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.
18 @c Don't need the other types of indices.
29 This file documents @sc{cln}, a Class Library for Numbers.
31 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
32 Richard Kreckel, @code{<kreckel@@ginac.de>}.
34 Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000.
36 Permission is granted to make and distribute verbatim copies of
37 this manual provided the copyright notice and this permission notice
38 are preserved on all copies.
41 Permission is granted to process this file through TeX and print the
42 results, provided the printed document carries copying permission
43 notice identical to this one except for the removal of this paragraph
44 (this paragraph not being relevant to the printed manual).
47 Permission is granted to copy and distribute modified versions of this
48 manual under the conditions for verbatim copying, provided that the entire
49 resulting derived work is distributed under the terms of a permission
50 notice identical to this one.
52 Permission is granted to copy and distribute translations of this manual
53 into another language, under the above conditions for modified versions,
54 except that this permission notice may be stated in a translation approved
60 @c prevent ugly black rectangles on overfull hbox lines:
63 @title CLN, a Class Library for Numbers
65 @author by Bruno Haible
67 @vskip 0pt plus 1filll
68 Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000.
71 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
72 Richard Kreckel, @code{<kreckel@@ginac.de>}.
74 Permission is granted to make and distribute verbatim copies of
75 this manual provided the copyright notice and this permission notice
76 are preserved on all copies.
78 Permission is granted to copy and distribute modified versions of this
79 manual under the conditions for verbatim copying, provided that the entire
80 resulting derived work is distributed under the terms of a permission
81 notice identical to this one.
83 Permission is granted to copy and distribute translations of this manual
84 into another language, under the above conditions for modified versions,
85 except that this permission notice may be stated in a translation approved
92 @node Top, Introduction, (dir), (dir)
95 @c * Introduction:: Introduction
99 @node Introduction, Top, Top, Top
100 @comment node-name, next, previous, up
101 @chapter Introduction
104 CLN is a library for computations with all kinds of numbers.
105 It has a rich set of number classes:
109 Integers (with unlimited precision),
115 Floating-point numbers:
125 Long float (with unlimited precision),
132 Modular integers (integers modulo a fixed integer),
135 Univariate polynomials.
139 The subtypes of the complex numbers among these are exactly the
140 types of numbers known to the Common Lisp language. Therefore
141 @code{CLN} can be used for Common Lisp implementations, giving
142 @samp{CLN} another meaning: it becomes an abbreviation of
143 ``Common Lisp Numbers''.
146 The CLN package implements
150 Elementary functions (@code{+}, @code{-}, @code{*}, @code{/}, @code{sqrt},
151 comparisons, @dots{}),
154 Logical functions (logical @code{and}, @code{or}, @code{not}, @dots{}),
157 Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
158 functions and their inverse functions).
162 CLN is a C++ library. Using C++ as an implementation language provides
166 efficiency: it compiles to machine code,
168 type safety: the C++ compiler knows about the number types and complains
169 if, for example, you try to assign a float to an integer variable.
171 algebraic syntax: You can use the @code{+}, @code{-}, @code{*}, @code{=},
172 @code{==}, @dots{} operators as in C or C++.
176 CLN is memory efficient:
180 Small integers and short floats are immediate, not heap allocated.
182 Heap-allocated memory is reclaimed through an automatic, non-interruptive
187 CLN is speed efficient:
191 The kernel of CLN has been written in assembly language for some CPUs
192 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
195 On all CPUs, CLN may be configured to use the superefficient low-level
196 routines from GNU GMP version 3.
198 It uses Karatsuba multiplication, which is significantly faster
199 for large numbers than the standard multiplication algorithm.
201 For very large numbers (more than 12000 decimal digits), it uses
203 Sch{@"o}nhage-Strassen
204 @cindex Sch{@"o}nhage-Strassen multiplication
208 @cindex Schönhage-Strassen multiplication
210 multiplication, which is an asymptotically optimal multiplication
211 algorithm, for multiplication, division and radix conversion.
215 CLN aims at being easily integrated into larger software packages:
219 The garbage collection imposes no burden on the main application.
221 The library provides hooks for memory allocation and exceptions.
224 All non-macro identifiers are hidden in namespace @code{cln} in
225 order to avoid name clashes.
229 @chapter Installation
231 This section describes how to install the CLN package on your system.
234 @section Prerequisites
236 @subsection C++ compiler
238 To build CLN, you need a C++ compiler.
239 Actually, you need GNU @code{g++ 2.90} or newer, the EGCS compilers will
241 I recommend GNU @code{g++ 2.95} or newer.
243 The following C++ features are used:
244 classes, member functions, overloading of functions and operators,
245 constructors and destructors, inline, const, multiple inheritance,
246 templates and namespaces.
248 The following C++ features are not used:
249 @code{new}, @code{delete}, virtual inheritance, exceptions.
251 CLN relies on semi-automatic ordering of initializations
252 of static and global variables, a feature which I could
253 implement for GNU g++ only.
256 @comment cl_modules.h requires g++
257 Therefore nearly any C++ compiler will do.
259 The following C++ compilers are known to compile CLN:
262 GNU @code{g++ 2.7.0}, @code{g++ 2.7.2}
267 The following C++ compilers are known to be unusable for CLN:
270 On SunOS 4, @code{CC 2.1}, because it doesn't grok @code{//} comments
271 in lines containing @code{#if} or @code{#elif} preprocessor commands.
273 On AIX 3.2.5, @code{xlC}, because it doesn't grok the template syntax
274 in @code{cl_SV.h} and @code{cl_GV.h}, because it forces most class types
275 to have default constructors, and because it probably miscompiles the
276 integer multiplication routines.
278 On AIX 4.1.4.0, @code{xlC}, because when optimizing, it sometimes converts
279 @code{short}s to @code{int}s by zero-extend.
283 On HPPA, GNU @code{g++ 2.7.x}, because the semi-automatic ordering of
284 initializations will not work.
288 @subsection Make utility
291 To build CLN, you also need to have GNU @code{make} installed.
293 @subsection Sed utility
296 To build CLN on HP-UX, you also need to have GNU @code{sed} installed.
297 This is because the libtool script, which creates the CLN library, relies
298 on @code{sed}, and the vendor's @code{sed} utility on these systems is too
302 @section Building the library
304 As with any autoconfiguring GNU software, installation is as easy as this:
312 If on your system, @samp{make} is not GNU @code{make}, you have to use
313 @samp{gmake} instead of @samp{make} above.
315 The @code{configure} command checks out some features of your system and
316 C++ compiler and builds the @code{Makefile}s. The @code{make} command
317 builds the library. This step may take 4 hours on an average workstation.
318 The @code{make check} runs some test to check that no important subroutine
319 has been miscompiled.
321 The @code{configure} command accepts options. To get a summary of them, try
327 Some of the options are explained in detail in the @samp{INSTALL.generic} file.
329 You can specify the C compiler, the C++ compiler and their options through
330 the following environment variables when running @code{configure}:
334 Specifies the C compiler.
337 Flags to be given to the C compiler when compiling programs (not when linking).
340 Specifies the C++ compiler.
343 Flags to be given to the C++ compiler when compiling programs (not when linking).
349 $ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
350 $ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
351 CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
352 $ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
353 CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
356 @comment cl_modules.h requires g++
357 You should not mix GNU and non-GNU compilers. So, if @code{CXX} is a non-GNU
358 compiler, @code{CC} should be set to a non-GNU compiler as well. Examples:
361 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O" ./configure
362 $ CC="gcc -V 2.7.0" CFLAGS="-g" CXX="g++ -V 2.7.0" CXXFLAGS="-g" ./configure
365 On SGI Irix 5, if you wish not to use @code{g++}:
368 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O -Olimit 16000" ./configure
371 On SGI Irix 6, if you wish not to use @code{g++}:
374 $ CC="cc -32" CFLAGS="-O" CXX="CC -32" CXXFLAGS="-O -Olimit 34000" \
375 ./configure --without-gmp
376 $ CC="cc -n32" CFLAGS="-O" CXX="CC -n32" CXXFLAGS="-O \
377 -OPT:const_copy_limit=32400 -OPT:global_limit=32400 -OPT:fprop_limit=4000" \
378 ./configure --without-gmp
382 Note that for these environment variables to take effect, you have to set
383 them (assuming a Bourne-compatible shell) on the same line as the
384 @code{configure} command. If you made the settings in earlier shell
385 commands, you have to @code{export} the environment variables before
386 calling @code{configure}. In a @code{csh} shell, you have to use the
387 @samp{setenv} command for setting each of the environment variables.
389 Currently CLN works only with the GNU @code{g++} compiler, and only in
390 optimizing mode. So you should specify at least @code{-O} in the CXXFLAGS,
391 or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use @code{-O}.)
393 If you use @code{g++} version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or
394 gcc-2.95.x, I recommend adding @samp{-fno-exceptions} to the CXXFLAGS.
395 This will likely generate better code.
397 If you use @code{g++} version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc,
398 add either @samp{-O}, @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the
399 CXXFLAGS. With full @samp{-O2}, @code{g++} miscompiles the division routines.
400 Also, if you have @code{g++} version egcs-1.1.1 or older on Sparc, you must
401 specify @samp{--disable-shared} because @code{g++} would miscompile parts of
404 By default, both a shared and a static library are built. You can build
405 CLN as a static (or shared) library only, by calling @code{configure} with
406 the option @samp{--disable-shared} (or @samp{--disable-static}). While
407 shared libraries are usually more convenient to use, they may not work
408 on all architectures. Try disabling them if you run into linker
409 problems. Also, they are generally somewhat slower than static
410 libraries so runtime-critical applications should be linked statically.
413 @subsection Using the GNU MP Library
416 Starting with version 1.1, CLN may be configured to make use of a
417 preinstalled @code{gmp} library. Please make sure that you have at
418 least @code{gmp} version 3.0 installed since earlier versions are
419 unsupported and likely not to work. Enabling this feature by calling
420 @code{configure} with the option @samp{--with-gmp} is known to be quite
421 a boost for CLN's performance.
423 If you have installed the @code{gmp} library and its header file in
424 some place where your compiler cannot find it by default, you must help
425 @code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
429 $ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
430 CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
434 @section Installing the library
437 As with any autoconfiguring GNU software, installation is as easy as this:
443 The @samp{make install} command installs the library and the include files
444 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
445 if you haven't specified a @code{--prefix} option to @code{configure}).
446 This step may require superuser privileges.
448 If you have already built the library and wish to install it, but didn't
449 specify @code{--prefix=@dots{}} at configure time, just re-run
450 @code{configure}, giving it the same options as the first time, plus
451 the @code{--prefix=@dots{}} option.
456 You can remove system-dependent files generated by @code{make} through
462 You can remove all files generated by @code{make}, thus reverting to a
463 virgin distribution of CLN, through
470 @chapter Ordinary number types
472 CLN implements the following class hierarchy:
480 Real or complex number
489 +-------------------+-------------------+
491 Rational number Floating-point number
493 <cln/rational.h> <cln/float.h>
495 | +--------------+--------------+--------------+
497 cl_I Short-Float Single-Float Double-Float Long-Float
498 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
499 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
502 @cindex @code{cl_number}
503 @cindex abstract class
504 The base class @code{cl_number} is an abstract base class.
505 It is not useful to declare a variable of this type except if you want
506 to completely disable compile-time type checking and use run-time type
511 @cindex complex number
512 The class @code{cl_N} comprises real and complex numbers. There is
513 no special class for complex numbers since complex numbers with imaginary
514 part @code{0} are automatically converted to real numbers.
517 The class @code{cl_R} comprises real numbers of different kinds. It is an
521 @cindex rational number
523 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
524 integers. There is no special class for non-integral rational numbers
525 since rational numbers with denominator @code{1} are automatically converted
529 The class @code{cl_F} implements floating-point approximations to real numbers.
530 It is an abstract class.
533 @section Exact numbers
536 Some numbers are represented as exact numbers: there is no loss of information
537 when such a number is converted from its mathematical value to its internal
538 representation. On exact numbers, the elementary operations (@code{+},
539 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
542 In CLN, the exact numbers are:
546 rational numbers (including integers),
548 complex numbers whose real and imaginary parts are both rational numbers.
