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.
779 @cindex @code{As()()}
780 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
781 @var{type} and returns it as such.
782 @cindex @code{The()()}
783 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
784 @var{type} and returns it as such. It is your responsibility to ensure
785 that this assumption is valid. Since macros and namespaces don't go
786 together well, there is an equivalent to @samp{The}: the template
794 if (!(x >= 0)) abort();
795 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
796 // In general, it would be a rational number.
797 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
802 @chapter Functions on numbers
804 Each of the number classes declares its mathematical operations in the
805 corresponding include file. For example, if your code operates with
806 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
809 @section Constructing numbers
811 Here is how to create number objects ``from nothing''.
814 @subsection Constructing integers
816 @code{cl_I} objects are most easily constructed from C integers and from
817 strings. See @ref{Conversions}.
820 @subsection Constructing rational numbers
822 @code{cl_RA} objects can be constructed from strings. The syntax
823 for rational numbers is described in @ref{Internal and printed representation}.
824 Another standard way to produce a rational number is through application
825 of @samp{operator /} or @samp{recip} on integers.
828 @subsection Constructing floating-point numbers
830 @code{cl_F} objects with low precision are most easily constructed from
831 C @samp{float} and @samp{double}. See @ref{Conversions}.
833 To construct a @code{cl_F} with high precision, you can use the conversion
834 from @samp{const char *}, but you have to specify the desired precision
835 within the string. (See @ref{Internal and printed representation}.)
838 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
840 will set @samp{e} to the given value, with a precision of 40 decimal digits.
842 The programmatic way to construct a @code{cl_F} with high precision is
843 through the @code{cl_float} conversion function, see
844 @ref{Conversion to floating-point numbers}. For example, to compute
845 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
846 and then apply the exponential function:
848 float_format_t precision = float_format(40);
849 cl_F e = exp(cl_float(1,precision));
853 @subsection Constructing complex numbers
855 Non-real @code{cl_N} objects are normally constructed through the function
857 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
859 See @ref{Elementary complex functions}.
862 @section Elementary functions
864 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
865 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
866 defines the following operations:
869 @item @var{type} operator + (const @var{type}&, const @var{type}&)
870 @cindex @code{operator + ()}
873 @item @var{type} operator - (const @var{type}&, const @var{type}&)
874 @cindex @code{operator - ()}
877 @item @var{type} operator - (const @var{type}&)
878 Returns the negative of the argument.
880 @item @var{type} plus1 (const @var{type}& x)
881 @cindex @code{plus1 ()}
882 Returns @code{x + 1}.
884 @item @var{type} minus1 (const @var{type}& x)
885 @cindex @code{minus1 ()}
886 Returns @code{x - 1}.
888 @item @var{type} operator * (const @var{type}&, const @var{type}&)
889 @cindex @code{operator * ()}
892 @item @var{type} square (const @var{type}& x)
893 @cindex @code{square ()}
894 Returns @code{x * x}.
897 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
898 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
899 defines the following operations:
902 @item @var{type} operator / (const @var{type}&, const @var{type}&)
903 @cindex @code{operator / ()}
906 @item @var{type} recip (const @var{type}&)
907 @cindex @code{recip ()}
908 Returns the reciprocal of the argument.
911 The class @code{cl_I} doesn't define a @samp{/} operation because
912 in the C/C++ language this operator, applied to integral types,
913 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
914 is implementation dependent). (@xref{Rounding functions}.)
915 Instead, @code{cl_I} defines an ``exact quotient'' function:
918 @item cl_I exquo (const cl_I& x, const cl_I& y)
919 @cindex @code{exquo ()}
920 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
923 The following exponentiation functions are defined:
926 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
927 @cindex @code{expt_pos ()}
928 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
929 @code{y} must be > 0. Returns @code{x^y}.
931 @item cl_RA expt (const cl_RA& x, const cl_I& y)
932 @cindex @code{expt ()}
933 @itemx cl_R expt (const cl_R& x, const cl_I& y)
934 @itemx cl_N expt (const cl_N& x, const cl_I& y)
938 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
939 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
940 defines the following operation:
943 @item @var{type} abs (const @var{type}& x)
944 @cindex @code{abs ()}
945 Returns the absolute value of @code{x}.
946 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
949 The class @code{cl_N} implements this as follows:
952 @item cl_R abs (const cl_N x)
953 Returns the absolute value of @code{x}.
956 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
957 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
958 defines the following operation:
961 @item @var{type} signum (const @var{type}& x)
962 @cindex @code{signum ()}
963 Returns the sign of @code{x}, in the same number format as @code{x}.
964 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
965 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
970 @section Elementary rational functions
972 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
975 @item cl_I numerator (const @var{type}& x)
976 @cindex @code{numerator ()}
977 Returns the numerator of @code{x}.
979 @item cl_I denominator (const @var{type}& x)
980 @cindex @code{denominator ()}
981 Returns the denominator of @code{x}.
984 The numerator and denominator of a rational number are normalized in such
985 a way that they have no factor in common and the denominator is positive.
988 @section Elementary complex functions
990 The class @code{cl_N} defines the following operation:
993 @item cl_N complex (const cl_R& a, const cl_R& b)
994 @cindex @code{complex ()}
995 Returns the complex number @code{a+bi}, that is, the complex number with
996 real part @code{a} and imaginary part @code{b}.
999 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
1002 @item cl_R realpart (const @var{type}& x)
1003 @cindex @code{realpart ()}
1004 Returns the real part of @code{x}.
1006 @item cl_R imagpart (const @var{type}& x)
1007 @cindex @code{imagpart ()}
1008 Returns the imaginary part of @code{x}.
1010 @item @var{type} conjugate (const @var{type}& x)
1011 @cindex @code{conjugate ()}
1012 Returns the complex conjugate of @code{x}.
1015 We have the relations
1019 @code{x = complex(realpart(x), imagpart(x))}
1021 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
1025 @section Comparisons
1028 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1029 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1030 defines the following operations:
1033 @item bool operator == (const @var{type}&, const @var{type}&)
1034 @cindex @code{operator == ()}
1035 @itemx bool operator != (const @var{type}&, const @var{type}&)
1036 @cindex @code{operator != ()}
1037 Comparison, as in C and C++.
1039 @item uint32 equal_hashcode (const @var{type}&)
1040 @cindex @code{equal_hashcode ()}
1041 Returns a 32-bit hash code that is the same for any two numbers which are
1042 the same according to @code{==}. This hash code depends on the number's value,
1043 not its type or precision.
1045 @item cl_boolean zerop (const @var{type}& x)
1046 @cindex @code{zerop ()}
1047 Compare against zero: @code{x == 0}
1050 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1051 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1052 defines the following operations:
1055 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1056 @cindex @code{compare ()}
1057 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1058 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1060 @item bool operator <= (const @var{type}&, const @var{type}&)
1061 @cindex @code{operator <= ()}
1062 @itemx bool operator < (const @var{type}&, const @var{type}&)
1063 @cindex @code{operator < ()}
1064 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1065 @cindex @code{operator >= ()}
1066 @itemx bool operator > (const @var{type}&, const @var{type}&)
1067 @cindex @code{operator > ()}
1068 Comparison, as in C and C++.
1070 @item cl_boolean minusp (const @var{type}& x)
1071 @cindex @code{minusp ()}
1072 Compare against zero: @code{x < 0}
1074 @item cl_boolean plusp (const @var{type}& x)
1075 @cindex @code{plusp ()}
1076 Compare against zero: @code{x > 0}
1078 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1079 @cindex @code{max ()}
1080 Return the maximum of @code{x} and @code{y}.
1082 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1083 @cindex @code{min ()}
1084 Return the minimum of @code{x} and @code{y}.
1087 When a floating point number and a rational number are compared, the float
1088 is first converted to a rational number using the function @code{rational}.
1089 Since a floating point number actually represents an interval of real numbers,
1090 the result might be surprising.
1091 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1092 there is no floating point number whose value is exactly @code{1/3}.
1095 @section Rounding functions
1098 When a real number is to be converted to an integer, there is no ``best''
1099 rounding. The desired rounding function depends on the application.
1100 The Common Lisp and ISO Lisp standards offer four rounding functions:
1104 This is the largest integer <=@code{x}.
1107 This is the smallest integer >=@code{x}.
1110 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1113 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1114 integers, choose the even one.
1117 These functions have different advantages:
1119 @code{floor} and @code{ceiling} are translation invariant:
1120 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1121 for every @code{x} and every integer @code{n}.
1123 On the other hand, @code{truncate} and @code{round} are symmetric:
1124 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1125 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1126 down exactly as often as it rounds up.
1128 The functions are related like this:
1132 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1133 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1135 @code{truncate(x) = sign(x) * floor(abs(x))}
1138 Each of the classes @code{cl_R}, @code{cl_RA},
1139 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1140 defines the following operations:
1143 @item cl_I floor1 (const @var{type}& x)
1144 @cindex @code{floor1 ()}
1145 Returns @code{floor(x)}.
1146 @item cl_I ceiling1 (const @var{type}& x)
1147 @cindex @code{ceiling1 ()}
1148 Returns @code{ceiling(x)}.
1149 @item cl_I truncate1 (const @var{type}& x)
1150 @cindex @code{truncate1 ()}
1151 Returns @code{truncate(x)}.
1152 @item cl_I round1 (const @var{type}& x)
1153 @cindex @code{round1 ()}
1154 Returns @code{round(x)}.
1157 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1158 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1159 defines the following operations:
1162 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1163 Returns @code{floor(x/y)}.
1164 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1165 Returns @code{ceiling(x/y)}.
1166 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1167 Returns @code{truncate(x/y)}.
1168 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1169 Returns @code{round(x/y)}.
1172 These functions are called @samp{floor1}, @dots{} here instead of
1173 @samp{floor}, @dots{}, because on some systems, system dependent include
1174 files define @samp{floor} and @samp{ceiling} as macros.
