}&, const @var{type}&)
869 @cindex @code{operator + ()}
872 @item @var{type} operator - (const @var{type}&, const @var{type}&)
873 @cindex @code{operator - ()}
876 @item @var{type} operator - (const @var{type}&)
877 Returns the negative of the argument.
879 @item @var{type} plus1 (const @var{type}& x)
880 @cindex @code{plus1 ()}
881 Returns @code{x + 1}.
883 @item @var{type} minus1 (const @var{type}& x)
884 @cindex @code{minus1 ()}
885 Returns @code{x - 1}.
887 @item @var{type} operator * (const @var{type}&, const @var{type}&)
888 @cindex @code{operator * ()}
891 @item @var{type} square (const @var{type}& x)
892 @cindex @code{square ()}
893 Returns @code{x * x}.
896 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
897 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
898 defines the following operations:
901 @item @var{type} operator / (const @var{type}&, const @var{type}&)
902 @cindex @code{operator / ()}
905 @item @var{type} recip (const @var{type}&)
906 @cindex @code{recip ()}
907 Returns the reciprocal of the argument.
910 The class @code{cl_I} doesn't define a @samp{/} operation because
911 in the C/C++ language this operator, applied to integral types,
912 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
913 is implementation dependent). (@xref{Rounding functions}.)
914 Instead, @code{cl_I} defines an ``exact quotient'' function:
917 @item cl_I exquo (const cl_I& x, const cl_I& y)
918 @cindex @code{exquo ()}
919 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
922 The following exponentiation functions are defined:
925 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
926 @cindex @code{expt_pos ()}
927 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
928 @code{y} must be > 0. Returns @code{x^y}.
930 @item cl_RA expt (const cl_RA& x, const cl_I& y)
931 @cindex @code{expt ()}
932 @itemx cl_R expt (const cl_R& x, const cl_I& y)
933 @itemx cl_N expt (const cl_N& x, const cl_I& y)
937 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
938 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
939 defines the following operation:
942 @item @var{type} abs (const @var{type}& x)
943 @cindex @code{abs ()}
944 Returns the absolute value of @code{x}.
945 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
948 The class @code{cl_N} implements this as follows:
951 @item cl_R abs (const cl_N x)
952 Returns the absolute value of @code{x}.
955 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
956 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
957 defines the following operation:
960 @item @var{type} signum (const @var{type}& x)
961 @cindex @code{signum ()}
962 Returns the sign of @code{x}, in the same number format as @code{x}.
963 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
964 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
969 @section Elementary rational functions
971 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
974 @item cl_I numerator (const @var{type}& x)
975 @cindex @code{numerator ()}
976 Returns the numerator of @code{x}.
978 @item cl_I denominator (const @var{type}& x)
979 @cindex @code{denominator ()}
980 Returns the denominator of @code{x}.
983 The numerator and denominator of a rational number are normalized in such
984 a way that they have no factor in common and the denominator is positive.
987 @section Elementary complex functions
989 The class @code{cl_N} defines the following operation:
992 @item cl_N complex (const cl_R& a, const cl_R& b)
993 @cindex @code{complex ()}
994 Returns the complex number @code{a+bi}, that is, the complex number with
995 real part @code{a} and imaginary part @code{b}.
998 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
1001 @item cl_R realpart (const @var{type}& x)
1002 @cindex @code{realpart ()}
1003 Returns the real part of @code{x}.
1005 @item cl_R imagpart (const @var{type}& x)
1006 @cindex @code{imagpart ()}
1007 Returns the imaginary part of @code{x}.
1009 @item @var{type} conjugate (const @var{type}& x)
1010 @cindex @code{conjugate ()}
1011 Returns the complex conjugate of @code{x}.
1014 We have the relations
1018 @code{x = complex(realpart(x), imagpart(x))}
1020 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
1024 @section Comparisons
1027 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1028 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1029 defines the following operations:
1032 @item bool operator == (const @var{type}&, const @var{type}&)
1033 @cindex @code{operator == ()}
1034 @itemx bool operator != (const @var{type}&, const @var{type}&)
1035 @cindex @code{operator != ()}
1036 Comparison, as in C and C++.
1038 @item uint32 equal_hashcode (const @var{type}&)
1039 @cindex @code{equal_hashcode ()}
1040 Returns a 32-bit hash code that is the same for any two numbers which are
1041 the same according to @code{==}. This hash code depends on the number's value,
1042 not its type or precision.
1044 @item cl_boolean zerop (const @var{type}& x)
1045 @cindex @code{zerop ()}
1046 Compare against zero: @code{x == 0}
1049 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1050 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1051 defines the following operations:
1054 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1055 @cindex @code{compare ()}
1056 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1057 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1059 @item bool operator <= (const @var{type}&, const @var{type}&)
1060 @cindex @code{operator <= ()}
1061 @itemx bool operator < (const @var{type}&, const @var{type}&)
1062 @cindex @code{operator < ()}
1063 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1064 @cindex @code{operator >= ()}
1065 @itemx bool operator > (const @var{type}&, const @var{type}&)
1066 @cindex @code{operator > ()}
1067 Comparison, as in C and C++.
1069 @item cl_boolean minusp (const @var{type}& x)
1070 @cindex @code{minusp ()}
1071 Compare against zero: @code{x < 0}
1073 @item cl_boolean plusp (const @var{type}& x)
1074 @cindex @code{plusp ()}
1075 Compare against zero: @code{x > 0}
1077 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1078 @cindex @code{max ()}
1079 Return the maximum of @code{x} and @code{y}.
1081 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1082 @cindex @code{min ()}
1083 Return the minimum of @code{x} and @code{y}.
1086 When a floating point number and a rational number are compared, the float
1087 is first converted to a rational number using the function @code{rational}.
1088 Since a floating point number actually represents an interval of real numbers,
1089 the result might be surprising.
1090 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1091 there is no floating point number whose value is exactly @code{1/3}.
1094 @section Rounding functions
1097 When a real number is to be converted to an integer, there is no ``best''
1098 rounding. The desired rounding function depends on the application.
1099 The Common Lisp and ISO Lisp standards offer four rounding functions:
1103 This is the largest integer <=@code{x}.
1106 This is the smallest integer >=@code{x}.
1109 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1112 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1113 integers, choose the even one.
1116 These functions have different advantages:
1118 @code{floor} and @code{ceiling} are translation invariant:
1119 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1120 for every @code{x} and every integer @code{n}.
1122 On the other hand, @code{truncate} and @code{round} are symmetric:
1123 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1124 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1125 down exactly as often as it rounds up.
1127 The functions are related like this:
1131 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1132 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1134 @code{truncate(x) = sign(x) * floor(abs(x))}
1137 Each of the classes @code{cl_R}, @code{cl_RA},
1138 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1139 defines the following operations:
1142 @item cl_I floor1 (const @var{type}& x)
1143 @cindex @code{floor1 ()}
1144 Returns @code{floor(x)}.
1145 @item cl_I ceiling1 (const @var{type}& x)
1146 @cindex @code{ceiling1 ()}
1147 Returns @code{ceiling(x)}.
1148 @item cl_I truncate1 (const @var{type}& x)
1149 @cindex @code{truncate1 ()}
1150 Returns @code{truncate(x)}.
1151 @item cl_I round1 (const @var{type}& x)
1152 @cindex @code{round1 ()}
1153 Returns @code{round(x)}.
1156 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1157 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1158 defines the following operations:
1161 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1162 Returns @code{floor(x/y)}.
1163 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1164 Returns @code{ceiling(x/y)}.
1165 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1166 Returns @code{truncate(x/y)}.
1167 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1168 Returns @code{round(x/y)}.
1171 These functions are called @samp{floor1}, @dots{} here instead of
1172 @samp{floor}, @dots{}, because on some systems, system dependent include
1173 files define @samp{floor} and @samp{ceiling} as macros.
1175 In many cases, one needs both the quotient and the remainder of a division.
1176 It is more efficient to compute both at the same time than to perform
1177 two divisions, one for quotient and the next one for the remainder.
1178 The following functions therefore return a structure containing both
1179 the quotient and the remainder. The suffix @samp{2} indicates the number
1180 of ``return values''. The remainder is defined as follows:
1184 for the computation of @code{quotient = floor(x)},
1185 @code{remainder = x - quotient},
1187 for the computation of @code{quotient = floor(x,y)},
1188 @code{remainder = x - quotient*y},
1191 and similarly for the other three operations.
1193 Each of the classes @code{cl_R}, @code{cl_RA},
1194 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1195 defines the following operations:
1198 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1199 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1200 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1201 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1202 @itemx @var{type}_div_t round2 (const @var{type}& x)
1205 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1206 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1207 defines the following operations:
1210 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1211 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1212 @cindex @code{floor2 ()}
1213 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1214 @cindex @code{ceiling2 ()}
1215 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1216 @cindex @code{truncate2 ()}
1217 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1218 @cindex @code{round2 ()}
1221 Sometimes, one wants the quotient as a floating-point number (of the
1222 same format as the argument, if the argument is a float) instead of as
1223 an integer. The prefix @samp{f} indicates this.
1226 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1227 defines the following operations:
1230 @item @var{type} ffloor (const @var{type}& x)
1231 @cindex @code{ffloor ()}
1232 @itemx @var{type} fceiling (const @var{type}& x)
1233 @cindex @code{fceiling ()}
1234 @itemx @var{type} ftruncate (const @var{type}& x)
1235 @cindex @code{ftruncate ()}
1236 @itemx @var{type} fround (const @var{type}& x)
1237 @cindex @code{fround ()}
1240 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1242 The class @code{cl_R} defines the following operations:
1245 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1246 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1247 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1248 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1251 These functions also exist in versions which return both the quotient
1252 and the remainder. The suffix @samp{2} indicates this.
