1 \input texinfo @c -*-texinfo-*-
4 @settitle CLN, a Class Library for Numbers
5 @c @setchapternewpage off
6 @c I hate putting "@noindent" in front of every paragraph.
7 @c For `info' and TeX only.
11 @dircategory Mathematics
13 * CLN: (cln). Class Library for Numbers (C++).
18 @c Don't need the other types of indices.
29 This file documents @sc{cln}, a Class Library for Numbers.
31 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
32 Richard B. Kreckel, @code{<kreckel@@ginac.de>}.
34 Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008.
35 Copyright (C) Richard B. Kreckel 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008.
37 Permission is granted to make and distribute verbatim copies of
38 this manual provided the copyright notice and this permission notice
39 are preserved on all copies.
42 Permission is granted to process this file through TeX and print the
43 results, provided the printed document carries copying permission
44 notice identical to this one except for the removal of this paragraph
45 (this paragraph not being relevant to the printed manual).
48 Permission is granted to copy and distribute modified versions of this
49 manual under the conditions for verbatim copying, provided that the entire
50 resulting derived work is distributed under the terms of a permission
51 notice identical to this one.
53 Permission is granted to copy and distribute translations of this manual
54 into another language, under the above conditions for modified versions,
55 except that this permission notice may be stated in a translation approved
61 @c prevent ugly black rectangles on overfull hbox lines:
64 @title CLN, a Class Library for Numbers
66 @author @uref{http://www.ginac.de/CLN}
68 @vskip 0pt plus 1filll
69 Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008.
71 Copyright @copyright{} Richard B. Kreckel 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008.
74 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
75 Richard B. Kreckel, @code{<kreckel@@ginac.de>}.
77 Permission is granted to make and distribute verbatim copies of
78 this manual provided the copyright notice and this permission notice
79 are preserved on all copies.
81 Permission is granted to copy and distribute modified versions of this
82 manual under the conditions for verbatim copying, provided that the entire
83 resulting derived work is distributed under the terms of a permission
84 notice identical to this one.
86 Permission is granted to copy and distribute translations of this manual
87 into another language, under the above conditions for modified versions,
88 except that this permission notice may be stated in a translation approved
108 * Ordinary number types::
109 * Functions on numbers::
113 * Symbolic data types::
114 * Univariate polynomials::
116 * Using the library::
120 --- The Detailed Node Listing ---
125 * Building the library::
126 * Installing the library::
137 * Using the GNU MP Library::
139 Ordinary number types
142 * Floating-point numbers::
148 * Constructing numbers::
149 * Elementary functions::
150 * Elementary rational functions::
151 * Elementary complex functions::
153 * Rounding functions::
155 * Transcendental functions::
156 * Functions on integers::
157 * Functions on floating-point numbers::
158 * Conversion functions::
159 * Random number generators::
160 * Modifying operators::
164 * Constructing integers::
165 * Constructing rational numbers::
166 * Constructing floating-point numbers::
167 * Constructing complex numbers::
169 Transcendental functions
171 * Exponential and logarithmic functions::
172 * Trigonometric functions::
173 * Hyperbolic functions::
177 Functions on integers
179 * Logical functions::
180 * Number theoretic functions::
181 * Combinatorial functions::
185 * Conversion to floating-point numbers::
186 * Conversion to rational numbers::
190 * Internal and printed representation::
196 * Modular integer rings::
197 * Functions on modular integers::
204 Univariate polynomials
206 * Univariate polynomial rings::
207 * Functions on univariate polynomials::
208 * Special polynomials::
213 * Memory efficiency::
215 * Garbage collection::
222 * Debugging support::
223 * Reporting Problems::
228 * Floating-point underflow::
230 * Customizing the memory allocator::
235 @chapter Introduction
238 CLN is a library for computations with all kinds of numbers.
239 It has a rich set of number classes:
243 Integers (with unlimited precision),
249 Floating-point numbers:
259 Long float (with unlimited precision),
266 Modular integers (integers modulo a fixed integer),
269 Univariate polynomials.
273 The subtypes of the complex numbers among these are exactly the
274 types of numbers known to the Common Lisp language. Therefore
275 @code{CLN} can be used for Common Lisp implementations, giving
276 @samp{CLN} another meaning: it becomes an abbreviation of
277 ``Common Lisp Numbers''.
280 The CLN package implements
284 Elementary functions (@code{+}, @code{-}, @code{*}, @code{/}, @code{sqrt},
285 comparisons, @dots{}),
288 Logical functions (logical @code{and}, @code{or}, @code{not}, @dots{}),
291 Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
292 functions and their inverse functions).
296 CLN is a C++ library. Using C++ as an implementation language provides
300 efficiency: it compiles to machine code,
302 type safety: the C++ compiler knows about the number types and complains
303 if, for example, you try to assign a float to an integer variable.
305 algebraic syntax: You can use the @code{+}, @code{-}, @code{*}, @code{=},
306 @code{==}, @dots{} operators as in C or C++.
310 CLN is memory efficient:
314 Small integers and short floats are immediate, not heap allocated.
316 Heap-allocated memory is reclaimed through an automatic, non-interruptive
321 CLN is speed efficient:
325 The kernel of CLN has been written in assembly language for some CPUs
326 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
329 On all CPUs, CLN may be configured to use the superefficient low-level
330 routines from GNU GMP version 3.
332 It uses Karatsuba multiplication, which is significantly faster
333 for large numbers than the standard multiplication algorithm.
335 For very large numbers (more than 12000 decimal digits), it uses
337 Sch{@"o}nhage-Strassen
338 @cindex Sch{@"o}nhage-Strassen multiplication
342 @cindex Schoenhage-Strassen multiplication
344 multiplication, which is an asymptotically optimal multiplication
345 algorithm, for multiplication, division and radix conversion.
347 @cindex binary splitting
348 It uses binary splitting for fast evaluation of series of rational
349 numbers as they occur in the evaluation of elementary functions and some
354 CLN aims at being easily integrated into larger software packages:
358 The garbage collection imposes no burden on the main application.
360 The library provides hooks for memory allocation and throws exceptions
364 All non-macro identifiers are hidden in namespace @code{cln} in
365 order to avoid name clashes.
370 @chapter Installation
372 This section describes how to install the CLN package on your system.
377 * Building the library::
378 * Installing the library::
382 @node Prerequisites, Building the library, Installation, Installation
383 @section Prerequisites
392 @subsection C++ compiler
394 To build CLN, you need a C++ compiler.
395 Actually, you need GNU @code{g++ 3.0.0} or newer.
397 The following C++ features are used:
398 classes, member functions, overloading of functions and operators,
399 constructors and destructors, inline, const, multiple inheritance,
400 templates and namespaces.
402 The following C++ features are not used:
403 @code{new}, @code{delete}, virtual inheritance.
405 CLN relies on semi-automatic ordering of initializations of static and
406 global variables, a feature which I could implement for GNU g++
407 only. Also, it is not known whether this semi-automatic ordering works
408 on all platforms when a non-GNU assembler is being used.
411 @subsection Make utility
414 To build CLN, you also need to have GNU @code{make} installed.
416 Only GNU @code{make} 3.77 is unusable for CLN; other versions work fine.
419 @subsection Sed utility
422 To build CLN on HP-UX, you also need to have GNU @code{sed} installed.
423 This is because the libtool script, which creates the CLN library, relies
424 on @code{sed}, and the vendor's @code{sed} utility on these systems is too
428 @node Building the library
429 @section Building the library
431 As with any autoconfiguring GNU software, installation is as easy as this:
439 If on your system, @samp{make} is not GNU @code{make}, you have to use
440 @samp{gmake} instead of @samp{make} above.
442 The @code{configure} command checks out some features of your system and
443 C++ compiler and builds the @code{Makefile}s. The @code{make} command
444 builds the library. This step may take about half an hour on an average
445 workstation. The @code{make check} runs some test to check that no
446 important subroutine has been miscompiled.
448 The @code{configure} command accepts options. To get a summary of them, try
454 Some of the options are explained in detail in the @samp{INSTALL.generic} file.
456 You can specify the C compiler, the C++ compiler and their options through
457 the following environment variables when running @code{configure}:
461 Specifies the C compiler.
464 Flags to be given to the C compiler when compiling programs (not when linking).
467 Specifies the C++ compiler.
470 Flags to be given to the C++ compiler when compiling programs (not when linking).
473 Flags to be given to the C/C++ preprocessor.
479 $ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
482 $ CC="gcc -V 3.2.3" CFLAGS="-O2 -finline-limit=1000" \
483 CXX="g++ -V 3.2.3" CXXFLAGS="-O2 -finline-limit=1000" \
484 CPPFLAGS="-DNO_ASM" ./configure
487 $ CC="gcc-4.2" CFLAGS="-O2" CXX="g++-4.2" CXXFLAGS="-O2" ./configure
490 Note that for these environment variables to take effect, you have to set
491 them (assuming a Bourne-compatible shell) on the same line as the
492 @code{configure} command. If you made the settings in earlier shell
493 commands, you have to @code{export} the environment variables before
494 calling @code{configure}. In a @code{csh} shell, you have to use the
495 @samp{setenv} command for setting each of the environment variables.
497 Currently CLN works only with the GNU @code{g++} compiler, and only in
498 optimizing mode. So you should specify at least @code{-O} in the
499 CXXFLAGS, or no CXXFLAGS at all. If CXXFLAGS is not set, CLN will be
500 compiled with @code{-O}.
502 The assembler language kernel can be turned off by specifying
503 @code{-DNO_ASM} in the CPPFLAGS. If @code{make check} reports any
504 problems, you may try to clean up (see @ref{Cleaning up}) and configure
505 and compile again, this time with @code{-DNO_ASM}.
507 If you use @code{g++} 3.2.x or earlier, I recommend adding
508 @samp{-finline-limit=1000} to the CXXFLAGS. This is essential for good
511 If you use @code{g++} from gcc-3.0.4 or older on Sparc, add either
512 @samp{-O}, @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the
513 CXXFLAGS. With full @samp{-O2}, @code{g++} miscompiles the division
514 routines. Also, do not use gcc-3.0 on Sparc for compiling CLN, it
517 Also, please do not compile CLN with @code{g++} using the @code{-O3}
518 optimization level. This leads to inferior code quality.
520 Some newer versions of @code{g++} require quite an amount of memory.
521 You might need some swap space if your machine doesn't have 512 MB of
524 By default, both a shared and a static library are built. You can build
525 CLN as a static (or shared) library only, by calling @code{configure}
526 with the option @samp{--disable-shared} (or @samp{--disable-static}).
527 While shared libraries are usually more convenient to use, they may not
528 work on all architectures. Try disabling them if you run into linker
529 problems. Also, they are generally slightly slower than static
530 libraries so runtime-critical applications should be linked statically.
534 * Using the GNU MP Library::
537 @node Using the GNU MP Library
538 @subsection Using the GNU MP Library
541 Starting with version 1.1, CLN may be configured to make use of a
542 preinstalled @code{gmp} library for some low-level routines. Please
543 make sure that you have at least @code{gmp} version 3.0 installed
544 since earlier versions are unsupported and likely not to work. This
545 feature is known to be quite a boost for CLN's performance.
547 By default, CLN will autodetect @code{gmp} and use it. But if you have
548 installed the @code{gmp} library and its header file in some place where
549 your compiler cannot find it by default, you must help @code{configure}
550 by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is an example:
553 $ CFLAGS="-O2" CXXFLAGS="-O2" CPPFLAGS="-I/opt/gmp/include" \
554 LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
557 If you do not want CLN to make use of a preinstalled @code{gmp}
558 library, then you can explicitly specify so by calling
559 @code{configure} with the option @samp{--without-gmp}.
562 @node Installing the library
563 @section Installing the library
566 As with any autoconfiguring GNU software, installation is as easy as this:
572 The @samp{make install} command installs the library and the include files
573 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
574 if you haven't specified a @code{--prefix} option to @code{configure}).
575 This step may require superuser privileges.
577 If you have already built the library and wish to install it, but didn't
578 specify @code{--prefix=@dots{}} at configure time, just re-run
579 @code{configure}, giving it the same options as the first time, plus
580 the @code{--prefix=@dots{}} option.
586 You can remove system-dependent files generated by @code{make} through
592 You can remove all files generated by @code{make}, thus reverting to a
593 virgin distribution of CLN, through
600 @node Ordinary number types
601 @chapter Ordinary number types
603 CLN implements the following class hierarchy:
611 Real or complex number
620 +-------------------+-------------------+
622 Rational number Floating-point number
624 <cln/rational.h> <cln/float.h>
626 | +--------------+--------------+--------------+
628 cl_I Short-Float Single-Float Double-Float Long-Float
629 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
630 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
633 @cindex @code{cl_number}
634 @cindex abstract class
635 The base class @code{cl_number} is an abstract base class.
636 It is not useful to declare a variable of this type except if you want
637 to completely disable compile-time type checking and use run-time type
642 @cindex complex number
643 The class @code{cl_N} comprises real and complex numbers. There is
644 no special class for complex numbers since complex numbers with imaginary
645 part @code{0} are automatically converted to real numbers.
