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 @cindex immediate numbers
557 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
558 for 32-bit machines) are especially efficient, because they consume no heap
559 allocation. Otherwise the distinction between these immediate integers
560 (called ``fixnums'') and heap allocated integers (called ``bignums'')
561 is completely transparent.
564 @section Floating-point numbers
565 @cindex floating-point number
567 Not all real numbers can be represented exactly. (There is an easy mathematical
568 proof for this: Only a countable set of numbers can be stored exactly in
569 a computer, even if one assumes that it has unlimited storage. But there
570 are uncountably many real numbers.) So some approximation is needed.
571 CLN implements ordinary floating-point numbers, with mantissa and exponent.
573 @cindex rounding error
574 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
575 only return approximate results. For example, the value of the expression
576 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
577 @samp{0.7}. Rounding errors like this one are inevitable when computing
578 with floating-point numbers.
580 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
581 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
582 rule: It first computes the exact mathematical result and then returns the
583 floating-point number which is nearest to this. If two floating-point numbers
584 are equally distant from the ideal result, the one with a @code{0} in its least
585 significant mantissa bit is chosen.
587 Similarly, testing floating point numbers for equality @samp{x == y}
588 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
589 for some well-chosen @code{epsilon}.
591 Floating point numbers come in four flavors:
596 Short floats, type @code{cl_SF}.
597 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
598 and 17 mantissa bits (including the ``hidden'' bit).
599 They don't consume heap allocation.
603 Single floats, type @code{cl_FF}.
604 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
605 and 24 mantissa bits (including the ``hidden'' bit).
606 In CLN, they are represented as IEEE single-precision floating point numbers.
607 This corresponds closely to the C/C++ type @samp{float}.
611 Double floats, type @code{cl_DF}.
612 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
613 and 53 mantissa bits (including the ``hidden'' bit).
614 In CLN, they are represented as IEEE double-precision floating point numbers.
615 This corresponds closely to the C/C++ type @samp{double}.
619 Long floats, type @code{cl_LF}.
620 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
621 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
622 The precision of a long float is unlimited, but once created, a long float
623 has a fixed precision. (No ``lazy recomputation''.)
626 Of course, computations with long floats are more expensive than those
627 with smaller floating-point formats.
629 CLN does not implement features like NaNs, denormalized numbers and
630 gradual underflow. If the exponent range of some floating-point type
631 is too limited for your application, choose another floating-point type
632 with larger exponent range.
635 As a user of CLN, you can forget about the differences between the
636 four floating-point types and just declare all your floating-point
637 variables as being of type @code{cl_F}. This has the advantage that
638 when you change the precision of some computation (say, from @code{cl_DF}
639 to @code{cl_LF}), you don't have to change the code, only the precision
640 of the initial values. Also, many transcendental functions have been
641 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
642 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
643 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
644 the floating point contagion rule happened to change in the future.)
647 @section Complex numbers
648 @cindex complex number
650 Complex numbers, as implemented by the class @code{cl_N}, have a real
651 part and an imaginary part, both real numbers. A complex number whose
652 imaginary part is the exact number @code{0} is automatically converted
655 Complex numbers can arise from real numbers alone, for example
656 through application of @code{sqrt} or transcendental functions.
662 Conversions from any class to any its superclasses (``base classes'' in
663 C++ terminology) is done automatically.
665 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
666 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
667 @code{cl_N} and @code{cl_number}.
669 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
670 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
671 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
672 efficiency. Their range is therefore limited:
676 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
678 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
681 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
682 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
683 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
685 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
686 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
687 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
688 @samp{cl_I}, first convert it to an @samp{unsigned long}.
690 Conversions from the C built-in type @samp{float} are provided for the classes
691 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
693 Conversions from the C built-in type @samp{double} are provided for the classes
694 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
696 Conversions from @samp{const char *} are provided for the classes
697 @code{cl_I}, @code{cl_RA},
698 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
699 @code{cl_R}, @code{cl_N}.
700 The easiest way to specify a value which is outside of the range of the
701 C++ built-in types is therefore to specify it as a string, like this:
704 cl_I order_of_rubiks_cube_group = "43252003274489856000";
706 Note that this conversion is done at runtime, not at compile-time.
708 Conversions from @code{cl_I} to the C built-in types @samp{int},
709 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
713 @item int cl_I_to_int (const cl_I& x)
714 @cindex @code{cl_I_to_int ()}
715 @itemx unsigned int cl_I_to_uint (const cl_I& x)
716 @cindex @code{cl_I_to_uint ()}
717 @itemx long cl_I_to_long (const cl_I& x)
718 @cindex @code{cl_I_to_long ()}
719 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
720 @cindex @code{cl_I_to_ulong ()}
721 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
722 representable in the range of @var{ctype}, a runtime error occurs.
725 Conversions from the classes @code{cl_I}, @code{cl_RA},
726 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
728 to the C built-in types @samp{float} and @samp{double} are provided through
732 @item float float_approx (const @var{type}& x)
733 @cindex @code{float_approx ()}
734 @itemx double double_approx (const @var{type}& x)
735 @cindex @code{double_approx ()}
736 Returns an approximation of @code{x} of C type @var{ctype}.
737 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
738 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
741 Conversions from any class to any of its subclasses (``derived classes'' in
742 C++ terminology) are not provided. Instead, you can assert and check
743 that a value belongs to a certain subclass, and return it as element of that
744 class, using the @samp{As} and @samp{The} macros.
745 @cindex @code{As()()}
746 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
747 @var{type} and returns it as such.
748 @cindex @code{The()()}
749 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
750 @var{type} and returns it as such. It is your responsibility to ensure
751 that this assumption is valid. Since macros and namespaces don't go
752 together well, there is an equivalent to @samp{The}: the template
760 if (!(x >= 0)) abort();
761 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
762 // In general, it would be a rational number.
763 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
768 @chapter Functions on numbers
770 Each of the number classes declares its mathematical operations in the
771 corresponding include file. For example, if your code operates with
772 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
775 @section Constructing numbers
777 Here is how to create number objects ``from nothing''.
780 @subsection Constructing integers
782 @code{cl_I} objects are most easily constructed from C integers and from
783 strings. See @ref{Conversions}.
786 @subsection Constructing rational numbers
788 @code{cl_RA} objects can be constructed from strings. The syntax
789 for rational numbers is described in @ref{Internal and printed representation}.
790 Another standard way to produce a rational number is through application
791 of @samp{operator /} or @samp{recip} on integers.
794 @subsection Constructing floating-point numbers
796 @code{cl_F} objects with low precision are most easily constructed from
797 C @samp{float} and @samp{double}. See @ref{Conversions}.
799 To construct a @code{cl_F} with high precision, you can use the conversion
800 from @samp{const char *}, but you have to specify the desired precision
801 within the string. (See @ref{Internal and printed representation}.)
804 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
806 will set @samp{e} to the given value, with a precision of 40 decimal digits.
808 The programmatic way to construct a @code{cl_F} with high precision is
809 through the @code{cl_float} conversion function, see
810 @ref{Conversion to floating-point numbers}. For example, to compute
811 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
812 and then apply the exponential function:
814 float_format_t precision = float_format(40);
815 cl_F e = exp(cl_float(1,precision));
819 @subsection Constructing complex numbers
821 Non-real @code{cl_N} objects are normally constructed through the function
823 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
825 See @ref{Elementary complex functions}.
828 @section Elementary functions
830 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
831 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
832 defines the following operations:
835 @item @var{type} operator + (const @var{type}&, const @var{type}&)
836 @cindex @code{operator + ()}
839 @item @var{type} operator - (const @var{type}&, const @var{type}&)
840 @cindex @code{operator - ()}
843 @item @var{type} operator - (const @var{type}&)
844 Returns the negative of the argument.
846 @item @var{type} plus1 (const @var{type}& x)
847 @cindex @code{plus1 ()}
848 Returns @code{x + 1}.
850 @item @var{type} minus1 (const @var{type}& x)
851 @cindex @code{minus1 ()}
852 Returns @code{x - 1}.
854 @item @var{type} operator * (const @var{type}&, const @var{type}&)
855 @cindex @code{operator * ()}
858 @item @var{type} square (const @var{type}& x)
859 @cindex @code{square ()}
860 Returns @code{x * x}.
863 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
864 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
865 defines the following operations:
868 @item @var{type} operator / (const @var{type}&, const @var{type}&)
869 @cindex @code{operator / ()}
872 @item @var{type} recip (const @var{type}&)
873 @cindex @code{recip ()}
874 Returns the reciprocal of the argument.
877 The class @code{cl_I} doesn't define a @samp{/} operation because
878 in the C/C++ language this operator, applied to integral types,
879 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
880 is implementation dependent). (@xref{Rounding functions}.)
881 Instead, @code{cl_I} defines an ``exact quotient'' function:
884 @item cl_I exquo (const cl_I& x, const cl_I& y)
885 @cindex @code{exquo ()}
886 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
889 The following exponentiation functions are defined:
892 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
893 @cindex @code{expt_pos ()}
894 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
895 @code{y} must be > 0. Returns @code{x^y}.
897 @item cl_RA expt (const cl_RA& x, const cl_I& y)
898 @cindex @code{expt ()}
899 @itemx cl_R expt (const cl_R& x, const cl_I& y)
900 @itemx cl_N expt (const cl_N& x, const cl_I& y)
904 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
905 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
906 defines the following operation:
909 @item @var{type} abs (const @var{type}& x)
910 @cindex @code{abs ()}
911 Returns the absolute value of @code{x}.
912 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
915 The class @code{cl_N} implements this as follows:
918 @item cl_R abs (const cl_N x)
919 Returns the absolute value of @code{x}.
922 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
923 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
924 defines the following operation:
927 @item @var{type} signum (const @var{type}& x)
928 @cindex @code{signum ()}
929 Returns the sign of @code{x}, in the same number format as @code{x}.
930 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
931 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
936 @section Elementary rational functions
938 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
941 @item cl_I numerator (const @var{type}& x)
942 @cindex @code{numerator ()}
943 Returns the numerator of @code{x}.
945 @item cl_I denominator (const @var{type}& x)
946 @cindex @code{denominator ()}
947 Returns the denominator of @code{x}.
950 The numerator and denominator of a rational number are normalized in such
951 a way that they have no factor in common and the denominator is positive.
954 @section Elementary complex functions
956 The class @code{cl_N} defines the following operation:
959 @item cl_N complex (const cl_R& a, const cl_R& b)
960 @cindex @code{complex ()}
961 Returns the complex number @code{a+bi}, that is, the complex number with
962 real part @code{a} and imaginary part @code{b}.