551 Rational numbers are always normalized to the form
552 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
553 are coprime integers and the denominator is positive. If the resulting
554 denominator is @code{1}, the rational number is converted to an integer.
556 Small integers (typically in the range @code{-2^30}@dots{}@code{2^30-1},
557 for 32-bit machines) are especially efficient, because they consume no heap
558 allocation. Otherwise the distinction between these immediate integers
559 (called ``fixnums'') and heap allocated integers (called ``bignums'')
560 is completely transparent.
563 @section Floating-point numbers
564 @cindex floating-point number
566 Not all real numbers can be represented exactly. (There is an easy mathematical
567 proof for this: Only a countable set of numbers can be stored exactly in
568 a computer, even if one assumes that it has unlimited storage. But there
569 are uncountably many real numbers.) So some approximation is needed.
570 CLN implements ordinary floating-point numbers, with mantissa and exponent.
572 @cindex rounding error
573 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
574 only return approximate results. For example, the value of the expression
575 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
576 @samp{0.7}. Rounding errors like this one are inevitable when computing
577 with floating-point numbers.
579 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
580 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
581 rule: It first computes the exact mathematical result and then returns the
582 floating-point number which is nearest to this. If two floating-point numbers
583 are equally distant from the ideal result, the one with a @code{0} in its least
584 significant mantissa bit is chosen.
586 Similarly, testing floating point numbers for equality @samp{x == y}
587 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
588 for some well-chosen @code{epsilon}.
590 Floating point numbers come in four flavors:
595 Short floats, type @code{cl_SF}.
596 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
597 and 17 mantissa bits (including the ``hidden'' bit).
598 They don't consume heap allocation.
602 Single floats, type @code{cl_FF}.
603 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
604 and 24 mantissa bits (including the ``hidden'' bit).
605 In CLN, they are represented as IEEE single-precision floating point numbers.
606 This corresponds closely to the C/C++ type @samp{float}.
610 Double floats, type @code{cl_DF}.
611 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
612 and 53 mantissa bits (including the ``hidden'' bit).
613 In CLN, they are represented as IEEE double-precision floating point numbers.
614 This corresponds closely to the C/C++ type @samp{double}.
618 Long floats, type @code{cl_LF}.
619 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
620 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
621 The precision of a long float is unlimited, but once created, a long float
622 has a fixed precision. (No ``lazy recomputation''.)
625 Of course, computations with long floats are more expensive than those
626 with smaller floating-point formats.
628 CLN does not implement features like NaNs, denormalized numbers and
629 gradual underflow. If the exponent range of some floating-point type
630 is too limited for your application, choose another floating-point type
631 with larger exponent range.
634 As a user of CLN, you can forget about the differences between the
635 four floating-point types and just declare all your floating-point
636 variables as being of type @code{cl_F}. This has the advantage that
637 when you change the precision of some computation (say, from @code{cl_DF}
638 to @code{cl_LF}), you don't have to change the code, only the precision
639 of the initial values. Also, many transcendental functions have been
640 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
641 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
642 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
643 the floating point contagion rule happened to change in the future.)
646 @section Complex numbers
647 @cindex complex number
649 Complex numbers, as implemented by the class @code{cl_N}, have a real
650 part and an imaginary part, both real numbers. A complex number whose
651 imaginary part is the exact number @code{0} is automatically converted
654 Complex numbers can arise from real numbers alone, for example
655 through application of @code{sqrt} or transcendental functions.
661 Conversions from any class to any its superclasses (``base classes'' in
662 C++ terminology) is done automatically.
664 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
665 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
666 @code{cl_N} and @code{cl_number}.
668 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
669 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
670 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
671 efficiency. Their range is therefore limited:
675 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
677 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
680 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
681 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
682 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
684 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
685 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
686 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
687 @samp{cl_I}, first convert it to an @samp{unsigned long}.
689 Conversions from the C built-in type @samp{float} are provided for the classes
690 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
692 Conversions from the C built-in type @samp{double} are provided for the classes
693 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
695 Conversions from @samp{const char *} are provided for the classes
696 @code{cl_I}, @code{cl_RA},
697 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
698 @code{cl_R}, @code{cl_N}.
699 The easiest way to specify a value which is outside of the range of the
700 C++ built-in types is therefore to specify it as a string, like this:
703 cl_I order_of_rubiks_cube_group = "43252003274489856000";
705 Note that this conversion is done at runtime, not at compile-time.
707 Conversions from @code{cl_I} to the C built-in types @samp{int},
708 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
712 @item int cl_I_to_int (const cl_I& x)
713 @cindex @code{cl_I_to_int ()}
714 @itemx unsigned int cl_I_to_uint (const cl_I& x)
715 @cindex @code{cl_I_to_uint ()}
716 @itemx long cl_I_to_long (const cl_I& x)
717 @cindex @code{cl_I_to_long ()}
718 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
719 @cindex @code{cl_I_to_ulong ()}
720 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
721 representable in the range of @var{ctype}, a runtime error occurs.
724 Conversions from the classes @code{cl_I}, @code{cl_RA},
725 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
727 to the C built-in types @samp{float} and @samp{double} are provided through
731 @item float float_approx (const @var{type}& x)
732 @cindex @code{float_approx ()}
733 @itemx double double_approx (const @var{type}& x)
734 @cindex @code{double_approx ()}
735 Returns an approximation of @code{x} of C type @var{ctype}.
736 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
737 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
740 Conversions from any class to any of its subclasses (``derived classes'' in
741 C++ terminology) are not provided. Instead, you can assert and check
742 that a value belongs to a certain subclass, and return it as element of that
743 class, using the @samp{As} and @samp{The} macros.
744 @cindex @code{As()()}
745 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
746 @var{type} and returns it as such.
747 @cindex @code{The()()}
748 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
749 @var{type} and returns it as such. It is your responsibility to ensure
750 that this assumption is valid.
756 if (!(x >= 0)) abort();
757 cl_I ten_x = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
758 // In general, it would be a rational number.
763 @chapter Functions on numbers
765 Each of the number classes declares its mathematical operations in the
766 corresponding include file. For example, if your code operates with
767 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
770 @section Constructing numbers
772 Here is how to create number objects ``from nothing''.
775 @subsection Constructing integers
777 @code{cl_I} objects are most easily constructed from C integers and from
778 strings. See @ref{Conversions}.
781 @subsection Constructing rational numbers
783 @code{cl_RA} objects can be constructed from strings. The syntax
784 for rational numbers is described in @ref{Internal and printed representation}.
785 Another standard way to produce a rational number is through application
786 of @samp{operator /} or @samp{recip} on integers.
789 @subsection Constructing floating-point numbers
791 @code{cl_F} objects with low precision are most easily constructed from
792 C @samp{float} and @samp{double}. See @ref{Conversions}.
794 To construct a @code{cl_F} with high precision, you can use the conversion
795 from @samp{const char *}, but you have to specify the desired precision
796 within the string. (See @ref{Internal and printed representation}.)
799 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
801 will set @samp{e} to the given value, with a precision of 40 decimal digits.
803 The programmatic way to construct a @code{cl_F} with high precision is
804 through the @code{cl_float} conversion function, see
805 @ref{Conversion to floating-point numbers}. For example, to compute
806 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
807 and then apply the exponential function:
809 cl_float_format_t precision = cl_float_format(40);
810 cl_F e = exp(cl_float(1,precision));
814 @subsection Constructing complex numbers
816 Non-real @code{cl_N} objects are normally constructed through the function
818 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
820 See @ref{Elementary complex functions}.
823 @section Elementary functions
825 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
826 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
827 defines the following operations:
830 @item @var{type} operator + (const @var{type}&, const @var{type}&)
831 @cindex @code{operator + ()}
834 @item @var{type} operator - (const @var{type}&, const @var{type}&)
835 @cindex @code{operator - ()}
838 @item @var{type} operator - (const @var{type}&)
839 Returns the negative of the argument.
841 @item @var{type} plus1 (const @var{type}& x)
842 @cindex @code{plus1 ()}
843 Returns @code{x + 1}.
845 @item @var{type} minus1 (const @var{type}& x)
846 @cindex @code{minus1 ()}
847 Returns @code{x - 1}.
849 @item @var{type} operator * (const @var{type}&, const @var{type}&)
850 @cindex @code{operator * ()}
853 @item @var{type} square (const @var{type}& x)
854 @cindex @code{square ()}
855 Returns @code{x * x}.
858 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
859 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
860 defines the following operations:
863 @item @var{type} operator / (const @var{type}&, const @var{type}&)
864 @cindex @code{operator / ()}
867 @item @var{type} recip (const @var{type}&)
868 @cindex @code{recip ()}
869 Returns the reciprocal of the argument.
872 The class @code{cl_I} doesn't define a @samp{/} operation because
873 in the C/C++ language this operator, applied to integral types,
874 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
875 is implementation dependent). (@xref{Rounding functions}.)
876 Instead, @code{cl_I} defines an ``exact quotient'' function:
879 @item cl_I exquo (const cl_I& x, const cl_I& y)
880 @cindex @code{exquo ()}
881 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
884 The following exponentiation functions are defined:
887 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
888 @cindex @code{expt_pos ()}
889 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
890 @code{y} must be > 0. Returns @code{x^y}.
892 @item cl_RA expt (const cl_RA& x, const cl_I& y)
893 @cindex @code{expt ()}
894 @itemx cl_R expt (const cl_R& x, const cl_I& y)
895 @itemx cl_N expt (const cl_N& x, const cl_I& y)
899 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
900 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
901 defines the following operation:
904 @item @var{type} abs (const @var{type}& x)
905 @cindex @code{abs ()}
906 Returns the absolute value of @code{x}.
907 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
910 The class @code{cl_N} implements this as follows:
913 @item cl_R abs (const cl_N x)
914 Returns the absolute value of @code{x}.
917 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
918 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
919 defines the following operation:
922 @item @var{type} signum (const @var{type}& x)
923 @cindex @code{signum ()}
924 Returns the sign of @code{x}, in the same number format as @code{x}.
925 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
926 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
931 @section Elementary rational functions
933 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
936 @item cl_I numerator (const @var{type}& x)
937 @cindex @code{numerator ()}
938 Returns the numerator of @code{x}.
940 @item cl_I denominator (const @var{type}& x)
941 @cindex @code{denominator ()}
942 Returns the denominator of @code{x}.
945 The numerator and denominator of a rational number are normalized in such
946 a way that they have no factor in common and the denominator is positive.
949 @section Elementary complex functions
951 The class @code{cl_N} defines the following operation:
954 @item cl_N complex (const cl_R& a, const cl_R& b)
955 @cindex @code{complex ()}
956 Returns the complex number @code{a+bi}, that is, the complex number with
957 real part @code{a} and imaginary part @code{b}.
960 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
963 @item cl_R realpart (const @var{type}& x)
964 @cindex @code{realpart ()}
965 Returns the real part of @code{x}.
967 @item cl_R imagpart (const @var{type}& x)
968 @cindex @code{imagpart ()}
969 Returns the imaginary part of @code{x}.
971 @item @var{type} conjugate (const @var{type}& x)
972 @cindex @code{conjugate ()}
973 Returns the complex conjugate of @code{x}.
976 We have the relations
980 @code{x = complex(realpart(x), imagpart(x))}
982 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
989 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
990 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
991 defines the following operations:
994 @item bool operator == (const @var{type}&, const @var{type}&)
995 @cindex @code{operator == ()}
996 @itemx bool operator != (const @var{type}&, const @var{type}&)
997 @cindex @code{operator != ()}
998 Comparison, as in C and C++.
1000 @item uint32 equal_hashcode (const @var{type}&)
1001 @cindex @code{equal_hashcode ()}
1002 Returns a 32-bit hash code that is the same for any two numbers which are
1003 the same according to @code{==}. This hash code depends on the number's value,
1004 not its type or precision.