648 The class @code{cl_R} comprises real numbers of different kinds. It is an
652 @cindex rational number
654 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
655 integers. There is no special class for non-integral rational numbers
656 since rational numbers with denominator @code{1} are automatically converted
660 The class @code{cl_F} implements floating-point approximations to real numbers.
661 It is an abstract class.
666 * Floating-point numbers::
672 @section Exact numbers
675 Some numbers are represented as exact numbers: there is no loss of information
676 when such a number is converted from its mathematical value to its internal
677 representation. On exact numbers, the elementary operations (@code{+},
678 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
681 In CLN, the exact numbers are:
685 rational numbers (including integers),
687 complex numbers whose real and imaginary parts are both rational numbers.
690 Rational numbers are always normalized to the form
691 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
692 are coprime integers and the denominator is positive. If the resulting
693 denominator is @code{1}, the rational number is converted to an integer.
695 @cindex immediate numbers
696 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
697 for 32-bit machines) are especially efficient, because they consume no heap
698 allocation. Otherwise the distinction between these immediate integers
699 (called ``fixnums'') and heap allocated integers (called ``bignums'')
700 is completely transparent.
703 @node Floating-point numbers
704 @section Floating-point numbers
705 @cindex floating-point number
707 Not all real numbers can be represented exactly. (There is an easy mathematical
708 proof for this: Only a countable set of numbers can be stored exactly in
709 a computer, even if one assumes that it has unlimited storage. But there
710 are uncountably many real numbers.) So some approximation is needed.
711 CLN implements ordinary floating-point numbers, with mantissa and exponent.
713 @cindex rounding error
714 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
715 only return approximate results. For example, the value of the expression
716 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
717 @samp{0.7}. Rounding errors like this one are inevitable when computing
718 with floating-point numbers.
720 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
721 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
722 rule: It first computes the exact mathematical result and then returns the
723 floating-point number which is nearest to this. If two floating-point numbers
724 are equally distant from the ideal result, the one with a @code{0} in its least
725 significant mantissa bit is chosen.
727 Similarly, testing floating point numbers for equality @samp{x == y}
728 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
729 for some well-chosen @code{epsilon}.
731 Floating point numbers come in four flavors:
736 Short floats, type @code{cl_SF}.
737 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
738 and 17 mantissa bits (including the ``hidden'' bit).
739 They don't consume heap allocation.
743 Single floats, type @code{cl_FF}.
744 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
745 and 24 mantissa bits (including the ``hidden'' bit).
746 In CLN, they are represented as IEEE single-precision floating point numbers.
747 This corresponds closely to the C/C++ type @samp{float}.
751 Double floats, type @code{cl_DF}.
752 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
753 and 53 mantissa bits (including the ``hidden'' bit).
754 In CLN, they are represented as IEEE double-precision floating point numbers.
755 This corresponds closely to the C/C++ type @samp{double}.
759 Long floats, type @code{cl_LF}.
760 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
761 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
762 The precision of a long float is unlimited, but once created, a long float
763 has a fixed precision. (No ``lazy recomputation''.)
766 Of course, computations with long floats are more expensive than those
767 with smaller floating-point formats.
769 CLN does not implement features like NaNs, denormalized numbers and
770 gradual underflow. If the exponent range of some floating-point type
771 is too limited for your application, choose another floating-point type
772 with larger exponent range.
775 As a user of CLN, you can forget about the differences between the
776 four floating-point types and just declare all your floating-point
777 variables as being of type @code{cl_F}. This has the advantage that
778 when you change the precision of some computation (say, from @code{cl_DF}
779 to @code{cl_LF}), you don't have to change the code, only the precision
780 of the initial values. Also, many transcendental functions have been
781 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
782 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
783 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
784 the floating point contagion rule happened to change in the future.)
787 @node Complex numbers
788 @section Complex numbers
789 @cindex complex number
791 Complex numbers, as implemented by the class @code{cl_N}, have a real
792 part and an imaginary part, both real numbers. A complex number whose
793 imaginary part is the exact number @code{0} is automatically converted
796 Complex numbers can arise from real numbers alone, for example
797 through application of @code{sqrt} or transcendental functions.
804 Conversions from any class to any its superclasses (``base classes'' in
805 C++ terminology) is done automatically.
807 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
808 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
809 @code{cl_N} and @code{cl_number}.
811 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
812 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
813 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
814 efficiency. On 32-bit systems, their range is therefore limited:
818 The conversion from @samp{int} works only if the argument is < 2^29 and >= -2^29.
820 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
823 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
824 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
825 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
826 in error on 32-bit machines.
827 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
828 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
829 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
830 @samp{cl_I}, first convert it to an @samp{unsigned long}. On 64-bit machines
831 there is no such restriction. There, conversions from arbitrary 32-bit @samp{int}
832 values always works correctly.
834 Conversions from the C built-in type @samp{float} are provided for the classes
835 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
837 Conversions from the C built-in type @samp{double} are provided for the classes
838 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
840 Conversions from @samp{const char *} are provided for the classes
841 @code{cl_I}, @code{cl_RA},
842 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
843 @code{cl_R}, @code{cl_N}.
844 The easiest way to specify a value which is outside of the range of the
845 C++ built-in types is therefore to specify it as a string, like this:
848 cl_I order_of_rubiks_cube_group = "43252003274489856000";
850 Note that this conversion is done at runtime, not at compile-time.
852 Conversions from @code{cl_I} to the C built-in types @samp{int},
853 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
857 @item int cl_I_to_int (const cl_I& x)
858 @cindex @code{cl_I_to_int ()}
859 @itemx unsigned int cl_I_to_uint (const cl_I& x)
860 @cindex @code{cl_I_to_uint ()}
861 @itemx long cl_I_to_long (const cl_I& x)
862 @cindex @code{cl_I_to_long ()}
863 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
864 @cindex @code{cl_I_to_ulong ()}
865 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
866 representable in the range of @var{ctype}, a runtime error occurs.
869 Conversions from the classes @code{cl_I}, @code{cl_RA},
870 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
872 to the C built-in types @samp{float} and @samp{double} are provided through
876 @item float float_approx (const @var{type}& x)
877 @cindex @code{float_approx ()}
878 @itemx double double_approx (const @var{type}& x)
879 @cindex @code{double_approx ()}
880 Returns an approximation of @code{x} of C type @var{ctype}.
881 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
882 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
885 Conversions from any class to any of its subclasses (``derived classes'' in
886 C++ terminology) are not provided. Instead, you can assert and check
887 that a value belongs to a certain subclass, and return it as element of that
888 class, using the @samp{As} and @samp{The} macros.
890 @cindex @code{As()()}
891 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
892 @var{type} and returns it as such.
893 @cindex @code{The()()}
894 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
895 @var{type} and returns it as such. It is your responsibility to ensure
896 that this assumption is valid. Since macros and namespaces don't go
897 together well, there is an equivalent to @samp{The}: the template
905 if (!(x >= 0)) abort();
906 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
907 // In general, it would be a rational number.
908 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
913 @node Functions on numbers
914 @chapter Functions on numbers
916 Each of the number classes declares its mathematical operations in the
917 corresponding include file. For example, if your code operates with
918 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
922 * Constructing numbers::
923 * Elementary functions::
924 * Elementary rational functions::
925 * Elementary complex functions::
927 * Rounding functions::
929 * Transcendental functions::
930 * Functions on integers::
931 * Functions on floating-point numbers::
932 * Conversion functions::
933 * Random number generators::
934 * Modifying operators::
937 @node Constructing numbers
938 @section Constructing numbers
940 Here is how to create number objects ``from nothing''.
944 * Constructing integers::
945 * Constructing rational numbers::
946 * Constructing floating-point numbers::
947 * Constructing complex numbers::
950 @node Constructing integers
951 @subsection Constructing integers
953 @code{cl_I} objects are most easily constructed from C integers and from
954 strings. See @ref{Conversions}.
957 @node Constructing rational numbers
958 @subsection Constructing rational numbers
960 @code{cl_RA} objects can be constructed from strings. The syntax
961 for rational numbers is described in @ref{Internal and printed representation}.
962 Another standard way to produce a rational number is through application
963 of @samp{operator /} or @samp{recip} on integers.
966 @node Constructing floating-point numbers
967 @subsection Constructing floating-point numbers
969 @code{cl_F} objects with low precision are most easily constructed from
970 C @samp{float} and @samp{double}. See @ref{Conversions}.
972 To construct a @code{cl_F} with high precision, you can use the conversion
973 from @samp{const char *}, but you have to specify the desired precision
974 within the string. (See @ref{Internal and printed representation}.)
977 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
979 will set @samp{e} to the given value, with a precision of 40 decimal digits.
981 The programmatic way to construct a @code{cl_F} with high precision is
982 through the @code{cl_float} conversion function, see
983 @ref{Conversion to floating-point numbers}. For example, to compute
984 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
985 and then apply the exponential function:
987 float_format_t precision = float_format(40);
988 cl_F e = exp(cl_float(1,precision));
992 @node Constructing complex numbers
993 @subsection Constructing complex numbers
995 Non-real @code{cl_N} objects are normally constructed through the function
997 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
999 See @ref{Elementary complex functions}.
1002 @node Elementary functions
1003 @section Elementary functions
1005 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1006 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1007 defines the following operations:
1010 @item @var{type} operator + (const @var{type}&, const @var{type}&)
1011 @cindex @code{operator + ()}
1014 @item @var{type} operator - (const @var{type}&, const @var{type}&)
1015 @cindex @code{operator - ()}
1018 @item @var{type} operator - (const @var{type}&)
1019 Returns the negative of the argument.
1021 @item @var{type} plus1 (const @var{type}& x)
1022 @cindex @code{plus1 ()}
1023 Returns @code{x + 1}.
1025 @item @var{type} minus1 (const @var{type}& x)
1026 @cindex @code{minus1 ()}
1027 Returns @code{x - 1}.
1029 @item @var{type} operator * (const @var{type}&, const @var{type}&)
1030 @cindex @code{operator * ()}
1033 @item @var{type} square (const @var{type}& x)
1034 @cindex @code{square ()}
1035 Returns @code{x * x}.
1038 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
1039 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1040 defines the following operations:
1043 @item @var{type} operator / (const @var{type}&, const @var{type}&)
1044 @cindex @code{operator / ()}
1047 @item @var{type} recip (const @var{type}&)
1048 @cindex @code{recip ()}
1049 Returns the reciprocal of the argument.
1052 The class @code{cl_I} doesn't define a @samp{/} operation because
1053 in the C/C++ language this operator, applied to integral types,
1054 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
1055 is implementation dependent). (@xref{Rounding functions}.)
1056 Instead, @code{cl_I} defines an ``exact quotient'' function:
1059 @item cl_I exquo (const cl_I& x, const cl_I& y)
1060 @cindex @code{exquo ()}
1061 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
1064 The following exponentiation functions are defined:
1067 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
1068 @cindex @code{expt_pos ()}
1069 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
1070 @code{y} must be > 0. Returns @code{x^y}.
1072 @item cl_RA expt (const cl_RA& x, const cl_I& y)
1073 @cindex @code{expt ()}
1074 @itemx cl_R expt (const cl_R& x, const cl_I& y)
1075 @itemx cl_N expt (const cl_N& x, const cl_I& y)
1079 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1080 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1081 defines the following operation:
1084 @item @var{type} abs (const @var{type}& x)
1085 @cindex @code{abs ()}
1086 Returns the absolute value of @code{x}.
1087 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
1090 The class @code{cl_N} implements this as follows:
1093 @item cl_R abs (const cl_N x)
1094 Returns the absolute value of @code{x}.
1097 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1098 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1099 defines the following operation:
1102 @item @var{type} signum (const @var{type}& x)
1103 @cindex @code{signum ()}
1104 Returns the sign of @code{x}, in the same number format as @code{x}.
1105 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
1106 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
1111 @node Elementary rational functions
1112 @section Elementary rational functions
1114 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
1117 @item cl_I numerator (const @var{type}& x)
1118 @cindex @code{numerator ()}
1119 Returns the numerator of @code{x}.
1121 @item cl_I denominator (const @var{type}& x)
1122 @cindex @code{denominator ()}
1123 Returns the denominator of @code{x}.
1126 The numerator and denominator of a rational number are normalized in such
1127 a way that they have no factor in common and the denominator is positive.
1130 @node Elementary complex functions
1131 @section Elementary complex functions
1133 The class @code{cl_N} defines the following operation:
1136 @item cl_N complex (const cl_R& a, const cl_R& b)
1137 @cindex @code{complex ()}
1138 Returns the complex number @code{a+bi}, that is, the complex number with
1139 real part @code{a} and imaginary part @code{b}.
