1 \input texinfo @c -*-texinfo-*-
4 @settitle CLN, a Class Library for Numbers
5 @c @setchapternewpage off
10 @c I hate putting "@noindent" in front of every paragraph.
16 * CLN: (cln). Class Library for Numbers (C++).
21 @c Don't need the other types of indices.
32 This file documents @sc{cln}, a Class Library for Numbers.
34 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
35 Richard Kreckel, @code{<kreckel@@ginac.de>}.
37 Copyright (C) Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001.
38 Copyright (C) Richard Kreckel 2000, 2001.
40 Permission is granted to make and distribute verbatim copies of
41 this manual provided the copyright notice and this permission notice
42 are preserved on all copies.
45 Permission is granted to process this file through TeX and print the
46 results, provided the printed document carries copying permission
47 notice identical to this one except for the removal of this paragraph
48 (this paragraph not being relevant to the printed manual).
51 Permission is granted to copy and distribute modified versions of this
52 manual under the conditions for verbatim copying, provided that the entire
53 resulting derived work is distributed under the terms of a permission
54 notice identical to this one.
56 Permission is granted to copy and distribute translations of this manual
57 into another language, under the above conditions for modified versions,
58 except that this permission notice may be stated in a translation approved
64 @c prevent ugly black rectangles on overfull hbox lines:
67 @title CLN, a Class Library for Numbers
69 @author by Bruno Haible
71 @vskip 0pt plus 1filll
72 Copyright @copyright{} Bruno Haible 1995, 1996, 1997, 1998, 1999, 2000, 2001.
74 Copyright @copyright{} Richard Kreckel 2000, 2001.
77 Published by Bruno Haible, @code{<haible@@clisp.cons.org>} and
78 Richard Kreckel, @code{<kreckel@@ginac.de>}.
80 Permission is granted to make and distribute verbatim copies of
81 this manual provided the copyright notice and this permission notice
82 are preserved on all copies.
84 Permission is granted to copy and distribute modified versions of this
85 manual under the conditions for verbatim copying, provided that the entire
86 resulting derived work is distributed under the terms of a permission
87 notice identical to this one.
89 Permission is granted to copy and distribute translations of this manual
90 into another language, under the above conditions for modified versions,
91 except that this permission notice may be stated in a translation approved
98 @node Top, Introduction, (dir), (dir)
101 @c * Introduction:: Introduction
105 @node Introduction, Top, Top, Top
106 @comment node-name, next, previous, up
107 @chapter Introduction
110 CLN is a library for computations with all kinds of numbers.
111 It has a rich set of number classes:
115 Integers (with unlimited precision),
121 Floating-point numbers:
131 Long float (with unlimited precision),
138 Modular integers (integers modulo a fixed integer),
141 Univariate polynomials.
145 The subtypes of the complex numbers among these are exactly the
146 types of numbers known to the Common Lisp language. Therefore
147 @code{CLN} can be used for Common Lisp implementations, giving
148 @samp{CLN} another meaning: it becomes an abbreviation of
149 ``Common Lisp Numbers''.
152 The CLN package implements
156 Elementary functions (@code{+}, @code{-}, @code{*}, @code{/}, @code{sqrt},
157 comparisons, @dots{}),
160 Logical functions (logical @code{and}, @code{or}, @code{not}, @dots{}),
163 Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
164 functions and their inverse functions).
168 CLN is a C++ library. Using C++ as an implementation language provides
172 efficiency: it compiles to machine code,
174 type safety: the C++ compiler knows about the number types and complains
175 if, for example, you try to assign a float to an integer variable.
177 algebraic syntax: You can use the @code{+}, @code{-}, @code{*}, @code{=},
178 @code{==}, @dots{} operators as in C or C++.
182 CLN is memory efficient:
186 Small integers and short floats are immediate, not heap allocated.
188 Heap-allocated memory is reclaimed through an automatic, non-interruptive
193 CLN is speed efficient:
197 The kernel of CLN has been written in assembly language for some CPUs
198 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
201 On all CPUs, CLN may be configured to use the superefficient low-level
202 routines from GNU GMP version 3.
204 It uses Karatsuba multiplication, which is significantly faster
205 for large numbers than the standard multiplication algorithm.
207 For very large numbers (more than 12000 decimal digits), it uses
209 Sch{@"o}nhage-Strassen
210 @cindex Sch{@"o}nhage-Strassen multiplication
214 @cindex Schönhage-Strassen multiplication
216 multiplication, which is an asymptotically optimal multiplication
217 algorithm, for multiplication, division and radix conversion.
221 CLN aims at being easily integrated into larger software packages:
225 The garbage collection imposes no burden on the main application.
227 The library provides hooks for memory allocation and exceptions.
230 All non-macro identifiers are hidden in namespace @code{cln} in
231 order to avoid name clashes.
235 @chapter Installation
237 This section describes how to install the CLN package on your system.
240 @section Prerequisites
242 @subsection C++ compiler
244 To build CLN, you need a C++ compiler.
245 Actually, you need GNU @code{g++ 2.95} or newer.
247 The following C++ features are used:
248 classes, member functions, overloading of functions and operators,
249 constructors and destructors, inline, const, multiple inheritance,
250 templates and namespaces.
252 The following C++ features are not used:
253 @code{new}, @code{delete}, virtual inheritance, exceptions.
255 CLN relies on semi-automatic ordering of initializations
256 of static and global variables, a feature which I could
257 implement for GNU g++ only.
260 @comment cl_modules.h requires g++
261 Therefore nearly any C++ compiler will do.
263 The following C++ compilers are known to compile CLN:
266 GNU @code{g++ 2.7.0}, @code{g++ 2.7.2}
271 The following C++ compilers are known to be unusable for CLN:
274 On SunOS 4, @code{CC 2.1}, because it doesn't grok @code{//} comments
275 in lines containing @code{#if} or @code{#elif} preprocessor commands.
277 On AIX 3.2.5, @code{xlC}, because it doesn't grok the template syntax
278 in @code{cl_SV.h} and @code{cl_GV.h}, because it forces most class types
279 to have default constructors, and because it probably miscompiles the
280 integer multiplication routines.
282 On AIX 4.1.4.0, @code{xlC}, because when optimizing, it sometimes converts
283 @code{short}s to @code{int}s by zero-extend.
287 On HPPA, GNU @code{g++ 2.7.x}, because the semi-automatic ordering of
288 initializations will not work.
292 @subsection Make utility
295 To build CLN, you also need to have GNU @code{make} installed.
297 @subsection Sed utility
300 To build CLN on HP-UX, you also need to have GNU @code{sed} installed.
301 This is because the libtool script, which creates the CLN library, relies
302 on @code{sed}, and the vendor's @code{sed} utility on these systems is too
306 @section Building the library
308 As with any autoconfiguring GNU software, installation is as easy as this:
316 If on your system, @samp{make} is not GNU @code{make}, you have to use
317 @samp{gmake} instead of @samp{make} above.
319 The @code{configure} command checks out some features of your system and
320 C++ compiler and builds the @code{Makefile}s. The @code{make} command
321 builds the library. This step may take about an hour on an average workstation.
322 The @code{make check} runs some test to check that no important subroutine
323 has been miscompiled.
325 The @code{configure} command accepts options. To get a summary of them, try
331 Some of the options are explained in detail in the @samp{INSTALL.generic} file.
333 You can specify the C compiler, the C++ compiler and their options through
334 the following environment variables when running @code{configure}:
338 Specifies the C compiler.
341 Flags to be given to the C compiler when compiling programs (not when linking).
344 Specifies the C++ compiler.
347 Flags to be given to the C++ compiler when compiling programs (not when linking).
353 $ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
354 $ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
355 CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
356 $ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
357 CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
360 @comment cl_modules.h requires g++
361 You should not mix GNU and non-GNU compilers. So, if @code{CXX} is a non-GNU
362 compiler, @code{CC} should be set to a non-GNU compiler as well. Examples:
365 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O" ./configure
366 $ CC="gcc -V 2.7.0" CFLAGS="-g" CXX="g++ -V 2.7.0" CXXFLAGS="-g" ./configure
369 On SGI Irix 5, if you wish not to use @code{g++}:
372 $ CC="cc" CFLAGS="-O" CXX="CC" CXXFLAGS="-O -Olimit 16000" ./configure
375 On SGI Irix 6, if you wish not to use @code{g++}:
378 $ CC="cc -32" CFLAGS="-O" CXX="CC -32" CXXFLAGS="-O -Olimit 34000" \
379 ./configure --without-gmp
380 $ CC="cc -n32" CFLAGS="-O" CXX="CC -n32" CXXFLAGS="-O \
381 -OPT:const_copy_limit=32400 -OPT:global_limit=32400 -OPT:fprop_limit=4000" \
382 ./configure --without-gmp
386 Note that for these environment variables to take effect, you have to set
387 them (assuming a Bourne-compatible shell) on the same line as the
388 @code{configure} command. If you made the settings in earlier shell
389 commands, you have to @code{export} the environment variables before
390 calling @code{configure}. In a @code{csh} shell, you have to use the
391 @samp{setenv} command for setting each of the environment variables.
393 Currently CLN works only with the GNU @code{g++} compiler, and only in
394 optimizing mode. So you should specify at least @code{-O} in the CXXFLAGS,
395 or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use @code{-O}.)
397 If you use @code{g++} gcc-2.95.x or gcc-3.0, I recommend adding
398 @samp{-fno-exceptions} to the CXXFLAGS. This will likely generate better code.
400 If you use @code{g++} from gcc-2.95.x on Sparc, add either @samp{-O},
401 @samp{-O1} or @samp{-O2 -fno-schedule-insns} to the CXXFLAGS. With full
402 @samp{-O2}, @code{g++} miscompiles the division routines. If you use
403 @code{g++} older than 2.95.3 on Sparc you should also specify
404 @samp{--disable-shared} because of bad code produced in the shared
405 library. Also, on OSF/1 or Tru64 using gcc-2.95.x, you should specify
406 @samp{--disable-shared} because of linker problems with duplicate symbols
409 By default, both a shared and a static library are built. You can build
410 CLN as a static (or shared) library only, by calling @code{configure} with
411 the option @samp{--disable-shared} (or @samp{--disable-static}). While
412 shared libraries are usually more convenient to use, they may not work
413 on all architectures. Try disabling them if you run into linker
414 problems. Also, they are generally somewhat slower than static
415 libraries so runtime-critical applications should be linked statically.
418 @subsection Using the GNU MP Library
421 Starting with version 1.1, CLN may be configured to make use of a
422 preinstalled @code{gmp} library. Please make sure that you have at
423 least @code{gmp} version 3.0 installed since earlier versions are
424 unsupported and likely not to work. Enabling this feature by calling
425 @code{configure} with the option @samp{--with-gmp} is known to be quite
426 a boost for CLN's performance.
428 If you have installed the @code{gmp} library and its header file in
429 some place where your compiler cannot find it by default, you must help
430 @code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
434 $ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
435 CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
439 @section Installing the library
442 As with any autoconfiguring GNU software, installation is as easy as this:
448 The @samp{make install} command installs the library and the include files
449 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
450 if you haven't specified a @code{--prefix} option to @code{configure}).
451 This step may require superuser privileges.
453 If you have already built the library and wish to install it, but didn't
454 specify @code{--prefix=@dots{}} at configure time, just re-run
455 @code{configure}, giving it the same options as the first time, plus
456 the @code{--prefix=@dots{}} option.
461 You can remove system-dependent files generated by @code{make} through
467 You can remove all files generated by @code{make}, thus reverting to a
468 virgin distribution of CLN, through
475 @chapter Ordinary number types
477 CLN implements the following class hierarchy:
485 Real or complex number
494 +-------------------+-------------------+
496 Rational number Floating-point number
498 <cln/rational.h> <cln/float.h>
500 | +--------------+--------------+--------------+
502 cl_I Short-Float Single-Float Double-Float Long-Float
503 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
504 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
507 @cindex @code{cl_number}
508 @cindex abstract class
509 The base class @code{cl_number} is an abstract base class.
510 It is not useful to declare a variable of this type except if you want
511 to completely disable compile-time type checking and use run-time type
516 @cindex complex number
517 The class @code{cl_N} comprises real and complex numbers. There is
518 no special class for complex numbers since complex numbers with imaginary
519 part @code{0} are automatically converted to real numbers.
