1 \input texinfo @c -*-texinfo-*-
4 @settitle CLN, a Class Library for Numbers
5 @c @setchapternewpage off
10 @c I hate putting "@noindent" in front of every paragraph.
16 * CLN: (cln). Class Library for Numbers (C++).
21 @c Don't need the other types of indices.
32 This file documents @sc{cln}, a Class Library for Numbers.
34 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
35 Richard Kreckel, @code{<kreckel@@ginac.de>}.
37 Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001.
38 Copyright (C) Richard Kreckel 2000, 2001, 2002.
40 Permission is granted to make and distribute verbatim copies of
41 this manual provided the copyright notice and this permission notice
42 are preserved on all copies.
45 Permission is granted to process this file through TeX and print the
46 results, provided the printed document carries copying permission
47 notice identical to this one except for the removal of this paragraph
48 (this paragraph not being relevant to the printed manual).
51 Permission is granted to copy and distribute modified versions of this
52 manual under the conditions for verbatim copying, provided that the entire
53 resulting derived work is distributed under the terms of a permission
54 notice identical to this one.
56 Permission is granted to copy and distribute translations of this manual
57 into another language, under the above conditions for modified versions,
58 except that this permission notice may be stated in a translation approved
64 @c prevent ugly black rectangles on overfull hbox lines:
67 @title CLN, a Class Library for Numbers
69 @author by Bruno Haible
71 @vskip 0pt plus 1filll
72 Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001.
74 Copyright @copyright{} Richard Kreckel 2000, 2001.
77 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
78 Richard Kreckel, @code{<kreckel@@ginac.de>}.
80 Permission is granted to make and distribute verbatim copies of
81 this manual provided the copyright notice and this permission notice
82 are preserved on all copies.
84 Permission is granted to copy and distribute modified versions of this
85 manual under the conditions for verbatim copying, provided that the entire
86 resulting derived work is distributed under the terms of a permission
87 notice identical to this one.
89 Permission is granted to copy and distribute translations of this manual
90 into another language, under the above conditions for modified versions,
91 except that this permission notice may be stated in a translation approved
98 @node Top, Introduction, (dir), (dir)
101 @c * Introduction:: Introduction
105 @node Introduction, Top, Top, Top
106 @comment node-name, next, previous, up
107 @chapter Introduction
110 CLN is a library for computations with all kinds of numbers.
111 It has a rich set of number classes:
115 Integers (with unlimited precision),
121 Floating-point numbers:
131 Long float (with unlimited precision),
138 Modular integers (integers modulo a fixed integer),
141 Univariate polynomials.
145 The subtypes of the complex numbers among these are exactly the
146 types of numbers known to the Common Lisp language. Therefore
147 @code{CLN} can be used for Common Lisp implementations, giving
148 @samp{CLN} another meaning: it becomes an abbreviation of
149 ``Common Lisp Numbers''.
152 The CLN package implements
156 Elementary functions (@code{+}, @code{-}, @code{*}, @code{/}, @code{sqrt},
157 comparisons, @dots{}),
160 Logical functions (logical @code{and}, @code{or}, @code{not}, @dots{}),
163 Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
164 functions and their inverse functions).
168 CLN is a C++ library. Using C++ as an implementation language provides
172 efficiency: it compiles to machine code,
174 type safety: the C++ compiler knows about the number types and complains
175 if, for example, you try to assign a float to an integer variable.
177 algebraic syntax: You can use the @code{+}, @code{-}, @code{*}, @code{=},
178 @code{==}, @dots{} operators as in C or C++.
182 CLN is memory efficient:
186 Small integers and short floats are immediate, not heap allocated.
188 Heap-allocated memory is reclaimed through an automatic, non-interruptive
193 CLN is speed efficient:
197 The kernel of CLN has been written in assembly language for some CPUs
198 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
201 On all CPUs, CLN may be configured to use the superefficient low-level
202 routines from GNU GMP version 3.
204 It uses Karatsuba multiplication, which is significantly faster
205 for large numbers than the standard multiplication algorithm.
207 For very large numbers (more than 12000 decimal digits), it uses
209 Sch{@"o}nhage-Strassen
210 @cindex Sch{@"o}nhage-Strassen multiplication
214 @cindex Schönhage-Strassen multiplication
216 multiplication, which is an asymptotically optimal multiplication
217 algorithm, for multiplication, division and radix conversion.
221 CLN aims at being easily integrated into larger software packages:
225 The garbage collection imposes no burden on the main application.
227 The library provides hooks for memory allocation and exceptions.
230 All non-macro identifiers are hidden in namespace @code{cln} in
231 order to avoid name clashes.
235 @chapter Installation
237 This section describes how to install the CLN package on your system.
240 @section Prerequisites
242 @subsection C++ compiler
244 To build CLN, you need a C++ compiler.
245 Actually, you need GNU @code{g++ 2.95} or newer.
247 The following C++ features are used:
248 classes, member functions, overloading of functions and operators,
249 constructors and destructors, inline, const, multiple inheritance,
250 templates and namespaces.
252 The following C++ features are not used:
253 @code{new}, @code{delete}, virtual inheritance, exceptions.
255 CLN relies on semi-automatic ordering of initializations
256 of static and global variables, a feature which I could
257 implement for GNU g++ only.
260 @comment cl_modules.h requires g++
261 Therefore nearly any C++ compiler will do.
263 The following C++ compilers are known to compile CLN:
266 GNU @code{g++ 2.7.0}, @code{g++ 2.7.2}
271 The following C++ compilers are known to be unusable for CLN:
274 On SunOS 4, @code{CC 2.1}, because it doesn't grok @code{//} comments
275 in lines containing @code{#if} or @code{#elif} preprocessor commands.
277 On AIX 3.2.5, @code{xlC}, because it doesn't grok the template syntax
278 in @code{cl_SV.h} and @code{cl_GV.h}, because it forces most class types
279 to have default constructors, and because it probably miscompiles the
280 integer multiplication routines.
282 On AIX 4.1.4.0, @code{xlC}, because when optimizing, it sometimes converts
283 @code{short}s to @code{int}s by zero-extend.
287 On HPPA, GNU @code{g++ 2.7.x}, because the semi-automatic ordering of
288 initializations will not work.
292 @subsection Make utility
295 To build CLN, you also need to have GNU @code{make} installed.
297 Only GNU @code{make} 3.77 is unusable for CLN; other versions work fine.
299 @subsection Sed utility
302 To build CLN on HP-UX, you also need to have GNU @code{sed} installed.
303 This is because the libtool script, which creates the CLN library, relies
304 on @code{sed}, and the vendor's @code{sed} utility on these systems is too
308 @section Building the library
310 As with any autoconfiguring GNU software, installation is as easy as this:
318 If on your system, @samp{make} is not GNU @code{make}, you have to use
319 @samp{gmake} instead of @samp{make} above.
321 The @code{configure} command checks out some features of your system and
322 C++ compiler and builds the @code{Makefile}s. The @code{make} command
323 builds the library. This step may take about an hour on an average workstation.
324 The @code{make check} runs some test to check that no important subroutine
325 has been miscompiled.
327 The @code{configure} command accepts options. To get a summary of them, try
333 Some of the options are explained in detail in the @samp{INSTALL.generic} file.
335 You can specify the C compiler, the C++ compiler and their options through
336 the following environment variables when running @code{configure}:
340 Specifies the C compiler.
343 Flags to be given to the C compiler when compiling programs (not when linking).
346 Specifies the C++ compiler.
349 Flags to be given to the C++ compiler when compiling programs (not when linking).
355 $ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
356 $ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
357 CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
358 $ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
359 CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
360 $ CC="gcc -V 3.0.4" CFLAGS="-O2 -finline-limit=1000 -fno-exceptions" \
361 CXX="g++ -V 3.0.4" CFLAGS="-O2 -finline-limit=1000 -fno-exceptions" \
365 @comment cl_modules.h requires g++
366 You should not mix GNU and non-GNU compilers. So, if @code{CXX} is a non-GNU
367 compiler, @code{CC} should be set to a non-GNU compiler as well. Examples:
370 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O" ./configure
371 $ CC="gcc -V 2.7.0" CFLAGS="-g" CXX="g++ -V 2.7.0" CXXFLAGS="-g" ./configure
374 On SGI Irix 5, if you wish not to use @code{g++}:
377 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O -Olimit 16000" ./configure
380 On SGI Irix 6, if you wish not to use @code{g++}:
383 $ CC="cc -32" CFLAGS="-O" CXX="CC -32" CXXFLAGS="-O -Olimit 34000" \
384 ./configure --without-gmp
385 $ CC="cc -n32" CFLAGS="-O" CXX="CC -n32" CXXFLAGS="-O \
386 -OPT:const_copy_limit=32400 -OPT:global_limit=32400 -OPT:fprop_limit=4000" \
387 ./configure --without-gmp
391 Note that for these environment variables to take effect, you have to set
392 them (assuming a Bourne-compatible shell) on the same line as the
393 @code{configure} command. If you made the settings in earlier shell
394 commands, you have to @code{export} the environment variables before
395 calling @code{configure}. In a @code{csh} shell, you have to use the
396 @samp{setenv} command for setting each of the environment variables.
398 Currently CLN works only with the GNU @code{g++} compiler, and only in
399 optimizing mode. So you should specify at least @code{-O} in the CXXFLAGS,
400 or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use @code{-O}.)
402 If you use @code{g++} 3.0.x or 3.1, I recommend adding
403 @samp{-finline-limit=1000} to the CXXFLAGS. This is essential for good code.
405 If you use @code{g++} gcc-2.95.x or gcc-3.0.x , I recommend adding
406 @samp{-fno-exceptions} to the CXXFLAGS. This will likely generate better code.
408 If you use @code{g++} from gcc-2.95.x on Sparc, add either @samp{-O},
409 @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the CXXFLAGS. With full
410 @samp{-O2}, @code{g++} miscompiles the division routines. If you use
411 @code{g++} older than 2.95.3 on Sparc you should also specify
412 @samp{--disable-shared} because of bad code produced in the shared
415 If you use @code{g++} on OSF/1 or Tru64 using gcc-2.95.x, you should
416 specify @samp{--disable-shared} because of linker problems with
417 duplicate symbols in shared libraries. If you use @code{g++} from
418 gcc-3.0.n, with n larger than 1, you should @emph{not} add
419 @samp{-fno-exceptions} to the CXXFLAGS, since that will generate wrong
420 code (gcc-3.1.0 is okay again, as is gcc-3.0.0).
422 If you use @code{g++} from gcc-3.1, it will need 235 MB of virtual memory.
423 You might need some swap space if your machine doesn't have 512 MB of RAM.
425 By default, both a shared and a static library are built. You can build
426 CLN as a static (or shared) library only, by calling @code{configure} with
427 the option @samp{--disable-shared} (or @samp{--disable-static}). While
428 shared libraries are usually more convenient to use, they may not work
429 on all architectures. Try disabling them if you run into linker
430 problems. Also, they are generally somewhat slower than static
431 libraries so runtime-critical applications should be linked statically.
433 If you use @code{g++} from gcc-3.1 with option @samp{-g}, you will need
434 some disk space: 335 MB for building as both a shared and a static library,
435 or 130 MB when building as a shared library only.
438 @subsection Using the GNU MP Library
441 Starting with version 1.1, CLN may be configured to make use of a
442 preinstalled @code{gmp} library. Please make sure that you have at
443 least @code{gmp} version 3.0 installed since earlier versions are
444 unsupported and likely not to work. Enabling this feature by calling
445 @code{configure} with the option @samp{--with-gmp} is known to be quite
446 a boost for CLN's performance.
448 If you have installed the @code{gmp} library and its header file in
449 some place where your compiler cannot find it by default, you must help
450 @code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
454 $ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
455 CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
459 @section Installing the library
462 As with any autoconfiguring GNU software, installation is as easy as this:
468 The @samp{make install} command installs the library and the include files
469 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
470 if you haven't specified a @code{--prefix} option to @code{configure}).
471 This step may require superuser privileges.
473 If you have already built the library and wish to install it, but didn't
474 specify @code{--prefix=@dots{}} at configure time, just re-run
475 @code{configure}, giving it the same options as the first time, plus
476 the @code{--prefix=@dots{}} option.