1255 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1256 defines the following operations:
1257 @cindex @code{cl_F_fdiv_t}
1258 @cindex @code{cl_SF_fdiv_t}
1259 @cindex @code{cl_FF_fdiv_t}
1260 @cindex @code{cl_DF_fdiv_t}
1261 @cindex @code{cl_LF_fdiv_t}
1264 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1265 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1266 @cindex @code{ffloor2 ()}
1267 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1268 @cindex @code{fceiling2 ()}
1269 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1270 @cindex @code{ftruncate2 ()}
1271 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1272 @cindex @code{fround2 ()}
1274 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1275 @cindex @code{cl_R_fdiv_t}
1277 The class @code{cl_R} defines the following operations:
1280 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1281 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1282 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1283 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1284 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1287 Other applications need only the remainder of a division.
1288 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1289 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1290 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1294 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1296 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1299 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1300 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1301 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1303 The classes @code{cl_R}, @code{cl_I} define the following operations:
1306 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1307 @cindex @code{mod ()}
1308 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1309 @cindex @code{rem ()}
1315 Each of the classes @code{cl_R},
1316 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1317 defines the following operation:
1320 @item @var{type} sqrt (const @var{type}& x)
1321 @cindex @code{sqrt ()}
1322 @code{x} must be >= 0. This function returns the square root of @code{x},
1323 normalized to be >= 0. If @code{x} is the square of a rational number,
1324 @code{sqrt(x)} will be a rational number, else it will return a
1325 floating-point approximation.
1328 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1331 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1332 @cindex @code{sqrtp ()}
1333 This tests whether @code{x} is a perfect square. If so, it returns true
1334 and the exact square root in @code{*root}, else it returns false.
1337 Furthermore, for integers, similarly:
1340 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1341 @cindex @code{isqrt ()}
1342 @code{x} should be >= 0. This function sets @code{*root} to
1343 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1344 the boolean value @code{(expt(*root,2) == x)}.
1347 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1348 define the following operation:
1351 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1352 @cindex @code{rootp ()}
1353 @code{x} must be >= 0. @code{n} must be > 0.
1354 This tests whether @code{x} is an @code{n}th power of a rational number.
1355 If so, it returns true and the exact root in @code{*root}, else it returns
1359 The only square root function which accepts negative numbers is the one
1360 for class @code{cl_N}:
1363 @item cl_N sqrt (const cl_N& z)
1364 @cindex @code{sqrt ()}
1365 Returns the square root of @code{z}, as defined by the formula
1366 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1367 or to a complex number are done if necessary. The range of the result is the
1368 right half plane @code{realpart(sqrt(z)) >= 0}
1369 including the positive imaginary axis and 0, but excluding
1370 the negative imaginary axis.
1371 The result is an exact number only if @code{z} is an exact number.
1375 @section Transcendental functions
1376 @cindex transcendental functions
1378 The transcendental functions return an exact result if the argument
1379 is exact and the result is exact as well. Otherwise they must return
1380 inexact numbers even if the argument is exact.
1381 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1384 @subsection Exponential and logarithmic functions
1387 @item cl_R exp (const cl_R& x)
1388 @cindex @code{exp ()}
1389 @itemx cl_N exp (const cl_N& x)
1390 Returns the exponential function of @code{x}. This is @code{e^x} where
1391 @code{e} is the base of the natural logarithms. The range of the result
1392 is the entire complex plane excluding 0.
1394 @item cl_R ln (const cl_R& x)
1395 @cindex @code{ln ()}
1396 @code{x} must be > 0. Returns the (natural) logarithm of x.
1398 @item cl_N log (const cl_N& x)
1399 @cindex @code{log ()}
1400 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1401 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1402 The range of the result is the strip in the complex plane
1403 @code{-pi < imagpart(log(x)) <= pi}.
1405 @item cl_R phase (const cl_N& x)
1406 @cindex @code{phase ()}
1407 Returns the angle part of @code{x} in its polar representation as a
1408 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1409 This is also the imaginary part of @code{log(x)}.
1410 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1411 The result will be an exact number only if @code{zerop(x)} or
1412 if @code{x} is real and positive.
1414 @item cl_R log (const cl_R& a, const cl_R& b)
1415 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1416 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1417 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1420 @item cl_N log (const cl_N& a, const cl_N& b)
1421 Returns the logarithm of @code{a} with respect to base @code{b}.
1422 @code{log(a,b) = log(a)/log(b)}.
1424 @item cl_N expt (const cl_N& x, const cl_N& y)
1425 @cindex @code{expt ()}
1426 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1429 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1432 @item cl_F exp1 (float_format_t f)
1433 @cindex @code{exp1 ()}
1434 Returns e as a float of format @code{f}.
1436 @item cl_F exp1 (const cl_F& y)
1437 Returns e in the float format of @code{y}.
1439 @item cl_F exp1 (void)
1440 Returns e as a float of format @code{default_float_format}.
1444 @subsection Trigonometric functions
1447 @item cl_R sin (const cl_R& x)
1448 @cindex @code{sin ()}
1449 Returns @code{sin(x)}. The range of the result is the interval
1450 @code{-1 <= sin(x) <= 1}.
1452 @item cl_N sin (const cl_N& z)
1453 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1455 @item cl_R cos (const cl_R& x)
1456 @cindex @code{cos ()}
1457 Returns @code{cos(x)}. The range of the result is the interval
1458 @code{-1 <= cos(x) <= 1}.
1460 @item cl_N cos (const cl_N& x)
1461 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1463 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1464 @cindex @code{cos_sin_t}
1465 @itemx cos_sin_t cos_sin (const cl_R& x)
1466 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1467 @cindex @code{cos_sin ()}
1468 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1469 hold only approximately.
1471 @item cl_R tan (const cl_R& x)
1472 @cindex @code{tan ()}
1473 @itemx cl_N tan (const cl_N& x)
1474 Returns @code{tan(x) = sin(x)/cos(x)}.
1476 @item cl_N cis (const cl_R& x)
1477 @cindex @code{cis ()}
1478 @itemx cl_N cis (const cl_N& x)
1479 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1480 @code{e^(i*x) = cos(x) + i*sin(x)}.
1483 @cindex @code{asin ()}
1484 @item cl_N asin (const cl_N& z)
1485 Returns @code{arcsin(z)}. This is defined as
1486 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1487 @code{arcsin(-z) = -arcsin(z)}.
1488 The range of the result is the strip in the complex domain
1489 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1490 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1491 with @code{realpart = pi/2} and @code{imagpart > 0}.
1493 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1497 @item cl_N acos (const cl_N& z)
1498 @cindex @code{acos ()}
1499 Returns @code{arccos(z)}. This is defined as
1500 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1503 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1505 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1506 The range of the result is the strip in the complex domain
1507 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1508 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1509 with @code{realpart = pi} and @code{imagpart > 0}.
1511 Proof: This follows from the results about arcsin.
1515 @cindex @code{atan ()}
1516 @item cl_R atan (const cl_R& x, const cl_R& y)
1517 Returns the angle of the polar representation of the complex number
1518 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1519 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1520 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1521 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1522 with reversed order of arguments.
1524 @item cl_R atan (const cl_R& x)
1525 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1526 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1527 will be an exact number only if @code{x} is the exact @code{0}.
1529 @item cl_N atan (const cl_N& z)
1530 Returns @code{arctan(z)}. This is defined as
1531 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1532 @code{arctan(-z) = -arctan(z)}. The range of the result is
1533 the strip in the complex domain
1534 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1535 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1536 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1538 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1544 @cindex Archimedes' constant
1545 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1548 @item cl_F pi (float_format_t f)
1549 @cindex @code{pi ()}
1550 Returns pi as a float of format @code{f}.
1552 @item cl_F pi (const cl_F& y)
1553 Returns pi in the float format of @code{y}.
1555 @item cl_F pi (void)
1556 Returns pi as a float of format @code{default_float_format}.
1560 @subsection Hyperbolic functions
1563 @item cl_R sinh (const cl_R& x)
1564 @cindex @code{sinh ()}
1565 Returns @code{sinh(x)}.
1567 @item cl_N sinh (const cl_N& z)
1568 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1570 @item cl_R cosh (const cl_R& x)
1571 @cindex @code{cosh ()}
1572 Returns @code{cosh(x)}. The range of the result is the interval
1573 @code{cosh(x) >= 1}.
1575 @item cl_N cosh (const cl_N& z)
1576 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1578 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1579 @cindex @code{cosh_sinh_t}
1580 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1581 @cindex @code{cosh_sinh ()}
1582 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1583 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1584 hold only approximately.
1586 @item cl_R tanh (const cl_R& x)
1587 @cindex @code{tanh ()}
1588 @itemx cl_N tanh (const cl_N& x)
1589 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1591 @item cl_N asinh (const cl_N& z)
1592 @cindex @code{asinh ()}
1593 Returns @code{arsinh(z)}. This is defined as
1594 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1595 @code{arsinh(-z) = -arsinh(z)}.
1597 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1598 Actually, z+sqrt(1+z^2) can never be real and <0, so
1599 -pi < imagpart(arsinh(z)) < pi.
1600 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1601 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1603 The range of the result is the strip in the complex domain
1604 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1605 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1606 with @code{imagpart = pi/2} and @code{realpart < 0}.
1608 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1609 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1610 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1611 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1612 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1613 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1614 If y >= 1, the imagpart is pi/2 and the realpart is
1615 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1618 Moreover, if z is in Range(sqrt),
1619 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1620 (for a proof, see file src/cl_C_asinh.cc).