522 The class @code{cl_R} comprises real numbers of different kinds. It is an
526 @cindex rational number
528 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
529 integers. There is no special class for non-integral rational numbers
530 since rational numbers with denominator @code{1} are automatically converted
534 The class @code{cl_F} implements floating-point approximations to real numbers.
535 It is an abstract class.
538 @section Exact numbers
541 Some numbers are represented as exact numbers: there is no loss of information
542 when such a number is converted from its mathematical value to its internal
543 representation. On exact numbers, the elementary operations (@code{+},
544 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
547 In CLN, the exact numbers are:
551 rational numbers (including integers),
553 complex numbers whose real and imaginary parts are both rational numbers.
556 Rational numbers are always normalized to the form
557 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
558 are coprime integers and the denominator is positive. If the resulting
559 denominator is @code{1}, the rational number is converted to an integer.
561 @cindex immediate numbers
562 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
563 for 32-bit machines) are especially efficient, because they consume no heap
564 allocation. Otherwise the distinction between these immediate integers
565 (called ``fixnums'') and heap allocated integers (called ``bignums'')
566 is completely transparent.
569 @section Floating-point numbers
570 @cindex floating-point number
572 Not all real numbers can be represented exactly. (There is an easy mathematical
573 proof for this: Only a countable set of numbers can be stored exactly in
574 a computer, even if one assumes that it has unlimited storage. But there
575 are uncountably many real numbers.) So some approximation is needed.
576 CLN implements ordinary floating-point numbers, with mantissa and exponent.
578 @cindex rounding error
579 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
580 only return approximate results. For example, the value of the expression
581 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
582 @samp{0.7}. Rounding errors like this one are inevitable when computing
583 with floating-point numbers.
585 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
586 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
587 rule: It first computes the exact mathematical result and then returns the
588 floating-point number which is nearest to this. If two floating-point numbers
589 are equally distant from the ideal result, the one with a @code{0} in its least
590 significant mantissa bit is chosen.
592 Similarly, testing floating point numbers for equality @samp{x == y}
593 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
594 for some well-chosen @code{epsilon}.
596 Floating point numbers come in four flavors:
601 Short floats, type @code{cl_SF}.
602 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
603 and 17 mantissa bits (including the ``hidden'' bit).
604 They don't consume heap allocation.
608 Single floats, type @code{cl_FF}.
609 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
610 and 24 mantissa bits (including the ``hidden'' bit).
611 In CLN, they are represented as IEEE single-precision floating point numbers.
612 This corresponds closely to the C/C++ type @samp{float}.
616 Double floats, type @code{cl_DF}.
617 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
618 and 53 mantissa bits (including the ``hidden'' bit).
619 In CLN, they are represented as IEEE double-precision floating point numbers.
620 This corresponds closely to the C/C++ type @samp{double}.
624 Long floats, type @code{cl_LF}.
625 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
626 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
627 The precision of a long float is unlimited, but once created, a long float
628 has a fixed precision. (No ``lazy recomputation''.)
631 Of course, computations with long floats are more expensive than those
632 with smaller floating-point formats.
634 CLN does not implement features like NaNs, denormalized numbers and
635 gradual underflow. If the exponent range of some floating-point type
636 is too limited for your application, choose another floating-point type
637 with larger exponent range.
640 As a user of CLN, you can forget about the differences between the
641 four floating-point types and just declare all your floating-point
642 variables as being of type @code{cl_F}. This has the advantage that
643 when you change the precision of some computation (say, from @code{cl_DF}
644 to @code{cl_LF}), you don't have to change the code, only the precision
645 of the initial values. Also, many transcendental functions have been
646 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
647 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
648 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
649 the floating point contagion rule happened to change in the future.)
652 @section Complex numbers
653 @cindex complex number
655 Complex numbers, as implemented by the class @code{cl_N}, have a real
656 part and an imaginary part, both real numbers. A complex number whose
657 imaginary part is the exact number @code{0} is automatically converted
660 Complex numbers can arise from real numbers alone, for example
661 through application of @code{sqrt} or transcendental functions.
667 Conversions from any class to any its superclasses (``base classes'' in
668 C++ terminology) is done automatically.
670 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
671 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
672 @code{cl_N} and @code{cl_number}.
674 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
675 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
676 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
677 efficiency. Their range is therefore limited:
681 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
683 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
686 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
687 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
688 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
690 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
691 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
692 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
693 @samp{cl_I}, first convert it to an @samp{unsigned long}.
695 Conversions from the C built-in type @samp{float} are provided for the classes
696 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
698 Conversions from the C built-in type @samp{double} are provided for the classes
699 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
701 Conversions from @samp{const char *} are provided for the classes
702 @code{cl_I}, @code{cl_RA},
703 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
704 @code{cl_R}, @code{cl_N}.
705 The easiest way to specify a value which is outside of the range of the
706 C++ built-in types is therefore to specify it as a string, like this:
709 cl_I order_of_rubiks_cube_group = "43252003274489856000";
711 Note that this conversion is done at runtime, not at compile-time.
713 Conversions from @code{cl_I} to the C built-in types @samp{int},
714 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
718 @item int cl_I_to_int (const cl_I& x)
719 @cindex @code{cl_I_to_int ()}
720 @itemx unsigned int cl_I_to_uint (const cl_I& x)
721 @cindex @code{cl_I_to_uint ()}
722 @itemx long cl_I_to_long (const cl_I& x)
723 @cindex @code{cl_I_to_long ()}
724 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
725 @cindex @code{cl_I_to_ulong ()}
726 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
727 representable in the range of @var{ctype}, a runtime error occurs.
730 Conversions from the classes @code{cl_I}, @code{cl_RA},
731 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
733 to the C built-in types @samp{float} and @samp{double} are provided through
737 @item float float_approx (const @var{type}& x)
738 @cindex @code{float_approx ()}
739 @itemx double double_approx (const @var{type}& x)
740 @cindex @code{double_approx ()}
741 Returns an approximation of @code{x} of C type @var{ctype}.
742 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
743 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
746 Conversions from any class to any of its subclasses (``derived classes'' in
747 C++ terminology) are not provided. Instead, you can assert and check
748 that a value belongs to a certain subclass, and return it as element of that
749 class, using the @samp{As} and @samp{The} macros.
750 @cindex @code{As()()}
751 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
752 @var{type} and returns it as such.
753 @cindex @code{The()()}
754 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
755 @var{type} and returns it as such. It is your responsibility to ensure
756 that this assumption is valid. Since macros and namespaces don't go
757 together well, there is an equivalent to @samp{The}: the template
765 if (!(x >= 0)) abort();
766 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
767 // In general, it would be a rational number.
768 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
773 @chapter Functions on numbers
775 Each of the number classes declares its mathematical operations in the
776 corresponding include file. For example, if your code operates with
777 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
780 @section Constructing numbers
782 Here is how to create number objects ``from nothing''.
785 @subsection Constructing integers
787 @code{cl_I} objects are most easily constructed from C integers and from
788 strings. See @ref{Conversions}.
791 @subsection Constructing rational numbers
793 @code{cl_RA} objects can be constructed from strings. The syntax
794 for rational numbers is described in @ref{Internal and printed representation}.
795 Another standard way to produce a rational number is through application
796 of @samp{operator /} or @samp{recip} on integers.
799 @subsection Constructing floating-point numbers
801 @code{cl_F} objects with low precision are most easily constructed from
802 C @samp{float} and @samp{double}. See @ref{Conversions}.
804 To construct a @code{cl_F} with high precision, you can use the conversion
805 from @samp{const char *}, but you have to specify the desired precision
806 within the string. (See @ref{Internal and printed representation}.)
809 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
811 will set @samp{e} to the given value, with a precision of 40 decimal digits.
813 The programmatic way to construct a @code{cl_F} with high precision is
814 through the @code{cl_float} conversion function, see
815 @ref{Conversion to floating-point numbers}. For example, to compute
816 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
817 and then apply the exponential function:
819 float_format_t precision = float_format(40);
820 cl_F e = exp(cl_float(1,precision));
824 @subsection Constructing complex numbers
826 Non-real @code{cl_N} objects are normally constructed through the function
828 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
830 See @ref{Elementary complex functions}.
833 @section Elementary functions
835 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
836 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
837 defines the following operations:
840 @item @var{type} operator + (const @var{type}&, const @var{type}&)
841 @cindex @code{operator + ()}
844 @item @var{type} operator - (const @var{type}&, const @var{type}&)
845 @cindex @code{operator - ()}
848 @item @var{type} operator - (const @var{type}&)
849 Returns the negative of the argument.
851 @item @var{type} plus1 (const @var{type}& x)
852 @cindex @code{plus1 ()}
853 Returns @code{x + 1}.
855 @item @var{type} minus1 (const @var{type}& x)
856 @cindex @code{minus1 ()}
857 Returns @code{x - 1}.
859 @item @var{type} operator * (const @var{type}&, const @var{type}&)
860 @cindex @code{operator * ()}
863 @item @var{type} square (const @var{type}& x)
864 @cindex @code{square ()}
865 Returns @code{x * x}.
868 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
869 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
870 defines the following operations:
873 @item @var{type} operator / (const @var{type}&, const @var{type}&)
874 @cindex @code{operator / ()}
877 @item @var{type} recip (const @var{type}&)
878 @cindex @code{recip ()}
879 Returns the reciprocal of the argument.
882 The class @code{cl_I} doesn't define a @samp{/} operation because
883 in the C/C++ language this operator, applied to integral types,
884 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
885 is implementation dependent). (@xref{Rounding functions}.)
886 Instead, @code{cl_I} defines an ``exact quotient'' function:
889 @item cl_I exquo (const cl_I& x, const cl_I& y)
890 @cindex @code{exquo ()}
891 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
894 The following exponentiation functions are defined:
897 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
898 @cindex @code{expt_pos ()}
899 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
900 @code{y} must be > 0. Returns @code{x^y}.
902 @item cl_RA expt (const cl_RA& x, const cl_I& y)
903 @cindex @code{expt ()}
904 @itemx cl_R expt (const cl_R& x, const cl_I& y)
905 @itemx cl_N expt (const cl_N& x, const cl_I& y)
909 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
910 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
911 defines the following operation:
914 @item @var{type} abs (const @var{type}& x)
915 @cindex @code{abs ()}
916 Returns the absolute value of @code{x}.
917 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
920 The class @code{cl_N} implements this as follows:
923 @item cl_R abs (const cl_N x)
924 Returns the absolute value of @code{x}.
927 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
928 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
929 defines the following operation:
932 @item @var{type} signum (const @var{type}& x)
933 @cindex @code{signum ()}
934 Returns the sign of @code{x}, in the same number format as @code{x}.
935 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
936 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
941 @section Elementary rational functions
943 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
946 @item cl_I numerator (const @var{type}& x)
947 @cindex @code{numerator ()}
948 Returns the numerator of @code{x}.
950 @item cl_I denominator (const @var{type}& x)
951 @cindex @code{denominator ()}
952 Returns the denominator of @code{x}.
955 The numerator and denominator of a rational number are normalized in such
956 a way that they have no factor in common and the denominator is positive.
959 @section Elementary complex functions
961 The class @code{cl_N} defines the following operation:
964 @item cl_N complex (const cl_R& a, const cl_R& b)
965 @cindex @code{complex ()}
966 Returns the complex number @code{a+bi}, that is, the complex number with
967 real part @code{a} and imaginary part @code{b}.
