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
362 @comment cl_modules.h requires g++
363 You should not mix GNU and non-GNU compilers. So, if @code{CXX} is a non-GNU
364 compiler, @code{CC} should be set to a non-GNU compiler as well. Examples:
367 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O" ./configure
368 $ CC="gcc -V 2.7.0" CFLAGS="-g" CXX="g++ -V 2.7.0" CXXFLAGS="-g" ./configure
371 On SGI Irix 5, if you wish not to use @code{g++}:
374 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O -Olimit 16000" ./configure
377 On SGI Irix 6, if you wish not to use @code{g++}:
380 $ CC="cc -32" CFLAGS="-O" CXX="CC -32" CXXFLAGS="-O -Olimit 34000" \
381 ./configure --without-gmp
382 $ CC="cc -n32" CFLAGS="-O" CXX="CC -n32" CXXFLAGS="-O \
383 -OPT:const_copy_limit=32400 -OPT:global_limit=32400 -OPT:fprop_limit=4000" \
384 ./configure --without-gmp
388 Note that for these environment variables to take effect, you have to set
389 them (assuming a Bourne-compatible shell) on the same line as the
390 @code{configure} command. If you made the settings in earlier shell
391 commands, you have to @code{export} the environment variables before
392 calling @code{configure}. In a @code{csh} shell, you have to use the
393 @samp{setenv} command for setting each of the environment variables.
395 Currently CLN works only with the GNU @code{g++} compiler, and only in
396 optimizing mode. So you should specify at least @code{-O} in the CXXFLAGS,
397 or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use @code{-O}.)
399 If you use @code{g++} gcc-2.95.x or gcc-3.0, I recommend adding
400 @samp{-fno-exceptions} to the CXXFLAGS. This will likely generate better code.
402 If you use @code{g++} from gcc-2.95.x on Sparc, add either @samp{-O},
403 @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the CXXFLAGS. With full
404 @samp{-O2}, @code{g++} miscompiles the division routines. If you use
405 @code{g++} older than 2.95.3 on Sparc you should also specify
406 @samp{--disable-shared} because of bad code produced in the shared
407 library. Also, on OSF/1 or Tru64 using gcc-2.95.x, you should specify
408 @samp{--disable-shared} because of linker problems with duplicate symbols
411 By default, both a shared and a static library are built. You can build
412 CLN as a static (or shared) library only, by calling @code{configure} with
413 the option @samp{--disable-shared} (or @samp{--disable-static}). While
414 shared libraries are usually more convenient to use, they may not work
415 on all architectures. Try disabling them if you run into linker
416 problems. Also, they are generally somewhat slower than static
417 libraries so runtime-critical applications should be linked statically.
420 @subsection Using the GNU MP Library
423 Starting with version 1.1, CLN may be configured to make use of a
424 preinstalled @code{gmp} library. Please make sure that you have at
425 least @code{gmp} version 3.0 installed since earlier versions are
426 unsupported and likely not to work. Enabling this feature by calling
427 @code{configure} with the option @samp{--with-gmp} is known to be quite
428 a boost for CLN's performance.
430 If you have installed the @code{gmp} library and its header file in
431 some place where your compiler cannot find it by default, you must help
432 @code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
436 $ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
437 CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
441 @section Installing the library
444 As with any autoconfiguring GNU software, installation is as easy as this:
450 The @samp{make install} command installs the library and the include files
451 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
452 if you haven't specified a @code{--prefix} option to @code{configure}).
453 This step may require superuser privileges.
455 If you have already built the library and wish to install it, but didn't
456 specify @code{--prefix=@dots{}} at configure time, just re-run
457 @code{configure}, giving it the same options as the first time, plus
458 the @code{--prefix=@dots{}} option.
463 You can remove system-dependent files generated by @code{make} through
469 You can remove all files generated by @code{make}, thus reverting to a
470 virgin distribution of CLN, through
477 @chapter Ordinary number types
479 CLN implements the following class hierarchy:
487 Real or complex number
496 +-------------------+-------------------+
498 Rational number Floating-point number
500 <cln/rational.h> <cln/float.h>
502 | +--------------+--------------+--------------+
504 cl_I Short-Float Single-Float Double-Float Long-Float
505 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
506 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
509 @cindex @code{cl_number}
510 @cindex abstract class
511 The base class @code{cl_number} is an abstract base class.
512 It is not useful to declare a variable of this type except if you want
513 to completely disable compile-time type checking and use run-time type
518 @cindex complex number
519 The class @code{cl_N} comprises real and complex numbers. There is
520 no special class for complex numbers since complex numbers with imaginary
521 part @code{0} are automatically converted to real numbers.
524 The class @code{cl_R} comprises real numbers of different kinds. It is an
528 @cindex rational number
530 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
531 integers. There is no special class for non-integral rational numbers
532 since rational numbers with denominator @code{1} are automatically converted
536 The class @code{cl_F} implements floating-point approximations to real numbers.
537 It is an abstract class.
540 @section Exact numbers
543 Some numbers are represented as exact numbers: there is no loss of information
544 when such a number is converted from its mathematical value to its internal
545 representation. On exact numbers, the elementary operations (@code{+},
546 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
549 In CLN, the exact numbers are:
553 rational numbers (including integers),
555 complex numbers whose real and imaginary parts are both rational numbers.
558 Rational numbers are always normalized to the form
559 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
560 are coprime integers and the denominator is positive. If the resulting
561 denominator is @code{1}, the rational number is converted to an integer.
563 @cindex immediate numbers
564 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
565 for 32-bit machines) are especially efficient, because they consume no heap
566 allocation. Otherwise the distinction between these immediate integers
567 (called ``fixnums'') and heap allocated integers (called ``bignums'')
568 is completely transparent.
571 @section Floating-point numbers
572 @cindex floating-point number
574 Not all real numbers can be represented exactly. (There is an easy mathematical
575 proof for this: Only a countable set of numbers can be stored exactly in
576 a computer, even if one assumes that it has unlimited storage. But there
577 are uncountably many real numbers.) So some approximation is needed.
578 CLN implements ordinary floating-point numbers, with mantissa and exponent.
580 @cindex rounding error
581 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
582 only return approximate results. For example, the value of the expression
583 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
584 @samp{0.7}. Rounding errors like this one are inevitable when computing
585 with floating-point numbers.
587 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
588 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
589 rule: It first computes the exact mathematical result and then returns the
590 floating-point number which is nearest to this. If two floating-point numbers
591 are equally distant from the ideal result, the one with a @code{0} in its least
592 significant mantissa bit is chosen.
594 Similarly, testing floating point numbers for equality @samp{x == y}
595 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
596 for some well-chosen @code{epsilon}.
598 Floating point numbers come in four flavors:
603 Short floats, type @code{cl_SF}.
604 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
605 and 17 mantissa bits (including the ``hidden'' bit).
606 They don't consume heap allocation.
610 Single floats, type @code{cl_FF}.
611 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
612 and 24 mantissa bits (including the ``hidden'' bit).
613 In CLN, they are represented as IEEE single-precision floating point numbers.
614 This corresponds closely to the C/C++ type @samp{float}.
618 Double floats, type @code{cl_DF}.
619 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
620 and 53 mantissa bits (including the ``hidden'' bit).
621 In CLN, they are represented as IEEE double-precision floating point numbers.
622 This corresponds closely to the C/C++ type @samp{double}.
626 Long floats, type @code{cl_LF}.
627 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
628 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
629 The precision of a long float is unlimited, but once created, a long float
630 has a fixed precision. (No ``lazy recomputation''.)
633 Of course, computations with long floats are more expensive than those
634 with smaller floating-point formats.
636 CLN does not implement features like NaNs, denormalized numbers and
637 gradual underflow. If the exponent range of some floating-point type
638 is too limited for your application, choose another floating-point type
639 with larger exponent range.
642 As a user of CLN, you can forget about the differences between the
643 four floating-point types and just declare all your floating-point
644 variables as being of type @code{cl_F}. This has the advantage that
645 when you change the precision of some computation (say, from @code{cl_DF}
646 to @code{cl_LF}), you don't have to change the code, only the precision
647 of the initial values. Also, many transcendental functions have been
648 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
649 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
650 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
651 the floating point contagion rule happened to change in the future.)
654 @section Complex numbers
655 @cindex complex number
657 Complex numbers, as implemented by the class @code{cl_N}, have a real
658 part and an imaginary part, both real numbers. A complex number whose
659 imaginary part is the exact number @code{0} is automatically converted
662 Complex numbers can arise from real numbers alone, for example
663 through application of @code{sqrt} or transcendental functions.
669 Conversions from any class to any its superclasses (``base classes'' in
670 C++ terminology) is done automatically.
672 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
673 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
674 @code{cl_N} and @code{cl_number}.
676 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
677 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
678 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
679 efficiency. Their range is therefore limited:
683 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
685 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
688 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
689 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
690 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
692 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
693 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
694 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
695 @samp{cl_I}, first convert it to an @samp{unsigned long}.
697 Conversions from the C built-in type @samp{float} are provided for the classes
698 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
700 Conversions from the C built-in type @samp{double} are provided for the classes
701 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
703 Conversions from @samp{const char *} are provided for the classes
704 @code{cl_I}, @code{cl_RA},
705 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
706 @code{cl_R}, @code{cl_N}.
707 The easiest way to specify a value which is outside of the range of the
708 C++ built-in types is therefore to specify it as a string, like this:
711 cl_I order_of_rubiks_cube_group = "43252003274489856000";
713 Note that this conversion is done at runtime, not at compile-time.
715 Conversions from @code{cl_I} to the C built-in types @samp{int},
716 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
720 @item int cl_I_to_int (const cl_I& x)
721 @cindex @code{cl_I_to_int ()}
722 @itemx unsigned int cl_I_to_uint (const cl_I& x)
723 @cindex @code{cl_I_to_uint ()}
724 @itemx long cl_I_to_long (const cl_I& x)
725 @cindex @code{cl_I_to_long ()}
726 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
727 @cindex @code{cl_I_to_ulong ()}
728 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
729 representable in the range of @var{ctype}, a runtime error occurs.
732 Conversions from the classes @code{cl_I}, @code{cl_RA},
733 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
735 to the C built-in types @samp{float} and @samp{double} are provided through
739 @item float float_approx (const @var{type}& x)
740 @cindex @code{float_approx ()}
741 @itemx double double_approx (const @var{type}& x)
742 @cindex @code{double_approx ()}
743 Returns an approximation of @code{x} of C type @var{ctype}.
744 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
745 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
748 Conversions from any class to any of its subclasses (``derived classes'' in
749 C++ terminology) are not provided. Instead, you can assert and check
750 that a value belongs to a certain subclass, and return it as element of that
751 class, using the @samp{As} and @samp{The} macros.
752 @cindex @code{As()()}
753 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
754 @var{type} and returns it as such.
755 @cindex @code{The()()}
756 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
757 @var{type} and returns it as such. It is your responsibility to ensure
758 that this assumption is valid. Since macros and namespaces don't go
759 together well, there is an equivalent to @samp{The}: the template
767 if (!(x >= 0)) abort();
768 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
769 // In general, it would be a rational number.