1176 In many cases, one needs both the quotient and the remainder of a division.
1177 It is more efficient to compute both at the same time than to perform
1178 two divisions, one for quotient and the next one for the remainder.
1179 The following functions therefore return a structure containing both
1180 the quotient and the remainder. The suffix @samp{2} indicates the number
1181 of ``return values''. The remainder is defined as follows:
1185 for the computation of @code{quotient = floor(x)},
1186 @code{remainder = x - quotient},
1188 for the computation of @code{quotient = floor(x,y)},
1189 @code{remainder = x - quotient*y},
1192 and similarly for the other three operations.
1194 Each of the classes @code{cl_R}, @code{cl_RA},
1195 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1196 defines the following operations:
1199 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1200 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1201 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1202 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1203 @itemx @var{type}_div_t round2 (const @var{type}& x)
1206 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1207 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1208 defines the following operations:
1211 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1212 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1213 @cindex @code{floor2 ()}
1214 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1215 @cindex @code{ceiling2 ()}
1216 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1217 @cindex @code{truncate2 ()}
1218 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1219 @cindex @code{round2 ()}
1222 Sometimes, one wants the quotient as a floating-point number (of the
1223 same format as the argument, if the argument is a float) instead of as
1224 an integer. The prefix @samp{f} indicates this.
1227 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1228 defines the following operations:
1231 @item @var{type} ffloor (const @var{type}& x)
1232 @cindex @code{ffloor ()}
1233 @itemx @var{type} fceiling (const @var{type}& x)
1234 @cindex @code{fceiling ()}
1235 @itemx @var{type} ftruncate (const @var{type}& x)
1236 @cindex @code{ftruncate ()}
1237 @itemx @var{type} fround (const @var{type}& x)
1238 @cindex @code{fround ()}
1241 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1243 The class @code{cl_R} defines the following operations:
1246 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1247 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1248 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1249 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1252 These functions also exist in versions which return both the quotient
1253 and the remainder. The suffix @samp{2} indicates this.
1256 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1257 defines the following operations:
1258 @cindex @code{cl_F_fdiv_t}
1259 @cindex @code{cl_SF_fdiv_t}
1260 @cindex @code{cl_FF_fdiv_t}
1261 @cindex @code{cl_DF_fdiv_t}
1262 @cindex @code{cl_LF_fdiv_t}
1265 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1266 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1267 @cindex @code{ffloor2 ()}
1268 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1269 @cindex @code{fceiling2 ()}
1270 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1271 @cindex @code{ftruncate2 ()}
1272 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1273 @cindex @code{fround2 ()}
1275 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1276 @cindex @code{cl_R_fdiv_t}
1278 The class @code{cl_R} defines the following operations:
1281 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1282 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1283 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1284 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1285 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1288 Other applications need only the remainder of a division.
1289 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1290 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1291 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1295 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1297 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1300 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1301 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1302 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1304 The classes @code{cl_R}, @code{cl_I} define the following operations:
1307 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1308 @cindex @code{mod ()}
1309 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1310 @cindex @code{rem ()}
1316 Each of the classes @code{cl_R},
1317 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1318 defines the following operation:
1321 @item @var{type} sqrt (const @var{type}& x)
1322 @cindex @code{sqrt ()}
1323 @code{x} must be >= 0. This function returns the square root of @code{x},
1324 normalized to be >= 0. If @code{x} is the square of a rational number,
1325 @code{sqrt(x)} will be a rational number, else it will return a
1326 floating-point approximation.
1329 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1332 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1333 @cindex @code{sqrtp ()}
1334 This tests whether @code{x} is a perfect square. If so, it returns true
1335 and the exact square root in @code{*root}, else it returns false.
1338 Furthermore, for integers, similarly:
1341 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1342 @cindex @code{isqrt ()}
1343 @code{x} should be >= 0. This function sets @code{*root} to
1344 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1345 the boolean value @code{(expt(*root,2) == x)}.
1348 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1349 define the following operation:
1352 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1353 @cindex @code{rootp ()}
1354 @code{x} must be >= 0. @code{n} must be > 0.
1355 This tests whether @code{x} is an @code{n}th power of a rational number.
1356 If so, it returns true and the exact root in @code{*root}, else it returns
1360 The only square root function which accepts negative numbers is the one
1361 for class @code{cl_N}:
1364 @item cl_N sqrt (const cl_N& z)
1365 @cindex @code{sqrt ()}
1366 Returns the square root of @code{z}, as defined by the formula
1367 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1368 or to a complex number are done if necessary. The range of the result is the
1369 right half plane @code{realpart(sqrt(z)) >= 0}
1370 including the positive imaginary axis and 0, but excluding
1371 the negative imaginary axis.
1372 The result is an exact number only if @code{z} is an exact number.
1376 @section Transcendental functions
1377 @cindex transcendental functions
1379 The transcendental functions return an exact result if the argument
1380 is exact and the result is exact as well. Otherwise they must return
1381 inexact numbers even if the argument is exact.
1382 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1385 @subsection Exponential and logarithmic functions
1388 @item cl_R exp (const cl_R& x)
1389 @cindex @code{exp ()}
1390 @itemx cl_N exp (const cl_N& x)
1391 Returns the exponential function of @code{x}. This is @code{e^x} where
1392 @code{e} is the base of the natural logarithms. The range of the result
1393 is the entire complex plane excluding 0.
1395 @item cl_R ln (const cl_R& x)
1396 @cindex @code{ln ()}
1397 @code{x} must be > 0. Returns the (natural) logarithm of x.
1399 @item cl_N log (const cl_N& x)
1400 @cindex @code{log ()}
1401 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1402 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1403 The range of the result is the strip in the complex plane
1404 @code{-pi < imagpart(log(x)) <= pi}.
1406 @item cl_R phase (const cl_N& x)
1407 @cindex @code{phase ()}
1408 Returns the angle part of @code{x} in its polar representation as a
1409 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1410 This is also the imaginary part of @code{log(x)}.
1411 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1412 The result will be an exact number only if @code{zerop(x)} or
1413 if @code{x} is real and positive.
1415 @item cl_R log (const cl_R& a, const cl_R& b)
1416 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1417 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1418 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1421 @item cl_N log (const cl_N& a, const cl_N& b)
1422 Returns the logarithm of @code{a} with respect to base @code{b}.
1423 @code{log(a,b) = log(a)/log(b)}.
1425 @item cl_N expt (const cl_N& x, const cl_N& y)
1426 @cindex @code{expt ()}
1427 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1430 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1433 @item cl_F exp1 (float_format_t f)
1434 @cindex @code{exp1 ()}
1435 Returns e as a float of format @code{f}.
1437 @item cl_F exp1 (const cl_F& y)
1438 Returns e in the float format of @code{y}.
1440 @item cl_F exp1 (void)
1441 Returns e as a float of format @code{default_float_format}.
1445 @subsection Trigonometric functions
1448 @item cl_R sin (const cl_R& x)
1449 @cindex @code{sin ()}
1450 Returns @code{sin(x)}. The range of the result is the interval
1451 @code{-1 <= sin(x) <= 1}.
1453 @item cl_N sin (const cl_N& z)
1454 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1456 @item cl_R cos (const cl_R& x)
1457 @cindex @code{cos ()}
1458 Returns @code{cos(x)}. The range of the result is the interval
1459 @code{-1 <= cos(x) <= 1}.
1461 @item cl_N cos (const cl_N& x)
1462 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1464 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1465 @cindex @code{cos_sin_t}
1466 @itemx cos_sin_t cos_sin (const cl_R& x)
1467 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1468 @cindex @code{cos_sin ()}
1469 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1470 hold only approximately.
1472 @item cl_R tan (const cl_R& x)
1473 @cindex @code{tan ()}
1474 @itemx cl_N tan (const cl_N& x)
1475 Returns @code{tan(x) = sin(x)/cos(x)}.
1477 @item cl_N cis (const cl_R& x)
1478 @cindex @code{cis ()}
1479 @itemx cl_N cis (const cl_N& x)
1480 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1481 @code{e^(i*x) = cos(x) + i*sin(x)}.
1484 @cindex @code{asin ()}
1485 @item cl_N asin (const cl_N& z)
1486 Returns @code{arcsin(z)}. This is defined as
1487 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1488 @code{arcsin(-z) = -arcsin(z)}.
1489 The range of the result is the strip in the complex domain
1490 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1491 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1492 with @code{realpart = pi/2} and @code{imagpart > 0}.
1494 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1498 @item cl_N acos (const cl_N& z)
1499 @cindex @code{acos ()}
1500 Returns @code{arccos(z)}. This is defined as
1501 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1504 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1506 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1507 The range of the result is the strip in the complex domain
1508 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1509 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1510 with @code{realpart = pi} and @code{imagpart > 0}.
1512 Proof: This follows from the results about arcsin.
1516 @cindex @code{atan ()}
1517 @item cl_R atan (const cl_R& x, const cl_R& y)
1518 Returns the angle of the polar representation of the complex number
1519 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1520 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1521 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1522 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1523 with reversed order of arguments.
1525 @item cl_R atan (const cl_R& x)
1526 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1527 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1528 will be an exact number only if @code{x} is the exact @code{0}.
1530 @item cl_N atan (const cl_N& z)
1531 Returns @code{arctan(z)}. This is defined as
1532 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1533 @code{arctan(-z) = -arctan(z)}. The range of the result is
1534 the strip in the complex domain
1535 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1536 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1537 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1539 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1545 @cindex Archimedes' constant
1546 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1549 @item cl_F pi (float_format_t f)
1550 @cindex @code{pi ()}
1551 Returns pi as a float of format @code{f}.
1553 @item cl_F pi (const cl_F& y)
1554 Returns pi in the float format of @code{y}.
1556 @item cl_F pi (void)
1557 Returns pi as a float of format @code{default_float_format}.
1561 @subsection Hyperbolic functions
1564 @item cl_R sinh (const cl_R& x)
1565 @cindex @code{sinh ()}
1566 Returns @code{sinh(x)}.
1568 @item cl_N sinh (const cl_N& z)
1569 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1571 @item cl_R cosh (const cl_R& x)
1572 @cindex @code{cosh ()}
1573 Returns @code{cosh(x)}. The range of the result is the interval
1574 @code{cosh(x) >= 1}.
