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 4 hours 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. Also, on OSF/1 or
403 Tru64 using gcc-2.95.x, you should specify @samp{--disable-shared} because of
404 linker problems with duplicate symbols in shared libraries.
406 By default, both a shared and a static library are built. You can build
407 CLN as a static (or shared) library only, by calling @code{configure} with
408 the option @samp{--disable-shared} (or @samp{--disable-static}). While
409 shared libraries are usually more convenient to use, they may not work
410 on all architectures. Try disabling them if you run into linker
411 problems. Also, they are generally somewhat slower than static
412 libraries so runtime-critical applications should be linked statically.
415 @subsection Using the GNU MP Library
418 Starting with version 1.1, CLN may be configured to make use of a
419 preinstalled @code{gmp} library. Please make sure that you have at
420 least @code{gmp} version 3.0 installed since earlier versions are
421 unsupported and likely not to work. Enabling this feature by calling
422 @code{configure} with the option @samp{--with-gmp} is known to be quite
423 a boost for CLN's performance.
425 If you have installed the @code{gmp} library and its header file in
426 some place where your compiler cannot find it by default, you must help
427 @code{configure} by setting @code{CPPFLAGS} and @code{LDFLAGS}. Here is
431 $ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
432 CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
436 @section Installing the library
439 As with any autoconfiguring GNU software, installation is as easy as this:
445 The @samp{make install} command installs the library and the include files
446 into public places (@file{/usr/local/lib/} and @file{/usr/local/include/},
447 if you haven't specified a @code{--prefix} option to @code{configure}).
448 This step may require superuser privileges.
450 If you have already built the library and wish to install it, but didn't
451 specify @code{--prefix=@dots{}} at configure time, just re-run
452 @code{configure}, giving it the same options as the first time, plus
453 the @code{--prefix=@dots{}} option.
458 You can remove system-dependent files generated by @code{make} through
464 You can remove all files generated by @code{make}, thus reverting to a
465 virgin distribution of CLN, through
472 @chapter Ordinary number types
474 CLN implements the following class hierarchy:
482 Real or complex number
491 +-------------------+-------------------+
493 Rational number Floating-point number
495 <cln/rational.h> <cln/float.h>
497 | +--------------+--------------+--------------+
499 cl_I Short-Float Single-Float Double-Float Long-Float
500 <cln/integer.h> cl_SF cl_FF cl_DF cl_LF
501 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
504 @cindex @code{cl_number}
505 @cindex abstract class
506 The base class @code{cl_number} is an abstract base class.
507 It is not useful to declare a variable of this type except if you want
508 to completely disable compile-time type checking and use run-time type
513 @cindex complex number
514 The class @code{cl_N} comprises real and complex numbers. There is
515 no special class for complex numbers since complex numbers with imaginary
516 part @code{0} are automatically converted to real numbers.
519 The class @code{cl_R} comprises real numbers of different kinds. It is an
523 @cindex rational number
525 The class @code{cl_RA} comprises exact real numbers: rational numbers, including
526 integers. There is no special class for non-integral rational numbers
527 since rational numbers with denominator @code{1} are automatically converted
531 The class @code{cl_F} implements floating-point approximations to real numbers.
532 It is an abstract class.
535 @section Exact numbers
538 Some numbers are represented as exact numbers: there is no loss of information
539 when such a number is converted from its mathematical value to its internal
540 representation. On exact numbers, the elementary operations (@code{+},
541 @code{-}, @code{*}, @code{/}, comparisons, @dots{}) compute the completely
544 In CLN, the exact numbers are:
548 rational numbers (including integers),
550 complex numbers whose real and imaginary parts are both rational numbers.
553 Rational numbers are always normalized to the form
554 @code{@var{numerator}/@var{denominator}} where the numerator and denominator
555 are coprime integers and the denominator is positive. If the resulting
556 denominator is @code{1}, the rational number is converted to an integer.
558 @cindex immediate numbers
559 Small integers (typically in the range @code{-2^29}@dots{}@code{2^29-1},
560 for 32-bit machines) are especially efficient, because they consume no heap
561 allocation. Otherwise the distinction between these immediate integers
562 (called ``fixnums'') and heap allocated integers (called ``bignums'')
563 is completely transparent.
566 @section Floating-point numbers
567 @cindex floating-point number
569 Not all real numbers can be represented exactly. (There is an easy mathematical
570 proof for this: Only a countable set of numbers can be stored exactly in
571 a computer, even if one assumes that it has unlimited storage. But there
572 are uncountably many real numbers.) So some approximation is needed.
573 CLN implements ordinary floating-point numbers, with mantissa and exponent.
575 @cindex rounding error
576 The elementary operations (@code{+}, @code{-}, @code{*}, @code{/}, @dots{})
577 only return approximate results. For example, the value of the expression
578 @code{(cl_F) 0.3 + (cl_F) 0.4} prints as @samp{0.70000005}, not as
579 @samp{0.7}. Rounding errors like this one are inevitable when computing
580 with floating-point numbers.
582 Nevertheless, CLN rounds the floating-point results of the operations @code{+},
583 @code{-}, @code{*}, @code{/}, @code{sqrt} according to the ``round-to-even''
584 rule: It first computes the exact mathematical result and then returns the
585 floating-point number which is nearest to this. If two floating-point numbers
586 are equally distant from the ideal result, the one with a @code{0} in its least
587 significant mantissa bit is chosen.