965 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
968 @item cl_R realpart (const @var{type}& x)
969 @cindex @code{realpart ()}
970 Returns the real part of @code{x}.
972 @item cl_R imagpart (const @var{type}& x)
973 @cindex @code{imagpart ()}
974 Returns the imaginary part of @code{x}.
976 @item @var{type} conjugate (const @var{type}& x)
977 @cindex @code{conjugate ()}
978 Returns the complex conjugate of @code{x}.
981 We have the relations
985 @code{x = complex(realpart(x), imagpart(x))}
987 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
994 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
995 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
996 defines the following operations:
999 @item bool operator == (const @var{type}&, const @var{type}&)
1000 @cindex @code{operator == ()}
1001 @itemx bool operator != (const @var{type}&, const @var{type}&)
1002 @cindex @code{operator != ()}
1003 Comparison, as in C and C++.
1005 @item uint32 equal_hashcode (const @var{type}&)
1006 @cindex @code{equal_hashcode ()}
1007 Returns a 32-bit hash code that is the same for any two numbers which are
1008 the same according to @code{==}. This hash code depends on the number's value,
1009 not its type or precision.
1011 @item cl_boolean zerop (const @var{type}& x)
1012 @cindex @code{zerop ()}
1013 Compare against zero: @code{x == 0}
1016 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1017 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1018 defines the following operations:
1021 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1022 @cindex @code{compare ()}
1023 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1024 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1026 @item bool operator <= (const @var{type}&, const @var{type}&)
1027 @cindex @code{operator <= ()}
1028 @itemx bool operator < (const @var{type}&, const @var{type}&)
1029 @cindex @code{operator < ()}
1030 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1031 @cindex @code{operator >= ()}
1032 @itemx bool operator > (const @var{type}&, const @var{type}&)
1033 @cindex @code{operator > ()}
1034 Comparison, as in C and C++.
1036 @item cl_boolean minusp (const @var{type}& x)
1037 @cindex @code{minusp ()}
1038 Compare against zero: @code{x < 0}
1040 @item cl_boolean plusp (const @var{type}& x)
1041 @cindex @code{plusp ()}
1042 Compare against zero: @code{x > 0}
1044 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1045 @cindex @code{max ()}
1046 Return the maximum of @code{x} and @code{y}.
1048 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1049 @cindex @code{min ()}
1050 Return the minimum of @code{x} and @code{y}.
1053 When a floating point number and a rational number are compared, the float
1054 is first converted to a rational number using the function @code{rational}.
1055 Since a floating point number actually represents an interval of real numbers,
1056 the result might be surprising.
1057 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1058 there is no floating point number whose value is exactly @code{1/3}.
1061 @section Rounding functions
1064 When a real number is to be converted to an integer, there is no ``best''
1065 rounding. The desired rounding function depends on the application.
1066 The Common Lisp and ISO Lisp standards offer four rounding functions:
1070 This is the largest integer <=@code{x}.
1073 This is the smallest integer >=@code{x}.
1076 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1079 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1080 integers, choose the even one.
1083 These functions have different advantages:
1085 @code{floor} and @code{ceiling} are translation invariant:
1086 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1087 for every @code{x} and every integer @code{n}.
1089 On the other hand, @code{truncate} and @code{round} are symmetric:
1090 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1091 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1092 down exactly as often as it rounds up.
1094 The functions are related like this:
1098 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1099 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1101 @code{truncate(x) = sign(x) * floor(abs(x))}
1104 Each of the classes @code{cl_R}, @code{cl_RA},
1105 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1106 defines the following operations:
1109 @item cl_I floor1 (const @var{type}& x)
1110 @cindex @code{floor1 ()}
1111 Returns @code{floor(x)}.
1112 @item cl_I ceiling1 (const @var{type}& x)
1113 @cindex @code{ceiling1 ()}
1114 Returns @code{ceiling(x)}.
1115 @item cl_I truncate1 (const @var{type}& x)
1116 @cindex @code{truncate1 ()}
1117 Returns @code{truncate(x)}.
1118 @item cl_I round1 (const @var{type}& x)
1119 @cindex @code{round1 ()}
1120 Returns @code{round(x)}.
1123 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1124 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1125 defines the following operations:
1128 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1129 Returns @code{floor(x/y)}.
1130 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1131 Returns @code{ceiling(x/y)}.
1132 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1133 Returns @code{truncate(x/y)}.
1134 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1135 Returns @code{round(x/y)}.
1138 These functions are called @samp{floor1}, @dots{} here instead of
1139 @samp{floor}, @dots{}, because on some systems, system dependent include
1140 files define @samp{floor} and @samp{ceiling} as macros.
1142 In many cases, one needs both the quotient and the remainder of a division.
1143 It is more efficient to compute both at the same time than to perform
1144 two divisions, one for quotient and the next one for the remainder.
1145 The following functions therefore return a structure containing both
1146 the quotient and the remainder. The suffix @samp{2} indicates the number
1147 of ``return values''. The remainder is defined as follows:
1151 for the computation of @code{quotient = floor(x)},
1152 @code{remainder = x - quotient},
1154 for the computation of @code{quotient = floor(x,y)},
1155 @code{remainder = x - quotient*y},
1158 and similarly for the other three operations.
1160 Each of the classes @code{cl_R}, @code{cl_RA},
1161 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1162 defines the following operations:
1165 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1166 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1167 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1168 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1169 @itemx @var{type}_div_t round2 (const @var{type}& x)
1172 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1173 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1174 defines the following operations:
1177 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1178 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1179 @cindex @code{floor2 ()}
1180 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1181 @cindex @code{ceiling2 ()}
1182 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1183 @cindex @code{truncate2 ()}
1184 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1185 @cindex @code{round2 ()}
1188 Sometimes, one wants the quotient as a floating-point number (of the
1189 same format as the argument, if the argument is a float) instead of as
1190 an integer. The prefix @samp{f} indicates this.
1193 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1194 defines the following operations:
1197 @item @var{type} ffloor (const @var{type}& x)
1198 @cindex @code{ffloor ()}
1199 @itemx @var{type} fceiling (const @var{type}& x)
1200 @cindex @code{fceiling ()}
1201 @itemx @var{type} ftruncate (const @var{type}& x)
1202 @cindex @code{ftruncate ()}
1203 @itemx @var{type} fround (const @var{type}& x)
1204 @cindex @code{fround ()}
1207 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1209 The class @code{cl_R} defines the following operations:
1212 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1213 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1214 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1215 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1218 These functions also exist in versions which return both the quotient
1219 and the remainder. The suffix @samp{2} indicates this.
1222 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1223 defines the following operations:
1224 @cindex @code{cl_F_fdiv_t}
1225 @cindex @code{cl_SF_fdiv_t}
1226 @cindex @code{cl_FF_fdiv_t}
1227 @cindex @code{cl_DF_fdiv_t}
1228 @cindex @code{cl_LF_fdiv_t}
1231 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1232 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1233 @cindex @code{ffloor2 ()}
1234 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1235 @cindex @code{fceiling2 ()}
1236 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1237 @cindex @code{ftruncate2 ()}
1238 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1239 @cindex @code{fround2 ()}
1241 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1242 @cindex @code{cl_R_fdiv_t}
1244 The class @code{cl_R} defines the following operations:
1247 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1248 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1249 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1250 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1251 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1254 Other applications need only the remainder of a division.
1255 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1256 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1257 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1261 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1263 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1266 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1267 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1268 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1270 The classes @code{cl_R}, @code{cl_I} define the following operations:
1273 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1274 @cindex @code{mod ()}
1275 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1276 @cindex @code{rem ()}
1282 Each of the classes @code{cl_R},
1283 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1284 defines the following operation:
1287 @item @var{type} sqrt (const @var{type}& x)
1288 @cindex @code{sqrt ()}
1289 @code{x} must be >= 0. This function returns the square root of @code{x},
1290 normalized to be >= 0. If @code{x} is the square of a rational number,
1291 @code{sqrt(x)} will be a rational number, else it will return a
1292 floating-point approximation.
1295 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1298 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1299 @cindex @code{sqrtp ()}
1300 This tests whether @code{x} is a perfect square. If so, it returns true
1301 and the exact square root in @code{*root}, else it returns false.
1304 Furthermore, for integers, similarly:
1307 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1308 @cindex @code{isqrt ()}
1309 @code{x} should be >= 0. This function sets @code{*root} to
1310 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1311 the boolean value @code{(expt(*root,2) == x)}.
1314 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1315 define the following operation:
1318 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1319 @cindex @code{rootp ()}
1320 @code{x} must be >= 0. @code{n} must be > 0.
1321 This tests whether @code{x} is an @code{n}th power of a rational number.
1322 If so, it returns true and the exact root in @code{*root}, else it returns
1326 The only square root function which accepts negative numbers is the one
1327 for class @code{cl_N}:
1330 @item cl_N sqrt (const cl_N& z)
1331 @cindex @code{sqrt ()}
1332 Returns the square root of @code{z}, as defined by the formula
1333 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1334 or to a complex number are done if necessary. The range of the result is the
1335 right half plane @code{realpart(sqrt(z)) >= 0}
1336 including the positive imaginary axis and 0, but excluding
1337 the negative imaginary axis.
1338 The result is an exact number only if @code{z} is an exact number.
1342 @section Transcendental functions
1343 @cindex transcendental functions
1345 The transcendental functions return an exact result if the argument
1346 is exact and the result is exact as well. Otherwise they must return
1347 inexact numbers even if the argument is exact.
1348 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1351 @subsection Exponential and logarithmic functions
1354 @item cl_R exp (const cl_R& x)
1355 @cindex @code{exp ()}
1356 @itemx cl_N exp (const cl_N& x)
1357 Returns the exponential function of @code{x}. This is @code{e^x} where
1358 @code{e} is the base of the natural logarithms. The range of the result
1359 is the entire complex plane excluding 0.
1361 @item cl_R ln (const cl_R& x)
1362 @cindex @code{ln ()}
1363 @code{x} must be > 0. Returns the (natural) logarithm of x.
1365 @item cl_N log (const cl_N& x)
1366 @cindex @code{log ()}
1367 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1368 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1369 The range of the result is the strip in the complex plane
1370 @code{-pi < imagpart(log(x)) <= pi}.
1372 @item cl_R phase (const cl_N& x)
1373 @cindex @code{phase ()}
1374 Returns the angle part of @code{x} in its polar representation as a
1375 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1376 This is also the imaginary part of @code{log(x)}.
1377 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1378 The result will be an exact number only if @code{zerop(x)} or
1379 if @code{x} is real and positive.