970 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
973 @item cl_R realpart (const @var{type}& x)
974 @cindex @code{realpart ()}
975 Returns the real part of @code{x}.
977 @item cl_R imagpart (const @var{type}& x)
978 @cindex @code{imagpart ()}
979 Returns the imaginary part of @code{x}.
981 @item @var{type} conjugate (const @var{type}& x)
982 @cindex @code{conjugate ()}
983 Returns the complex conjugate of @code{x}.
986 We have the relations
990 @code{x = complex(realpart(x), imagpart(x))}
992 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
999 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
1000 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1001 defines the following operations:
1004 @item bool operator == (const @var{type}&, const @var{type}&)
1005 @cindex @code{operator == ()}
1006 @itemx bool operator != (const @var{type}&, const @var{type}&)
1007 @cindex @code{operator != ()}
1008 Comparison, as in C and C++.
1010 @item uint32 equal_hashcode (const @var{type}&)
1011 @cindex @code{equal_hashcode ()}
1012 Returns a 32-bit hash code that is the same for any two numbers which are
1013 the same according to @code{==}. This hash code depends on the number's value,
1014 not its type or precision.
1016 @item cl_boolean zerop (const @var{type}& x)
1017 @cindex @code{zerop ()}
1018 Compare against zero: @code{x == 0}
1021 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1022 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1023 defines the following operations:
1026 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1027 @cindex @code{compare ()}
1028 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1029 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1031 @item bool operator <= (const @var{type}&, const @var{type}&)
1032 @cindex @code{operator <= ()}
1033 @itemx bool operator < (const @var{type}&, const @var{type}&)
1034 @cindex @code{operator < ()}
1035 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1036 @cindex @code{operator >= ()}
1037 @itemx bool operator > (const @var{type}&, const @var{type}&)
1038 @cindex @code{operator > ()}
1039 Comparison, as in C and C++.
1041 @item cl_boolean minusp (const @var{type}& x)
1042 @cindex @code{minusp ()}
1043 Compare against zero: @code{x < 0}
1045 @item cl_boolean plusp (const @var{type}& x)
1046 @cindex @code{plusp ()}
1047 Compare against zero: @code{x > 0}
1049 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1050 @cindex @code{max ()}
1051 Return the maximum of @code{x} and @code{y}.
1053 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1054 @cindex @code{min ()}
1055 Return the minimum of @code{x} and @code{y}.
1058 When a floating point number and a rational number are compared, the float
1059 is first converted to a rational number using the function @code{rational}.
1060 Since a floating point number actually represents an interval of real numbers,
1061 the result might be surprising.
1062 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1063 there is no floating point number whose value is exactly @code{1/3}.
1066 @section Rounding functions
1069 When a real number is to be converted to an integer, there is no ``best''
1070 rounding. The desired rounding function depends on the application.
1071 The Common Lisp and ISO Lisp standards offer four rounding functions:
1075 This is the largest integer <=@code{x}.
1078 This is the smallest integer >=@code{x}.
1081 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1084 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1085 integers, choose the even one.
1088 These functions have different advantages:
1090 @code{floor} and @code{ceiling} are translation invariant:
1091 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1092 for every @code{x} and every integer @code{n}.
1094 On the other hand, @code{truncate} and @code{round} are symmetric:
1095 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1096 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1097 down exactly as often as it rounds up.
1099 The functions are related like this:
1103 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1104 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1106 @code{truncate(x) = sign(x) * floor(abs(x))}
1109 Each of the classes @code{cl_R}, @code{cl_RA},
1110 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1111 defines the following operations:
1114 @item cl_I floor1 (const @var{type}& x)
1115 @cindex @code{floor1 ()}
1116 Returns @code{floor(x)}.
1117 @item cl_I ceiling1 (const @var{type}& x)
1118 @cindex @code{ceiling1 ()}
1119 Returns @code{ceiling(x)}.
1120 @item cl_I truncate1 (const @var{type}& x)
1121 @cindex @code{truncate1 ()}
1122 Returns @code{truncate(x)}.
1123 @item cl_I round1 (const @var{type}& x)
1124 @cindex @code{round1 ()}
1125 Returns @code{round(x)}.
1128 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1129 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1130 defines the following operations:
1133 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1134 Returns @code{floor(x/y)}.
1135 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1136 Returns @code{ceiling(x/y)}.
1137 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1138 Returns @code{truncate(x/y)}.
1139 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1140 Returns @code{round(x/y)}.
1143 These functions are called @samp{floor1}, @dots{} here instead of
1144 @samp{floor}, @dots{}, because on some systems, system dependent include
1145 files define @samp{floor} and @samp{ceiling} as macros.
1147 In many cases, one needs both the quotient and the remainder of a division.
1148 It is more efficient to compute both at the same time than to perform
1149 two divisions, one for quotient and the next one for the remainder.
1150 The following functions therefore return a structure containing both
1151 the quotient and the remainder. The suffix @samp{2} indicates the number
1152 of ``return values''. The remainder is defined as follows:
1156 for the computation of @code{quotient = floor(x)},
1157 @code{remainder = x - quotient},
1159 for the computation of @code{quotient = floor(x,y)},
1160 @code{remainder = x - quotient*y},
1163 and similarly for the other three operations.
1165 Each of the classes @code{cl_R}, @code{cl_RA},
1166 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1167 defines the following operations:
1170 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1171 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1172 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1173 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1174 @itemx @var{type}_div_t round2 (const @var{type}& x)
1177 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1178 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1179 defines the following operations:
1182 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1183 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1184 @cindex @code{floor2 ()}
1185 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1186 @cindex @code{ceiling2 ()}
1187 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1188 @cindex @code{truncate2 ()}
1189 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1190 @cindex @code{round2 ()}
1193 Sometimes, one wants the quotient as a floating-point number (of the
1194 same format as the argument, if the argument is a float) instead of as
1195 an integer. The prefix @samp{f} indicates this.
1198 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1199 defines the following operations:
1202 @item @var{type} ffloor (const @var{type}& x)
1203 @cindex @code{ffloor ()}
1204 @itemx @var{type} fceiling (const @var{type}& x)
1205 @cindex @code{fceiling ()}
1206 @itemx @var{type} ftruncate (const @var{type}& x)
1207 @cindex @code{ftruncate ()}
1208 @itemx @var{type} fround (const @var{type}& x)
1209 @cindex @code{fround ()}
1212 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1214 The class @code{cl_R} defines the following operations:
1217 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1218 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1219 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1220 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1223 These functions also exist in versions which return both the quotient
1224 and the remainder. The suffix @samp{2} indicates this.
1227 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1228 defines the following operations:
1229 @cindex @code{cl_F_fdiv_t}
1230 @cindex @code{cl_SF_fdiv_t}
1231 @cindex @code{cl_FF_fdiv_t}
1232 @cindex @code{cl_DF_fdiv_t}
1233 @cindex @code{cl_LF_fdiv_t}
1236 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1237 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1238 @cindex @code{ffloor2 ()}
1239 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1240 @cindex @code{fceiling2 ()}
1241 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1242 @cindex @code{ftruncate2 ()}
1243 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1244 @cindex @code{fround2 ()}
1246 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1247 @cindex @code{cl_R_fdiv_t}
1249 The class @code{cl_R} defines the following operations:
1252 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1253 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1254 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1255 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1256 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1259 Other applications need only the remainder of a division.
1260 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1261 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1262 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1266 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1268 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1271 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1272 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1273 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1275 The classes @code{cl_R}, @code{cl_I} define the following operations:
1278 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1279 @cindex @code{mod ()}
1280 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1281 @cindex @code{rem ()}
1287 Each of the classes @code{cl_R},
1288 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1289 defines the following operation:
1292 @item @var{type} sqrt (const @var{type}& x)
1293 @cindex @code{sqrt ()}
1294 @code{x} must be >= 0. This function returns the square root of @code{x},
1295 normalized to be >= 0. If @code{x} is the square of a rational number,
1296 @code{sqrt(x)} will be a rational number, else it will return a
1297 floating-point approximation.
1300 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1303 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1304 @cindex @code{sqrtp ()}
1305 This tests whether @code{x} is a perfect square. If so, it returns true
1306 and the exact square root in @code{*root}, else it returns false.
1309 Furthermore, for integers, similarly:
1312 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1313 @cindex @code{isqrt ()}
1314 @code{x} should be >= 0. This function sets @code{*root} to
1315 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1316 the boolean value @code{(expt(*root,2) == x)}.
1319 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1320 define the following operation:
1323 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1324 @cindex @code{rootp ()}
1325 @code{x} must be >= 0. @code{n} must be > 0.
1326 This tests whether @code{x} is an @code{n}th power of a rational number.
1327 If so, it returns true and the exact root in @code{*root}, else it returns
1331 The only square root function which accepts negative numbers is the one
1332 for class @code{cl_N}:
1335 @item cl_N sqrt (const cl_N& z)
1336 @cindex @code{sqrt ()}
1337 Returns the square root of @code{z}, as defined by the formula
1338 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1339 or to a complex number are done if necessary. The range of the result is the
1340 right half plane @code{realpart(sqrt(z)) >= 0}
1341 including the positive imaginary axis and 0, but excluding
1342 the negative imaginary axis.
1343 The result is an exact number only if @code{z} is an exact number.
1347 @section Transcendental functions
1348 @cindex transcendental functions
1350 The transcendental functions return an exact result if the argument
1351 is exact and the result is exact as well. Otherwise they must return
1352 inexact numbers even if the argument is exact.
1353 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1356 @subsection Exponential and logarithmic functions
1359 @item cl_R exp (const cl_R& x)
1360 @cindex @code{exp ()}
1361 @itemx cl_N exp (const cl_N& x)
1362 Returns the exponential function of @code{x}. This is @code{e^x} where
1363 @code{e} is the base of the natural logarithms. The range of the result
1364 is the entire complex plane excluding 0.
1366 @item cl_R ln (const cl_R& x)
1367 @cindex @code{ln ()}
1368 @code{x} must be > 0. Returns the (natural) logarithm of x.
1370 @item cl_N log (const cl_N& x)
1371 @cindex @code{log ()}
1372 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1373 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1374 The range of the result is the strip in the complex plane
1375 @code{-pi < imagpart(log(x)) <= pi}.
1377 @item cl_R phase (const cl_N& x)
1378 @cindex @code{phase ()}
1379 Returns the angle part of @code{x} in its polar representation as a
1380 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1381 This is also the imaginary part of @code{log(x)}.
1382 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1383 The result will be an exact number only if @code{zerop(x)} or
1384 if @code{x} is real and positive.
1386 @item cl_R log (const cl_R& a, const cl_R& b)
1387 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1388 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1389 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1392 @item cl_N log (const cl_N& a, const cl_N& b)
1393 Returns the logarithm of @code{a} with respect to base @code{b}.
1394 @code{log(a,b) = log(a)/log(b)}.
1396 @item cl_N expt (const cl_N& x, const cl_N& y)
1397 @cindex @code{expt ()}
1398 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1401 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1404 @item cl_F exp1 (float_format_t f)
1405 @cindex @code{exp1 ()}
1406 Returns e as a float of format @code{f}.
1408 @item cl_F exp1 (const cl_F& y)
1409 Returns e in the float format of @code{y}.
1411 @item cl_F exp1 (void)
1412 Returns e as a float of format @code{default_float_format}.
1416 @subsection Trigonometric functions
1419 @item cl_R sin (const cl_R& x)
1420 @cindex @code{sin ()}
1421 Returns @code{sin(x)}. The range of the result is the interval
1422 @code{-1 <= sin(x) <= 1}.
1424 @item cl_N sin (const cl_N& z)
1425 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1427 @item cl_R cos (const cl_R& x)
1428 @cindex @code{cos ()}
1429 Returns @code{cos(x)}. The range of the result is the interval
1430 @code{-1 <= cos(x) <= 1}.
1432 @item cl_N cos (const cl_N& x)
1433 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1435 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1436 @cindex @code{cos_sin_t}
1437 @itemx cos_sin_t cos_sin (const cl_R& x)
1438 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1439 @cindex @code{cos_sin ()}
1440 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1441 hold only approximately.
1443 @item cl_R tan (const cl_R& x)
1444 @cindex @code{tan ()}
1445 @itemx cl_N tan (const cl_N& x)
1446 Returns @code{tan(x) = sin(x)/cos(x)}.
1448 @item cl_N cis (const cl_R& x)
1449 @cindex @code{cis ()}
1450 @itemx cl_N cis (const cl_N& x)
1451 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1452 @code{e^(i*x) = cos(x) + i*sin(x)}.