481 You can remove system-dependent files generated by @code{make} through
487 You can remove all files generated by @code{make}, thus reverting to a
488 virgin distribution of CLN, through
495 @chapter Ordinary number types
497 CLN implements the following class hierarchy:
505 Real or complex number
514 +-------------------+-------------------+
516 Rational number Floating-point number
518 <cln/rational.h> <cln/float.h>
520 | +--------------+--------------+--------------+
522 cl_I Short-Float Single-Float Double-Float Long-Float
523 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
524 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
527 @cindex @code{cl_number}
528 @cindex abstract class
529 The base class @code{cl_number} is an abstract base class.
530 It is not useful to declare a variable of this type except if you want
531 to completely disable compile-time type checking and use run-time type
536 @cindex complex number
537 The class @code{cl_N} comprises real and complex numbers. There is
538 no special class for complex numbers since complex numbers with imaginary
539 part @code{0} are automatically converted to real numbers.
542 The class @code{cl_R} comprises real numbers of different kinds. It is an
546 @cindex rational number
548 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
549 integers. There is no special class for non-integral rational numbers
550 since rational numbers with denominator @code{1} are automatically converted
554 The class @code{cl_F} implements floating-point approximations to real numbers.
555 It is an abstract class.
558 @section Exact numbers
561 Some numbers are represented as exact numbers: there is no loss of information
562 when such a number is converted from its mathematical value to its internal
563 representation. On exact numbers, the elementary operations (@code{+},
564 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
567 In CLN, the exact numbers are:
571 rational numbers (including integers),
573 complex numbers whose real and imaginary parts are both rational numbers.
576 Rational numbers are always normalized to the form
577 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
578 are coprime integers and the denominator is positive. If the resulting
579 denominator is @code{1}, the rational number is converted to an integer.
581 @cindex immediate numbers
582 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
583 for 32-bit machines) are especially efficient, because they consume no heap
584 allocation. Otherwise the distinction between these immediate integers
585 (called ``fixnums'') and heap allocated integers (called ``bignums'')
586 is completely transparent.
589 @section Floating-point numbers
590 @cindex floating-point number
592 Not all real numbers can be represented exactly. (There is an easy mathematical
593 proof for this: Only a countable set of numbers can be stored exactly in
594 a computer, even if one assumes that it has unlimited storage. But there
595 are uncountably many real numbers.) So some approximation is needed.
596 CLN implements ordinary floating-point numbers, with mantissa and exponent.
598 @cindex rounding error
599 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
600 only return approximate results. For example, the value of the expression
601 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
602 @samp{0.7}. Rounding errors like this one are inevitable when computing
603 with floating-point numbers.
605 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
606 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
607 rule: It first computes the exact mathematical result and then returns the
608 floating-point number which is nearest to this. If two floating-point numbers
609 are equally distant from the ideal result, the one with a @code{0} in its least
610 significant mantissa bit is chosen.
612 Similarly, testing floating point numbers for equality @samp{x == y}
613 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
614 for some well-chosen @code{epsilon}.
616 Floating point numbers come in four flavors:
621 Short floats, type @code{cl_SF}.
622 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
623 and 17 mantissa bits (including the ``hidden'' bit).
624 They don't consume heap allocation.
628 Single floats, type @code{cl_FF}.
629 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
630 and 24 mantissa bits (including the ``hidden'' bit).
631 In CLN, they are represented as IEEE single-precision floating point numbers.
632 This corresponds closely to the C/C++ type @samp{float}.
636 Double floats, type @code{cl_DF}.
637 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
638 and 53 mantissa bits (including the ``hidden'' bit).
639 In CLN, they are represented as IEEE double-precision floating point numbers.
640 This corresponds closely to the C/C++ type @samp{double}.
644 Long floats, type @code{cl_LF}.
645 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
646 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
647 The precision of a long float is unlimited, but once created, a long float
648 has a fixed precision. (No ``lazy recomputation''.)
651 Of course, computations with long floats are more expensive than those
652 with smaller floating-point formats.
654 CLN does not implement features like NaNs, denormalized numbers and
655 gradual underflow. If the exponent range of some floating-point type
656 is too limited for your application, choose another floating-point type
657 with larger exponent range.
660 As a user of CLN, you can forget about the differences between the
661 four floating-point types and just declare all your floating-point
662 variables as being of type @code{cl_F}. This has the advantage that
663 when you change the precision of some computation (say, from @code{cl_DF}
664 to @code{cl_LF}), you don't have to change the code, only the precision
665 of the initial values. Also, many transcendental functions have been
666 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
667 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
668 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
669 the floating point contagion rule happened to change in the future.)
672 @section Complex numbers
673 @cindex complex number
675 Complex numbers, as implemented by the class @code{cl_N}, have a real
676 part and an imaginary part, both real numbers. A complex number whose
677 imaginary part is the exact number @code{0} is automatically converted
680 Complex numbers can arise from real numbers alone, for example
681 through application of @code{sqrt} or transcendental functions.
687 Conversions from any class to any its superclasses (``base classes'' in
688 C++ terminology) is done automatically.
690 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
691 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
692 @code{cl_N} and @code{cl_number}.
694 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
695 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
696 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
697 efficiency. Their range is therefore limited:
701 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
703 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
706 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
707 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
708 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
710 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
711 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
712 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
713 @samp{cl_I}, first convert it to an @samp{unsigned long}.
715 Conversions from the C built-in type @samp{float} are provided for the classes
716 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
718 Conversions from the C built-in type @samp{double} are provided for the classes
719 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
721 Conversions from @samp{const char *} are provided for the classes
722 @code{cl_I}, @code{cl_RA},
723 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
724 @code{cl_R}, @code{cl_N}.
725 The easiest way to specify a value which is outside of the range of the
726 C++ built-in types is therefore to specify it as a string, like this:
729 cl_I order_of_rubiks_cube_group = "43252003274489856000";
731 Note that this conversion is done at runtime, not at compile-time.
733 Conversions from @code{cl_I} to the C built-in types @samp{int},
734 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
738 @item int cl_I_to_int (const cl_I& x)
739 @cindex @code{cl_I_to_int ()}
740 @itemx unsigned int cl_I_to_uint (const cl_I& x)
741 @cindex @code{cl_I_to_uint ()}
742 @itemx long cl_I_to_long (const cl_I& x)
743 @cindex @code{cl_I_to_long ()}
744 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
745 @cindex @code{cl_I_to_ulong ()}
746 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
747 representable in the range of @var{ctype}, a runtime error occurs.
750 Conversions from the classes @code{cl_I}, @code{cl_RA},
751 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
753 to the C built-in types @samp{float} and @samp{double} are provided through
757 @item float float_approx (const @var{type}& x)
758 @cindex @code{float_approx ()}
759 @itemx double double_approx (const @var{type}& x)
760 @cindex @code{double_approx ()}
761 Returns an approximation of @code{x} of C type @var{ctype}.
762 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
763 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
766 Conversions from any class to any of its subclasses (``derived classes'' in
767 C++ terminology) are not provided. Instead, you can assert and check
768 that a value belongs to a certain subclass, and return it as element of that
769 class, using the @samp{As} and @samp{The} macros.
770 @cindex @code{As()()}
771 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
772 @var{type} and returns it as such.
773 @cindex @code{The()()}
774 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
775 @var{type} and returns it as such. It is your responsibility to ensure
776 that this assumption is valid. Since macros and namespaces don't go
777 together well, there is an equivalent to @samp{The}: the template
785 if (!(x >= 0)) abort();
786 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
787 // In general, it would be a rational number.
788 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
793 @chapter Functions on numbers
795 Each of the number classes declares its mathematical operations in the
796 corresponding include file. For example, if your code operates with
797 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
800 @section Constructing numbers
802 Here is how to create number objects ``from nothing''.
805 @subsection Constructing integers
807 @code{cl_I} objects are most easily constructed from C integers and from
808 strings. See @ref{Conversions}.
811 @subsection Constructing rational numbers
813 @code{cl_RA} objects can be constructed from strings. The syntax
814 for rational numbers is described in @ref{Internal and printed representation}.
815 Another standard way to produce a rational number is through application
816 of @samp{operator /} or @samp{recip} on integers.
819 @subsection Constructing floating-point numbers
821 @code{cl_F} objects with low precision are most easily constructed from
822 C @samp{float} and @samp{double}. See @ref{Conversions}.
824 To construct a @code{cl_F} with high precision, you can use the conversion
825 from @samp{const char *}, but you have to specify the desired precision
826 within the string. (See @ref{Internal and printed representation}.)
829 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
831 will set @samp{e} to the given value, with a precision of 40 decimal digits.
833 The programmatic way to construct a @code{cl_F} with high precision is
834 through the @code{cl_float} conversion function, see
835 @ref{Conversion to floating-point numbers}. For example, to compute
836 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
837 and then apply the exponential function:
839 float_format_t precision = float_format(40);
840 cl_F e = exp(cl_float(1,precision));
844 @subsection Constructing complex numbers
846 Non-real @code{cl_N} objects are normally constructed through the function
848 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
850 See @ref{Elementary complex functions}.
853 @section Elementary functions
855 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
856 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
857 defines the following operations:
860 @item @var{type} operator + (const @var{type}&, const @var{type}&)
861 @cindex @code{operator + ()}
864 @item @var{type} operator - (const @var{type}&, const @var{type}&)
865 @cindex @code{operator - ()}
868 @item @var{type} operator - (const @var{type}&)
869 Returns the negative of the argument.
871 @item @var{type} plus1 (const @var{type}& x)
872 @cindex @code{plus1 ()}
873 Returns @code{x + 1}.
875 @item @var{type} minus1 (const @var{type}& x)
876 @cindex @code{minus1 ()}
877 Returns @code{x - 1}.
879 @item @var{type} operator * (const @var{type}&, const @var{type}&)
880 @cindex @code{operator * ()}
883 @item @var{type} square (const @var{type}& x)
884 @cindex @code{square ()}
885 Returns @code{x * x}.
888 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
889 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
890 defines the following operations:
893 @item @var{type} operator / (const @var{type}&, const @var{type}&)
894 @cindex @code{operator / ()}
897 @item @var{type} recip (const @var{type}&)
898 @cindex @code{recip ()}
899 Returns the reciprocal of the argument.
902 The class @code{cl_I} doesn't define a @samp{/} operation because
903 in the C/C++ language this operator, applied to integral types,
904 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
905 is implementation dependent). (@xref{Rounding functions}.)
906 Instead, @code{cl_I} defines an ``exact quotient'' function:
909 @item cl_I exquo (const cl_I& x, const cl_I& y)
910 @cindex @code{exquo ()}
911 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
914 The following exponentiation functions are defined:
917 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
918 @cindex @code{expt_pos ()}
919 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
920 @code{y} must be > 0. Returns @code{x^y}.
922 @item cl_RA expt (const cl_RA& x, const cl_I& y)
923 @cindex @code{expt ()}
924 @itemx cl_R expt (const cl_R& x, const cl_I& y)
925 @itemx cl_N expt (const cl_N& x, const cl_I& y)
929 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
930 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
931 defines the following operation:
934 @item @var{type} abs (const @var{type}& x)
935 @cindex @code{abs ()}
936 Returns the absolute value of @code{x}.
937 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
940 The class @code{cl_N} implements this as follows:
943 @item cl_R abs (const cl_N x)
944 Returns the absolute value of @code{x}.
947 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
948 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
949 defines the following operation:
952 @item @var{type} signum (const @var{type}& x)
953 @cindex @code{signum ()}
954 Returns the sign of @code{x}, in the same number format as @code{x}.
955 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
956 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
961 @section Elementary rational functions
963 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
966 @item cl_I numerator (const @var{type}& x)
967 @cindex @code{numerator ()}
968 Returns the numerator of @code{x}.
970 @item cl_I denominator (const @var{type}& x)
971 @cindex @code{denominator ()}
972 Returns the denominator of @code{x}.
975 The numerator and denominator of a rational number are normalized in such
976 a way that they have no factor in common and the denominator is positive.