1142 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
1145 @item cl_R realpart (const @var{type}& x)
1146 @cindex @code{realpart ()}
1147 Returns the real part of @code{x}.
1149 @item cl_R imagpart (const @var{type}& x)
1150 @cindex @code{imagpart ()}
1151 Returns the imaginary part of @code{x}.
1153 @item @var{type} conjugate (const @var{type}& x)
1154 @cindex @code{conjugate ()}
1155 Returns the complex conjugate of @code{x}.
1158 We have the relations
1162 @code{x = complex(realpart(x), imagpart(x))}
1164 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
1169 @section Comparisons
1172 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1173 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1174 defines the following operations:
1177 @item bool operator == (const @var{type}&, const @var{type}&)
1178 @cindex @code{operator == ()}
1179 @itemx bool operator != (const @var{type}&, const @var{type}&)
1180 @cindex @code{operator != ()}
1181 Comparison, as in C and C++.
1183 @item uint32 equal_hashcode (const @var{type}&)
1184 @cindex @code{equal_hashcode ()}
1185 Returns a 32-bit hash code that is the same for any two numbers which are
1186 the same according to @code{==}. This hash code depends on the number's value,
1187 not its type or precision.
1189 @item bool zerop (const @var{type}& x)
1190 @cindex @code{zerop ()}
1191 Compare against zero: @code{x == 0}
1194 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1195 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1196 defines the following operations:
1199 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1200 @cindex @code{compare ()}
1201 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1202 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1204 @item bool operator <= (const @var{type}&, const @var{type}&)
1205 @cindex @code{operator <= ()}
1206 @itemx bool operator < (const @var{type}&, const @var{type}&)
1207 @cindex @code{operator < ()}
1208 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1209 @cindex @code{operator >= ()}
1210 @itemx bool operator > (const @var{type}&, const @var{type}&)
1211 @cindex @code{operator > ()}
1212 Comparison, as in C and C++.
1214 @item bool minusp (const @var{type}& x)
1215 @cindex @code{minusp ()}
1216 Compare against zero: @code{x < 0}
1218 @item bool plusp (const @var{type}& x)
1219 @cindex @code{plusp ()}
1220 Compare against zero: @code{x > 0}
1222 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1223 @cindex @code{max ()}
1224 Return the maximum of @code{x} and @code{y}.
1226 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1227 @cindex @code{min ()}
1228 Return the minimum of @code{x} and @code{y}.
1231 When a floating point number and a rational number are compared, the float
1232 is first converted to a rational number using the function @code{rational}.
1233 Since a floating point number actually represents an interval of real numbers,
1234 the result might be surprising.
1235 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1236 there is no floating point number whose value is exactly @code{1/3}.
1239 @node Rounding functions
1240 @section Rounding functions
1243 When a real number is to be converted to an integer, there is no ``best''
1244 rounding. The desired rounding function depends on the application.
1245 The Common Lisp and ISO Lisp standards offer four rounding functions:
1249 This is the largest integer <=@code{x}.
1252 This is the smallest integer >=@code{x}.
1255 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1258 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1259 integers, choose the even one.
1262 These functions have different advantages:
1264 @code{floor} and @code{ceiling} are translation invariant:
1265 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1266 for every @code{x} and every integer @code{n}.
1268 On the other hand, @code{truncate} and @code{round} are symmetric:
1269 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1270 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1271 down exactly as often as it rounds up.
1273 The functions are related like this:
1277 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1278 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1280 @code{truncate(x) = sign(x) * floor(abs(x))}
1283 Each of the classes @code{cl_R}, @code{cl_RA},
1284 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1285 defines the following operations:
1288 @item cl_I floor1 (const @var{type}& x)
1289 @cindex @code{floor1 ()}
1290 Returns @code{floor(x)}.
1291 @item cl_I ceiling1 (const @var{type}& x)
1292 @cindex @code{ceiling1 ()}
1293 Returns @code{ceiling(x)}.
1294 @item cl_I truncate1 (const @var{type}& x)
1295 @cindex @code{truncate1 ()}
1296 Returns @code{truncate(x)}.
1297 @item cl_I round1 (const @var{type}& x)
1298 @cindex @code{round1 ()}
1299 Returns @code{round(x)}.
1302 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1303 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1304 defines the following operations:
1307 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1308 Returns @code{floor(x/y)}.
1309 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1310 Returns @code{ceiling(x/y)}.
1311 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1312 Returns @code{truncate(x/y)}.
1313 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1314 Returns @code{round(x/y)}.
1317 These functions are called @samp{floor1}, @dots{} here instead of
1318 @samp{floor}, @dots{}, because on some systems, system dependent include
1319 files define @samp{floor} and @samp{ceiling} as macros.
1321 In many cases, one needs both the quotient and the remainder of a division.
1322 It is more efficient to compute both at the same time than to perform
1323 two divisions, one for quotient and the next one for the remainder.
1324 The following functions therefore return a structure containing both
1325 the quotient and the remainder. The suffix @samp{2} indicates the number
1326 of ``return values''. The remainder is defined as follows:
1330 for the computation of @code{quotient = floor(x)},
1331 @code{remainder = x - quotient},
1333 for the computation of @code{quotient = floor(x,y)},
1334 @code{remainder = x - quotient*y},
1337 and similarly for the other three operations.
1339 Each of the classes @code{cl_R}, @code{cl_RA},
1340 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1341 defines the following operations:
1344 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1345 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1346 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1347 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1348 @itemx @var{type}_div_t round2 (const @var{type}& x)
1351 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1352 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1353 defines the following operations:
1356 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1357 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1358 @cindex @code{floor2 ()}
1359 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1360 @cindex @code{ceiling2 ()}
1361 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1362 @cindex @code{truncate2 ()}
1363 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1364 @cindex @code{round2 ()}
1367 Sometimes, one wants the quotient as a floating-point number (of the
1368 same format as the argument, if the argument is a float) instead of as
1369 an integer. The prefix @samp{f} indicates this.
1372 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1373 defines the following operations:
1376 @item @var{type} ffloor (const @var{type}& x)
1377 @cindex @code{ffloor ()}
1378 @itemx @var{type} fceiling (const @var{type}& x)
1379 @cindex @code{fceiling ()}
1380 @itemx @var{type} ftruncate (const @var{type}& x)
1381 @cindex @code{ftruncate ()}
1382 @itemx @var{type} fround (const @var{type}& x)
1383 @cindex @code{fround ()}
1386 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1388 The class @code{cl_R} defines the following operations:
1391 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1392 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1393 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1394 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1397 These functions also exist in versions which return both the quotient
1398 and the remainder. The suffix @samp{2} indicates this.
1401 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1402 defines the following operations:
1403 @cindex @code{cl_F_fdiv_t}
1404 @cindex @code{cl_SF_fdiv_t}
1405 @cindex @code{cl_FF_fdiv_t}
1406 @cindex @code{cl_DF_fdiv_t}
1407 @cindex @code{cl_LF_fdiv_t}
1410 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1411 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1412 @cindex @code{ffloor2 ()}
1413 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1414 @cindex @code{fceiling2 ()}
1415 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1416 @cindex @code{ftruncate2 ()}
1417 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1418 @cindex @code{fround2 ()}
1420 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1421 @cindex @code{cl_R_fdiv_t}
1423 The class @code{cl_R} defines the following operations:
1426 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1427 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1428 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1429 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1430 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1433 Other applications need only the remainder of a division.
1434 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1435 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1436 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1440 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1442 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1445 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1446 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1447 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1449 The classes @code{cl_R}, @code{cl_I} define the following operations:
1452 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1453 @cindex @code{mod ()}
1454 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1455 @cindex @code{rem ()}
1462 Each of the classes @code{cl_R},
1463 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1464 defines the following operation:
1467 @item @var{type} sqrt (const @var{type}& x)
1468 @cindex @code{sqrt ()}
1469 @code{x} must be >= 0. This function returns the square root of @code{x},
1470 normalized to be >= 0. If @code{x} is the square of a rational number,
1471 @code{sqrt(x)} will be a rational number, else it will return a
1472 floating-point approximation.
1475 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1478 @item bool sqrtp (const @var{type}& x, @var{type}* root)
1479 @cindex @code{sqrtp ()}
1480 This tests whether @code{x} is a perfect square. If so, it returns true
1481 and the exact square root in @code{*root}, else it returns false.
1484 Furthermore, for integers, similarly:
1487 @item bool isqrt (const @var{type}& x, @var{type}* root)
1488 @cindex @code{isqrt ()}
1489 @code{x} should be >= 0. This function sets @code{*root} to
1490 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1491 the boolean value @code{(expt(*root,2) == x)}.
1494 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1495 define the following operation:
1498 @item bool rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1499 @cindex @code{rootp ()}
1500 @code{x} must be >= 0. @code{n} must be > 0.
1501 This tests whether @code{x} is an @code{n}th power of a rational number.
1502 If so, it returns true and the exact root in @code{*root}, else it returns
1506 The only square root function which accepts negative numbers is the one
1507 for class @code{cl_N}:
1510 @item cl_N sqrt (const cl_N& z)
1511 @cindex @code{sqrt ()}
1512 Returns the square root of @code{z}, as defined by the formula
1513 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1514 or to a complex number are done if necessary. The range of the result is the
1515 right half plane @code{realpart(sqrt(z)) >= 0}
1516 including the positive imaginary axis and 0, but excluding
1517 the negative imaginary axis.
1518 The result is an exact number only if @code{z} is an exact number.
1522 @node Transcendental functions
1523 @section Transcendental functions
1524 @cindex transcendental functions
1526 The transcendental functions return an exact result if the argument
1527 is exact and the result is exact as well. Otherwise they must return
1528 inexact numbers even if the argument is exact.
1529 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1533 * Exponential and logarithmic functions::
1534 * Trigonometric functions::
1535 * Hyperbolic functions::
1540 @node Exponential and logarithmic functions
1541 @subsection Exponential and logarithmic functions
1544 @item cl_R exp (const cl_R& x)
1545 @cindex @code{exp ()}
1546 @itemx cl_N exp (const cl_N& x)
1547 Returns the exponential function of @code{x}. This is @code{e^x} where
1548 @code{e} is the base of the natural logarithms. The range of the result
1549 is the entire complex plane excluding 0.
1551 @item cl_R ln (const cl_R& x)
1552 @cindex @code{ln ()}
1553 @code{x} must be > 0. Returns the (natural) logarithm of x.
1555 @item cl_N log (const cl_N& x)
1556 @cindex @code{log ()}
1557 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1558 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1559 The range of the result is the strip in the complex plane
1560 @code{-pi < imagpart(log(x)) <= pi}.
1562 @item cl_R phase (const cl_N& x)
1563 @cindex @code{phase ()}
1564 Returns the angle part of @code{x} in its polar representation as a
1565 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1566 This is also the imaginary part of @code{log(x)}.
1567 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1568 The result will be an exact number only if @code{zerop(x)} or
1569 if @code{x} is real and positive.
1571 @item cl_R log (const cl_R& a, const cl_R& b)
1572 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1573 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1574 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1577 @item cl_N log (const cl_N& a, const cl_N& b)
1578 Returns the logarithm of @code{a} with respect to base @code{b}.
1579 @code{log(a,b) = log(a)/log(b)}.
1581 @item cl_N expt (const cl_N& x, const cl_N& y)
1582 @cindex @code{expt ()}
1583 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1586 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1589 @item cl_F exp1 (float_format_t f)
1590 @cindex @code{exp1 ()}
1591 Returns e as a float of format @code{f}.
1593 @item cl_F exp1 (const cl_F& y)
1594 Returns e in the float format of @code{y}.
1596 @item cl_F exp1 (void)
1597 Returns e as a float of format @code{default_float_format}.
1601 @node Trigonometric functions
1602 @subsection Trigonometric functions
1605 @item cl_R sin (const cl_R& x)
1606 @cindex @code{sin ()}
1607 Returns @code{sin(x)}. The range of the result is the interval
1608 @code{-1 <= sin(x) <= 1}.
1610 @item cl_N sin (const cl_N& z)
1611 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1613 @item cl_R cos (const cl_R& x)
1614 @cindex @code{cos ()}
1615 Returns @code{cos(x)}. The range of the result is the interval
1616 @code{-1 <= cos(x) <= 1}.
1618 @item cl_N cos (const cl_N& x)
1619 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1621 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1622 @cindex @code{cos_sin_t}
1623 @itemx cos_sin_t cos_sin (const cl_R& x)
1624 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1625 @cindex @code{cos_sin ()}
1626 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1627 hold only approximately.
1629 @item cl_R tan (const cl_R& x)
1630 @cindex @code{tan ()}
1631 @itemx cl_N tan (const cl_N& x)
1632 Returns @code{tan(x) = sin(x)/cos(x)}.
1634 @item cl_N cis (const cl_R& x)
1635 @cindex @code{cis ()}
1636 @itemx cl_N cis (const cl_N& x)
1637 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1638 @code{e^(i*x) = cos(x) + i*sin(x)}.
1641 @cindex @code{asin ()}
1642 @item cl_N asin (const cl_N& z)
1643 Returns @code{arcsin(z)}. This is defined as
1644 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1645 @code{arcsin(-z) = -arcsin(z)}.