1455 @cindex @code{asin ()}
1456 @item cl_N asin (const cl_N& z)
1457 Returns @code{arcsin(z)}. This is defined as
1458 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1459 @code{arcsin(-z) = -arcsin(z)}.
1460 The range of the result is the strip in the complex domain
1461 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1462 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1463 with @code{realpart = pi/2} and @code{imagpart > 0}.
1465 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1469 @item cl_N acos (const cl_N& z)
1470 @cindex @code{acos ()}
1471 Returns @code{arccos(z)}. This is defined as
1472 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1475 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1477 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1478 The range of the result is the strip in the complex domain
1479 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1480 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1481 with @code{realpart = pi} and @code{imagpart > 0}.
1483 Proof: This follows from the results about arcsin.
1487 @cindex @code{atan ()}
1488 @item cl_R atan (const cl_R& x, const cl_R& y)
1489 Returns the angle of the polar representation of the complex number
1490 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1491 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1492 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1493 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1494 with reversed order of arguments.
1496 @item cl_R atan (const cl_R& x)
1497 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1498 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1499 will be an exact number only if @code{x} is the exact @code{0}.
1501 @item cl_N atan (const cl_N& z)
1502 Returns @code{arctan(z)}. This is defined as
1503 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1504 @code{arctan(-z) = -arctan(z)}. The range of the result is
1505 the strip in the complex domain
1506 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1507 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1508 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1510 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1516 @cindex Archimedes' constant
1517 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1520 @item cl_F pi (float_format_t f)
1521 @cindex @code{pi ()}
1522 Returns pi as a float of format @code{f}.
1524 @item cl_F pi (const cl_F& y)
1525 Returns pi in the float format of @code{y}.
1527 @item cl_F pi (void)
1528 Returns pi as a float of format @code{default_float_format}.
1532 @subsection Hyperbolic functions
1535 @item cl_R sinh (const cl_R& x)
1536 @cindex @code{sinh ()}
1537 Returns @code{sinh(x)}.
1539 @item cl_N sinh (const cl_N& z)
1540 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1542 @item cl_R cosh (const cl_R& x)
1543 @cindex @code{cosh ()}
1544 Returns @code{cosh(x)}. The range of the result is the interval
1545 @code{cosh(x) >= 1}.
1547 @item cl_N cosh (const cl_N& z)
1548 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1550 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1551 @cindex @code{cosh_sinh_t}
1552 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1553 @cindex @code{cosh_sinh ()}
1554 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1555 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1556 hold only approximately.
1558 @item cl_R tanh (const cl_R& x)
1559 @cindex @code{tanh ()}
1560 @itemx cl_N tanh (const cl_N& x)
1561 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1563 @item cl_N asinh (const cl_N& z)
1564 @cindex @code{asinh ()}
1565 Returns @code{arsinh(z)}. This is defined as
1566 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1567 @code{arsinh(-z) = -arsinh(z)}.
1569 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1570 Actually, z+sqrt(1+z^2) can never be real and <0, so
1571 -pi < imagpart(arsinh(z)) < pi.
1572 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1573 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1575 The range of the result is the strip in the complex domain
1576 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1577 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1578 with @code{imagpart = pi/2} and @code{realpart < 0}.
1580 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1581 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1582 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1583 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1584 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1585 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1586 If y >= 1, the imagpart is pi/2 and the realpart is
1587 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1590 Moreover, if z is in Range(sqrt),
1591 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1592 (for a proof, see file src/cl_C_asinh.cc).
1595 @item cl_N acosh (const cl_N& z)
1596 @cindex @code{acosh ()}
1597 Returns @code{arcosh(z)}. This is defined as
1598 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1599 The range of the result is the half-strip in the complex domain
1600 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1601 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1603 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1604 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1605 If z is in Range(sqrt), we have
1606 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1607 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1609 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1610 and since the imagpart of both expressions is > -pi, <= pi
1611 ==> arcosh(z) = log(z+sqrt(z^2-1))
1612 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1613 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1614 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1615 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1616 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1618 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1619 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1620 = x^2 + y^2 + sqrt(u^2+v^2)
1625 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1626 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1627 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1628 Otherwise, -z is in Range(sqrt).
1629 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1630 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1631 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1632 and this has realpart > 0.
1633 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1634 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1635 has realpart = 0 and imagpart > 0.
1636 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1637 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1638 This has realpart >= 0 and imagpart = pi.
1641 @item cl_N atanh (const cl_N& z)
1642 @cindex @code{atanh ()}
1643 Returns @code{artanh(z)}. This is defined as
1644 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1645 @code{artanh(-z) = -artanh(z)}. The range of the result is
1646 the strip in the complex domain
1647 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1648 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1649 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1651 Proof: Write z = x+iy. Examine
1652 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1654 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1655 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1656 |x| < 1 ==> imagpart = 0
1659 = (atan(1+x,y) - atan(1-x,-y))/2
1660 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1661 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1662 > (pi - pi/2 - pi/2 )/2 = 0
1663 and (1+x)/y > (1-x)/y
1664 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1665 ==> imagpart < pi/2.
1666 Hence 0 < imagpart < pi/2.
1668 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1673 @subsection Euler gamma
1674 @cindex Euler's constant
1676 Euler's constant C = 0.577@dots{} is returned by the following functions:
1679 @item cl_F eulerconst (float_format_t f)
1680 @cindex @code{eulerconst ()}
1681 Returns Euler's constant as a float of format @code{f}.
1683 @item cl_F eulerconst (const cl_F& y)
1684 Returns Euler's constant in the float format of @code{y}.
1686 @item cl_F eulerconst (void)
1687 Returns Euler's constant as a float of format @code{default_float_format}.
1690 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1691 @cindex Catalan's constant
1694 @item cl_F catalanconst (float_format_t f)
1695 @cindex @code{catalanconst ()}
1696 Returns Catalan's constant as a float of format @code{f}.
1698 @item cl_F catalanconst (const cl_F& y)
1699 Returns Catalan's constant in the float format of @code{y}.
1701 @item cl_F catalanconst (void)
1702 Returns Catalan's constant as a float of format @code{default_float_format}.
1706 @subsection Riemann zeta
1707 @cindex Riemann's zeta
1709 Riemann's zeta function at an integral point @code{s>1} is returned by the
1710 following functions:
1713 @item cl_F zeta (int s, float_format_t f)
1714 @cindex @code{zeta ()}
1715 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1717 @item cl_F zeta (int s, const cl_F& y)
1718 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1720 @item cl_F zeta (int s)
1721 Returns Riemann's zeta function at @code{s} as a float of format
1722 @code{default_float_format}.
1726 @section Functions on integers
1728 @subsection Logical functions
1730 Integers, when viewed as in two's complement notation, can be thought as
1731 infinite bit strings where the bits' values eventually are constant.
1738 The logical operations view integers as such bit strings and operate
1739 on each of the bit positions in parallel.
1742 @item cl_I lognot (const cl_I& x)
1743 @cindex @code{lognot ()}
1744 @itemx cl_I operator ~ (const cl_I& x)
1745 @cindex @code{operator ~ ()}
1746 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1748 @item cl_I logand (const cl_I& x, const cl_I& y)
1749 @cindex @code{logand ()}
1750 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1751 @cindex @code{operator & ()}
1752 Logical and, like @code{x & y} in C.
1754 @item cl_I logior (const cl_I& x, const cl_I& y)
1755 @cindex @code{logior ()}
1756 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1757 @cindex @code{operator | ()}
1758 Logical (inclusive) or, like @code{x | y} in C.
1760 @item cl_I logxor (const cl_I& x, const cl_I& y)
1761 @cindex @code{logxor ()}
1762 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1763 @cindex @code{operator ^ ()}
1764 Exclusive or, like @code{x ^ y} in C.
1766 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1767 @cindex @code{logeqv ()}
1768 Bitwise equivalence, like @code{~(x ^ y)} in C.
1770 @item cl_I lognand (const cl_I& x, const cl_I& y)
1771 @cindex @code{lognand ()}
1772 Bitwise not and, like @code{~(x & y)} in C.
1774 @item cl_I lognor (const cl_I& x, const cl_I& y)
1775 @cindex @code{lognor ()}
1776 Bitwise not or, like @code{~(x | y)} in C.
1778 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1779 @cindex @code{logandc1 ()}
1780 Logical and, complementing the first argument, like @code{~x & y} in C.
1782 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1783 @cindex @code{logandc2 ()}
1784 Logical and, complementing the second argument, like @code{x & ~y} in C.
1786 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1787 @cindex @code{logorc1 ()}
1788 Logical or, complementing the first argument, like @code{~x | y} in C.
1790 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1791 @cindex @code{logorc2 ()}
1792 Logical or, complementing the second argument, like @code{x | ~y} in C.
1795 These operations are all available though the function
1797 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1798 @cindex @code{boole ()}
1800 where @code{op} must have one of the 16 values (each one stands for a function
1801 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1802 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1803 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1804 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1805 @code{boole_orc1}, @code{boole_orc2}.
1806 @cindex @code{boole_clr}
1807 @cindex @code{boole_set}
1808 @cindex @code{boole_1}
1809 @cindex @code{boole_2}
1810 @cindex @code{boole_c1}
1811 @cindex @code{boole_c2}
1812 @cindex @code{boole_and}
1813 @cindex @code{boole_xor}
1814 @cindex @code{boole_eqv}
1815 @cindex @code{boole_nand}
1816 @cindex @code{boole_nor}
1817 @cindex @code{boole_andc1}
1818 @cindex @code{boole_andc2}
1819 @cindex @code{boole_orc1}
1820 @cindex @code{boole_orc2}
1823 Other functions that view integers as bit strings:
1826 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1827 @cindex @code{logtest ()}
1828 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1829 @code{logand(x,y) != 0}.
1831 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1832 @cindex @code{logbitp ()}
1833 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1834 Bit 0 is the least significant bit.
1836 @item uintL logcount (const cl_I& x)
1837 @cindex @code{logcount ()}
1838 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1839 the number of zero bits in @code{x}, if @code{x} < 0.
1842 The following functions operate on intervals of bits in integers.
1845 struct cl_byte @{ uintL size; uintL position; @};
1847 @cindex @code{cl_byte}
1848 represents the bit interval containing the bits
1849 @code{position}@dots{}@code{position+size-1} of an integer.
1850 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1853 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1854 @cindex @code{ldb ()}
1855 extracts the bits of @code{n} described by the bit interval @code{b}
1856 and returns them as a nonnegative integer with @code{b.size} bits.
1858 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1859 @cindex @code{ldb_test ()}
1860 Returns true if some bit described by the bit interval @code{b} is set in
1863 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1864 @cindex @code{dpb ()}
1865 Returns @code{n}, with the bits described by the bit interval @code{b}
1866 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1867 @code{newbyte} are relevant.
1870 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1871 functions are their counterparts without shifting:
1874 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1875 @cindex @code{mask_field ()}
1876 returns an integer with the bits described by the bit interval @code{b}
1877 copied from the corresponding bits in @code{n}, the other bits zero.
1879 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1880 @cindex @code{deposit_field ()}
1881 returns an integer where the bits described by the bit interval @code{b}
1882 come from @code{newbyte} and the other bits come from @code{n}.
1885 The following relations hold:
1889 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1891 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1893 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1896 The following operations on integers as bit strings are efficient shortcuts
1897 for common arithmetic operations:
1900 @item cl_boolean oddp (const cl_I& x)
1901 @cindex @code{oddp ()}
1902 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1903 @code{mod(x,2) != 0}.
1905 @item cl_boolean evenp (const cl_I& x)
1906 @cindex @code{evenp ()}
1907 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1908 @code{mod(x,2) == 0}.
1910 @item cl_I operator << (const cl_I& x, const cl_I& n)
1911 @cindex @code{operator << ()}
1912 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1913 Equivalent to @code{x * expt(2,n)}.
1915 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1916 @cindex @code{operator >> ()}
1917 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1918 Bits shifted out to the right are thrown away.