1623 @item cl_N acosh (const cl_N& z)
1624 @cindex @code{acosh ()}
1625 Returns @code{arcosh(z)}. This is defined as
1626 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1627 The range of the result is the half-strip in the complex domain
1628 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1629 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1631 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1632 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1633 If z is in Range(sqrt), we have
1634 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1635 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1637 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1638 and since the imagpart of both expressions is > -pi, <= pi
1639 ==> arcosh(z) = log(z+sqrt(z^2-1))
1640 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1641 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1642 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1643 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1644 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1646 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1647 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1648 = x^2 + y^2 + sqrt(u^2+v^2)
1653 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1654 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1655 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1656 Otherwise, -z is in Range(sqrt).
1657 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1658 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1659 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1660 and this has realpart > 0.
1661 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1662 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1663 has realpart = 0 and imagpart > 0.
1664 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1665 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1666 This has realpart >= 0 and imagpart = pi.
1669 @item cl_N atanh (const cl_N& z)
1670 @cindex @code{atanh ()}
1671 Returns @code{artanh(z)}. This is defined as
1672 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1673 @code{artanh(-z) = -artanh(z)}. The range of the result is
1674 the strip in the complex domain
1675 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1676 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1677 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1679 Proof: Write z = x+iy. Examine
1680 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1682 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1683 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1684 |x| < 1 ==> imagpart = 0
1687 = (atan(1+x,y) - atan(1-x,-y))/2
1688 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1689 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1690 > (pi - pi/2 - pi/2 )/2 = 0
1691 and (1+x)/y > (1-x)/y
1692 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1693 ==> imagpart < pi/2.
1694 Hence 0 < imagpart < pi/2.
1696 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1701 @subsection Euler gamma
1702 @cindex Euler's constant
1704 Euler's constant C = 0.577@dots{} is returned by the following functions:
1707 @item cl_F eulerconst (float_format_t f)
1708 @cindex @code{eulerconst ()}
1709 Returns Euler's constant as a float of format @code{f}.
1711 @item cl_F eulerconst (const cl_F& y)
1712 Returns Euler's constant in the float format of @code{y}.
1714 @item cl_F eulerconst (void)
1715 Returns Euler's constant as a float of format @code{default_float_format}.
1718 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1719 @cindex Catalan's constant
1722 @item cl_F catalanconst (float_format_t f)
1723 @cindex @code{catalanconst ()}
1724 Returns Catalan's constant as a float of format @code{f}.
1726 @item cl_F catalanconst (const cl_F& y)
1727 Returns Catalan's constant in the float format of @code{y}.
1729 @item cl_F catalanconst (void)
1730 Returns Catalan's constant as a float of format @code{default_float_format}.
1734 @subsection Riemann zeta
1735 @cindex Riemann's zeta
1737 Riemann's zeta function at an integral point @code{s>1} is returned by the
1738 following functions:
1741 @item cl_F zeta (int s, float_format_t f)
1742 @cindex @code{zeta ()}
1743 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1745 @item cl_F zeta (int s, const cl_F& y)
1746 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1748 @item cl_F zeta (int s)
1749 Returns Riemann's zeta function at @code{s} as a float of format
1750 @code{default_float_format}.
1754 @section Functions on integers
1756 @subsection Logical functions
1758 Integers, when viewed as in two's complement notation, can be thought as
1759 infinite bit strings where the bits' values eventually are constant.
1766 The logical operations view integers as such bit strings and operate
1767 on each of the bit positions in parallel.
1770 @item cl_I lognot (const cl_I& x)
1771 @cindex @code{lognot ()}
1772 @itemx cl_I operator ~ (const cl_I& x)
1773 @cindex @code{operator ~ ()}
1774 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1776 @item cl_I logand (const cl_I& x, const cl_I& y)
1777 @cindex @code{logand ()}
1778 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1779 @cindex @code{operator & ()}
1780 Logical and, like @code{x & y} in C.
1782 @item cl_I logior (const cl_I& x, const cl_I& y)
1783 @cindex @code{logior ()}
1784 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1785 @cindex @code{operator | ()}
1786 Logical (inclusive) or, like @code{x | y} in C.
1788 @item cl_I logxor (const cl_I& x, const cl_I& y)
1789 @cindex @code{logxor ()}
1790 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1791 @cindex @code{operator ^ ()}
1792 Exclusive or, like @code{x ^ y} in C.
1794 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1795 @cindex @code{logeqv ()}
1796 Bitwise equivalence, like @code{~(x ^ y)} in C.
1798 @item cl_I lognand (const cl_I& x, const cl_I& y)
1799 @cindex @code{lognand ()}
1800 Bitwise not and, like @code{~(x & y)} in C.
1802 @item cl_I lognor (const cl_I& x, const cl_I& y)
1803 @cindex @code{lognor ()}
1804 Bitwise not or, like @code{~(x | y)} in C.
1806 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1807 @cindex @code{logandc1 ()}
1808 Logical and, complementing the first argument, like @code{~x & y} in C.
1810 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1811 @cindex @code{logandc2 ()}
1812 Logical and, complementing the second argument, like @code{x & ~y} in C.
1814 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1815 @cindex @code{logorc1 ()}
1816 Logical or, complementing the first argument, like @code{~x | y} in C.
1818 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1819 @cindex @code{logorc2 ()}
1820 Logical or, complementing the second argument, like @code{x | ~y} in C.
1823 These operations are all available though the function
1825 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1826 @cindex @code{boole ()}
1828 where @code{op} must have one of the 16 values (each one stands for a function
1829 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1830 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1831 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1832 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1833 @code{boole_orc1}, @code{boole_orc2}.
1834 @cindex @code{boole_clr}
1835 @cindex @code{boole_set}
1836 @cindex @code{boole_1}
1837 @cindex @code{boole_2}
1838 @cindex @code{boole_c1}
1839 @cindex @code{boole_c2}
1840 @cindex @code{boole_and}
1841 @cindex @code{boole_xor}
1842 @cindex @code{boole_eqv}
1843 @cindex @code{boole_nand}
1844 @cindex @code{boole_nor}
1845 @cindex @code{boole_andc1}
1846 @cindex @code{boole_andc2}
1847 @cindex @code{boole_orc1}
1848 @cindex @code{boole_orc2}
1851 Other functions that view integers as bit strings:
1854 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1855 @cindex @code{logtest ()}
1856 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1857 @code{logand(x,y) != 0}.
1859 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1860 @cindex @code{logbitp ()}
1861 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1862 Bit 0 is the least significant bit.
1864 @item uintL logcount (const cl_I& x)
1865 @cindex @code{logcount ()}
1866 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1867 the number of zero bits in @code{x}, if @code{x} < 0.
1870 The following functions operate on intervals of bits in integers.
1873 struct cl_byte @{ uintL size; uintL position; @};
1875 @cindex @code{cl_byte}
1876 represents the bit interval containing the bits
1877 @code{position}@dots{}@code{position+size-1} of an integer.
1878 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1881 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1882 @cindex @code{ldb ()}
1883 extracts the bits of @code{n} described by the bit interval @code{b}
1884 and returns them as a nonnegative integer with @code{b.size} bits.
1886 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1887 @cindex @code{ldb_test ()}
1888 Returns true if some bit described by the bit interval @code{b} is set in
1891 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1892 @cindex @code{dpb ()}
1893 Returns @code{n}, with the bits described by the bit interval @code{b}
1894 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1895 @code{newbyte} are relevant.
1898 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1899 functions are their counterparts without shifting:
1902 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1903 @cindex @code{mask_field ()}
1904 returns an integer with the bits described by the bit interval @code{b}
1905 copied from the corresponding bits in @code{n}, the other bits zero.
1907 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1908 @cindex @code{deposit_field ()}
1909 returns an integer where the bits described by the bit interval @code{b}
1910 come from @code{newbyte} and the other bits come from @code{n}.
1913 The following relations hold:
1917 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1919 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1921 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1924 The following operations on integers as bit strings are efficient shortcuts
1925 for common arithmetic operations:
1928 @item cl_boolean oddp (const cl_I& x)
1929 @cindex @code{oddp ()}
1930 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1931 @code{mod(x,2) != 0}.
1933 @item cl_boolean evenp (const cl_I& x)
1934 @cindex @code{evenp ()}
1935 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1936 @code{mod(x,2) == 0}.
1938 @item cl_I operator << (const cl_I& x, const cl_I& n)
1939 @cindex @code{operator << ()}
1940 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1941 Equivalent to @code{x * expt(2,n)}.
1943 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1944 @cindex @code{operator >> ()}
1945 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1946 Bits shifted out to the right are thrown away.
1947 Equivalent to @code{floor(x / expt(2,n))}.
1949 @item cl_I ash (const cl_I& x, const cl_I& y)
1950 @cindex @code{ash ()}
1951 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1952 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1953 returns @code{floor(x * expt(2,y))}.
1955 @item uintL integer_length (const cl_I& x)
1956 @cindex @code{integer_length ()}
1957 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1958 in two's complement notation. This is the smallest n >= 0 such that
1959 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1962 @item uintL ord2 (const cl_I& x)
1963 @cindex @code{ord2 ()}
1964 @code{x} must be non-zero. This function returns the number of 0 bits at the
1965 right of @code{x} in two's complement notation. This is the largest n >= 0
1966 such that 2^n divides @code{x}.
1968 @item uintL power2p (const cl_I& x)
1969 @cindex @code{power2p ()}
1970 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1971 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1972 (See also the function @code{logp}.)
1976 @subsection Number theoretic functions
1979 @item uint32 gcd (uint32 a, uint32 b)
1980 @cindex @code{gcd ()}
1981 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1982 This function returns the greatest common divisor of @code{a} and @code{b},
1983 normalized to be >= 0.