1576 @item cl_N cosh (const cl_N& z)
1577 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1579 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1580 @cindex @code{cosh_sinh_t}
1581 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1582 @cindex @code{cosh_sinh ()}
1583 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1584 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1585 hold only approximately.
1587 @item cl_R tanh (const cl_R& x)
1588 @cindex @code{tanh ()}
1589 @itemx cl_N tanh (const cl_N& x)
1590 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1592 @item cl_N asinh (const cl_N& z)
1593 @cindex @code{asinh ()}
1594 Returns @code{arsinh(z)}. This is defined as
1595 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1596 @code{arsinh(-z) = -arsinh(z)}.
1598 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1599 Actually, z+sqrt(1+z^2) can never be real and <0, so
1600 -pi < imagpart(arsinh(z)) < pi.
1601 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1602 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1604 The range of the result is the strip in the complex domain
1605 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1606 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1607 with @code{imagpart = pi/2} and @code{realpart < 0}.
1609 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1610 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1611 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1612 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1613 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1614 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1615 If y >= 1, the imagpart is pi/2 and the realpart is
1616 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1619 Moreover, if z is in Range(sqrt),
1620 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1621 (for a proof, see file src/cl_C_asinh.cc).
1624 @item cl_N acosh (const cl_N& z)
1625 @cindex @code{acosh ()}
1626 Returns @code{arcosh(z)}. This is defined as
1627 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1628 The range of the result is the half-strip in the complex domain
1629 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1630 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1632 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1633 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1634 If z is in Range(sqrt), we have
1635 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1636 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1638 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1639 and since the imagpart of both expressions is > -pi, <= pi
1640 ==> arcosh(z) = log(z+sqrt(z^2-1))
1641 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1642 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1643 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1644 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1645 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1647 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1648 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1649 = x^2 + y^2 + sqrt(u^2+v^2)
1654 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1655 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1656 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1657 Otherwise, -z is in Range(sqrt).
1658 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1659 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1660 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1661 and this has realpart > 0.
1662 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1663 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1664 has realpart = 0 and imagpart > 0.
1665 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1666 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1667 This has realpart >= 0 and imagpart = pi.
1670 @item cl_N atanh (const cl_N& z)
1671 @cindex @code{atanh ()}
1672 Returns @code{artanh(z)}. This is defined as
1673 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1674 @code{artanh(-z) = -artanh(z)}. The range of the result is
1675 the strip in the complex domain
1676 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1677 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1678 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1680 Proof: Write z = x+iy. Examine
1681 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1683 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1684 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1685 |x| < 1 ==> imagpart = 0
1688 = (atan(1+x,y) - atan(1-x,-y))/2
1689 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1690 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1691 > (pi - pi/2 - pi/2 )/2 = 0
1692 and (1+x)/y > (1-x)/y
1693 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1694 ==> imagpart < pi/2.
1695 Hence 0 < imagpart < pi/2.
1697 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1702 @subsection Euler gamma
1703 @cindex Euler's constant
1705 Euler's constant C = 0.577@dots{} is returned by the following functions:
1708 @item cl_F eulerconst (float_format_t f)
1709 @cindex @code{eulerconst ()}
1710 Returns Euler's constant as a float of format @code{f}.
1712 @item cl_F eulerconst (const cl_F& y)
1713 Returns Euler's constant in the float format of @code{y}.
1715 @item cl_F eulerconst (void)
1716 Returns Euler's constant as a float of format @code{default_float_format}.
1719 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1720 @cindex Catalan's constant
1723 @item cl_F catalanconst (float_format_t f)
1724 @cindex @code{catalanconst ()}
1725 Returns Catalan's constant as a float of format @code{f}.
1727 @item cl_F catalanconst (const cl_F& y)
1728 Returns Catalan's constant in the float format of @code{y}.
1730 @item cl_F catalanconst (void)
1731 Returns Catalan's constant as a float of format @code{default_float_format}.
1735 @subsection Riemann zeta
1736 @cindex Riemann's zeta
1738 Riemann's zeta function at an integral point @code{s>1} is returned by the
1739 following functions:
1742 @item cl_F zeta (int s, float_format_t f)
1743 @cindex @code{zeta ()}
1744 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1746 @item cl_F zeta (int s, const cl_F& y)
1747 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1749 @item cl_F zeta (int s)
1750 Returns Riemann's zeta function at @code{s} as a float of format
1751 @code{default_float_format}.
1755 @section Functions on integers
1757 @subsection Logical functions
1759 Integers, when viewed as in two's complement notation, can be thought as
1760 infinite bit strings where the bits' values eventually are constant.
1767 The logical operations view integers as such bit strings and operate
1768 on each of the bit positions in parallel.
1771 @item cl_I lognot (const cl_I& x)
1772 @cindex @code{lognot ()}
1773 @itemx cl_I operator ~ (const cl_I& x)
1774 @cindex @code{operator ~ ()}
1775 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1777 @item cl_I logand (const cl_I& x, const cl_I& y)
1778 @cindex @code{logand ()}
1779 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1780 @cindex @code{operator & ()}
1781 Logical and, like @code{x & y} in C.
1783 @item cl_I logior (const cl_I& x, const cl_I& y)
1784 @cindex @code{logior ()}
1785 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1786 @cindex @code{operator | ()}
1787 Logical (inclusive) or, like @code{x | y} in C.
1789 @item cl_I logxor (const cl_I& x, const cl_I& y)
1790 @cindex @code{logxor ()}
1791 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1792 @cindex @code{operator ^ ()}
1793 Exclusive or, like @code{x ^ y} in C.
1795 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1796 @cindex @code{logeqv ()}
1797 Bitwise equivalence, like @code{~(x ^ y)} in C.
1799 @item cl_I lognand (const cl_I& x, const cl_I& y)
1800 @cindex @code{lognand ()}
1801 Bitwise not and, like @code{~(x & y)} in C.
1803 @item cl_I lognor (const cl_I& x, const cl_I& y)
1804 @cindex @code{lognor ()}
1805 Bitwise not or, like @code{~(x | y)} in C.
1807 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1808 @cindex @code{logandc1 ()}
1809 Logical and, complementing the first argument, like @code{~x & y} in C.
1811 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1812 @cindex @code{logandc2 ()}
1813 Logical and, complementing the second argument, like @code{x & ~y} in C.
1815 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1816 @cindex @code{logorc1 ()}
1817 Logical or, complementing the first argument, like @code{~x | y} in C.
1819 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1820 @cindex @code{logorc2 ()}
1821 Logical or, complementing the second argument, like @code{x | ~y} in C.
1824 These operations are all available though the function
1826 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1827 @cindex @code{boole ()}
1829 where @code{op} must have one of the 16 values (each one stands for a function
1830 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1831 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1832 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1833 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1834 @code{boole_orc1}, @code{boole_orc2}.
1835 @cindex @code{boole_clr}
1836 @cindex @code{boole_set}
1837 @cindex @code{boole_1}
1838 @cindex @code{boole_2}
1839 @cindex @code{boole_c1}
1840 @cindex @code{boole_c2}
1841 @cindex @code{boole_and}
1842 @cindex @code{boole_xor}
1843 @cindex @code{boole_eqv}
1844 @cindex @code{boole_nand}
1845 @cindex @code{boole_nor}
1846 @cindex @code{boole_andc1}
1847 @cindex @code{boole_andc2}
1848 @cindex @code{boole_orc1}
1849 @cindex @code{boole_orc2}
1852 Other functions that view integers as bit strings:
1855 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1856 @cindex @code{logtest ()}
1857 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1858 @code{logand(x,y) != 0}.
1860 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1861 @cindex @code{logbitp ()}
1862 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1863 Bit 0 is the least significant bit.
1865 @item uintL logcount (const cl_I& x)
1866 @cindex @code{logcount ()}
1867 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1868 the number of zero bits in @code{x}, if @code{x} < 0.
1871 The following functions operate on intervals of bits in integers.
1874 struct cl_byte @{ uintL size; uintL position; @};
1876 @cindex @code{cl_byte}
1877 represents the bit interval containing the bits
1878 @code{position}@dots{}@code{position+size-1} of an integer.
1879 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1882 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1883 @cindex @code{ldb ()}
1884 extracts the bits of @code{n} described by the bit interval @code{b}
1885 and returns them as a nonnegative integer with @code{b.size} bits.
1887 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1888 @cindex @code{ldb_test ()}
1889 Returns true if some bit described by the bit interval @code{b} is set in
1892 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1893 @cindex @code{dpb ()}
1894 Returns @code{n}, with the bits described by the bit interval @code{b}
1895 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1896 @code{newbyte} are relevant.
1899 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1900 functions are their counterparts without shifting:
1903 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1904 @cindex @code{mask_field ()}
1905 returns an integer with the bits described by the bit interval @code{b}
1906 copied from the corresponding bits in @code{n}, the other bits zero.
1908 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1909 @cindex @code{deposit_field ()}
1910 returns an integer where the bits described by the bit interval @code{b}
1911 come from @code{newbyte} and the other bits come from @code{n}.
1914 The following relations hold:
1918 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1920 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1922 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1925 The following operations on integers as bit strings are efficient shortcuts
1926 for common arithmetic operations:
1929 @item cl_boolean oddp (const cl_I& x)
1930 @cindex @code{oddp ()}
1931 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1932 @code{mod(x,2) != 0}.
1934 @item cl_boolean evenp (const cl_I& x)
1935 @cindex @code{evenp ()}
1936 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1937 @code{mod(x,2) == 0}.
1939 @item cl_I operator << (const cl_I& x, const cl_I& n)
1940 @cindex @code{operator << ()}
1941 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1942 Equivalent to @code{x * expt(2,n)}.