979 @section Elementary complex functions
981 The class @code{cl_N} defines the following operation:
984 @item cl_N complex (const cl_R& a, const cl_R& b)
985 @cindex @code{complex ()}
986 Returns the complex number @code{a+bi}, that is, the complex number with
987 real part @code{a} and imaginary part @code{b}.
990 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
993 @item cl_R realpart (const @var{type}& x)
994 @cindex @code{realpart ()}
995 Returns the real part of @code{x}.
997 @item cl_R imagpart (const @var{type}& x)
998 @cindex @code{imagpart ()}
999 Returns the imaginary part of @code{x}.
1001 @item @var{type} conjugate (const @var{type}& x)
1002 @cindex @code{conjugate ()}
1003 Returns the complex conjugate of @code{x}.
1006 We have the relations
1010 @code{x = complex(realpart(x), imagpart(x))}
1012 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
1016 @section Comparisons
1019 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1020 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1021 defines the following operations:
1024 @item bool operator == (const @var{type}&, const @var{type}&)
1025 @cindex @code{operator == ()}
1026 @itemx bool operator != (const @var{type}&, const @var{type}&)
1027 @cindex @code{operator != ()}
1028 Comparison, as in C and C++.
1030 @item uint32 equal_hashcode (const @var{type}&)
1031 @cindex @code{equal_hashcode ()}
1032 Returns a 32-bit hash code that is the same for any two numbers which are
1033 the same according to @code{==}. This hash code depends on the number's value,
1034 not its type or precision.
1036 @item cl_boolean zerop (const @var{type}& x)
1037 @cindex @code{zerop ()}
1038 Compare against zero: @code{x == 0}
1041 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1042 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1043 defines the following operations:
1046 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1047 @cindex @code{compare ()}
1048 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1049 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1051 @item bool operator <= (const @var{type}&, const @var{type}&)
1052 @cindex @code{operator <= ()}
1053 @itemx bool operator < (const @var{type}&, const @var{type}&)
1054 @cindex @code{operator < ()}
1055 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1056 @cindex @code{operator >= ()}
1057 @itemx bool operator > (const @var{type}&, const @var{type}&)
1058 @cindex @code{operator > ()}
1059 Comparison, as in C and C++.
1061 @item cl_boolean minusp (const @var{type}& x)
1062 @cindex @code{minusp ()}
1063 Compare against zero: @code{x < 0}
1065 @item cl_boolean plusp (const @var{type}& x)
1066 @cindex @code{plusp ()}
1067 Compare against zero: @code{x > 0}
1069 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1070 @cindex @code{max ()}
1071 Return the maximum of @code{x} and @code{y}.
1073 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1074 @cindex @code{min ()}
1075 Return the minimum of @code{x} and @code{y}.
1078 When a floating point number and a rational number are compared, the float
1079 is first converted to a rational number using the function @code{rational}.
1080 Since a floating point number actually represents an interval of real numbers,
1081 the result might be surprising.
1082 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1083 there is no floating point number whose value is exactly @code{1/3}.
1086 @section Rounding functions
1089 When a real number is to be converted to an integer, there is no ``best''
1090 rounding. The desired rounding function depends on the application.
1091 The Common Lisp and ISO Lisp standards offer four rounding functions:
1095 This is the largest integer <=@code{x}.
1098 This is the smallest integer >=@code{x}.
1101 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1104 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1105 integers, choose the even one.
1108 These functions have different advantages:
1110 @code{floor} and @code{ceiling} are translation invariant:
1111 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1112 for every @code{x} and every integer @code{n}.
1114 On the other hand, @code{truncate} and @code{round} are symmetric:
1115 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1116 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1117 down exactly as often as it rounds up.
1119 The functions are related like this:
1123 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1124 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1126 @code{truncate(x) = sign(x) * floor(abs(x))}
1129 Each of the classes @code{cl_R}, @code{cl_RA},
1130 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1131 defines the following operations:
1134 @item cl_I floor1 (const @var{type}& x)
1135 @cindex @code{floor1 ()}
1136 Returns @code{floor(x)}.
1137 @item cl_I ceiling1 (const @var{type}& x)
1138 @cindex @code{ceiling1 ()}
1139 Returns @code{ceiling(x)}.
1140 @item cl_I truncate1 (const @var{type}& x)
1141 @cindex @code{truncate1 ()}
1142 Returns @code{truncate(x)}.
1143 @item cl_I round1 (const @var{type}& x)
1144 @cindex @code{round1 ()}
1145 Returns @code{round(x)}.
1148 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1149 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1150 defines the following operations:
1153 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1154 Returns @code{floor(x/y)}.
1155 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1156 Returns @code{ceiling(x/y)}.
1157 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1158 Returns @code{truncate(x/y)}.
1159 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1160 Returns @code{round(x/y)}.
1163 These functions are called @samp{floor1}, @dots{} here instead of
1164 @samp{floor}, @dots{}, because on some systems, system dependent include
1165 files define @samp{floor} and @samp{ceiling} as macros.
1167 In many cases, one needs both the quotient and the remainder of a division.
1168 It is more efficient to compute both at the same time than to perform
1169 two divisions, one for quotient and the next one for the remainder.
1170 The following functions therefore return a structure containing both
1171 the quotient and the remainder. The suffix @samp{2} indicates the number
1172 of ``return values''. The remainder is defined as follows:
1176 for the computation of @code{quotient = floor(x)},
1177 @code{remainder = x - quotient},
1179 for the computation of @code{quotient = floor(x,y)},
1180 @code{remainder = x - quotient*y},
1183 and similarly for the other three operations.
1185 Each of the classes @code{cl_R}, @code{cl_RA},
1186 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1187 defines the following operations:
1190 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1191 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1192 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1193 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1194 @itemx @var{type}_div_t round2 (const @var{type}& x)
1197 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1198 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1199 defines the following operations:
1202 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1203 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1204 @cindex @code{floor2 ()}
1205 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1206 @cindex @code{ceiling2 ()}
1207 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1208 @cindex @code{truncate2 ()}
1209 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1210 @cindex @code{round2 ()}
1213 Sometimes, one wants the quotient as a floating-point number (of the
1214 same format as the argument, if the argument is a float) instead of as
1215 an integer. The prefix @samp{f} indicates this.
1218 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1219 defines the following operations:
1222 @item @var{type} ffloor (const @var{type}& x)
1223 @cindex @code{ffloor ()}
1224 @itemx @var{type} fceiling (const @var{type}& x)
1225 @cindex @code{fceiling ()}
1226 @itemx @var{type} ftruncate (const @var{type}& x)
1227 @cindex @code{ftruncate ()}
1228 @itemx @var{type} fround (const @var{type}& x)
1229 @cindex @code{fround ()}
1232 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1234 The class @code{cl_R} defines the following operations:
1237 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1238 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1239 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1240 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1243 These functions also exist in versions which return both the quotient
1244 and the remainder. The suffix @samp{2} indicates this.
1247 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1248 defines the following operations:
1249 @cindex @code{cl_F_fdiv_t}
1250 @cindex @code{cl_SF_fdiv_t}
1251 @cindex @code{cl_FF_fdiv_t}
1252 @cindex @code{cl_DF_fdiv_t}
1253 @cindex @code{cl_LF_fdiv_t}
1256 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1257 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1258 @cindex @code{ffloor2 ()}
1259 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1260 @cindex @code{fceiling2 ()}
1261 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1262 @cindex @code{ftruncate2 ()}
1263 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1264 @cindex @code{fround2 ()}
1266 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1267 @cindex @code{cl_R_fdiv_t}
1269 The class @code{cl_R} defines the following operations:
1272 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1273 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1274 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1275 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1276 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1279 Other applications need only the remainder of a division.
1280 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1281 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1282 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1286 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1288 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1291 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1292 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1293 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1295 The classes @code{cl_R}, @code{cl_I} define the following operations:
1298 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1299 @cindex @code{mod ()}
1300 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1301 @cindex @code{rem ()}
1307 Each of the classes @code{cl_R},
1308 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1309 defines the following operation:
1312 @item @var{type} sqrt (const @var{type}& x)
1313 @cindex @code{sqrt ()}
1314 @code{x} must be >= 0. This function returns the square root of @code{x},
1315 normalized to be >= 0. If @code{x} is the square of a rational number,
1316 @code{sqrt(x)} will be a rational number, else it will return a
1317 floating-point approximation.
1320 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1323 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1324 @cindex @code{sqrtp ()}
1325 This tests whether @code{x} is a perfect square. If so, it returns true
1326 and the exact square root in @code{*root}, else it returns false.
1329 Furthermore, for integers, similarly:
1332 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1333 @cindex @code{isqrt ()}
1334 @code{x} should be >= 0. This function sets @code{*root} to
1335 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1336 the boolean value @code{(expt(*root,2) == x)}.
1339 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1340 define the following operation:
1343 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1344 @cindex @code{rootp ()}
1345 @code{x} must be >= 0. @code{n} must be > 0.
1346 This tests whether @code{x} is an @code{n}th power of a rational number.
1347 If so, it returns true and the exact root in @code{*root}, else it returns
1351 The only square root function which accepts negative numbers is the one
1352 for class @code{cl_N}:
1355 @item cl_N sqrt (const cl_N& z)
1356 @cindex @code{sqrt ()}
1357 Returns the square root of @code{z}, as defined by the formula
1358 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1359 or to a complex number are done if necessary. The range of the result is the
1360 right half plane @code{realpart(sqrt(z)) >= 0}
1361 including the positive imaginary axis and 0, but excluding
1362 the negative imaginary axis.
1363 The result is an exact number only if @code{z} is an exact number.
1367 @section Transcendental functions
1368 @cindex transcendental functions
1370 The transcendental functions return an exact result if the argument
1371 is exact and the result is exact as well. Otherwise they must return
1372 inexact numbers even if the argument is exact.
1373 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1376 @subsection Exponential and logarithmic functions
1379 @item cl_R exp (const cl_R& x)
1380 @cindex @code{exp ()}
1381 @itemx cl_N exp (const cl_N& x)
1382 Returns the exponential function of @code{x}. This is @code{e^x} where
1383 @code{e} is the base of the natural logarithms. The range of the result
1384 is the entire complex plane excluding 0.
1386 @item cl_R ln (const cl_R& x)
1387 @cindex @code{ln ()}
1388 @code{x} must be > 0. Returns the (natural) logarithm of x.
1390 @item cl_N log (const cl_N& x)
1391 @cindex @code{log ()}
1392 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1393 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1394 The range of the result is the strip in the complex plane
1395 @code{-pi < imagpart(log(x)) <= pi}.
1397 @item cl_R phase (const cl_N& x)
1398 @cindex @code{phase ()}
1399 Returns the angle part of @code{x} in its polar representation as a
1400 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1401 This is also the imaginary part of @code{log(x)}.
1402 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1403 The result will be an exact number only if @code{zerop(x)} or
1404 if @code{x} is real and positive.
1406 @item cl_R log (const cl_R& a, const cl_R& b)
1407 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1408 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1409 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1412 @item cl_N log (const cl_N& a, const cl_N& b)
1413 Returns the logarithm of @code{a} with respect to base @code{b}.
1414 @code{log(a,b) = log(a)/log(b)}.
1416 @item cl_N expt (const cl_N& x, const cl_N& y)
1417 @cindex @code{expt ()}
1418 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1421 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1424 @item cl_F exp1 (float_format_t f)
1425 @cindex @code{exp1 ()}
1426 Returns e as a float of format @code{f}.
1428 @item cl_F exp1 (const cl_F& y)
1429 Returns e in the float format of @code{y}.
1431 @item cl_F exp1 (void)
1432 Returns e as a float of format @code{default_float_format}.
1436 @subsection Trigonometric functions
1439 @item cl_R sin (const cl_R& x)
1440 @cindex @code{sin ()}
1441 Returns @code{sin(x)}. The range of the result is the interval
1442 @code{-1 <= sin(x) <= 1}.
1444 @item cl_N sin (const cl_N& z)
1445 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1447 @item cl_R cos (const cl_R& x)
1448 @cindex @code{cos ()}
1449 Returns @code{cos(x)}. The range of the result is the interval
1450 @code{-1 <= cos(x) <= 1}.
1452 @item cl_N cos (const cl_N& x)
1453 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1455 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1456 @cindex @code{cos_sin_t}
1457 @itemx cos_sin_t cos_sin (const cl_R& x)
1458 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1459 @cindex @code{cos_sin ()}
1460 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1461 hold only approximately.