1985 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1986 @cindex @code{xgcd ()}
1987 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1988 @code{a} and @code{b} and at the same time the representation of @code{g}
1989 as an integral linear combination of @code{a} and @code{b}:
1990 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1991 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1992 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1993 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1994 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1996 @item cl_I lcm (const cl_I& a, const cl_I& b)
1997 @cindex @code{lcm ()}
1998 This function returns the least common multiple of @code{a} and @code{b},
1999 normalized to be >= 0.
2001 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
2002 @cindex @code{logp ()}
2003 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
2004 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
2005 rational number, this function returns true and sets *l = log(a,b), else
2008 @item int jacobi (sint32 a, sint32 b)
2009 @cindex @code{jacobi()}
2010 @itemx int jacobi (const cl_I& a, const cl_I& b)
2011 Returns the Jacobi symbol
2013 $\left({a\over b}\right)$,
2018 @code{a,b} must be integers, @code{b>0} and odd. The result is 0
2021 @item cl_boolean isprobprime (const cl_I& n)
2023 @cindex @code{isprobprime()}
2024 Returns true if @code{n} is a small prime or passes the Miller-Rabin
2025 primality test. The probability of a false positive is 1:10^30.
2027 @item cl_I nextprobprime (const cl_R& x)
2028 @cindex @code{nextprobprime()}
2029 Returns the smallest probable prime >=@code{x}.
2033 @subsection Combinatorial functions
2036 @item cl_I factorial (uintL n)
2037 @cindex @code{factorial ()}
2038 @code{n} must be a small integer >= 0. This function returns the factorial
2039 @code{n}! = @code{1*2*@dots{}*n}.
2041 @item cl_I doublefactorial (uintL n)
2042 @cindex @code{doublefactorial ()}
2043 @code{n} must be a small integer >= 0. This function returns the
2044 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
2045 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
2047 @item cl_I binomial (uintL n, uintL k)
2048 @cindex @code{binomial ()}
2049 @code{n} and @code{k} must be small integers >= 0. This function returns the
2050 binomial coefficient
2052 ${n \choose k} = {n! \over n! (n-k)!}$
2055 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2057 for 0 <= k <= n, 0 else.
2061 @section Functions on floating-point numbers
2063 Recall that a floating-point number consists of a sign @code{s}, an
2064 exponent @code{e} and a mantissa @code{m}. The value of the number is
2065 @code{(-1)^s * 2^e * m}.
2068 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2069 defines the following operations.
2072 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2073 @cindex @code{scale_float ()}
2074 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2075 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2076 because it copies @code{x} and modifies the exponent.
2079 The following functions provide an abstract interface to the underlying
2080 representation of floating-point numbers.
2083 @item sintL float_exponent (const @var{type}& x)
2084 @cindex @code{float_exponent ()}
2085 Returns the exponent @code{e} of @code{x}.
2086 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2087 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2089 @item sintL float_radix (const @var{type}& x)
2090 @cindex @code{float_radix ()}
2091 Returns the base of the floating-point representation. This is always @code{2}.
2093 @item @var{type} float_sign (const @var{type}& x)
2094 @cindex @code{float_sign ()}
2095 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2096 @code{x} >= 0, -1 for @code{x} < 0.
2098 @item uintL float_digits (const @var{type}& x)
2099 @cindex @code{float_digits ()}
2100 Returns the number of mantissa bits in the floating-point representation
2101 of @code{x}, including the hidden bit. The value only depends on the type
2102 of @code{x}, not on its value.
2104 @item uintL float_precision (const @var{type}& x)
2105 @cindex @code{float_precision ()}
2106 Returns the number of significant mantissa bits in the floating-point
2107 representation of @code{x}. Since denormalized numbers are not supported,
2108 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2112 The complete internal representation of a float is encoded in the type
2113 @cindex @code{decoded_float}
2114 @cindex @code{decoded_sfloat}
2115 @cindex @code{decoded_ffloat}
2116 @cindex @code{decoded_dfloat}
2117 @cindex @code{decoded_lfloat}
2118 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2119 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2121 struct decoded_@var{type}float @{
2122 @var{type} mantissa; cl_I exponent; @var{type} sign;
2126 and returned by the function
2129 @item decoded_@var{type}float decode_float (const @var{type}& x)
2130 @cindex @code{decode_float ()}
2131 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2132 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2133 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2134 @code{e} is the same as returned by the function @code{float_exponent}.
2137 A complete decoding in terms of integers is provided as type
2138 @cindex @code{cl_idecoded_float}
2140 struct cl_idecoded_float @{
2141 cl_I mantissa; cl_I exponent; cl_I sign;
2144 by the following function:
2147 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2148 @cindex @code{integer_decode_float ()}
2149 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2150 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2151 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2152 WARNING: The exponent @code{e} is not the same as the one returned by
2153 the functions @code{decode_float} and @code{float_exponent}.
2156 Some other function, implemented only for class @code{cl_F}:
2159 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2160 @cindex @code{float_sign ()}
2161 This returns a floating point number whose precision and absolute value
2162 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2163 zero, it is treated as positive. Same for @code{y}.
2167 @section Conversion functions
2170 @subsection Conversion to floating-point numbers
2172 The type @code{float_format_t} describes a floating-point format.
2173 @cindex @code{float_format_t}
2176 @item float_format_t float_format (uintL n)
2177 @cindex @code{float_format ()}
2178 Returns the smallest float format which guarantees at least @code{n}
2179 decimal digits in the mantissa (after the decimal point).
2181 @item float_format_t float_format (const cl_F& x)
2182 Returns the floating point format of @code{x}.
2184 @item float_format_t default_float_format
2185 @cindex @code{default_float_format}
2186 Global variable: the default float format used when converting rational numbers
2190 To convert a real number to a float, each of the types
2191 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2192 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2193 defines the following operations:
2196 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2197 @cindex @code{cl_float ()}
2198 Returns @code{x} as a float of format @code{f}.
2199 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2200 Returns @code{x} in the float format of @code{y}.
2201 @item cl_F cl_float (const @var{type}&x)
2202 Returns @code{x} as a float of format @code{default_float_format} if
2203 it is an exact number, or @code{x} itself if it is already a float.
2206 Of course, converting a number to a float can lose precision.
2208 Every floating-point format has some characteristic numbers:
2211 @item cl_F most_positive_float (float_format_t f)
2212 @cindex @code{most_positive_float ()}
2213 Returns the largest (most positive) floating point number in float format @code{f}.
2215 @item cl_F most_negative_float (float_format_t f)
2216 @cindex @code{most_negative_float ()}
2217 Returns the smallest (most negative) floating point number in float format @code{f}.
2219 @item cl_F least_positive_float (float_format_t f)
2220 @cindex @code{least_positive_float ()}
2221 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2222 in float format @code{f}.
2224 @item cl_F least_negative_float (float_format_t f)
2225 @cindex @code{least_negative_float ()}
2226 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2227 in float format @code{f}.
2229 @item cl_F float_epsilon (float_format_t f)
2230 @cindex @code{float_epsilon ()}
2231 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2233 @item cl_F float_negative_epsilon (float_format_t f)
2234 @cindex @code{float_negative_epsilon ()}
2235 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2239 @subsection Conversion to rational numbers
2241 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2242 defines the following operation:
2245 @item cl_RA rational (const @var{type}& x)
2246 @cindex @code{rational ()}
2247 Returns the value of @code{x} as an exact number. If @code{x} is already
2248 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2249 the value is a rational number whose denominator is a power of 2.
2252 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2256 @item cl_RA rationalize (const cl_R& x)
2257 @cindex @code{rationalize ()}
2258 If @code{x} is a floating-point number, it actually represents an interval
2259 of real numbers, and this function returns the rational number with
2260 smallest denominator (and smallest numerator, in magnitude)
2261 which lies in this interval.
2262 If @code{x} is already an exact number, this function returns @code{x}.
2265 If @code{x} is any float, one has
2269 @code{cl_float(rational(x),x) = x}
2271 @code{cl_float(rationalize(x),x) = x}
2275 @section Random number generators
2278 A random generator is a machine which produces (pseudo-)random numbers.
2279 The include file @code{<cln/random.h>} defines a class @code{random_state}
2280 which contains the state of a random generator. If you make a copy
2281 of the random number generator, the original one and the copy will produce
2282 the same sequence of random numbers.
2284 The following functions return (pseudo-)random numbers in different formats.
2285 Calling one of these modifies the state of the random number generator in
2286 a complicated but deterministic way.
2289 @cindex @code{random_state}
2290 @cindex @code{default_random_state}
2292 random_state default_random_state
2294 contains a default random number generator. It is used when the functions
2295 below are called without @code{random_state} argument.
2298 @item uint32 random32 (random_state& randomstate)
2299 @itemx uint32 random32 ()
2300 @cindex @code{random32 ()}
2301 Returns a random unsigned 32-bit number. All bits are equally random.
2303 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2304 @itemx cl_I random_I (const cl_I& n)
2305 @cindex @code{random_I ()}
2306 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2307 in the range @code{0 <= x < n}.
2309 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2310 @itemx cl_F random_F (const cl_F& n)
2311 @cindex @code{random_F ()}
2312 @code{n} must be a float > 0. This function returns a random floating-point
2313 number of the same format as @code{n} in the range @code{0 <= x < n}.
2315 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2316 @itemx cl_R random_R (const cl_R& n)
2317 @cindex @code{random_R ()}
2318 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2319 if @code{n} is a float.