1944 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1945 @cindex @code{operator >> ()}
1946 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1947 Bits shifted out to the right are thrown away.
1948 Equivalent to @code{floor(x / expt(2,n))}.
1950 @item cl_I ash (const cl_I& x, const cl_I& y)
1951 @cindex @code{ash ()}
1952 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1953 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1954 returns @code{floor(x * expt(2,y))}.
1956 @item uintL integer_length (const cl_I& x)
1957 @cindex @code{integer_length ()}
1958 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1959 in two's complement notation. This is the smallest n >= 0 such that
1960 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1963 @item uintL ord2 (const cl_I& x)
1964 @cindex @code{ord2 ()}
1965 @code{x} must be non-zero. This function returns the number of 0 bits at the
1966 right of @code{x} in two's complement notation. This is the largest n >= 0
1967 such that 2^n divides @code{x}.
1969 @item uintL power2p (const cl_I& x)
1970 @cindex @code{power2p ()}
1971 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1972 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1973 (See also the function @code{logp}.)
1977 @subsection Number theoretic functions
1980 @item uint32 gcd (uint32 a, uint32 b)
1981 @cindex @code{gcd ()}
1982 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1983 This function returns the greatest common divisor of @code{a} and @code{b},
1984 normalized to be >= 0.
1986 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1987 @cindex @code{xgcd ()}
1988 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1989 @code{a} and @code{b} and at the same time the representation of @code{g}
1990 as an integral linear combination of @code{a} and @code{b}:
1991 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1992 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1993 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1994 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1995 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1997 @item cl_I lcm (const cl_I& a, const cl_I& b)
1998 @cindex @code{lcm ()}
1999 This function returns the least common multiple of @code{a} and @code{b},
2000 normalized to be >= 0.
2002 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
2003 @cindex @code{logp ()}
2004 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
2005 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
2006 rational number, this function returns true and sets *l = log(a,b), else
2009 @item int jacobi (sint32 a, sint32 b)
2010 @cindex @code{jacobi()}
2011 @itemx int jacobi (const cl_I& a, const cl_I& b)
2012 Returns the Jacobi symbol
2014 $\left({a\over b}\right)$,
2019 @code{a,b} must be integers, @code{b>0} and odd. The result is 0
2022 @item cl_boolean isprobprime (const cl_I& n)
2024 @cindex @code{isprobprime()}
2025 Returns true if @code{n} is a small prime or passes the Miller-Rabin
2026 primality test. The probability of a false positive is 1:10^30.
2028 @item cl_I nextprobprime (const cl_R& x)
2029 @cindex @code{nextprobprime()}
2030 Returns the smallest probable prime >=@code{x}.
2034 @subsection Combinatorial functions
2037 @item cl_I factorial (uintL n)
2038 @cindex @code{factorial ()}
2039 @code{n} must be a small integer >= 0. This function returns the factorial
2040 @code{n}! = @code{1*2*@dots{}*n}.
2042 @item cl_I doublefactorial (uintL n)
2043 @cindex @code{doublefactorial ()}
2044 @code{n} must be a small integer >= 0. This function returns the
2045 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
2046 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
2048 @item cl_I binomial (uintL n, uintL k)
2049 @cindex @code{binomial ()}
2050 @code{n} and @code{k} must be small integers >= 0. This function returns the
2051 binomial coefficient
2053 ${n \choose k} = {n! \over n! (n-k)!}$
2056 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2058 for 0 <= k <= n, 0 else.
2062 @section Functions on floating-point numbers
2064 Recall that a floating-point number consists of a sign @code{s}, an
2065 exponent @code{e} and a mantissa @code{m}. The value of the number is
2066 @code{(-1)^s * 2^e * m}.
2069 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2070 defines the following operations.
2073 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2074 @cindex @code{scale_float ()}
2075 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2076 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2077 because it copies @code{x} and modifies the exponent.
2080 The following functions provide an abstract interface to the underlying
2081 representation of floating-point numbers.
2084 @item sintL float_exponent (const @var{type}& x)
2085 @cindex @code{float_exponent ()}
2086 Returns the exponent @code{e} of @code{x}.
2087 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2088 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2090 @item sintL float_radix (const @var{type}& x)
2091 @cindex @code{float_radix ()}
2092 Returns the base of the floating-point representation. This is always @code{2}.
2094 @item @var{type} float_sign (const @var{type}& x)
2095 @cindex @code{float_sign ()}
2096 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2097 @code{x} >= 0, -1 for @code{x} < 0.
2099 @item uintL float_digits (const @var{type}& x)
2100 @cindex @code{float_digits ()}
2101 Returns the number of mantissa bits in the floating-point representation
2102 of @code{x}, including the hidden bit. The value only depends on the type
2103 of @code{x}, not on its value.
2105 @item uintL float_precision (const @var{type}& x)
2106 @cindex @code{float_precision ()}
2107 Returns the number of significant mantissa bits in the floating-point
2108 representation of @code{x}. Since denormalized numbers are not supported,
2109 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2113 The complete internal representation of a float is encoded in the type
2114 @cindex @code{decoded_float}
2115 @cindex @code{decoded_sfloat}
2116 @cindex @code{decoded_ffloat}
2117 @cindex @code{decoded_dfloat}
2118 @cindex @code{decoded_lfloat}
2119 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2120 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2122 struct decoded_@var{type}float @{
2123 @var{type} mantissa; cl_I exponent; @var{type} sign;
2127 and returned by the function
2130 @item decoded_@var{type}float decode_float (const @var{type}& x)
2131 @cindex @code{decode_float ()}
2132 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2133 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2134 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2135 @code{e} is the same as returned by the function @code{float_exponent}.
2138 A complete decoding in terms of integers is provided as type
2139 @cindex @code{cl_idecoded_float}
2141 struct cl_idecoded_float @{
2142 cl_I mantissa; cl_I exponent; cl_I sign;
2145 by the following function:
2148 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2149 @cindex @code{integer_decode_float ()}
2150 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2151 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2152 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2153 WARNING: The exponent @code{e} is not the same as the one returned by
2154 the functions @code{decode_float} and @code{float_exponent}.
2157 Some other function, implemented only for class @code{cl_F}:
2160 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2161 @cindex @code{float_sign ()}
2162 This returns a floating point number whose precision and absolute value
2163 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2164 zero, it is treated as positive. Same for @code{y}.
2168 @section Conversion functions
2171 @subsection Conversion to floating-point numbers
2173 The type @code{float_format_t} describes a floating-point format.
2174 @cindex @code{float_format_t}
2177 @item float_format_t float_format (uintL n)
2178 @cindex @code{float_format ()}
2179 Returns the smallest float format which guarantees at least @code{n}
2180 decimal digits in the mantissa (after the decimal point).
2182 @item float_format_t float_format (const cl_F& x)
2183 Returns the floating point format of @code{x}.
2185 @item float_format_t default_float_format
2186 @cindex @code{default_float_format}
2187 Global variable: the default float format used when converting rational numbers
2191 To convert a real number to a float, each of the types
2192 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2193 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2194 defines the following operations:
2197 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2198 @cindex @code{cl_float ()}
2199 Returns @code{x} as a float of format @code{f}.
2200 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2201 Returns @code{x} in the float format of @code{y}.
2202 @item cl_F cl_float (const @var{type}&x)
2203 Returns @code{x} as a float of format @code{default_float_format} if
2204 it is an exact number, or @code{x} itself if it is already a float.
2207 Of course, converting a number to a float can lose precision.
2209 Every floating-point format has some characteristic numbers:
2212 @item cl_F most_positive_float (float_format_t f)
2213 @cindex @code{most_positive_float ()}
2214 Returns the largest (most positive) floating point number in float format @code{f}.
2216 @item cl_F most_negative_float (float_format_t f)
2217 @cindex @code{most_negative_float ()}
2218 Returns the smallest (most negative) floating point number in float format @code{f}.
2220 @item cl_F least_positive_float (float_format_t f)
2221 @cindex @code{least_positive_float ()}
2222 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2223 in float format @code{f}.
2225 @item cl_F least_negative_float (float_format_t f)
2226 @cindex @code{least_negative_float ()}
2227 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2228 in float format @code{f}.
2230 @item cl_F float_epsilon (float_format_t f)
2231 @cindex @code{float_epsilon ()}
2232 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2234 @item cl_F float_negative_epsilon (float_format_t f)
2235 @cindex @code{float_negative_epsilon ()}
2236 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2240 @subsection Conversion to rational numbers
2242 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2243 defines the following operation:
2246 @item cl_RA rational (const @var{type}& x)
2247 @cindex @code{rational ()}
2248 Returns the value of @code{x} as an exact number. If @code{x} is already
2249 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2250 the value is a rational number whose denominator is a power of 2.
2253 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2257 @item cl_RA rationalize (const cl_R& x)
2258 @cindex @code{rationalize ()}
2259 If @code{x} is a floating-point number, it actually represents an interval
2260 of real numbers, and this function returns the rational number with
2261 smallest denominator (and smallest numerator, in magnitude)
2262 which lies in this interval.
2263 If @code{x} is already an exact number, this function returns @code{x}.
2266 If @code{x} is any float, one has
2270 @code{cl_float(rational(x),x) = x}
2272 @code{cl_float(rationalize(x),x) = x}
2276 @section Random number generators
2279 A random generator is a machine which produces (pseudo-)random numbers.
2280 The include file @code{<cln/random.h>} defines a class @code{random_state}
2281 which contains the state of a random generator. If you make a copy
2282 of the random number generator, the original one and the copy will produce
2283 the same sequence of random numbers.
2285 The following functions return (pseudo-)random numbers in different formats.
2286 Calling one of these modifies the state of the random number generator in
2287 a complicated but deterministic way.
2290 @cindex @code{random_state}
2291 @cindex @code{default_random_state}
2293 random_state default_random_state
2295 contains a default random number generator. It is used when the functions
2296 below are called without @code{random_state} argument.