589 Similarly, testing floating point numbers for equality @samp{x == y}
590 is gambling with random errors. Better check for @samp{abs(x - y) < epsilon}
591 for some well-chosen @code{epsilon}.
593 Floating point numbers come in four flavors:
598 Short floats, type @code{cl_SF}.
599 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
600 and 17 mantissa bits (including the ``hidden'' bit).
601 They don't consume heap allocation.
605 Single floats, type @code{cl_FF}.
606 They have 1 sign bit, 8 exponent bits (including the exponent's sign),
607 and 24 mantissa bits (including the ``hidden'' bit).
608 In CLN, they are represented as IEEE single-precision floating point numbers.
609 This corresponds closely to the C/C++ type @samp{float}.
613 Double floats, type @code{cl_DF}.
614 They have 1 sign bit, 11 exponent bits (including the exponent's sign),
615 and 53 mantissa bits (including the ``hidden'' bit).
616 In CLN, they are represented as IEEE double-precision floating point numbers.
617 This corresponds closely to the C/C++ type @samp{double}.
621 Long floats, type @code{cl_LF}.
622 They have 1 sign bit, 32 exponent bits (including the exponent's sign),
623 and n mantissa bits (including the ``hidden'' bit), where n >= 64.
624 The precision of a long float is unlimited, but once created, a long float
625 has a fixed precision. (No ``lazy recomputation''.)
628 Of course, computations with long floats are more expensive than those
629 with smaller floating-point formats.
631 CLN does not implement features like NaNs, denormalized numbers and
632 gradual underflow. If the exponent range of some floating-point type
633 is too limited for your application, choose another floating-point type
634 with larger exponent range.
637 As a user of CLN, you can forget about the differences between the
638 four floating-point types and just declare all your floating-point
639 variables as being of type @code{cl_F}. This has the advantage that
640 when you change the precision of some computation (say, from @code{cl_DF}
641 to @code{cl_LF}), you don't have to change the code, only the precision
642 of the initial values. Also, many transcendental functions have been
643 declared as returning a @code{cl_F} when the argument is a @code{cl_F},
644 but such declarations are missing for the types @code{cl_SF}, @code{cl_FF},
645 @code{cl_DF}, @code{cl_LF}. (Such declarations would be wrong if
646 the floating point contagion rule happened to change in the future.)
649 @section Complex numbers
650 @cindex complex number
652 Complex numbers, as implemented by the class @code{cl_N}, have a real
653 part and an imaginary part, both real numbers. A complex number whose
654 imaginary part is the exact number @code{0} is automatically converted
657 Complex numbers can arise from real numbers alone, for example
658 through application of @code{sqrt} or transcendental functions.
664 Conversions from any class to any its superclasses (``base classes'' in
665 C++ terminology) is done automatically.
667 Conversions from the C built-in types @samp{long} and @samp{unsigned long}
668 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
669 @code{cl_N} and @code{cl_number}.
671 Conversions from the C built-in types @samp{int} and @samp{unsigned int}
672 are provided for the classes @code{cl_I}, @code{cl_RA}, @code{cl_R},
673 @code{cl_N} and @code{cl_number}. However, these conversions emphasize
674 efficiency. Their range is therefore limited:
678 The conversion from @samp{int} works only if the argument is < 2^29 and > -2^29.
680 The conversion from @samp{unsigned int} works only if the argument is < 2^29.
683 In a declaration like @samp{cl_I x = 10;} the C++ compiler is able to
684 do the conversion of @code{10} from @samp{int} to @samp{cl_I} at compile time
685 already. On the other hand, code like @samp{cl_I x = 1000000000;} is
687 So, if you want to be sure that an @samp{int} whose magnitude is not guaranteed
688 to be < 2^29 is correctly converted to a @samp{cl_I}, first convert it to a
689 @samp{long}. Similarly, if a large @samp{unsigned int} is to be converted to a
690 @samp{cl_I}, first convert it to an @samp{unsigned long}.
692 Conversions from the C built-in type @samp{float} are provided for the classes
693 @code{cl_FF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
695 Conversions from the C built-in type @samp{double} are provided for the classes
696 @code{cl_DF}, @code{cl_F}, @code{cl_R}, @code{cl_N} and @code{cl_number}.
698 Conversions from @samp{const char *} are provided for the classes
699 @code{cl_I}, @code{cl_RA},
700 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F},
701 @code{cl_R}, @code{cl_N}.
702 The easiest way to specify a value which is outside of the range of the
703 C++ built-in types is therefore to specify it as a string, like this:
706 cl_I order_of_rubiks_cube_group = "43252003274489856000";
708 Note that this conversion is done at runtime, not at compile-time.
710 Conversions from @code{cl_I} to the C built-in types @samp{int},
711 @samp{unsigned int}, @samp{long}, @samp{unsigned long} are provided through
715 @item int cl_I_to_int (const cl_I& x)
716 @cindex @code{cl_I_to_int ()}
717 @itemx unsigned int cl_I_to_uint (const cl_I& x)
718 @cindex @code{cl_I_to_uint ()}
719 @itemx long cl_I_to_long (const cl_I& x)
720 @cindex @code{cl_I_to_long ()}
721 @itemx unsigned long cl_I_to_ulong (const cl_I& x)
722 @cindex @code{cl_I_to_ulong ()}
723 Returns @code{x} as element of the C type @var{ctype}. If @code{x} is not
724 representable in the range of @var{ctype}, a runtime error occurs.