1006 @item cl_boolean zerop (const @var{type}& x)
1007 @cindex @code{zerop ()}
1008 Compare against zero: @code{x == 0}
1011 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1012 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1013 defines the following operations:
1016 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1017 @cindex @code{compare ()}
1018 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1019 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1021 @item bool operator <= (const @var{type}&, const @var{type}&)
1022 @cindex @code{operator <= ()}
1023 @itemx bool operator < (const @var{type}&, const @var{type}&)
1024 @cindex @code{operator < ()}
1025 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1026 @cindex @code{operator >= ()}
1027 @itemx bool operator > (const @var{type}&, const @var{type}&)
1028 @cindex @code{operator > ()}
1029 Comparison, as in C and C++.
1031 @item cl_boolean minusp (const @var{type}& x)
1032 @cindex @code{minusp ()}
1033 Compare against zero: @code{x < 0}
1035 @item cl_boolean plusp (const @var{type}& x)
1036 @cindex @code{plusp ()}
1037 Compare against zero: @code{x > 0}
1039 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1040 @cindex @code{max ()}
1041 Return the maximum of @code{x} and @code{y}.
1043 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1044 @cindex @code{min ()}
1045 Return the minimum of @code{x} and @code{y}.
1048 When a floating point number and a rational number are compared, the float
1049 is first converted to a rational number using the function @code{rational}.
1050 Since a floating point number actually represents an interval of real numbers,
1051 the result might be surprising.
1052 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1053 there is no floating point number whose value is exactly @code{1/3}.
1056 @section Rounding functions
1059 When a real number is to be converted to an integer, there is no ``best''
1060 rounding. The desired rounding function depends on the application.
1061 The Common Lisp and ISO Lisp standards offer four rounding functions:
1065 This is the largest integer <=@code{x}.
1068 This is the smallest integer >=@code{x}.
1071 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1074 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1075 integers, choose the even one.
1078 These functions have different advantages:
1080 @code{floor} and @code{ceiling} are translation invariant:
1081 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1082 for every @code{x} and every integer @code{n}.
1084 On the other hand, @code{truncate} and @code{round} are symmetric:
1085 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1086 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1087 down exactly as often as it rounds up.
1089 The functions are related like this:
1093 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1094 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1096 @code{truncate(x) = sign(x) * floor(abs(x))}
1099 Each of the classes @code{cl_R}, @code{cl_RA},
1100 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1101 defines the following operations:
1104 @item cl_I floor1 (const @var{type}& x)
1105 @cindex @code{floor1 ()}
1106 Returns @code{floor(x)}.
1107 @item cl_I ceiling1 (const @var{type}& x)
1108 @cindex @code{ceiling1 ()}
1109 Returns @code{ceiling(x)}.
1110 @item cl_I truncate1 (const @var{type}& x)
1111 @cindex @code{truncate1 ()}
1112 Returns @code{truncate(x)}.
1113 @item cl_I round1 (const @var{type}& x)
1114 @cindex @code{round1 ()}
1115 Returns @code{round(x)}.
1118 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1119 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1120 defines the following operations:
1123 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1124 Returns @code{floor(x/y)}.
1125 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1126 Returns @code{ceiling(x/y)}.
1127 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1128 Returns @code{truncate(x/y)}.
1129 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1130 Returns @code{round(x/y)}.
1133 These functions are called @samp{floor1}, @dots{} here instead of
1134 @samp{floor}, @dots{}, because on some systems, system dependent include
1135 files define @samp{floor} and @samp{ceiling} as macros.
1137 In many cases, one needs both the quotient and the remainder of a division.
1138 It is more efficient to compute both at the same time than to perform
1139 two divisions, one for quotient and the next one for the remainder.
1140 The following functions therefore return a structure containing both
1141 the quotient and the remainder. The suffix @samp{2} indicates the number
1142 of ``return values''. The remainder is defined as follows:
1146 for the computation of @code{quotient = floor(x)},
1147 @code{remainder = x - quotient},
1149 for the computation of @code{quotient = floor(x,y)},
1150 @code{remainder = x - quotient*y},
1153 and similarly for the other three operations.
1155 Each of the classes @code{cl_R}, @code{cl_RA},
1156 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1157 defines the following operations:
1160 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1161 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1162 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1163 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1164 @itemx @var{type}_div_t round2 (const @var{type}& x)
1167 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1168 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1169 defines the following operations:
1172 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1173 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1174 @cindex @code{floor2 ()}
1175 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1176 @cindex @code{ceiling2 ()}
1177 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1178 @cindex @code{truncate2 ()}
1179 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1180 @cindex @code{round2 ()}
1183 Sometimes, one wants the quotient as a floating-point number (of the
1184 same format as the argument, if the argument is a float) instead of as
1185 an integer. The prefix @samp{f} indicates this.
1188 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1189 defines the following operations:
1192 @item @var{type} ffloor (const @var{type}& x)
1193 @cindex @code{ffloor ()}
1194 @itemx @var{type} fceiling (const @var{type}& x)
1195 @cindex @code{fceiling ()}
1196 @itemx @var{type} ftruncate (const @var{type}& x)
1197 @cindex @code{ftruncate ()}
1198 @itemx @var{type} fround (const @var{type}& x)
1199 @cindex @code{fround ()}
1202 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1204 The class @code{cl_R} defines the following operations:
1207 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1208 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1209 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1210 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1213 These functions also exist in versions which return both the quotient
1214 and the remainder. The suffix @samp{2} indicates this.
1217 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1218 defines the following operations:
1219 @cindex @code{cl_F_fdiv_t}
1220 @cindex @code{cl_SF_fdiv_t}
1221 @cindex @code{cl_FF_fdiv_t}
1222 @cindex @code{cl_DF_fdiv_t}
1223 @cindex @code{cl_LF_fdiv_t}
1226 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1227 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1228 @cindex @code{ffloor2 ()}
1229 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1230 @cindex @code{fceiling2 ()}
1231 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1232 @cindex @code{ftruncate2 ()}
1233 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1234 @cindex @code{fround2 ()}
1236 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1237 @cindex @code{cl_R_fdiv_t}
1239 The class @code{cl_R} defines the following operations:
1242 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1243 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1244 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1245 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1246 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1249 Other applications need only the remainder of a division.
1250 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1251 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1252 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1256 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1258 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1261 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1262 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1263 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1265 The classes @code{cl_R}, @code{cl_I} define the following operations:
1268 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1269 @cindex @code{mod ()}
1270 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1271 @cindex @code{rem ()}
1277 Each of the classes @code{cl_R},
1278 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1279 defines the following operation:
1282 @item @var{type} sqrt (const @var{type}& x)
1283 @cindex @code{sqrt ()}
1284 @code{x} must be >= 0. This function returns the square root of @code{x},
1285 normalized to be >= 0. If @code{x} is the square of a rational number,
1286 @code{sqrt(x)} will be a rational number, else it will return a
1287 floating-point approximation.
1290 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1293 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1294 @cindex @code{sqrtp ()}
1295 This tests whether @code{x} is a perfect square. If so, it returns true
1296 and the exact square root in @code{*root}, else it returns false.
1299 Furthermore, for integers, similarly:
1302 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1303 @cindex @code{isqrt ()}
1304 @code{x} should be >= 0. This function sets @code{*root} to
1305 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1306 the boolean value @code{(expt(*root,2) == x)}.
1309 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1310 define the following operation:
1313 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1314 @cindex @code{rootp ()}
1315 @code{x} must be >= 0. @code{n} must be > 0.
1316 This tests whether @code{x} is an @code{n}th power of a rational number.
1317 If so, it returns true and the exact root in @code{*root}, else it returns
1321 The only square root function which accepts negative numbers is the one
1322 for class @code{cl_N}:
1325 @item cl_N sqrt (const cl_N& z)
1326 @cindex @code{sqrt ()}
1327 Returns the square root of @code{z}, as defined by the formula
1328 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1329 or to a complex number are done if necessary. The range of the result is the
1330 right half plane @code{realpart(sqrt(z)) >= 0}
1331 including the positive imaginary axis and 0, but excluding
1332 the negative imaginary axis.
1333 The result is an exact number only if @code{z} is an exact number.
1337 @section Transcendental functions
1338 @cindex transcendental functions
1340 The transcendental functions return an exact result if the argument
1341 is exact and the result is exact as well. Otherwise they must return
1342 inexact numbers even if the argument is exact.
1343 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1346 @subsection Exponential and logarithmic functions
1349 @item cl_R exp (const cl_R& x)
1350 @cindex @code{exp ()}
1351 @itemx cl_N exp (const cl_N& x)
1352 Returns the exponential function of @code{x}. This is @code{e^x} where
1353 @code{e} is the base of the natural logarithms. The range of the result
1354 is the entire complex plane excluding 0.
1356 @item cl_R ln (const cl_R& x)
1357 @cindex @code{ln ()}
1358 @code{x} must be > 0. Returns the (natural) logarithm of x.
1360 @item cl_N log (const cl_N& x)
1361 @cindex @code{log ()}
1362 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1363 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1364 The range of the result is the strip in the complex plane
1365 @code{-pi < imagpart(log(x)) <= pi}.
1367 @item cl_R phase (const cl_N& x)
1368 @cindex @code{phase ()}
1369 Returns the angle part of @code{x} in its polar representation as a
1370 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1371 This is also the imaginary part of @code{log(x)}.
1372 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1373 The result will be an exact number only if @code{zerop(x)} or
1374 if @code{x} is real and positive.
1376 @item cl_R log (const cl_R& a, const cl_R& b)
1377 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1378 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1379 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1382 @item cl_N log (const cl_N& a, const cl_N& b)
1383 Returns the logarithm of @code{a} with respect to base @code{b}.
1384 @code{log(a,b) = log(a)/log(b)}.
1386 @item cl_N expt (const cl_N& x, const cl_N& y)
1387 @cindex @code{expt ()}
1388 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1391 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1394 @item cl_F exp1 (cl_float_format_t f)
1395 @cindex @code{exp1 ()}
1396 Returns e as a float of format @code{f}.
1398 @item cl_F exp1 (const cl_F& y)
1399 Returns e in the float format of @code{y}.
1401 @item cl_F exp1 (void)
1402 Returns e as a float of format @code{default_float_format}.
1406 @subsection Trigonometric functions
1409 @item cl_R sin (const cl_R& x)
1410 @cindex @code{sin ()}
1411 Returns @code{sin(x)}. The range of the result is the interval
1412 @code{-1 <= sin(x) <= 1}.
1414 @item cl_N sin (const cl_N& z)
1415 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1417 @item cl_R cos (const cl_R& x)
1418 @cindex @code{cos ()}
1419 Returns @code{cos(x)}. The range of the result is the interval
1420 @code{-1 <= cos(x) <= 1}.
1422 @item cl_N cos (const cl_N& x)
1423 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1425 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1426 @cindex @code{cos_sin_t}
1427 @itemx cos_sin_t cos_sin (const cl_R& x)
1428 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1429 @cindex @code{cos_sin ()}
1430 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1431 hold only approximately.
1433 @item cl_R tan (const cl_R& x)
1434 @cindex @code{tan ()}
1435 @itemx cl_N tan (const cl_N& x)
1436 Returns @code{tan(x) = sin(x)/cos(x)}.
1438 @item cl_N cis (const cl_R& x)
1439 @cindex @code{cis ()}
1440 @itemx cl_N cis (const cl_N& x)
1441 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1442 @code{e^(i*x) = cos(x) + i*sin(x)}.
1445 @cindex @code{asin ()}
1446 @item cl_N asin (const cl_N& z)
1447 Returns @code{arcsin(z)}. This is defined as
1448 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1449 @code{arcsin(-z) = -arcsin(z)}.
1450 The range of the result is the strip in the complex domain
1451 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1452 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1453 with @code{realpart = pi/2} and @code{imagpart > 0}.
1455 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1459 @item cl_N acos (const cl_N& z)
1460 @cindex @code{acos ()}
1461 Returns @code{arccos(z)}. This is defined as
1462 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1465 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1467 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1468 The range of the result is the strip in the complex domain
1469 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1470 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1471 with @code{realpart = pi} and @code{imagpart > 0}.