1381 @item cl_R log (const cl_R& a, const cl_R& b)
1382 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1383 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1384 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1387 @item cl_N log (const cl_N& a, const cl_N& b)
1388 Returns the logarithm of @code{a} with respect to base @code{b}.
1389 @code{log(a,b) = log(a)/log(b)}.
1391 @item cl_N expt (const cl_N& x, const cl_N& y)
1392 @cindex @code{expt ()}
1393 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1396 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1399 @item cl_F exp1 (float_format_t f)
1400 @cindex @code{exp1 ()}
1401 Returns e as a float of format @code{f}.
1403 @item cl_F exp1 (const cl_F& y)
1404 Returns e in the float format of @code{y}.
1406 @item cl_F exp1 (void)
1407 Returns e as a float of format @code{default_float_format}.
1411 @subsection Trigonometric functions
1414 @item cl_R sin (const cl_R& x)
1415 @cindex @code{sin ()}
1416 Returns @code{sin(x)}. The range of the result is the interval
1417 @code{-1 <= sin(x) <= 1}.
1419 @item cl_N sin (const cl_N& z)
1420 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1422 @item cl_R cos (const cl_R& x)
1423 @cindex @code{cos ()}
1424 Returns @code{cos(x)}. The range of the result is the interval
1425 @code{-1 <= cos(x) <= 1}.
1427 @item cl_N cos (const cl_N& x)
1428 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1430 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1431 @cindex @code{cos_sin_t}
1432 @itemx cos_sin_t cos_sin (const cl_R& x)
1433 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1434 @cindex @code{cos_sin ()}
1435 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1436 hold only approximately.
1438 @item cl_R tan (const cl_R& x)
1439 @cindex @code{tan ()}
1440 @itemx cl_N tan (const cl_N& x)
1441 Returns @code{tan(x) = sin(x)/cos(x)}.
1443 @item cl_N cis (const cl_R& x)
1444 @cindex @code{cis ()}
1445 @itemx cl_N cis (const cl_N& x)
1446 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1447 @code{e^(i*x) = cos(x) + i*sin(x)}.
1450 @cindex @code{asin ()}
1451 @item cl_N asin (const cl_N& z)
1452 Returns @code{arcsin(z)}. This is defined as
1453 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1454 @code{arcsin(-z) = -arcsin(z)}.
1455 The range of the result is the strip in the complex domain
1456 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1457 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1458 with @code{realpart = pi/2} and @code{imagpart > 0}.
1460 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1464 @item cl_N acos (const cl_N& z)
1465 @cindex @code{acos ()}
1466 Returns @code{arccos(z)}. This is defined as
1467 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1470 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1472 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1473 The range of the result is the strip in the complex domain
1474 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1475 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1476 with @code{realpart = pi} and @code{imagpart > 0}.
1478 Proof: This follows from the results about arcsin.
1482 @cindex @code{atan ()}
1483 @item cl_R atan (const cl_R& x, const cl_R& y)
1484 Returns the angle of the polar representation of the complex number
1485 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1486 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1487 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1488 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1489 with reversed order of arguments.
1491 @item cl_R atan (const cl_R& x)
1492 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1493 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1494 will be an exact number only if @code{x} is the exact @code{0}.
1496 @item cl_N atan (const cl_N& z)
1497 Returns @code{arctan(z)}. This is defined as
1498 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1499 @code{arctan(-z) = -arctan(z)}. The range of the result is
1500 the strip in the complex domain
1501 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1502 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1503 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1505 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1511 @cindex Archimedes' constant
1512 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1515 @item cl_F pi (float_format_t f)
1516 @cindex @code{pi ()}
1517 Returns pi as a float of format @code{f}.
1519 @item cl_F pi (const cl_F& y)
1520 Returns pi in the float format of @code{y}.
1522 @item cl_F pi (void)
1523 Returns pi as a float of format @code{default_float_format}.
1527 @subsection Hyperbolic functions
1530 @item cl_R sinh (const cl_R& x)
1531 @cindex @code{sinh ()}
1532 Returns @code{sinh(x)}.
1534 @item cl_N sinh (const cl_N& z)
1535 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1537 @item cl_R cosh (const cl_R& x)
1538 @cindex @code{cosh ()}
1539 Returns @code{cosh(x)}. The range of the result is the interval
1540 @code{cosh(x) >= 1}.
1542 @item cl_N cosh (const cl_N& z)
1543 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1545 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1546 @cindex @code{cosh_sinh_t}
1547 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1548 @cindex @code{cosh_sinh ()}
1549 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1550 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1551 hold only approximately.
1553 @item cl_R tanh (const cl_R& x)
1554 @cindex @code{tanh ()}
1555 @itemx cl_N tanh (const cl_N& x)
1556 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1558 @item cl_N asinh (const cl_N& z)
1559 @cindex @code{asinh ()}
1560 Returns @code{arsinh(z)}. This is defined as
1561 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1562 @code{arsinh(-z) = -arsinh(z)}.
1564 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1565 Actually, z+sqrt(1+z^2) can never be real and <0, so
1566 -pi < imagpart(arsinh(z)) < pi.
1567 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1568 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1570 The range of the result is the strip in the complex domain
1571 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1572 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1573 with @code{imagpart = pi/2} and @code{realpart < 0}.
1575 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1576 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1577 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1578 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1579 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1580 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1581 If y >= 1, the imagpart is pi/2 and the realpart is
1582 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1585 Moreover, if z is in Range(sqrt),
1586 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1587 (for a proof, see file src/cl_C_asinh.cc).
1590 @item cl_N acosh (const cl_N& z)
1591 @cindex @code{acosh ()}
1592 Returns @code{arcosh(z)}. This is defined as
1593 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1594 The range of the result is the half-strip in the complex domain
1595 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1596 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1598 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1599 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1600 If z is in Range(sqrt), we have
1601 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1602 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1604 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1605 and since the imagpart of both expressions is > -pi, <= pi
1606 ==> arcosh(z) = log(z+sqrt(z^2-1))
1607 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1608 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1609 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1610 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1611 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1613 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1614 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1615 = x^2 + y^2 + sqrt(u^2+v^2)
1620 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1621 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1622 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1623 Otherwise, -z is in Range(sqrt).
1624 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1625 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1626 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1627 and this has realpart > 0.
1628 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1629 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1630 has realpart = 0 and imagpart > 0.
1631 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1632 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1633 This has realpart >= 0 and imagpart = pi.
1636 @item cl_N atanh (const cl_N& z)
1637 @cindex @code{atanh ()}
1638 Returns @code{artanh(z)}. This is defined as
1639 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1640 @code{artanh(-z) = -artanh(z)}. The range of the result is
1641 the strip in the complex domain
1642 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1643 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1644 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1646 Proof: Write z = x+iy. Examine
1647 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1649 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1650 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1651 |x| < 1 ==> imagpart = 0
1654 = (atan(1+x,y) - atan(1-x,-y))/2
1655 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1656 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1657 > (pi - pi/2 - pi/2 )/2 = 0
1658 and (1+x)/y > (1-x)/y
1659 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1660 ==> imagpart < pi/2.
1661 Hence 0 < imagpart < pi/2.
1663 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1668 @subsection Euler gamma
1669 @cindex Euler's constant
1671 Euler's constant C = 0.577@dots{} is returned by the following functions:
1674 @item cl_F eulerconst (float_format_t f)
1675 @cindex @code{eulerconst ()}
1676 Returns Euler's constant as a float of format @code{f}.
1678 @item cl_F eulerconst (const cl_F& y)
1679 Returns Euler's constant in the float format of @code{y}.
1681 @item cl_F eulerconst (void)
1682 Returns Euler's constant as a float of format @code{default_float_format}.
1685 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1686 @cindex Catalan's constant
1689 @item cl_F catalanconst (float_format_t f)
1690 @cindex @code{catalanconst ()}
1691 Returns Catalan's constant as a float of format @code{f}.
1693 @item cl_F catalanconst (const cl_F& y)
1694 Returns Catalan's constant in the float format of @code{y}.
1696 @item cl_F catalanconst (void)
1697 Returns Catalan's constant as a float of format @code{default_float_format}.
1701 @subsection Riemann zeta
1702 @cindex Riemann's zeta
1704 Riemann's zeta function at an integral point @code{s>1} is returned by the
1705 following functions:
1708 @item cl_F zeta (int s, float_format_t f)
1709 @cindex @code{zeta ()}
1710 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1712 @item cl_F zeta (int s, const cl_F& y)
1713 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1715 @item cl_F zeta (int s)
1716 Returns Riemann's zeta function at @code{s} as a float of format
1717 @code{default_float_format}.
1721 @section Functions on integers
1723 @subsection Logical functions
1725 Integers, when viewed as in two's complement notation, can be thought as
1726 infinite bit strings where the bits' values eventually are constant.
1733 The logical operations view integers as such bit strings and operate
1734 on each of the bit positions in parallel.
1737 @item cl_I lognot (const cl_I& x)
1738 @cindex @code{lognot ()}
1739 @itemx cl_I operator ~ (const cl_I& x)
1740 @cindex @code{operator ~ ()}
1741 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1743 @item cl_I logand (const cl_I& x, const cl_I& y)
1744 @cindex @code{logand ()}
1745 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1746 @cindex @code{operator & ()}
1747 Logical and, like @code{x & y} in C.
1749 @item cl_I logior (const cl_I& x, const cl_I& y)
1750 @cindex @code{logior ()}
1751 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1752 @cindex @code{operator | ()}
1753 Logical (inclusive) or, like @code{x | y} in C.
1755 @item cl_I logxor (const cl_I& x, const cl_I& y)
1756 @cindex @code{logxor ()}
1757 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1758 @cindex @code{operator ^ ()}
1759 Exclusive or, like @code{x ^ y} in C.
1761 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1762 @cindex @code{logeqv ()}
1763 Bitwise equivalence, like @code{~(x ^ y)} in C.
1765 @item cl_I lognand (const cl_I& x, const cl_I& y)
1766 @cindex @code{lognand ()}
1767 Bitwise not and, like @code{~(x & y)} in C.
1769 @item cl_I lognor (const cl_I& x, const cl_I& y)
1770 @cindex @code{lognor ()}
1771 Bitwise not or, like @code{~(x | y)} in C.
1773 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1774 @cindex @code{logandc1 ()}
1775 Logical and, complementing the first argument, like @code{~x & y} in C.
1777 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1778 @cindex @code{logandc2 ()}
1779 Logical and, complementing the second argument, like @code{x & ~y} in C.
1781 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1782 @cindex @code{logorc1 ()}
1783 Logical or, complementing the first argument, like @code{~x | y} in C.
1785 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1786 @cindex @code{logorc2 ()}
1787 Logical or, complementing the second argument, like @code{x | ~y} in C.
1790 These operations are all available though the function
1792 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1793 @cindex @code{boole ()}
1795 where @code{op} must have one of the 16 values (each one stands for a function
1796 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1797 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1798 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1799 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1800 @code{boole_orc1}, @code{boole_orc2}.