727 Conversions from the classes @code{cl_I}, @code{cl_RA},
728 @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}, @code{cl_F} and
730 to the C built-in types @samp{float} and @samp{double} are provided through
734 @item float float_approx (const @var{type}& x)
735 @cindex @code{float_approx ()}
736 @itemx double double_approx (const @var{type}& x)
737 @cindex @code{double_approx ()}
738 Returns an approximation of @code{x} of C type @var{ctype}.
739 If @code{abs(x)} is too close to 0 (underflow), 0 is returned.
740 If @code{abs(x)} is too large (overflow), an IEEE infinity is returned.
743 Conversions from any class to any of its subclasses (``derived classes'' in
744 C++ terminology) are not provided. Instead, you can assert and check
745 that a value belongs to a certain subclass, and return it as element of that
746 class, using the @samp{As} and @samp{The} macros.
747 @cindex @code{As()()}
748 @code{As(@var{type})(@var{value})} checks that @var{value} belongs to
749 @var{type} and returns it as such.
750 @cindex @code{The()()}
751 @code{The(@var{type})(@var{value})} assumes that @var{value} belongs to
752 @var{type} and returns it as such. It is your responsibility to ensure
753 that this assumption is valid. Since macros and namespaces don't go
754 together well, there is an equivalent to @samp{The}: the template
762 if (!(x >= 0)) abort();
763 cl_I ten_x_a = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
764 // In general, it would be a rational number.
765 cl_I ten_x_b = the<cl_I>(expt(10,x)); // The same as above.
770 @chapter Functions on numbers
772 Each of the number classes declares its mathematical operations in the
773 corresponding include file. For example, if your code operates with
774 objects of type @code{cl_I}, it should @code{#include <cln/integer.h>}.
777 @section Constructing numbers
779 Here is how to create number objects ``from nothing''.
782 @subsection Constructing integers
784 @code{cl_I} objects are most easily constructed from C integers and from
785 strings. See @ref{Conversions}.
788 @subsection Constructing rational numbers
790 @code{cl_RA} objects can be constructed from strings. The syntax
791 for rational numbers is described in @ref{Internal and printed representation}.
792 Another standard way to produce a rational number is through application
793 of @samp{operator /} or @samp{recip} on integers.
796 @subsection Constructing floating-point numbers
798 @code{cl_F} objects with low precision are most easily constructed from
799 C @samp{float} and @samp{double}. See @ref{Conversions}.
801 To construct a @code{cl_F} with high precision, you can use the conversion
802 from @samp{const char *}, but you have to specify the desired precision
803 within the string. (See @ref{Internal and printed representation}.)
806 cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
808 will set @samp{e} to the given value, with a precision of 40 decimal digits.
810 The programmatic way to construct a @code{cl_F} with high precision is
811 through the @code{cl_float} conversion function, see
812 @ref{Conversion to floating-point numbers}. For example, to compute
813 @code{e} to 40 decimal places, first construct 1.0 to 40 decimal places
814 and then apply the exponential function:
816 float_format_t precision = float_format(40);
817 cl_F e = exp(cl_float(1,precision));
821 @subsection Constructing complex numbers
823 Non-real @code{cl_N} objects are normally constructed through the function
825 cl_N complex (const cl_R& realpart, const cl_R& imagpart)
827 See @ref{Elementary complex functions}.
830 @section Elementary functions
832 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
833 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
834 defines the following operations:
837 @item @var{type} operator + (const @var{type}&, const @var{type}&)
838 @cindex @code{operator + ()}
841 @item @var{type} operator - (const @var{type}&, const @var{type}&)
842 @cindex @code{operator - ()}
845 @item @var{type} operator - (const @var{type}&)
846 Returns the negative of the argument.
848 @item @var{type} plus1 (const @var{type}& x)
849 @cindex @code{plus1 ()}
850 Returns @code{x + 1}.
852 @item @var{type} minus1 (const @var{type}& x)
853 @cindex @code{minus1 ()}
854 Returns @code{x - 1}.
856 @item @var{type} operator * (const @var{type}&, const @var{type}&)
857 @cindex @code{operator * ()}
860 @item @var{type} square (const @var{type}& x)
861 @cindex @code{square ()}
862 Returns @code{x * x}.
865 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
866 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
867 defines the following operations:
870 @item @var{type} operator / (const @var{type}&, const @var{type}&)
871 @cindex @code{operator / ()}
874 @item @var{type} recip (const @var{type}&)
875 @cindex @code{recip ()}
876 Returns the reciprocal of the argument.
879 The class @code{cl_I} doesn't define a @samp{/} operation because
880 in the C/C++ language this operator, applied to integral types,
881 denotes the @samp{floor} or @samp{truncate} operation (which one of these,
882 is implementation dependent). (@xref{Rounding functions}.)
883 Instead, @code{cl_I} defines an ``exact quotient'' function:
886 @item cl_I exquo (const cl_I& x, const cl_I& y)
887 @cindex @code{exquo ()}
888 Checks that @code{y} divides @code{x}, and returns the quotient @code{x}/@code{y}.
891 The following exponentiation functions are defined:
894 @item cl_I expt_pos (const cl_I& x, const cl_I& y)
895 @cindex @code{expt_pos ()}
896 @itemx cl_RA expt_pos (const cl_RA& x, const cl_I& y)
897 @code{y} must be > 0. Returns @code{x^y}.