2323 @section Obfuscating operators
2324 @cindex modifying operators
2326 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2327 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2328 are not available by default because their
2329 use tends to make programs unreadable. It is trivial to get away without
2330 them. However, if you feel that you absolutely need these operators
2331 to get happy, then add
2333 #define WANT_OBFUSCATING_OPERATORS
2335 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2336 to the beginning of your source files, before the inclusion of any CLN
2337 include files. This flag will enable the following operators:
2339 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2340 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2343 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2344 @cindex @code{operator += ()}
2345 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2346 @cindex @code{operator -= ()}
2347 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2348 @cindex @code{operator *= ()}
2349 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2350 @cindex @code{operator /= ()}
2353 For the class @code{cl_I}:
2356 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2357 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2358 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2359 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2360 @cindex @code{operator &= ()}
2361 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2362 @cindex @code{operator |= ()}
2363 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2364 @cindex @code{operator ^= ()}
2365 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2366 @cindex @code{operator <<= ()}
2367 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2368 @cindex @code{operator >>= ()}
2371 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2372 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2375 @item @var{type}& operator ++ (@var{type}& x)
2376 @cindex @code{operator ++ ()}
2377 The prefix operator @code{++x}.
2379 @item void operator ++ (@var{type}& x, int)
2380 The postfix operator @code{x++}.
2382 @item @var{type}& operator -- (@var{type}& x)
2383 @cindex @code{operator -- ()}
2384 The prefix operator @code{--x}.
2386 @item void operator -- (@var{type}& x, int)
2387 The postfix operator @code{x--}.
2390 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2391 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2395 @chapter Input/Output
2396 @cindex Input/Output
2398 @section Internal and printed representation
2399 @cindex representation
2401 All computations deal with the internal representations of the numbers.
2403 Every number has an external representation as a sequence of ASCII characters.
2404 Several external representations may denote the same number, for example,
2405 "20.0" and "20.000".
2407 Converting an internal to an external representation is called ``printing'',
2409 converting an external to an internal representation is called ``reading''.
2411 In CLN, it is always true that conversion of an internal to an external
2412 representation and then back to an internal representation will yield the
2413 same internal representation. Symbolically: @code{read(print(x)) == x}.
2414 This is called ``print-read consistency''.
2416 Different types of numbers have different external representations (case
2421 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2422 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2423 for decimal integers
2424 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2426 @item Rational numbers
2427 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2428 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2431 @item Floating-point numbers
2432 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2433 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2434 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2435 of the form _@var{prec} may be appended. There must be at least
2436 one digit in the non-exponent part. The exponent has the syntax
2437 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2438 The exponent marker is
2442 @samp{s} for short-floats,
2444 @samp{f} for single-floats,
2446 @samp{d} for double-floats,
2448 @samp{L} for long-floats,
2451 or @samp{e}, which denotes a default float format. The precision specifying
2452 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2453 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2454 function @samp{float_format}.
2456 @item Complex numbers
2457 External representation:
2460 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2461 if @var{imagpart} is negative, its printed representation begins with
2462 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2463 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2464 is rational and the rational number's base is >18, because the @samp{i}
2465 is then read as a digit.
2467 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2472 @section Input functions
2474 Including @code{<cln/io.h>} defines a number of simple input functions
2475 that read from @code{std::istream&}:
2478 @item int freadchar (std::istream& stream)
2479 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2480 if the end of stream was encountered or an error occurred.
2482 @item int funreadchar (std::istream& stream, int c)
2483 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2484 last @code{freadchar} operation on @code{stream}.
2487 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2488 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2489 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2492 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2493 Reads a number from @code{stream} and stores it in the @code{result}.
2496 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2500 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2501 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2502 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2503 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2504 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2505 Reads a number from @code{stream}. The @code{flags} are parameters which
2506 affect the input syntax. Whitespace before the number is silently skipped.
2508 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2509 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2510 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2511 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2512 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2513 Reads a number from a string in memory. The @code{flags} are parameters which
2514 affect the input syntax. The string starts at @code{string} and ends at
2515 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2516 @code{NULL}, denoting the entire string, i.e. equivalent to
2517 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2518 @code{NULL}, the string in memory must contain exactly one number and nothing
2519 more, else a fatal error will be signalled. If @code{end_of_parse}
2520 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2521 the last parsed character (i.e. @code{string_limit} if nothing came after
2522 the number). Whitespace is not allowed.
2525 The structure @code{cl_read_flags} contains the following fields:
2528 @item cl_read_syntax_t syntax
2529 The possible results of the read operation. Possible values are
2530 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2531 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2532 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2534 @item cl_read_lsyntax_t lsyntax
2535 Specifies the language-dependent syntax variant for the read operation.
2539 @item lsyntax_standard
2540 accept standard algebraic notation only, no complex numbers,
2541 @item lsyntax_algebraic
2542 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2543 @item lsyntax_commonlisp
2544 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2545 hexadecimal numbers,
2546 @code{#@var{base}R} for rational numbers in a given base,
2547 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2549 accept all of these extensions.
2552 @item unsigned int rational_base
2553 The base in which rational numbers are read.
2555 @item float_format_t float_flags.default_float_format
2556 The float format used when reading floats with exponent marker @samp{e}.
2558 @item float_format_t float_flags.default_lfloat_format
2559 The float format used when reading floats with exponent marker @samp{l}.
2561 @item cl_boolean float_flags.mantissa_dependent_float_format
2562 When this flag is true, floats specified with more digits than corresponding
2563 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2564 precision corresponding to their number of significant digits.
2568 @section Output functions
2570 Including @code{<cln/io.h>} defines a number of simple output functions
2571 that write to @code{std::ostream&}:
2574 @item void fprintchar (std::ostream& stream, char c)
2575 Prints the character @code{x} literally on the @code{stream}.
2577 @item void fprint (std::ostream& stream, const char * string)
2578 Prints the @code{string} literally on the @code{stream}.
2580 @item void fprintdecimal (std::ostream& stream, int x)
2581 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2582 Prints the integer @code{x} in decimal on the @code{stream}.
2584 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2585 Prints the integer @code{x} in binary (base 2, without prefix)
2586 on the @code{stream}.
2588 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2589 Prints the integer @code{x} in octal (base 8, without prefix)
2590 on the @code{stream}.
2592 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2593 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2594 on the @code{stream}.
2597 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2598 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2599 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2602 @item void fprint (std::ostream& stream, const @var{type}& x)
2603 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2604 Prints the number @code{x} on the @code{stream}. The output may depend
2605 on the global printer settings in the variable @code{default_print_flags}.
2606 The @code{ostream} flags and settings (flags, width and locale) are
2610 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2613 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2615 void print_real (std::ostream& stream, const cl_print_flags& flags,
2617 void print_float (std::ostream& stream, const cl_print_flags& flags,
2619 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2621 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2624 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2625 parameters which affect the output.
2627 The structure type @code{cl_print_flags} contains the following fields:
2630 @item unsigned int rational_base
2631 The base in which rational numbers are printed. Default is @code{10}.
2633 @item cl_boolean rational_readably
2634 If this flag is true, rational numbers are printed with radix specifiers in
2635 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2636 prefixes, trailing dot). Default is false.
2638 @item cl_boolean float_readably
2639 If this flag is true, type specific exponent markers have precedence over 'E'.
2642 @item float_format_t default_float_format
2643 Floating point numbers of this format will be printed using the 'E' exponent
2644 marker. Default is @code{float_format_ffloat}.
2646 @item cl_boolean complex_readably
2647 If this flag is true, complex numbers will be printed using the Common Lisp
2648 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2650 @item cl_string univpoly_varname
2651 Univariate polynomials with no explicit indeterminate name will be printed
2652 using this variable name. Default is @code{"x"}.
2655 The global variable @code{default_print_flags} contains the default values,
2656 used by the function @code{fprint}.
2661 CLN has a class of abstract rings.
2669 Rings can be compared for equality:
2672 @item bool operator== (const cl_ring&, const cl_ring&)
2673 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2674 These compare two rings for equality.
2677 Given a ring @code{R}, the following members can be used.
2680 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2681 @cindex @code{fprint ()}
2682 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2683 @cindex @code{equal ()}
2684 @itemx cl_ring_element R->zero ()
2685 @cindex @code{zero ()}
2686 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2687 @cindex @code{zerop ()}
2688 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2689 @cindex @code{plus ()}
2690 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2691 @cindex @code{minus ()}
2692 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2693 @cindex @code{uminus ()}
2694 @itemx cl_ring_element R->one ()
2695 @cindex @code{one ()}
2696 @itemx cl_ring_element R->canonhom (const cl_I& x)
2697 @cindex @code{canonhom ()}
2698 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2699 @cindex @code{mul ()}
2700 @itemx cl_ring_element R->square (const cl_ring_element& x)
2701 @cindex @code{square ()}
2702 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2703 @cindex @code{expt_pos ()}
2706 The following rings are built-in.
2709 @item cl_null_ring cl_0_ring
2710 The null ring, containing only zero.
2712 @item cl_complex_ring cl_C_ring
2713 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2715 @item cl_real_ring cl_R_ring
2716 The ring of real numbers. This corresponds to the type @code{cl_R}.
2718 @item cl_rational_ring cl_RA_ring
2719 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2721 @item cl_integer_ring cl_I_ring
2722 The ring of integers. This corresponds to the type @code{cl_I}.
2725 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2726 @code{cl_RA_ring}, @code{cl_I_ring}:
2729 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2730 @cindex @code{instanceof ()}
2731 Tests whether the given number is an element of the number ring R.
2735 @chapter Modular integers
2736 @cindex modular integer
2738 @section Modular integer rings
2741 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2742 The modulus is explicitly part of every modular integer. CLN doesn't
2743 allow you to (accidentally) mix elements of different modular rings,
2744 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2745 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2746 doesn't have generic types. So one has to live with runtime checks.)