1801 @cindex @code{boole_clr}
1802 @cindex @code{boole_set}
1803 @cindex @code{boole_1}
1804 @cindex @code{boole_2}
1805 @cindex @code{boole_c1}
1806 @cindex @code{boole_c2}
1807 @cindex @code{boole_and}
1808 @cindex @code{boole_xor}
1809 @cindex @code{boole_eqv}
1810 @cindex @code{boole_nand}
1811 @cindex @code{boole_nor}
1812 @cindex @code{boole_andc1}
1813 @cindex @code{boole_andc2}
1814 @cindex @code{boole_orc1}
1815 @cindex @code{boole_orc2}
1818 Other functions that view integers as bit strings:
1821 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1822 @cindex @code{logtest ()}
1823 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1824 @code{logand(x,y) != 0}.
1826 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1827 @cindex @code{logbitp ()}
1828 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1829 Bit 0 is the least significant bit.
1831 @item uintL logcount (const cl_I& x)
1832 @cindex @code{logcount ()}
1833 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1834 the number of zero bits in @code{x}, if @code{x} < 0.
1837 The following functions operate on intervals of bits in integers.
1840 struct cl_byte @{ uintL size; uintL position; @};
1842 @cindex @code{cl_byte}
1843 represents the bit interval containing the bits
1844 @code{position}@dots{}@code{position+size-1} of an integer.
1845 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1848 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1849 @cindex @code{ldb ()}
1850 extracts the bits of @code{n} described by the bit interval @code{b}
1851 and returns them as a nonnegative integer with @code{b.size} bits.
1853 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1854 @cindex @code{ldb_test ()}
1855 Returns true if some bit described by the bit interval @code{b} is set in
1858 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1859 @cindex @code{dpb ()}
1860 Returns @code{n}, with the bits described by the bit interval @code{b}
1861 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1862 @code{newbyte} are relevant.
1865 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1866 functions are their counterparts without shifting:
1869 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1870 @cindex @code{mask_field ()}
1871 returns an integer with the bits described by the bit interval @code{b}
1872 copied from the corresponding bits in @code{n}, the other bits zero.
1874 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1875 @cindex @code{deposit_field ()}
1876 returns an integer where the bits described by the bit interval @code{b}
1877 come from @code{newbyte} and the other bits come from @code{n}.
1880 The following relations hold:
1884 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1886 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1888 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1891 The following operations on integers as bit strings are efficient shortcuts
1892 for common arithmetic operations:
1895 @item cl_boolean oddp (const cl_I& x)
1896 @cindex @code{oddp ()}
1897 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1898 @code{mod(x,2) != 0}.
1900 @item cl_boolean evenp (const cl_I& x)
1901 @cindex @code{evenp ()}
1902 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1903 @code{mod(x,2) == 0}.
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 left. @code{n} should be >=0.
1908 Equivalent to @code{x * expt(2,n)}.
1910 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1911 @cindex @code{operator >> ()}
1912 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1913 Bits shifted out to the right are thrown away.
1914 Equivalent to @code{floor(x / expt(2,n))}.
1916 @item cl_I ash (const cl_I& x, const cl_I& y)
1917 @cindex @code{ash ()}
1918 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1919 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1920 returns @code{floor(x * expt(2,y))}.
1922 @item uintL integer_length (const cl_I& x)
1923 @cindex @code{integer_length ()}
1924 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1925 in two's complement notation. This is the smallest n >= 0 such that
1926 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1929 @item uintL ord2 (const cl_I& x)
1930 @cindex @code{ord2 ()}
1931 @code{x} must be non-zero. This function returns the number of 0 bits at the
1932 right of @code{x} in two's complement notation. This is the largest n >= 0
1933 such that 2^n divides @code{x}.
1935 @item uintL power2p (const cl_I& x)
1936 @cindex @code{power2p ()}
1937 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1938 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1939 (See also the function @code{logp}.)
1943 @subsection Number theoretic functions
1946 @item uint32 gcd (uint32 a, uint32 b)
1947 @cindex @code{gcd ()}
1948 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1949 This function returns the greatest common divisor of @code{a} and @code{b},
1950 normalized to be >= 0.
1952 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1953 @cindex @code{xgcd ()}
1954 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1955 @code{a} and @code{b} and at the same time the representation of @code{g}
1956 as an integral linear combination of @code{a} and @code{b}:
1957 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1958 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1959 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1960 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1961 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1963 @item cl_I lcm (const cl_I& a, const cl_I& b)
1964 @cindex @code{lcm ()}
1965 This function returns the least common multiple of @code{a} and @code{b},
1966 normalized to be >= 0.
1968 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
1969 @cindex @code{logp ()}
1970 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
1971 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
1972 rational number, this function returns true and sets *l = log(a,b), else
1977 @subsection Combinatorial functions
1980 @item cl_I factorial (uintL n)
1981 @cindex @code{factorial ()}
1982 @code{n} must be a small integer >= 0. This function returns the factorial
1983 @code{n}! = @code{1*2*@dots{}*n}.
1985 @item cl_I doublefactorial (uintL n)
1986 @cindex @code{doublefactorial ()}
1987 @code{n} must be a small integer >= 0. This function returns the
1988 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
1989 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
1991 @item cl_I binomial (uintL n, uintL k)
1992 @cindex @code{binomial ()}
1993 @code{n} and @code{k} must be small integers >= 0. This function returns the
1994 binomial coefficient
1996 ${n \choose k} = {n! \over n! (n-k)!}$
1999 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2001 for 0 <= k <= n, 0 else.
2005 @section Functions on floating-point numbers
2007 Recall that a floating-point number consists of a sign @code{s}, an
2008 exponent @code{e} and a mantissa @code{m}. The value of the number is
2009 @code{(-1)^s * 2^e * m}.
2012 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2013 defines the following operations.
2016 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2017 @cindex @code{scale_float ()}
2018 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2019 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2020 because it copies @code{x} and modifies the exponent.
2023 The following functions provide an abstract interface to the underlying
2024 representation of floating-point numbers.
2027 @item sintL float_exponent (const @var{type}& x)
2028 @cindex @code{float_exponent ()}
2029 Returns the exponent @code{e} of @code{x}.
2030 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2031 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2033 @item sintL float_radix (const @var{type}& x)
2034 @cindex @code{float_radix ()}
2035 Returns the base of the floating-point representation. This is always @code{2}.
2037 @item @var{type} float_sign (const @var{type}& x)
2038 @cindex @code{float_sign ()}
2039 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2040 @code{x} >= 0, -1 for @code{x} < 0.
2042 @item uintL float_digits (const @var{type}& x)
2043 @cindex @code{float_digits ()}
2044 Returns the number of mantissa bits in the floating-point representation
2045 of @code{x}, including the hidden bit. The value only depends on the type
2046 of @code{x}, not on its value.
2048 @item uintL float_precision (const @var{type}& x)
2049 @cindex @code{float_precision ()}
2050 Returns the number of significant mantissa bits in the floating-point
2051 representation of @code{x}. Since denormalized numbers are not supported,
2052 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2056 The complete internal representation of a float is encoded in the type
2057 @cindex @code{decoded_float}
2058 @cindex @code{decoded_sfloat}
2059 @cindex @code{decoded_ffloat}
2060 @cindex @code{decoded_dfloat}
2061 @cindex @code{decoded_lfloat}
2062 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2063 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2065 struct decoded_@var{type}float @{
2066 @var{type} mantissa; cl_I exponent; @var{type} sign;
2070 and returned by the function
2073 @item decoded_@var{type}float decode_float (const @var{type}& x)
2074 @cindex @code{decode_float ()}
2075 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2076 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2077 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2078 @code{e} is the same as returned by the function @code{float_exponent}.
2081 A complete decoding in terms of integers is provided as type
2082 @cindex @code{cl_idecoded_float}
2084 struct cl_idecoded_float @{
2085 cl_I mantissa; cl_I exponent; cl_I sign;
2088 by the following function:
2091 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2092 @cindex @code{integer_decode_float ()}
2093 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2094 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2095 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2096 WARNING: The exponent @code{e} is not the same as the one returned by
2097 the functions @code{decode_float} and @code{float_exponent}.
2100 Some other function, implemented only for class @code{cl_F}:
2103 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2104 @cindex @code{float_sign ()}
2105 This returns a floating point number whose precision and absolute value
2106 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2107 zero, it is treated as positive. Same for @code{y}.
2111 @section Conversion functions
2114 @subsection Conversion to floating-point numbers
2116 The type @code{float_format_t} describes a floating-point format.
2117 @cindex @code{float_format_t}
2120 @item float_format_t float_format (uintL n)
2121 @cindex @code{float_format ()}
2122 Returns the smallest float format which guarantees at least @code{n}
2123 decimal digits in the mantissa (after the decimal point).
2125 @item float_format_t float_format (const cl_F& x)
2126 Returns the floating point format of @code{x}.
2128 @item float_format_t default_float_format
2129 @cindex @code{default_float_format}
2130 Global variable: the default float format used when converting rational numbers
2134 To convert a real number to a float, each of the types
2135 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2136 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2137 defines the following operations:
2140 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2141 @cindex @code{cl_float ()}
2142 Returns @code{x} as a float of format @code{f}.
2143 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2144 Returns @code{x} in the float format of @code{y}.
2145 @item cl_F cl_float (const @var{type}&x)
2146 Returns @code{x} as a float of format @code{default_float_format} if
2147 it is an exact number, or @code{x} itself if it is already a float.
2150 Of course, converting a number to a float can lose precision.
2152 Every floating-point format has some characteristic numbers:
2155 @item cl_F most_positive_float (float_format_t f)
2156 @cindex @code{most_positive_float ()}
2157 Returns the largest (most positive) floating point number in float format @code{f}.
2159 @item cl_F most_negative_float (float_format_t f)
2160 @cindex @code{most_negative_float ()}
2161 Returns the smallest (most negative) floating point number in float format @code{f}.
2163 @item cl_F least_positive_float (float_format_t f)
2164 @cindex @code{least_positive_float ()}
2165 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2166 in float format @code{f}.
2168 @item cl_F least_negative_float (float_format_t f)
2169 @cindex @code{least_negative_float ()}
2170 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2171 in float format @code{f}.