899 @item cl_RA expt (const cl_RA& x, const cl_I& y)
900 @cindex @code{expt ()}
901 @itemx cl_R expt (const cl_R& x, const cl_I& y)
902 @itemx cl_N expt (const cl_N& x, const cl_I& y)
906 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
907 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
908 defines the following operation:
911 @item @var{type} abs (const @var{type}& x)
912 @cindex @code{abs ()}
913 Returns the absolute value of @code{x}.
914 This is @code{x} if @code{x >= 0}, and @code{-x} if @code{x <= 0}.
917 The class @code{cl_N} implements this as follows:
920 @item cl_R abs (const cl_N x)
921 Returns the absolute value of @code{x}.
924 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
925 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
926 defines the following operation:
929 @item @var{type} signum (const @var{type}& x)
930 @cindex @code{signum ()}
931 Returns the sign of @code{x}, in the same number format as @code{x}.
932 This is defined as @code{x / abs(x)} if @code{x} is non-zero, and
933 @code{x} if @code{x} is zero. If @code{x} is real, the value is either
938 @section Elementary rational functions
940 Each of the classes @code{cl_RA}, @code{cl_I} defines the following operations:
943 @item cl_I numerator (const @var{type}& x)
944 @cindex @code{numerator ()}
945 Returns the numerator of @code{x}.
947 @item cl_I denominator (const @var{type}& x)
948 @cindex @code{denominator ()}
949 Returns the denominator of @code{x}.
952 The numerator and denominator of a rational number are normalized in such
953 a way that they have no factor in common and the denominator is positive.
956 @section Elementary complex functions
958 The class @code{cl_N} defines the following operation:
961 @item cl_N complex (const cl_R& a, const cl_R& b)
962 @cindex @code{complex ()}
963 Returns the complex number @code{a+bi}, that is, the complex number with
964 real part @code{a} and imaginary part @code{b}.
967 Each of the classes @code{cl_N}, @code{cl_R} defines the following operations:
970 @item cl_R realpart (const @var{type}& x)
971 @cindex @code{realpart ()}
972 Returns the real part of @code{x}.
974 @item cl_R imagpart (const @var{type}& x)
975 @cindex @code{imagpart ()}
976 Returns the imaginary part of @code{x}.
978 @item @var{type} conjugate (const @var{type}& x)
979 @cindex @code{conjugate ()}
980 Returns the complex conjugate of @code{x}.
983 We have the relations
987 @code{x = complex(realpart(x), imagpart(x))}
989 @code{conjugate(x) = complex(realpart(x), -imagpart(x))}
996 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
997 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
998 defines the following operations:
1001 @item bool operator == (const @var{type}&, const @var{type}&)
1002 @cindex @code{operator == ()}
1003 @itemx bool operator != (const @var{type}&, const @var{type}&)
1004 @cindex @code{operator != ()}
1005 Comparison, as in C and C++.
1007 @item uint32 equal_hashcode (const @var{type}&)
1008 @cindex @code{equal_hashcode ()}
1009 Returns a 32-bit hash code that is the same for any two numbers which are
1010 the same according to @code{==}. This hash code depends on the number's value,
1011 not its type or precision.
1013 @item cl_boolean zerop (const @var{type}& x)
1014 @cindex @code{zerop ()}
1015 Compare against zero: @code{x == 0}
1018 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1019 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1020 defines the following operations:
1023 @item cl_signean compare (const @var{type}& x, const @var{type}& y)
1024 @cindex @code{compare ()}
1025 Compares @code{x} and @code{y}. Returns +1 if @code{x}>@code{y},
1026 -1 if @code{x}<@code{y}, 0 if @code{x}=@code{y}.
1028 @item bool operator <= (const @var{type}&, const @var{type}&)
1029 @cindex @code{operator <= ()}
1030 @itemx bool operator < (const @var{type}&, const @var{type}&)
1031 @cindex @code{operator < ()}
1032 @itemx bool operator >= (const @var{type}&, const @var{type}&)
1033 @cindex @code{operator >= ()}
1034 @itemx bool operator > (const @var{type}&, const @var{type}&)
1035 @cindex @code{operator > ()}
1036 Comparison, as in C and C++.
1038 @item cl_boolean minusp (const @var{type}& x)
1039 @cindex @code{minusp ()}
1040 Compare against zero: @code{x < 0}
1042 @item cl_boolean plusp (const @var{type}& x)
1043 @cindex @code{plusp ()}
1044 Compare against zero: @code{x > 0}
1046 @item @var{type} max (const @var{type}& x, const @var{type}& y)
1047 @cindex @code{max ()}
1048 Return the maximum of @code{x} and @code{y}.
1050 @item @var{type} min (const @var{type}& x, const @var{type}& y)
1051 @cindex @code{min ()}
1052 Return the minimum of @code{x} and @code{y}.
1055 When a floating point number and a rational number are compared, the float
1056 is first converted to a rational number using the function @code{rational}.
1057 Since a floating point number actually represents an interval of real numbers,
1058 the result might be surprising.
1059 For example, @code{(cl_F)(cl_R)"1/3" == (cl_R)"1/3"} returns false because
1060 there is no floating point number whose value is exactly @code{1/3}.
1063 @section Rounding functions
1066 When a real number is to be converted to an integer, there is no ``best''
1067 rounding. The desired rounding function depends on the application.