1473 Proof: This follows from the results about arcsin.
1477 @cindex @code{atan ()}
1478 @item cl_R atan (const cl_R& x, const cl_R& y)
1479 Returns the angle of the polar representation of the complex number
1480 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1481 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1482 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1483 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1484 with reversed order of arguments.
1486 @item cl_R atan (const cl_R& x)
1487 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1488 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1489 will be an exact number only if @code{x} is the exact @code{0}.
1491 @item cl_N atan (const cl_N& z)
1492 Returns @code{arctan(z)}. This is defined as
1493 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1494 @code{arctan(-z) = -arctan(z)}. The range of the result is
1495 the strip in the complex domain
1496 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1497 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1498 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1500 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1506 @cindex Archimedes' constant
1507 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1510 @item cl_F pi (cl_float_format_t f)
1511 @cindex @code{pi ()}
1512 Returns pi as a float of format @code{f}.
1514 @item cl_F pi (const cl_F& y)
1515 Returns pi in the float format of @code{y}.
1517 @item cl_F pi (void)
1518 Returns pi as a float of format @code{default_float_format}.
1522 @subsection Hyperbolic functions
1525 @item cl_R sinh (const cl_R& x)
1526 @cindex @code{sinh ()}
1527 Returns @code{sinh(x)}.
1529 @item cl_N sinh (const cl_N& z)
1530 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1532 @item cl_R cosh (const cl_R& x)
1533 @cindex @code{cosh ()}
1534 Returns @code{cosh(x)}. The range of the result is the interval
1535 @code{cosh(x) >= 1}.
1537 @item cl_N cosh (const cl_N& z)
1538 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1540 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1541 @cindex @code{cosh_sinh_t}
1542 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1543 @cindex @code{cosh_sinh ()}
1544 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1545 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1546 hold only approximately.
1548 @item cl_R tanh (const cl_R& x)
1549 @cindex @code{tanh ()}
1550 @itemx cl_N tanh (const cl_N& x)
1551 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1553 @item cl_N asinh (const cl_N& z)
1554 @cindex @code{asinh ()}
1555 Returns @code{arsinh(z)}. This is defined as
1556 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1557 @code{arsinh(-z) = -arsinh(z)}.
1559 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1560 Actually, z+sqrt(1+z^2) can never be real and <0, so
1561 -pi < imagpart(arsinh(z)) < pi.
1562 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1563 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1565 The range of the result is the strip in the complex domain
1566 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1567 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1568 with @code{imagpart = pi/2} and @code{realpart < 0}.
1570 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1571 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1572 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1573 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1574 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1575 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1576 If y >= 1, the imagpart is pi/2 and the realpart is
1577 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1580 Moreover, if z is in Range(sqrt),
1581 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1582 (for a proof, see file src/cl_C_asinh.cc).
1585 @item cl_N acosh (const cl_N& z)
1586 @cindex @code{acosh ()}
1587 Returns @code{arcosh(z)}. This is defined as
1588 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1589 The range of the result is the half-strip in the complex domain
1590 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1591 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1593 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1594 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1595 If z is in Range(sqrt), we have
1596 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1597 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1599 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1600 and since the imagpart of both expressions is > -pi, <= pi
1601 ==> arcosh(z) = log(z+sqrt(z^2-1))
1602 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1603 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1604 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1605 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1606 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1608 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1609 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1610 = x^2 + y^2 + sqrt(u^2+v^2)
1615 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1616 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1617 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1618 Otherwise, -z is in Range(sqrt).
1619 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1620 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1621 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1622 and this has realpart > 0.
1623 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1624 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1625 has realpart = 0 and imagpart > 0.
1626 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1627 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1628 This has realpart >= 0 and imagpart = pi.
1631 @item cl_N atanh (const cl_N& z)
1632 @cindex @code{atanh ()}
1633 Returns @code{artanh(z)}. This is defined as
1634 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1635 @code{artanh(-z) = -artanh(z)}. The range of the result is
1636 the strip in the complex domain
1637 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1638 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1639 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1641 Proof: Write z = x+iy. Examine
1642 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1644 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1645 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1646 |x| < 1 ==> imagpart = 0
1649 = (atan(1+x,y) - atan(1-x,-y))/2
1650 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1651 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1652 > (pi - pi/2 - pi/2 )/2 = 0
1653 and (1+x)/y > (1-x)/y
1654 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1655 ==> imagpart < pi/2.
1656 Hence 0 < imagpart < pi/2.
1658 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1663 @subsection Euler gamma
1664 @cindex Euler's constant
1666 Euler's constant C = 0.577@dots{} is returned by the following functions:
1669 @item cl_F eulerconst (cl_float_format_t f)
1670 @cindex @code{eulerconst ()}
1671 Returns Euler's constant as a float of format @code{f}.
1673 @item cl_F eulerconst (const cl_F& y)
1674 Returns Euler's constant in the float format of @code{y}.
1676 @item cl_F eulerconst (void)
1677 Returns Euler's constant as a float of format @code{default_float_format}.
1680 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1681 @cindex Catalan's constant
1684 @item cl_F catalanconst (cl_float_format_t f)
1685 @cindex @code{catalanconst ()}
1686 Returns Catalan's constant as a float of format @code{f}.
1688 @item cl_F catalanconst (const cl_F& y)
1689 Returns Catalan's constant in the float format of @code{y}.
1691 @item cl_F catalanconst (void)
1692 Returns Catalan's constant as a float of format @code{default_float_format}.
1696 @subsection Riemann zeta
1697 @cindex Riemann's zeta
1699 Riemann's zeta function at an integral point @code{s>1} is returned by the
1700 following functions:
1703 @item cl_F zeta (int s, cl_float_format_t f)
1704 @cindex @code{zeta ()}
1705 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1707 @item cl_F zeta (int s, const cl_F& y)
1708 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1710 @item cl_F zeta (int s)
1711 Returns Riemann's zeta function at @code{s} as a float of format
1712 @code{default_float_format}.
1716 @section Functions on integers
1718 @subsection Logical functions
1720 Integers, when viewed as in two's complement notation, can be thought as
1721 infinite bit strings where the bits' values eventually are constant.
1728 The logical operations view integers as such bit strings and operate
1729 on each of the bit positions in parallel.
1732 @item cl_I lognot (const cl_I& x)
1733 @cindex @code{lognot ()}
1734 @itemx cl_I operator ~ (const cl_I& x)
1735 @cindex @code{operator ~ ()}
1736 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1738 @item cl_I logand (const cl_I& x, const cl_I& y)
1739 @cindex @code{logand ()}
1740 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1741 @cindex @code{operator & ()}
1742 Logical and, like @code{x & y} in C.
1744 @item cl_I logior (const cl_I& x, const cl_I& y)
1745 @cindex @code{logior ()}
1746 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1747 @cindex @code{operator | ()}
1748 Logical (inclusive) or, like @code{x | y} in C.
1750 @item cl_I logxor (const cl_I& x, const cl_I& y)
1751 @cindex @code{logxor ()}
1752 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1753 @cindex @code{operator ^ ()}
1754 Exclusive or, like @code{x ^ y} in C.
1756 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1757 @cindex @code{logeqv ()}
1758 Bitwise equivalence, like @code{~(x ^ y)} in C.
1760 @item cl_I lognand (const cl_I& x, const cl_I& y)
1761 @cindex @code{lognand ()}
1762 Bitwise not and, like @code{~(x & y)} in C.
1764 @item cl_I lognor (const cl_I& x, const cl_I& y)
1765 @cindex @code{lognor ()}
1766 Bitwise not or, like @code{~(x | y)} in C.
1768 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1769 @cindex @code{logandc1 ()}
1770 Logical and, complementing the first argument, like @code{~x & y} in C.
1772 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1773 @cindex @code{logandc2 ()}
1774 Logical and, complementing the second argument, like @code{x & ~y} in C.
1776 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1777 @cindex @code{logorc1 ()}
1778 Logical or, complementing the first argument, like @code{~x | y} in C.
1780 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1781 @cindex @code{logorc2 ()}
1782 Logical or, complementing the second argument, like @code{x | ~y} in C.
1785 These operations are all available though the function
1787 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1788 @cindex @code{boole ()}
1790 where @code{op} must have one of the 16 values (each one stands for a function
1791 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1792 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1793 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1794 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1795 @code{boole_orc1}, @code{boole_orc2}.
1796 @cindex @code{boole_clr}
1797 @cindex @code{boole_set}
1798 @cindex @code{boole_1}
1799 @cindex @code{boole_2}
1800 @cindex @code{boole_c1}
1801 @cindex @code{boole_c2}
1802 @cindex @code{boole_and}
1803 @cindex @code{boole_xor}
1804 @cindex @code{boole_eqv}
1805 @cindex @code{boole_nand}
1806 @cindex @code{boole_nor}
1807 @cindex @code{boole_andc1}
1808 @cindex @code{boole_andc2}
1809 @cindex @code{boole_orc1}
1810 @cindex @code{boole_orc2}
1813 Other functions that view integers as bit strings:
1816 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1817 @cindex @code{logtest ()}
1818 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1819 @code{logand(x,y) != 0}.
1821 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1822 @cindex @code{logbitp ()}
1823 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1824 Bit 0 is the least significant bit.
1826 @item uintL logcount (const cl_I& x)
1827 @cindex @code{logcount ()}
1828 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1829 the number of zero bits in @code{x}, if @code{x} < 0.
1832 The following functions operate on intervals of bits in integers.
1835 struct cl_byte @{ uintL size; uintL position; @};
1837 @cindex @code{cl_byte}
1838 represents the bit interval containing the bits
1839 @code{position}@dots{}@code{position+size-1} of an integer.
1840 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1843 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1844 @cindex @code{ldb ()}
1845 extracts the bits of @code{n} described by the bit interval @code{b}
1846 and returns them as a nonnegative integer with @code{b.size} bits.
1848 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1849 @cindex @code{ldb_test ()}
1850 Returns true if some bit described by the bit interval @code{b} is set in
1853 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1854 @cindex @code{dpb ()}
1855 Returns @code{n}, with the bits described by the bit interval @code{b}
1856 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1857 @code{newbyte} are relevant.
1860 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1861 functions are their counterparts without shifting:
1864 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1865 @cindex @code{mask_field ()}
1866 returns an integer with the bits described by the bit interval @code{b}
1867 copied from the corresponding bits in @code{n}, the other bits zero.
1869 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1870 @cindex @code{deposit_field ()}
1871 returns an integer where the bits described by the bit interval @code{b}
1872 come from @code{newbyte} and the other bits come from @code{n}.
1875 The following relations hold:
1879 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1881 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1883 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1886 The following operations on integers as bit strings are efficient shortcuts
1887 for common arithmetic operations:
1890 @item cl_boolean oddp (const cl_I& x)
1891 @cindex @code{oddp ()}
1892 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1893 @code{mod(x,2) != 0}.
1895 @item cl_boolean evenp (const cl_I& x)
1896 @cindex @code{evenp ()}
1897 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1898 @code{mod(x,2) == 0}.
1900 @item cl_I operator << (const cl_I& x, const cl_I& n)
1901 @cindex @code{operator << ()}
1902 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1903 Equivalent to @code{x * expt(2,n)}.
1905 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1906 @cindex @code{operator >> ()}
1907 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1908 Bits shifted out to the right are thrown away.
1909 Equivalent to @code{floor(x / expt(2,n))}.
1911 @item cl_I ash (const cl_I& x, const cl_I& y)
1912 @cindex @code{ash ()}
1913 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1914 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1915 returns @code{floor(x * expt(2,y))}.
1917 @item uintL integer_length (const cl_I& x)
1918 @cindex @code{integer_length ()}
1919 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1920 in two's complement notation. This is the smallest n >= 0 such that
1921 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1924 @item uintL ord2 (const cl_I& x)
1925 @cindex @code{ord2 ()}
1926 @code{x} must be non-zero. This function returns the number of 0 bits at the
1927 right of @code{x} in two's complement notation. This is the largest n >= 0
1928 such that 2^n divides @code{x}.
1930 @item uintL power2p (const cl_I& x)
1931 @cindex @code{power2p ()}
1932 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1933 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1934 (See also the function @code{logp}.)