2173 @item cl_F float_epsilon (float_format_t f)
2174 @cindex @code{float_epsilon ()}
2175 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2177 @item cl_F float_negative_epsilon (float_format_t f)
2178 @cindex @code{float_negative_epsilon ()}
2179 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2183 @subsection Conversion to rational numbers
2185 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2186 defines the following operation:
2189 @item cl_RA rational (const @var{type}& x)
2190 @cindex @code{rational ()}
2191 Returns the value of @code{x} as an exact number. If @code{x} is already
2192 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2193 the value is a rational number whose denominator is a power of 2.
2196 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2200 @item cl_RA rationalize (const cl_R& x)
2201 @cindex @code{rationalize ()}
2202 If @code{x} is a floating-point number, it actually represents an interval
2203 of real numbers, and this function returns the rational number with
2204 smallest denominator (and smallest numerator, in magnitude)
2205 which lies in this interval.
2206 If @code{x} is already an exact number, this function returns @code{x}.
2209 If @code{x} is any float, one has
2213 @code{cl_float(rational(x),x) = x}
2215 @code{cl_float(rationalize(x),x) = x}
2219 @section Random number generators
2222 A random generator is a machine which produces (pseudo-)random numbers.
2223 The include file @code{<cln/random.h>} defines a class @code{random_state}
2224 which contains the state of a random generator. If you make a copy
2225 of the random number generator, the original one and the copy will produce
2226 the same sequence of random numbers.
2228 The following functions return (pseudo-)random numbers in different formats.
2229 Calling one of these modifies the state of the random number generator in
2230 a complicated but deterministic way.
2233 @cindex @code{random_state}
2234 @cindex @code{default_random_state}
2236 random_state default_random_state
2238 contains a default random number generator. It is used when the functions
2239 below are called without @code{random_state} argument.
2242 @item uint32 random32 (random_state& randomstate)
2243 @itemx uint32 random32 ()
2244 @cindex @code{random32 ()}
2245 Returns a random unsigned 32-bit number. All bits are equally random.
2247 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2248 @itemx cl_I random_I (const cl_I& n)
2249 @cindex @code{random_I ()}
2250 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2251 in the range @code{0 <= x < n}.
2253 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2254 @itemx cl_F random_F (const cl_F& n)
2255 @cindex @code{random_F ()}
2256 @code{n} must be a float > 0. This function returns a random floating-point
2257 number of the same format as @code{n} in the range @code{0 <= x < n}.
2259 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2260 @itemx cl_R random_R (const cl_R& n)
2261 @cindex @code{random_R ()}
2262 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2263 if @code{n} is a float.
2267 @section Obfuscating operators
2268 @cindex modifying operators
2270 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2271 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2272 are not available by default because their
2273 use tends to make programs unreadable. It is trivial to get away without
2274 them. However, if you feel that you absolutely need these operators
2275 to get happy, then add
2277 #define WANT_OBFUSCATING_OPERATORS
2279 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2280 to the beginning of your source files, before the inclusion of any CLN
2281 include files. This flag will enable the following operators:
2283 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2284 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2287 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2288 @cindex @code{operator += ()}
2289 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2290 @cindex @code{operator -= ()}
2291 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2292 @cindex @code{operator *= ()}
2293 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2294 @cindex @code{operator /= ()}
2297 For the class @code{cl_I}:
2300 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2301 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2302 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2303 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2304 @cindex @code{operator &= ()}
2305 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2306 @cindex @code{operator |= ()}
2307 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2308 @cindex @code{operator ^= ()}
2309 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2310 @cindex @code{operator <<= ()}
2311 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2312 @cindex @code{operator >>= ()}
2315 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2316 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2319 @item @var{type}& operator ++ (@var{type}& x)
2320 @cindex @code{operator ++ ()}
2321 The prefix operator @code{++x}.
2323 @item void operator ++ (@var{type}& x, int)
2324 The postfix operator @code{x++}.
2326 @item @var{type}& operator -- (@var{type}& x)
2327 @cindex @code{operator -- ()}
2328 The prefix operator @code{--x}.
2330 @item void operator -- (@var{type}& x, int)
2331 The postfix operator @code{x--}.
2334 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2335 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2339 @chapter Input/Output
2340 @cindex Input/Output
2342 @section Internal and printed representation
2343 @cindex representation
2345 All computations deal with the internal representations of the numbers.
2347 Every number has an external representation as a sequence of ASCII characters.
2348 Several external representations may denote the same number, for example,
2349 "20.0" and "20.000".
2351 Converting an internal to an external representation is called ``printing'',
2353 converting an external to an internal representation is called ``reading''.
2355 In CLN, it is always true that conversion of an internal to an external
2356 representation and then back to an internal representation will yield the
2357 same internal representation. Symbolically: @code{read(print(x)) == x}.
2358 This is called ``print-read consistency''.
2360 Different types of numbers have different external representations (case
2365 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2366 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2367 for decimal integers
2368 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2370 @item Rational numbers
2371 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2372 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2375 @item Floating-point numbers
2376 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2377 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2378 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2379 of the form _@var{prec} may be appended. There must be at least
2380 one digit in the non-exponent part. The exponent has the syntax
2381 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2382 The exponent marker is
2386 @samp{s} for short-floats,
2388 @samp{f} for single-floats,
2390 @samp{d} for double-floats,
2392 @samp{L} for long-floats,
2395 or @samp{e}, which denotes a default float format. The precision specifying
2396 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2397 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2398 function @samp{float_format}.
2400 @item Complex numbers
2401 External representation:
2404 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2405 if @var{imagpart} is negative, its printed representation begins with
2406 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2407 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2408 is rational and the rational number's base is >18, because the @samp{i}
2409 is then read as a digit.
2411 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2416 @section Input functions
2418 Including @code{<cln/io.h>} defines a type @code{cl_istream}, which is
2419 the type of the first argument to all input functions. @code{cl_istream}
2420 is the same as @code{std::istream&}.
2422 These are the simple input functions:
2425 @item int freadchar (cl_istream stream)
2426 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2427 if the end of stream was encountered or an error occurred.
2429 @item int funreadchar (cl_istream stream, int c)
2430 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2431 last @code{freadchar} operation on @code{stream}.
2434 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2435 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2436 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2439 @item cl_istream operator>> (cl_istream stream, @var{type}& result)
2440 Reads a number from @code{stream} and stores it in the @code{result}.
2443 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2447 @item cl_N read_complex (cl_istream stream, const cl_read_flags& flags)
2448 @itemx cl_R read_real (cl_istream stream, const cl_read_flags& flags)
2449 @itemx cl_F read_float (cl_istream stream, const cl_read_flags& flags)
2450 @itemx cl_RA read_rational (cl_istream stream, const cl_read_flags& flags)
2451 @itemx cl_I read_integer (cl_istream stream, const cl_read_flags& flags)
2452 Reads a number from @code{stream}. The @code{flags} are parameters which
2453 affect the input syntax. Whitespace before the number is silently skipped.
2455 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2456 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2457 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2458 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2459 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2460 Reads a number from a string in memory. The @code{flags} are parameters which
2461 affect the input syntax. The string starts at @code{string} and ends at
2462 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2463 @code{NULL}, denoting the entire string, i.e. equivalent to
2464 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2465 @code{NULL}, the string in memory must contain exactly one number and nothing
2466 more, else a fatal error will be signalled. If @code{end_of_parse}
2467 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2468 the last parsed character (i.e. @code{string_limit} if nothing came after
2469 the number). Whitespace is not allowed.
2472 The structure @code{cl_read_flags} contains the following fields:
2475 @item cl_read_syntax_t syntax
2476 The possible results of the read operation. Possible values are
2477 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2478 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2479 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2481 @item cl_read_lsyntax_t lsyntax
2482 Specifies the language-dependent syntax variant for the read operation.
2486 @item lsyntax_standard
2487 accept standard algebraic notation only, no complex numbers,
2488 @item lsyntax_algebraic
2489 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2490 @item lsyntax_commonlisp
2491 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2492 hexadecimal numbers,
2493 @code{#@var{base}R} for rational numbers in a given base,
2494 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2496 accept all of these extensions.
2499 @item unsigned int rational_base
2500 The base in which rational numbers are read.
2502 @item float_format_t float_flags.default_float_format
2503 The float format used when reading floats with exponent marker @samp{e}.
2505 @item float_format_t float_flags.default_lfloat_format
2506 The float format used when reading floats with exponent marker @samp{l}.
2508 @item cl_boolean float_flags.mantissa_dependent_float_format
2509 When this flag is true, floats specified with more digits than corresponding
2510 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2511 precision corresponding to their number of significant digits.
2515 @section Output functions
2517 Including @code{<cln/io.h>} defines a type @code{cl_ostream}, which is
2518 the type of the first argument to all output functions. @code{cl_ostream}
2519 is the same as @code{std::ostream&}.
2521 These are the simple output functions:
2524 @item void fprintchar (cl_ostream stream, char c)
2525 Prints the character @code{x} literally on the @code{stream}.
2527 @item void fprint (cl_ostream stream, const char * string)
2528 Prints the @code{string} literally on the @code{stream}.
2530 @item void fprintdecimal (cl_ostream stream, int x)
2531 @itemx void fprintdecimal (cl_ostream stream, const cl_I& x)
2532 Prints the integer @code{x} in decimal on the @code{stream}.
2534 @item void fprintbinary (cl_ostream stream, const cl_I& x)
2535 Prints the integer @code{x} in binary (base 2, without prefix)
2536 on the @code{stream}.
2538 @item void fprintoctal (cl_ostream stream, const cl_I& x)
2539 Prints the integer @code{x} in octal (base 8, without prefix)
2540 on the @code{stream}.
2542 @item void fprinthexadecimal (cl_ostream stream, const cl_I& x)
2543 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2544 on the @code{stream}.
2547 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2548 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2549 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2552 @item void fprint (cl_ostream stream, const @var{type}& x)
2553 @itemx cl_ostream operator<< (cl_ostream stream, const @var{type}& x)
2554 Prints the number @code{x} on the @code{stream}. The output may depend
2555 on the global printer settings in the variable @code{default_print_flags}.
2556 The @code{ostream} flags and settings (flags, width and locale) are
2560 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2563 void print_complex (cl_ostream stream, const cl_print_flags& flags,
2565 void print_real (cl_ostream stream, const cl_print_flags& flags,
2567 void print_float (cl_ostream stream, const cl_print_flags& flags,
2569 void print_rational (cl_ostream stream, const cl_print_flags& flags,
2571 void print_integer (cl_ostream stream, const cl_print_flags& flags,
2574 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2575 parameters which affect the output.
2577 The structure type @code{cl_print_flags} contains the following fields:
2580 @item unsigned int rational_base
2581 The base in which rational numbers are printed. Default is @code{10}.
2583 @item cl_boolean rational_readably
2584 If this flag is true, rational numbers are printed with radix specifiers in
2585 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2586 prefixes, trailing dot). Default is false.