2299 @item uint32 random32 (random_state& randomstate)
2300 @itemx uint32 random32 ()
2301 @cindex @code{random32 ()}
2302 Returns a random unsigned 32-bit number. All bits are equally random.
2304 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2305 @itemx cl_I random_I (const cl_I& n)
2306 @cindex @code{random_I ()}
2307 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2308 in the range @code{0 <= x < n}.
2310 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2311 @itemx cl_F random_F (const cl_F& n)
2312 @cindex @code{random_F ()}
2313 @code{n} must be a float > 0. This function returns a random floating-point
2314 number of the same format as @code{n} in the range @code{0 <= x < n}.
2316 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2317 @itemx cl_R random_R (const cl_R& n)
2318 @cindex @code{random_R ()}
2319 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2320 if @code{n} is a float.
2324 @section Obfuscating operators
2325 @cindex modifying operators
2327 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2328 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2329 are not available by default because their
2330 use tends to make programs unreadable. It is trivial to get away without
2331 them. However, if you feel that you absolutely need these operators
2332 to get happy, then add
2334 #define WANT_OBFUSCATING_OPERATORS
2336 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2337 to the beginning of your source files, before the inclusion of any CLN
2338 include files. This flag will enable the following operators:
2340 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2341 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2344 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2345 @cindex @code{operator += ()}
2346 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2347 @cindex @code{operator -= ()}
2348 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2349 @cindex @code{operator *= ()}
2350 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2351 @cindex @code{operator /= ()}
2354 For the class @code{cl_I}:
2357 @item @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 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2361 @cindex @code{operator &= ()}
2362 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2363 @cindex @code{operator |= ()}
2364 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2365 @cindex @code{operator ^= ()}
2366 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2367 @cindex @code{operator <<= ()}
2368 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2369 @cindex @code{operator >>= ()}
2372 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2373 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2376 @item @var{type}& operator ++ (@var{type}& x)
2377 @cindex @code{operator ++ ()}
2378 The prefix operator @code{++x}.
2380 @item void operator ++ (@var{type}& x, int)
2381 The postfix operator @code{x++}.
2383 @item @var{type}& operator -- (@var{type}& x)
2384 @cindex @code{operator -- ()}
2385 The prefix operator @code{--x}.
2387 @item void operator -- (@var{type}& x, int)
2388 The postfix operator @code{x--}.
2391 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2392 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2396 @chapter Input/Output
2397 @cindex Input/Output
2399 @section Internal and printed representation
2400 @cindex representation
2402 All computations deal with the internal representations of the numbers.
2404 Every number has an external representation as a sequence of ASCII characters.
2405 Several external representations may denote the same number, for example,
2406 "20.0" and "20.000".
2408 Converting an internal to an external representation is called ``printing'',
2410 converting an external to an internal representation is called ``reading''.
2412 In CLN, it is always true that conversion of an internal to an external
2413 representation and then back to an internal representation will yield the
2414 same internal representation. Symbolically: @code{read(print(x)) == x}.
2415 This is called ``print-read consistency''.
2417 Different types of numbers have different external representations (case
2422 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2423 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2424 for decimal integers
2425 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2427 @item Rational numbers
2428 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2429 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2432 @item Floating-point numbers
2433 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2434 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2435 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2436 of the form _@var{prec} may be appended. There must be at least
2437 one digit in the non-exponent part. The exponent has the syntax
2438 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2439 The exponent marker is
2443 @samp{s} for short-floats,
2445 @samp{f} for single-floats,
2447 @samp{d} for double-floats,
2449 @samp{L} for long-floats,
2452 or @samp{e}, which denotes a default float format. The precision specifying
2453 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2454 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2455 function @samp{float_format}.
2457 @item Complex numbers
2458 External representation:
2461 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2462 if @var{imagpart} is negative, its printed representation begins with
2463 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2464 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2465 is rational and the rational number's base is >18, because the @samp{i}
2466 is then read as a digit.
2468 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2473 @section Input functions
2475 Including @code{<cln/io.h>} defines a number of simple input functions
2476 that read from @code{std::istream&}:
2479 @item int freadchar (std::istream& stream)
2480 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2481 if the end of stream was encountered or an error occurred.
2483 @item int funreadchar (std::istream& stream, int c)
2484 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2485 last @code{freadchar} operation on @code{stream}.
2488 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2489 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2490 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2493 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2494 Reads a number from @code{stream} and stores it in the @code{result}.
2497 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2501 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2502 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2503 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2504 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2505 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2506 Reads a number from @code{stream}. The @code{flags} are parameters which
2507 affect the input syntax. Whitespace before the number is silently skipped.
2509 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2510 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2511 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2512 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2513 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2514 Reads a number from a string in memory. The @code{flags} are parameters which
2515 affect the input syntax. The string starts at @code{string} and ends at
2516 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2517 @code{NULL}, denoting the entire string, i.e. equivalent to
2518 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2519 @code{NULL}, the string in memory must contain exactly one number and nothing
2520 more, else a fatal error will be signalled. If @code{end_of_parse}
2521 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2522 the last parsed character (i.e. @code{string_limit} if nothing came after
2523 the number). Whitespace is not allowed.
2526 The structure @code{cl_read_flags} contains the following fields:
2529 @item cl_read_syntax_t syntax
2530 The possible results of the read operation. Possible values are
2531 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2532 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2533 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2535 @item cl_read_lsyntax_t lsyntax
2536 Specifies the language-dependent syntax variant for the read operation.
2540 @item lsyntax_standard
2541 accept standard algebraic notation only, no complex numbers,
2542 @item lsyntax_algebraic
2543 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2544 @item lsyntax_commonlisp
2545 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2546 hexadecimal numbers,
2547 @code{#@var{base}R} for rational numbers in a given base,
2548 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2550 accept all of these extensions.
2553 @item unsigned int rational_base
2554 The base in which rational numbers are read.
2556 @item float_format_t float_flags.default_float_format
2557 The float format used when reading floats with exponent marker @samp{e}.
2559 @item float_format_t float_flags.default_lfloat_format
2560 The float format used when reading floats with exponent marker @samp{l}.
2562 @item cl_boolean float_flags.mantissa_dependent_float_format
2563 When this flag is true, floats specified with more digits than corresponding
2564 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2565 precision corresponding to their number of significant digits.
2569 @section Output functions
2571 Including @code{<cln/io.h>} defines a number of simple output functions
2572 that write to @code{std::ostream&}:
2575 @item void fprintchar (std::ostream& stream, char c)
2576 Prints the character @code{x} literally on the @code{stream}.
2578 @item void fprint (std::ostream& stream, const char * string)
2579 Prints the @code{string} literally on the @code{stream}.
2581 @item void fprintdecimal (std::ostream& stream, int x)
2582 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2583 Prints the integer @code{x} in decimal on the @code{stream}.
2585 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2586 Prints the integer @code{x} in binary (base 2, without prefix)
2587 on the @code{stream}.
2589 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2590 Prints the integer @code{x} in octal (base 8, without prefix)
2591 on the @code{stream}.
2593 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2594 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2595 on the @code{stream}.
2598 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2599 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2600 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2603 @item void fprint (std::ostream& stream, const @var{type}& x)
2604 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2605 Prints the number @code{x} on the @code{stream}. The output may depend
2606 on the global printer settings in the variable @code{default_print_flags}.
2607 The @code{ostream} flags and settings (flags, width and locale) are
2611 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2614 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2616 void print_real (std::ostream& stream, const cl_print_flags& flags,
2618 void print_float (std::ostream& stream, const cl_print_flags& flags,
2620 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2622 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2625 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2626 parameters which affect the output.
2628 The structure type @code{cl_print_flags} contains the following fields:
2631 @item unsigned int rational_base
2632 The base in which rational numbers are printed. Default is @code{10}.
2634 @item cl_boolean rational_readably
2635 If this flag is true, rational numbers are printed with radix specifiers in
2636 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2637 prefixes, trailing dot). Default is false.
2639 @item cl_boolean float_readably
2640 If this flag is true, type specific exponent markers have precedence over 'E'.
2643 @item float_format_t default_float_format
2644 Floating point numbers of this format will be printed using the 'E' exponent
2645 marker. Default is @code{float_format_ffloat}.
2647 @item cl_boolean complex_readably
2648 If this flag is true, complex numbers will be printed using the Common Lisp
2649 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2651 @item cl_string univpoly_varname
2652 Univariate polynomials with no explicit indeterminate name will be printed
2653 using this variable name. Default is @code{"x"}.
2656 The global variable @code{default_print_flags} contains the default values,
2657 used by the function @code{fprint}.
2662 CLN has a class of abstract rings.
2670 Rings can be compared for equality:
2673 @item bool operator== (const cl_ring&, const cl_ring&)
2674 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2675 These compare two rings for equality.
2678 Given a ring @code{R}, the following members can be used.
2681 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2682 @cindex @code{fprint ()}
2683 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2684 @cindex @code{equal ()}
2685 @itemx cl_ring_element R->zero ()
2686 @cindex @code{zero ()}
2687 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2688 @cindex @code{zerop ()}
2689 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2690 @cindex @code{plus ()}
2691 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2692 @cindex @code{minus ()}
2693 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2694 @cindex @code{uminus ()}
2695 @itemx cl_ring_element R->one ()
2696 @cindex @code{one ()}
2697 @itemx cl_ring_element R->canonhom (const cl_I& x)
2698 @cindex @code{canonhom ()}
2699 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2700 @cindex @code{mul ()}
2701 @itemx cl_ring_element R->square (const cl_ring_element& x)
2702 @cindex @code{square ()}
2703 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2704 @cindex @code{expt_pos ()}
2707 The following rings are built-in.
2710 @item cl_null_ring cl_0_ring
2711 The null ring, containing only zero.
2713 @item cl_complex_ring cl_C_ring
2714 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2716 @item cl_real_ring cl_R_ring
2717 The ring of real numbers. This corresponds to the type @code{cl_R}.
2719 @item cl_rational_ring cl_RA_ring
2720 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2722 @item cl_integer_ring cl_I_ring
2723 The ring of integers. This corresponds to the type @code{cl_I}.
2726 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2727 @code{cl_RA_ring}, @code{cl_I_ring}:
2730 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2731 @cindex @code{instanceof ()}
2732 Tests whether the given number is an element of the number ring R.
2736 @chapter Modular integers
2737 @cindex modular integer
2739 @section Modular integer rings
2742 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2743 The modulus is explicitly part of every modular integer. CLN doesn't
2744 allow you to (accidentally) mix elements of different modular rings,
2745 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2746 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2747 doesn't have generic types. So one has to live with runtime checks.)