770 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
775 @chapter Functions on numbers
777 Each of the number classes declares its mathematical operations in the
778 corresponding include file. For example, if your code operates with
779 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
782 @section Constructing numbers
784 Here is how to create number objects ``from nothing''.
787 @subsection Constructing integers
789 @code{cl_I} objects are most easily constructed from C integers and from
790 strings. See @ref{Conversions}.
793 @subsection Constructing rational numbers
795 @code{cl_RA} objects can be constructed from strings. The syntax
796 for rational numbers is described in @ref{Internal and printed representation}.
797 Another standard way to produce a rational number is through application
798 of @samp{operator /} or @samp{recip} on integers.
801 @subsection Constructing floating-point numbers
803 @code{cl_F} objects with low precision are most easily constructed from
804 C @samp{float} and @samp{double}. See @ref{Conversions}.
806 To construct a @code{cl_F} with high precision, you can use the conversion
807 from @samp{const char *}, but you have to specify the desired precision
808 within the string. (See @ref{Internal and printed representation}.)
811 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
813 will set @samp{e} to the given value, with a precision of 40 decimal digits.
815 The programmatic way to construct a @code{cl_F} with high precision is
816 through the @code{cl_float} conversion function, see
817 @ref{Conversion to floating-point numbers}. For example, to compute
818 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
819 and then apply the exponential function:
821 float_format_t precision = float_format(40);
822 cl_F e = exp(cl_float(1,precision));
826 @subsection Constructing complex numbers
828 Non-real @code{cl_N} objects are normally constructed through the function
830 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
832 See @ref{Elementary complex functions}.
835 @section Elementary functions
837 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
838 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
839 defines the following operations:
842 @item @var{type} operator + (const @var{type}&, const @var{type}&)
843 @cindex @code{operator + ()}
846 @item @var{type} operator - (const @var{type}&, const @var{type}&)
847 @cindex @code{operator - ()}
850 @item @var{type} operator - (const @var{type}&)
851 Returns the negative of the argument.
853 @item @var{type} plus1 (const @var{type}& x)
854 @cindex @code{plus1 ()}
855 Returns @code{x + 1}.
857 @item @var{type} minus1 (const @var{type}& x)
858 @cindex @code{minus1 ()}
859 Returns @code{x - 1}.
861 @item @var{type} operator * (const @var{type}&, const @var{type}&)
862 @cindex @code{operator * ()}
865 @item @var{type} square (const @var{type}& x)
866 @cindex @code{square ()}
867 Returns @code{x * x}.
870 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
871 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
872 defines the following operations:
875 @item @var{type} operator / (const @var{type}&, const @var{type}&)
876 @cindex @code{operator / ()}
879 @item @var{type} recip (const @var{type}&)
880 @cindex @code{recip ()}
881 Returns the reciprocal of the argument.
884 The class @code{cl_I} doesn't define a @samp{/} operation because
885 in the C/C++ language this operator, applied to integral types,
886 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
887 is implementation dependent). (@xref{Rounding functions}.)
888 Instead, @code{cl_I} defines an ``exact quotient'' function:
891 @item cl_I exquo (const cl_I& x, const cl_I& y)
892 @cindex @code{exquo ()}
893 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
896 The following exponentiation functions are defined:
899 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
900 @cindex @code{expt_pos ()}
901 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
902 @code{y} must be > 0. Returns @code{x^y}.
904 @item cl_RA expt (const cl_RA& x, const cl_I& y)
905 @cindex @code{expt ()}
906 @itemx cl_R expt (const cl_R& x, const cl_I& y)
907 @itemx cl_N expt (const cl_N& x, const cl_I& y)
911 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
912 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
913 defines the following operation:
916 @item @var{type} abs (const @var{type}& x)
917 @cindex @code{abs ()}
918 Returns the absolute value of @code{x}.
919 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
922 The class @code{cl_N} implements this as follows:
925 @item cl_R abs (const cl_N x)
926 Returns the absolute value of @code{x}.
929 Each of the classes @code{cl_N}, @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} signum (const @var{type}& x)
935 @cindex @code{signum ()}
936 Returns the sign of @code{x}, in the same number format as @code{x}.
937 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
938 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
943 @section Elementary rational functions
945 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
948 @item cl_I numerator (const @var{type}& x)
949 @cindex @code{numerator ()}
950 Returns the numerator of @code{x}.
952 @item cl_I denominator (const @var{type}& x)
953 @cindex @code{denominator ()}
954 Returns the denominator of @code{x}.
957 The numerator and denominator of a rational number are normalized in such
958 a way that they have no factor in common and the denominator is positive.
961 @section Elementary complex functions
963 The class @code{cl_N} defines the following operation:
966 @item cl_N complex (const cl_R& a, const cl_R& b)
967 @cindex @code{complex ()}
968 Returns the complex number @code{a+bi}, that is, the complex number with
969 real part @code{a} and imaginary part @code{b}.
972 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
975 @item cl_R realpart (const @var{type}& x)
976 @cindex @code{realpart ()}
977 Returns the real part of @code{x}.
979 @item cl_R imagpart (const @var{type}& x)
980 @cindex @code{imagpart ()}
981 Returns the imaginary part of @code{x}.
983 @item @var{type} conjugate (const @var{type}& x)
984 @cindex @code{conjugate ()}
985 Returns the complex conjugate of @code{x}.
988 We have the relations
992 @code{x = complex(realpart(x), imagpart(x))}
994 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
1001 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1002 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1003 defines the following operations:
1006 @item bool operator == (const @var{type}&, const @var{type}&)
1007 @cindex @code{operator == ()}
1008 @itemx bool operator != (const @var{type}&, const @var{type}&)
1009 @cindex @code{operator != ()}
1010 Comparison, as in C and C++.
1012 @item uint32 equal_hashcode (const @var{type}&)
1013 @cindex @code{equal_hashcode ()}
1014 Returns a 32-bit hash code that is the same for any two numbers which are
1015 the same according to @code{==}. This hash code depends on the number's value,
1016 not its type or precision.
1018 @item cl_boolean zerop (const @var{type}& x)
1019 @cindex @code{zerop ()}
1020 Compare against zero: @code{x == 0}
1023 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1024 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1025 defines the following operations:
1028 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1029 @cindex @code{compare ()}
1030 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1031 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1033 @item bool operator <= (const @var{type}&, const @var{type}&)
1034 @cindex @code{operator <= ()}
1035 @itemx bool operator < (const @var{type}&, const @var{type}&)
1036 @cindex @code{operator < ()}
1037 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1038 @cindex @code{operator >= ()}
1039 @itemx bool operator > (const @var{type}&, const @var{type}&)
1040 @cindex @code{operator > ()}
1041 Comparison, as in C and C++.
1043 @item cl_boolean minusp (const @var{type}& x)
1044 @cindex @code{minusp ()}
1045 Compare against zero: @code{x < 0}
1047 @item cl_boolean plusp (const @var{type}& x)
1048 @cindex @code{plusp ()}
1049 Compare against zero: @code{x > 0}
1051 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1052 @cindex @code{max ()}
1053 Return the maximum of @code{x} and @code{y}.
1055 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1056 @cindex @code{min ()}
1057 Return the minimum of @code{x} and @code{y}.
1060 When a floating point number and a rational number are compared, the float
1061 is first converted to a rational number using the function @code{rational}.
1062 Since a floating point number actually represents an interval of real numbers,
1063 the result might be surprising.
1064 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1065 there is no floating point number whose value is exactly @code{1/3}.
1068 @section Rounding functions
1071 When a real number is to be converted to an integer, there is no ``best''
1072 rounding. The desired rounding function depends on the application.
1073 The Common Lisp and ISO Lisp standards offer four rounding functions:
1077 This is the largest integer <=@code{x}.
1080 This is the smallest integer >=@code{x}.
1083 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1086 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1087 integers, choose the even one.
1090 These functions have different advantages:
1092 @code{floor} and @code{ceiling} are translation invariant:
1093 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1094 for every @code{x} and every integer @code{n}.
1096 On the other hand, @code{truncate} and @code{round} are symmetric:
1097 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1098 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1099 down exactly as often as it rounds up.
1101 The functions are related like this:
1105 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1106 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1108 @code{truncate(x) = sign(x) * floor(abs(x))}
1111 Each of the classes @code{cl_R}, @code{cl_RA},
1112 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1113 defines the following operations:
1116 @item cl_I floor1 (const @var{type}& x)
1117 @cindex @code{floor1 ()}
1118 Returns @code{floor(x)}.
1119 @item cl_I ceiling1 (const @var{type}& x)
1120 @cindex @code{ceiling1 ()}
1121 Returns @code{ceiling(x)}.
1122 @item cl_I truncate1 (const @var{type}& x)
1123 @cindex @code{truncate1 ()}
1124 Returns @code{truncate(x)}.
1125 @item cl_I round1 (const @var{type}& x)
1126 @cindex @code{round1 ()}
1127 Returns @code{round(x)}.
1130 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1131 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1132 defines the following operations:
1135 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1136 Returns @code{floor(x/y)}.
1137 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1138 Returns @code{ceiling(x/y)}.
1139 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1140 Returns @code{truncate(x/y)}.
1141 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1142 Returns @code{round(x/y)}.
1145 These functions are called @samp{floor1}, @dots{} here instead of
1146 @samp{floor}, @dots{}, because on some systems, system dependent include
1147 files define @samp{floor} and @samp{ceiling} as macros.
1149 In many cases, one needs both the quotient and the remainder of a division.
1150 It is more efficient to compute both at the same time than to perform
1151 two divisions, one for quotient and the next one for the remainder.
1152 The following functions therefore return a structure containing both
1153 the quotient and the remainder. The suffix @samp{2} indicates the number
1154 of ``return values''. The remainder is defined as follows:
1158 for the computation of @code{quotient = floor(x)},
1159 @code{remainder = x - quotient},
1161 for the computation of @code{quotient = floor(x,y)},
1162 @code{remainder = x - quotient*y},
1165 and similarly for the other three operations.
1167 Each of the classes @code{cl_R}, @code{cl_RA},
1168 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1169 defines the following operations:
1172 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1173 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1174 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1175 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1176 @itemx @var{type}_div_t round2 (const @var{type}& x)
1179 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1180 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1181 defines the following operations:
1184 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1185 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1186 @cindex @code{floor2 ()}
1187 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1188 @cindex @code{ceiling2 ()}
1189 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1190 @cindex @code{truncate2 ()}
1191 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1192 @cindex @code{round2 ()}
1195 Sometimes, one wants the quotient as a floating-point number (of the
1196 same format as the argument, if the argument is a float) instead of as
1197 an integer. The prefix @samp{f} indicates this.
1200 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1201 defines the following operations:
1204 @item @var{type} ffloor (const @var{type}& x)
1205 @cindex @code{ffloor ()}
1206 @itemx @var{type} fceiling (const @var{type}& x)
1207 @cindex @code{fceiling ()}
1208 @itemx @var{type} ftruncate (const @var{type}& x)
1209 @cindex @code{ftruncate ()}
1210 @itemx @var{type} fround (const @var{type}& x)
1211 @cindex @code{fround ()}
1214 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1216 The class @code{cl_R} defines the following operations:
1219 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1220 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1221 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1222 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1225 These functions also exist in versions which return both the quotient
1226 and the remainder. The suffix @samp{2} indicates this.