1919 Equivalent to @code{floor(x / expt(2,n))}.
1921 @item cl_I ash (const cl_I& x, const cl_I& y)
1922 @cindex @code{ash ()}
1923 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1924 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1925 returns @code{floor(x * expt(2,y))}.
1927 @item uintL integer_length (const cl_I& x)
1928 @cindex @code{integer_length ()}
1929 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1930 in two's complement notation. This is the smallest n >= 0 such that
1931 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1934 @item uintL ord2 (const cl_I& x)
1935 @cindex @code{ord2 ()}
1936 @code{x} must be non-zero. This function returns the number of 0 bits at the
1937 right of @code{x} in two's complement notation. This is the largest n >= 0
1938 such that 2^n divides @code{x}.
1940 @item uintL power2p (const cl_I& x)
1941 @cindex @code{power2p ()}
1942 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1943 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1944 (See also the function @code{logp}.)
1948 @subsection Number theoretic functions
1951 @item uint32 gcd (uint32 a, uint32 b)
1952 @cindex @code{gcd ()}
1953 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1954 This function returns the greatest common divisor of @code{a} and @code{b},
1955 normalized to be >= 0.
1957 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1958 @cindex @code{xgcd ()}
1959 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1960 @code{a} and @code{b} and at the same time the representation of @code{g}
1961 as an integral linear combination of @code{a} and @code{b}:
1962 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1963 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1964 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1965 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1966 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1968 @item cl_I lcm (const cl_I& a, const cl_I& b)
1969 @cindex @code{lcm ()}
1970 This function returns the least common multiple of @code{a} and @code{b},
1971 normalized to be >= 0.
1973 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
1974 @cindex @code{logp ()}
1975 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
1976 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
1977 rational number, this function returns true and sets *l = log(a,b), else
1982 @subsection Combinatorial functions
1985 @item cl_I factorial (uintL n)
1986 @cindex @code{factorial ()}
1987 @code{n} must be a small integer >= 0. This function returns the factorial
1988 @code{n}! = @code{1*2*@dots{}*n}.
1990 @item cl_I doublefactorial (uintL n)
1991 @cindex @code{doublefactorial ()}
1992 @code{n} must be a small integer >= 0. This function returns the
1993 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
1994 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
1996 @item cl_I binomial (uintL n, uintL k)
1997 @cindex @code{binomial ()}
1998 @code{n} and @code{k} must be small integers >= 0. This function returns the
1999 binomial coefficient
2001 ${n \choose k} = {n! \over n! (n-k)!}$
2004 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2006 for 0 <= k <= n, 0 else.
2010 @section Functions on floating-point numbers
2012 Recall that a floating-point number consists of a sign @code{s}, an
2013 exponent @code{e} and a mantissa @code{m}. The value of the number is
2014 @code{(-1)^s * 2^e * m}.
2017 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2018 defines the following operations.
2021 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2022 @cindex @code{scale_float ()}
2023 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2024 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2025 because it copies @code{x} and modifies the exponent.
2028 The following functions provide an abstract interface to the underlying
2029 representation of floating-point numbers.
2032 @item sintL float_exponent (const @var{type}& x)
2033 @cindex @code{float_exponent ()}
2034 Returns the exponent @code{e} of @code{x}.
2035 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2036 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2038 @item sintL float_radix (const @var{type}& x)
2039 @cindex @code{float_radix ()}
2040 Returns the base of the floating-point representation. This is always @code{2}.
2042 @item @var{type} float_sign (const @var{type}& x)
2043 @cindex @code{float_sign ()}
2044 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2045 @code{x} >= 0, -1 for @code{x} < 0.
2047 @item uintL float_digits (const @var{type}& x)
2048 @cindex @code{float_digits ()}
2049 Returns the number of mantissa bits in the floating-point representation
2050 of @code{x}, including the hidden bit. The value only depends on the type
2051 of @code{x}, not on its value.
2053 @item uintL float_precision (const @var{type}& x)
2054 @cindex @code{float_precision ()}
2055 Returns the number of significant mantissa bits in the floating-point
2056 representation of @code{x}. Since denormalized numbers are not supported,
2057 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2061 The complete internal representation of a float is encoded in the type
2062 @cindex @code{decoded_float}
2063 @cindex @code{decoded_sfloat}
2064 @cindex @code{decoded_ffloat}
2065 @cindex @code{decoded_dfloat}
2066 @cindex @code{decoded_lfloat}
2067 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2068 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2070 struct decoded_@var{type}float @{
2071 @var{type} mantissa; cl_I exponent; @var{type} sign;
2075 and returned by the function
2078 @item decoded_@var{type}float decode_float (const @var{type}& x)
2079 @cindex @code{decode_float ()}
2080 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2081 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2082 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2083 @code{e} is the same as returned by the function @code{float_exponent}.
2086 A complete decoding in terms of integers is provided as type
2087 @cindex @code{cl_idecoded_float}
2089 struct cl_idecoded_float @{
2090 cl_I mantissa; cl_I exponent; cl_I sign;
2093 by the following function:
2096 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2097 @cindex @code{integer_decode_float ()}
2098 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2099 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2100 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2101 WARNING: The exponent @code{e} is not the same as the one returned by
2102 the functions @code{decode_float} and @code{float_exponent}.
2105 Some other function, implemented only for class @code{cl_F}:
2108 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2109 @cindex @code{float_sign ()}
2110 This returns a floating point number whose precision and absolute value
2111 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2112 zero, it is treated as positive. Same for @code{y}.
2116 @section Conversion functions
2119 @subsection Conversion to floating-point numbers
2121 The type @code{float_format_t} describes a floating-point format.
2122 @cindex @code{float_format_t}
2125 @item float_format_t float_format (uintL n)
2126 @cindex @code{float_format ()}
2127 Returns the smallest float format which guarantees at least @code{n}
2128 decimal digits in the mantissa (after the decimal point).
2130 @item float_format_t float_format (const cl_F& x)
2131 Returns the floating point format of @code{x}.
2133 @item float_format_t default_float_format
2134 @cindex @code{default_float_format}
2135 Global variable: the default float format used when converting rational numbers
2139 To convert a real number to a float, each of the types
2140 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2141 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2142 defines the following operations:
2145 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2146 @cindex @code{cl_float ()}
2147 Returns @code{x} as a float of format @code{f}.
2148 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2149 Returns @code{x} in the float format of @code{y}.
2150 @item cl_F cl_float (const @var{type}&x)
2151 Returns @code{x} as a float of format @code{default_float_format} if
2152 it is an exact number, or @code{x} itself if it is already a float.
2155 Of course, converting a number to a float can lose precision.
2157 Every floating-point format has some characteristic numbers:
2160 @item cl_F most_positive_float (float_format_t f)
2161 @cindex @code{most_positive_float ()}
2162 Returns the largest (most positive) floating point number in float format @code{f}.
2164 @item cl_F most_negative_float (float_format_t f)
2165 @cindex @code{most_negative_float ()}
2166 Returns the smallest (most negative) floating point number in float format @code{f}.
2168 @item cl_F least_positive_float (float_format_t f)
2169 @cindex @code{least_positive_float ()}
2170 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2171 in float format @code{f}.
2173 @item cl_F least_negative_float (float_format_t f)
2174 @cindex @code{least_negative_float ()}
2175 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2176 in float format @code{f}.
2178 @item cl_F float_epsilon (float_format_t f)
2179 @cindex @code{float_epsilon ()}
2180 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2182 @item cl_F float_negative_epsilon (float_format_t f)
2183 @cindex @code{float_negative_epsilon ()}
2184 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2188 @subsection Conversion to rational numbers
2190 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2191 defines the following operation:
2194 @item cl_RA rational (const @var{type}& x)
2195 @cindex @code{rational ()}
2196 Returns the value of @code{x} as an exact number. If @code{x} is already
2197 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2198 the value is a rational number whose denominator is a power of 2.
2201 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2205 @item cl_RA rationalize (const cl_R& x)
2206 @cindex @code{rationalize ()}
2207 If @code{x} is a floating-point number, it actually represents an interval
2208 of real numbers, and this function returns the rational number with
2209 smallest denominator (and smallest numerator, in magnitude)
2210 which lies in this interval.
2211 If @code{x} is already an exact number, this function returns @code{x}.
2214 If @code{x} is any float, one has
2218 @code{cl_float(rational(x),x) = x}
2220 @code{cl_float(rationalize(x),x) = x}
2224 @section Random number generators
2227 A random generator is a machine which produces (pseudo-)random numbers.
2228 The include file @code{<cln/random.h>} defines a class @code{random_state}
2229 which contains the state of a random generator. If you make a copy
2230 of the random number generator, the original one and the copy will produce
2231 the same sequence of random numbers.
2233 The following functions return (pseudo-)random numbers in different formats.
2234 Calling one of these modifies the state of the random number generator in
2235 a complicated but deterministic way.
2238 @cindex @code{random_state}
2239 @cindex @code{default_random_state}
2241 random_state default_random_state
2243 contains a default random number generator. It is used when the functions
2244 below are called without @code{random_state} argument.
2247 @item uint32 random32 (random_state& randomstate)
2248 @itemx uint32 random32 ()
2249 @cindex @code{random32 ()}
2250 Returns a random unsigned 32-bit number. All bits are equally random.
2252 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2253 @itemx cl_I random_I (const cl_I& n)
2254 @cindex @code{random_I ()}
2255 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2256 in the range @code{0 <= x < n}.
2258 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2259 @itemx cl_F random_F (const cl_F& n)
2260 @cindex @code{random_F ()}
2261 @code{n} must be a float > 0. This function returns a random floating-point
2262 number of the same format as @code{n} in the range @code{0 <= x < n}.
2264 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2265 @itemx cl_R random_R (const cl_R& n)
2266 @cindex @code{random_R ()}
2267 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2268 if @code{n} is a float.
2272 @section Obfuscating operators
2273 @cindex modifying operators
2275 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2276 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2277 are not available by default because their
2278 use tends to make programs unreadable. It is trivial to get away without
2279 them. However, if you feel that you absolutely need these operators
2280 to get happy, then add
2282 #define WANT_OBFUSCATING_OPERATORS
2284 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2285 to the beginning of your source files, before the inclusion of any CLN
2286 include files. This flag will enable the following operators:
2288 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2289 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2292 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2293 @cindex @code{operator += ()}
2294 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2295 @cindex @code{operator -= ()}
2296 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2297 @cindex @code{operator *= ()}
2298 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2299 @cindex @code{operator /= ()}
2302 For the class @code{cl_I}:
2305 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2306 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2307 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2308 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2309 @cindex @code{operator &= ()}
2310 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2311 @cindex @code{operator |= ()}
2312 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2313 @cindex @code{operator ^= ()}
2314 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2315 @cindex @code{operator <<= ()}
2316 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2317 @cindex @code{operator >>= ()}
2320 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2321 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2324 @item @var{type}& operator ++ (@var{type}& x)
2325 @cindex @code{operator ++ ()}
2326 The prefix operator @code{++x}.
2328 @item void operator ++ (@var{type}& x, int)
2329 The postfix operator @code{x++}.
2331 @item @var{type}& operator -- (@var{type}& x)
2332 @cindex @code{operator -- ()}
2333 The prefix operator @code{--x}.
2335 @item void operator -- (@var{type}& x, int)
2336 The postfix operator @code{x--}.
2339 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2340 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2344 @chapter Input/Output
2345 @cindex Input/Output
2347 @section Internal and printed representation
2348 @cindex representation
2350 All computations deal with the internal representations of the numbers.
2352 Every number has an external representation as a sequence of ASCII characters.
2353 Several external representations may denote the same number, for example,
2354 "20.0" and "20.000".
2356 Converting an internal to an external representation is called ``printing'',
2358 converting an external to an internal representation is called ``reading''.
2360 In CLN, it is always true that conversion of an internal to an external
2361 representation and then back to an internal representation will yield the
2362 same internal representation. Symbolically: @code{read(print(x)) == x}.