1646 The range of the result is the strip in the complex domain
1647 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1648 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1649 with @code{realpart = pi/2} and @code{imagpart > 0}.
1651 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1655 @item cl_N acos (const cl_N& z)
1656 @cindex @code{acos ()}
1657 Returns @code{arccos(z)}. This is defined as
1658 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1661 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1663 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1664 The range of the result is the strip in the complex domain
1665 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1666 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1667 with @code{realpart = pi} and @code{imagpart > 0}.
1669 Proof: This follows from the results about arcsin.
1673 @cindex @code{atan ()}
1674 @item cl_R atan (const cl_R& x, const cl_R& y)
1675 Returns the angle of the polar representation of the complex number
1676 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1677 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1678 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1679 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1680 with reversed order of arguments.
1682 @item cl_R atan (const cl_R& x)
1683 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1684 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1685 will be an exact number only if @code{x} is the exact @code{0}.
1687 @item cl_N atan (const cl_N& z)
1688 Returns @code{arctan(z)}. This is defined as
1689 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1690 @code{arctan(-z) = -arctan(z)}. The range of the result is
1691 the strip in the complex domain
1692 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1693 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1694 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1696 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1702 @cindex Archimedes' constant
1703 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1706 @item cl_F pi (float_format_t f)
1707 @cindex @code{pi ()}
1708 Returns pi as a float of format @code{f}.
1710 @item cl_F pi (const cl_F& y)
1711 Returns pi in the float format of @code{y}.
1713 @item cl_F pi (void)
1714 Returns pi as a float of format @code{default_float_format}.
1718 @node Hyperbolic functions
1719 @subsection Hyperbolic functions
1722 @item cl_R sinh (const cl_R& x)
1723 @cindex @code{sinh ()}
1724 Returns @code{sinh(x)}.
1726 @item cl_N sinh (const cl_N& z)
1727 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1729 @item cl_R cosh (const cl_R& x)
1730 @cindex @code{cosh ()}
1731 Returns @code{cosh(x)}. The range of the result is the interval
1732 @code{cosh(x) >= 1}.
1734 @item cl_N cosh (const cl_N& z)
1735 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1737 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1738 @cindex @code{cosh_sinh_t}
1739 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1740 @cindex @code{cosh_sinh ()}
1741 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1742 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1743 hold only approximately.
1745 @item cl_R tanh (const cl_R& x)
1746 @cindex @code{tanh ()}
1747 @itemx cl_N tanh (const cl_N& x)
1748 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1750 @item cl_N asinh (const cl_N& z)
1751 @cindex @code{asinh ()}
1752 Returns @code{arsinh(z)}. This is defined as
1753 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1754 @code{arsinh(-z) = -arsinh(z)}.
1756 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1757 Actually, z+sqrt(1+z^2) can never be real and <0, so
1758 -pi < imagpart(arsinh(z)) < pi.
1759 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1760 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1762 The range of the result is the strip in the complex domain
1763 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1764 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1765 with @code{imagpart = pi/2} and @code{realpart < 0}.
1767 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1768 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1769 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1770 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1771 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1772 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1773 If y >= 1, the imagpart is pi/2 and the realpart is
1774 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1777 Moreover, if z is in Range(sqrt),
1778 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1779 (for a proof, see file src/cl_C_asinh.cc).
1782 @item cl_N acosh (const cl_N& z)
1783 @cindex @code{acosh ()}
1784 Returns @code{arcosh(z)}. This is defined as
1785 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1786 The range of the result is the half-strip in the complex domain
1787 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1788 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1790 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1791 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1792 If z is in Range(sqrt), we have
1793 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1794 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1796 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1797 and since the imagpart of both expressions is > -pi, <= pi
1798 ==> arcosh(z) = log(z+sqrt(z^2-1))
1799 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1800 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1801 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1802 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1803 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1805 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1806 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1807 = x^2 + y^2 + sqrt(u^2+v^2)
1812 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1813 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1814 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1815 Otherwise, -z is in Range(sqrt).
1816 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1817 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1818 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1819 and this has realpart > 0.
1820 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1821 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1822 has realpart = 0 and imagpart > 0.
1823 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1824 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1825 This has realpart >= 0 and imagpart = pi.
1828 @item cl_N atanh (const cl_N& z)
1829 @cindex @code{atanh ()}
1830 Returns @code{artanh(z)}. This is defined as
1831 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1832 @code{artanh(-z) = -artanh(z)}. The range of the result is
1833 the strip in the complex domain
1834 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1835 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1836 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1838 Proof: Write z = x+iy. Examine
1839 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1841 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1842 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1843 |x| < 1 ==> imagpart = 0
1846 = (atan(1+x,y) - atan(1-x,-y))/2
1847 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1848 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1849 > (pi - pi/2 - pi/2 )/2 = 0
1850 and (1+x)/y > (1-x)/y
1851 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1852 ==> imagpart < pi/2.
1853 Hence 0 < imagpart < pi/2.
1855 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1861 @subsection Euler gamma
1862 @cindex Euler's constant
1864 Euler's constant C = 0.577@dots{} is returned by the following functions:
1867 @item cl_F eulerconst (float_format_t f)
1868 @cindex @code{eulerconst ()}
1869 Returns Euler's constant as a float of format @code{f}.
1871 @item cl_F eulerconst (const cl_F& y)
1872 Returns Euler's constant in the float format of @code{y}.
1874 @item cl_F eulerconst (void)
1875 Returns Euler's constant as a float of format @code{default_float_format}.
1878 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1879 @cindex Catalan's constant
1882 @item cl_F catalanconst (float_format_t f)
1883 @cindex @code{catalanconst ()}
1884 Returns Catalan's constant as a float of format @code{f}.
1886 @item cl_F catalanconst (const cl_F& y)
1887 Returns Catalan's constant in the float format of @code{y}.
1889 @item cl_F catalanconst (void)
1890 Returns Catalan's constant as a float of format @code{default_float_format}.
1895 @subsection Riemann zeta
1896 @cindex Riemann's zeta
1898 Riemann's zeta function at an integral point @code{s>1} is returned by the
1899 following functions:
1902 @item cl_F zeta (int s, float_format_t f)
1903 @cindex @code{zeta ()}
1904 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1906 @item cl_F zeta (int s, const cl_F& y)
1907 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1909 @item cl_F zeta (int s)
1910 Returns Riemann's zeta function at @code{s} as a float of format
1911 @code{default_float_format}.
1915 @node Functions on integers
1916 @section Functions on integers
1919 * Logical functions::
1920 * Number theoretic functions::
1921 * Combinatorial functions::
1924 @node Logical functions
1925 @subsection Logical functions
1927 Integers, when viewed as in two's complement notation, can be thought as
1928 infinite bit strings where the bits' values eventually are constant.
1935 The logical operations view integers as such bit strings and operate
1936 on each of the bit positions in parallel.
1939 @item cl_I lognot (const cl_I& x)
1940 @cindex @code{lognot ()}
1941 @itemx cl_I operator ~ (const cl_I& x)
1942 @cindex @code{operator ~ ()}
1943 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1945 @item cl_I logand (const cl_I& x, const cl_I& y)
1946 @cindex @code{logand ()}
1947 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1948 @cindex @code{operator & ()}
1949 Logical and, like @code{x & y} in C.
1951 @item cl_I logior (const cl_I& x, const cl_I& y)
1952 @cindex @code{logior ()}
1953 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1954 @cindex @code{operator | ()}
1955 Logical (inclusive) or, like @code{x | y} in C.
1957 @item cl_I logxor (const cl_I& x, const cl_I& y)
1958 @cindex @code{logxor ()}
1959 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1960 @cindex @code{operator ^ ()}
1961 Exclusive or, like @code{x ^ y} in C.
1963 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1964 @cindex @code{logeqv ()}
1965 Bitwise equivalence, like @code{~(x ^ y)} in C.
1967 @item cl_I lognand (const cl_I& x, const cl_I& y)
1968 @cindex @code{lognand ()}
1969 Bitwise not and, like @code{~(x & y)} in C.
1971 @item cl_I lognor (const cl_I& x, const cl_I& y)
1972 @cindex @code{lognor ()}
1973 Bitwise not or, like @code{~(x | y)} in C.
1975 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1976 @cindex @code{logandc1 ()}
1977 Logical and, complementing the first argument, like @code{~x & y} in C.
1979 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1980 @cindex @code{logandc2 ()}
1981 Logical and, complementing the second argument, like @code{x & ~y} in C.
1983 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1984 @cindex @code{logorc1 ()}
1985 Logical or, complementing the first argument, like @code{~x | y} in C.
1987 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1988 @cindex @code{logorc2 ()}
1989 Logical or, complementing the second argument, like @code{x | ~y} in C.
1992 These operations are all available though the function
1994 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1995 @cindex @code{boole ()}
1997 where @code{op} must have one of the 16 values (each one stands for a function
1998 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1999 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
2000 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
2001 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
2002 @code{boole_orc1}, @code{boole_orc2}.
2003 @cindex @code{boole_clr}
2004 @cindex @code{boole_set}
2005 @cindex @code{boole_1}
2006 @cindex @code{boole_2}
2007 @cindex @code{boole_c1}
2008 @cindex @code{boole_c2}
2009 @cindex @code{boole_and}
2010 @cindex @code{boole_xor}
2011 @cindex @code{boole_eqv}
2012 @cindex @code{boole_nand}
2013 @cindex @code{boole_nor}
2014 @cindex @code{boole_andc1}
2015 @cindex @code{boole_andc2}
2016 @cindex @code{boole_orc1}
2017 @cindex @code{boole_orc2}
2020 Other functions that view integers as bit strings:
2023 @item bool logtest (const cl_I& x, const cl_I& y)
2024 @cindex @code{logtest ()}
2025 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
2026 @code{logand(x,y) != 0}.
2028 @item bool logbitp (const cl_I& n, const cl_I& x)
2029 @cindex @code{logbitp ()}
2030 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
2031 Bit 0 is the least significant bit.
2033 @item uintC logcount (const cl_I& x)
2034 @cindex @code{logcount ()}
2035 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
2036 the number of zero bits in @code{x}, if @code{x} < 0.
2039 The following functions operate on intervals of bits in integers.
2042 struct cl_byte @{ uintC size; uintC position; @};
2044 @cindex @code{cl_byte}
2045 represents the bit interval containing the bits
2046 @code{position}@dots{}@code{position+size-1} of an integer.
2047 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
2050 @item cl_I ldb (const cl_I& n, const cl_byte& b)
2051 @cindex @code{ldb ()}
2052 extracts the bits of @code{n} described by the bit interval @code{b}
2053 and returns them as a nonnegative integer with @code{b.size} bits.
2055 @item bool ldb_test (const cl_I& n, const cl_byte& b)
2056 @cindex @code{ldb_test ()}
2057 Returns true if some bit described by the bit interval @code{b} is set in
2060 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
2061 @cindex @code{dpb ()}
2062 Returns @code{n}, with the bits described by the bit interval @code{b}
2063 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
2064 @code{newbyte} are relevant.
2067 The functions @code{ldb} and @code{dpb} implicitly shift. The following
2068 functions are their counterparts without shifting:
2071 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
2072 @cindex @code{mask_field ()}
2073 returns an integer with the bits described by the bit interval @code{b}
2074 copied from the corresponding bits in @code{n}, the other bits zero.
2076 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
2077 @cindex @code{deposit_field ()}
2078 returns an integer where the bits described by the bit interval @code{b}
2079 come from @code{newbyte} and the other bits come from @code{n}.
2082 The following relations hold:
2086 @code{ldb (n, b) = mask_field(n, b) >> b.position},
2088 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
2090 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
2093 The following operations on integers as bit strings are efficient shortcuts
2094 for common arithmetic operations:
2097 @item bool oddp (const cl_I& x)
2098 @cindex @code{oddp ()}
2099 Returns true if the least significant bit of @code{x} is 1. Equivalent to
2100 @code{mod(x,2) != 0}.
2102 @item bool evenp (const cl_I& x)
2103 @cindex @code{evenp ()}
2104 Returns true if the least significant bit of @code{x} is 0. Equivalent to
2105 @code{mod(x,2) == 0}.
2107 @item cl_I operator << (const cl_I& x, const cl_I& n)
2108 @cindex @code{operator << ()}
2109 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
2110 Equivalent to @code{x * expt(2,n)}.
2112 @item cl_I operator >> (const cl_I& x, const cl_I& n)
2113 @cindex @code{operator >> ()}
2114 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
2115 Bits shifted out to the right are thrown away.
2116 Equivalent to @code{floor(x / expt(2,n))}.
2118 @item cl_I ash (const cl_I& x, const cl_I& y)
2119 @cindex @code{ash ()}
2120 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
2121 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
2122 returns @code{floor(x * expt(2,y))}.
2124 @item uintC integer_length (const cl_I& x)
2125 @cindex @code{integer_length ()}
2126 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
2127 in two's complement notation. This is the smallest n >= 0 such that
2128 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
2131 @item uintC ord2 (const cl_I& x)
2132 @cindex @code{ord2 ()}
2133 @code{x} must be non-zero. This function returns the number of 0 bits at the
2134 right of @code{x} in two's complement notation. This is the largest n >= 0
2135 such that 2^n divides @code{x}.