1229 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1230 defines the following operations:
1231 @cindex @code{cl_F_fdiv_t}
1232 @cindex @code{cl_SF_fdiv_t}
1233 @cindex @code{cl_FF_fdiv_t}
1234 @cindex @code{cl_DF_fdiv_t}
1235 @cindex @code{cl_LF_fdiv_t}
1238 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1239 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1240 @cindex @code{ffloor2 ()}
1241 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1242 @cindex @code{fceiling2 ()}
1243 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1244 @cindex @code{ftruncate2 ()}
1245 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1246 @cindex @code{fround2 ()}
1248 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1249 @cindex @code{cl_R_fdiv_t}
1251 The class @code{cl_R} defines the following operations:
1254 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1255 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1256 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1257 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1258 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1261 Other applications need only the remainder of a division.
1262 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1263 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1264 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1268 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1270 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1273 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1274 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1275 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1277 The classes @code{cl_R}, @code{cl_I} define the following operations:
1280 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1281 @cindex @code{mod ()}
1282 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1283 @cindex @code{rem ()}
1289 Each of the classes @code{cl_R},
1290 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1291 defines the following operation:
1294 @item @var{type} sqrt (const @var{type}& x)
1295 @cindex @code{sqrt ()}
1296 @code{x} must be >= 0. This function returns the square root of @code{x},
1297 normalized to be >= 0. If @code{x} is the square of a rational number,
1298 @code{sqrt(x)} will be a rational number, else it will return a
1299 floating-point approximation.
1302 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1305 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1306 @cindex @code{sqrtp ()}
1307 This tests whether @code{x} is a perfect square. If so, it returns true
1308 and the exact square root in @code{*root}, else it returns false.
1311 Furthermore, for integers, similarly:
1314 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1315 @cindex @code{isqrt ()}
1316 @code{x} should be >= 0. This function sets @code{*root} to
1317 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1318 the boolean value @code{(expt(*root,2) == x)}.
1321 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1322 define the following operation:
1325 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1326 @cindex @code{rootp ()}
1327 @code{x} must be >= 0. @code{n} must be > 0.
1328 This tests whether @code{x} is an @code{n}th power of a rational number.
1329 If so, it returns true and the exact root in @code{*root}, else it returns
1333 The only square root function which accepts negative numbers is the one
1334 for class @code{cl_N}:
1337 @item cl_N sqrt (const cl_N& z)
1338 @cindex @code{sqrt ()}
1339 Returns the square root of @code{z}, as defined by the formula
1340 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1341 or to a complex number are done if necessary. The range of the result is the
1342 right half plane @code{realpart(sqrt(z)) >= 0}
1343 including the positive imaginary axis and 0, but excluding
1344 the negative imaginary axis.
1345 The result is an exact number only if @code{z} is an exact number.
1349 @section Transcendental functions
1350 @cindex transcendental functions
1352 The transcendental functions return an exact result if the argument
1353 is exact and the result is exact as well. Otherwise they must return
1354 inexact numbers even if the argument is exact.
1355 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1358 @subsection Exponential and logarithmic functions
1361 @item cl_R exp (const cl_R& x)
1362 @cindex @code{exp ()}
1363 @itemx cl_N exp (const cl_N& x)
1364 Returns the exponential function of @code{x}. This is @code{e^x} where
1365 @code{e} is the base of the natural logarithms. The range of the result
1366 is the entire complex plane excluding 0.
1368 @item cl_R ln (const cl_R& x)
1369 @cindex @code{ln ()}
1370 @code{x} must be > 0. Returns the (natural) logarithm of x.
1372 @item cl_N log (const cl_N& x)
1373 @cindex @code{log ()}
1374 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1375 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1376 The range of the result is the strip in the complex plane
1377 @code{-pi < imagpart(log(x)) <= pi}.
1379 @item cl_R phase (const cl_N& x)
1380 @cindex @code{phase ()}
1381 Returns the angle part of @code{x} in its polar representation as a
1382 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1383 This is also the imaginary part of @code{log(x)}.
1384 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1385 The result will be an exact number only if @code{zerop(x)} or
1386 if @code{x} is real and positive.
1388 @item cl_R log (const cl_R& a, const cl_R& b)
1389 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1390 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1391 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1394 @item cl_N log (const cl_N& a, const cl_N& b)
1395 Returns the logarithm of @code{a} with respect to base @code{b}.
1396 @code{log(a,b) = log(a)/log(b)}.
1398 @item cl_N expt (const cl_N& x, const cl_N& y)
1399 @cindex @code{expt ()}
1400 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1403 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1406 @item cl_F exp1 (float_format_t f)
1407 @cindex @code{exp1 ()}
1408 Returns e as a float of format @code{f}.
1410 @item cl_F exp1 (const cl_F& y)
1411 Returns e in the float format of @code{y}.
1413 @item cl_F exp1 (void)
1414 Returns e as a float of format @code{default_float_format}.
1418 @subsection Trigonometric functions
1421 @item cl_R sin (const cl_R& x)
1422 @cindex @code{sin ()}
1423 Returns @code{sin(x)}. The range of the result is the interval
1424 @code{-1 <= sin(x) <= 1}.
1426 @item cl_N sin (const cl_N& z)
1427 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1429 @item cl_R cos (const cl_R& x)
1430 @cindex @code{cos ()}
1431 Returns @code{cos(x)}. The range of the result is the interval
1432 @code{-1 <= cos(x) <= 1}.
1434 @item cl_N cos (const cl_N& x)
1435 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1437 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1438 @cindex @code{cos_sin_t}
1439 @itemx cos_sin_t cos_sin (const cl_R& x)
1440 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1441 @cindex @code{cos_sin ()}
1442 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1443 hold only approximately.
1445 @item cl_R tan (const cl_R& x)
1446 @cindex @code{tan ()}
1447 @itemx cl_N tan (const cl_N& x)
1448 Returns @code{tan(x) = sin(x)/cos(x)}.
1450 @item cl_N cis (const cl_R& x)
1451 @cindex @code{cis ()}
1452 @itemx cl_N cis (const cl_N& x)
1453 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1454 @code{e^(i*x) = cos(x) + i*sin(x)}.
1457 @cindex @code{asin ()}
1458 @item cl_N asin (const cl_N& z)
1459 Returns @code{arcsin(z)}. This is defined as
1460 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1461 @code{arcsin(-z) = -arcsin(z)}.
1462 The range of the result is the strip in the complex domain
1463 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1464 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1465 with @code{realpart = pi/2} and @code{imagpart > 0}.
1467 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1471 @item cl_N acos (const cl_N& z)
1472 @cindex @code{acos ()}
1473 Returns @code{arccos(z)}. This is defined as
1474 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1477 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1479 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1480 The range of the result is the strip in the complex domain
1481 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1482 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1483 with @code{realpart = pi} and @code{imagpart > 0}.
1485 Proof: This follows from the results about arcsin.
1489 @cindex @code{atan ()}
1490 @item cl_R atan (const cl_R& x, const cl_R& y)
1491 Returns the angle of the polar representation of the complex number
1492 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1493 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1494 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1495 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1496 with reversed order of arguments.
1498 @item cl_R atan (const cl_R& x)
1499 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1500 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1501 will be an exact number only if @code{x} is the exact @code{0}.
1503 @item cl_N atan (const cl_N& z)
1504 Returns @code{arctan(z)}. This is defined as
1505 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1506 @code{arctan(-z) = -arctan(z)}. The range of the result is
1507 the strip in the complex domain
1508 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1509 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1510 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1512 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1518 @cindex Archimedes' constant
1519 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1522 @item cl_F pi (float_format_t f)
1523 @cindex @code{pi ()}
1524 Returns pi as a float of format @code{f}.
1526 @item cl_F pi (const cl_F& y)
1527 Returns pi in the float format of @code{y}.
1529 @item cl_F pi (void)
1530 Returns pi as a float of format @code{default_float_format}.
1534 @subsection Hyperbolic functions
1537 @item cl_R sinh (const cl_R& x)
1538 @cindex @code{sinh ()}
1539 Returns @code{sinh(x)}.
1541 @item cl_N sinh (const cl_N& z)
1542 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1544 @item cl_R cosh (const cl_R& x)
1545 @cindex @code{cosh ()}
1546 Returns @code{cosh(x)}. The range of the result is the interval
1547 @code{cosh(x) >= 1}.
1549 @item cl_N cosh (const cl_N& z)
1550 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1552 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1553 @cindex @code{cosh_sinh_t}
1554 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1555 @cindex @code{cosh_sinh ()}
1556 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1557 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1558 hold only approximately.
1560 @item cl_R tanh (const cl_R& x)
1561 @cindex @code{tanh ()}
1562 @itemx cl_N tanh (const cl_N& x)
1563 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1565 @item cl_N asinh (const cl_N& z)
1566 @cindex @code{asinh ()}
1567 Returns @code{arsinh(z)}. This is defined as
1568 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1569 @code{arsinh(-z) = -arsinh(z)}.
1571 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1572 Actually, z+sqrt(1+z^2) can never be real and <0, so
1573 -pi < imagpart(arsinh(z)) < pi.
1574 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1575 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1577 The range of the result is the strip in the complex domain
1578 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1579 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1580 with @code{imagpart = pi/2} and @code{realpart < 0}.
1582 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1583 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1584 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1585 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1586 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1587 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1588 If y >= 1, the imagpart is pi/2 and the realpart is
1589 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1592 Moreover, if z is in Range(sqrt),
1593 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1594 (for a proof, see file src/cl_C_asinh.cc).
1597 @item cl_N acosh (const cl_N& z)
1598 @cindex @code{acosh ()}
1599 Returns @code{arcosh(z)}. This is defined as
1600 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1601 The range of the result is the half-strip in the complex domain
1602 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1603 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1605 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1606 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1607 If z is in Range(sqrt), we have
1608 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1609 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1611 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1612 and since the imagpart of both expressions is > -pi, <= pi
1613 ==> arcosh(z) = log(z+sqrt(z^2-1))
1614 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1615 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1616 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1617 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1618 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1620 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1621 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1622 = x^2 + y^2 + sqrt(u^2+v^2)
1627 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1628 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1629 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1630 Otherwise, -z is in Range(sqrt).
1631 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1632 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1633 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1634 and this has realpart > 0.
1635 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1636 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1637 has realpart = 0 and imagpart > 0.
1638 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1639 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1640 This has realpart >= 0 and imagpart = pi.
1643 @item cl_N atanh (const cl_N& z)
1644 @cindex @code{atanh ()}
1645 Returns @code{artanh(z)}. This is defined as
1646 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1647 @code{artanh(-z) = -artanh(z)}. The range of the result is
1648 the strip in the complex domain
1649 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1650 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1651 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1653 Proof: Write z = x+iy. Examine
1654 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1656 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1657 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1658 |x| < 1 ==> imagpart = 0
1661 = (atan(1+x,y) - atan(1-x,-y))/2
1662 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1663 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1664 > (pi - pi/2 - pi/2 )/2 = 0
1665 and (1+x)/y > (1-x)/y
1666 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1667 ==> imagpart < pi/2.