1463 @item cl_R tan (const cl_R& x)
1464 @cindex @code{tan ()}
1465 @itemx cl_N tan (const cl_N& x)
1466 Returns @code{tan(x) = sin(x)/cos(x)}.
1468 @item cl_N cis (const cl_R& x)
1469 @cindex @code{cis ()}
1470 @itemx cl_N cis (const cl_N& x)
1471 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1472 @code{e^(i*x) = cos(x) + i*sin(x)}.
1475 @cindex @code{asin ()}
1476 @item cl_N asin (const cl_N& z)
1477 Returns @code{arcsin(z)}. This is defined as
1478 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1479 @code{arcsin(-z) = -arcsin(z)}.
1480 The range of the result is the strip in the complex domain
1481 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1482 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1483 with @code{realpart = pi/2} and @code{imagpart > 0}.
1485 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1489 @item cl_N acos (const cl_N& z)
1490 @cindex @code{acos ()}
1491 Returns @code{arccos(z)}. This is defined as
1492 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1495 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1497 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1498 The range of the result is the strip in the complex domain
1499 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1500 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1501 with @code{realpart = pi} and @code{imagpart > 0}.
1503 Proof: This follows from the results about arcsin.
1507 @cindex @code{atan ()}
1508 @item cl_R atan (const cl_R& x, const cl_R& y)
1509 Returns the angle of the polar representation of the complex number
1510 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1511 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1512 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1513 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1514 with reversed order of arguments.
1516 @item cl_R atan (const cl_R& x)
1517 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1518 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1519 will be an exact number only if @code{x} is the exact @code{0}.
1521 @item cl_N atan (const cl_N& z)
1522 Returns @code{arctan(z)}. This is defined as
1523 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1524 @code{arctan(-z) = -arctan(z)}. The range of the result is
1525 the strip in the complex domain
1526 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1527 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1528 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1530 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1536 @cindex Archimedes' constant
1537 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1540 @item cl_F pi (float_format_t f)
1541 @cindex @code{pi ()}
1542 Returns pi as a float of format @code{f}.
1544 @item cl_F pi (const cl_F& y)
1545 Returns pi in the float format of @code{y}.
1547 @item cl_F pi (void)
1548 Returns pi as a float of format @code{default_float_format}.
1552 @subsection Hyperbolic functions
1555 @item cl_R sinh (const cl_R& x)
1556 @cindex @code{sinh ()}
1557 Returns @code{sinh(x)}.
1559 @item cl_N sinh (const cl_N& z)
1560 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1562 @item cl_R cosh (const cl_R& x)
1563 @cindex @code{cosh ()}
1564 Returns @code{cosh(x)}. The range of the result is the interval
1565 @code{cosh(x) >= 1}.
1567 @item cl_N cosh (const cl_N& z)
1568 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1570 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1571 @cindex @code{cosh_sinh_t}
1572 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1573 @cindex @code{cosh_sinh ()}
1574 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1575 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1576 hold only approximately.
1578 @item cl_R tanh (const cl_R& x)
1579 @cindex @code{tanh ()}
1580 @itemx cl_N tanh (const cl_N& x)
1581 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1583 @item cl_N asinh (const cl_N& z)
1584 @cindex @code{asinh ()}
1585 Returns @code{arsinh(z)}. This is defined as
1586 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1587 @code{arsinh(-z) = -arsinh(z)}.
1589 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1590 Actually, z+sqrt(1+z^2) can never be real and <0, so
1591 -pi < imagpart(arsinh(z)) < pi.
1592 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1593 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1595 The range of the result is the strip in the complex domain
1596 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1597 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1598 with @code{imagpart = pi/2} and @code{realpart < 0}.
1600 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1601 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1602 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1603 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1604 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1605 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1606 If y >= 1, the imagpart is pi/2 and the realpart is
1607 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1610 Moreover, if z is in Range(sqrt),
1611 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1612 (for a proof, see file src/cl_C_asinh.cc).
1615 @item cl_N acosh (const cl_N& z)
1616 @cindex @code{acosh ()}
1617 Returns @code{arcosh(z)}. This is defined as
1618 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1619 The range of the result is the half-strip in the complex domain
1620 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1621 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1623 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1624 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1625 If z is in Range(sqrt), we have
1626 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1627 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1629 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1630 and since the imagpart of both expressions is > -pi, <= pi
1631 ==> arcosh(z) = log(z+sqrt(z^2-1))
1632 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1633 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1634 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1635 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1636 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1638 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1639 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1640 = x^2 + y^2 + sqrt(u^2+v^2)
1645 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1646 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1647 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1648 Otherwise, -z is in Range(sqrt).
1649 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1650 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1651 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1652 and this has realpart > 0.
1653 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1654 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1655 has realpart = 0 and imagpart > 0.
1656 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1657 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1658 This has realpart >= 0 and imagpart = pi.
1661 @item cl_N atanh (const cl_N& z)
1662 @cindex @code{atanh ()}
1663 Returns @code{artanh(z)}. This is defined as
1664 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1665 @code{artanh(-z) = -artanh(z)}. The range of the result is
1666 the strip in the complex domain
1667 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1668 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1669 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1671 Proof: Write z = x+iy. Examine
1672 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1674 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1675 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1676 |x| < 1 ==> imagpart = 0
1679 = (atan(1+x,y) - atan(1-x,-y))/2
1680 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1681 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1682 > (pi - pi/2 - pi/2 )/2 = 0
1683 and (1+x)/y > (1-x)/y
1684 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1685 ==> imagpart < pi/2.
1686 Hence 0 < imagpart < pi/2.
1688 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1693 @subsection Euler gamma
1694 @cindex Euler's constant
1696 Euler's constant C = 0.577@dots{} is returned by the following functions:
1699 @item cl_F eulerconst (float_format_t f)
1700 @cindex @code{eulerconst ()}
1701 Returns Euler's constant as a float of format @code{f}.
1703 @item cl_F eulerconst (const cl_F& y)
1704 Returns Euler's constant in the float format of @code{y}.
1706 @item cl_F eulerconst (void)
1707 Returns Euler's constant as a float of format @code{default_float_format}.
1710 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1711 @cindex Catalan's constant
1714 @item cl_F catalanconst (float_format_t f)
1715 @cindex @code{catalanconst ()}
1716 Returns Catalan's constant as a float of format @code{f}.
1718 @item cl_F catalanconst (const cl_F& y)
1719 Returns Catalan's constant in the float format of @code{y}.
1721 @item cl_F catalanconst (void)
1722 Returns Catalan's constant as a float of format @code{default_float_format}.
1726 @subsection Riemann zeta
1727 @cindex Riemann's zeta
1729 Riemann's zeta function at an integral point @code{s>1} is returned by the
1730 following functions:
1733 @item cl_F zeta (int s, float_format_t f)
1734 @cindex @code{zeta ()}
1735 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1737 @item cl_F zeta (int s, const cl_F& y)
1738 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1740 @item cl_F zeta (int s)
1741 Returns Riemann's zeta function at @code{s} as a float of format
1742 @code{default_float_format}.
1746 @section Functions on integers
1748 @subsection Logical functions
1750 Integers, when viewed as in two's complement notation, can be thought as
1751 infinite bit strings where the bits' values eventually are constant.
1758 The logical operations view integers as such bit strings and operate
1759 on each of the bit positions in parallel.
1762 @item cl_I lognot (const cl_I& x)
1763 @cindex @code{lognot ()}
1764 @itemx cl_I operator ~ (const cl_I& x)
1765 @cindex @code{operator ~ ()}
1766 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1768 @item cl_I logand (const cl_I& x, const cl_I& y)
1769 @cindex @code{logand ()}
1770 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1771 @cindex @code{operator & ()}
1772 Logical and, like @code{x & y} in C.
1774 @item cl_I logior (const cl_I& x, const cl_I& y)
1775 @cindex @code{logior ()}
1776 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1777 @cindex @code{operator | ()}
1778 Logical (inclusive) or, like @code{x | y} in C.
1780 @item cl_I logxor (const cl_I& x, const cl_I& y)
1781 @cindex @code{logxor ()}
1782 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1783 @cindex @code{operator ^ ()}
1784 Exclusive or, like @code{x ^ y} in C.
1786 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1787 @cindex @code{logeqv ()}
1788 Bitwise equivalence, like @code{~(x ^ y)} in C.
1790 @item cl_I lognand (const cl_I& x, const cl_I& y)
1791 @cindex @code{lognand ()}
1792 Bitwise not and, like @code{~(x & y)} in C.
1794 @item cl_I lognor (const cl_I& x, const cl_I& y)
1795 @cindex @code{lognor ()}
1796 Bitwise not or, like @code{~(x | y)} in C.
1798 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1799 @cindex @code{logandc1 ()}
1800 Logical and, complementing the first argument, like @code{~x & y} in C.
1802 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1803 @cindex @code{logandc2 ()}
1804 Logical and, complementing the second argument, like @code{x & ~y} in C.
1806 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1807 @cindex @code{logorc1 ()}
1808 Logical or, complementing the first argument, like @code{~x | y} in C.
1810 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1811 @cindex @code{logorc2 ()}
1812 Logical or, complementing the second argument, like @code{x | ~y} in C.
1815 These operations are all available though the function
1817 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1818 @cindex @code{boole ()}
1820 where @code{op} must have one of the 16 values (each one stands for a function
1821 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1822 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1823 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1824 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1825 @code{boole_orc1}, @code{boole_orc2}.
1826 @cindex @code{boole_clr}
1827 @cindex @code{boole_set}
1828 @cindex @code{boole_1}
1829 @cindex @code{boole_2}
1830 @cindex @code{boole_c1}
1831 @cindex @code{boole_c2}
1832 @cindex @code{boole_and}
1833 @cindex @code{boole_xor}
1834 @cindex @code{boole_eqv}
1835 @cindex @code{boole_nand}
1836 @cindex @code{boole_nor}
1837 @cindex @code{boole_andc1}
1838 @cindex @code{boole_andc2}
1839 @cindex @code{boole_orc1}
1840 @cindex @code{boole_orc2}
1843 Other functions that view integers as bit strings:
1846 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1847 @cindex @code{logtest ()}
1848 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1849 @code{logand(x,y) != 0}.
1851 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1852 @cindex @code{logbitp ()}
1853 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1854 Bit 0 is the least significant bit.
1856 @item uintL logcount (const cl_I& x)
1857 @cindex @code{logcount ()}
1858 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1859 the number of zero bits in @code{x}, if @code{x} < 0.
1862 The following functions operate on intervals of bits in integers.
1865 struct cl_byte @{ uintL size; uintL position; @};
1867 @cindex @code{cl_byte}
1868 represents the bit interval containing the bits
1869 @code{position}@dots{}@code{position+size-1} of an integer.
1870 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1873 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1874 @cindex @code{ldb ()}
1875 extracts the bits of @code{n} described by the bit interval @code{b}
1876 and returns them as a nonnegative integer with @code{b.size} bits.
1878 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1879 @cindex @code{ldb_test ()}
1880 Returns true if some bit described by the bit interval @code{b} is set in
1883 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1884 @cindex @code{dpb ()}
1885 Returns @code{n}, with the bits described by the bit interval @code{b}
1886 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1887 @code{newbyte} are relevant.
1890 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1891 functions are their counterparts without shifting:
1894 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1895 @cindex @code{mask_field ()}
1896 returns an integer with the bits described by the bit interval @code{b}
1897 copied from the corresponding bits in @code{n}, the other bits zero.
1899 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1900 @cindex @code{deposit_field ()}
1901 returns an integer where the bits described by the bit interval @code{b}
1902 come from @code{newbyte} and the other bits come from @code{n}.
1905 The following relations hold:
1909 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1911 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1913 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1916 The following operations on integers as bit strings are efficient shortcuts
1917 for common arithmetic operations:
1920 @item cl_boolean oddp (const cl_I& x)
1921 @cindex @code{oddp ()}
1922 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1923 @code{mod(x,2) != 0}.
1925 @item cl_boolean evenp (const cl_I& x)
1926 @cindex @code{evenp ()}
1927 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1928 @code{mod(x,2) == 0}.