2137 @item uintC power2p (const cl_I& x)
2138 @cindex @code{power2p ()}
2139 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
2140 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
2141 (See also the function @code{logp}.)
2145 @node Number theoretic functions
2146 @subsection Number theoretic functions
2149 @item uint32 gcd (unsigned long a, unsigned long b)
2150 @cindex @code{gcd ()}
2151 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
2152 This function returns the greatest common divisor of @code{a} and @code{b},
2153 normalized to be >= 0.
2155 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
2156 @cindex @code{xgcd ()}
2157 This function (``extended gcd'') returns the greatest common divisor @code{g} of
2158 @code{a} and @code{b} and at the same time the representation of @code{g}
2159 as an integral linear combination of @code{a} and @code{b}:
2160 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
2161 @code{u} and @code{v} will be normalized to be of smallest possible absolute
2162 value, in the following sense: If @code{a} and @code{b} are non-zero, and
2163 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
2164 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
2166 @item cl_I lcm (const cl_I& a, const cl_I& b)
2167 @cindex @code{lcm ()}
2168 This function returns the least common multiple of @code{a} and @code{b},
2169 normalized to be >= 0.
2171 @item bool logp (const cl_I& a, const cl_I& b, cl_RA* l)
2172 @cindex @code{logp ()}
2173 @itemx bool logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
2174 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
2175 rational number, this function returns true and sets *l = log(a,b), else
2178 @item int jacobi (signed long a, signed long b)
2179 @cindex @code{jacobi()}
2180 @itemx int jacobi (const cl_I& a, const cl_I& b)
2181 Returns the Jacobi symbol
2183 $\left({a\over b}\right)$,
2188 @code{a,b} must be integers, @code{b>0} and odd. The result is 0
2191 @item bool isprobprime (const cl_I& n)
2193 @cindex @code{isprobprime()}
2194 Returns true if @code{n} is a small prime or passes the Miller-Rabin
2195 primality test. The probability of a false positive is 1:10^30.
2197 @item cl_I nextprobprime (const cl_R& x)
2198 @cindex @code{nextprobprime()}
2199 Returns the smallest probable prime >=@code{x}.
2203 @node Combinatorial functions
2204 @subsection Combinatorial functions
2207 @item cl_I factorial (uintL n)
2208 @cindex @code{factorial ()}
2209 @code{n} must be a small integer >= 0. This function returns the factorial
2210 @code{n}! = @code{1*2*@dots{}*n}.
2212 @item cl_I doublefactorial (uintL n)
2213 @cindex @code{doublefactorial ()}
2214 @code{n} must be a small integer >= 0. This function returns the
2215 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
2216 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
2218 @item cl_I binomial (uintL n, uintL k)
2219 @cindex @code{binomial ()}
2220 @code{n} and @code{k} must be small integers >= 0. This function returns the
2221 binomial coefficient
2223 ${n \choose k} = {n! \over n! (n-k)!}$
2226 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2228 for 0 <= k <= n, 0 else.
2232 @node Functions on floating-point numbers
2233 @section Functions on floating-point numbers
2235 Recall that a floating-point number consists of a sign @code{s}, an
2236 exponent @code{e} and a mantissa @code{m}. The value of the number is
2237 @code{(-1)^s * 2^e * m}.
2240 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2241 defines the following operations.
2244 @item @var{type} scale_float (const @var{type}& x, sintC delta)
2245 @cindex @code{scale_float ()}
2246 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2247 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2248 because it copies @code{x} and modifies the exponent.
2251 The following functions provide an abstract interface to the underlying
2252 representation of floating-point numbers.
2255 @item sintE float_exponent (const @var{type}& x)
2256 @cindex @code{float_exponent ()}
2257 Returns the exponent @code{e} of @code{x}.
2258 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2259 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2261 @item sintL float_radix (const @var{type}& x)
2262 @cindex @code{float_radix ()}
2263 Returns the base of the floating-point representation. This is always @code{2}.
2265 @item @var{type} float_sign (const @var{type}& x)
2266 @cindex @code{float_sign ()}
2267 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2268 @code{x} >= 0, -1 for @code{x} < 0.
2270 @item uintC float_digits (const @var{type}& x)
2271 @cindex @code{float_digits ()}
2272 Returns the number of mantissa bits in the floating-point representation
2273 of @code{x}, including the hidden bit. The value only depends on the type
2274 of @code{x}, not on its value.
2276 @item uintC float_precision (const @var{type}& x)
2277 @cindex @code{float_precision ()}
2278 Returns the number of significant mantissa bits in the floating-point
2279 representation of @code{x}. Since denormalized numbers are not supported,
2280 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2284 The complete internal representation of a float is encoded in the type
2285 @cindex @code{decoded_float}
2286 @cindex @code{decoded_sfloat}
2287 @cindex @code{decoded_ffloat}
2288 @cindex @code{decoded_dfloat}
2289 @cindex @code{decoded_lfloat}
2290 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2291 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2293 struct decoded_@var{type}float @{
2294 @var{type} mantissa; cl_I exponent; @var{type} sign;
2298 and returned by the function
2301 @item decoded_@var{type}float decode_float (const @var{type}& x)
2302 @cindex @code{decode_float ()}
2303 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2304 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2305 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2306 @code{e} is the same as returned by the function @code{float_exponent}.
2309 A complete decoding in terms of integers is provided as type
2310 @cindex @code{cl_idecoded_float}
2312 struct cl_idecoded_float @{
2313 cl_I mantissa; cl_I exponent; cl_I sign;
2316 by the following function:
2319 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2320 @cindex @code{integer_decode_float ()}
2321 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2322 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2323 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2324 WARNING: The exponent @code{e} is not the same as the one returned by
2325 the functions @code{decode_float} and @code{float_exponent}.
2328 Some other function, implemented only for class @code{cl_F}:
2331 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2332 @cindex @code{float_sign ()}
2333 This returns a floating point number whose precision and absolute value
2334 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2335 zero, it is treated as positive. Same for @code{y}.
2339 @node Conversion functions
2340 @section Conversion functions
2344 * Conversion to floating-point numbers::
2345 * Conversion to rational numbers::
2348 @node Conversion to floating-point numbers
2349 @subsection Conversion to floating-point numbers
2351 The type @code{float_format_t} describes a floating-point format.
2352 @cindex @code{float_format_t}
2355 @item float_format_t float_format (uintE n)
2356 @cindex @code{float_format ()}
2357 Returns the smallest float format which guarantees at least @code{n}
2358 decimal digits in the mantissa (after the decimal point).
2360 @item float_format_t float_format (const cl_F& x)
2361 Returns the floating point format of @code{x}.
2363 @item float_format_t default_float_format
2364 @cindex @code{default_float_format}
2365 Global variable: the default float format used when converting rational numbers
2369 To convert a real number to a float, each of the types
2370 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2371 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2372 defines the following operations:
2375 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2376 @cindex @code{cl_float ()}
2377 Returns @code{x} as a float of format @code{f}.
2378 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2379 Returns @code{x} in the float format of @code{y}.
2380 @item cl_F cl_float (const @var{type}&x)
2381 Returns @code{x} as a float of format @code{default_float_format} if
2382 it is an exact number, or @code{x} itself if it is already a float.
2385 Of course, converting a number to a float can lose precision.
2387 Every floating-point format has some characteristic numbers:
2390 @item cl_F most_positive_float (float_format_t f)
2391 @cindex @code{most_positive_float ()}
2392 Returns the largest (most positive) floating point number in float format @code{f}.
2394 @item cl_F most_negative_float (float_format_t f)
2395 @cindex @code{most_negative_float ()}
2396 Returns the smallest (most negative) floating point number in float format @code{f}.
2398 @item cl_F least_positive_float (float_format_t f)
2399 @cindex @code{least_positive_float ()}
2400 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2401 in float format @code{f}.
2403 @item cl_F least_negative_float (float_format_t f)
2404 @cindex @code{least_negative_float ()}
2405 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2406 in float format @code{f}.
2408 @item cl_F float_epsilon (float_format_t f)
2409 @cindex @code{float_epsilon ()}
2410 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2412 @item cl_F float_negative_epsilon (float_format_t f)
2413 @cindex @code{float_negative_epsilon ()}
2414 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2418 @node Conversion to rational numbers
2419 @subsection Conversion to rational numbers
2421 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2422 defines the following operation:
2425 @item cl_RA rational (const @var{type}& x)
2426 @cindex @code{rational ()}
2427 Returns the value of @code{x} as an exact number. If @code{x} is already
2428 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2429 the value is a rational number whose denominator is a power of 2.
2432 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2436 @item cl_RA rationalize (const cl_R& x)
2437 @cindex @code{rationalize ()}
2438 If @code{x} is a floating-point number, it actually represents an interval
2439 of real numbers, and this function returns the rational number with
2440 smallest denominator (and smallest numerator, in magnitude)
2441 which lies in this interval.
2442 If @code{x} is already an exact number, this function returns @code{x}.
2445 If @code{x} is any float, one has
2449 @code{cl_float(rational(x),x) = x}
2451 @code{cl_float(rationalize(x),x) = x}
2455 @node Random number generators
2456 @section Random number generators
2459 A random generator is a machine which produces (pseudo-)random numbers.
2460 The include file @code{<cln/random.h>} defines a class @code{random_state}
2461 which contains the state of a random generator. If you make a copy
2462 of the random number generator, the original one and the copy will produce
2463 the same sequence of random numbers.
2465 The following functions return (pseudo-)random numbers in different formats.
2466 Calling one of these modifies the state of the random number generator in
2467 a complicated but deterministic way.
2470 @cindex @code{random_state}
2471 @cindex @code{default_random_state}
2473 random_state default_random_state
2475 contains a default random number generator. It is used when the functions
2476 below are called without @code{random_state} argument.
2479 @item uint32 random32 (random_state& randomstate)
2480 @itemx uint32 random32 ()
2481 @cindex @code{random32 ()}
2482 Returns a random unsigned 32-bit number. All bits are equally random.
2484 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2485 @itemx cl_I random_I (const cl_I& n)
2486 @cindex @code{random_I ()}
2487 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2488 in the range @code{0 <= x < n}.
2490 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2491 @itemx cl_F random_F (const cl_F& n)
2492 @cindex @code{random_F ()}
2493 @code{n} must be a float > 0. This function returns a random floating-point
2494 number of the same format as @code{n} in the range @code{0 <= x < n}.
2496 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2497 @itemx cl_R random_R (const cl_R& n)
2498 @cindex @code{random_R ()}
2499 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2500 if @code{n} is a float.
2504 @node Modifying operators
2505 @section Modifying operators
2506 @cindex modifying operators
2508 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2509 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2512 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2513 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2516 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2517 @cindex @code{operator += ()}
2518 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2519 @cindex @code{operator -= ()}
2520 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2521 @cindex @code{operator *= ()}
2522 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2523 @cindex @code{operator /= ()}
2526 For the class @code{cl_I}:
2529 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2530 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2531 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2532 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2533 @cindex @code{operator &= ()}
2534 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2535 @cindex @code{operator |= ()}
2536 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2537 @cindex @code{operator ^= ()}
2538 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2539 @cindex @code{operator <<= ()}
2540 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2541 @cindex @code{operator >>= ()}
2544 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2545 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2548 @item @var{type}& operator ++ (@var{type}& x)
2549 @cindex @code{operator ++ ()}
2550 The prefix operator @code{++x}.
2552 @item void operator ++ (@var{type}& x, int)
2553 The postfix operator @code{x++}.
2555 @item @var{type}& operator -- (@var{type}& x)
2556 @cindex @code{operator -- ()}
2557 The prefix operator @code{--x}.
2559 @item void operator -- (@var{type}& x, int)
2560 The postfix operator @code{x--}.
2563 Note that by using these modifying operators, you don't gain efficiency:
2564 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2569 @chapter Input/Output
2570 @cindex Input/Output
2573 * Internal and printed representation::
2575 * Output functions::
2578 @node Internal and printed representation
2579 @section Internal and printed representation
2580 @cindex representation
2582 All computations deal with the internal representations of the numbers.
2584 Every number has an external representation as a sequence of ASCII characters.
2585 Several external representations may denote the same number, for example,
2586 "20.0" and "20.000".
2588 Converting an internal to an external representation is called ``printing'',
2590 converting an external to an internal representation is called ``reading''.
2592 In CLN, it is always true that conversion of an internal to an external
2593 representation and then back to an internal representation will yield the
2594 same internal representation. Symbolically: @code{read(print(x)) == x}.