1668 Hence 0 < imagpart < pi/2.
1670 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1675 @subsection Euler gamma
1676 @cindex Euler's constant
1678 Euler's constant C = 0.577@dots{} is returned by the following functions:
1681 @item cl_F eulerconst (float_format_t f)
1682 @cindex @code{eulerconst ()}
1683 Returns Euler's constant as a float of format @code{f}.
1685 @item cl_F eulerconst (const cl_F& y)
1686 Returns Euler's constant in the float format of @code{y}.
1688 @item cl_F eulerconst (void)
1689 Returns Euler's constant as a float of format @code{default_float_format}.
1692 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1693 @cindex Catalan's constant
1696 @item cl_F catalanconst (float_format_t f)
1697 @cindex @code{catalanconst ()}
1698 Returns Catalan's constant as a float of format @code{f}.
1700 @item cl_F catalanconst (const cl_F& y)
1701 Returns Catalan's constant in the float format of @code{y}.
1703 @item cl_F catalanconst (void)
1704 Returns Catalan's constant as a float of format @code{default_float_format}.
1708 @subsection Riemann zeta
1709 @cindex Riemann's zeta
1711 Riemann's zeta function at an integral point @code{s>1} is returned by the
1712 following functions:
1715 @item cl_F zeta (int s, float_format_t f)
1716 @cindex @code{zeta ()}
1717 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1719 @item cl_F zeta (int s, const cl_F& y)
1720 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1722 @item cl_F zeta (int s)
1723 Returns Riemann's zeta function at @code{s} as a float of format
1724 @code{default_float_format}.
1728 @section Functions on integers
1730 @subsection Logical functions
1732 Integers, when viewed as in two's complement notation, can be thought as
1733 infinite bit strings where the bits' values eventually are constant.
1740 The logical operations view integers as such bit strings and operate
1741 on each of the bit positions in parallel.
1744 @item cl_I lognot (const cl_I& x)
1745 @cindex @code{lognot ()}
1746 @itemx cl_I operator ~ (const cl_I& x)
1747 @cindex @code{operator ~ ()}
1748 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1750 @item cl_I logand (const cl_I& x, const cl_I& y)
1751 @cindex @code{logand ()}
1752 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1753 @cindex @code{operator & ()}
1754 Logical and, like @code{x & y} in C.
1756 @item cl_I logior (const cl_I& x, const cl_I& y)
1757 @cindex @code{logior ()}
1758 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1759 @cindex @code{operator | ()}
1760 Logical (inclusive) or, like @code{x | y} in C.
1762 @item cl_I logxor (const cl_I& x, const cl_I& y)
1763 @cindex @code{logxor ()}
1764 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1765 @cindex @code{operator ^ ()}
1766 Exclusive or, like @code{x ^ y} in C.
1768 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1769 @cindex @code{logeqv ()}
1770 Bitwise equivalence, like @code{~(x ^ y)} in C.
1772 @item cl_I lognand (const cl_I& x, const cl_I& y)
1773 @cindex @code{lognand ()}
1774 Bitwise not and, like @code{~(x & y)} in C.
1776 @item cl_I lognor (const cl_I& x, const cl_I& y)
1777 @cindex @code{lognor ()}
1778 Bitwise not or, like @code{~(x | y)} in C.
1780 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1781 @cindex @code{logandc1 ()}
1782 Logical and, complementing the first argument, like @code{~x & y} in C.
1784 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1785 @cindex @code{logandc2 ()}
1786 Logical and, complementing the second argument, like @code{x & ~y} in C.
1788 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1789 @cindex @code{logorc1 ()}
1790 Logical or, complementing the first argument, like @code{~x | y} in C.
1792 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1793 @cindex @code{logorc2 ()}
1794 Logical or, complementing the second argument, like @code{x | ~y} in C.
1797 These operations are all available though the function
1799 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1800 @cindex @code{boole ()}
1802 where @code{op} must have one of the 16 values (each one stands for a function
1803 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1804 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1805 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1806 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1807 @code{boole_orc1}, @code{boole_orc2}.
1808 @cindex @code{boole_clr}
1809 @cindex @code{boole_set}
1810 @cindex @code{boole_1}
1811 @cindex @code{boole_2}
1812 @cindex @code{boole_c1}
1813 @cindex @code{boole_c2}
1814 @cindex @code{boole_and}
1815 @cindex @code{boole_xor}
1816 @cindex @code{boole_eqv}
1817 @cindex @code{boole_nand}
1818 @cindex @code{boole_nor}
1819 @cindex @code{boole_andc1}
1820 @cindex @code{boole_andc2}
1821 @cindex @code{boole_orc1}
1822 @cindex @code{boole_orc2}
1825 Other functions that view integers as bit strings:
1828 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1829 @cindex @code{logtest ()}
1830 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1831 @code{logand(x,y) != 0}.
1833 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1834 @cindex @code{logbitp ()}
1835 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1836 Bit 0 is the least significant bit.
1838 @item uintL logcount (const cl_I& x)
1839 @cindex @code{logcount ()}
1840 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1841 the number of zero bits in @code{x}, if @code{x} < 0.
1844 The following functions operate on intervals of bits in integers.
1847 struct cl_byte @{ uintL size; uintL position; @};
1849 @cindex @code{cl_byte}
1850 represents the bit interval containing the bits
1851 @code{position}@dots{}@code{position+size-1} of an integer.
1852 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1855 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1856 @cindex @code{ldb ()}
1857 extracts the bits of @code{n} described by the bit interval @code{b}
1858 and returns them as a nonnegative integer with @code{b.size} bits.
1860 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1861 @cindex @code{ldb_test ()}
1862 Returns true if some bit described by the bit interval @code{b} is set in
1865 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1866 @cindex @code{dpb ()}
1867 Returns @code{n}, with the bits described by the bit interval @code{b}
1868 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1869 @code{newbyte} are relevant.
1872 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1873 functions are their counterparts without shifting:
1876 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1877 @cindex @code{mask_field ()}
1878 returns an integer with the bits described by the bit interval @code{b}
1879 copied from the corresponding bits in @code{n}, the other bits zero.
1881 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1882 @cindex @code{deposit_field ()}
1883 returns an integer where the bits described by the bit interval @code{b}
1884 come from @code{newbyte} and the other bits come from @code{n}.
1887 The following relations hold:
1891 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1893 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1895 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1898 The following operations on integers as bit strings are efficient shortcuts
1899 for common arithmetic operations:
1902 @item cl_boolean oddp (const cl_I& x)
1903 @cindex @code{oddp ()}
1904 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1905 @code{mod(x,2) != 0}.
1907 @item cl_boolean evenp (const cl_I& x)
1908 @cindex @code{evenp ()}
1909 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1910 @code{mod(x,2) == 0}.
1912 @item cl_I operator << (const cl_I& x, const cl_I& n)
1913 @cindex @code{operator << ()}
1914 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1915 Equivalent to @code{x * expt(2,n)}.
1917 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1918 @cindex @code{operator >> ()}
1919 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1920 Bits shifted out to the right are thrown away.
1921 Equivalent to @code{floor(x / expt(2,n))}.
1923 @item cl_I ash (const cl_I& x, const cl_I& y)
1924 @cindex @code{ash ()}
1925 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1926 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1927 returns @code{floor(x * expt(2,y))}.
1929 @item uintL integer_length (const cl_I& x)
1930 @cindex @code{integer_length ()}
1931 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1932 in two's complement notation. This is the smallest n >= 0 such that
1933 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1936 @item uintL ord2 (const cl_I& x)
1937 @cindex @code{ord2 ()}
1938 @code{x} must be non-zero. This function returns the number of 0 bits at the
1939 right of @code{x} in two's complement notation. This is the largest n >= 0
1940 such that 2^n divides @code{x}.
1942 @item uintL power2p (const cl_I& x)
1943 @cindex @code{power2p ()}
1944 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1945 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1946 (See also the function @code{logp}.)
1950 @subsection Number theoretic functions
1953 @item uint32 gcd (uint32 a, uint32 b)
1954 @cindex @code{gcd ()}
1955 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1956 This function returns the greatest common divisor of @code{a} and @code{b},
1957 normalized to be >= 0.
1959 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1960 @cindex @code{xgcd ()}
1961 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1962 @code{a} and @code{b} and at the same time the representation of @code{g}
1963 as an integral linear combination of @code{a} and @code{b}:
1964 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1965 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1966 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1967 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1968 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1970 @item cl_I lcm (const cl_I& a, const cl_I& b)
1971 @cindex @code{lcm ()}
1972 This function returns the least common multiple of @code{a} and @code{b},
1973 normalized to be >= 0.
1975 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
1976 @cindex @code{logp ()}
1977 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
1978 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
1979 rational number, this function returns true and sets *l = log(a,b), else
1984 @subsection Combinatorial functions
1987 @item cl_I factorial (uintL n)
1988 @cindex @code{factorial ()}
1989 @code{n} must be a small integer >= 0. This function returns the factorial
1990 @code{n}! = @code{1*2*@dots{}*n}.
1992 @item cl_I doublefactorial (uintL n)
1993 @cindex @code{doublefactorial ()}
1994 @code{n} must be a small integer >= 0. This function returns the
1995 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
1996 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
1998 @item cl_I binomial (uintL n, uintL k)
1999 @cindex @code{binomial ()}
2000 @code{n} and @code{k} must be small integers >= 0. This function returns the
2001 binomial coefficient
2003 ${n \choose k} = {n! \over n! (n-k)!}$
2006 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2008 for 0 <= k <= n, 0 else.
2012 @section Functions on floating-point numbers
2014 Recall that a floating-point number consists of a sign @code{s}, an
2015 exponent @code{e} and a mantissa @code{m}. The value of the number is
2016 @code{(-1)^s * 2^e * m}.
2019 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2020 defines the following operations.
2023 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2024 @cindex @code{scale_float ()}
2025 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2026 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2027 because it copies @code{x} and modifies the exponent.
2030 The following functions provide an abstract interface to the underlying
2031 representation of floating-point numbers.
2034 @item sintL float_exponent (const @var{type}& x)
2035 @cindex @code{float_exponent ()}
2036 Returns the exponent @code{e} of @code{x}.
2037 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2038 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2040 @item sintL float_radix (const @var{type}& x)
2041 @cindex @code{float_radix ()}
2042 Returns the base of the floating-point representation. This is always @code{2}.
2044 @item @var{type} float_sign (const @var{type}& x)
2045 @cindex @code{float_sign ()}
2046 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2047 @code{x} >= 0, -1 for @code{x} < 0.
2049 @item uintL float_digits (const @var{type}& x)
2050 @cindex @code{float_digits ()}
2051 Returns the number of mantissa bits in the floating-point representation
2052 of @code{x}, including the hidden bit. The value only depends on the type
2053 of @code{x}, not on its value.