2588 @item cl_boolean float_readably
2589 If this flag is true, type specific exponent markers have precedence over 'E'.
2592 @item float_format_t default_float_format
2593 Floating point numbers of this format will be printed using the 'E' exponent
2594 marker. Default is @code{float_format_ffloat}.
2596 @item cl_boolean complex_readably
2597 If this flag is true, complex numbers will be printed using the Common Lisp
2598 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2600 @item cl_string univpoly_varname
2601 Univariate polynomials with no explicit indeterminate name will be printed
2602 using this variable name. Default is @code{"x"}.
2605 The global variable @code{default_print_flags} contains the default values,
2606 used by the function @code{fprint}.
2611 CLN has a class of abstract rings.
2619 Rings can be compared for equality:
2622 @item bool operator== (const cl_ring&, const cl_ring&)
2623 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2624 These compare two rings for equality.
2627 Given a ring @code{R}, the following members can be used.
2630 @item void R->fprint (cl_ostream stream, const cl_ring_element& x)
2631 @cindex @code{fprint ()}
2632 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2633 @cindex @code{equal ()}
2634 @itemx cl_ring_element R->zero ()
2635 @cindex @code{zero ()}
2636 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2637 @cindex @code{zerop ()}
2638 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2639 @cindex @code{plus ()}
2640 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2641 @cindex @code{minus ()}
2642 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2643 @cindex @code{uminus ()}
2644 @itemx cl_ring_element R->one ()
2645 @cindex @code{one ()}
2646 @itemx cl_ring_element R->canonhom (const cl_I& x)
2647 @cindex @code{canonhom ()}
2648 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2649 @cindex @code{mul ()}
2650 @itemx cl_ring_element R->square (const cl_ring_element& x)
2651 @cindex @code{square ()}
2652 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2653 @cindex @code{expt_pos ()}
2656 The following rings are built-in.
2659 @item cl_null_ring cl_0_ring
2660 The null ring, containing only zero.
2662 @item cl_complex_ring cl_C_ring
2663 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2665 @item cl_real_ring cl_R_ring
2666 The ring of real numbers. This corresponds to the type @code{cl_R}.
2668 @item cl_rational_ring cl_RA_ring
2669 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2671 @item cl_integer_ring cl_I_ring
2672 The ring of integers. This corresponds to the type @code{cl_I}.
2675 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2676 @code{cl_RA_ring}, @code{cl_I_ring}:
2679 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2680 @cindex @code{instanceof ()}
2681 Tests whether the given number is an element of the number ring R.
2685 @chapter Modular integers
2686 @cindex modular integer
2688 @section Modular integer rings
2691 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2692 The modulus is explicitly part of every modular integer. CLN doesn't
2693 allow you to (accidentally) mix elements of different modular rings,
2694 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2695 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2696 doesn't have generic types. So one has to live with runtime checks.)
2698 The class of modular integer rings is
2706 Modular integer ring
2710 @cindex @code{cl_modint_ring}
2712 and the class of all modular integers (elements of modular integer rings) is
2720 Modular integer rings are constructed using the function
2723 @item cl_modint_ring find_modint_ring (const cl_I& N)
2724 @cindex @code{find_modint_ring ()}
2725 This function returns the modular ring @samp{Z/NZ}. It takes care
2726 of finding out about special cases of @code{N}, like powers of two
2727 and odd numbers for which Montgomery multiplication will be a win,
2728 @cindex Montgomery multiplication
2729 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2730 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2731 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2735 Modular integer rings can be compared for equality:
2738 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2739 @cindex @code{operator == ()}
2740 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2741 @cindex @code{operator != ()}
2742 These compare two modular integer rings for equality. Two different calls
2743 to @code{find_modint_ring} with the same argument necessarily return the
2744 same ring because it is memoized in the cache table.
2747 @section Functions on modular integers
2749 Given a modular integer ring @code{R}, the following members can be used.
2752 @item cl_I R->modulus
2753 @cindex @code{modulus}
2754 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2756 @item cl_MI R->zero()
2757 @cindex @code{zero ()}
2758 This returns @code{0 mod N}.
2760 @item cl_MI R->one()
2761 @cindex @code{one ()}
2762 This returns @code{1 mod N}.
2764 @item cl_MI R->canonhom (const cl_I& x)
2765 @cindex @code{canonhom ()}
2766 This returns @code{x mod N}.
2768 @item cl_I R->retract (const cl_MI& x)
2769 @cindex @code{retract ()}
2770 This is a partial inverse function to @code{R->canonhom}. It returns the
2771 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2773 @item cl_MI R->random(random_state& randomstate)
2774 @itemx cl_MI R->random()
2775 @cindex @code{random ()}
2776 This returns a random integer modulo @code{N}.
2779 The following operations are defined on modular integers.
2782 @item cl_modint_ring x.ring ()
2783 @cindex @code{ring ()}
2784 Returns the ring to which the modular integer @code{x} belongs.
2786 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2787 @cindex @code{operator + ()}
2788 Returns the sum of two modular integers. One of the arguments may also
2791 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2792 @cindex @code{operator - ()}
2793 Returns the difference of two modular integers. One of the arguments may also
2796 @item cl_MI operator- (const cl_MI&)
2797 Returns the negative of a modular integer.
2799 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2800 @cindex @code{operator * ()}
2801 Returns the product of two modular integers. One of the arguments may also
2804 @item cl_MI square (const cl_MI&)
2805 @cindex @code{square ()}
2806 Returns the square of a modular integer.
2808 @item cl_MI recip (const cl_MI& x)
2809 @cindex @code{recip ()}
2810 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2811 must be coprime to the modulus, otherwise an error message is issued.
2813 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2814 @cindex @code{div ()}
2815 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2816 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2818 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2819 @cindex @code{expt_pos ()}
2820 @code{y} must be > 0. Returns @code{x^y}.
2822 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2823 @cindex @code{expt ()}
2824 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2825 modulus, else an error message is issued.
2827 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2828 @cindex @code{operator << ()}
2829 Returns @code{x*2^y}.
2831 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2832 @cindex @code{operator >> ()}
2833 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2834 or an error message is issued.
2836 @item bool operator== (const cl_MI&, const cl_MI&)
2837 @cindex @code{operator == ()}
2838 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2839 @cindex @code{operator != ()}
2840 Compares two modular integers, belonging to the same modular integer ring,
2843 @item cl_boolean zerop (const cl_MI& x)
2844 @cindex @code{zerop ()}
2845 Returns true if @code{x} is @code{0 mod N}.
2848 The following output functions are defined (see also the chapter on
2852 @item void fprint (cl_ostream stream, const cl_MI& x)
2853 @cindex @code{fprint ()}
2854 @itemx cl_ostream operator<< (cl_ostream stream, const cl_MI& x)
2855 @cindex @code{operator << ()}
2856 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2857 on the global printer settings in the variable @code{default_print_flags}.
2861 @chapter Symbolic data types
2862 @cindex symbolic type
2864 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2868 @cindex @code{cl_string}
2878 implements immutable strings.
2880 Strings are constructed through the following constructors:
2883 @item cl_string (const char * s)
2884 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2886 @item cl_string (const char * ptr, unsigned long len)
2887 Returns an immutable copy of the @code{len} characters at
2888 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2891 The following functions are available on strings:
2895 Assignment from @code{cl_string} and @code{const char *}.
2898 @cindex @code{length ()}
2900 @cindex @code{strlen ()}
2901 Returns the length of the string @code{s}.
2904 @cindex @code{operator [] ()}
2905 Returns the @code{i}th character of the string @code{s}.
2906 @code{i} must be in the range @code{0 <= i < s.length()}.
2908 @item bool equal (const cl_string& s1, const cl_string& s2)
2909 @cindex @code{equal ()}
2910 Compares two strings for equality. One of the arguments may also be a
2911 plain @code{const char *}.
2916 @cindex @code{cl_symbol}
2918 Symbols are uniquified strings: all symbols with the same name are shared.
2919 This means that comparison of two symbols is fast (effectively just a pointer
2920 comparison), whereas comparison of two strings must in the worst case walk
2921 both strings until their end.
2922 Symbols are used, for example, as tags for properties, as names of variables
2923 in polynomial rings, etc.
2925 Symbols are constructed through the following constructor:
2928 @item cl_symbol (const cl_string& s)
2929 Looks up or creates a new symbol with a given name.
2932 The following operations are available on symbols:
2935 @item cl_string (const cl_symbol& sym)
2936 Conversion to @code{cl_string}: Returns the string which names the symbol
2939 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2940 @cindex @code{equal ()}
2941 Compares two symbols for equality. This is very fast.
2945 @chapter Univariate polynomials
2947 @cindex univariate polynomial
2949 @section Univariate polynomial rings
2951 CLN implements univariate polynomials (polynomials in one variable) over an
2952 arbitrary ring. The indeterminate variable may be either unnamed (and will be
2953 printed according to @code{default_print_flags.univpoly_varname}, which
2954 defaults to @samp{x}) or carry a given name. The base ring and the
2955 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
2956 (accidentally) mix elements of different polynomial rings, e.g.
2957 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
2958 return a multivariate polynomial, but they are not yet implemented in CLN.)
2960 The classes of univariate polynomial rings are
2968 Univariate polynomial ring
2972 +----------------+-------------------+
2974 Complex polynomial ring | Modular integer polynomial ring
2975 cl_univpoly_complex_ring | cl_univpoly_modint_ring
2976 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
2980 Real polynomial ring |
2981 cl_univpoly_real_ring |
2982 <cln/univpoly_real.h> |
2986 Rational polynomial ring |
2987 cl_univpoly_rational_ring |
2988 <cln/univpoly_rational.h> |
2992 Integer polynomial ring
2993 cl_univpoly_integer_ring
2994 <cln/univpoly_integer.h>
2997 and the corresponding classes of univariate polynomials are
3000 Univariate polynomial
3004 +----------------+-------------------+
3006 Complex polynomial | Modular integer polynomial
3008 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3014 <cln/univpoly_real.h> |
3018 Rational polynomial |
3020 <cln/univpoly_rational.h> |
3026 <cln/univpoly_integer.h>
3029 Univariate polynomial rings are constructed using the functions
3032 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3033 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3034 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3035 @code{R} may be an arbitrary ring. This function takes care of finding out
3036 about special cases of @code{R}, such as the rings of complex numbers,
3037 real numbers, rational numbers, integers, or modular integer rings.
3038 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3039 This ensures that two calls of this function with the same arguments will
3040 return the same polynomial ring.