1938 @subsection Number theoretic functions
1941 @item uint32 gcd (uint32 a, uint32 b)
1942 @cindex @code{gcd ()}
1943 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1944 This function returns the greatest common divisor of @code{a} and @code{b},
1945 normalized to be >= 0.
1947 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1948 @cindex @code{xgcd ()}
1949 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1950 @code{a} and @code{b} and at the same time the representation of @code{g}
1951 as an integral linear combination of @code{a} and @code{b}:
1952 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1953 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1954 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1955 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1956 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1958 @item cl_I lcm (const cl_I& a, const cl_I& b)
1959 @cindex @code{lcm ()}
1960 This function returns the least common multiple of @code{a} and @code{b},
1961 normalized to be >= 0.
1963 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
1964 @cindex @code{logp ()}
1965 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
1966 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
1967 rational number, this function returns true and sets *l = log(a,b), else
1972 @subsection Combinatorial functions
1975 @item cl_I factorial (uintL n)
1976 @cindex @code{factorial ()}
1977 @code{n} must be a small integer >= 0. This function returns the factorial
1978 @code{n}! = @code{1*2*@dots{}*n}.
1980 @item cl_I doublefactorial (uintL n)
1981 @cindex @code{doublefactorial ()}
1982 @code{n} must be a small integer >= 0. This function returns the
1983 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
1984 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
1986 @item cl_I binomial (uintL n, uintL k)
1987 @cindex @code{binomial ()}
1988 @code{n} and @code{k} must be small integers >= 0. This function returns the
1989 binomial coefficient
1991 ${n \choose k} = {n! \over n! (n-k)!}$
1994 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
1996 for 0 <= k <= n, 0 else.
2000 @section Functions on floating-point numbers
2002 Recall that a floating-point number consists of a sign @code{s}, an
2003 exponent @code{e} and a mantissa @code{m}. The value of the number is
2004 @code{(-1)^s * 2^e * m}.
2007 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2008 defines the following operations.
2011 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2012 @cindex @code{scale_float ()}
2013 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2014 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2015 because it copies @code{x} and modifies the exponent.
2018 The following functions provide an abstract interface to the underlying
2019 representation of floating-point numbers.
2022 @item sintL float_exponent (const @var{type}& x)
2023 @cindex @code{float_exponent ()}
2024 Returns the exponent @code{e} of @code{x}.
2025 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2026 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2028 @item sintL float_radix (const @var{type}& x)
2029 @cindex @code{float_radix ()}
2030 Returns the base of the floating-point representation. This is always @code{2}.
2032 @item @var{type} float_sign (const @var{type}& x)
2033 @cindex @code{float_sign ()}
2034 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2035 @code{x} >= 0, -1 for @code{x} < 0.
2037 @item uintL float_digits (const @var{type}& x)
2038 @cindex @code{float_digits ()}
2039 Returns the number of mantissa bits in the floating-point representation
2040 of @code{x}, including the hidden bit. The value only depends on the type
2041 of @code{x}, not on its value.
2043 @item uintL float_precision (const @var{type}& x)
2044 @cindex @code{float_precision ()}
2045 Returns the number of significant mantissa bits in the floating-point
2046 representation of @code{x}. Since denormalized numbers are not supported,
2047 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2051 The complete internal representation of a float is encoded in the type
2052 @cindex @code{decoded_float}
2053 @cindex @code{decoded_sfloat}
2054 @cindex @code{decoded_ffloat}
2055 @cindex @code{decoded_dfloat}
2056 @cindex @code{decoded_lfloat}
2057 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2058 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2060 struct decoded_@var{type}float @{
2061 @var{type} mantissa; cl_I exponent; @var{type} sign;
2065 and returned by the function
2068 @item decoded_@var{type}float decode_float (const @var{type}& x)
2069 @cindex @code{decode_float ()}
2070 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2071 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2072 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2073 @code{e} is the same as returned by the function @code{float_exponent}.
2076 A complete decoding in terms of integers is provided as type
2077 @cindex @code{cl_idecoded_float}
2079 struct cl_idecoded_float @{
2080 cl_I mantissa; cl_I exponent; cl_I sign;
2083 by the following function:
2086 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2087 @cindex @code{integer_decode_float ()}
2088 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2089 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2090 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2091 WARNING: The exponent @code{e} is not the same as the one returned by
2092 the functions @code{decode_float} and @code{float_exponent}.
2095 Some other function, implemented only for class @code{cl_F}:
2098 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2099 @cindex @code{float_sign ()}
2100 This returns a floating point number whose precision and absolute value
2101 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2102 zero, it is treated as positive. Same for @code{y}.
2106 @section Conversion functions
2109 @subsection Conversion to floating-point numbers
2111 The type @code{cl_float_format_t} describes a floating-point format.
2112 @cindex @code{cl_float_format_t}
2115 @item cl_float_format_t cl_float_format (uintL n)
2116 @cindex @code{cl_float_format ()}
2117 Returns the smallest float format which guarantees at least @code{n}
2118 decimal digits in the mantissa (after the decimal point).
2120 @item cl_float_format_t cl_float_format (const cl_F& x)
2121 Returns the floating point format of @code{x}.
2123 @item cl_float_format_t default_float_format
2124 @cindex @code{default_float_format}
2125 Global variable: the default float format used when converting rational numbers
2129 To convert a real number to a float, each of the types
2130 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2131 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2132 defines the following operations:
2135 @item cl_F cl_float (const @var{type}&x, cl_float_format_t f)
2136 @cindex @code{cl_float ()}
2137 Returns @code{x} as a float of format @code{f}.
2138 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2139 Returns @code{x} in the float format of @code{y}.
2140 @item cl_F cl_float (const @var{type}&x)
2141 Returns @code{x} as a float of format @code{default_float_format} if
2142 it is an exact number, or @code{x} itself if it is already a float.
2145 Of course, converting a number to a float can lose precision.
2147 Every floating-point format has some characteristic numbers:
2150 @item cl_F most_positive_float (cl_float_format_t f)
2151 @cindex @code{most_positive_float ()}
2152 Returns the largest (most positive) floating point number in float format @code{f}.
2154 @item cl_F most_negative_float (cl_float_format_t f)
2155 @cindex @code{most_negative_float ()}
2156 Returns the smallest (most negative) floating point number in float format @code{f}.
2158 @item cl_F least_positive_float (cl_float_format_t f)
2159 @cindex @code{least_positive_float ()}
2160 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2161 in float format @code{f}.
2163 @item cl_F least_negative_float (cl_float_format_t f)
2164 @cindex @code{least_negative_float ()}
2165 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2166 in float format @code{f}.
2168 @item cl_F float_epsilon (cl_float_format_t f)
2169 @cindex @code{float_epsilon ()}
2170 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2172 @item cl_F float_negative_epsilon (cl_float_format_t f)
2173 @cindex @code{float_negative_epsilon ()}
2174 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2178 @subsection Conversion to rational numbers
2180 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2181 defines the following operation:
2184 @item cl_RA rational (const @var{type}& x)
2185 @cindex @code{rational ()}
2186 Returns the value of @code{x} as an exact number. If @code{x} is already
2187 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2188 the value is a rational number whose denominator is a power of 2.
2191 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2195 @item cl_RA rationalize (const cl_R& x)
2196 @cindex @code{rationalize ()}
2197 If @code{x} is a floating-point number, it actually represents an interval
2198 of real numbers, and this function returns the rational number with
2199 smallest denominator (and smallest numerator, in magnitude)
2200 which lies in this interval.
2201 If @code{x} is already an exact number, this function returns @code{x}.
2204 If @code{x} is any float, one has
2208 @code{cl_float(rational(x),x) = x}
2210 @code{cl_float(rationalize(x),x) = x}
2214 @section Random number generators
2217 A random generator is a machine which produces (pseudo-)random numbers.
2218 The include file @code{<cln/random.h>} defines a class @code{random_state}
2219 which contains the state of a random generator. If you make a copy
2220 of the random number generator, the original one and the copy will produce
2221 the same sequence of random numbers.
2223 The following functions return (pseudo-)random numbers in different formats.
2224 Calling one of these modifies the state of the random number generator in
2225 a complicated but deterministic way.
2228 @cindex @code{random_state}
2229 @cindex @code{default_random_state}
2231 random_state default_random_state
2233 contains a default random number generator. It is used when the functions
2234 below are called without @code{random_state} argument.
2237 @item uint32 random32 (random_state& randomstate)
2238 @itemx uint32 random32 ()
2239 @cindex @code{random32 ()}
2240 Returns a random unsigned 32-bit number. All bits are equally random.
2242 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2243 @itemx cl_I random_I (const cl_I& n)
2244 @cindex @code{random_I ()}
2245 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2246 in the range @code{0 <= x < n}.
2248 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2249 @itemx cl_F random_F (const cl_F& n)
2250 @cindex @code{random_F ()}
2251 @code{n} must be a float > 0. This function returns a random floating-point
2252 number of the same format as @code{n} in the range @code{0 <= x < n}.
2254 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2255 @itemx cl_R random_R (const cl_R& n)
2256 @cindex @code{random_R ()}
2257 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2258 if @code{n} is a float.
2262 @section Obfuscating operators
2263 @cindex modifying operators
2265 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2266 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2267 are not available by default because their
2268 use tends to make programs unreadable. It is trivial to get away without
2269 them. However, if you feel that you absolutely need these operators
2270 to get happy, then add
2272 #define WANT_OBFUSCATING_OPERATORS
2274 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2275 to the beginning of your source files, before the inclusion of any CLN
2276 include files. This flag will enable the following operators:
2278 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2279 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2282 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2283 @cindex @code{operator += ()}
2284 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2285 @cindex @code{operator -= ()}
2286 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2287 @cindex @code{operator *= ()}
2288 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2289 @cindex @code{operator /= ()}
2292 For the class @code{cl_I}:
2295 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2296 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2297 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2298 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2299 @cindex @code{operator &= ()}
2300 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2301 @cindex @code{operator |= ()}
2302 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2303 @cindex @code{operator ^= ()}
2304 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2305 @cindex @code{operator <<= ()}
2306 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2307 @cindex @code{operator >>= ()}
2310 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2311 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2314 @item @var{type}& operator ++ (@var{type}& x)
2315 @cindex @code{operator ++ ()}
2316 The prefix operator @code{++x}.
2318 @item void operator ++ (@var{type}& x, int)
2319 The postfix operator @code{x++}.
2321 @item @var{type}& operator -- (@var{type}& x)
2322 @cindex @code{operator -- ()}
2323 The prefix operator @code{--x}.
2325 @item void operator -- (@var{type}& x, int)
2326 The postfix operator @code{x--}.
2329 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2330 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2334 @chapter Input/Output
2335 @cindex Input/Output
2337 @section Internal and printed representation
2338 @cindex representation
2340 All computations deal with the internal representations of the numbers.
2342 Every number has an external representation as a sequence of ASCII characters.
2343 Several external representations may denote the same number, for example,
2344 "20.0" and "20.000".
2346 Converting an internal to an external representation is called ``printing'',
2348 converting an external to an internal representation is called ``reading''.
2350 In CLN, it is always true that conversion of an internal to an external
2351 representation and then back to an internal representation will yield the
2352 same internal representation. Symbolically: @code{read(print(x)) == x}.
2353 This is called ``print-read consistency''.
2355 Different types of numbers have different external representations (case
2360 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2361 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2362 for decimal integers
2363 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2365 @item Rational numbers
2366 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2367 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2370 @item Floating-point numbers
2371 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2372 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2373 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2374 of the form _@var{prec} may be appended. There must be at least
2375 one digit in the non-exponent part. The exponent has the syntax
2376 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2377 The exponent marker is
2381 @samp{s} for short-floats,
2383 @samp{f} for single-floats,
2385 @samp{d} for double-floats,
2387 @samp{L} for long-floats,
2390 or @samp{e}, which denotes a default float format. The precision specifying
2391 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2392 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2393 function @samp{cl_float_format}.