2055 @item uintL float_precision (const @var{type}& x)
2056 @cindex @code{float_precision ()}
2057 Returns the number of significant mantissa bits in the floating-point
2058 representation of @code{x}. Since denormalized numbers are not supported,
2059 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2063 The complete internal representation of a float is encoded in the type
2064 @cindex @code{decoded_float}
2065 @cindex @code{decoded_sfloat}
2066 @cindex @code{decoded_ffloat}
2067 @cindex @code{decoded_dfloat}
2068 @cindex @code{decoded_lfloat}
2069 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2070 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2072 struct decoded_@var{type}float @{
2073 @var{type} mantissa; cl_I exponent; @var{type} sign;
2077 and returned by the function
2080 @item decoded_@var{type}float decode_float (const @var{type}& x)
2081 @cindex @code{decode_float ()}
2082 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2083 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2084 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2085 @code{e} is the same as returned by the function @code{float_exponent}.
2088 A complete decoding in terms of integers is provided as type
2089 @cindex @code{cl_idecoded_float}
2091 struct cl_idecoded_float @{
2092 cl_I mantissa; cl_I exponent; cl_I sign;
2095 by the following function:
2098 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2099 @cindex @code{integer_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{m} an integer with @code{float_digits(x)}
2102 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2103 WARNING: The exponent @code{e} is not the same as the one returned by
2104 the functions @code{decode_float} and @code{float_exponent}.
2107 Some other function, implemented only for class @code{cl_F}:
2110 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2111 @cindex @code{float_sign ()}
2112 This returns a floating point number whose precision and absolute value
2113 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2114 zero, it is treated as positive. Same for @code{y}.
2118 @section Conversion functions
2121 @subsection Conversion to floating-point numbers
2123 The type @code{float_format_t} describes a floating-point format.
2124 @cindex @code{float_format_t}
2127 @item float_format_t float_format (uintL n)
2128 @cindex @code{float_format ()}
2129 Returns the smallest float format which guarantees at least @code{n}
2130 decimal digits in the mantissa (after the decimal point).
2132 @item float_format_t float_format (const cl_F& x)
2133 Returns the floating point format of @code{x}.
2135 @item float_format_t default_float_format
2136 @cindex @code{default_float_format}
2137 Global variable: the default float format used when converting rational numbers
2141 To convert a real number to a float, each of the types
2142 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2143 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2144 defines the following operations:
2147 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2148 @cindex @code{cl_float ()}
2149 Returns @code{x} as a float of format @code{f}.
2150 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2151 Returns @code{x} in the float format of @code{y}.
2152 @item cl_F cl_float (const @var{type}&x)
2153 Returns @code{x} as a float of format @code{default_float_format} if
2154 it is an exact number, or @code{x} itself if it is already a float.
2157 Of course, converting a number to a float can lose precision.
2159 Every floating-point format has some characteristic numbers:
2162 @item cl_F most_positive_float (float_format_t f)
2163 @cindex @code{most_positive_float ()}
2164 Returns the largest (most positive) floating point number in float format @code{f}.
2166 @item cl_F most_negative_float (float_format_t f)
2167 @cindex @code{most_negative_float ()}
2168 Returns the smallest (most negative) floating point number in float format @code{f}.
2170 @item cl_F least_positive_float (float_format_t f)
2171 @cindex @code{least_positive_float ()}
2172 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2173 in float format @code{f}.
2175 @item cl_F least_negative_float (float_format_t f)
2176 @cindex @code{least_negative_float ()}
2177 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2178 in float format @code{f}.
2180 @item cl_F float_epsilon (float_format_t f)
2181 @cindex @code{float_epsilon ()}
2182 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2184 @item cl_F float_negative_epsilon (float_format_t f)
2185 @cindex @code{float_negative_epsilon ()}
2186 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2190 @subsection Conversion to rational numbers
2192 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2193 defines the following operation:
2196 @item cl_RA rational (const @var{type}& x)
2197 @cindex @code{rational ()}
2198 Returns the value of @code{x} as an exact number. If @code{x} is already
2199 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2200 the value is a rational number whose denominator is a power of 2.
2203 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2207 @item cl_RA rationalize (const cl_R& x)
2208 @cindex @code{rationalize ()}
2209 If @code{x} is a floating-point number, it actually represents an interval
2210 of real numbers, and this function returns the rational number with
2211 smallest denominator (and smallest numerator, in magnitude)
2212 which lies in this interval.
2213 If @code{x} is already an exact number, this function returns @code{x}.
2216 If @code{x} is any float, one has
2220 @code{cl_float(rational(x),x) = x}
2222 @code{cl_float(rationalize(x),x) = x}
2226 @section Random number generators
2229 A random generator is a machine which produces (pseudo-)random numbers.
2230 The include file @code{<cln/random.h>} defines a class @code{random_state}
2231 which contains the state of a random generator. If you make a copy
2232 of the random number generator, the original one and the copy will produce
2233 the same sequence of random numbers.
2235 The following functions return (pseudo-)random numbers in different formats.
2236 Calling one of these modifies the state of the random number generator in
2237 a complicated but deterministic way.
2240 @cindex @code{random_state}
2241 @cindex @code{default_random_state}
2243 random_state default_random_state
2245 contains a default random number generator. It is used when the functions
2246 below are called without @code{random_state} argument.
2249 @item uint32 random32 (random_state& randomstate)
2250 @itemx uint32 random32 ()
2251 @cindex @code{random32 ()}
2252 Returns a random unsigned 32-bit number. All bits are equally random.
2254 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2255 @itemx cl_I random_I (const cl_I& n)
2256 @cindex @code{random_I ()}
2257 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2258 in the range @code{0 <= x < n}.
2260 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2261 @itemx cl_F random_F (const cl_F& n)
2262 @cindex @code{random_F ()}
2263 @code{n} must be a float > 0. This function returns a random floating-point
2264 number of the same format as @code{n} in the range @code{0 <= x < n}.
2266 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2267 @itemx cl_R random_R (const cl_R& n)
2268 @cindex @code{random_R ()}
2269 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2270 if @code{n} is a float.
2274 @section Obfuscating operators
2275 @cindex modifying operators
2277 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2278 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2279 are not available by default because their
2280 use tends to make programs unreadable. It is trivial to get away without
2281 them. However, if you feel that you absolutely need these operators
2282 to get happy, then add
2284 #define WANT_OBFUSCATING_OPERATORS
2286 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2287 to the beginning of your source files, before the inclusion of any CLN
2288 include files. This flag will enable the following operators:
2290 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2291 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2294 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2295 @cindex @code{operator += ()}
2296 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2297 @cindex @code{operator -= ()}
2298 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2299 @cindex @code{operator *= ()}
2300 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2301 @cindex @code{operator /= ()}
2304 For the class @code{cl_I}:
2307 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2308 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2309 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2310 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2311 @cindex @code{operator &= ()}
2312 @itemx @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 classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2323 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2326 @item @var{type}& operator ++ (@var{type}& x)
2327 @cindex @code{operator ++ ()}
2328 The prefix operator @code{++x}.
2330 @item void operator ++ (@var{type}& x, int)
2331 The postfix operator @code{x++}.
2333 @item @var{type}& operator -- (@var{type}& x)
2334 @cindex @code{operator -- ()}
2335 The prefix operator @code{--x}.
2337 @item void operator -- (@var{type}& x, int)
2338 The postfix operator @code{x--}.
2341 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2342 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2346 @chapter Input/Output
2347 @cindex Input/Output
2349 @section Internal and printed representation
2350 @cindex representation
2352 All computations deal with the internal representations of the numbers.
2354 Every number has an external representation as a sequence of ASCII characters.
2355 Several external representations may denote the same number, for example,
2356 "20.0" and "20.000".
2358 Converting an internal to an external representation is called ``printing'',
2360 converting an external to an internal representation is called ``reading''.
2362 In CLN, it is always true that conversion of an internal to an external
2363 representation and then back to an internal representation will yield the
2364 same internal representation. Symbolically: @code{read(print(x)) == x}.
2365 This is called ``print-read consistency''.
2367 Different types of numbers have different external representations (case
2372 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2373 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2374 for decimal integers
2375 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2377 @item Rational numbers
2378 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2379 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2382 @item Floating-point numbers
2383 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2384 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2385 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2386 of the form _@var{prec} may be appended. There must be at least
2387 one digit in the non-exponent part. The exponent has the syntax
2388 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2389 The exponent marker is
2393 @samp{s} for short-floats,
2395 @samp{f} for single-floats,
2397 @samp{d} for double-floats,
2399 @samp{L} for long-floats,
2402 or @samp{e}, which denotes a default float format. The precision specifying
2403 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2404 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2405 function @samp{float_format}.
2407 @item Complex numbers
2408 External representation:
2411 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2412 if @var{imagpart} is negative, its printed representation begins with
2413 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2414 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2415 is rational and the rational number's base is >18, because the @samp{i}
2416 is then read as a digit.
2418 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2423 @section Input functions
2425 Including @code{<cln/io.h>} defines a number of simple input functions
2426 that read from @code{std::istream&}:
2429 @item int freadchar (std::istream& stream)
2430 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2431 if the end of stream was encountered or an error occurred.
2433 @item int funreadchar (std::istream& stream, int c)
2434 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2435 last @code{freadchar} operation on @code{stream}.
2438 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2439 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2440 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2443 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2444 Reads a number from @code{stream} and stores it in the @code{result}.
2447 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2451 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2452 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2453 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2454 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2455 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2456 Reads a number from @code{stream}. The @code{flags} are parameters which
2457 affect the input syntax. Whitespace before the number is silently skipped.
2459 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2460 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2461 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2462 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2463 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2464 Reads a number from a string in memory. The @code{flags} are parameters which
2465 affect the input syntax. The string starts at @code{string} and ends at
2466 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2467 @code{NULL}, denoting the entire string, i.e. equivalent to
2468 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2469 @code{NULL}, the string in memory must contain exactly one number and nothing
2470 more, else a fatal error will be signalled. If @code{end_of_parse}
2471 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2472 the last parsed character (i.e. @code{string_limit} if nothing came after
2473 the number). Whitespace is not allowed.
2476 The structure @code{cl_read_flags} contains the following fields:
2479 @item cl_read_syntax_t syntax
2480 The possible results of the read operation. Possible values are
2481 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2482 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2483 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2485 @item cl_read_lsyntax_t lsyntax
2486 Specifies the language-dependent syntax variant for the read operation.
2490 @item lsyntax_standard
2491 accept standard algebraic notation only, no complex numbers,
2492 @item lsyntax_algebraic
2493 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2494 @item lsyntax_commonlisp
2495 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2496 hexadecimal numbers,
2497 @code{#@var{base}R} for rational numbers in a given base,
2498 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2500 accept all of these extensions.
2503 @item unsigned int rational_base
2504 The base in which rational numbers are read.
2506 @item float_format_t float_flags.default_float_format
2507 The float format used when reading floats with exponent marker @samp{e}.
2509 @item float_format_t float_flags.default_lfloat_format
2510 The float format used when reading floats with exponent marker @samp{l}.
2512 @item cl_boolean float_flags.mantissa_dependent_float_format
2513 When this flag is true, floats specified with more digits than corresponding
2514 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2515 precision corresponding to their number of significant digits.