2363 This is called ``print-read consistency''.
2365 Different types of numbers have different external representations (case
2370 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2371 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2372 for decimal integers
2373 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2375 @item Rational numbers
2376 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2377 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2380 @item Floating-point numbers
2381 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2382 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2383 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2384 of the form _@var{prec} may be appended. There must be at least
2385 one digit in the non-exponent part. The exponent has the syntax
2386 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2387 The exponent marker is
2391 @samp{s} for short-floats,
2393 @samp{f} for single-floats,
2395 @samp{d} for double-floats,
2397 @samp{L} for long-floats,
2400 or @samp{e}, which denotes a default float format. The precision specifying
2401 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2402 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2403 function @samp{float_format}.
2405 @item Complex numbers
2406 External representation:
2409 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2410 if @var{imagpart} is negative, its printed representation begins with
2411 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2412 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2413 is rational and the rational number's base is >18, because the @samp{i}
2414 is then read as a digit.
2416 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2421 @section Input functions
2423 Including @code{<cln/io.h>} defines a number of simple input functions
2424 that read from @code{std::istream&}:
2427 @item int freadchar (std::istream& stream)
2428 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2429 if the end of stream was encountered or an error occurred.
2431 @item int funreadchar (std::istream& stream, int c)
2432 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2433 last @code{freadchar} operation on @code{stream}.
2436 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2437 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2438 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2441 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2442 Reads a number from @code{stream} and stores it in the @code{result}.
2445 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2449 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2450 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2451 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2452 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2453 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2454 Reads a number from @code{stream}. The @code{flags} are parameters which
2455 affect the input syntax. Whitespace before the number is silently skipped.
2457 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2458 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2459 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2460 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2461 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2462 Reads a number from a string in memory. The @code{flags} are parameters which
2463 affect the input syntax. The string starts at @code{string} and ends at
2464 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2465 @code{NULL}, denoting the entire string, i.e. equivalent to
2466 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2467 @code{NULL}, the string in memory must contain exactly one number and nothing
2468 more, else a fatal error will be signalled. If @code{end_of_parse}
2469 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2470 the last parsed character (i.e. @code{string_limit} if nothing came after
2471 the number). Whitespace is not allowed.
2474 The structure @code{cl_read_flags} contains the following fields:
2477 @item cl_read_syntax_t syntax
2478 The possible results of the read operation. Possible values are
2479 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2480 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2481 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2483 @item cl_read_lsyntax_t lsyntax
2484 Specifies the language-dependent syntax variant for the read operation.
2488 @item lsyntax_standard
2489 accept standard algebraic notation only, no complex numbers,
2490 @item lsyntax_algebraic
2491 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2492 @item lsyntax_commonlisp
2493 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2494 hexadecimal numbers,
2495 @code{#@var{base}R} for rational numbers in a given base,
2496 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2498 accept all of these extensions.
2501 @item unsigned int rational_base
2502 The base in which rational numbers are read.
2504 @item float_format_t float_flags.default_float_format
2505 The float format used when reading floats with exponent marker @samp{e}.
2507 @item float_format_t float_flags.default_lfloat_format
2508 The float format used when reading floats with exponent marker @samp{l}.
2510 @item cl_boolean float_flags.mantissa_dependent_float_format
2511 When this flag is true, floats specified with more digits than corresponding
2512 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2513 precision corresponding to their number of significant digits.
2517 @section Output functions
2519 Including @code{<cln/io.h>} defines a number of simple output functions
2520 that write to @code{std::ostream&}:
2523 @item void fprintchar (std::ostream& stream, char c)
2524 Prints the character @code{x} literally on the @code{stream}.
2526 @item void fprint (std::ostream& stream, const char * string)
2527 Prints the @code{string} literally on the @code{stream}.
2529 @item void fprintdecimal (std::ostream& stream, int x)
2530 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2531 Prints the integer @code{x} in decimal on the @code{stream}.
2533 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2534 Prints the integer @code{x} in binary (base 2, without prefix)
2535 on the @code{stream}.
2537 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2538 Prints the integer @code{x} in octal (base 8, without prefix)
2539 on the @code{stream}.
2541 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2542 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2543 on the @code{stream}.
2546 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2547 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2548 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2551 @item void fprint (std::ostream& stream, const @var{type}& x)
2552 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2553 Prints the number @code{x} on the @code{stream}. The output may depend
2554 on the global printer settings in the variable @code{default_print_flags}.
2555 The @code{ostream} flags and settings (flags, width and locale) are
2559 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2562 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2564 void print_real (std::ostream& stream, const cl_print_flags& flags,
2566 void print_float (std::ostream& stream, const cl_print_flags& flags,
2568 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2570 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2573 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2574 parameters which affect the output.
2576 The structure type @code{cl_print_flags} contains the following fields:
2579 @item unsigned int rational_base
2580 The base in which rational numbers are printed. Default is @code{10}.
2582 @item cl_boolean rational_readably
2583 If this flag is true, rational numbers are printed with radix specifiers in
2584 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2585 prefixes, trailing dot). Default is false.
2587 @item cl_boolean float_readably
2588 If this flag is true, type specific exponent markers have precedence over 'E'.
2591 @item float_format_t default_float_format
2592 Floating point numbers of this format will be printed using the 'E' exponent
2593 marker. Default is @code{float_format_ffloat}.
2595 @item cl_boolean complex_readably
2596 If this flag is true, complex numbers will be printed using the Common Lisp
2597 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2599 @item cl_string univpoly_varname
2600 Univariate polynomials with no explicit indeterminate name will be printed
2601 using this variable name. Default is @code{"x"}.
2604 The global variable @code{default_print_flags} contains the default values,
2605 used by the function @code{fprint}.
2610 CLN has a class of abstract rings.
2618 Rings can be compared for equality:
2621 @item bool operator== (const cl_ring&, const cl_ring&)
2622 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2623 These compare two rings for equality.
2626 Given a ring @code{R}, the following members can be used.
2629 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2630 @cindex @code{fprint ()}
2631 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2632 @cindex @code{equal ()}
2633 @itemx cl_ring_element R->zero ()
2634 @cindex @code{zero ()}
2635 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2636 @cindex @code{zerop ()}
2637 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2638 @cindex @code{plus ()}
2639 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2640 @cindex @code{minus ()}
2641 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2642 @cindex @code{uminus ()}
2643 @itemx cl_ring_element R->one ()
2644 @cindex @code{one ()}
2645 @itemx cl_ring_element R->canonhom (const cl_I& x)
2646 @cindex @code{canonhom ()}
2647 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2648 @cindex @code{mul ()}
2649 @itemx cl_ring_element R->square (const cl_ring_element& x)
2650 @cindex @code{square ()}
2651 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2652 @cindex @code{expt_pos ()}
2655 The following rings are built-in.
2658 @item cl_null_ring cl_0_ring
2659 The null ring, containing only zero.
2661 @item cl_complex_ring cl_C_ring
2662 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2664 @item cl_real_ring cl_R_ring
2665 The ring of real numbers. This corresponds to the type @code{cl_R}.
2667 @item cl_rational_ring cl_RA_ring
2668 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2670 @item cl_integer_ring cl_I_ring
2671 The ring of integers. This corresponds to the type @code{cl_I}.
2674 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2675 @code{cl_RA_ring}, @code{cl_I_ring}:
2678 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2679 @cindex @code{instanceof ()}
2680 Tests whether the given number is an element of the number ring R.
2684 @chapter Modular integers
2685 @cindex modular integer
2687 @section Modular integer rings
2690 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2691 The modulus is explicitly part of every modular integer. CLN doesn't
2692 allow you to (accidentally) mix elements of different modular rings,
2693 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2694 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2695 doesn't have generic types. So one has to live with runtime checks.)
2697 The class of modular integer rings is
2705 Modular integer ring
2709 @cindex @code{cl_modint_ring}
2711 and the class of all modular integers (elements of modular integer rings) is
2719 Modular integer rings are constructed using the function
2722 @item cl_modint_ring find_modint_ring (const cl_I& N)
2723 @cindex @code{find_modint_ring ()}
2724 This function returns the modular ring @samp{Z/NZ}. It takes care
2725 of finding out about special cases of @code{N}, like powers of two
2726 and odd numbers for which Montgomery multiplication will be a win,
2727 @cindex Montgomery multiplication
2728 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2729 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2730 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2734 Modular integer rings can be compared for equality:
2737 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2738 @cindex @code{operator == ()}
2739 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2740 @cindex @code{operator != ()}
2741 These compare two modular integer rings for equality. Two different calls
2742 to @code{find_modint_ring} with the same argument necessarily return the
2743 same ring because it is memoized in the cache table.
2746 @section Functions on modular integers
2748 Given a modular integer ring @code{R}, the following members can be used.
2751 @item cl_I R->modulus
2752 @cindex @code{modulus}
2753 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2755 @item cl_MI R->zero()
2756 @cindex @code{zero ()}
2757 This returns @code{0 mod N}.
2759 @item cl_MI R->one()
2760 @cindex @code{one ()}
2761 This returns @code{1 mod N}.
2763 @item cl_MI R->canonhom (const cl_I& x)
2764 @cindex @code{canonhom ()}
2765 This returns @code{x mod N}.
2767 @item cl_I R->retract (const cl_MI& x)
2768 @cindex @code{retract ()}
2769 This is a partial inverse function to @code{R->canonhom}. It returns the
2770 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2772 @item cl_MI R->random(random_state& randomstate)
2773 @itemx cl_MI R->random()
2774 @cindex @code{random ()}
2775 This returns a random integer modulo @code{N}.
2778 The following operations are defined on modular integers.
2781 @item cl_modint_ring x.ring ()
2782 @cindex @code{ring ()}
2783 Returns the ring to which the modular integer @code{x} belongs.
2785 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2786 @cindex @code{operator + ()}
2787 Returns the sum of two modular integers. One of the arguments may also
2790 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2791 @cindex @code{operator - ()}
2792 Returns the difference of two modular integers. One of the arguments may also
2795 @item cl_MI operator- (const cl_MI&)
2796 Returns the negative of a modular integer.
2798 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2799 @cindex @code{operator * ()}
2800 Returns the product of two modular integers. One of the arguments may also
2803 @item cl_MI square (const cl_MI&)
2804 @cindex @code{square ()}
2805 Returns the square of a modular integer.
2807 @item cl_MI recip (const cl_MI& x)
2808 @cindex @code{recip ()}
2809 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2810 must be coprime to the modulus, otherwise an error message is issued.
2812 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2813 @cindex @code{div ()}
2814 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2815 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2817 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2818 @cindex @code{expt_pos ()}
2819 @code{y} must be > 0. Returns @code{x^y}.
2821 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2822 @cindex @code{expt ()}
2823 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2824 modulus, else an error message is issued.
2826 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2827 @cindex @code{operator << ()}
2828 Returns @code{x*2^y}.
2830 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2831 @cindex @code{operator >> ()}
2832 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2833 or an error message is issued.
2835 @item bool operator== (const cl_MI&, const cl_MI&)
2836 @cindex @code{operator == ()}
2837 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2838 @cindex @code{operator != ()}
2839 Compares two modular integers, belonging to the same modular integer ring,
2842 @item cl_boolean zerop (const cl_MI& x)
2843 @cindex @code{zerop ()}
2844 Returns true if @code{x} is @code{0 mod N}.
2847 The following output functions are defined (see also the chapter on
2851 @item void fprint (std::ostream& stream, const cl_MI& x)
2852 @cindex @code{fprint ()}
2853 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
2854 @cindex @code{operator << ()}
2855 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2856 on the global printer settings in the variable @code{default_print_flags}.
2860 @chapter Symbolic data types
2861 @cindex symbolic type
2863 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2867 @cindex @code{cl_string}
2877 implements immutable strings.
2879 Strings are constructed through the following constructors:
2882 @item cl_string (const char * s)
2883 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2885 @item cl_string (const char * ptr, unsigned long len)
2886 Returns an immutable copy of the @code{len} characters at
2887 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2890 The following functions are available on strings:
2894 Assignment from @code{cl_string} and @code{const char *}.
2897 @cindex @code{length ()}
2899 @cindex @code{strlen ()}
2900 Returns the length of the string @code{s}.
2903 @cindex @code{operator [] ()}
2904 Returns the @code{i}th character of the string @code{s}.
2905 @code{i} must be in the range @code{0 <= i < s.length()}.