2395 @item Complex numbers
2396 External representation:
2399 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2400 if @var{imagpart} is negative, its printed representation begins with
2401 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2402 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2403 is rational and the rational number's base is >18, because the @samp{i}
2404 is then read as a digit.
2406 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2411 @section Input functions
2413 Including @code{<cln/io.h>} defines a type @code{cl_istream}, which is
2414 the type of the first argument to all input functions. @code{cl_istream}
2415 is the same as @code{std::istream&}.
2420 @code{cl_istream stdin}
2422 contains the standard input stream.
2424 These are the simple input functions:
2427 @item int freadchar (cl_istream stream)
2428 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2429 if the end of stream was encountered or an error occurred.
2431 @item int funreadchar (cl_istream stream, int c)
2432 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2433 last @code{freadchar} operation on @code{stream}.
2436 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2437 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2438 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2441 @item cl_istream operator>> (cl_istream stream, @var{type}& result)
2442 Reads a number from @code{stream} and stores it in the @code{result}.
2445 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2449 @item cl_N read_complex (cl_istream stream, const cl_read_flags& flags)
2450 @itemx cl_R read_real (cl_istream stream, const cl_read_flags& flags)
2451 @itemx cl_F read_float (cl_istream stream, const cl_read_flags& flags)
2452 @itemx cl_RA read_rational (cl_istream stream, const cl_read_flags& flags)
2453 @itemx cl_I read_integer (cl_istream stream, const cl_read_flags& flags)
2454 Reads a number from @code{stream}. The @code{flags} are parameters which
2455 affect the input syntax. Whitespace before the number is silently skipped.
2457 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2458 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2459 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2460 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2461 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2462 Reads a number from a string in memory. The @code{flags} are parameters which
2463 affect the input syntax. The string starts at @code{string} and ends at
2464 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2465 @code{NULL}, denoting the entire string, i.e. equivalent to
2466 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2467 @code{NULL}, the string in memory must contain exactly one number and nothing
2468 more, else a fatal error will be signalled. If @code{end_of_parse}
2469 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2470 the last parsed character (i.e. @code{string_limit} if nothing came after
2471 the number). Whitespace is not allowed.
2474 The structure @code{cl_read_flags} contains the following fields:
2477 @item cl_read_syntax_t syntax
2478 The possible results of the read operation. Possible values are
2479 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2480 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2481 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2483 @item cl_read_lsyntax_t lsyntax
2484 Specifies the language-dependent syntax variant for the read operation.
2488 @item lsyntax_standard
2489 accept standard algebraic notation only, no complex numbers,
2490 @item lsyntax_algebraic
2491 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2492 @item lsyntax_commonlisp
2493 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2494 hexadecimal numbers,
2495 @code{#@var{base}R} for rational numbers in a given base,
2496 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2498 accept all of these extensions.
2501 @item unsigned int rational_base
2502 The base in which rational numbers are read.
2504 @item cl_float_format_t float_flags.default_float_format
2505 The float format used when reading floats with exponent marker @samp{e}.
2507 @item cl_float_format_t float_flags.default_lfloat_format
2508 The float format used when reading floats with exponent marker @samp{l}.
2510 @item cl_boolean float_flags.mantissa_dependent_float_format
2511 When this flag is true, floats specified with more digits than corresponding
2512 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2513 precision corresponding to their number of significant digits.
2517 @section Output functions
2519 Including @code{<cln/io.h>} defines a type @code{cl_ostream}, which is
2520 the type of the first argument to all output functions. @code{cl_ostream}
2521 is the same as @code{std::ostream&}.
2526 @code{cl_ostream stdout}
2528 contains the standard output stream.
2533 @code{cl_ostream stderr}
2535 contains the standard error output stream.
2537 These are the simple output functions:
2540 @item void fprintchar (cl_ostream stream, char c)
2541 Prints the character @code{x} literally on the @code{stream}.
2543 @item void fprint (cl_ostream stream, const char * string)
2544 Prints the @code{string} literally on the @code{stream}.
2546 @item void fprintdecimal (cl_ostream stream, int x)
2547 @itemx void fprintdecimal (cl_ostream stream, const cl_I& x)
2548 Prints the integer @code{x} in decimal on the @code{stream}.
2550 @item void fprintbinary (cl_ostream stream, const cl_I& x)
2551 Prints the integer @code{x} in binary (base 2, without prefix)
2552 on the @code{stream}.
2554 @item void fprintoctal (cl_ostream stream, const cl_I& x)
2555 Prints the integer @code{x} in octal (base 8, without prefix)
2556 on the @code{stream}.
2558 @item void fprinthexadecimal (cl_ostream stream, const cl_I& x)
2559 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2560 on the @code{stream}.
2563 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2564 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2565 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2568 @item void fprint (cl_ostream stream, const @var{type}& x)
2569 @itemx cl_ostream operator<< (cl_ostream stream, const @var{type}& x)
2570 Prints the number @code{x} on the @code{stream}. The output may depend
2571 on the global printer settings in the variable @code{default_print_flags}.
2572 The @code{ostream} flags and settings (flags, width and locale) are
2576 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2579 void print_complex (cl_ostream stream, const cl_print_flags& flags,
2581 void print_real (cl_ostream stream, const cl_print_flags& flags,
2583 void print_float (cl_ostream stream, const cl_print_flags& flags,
2585 void print_rational (cl_ostream stream, const cl_print_flags& flags,
2587 void print_integer (cl_ostream stream, const cl_print_flags& flags,
2590 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2591 parameters which affect the output.
2593 The structure type @code{cl_print_flags} contains the following fields:
2596 @item unsigned int rational_base
2597 The base in which rational numbers are printed. Default is @code{10}.
2599 @item cl_boolean rational_readably
2600 If this flag is true, rational numbers are printed with radix specifiers in
2601 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2602 prefixes, trailing dot). Default is false.
2604 @item cl_boolean float_readably
2605 If this flag is true, type specific exponent markers have precedence over 'E'.
2608 @item cl_float_format_t default_float_format
2609 Floating point numbers of this format will be printed using the 'E' exponent
2610 marker. Default is @code{cl_float_format_ffloat}.
2612 @item cl_boolean complex_readably
2613 If this flag is true, complex numbers will be printed using the Common Lisp
2614 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2616 @item cl_string univpoly_varname
2617 Univariate polynomials with no explicit indeterminate name will be printed
2618 using this variable name. Default is @code{"x"}.
2621 The global variable @code{default_print_flags} contains the default values,
2622 used by the function @code{fprint}.
2627 CLN has a class of abstract rings.
2635 Rings can be compared for equality:
2638 @item bool operator== (const cl_ring&, const cl_ring&)
2639 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2640 These compare two rings for equality.
2643 Given a ring @code{R}, the following members can be used.
2646 @item void R->fprint (cl_ostream stream, const cl_ring_element& x)
2647 @cindex @code{fprint ()}
2648 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2649 @cindex @code{equal ()}
2650 @itemx cl_ring_element R->zero ()
2651 @cindex @code{zero ()}
2652 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2653 @cindex @code{zerop ()}
2654 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2655 @cindex @code{plus ()}
2656 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2657 @cindex @code{minus ()}
2658 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2659 @cindex @code{uminus ()}
2660 @itemx cl_ring_element R->one ()
2661 @cindex @code{one ()}
2662 @itemx cl_ring_element R->canonhom (const cl_I& x)
2663 @cindex @code{canonhom ()}
2664 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2665 @cindex @code{mul ()}
2666 @itemx cl_ring_element R->square (const cl_ring_element& x)
2667 @cindex @code{square ()}
2668 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2669 @cindex @code{expt_pos ()}
2672 The following rings are built-in.
2675 @item cl_null_ring cl_0_ring
2676 The null ring, containing only zero.
2678 @item cl_complex_ring cl_C_ring
2679 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2681 @item cl_real_ring cl_R_ring
2682 The ring of real numbers. This corresponds to the type @code{cl_R}.
2684 @item cl_rational_ring cl_RA_ring
2685 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2687 @item cl_integer_ring cl_I_ring
2688 The ring of integers. This corresponds to the type @code{cl_I}.
2691 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2692 @code{cl_RA_ring}, @code{cl_I_ring}:
2695 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2696 @cindex @code{instanceof ()}
2697 Tests whether the given number is an element of the number ring R.
2701 @chapter Modular integers
2702 @cindex modular integer
2704 @section Modular integer rings
2707 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2708 The modulus is explicitly part of every modular integer. CLN doesn't
2709 allow you to (accidentally) mix elements of different modular rings,
2710 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2711 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2712 doesn't have generic types. So one has to live with runtime checks.)
2714 The class of modular integer rings is
2722 Modular integer ring
2726 @cindex @code{cl_modint_ring}
2728 and the class of all modular integers (elements of modular integer rings) is
2736 Modular integer rings are constructed using the function
2739 @item cl_modint_ring find_modint_ring (const cl_I& N)
2740 @cindex @code{find_modint_ring ()}
2741 This function returns the modular ring @samp{Z/NZ}. It takes care
2742 of finding out about special cases of @code{N}, like powers of two
2743 and odd numbers for which Montgomery multiplication will be a win,
2744 @cindex Montgomery multiplication
2745 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2746 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2747 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2751 Modular integer rings can be compared for equality:
2754 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2755 @cindex @code{operator == ()}
2756 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2757 @cindex @code{operator != ()}
2758 These compare two modular integer rings for equality. Two different calls
2759 to @code{find_modint_ring} with the same argument necessarily return the
2760 same ring because it is memoized in the cache table.
2763 @section Functions on modular integers
2765 Given a modular integer ring @code{R}, the following members can be used.
2768 @item cl_I R->modulus
2769 @cindex @code{modulus}
2770 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2772 @item cl_MI R->zero()
2773 @cindex @code{zero ()}
2774 This returns @code{0 mod N}.
2776 @item cl_MI R->one()
2777 @cindex @code{one ()}
2778 This returns @code{1 mod N}.
2780 @item cl_MI R->canonhom (const cl_I& x)
2781 @cindex @code{canonhom ()}
2782 This returns @code{x mod N}.
2784 @item cl_I R->retract (const cl_MI& x)
2785 @cindex @code{retract ()}
2786 This is a partial inverse function to @code{R->canonhom}. It returns the
2787 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2789 @item cl_MI R->random(random_state& randomstate)
2790 @itemx cl_MI R->random()
2791 @cindex @code{random ()}
2792 This returns a random integer modulo @code{N}.
2795 The following operations are defined on modular integers.
2798 @item cl_modint_ring x.ring ()
2799 @cindex @code{ring ()}
2800 Returns the ring to which the modular integer @code{x} belongs.
2802 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2803 @cindex @code{operator + ()}
2804 Returns the sum of two modular integers. One of the arguments may also
2807 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2808 @cindex @code{operator - ()}
2809 Returns the difference of two modular integers. One of the arguments may also
2812 @item cl_MI operator- (const cl_MI&)
2813 Returns the negative of a modular integer.
2815 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2816 @cindex @code{operator * ()}
2817 Returns the product of two modular integers. One of the arguments may also
2820 @item cl_MI square (const cl_MI&)
2821 @cindex @code{square ()}
2822 Returns the square of a modular integer.
2824 @item cl_MI recip (const cl_MI& x)
2825 @cindex @code{recip ()}
2826 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2827 must be coprime to the modulus, otherwise an error message is issued.
2829 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2830 @cindex @code{div ()}
2831 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2832 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2834 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2835 @cindex @code{expt_pos ()}
2836 @code{y} must be > 0. Returns @code{x^y}.
2838 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2839 @cindex @code{expt ()}
2840 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2841 modulus, else an error message is issued.
2843 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2844 @cindex @code{operator << ()}
2845 Returns @code{x*2^y}.
2847 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2848 @cindex @code{operator >> ()}
2849 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2850 or an error message is issued.