2519 @section Output functions
2521 Including @code{<cln/io.h>} defines a number of simple output functions
2522 that write to @code{std::ostream&}:
2525 @item void fprintchar (std::ostream& stream, char c)
2526 Prints the character @code{x} literally on the @code{stream}.
2528 @item void fprint (std::ostream& stream, const char * string)
2529 Prints the @code{string} literally on the @code{stream}.
2531 @item void fprintdecimal (std::ostream& stream, int x)
2532 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2533 Prints the integer @code{x} in decimal on the @code{stream}.
2535 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2536 Prints the integer @code{x} in binary (base 2, without prefix)
2537 on the @code{stream}.
2539 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2540 Prints the integer @code{x} in octal (base 8, without prefix)
2541 on the @code{stream}.
2543 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2544 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2545 on the @code{stream}.
2548 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2549 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2550 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2553 @item void fprint (std::ostream& stream, const @var{type}& x)
2554 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2555 Prints the number @code{x} on the @code{stream}. The output may depend
2556 on the global printer settings in the variable @code{default_print_flags}.
2557 The @code{ostream} flags and settings (flags, width and locale) are
2561 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2564 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2566 void print_real (std::ostream& stream, const cl_print_flags& flags,
2568 void print_float (std::ostream& stream, const cl_print_flags& flags,
2570 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2572 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2575 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2576 parameters which affect the output.
2578 The structure type @code{cl_print_flags} contains the following fields:
2581 @item unsigned int rational_base
2582 The base in which rational numbers are printed. Default is @code{10}.
2584 @item cl_boolean rational_readably
2585 If this flag is true, rational numbers are printed with radix specifiers in
2586 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2587 prefixes, trailing dot). Default is false.
2589 @item cl_boolean float_readably
2590 If this flag is true, type specific exponent markers have precedence over 'E'.
2593 @item float_format_t default_float_format
2594 Floating point numbers of this format will be printed using the 'E' exponent
2595 marker. Default is @code{float_format_ffloat}.
2597 @item cl_boolean complex_readably
2598 If this flag is true, complex numbers will be printed using the Common Lisp
2599 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2601 @item cl_string univpoly_varname
2602 Univariate polynomials with no explicit indeterminate name will be printed
2603 using this variable name. Default is @code{"x"}.
2606 The global variable @code{default_print_flags} contains the default values,
2607 used by the function @code{fprint}.
2612 CLN has a class of abstract rings.
2620 Rings can be compared for equality:
2623 @item bool operator== (const cl_ring&, const cl_ring&)
2624 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2625 These compare two rings for equality.
2628 Given a ring @code{R}, the following members can be used.
2631 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2632 @cindex @code{fprint ()}
2633 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2634 @cindex @code{equal ()}
2635 @itemx cl_ring_element R->zero ()
2636 @cindex @code{zero ()}
2637 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2638 @cindex @code{zerop ()}
2639 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2640 @cindex @code{plus ()}
2641 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2642 @cindex @code{minus ()}
2643 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2644 @cindex @code{uminus ()}
2645 @itemx cl_ring_element R->one ()
2646 @cindex @code{one ()}
2647 @itemx cl_ring_element R->canonhom (const cl_I& x)
2648 @cindex @code{canonhom ()}
2649 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2650 @cindex @code{mul ()}
2651 @itemx cl_ring_element R->square (const cl_ring_element& x)
2652 @cindex @code{square ()}
2653 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2654 @cindex @code{expt_pos ()}
2657 The following rings are built-in.
2660 @item cl_null_ring cl_0_ring
2661 The null ring, containing only zero.
2663 @item cl_complex_ring cl_C_ring
2664 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2666 @item cl_real_ring cl_R_ring
2667 The ring of real numbers. This corresponds to the type @code{cl_R}.
2669 @item cl_rational_ring cl_RA_ring
2670 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2672 @item cl_integer_ring cl_I_ring
2673 The ring of integers. This corresponds to the type @code{cl_I}.
2676 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2677 @code{cl_RA_ring}, @code{cl_I_ring}:
2680 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2681 @cindex @code{instanceof ()}
2682 Tests whether the given number is an element of the number ring R.
2686 @chapter Modular integers
2687 @cindex modular integer
2689 @section Modular integer rings
2692 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2693 The modulus is explicitly part of every modular integer. CLN doesn't
2694 allow you to (accidentally) mix elements of different modular rings,
2695 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2696 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2697 doesn't have generic types. So one has to live with runtime checks.)
2699 The class of modular integer rings is
2707 Modular integer ring
2711 @cindex @code{cl_modint_ring}
2713 and the class of all modular integers (elements of modular integer rings) is
2721 Modular integer rings are constructed using the function
2724 @item cl_modint_ring find_modint_ring (const cl_I& N)
2725 @cindex @code{find_modint_ring ()}
2726 This function returns the modular ring @samp{Z/NZ}. It takes care
2727 of finding out about special cases of @code{N}, like powers of two
2728 and odd numbers for which Montgomery multiplication will be a win,
2729 @cindex Montgomery multiplication
2730 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2731 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2732 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2736 Modular integer rings can be compared for equality:
2739 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2740 @cindex @code{operator == ()}
2741 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2742 @cindex @code{operator != ()}
2743 These compare two modular integer rings for equality. Two different calls
2744 to @code{find_modint_ring} with the same argument necessarily return the
2745 same ring because it is memoized in the cache table.
2748 @section Functions on modular integers
2750 Given a modular integer ring @code{R}, the following members can be used.
2753 @item cl_I R->modulus
2754 @cindex @code{modulus}
2755 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2757 @item cl_MI R->zero()
2758 @cindex @code{zero ()}
2759 This returns @code{0 mod N}.
2761 @item cl_MI R->one()
2762 @cindex @code{one ()}
2763 This returns @code{1 mod N}.
2765 @item cl_MI R->canonhom (const cl_I& x)
2766 @cindex @code{canonhom ()}
2767 This returns @code{x mod N}.
2769 @item cl_I R->retract (const cl_MI& x)
2770 @cindex @code{retract ()}
2771 This is a partial inverse function to @code{R->canonhom}. It returns the
2772 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2774 @item cl_MI R->random(random_state& randomstate)
2775 @itemx cl_MI R->random()
2776 @cindex @code{random ()}
2777 This returns a random integer modulo @code{N}.
2780 The following operations are defined on modular integers.
2783 @item cl_modint_ring x.ring ()
2784 @cindex @code{ring ()}
2785 Returns the ring to which the modular integer @code{x} belongs.
2787 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2788 @cindex @code{operator + ()}
2789 Returns the sum of two modular integers. One of the arguments may also
2792 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2793 @cindex @code{operator - ()}
2794 Returns the difference of two modular integers. One of the arguments may also
2797 @item cl_MI operator- (const cl_MI&)
2798 Returns the negative of a modular integer.
2800 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2801 @cindex @code{operator * ()}
2802 Returns the product of two modular integers. One of the arguments may also
2805 @item cl_MI square (const cl_MI&)
2806 @cindex @code{square ()}
2807 Returns the square of a modular integer.
2809 @item cl_MI recip (const cl_MI& x)
2810 @cindex @code{recip ()}
2811 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2812 must be coprime to the modulus, otherwise an error message is issued.
2814 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2815 @cindex @code{div ()}
2816 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2817 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2819 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2820 @cindex @code{expt_pos ()}
2821 @code{y} must be > 0. Returns @code{x^y}.
2823 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2824 @cindex @code{expt ()}
2825 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2826 modulus, else an error message is issued.
2828 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2829 @cindex @code{operator << ()}
2830 Returns @code{x*2^y}.
2832 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2833 @cindex @code{operator >> ()}
2834 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2835 or an error message is issued.
2837 @item bool operator== (const cl_MI&, const cl_MI&)
2838 @cindex @code{operator == ()}
2839 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2840 @cindex @code{operator != ()}
2841 Compares two modular integers, belonging to the same modular integer ring,
2844 @item cl_boolean zerop (const cl_MI& x)
2845 @cindex @code{zerop ()}
2846 Returns true if @code{x} is @code{0 mod N}.
2849 The following output functions are defined (see also the chapter on
2853 @item void fprint (std::ostream& stream, const cl_MI& x)
2854 @cindex @code{fprint ()}
2855 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
2856 @cindex @code{operator << ()}
2857 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2858 on the global printer settings in the variable @code{default_print_flags}.
2862 @chapter Symbolic data types
2863 @cindex symbolic type
2865 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2869 @cindex @code{cl_string}
2879 implements immutable strings.
2881 Strings are constructed through the following constructors:
2884 @item cl_string (const char * s)
2885 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2887 @item cl_string (const char * ptr, unsigned long len)
2888 Returns an immutable copy of the @code{len} characters at
2889 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2892 The following functions are available on strings:
2896 Assignment from @code{cl_string} and @code{const char *}.
2899 @cindex @code{length ()}
2901 @cindex @code{strlen ()}
2902 Returns the length of the string @code{s}.
2905 @cindex @code{operator [] ()}
2906 Returns the @code{i}th character of the string @code{s}.
2907 @code{i} must be in the range @code{0 <= i < s.length()}.
2909 @item bool equal (const cl_string& s1, const cl_string& s2)
2910 @cindex @code{equal ()}
2911 Compares two strings for equality. One of the arguments may also be a
2912 plain @code{const char *}.
2917 @cindex @code{cl_symbol}
2919 Symbols are uniquified strings: all symbols with the same name are shared.
2920 This means that comparison of two symbols is fast (effectively just a pointer
2921 comparison), whereas comparison of two strings must in the worst case walk
2922 both strings until their end.
2923 Symbols are used, for example, as tags for properties, as names of variables
2924 in polynomial rings, etc.
2926 Symbols are constructed through the following constructor:
2929 @item cl_symbol (const cl_string& s)
2930 Looks up or creates a new symbol with a given name.
2933 The following operations are available on symbols:
2936 @item cl_string (const cl_symbol& sym)
2937 Conversion to @code{cl_string}: Returns the string which names the symbol
2940 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2941 @cindex @code{equal ()}
2942 Compares two symbols for equality. This is very fast.
2946 @chapter Univariate polynomials
2948 @cindex univariate polynomial
2950 @section Univariate polynomial rings
2952 CLN implements univariate polynomials (polynomials in one variable) over an
2953 arbitrary ring. The indeterminate variable may be either unnamed (and will be
2954 printed according to @code{default_print_flags.univpoly_varname}, which
2955 defaults to @samp{x}) or carry a given name. The base ring and the
2956 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
2957 (accidentally) mix elements of different polynomial rings, e.g.
2958 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
2959 return a multivariate polynomial, but they are not yet implemented in CLN.)