1930 @item cl_I operator << (const cl_I& x, const cl_I& n)
1931 @cindex @code{operator << ()}
1932 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1933 Equivalent to @code{x * expt(2,n)}.
1935 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1936 @cindex @code{operator >> ()}
1937 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1938 Bits shifted out to the right are thrown away.
1939 Equivalent to @code{floor(x / expt(2,n))}.
1941 @item cl_I ash (const cl_I& x, const cl_I& y)
1942 @cindex @code{ash ()}
1943 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1944 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1945 returns @code{floor(x * expt(2,y))}.
1947 @item uintL integer_length (const cl_I& x)
1948 @cindex @code{integer_length ()}
1949 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1950 in two's complement notation. This is the smallest n >= 0 such that
1951 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1954 @item uintL ord2 (const cl_I& x)
1955 @cindex @code{ord2 ()}
1956 @code{x} must be non-zero. This function returns the number of 0 bits at the
1957 right of @code{x} in two's complement notation. This is the largest n >= 0
1958 such that 2^n divides @code{x}.
1960 @item uintL power2p (const cl_I& x)
1961 @cindex @code{power2p ()}
1962 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1963 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1964 (See also the function @code{logp}.)
1968 @subsection Number theoretic functions
1971 @item uint32 gcd (uint32 a, uint32 b)
1972 @cindex @code{gcd ()}
1973 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1974 This function returns the greatest common divisor of @code{a} and @code{b},
1975 normalized to be >= 0.
1977 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1978 @cindex @code{xgcd ()}
1979 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1980 @code{a} and @code{b} and at the same time the representation of @code{g}
1981 as an integral linear combination of @code{a} and @code{b}:
1982 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1983 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1984 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1985 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1986 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1988 @item cl_I lcm (const cl_I& a, const cl_I& b)
1989 @cindex @code{lcm ()}
1990 This function returns the least common multiple of @code{a} and @code{b},
1991 normalized to be >= 0.
1993 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
1994 @cindex @code{logp ()}
1995 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
1996 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
1997 rational number, this function returns true and sets *l = log(a,b), else
2002 @subsection Combinatorial functions
2005 @item cl_I factorial (uintL n)
2006 @cindex @code{factorial ()}
2007 @code{n} must be a small integer >= 0. This function returns the factorial
2008 @code{n}! = @code{1*2*@dots{}*n}.
2010 @item cl_I doublefactorial (uintL n)
2011 @cindex @code{doublefactorial ()}
2012 @code{n} must be a small integer >= 0. This function returns the
2013 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
2014 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
2016 @item cl_I binomial (uintL n, uintL k)
2017 @cindex @code{binomial ()}
2018 @code{n} and @code{k} must be small integers >= 0. This function returns the
2019 binomial coefficient
2021 ${n \choose k} = {n! \over n! (n-k)!}$
2024 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2026 for 0 <= k <= n, 0 else.
2030 @section Functions on floating-point numbers
2032 Recall that a floating-point number consists of a sign @code{s}, an
2033 exponent @code{e} and a mantissa @code{m}. The value of the number is
2034 @code{(-1)^s * 2^e * m}.
2037 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2038 defines the following operations.
2041 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2042 @cindex @code{scale_float ()}
2043 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2044 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2045 because it copies @code{x} and modifies the exponent.
2048 The following functions provide an abstract interface to the underlying
2049 representation of floating-point numbers.
2052 @item sintL float_exponent (const @var{type}& x)
2053 @cindex @code{float_exponent ()}
2054 Returns the exponent @code{e} of @code{x}.
2055 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2056 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2058 @item sintL float_radix (const @var{type}& x)
2059 @cindex @code{float_radix ()}
2060 Returns the base of the floating-point representation. This is always @code{2}.
2062 @item @var{type} float_sign (const @var{type}& x)
2063 @cindex @code{float_sign ()}
2064 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2065 @code{x} >= 0, -1 for @code{x} < 0.
2067 @item uintL float_digits (const @var{type}& x)
2068 @cindex @code{float_digits ()}
2069 Returns the number of mantissa bits in the floating-point representation
2070 of @code{x}, including the hidden bit. The value only depends on the type
2071 of @code{x}, not on its value.
2073 @item uintL float_precision (const @var{type}& x)
2074 @cindex @code{float_precision ()}
2075 Returns the number of significant mantissa bits in the floating-point
2076 representation of @code{x}. Since denormalized numbers are not supported,
2077 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2081 The complete internal representation of a float is encoded in the type
2082 @cindex @code{decoded_float}
2083 @cindex @code{decoded_sfloat}
2084 @cindex @code{decoded_ffloat}
2085 @cindex @code{decoded_dfloat}
2086 @cindex @code{decoded_lfloat}
2087 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2088 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2090 struct decoded_@var{type}float @{
2091 @var{type} mantissa; cl_I exponent; @var{type} sign;
2095 and returned by the function
2098 @item decoded_@var{type}float decode_float (const @var{type}& x)
2099 @cindex @code{decode_float ()}
2100 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2101 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2102 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2103 @code{e} is the same as returned by the function @code{float_exponent}.
2106 A complete decoding in terms of integers is provided as type
2107 @cindex @code{cl_idecoded_float}
2109 struct cl_idecoded_float @{
2110 cl_I mantissa; cl_I exponent; cl_I sign;
2113 by the following function:
2116 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2117 @cindex @code{integer_decode_float ()}
2118 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2119 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2120 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2121 WARNING: The exponent @code{e} is not the same as the one returned by
2122 the functions @code{decode_float} and @code{float_exponent}.
2125 Some other function, implemented only for class @code{cl_F}:
2128 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2129 @cindex @code{float_sign ()}
2130 This returns a floating point number whose precision and absolute value
2131 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2132 zero, it is treated as positive. Same for @code{y}.
2136 @section Conversion functions
2139 @subsection Conversion to floating-point numbers
2141 The type @code{float_format_t} describes a floating-point format.
2142 @cindex @code{float_format_t}
2145 @item float_format_t float_format (uintL n)
2146 @cindex @code{float_format ()}
2147 Returns the smallest float format which guarantees at least @code{n}
2148 decimal digits in the mantissa (after the decimal point).
2150 @item float_format_t float_format (const cl_F& x)
2151 Returns the floating point format of @code{x}.
2153 @item float_format_t default_float_format
2154 @cindex @code{default_float_format}
2155 Global variable: the default float format used when converting rational numbers
2159 To convert a real number to a float, each of the types
2160 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2161 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2162 defines the following operations:
2165 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2166 @cindex @code{cl_float ()}
2167 Returns @code{x} as a float of format @code{f}.
2168 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2169 Returns @code{x} in the float format of @code{y}.
2170 @item cl_F cl_float (const @var{type}&x)
2171 Returns @code{x} as a float of format @code{default_float_format} if
2172 it is an exact number, or @code{x} itself if it is already a float.
2175 Of course, converting a number to a float can lose precision.
2177 Every floating-point format has some characteristic numbers:
2180 @item cl_F most_positive_float (float_format_t f)
2181 @cindex @code{most_positive_float ()}
2182 Returns the largest (most positive) floating point number in float format @code{f}.
2184 @item cl_F most_negative_float (float_format_t f)
2185 @cindex @code{most_negative_float ()}
2186 Returns the smallest (most negative) floating point number in float format @code{f}.
2188 @item cl_F least_positive_float (float_format_t f)
2189 @cindex @code{least_positive_float ()}
2190 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2191 in float format @code{f}.
2193 @item cl_F least_negative_float (float_format_t f)
2194 @cindex @code{least_negative_float ()}
2195 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2196 in float format @code{f}.
2198 @item cl_F float_epsilon (float_format_t f)
2199 @cindex @code{float_epsilon ()}
2200 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2202 @item cl_F float_negative_epsilon (float_format_t f)
2203 @cindex @code{float_negative_epsilon ()}
2204 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2208 @subsection Conversion to rational numbers
2210 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2211 defines the following operation:
2214 @item cl_RA rational (const @var{type}& x)
2215 @cindex @code{rational ()}
2216 Returns the value of @code{x} as an exact number. If @code{x} is already
2217 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2218 the value is a rational number whose denominator is a power of 2.
2221 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2225 @item cl_RA rationalize (const cl_R& x)
2226 @cindex @code{rationalize ()}
2227 If @code{x} is a floating-point number, it actually represents an interval
2228 of real numbers, and this function returns the rational number with
2229 smallest denominator (and smallest numerator, in magnitude)
2230 which lies in this interval.
2231 If @code{x} is already an exact number, this function returns @code{x}.
2234 If @code{x} is any float, one has
2238 @code{cl_float(rational(x),x) = x}
2240 @code{cl_float(rationalize(x),x) = x}
2244 @section Random number generators
2247 A random generator is a machine which produces (pseudo-)random numbers.
2248 The include file @code{<cln/random.h>} defines a class @code{random_state}
2249 which contains the state of a random generator. If you make a copy
2250 of the random number generator, the original one and the copy will produce
2251 the same sequence of random numbers.
2253 The following functions return (pseudo-)random numbers in different formats.
2254 Calling one of these modifies the state of the random number generator in
2255 a complicated but deterministic way.
2258 @cindex @code{random_state}
2259 @cindex @code{default_random_state}
2261 random_state default_random_state
2263 contains a default random number generator. It is used when the functions
2264 below are called without @code{random_state} argument.
2267 @item uint32 random32 (random_state& randomstate)
2268 @itemx uint32 random32 ()
2269 @cindex @code{random32 ()}
2270 Returns a random unsigned 32-bit number. All bits are equally random.
2272 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2273 @itemx cl_I random_I (const cl_I& n)
2274 @cindex @code{random_I ()}
2275 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2276 in the range @code{0 <= x < n}.
2278 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2279 @itemx cl_F random_F (const cl_F& n)
2280 @cindex @code{random_F ()}
2281 @code{n} must be a float > 0. This function returns a random floating-point
2282 number of the same format as @code{n} in the range @code{0 <= x < n}.
2284 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2285 @itemx cl_R random_R (const cl_R& n)
2286 @cindex @code{random_R ()}
2287 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2288 if @code{n} is a float.
2292 @section Obfuscating operators
2293 @cindex modifying operators
2295 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2296 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2297 are not available by default because their
2298 use tends to make programs unreadable. It is trivial to get away without
2299 them. However, if you feel that you absolutely need these operators
2300 to get happy, then add
2302 #define WANT_OBFUSCATING_OPERATORS
2304 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2305 to the beginning of your source files, before the inclusion of any CLN
2306 include files. This flag will enable the following operators:
2308 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2309 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2312 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2313 @cindex @code{operator += ()}
2314 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2315 @cindex @code{operator -= ()}
2316 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2317 @cindex @code{operator *= ()}
2318 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2319 @cindex @code{operator /= ()}
2322 For the class @code{cl_I}:
2325 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2326 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2327 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2328 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2329 @cindex @code{operator &= ()}
2330 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2331 @cindex @code{operator |= ()}
2332 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2333 @cindex @code{operator ^= ()}
2334 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2335 @cindex @code{operator <<= ()}
2336 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2337 @cindex @code{operator >>= ()}
2340 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2341 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2344 @item @var{type}& operator ++ (@var{type}& x)
2345 @cindex @code{operator ++ ()}
2346 The prefix operator @code{++x}.
2348 @item void operator ++ (@var{type}& x, int)
2349 The postfix operator @code{x++}.
2351 @item @var{type}& operator -- (@var{type}& x)
2352 @cindex @code{operator -- ()}
2353 The prefix operator @code{--x}.
2355 @item void operator -- (@var{type}& x, int)
2356 The postfix operator @code{x--}.
2359 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2360 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2364 @chapter Input/Output
2365 @cindex Input/Output
2367 @section Internal and printed representation
2368 @cindex representation
2370 All computations deal with the internal representations of the numbers.
2372 Every number has an external representation as a sequence of ASCII characters.
2373 Several external representations may denote the same number, for example,
2374 "20.0" and "20.000".
2376 Converting an internal to an external representation is called ``printing'',
2378 converting an external to an internal representation is called ``reading''.