2748 The class of modular integer rings is
2756 Modular integer ring
2760 @cindex @code{cl_modint_ring}
2762 and the class of all modular integers (elements of modular integer rings) is
2770 Modular integer rings are constructed using the function
2773 @item cl_modint_ring find_modint_ring (const cl_I& N)
2774 @cindex @code{find_modint_ring ()}
2775 This function returns the modular ring @samp{Z/NZ}. It takes care
2776 of finding out about special cases of @code{N}, like powers of two
2777 and odd numbers for which Montgomery multiplication will be a win,
2778 @cindex Montgomery multiplication
2779 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2780 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2781 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2785 Modular integer rings can be compared for equality:
2788 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2789 @cindex @code{operator == ()}
2790 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2791 @cindex @code{operator != ()}
2792 These compare two modular integer rings for equality. Two different calls
2793 to @code{find_modint_ring} with the same argument necessarily return the
2794 same ring because it is memoized in the cache table.
2797 @section Functions on modular integers
2799 Given a modular integer ring @code{R}, the following members can be used.
2802 @item cl_I R->modulus
2803 @cindex @code{modulus}
2804 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2806 @item cl_MI R->zero()
2807 @cindex @code{zero ()}
2808 This returns @code{0 mod N}.
2810 @item cl_MI R->one()
2811 @cindex @code{one ()}
2812 This returns @code{1 mod N}.
2814 @item cl_MI R->canonhom (const cl_I& x)
2815 @cindex @code{canonhom ()}
2816 This returns @code{x mod N}.
2818 @item cl_I R->retract (const cl_MI& x)
2819 @cindex @code{retract ()}
2820 This is a partial inverse function to @code{R->canonhom}. It returns the
2821 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2823 @item cl_MI R->random(random_state& randomstate)
2824 @itemx cl_MI R->random()
2825 @cindex @code{random ()}
2826 This returns a random integer modulo @code{N}.
2829 The following operations are defined on modular integers.
2832 @item cl_modint_ring x.ring ()
2833 @cindex @code{ring ()}
2834 Returns the ring to which the modular integer @code{x} belongs.
2836 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2837 @cindex @code{operator + ()}
2838 Returns the sum of two modular integers. One of the arguments may also
2841 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2842 @cindex @code{operator - ()}
2843 Returns the difference of two modular integers. One of the arguments may also
2846 @item cl_MI operator- (const cl_MI&)
2847 Returns the negative of a modular integer.
2849 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2850 @cindex @code{operator * ()}
2851 Returns the product of two modular integers. One of the arguments may also
2854 @item cl_MI square (const cl_MI&)
2855 @cindex @code{square ()}
2856 Returns the square of a modular integer.
2858 @item cl_MI recip (const cl_MI& x)
2859 @cindex @code{recip ()}
2860 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2861 must be coprime to the modulus, otherwise an error message is issued.
2863 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2864 @cindex @code{div ()}
2865 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2866 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2868 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2869 @cindex @code{expt_pos ()}
2870 @code{y} must be > 0. Returns @code{x^y}.
2872 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2873 @cindex @code{expt ()}
2874 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2875 modulus, else an error message is issued.
2877 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2878 @cindex @code{operator << ()}
2879 Returns @code{x*2^y}.
2881 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2882 @cindex @code{operator >> ()}
2883 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2884 or an error message is issued.
2886 @item bool operator== (const cl_MI&, const cl_MI&)
2887 @cindex @code{operator == ()}
2888 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2889 @cindex @code{operator != ()}
2890 Compares two modular integers, belonging to the same modular integer ring,
2893 @item cl_boolean zerop (const cl_MI& x)
2894 @cindex @code{zerop ()}
2895 Returns true if @code{x} is @code{0 mod N}.
2898 The following output functions are defined (see also the chapter on
2902 @item void fprint (std::ostream& stream, const cl_MI& x)
2903 @cindex @code{fprint ()}
2904 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
2905 @cindex @code{operator << ()}
2906 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2907 on the global printer settings in the variable @code{default_print_flags}.
2911 @chapter Symbolic data types
2912 @cindex symbolic type
2914 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2918 @cindex @code{cl_string}
2928 implements immutable strings.
2930 Strings are constructed through the following constructors:
2933 @item cl_string (const char * s)
2934 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2936 @item cl_string (const char * ptr, unsigned long len)
2937 Returns an immutable copy of the @code{len} characters at
2938 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2941 The following functions are available on strings:
2945 Assignment from @code{cl_string} and @code{const char *}.
2948 @cindex @code{length ()}
2950 @cindex @code{strlen ()}
2951 Returns the length of the string @code{s}.
2954 @cindex @code{operator [] ()}
2955 Returns the @code{i}th character of the string @code{s}.
2956 @code{i} must be in the range @code{0 <= i < s.length()}.
2958 @item bool equal (const cl_string& s1, const cl_string& s2)
2959 @cindex @code{equal ()}
2960 Compares two strings for equality. One of the arguments may also be a
2961 plain @code{const char *}.
2966 @cindex @code{cl_symbol}
2968 Symbols are uniquified strings: all symbols with the same name are shared.
2969 This means that comparison of two symbols is fast (effectively just a pointer
2970 comparison), whereas comparison of two strings must in the worst case walk
2971 both strings until their end.
2972 Symbols are used, for example, as tags for properties, as names of variables
2973 in polynomial rings, etc.
2975 Symbols are constructed through the following constructor:
2978 @item cl_symbol (const cl_string& s)
2979 Looks up or creates a new symbol with a given name.
2982 The following operations are available on symbols:
2985 @item cl_string (const cl_symbol& sym)
2986 Conversion to @code{cl_string}: Returns the string which names the symbol
2989 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2990 @cindex @code{equal ()}
2991 Compares two symbols for equality. This is very fast.
2995 @chapter Univariate polynomials
2997 @cindex univariate polynomial
2999 @section Univariate polynomial rings
3001 CLN implements univariate polynomials (polynomials in one variable) over an
3002 arbitrary ring. The indeterminate variable may be either unnamed (and will be
3003 printed according to @code{default_print_flags.univpoly_varname}, which
3004 defaults to @samp{x}) or carry a given name. The base ring and the
3005 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
3006 (accidentally) mix elements of different polynomial rings, e.g.
3007 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
3008 return a multivariate polynomial, but they are not yet implemented in CLN.)
3010 The classes of univariate polynomial rings are
3018 Univariate polynomial ring
3022 +----------------+-------------------+
3024 Complex polynomial ring | Modular integer polynomial ring
3025 cl_univpoly_complex_ring | cl_univpoly_modint_ring
3026 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3030 Real polynomial ring |
3031 cl_univpoly_real_ring |
3032 <cln/univpoly_real.h> |
3036 Rational polynomial ring |
3037 cl_univpoly_rational_ring |
3038 <cln/univpoly_rational.h> |
3042 Integer polynomial ring
3043 cl_univpoly_integer_ring
3044 <cln/univpoly_integer.h>
3047 and the corresponding classes of univariate polynomials are
3050 Univariate polynomial
3054 +----------------+-------------------+
3056 Complex polynomial | Modular integer polynomial
3058 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3064 <cln/univpoly_real.h> |
3068 Rational polynomial |
3070 <cln/univpoly_rational.h> |
3076 <cln/univpoly_integer.h>
3079 Univariate polynomial rings are constructed using the functions
3082 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3083 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3084 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3085 @code{R} may be an arbitrary ring. This function takes care of finding out
3086 about special cases of @code{R}, such as the rings of complex numbers,
3087 real numbers, rational numbers, integers, or modular integer rings.
3088 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3089 This ensures that two calls of this function with the same arguments will
3090 return the same polynomial ring.
3092 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3093 @cindex @code{find_univpoly_ring ()}
3094 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3095 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3096 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3097 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3098 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3099 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3100 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3101 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3102 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3103 These functions are equivalent to the general @code{find_univpoly_ring},
3104 only the return type is more specific, according to the base ring's type.
3107 @section Functions on univariate polynomials
3109 Given a univariate polynomial ring @code{R}, the following members can be used.
3112 @item cl_ring R->basering()
3113 @cindex @code{basering ()}
3114 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3116 @item cl_UP R->zero()
3117 @cindex @code{zero ()}
3118 This returns @code{0 in R}, a polynomial of degree -1.
3120 @item cl_UP R->one()
3121 @cindex @code{one ()}
3122 This returns @code{1 in R}, a polynomial of degree == 0.
3124 @item cl_UP R->canonhom (const cl_I& x)
3125 @cindex @code{canonhom ()}
3126 This returns @code{x in R}, a polynomial of degree <= 0.
3128 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3129 @cindex @code{monomial ()}
3130 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3133 @item cl_UP R->create (sintL degree)
3134 @cindex @code{create ()}
3135 Creates a new polynomial with a given degree. The zero polynomial has degree
3136 @code{-1}. After creating the polynomial, you should put in the coefficients,
3137 using the @code{set_coeff} member function, and then call the @code{finalize}
3141 The following are the only destructive operations on univariate polynomials.
3144 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3145 @cindex @code{set_coeff ()}
3146 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3147 After changing a polynomial and before applying any "normal" operation on it,
3148 you should call its @code{finalize} member function.
3150 @item void finalize (cl_UP& x)
3151 @cindex @code{finalize ()}
3152 This function marks the endpoint of destructive modifications of a polynomial.
3153 It normalizes the internal representation so that subsequent computations have
3154 less overhead. Doing normal computations on unnormalized polynomials may
3155 produce wrong results or crash the program.