2961 The classes of univariate polynomial rings are
2969 Univariate polynomial ring
2973 +----------------+-------------------+
2975 Complex polynomial ring | Modular integer polynomial ring
2976 cl_univpoly_complex_ring | cl_univpoly_modint_ring
2977 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
2981 Real polynomial ring |
2982 cl_univpoly_real_ring |
2983 <cln/univpoly_real.h> |
2987 Rational polynomial ring |
2988 cl_univpoly_rational_ring |
2989 <cln/univpoly_rational.h> |
2993 Integer polynomial ring
2994 cl_univpoly_integer_ring
2995 <cln/univpoly_integer.h>
2998 and the corresponding classes of univariate polynomials are
3001 Univariate polynomial
3005 +----------------+-------------------+
3007 Complex polynomial | Modular integer polynomial
3009 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3015 <cln/univpoly_real.h> |
3019 Rational polynomial |
3021 <cln/univpoly_rational.h> |
3027 <cln/univpoly_integer.h>
3030 Univariate polynomial rings are constructed using the functions
3033 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3034 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3035 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3036 @code{R} may be an arbitrary ring. This function takes care of finding out
3037 about special cases of @code{R}, such as the rings of complex numbers,
3038 real numbers, rational numbers, integers, or modular integer rings.
3039 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3040 This ensures that two calls of this function with the same arguments will
3041 return the same polynomial ring.
3043 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3044 @cindex @code{find_univpoly_ring ()}
3045 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3046 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3047 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3048 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3049 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3050 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3051 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3052 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3053 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3054 These functions are equivalent to the general @code{find_univpoly_ring},
3055 only the return type is more specific, according to the base ring's type.
3058 @section Functions on univariate polynomials
3060 Given a univariate polynomial ring @code{R}, the following members can be used.
3063 @item cl_ring R->basering()
3064 @cindex @code{basering ()}
3065 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3067 @item cl_UP R->zero()
3068 @cindex @code{zero ()}
3069 This returns @code{0 in R}, a polynomial of degree -1.
3071 @item cl_UP R->one()
3072 @cindex @code{one ()}
3073 This returns @code{1 in R}, a polynomial of degree <= 0.
3075 @item cl_UP R->canonhom (const cl_I& x)
3076 @cindex @code{canonhom ()}
3077 This returns @code{x in R}, a polynomial of degree <= 0.
3079 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3080 @cindex @code{monomial ()}
3081 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3084 @item cl_UP R->create (sintL degree)
3085 @cindex @code{create ()}
3086 Creates a new polynomial with a given degree. The zero polynomial has degree
3087 @code{-1}. After creating the polynomial, you should put in the coefficients,
3088 using the @code{set_coeff} member function, and then call the @code{finalize}
3092 The following are the only destructive operations on univariate polynomials.
3095 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3096 @cindex @code{set_coeff ()}
3097 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3098 After changing a polynomial and before applying any "normal" operation on it,
3099 you should call its @code{finalize} member function.
3101 @item void finalize (cl_UP& x)
3102 @cindex @code{finalize ()}
3103 This function marks the endpoint of destructive modifications of a polynomial.
3104 It normalizes the internal representation so that subsequent computations have
3105 less overhead. Doing normal computations on unnormalized polynomials may
3106 produce wrong results or crash the program.
3109 The following operations are defined on univariate polynomials.
3112 @item cl_univpoly_ring x.ring ()
3113 @cindex @code{ring ()}
3114 Returns the ring to which the univariate polynomial @code{x} belongs.
3116 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3117 @cindex @code{operator + ()}
3118 Returns the sum of two univariate polynomials.
3120 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3121 @cindex @code{operator - ()}
3122 Returns the difference of two univariate polynomials.
3124 @item cl_UP operator- (const cl_UP&)
3125 Returns the negative of a univariate polynomial.
3127 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3128 @cindex @code{operator * ()}
3129 Returns the product of two univariate polynomials. One of the arguments may
3130 also be a plain integer or an element of the base ring.
3132 @item cl_UP square (const cl_UP&)
3133 @cindex @code{square ()}
3134 Returns the square of a univariate polynomial.
3136 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3137 @cindex @code{expt_pos ()}
3138 @code{y} must be > 0. Returns @code{x^y}.
3140 @item bool operator== (const cl_UP&, const cl_UP&)
3141 @cindex @code{operator == ()}
3142 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3143 @cindex @code{operator != ()}
3144 Compares two univariate polynomials, belonging to the same univariate
3145 polynomial ring, for equality.
3147 @item cl_boolean zerop (const cl_UP& x)
3148 @cindex @code{zerop ()}
3149 Returns true if @code{x} is @code{0 in R}.
3151 @item sintL degree (const cl_UP& x)
3152 @cindex @code{degree ()}
3153 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3155 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3156 @cindex @code{coeff ()}
3157 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3159 @item cl_ring_element x (const cl_ring_element& y)
3160 @cindex @code{operator () ()}
3161 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3162 then @samp{x(y)} returns the value of the substitution of @code{y} into
3165 @item cl_UP deriv (const cl_UP& x)
3166 @cindex @code{deriv ()}
3167 Returns the derivative of the polynomial @code{x} with respect to the
3168 indeterminate @code{X}.
3171 The following output functions are defined (see also the chapter on
3175 @item void fprint (std::ostream& stream, const cl_UP& x)
3176 @cindex @code{fprint ()}
3177 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3178 @cindex @code{operator << ()}
3179 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3180 depend on the global printer settings in the variable
3181 @code{default_print_flags}.
3184 @section Special polynomials
3186 The following functions return special polynomials.
3189 @item cl_UP_I tschebychev (sintL n)
3190 @cindex @code{tschebychev ()}
3191 @cindex Chebyshev polynomial
3192 Returns the n-th Chebyshev polynomial (n >= 0).
3194 @item cl_UP_I hermite (sintL n)
3195 @cindex @code{hermite ()}
3196 @cindex Hermite polynomial
3197 Returns the n-th Hermite polynomial (n >= 0).
3199 @item cl_UP_RA legendre (sintL n)
3200 @cindex @code{legendre ()}
3201 @cindex Legende polynomial
3202 Returns the n-th Legendre polynomial (n >= 0).
3204 @item cl_UP_I laguerre (sintL n)
3205 @cindex @code{laguerre ()}
3206 @cindex Laguerre polynomial
3207 Returns the n-th Laguerre polynomial (n >= 0).
3210 Information how to derive the differential equation satisfied by each
3211 of these polynomials from their definition can be found in the
3212 @code{doc/polynomial/} directory.
3220 Using C++ as an implementation language provides
3224 Efficiency: It compiles to machine code.
3228 Portability: It runs on all platforms supporting a C++ compiler. Because
3229 of the availability of GNU C++, this includes all currently used 32-bit and
3230 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3233 Type safety: The C++ compilers knows about the number types and complains if,
3234 for example, you try to assign a float to an integer variable. However,
3235 a drawback is that C++ doesn't know about generic types, hence a restriction
3236 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3237 arguments belong to the same modular ring cannot be expressed as a compile-time
3241 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3242 @code{=}, @code{==}, ... can be used in infix notation, which is more
3243 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3246 With these language features, there is no need for two separate languages,
3247 one for the implementation of the library and one in which the library's users
3248 can program. This means that a prototype implementation of an algorithm
3249 can be integrated into the library immediately after it has been tested and
3250 debugged. No need to rewrite it in a low-level language after having prototyped
3251 in a high-level language.
3254 @section Memory efficiency
3256 In order to save memory allocations, CLN implements:
3260 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3263 @cindex garbage collection
3264 @cindex reference counting
3265 Garbage collection: A reference counting mechanism makes sure that any
3266 number object's storage is freed immediately when the last reference to the
3269 @cindex immediate numbers
3270 Small integers are represented as immediate values instead of pointers
3271 to heap allocated storage. This means that integers @code{> -2^29},
3272 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3277 @section Speed efficiency
3279 Speed efficiency is obtained by the combination of the following tricks
3284 Small integers, being represented as immediate values, don't require
3285 memory access, just a couple of instructions for each elementary operation.
3287 The kernel of CLN has been written in assembly language for some CPUs
3288 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3290 On all CPUs, CLN may be configured to use the superefficient low-level
3291 routines from GNU GMP version 3.
3293 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3294 algorithm, the Karatsuba multiplication, which is an
3305 For very large numbers (more than 12000 decimal digits), CLN uses
3307 Sch{@"o}nhage-Strassen
3308 @cindex Sch{@"o}nhage-Strassen multiplication
3312 @cindex Schönhage-Strassen multiplication
3314 multiplication, which is an asymptotically optimal multiplication
3317 These fast multiplication algorithms also give improvements in the speed
3318 of division and radix conversion.
3322 @section Garbage collection
3323 @cindex garbage collection
3325 All the number classes are reference count classes: They only contain a pointer
3326 to an object in the heap. Upon construction, assignment and destruction of
3327 number objects, only the objects' reference count are manipulated.
3329 Memory occupied by number objects are automatically reclaimed as soon as
3330 their reference count drops to zero.
3332 For number rings, another strategy is implemented: There is a cache of,
3333 for example, the modular integer rings. A modular integer ring is destroyed
3334 only if its reference count dropped to zero and the cache is about to be
3335 resized. The effect of this strategy is that recently used rings remain
3336 cached, whereas undue memory consumption through cached rings is avoided.
3339 @chapter Using the library
3341 For the following discussion, we will assume that you have installed
3342 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3343 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3344 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3345 environment variables, or directly substitute the appropriate values.
3348 @section Compiler options
3349 @cindex compiler options
3351 Until you have installed CLN in a public place, the following options are
3354 When you compile CLN application code, add the flags
3356 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3358 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3359 When you link CLN application code to form an executable, add the flags
3361 $CLN_TARGETDIR/src/libcln.a
3363 to the C/C++ compiler's command line (@code{make} variable LIBS).
3365 If you did a @code{make install}, the include files are installed in a
3366 public directory (normally @code{/usr/local/include}), hence you don't
3367 need special flags for compiling. The library has been installed to a
3368 public directory as well (normally @code{/usr/local/lib}), hence when
3369 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3371 Since CLN version 1.1, there are two tools to make the creation of
3372 software packages that use CLN easier:
3375 @cindex @code{cln-config}
3376 @code{cln-config} is a shell script that you can use to determine the
3377 compiler and linker command line options required to compile and link a
3378 program with CLN. Start it with @code{--help} to learn about its options
3379 or consult the manpage that comes with it.
3381 @cindex @code{AC_PATH_CLN}
3382 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3385 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3387 This macro determines the location of CLN using @code{cln-config}, which
3388 is either found in the user's path, or from the environment variable
3389 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3390 their version is not earlier than @var{MIN-VERSION} (a default version
3391 will be used if not specified). If the required version was found, sets
3392 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3393 macro is in the file @file{cln.m4} which is installed in
3394 @file{$datadir/aclocal}. Note that if automake was installed with a
3395 different @samp{--prefix} than CLN, you will either have to manually
3396 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3397 aclocal the @samp{-I} option when running it. Here is a possible example
3398 to be included in your package's @file{configure.ac}:
3400 AC_PATH_CLN(1.1.0, [
3401 LIBS="$LIBS $CLN_LIBS"
3402 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3403 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3408 @section Compatibility to old CLN versions
3410 @cindex compatibility
3412 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3413 @code{cln} in order to avoid potential name clashes with other C++
3414 libraries. If you have an old application, you will have to manually
3415 port it to the new scheme. The following principles will help during
3419 All headers are now in a separate subdirectory. Instead of including
3420 @code{cl_}@var{something}@code{.h}, include
3421 @code{cln/}@var{something}@code{.h} now.