2595 This is called ``print-read consistency''.
2597 Different types of numbers have different external representations (case
2602 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2603 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2604 for decimal integers
2605 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2607 @item Rational numbers
2608 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2609 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2612 @item Floating-point numbers
2613 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2614 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2615 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2616 of the form _@var{prec} may be appended. There must be at least
2617 one digit in the non-exponent part. The exponent has the syntax
2618 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2619 The exponent marker is
2623 @samp{s} for short-floats,
2625 @samp{f} for single-floats,
2627 @samp{d} for double-floats,
2629 @samp{L} for long-floats,
2632 or @samp{e}, which denotes a default float format. The precision specifying
2633 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2634 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2635 function @samp{float_format}.
2637 @item Complex numbers
2638 External representation:
2641 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2642 if @var{imagpart} is negative, its printed representation begins with
2643 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2644 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2645 is rational and the rational number's base is >18, because the @samp{i}
2646 is then read as a digit.
2648 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2653 @node Input functions
2654 @section Input functions
2656 Including @code{<cln/io.h>} defines a number of simple input functions
2657 that read from @code{std::istream&}:
2660 @item int freadchar (std::istream& stream)
2661 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2662 if the end of stream was encountered or an error occurred.
2664 @item int funreadchar (std::istream& stream, int c)
2665 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2666 last @code{freadchar} operation on @code{stream}.
2669 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2670 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2671 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2674 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2675 Reads a number from @code{stream} and stores it in the @code{result}.
2678 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2682 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2683 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2684 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2685 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2686 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2687 Reads a number from @code{stream}. The @code{flags} are parameters which
2688 affect the input syntax. Whitespace before the number is silently skipped.
2690 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2691 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2692 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2693 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2694 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2695 Reads a number from a string in memory. The @code{flags} are parameters which
2696 affect the input syntax. The string starts at @code{string} and ends at
2697 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2698 @code{NULL}, denoting the entire string, i.e. equivalent to
2699 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2700 @code{NULL}, the string in memory must contain exactly one number and nothing
2701 more, else an exception will be thrown. If @code{end_of_parse}
2702 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2703 the last parsed character (i.e. @code{string_limit} if nothing came after
2704 the number). Whitespace is not allowed.
2707 The structure @code{cl_read_flags} contains the following fields:
2710 @item cl_read_syntax_t syntax
2711 The possible results of the read operation. Possible values are
2712 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2713 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2714 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2716 @item cl_read_lsyntax_t lsyntax
2717 Specifies the language-dependent syntax variant for the read operation.
2721 @item lsyntax_standard
2722 accept standard algebraic notation only, no complex numbers,
2723 @item lsyntax_algebraic
2724 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2725 @item lsyntax_commonlisp
2726 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2727 hexadecimal numbers,
2728 @code{#@var{base}R} for rational numbers in a given base,
2729 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2731 accept all of these extensions.
2734 @item unsigned int rational_base
2735 The base in which rational numbers are read.
2737 @item float_format_t float_flags.default_float_format
2738 The float format used when reading floats with exponent marker @samp{e}.
2740 @item float_format_t float_flags.default_lfloat_format
2741 The float format used when reading floats with exponent marker @samp{l}.
2743 @item bool float_flags.mantissa_dependent_float_format
2744 When this flag is true, floats specified with more digits than corresponding
2745 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2746 precision corresponding to their number of significant digits.
2750 @node Output functions
2751 @section Output functions
2753 Including @code{<cln/io.h>} defines a number of simple output functions
2754 that write to @code{std::ostream&}:
2757 @item void fprintchar (std::ostream& stream, char c)
2758 Prints the character @code{x} literally on the @code{stream}.
2760 @item void fprint (std::ostream& stream, const char * string)
2761 Prints the @code{string} literally on the @code{stream}.
2763 @item void fprintdecimal (std::ostream& stream, int x)
2764 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2765 Prints the integer @code{x} in decimal on the @code{stream}.
2767 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2768 Prints the integer @code{x} in binary (base 2, without prefix)
2769 on the @code{stream}.
2771 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2772 Prints the integer @code{x} in octal (base 8, without prefix)
2773 on the @code{stream}.
2775 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2776 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2777 on the @code{stream}.
2780 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2781 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2782 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2785 @item void fprint (std::ostream& stream, const @var{type}& x)
2786 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2787 Prints the number @code{x} on the @code{stream}. The output may depend
2788 on the global printer settings in the variable @code{default_print_flags}.
2789 The @code{ostream} flags and settings (flags, width and locale) are
2793 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2796 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2798 void print_real (std::ostream& stream, const cl_print_flags& flags,
2800 void print_float (std::ostream& stream, const cl_print_flags& flags,
2802 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2804 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2807 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2808 parameters which affect the output.
2810 The structure type @code{cl_print_flags} contains the following fields:
2813 @item unsigned int rational_base
2814 The base in which rational numbers are printed. Default is @code{10}.
2816 @item bool rational_readably
2817 If this flag is true, rational numbers are printed with radix specifiers in
2818 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2819 prefixes, trailing dot). Default is false.
2821 @item bool float_readably
2822 If this flag is true, type specific exponent markers have precedence over 'E'.
2825 @item float_format_t default_float_format
2826 Floating point numbers of this format will be printed using the 'E' exponent
2827 marker. Default is @code{float_format_ffloat}.
2829 @item bool complex_readably
2830 If this flag is true, complex numbers will be printed using the Common Lisp
2831 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2833 @item cl_string univpoly_varname
2834 Univariate polynomials with no explicit indeterminate name will be printed
2835 using this variable name. Default is @code{"x"}.
2838 The global variable @code{default_print_flags} contains the default values,
2839 used by the function @code{fprint}.
2845 CLN has a class of abstract rings.
2853 Rings can be compared for equality:
2856 @item bool operator== (const cl_ring&, const cl_ring&)
2857 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2858 These compare two rings for equality.
2861 Given a ring @code{R}, the following members can be used.
2864 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2865 @cindex @code{fprint ()}
2866 @itemx bool R->equal (const cl_ring_element& x, const cl_ring_element& y)
2867 @cindex @code{equal ()}
2868 @itemx cl_ring_element R->zero ()
2869 @cindex @code{zero ()}
2870 @itemx bool R->zerop (const cl_ring_element& x)
2871 @cindex @code{zerop ()}
2872 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2873 @cindex @code{plus ()}
2874 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2875 @cindex @code{minus ()}
2876 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2877 @cindex @code{uminus ()}
2878 @itemx cl_ring_element R->one ()
2879 @cindex @code{one ()}
2880 @itemx cl_ring_element R->canonhom (const cl_I& x)
2881 @cindex @code{canonhom ()}
2882 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2883 @cindex @code{mul ()}
2884 @itemx cl_ring_element R->square (const cl_ring_element& x)
2885 @cindex @code{square ()}
2886 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2887 @cindex @code{expt_pos ()}
2890 The following rings are built-in.
2893 @item cl_null_ring cl_0_ring
2894 The null ring, containing only zero.
2896 @item cl_complex_ring cl_C_ring
2897 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2899 @item cl_real_ring cl_R_ring
2900 The ring of real numbers. This corresponds to the type @code{cl_R}.
2902 @item cl_rational_ring cl_RA_ring
2903 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2905 @item cl_integer_ring cl_I_ring
2906 The ring of integers. This corresponds to the type @code{cl_I}.
2909 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2910 @code{cl_RA_ring}, @code{cl_I_ring}:
2913 @item bool instanceof (const cl_number& x, const cl_number_ring& R)
2914 @cindex @code{instanceof ()}
2915 Tests whether the given number is an element of the number ring R.
2919 @node Modular integers
2920 @chapter Modular integers
2921 @cindex modular integer
2924 * Modular integer rings::
2925 * Functions on modular integers::
2928 @node Modular integer rings
2929 @section Modular integer rings
2932 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2933 The modulus is explicitly part of every modular integer. CLN doesn't
2934 allow you to (accidentally) mix elements of different modular rings,
2935 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2936 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2937 doesn't have generic types. So one has to live with runtime checks.)
2939 The class of modular integer rings is
2947 Modular integer ring
2951 @cindex @code{cl_modint_ring}
2953 and the class of all modular integers (elements of modular integer rings) is
2961 Modular integer rings are constructed using the function
2964 @item cl_modint_ring find_modint_ring (const cl_I& N)
2965 @cindex @code{find_modint_ring ()}
2966 This function returns the modular ring @samp{Z/NZ}. It takes care
2967 of finding out about special cases of @code{N}, like powers of two
2968 and odd numbers for which Montgomery multiplication will be a win,
2969 @cindex Montgomery multiplication
2970 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2971 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2972 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2976 Modular integer rings can be compared for equality:
2979 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2980 @cindex @code{operator == ()}
2981 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2982 @cindex @code{operator != ()}
2983 These compare two modular integer rings for equality. Two different calls
2984 to @code{find_modint_ring} with the same argument necessarily return the
2985 same ring because it is memoized in the cache table.
2988 @node Functions on modular integers
2989 @section Functions on modular integers
2991 Given a modular integer ring @code{R}, the following members can be used.
2994 @item cl_I R->modulus
2995 @cindex @code{modulus}
2996 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2998 @item cl_MI R->zero()
2999 @cindex @code{zero ()}
3000 This returns @code{0 mod N}.
3002 @item cl_MI R->one()
3003 @cindex @code{one ()}
3004 This returns @code{1 mod N}.
3006 @item cl_MI R->canonhom (const cl_I& x)
3007 @cindex @code{canonhom ()}
3008 This returns @code{x mod N}.
3010 @item cl_I R->retract (const cl_MI& x)
3011 @cindex @code{retract ()}
3012 This is a partial inverse function to @code{R->canonhom}. It returns the
3013 standard representative (@code{>=0}, @code{<N}) of @code{x}.
3015 @item cl_MI R->random(random_state& randomstate)
3016 @itemx cl_MI R->random()
3017 @cindex @code{random ()}
3018 This returns a random integer modulo @code{N}.
3021 The following operations are defined on modular integers.
3024 @item cl_modint_ring x.ring ()
3025 @cindex @code{ring ()}
3026 Returns the ring to which the modular integer @code{x} belongs.
3028 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
3029 @cindex @code{operator + ()}
3030 Returns the sum of two modular integers. One of the arguments may also
3033 @item cl_MI operator- (const cl_MI&, const cl_MI&)
3034 @cindex @code{operator - ()}
3035 Returns the difference of two modular integers. One of the arguments may also
3038 @item cl_MI operator- (const cl_MI&)
3039 Returns the negative of a modular integer.
3041 @item cl_MI operator* (const cl_MI&, const cl_MI&)
3042 @cindex @code{operator * ()}
3043 Returns the product of two modular integers. One of the arguments may also
3046 @item cl_MI square (const cl_MI&)
3047 @cindex @code{square ()}
3048 Returns the square of a modular integer.
3050 @item cl_MI recip (const cl_MI& x)
3051 @cindex @code{recip ()}
3052 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
3053 must be coprime to the modulus, otherwise an error message is issued.
3055 @item cl_MI div (const cl_MI& x, const cl_MI& y)
3056 @cindex @code{div ()}
3057 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
3058 @code{y} must be coprime to the modulus, otherwise an error message is issued.
3060 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
3061 @cindex @code{expt_pos ()}
3062 @code{y} must be > 0. Returns @code{x^y}.
3064 @item cl_MI expt (const cl_MI& x, const cl_I& y)
3065 @cindex @code{expt ()}
3066 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
3067 modulus, else an error message is issued.
3069 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
3070 @cindex @code{operator << ()}
3071 Returns @code{x*2^y}.
3073 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
3074 @cindex @code{operator >> ()}
3075 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
3076 or an error message is issued.
3078 @item bool operator== (const cl_MI&, const cl_MI&)
3079 @cindex @code{operator == ()}
3080 @itemx bool operator!= (const cl_MI&, const cl_MI&)
3081 @cindex @code{operator != ()}
3082 Compares two modular integers, belonging to the same modular integer ring,
3085 @item bool zerop (const cl_MI& x)
3086 @cindex @code{zerop ()}
3087 Returns true if @code{x} is @code{0 mod N}.
3090 The following output functions are defined (see also the chapter on
3094 @item void fprint (std::ostream& stream, const cl_MI& x)
3095 @cindex @code{fprint ()}
3096 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
3097 @cindex @code{operator << ()}
3098 Prints the modular integer @code{x} on the @code{stream}. The output may depend
3099 on the global printer settings in the variable @code{default_print_flags}.
3103 @node Symbolic data types
3104 @chapter Symbolic data types
3105 @cindex symbolic type
3107 CLN implements two symbolic (non-numeric) data types: strings and symbols.
3117 @cindex @code{cl_string}
3127 implements immutable strings.
3129 Strings are constructed through the following constructors:
3132 @item cl_string (const char * s)
3133 Returns an immutable copy of the (zero-terminated) C string @code{s}.
3135 @item cl_string (const char * ptr, unsigned long len)
3136 Returns an immutable copy of the @code{len} characters at
3137 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
3140 The following functions are available on strings:
3144 Assignment from @code{cl_string} and @code{const char *}.
3147 @cindex @code{length ()}
3149 @cindex @code{strlen ()}
3150 Returns the length of the string @code{s}.
3153 @cindex @code{operator [] ()}
3154 Returns the @code{i}th character of the string @code{s}.
3155 @code{i} must be in the range @code{0 <= i < s.length()}.