1068 The Common Lisp and ISO Lisp standards offer four rounding functions:
1072 This is the largest integer <=@code{x}.
1075 This is the smallest integer >=@code{x}.
1078 Among the integers between 0 and @code{x} (inclusive) the one nearest to @code{x}.
1081 The integer nearest to @code{x}. If @code{x} is exactly halfway between two
1082 integers, choose the even one.
1085 These functions have different advantages:
1087 @code{floor} and @code{ceiling} are translation invariant:
1088 @code{floor(x+n) = floor(x) + n} and @code{ceiling(x+n) = ceiling(x) + n}
1089 for every @code{x} and every integer @code{n}.
1091 On the other hand, @code{truncate} and @code{round} are symmetric:
1092 @code{truncate(-x) = -truncate(x)} and @code{round(-x) = -round(x)},
1093 and furthermore @code{round} is unbiased: on the ``average'', it rounds
1094 down exactly as often as it rounds up.
1096 The functions are related like this:
1100 @code{ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1}
1101 for rational numbers @code{m/n} (@code{m}, @code{n} integers, @code{n}>0), and
1103 @code{truncate(x) = sign(x) * floor(abs(x))}
1106 Each of the classes @code{cl_R}, @code{cl_RA},
1107 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1108 defines the following operations:
1111 @item cl_I floor1 (const @var{type}& x)
1112 @cindex @code{floor1 ()}
1113 Returns @code{floor(x)}.
1114 @item cl_I ceiling1 (const @var{type}& x)
1115 @cindex @code{ceiling1 ()}
1116 Returns @code{ceiling(x)}.
1117 @item cl_I truncate1 (const @var{type}& x)
1118 @cindex @code{truncate1 ()}
1119 Returns @code{truncate(x)}.
1120 @item cl_I round1 (const @var{type}& x)
1121 @cindex @code{round1 ()}
1122 Returns @code{round(x)}.
1125 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1126 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1127 defines the following operations:
1130 @item cl_I floor1 (const @var{type}& x, const @var{type}& y)
1131 Returns @code{floor(x/y)}.
1132 @item cl_I ceiling1 (const @var{type}& x, const @var{type}& y)
1133 Returns @code{ceiling(x/y)}.
1134 @item cl_I truncate1 (const @var{type}& x, const @var{type}& y)
1135 Returns @code{truncate(x/y)}.
1136 @item cl_I round1 (const @var{type}& x, const @var{type}& y)
1137 Returns @code{round(x/y)}.
1140 These functions are called @samp{floor1}, @dots{} here instead of
1141 @samp{floor}, @dots{}, because on some systems, system dependent include
1142 files define @samp{floor} and @samp{ceiling} as macros.
1144 In many cases, one needs both the quotient and the remainder of a division.
1145 It is more efficient to compute both at the same time than to perform
1146 two divisions, one for quotient and the next one for the remainder.
1147 The following functions therefore return a structure containing both
1148 the quotient and the remainder. The suffix @samp{2} indicates the number
1149 of ``return values''. The remainder is defined as follows:
1153 for the computation of @code{quotient = floor(x)},
1154 @code{remainder = x - quotient},
1156 for the computation of @code{quotient = floor(x,y)},
1157 @code{remainder = x - quotient*y},
1160 and similarly for the other three operations.
1162 Each of the classes @code{cl_R}, @code{cl_RA},
1163 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1164 defines the following operations:
1167 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1168 @itemx @var{type}_div_t floor2 (const @var{type}& x)
1169 @itemx @var{type}_div_t ceiling2 (const @var{type}& x)
1170 @itemx @var{type}_div_t truncate2 (const @var{type}& x)
1171 @itemx @var{type}_div_t round2 (const @var{type}& x)
1174 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_I},
1175 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1176 defines the following operations:
1179 @item struct @var{type}_div_t @{ cl_I quotient; @var{type} remainder; @};
1180 @itemx @var{type}_div_t floor2 (const @var{type}& x, const @var{type}& y)
1181 @cindex @code{floor2 ()}
1182 @itemx @var{type}_div_t ceiling2 (const @var{type}& x, const @var{type}& y)
1183 @cindex @code{ceiling2 ()}
1184 @itemx @var{type}_div_t truncate2 (const @var{type}& x, const @var{type}& y)
1185 @cindex @code{truncate2 ()}
1186 @itemx @var{type}_div_t round2 (const @var{type}& x, const @var{type}& y)
1187 @cindex @code{round2 ()}
1190 Sometimes, one wants the quotient as a floating-point number (of the
1191 same format as the argument, if the argument is a float) instead of as
1192 an integer. The prefix @samp{f} indicates this.
1195 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1196 defines the following operations:
1199 @item @var{type} ffloor (const @var{type}& x)
1200 @cindex @code{ffloor ()}
1201 @itemx @var{type} fceiling (const @var{type}& x)
1202 @cindex @code{fceiling ()}
1203 @itemx @var{type} ftruncate (const @var{type}& x)
1204 @cindex @code{ftruncate ()}
1205 @itemx @var{type} fround (const @var{type}& x)
1206 @cindex @code{fround ()}
1209 and similarly for class @code{cl_R}, but with return type @code{cl_F}.