2852 @item bool operator== (const cl_MI&, const cl_MI&)
2853 @cindex @code{operator == ()}
2854 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2855 @cindex @code{operator != ()}
2856 Compares two modular integers, belonging to the same modular integer ring,
2859 @item cl_boolean zerop (const cl_MI& x)
2860 @cindex @code{zerop ()}
2861 Returns true if @code{x} is @code{0 mod N}.
2864 The following output functions are defined (see also the chapter on
2868 @item void fprint (cl_ostream stream, const cl_MI& x)
2869 @cindex @code{fprint ()}
2870 @itemx cl_ostream operator<< (cl_ostream stream, const cl_MI& x)
2871 @cindex @code{operator << ()}
2872 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2873 on the global printer settings in the variable @code{default_print_flags}.
2877 @chapter Symbolic data types
2878 @cindex symbolic type
2880 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2884 @cindex @code{cl_string}
2894 implements immutable strings.
2896 Strings are constructed through the following constructors:
2899 @item cl_string (const char * s)
2900 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2902 @item cl_string (const char * ptr, unsigned long len)
2903 Returns an immutable copy of the @code{len} characters at
2904 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2907 The following functions are available on strings:
2911 Assignment from @code{cl_string} and @code{const char *}.
2914 @cindex @code{length ()}
2916 @cindex @code{strlen ()}
2917 Returns the length of the string @code{s}.
2920 @cindex @code{operator [] ()}
2921 Returns the @code{i}th character of the string @code{s}.
2922 @code{i} must be in the range @code{0 <= i < s.length()}.
2924 @item bool equal (const cl_string& s1, const cl_string& s2)
2925 @cindex @code{equal ()}
2926 Compares two strings for equality. One of the arguments may also be a
2927 plain @code{const char *}.
2932 @cindex @code{cl_symbol}
2934 Symbols are uniquified strings: all symbols with the same name are shared.
2935 This means that comparison of two symbols is fast (effectively just a pointer
2936 comparison), whereas comparison of two strings must in the worst case walk
2937 both strings until their end.
2938 Symbols are used, for example, as tags for properties, as names of variables
2939 in polynomial rings, etc.
2941 Symbols are constructed through the following constructor:
2944 @item cl_symbol (const cl_string& s)
2945 Looks up or creates a new symbol with a given name.
2948 The following operations are available on symbols:
2951 @item cl_string (const cl_symbol& sym)
2952 Conversion to @code{cl_string}: Returns the string which names the symbol
2955 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2956 @cindex @code{equal ()}
2957 Compares two symbols for equality. This is very fast.
2961 @chapter Univariate polynomials
2963 @cindex univariate polynomial
2965 @section Univariate polynomial rings
2967 CLN implements univariate polynomials (polynomials in one variable) over an
2968 arbitrary ring. The indeterminate variable may be either unnamed (and will be
2969 printed according to @code{default_print_flags.univpoly_varname}, which
2970 defaults to @samp{x}) or carry a given name. The base ring and the
2971 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
2972 (accidentally) mix elements of different polynomial rings, e.g.
2973 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
2974 return a multivariate polynomial, but they are not yet implemented in CLN.)
2976 The classes of univariate polynomial rings are
2984 Univariate polynomial ring
2988 +----------------+-------------------+
2990 Complex polynomial ring | Modular integer polynomial ring
2991 cl_univpoly_complex_ring | cl_univpoly_modint_ring
2992 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
2996 Real polynomial ring |
2997 cl_univpoly_real_ring |
2998 <cln/univpoly_real.h> |
3002 Rational polynomial ring |
3003 cl_univpoly_rational_ring |
3004 <cln/univpoly_rational.h> |
3008 Integer polynomial ring
3009 cl_univpoly_integer_ring
3010 <cln/univpoly_integer.h>
3013 and the corresponding classes of univariate polynomials are
3016 Univariate polynomial
3020 +----------------+-------------------+
3022 Complex polynomial | Modular integer polynomial
3024 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3030 <cln/univpoly_real.h> |
3034 Rational polynomial |
3036 <cln/univpoly_rational.h> |
3042 <cln/univpoly_integer.h>
3045 Univariate polynomial rings are constructed using the functions
3048 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3049 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3050 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3051 @code{R} may be an arbitrary ring. This function takes care of finding out
3052 about special cases of @code{R}, such as the rings of complex numbers,
3053 real numbers, rational numbers, integers, or modular integer rings.
3054 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3055 This ensures that two calls of this function with the same arguments will
3056 return the same polynomial ring.
3058 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3059 @cindex @code{find_univpoly_ring ()}
3060 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3061 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3062 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3063 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3064 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3065 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3066 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3067 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3068 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3069 These functions are equivalent to the general @code{find_univpoly_ring},
3070 only the return type is more specific, according to the base ring's type.
3073 @section Functions on univariate polynomials
3075 Given a univariate polynomial ring @code{R}, the following members can be used.
3078 @item cl_ring R->basering()
3079 @cindex @code{basering ()}
3080 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3082 @item cl_UP R->zero()
3083 @cindex @code{zero ()}
3084 This returns @code{0 in R}, a polynomial of degree -1.
3086 @item cl_UP R->one()
3087 @cindex @code{one ()}
3088 This returns @code{1 in R}, a polynomial of degree <= 0.
3090 @item cl_UP R->canonhom (const cl_I& x)
3091 @cindex @code{canonhom ()}
3092 This returns @code{x in R}, a polynomial of degree <= 0.
3094 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3095 @cindex @code{monomial ()}
3096 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3099 @item cl_UP R->create (sintL degree)
3100 @cindex @code{create ()}
3101 Creates a new polynomial with a given degree. The zero polynomial has degree
3102 @code{-1}. After creating the polynomial, you should put in the coefficients,
3103 using the @code{set_coeff} member function, and then call the @code{finalize}
3107 The following are the only destructive operations on univariate polynomials.
3110 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3111 @cindex @code{set_coeff ()}
3112 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3113 After changing a polynomial and before applying any "normal" operation on it,
3114 you should call its @code{finalize} member function.
3116 @item void finalize (cl_UP& x)
3117 @cindex @code{finalize ()}
3118 This function marks the endpoint of destructive modifications of a polynomial.
3119 It normalizes the internal representation so that subsequent computations have
3120 less overhead. Doing normal computations on unnormalized polynomials may
3121 produce wrong results or crash the program.
3124 The following operations are defined on univariate polynomials.
3127 @item cl_univpoly_ring x.ring ()
3128 @cindex @code{ring ()}
3129 Returns the ring to which the univariate polynomial @code{x} belongs.
3131 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3132 @cindex @code{operator + ()}
3133 Returns the sum of two univariate polynomials.
3135 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3136 @cindex @code{operator - ()}
3137 Returns the difference of two univariate polynomials.
3139 @item cl_UP operator- (const cl_UP&)
3140 Returns the negative of a univariate polynomial.
3142 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3143 @cindex @code{operator * ()}
3144 Returns the product of two univariate polynomials. One of the arguments may
3145 also be a plain integer or an element of the base ring.
3147 @item cl_UP square (const cl_UP&)
3148 @cindex @code{square ()}
3149 Returns the square of a univariate polynomial.
3151 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3152 @cindex @code{expt_pos ()}
3153 @code{y} must be > 0. Returns @code{x^y}.
3155 @item bool operator== (const cl_UP&, const cl_UP&)
3156 @cindex @code{operator == ()}
3157 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3158 @cindex @code{operator != ()}
3159 Compares two univariate polynomials, belonging to the same univariate
3160 polynomial ring, for equality.
3162 @item cl_boolean zerop (const cl_UP& x)
3163 @cindex @code{zerop ()}
3164 Returns true if @code{x} is @code{0 in R}.
3166 @item sintL degree (const cl_UP& x)
3167 @cindex @code{degree ()}
3168 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3170 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3171 @cindex @code{coeff ()}
3172 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3174 @item cl_ring_element x (const cl_ring_element& y)
3175 @cindex @code{operator () ()}
3176 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3177 then @samp{x(y)} returns the value of the substitution of @code{y} into
3180 @item cl_UP deriv (const cl_UP& x)
3181 @cindex @code{deriv ()}
3182 Returns the derivative of the polynomial @code{x} with respect to the
3183 indeterminate @code{X}.
3186 The following output functions are defined (see also the chapter on
3190 @item void fprint (cl_ostream stream, const cl_UP& x)
3191 @cindex @code{fprint ()}
3192 @itemx cl_ostream operator<< (cl_ostream stream, const cl_UP& x)
3193 @cindex @code{operator << ()}
3194 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3195 depend on the global printer settings in the variable
3196 @code{default_print_flags}.
3199 @section Special polynomials
3201 The following functions return special polynomials.
3204 @item cl_UP_I tschebychev (sintL n)
3205 @cindex @code{tschebychev ()}
3206 @cindex Chebyshev polynomial
3207 Returns the n-th Chebyshev polynomial (n >= 0).
3209 @item cl_UP_I hermite (sintL n)
3210 @cindex @code{hermite ()}
3211 @cindex Hermite polynomial
3212 Returns the n-th Hermite polynomial (n >= 0).
3214 @item cl_UP_RA legendre (sintL n)
3215 @cindex @code{legendre ()}
3216 @cindex Legende polynomial
3217 Returns the n-th Legendre polynomial (n >= 0).
3219 @item cl_UP_I laguerre (sintL n)
3220 @cindex @code{laguerre ()}
3221 @cindex Laguerre polynomial
3222 Returns the n-th Laguerre polynomial (n >= 0).
3225 Information how to derive the differential equation satisfied by each
3226 of these polynomials from their definition can be found in the
3227 @code{doc/polynomial/} directory.
3235 Using C++ as an implementation language provides
3239 Efficiency: It compiles to machine code.
3243 Portability: It runs on all platforms supporting a C++ compiler. Because
3244 of the availability of GNU C++, this includes all currently used 32-bit and
3245 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3248 Type safety: The C++ compilers knows about the number types and complains if,
3249 for example, you try to assign a float to an integer variable. However,
3250 a drawback is that C++ doesn't know about generic types, hence a restriction
3251 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3252 arguments belong to the same modular ring cannot be expressed as a compile-time
3256 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3257 @code{=}, @code{==}, ... can be used in infix notation, which is more
3258 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3261 With these language features, there is no need for two separate languages,
3262 one for the implementation of the library and one in which the library's users
3263 can program. This means that a prototype implementation of an algorithm
3264 can be integrated into the library immediately after it has been tested and
3265 debugged. No need to rewrite it in a low-level language after having prototyped
3266 in a high-level language.
3269 @section Memory efficiency
3271 In order to save memory allocations, CLN implements:
3275 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3278 @cindex garbage collection
3279 @cindex reference counting
3280 Garbage collection: A reference counting mechanism makes sure that any
3281 number object's storage is freed immediately when the last reference to the
3284 Small integers are represented as immediate values instead of pointers
3285 to heap allocated storage. This means that integers @code{> -2^29},
3286 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3291 @section Speed efficiency
3293 Speed efficiency is obtained by the combination of the following tricks
3298 Small integers, being represented as immediate values, don't require
3299 memory access, just a couple of instructions for each elementary operation.
3301 The kernel of CLN has been written in assembly language for some CPUs
3302 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3304 On all CPUs, CLN may be configured to use the superefficient low-level
3305 routines from GNU GMP version 3.
3307 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3308 algorithm, the Karatsuba multiplication, which is an
3319 For very large numbers (more than 12000 decimal digits), CLN uses
3321 Sch{@"o}nhage-Strassen
3322 @cindex Sch{@"o}nhage-Strassen multiplication
3326 @cindex Schönhage-Strassen multiplication
3328 multiplication, which is an asymptotically optimal multiplication
3331 These fast multiplication algorithms also give improvements in the speed
3332 of division and radix conversion.
3336 @section Garbage collection
3337 @cindex garbage collection
3339 All the number classes are reference count classes: They only contain a pointer
3340 to an object in the heap. Upon construction, assignment and destruction of
3341 number objects, only the objects' reference count are manipulated.
3343 Memory occupied by number objects are automatically reclaimed as soon as
3344 their reference count drops to zero.
3346 For number rings, another strategy is implemented: There is a cache of,
3347 for example, the modular integer rings. A modular integer ring is destroyed
3348 only if its reference count dropped to zero and the cache is about to be
3349 resized. The effect of this strategy is that recently used rings remain
3350 cached, whereas undue memory consumption through cached rings is avoided.