2380 In CLN, it is always true that conversion of an internal to an external
2381 representation and then back to an internal representation will yield the
2382 same internal representation. Symbolically: @code{read(print(x)) == x}.
2383 This is called ``print-read consistency''.
2385 Different types of numbers have different external representations (case
2390 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2391 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2392 for decimal integers
2393 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2395 @item Rational numbers
2396 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2397 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2400 @item Floating-point numbers
2401 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2402 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2403 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2404 of the form _@var{prec} may be appended. There must be at least
2405 one digit in the non-exponent part. The exponent has the syntax
2406 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2407 The exponent marker is
2411 @samp{s} for short-floats,
2413 @samp{f} for single-floats,
2415 @samp{d} for double-floats,
2417 @samp{L} for long-floats,
2420 or @samp{e}, which denotes a default float format. The precision specifying
2421 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2422 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2423 function @samp{float_format}.
2425 @item Complex numbers
2426 External representation:
2429 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2430 if @var{imagpart} is negative, its printed representation begins with
2431 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2432 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2433 is rational and the rational number's base is >18, because the @samp{i}
2434 is then read as a digit.
2436 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2441 @section Input functions
2443 Including @code{<cln/io.h>} defines a number of simple input functions
2444 that read from @code{std::istream&}:
2447 @item int freadchar (std::istream& stream)
2448 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2449 if the end of stream was encountered or an error occurred.
2451 @item int funreadchar (std::istream& stream, int c)
2452 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2453 last @code{freadchar} operation on @code{stream}.
2456 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2457 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2458 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2461 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2462 Reads a number from @code{stream} and stores it in the @code{result}.
2465 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2469 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2470 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2471 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2472 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2473 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2474 Reads a number from @code{stream}. The @code{flags} are parameters which
2475 affect the input syntax. Whitespace before the number is silently skipped.
2477 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2478 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2479 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2480 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2481 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2482 Reads a number from a string in memory. The @code{flags} are parameters which
2483 affect the input syntax. The string starts at @code{string} and ends at
2484 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2485 @code{NULL}, denoting the entire string, i.e. equivalent to
2486 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2487 @code{NULL}, the string in memory must contain exactly one number and nothing
2488 more, else a fatal error will be signalled. If @code{end_of_parse}
2489 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2490 the last parsed character (i.e. @code{string_limit} if nothing came after
2491 the number). Whitespace is not allowed.
2494 The structure @code{cl_read_flags} contains the following fields:
2497 @item cl_read_syntax_t syntax
2498 The possible results of the read operation. Possible values are
2499 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2500 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2501 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2503 @item cl_read_lsyntax_t lsyntax
2504 Specifies the language-dependent syntax variant for the read operation.
2508 @item lsyntax_standard
2509 accept standard algebraic notation only, no complex numbers,
2510 @item lsyntax_algebraic
2511 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2512 @item lsyntax_commonlisp
2513 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2514 hexadecimal numbers,
2515 @code{#@var{base}R} for rational numbers in a given base,
2516 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2518 accept all of these extensions.
2521 @item unsigned int rational_base
2522 The base in which rational numbers are read.
2524 @item float_format_t float_flags.default_float_format
2525 The float format used when reading floats with exponent marker @samp{e}.
2527 @item float_format_t float_flags.default_lfloat_format
2528 The float format used when reading floats with exponent marker @samp{l}.
2530 @item cl_boolean float_flags.mantissa_dependent_float_format
2531 When this flag is true, floats specified with more digits than corresponding
2532 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2533 precision corresponding to their number of significant digits.
2537 @section Output functions
2539 Including @code{<cln/io.h>} defines a number of simple output functions
2540 that write to @code{std::ostream&}:
2543 @item void fprintchar (std::ostream& stream, char c)
2544 Prints the character @code{x} literally on the @code{stream}.
2546 @item void fprint (std::ostream& stream, const char * string)
2547 Prints the @code{string} literally on the @code{stream}.
2549 @item void fprintdecimal (std::ostream& stream, int x)
2550 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2551 Prints the integer @code{x} in decimal on the @code{stream}.
2553 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2554 Prints the integer @code{x} in binary (base 2, without prefix)
2555 on the @code{stream}.
2557 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2558 Prints the integer @code{x} in octal (base 8, without prefix)
2559 on the @code{stream}.
2561 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2562 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2563 on the @code{stream}.
2566 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2567 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2568 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2571 @item void fprint (std::ostream& stream, const @var{type}& x)
2572 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2573 Prints the number @code{x} on the @code{stream}. The output may depend
2574 on the global printer settings in the variable @code{default_print_flags}.
2575 The @code{ostream} flags and settings (flags, width and locale) are
2579 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2582 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2584 void print_real (std::ostream& stream, const cl_print_flags& flags,
2586 void print_float (std::ostream& stream, const cl_print_flags& flags,
2588 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2590 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2593 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2594 parameters which affect the output.
2596 The structure type @code{cl_print_flags} contains the following fields:
2599 @item unsigned int rational_base
2600 The base in which rational numbers are printed. Default is @code{10}.
2602 @item cl_boolean rational_readably
2603 If this flag is true, rational numbers are printed with radix specifiers in
2604 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2605 prefixes, trailing dot). Default is false.
2607 @item cl_boolean float_readably
2608 If this flag is true, type specific exponent markers have precedence over 'E'.
2611 @item float_format_t default_float_format
2612 Floating point numbers of this format will be printed using the 'E' exponent
2613 marker. Default is @code{float_format_ffloat}.
2615 @item cl_boolean complex_readably
2616 If this flag is true, complex numbers will be printed using the Common Lisp
2617 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2619 @item cl_string univpoly_varname
2620 Univariate polynomials with no explicit indeterminate name will be printed
2621 using this variable name. Default is @code{"x"}.
2624 The global variable @code{default_print_flags} contains the default values,
2625 used by the function @code{fprint}.
2630 CLN has a class of abstract rings.
2638 Rings can be compared for equality:
2641 @item bool operator== (const cl_ring&, const cl_ring&)
2642 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2643 These compare two rings for equality.
2646 Given a ring @code{R}, the following members can be used.
2649 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2650 @cindex @code{fprint ()}
2651 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2652 @cindex @code{equal ()}
2653 @itemx cl_ring_element R->zero ()
2654 @cindex @code{zero ()}
2655 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2656 @cindex @code{zerop ()}
2657 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2658 @cindex @code{plus ()}
2659 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2660 @cindex @code{minus ()}
2661 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2662 @cindex @code{uminus ()}
2663 @itemx cl_ring_element R->one ()
2664 @cindex @code{one ()}
2665 @itemx cl_ring_element R->canonhom (const cl_I& x)
2666 @cindex @code{canonhom ()}
2667 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2668 @cindex @code{mul ()}
2669 @itemx cl_ring_element R->square (const cl_ring_element& x)
2670 @cindex @code{square ()}
2671 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2672 @cindex @code{expt_pos ()}
2675 The following rings are built-in.
2678 @item cl_null_ring cl_0_ring
2679 The null ring, containing only zero.
2681 @item cl_complex_ring cl_C_ring
2682 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2684 @item cl_real_ring cl_R_ring
2685 The ring of real numbers. This corresponds to the type @code{cl_R}.
2687 @item cl_rational_ring cl_RA_ring
2688 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2690 @item cl_integer_ring cl_I_ring
2691 The ring of integers. This corresponds to the type @code{cl_I}.
2694 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2695 @code{cl_RA_ring}, @code{cl_I_ring}:
2698 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2699 @cindex @code{instanceof ()}
2700 Tests whether the given number is an element of the number ring R.
2704 @chapter Modular integers
2705 @cindex modular integer
2707 @section Modular integer rings
2710 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2711 The modulus is explicitly part of every modular integer. CLN doesn't
2712 allow you to (accidentally) mix elements of different modular rings,
2713 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2714 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2715 doesn't have generic types. So one has to live with runtime checks.)
2717 The class of modular integer rings is
2725 Modular integer ring
2729 @cindex @code{cl_modint_ring}
2731 and the class of all modular integers (elements of modular integer rings) is
2739 Modular integer rings are constructed using the function
2742 @item cl_modint_ring find_modint_ring (const cl_I& N)
2743 @cindex @code{find_modint_ring ()}
2744 This function returns the modular ring @samp{Z/NZ}. It takes care
2745 of finding out about special cases of @code{N}, like powers of two
2746 and odd numbers for which Montgomery multiplication will be a win,
2747 @cindex Montgomery multiplication
2748 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2749 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2750 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2754 Modular integer rings can be compared for equality:
2757 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2758 @cindex @code{operator == ()}
2759 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2760 @cindex @code{operator != ()}
2761 These compare two modular integer rings for equality. Two different calls
2762 to @code{find_modint_ring} with the same argument necessarily return the
2763 same ring because it is memoized in the cache table.
2766 @section Functions on modular integers
2768 Given a modular integer ring @code{R}, the following members can be used.
2771 @item cl_I R->modulus
2772 @cindex @code{modulus}
2773 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2775 @item cl_MI R->zero()
2776 @cindex @code{zero ()}
2777 This returns @code{0 mod N}.
2779 @item cl_MI R->one()
2780 @cindex @code{one ()}
2781 This returns @code{1 mod N}.
2783 @item cl_MI R->canonhom (const cl_I& x)
2784 @cindex @code{canonhom ()}
2785 This returns @code{x mod N}.
2787 @item cl_I R->retract (const cl_MI& x)
2788 @cindex @code{retract ()}
2789 This is a partial inverse function to @code{R->canonhom}. It returns the
2790 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2792 @item cl_MI R->random(random_state& randomstate)
2793 @itemx cl_MI R->random()
2794 @cindex @code{random ()}
2795 This returns a random integer modulo @code{N}.
2798 The following operations are defined on modular integers.
2801 @item cl_modint_ring x.ring ()
2802 @cindex @code{ring ()}
2803 Returns the ring to which the modular integer @code{x} belongs.
2805 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2806 @cindex @code{operator + ()}
2807 Returns the sum of two modular integers. One of the arguments may also
2810 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2811 @cindex @code{operator - ()}
2812 Returns the difference of two modular integers. One of the arguments may also
2815 @item cl_MI operator- (const cl_MI&)
2816 Returns the negative of a modular integer.
2818 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2819 @cindex @code{operator * ()}
2820 Returns the product of two modular integers. One of the arguments may also
2823 @item cl_MI square (const cl_MI&)
2824 @cindex @code{square ()}
2825 Returns the square of a modular integer.
2827 @item cl_MI recip (const cl_MI& x)
2828 @cindex @code{recip ()}
2829 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2830 must be coprime to the modulus, otherwise an error message is issued.
2832 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2833 @cindex @code{div ()}
2834 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2835 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2837 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2838 @cindex @code{expt_pos ()}
2839 @code{y} must be > 0. Returns @code{x^y}.
2841 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2842 @cindex @code{expt ()}
2843 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2844 modulus, else an error message is issued.
2846 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2847 @cindex @code{operator << ()}
2848 Returns @code{x*2^y}.
2850 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2851 @cindex @code{operator >> ()}
2852 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2853 or an error message is issued.
2855 @item bool operator== (const cl_MI&, const cl_MI&)
2856 @cindex @code{operator == ()}
2857 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2858 @cindex @code{operator != ()}
2859 Compares two modular integers, belonging to the same modular integer ring,
2862 @item cl_boolean zerop (const cl_MI& x)
2863 @cindex @code{zerop ()}
2864 Returns true if @code{x} is @code{0 mod N}.
2867 The following output functions are defined (see also the chapter on
2871 @item void fprint (std::ostream& stream, const cl_MI& x)
2872 @cindex @code{fprint ()}
2873 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
2874 @cindex @code{operator << ()}
2875 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2876 on the global printer settings in the variable @code{default_print_flags}.
2880 @chapter Symbolic data types
2881 @cindex symbolic type
2883 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2887 @cindex @code{cl_string}
2897 implements immutable strings.
2899 Strings are constructed through the following constructors:
2902 @item cl_string (const char * s)
2903 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2905 @item cl_string (const char * ptr, unsigned long len)
2906 Returns an immutable copy of the @code{len} characters at
2907 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2910 The following functions are available on strings:
2914 Assignment from @code{cl_string} and @code{const char *}.
2917 @cindex @code{length ()}
2919 @cindex @code{strlen ()}
2920 Returns the length of the string @code{s}.
2923 @cindex @code{operator [] ()}
2924 Returns the @code{i}th character of the string @code{s}.
2925 @code{i} must be in the range @code{0 <= i < s.length()}.