1211 The class @code{cl_R} defines the following operations:
1214 @item cl_F ffloor (const @var{type}& x, const @var{type}& y)
1215 @itemx cl_F fceiling (const @var{type}& x, const @var{type}& y)
1216 @itemx cl_F ftruncate (const @var{type}& x, const @var{type}& y)
1217 @itemx cl_F fround (const @var{type}& x, const @var{type}& y)
1220 These functions also exist in versions which return both the quotient
1221 and the remainder. The suffix @samp{2} indicates this.
1224 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1225 defines the following operations:
1226 @cindex @code{cl_F_fdiv_t}
1227 @cindex @code{cl_SF_fdiv_t}
1228 @cindex @code{cl_FF_fdiv_t}
1229 @cindex @code{cl_DF_fdiv_t}
1230 @cindex @code{cl_LF_fdiv_t}
1233 @item struct @var{type}_fdiv_t @{ @var{type} quotient; @var{type} remainder; @};
1234 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x)
1235 @cindex @code{ffloor2 ()}
1236 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x)
1237 @cindex @code{fceiling2 ()}
1238 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x)
1239 @cindex @code{ftruncate2 ()}
1240 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x)
1241 @cindex @code{fround2 ()}
1243 and similarly for class @code{cl_R}, but with quotient type @code{cl_F}.
1244 @cindex @code{cl_R_fdiv_t}
1246 The class @code{cl_R} defines the following operations:
1249 @item struct @var{type}_fdiv_t @{ cl_F quotient; cl_R remainder; @};
1250 @itemx @var{type}_fdiv_t ffloor2 (const @var{type}& x, const @var{type}& y)
1251 @itemx @var{type}_fdiv_t fceiling2 (const @var{type}& x, const @var{type}& y)
1252 @itemx @var{type}_fdiv_t ftruncate2 (const @var{type}& x, const @var{type}& y)
1253 @itemx @var{type}_fdiv_t fround2 (const @var{type}& x, const @var{type}& y)
1256 Other applications need only the remainder of a division.
1257 The remainder of @samp{floor} and @samp{ffloor} is called @samp{mod}
1258 (abbreviation of ``modulo''). The remainder @samp{truncate} and
1259 @samp{ftruncate} is called @samp{rem} (abbreviation of ``remainder'').
1263 @code{mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y}
1265 @code{rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y}
1268 If @code{x} and @code{y} are both >= 0, @code{mod(x,y) = rem(x,y) >= 0}.
1269 In general, @code{mod(x,y)} has the sign of @code{y} or is zero,
1270 and @code{rem(x,y)} has the sign of @code{x} or is zero.
1272 The classes @code{cl_R}, @code{cl_I} define the following operations:
1275 @item @var{type} mod (const @var{type}& x, const @var{type}& y)
1276 @cindex @code{mod ()}
1277 @itemx @var{type} rem (const @var{type}& x, const @var{type}& y)
1278 @cindex @code{rem ()}
1284 Each of the classes @code{cl_R},
1285 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
1286 defines the following operation:
1289 @item @var{type} sqrt (const @var{type}& x)
1290 @cindex @code{sqrt ()}
1291 @code{x} must be >= 0. This function returns the square root of @code{x},
1292 normalized to be >= 0. If @code{x} is the square of a rational number,
1293 @code{sqrt(x)} will be a rational number, else it will return a
1294 floating-point approximation.
1297 The classes @code{cl_RA}, @code{cl_I} define the following operation:
1300 @item cl_boolean sqrtp (const @var{type}& x, @var{type}* root)
1301 @cindex @code{sqrtp ()}
1302 This tests whether @code{x} is a perfect square. If so, it returns true
1303 and the exact square root in @code{*root}, else it returns false.
1306 Furthermore, for integers, similarly:
1309 @item cl_boolean isqrt (const @var{type}& x, @var{type}* root)
1310 @cindex @code{isqrt ()}
1311 @code{x} should be >= 0. This function sets @code{*root} to
1312 @code{floor(sqrt(x))} and returns the same value as @code{sqrtp}:
1313 the boolean value @code{(expt(*root,2) == x)}.
1316 For @code{n}th roots, the classes @code{cl_RA}, @code{cl_I}
1317 define the following operation:
1320 @item cl_boolean rootp (const @var{type}& x, const cl_I& n, @var{type}* root)
1321 @cindex @code{rootp ()}
1322 @code{x} must be >= 0. @code{n} must be > 0.
1323 This tests whether @code{x} is an @code{n}th power of a rational number.
1324 If so, it returns true and the exact root in @code{*root}, else it returns
1328 The only square root function which accepts negative numbers is the one
1329 for class @code{cl_N}:
1332 @item cl_N sqrt (const cl_N& z)
1333 @cindex @code{sqrt ()}
1334 Returns the square root of @code{z}, as defined by the formula
1335 @code{sqrt(z) = exp(log(z)/2)}. Conversion to a floating-point type
1336 or to a complex number are done if necessary. The range of the result is the
1337 right half plane @code{realpart(sqrt(z)) >= 0}
1338 including the positive imaginary axis and 0, but excluding
1339 the negative imaginary axis.
1340 The result is an exact number only if @code{z} is an exact number.
1344 @section Transcendental functions
1345 @cindex transcendental functions
1347 The transcendental functions return an exact result if the argument
1348 is exact and the result is exact as well. Otherwise they must return
1349 inexact numbers even if the argument is exact.