3042 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3043 @cindex @code{find_univpoly_ring ()}
3044 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3045 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3046 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3047 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3048 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3049 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3050 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3051 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3052 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3053 These functions are equivalent to the general @code{find_univpoly_ring},
3054 only the return type is more specific, according to the base ring's type.
3057 @section Functions on univariate polynomials
3059 Given a univariate polynomial ring @code{R}, the following members can be used.
3062 @item cl_ring R->basering()
3063 @cindex @code{basering ()}
3064 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3066 @item cl_UP R->zero()
3067 @cindex @code{zero ()}
3068 This returns @code{0 in R}, a polynomial of degree -1.
3070 @item cl_UP R->one()
3071 @cindex @code{one ()}
3072 This returns @code{1 in R}, a polynomial of degree <= 0.
3074 @item cl_UP R->canonhom (const cl_I& x)
3075 @cindex @code{canonhom ()}
3076 This returns @code{x in R}, a polynomial of degree <= 0.
3078 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3079 @cindex @code{monomial ()}
3080 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3083 @item cl_UP R->create (sintL degree)
3084 @cindex @code{create ()}
3085 Creates a new polynomial with a given degree. The zero polynomial has degree
3086 @code{-1}. After creating the polynomial, you should put in the coefficients,
3087 using the @code{set_coeff} member function, and then call the @code{finalize}
3091 The following are the only destructive operations on univariate polynomials.
3094 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3095 @cindex @code{set_coeff ()}
3096 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3097 After changing a polynomial and before applying any "normal" operation on it,
3098 you should call its @code{finalize} member function.
3100 @item void finalize (cl_UP& x)
3101 @cindex @code{finalize ()}
3102 This function marks the endpoint of destructive modifications of a polynomial.
3103 It normalizes the internal representation so that subsequent computations have
3104 less overhead. Doing normal computations on unnormalized polynomials may
3105 produce wrong results or crash the program.
3108 The following operations are defined on univariate polynomials.
3111 @item cl_univpoly_ring x.ring ()
3112 @cindex @code{ring ()}
3113 Returns the ring to which the univariate polynomial @code{x} belongs.
3115 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3116 @cindex @code{operator + ()}
3117 Returns the sum of two univariate polynomials.
3119 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3120 @cindex @code{operator - ()}
3121 Returns the difference of two univariate polynomials.
3123 @item cl_UP operator- (const cl_UP&)
3124 Returns the negative of a univariate polynomial.
3126 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3127 @cindex @code{operator * ()}
3128 Returns the product of two univariate polynomials. One of the arguments may
3129 also be a plain integer or an element of the base ring.
3131 @item cl_UP square (const cl_UP&)
3132 @cindex @code{square ()}
3133 Returns the square of a univariate polynomial.
3135 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3136 @cindex @code{expt_pos ()}
3137 @code{y} must be > 0. Returns @code{x^y}.
3139 @item bool operator== (const cl_UP&, const cl_UP&)
3140 @cindex @code{operator == ()}
3141 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3142 @cindex @code{operator != ()}
3143 Compares two univariate polynomials, belonging to the same univariate
3144 polynomial ring, for equality.
3146 @item cl_boolean zerop (const cl_UP& x)
3147 @cindex @code{zerop ()}
3148 Returns true if @code{x} is @code{0 in R}.
3150 @item sintL degree (const cl_UP& x)
3151 @cindex @code{degree ()}
3152 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3154 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3155 @cindex @code{coeff ()}
3156 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3158 @item cl_ring_element x (const cl_ring_element& y)
3159 @cindex @code{operator () ()}
3160 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3161 then @samp{x(y)} returns the value of the substitution of @code{y} into
3164 @item cl_UP deriv (const cl_UP& x)
3165 @cindex @code{deriv ()}
3166 Returns the derivative of the polynomial @code{x} with respect to the
3167 indeterminate @code{X}.
3170 The following output functions are defined (see also the chapter on
3174 @item void fprint (cl_ostream stream, const cl_UP& x)
3175 @cindex @code{fprint ()}
3176 @itemx cl_ostream operator<< (cl_ostream stream, const cl_UP& x)
3177 @cindex @code{operator << ()}
3178 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3179 depend on the global printer settings in the variable
3180 @code{default_print_flags}.
3183 @section Special polynomials
3185 The following functions return special polynomials.
3188 @item cl_UP_I tschebychev (sintL n)
3189 @cindex @code{tschebychev ()}
3190 @cindex Chebyshev polynomial
3191 Returns the n-th Chebyshev polynomial (n >= 0).
3193 @item cl_UP_I hermite (sintL n)
3194 @cindex @code{hermite ()}
3195 @cindex Hermite polynomial
3196 Returns the n-th Hermite polynomial (n >= 0).
3198 @item cl_UP_RA legendre (sintL n)
3199 @cindex @code{legendre ()}
3200 @cindex Legende polynomial
3201 Returns the n-th Legendre polynomial (n >= 0).
3203 @item cl_UP_I laguerre (sintL n)
3204 @cindex @code{laguerre ()}
3205 @cindex Laguerre polynomial
3206 Returns the n-th Laguerre polynomial (n >= 0).
3209 Information how to derive the differential equation satisfied by each
3210 of these polynomials from their definition can be found in the
3211 @code{doc/polynomial/} directory.
3219 Using C++ as an implementation language provides
3223 Efficiency: It compiles to machine code.
3227 Portability: It runs on all platforms supporting a C++ compiler. Because
3228 of the availability of GNU C++, this includes all currently used 32-bit and
3229 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3232 Type safety: The C++ compilers knows about the number types and complains if,
3233 for example, you try to assign a float to an integer variable. However,
3234 a drawback is that C++ doesn't know about generic types, hence a restriction
3235 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3236 arguments belong to the same modular ring cannot be expressed as a compile-time
3240 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3241 @code{=}, @code{==}, ... can be used in infix notation, which is more
3242 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3245 With these language features, there is no need for two separate languages,
3246 one for the implementation of the library and one in which the library's users
3247 can program. This means that a prototype implementation of an algorithm
3248 can be integrated into the library immediately after it has been tested and
3249 debugged. No need to rewrite it in a low-level language after having prototyped
3250 in a high-level language.
3253 @section Memory efficiency
3255 In order to save memory allocations, CLN implements:
3259 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3262 @cindex garbage collection
3263 @cindex reference counting
3264 Garbage collection: A reference counting mechanism makes sure that any
3265 number object's storage is freed immediately when the last reference to the
3268 @cindex immediate numbers
3269 Small integers are represented as immediate values instead of pointers
3270 to heap allocated storage. This means that integers @code{> -2^29},
3271 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3276 @section Speed efficiency
3278 Speed efficiency is obtained by the combination of the following tricks
3283 Small integers, being represented as immediate values, don't require
3284 memory access, just a couple of instructions for each elementary operation.
3286 The kernel of CLN has been written in assembly language for some CPUs
3287 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3289 On all CPUs, CLN may be configured to use the superefficient low-level
3290 routines from GNU GMP version 3.
3292 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3293 algorithm, the Karatsuba multiplication, which is an
3304 For very large numbers (more than 12000 decimal digits), CLN uses
3306 Sch{@"o}nhage-Strassen
3307 @cindex Sch{@"o}nhage-Strassen multiplication
3311 @cindex Schönhage-Strassen multiplication
3313 multiplication, which is an asymptotically optimal multiplication
3316 These fast multiplication algorithms also give improvements in the speed
3317 of division and radix conversion.
3321 @section Garbage collection
3322 @cindex garbage collection
3324 All the number classes are reference count classes: They only contain a pointer
3325 to an object in the heap. Upon construction, assignment and destruction of
3326 number objects, only the objects' reference count are manipulated.
3328 Memory occupied by number objects are automatically reclaimed as soon as
3329 their reference count drops to zero.
3331 For number rings, another strategy is implemented: There is a cache of,
3332 for example, the modular integer rings. A modular integer ring is destroyed
3333 only if its reference count dropped to zero and the cache is about to be
3334 resized. The effect of this strategy is that recently used rings remain
3335 cached, whereas undue memory consumption through cached rings is avoided.
3338 @chapter Using the library
3340 For the following discussion, we will assume that you have installed
3341 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3342 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3343 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3344 environment variables, or directly substitute the appropriate values.
3347 @section Compiler options
3348 @cindex compiler options
3350 Until you have installed CLN in a public place, the following options are
3353 When you compile CLN application code, add the flags
3355 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3357 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3358 When you link CLN application code to form an executable, add the flags
3360 $CLN_TARGETDIR/src/libcln.a
3362 to the C/C++ compiler's command line (@code{make} variable LIBS).
3364 If you did a @code{make install}, the include files are installed in a
3365 public directory (normally @code{/usr/local/include}), hence you don't
3366 need special flags for compiling. The library has been installed to a
3367 public directory as well (normally @code{/usr/local/lib}), hence when
3368 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3370 Since CLN version 1.1, there are two tools to make the creation of
3371 software packages that use CLN easier:
3374 @cindex @code{cln-config}
3375 @code{cln-config} is a shell script that you can use to determine the
3376 compiler and linker command line options required to compile and link a
3377 program with CLN. Start it with @code{--help} to learn about its options
3378 or consult the manpage that comes with it.
3380 @cindex @code{AC_PATH_CLN}
3381 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3384 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3386 This macro determines the location of CLN using @code{cln-config}, which
3387 is either found in the user's path, or from the environment variable
3388 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3389 their version is not earlier than @var{MIN-VERSION} (a default version
3390 will be used if not specified). If the required version was found, sets
3391 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3392 macro is in the file @file{cln.m4} which is installed in
3393 @file{$datadir/aclocal}. Note that if automake was installed with a
3394 different @samp{--prefix} than CLN, you will either have to manually
3395 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3396 aclocal the @samp{-I} option when running it. Here is a possible example
3397 to be included in your package's @file{configure.in}:
3399 AC_PATH_CLN(1.1.0, [
3400 LIBS="$LIBS $CLN_LIBS"
3401 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3402 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3407 @section Compatibility to old CLN versions
3409 @cindex compatibility
3411 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3412 @code{cln} in order to avoid potential name clashes with other C++
3413 libraries. If you have an old application, you will have to manually
3414 port it to the new scheme. The following principles will help during
3418 All headers are now in a separate subdirectory. Instead of including
3419 @code{cl_}@var{something}@code{.h}, include
3420 @code{cln/}@var{something}@code{.h} now.
3422 All public identifiers (typenames and functions) have lost their
3423 @code{cl_} prefix. Exceptions are all the typenames of number types,
3424 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3425 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3426 names would not be mnemonic enough once the namespace @code{cln} is
3427 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3429 All public @emph{functions} that had by a @code{cl_} in their name still
3430 carry that @code{cl_} if it is intrinsic part of a typename (as in
3431 @code{cl_I_to_int ()}).