2927 @item bool equal (const cl_string& s1, const cl_string& s2)
2928 @cindex @code{equal ()}
2929 Compares two strings for equality. One of the arguments may also be a
2930 plain @code{const char *}.
2935 @cindex @code{cl_symbol}
2937 Symbols are uniquified strings: all symbols with the same name are shared.
2938 This means that comparison of two symbols is fast (effectively just a pointer
2939 comparison), whereas comparison of two strings must in the worst case walk
2940 both strings until their end.
2941 Symbols are used, for example, as tags for properties, as names of variables
2942 in polynomial rings, etc.
2944 Symbols are constructed through the following constructor:
2947 @item cl_symbol (const cl_string& s)
2948 Looks up or creates a new symbol with a given name.
2951 The following operations are available on symbols:
2954 @item cl_string (const cl_symbol& sym)
2955 Conversion to @code{cl_string}: Returns the string which names the symbol
2958 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2959 @cindex @code{equal ()}
2960 Compares two symbols for equality. This is very fast.
2964 @chapter Univariate polynomials
2966 @cindex univariate polynomial
2968 @section Univariate polynomial rings
2970 CLN implements univariate polynomials (polynomials in one variable) over an
2971 arbitrary ring. The indeterminate variable may be either unnamed (and will be
2972 printed according to @code{default_print_flags.univpoly_varname}, which
2973 defaults to @samp{x}) or carry a given name. The base ring and the
2974 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
2975 (accidentally) mix elements of different polynomial rings, e.g.
2976 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
2977 return a multivariate polynomial, but they are not yet implemented in CLN.)
2979 The classes of univariate polynomial rings are
2987 Univariate polynomial ring
2991 +----------------+-------------------+
2993 Complex polynomial ring | Modular integer polynomial ring
2994 cl_univpoly_complex_ring | cl_univpoly_modint_ring
2995 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
2999 Real polynomial ring |
3000 cl_univpoly_real_ring |
3001 <cln/univpoly_real.h> |
3005 Rational polynomial ring |
3006 cl_univpoly_rational_ring |
3007 <cln/univpoly_rational.h> |
3011 Integer polynomial ring
3012 cl_univpoly_integer_ring
3013 <cln/univpoly_integer.h>
3016 and the corresponding classes of univariate polynomials are
3019 Univariate polynomial
3023 +----------------+-------------------+
3025 Complex polynomial | Modular integer polynomial
3027 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3033 <cln/univpoly_real.h> |
3037 Rational polynomial |
3039 <cln/univpoly_rational.h> |
3045 <cln/univpoly_integer.h>
3048 Univariate polynomial rings are constructed using the functions
3051 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3052 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3053 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3054 @code{R} may be an arbitrary ring. This function takes care of finding out
3055 about special cases of @code{R}, such as the rings of complex numbers,
3056 real numbers, rational numbers, integers, or modular integer rings.
3057 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3058 This ensures that two calls of this function with the same arguments will
3059 return the same polynomial ring.
3061 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3062 @cindex @code{find_univpoly_ring ()}
3063 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3064 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3065 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3066 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3067 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3068 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3069 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3070 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3071 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3072 These functions are equivalent to the general @code{find_univpoly_ring},
3073 only the return type is more specific, according to the base ring's type.
3076 @section Functions on univariate polynomials
3078 Given a univariate polynomial ring @code{R}, the following members can be used.
3081 @item cl_ring R->basering()
3082 @cindex @code{basering ()}
3083 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3085 @item cl_UP R->zero()
3086 @cindex @code{zero ()}
3087 This returns @code{0 in R}, a polynomial of degree -1.
3089 @item cl_UP R->one()
3090 @cindex @code{one ()}
3091 This returns @code{1 in R}, a polynomial of degree <= 0.
3093 @item cl_UP R->canonhom (const cl_I& x)
3094 @cindex @code{canonhom ()}
3095 This returns @code{x in R}, a polynomial of degree <= 0.
3097 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3098 @cindex @code{monomial ()}
3099 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3102 @item cl_UP R->create (sintL degree)
3103 @cindex @code{create ()}
3104 Creates a new polynomial with a given degree. The zero polynomial has degree
3105 @code{-1}. After creating the polynomial, you should put in the coefficients,
3106 using the @code{set_coeff} member function, and then call the @code{finalize}
3110 The following are the only destructive operations on univariate polynomials.
3113 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3114 @cindex @code{set_coeff ()}
3115 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3116 After changing a polynomial and before applying any "normal" operation on it,
3117 you should call its @code{finalize} member function.
3119 @item void finalize (cl_UP& x)
3120 @cindex @code{finalize ()}
3121 This function marks the endpoint of destructive modifications of a polynomial.
3122 It normalizes the internal representation so that subsequent computations have
3123 less overhead. Doing normal computations on unnormalized polynomials may
3124 produce wrong results or crash the program.
3127 The following operations are defined on univariate polynomials.
3130 @item cl_univpoly_ring x.ring ()
3131 @cindex @code{ring ()}
3132 Returns the ring to which the univariate polynomial @code{x} belongs.
3134 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3135 @cindex @code{operator + ()}
3136 Returns the sum of two univariate polynomials.
3138 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3139 @cindex @code{operator - ()}
3140 Returns the difference of two univariate polynomials.
3142 @item cl_UP operator- (const cl_UP&)
3143 Returns the negative of a univariate polynomial.
3145 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3146 @cindex @code{operator * ()}
3147 Returns the product of two univariate polynomials. One of the arguments may
3148 also be a plain integer or an element of the base ring.
3150 @item cl_UP square (const cl_UP&)
3151 @cindex @code{square ()}
3152 Returns the square of a univariate polynomial.
3154 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3155 @cindex @code{expt_pos ()}
3156 @code{y} must be > 0. Returns @code{x^y}.
3158 @item bool operator== (const cl_UP&, const cl_UP&)
3159 @cindex @code{operator == ()}
3160 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3161 @cindex @code{operator != ()}
3162 Compares two univariate polynomials, belonging to the same univariate
3163 polynomial ring, for equality.
3165 @item cl_boolean zerop (const cl_UP& x)
3166 @cindex @code{zerop ()}
3167 Returns true if @code{x} is @code{0 in R}.
3169 @item sintL degree (const cl_UP& x)
3170 @cindex @code{degree ()}
3171 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3173 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3174 @cindex @code{coeff ()}
3175 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3177 @item cl_ring_element x (const cl_ring_element& y)
3178 @cindex @code{operator () ()}
3179 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3180 then @samp{x(y)} returns the value of the substitution of @code{y} into
3183 @item cl_UP deriv (const cl_UP& x)
3184 @cindex @code{deriv ()}
3185 Returns the derivative of the polynomial @code{x} with respect to the
3186 indeterminate @code{X}.
3189 The following output functions are defined (see also the chapter on
3193 @item void fprint (std::ostream& stream, const cl_UP& x)
3194 @cindex @code{fprint ()}
3195 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3196 @cindex @code{operator << ()}
3197 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3198 depend on the global printer settings in the variable
3199 @code{default_print_flags}.
3202 @section Special polynomials
3204 The following functions return special polynomials.
3207 @item cl_UP_I tschebychev (sintL n)
3208 @cindex @code{tschebychev ()}
3209 @cindex Chebyshev polynomial
3210 Returns the n-th Chebyshev polynomial (n >= 0).
3212 @item cl_UP_I hermite (sintL n)
3213 @cindex @code{hermite ()}
3214 @cindex Hermite polynomial
3215 Returns the n-th Hermite polynomial (n >= 0).
3217 @item cl_UP_RA legendre (sintL n)
3218 @cindex @code{legendre ()}
3219 @cindex Legende polynomial
3220 Returns the n-th Legendre polynomial (n >= 0).
3222 @item cl_UP_I laguerre (sintL n)
3223 @cindex @code{laguerre ()}
3224 @cindex Laguerre polynomial
3225 Returns the n-th Laguerre polynomial (n >= 0).
3228 Information how to derive the differential equation satisfied by each
3229 of these polynomials from their definition can be found in the
3230 @code{doc/polynomial/} directory.
3238 Using C++ as an implementation language provides
3242 Efficiency: It compiles to machine code.
3246 Portability: It runs on all platforms supporting a C++ compiler. Because
3247 of the availability of GNU C++, this includes all currently used 32-bit and
3248 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3251 Type safety: The C++ compilers knows about the number types and complains if,
3252 for example, you try to assign a float to an integer variable. However,
3253 a drawback is that C++ doesn't know about generic types, hence a restriction
3254 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3255 arguments belong to the same modular ring cannot be expressed as a compile-time
3259 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3260 @code{=}, @code{==}, ... can be used in infix notation, which is more
3261 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3264 With these language features, there is no need for two separate languages,
3265 one for the implementation of the library and one in which the library's users
3266 can program. This means that a prototype implementation of an algorithm
3267 can be integrated into the library immediately after it has been tested and
3268 debugged. No need to rewrite it in a low-level language after having prototyped
3269 in a high-level language.
3272 @section Memory efficiency
3274 In order to save memory allocations, CLN implements:
3278 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3281 @cindex garbage collection
3282 @cindex reference counting
3283 Garbage collection: A reference counting mechanism makes sure that any
3284 number object's storage is freed immediately when the last reference to the
3287 @cindex immediate numbers
3288 Small integers are represented as immediate values instead of pointers
3289 to heap allocated storage. This means that integers @code{> -2^29},
3290 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3295 @section Speed efficiency
3297 Speed efficiency is obtained by the combination of the following tricks
3302 Small integers, being represented as immediate values, don't require
3303 memory access, just a couple of instructions for each elementary operation.
3305 The kernel of CLN has been written in assembly language for some CPUs
3306 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3308 On all CPUs, CLN may be configured to use the superefficient low-level
3309 routines from GNU GMP version 3.
3311 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3312 algorithm, the Karatsuba multiplication, which is an
3323 For very large numbers (more than 12000 decimal digits), CLN uses
3325 Sch{@"o}nhage-Strassen
3326 @cindex Sch{@"o}nhage-Strassen multiplication
3330 @cindex Schönhage-Strassen multiplication
3332 multiplication, which is an asymptotically optimal multiplication
3335 These fast multiplication algorithms also give improvements in the speed
3336 of division and radix conversion.
3340 @section Garbage collection
3341 @cindex garbage collection
3343 All the number classes are reference count classes: They only contain a pointer
3344 to an object in the heap. Upon construction, assignment and destruction of
3345 number objects, only the objects' reference count are manipulated.
3347 Memory occupied by number objects are automatically reclaimed as soon as
3348 their reference count drops to zero.
3350 For number rings, another strategy is implemented: There is a cache of,
3351 for example, the modular integer rings. A modular integer ring is destroyed
3352 only if its reference count dropped to zero and the cache is about to be
3353 resized. The effect of this strategy is that recently used rings remain
3354 cached, whereas undue memory consumption through cached rings is avoided.
3357 @chapter Using the library
3359 For the following discussion, we will assume that you have installed
3360 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3361 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3362 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3363 environment variables, or directly substitute the appropriate values.
3366 @section Compiler options
3367 @cindex compiler options
3369 Until you have installed CLN in a public place, the following options are
3372 When you compile CLN application code, add the flags
3374 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3376 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3377 When you link CLN application code to form an executable, add the flags
3379 $CLN_TARGETDIR/src/libcln.a
3381 to the C/C++ compiler's command line (@code{make} variable LIBS).
3383 If you did a @code{make install}, the include files are installed in a
3384 public directory (normally @code{/usr/local/include}), hence you don't
3385 need special flags for compiling. The library has been installed to a
3386 public directory as well (normally @code{/usr/local/lib}), hence when
3387 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3389 Since CLN version 1.1, there are two tools to make the creation of
3390 software packages that use CLN easier:
3393 @cindex @code{cln-config}
3394 @code{cln-config} is a shell script that you can use to determine the
3395 compiler and linker command line options required to compile and link a
3396 program with CLN. Start it with @code{--help} to learn about its options
3397 or consult the manpage that comes with it.