1350 For example, @code{cos(0) = 1} returns the rational number @code{1}.
1353 @subsection Exponential and logarithmic functions
1356 @item cl_R exp (const cl_R& x)
1357 @cindex @code{exp ()}
1358 @itemx cl_N exp (const cl_N& x)
1359 Returns the exponential function of @code{x}. This is @code{e^x} where
1360 @code{e} is the base of the natural logarithms. The range of the result
1361 is the entire complex plane excluding 0.
1363 @item cl_R ln (const cl_R& x)
1364 @cindex @code{ln ()}
1365 @code{x} must be > 0. Returns the (natural) logarithm of x.
1367 @item cl_N log (const cl_N& x)
1368 @cindex @code{log ()}
1369 Returns the (natural) logarithm of x. If @code{x} is real and positive,
1370 this is @code{ln(x)}. In general, @code{log(x) = log(abs(x)) + i*phase(x)}.
1371 The range of the result is the strip in the complex plane
1372 @code{-pi < imagpart(log(x)) <= pi}.
1374 @item cl_R phase (const cl_N& x)
1375 @cindex @code{phase ()}
1376 Returns the angle part of @code{x} in its polar representation as a
1377 complex number. That is, @code{phase(x) = atan(realpart(x),imagpart(x))}.
1378 This is also the imaginary part of @code{log(x)}.
1379 The range of the result is the interval @code{-pi < phase(x) <= pi}.
1380 The result will be an exact number only if @code{zerop(x)} or
1381 if @code{x} is real and positive.
1383 @item cl_R log (const cl_R& a, const cl_R& b)
1384 @code{a} and @code{b} must be > 0. Returns the logarithm of @code{a} with
1385 respect to base @code{b}. @code{log(a,b) = ln(a)/ln(b)}.
1386 The result can be exact only if @code{a = 1} or if @code{a} and @code{b}
1389 @item cl_N log (const cl_N& a, const cl_N& b)
1390 Returns the logarithm of @code{a} with respect to base @code{b}.
1391 @code{log(a,b) = log(a)/log(b)}.
1393 @item cl_N expt (const cl_N& x, const cl_N& y)
1394 @cindex @code{expt ()}
1395 Exponentiation: Returns @code{x^y = exp(y*log(x))}.
1398 The constant e = exp(1) = 2.71828@dots{} is returned by the following functions:
1401 @item cl_F exp1 (float_format_t f)
1402 @cindex @code{exp1 ()}
1403 Returns e as a float of format @code{f}.
1405 @item cl_F exp1 (const cl_F& y)
1406 Returns e in the float format of @code{y}.
1408 @item cl_F exp1 (void)
1409 Returns e as a float of format @code{default_float_format}.
1413 @subsection Trigonometric functions
1416 @item cl_R sin (const cl_R& x)
1417 @cindex @code{sin ()}
1418 Returns @code{sin(x)}. The range of the result is the interval
1419 @code{-1 <= sin(x) <= 1}.
1421 @item cl_N sin (const cl_N& z)
1422 Returns @code{sin(z)}. The range of the result is the entire complex plane.
1424 @item cl_R cos (const cl_R& x)
1425 @cindex @code{cos ()}
1426 Returns @code{cos(x)}. The range of the result is the interval
1427 @code{-1 <= cos(x) <= 1}.
1429 @item cl_N cos (const cl_N& x)
1430 Returns @code{cos(z)}. The range of the result is the entire complex plane.
1432 @item struct cos_sin_t @{ cl_R cos; cl_R sin; @};
1433 @cindex @code{cos_sin_t}
1434 @itemx cos_sin_t cos_sin (const cl_R& x)
1435 Returns both @code{sin(x)} and @code{cos(x)}. This is more efficient than
1436 @cindex @code{cos_sin ()}
1437 computing them separately. The relation @code{cos^2 + sin^2 = 1} will
1438 hold only approximately.
1440 @item cl_R tan (const cl_R& x)
1441 @cindex @code{tan ()}
1442 @itemx cl_N tan (const cl_N& x)
1443 Returns @code{tan(x) = sin(x)/cos(x)}.
1445 @item cl_N cis (const cl_R& x)
1446 @cindex @code{cis ()}
1447 @itemx cl_N cis (const cl_N& x)
1448 Returns @code{exp(i*x)}. The name @samp{cis} means ``cos + i sin'', because
1449 @code{e^(i*x) = cos(x) + i*sin(x)}.
1452 @cindex @code{asin ()}
1453 @item cl_N asin (const cl_N& z)
1454 Returns @code{arcsin(z)}. This is defined as
1455 @code{arcsin(z) = log(iz+sqrt(1-z^2))/i} and satisfies
1456 @code{arcsin(-z) = -arcsin(z)}.
1457 The range of the result is the strip in the complex domain
1458 @code{-pi/2 <= realpart(arcsin(z)) <= pi/2}, excluding the numbers
1459 with @code{realpart = -pi/2} and @code{imagpart < 0} and the numbers
1460 with @code{realpart = pi/2} and @code{imagpart > 0}.
1462 Proof: This follows from arcsin(z) = arsinh(iz)/i and the corresponding
1466 @item cl_N acos (const cl_N& z)
1467 @cindex @code{acos ()}
1468 Returns @code{arccos(z)}. This is defined as
1469 @code{arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i}
1472 @code{arccos(z) = 2*log(sqrt((1+z)/2)+i*sqrt((1-z)/2))/i}
1474 and satisfies @code{arccos(-z) = pi - arccos(z)}.