3157 @item bool equal (const cl_string& s1, const cl_string& s2)
3158 @cindex @code{equal ()}
3159 Compares two strings for equality. One of the arguments may also be a
3160 plain @code{const char *}.
3166 @cindex @code{cl_symbol}
3168 Symbols are uniquified strings: all symbols with the same name are shared.
3169 This means that comparison of two symbols is fast (effectively just a pointer
3170 comparison), whereas comparison of two strings must in the worst case walk
3171 both strings until their end.
3172 Symbols are used, for example, as tags for properties, as names of variables
3173 in polynomial rings, etc.
3175 Symbols are constructed through the following constructor:
3178 @item cl_symbol (const cl_string& s)
3179 Looks up or creates a new symbol with a given name.
3182 The following operations are available on symbols:
3185 @item cl_string (const cl_symbol& sym)
3186 Conversion to @code{cl_string}: Returns the string which names the symbol
3189 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
3190 @cindex @code{equal ()}
3191 Compares two symbols for equality. This is very fast.
3195 @node Univariate polynomials
3196 @chapter Univariate polynomials
3198 @cindex univariate polynomial
3201 * Univariate polynomial rings::
3202 * Functions on univariate polynomials::
3203 * Special polynomials::
3206 @node Univariate polynomial rings
3207 @section Univariate polynomial rings
3209 CLN implements univariate polynomials (polynomials in one variable) over an
3210 arbitrary ring. The indeterminate variable may be either unnamed (and will be
3211 printed according to @code{default_print_flags.univpoly_varname}, which
3212 defaults to @samp{x}) or carry a given name. The base ring and the
3213 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
3214 (accidentally) mix elements of different polynomial rings, e.g.
3215 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
3216 return a multivariate polynomial, but they are not yet implemented in CLN.)
3218 The classes of univariate polynomial rings are
3226 Univariate polynomial ring
3230 +----------------+-------------------+
3232 Complex polynomial ring | Modular integer polynomial ring
3233 cl_univpoly_complex_ring | cl_univpoly_modint_ring
3234 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3238 Real polynomial ring |
3239 cl_univpoly_real_ring |
3240 <cln/univpoly_real.h> |
3244 Rational polynomial ring |
3245 cl_univpoly_rational_ring |
3246 <cln/univpoly_rational.h> |
3250 Integer polynomial ring
3251 cl_univpoly_integer_ring
3252 <cln/univpoly_integer.h>
3255 and the corresponding classes of univariate polynomials are
3258 Univariate polynomial
3262 +----------------+-------------------+
3264 Complex polynomial | Modular integer polynomial
3266 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3272 <cln/univpoly_real.h> |
3276 Rational polynomial |
3278 <cln/univpoly_rational.h> |
3284 <cln/univpoly_integer.h>
3287 Univariate polynomial rings are constructed using the functions
3290 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3291 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3292 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3293 @code{R} may be an arbitrary ring. This function takes care of finding out
3294 about special cases of @code{R}, such as the rings of complex numbers,
3295 real numbers, rational numbers, integers, or modular integer rings.
3296 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3297 This ensures that two calls of this function with the same arguments will
3298 return the same polynomial ring.
3300 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3301 @cindex @code{find_univpoly_ring ()}
3302 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3303 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3304 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3305 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3306 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3307 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3308 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3309 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3310 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3311 These functions are equivalent to the general @code{find_univpoly_ring},
3312 only the return type is more specific, according to the base ring's type.
3315 @node Functions on univariate polynomials
3316 @section Functions on univariate polynomials
3318 Given a univariate polynomial ring @code{R}, the following members can be used.
3321 @item cl_ring R->basering()
3322 @cindex @code{basering ()}
3323 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3325 @item cl_UP R->zero()
3326 @cindex @code{zero ()}
3327 This returns @code{0 in R}, a polynomial of degree -1.
3329 @item cl_UP R->one()
3330 @cindex @code{one ()}
3331 This returns @code{1 in R}, a polynomial of degree == 0.
3333 @item cl_UP R->canonhom (const cl_I& x)
3334 @cindex @code{canonhom ()}
3335 This returns @code{x in R}, a polynomial of degree <= 0.
3337 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3338 @cindex @code{monomial ()}
3339 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3342 @item cl_UP R->create (sintL degree)
3343 @cindex @code{create ()}
3344 Creates a new polynomial with a given degree. The zero polynomial has degree
3345 @code{-1}. After creating the polynomial, you should put in the coefficients,
3346 using the @code{set_coeff} member function, and then call the @code{finalize}
3350 The following are the only destructive operations on univariate polynomials.
3353 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3354 @cindex @code{set_coeff ()}
3355 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3356 After changing a polynomial and before applying any "normal" operation on it,
3357 you should call its @code{finalize} member function.
3359 @item void finalize (cl_UP& x)
3360 @cindex @code{finalize ()}
3361 This function marks the endpoint of destructive modifications of a polynomial.
3362 It normalizes the internal representation so that subsequent computations have
3363 less overhead. Doing normal computations on unnormalized polynomials may
3364 produce wrong results or crash the program.
3367 The following operations are defined on univariate polynomials.
3370 @item cl_univpoly_ring x.ring ()
3371 @cindex @code{ring ()}
3372 Returns the ring to which the univariate polynomial @code{x} belongs.
3374 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3375 @cindex @code{operator + ()}
3376 Returns the sum of two univariate polynomials.
3378 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3379 @cindex @code{operator - ()}
3380 Returns the difference of two univariate polynomials.
3382 @item cl_UP operator- (const cl_UP&)
3383 Returns the negative of a univariate polynomial.
3385 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3386 @cindex @code{operator * ()}
3387 Returns the product of two univariate polynomials. One of the arguments may
3388 also be a plain integer or an element of the base ring.
3390 @item cl_UP square (const cl_UP&)
3391 @cindex @code{square ()}
3392 Returns the square of a univariate polynomial.
3394 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3395 @cindex @code{expt_pos ()}
3396 @code{y} must be > 0. Returns @code{x^y}.
3398 @item bool operator== (const cl_UP&, const cl_UP&)
3399 @cindex @code{operator == ()}
3400 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3401 @cindex @code{operator != ()}
3402 Compares two univariate polynomials, belonging to the same univariate
3403 polynomial ring, for equality.
3405 @item bool zerop (const cl_UP& x)
3406 @cindex @code{zerop ()}
3407 Returns true if @code{x} is @code{0 in R}.
3409 @item sintL degree (const cl_UP& x)
3410 @cindex @code{degree ()}
3411 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3413 @item sintL ldegree (const cl_UP& x)
3414 @cindex @code{degree ()}
3415 Returns the low degree of the polynomial. This is the degree of the first
3416 non-vanishing polynomial coefficient. The zero polynomial has ldegree @code{-1}.
3418 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3419 @cindex @code{coeff ()}
3420 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3422 @item cl_ring_element x (const cl_ring_element& y)
3423 @cindex @code{operator () ()}
3424 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3425 then @samp{x(y)} returns the value of the substitution of @code{y} into
3428 @item cl_UP deriv (const cl_UP& x)
3429 @cindex @code{deriv ()}
3430 Returns the derivative of the polynomial @code{x} with respect to the
3431 indeterminate @code{X}.
3434 The following output functions are defined (see also the chapter on
3438 @item void fprint (std::ostream& stream, const cl_UP& x)
3439 @cindex @code{fprint ()}
3440 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3441 @cindex @code{operator << ()}
3442 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3443 depend on the global printer settings in the variable
3444 @code{default_print_flags}.
3447 @node Special polynomials
3448 @section Special polynomials
3450 The following functions return special polynomials.
3453 @item cl_UP_I tschebychev (sintL n)
3454 @cindex @code{tschebychev ()}
3455 @cindex Chebyshev polynomial
3456 Returns the n-th Chebyshev polynomial (n >= 0).
3458 @item cl_UP_I hermite (sintL n)
3459 @cindex @code{hermite ()}
3460 @cindex Hermite polynomial
3461 Returns the n-th Hermite polynomial (n >= 0).
3463 @item cl_UP_RA legendre (sintL n)
3464 @cindex @code{legendre ()}
3465 @cindex Legende polynomial
3466 Returns the n-th Legendre polynomial (n >= 0).
3468 @item cl_UP_I laguerre (sintL n)
3469 @cindex @code{laguerre ()}
3470 @cindex Laguerre polynomial
3471 Returns the n-th Laguerre polynomial (n >= 0).
3474 Information how to derive the differential equation satisfied by each
3475 of these polynomials from their definition can be found in the
3476 @code{doc/polynomial/} directory.
3484 * Memory efficiency::
3485 * Speed efficiency::
3486 * Garbage collection::
3493 Using C++ as an implementation language provides
3497 Efficiency: It compiles to machine code.
3501 Portability: It runs on all platforms supporting a C++ compiler. Because
3502 of the availability of GNU C++, this includes all currently used 32-bit and
3503 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3506 Type safety: The C++ compilers knows about the number types and complains if,
3507 for example, you try to assign a float to an integer variable. However,
3508 a drawback is that C++ doesn't know about generic types, hence a restriction
3509 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3510 arguments belong to the same modular ring cannot be expressed as a compile-time
3514 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3515 @code{=}, @code{==}, ... can be used in infix notation, which is more
3516 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3519 With these language features, there is no need for two separate languages,
3520 one for the implementation of the library and one in which the library's users
3521 can program. This means that a prototype implementation of an algorithm
3522 can be integrated into the library immediately after it has been tested and
3523 debugged. No need to rewrite it in a low-level language after having prototyped
3524 in a high-level language.
3527 @node Memory efficiency
3528 @section Memory efficiency
3530 In order to save memory allocations, CLN implements:
3534 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3537 @cindex garbage collection
3538 @cindex reference counting
3539 Garbage collection: A reference counting mechanism makes sure that any
3540 number object's storage is freed immediately when the last reference to the
3543 @cindex immediate numbers
3544 Small integers are represented as immediate values instead of pointers
3545 to heap allocated storage. This means that integers @code{>= -2^29},
3546 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3551 @node Speed efficiency
3552 @section Speed efficiency
3554 Speed efficiency is obtained by the combination of the following tricks
3559 Small integers, being represented as immediate values, don't require
3560 memory access, just a couple of instructions for each elementary operation.
3562 The kernel of CLN has been written in assembly language for some CPUs
3563 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3565 On all CPUs, CLN may be configured to use the superefficient low-level
3566 routines from GNU GMP version 3.
3568 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3569 algorithm, the Karatsuba multiplication, which is an
3580 For very large numbers (more than 12000 decimal digits), CLN uses
3582 Sch{@"o}nhage-Strassen
3583 @cindex Sch{@"o}nhage-Strassen multiplication
3587 @cindex Schoenhage-Strassen multiplication
3589 multiplication, which is an asymptotically optimal multiplication
3592 These fast multiplication algorithms also give improvements in the speed
3593 of division and radix conversion.
3597 @node Garbage collection
3598 @section Garbage collection
3599 @cindex garbage collection
3601 All the number classes are reference count classes: They only contain a pointer
3602 to an object in the heap. Upon construction, assignment and destruction of
3603 number objects, only the objects' reference count are manipulated.
3605 Memory occupied by number objects are automatically reclaimed as soon as
3606 their reference count drops to zero.
3608 For number rings, another strategy is implemented: There is a cache of,
3609 for example, the modular integer rings. A modular integer ring is destroyed
3610 only if its reference count dropped to zero and the cache is about to be
3611 resized. The effect of this strategy is that recently used rings remain
3612 cached, whereas undue memory consumption through cached rings is avoided.
3615 @node Using the library
3616 @chapter Using the library
3618 For the following discussion, we will assume that you have installed
3619 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3620 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3621 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3622 environment variables, or directly substitute the appropriate values.
3626 * Compiler options::
3629 * Debugging support::
3630 * Reporting Problems::
3633 @node Compiler options
3634 @section Compiler options
3635 @cindex compiler options
3637 Until you have installed CLN in a public place, the following options are
3640 When you compile CLN application code, add the flags
3642 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3644 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3645 When you link CLN application code to form an executable, add the flags
3647 $CLN_TARGETDIR/src/libcln.a
3649 to the C/C++ compiler's command line (@code{make} variable LIBS).
3651 If you did a @code{make install}, the include files are installed in a
3652 public directory (normally @code{/usr/local/include}), hence you don't
3653 need special flags for compiling. The library has been installed to a
3654 public directory as well (normally @code{/usr/local/lib}), hence when
3655 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3657 @cindex @code{pkg-config}
3658 To make the creation of software packages that use CLN easier, the
3659 @code{pkg-config} utility can be used. CLN provides all the necessary
3660 metainformation in a file called @code{cln.pc} (installed in
3661 @code{/usr/local/lib/pkgconfig} by default). A program using CLN can
3662 be compiled and linked using @footnote{If you installed CLN to
3663 non-standard location @var{prefix}, you need to set the
3664 @env{PKG_CONFIG_PATH} environment variable to @var{prefix}/lib/pkgconfig
3667 g++ `pkg-config --libs cln` `pkg-config --cflags cln` prog.cc -o prog
3670 Software using GNU autoconf can check for CLN with the
3671 @code{PKG_CHECK_MODULES} macro supplied with @code{pkg-config}.