3158 The following operations are defined on univariate polynomials.
3161 @item cl_univpoly_ring x.ring ()
3162 @cindex @code{ring ()}
3163 Returns the ring to which the univariate polynomial @code{x} belongs.
3165 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3166 @cindex @code{operator + ()}
3167 Returns the sum of two univariate polynomials.
3169 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3170 @cindex @code{operator - ()}
3171 Returns the difference of two univariate polynomials.
3173 @item cl_UP operator- (const cl_UP&)
3174 Returns the negative of a univariate polynomial.
3176 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3177 @cindex @code{operator * ()}
3178 Returns the product of two univariate polynomials. One of the arguments may
3179 also be a plain integer or an element of the base ring.
3181 @item cl_UP square (const cl_UP&)
3182 @cindex @code{square ()}
3183 Returns the square of a univariate polynomial.
3185 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3186 @cindex @code{expt_pos ()}
3187 @code{y} must be > 0. Returns @code{x^y}.
3189 @item bool operator== (const cl_UP&, const cl_UP&)
3190 @cindex @code{operator == ()}
3191 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3192 @cindex @code{operator != ()}
3193 Compares two univariate polynomials, belonging to the same univariate
3194 polynomial ring, for equality.
3196 @item cl_boolean zerop (const cl_UP& x)
3197 @cindex @code{zerop ()}
3198 Returns true if @code{x} is @code{0 in R}.
3200 @item sintL degree (const cl_UP& x)
3201 @cindex @code{degree ()}
3202 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3204 @item sintL ldegree (const cl_UP& x)
3205 @cindex @code{degree ()}
3206 Returns the low degree of the polynomial. This is the degree of the first
3207 non-vanishing polynomial coefficient. The zero polynomial has ldegree @code{-1}.
3209 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3210 @cindex @code{coeff ()}
3211 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3213 @item cl_ring_element x (const cl_ring_element& y)
3214 @cindex @code{operator () ()}
3215 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3216 then @samp{x(y)} returns the value of the substitution of @code{y} into
3219 @item cl_UP deriv (const cl_UP& x)
3220 @cindex @code{deriv ()}
3221 Returns the derivative of the polynomial @code{x} with respect to the
3222 indeterminate @code{X}.
3225 The following output functions are defined (see also the chapter on
3229 @item void fprint (std::ostream& stream, const cl_UP& x)
3230 @cindex @code{fprint ()}
3231 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3232 @cindex @code{operator << ()}
3233 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3234 depend on the global printer settings in the variable
3235 @code{default_print_flags}.
3238 @section Special polynomials
3240 The following functions return special polynomials.
3243 @item cl_UP_I tschebychev (sintL n)
3244 @cindex @code{tschebychev ()}
3245 @cindex Chebyshev polynomial
3246 Returns the n-th Chebyshev polynomial (n >= 0).
3248 @item cl_UP_I hermite (sintL n)
3249 @cindex @code{hermite ()}
3250 @cindex Hermite polynomial
3251 Returns the n-th Hermite polynomial (n >= 0).
3253 @item cl_UP_RA legendre (sintL n)
3254 @cindex @code{legendre ()}
3255 @cindex Legende polynomial
3256 Returns the n-th Legendre polynomial (n >= 0).
3258 @item cl_UP_I laguerre (sintL n)
3259 @cindex @code{laguerre ()}
3260 @cindex Laguerre polynomial
3261 Returns the n-th Laguerre polynomial (n >= 0).
3264 Information how to derive the differential equation satisfied by each
3265 of these polynomials from their definition can be found in the
3266 @code{doc/polynomial/} directory.
3274 Using C++ as an implementation language provides
3278 Efficiency: It compiles to machine code.
3282 Portability: It runs on all platforms supporting a C++ compiler. Because
3283 of the availability of GNU C++, this includes all currently used 32-bit and
3284 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3287 Type safety: The C++ compilers knows about the number types and complains if,
3288 for example, you try to assign a float to an integer variable. However,
3289 a drawback is that C++ doesn't know about generic types, hence a restriction
3290 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3291 arguments belong to the same modular ring cannot be expressed as a compile-time
3295 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3296 @code{=}, @code{==}, ... can be used in infix notation, which is more
3297 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3300 With these language features, there is no need for two separate languages,
3301 one for the implementation of the library and one in which the library's users
3302 can program. This means that a prototype implementation of an algorithm
3303 can be integrated into the library immediately after it has been tested and
3304 debugged. No need to rewrite it in a low-level language after having prototyped
3305 in a high-level language.
3308 @section Memory efficiency
3310 In order to save memory allocations, CLN implements:
3314 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3317 @cindex garbage collection
3318 @cindex reference counting
3319 Garbage collection: A reference counting mechanism makes sure that any
3320 number object's storage is freed immediately when the last reference to the
3323 @cindex immediate numbers
3324 Small integers are represented as immediate values instead of pointers
3325 to heap allocated storage. This means that integers @code{> -2^29},
3326 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3331 @section Speed efficiency
3333 Speed efficiency is obtained by the combination of the following tricks
3338 Small integers, being represented as immediate values, don't require
3339 memory access, just a couple of instructions for each elementary operation.
3341 The kernel of CLN has been written in assembly language for some CPUs
3342 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3344 On all CPUs, CLN may be configured to use the superefficient low-level
3345 routines from GNU GMP version 3.
3347 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3348 algorithm, the Karatsuba multiplication, which is an
3359 For very large numbers (more than 12000 decimal digits), CLN uses
3361 Sch{@"o}nhage-Strassen
3362 @cindex Sch{@"o}nhage-Strassen multiplication
3366 @cindex Schnhage-Strassen multiplication
3368 multiplication, which is an asymptotically optimal multiplication
3371 These fast multiplication algorithms also give improvements in the speed
3372 of division and radix conversion.
3376 @section Garbage collection
3377 @cindex garbage collection
3379 All the number classes are reference count classes: They only contain a pointer
3380 to an object in the heap. Upon construction, assignment and destruction of
3381 number objects, only the objects' reference count are manipulated.
3383 Memory occupied by number objects are automatically reclaimed as soon as
3384 their reference count drops to zero.
3386 For number rings, another strategy is implemented: There is a cache of,
3387 for example, the modular integer rings. A modular integer ring is destroyed
3388 only if its reference count dropped to zero and the cache is about to be
3389 resized. The effect of this strategy is that recently used rings remain
3390 cached, whereas undue memory consumption through cached rings is avoided.
3393 @chapter Using the library
3395 For the following discussion, we will assume that you have installed
3396 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3397 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3398 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3399 environment variables, or directly substitute the appropriate values.
3402 @section Compiler options
3403 @cindex compiler options
3405 Until you have installed CLN in a public place, the following options are
3408 When you compile CLN application code, add the flags
3410 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3412 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3413 When you link CLN application code to form an executable, add the flags
3415 $CLN_TARGETDIR/src/libcln.a
3417 to the C/C++ compiler's command line (@code{make} variable LIBS).
3419 If you did a @code{make install}, the include files are installed in a
3420 public directory (normally @code{/usr/local/include}), hence you don't
3421 need special flags for compiling. The library has been installed to a
3422 public directory as well (normally @code{/usr/local/lib}), hence when
3423 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3425 Since CLN version 1.1, there are two tools to make the creation of
3426 software packages that use CLN easier:
3429 @cindex @code{cln-config}
3430 @code{cln-config} is a shell script that you can use to determine the
3431 compiler and linker command line options required to compile and link a
3432 program with CLN. Start it with @code{--help} to learn about its options
3433 or consult the manpage that comes with it.
3435 @cindex @code{AC_PATH_CLN}
3436 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3439 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3441 This macro determines the location of CLN using @code{cln-config}, which
3442 is either found in the user's path, or from the environment variable
3443 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3444 their version is not earlier than @var{MIN-VERSION} (a default version
3445 will be used if not specified). If the required version was found, sets
3446 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3447 macro is in the file @file{cln.m4} which is installed in
3448 @file{$datadir/aclocal}. Note that if automake was installed with a
3449 different @samp{--prefix} than CLN, you will either have to manually
3450 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3451 aclocal the @samp{-I} option when running it. Here is a possible example
3452 to be included in your package's @file{configure.ac}:
3454 AC_PATH_CLN(1.1.0, [
3455 LIBS="$LIBS $CLN_LIBS"
3456 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3457 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3462 @section Compatibility to old CLN versions
3464 @cindex compatibility
3466 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3467 @code{cln} in order to avoid potential name clashes with other C++
3468 libraries. If you have an old application, you will have to manually
3469 port it to the new scheme. The following principles will help during
3473 All headers are now in a separate subdirectory. Instead of including
3474 @code{cl_}@var{something}@code{.h}, include
3475 @code{cln/}@var{something}@code{.h} now.
3477 All public identifiers (typenames and functions) have lost their
3478 @code{cl_} prefix. Exceptions are all the typenames of number types,
3479 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3480 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3481 names would not be mnemonic enough once the namespace @code{cln} is
3482 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3484 All public @emph{functions} that had by a @code{cl_} in their name still
3485 carry that @code{cl_} if it is intrinsic part of a typename (as in
3486 @code{cl_I_to_int ()}).
3488 When developing other libraries, please keep in mind not to import the
3489 namespace @code{cln} in one of your public header files by saying
3490 @code{using namespace cln;}. This would propagate to other applications
3491 and can cause name clashes there.