2907 @item bool equal (const cl_string& s1, const cl_string& s2)
2908 @cindex @code{equal ()}
2909 Compares two strings for equality. One of the arguments may also be a
2910 plain @code{const char *}.
2915 @cindex @code{cl_symbol}
2917 Symbols are uniquified strings: all symbols with the same name are shared.
2918 This means that comparison of two symbols is fast (effectively just a pointer
2919 comparison), whereas comparison of two strings must in the worst case walk
2920 both strings until their end.
2921 Symbols are used, for example, as tags for properties, as names of variables
2922 in polynomial rings, etc.
2924 Symbols are constructed through the following constructor:
2927 @item cl_symbol (const cl_string& s)
2928 Looks up or creates a new symbol with a given name.
2931 The following operations are available on symbols:
2934 @item cl_string (const cl_symbol& sym)
2935 Conversion to @code{cl_string}: Returns the string which names the symbol
2938 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2939 @cindex @code{equal ()}
2940 Compares two symbols for equality. This is very fast.
2944 @chapter Univariate polynomials
2946 @cindex univariate polynomial
2948 @section Univariate polynomial rings
2950 CLN implements univariate polynomials (polynomials in one variable) over an
2951 arbitrary ring. The indeterminate variable may be either unnamed (and will be
2952 printed according to @code{default_print_flags.univpoly_varname}, which
2953 defaults to @samp{x}) or carry a given name. The base ring and the
2954 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
2955 (accidentally) mix elements of different polynomial rings, e.g.
2956 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
2957 return a multivariate polynomial, but they are not yet implemented in CLN.)
2959 The classes of univariate polynomial rings are
2967 Univariate polynomial ring
2971 +----------------+-------------------+
2973 Complex polynomial ring | Modular integer polynomial ring
2974 cl_univpoly_complex_ring | cl_univpoly_modint_ring
2975 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
2979 Real polynomial ring |
2980 cl_univpoly_real_ring |
2981 <cln/univpoly_real.h> |
2985 Rational polynomial ring |
2986 cl_univpoly_rational_ring |
2987 <cln/univpoly_rational.h> |
2991 Integer polynomial ring
2992 cl_univpoly_integer_ring
2993 <cln/univpoly_integer.h>
2996 and the corresponding classes of univariate polynomials are
2999 Univariate polynomial
3003 +----------------+-------------------+
3005 Complex polynomial | Modular integer polynomial
3007 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3013 <cln/univpoly_real.h> |
3017 Rational polynomial |
3019 <cln/univpoly_rational.h> |
3025 <cln/univpoly_integer.h>
3028 Univariate polynomial rings are constructed using the functions
3031 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3032 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3033 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3034 @code{R} may be an arbitrary ring. This function takes care of finding out
3035 about special cases of @code{R}, such as the rings of complex numbers,
3036 real numbers, rational numbers, integers, or modular integer rings.
3037 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3038 This ensures that two calls of this function with the same arguments will
3039 return the same polynomial ring.
3041 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3042 @cindex @code{find_univpoly_ring ()}
3043 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3044 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3045 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3046 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3047 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3048 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3049 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3050 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3051 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3052 These functions are equivalent to the general @code{find_univpoly_ring},
3053 only the return type is more specific, according to the base ring's type.
3056 @section Functions on univariate polynomials
3058 Given a univariate polynomial ring @code{R}, the following members can be used.
3061 @item cl_ring R->basering()
3062 @cindex @code{basering ()}
3063 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3065 @item cl_UP R->zero()
3066 @cindex @code{zero ()}
3067 This returns @code{0 in R}, a polynomial of degree -1.
3069 @item cl_UP R->one()
3070 @cindex @code{one ()}
3071 This returns @code{1 in R}, a polynomial of degree <= 0.
3073 @item cl_UP R->canonhom (const cl_I& x)
3074 @cindex @code{canonhom ()}
3075 This returns @code{x in R}, a polynomial of degree <= 0.
3077 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3078 @cindex @code{monomial ()}
3079 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3082 @item cl_UP R->create (sintL degree)
3083 @cindex @code{create ()}
3084 Creates a new polynomial with a given degree. The zero polynomial has degree
3085 @code{-1}. After creating the polynomial, you should put in the coefficients,
3086 using the @code{set_coeff} member function, and then call the @code{finalize}
3090 The following are the only destructive operations on univariate polynomials.
3093 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3094 @cindex @code{set_coeff ()}
3095 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3096 After changing a polynomial and before applying any "normal" operation on it,
3097 you should call its @code{finalize} member function.
3099 @item void finalize (cl_UP& x)
3100 @cindex @code{finalize ()}
3101 This function marks the endpoint of destructive modifications of a polynomial.
3102 It normalizes the internal representation so that subsequent computations have
3103 less overhead. Doing normal computations on unnormalized polynomials may
3104 produce wrong results or crash the program.
3107 The following operations are defined on univariate polynomials.
3110 @item cl_univpoly_ring x.ring ()
3111 @cindex @code{ring ()}
3112 Returns the ring to which the univariate polynomial @code{x} belongs.
3114 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3115 @cindex @code{operator + ()}
3116 Returns the sum of two univariate polynomials.
3118 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3119 @cindex @code{operator - ()}
3120 Returns the difference of two univariate polynomials.
3122 @item cl_UP operator- (const cl_UP&)
3123 Returns the negative of a univariate polynomial.
3125 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3126 @cindex @code{operator * ()}
3127 Returns the product of two univariate polynomials. One of the arguments may
3128 also be a plain integer or an element of the base ring.
3130 @item cl_UP square (const cl_UP&)
3131 @cindex @code{square ()}
3132 Returns the square of a univariate polynomial.
3134 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3135 @cindex @code{expt_pos ()}
3136 @code{y} must be > 0. Returns @code{x^y}.
3138 @item bool operator== (const cl_UP&, const cl_UP&)
3139 @cindex @code{operator == ()}
3140 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3141 @cindex @code{operator != ()}
3142 Compares two univariate polynomials, belonging to the same univariate
3143 polynomial ring, for equality.
3145 @item cl_boolean zerop (const cl_UP& x)
3146 @cindex @code{zerop ()}
3147 Returns true if @code{x} is @code{0 in R}.
3149 @item sintL degree (const cl_UP& x)
3150 @cindex @code{degree ()}
3151 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3153 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3154 @cindex @code{coeff ()}
3155 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3157 @item cl_ring_element x (const cl_ring_element& y)
3158 @cindex @code{operator () ()}
3159 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3160 then @samp{x(y)} returns the value of the substitution of @code{y} into
3163 @item cl_UP deriv (const cl_UP& x)
3164 @cindex @code{deriv ()}
3165 Returns the derivative of the polynomial @code{x} with respect to the
3166 indeterminate @code{X}.
3169 The following output functions are defined (see also the chapter on
3173 @item void fprint (std::ostream& stream, const cl_UP& x)
3174 @cindex @code{fprint ()}
3175 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3176 @cindex @code{operator << ()}
3177 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3178 depend on the global printer settings in the variable
3179 @code{default_print_flags}.
3182 @section Special polynomials
3184 The following functions return special polynomials.
3187 @item cl_UP_I tschebychev (sintL n)
3188 @cindex @code{tschebychev ()}
3189 @cindex Chebyshev polynomial
3190 Returns the n-th Chebyshev polynomial (n >= 0).
3192 @item cl_UP_I hermite (sintL n)
3193 @cindex @code{hermite ()}
3194 @cindex Hermite polynomial
3195 Returns the n-th Hermite polynomial (n >= 0).
3197 @item cl_UP_RA legendre (sintL n)
3198 @cindex @code{legendre ()}
3199 @cindex Legende polynomial
3200 Returns the n-th Legendre polynomial (n >= 0).
3202 @item cl_UP_I laguerre (sintL n)
3203 @cindex @code{laguerre ()}
3204 @cindex Laguerre polynomial
3205 Returns the n-th Laguerre polynomial (n >= 0).
3208 Information how to derive the differential equation satisfied by each
3209 of these polynomials from their definition can be found in the
3210 @code{doc/polynomial/} directory.
3218 Using C++ as an implementation language provides
3222 Efficiency: It compiles to machine code.
3226 Portability: It runs on all platforms supporting a C++ compiler. Because
3227 of the availability of GNU C++, this includes all currently used 32-bit and
3228 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3231 Type safety: The C++ compilers knows about the number types and complains if,
3232 for example, you try to assign a float to an integer variable. However,
3233 a drawback is that C++ doesn't know about generic types, hence a restriction
3234 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3235 arguments belong to the same modular ring cannot be expressed as a compile-time
3239 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3240 @code{=}, @code{==}, ... can be used in infix notation, which is more
3241 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3244 With these language features, there is no need for two separate languages,
3245 one for the implementation of the library and one in which the library's users
3246 can program. This means that a prototype implementation of an algorithm
3247 can be integrated into the library immediately after it has been tested and
3248 debugged. No need to rewrite it in a low-level language after having prototyped
3249 in a high-level language.
3252 @section Memory efficiency
3254 In order to save memory allocations, CLN implements:
3258 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3261 @cindex garbage collection
3262 @cindex reference counting
3263 Garbage collection: A reference counting mechanism makes sure that any
3264 number object's storage is freed immediately when the last reference to the
3267 @cindex immediate numbers
3268 Small integers are represented as immediate values instead of pointers
3269 to heap allocated storage. This means that integers @code{> -2^29},
3270 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3275 @section Speed efficiency
3277 Speed efficiency is obtained by the combination of the following tricks
3282 Small integers, being represented as immediate values, don't require
3283 memory access, just a couple of instructions for each elementary operation.
3285 The kernel of CLN has been written in assembly language for some CPUs
3286 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3288 On all CPUs, CLN may be configured to use the superefficient low-level
3289 routines from GNU GMP version 3.
3291 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3292 algorithm, the Karatsuba multiplication, which is an
3303 For very large numbers (more than 12000 decimal digits), CLN uses
3305 Sch{@"o}nhage-Strassen
3306 @cindex Sch{@"o}nhage-Strassen multiplication
3310 @cindex Schönhage-Strassen multiplication
3312 multiplication, which is an asymptotically optimal multiplication
3315 These fast multiplication algorithms also give improvements in the speed
3316 of division and radix conversion.
3320 @section Garbage collection
3321 @cindex garbage collection
3323 All the number classes are reference count classes: They only contain a pointer
3324 to an object in the heap. Upon construction, assignment and destruction of
3325 number objects, only the objects' reference count are manipulated.
3327 Memory occupied by number objects are automatically reclaimed as soon as
3328 their reference count drops to zero.
3330 For number rings, another strategy is implemented: There is a cache of,
3331 for example, the modular integer rings. A modular integer ring is destroyed
3332 only if its reference count dropped to zero and the cache is about to be
3333 resized. The effect of this strategy is that recently used rings remain
3334 cached, whereas undue memory consumption through cached rings is avoided.
3337 @chapter Using the library
3339 For the following discussion, we will assume that you have installed
3340 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3341 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3342 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3343 environment variables, or directly substitute the appropriate values.
3346 @section Compiler options
3347 @cindex compiler options
3349 Until you have installed CLN in a public place, the following options are
3352 When you compile CLN application code, add the flags
3354 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3356 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3357 When you link CLN application code to form an executable, add the flags
3359 $CLN_TARGETDIR/src/libcln.a
3361 to the C/C++ compiler's command line (@code{make} variable LIBS).
3363 If you did a @code{make install}, the include files are installed in a
3364 public directory (normally @code{/usr/local/include}), hence you don't
3365 need special flags for compiling. The library has been installed to a
3366 public directory as well (normally @code{/usr/local/lib}), hence when
3367 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3369 Since CLN version 1.1, there are two tools to make the creation of
3370 software packages that use CLN easier:
3373 @cindex @code{cln-config}
3374 @code{cln-config} is a shell script that you can use to determine the
3375 compiler and linker command line options required to compile and link a
3376 program with CLN. Start it with @code{--help} to learn about its options
3377 or consult the manpage that comes with it.