3673 PKG_CHECK_MODULES([CLN], [cln >= @var{MIN-VERSION}])
3675 This will check for CLN version at least @var{MIN-VERSION}. If the
3676 required version was found, the variables @var{CLN_CFLAGS} and
3677 @var{CLN_LIBS} are set. Otherwise the configure script aborts. If this
3678 is not the desired behaviour, use the following code instead
3679 @footnote{See the @code{pkg-config} documentation for more details.}
3681 PKG_CHECK_MODULES([CLN], [cln >= @var{MIN-VERSION}], [],
3682 [AC_MSG_WARNING([No suitable version of CLN can be found])])
3687 @section Include files
3688 @cindex include files
3689 @cindex header files
3691 Here is a summary of the include files and their contents.
3694 @item <cln/object.h>
3695 General definitions, reference counting, garbage collection.
3696 @item <cln/number.h>
3697 The class cl_number.
3698 @item <cln/complex.h>
3699 Functions for class cl_N, the complex numbers.
3701 Functions for class cl_R, the real numbers.
3703 Functions for class cl_F, the floats.
3704 @item <cln/sfloat.h>
3705 Functions for class cl_SF, the short-floats.
3706 @item <cln/ffloat.h>
3707 Functions for class cl_FF, the single-floats.
3708 @item <cln/dfloat.h>
3709 Functions for class cl_DF, the double-floats.
3710 @item <cln/lfloat.h>
3711 Functions for class cl_LF, the long-floats.
3712 @item <cln/rational.h>
3713 Functions for class cl_RA, the rational numbers.
3714 @item <cln/integer.h>
3715 Functions for class cl_I, the integers.
3718 @item <cln/complex_io.h>
3719 Input/Output for class cl_N, the complex numbers.
3720 @item <cln/real_io.h>
3721 Input/Output for class cl_R, the real numbers.
3722 @item <cln/float_io.h>
3723 Input/Output for class cl_F, the floats.
3724 @item <cln/sfloat_io.h>
3725 Input/Output for class cl_SF, the short-floats.
3726 @item <cln/ffloat_io.h>
3727 Input/Output for class cl_FF, the single-floats.
3728 @item <cln/dfloat_io.h>
3729 Input/Output for class cl_DF, the double-floats.
3730 @item <cln/lfloat_io.h>
3731 Input/Output for class cl_LF, the long-floats.
3732 @item <cln/rational_io.h>
3733 Input/Output for class cl_RA, the rational numbers.
3734 @item <cln/integer_io.h>
3735 Input/Output for class cl_I, the integers.
3737 Flags for customizing input operations.
3738 @item <cln/output.h>
3739 Flags for customizing output operations.
3740 @item <cln/malloc.h>
3741 @code{malloc_hook}, @code{free_hook}.
3742 @item <cln/exception.h>
3743 Exception base class.
3744 @item <cln/condition.h>
3746 @item <cln/string.h>
3748 @item <cln/symbol.h>
3750 @item <cln/proplist.h>
3754 @item <cln/null_ring.h>
3756 @item <cln/complex_ring.h>
3757 The ring of complex numbers.
3758 @item <cln/real_ring.h>
3759 The ring of real numbers.
3760 @item <cln/rational_ring.h>
3761 The ring of rational numbers.
3762 @item <cln/integer_ring.h>
3763 The ring of integers.
3764 @item <cln/numtheory.h>
3765 Number threory functions.
3766 @item <cln/modinteger.h>
3772 @item <cln/GV_number.h>
3773 General vectors over cl_number.
3774 @item <cln/GV_complex.h>
3775 General vectors over cl_N.
3776 @item <cln/GV_real.h>
3777 General vectors over cl_R.
3778 @item <cln/GV_rational.h>
3779 General vectors over cl_RA.
3780 @item <cln/GV_integer.h>
3781 General vectors over cl_I.
3782 @item <cln/GV_modinteger.h>
3783 General vectors of modular integers.
3786 @item <cln/SV_number.h>
3787 Simple vectors over cl_number.
3788 @item <cln/SV_complex.h>
3789 Simple vectors over cl_N.
3790 @item <cln/SV_real.h>
3791 Simple vectors over cl_R.
3792 @item <cln/SV_rational.h>
3793 Simple vectors over cl_RA.
3794 @item <cln/SV_integer.h>
3795 Simple vectors over cl_I.
3796 @item <cln/SV_ringelt.h>
3797 Simple vectors of general ring elements.
3798 @item <cln/univpoly.h>
3799 Univariate polynomials.
3800 @item <cln/univpoly_integer.h>
3801 Univariate polynomials over the integers.
3802 @item <cln/univpoly_rational.h>
3803 Univariate polynomials over the rational numbers.
3804 @item <cln/univpoly_real.h>
3805 Univariate polynomials over the real numbers.
3806 @item <cln/univpoly_complex.h>
3807 Univariate polynomials over the complex numbers.
3808 @item <cln/univpoly_modint.h>
3809 Univariate polynomials over modular integer rings.
3810 @item <cln/timing.h>
3813 Includes all of the above.
3820 A function which computes the nth Fibonacci number can be written as follows.
3821 @cindex Fibonacci number
3824 #include <cln/integer.h>
3825 #include <cln/real.h>
3826 using namespace cln;
3828 // Returns F_n, computed as the nearest integer to
3829 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3830 const cl_I fibonacci (int n)
3832 // Need a precision of ((1+sqrt(5))/2)^-n.
3833 float_format_t prec = float_format((int)(0.208987641*n+5));
3834 cl_R sqrt5 = sqrt(cl_float(5,prec));
3835 cl_R phi = (1+sqrt5)/2;
3836 return round1( expt(phi,n)/sqrt5 );
3840 Let's explain what is going on in detail.
3842 The include file @code{<cln/integer.h>} is necessary because the type
3843 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3844 is needed for the type @code{cl_R} and the floating point number functions.
3845 The order of the include files does not matter. In order not to write
3846 out @code{cln::}@var{foo} in this simple example we can safely import
3847 the whole namespace @code{cln}.
3849 Then comes the function declaration. The argument is an @code{int}, the
3850 result an integer. The return type is defined as @samp{const cl_I}, not
3851 simply @samp{cl_I}, because that allows the compiler to detect typos like
3852 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3853 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3854 number). We use the most specialized possible return type because functions
3855 which call @samp{fibonacci} will be able to profit from the compiler's type
3856 analysis: Adding two integers is slightly more efficient than adding the
3857 same objects declared as complex numbers, because it needs less type
3858 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3859 the size of the resulting executable program.
3861 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3862 integer. In order to get a correct result, the absolute error should be less
3863 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3864 To this end, the first line computes a floating point precision for sqrt(5)
3867 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3868 number and than taking the square root. The converse, first taking the square
3869 root of 5, and then converting to the desired precision, would not work in
3870 CLN: The square root would be computed to a default precision (normally
3871 single-float precision), and the following conversion could not help about
3872 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3873 system and does not represent sqrt(5) in a non-numeric way.
3875 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3876 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3877 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3878 @code{cl_N} because a @samp{round1} does not exist for general complex
3881 When the function returns, all the local variables in the function are
3882 automatically reclaimed (garbage collected). Only the result survives and
3883 gets passed to the caller.
3885 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3886 contains this implementation together with an even faster algorithm.
3888 @node Debugging support
3889 @section Debugging support
3892 When debugging a CLN application with GNU @code{gdb}, two facilities are
3893 available from the library:
3896 @item The library does type checks, range checks, consistency checks at
3897 many places. When one of these fails, an exception of a type derived from
3898 @code{runtime_exception} is thrown. When an exception is cought, the stack
3899 has already been unwound, so it is may not be possible to tell at which
3900 point the exception was thrown. For debugging, it is best to set up a
3901 catchpoint at the event of throwning a C++ exception:
3905 When this catchpoint is hit, look at the stack's backtrace:
3909 When control over the type of exception is required, it may be possible
3910 to set a breakpoint at the @code{g++} runtime library function
3911 @code{__raise_exception}. Refer to the documentation of GNU @code{gdb}
3914 @item The debugger's normal @code{print} command doesn't know about
3915 CLN's types and therefore prints mostly useless hexadecimal addresses.
3916 CLN offers a function @code{cl_print}, callable from the debugger,
3917 for printing number objects. In order to get this function, you have
3918 to define the macro @samp{CL_DEBUG} and then include all the header files
3919 for which you want @code{cl_print} debugging support. For example:
3920 @cindex @code{CL_DEBUG}
3923 #include <cln/string.h>
3925 Now, if you have in your program a variable @code{cl_string s}, and
3926 inspect it under @code{gdb}, the output may look like this:
3929 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3930 word = 134568800@}@}, @}
3931 (gdb) call cl_print(s)
3935 Note that the output of @code{cl_print} goes to the program's error output,
3936 not to gdb's standard output.
3938 Note, however, that the above facility does not work with all CLN types,
3939 only with number objects and similar. Therefore CLN offers a member function
3940 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3941 is needed for this member function to be implemented. Under @code{gdb},
3942 you call it like this:
3943 @cindex @code{debug_print ()}
3946 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3947 word = 134568800@}@}, @}
3948 (gdb) call s.debug_print()
3951 >call ($1).debug_print()
3956 Unfortunately, this feature does not seem to work under all circumstances.
3959 @node Reporting Problems
3960 @section Reporting Problems
3962 @cindex mailing list
3964 If you encounter any problem, please don't hesitate to send a detailed
3965 bugreport to the @code{cln-list@@ginac.de} mailing list. Please think
3966 about your bug: consider including a short description of your operating
3967 system and compilation environment with corresponding version numbers. A
3968 description of your configuration options may also be helpful. Also, a
3969 short test program together with the output you get and the output you
3970 expect will help us to reproduce it quickly. Finally, do not forget to
3971 report the version number of CLN.
3975 @chapter Customizing
3980 * Floating-point underflow::
3982 * Customizing the memory allocator::
3985 @node Error handling
3986 @section Error handling
3988 @cindex error handling
3990 @cindex @code{runtime_exception}
3991 CLN signals abnormal situations by throwning exceptions. All exceptions
3992 thrown by the library are of type @code{runtime_exception} or of a
3993 derived type. Class @code{cln::runtime_exception} in turn is derived
3994 from the C++ standard library class @code{std::runtime_error} and
3995 inherits the @code{.what()} member function that can be used to query
3996 details about the cause of error.
3998 The most important classes thrown by the library are
4000 @cindex @code{floating_point_exception}
4001 @cindex @code{read_number_exception}
4003 Exception base class
4007 +----------------+----------------+
4009 Malformed number input Floating-point error
4010 read_number_exception floating_poing_exception
4011 <cln/number_io.h> <cln/float.h>
4014 CLN has many more exception classes that allow for more fine-grained
4015 control but I refrain from documenting them all here. They are all
4016 declared in the public header files and they are all subclasses of the
4017 above exceptions, so catching those you are always on the safe side.
4020 @node Floating-point underflow
4021 @section Floating-point underflow
4024 @cindex @code{floating_point_underflow_exception}
4025 Floating point underflow denotes the situation when a floating-point
4026 number is to be created which is so close to @code{0} that its exponent
4027 is too low to be represented internally. By default, this causes the
4028 exception @code{floating_point_underflow_exception} (subclass of
4029 @code{floating_point_exception}) to be thrown. If you set the global
4032 bool cl_inhibit_floating_point_underflow
4034 to @code{true}, the exception will be inhibited, and a floating-point
4035 zero will be generated instead. The default value of
4036 @code{cl_inhibit_floating_point_underflow} is @code{false}.
4039 @node Customizing I/O
4040 @section Customizing I/O
4042 The output of the function @code{fprint} may be customized by changing the
4043 value of the global variable @code{default_print_flags}.
4044 @cindex @code{default_print_flags}
4047 @node Customizing the memory allocator
4048 @section Customizing the memory allocator
4050 Every memory allocation of CLN is done through the function pointer
4051 @code{malloc_hook}. Freeing of this memory is done through the function
4052 pointer @code{free_hook}. The default versions of these functions,
4053 provided in the library, call @code{malloc} and @code{free} and check
4054 the @code{malloc} result against @code{NULL}.
4055 If you want to provide another memory allocator, you need to define
4056 the variables @code{malloc_hook} and @code{free_hook} yourself,
4059 #include <cln/malloc.h>
4061 void* (*malloc_hook) (size_t size) = @dots{};
4062 void (*free_hook) (void* ptr) = @dots{};
4065 @cindex @code{malloc_hook ()}
4066 @cindex @code{free_hook ()}
4067 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
4069 It is not possible to change the memory allocator at runtime, because
4070 it is already called at program startup by the constructors of some
4078 @node Index, , Customizing, Top