3494 @section Include files
3495 @cindex include files
3496 @cindex header files
3498 Here is a summary of the include files and their contents.
3501 @item <cln/object.h>
3502 General definitions, reference counting, garbage collection.
3503 @item <cln/number.h>
3504 The class cl_number.
3505 @item <cln/complex.h>
3506 Functions for class cl_N, the complex numbers.
3508 Functions for class cl_R, the real numbers.
3510 Functions for class cl_F, the floats.
3511 @item <cln/sfloat.h>
3512 Functions for class cl_SF, the short-floats.
3513 @item <cln/ffloat.h>
3514 Functions for class cl_FF, the single-floats.
3515 @item <cln/dfloat.h>
3516 Functions for class cl_DF, the double-floats.
3517 @item <cln/lfloat.h>
3518 Functions for class cl_LF, the long-floats.
3519 @item <cln/rational.h>
3520 Functions for class cl_RA, the rational numbers.
3521 @item <cln/integer.h>
3522 Functions for class cl_I, the integers.
3525 @item <cln/complex_io.h>
3526 Input/Output for class cl_N, the complex numbers.
3527 @item <cln/real_io.h>
3528 Input/Output for class cl_R, the real numbers.
3529 @item <cln/float_io.h>
3530 Input/Output for class cl_F, the floats.
3531 @item <cln/sfloat_io.h>
3532 Input/Output for class cl_SF, the short-floats.
3533 @item <cln/ffloat_io.h>
3534 Input/Output for class cl_FF, the single-floats.
3535 @item <cln/dfloat_io.h>
3536 Input/Output for class cl_DF, the double-floats.
3537 @item <cln/lfloat_io.h>
3538 Input/Output for class cl_LF, the long-floats.
3539 @item <cln/rational_io.h>
3540 Input/Output for class cl_RA, the rational numbers.
3541 @item <cln/integer_io.h>
3542 Input/Output for class cl_I, the integers.
3544 Flags for customizing input operations.
3545 @item <cln/output.h>
3546 Flags for customizing output operations.
3547 @item <cln/malloc.h>
3548 @code{malloc_hook}, @code{free_hook}.
3551 @item <cln/condition.h>
3552 Conditions/exceptions.
3553 @item <cln/string.h>
3555 @item <cln/symbol.h>
3557 @item <cln/proplist.h>
3561 @item <cln/null_ring.h>
3563 @item <cln/complex_ring.h>
3564 The ring of complex numbers.
3565 @item <cln/real_ring.h>
3566 The ring of real numbers.
3567 @item <cln/rational_ring.h>
3568 The ring of rational numbers.
3569 @item <cln/integer_ring.h>
3570 The ring of integers.
3571 @item <cln/numtheory.h>
3572 Number threory functions.
3573 @item <cln/modinteger.h>
3579 @item <cln/GV_number.h>
3580 General vectors over cl_number.
3581 @item <cln/GV_complex.h>
3582 General vectors over cl_N.
3583 @item <cln/GV_real.h>
3584 General vectors over cl_R.
3585 @item <cln/GV_rational.h>
3586 General vectors over cl_RA.
3587 @item <cln/GV_integer.h>
3588 General vectors over cl_I.
3589 @item <cln/GV_modinteger.h>
3590 General vectors of modular integers.
3593 @item <cln/SV_number.h>
3594 Simple vectors over cl_number.
3595 @item <cln/SV_complex.h>
3596 Simple vectors over cl_N.
3597 @item <cln/SV_real.h>
3598 Simple vectors over cl_R.
3599 @item <cln/SV_rational.h>
3600 Simple vectors over cl_RA.
3601 @item <cln/SV_integer.h>
3602 Simple vectors over cl_I.
3603 @item <cln/SV_ringelt.h>
3604 Simple vectors of general ring elements.
3605 @item <cln/univpoly.h>
3606 Univariate polynomials.
3607 @item <cln/univpoly_integer.h>
3608 Univariate polynomials over the integers.
3609 @item <cln/univpoly_rational.h>
3610 Univariate polynomials over the rational numbers.
3611 @item <cln/univpoly_real.h>
3612 Univariate polynomials over the real numbers.
3613 @item <cln/univpoly_complex.h>
3614 Univariate polynomials over the complex numbers.
3615 @item <cln/univpoly_modint.h>
3616 Univariate polynomials over modular integer rings.
3617 @item <cln/timing.h>
3620 Includes all of the above.
3626 A function which computes the nth Fibonacci number can be written as follows.
3627 @cindex Fibonacci number
3630 #include <cln/integer.h>
3631 #include <cln/real.h>
3632 using namespace cln;
3634 // Returns F_n, computed as the nearest integer to
3635 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3636 const cl_I fibonacci (int n)
3638 // Need a precision of ((1+sqrt(5))/2)^-n.
3639 float_format_t prec = float_format((int)(0.208987641*n+5));
3640 cl_R sqrt5 = sqrt(cl_float(5,prec));
3641 cl_R phi = (1+sqrt5)/2;
3642 return round1( expt(phi,n)/sqrt5 );
3646 Let's explain what is going on in detail.
3648 The include file @code{<cln/integer.h>} is necessary because the type
3649 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3650 is needed for the type @code{cl_R} and the floating point number functions.
3651 The order of the include files does not matter. In order not to write
3652 out @code{cln::}@var{foo} in this simple example we can safely import
3653 the whole namespace @code{cln}.
3655 Then comes the function declaration. The argument is an @code{int}, the
3656 result an integer. The return type is defined as @samp{const cl_I}, not
3657 simply @samp{cl_I}, because that allows the compiler to detect typos like
3658 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3659 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3660 number). We use the most specialized possible return type because functions
3661 which call @samp{fibonacci} will be able to profit from the compiler's type
3662 analysis: Adding two integers is slightly more efficient than adding the
3663 same objects declared as complex numbers, because it needs less type
3664 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3665 the size of the resulting executable program.
3667 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3668 integer. In order to get a correct result, the absolute error should be less
3669 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3670 To this end, the first line computes a floating point precision for sqrt(5)
3673 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3674 number and than taking the square root. The converse, first taking the square
3675 root of 5, and then converting to the desired precision, would not work in
3676 CLN: The square root would be computed to a default precision (normally
3677 single-float precision), and the following conversion could not help about
3678 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3679 system and does not represent sqrt(5) in a non-numeric way.
3681 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3682 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3683 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3684 @code{cl_N} because a @samp{round1} does not exist for general complex
3687 When the function returns, all the local variables in the function are
3688 automatically reclaimed (garbage collected). Only the result survives and
3689 gets passed to the caller.
3691 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3692 contains this implementation together with an even faster algorithm.
3694 @section Debugging support
3697 When debugging a CLN application with GNU @code{gdb}, two facilities are
3698 available from the library:
3701 @item The library does type checks, range checks, consistency checks at
3702 many places. When one of these fails, the function @code{cl_abort()} is
3703 called. Its default implementation is to perform an @code{exit(1)}, so
3704 you won't have a core dump. But for debugging, it is best to set a
3705 breakpoint at this function:
3707 (gdb) break cl_abort
3709 When this breakpoint is hit, look at the stack's backtrace:
3714 @item The debugger's normal @code{print} command doesn't know about
3715 CLN's types and therefore prints mostly useless hexadecimal addresses.
3716 CLN offers a function @code{cl_print}, callable from the debugger,
3717 for printing number objects. In order to get this function, you have
3718 to define the macro @samp{CL_DEBUG} and then include all the header files
3719 for which you want @code{cl_print} debugging support. For example:
3720 @cindex @code{CL_DEBUG}
3723 #include <cln/string.h>
3725 Now, if you have in your program a variable @code{cl_string s}, and
3726 inspect it under @code{gdb}, the output may look like this:
3729 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3730 word = 134568800@}@}, @}
3731 (gdb) call cl_print(s)
3735 Note that the output of @code{cl_print} goes to the program's error output,
3736 not to gdb's standard output.
3738 Note, however, that the above facility does not work with all CLN types,
3739 only with number objects and similar. Therefore CLN offers a member function
3740 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3741 is needed for this member function to be implemented. Under @code{gdb},
3742 you call it like this:
3743 @cindex @code{debug_print ()}
3746 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3747 word = 134568800@}@}, @}
3748 (gdb) call s.debug_print()
3751 >call ($1).debug_print()
3756 Unfortunately, this feature does not seem to work under all circumstances.
3760 @chapter Customizing
3763 @section Error handling
3765 When a fatal error occurs, an error message is output to the standard error
3766 output stream, and the function @code{cl_abort} is called. The default
3767 version of this function (provided in the library) terminates the application.
3768 To catch such a fatal error, you need to define the function @code{cl_abort}
3769 yourself, with the prototype
3771 #include <cln/abort.h>
3772 void cl_abort (void);
3774 @cindex @code{cl_abort ()}
3775 This function must not return control to its caller.
3778 @section Floating-point underflow
3781 Floating point underflow denotes the situation when a floating-point number
3782 is to be created which is so close to @code{0} that its exponent is too
3783 low to be represented internally. By default, this causes a fatal error.
3784 If you set the global variable
3786 cl_boolean cl_inhibit_floating_point_underflow
3788 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3789 will be generated instead. The default value of
3790 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3793 @section Customizing I/O
3795 The output of the function @code{fprint} may be customized by changing the
3796 value of the global variable @code{default_print_flags}.
3797 @cindex @code{default_print_flags}
3800 @section Customizing the memory allocator
3802 Every memory allocation of CLN is done through the function pointer
3803 @code{malloc_hook}. Freeing of this memory is done through the function
3804 pointer @code{free_hook}. The default versions of these functions,
3805 provided in the library, call @code{malloc} and @code{free} and check
3806 the @code{malloc} result against @code{NULL}.
3807 If you want to provide another memory allocator, you need to define
3808 the variables @code{malloc_hook} and @code{free_hook} yourself,
3811 #include <cln/malloc.h>
3813 void* (*malloc_hook) (size_t size) = @dots{};
3814 void (*free_hook) (void* ptr) = @dots{};
3817 @cindex @code{malloc_hook ()}
3818 @cindex @code{free_hook ()}
3819 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3821 It is not possible to change the memory allocator at runtime, because
3822 it is already called at program startup by the constructors of some