2749 The class of modular integer rings is
2757 Modular integer ring
2761 @cindex @code{cl_modint_ring}
2763 and the class of all modular integers (elements of modular integer rings) is
2771 Modular integer rings are constructed using the function
2774 @item cl_modint_ring find_modint_ring (const cl_I& N)
2775 @cindex @code{find_modint_ring ()}
2776 This function returns the modular ring @samp{Z/NZ}. It takes care
2777 of finding out about special cases of @code{N}, like powers of two
2778 and odd numbers for which Montgomery multiplication will be a win,
2779 @cindex Montgomery multiplication
2780 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2781 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2782 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2786 Modular integer rings can be compared for equality:
2789 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2790 @cindex @code{operator == ()}
2791 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2792 @cindex @code{operator != ()}
2793 These compare two modular integer rings for equality. Two different calls
2794 to @code{find_modint_ring} with the same argument necessarily return the
2795 same ring because it is memoized in the cache table.
2798 @section Functions on modular integers
2800 Given a modular integer ring @code{R}, the following members can be used.
2803 @item cl_I R->modulus
2804 @cindex @code{modulus}
2805 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2807 @item cl_MI R->zero()
2808 @cindex @code{zero ()}
2809 This returns @code{0 mod N}.
2811 @item cl_MI R->one()
2812 @cindex @code{one ()}
2813 This returns @code{1 mod N}.
2815 @item cl_MI R->canonhom (const cl_I& x)
2816 @cindex @code{canonhom ()}
2817 This returns @code{x mod N}.
2819 @item cl_I R->retract (const cl_MI& x)
2820 @cindex @code{retract ()}
2821 This is a partial inverse function to @code{R->canonhom}. It returns the
2822 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2824 @item cl_MI R->random(random_state& randomstate)
2825 @itemx cl_MI R->random()
2826 @cindex @code{random ()}
2827 This returns a random integer modulo @code{N}.
2830 The following operations are defined on modular integers.
2833 @item cl_modint_ring x.ring ()
2834 @cindex @code{ring ()}
2835 Returns the ring to which the modular integer @code{x} belongs.
2837 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2838 @cindex @code{operator + ()}
2839 Returns the sum of two modular integers. One of the arguments may also
2842 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2843 @cindex @code{operator - ()}
2844 Returns the difference of two modular integers. One of the arguments may also
2847 @item cl_MI operator- (const cl_MI&)
2848 Returns the negative of a modular integer.
2850 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2851 @cindex @code{operator * ()}
2852 Returns the product of two modular integers. One of the arguments may also
2855 @item cl_MI square (const cl_MI&)
2856 @cindex @code{square ()}
2857 Returns the square of a modular integer.
2859 @item cl_MI recip (const cl_MI& x)
2860 @cindex @code{recip ()}
2861 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2862 must be coprime to the modulus, otherwise an error message is issued.
2864 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2865 @cindex @code{div ()}
2866 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2867 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2869 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2870 @cindex @code{expt_pos ()}
2871 @code{y} must be > 0. Returns @code{x^y}.
2873 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2874 @cindex @code{expt ()}
2875 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2876 modulus, else an error message is issued.
2878 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2879 @cindex @code{operator << ()}
2880 Returns @code{x*2^y}.
2882 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2883 @cindex @code{operator >> ()}
2884 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2885 or an error message is issued.
2887 @item bool operator== (const cl_MI&, const cl_MI&)
2888 @cindex @code{operator == ()}
2889 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2890 @cindex @code{operator != ()}
2891 Compares two modular integers, belonging to the same modular integer ring,
2894 @item cl_boolean zerop (const cl_MI& x)
2895 @cindex @code{zerop ()}
2896 Returns true if @code{x} is @code{0 mod N}.
2899 The following output functions are defined (see also the chapter on
2903 @item void fprint (std::ostream& stream, const cl_MI& x)
2904 @cindex @code{fprint ()}
2905 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
2906 @cindex @code{operator << ()}
2907 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2908 on the global printer settings in the variable @code{default_print_flags}.
2912 @chapter Symbolic data types
2913 @cindex symbolic type
2915 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2919 @cindex @code{cl_string}
2929 implements immutable strings.
2931 Strings are constructed through the following constructors:
2934 @item cl_string (const char * s)
2935 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2937 @item cl_string (const char * ptr, unsigned long len)
2938 Returns an immutable copy of the @code{len} characters at
2939 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2942 The following functions are available on strings:
2946 Assignment from @code{cl_string} and @code{const char *}.
2949 @cindex @code{length ()}
2951 @cindex @code{strlen ()}
2952 Returns the length of the string @code{s}.
2955 @cindex @code{operator [] ()}
2956 Returns the @code{i}th character of the string @code{s}.
2957 @code{i} must be in the range @code{0 <= i < s.length()}.
2959 @item bool equal (const cl_string& s1, const cl_string& s2)
2960 @cindex @code{equal ()}
2961 Compares two strings for equality. One of the arguments may also be a
2962 plain @code{const char *}.
2967 @cindex @code{cl_symbol}
2969 Symbols are uniquified strings: all symbols with the same name are shared.
2970 This means that comparison of two symbols is fast (effectively just a pointer
2971 comparison), whereas comparison of two strings must in the worst case walk
2972 both strings until their end.
2973 Symbols are used, for example, as tags for properties, as names of variables
2974 in polynomial rings, etc.
2976 Symbols are constructed through the following constructor:
2979 @item cl_symbol (const cl_string& s)
2980 Looks up or creates a new symbol with a given name.
2983 The following operations are available on symbols:
2986 @item cl_string (const cl_symbol& sym)
2987 Conversion to @code{cl_string}: Returns the string which names the symbol
2990 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2991 @cindex @code{equal ()}
2992 Compares two symbols for equality. This is very fast.
2996 @chapter Univariate polynomials
2998 @cindex univariate polynomial
3000 @section Univariate polynomial rings
3002 CLN implements univariate polynomials (polynomials in one variable) over an
3003 arbitrary ring. The indeterminate variable may be either unnamed (and will be
3004 printed according to @code{default_print_flags.univpoly_varname}, which
3005 defaults to @samp{x}) or carry a given name. The base ring and the
3006 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
3007 (accidentally) mix elements of different polynomial rings, e.g.
3008 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
3009 return a multivariate polynomial, but they are not yet implemented in CLN.)
3011 The classes of univariate polynomial rings are
3019 Univariate polynomial ring
3023 +----------------+-------------------+
3025 Complex polynomial ring | Modular integer polynomial ring
3026 cl_univpoly_complex_ring | cl_univpoly_modint_ring
3027 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3031 Real polynomial ring |
3032 cl_univpoly_real_ring |
3033 <cln/univpoly_real.h> |
3037 Rational polynomial ring |
3038 cl_univpoly_rational_ring |
3039 <cln/univpoly_rational.h> |
3043 Integer polynomial ring
3044 cl_univpoly_integer_ring
3045 <cln/univpoly_integer.h>
3048 and the corresponding classes of univariate polynomials are
3051 Univariate polynomial
3055 +----------------+-------------------+
3057 Complex polynomial | Modular integer polynomial
3059 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3065 <cln/univpoly_real.h> |
3069 Rational polynomial |
3071 <cln/univpoly_rational.h> |
3077 <cln/univpoly_integer.h>
3080 Univariate polynomial rings are constructed using the functions
3083 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3084 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3085 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3086 @code{R} may be an arbitrary ring. This function takes care of finding out
3087 about special cases of @code{R}, such as the rings of complex numbers,
3088 real numbers, rational numbers, integers, or modular integer rings.
3089 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3090 This ensures that two calls of this function with the same arguments will
3091 return the same polynomial ring.
3093 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3094 @cindex @code{find_univpoly_ring ()}
3095 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3096 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3097 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3098 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3099 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3100 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3101 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3102 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3103 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3104 These functions are equivalent to the general @code{find_univpoly_ring},
3105 only the return type is more specific, according to the base ring's type.
3108 @section Functions on univariate polynomials
3110 Given a univariate polynomial ring @code{R}, the following members can be used.
3113 @item cl_ring R->basering()
3114 @cindex @code{basering ()}
3115 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3117 @item cl_UP R->zero()
3118 @cindex @code{zero ()}
3119 This returns @code{0 in R}, a polynomial of degree -1.
3121 @item cl_UP R->one()
3122 @cindex @code{one ()}
3123 This returns @code{1 in R}, a polynomial of degree == 0.
3125 @item cl_UP R->canonhom (const cl_I& x)
3126 @cindex @code{canonhom ()}
3127 This returns @code{x in R}, a polynomial of degree <= 0.
3129 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3130 @cindex @code{monomial ()}
3131 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3134 @item cl_UP R->create (sintL degree)
3135 @cindex @code{create ()}
3136 Creates a new polynomial with a given degree. The zero polynomial has degree
3137 @code{-1}. After creating the polynomial, you should put in the coefficients,
3138 using the @code{set_coeff} member function, and then call the @code{finalize}
3142 The following are the only destructive operations on univariate polynomials.
3145 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3146 @cindex @code{set_coeff ()}
3147 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3148 After changing a polynomial and before applying any "normal" operation on it,
3149 you should call its @code{finalize} member function.