3399 @cindex @code{AC_PATH_CLN}
3400 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3403 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3405 This macro determines the location of CLN using @code{cln-config}, which
3406 is either found in the user's path, or from the environment variable
3407 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3408 their version is not earlier than @var{MIN-VERSION} (a default version
3409 will be used if not specified). If the required version was found, sets
3410 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3411 macro is in the file @file{cln.m4} which is installed in
3412 @file{$datadir/aclocal}. Note that if automake was installed with a
3413 different @samp{--prefix} than CLN, you will either have to manually
3414 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3415 aclocal the @samp{-I} option when running it. Here is a possible example
3416 to be included in your package's @file{configure.ac}:
3418 AC_PATH_CLN(1.1.0, [
3419 LIBS="$LIBS $CLN_LIBS"
3420 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3421 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3426 @section Compatibility to old CLN versions
3428 @cindex compatibility
3430 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3431 @code{cln} in order to avoid potential name clashes with other C++
3432 libraries. If you have an old application, you will have to manually
3433 port it to the new scheme. The following principles will help during
3437 All headers are now in a separate subdirectory. Instead of including
3438 @code{cl_}@var{something}@code{.h}, include
3439 @code{cln/}@var{something}@code{.h} now.
3441 All public identifiers (typenames and functions) have lost their
3442 @code{cl_} prefix. Exceptions are all the typenames of number types,
3443 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3444 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3445 names would not be mnemonic enough once the namespace @code{cln} is
3446 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3448 All public @emph{functions} that had by a @code{cl_} in their name still
3449 carry that @code{cl_} if it is intrinsic part of a typename (as in
3450 @code{cl_I_to_int ()}).
3452 When developing other libraries, please keep in mind not to import the
3453 namespace @code{cln} in one of your public header files by saying
3454 @code{using namespace cln;}. This would propagate to other applications
3455 and can cause name clashes there.
3458 @section Include files
3459 @cindex include files
3460 @cindex header files
3462 Here is a summary of the include files and their contents.
3465 @item <cln/object.h>
3466 General definitions, reference counting, garbage collection.
3467 @item <cln/number.h>
3468 The class cl_number.
3469 @item <cln/complex.h>
3470 Functions for class cl_N, the complex numbers.
3472 Functions for class cl_R, the real numbers.
3474 Functions for class cl_F, the floats.
3475 @item <cln/sfloat.h>
3476 Functions for class cl_SF, the short-floats.
3477 @item <cln/ffloat.h>
3478 Functions for class cl_FF, the single-floats.
3479 @item <cln/dfloat.h>
3480 Functions for class cl_DF, the double-floats.
3481 @item <cln/lfloat.h>
3482 Functions for class cl_LF, the long-floats.
3483 @item <cln/rational.h>
3484 Functions for class cl_RA, the rational numbers.
3485 @item <cln/integer.h>
3486 Functions for class cl_I, the integers.
3489 @item <cln/complex_io.h>
3490 Input/Output for class cl_N, the complex numbers.
3491 @item <cln/real_io.h>
3492 Input/Output for class cl_R, the real numbers.
3493 @item <cln/float_io.h>
3494 Input/Output for class cl_F, the floats.
3495 @item <cln/sfloat_io.h>
3496 Input/Output for class cl_SF, the short-floats.
3497 @item <cln/ffloat_io.h>
3498 Input/Output for class cl_FF, the single-floats.
3499 @item <cln/dfloat_io.h>
3500 Input/Output for class cl_DF, the double-floats.
3501 @item <cln/lfloat_io.h>
3502 Input/Output for class cl_LF, the long-floats.
3503 @item <cln/rational_io.h>
3504 Input/Output for class cl_RA, the rational numbers.
3505 @item <cln/integer_io.h>
3506 Input/Output for class cl_I, the integers.
3508 Flags for customizing input operations.
3509 @item <cln/output.h>
3510 Flags for customizing output operations.
3511 @item <cln/malloc.h>
3512 @code{malloc_hook}, @code{free_hook}.
3515 @item <cln/condition.h>
3516 Conditions/exceptions.
3517 @item <cln/string.h>
3519 @item <cln/symbol.h>
3521 @item <cln/proplist.h>
3525 @item <cln/null_ring.h>
3527 @item <cln/complex_ring.h>
3528 The ring of complex numbers.
3529 @item <cln/real_ring.h>
3530 The ring of real numbers.
3531 @item <cln/rational_ring.h>
3532 The ring of rational numbers.
3533 @item <cln/integer_ring.h>
3534 The ring of integers.
3535 @item <cln/numtheory.h>
3536 Number threory functions.
3537 @item <cln/modinteger.h>
3543 @item <cln/GV_number.h>
3544 General vectors over cl_number.
3545 @item <cln/GV_complex.h>
3546 General vectors over cl_N.
3547 @item <cln/GV_real.h>
3548 General vectors over cl_R.
3549 @item <cln/GV_rational.h>
3550 General vectors over cl_RA.
3551 @item <cln/GV_integer.h>
3552 General vectors over cl_I.
3553 @item <cln/GV_modinteger.h>
3554 General vectors of modular integers.
3557 @item <cln/SV_number.h>
3558 Simple vectors over cl_number.
3559 @item <cln/SV_complex.h>
3560 Simple vectors over cl_N.
3561 @item <cln/SV_real.h>
3562 Simple vectors over cl_R.
3563 @item <cln/SV_rational.h>
3564 Simple vectors over cl_RA.
3565 @item <cln/SV_integer.h>
3566 Simple vectors over cl_I.
3567 @item <cln/SV_ringelt.h>
3568 Simple vectors of general ring elements.
3569 @item <cln/univpoly.h>
3570 Univariate polynomials.
3571 @item <cln/univpoly_integer.h>
3572 Univariate polynomials over the integers.
3573 @item <cln/univpoly_rational.h>
3574 Univariate polynomials over the rational numbers.
3575 @item <cln/univpoly_real.h>
3576 Univariate polynomials over the real numbers.
3577 @item <cln/univpoly_complex.h>
3578 Univariate polynomials over the complex numbers.
3579 @item <cln/univpoly_modint.h>
3580 Univariate polynomials over modular integer rings.
3581 @item <cln/timing.h>
3584 Includes all of the above.
3590 A function which computes the nth Fibonacci number can be written as follows.
3591 @cindex Fibonacci number
3594 #include <cln/integer.h>
3595 #include <cln/real.h>
3596 using namespace cln;
3598 // Returns F_n, computed as the nearest integer to
3599 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3600 const cl_I fibonacci (int n)
3602 // Need a precision of ((1+sqrt(5))/2)^-n.
3603 float_format_t prec = float_format((int)(0.208987641*n+5));
3604 cl_R sqrt5 = sqrt(cl_float(5,prec));
3605 cl_R phi = (1+sqrt5)/2;
3606 return round1( expt(phi,n)/sqrt5 );
3610 Let's explain what is going on in detail.
3612 The include file @code{<cln/integer.h>} is necessary because the type
3613 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3614 is needed for the type @code{cl_R} and the floating point number functions.
3615 The order of the include files does not matter. In order not to write
3616 out @code{cln::}@var{foo} in this simple example we can safely import
3617 the whole namespace @code{cln}.
3619 Then comes the function declaration. The argument is an @code{int}, the
3620 result an integer. The return type is defined as @samp{const cl_I}, not
3621 simply @samp{cl_I}, because that allows the compiler to detect typos like
3622 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3623 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3624 number). We use the most specialized possible return type because functions
3625 which call @samp{fibonacci} will be able to profit from the compiler's type
3626 analysis: Adding two integers is slightly more efficient than adding the
3627 same objects declared as complex numbers, because it needs less type
3628 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3629 the size of the resulting executable program.
3631 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3632 integer. In order to get a correct result, the absolute error should be less
3633 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3634 To this end, the first line computes a floating point precision for sqrt(5)
3637 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3638 number and than taking the square root. The converse, first taking the square
3639 root of 5, and then converting to the desired precision, would not work in
3640 CLN: The square root would be computed to a default precision (normally
3641 single-float precision), and the following conversion could not help about
3642 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3643 system and does not represent sqrt(5) in a non-numeric way.
3645 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3646 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3647 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3648 @code{cl_N} because a @samp{round1} does not exist for general complex
3651 When the function returns, all the local variables in the function are
3652 automatically reclaimed (garbage collected). Only the result survives and
3653 gets passed to the caller.
3655 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3656 contains this implementation together with an even faster algorithm.
3658 @section Debugging support
3661 When debugging a CLN application with GNU @code{gdb}, two facilities are
3662 available from the library:
3665 @item The library does type checks, range checks, consistency checks at
3666 many places. When one of these fails, the function @code{cl_abort()} is
3667 called. Its default implementation is to perform an @code{exit(1)}, so
3668 you won't have a core dump. But for debugging, it is best to set a
3669 breakpoint at this function:
3671 (gdb) break cl_abort
3673 When this breakpoint is hit, look at the stack's backtrace:
3678 @item The debugger's normal @code{print} command doesn't know about
3679 CLN's types and therefore prints mostly useless hexadecimal addresses.
3680 CLN offers a function @code{cl_print}, callable from the debugger,
3681 for printing number objects. In order to get this function, you have
3682 to define the macro @samp{CL_DEBUG} and then include all the header files
3683 for which you want @code{cl_print} debugging support. For example:
3684 @cindex @code{CL_DEBUG}
3687 #include <cln/string.h>
3689 Now, if you have in your program a variable @code{cl_string s}, and
3690 inspect it under @code{gdb}, the output may look like this:
3693 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3694 word = 134568800@}@}, @}
3695 (gdb) call cl_print(s)
3699 Note that the output of @code{cl_print} goes to the program's error output,
3700 not to gdb's standard output.
3702 Note, however, that the above facility does not work with all CLN types,
3703 only with number objects and similar. Therefore CLN offers a member function
3704 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3705 is needed for this member function to be implemented. Under @code{gdb},
3706 you call it like this:
3707 @cindex @code{debug_print ()}
3710 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3711 word = 134568800@}@}, @}
3712 (gdb) call s.debug_print()
3715 >call ($1).debug_print()
3720 Unfortunately, this feature does not seem to work under all circumstances.
3724 @chapter Customizing
3727 @section Error handling
3729 When a fatal error occurs, an error message is output to the standard error
3730 output stream, and the function @code{cl_abort} is called. The default
3731 version of this function (provided in the library) terminates the application.
3732 To catch such a fatal error, you need to define the function @code{cl_abort}
3733 yourself, with the prototype
3735 #include <cln/abort.h>
3736 void cl_abort (void);
3738 @cindex @code{cl_abort ()}
3739 This function must not return control to its caller.
3742 @section Floating-point underflow
3745 Floating point underflow denotes the situation when a floating-point number
3746 is to be created which is so close to @code{0} that its exponent is too
3747 low to be represented internally. By default, this causes a fatal error.
3748 If you set the global variable
3750 cl_boolean cl_inhibit_floating_point_underflow
3752 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3753 will be generated instead. The default value of
3754 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3757 @section Customizing I/O
3759 The output of the function @code{fprint} may be customized by changing the
3760 value of the global variable @code{default_print_flags}.
3761 @cindex @code{default_print_flags}
3764 @section Customizing the memory allocator
3766 Every memory allocation of CLN is done through the function pointer
3767 @code{malloc_hook}. Freeing of this memory is done through the function
3768 pointer @code{free_hook}. The default versions of these functions,
3769 provided in the library, call @code{malloc} and @code{free} and check
3770 the @code{malloc} result against @code{NULL}.
3771 If you want to provide another memory allocator, you need to define
3772 the variables @code{malloc_hook} and @code{free_hook} yourself,
3775 #include <cln/malloc.h>
3777 void* (*malloc_hook) (size_t size) = @dots{};
3778 void (*free_hook) (void* ptr) = @dots{};
3781 @cindex @code{malloc_hook ()}
3782 @cindex @code{free_hook ()}
3783 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3785 It is not possible to change the memory allocator at runtime, because
3786 it is already called at program startup by the constructors of some
3799 @c Table of contents