3423 All public identifiers (typenames and functions) have lost their
3424 @code{cl_} prefix. Exceptions are all the typenames of number types,
3425 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3426 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3427 names would not be mnemonic enough once the namespace @code{cln} is
3428 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3430 All public @emph{functions} that had by a @code{cl_} in their name still
3431 carry that @code{cl_} if it is intrinsic part of a typename (as in
3432 @code{cl_I_to_int ()}).
3434 When developing other libraries, please keep in mind not to import the
3435 namespace @code{cln} in one of your public header files by saying
3436 @code{using namespace cln;}. This would propagate to other applications
3437 and can cause name clashes there.
3440 @section Include files
3441 @cindex include files
3442 @cindex header files
3444 Here is a summary of the include files and their contents.
3447 @item <cln/object.h>
3448 General definitions, reference counting, garbage collection.
3449 @item <cln/number.h>
3450 The class cl_number.
3451 @item <cln/complex.h>
3452 Functions for class cl_N, the complex numbers.
3454 Functions for class cl_R, the real numbers.
3456 Functions for class cl_F, the floats.
3457 @item <cln/sfloat.h>
3458 Functions for class cl_SF, the short-floats.
3459 @item <cln/ffloat.h>
3460 Functions for class cl_FF, the single-floats.
3461 @item <cln/dfloat.h>
3462 Functions for class cl_DF, the double-floats.
3463 @item <cln/lfloat.h>
3464 Functions for class cl_LF, the long-floats.
3465 @item <cln/rational.h>
3466 Functions for class cl_RA, the rational numbers.
3467 @item <cln/integer.h>
3468 Functions for class cl_I, the integers.
3471 @item <cln/complex_io.h>
3472 Input/Output for class cl_N, the complex numbers.
3473 @item <cln/real_io.h>
3474 Input/Output for class cl_R, the real numbers.
3475 @item <cln/float_io.h>
3476 Input/Output for class cl_F, the floats.
3477 @item <cln/sfloat_io.h>
3478 Input/Output for class cl_SF, the short-floats.
3479 @item <cln/ffloat_io.h>
3480 Input/Output for class cl_FF, the single-floats.
3481 @item <cln/dfloat_io.h>
3482 Input/Output for class cl_DF, the double-floats.
3483 @item <cln/lfloat_io.h>
3484 Input/Output for class cl_LF, the long-floats.
3485 @item <cln/rational_io.h>
3486 Input/Output for class cl_RA, the rational numbers.
3487 @item <cln/integer_io.h>
3488 Input/Output for class cl_I, the integers.
3490 Flags for customizing input operations.
3491 @item <cln/output.h>
3492 Flags for customizing output operations.
3493 @item <cln/malloc.h>
3494 @code{malloc_hook}, @code{free_hook}.
3497 @item <cln/condition.h>
3498 Conditions/exceptions.
3499 @item <cln/string.h>
3501 @item <cln/symbol.h>
3503 @item <cln/proplist.h>
3507 @item <cln/null_ring.h>
3509 @item <cln/complex_ring.h>
3510 The ring of complex numbers.
3511 @item <cln/real_ring.h>
3512 The ring of real numbers.
3513 @item <cln/rational_ring.h>
3514 The ring of rational numbers.
3515 @item <cln/integer_ring.h>
3516 The ring of integers.
3517 @item <cln/numtheory.h>
3518 Number threory functions.
3519 @item <cln/modinteger.h>
3525 @item <cln/GV_number.h>
3526 General vectors over cl_number.
3527 @item <cln/GV_complex.h>
3528 General vectors over cl_N.
3529 @item <cln/GV_real.h>
3530 General vectors over cl_R.
3531 @item <cln/GV_rational.h>
3532 General vectors over cl_RA.
3533 @item <cln/GV_integer.h>
3534 General vectors over cl_I.
3535 @item <cln/GV_modinteger.h>
3536 General vectors of modular integers.
3539 @item <cln/SV_number.h>
3540 Simple vectors over cl_number.
3541 @item <cln/SV_complex.h>
3542 Simple vectors over cl_N.
3543 @item <cln/SV_real.h>
3544 Simple vectors over cl_R.
3545 @item <cln/SV_rational.h>
3546 Simple vectors over cl_RA.
3547 @item <cln/SV_integer.h>
3548 Simple vectors over cl_I.
3549 @item <cln/SV_ringelt.h>
3550 Simple vectors of general ring elements.
3551 @item <cln/univpoly.h>
3552 Univariate polynomials.
3553 @item <cln/univpoly_integer.h>
3554 Univariate polynomials over the integers.
3555 @item <cln/univpoly_rational.h>
3556 Univariate polynomials over the rational numbers.
3557 @item <cln/univpoly_real.h>
3558 Univariate polynomials over the real numbers.
3559 @item <cln/univpoly_complex.h>
3560 Univariate polynomials over the complex numbers.
3561 @item <cln/univpoly_modint.h>
3562 Univariate polynomials over modular integer rings.
3563 @item <cln/timing.h>
3566 Includes all of the above.
3572 A function which computes the nth Fibonacci number can be written as follows.
3573 @cindex Fibonacci number
3576 #include <cln/integer.h>
3577 #include <cln/real.h>
3578 using namespace cln;
3580 // Returns F_n, computed as the nearest integer to
3581 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3582 const cl_I fibonacci (int n)
3584 // Need a precision of ((1+sqrt(5))/2)^-n.
3585 float_format_t prec = float_format((int)(0.208987641*n+5));
3586 cl_R sqrt5 = sqrt(cl_float(5,prec));
3587 cl_R phi = (1+sqrt5)/2;
3588 return round1( expt(phi,n)/sqrt5 );
3592 Let's explain what is going on in detail.
3594 The include file @code{<cln/integer.h>} is necessary because the type
3595 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3596 is needed for the type @code{cl_R} and the floating point number functions.
3597 The order of the include files does not matter. In order not to write
3598 out @code{cln::}@var{foo} in this simple example we can safely import
3599 the whole namespace @code{cln}.
3601 Then comes the function declaration. The argument is an @code{int}, the
3602 result an integer. The return type is defined as @samp{const cl_I}, not
3603 simply @samp{cl_I}, because that allows the compiler to detect typos like
3604 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3605 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3606 number). We use the most specialized possible return type because functions
3607 which call @samp{fibonacci} will be able to profit from the compiler's type
3608 analysis: Adding two integers is slightly more efficient than adding the
3609 same objects declared as complex numbers, because it needs less type
3610 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3611 the size of the resulting executable program.
3613 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3614 integer. In order to get a correct result, the absolute error should be less
3615 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3616 To this end, the first line computes a floating point precision for sqrt(5)
3619 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3620 number and than taking the square root. The converse, first taking the square
3621 root of 5, and then converting to the desired precision, would not work in
3622 CLN: The square root would be computed to a default precision (normally
3623 single-float precision), and the following conversion could not help about
3624 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3625 system and does not represent sqrt(5) in a non-numeric way.
3627 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3628 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3629 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3630 @code{cl_N} because a @samp{round1} does not exist for general complex
3633 When the function returns, all the local variables in the function are
3634 automatically reclaimed (garbage collected). Only the result survives and
3635 gets passed to the caller.
3637 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3638 contains this implementation together with an even faster algorithm.
3640 @section Debugging support
3643 When debugging a CLN application with GNU @code{gdb}, two facilities are
3644 available from the library:
3647 @item The library does type checks, range checks, consistency checks at
3648 many places. When one of these fails, the function @code{cl_abort()} is
3649 called. Its default implementation is to perform an @code{exit(1)}, so
3650 you won't have a core dump. But for debugging, it is best to set a
3651 breakpoint at this function:
3653 (gdb) break cl_abort
3655 When this breakpoint is hit, look at the stack's backtrace:
3660 @item The debugger's normal @code{print} command doesn't know about
3661 CLN's types and therefore prints mostly useless hexadecimal addresses.
3662 CLN offers a function @code{cl_print}, callable from the debugger,
3663 for printing number objects. In order to get this function, you have
3664 to define the macro @samp{CL_DEBUG} and then include all the header files
3665 for which you want @code{cl_print} debugging support. For example:
3666 @cindex @code{CL_DEBUG}
3669 #include <cln/string.h>
3671 Now, if you have in your program a variable @code{cl_string s}, and
3672 inspect it under @code{gdb}, the output may look like this:
3675 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3676 word = 134568800@}@}, @}
3677 (gdb) call cl_print(s)
3681 Note that the output of @code{cl_print} goes to the program's error output,
3682 not to gdb's standard output.
3684 Note, however, that the above facility does not work with all CLN types,
3685 only with number objects and similar. Therefore CLN offers a member function
3686 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3687 is needed for this member function to be implemented. Under @code{gdb},
3688 you call it like this:
3689 @cindex @code{debug_print ()}
3692 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3693 word = 134568800@}@}, @}
3694 (gdb) call s.debug_print()
3697 >call ($1).debug_print()
3702 Unfortunately, this feature does not seem to work under all circumstances.
3706 @chapter Customizing
3709 @section Error handling
3711 When a fatal error occurs, an error message is output to the standard error
3712 output stream, and the function @code{cl_abort} is called. The default
3713 version of this function (provided in the library) terminates the application.
3714 To catch such a fatal error, you need to define the function @code{cl_abort}
3715 yourself, with the prototype
3717 #include <cln/abort.h>
3718 void cl_abort (void);
3720 @cindex @code{cl_abort ()}
3721 This function must not return control to its caller.
3724 @section Floating-point underflow
3727 Floating point underflow denotes the situation when a floating-point number
3728 is to be created which is so close to @code{0} that its exponent is too
3729 low to be represented internally. By default, this causes a fatal error.
3730 If you set the global variable
3732 cl_boolean cl_inhibit_floating_point_underflow
3734 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3735 will be generated instead. The default value of
3736 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3739 @section Customizing I/O
3741 The output of the function @code{fprint} may be customized by changing the
3742 value of the global variable @code{default_print_flags}.
3743 @cindex @code{default_print_flags}
3746 @section Customizing the memory allocator
3748 Every memory allocation of CLN is done through the function pointer
3749 @code{malloc_hook}. Freeing of this memory is done through the function
3750 pointer @code{free_hook}. The default versions of these functions,
3751 provided in the library, call @code{malloc} and @code{free} and check
3752 the @code{malloc} result against @code{NULL}.
3753 If you want to provide another memory allocator, you need to define
3754 the variables @code{malloc_hook} and @code{free_hook} yourself,
3757 #include <cln/malloc.h>
3759 void* (*malloc_hook) (size_t size) = @dots{};
3760 void (*free_hook) (void* ptr) = @dots{};
3763 @cindex @code{malloc_hook ()}
3764 @cindex @code{free_hook ()}
3765 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3767 It is not possible to change the memory allocator at runtime, because
3768 it is already called at program startup by the constructors of some
3781 @c Table of contents