3379 @cindex @code{AC_PATH_CLN}
3380 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3383 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3385 This macro determines the location of CLN using @code{cln-config}, which
3386 is either found in the user's path, or from the environment variable
3387 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3388 their version is not earlier than @var{MIN-VERSION} (a default version
3389 will be used if not specified). If the required version was found, sets
3390 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3391 macro is in the file @file{cln.m4} which is installed in
3392 @file{$datadir/aclocal}. Note that if automake was installed with a
3393 different @samp{--prefix} than CLN, you will either have to manually
3394 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3395 aclocal the @samp{-I} option when running it. Here is a possible example
3396 to be included in your package's @file{configure.ac}:
3398 AC_PATH_CLN(1.1.0, [
3399 LIBS="$LIBS $CLN_LIBS"
3400 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3401 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3406 @section Compatibility to old CLN versions
3408 @cindex compatibility
3410 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3411 @code{cln} in order to avoid potential name clashes with other C++
3412 libraries. If you have an old application, you will have to manually
3413 port it to the new scheme. The following principles will help during
3417 All headers are now in a separate subdirectory. Instead of including
3418 @code{cl_}@var{something}@code{.h}, include
3419 @code{cln/}@var{something}@code{.h} now.
3421 All public identifiers (typenames and functions) have lost their
3422 @code{cl_} prefix. Exceptions are all the typenames of number types,
3423 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3424 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3425 names would not be mnemonic enough once the namespace @code{cln} is
3426 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3428 All public @emph{functions} that had by a @code{cl_} in their name still
3429 carry that @code{cl_} if it is intrinsic part of a typename (as in
3430 @code{cl_I_to_int ()}).
3432 When developing other libraries, please keep in mind not to import the
3433 namespace @code{cln} in one of your public header files by saying
3434 @code{using namespace cln;}. This would propagate to other applications
3435 and can cause name clashes there.
3438 @section Include files
3439 @cindex include files
3440 @cindex header files
3442 Here is a summary of the include files and their contents.
3445 @item <cln/object.h>
3446 General definitions, reference counting, garbage collection.
3447 @item <cln/number.h>
3448 The class cl_number.
3449 @item <cln/complex.h>
3450 Functions for class cl_N, the complex numbers.
3452 Functions for class cl_R, the real numbers.
3454 Functions for class cl_F, the floats.
3455 @item <cln/sfloat.h>
3456 Functions for class cl_SF, the short-floats.
3457 @item <cln/ffloat.h>
3458 Functions for class cl_FF, the single-floats.
3459 @item <cln/dfloat.h>
3460 Functions for class cl_DF, the double-floats.
3461 @item <cln/lfloat.h>
3462 Functions for class cl_LF, the long-floats.
3463 @item <cln/rational.h>
3464 Functions for class cl_RA, the rational numbers.
3465 @item <cln/integer.h>
3466 Functions for class cl_I, the integers.
3469 @item <cln/complex_io.h>
3470 Input/Output for class cl_N, the complex numbers.
3471 @item <cln/real_io.h>
3472 Input/Output for class cl_R, the real numbers.
3473 @item <cln/float_io.h>
3474 Input/Output for class cl_F, the floats.
3475 @item <cln/sfloat_io.h>
3476 Input/Output for class cl_SF, the short-floats.
3477 @item <cln/ffloat_io.h>
3478 Input/Output for class cl_FF, the single-floats.
3479 @item <cln/dfloat_io.h>
3480 Input/Output for class cl_DF, the double-floats.
3481 @item <cln/lfloat_io.h>
3482 Input/Output for class cl_LF, the long-floats.
3483 @item <cln/rational_io.h>
3484 Input/Output for class cl_RA, the rational numbers.
3485 @item <cln/integer_io.h>
3486 Input/Output for class cl_I, the integers.
3488 Flags for customizing input operations.
3489 @item <cln/output.h>
3490 Flags for customizing output operations.
3491 @item <cln/malloc.h>
3492 @code{malloc_hook}, @code{free_hook}.
3495 @item <cln/condition.h>
3496 Conditions/exceptions.
3497 @item <cln/string.h>
3499 @item <cln/symbol.h>
3501 @item <cln/proplist.h>
3505 @item <cln/null_ring.h>
3507 @item <cln/complex_ring.h>
3508 The ring of complex numbers.
3509 @item <cln/real_ring.h>
3510 The ring of real numbers.
3511 @item <cln/rational_ring.h>
3512 The ring of rational numbers.
3513 @item <cln/integer_ring.h>
3514 The ring of integers.
3515 @item <cln/numtheory.h>
3516 Number threory functions.
3517 @item <cln/modinteger.h>
3523 @item <cln/GV_number.h>
3524 General vectors over cl_number.
3525 @item <cln/GV_complex.h>
3526 General vectors over cl_N.
3527 @item <cln/GV_real.h>
3528 General vectors over cl_R.
3529 @item <cln/GV_rational.h>
3530 General vectors over cl_RA.
3531 @item <cln/GV_integer.h>
3532 General vectors over cl_I.
3533 @item <cln/GV_modinteger.h>
3534 General vectors of modular integers.
3537 @item <cln/SV_number.h>
3538 Simple vectors over cl_number.
3539 @item <cln/SV_complex.h>
3540 Simple vectors over cl_N.
3541 @item <cln/SV_real.h>
3542 Simple vectors over cl_R.
3543 @item <cln/SV_rational.h>
3544 Simple vectors over cl_RA.
3545 @item <cln/SV_integer.h>
3546 Simple vectors over cl_I.
3547 @item <cln/SV_ringelt.h>
3548 Simple vectors of general ring elements.
3549 @item <cln/univpoly.h>
3550 Univariate polynomials.
3551 @item <cln/univpoly_integer.h>
3552 Univariate polynomials over the integers.
3553 @item <cln/univpoly_rational.h>
3554 Univariate polynomials over the rational numbers.
3555 @item <cln/univpoly_real.h>
3556 Univariate polynomials over the real numbers.
3557 @item <cln/univpoly_complex.h>
3558 Univariate polynomials over the complex numbers.
3559 @item <cln/univpoly_modint.h>
3560 Univariate polynomials over modular integer rings.
3561 @item <cln/timing.h>
3564 Includes all of the above.
3570 A function which computes the nth Fibonacci number can be written as follows.
3571 @cindex Fibonacci number
3574 #include <cln/integer.h>
3575 #include <cln/real.h>
3576 using namespace cln;
3578 // Returns F_n, computed as the nearest integer to
3579 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3580 const cl_I fibonacci (int n)
3582 // Need a precision of ((1+sqrt(5))/2)^-n.
3583 float_format_t prec = float_format((int)(0.208987641*n+5));
3584 cl_R sqrt5 = sqrt(cl_float(5,prec));
3585 cl_R phi = (1+sqrt5)/2;
3586 return round1( expt(phi,n)/sqrt5 );
3590 Let's explain what is going on in detail.
3592 The include file @code{<cln/integer.h>} is necessary because the type
3593 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3594 is needed for the type @code{cl_R} and the floating point number functions.
3595 The order of the include files does not matter. In order not to write
3596 out @code{cln::}@var{foo} in this simple example we can safely import
3597 the whole namespace @code{cln}.
3599 Then comes the function declaration. The argument is an @code{int}, the
3600 result an integer. The return type is defined as @samp{const cl_I}, not
3601 simply @samp{cl_I}, because that allows the compiler to detect typos like
3602 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3603 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3604 number). We use the most specialized possible return type because functions
3605 which call @samp{fibonacci} will be able to profit from the compiler's type
3606 analysis: Adding two integers is slightly more efficient than adding the
3607 same objects declared as complex numbers, because it needs less type
3608 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3609 the size of the resulting executable program.
3611 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3612 integer. In order to get a correct result, the absolute error should be less
3613 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3614 To this end, the first line computes a floating point precision for sqrt(5)
3617 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3618 number and than taking the square root. The converse, first taking the square
3619 root of 5, and then converting to the desired precision, would not work in
3620 CLN: The square root would be computed to a default precision (normally
3621 single-float precision), and the following conversion could not help about
3622 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3623 system and does not represent sqrt(5) in a non-numeric way.
3625 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3626 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3627 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3628 @code{cl_N} because a @samp{round1} does not exist for general complex
3631 When the function returns, all the local variables in the function are
3632 automatically reclaimed (garbage collected). Only the result survives and
3633 gets passed to the caller.
3635 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3636 contains this implementation together with an even faster algorithm.
3638 @section Debugging support
3641 When debugging a CLN application with GNU @code{gdb}, two facilities are
3642 available from the library:
3645 @item The library does type checks, range checks, consistency checks at
3646 many places. When one of these fails, the function @code{cl_abort()} is
3647 called. Its default implementation is to perform an @code{exit(1)}, so
3648 you won't have a core dump. But for debugging, it is best to set a
3649 breakpoint at this function:
3651 (gdb) break cl_abort
3653 When this breakpoint is hit, look at the stack's backtrace:
3658 @item The debugger's normal @code{print} command doesn't know about
3659 CLN's types and therefore prints mostly useless hexadecimal addresses.
3660 CLN offers a function @code{cl_print}, callable from the debugger,
3661 for printing number objects. In order to get this function, you have
3662 to define the macro @samp{CL_DEBUG} and then include all the header files
3663 for which you want @code{cl_print} debugging support. For example:
3664 @cindex @code{CL_DEBUG}
3667 #include <cln/string.h>
3669 Now, if you have in your program a variable @code{cl_string s}, and
3670 inspect it under @code{gdb}, the output may look like this:
3673 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3674 word = 134568800@}@}, @}
3675 (gdb) call cl_print(s)
3679 Note that the output of @code{cl_print} goes to the program's error output,
3680 not to gdb's standard output.
3682 Note, however, that the above facility does not work with all CLN types,
3683 only with number objects and similar. Therefore CLN offers a member function
3684 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3685 is needed for this member function to be implemented. Under @code{gdb},
3686 you call it like this:
3687 @cindex @code{debug_print ()}
3690 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3691 word = 134568800@}@}, @}
3692 (gdb) call s.debug_print()
3695 >call ($1).debug_print()
3700 Unfortunately, this feature does not seem to work under all circumstances.
3704 @chapter Customizing
3707 @section Error handling
3709 When a fatal error occurs, an error message is output to the standard error
3710 output stream, and the function @code{cl_abort} is called. The default
3711 version of this function (provided in the library) terminates the application.
3712 To catch such a fatal error, you need to define the function @code{cl_abort}
3713 yourself, with the prototype
3715 #include <cln/abort.h>
3716 void cl_abort (void);
3718 @cindex @code{cl_abort ()}
3719 This function must not return control to its caller.
3722 @section Floating-point underflow
3725 Floating point underflow denotes the situation when a floating-point number
3726 is to be created which is so close to @code{0} that its exponent is too
3727 low to be represented internally. By default, this causes a fatal error.
3728 If you set the global variable
3730 cl_boolean cl_inhibit_floating_point_underflow
3732 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3733 will be generated instead. The default value of
3734 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3737 @section Customizing I/O
3739 The output of the function @code{fprint} may be customized by changing the
3740 value of the global variable @code{default_print_flags}.
3741 @cindex @code{default_print_flags}
3744 @section Customizing the memory allocator
3746 Every memory allocation of CLN is done through the function pointer
3747 @code{malloc_hook}. Freeing of this memory is done through the function
3748 pointer @code{free_hook}. The default versions of these functions,
3749 provided in the library, call @code{malloc} and @code{free} and check
3750 the @code{malloc} result against @code{NULL}.
3751 If you want to provide another memory allocator, you need to define
3752 the variables @code{malloc_hook} and @code{free_hook} yourself,
3755 #include <cln/malloc.h>
3757 void* (*malloc_hook) (size_t size) = @dots{};
3758 void (*free_hook) (void* ptr) = @dots{};
3761 @cindex @code{malloc_hook ()}
3762 @cindex @code{free_hook ()}
3763 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3765 It is not possible to change the memory allocator at runtime, because
3766 it is already called at program startup by the constructors of some
3779 @c Table of contents