3353 @chapter Using the library
3355 For the following discussion, we will assume that you have installed
3356 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3357 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3358 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3359 environment variables, or directly substitute the appropriate values.
3362 @section Compiler options
3363 @cindex compiler options
3365 Until you have installed CLN in a public place, the following options are
3368 When you compile CLN application code, add the flags
3370 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3372 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3373 When you link CLN application code to form an executable, add the flags
3375 $CLN_TARGETDIR/src/libcln.a
3377 to the C/C++ compiler's command line (@code{make} variable LIBS).
3379 If you did a @code{make install}, the include files are installed in a
3380 public directory (normally @code{/usr/local/include}), hence you don't
3381 need special flags for compiling. The library has been installed to a
3382 public directory as well (normally @code{/usr/local/lib}), hence when
3383 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3386 @section Compatibility to old CLN versions
3388 @cindex compatibility
3390 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3391 @code{cln} in order to avoid potential name clashes with other C++
3392 libraries. If you have an old application, you will have to manually
3393 port it to the new scheme. The following principles will help during
3397 All headers are now in a separate subdirectory. Instead of including
3398 @code{cl_}@var{something}@code{.h}, include
3399 @code{cln/}@var{something}@code{.h} now.
3401 All public identifiers (typenames and functions) have lost their
3402 @code{cl_} prefix. Exceptions are all the typenames of number types,
3403 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3404 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3405 names would not be mnemonic enough once the namespace @code{cln} is
3406 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3408 All public @emph{functions} that had by a @code{cl_} in their name still
3409 carry that @code{cl_} if it is intrinsic part of a typename (as in
3410 @code{cl_I_to_int ()}).
3412 When developing other libraries, please keep in mind not to import the
3413 namespace @code{cln} in one of your public header files by saying
3414 @code{using namespace cln;}. This would propagate to other applications
3415 and can cause name clashes there.
3418 @section Include files
3419 @cindex include files
3420 @cindex header files
3422 Here is a summary of the include files and their contents.
3425 @item <cln/object.h>
3426 General definitions, reference counting, garbage collection.
3427 @item <cln/number.h>
3428 The class cl_number.
3429 @item <cln/complex.h>
3430 Functions for class cl_N, the complex numbers.
3432 Functions for class cl_R, the real numbers.
3434 Functions for class cl_F, the floats.
3435 @item <cln/sfloat.h>
3436 Functions for class cl_SF, the short-floats.
3437 @item <cln/ffloat.h>
3438 Functions for class cl_FF, the single-floats.
3439 @item <cln/dfloat.h>
3440 Functions for class cl_DF, the double-floats.
3441 @item <cln/lfloat.h>
3442 Functions for class cl_LF, the long-floats.
3443 @item <cln/rational.h>
3444 Functions for class cl_RA, the rational numbers.
3445 @item <cln/integer.h>
3446 Functions for class cl_I, the integers.
3449 @item <cln/complex_io.h>
3450 Input/Output for class cl_N, the complex numbers.
3451 @item <cln/real_io.h>
3452 Input/Output for class cl_R, the real numbers.
3453 @item <cln/float_io.h>
3454 Input/Output for class cl_F, the floats.
3455 @item <cln/sfloat_io.h>
3456 Input/Output for class cl_SF, the short-floats.
3457 @item <cln/ffloat_io.h>
3458 Input/Output for class cl_FF, the single-floats.
3459 @item <cln/dfloat_io.h>
3460 Input/Output for class cl_DF, the double-floats.
3461 @item <cln/lfloat_io.h>
3462 Input/Output for class cl_LF, the long-floats.
3463 @item <cln/rational_io.h>
3464 Input/Output for class cl_RA, the rational numbers.
3465 @item <cln/integer_io.h>
3466 Input/Output for class cl_I, the integers.
3468 Flags for customizing input operations.
3469 @item <cln/output.h>
3470 Flags for customizing output operations.
3471 @item <cln/malloc.h>
3472 @code{malloc_hook}, @code{free_hook}.
3475 @item <cln/condition.h>
3476 Conditions/exceptions.
3477 @item <cln/string.h>
3479 @item <cln/symbol.h>
3481 @item <cln/proplist.h>
3485 @item <cln/null_ring.h>
3487 @item <cln/complex_ring.h>
3488 The ring of complex numbers.
3489 @item <cln/real_ring.h>
3490 The ring of real numbers.
3491 @item <cln/rational_ring.h>
3492 The ring of rational numbers.
3493 @item <cln/integer_ring.h>
3494 The ring of integers.
3495 @item <cln/numtheory.h>
3496 Number threory functions.
3497 @item <cln/modinteger.h>
3503 @item <cln/GV_number.h>
3504 General vectors over cl_number.
3505 @item <cln/GV_complex.h>
3506 General vectors over cl_N.
3507 @item <cln/GV_real.h>
3508 General vectors over cl_R.
3509 @item <cln/GV_rational.h>
3510 General vectors over cl_RA.
3511 @item <cln/GV_integer.h>
3512 General vectors over cl_I.
3513 @item <cln/GV_modinteger.h>
3514 General vectors of modular integers.
3517 @item <cln/SV_number.h>
3518 Simple vectors over cl_number.
3519 @item <cln/SV_complex.h>
3520 Simple vectors over cl_N.
3521 @item <cln/SV_real.h>
3522 Simple vectors over cl_R.
3523 @item <cln/SV_rational.h>
3524 Simple vectors over cl_RA.
3525 @item <cln/SV_integer.h>
3526 Simple vectors over cl_I.
3527 @item <cln/SV_ringelt.h>
3528 Simple vectors of general ring elements.
3529 @item <cln/univpoly.h>
3530 Univariate polynomials.
3531 @item <cln/univpoly_integer.h>
3532 Univariate polynomials over the integers.
3533 @item <cln/univpoly_rational.h>
3534 Univariate polynomials over the rational numbers.
3535 @item <cln/univpoly_real.h>
3536 Univariate polynomials over the real numbers.
3537 @item <cln/univpoly_complex.h>
3538 Univariate polynomials over the complex numbers.
3539 @item <cln/univpoly_modint.h>
3540 Univariate polynomials over modular integer rings.
3541 @item <cln/timing.h>
3544 Includes all of the above.
3550 A function which computes the nth Fibonacci number can be written as follows.
3551 @cindex Fibonacci number
3554 #include <cln/integer.h>
3555 #include <cln/real.h>
3556 using namespace cln;
3558 // Returns F_n, computed as the nearest integer to
3559 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3560 const cl_I fibonacci (int n)
3562 // Need a precision of ((1+sqrt(5))/2)^-n.
3563 cl_float_format_t prec = cl_float_format((int)(0.208987641*n+5));
3564 cl_R sqrt5 = sqrt(cl_float(5,prec));
3565 cl_R phi = (1+sqrt5)/2;
3566 return round1( expt(phi,n)/sqrt5 );
3570 Let's explain what is going on in detail.
3572 The include file @code{<cln/integer.h>} is necessary because the type
3573 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3574 is needed for the type @code{cl_R} and the floating point number functions.
3575 The order of the include files does not matter. In order not to write out
3576 @code{cln::}@var{foo} we can safely import the whole namespace @code{cln}.
3578 Then comes the function declaration. The argument is an @code{int}, the
3579 result an integer. The return type is defined as @samp{const cl_I}, not
3580 simply @samp{cl_I}, because that allows the compiler to detect typos like
3581 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3582 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3583 number). We use the most specialized possible return type because functions
3584 which call @samp{fibonacci} will be able to profit from the compiler's type
3585 analysis: Adding two integers is slightly more efficient than adding the
3586 same objects declared as complex numbers, because it needs less type
3587 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3588 the size of the resulting executable program.
3590 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3591 integer. In order to get a correct result, the absolute error should be less
3592 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3593 To this end, the first line computes a floating point precision for sqrt(5)
3596 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3597 number and than taking the square root. The converse, first taking the square
3598 root of 5, and then converting to the desired precision, would not work in
3599 CLN: The square root would be computed to a default precision (normally
3600 single-float precision), and the following conversion could not help about
3601 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3602 system and does not represent sqrt(5) in a non-numeric way.
3604 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3605 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3606 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3607 @code{cl_N} because a @samp{round1} does not exist for general complex
3610 When the function returns, all the local variables in the function are
3611 automatically reclaimed (garbage collected). Only the result survives and
3612 gets passed to the caller.
3614 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3615 contains this implementation together with an even faster algorithm.
3617 @section Debugging support
3620 When debugging a CLN application with GNU @code{gdb}, two facilities are
3621 available from the library:
3624 @item The library does type checks, range checks, consistency checks at
3625 many places. When one of these fails, the function @code{cl_abort()} is
3626 called. Its default implementation is to perform an @code{exit(1)}, so
3627 you won't have a core dump. But for debugging, it is best to set a
3628 breakpoint at this function:
3630 (gdb) break cl_abort
3632 When this breakpoint is hit, look at the stack's backtrace:
3637 @item The debugger's normal @code{print} command doesn't know about
3638 CLN's types and therefore prints mostly useless hexadecimal addresses.
3639 CLN offers a function @code{cl_print}, callable from the debugger,
3640 for printing number objects. In order to get this function, you have
3641 to define the macro @samp{CL_DEBUG} and then include all the header files
3642 for which you want @code{cl_print} debugging support. For example:
3643 @cindex @code{CL_DEBUG}
3646 #include <cln/string.h>
3648 Now, if you have in your program a variable @code{cl_string s}, and
3649 inspect it under @code{gdb}, the output may look like this:
3652 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3653 word = 134568800@}@}, @}
3654 (gdb) call cl_print(s)
3658 Note that the output of @code{cl_print} goes to the program's error output,
3659 not to gdb's standard output.
3661 Note, however, that the above facility does not work with all CLN types,
3662 only with number objects and similar. Therefore CLN offers a member function
3663 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3664 is needed for this member function to be implemented. Under @code{gdb},
3665 you call it like this:
3666 @cindex @code{debug_print ()}
3669 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3670 word = 134568800@}@}, @}
3671 (gdb) call s.debug_print()
3674 >call ($1).debug_print()
3679 Unfortunately, this feature does not seem to work under all circumstances.
3683 @chapter Customizing
3686 @section Error handling
3688 When a fatal error occurs, an error message is output to the standard error
3689 output stream, and the function @code{cl_abort} is called. The default
3690 version of this function (provided in the library) terminates the application.
3691 To catch such a fatal error, you need to define the function @code{cl_abort}
3692 yourself, with the prototype
3694 #include <cln/abort.h>
3695 void cl_abort (void);
3697 @cindex @code{cl_abort ()}
3698 This function must not return control to its caller.
3701 @section Floating-point underflow
3704 Floating point underflow denotes the situation when a floating-point number
3705 is to be created which is so close to @code{0} that its exponent is too
3706 low to be represented internally. By default, this causes a fatal error.
3707 If you set the global variable
3709 cl_boolean cl_inhibit_floating_point_underflow
3711 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3712 will be generated instead. The default value of
3713 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3716 @section Customizing I/O
3718 The output of the function @code{fprint} may be customized by changing the
3719 value of the global variable @code{default_print_flags}.
3720 @cindex @code{default_print_flags}
3723 @section Customizing the memory allocator
3725 Every memory allocation of CLN is done through the function pointer
3726 @code{malloc_hook}. Freeing of this memory is done through the function
3727 pointer @code{free_hook}. The default versions of these functions,
3728 provided in the library, call @code{malloc} and @code{free} and check
3729 the @code{malloc} result against @code{NULL}.
3730 If you want to provide another memory allocator, you need to define
3731 the variables @code{malloc_hook} and @code{free_hook} yourself,
3734 #include <cln/malloc.h>
3736 void* (*malloc_hook) (size_t size) = @dots{};
3737 void (*free_hook) (void* ptr) = @dots{};
3740 @cindex @code{malloc_hook ()}
3741 @cindex @code{free_hook ()}
3742 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3744 It is not possible to change the memory allocator at runtime, because
3745 it is already called at program startup by the constructors of some
3758 @c Table of contents