3151 @item void finalize (cl_UP& x)
3152 @cindex @code{finalize ()}
3153 This function marks the endpoint of destructive modifications of a polynomial.
3154 It normalizes the internal representation so that subsequent computations have
3155 less overhead. Doing normal computations on unnormalized polynomials may
3156 produce wrong results or crash the program.
3159 The following operations are defined on univariate polynomials.
3162 @item cl_univpoly_ring x.ring ()
3163 @cindex @code{ring ()}
3164 Returns the ring to which the univariate polynomial @code{x} belongs.
3166 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3167 @cindex @code{operator + ()}
3168 Returns the sum of two univariate polynomials.
3170 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3171 @cindex @code{operator - ()}
3172 Returns the difference of two univariate polynomials.
3174 @item cl_UP operator- (const cl_UP&)
3175 Returns the negative of a univariate polynomial.
3177 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3178 @cindex @code{operator * ()}
3179 Returns the product of two univariate polynomials. One of the arguments may
3180 also be a plain integer or an element of the base ring.
3182 @item cl_UP square (const cl_UP&)
3183 @cindex @code{square ()}
3184 Returns the square of a univariate polynomial.
3186 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3187 @cindex @code{expt_pos ()}
3188 @code{y} must be > 0. Returns @code{x^y}.
3190 @item bool operator== (const cl_UP&, const cl_UP&)
3191 @cindex @code{operator == ()}
3192 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3193 @cindex @code{operator != ()}
3194 Compares two univariate polynomials, belonging to the same univariate
3195 polynomial ring, for equality.
3197 @item cl_boolean zerop (const cl_UP& x)
3198 @cindex @code{zerop ()}
3199 Returns true if @code{x} is @code{0 in R}.
3201 @item sintL degree (const cl_UP& x)
3202 @cindex @code{degree ()}
3203 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3205 @item sintL ldegree (const cl_UP& x)
3206 @cindex @code{degree ()}
3207 Returns the low degree of the polynomial. This is the degree of the first
3208 non-vanishing polynomial coefficient. The zero polynomial has ldegree @code{-1}.
3210 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3211 @cindex @code{coeff ()}
3212 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3214 @item cl_ring_element x (const cl_ring_element& y)
3215 @cindex @code{operator () ()}
3216 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3217 then @samp{x(y)} returns the value of the substitution of @code{y} into
3220 @item cl_UP deriv (const cl_UP& x)
3221 @cindex @code{deriv ()}
3222 Returns the derivative of the polynomial @code{x} with respect to the
3223 indeterminate @code{X}.
3226 The following output functions are defined (see also the chapter on
3230 @item void fprint (std::ostream& stream, const cl_UP& x)
3231 @cindex @code{fprint ()}
3232 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3233 @cindex @code{operator << ()}
3234 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3235 depend on the global printer settings in the variable
3236 @code{default_print_flags}.
3239 @section Special polynomials
3241 The following functions return special polynomials.
3244 @item cl_UP_I tschebychev (sintL n)
3245 @cindex @code{tschebychev ()}
3246 @cindex Chebyshev polynomial
3247 Returns the n-th Chebyshev polynomial (n >= 0).
3249 @item cl_UP_I hermite (sintL n)
3250 @cindex @code{hermite ()}
3251 @cindex Hermite polynomial
3252 Returns the n-th Hermite polynomial (n >= 0).
3254 @item cl_UP_RA legendre (sintL n)
3255 @cindex @code{legendre ()}
3256 @cindex Legende polynomial
3257 Returns the n-th Legendre polynomial (n >= 0).
3259 @item cl_UP_I laguerre (sintL n)
3260 @cindex @code{laguerre ()}
3261 @cindex Laguerre polynomial
3262 Returns the n-th Laguerre polynomial (n >= 0).
3265 Information how to derive the differential equation satisfied by each
3266 of these polynomials from their definition can be found in the
3267 @code{doc/polynomial/} directory.
3275 Using C++ as an implementation language provides
3279 Efficiency: It compiles to machine code.
3283 Portability: It runs on all platforms supporting a C++ compiler. Because
3284 of the availability of GNU C++, this includes all currently used 32-bit and
3285 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3288 Type safety: The C++ compilers knows about the number types and complains if,
3289 for example, you try to assign a float to an integer variable. However,
3290 a drawback is that C++ doesn't know about generic types, hence a restriction
3291 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3292 arguments belong to the same modular ring cannot be expressed as a compile-time
3296 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3297 @code{=}, @code{==}, ... can be used in infix notation, which is more
3298 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3301 With these language features, there is no need for two separate languages,
3302 one for the implementation of the library and one in which the library's users
3303 can program. This means that a prototype implementation of an algorithm
3304 can be integrated into the library immediately after it has been tested and
3305 debugged. No need to rewrite it in a low-level language after having prototyped
3306 in a high-level language.
3309 @section Memory efficiency
3311 In order to save memory allocations, CLN implements:
3315 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3318 @cindex garbage collection
3319 @cindex reference counting
3320 Garbage collection: A reference counting mechanism makes sure that any
3321 number object's storage is freed immediately when the last reference to the
3324 @cindex immediate numbers
3325 Small integers are represented as immediate values instead of pointers
3326 to heap allocated storage. This means that integers @code{> -2^29},
3327 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3332 @section Speed efficiency
3334 Speed efficiency is obtained by the combination of the following tricks
3339 Small integers, being represented as immediate values, don't require
3340 memory access, just a couple of instructions for each elementary operation.
3342 The kernel of CLN has been written in assembly language for some CPUs
3343 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3345 On all CPUs, CLN may be configured to use the superefficient low-level
3346 routines from GNU GMP version 3.
3348 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3349 algorithm, the Karatsuba multiplication, which is an
3360 For very large numbers (more than 12000 decimal digits), CLN uses
3362 Sch{@"o}nhage-Strassen
3363 @cindex Sch{@"o}nhage-Strassen multiplication
3367 @cindex Schnhage-Strassen multiplication
3369 multiplication, which is an asymptotically optimal multiplication
3372 These fast multiplication algorithms also give improvements in the speed
3373 of division and radix conversion.
3377 @section Garbage collection
3378 @cindex garbage collection
3380 All the number classes are reference count classes: They only contain a pointer
3381 to an object in the heap. Upon construction, assignment and destruction of
3382 number objects, only the objects' reference count are manipulated.
3384 Memory occupied by number objects are automatically reclaimed as soon as
3385 their reference count drops to zero.
3387 For number rings, another strategy is implemented: There is a cache of,
3388 for example, the modular integer rings. A modular integer ring is destroyed
3389 only if its reference count dropped to zero and the cache is about to be
3390 resized. The effect of this strategy is that recently used rings remain
3391 cached, whereas undue memory consumption through cached rings is avoided.
3394 @chapter Using the library
3396 For the following discussion, we will assume that you have installed
3397 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3398 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3399 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3400 environment variables, or directly substitute the appropriate values.
3403 @section Compiler options
3404 @cindex compiler options
3406 Until you have installed CLN in a public place, the following options are
3409 When you compile CLN application code, add the flags
3411 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3413 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3414 When you link CLN application code to form an executable, add the flags
3416 $CLN_TARGETDIR/src/libcln.a
3418 to the C/C++ compiler's command line (@code{make} variable LIBS).
3420 If you did a @code{make install}, the include files are installed in a
3421 public directory (normally @code{/usr/local/include}), hence you don't
3422 need special flags for compiling. The library has been installed to a
3423 public directory as well (normally @code{/usr/local/lib}), hence when
3424 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3426 Since CLN version 1.1, there are two tools to make the creation of
3427 software packages that use CLN easier:
3430 @cindex @code{cln-config}
3431 @code{cln-config} is a shell script that you can use to determine the
3432 compiler and linker command line options required to compile and link a
3433 program with CLN. Start it with @code{--help} to learn about its options
3434 or consult the manpage that comes with it.
3436 @cindex @code{AC_PATH_CLN}
3437 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3440 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3442 This macro determines the location of CLN using @code{cln-config}, which
3443 is either found in the user's path, or from the environment variable
3444 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3445 their version is not earlier than @var{MIN-VERSION} (a default version
3446 will be used if not specified). If the required version was found, sets
3447 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3448 macro is in the file @file{cln.m4} which is installed in
3449 @file{$datadir/aclocal}. Note that if automake was installed with a
3450 different @samp{--prefix} than CLN, you will either have to manually
3451 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3452 aclocal the @samp{-I} option when running it. Here is a possible example
3453 to be included in your package's @file{configure.ac}:
3455 AC_PATH_CLN(1.1.0, [
3456 LIBS="$LIBS $CLN_LIBS"
3457 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3458 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3463 @section Compatibility to old CLN versions
3465 @cindex compatibility
3467 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3468 @code{cln} in order to avoid potential name clashes with other C++
3469 libraries. If you have an old application, you will have to manually
3470 port it to the new scheme. The following principles will help during
3474 All headers are now in a separate subdirectory. Instead of including
3475 @code{cl_}@var{something}@code{.h}, include
3476 @code{cln/}@var{something}@code{.h} now.
3478 All public identifiers (typenames and functions) have lost their
3479 @code{cl_} prefix. Exceptions are all the typenames of number types,
3480 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3481 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3482 names would not be mnemonic enough once the namespace @code{cln} is
3483 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3485 All public @emph{functions} that had by a @code{cl_} in their name still
3486 carry that @code{cl_} if it is intrinsic part of a typename (as in
3487 @code{cl_I_to_int ()}).
3489 When developing other libraries, please keep in mind not to import the
3490 namespace @code{cln} in one of your public header files by saying
3491 @code{using namespace cln;}. This would propagate to other applications
3492 and can cause name clashes there.