3433 When developing other libraries, please keep in mind not to import the
3434 namespace @code{cln} in one of your public header files by saying
3435 @code{using namespace cln;}. This would propagate to other applications
3436 and can cause name clashes there.
3439 @section Include files
3440 @cindex include files
3441 @cindex header files
3443 Here is a summary of the include files and their contents.
3446 @item <cln/object.h>
3447 General definitions, reference counting, garbage collection.
3448 @item <cln/number.h>
3449 The class cl_number.
3450 @item <cln/complex.h>
3451 Functions for class cl_N, the complex numbers.
3453 Functions for class cl_R, the real numbers.
3455 Functions for class cl_F, the floats.
3456 @item <cln/sfloat.h>
3457 Functions for class cl_SF, the short-floats.
3458 @item <cln/ffloat.h>
3459 Functions for class cl_FF, the single-floats.
3460 @item <cln/dfloat.h>
3461 Functions for class cl_DF, the double-floats.
3462 @item <cln/lfloat.h>
3463 Functions for class cl_LF, the long-floats.
3464 @item <cln/rational.h>
3465 Functions for class cl_RA, the rational numbers.
3466 @item <cln/integer.h>
3467 Functions for class cl_I, the integers.
3470 @item <cln/complex_io.h>
3471 Input/Output for class cl_N, the complex numbers.
3472 @item <cln/real_io.h>
3473 Input/Output for class cl_R, the real numbers.
3474 @item <cln/float_io.h>
3475 Input/Output for class cl_F, the floats.
3476 @item <cln/sfloat_io.h>
3477 Input/Output for class cl_SF, the short-floats.
3478 @item <cln/ffloat_io.h>
3479 Input/Output for class cl_FF, the single-floats.
3480 @item <cln/dfloat_io.h>
3481 Input/Output for class cl_DF, the double-floats.
3482 @item <cln/lfloat_io.h>
3483 Input/Output for class cl_LF, the long-floats.
3484 @item <cln/rational_io.h>
3485 Input/Output for class cl_RA, the rational numbers.
3486 @item <cln/integer_io.h>
3487 Input/Output for class cl_I, the integers.
3489 Flags for customizing input operations.
3490 @item <cln/output.h>
3491 Flags for customizing output operations.
3492 @item <cln/malloc.h>
3493 @code{malloc_hook}, @code{free_hook}.
3496 @item <cln/condition.h>
3497 Conditions/exceptions.
3498 @item <cln/string.h>
3500 @item <cln/symbol.h>
3502 @item <cln/proplist.h>
3506 @item <cln/null_ring.h>
3508 @item <cln/complex_ring.h>
3509 The ring of complex numbers.
3510 @item <cln/real_ring.h>
3511 The ring of real numbers.
3512 @item <cln/rational_ring.h>
3513 The ring of rational numbers.
3514 @item <cln/integer_ring.h>
3515 The ring of integers.
3516 @item <cln/numtheory.h>
3517 Number threory functions.
3518 @item <cln/modinteger.h>
3524 @item <cln/GV_number.h>
3525 General vectors over cl_number.
3526 @item <cln/GV_complex.h>
3527 General vectors over cl_N.
3528 @item <cln/GV_real.h>
3529 General vectors over cl_R.
3530 @item <cln/GV_rational.h>
3531 General vectors over cl_RA.
3532 @item <cln/GV_integer.h>
3533 General vectors over cl_I.
3534 @item <cln/GV_modinteger.h>
3535 General vectors of modular integers.
3538 @item <cln/SV_number.h>
3539 Simple vectors over cl_number.
3540 @item <cln/SV_complex.h>
3541 Simple vectors over cl_N.
3542 @item <cln/SV_real.h>
3543 Simple vectors over cl_R.
3544 @item <cln/SV_rational.h>
3545 Simple vectors over cl_RA.
3546 @item <cln/SV_integer.h>
3547 Simple vectors over cl_I.
3548 @item <cln/SV_ringelt.h>
3549 Simple vectors of general ring elements.
3550 @item <cln/univpoly.h>
3551 Univariate polynomials.
3552 @item <cln/univpoly_integer.h>
3553 Univariate polynomials over the integers.
3554 @item <cln/univpoly_rational.h>
3555 Univariate polynomials over the rational numbers.
3556 @item <cln/univpoly_real.h>
3557 Univariate polynomials over the real numbers.
3558 @item <cln/univpoly_complex.h>
3559 Univariate polynomials over the complex numbers.
3560 @item <cln/univpoly_modint.h>
3561 Univariate polynomials over modular integer rings.
3562 @item <cln/timing.h>
3565 Includes all of the above.
3571 A function which computes the nth Fibonacci number can be written as follows.
3572 @cindex Fibonacci number
3575 #include <cln/integer.h>
3576 #include <cln/real.h>
3577 using namespace cln;
3579 // Returns F_n, computed as the nearest integer to
3580 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3581 const cl_I fibonacci (int n)
3583 // Need a precision of ((1+sqrt(5))/2)^-n.
3584 float_format_t prec = float_format((int)(0.208987641*n+5));
3585 cl_R sqrt5 = sqrt(cl_float(5,prec));
3586 cl_R phi = (1+sqrt5)/2;
3587 return round1( expt(phi,n)/sqrt5 );
3591 Let's explain what is going on in detail.
3593 The include file @code{<cln/integer.h>} is necessary because the type
3594 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3595 is needed for the type @code{cl_R} and the floating point number functions.
3596 The order of the include files does not matter. In order not to write
3597 out @code{cln::}@var{foo} in this simple example we can safely import
3598 the whole namespace @code{cln}.
3600 Then comes the function declaration. The argument is an @code{int}, the
3601 result an integer. The return type is defined as @samp{const cl_I}, not
3602 simply @samp{cl_I}, because that allows the compiler to detect typos like
3603 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3604 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3605 number). We use the most specialized possible return type because functions
3606 which call @samp{fibonacci} will be able to profit from the compiler's type
3607 analysis: Adding two integers is slightly more efficient than adding the
3608 same objects declared as complex numbers, because it needs less type
3609 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3610 the size of the resulting executable program.
3612 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3613 integer. In order to get a correct result, the absolute error should be less
3614 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3615 To this end, the first line computes a floating point precision for sqrt(5)
3618 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3619 number and than taking the square root. The converse, first taking the square
3620 root of 5, and then converting to the desired precision, would not work in
3621 CLN: The square root would be computed to a default precision (normally
3622 single-float precision), and the following conversion could not help about
3623 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3624 system and does not represent sqrt(5) in a non-numeric way.
3626 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3627 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3628 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3629 @code{cl_N} because a @samp{round1} does not exist for general complex
3632 When the function returns, all the local variables in the function are
3633 automatically reclaimed (garbage collected). Only the result survives and
3634 gets passed to the caller.
3636 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3637 contains this implementation together with an even faster algorithm.
3639 @section Debugging support
3642 When debugging a CLN application with GNU @code{gdb}, two facilities are
3643 available from the library:
3646 @item The library does type checks, range checks, consistency checks at
3647 many places. When one of these fails, the function @code{cl_abort()} is
3648 called. Its default implementation is to perform an @code{exit(1)}, so
3649 you won't have a core dump. But for debugging, it is best to set a
3650 breakpoint at this function:
3652 (gdb) break cl_abort
3654 When this breakpoint is hit, look at the stack's backtrace:
3659 @item The debugger's normal @code{print} command doesn't know about
3660 CLN's types and therefore prints mostly useless hexadecimal addresses.
3661 CLN offers a function @code{cl_print}, callable from the debugger,
3662 for printing number objects. In order to get this function, you have
3663 to define the macro @samp{CL_DEBUG} and then include all the header files
3664 for which you want @code{cl_print} debugging support. For example:
3665 @cindex @code{CL_DEBUG}
3668 #include <cln/string.h>
3670 Now, if you have in your program a variable @code{cl_string s}, and
3671 inspect it under @code{gdb}, the output may look like this:
3674 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3675 word = 134568800@}@}, @}
3676 (gdb) call cl_print(s)
3680 Note that the output of @code{cl_print} goes to the program's error output,
3681 not to gdb's standard output.
3683 Note, however, that the above facility does not work with all CLN types,
3684 only with number objects and similar. Therefore CLN offers a member function
3685 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3686 is needed for this member function to be implemented. Under @code{gdb},
3687 you call it like this:
3688 @cindex @code{debug_print ()}
3691 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3692 word = 134568800@}@}, @}
3693 (gdb) call s.debug_print()
3696 >call ($1).debug_print()
3701 Unfortunately, this feature does not seem to work under all circumstances.
3705 @chapter Customizing
3708 @section Error handling
3710 When a fatal error occurs, an error message is output to the standard error
3711 output stream, and the function @code{cl_abort} is called. The default
3712 version of this function (provided in the library) terminates the application.
3713 To catch such a fatal error, you need to define the function @code{cl_abort}
3714 yourself, with the prototype
3716 #include <cln/abort.h>
3717 void cl_abort (void);
3719 @cindex @code{cl_abort ()}
3720 This function must not return control to its caller.
3723 @section Floating-point underflow
3726 Floating point underflow denotes the situation when a floating-point number
3727 is to be created which is so close to @code{0} that its exponent is too
3728 low to be represented internally. By default, this causes a fatal error.
3729 If you set the global variable
3731 cl_boolean cl_inhibit_floating_point_underflow
3733 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3734 will be generated instead. The default value of
3735 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3738 @section Customizing I/O
3740 The output of the function @code{fprint} may be customized by changing the
3741 value of the global variable @code{default_print_flags}.
3742 @cindex @code{default_print_flags}
3745 @section Customizing the memory allocator
3747 Every memory allocation of CLN is done through the function pointer
3748 @code{malloc_hook}. Freeing of this memory is done through the function
3749 pointer @code{free_hook}. The default versions of these functions,
3750 provided in the library, call @code{malloc} and @code{free} and check
3751 the @code{malloc} result against @code{NULL}.
3752 If you want to provide another memory allocator, you need to define
3753 the variables @code{malloc_hook} and @code{free_hook} yourself,
3756 #include <cln/malloc.h>
3758 void* (*malloc_hook) (size_t size) = @dots{};
3759 void (*free_hook) (void* ptr) = @dots{};
3762 @cindex @code{malloc_hook ()}
3763 @cindex @code{free_hook ()}
3764 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3766 It is not possible to change the memory allocator at runtime, because
3767 it is already called at program startup by the constructors of some
3780 @c Table of contents