3495 @section Include files
3496 @cindex include files
3497 @cindex header files
3499 Here is a summary of the include files and their contents.
3502 @item <cln/object.h>
3503 General definitions, reference counting, garbage collection.
3504 @item <cln/number.h>
3505 The class cl_number.
3506 @item <cln/complex.h>
3507 Functions for class cl_N, the complex numbers.
3509 Functions for class cl_R, the real numbers.
3511 Functions for class cl_F, the floats.
3512 @item <cln/sfloat.h>
3513 Functions for class cl_SF, the short-floats.
3514 @item <cln/ffloat.h>
3515 Functions for class cl_FF, the single-floats.
3516 @item <cln/dfloat.h>
3517 Functions for class cl_DF, the double-floats.
3518 @item <cln/lfloat.h>
3519 Functions for class cl_LF, the long-floats.
3520 @item <cln/rational.h>
3521 Functions for class cl_RA, the rational numbers.
3522 @item <cln/integer.h>
3523 Functions for class cl_I, the integers.
3526 @item <cln/complex_io.h>
3527 Input/Output for class cl_N, the complex numbers.
3528 @item <cln/real_io.h>
3529 Input/Output for class cl_R, the real numbers.
3530 @item <cln/float_io.h>
3531 Input/Output for class cl_F, the floats.
3532 @item <cln/sfloat_io.h>
3533 Input/Output for class cl_SF, the short-floats.
3534 @item <cln/ffloat_io.h>
3535 Input/Output for class cl_FF, the single-floats.
3536 @item <cln/dfloat_io.h>
3537 Input/Output for class cl_DF, the double-floats.
3538 @item <cln/lfloat_io.h>
3539 Input/Output for class cl_LF, the long-floats.
3540 @item <cln/rational_io.h>
3541 Input/Output for class cl_RA, the rational numbers.
3542 @item <cln/integer_io.h>
3543 Input/Output for class cl_I, the integers.
3545 Flags for customizing input operations.
3546 @item <cln/output.h>
3547 Flags for customizing output operations.
3548 @item <cln/malloc.h>
3549 @code{malloc_hook}, @code{free_hook}.
3552 @item <cln/condition.h>
3553 Conditions/exceptions.
3554 @item <cln/string.h>
3556 @item <cln/symbol.h>
3558 @item <cln/proplist.h>
3562 @item <cln/null_ring.h>
3564 @item <cln/complex_ring.h>
3565 The ring of complex numbers.
3566 @item <cln/real_ring.h>
3567 The ring of real numbers.
3568 @item <cln/rational_ring.h>
3569 The ring of rational numbers.
3570 @item <cln/integer_ring.h>
3571 The ring of integers.
3572 @item <cln/numtheory.h>
3573 Number threory functions.
3574 @item <cln/modinteger.h>
3580 @item <cln/GV_number.h>
3581 General vectors over cl_number.
3582 @item <cln/GV_complex.h>
3583 General vectors over cl_N.
3584 @item <cln/GV_real.h>
3585 General vectors over cl_R.
3586 @item <cln/GV_rational.h>
3587 General vectors over cl_RA.
3588 @item <cln/GV_integer.h>
3589 General vectors over cl_I.
3590 @item <cln/GV_modinteger.h>
3591 General vectors of modular integers.
3594 @item <cln/SV_number.h>
3595 Simple vectors over cl_number.
3596 @item <cln/SV_complex.h>
3597 Simple vectors over cl_N.
3598 @item <cln/SV_real.h>
3599 Simple vectors over cl_R.
3600 @item <cln/SV_rational.h>
3601 Simple vectors over cl_RA.
3602 @item <cln/SV_integer.h>
3603 Simple vectors over cl_I.
3604 @item <cln/SV_ringelt.h>
3605 Simple vectors of general ring elements.
3606 @item <cln/univpoly.h>
3607 Univariate polynomials.
3608 @item <cln/univpoly_integer.h>
3609 Univariate polynomials over the integers.
3610 @item <cln/univpoly_rational.h>
3611 Univariate polynomials over the rational numbers.
3612 @item <cln/univpoly_real.h>
3613 Univariate polynomials over the real numbers.
3614 @item <cln/univpoly_complex.h>
3615 Univariate polynomials over the complex numbers.
3616 @item <cln/univpoly_modint.h>
3617 Univariate polynomials over modular integer rings.
3618 @item <cln/timing.h>
3621 Includes all of the above.
3627 A function which computes the nth Fibonacci number can be written as follows.
3628 @cindex Fibonacci number
3631 #include <cln/integer.h>
3632 #include <cln/real.h>
3633 using namespace cln;
3635 // Returns F_n, computed as the nearest integer to
3636 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3637 const cl_I fibonacci (int n)
3639 // Need a precision of ((1+sqrt(5))/2)^-n.
3640 float_format_t prec = float_format((int)(0.208987641*n+5));
3641 cl_R sqrt5 = sqrt(cl_float(5,prec));
3642 cl_R phi = (1+sqrt5)/2;
3643 return round1( expt(phi,n)/sqrt5 );
3647 Let's explain what is going on in detail.
3649 The include file @code{<cln/integer.h>} is necessary because the type
3650 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3651 is needed for the type @code{cl_R} and the floating point number functions.
3652 The order of the include files does not matter. In order not to write
3653 out @code{cln::}@var{foo} in this simple example we can safely import
3654 the whole namespace @code{cln}.
3656 Then comes the function declaration. The argument is an @code{int}, the
3657 result an integer. The return type is defined as @samp{const cl_I}, not
3658 simply @samp{cl_I}, because that allows the compiler to detect typos like
3659 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3660 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3661 number). We use the most specialized possible return type because functions
3662 which call @samp{fibonacci} will be able to profit from the compiler's type
3663 analysis: Adding two integers is slightly more efficient than adding the
3664 same objects declared as complex numbers, because it needs less type
3665 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3666 the size of the resulting executable program.
3668 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3669 integer. In order to get a correct result, the absolute error should be less
3670 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3671 To this end, the first line computes a floating point precision for sqrt(5)
3674 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3675 number and than taking the square root. The converse, first taking the square
3676 root of 5, and then converting to the desired precision, would not work in
3677 CLN: The square root would be computed to a default precision (normally
3678 single-float precision), and the following conversion could not help about
3679 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3680 system and does not represent sqrt(5) in a non-numeric way.
3682 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3683 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3684 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3685 @code{cl_N} because a @samp{round1} does not exist for general complex
3688 When the function returns, all the local variables in the function are
3689 automatically reclaimed (garbage collected). Only the result survives and
3690 gets passed to the caller.
3692 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3693 contains this implementation together with an even faster algorithm.
3695 @section Debugging support
3698 When debugging a CLN application with GNU @code{gdb}, two facilities are
3699 available from the library:
3702 @item The library does type checks, range checks, consistency checks at
3703 many places. When one of these fails, the function @code{cl_abort()} is
3704 called. Its default implementation is to perform an @code{exit(1)}, so
3705 you won't have a core dump. But for debugging, it is best to set a
3706 breakpoint at this function:
3708 (gdb) break cl_abort
3710 When this breakpoint is hit, look at the stack's backtrace:
3715 @item The debugger's normal @code{print} command doesn't know about
3716 CLN's types and therefore prints mostly useless hexadecimal addresses.
3717 CLN offers a function @code{cl_print}, callable from the debugger,
3718 for printing number objects. In order to get this function, you have
3719 to define the macro @samp{CL_DEBUG} and then include all the header files
3720 for which you want @code{cl_print} debugging support. For example:
3721 @cindex @code{CL_DEBUG}
3724 #include <cln/string.h>
3726 Now, if you have in your program a variable @code{cl_string s}, and
3727 inspect it under @code{gdb}, the output may look like this:
3730 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3731 word = 134568800@}@}, @}
3732 (gdb) call cl_print(s)
3736 Note that the output of @code{cl_print} goes to the program's error output,
3737 not to gdb's standard output.
3739 Note, however, that the above facility does not work with all CLN types,
3740 only with number objects and similar. Therefore CLN offers a member function
3741 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3742 is needed for this member function to be implemented. Under @code{gdb},
3743 you call it like this:
3744 @cindex @code{debug_print ()}
3747 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3748 word = 134568800@}@}, @}
3749 (gdb) call s.debug_print()
3752 >call ($1).debug_print()
3757 Unfortunately, this feature does not seem to work under all circumstances.
3761 @chapter Customizing
3764 @section Error handling
3766 When a fatal error occurs, an error message is output to the standard error
3767 output stream, and the function @code{cl_abort} is called. The default
3768 version of this function (provided in the library) terminates the application.
3769 To catch such a fatal error, you need to define the function @code{cl_abort}
3770 yourself, with the prototype
3772 #include <cln/abort.h>
3773 void cl_abort (void);
3775 @cindex @code{cl_abort ()}
3776 This function must not return control to its caller.
3779 @section Floating-point underflow
3782 Floating point underflow denotes the situation when a floating-point number
3783 is to be created which is so close to @code{0} that its exponent is too
3784 low to be represented internally. By default, this causes a fatal error.
3785 If you set the global variable
3787 cl_boolean cl_inhibit_floating_point_underflow
3789 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3790 will be generated instead. The default value of
3791 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3794 @section Customizing I/O
3796 The output of the function @code{fprint} may be customized by changing the
3797 value of the global variable @code{default_print_flags}.
3798 @cindex @code{default_print_flags}
3801 @section Customizing the memory allocator
3803 Every memory allocation of CLN is done through the function pointer
3804 @code{malloc_hook}. Freeing of this memory is done through the function
3805 pointer @code{free_hook}. The default versions of these functions,
3806 provided in the library, call @code{malloc} and @code{free} and check
3807 the @code{malloc} result against @code{NULL}.
3808 If you want to provide another memory allocator, you need to define
3809 the variables @code{malloc_hook} and @code{free_hook} yourself,
3812 #include <cln/malloc.h>
3814 void* (*malloc_hook) (size_t size) = @dots{};
3815 void (*free_hook) (void* ptr) = @dots{};
3818 @cindex @code{malloc_hook ()}
3819 @cindex @code{free_hook ()}
3820 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3822 It is not possible to change the memory allocator at runtime, because
3823 it is already called at program startup by the constructors of some