1475 The range of the result is the strip in the complex domain
1476 @code{0 <= realpart(arcsin(z)) <= pi}, excluding the numbers
1477 with @code{realpart = 0} and @code{imagpart < 0} and the numbers
1478 with @code{realpart = pi} and @code{imagpart > 0}.
1480 Proof: This follows from the results about arcsin.
1484 @cindex @code{atan ()}
1485 @item cl_R atan (const cl_R& x, const cl_R& y)
1486 Returns the angle of the polar representation of the complex number
1487 @code{x+iy}. This is @code{atan(y/x)} if @code{x>0}. The range of
1488 the result is the interval @code{-pi < atan(x,y) <= pi}. The result will
1489 be an exact number only if @code{x > 0} and @code{y} is the exact @code{0}.
1490 WARNING: In Common Lisp, this function is called as @code{(atan y x)},
1491 with reversed order of arguments.
1493 @item cl_R atan (const cl_R& x)
1494 Returns @code{arctan(x)}. This is the same as @code{atan(1,x)}. The range
1495 of the result is the interval @code{-pi/2 < atan(x) < pi/2}. The result
1496 will be an exact number only if @code{x} is the exact @code{0}.
1498 @item cl_N atan (const cl_N& z)
1499 Returns @code{arctan(z)}. This is defined as
1500 @code{arctan(z) = (log(1+iz)-log(1-iz)) / 2i} and satisfies
1501 @code{arctan(-z) = -arctan(z)}. The range of the result is
1502 the strip in the complex domain
1503 @code{-pi/2 <= realpart(arctan(z)) <= pi/2}, excluding the numbers
1504 with @code{realpart = -pi/2} and @code{imagpart >= 0} and the numbers
1505 with @code{realpart = pi/2} and @code{imagpart <= 0}.
1507 Proof: arctan(z) = artanh(iz)/i, we know the range of the artanh function.
1513 @cindex Archimedes' constant
1514 Archimedes' constant pi = 3.14@dots{} is returned by the following functions:
1517 @item cl_F pi (float_format_t f)
1518 @cindex @code{pi ()}
1519 Returns pi as a float of format @code{f}.
1521 @item cl_F pi (const cl_F& y)
1522 Returns pi in the float format of @code{y}.
1524 @item cl_F pi (void)
1525 Returns pi as a float of format @code{default_float_format}.
1529 @subsection Hyperbolic functions
1532 @item cl_R sinh (const cl_R& x)
1533 @cindex @code{sinh ()}
1534 Returns @code{sinh(x)}.
1536 @item cl_N sinh (const cl_N& z)
1537 Returns @code{sinh(z)}. The range of the result is the entire complex plane.
1539 @item cl_R cosh (const cl_R& x)
1540 @cindex @code{cosh ()}
1541 Returns @code{cosh(x)}. The range of the result is the interval
1542 @code{cosh(x) >= 1}.
1544 @item cl_N cosh (const cl_N& z)
1545 Returns @code{cosh(z)}. The range of the result is the entire complex plane.
1547 @item struct cosh_sinh_t @{ cl_R cosh; cl_R sinh; @};
1548 @cindex @code{cosh_sinh_t}
1549 @itemx cosh_sinh_t cosh_sinh (const cl_R& x)
1550 @cindex @code{cosh_sinh ()}
1551 Returns both @code{sinh(x)} and @code{cosh(x)}. This is more efficient than
1552 computing them separately. The relation @code{cosh^2 - sinh^2 = 1} will
1553 hold only approximately.
1555 @item cl_R tanh (const cl_R& x)
1556 @cindex @code{tanh ()}
1557 @itemx cl_N tanh (const cl_N& x)
1558 Returns @code{tanh(x) = sinh(x)/cosh(x)}.
1560 @item cl_N asinh (const cl_N& z)
1561 @cindex @code{asinh ()}
1562 Returns @code{arsinh(z)}. This is defined as
1563 @code{arsinh(z) = log(z+sqrt(1+z^2))} and satisfies
1564 @code{arsinh(-z) = -arsinh(z)}.
1566 Proof: Knowing the range of log, we know -pi < imagpart(arsinh(z)) <= pi.
1567 Actually, z+sqrt(1+z^2) can never be real and <0, so
1568 -pi < imagpart(arsinh(z)) < pi.
1569 We have (z+sqrt(1+z^2))*(-z+sqrt(1+(-z)^2)) = (1+z^2)-z^2 = 1, hence the
1570 logs of both factors sum up to 0 mod 2*pi*i, hence to 0.
1572 The range of the result is the strip in the complex domain
1573 @code{-pi/2 <= imagpart(arsinh(z)) <= pi/2}, excluding the numbers
1574 with @code{imagpart = -pi/2} and @code{realpart > 0} and the numbers
1575 with @code{imagpart = pi/2} and @code{realpart < 0}.
1577 Proof: Write z = x+iy. Because of arsinh(-z) = -arsinh(z), we may assume
1578 that z is in Range(sqrt), that is, x>=0 and, if x=0, then y>=0.
1579 If x > 0, then Re(z+sqrt(1+z^2)) = x + Re(sqrt(1+z^2)) >= x > 0,
1580 so -pi/2 < imagpart(log(z+sqrt(1+z^2))) < pi/2.
1581 If x = 0 and y >= 0, arsinh(z) = log(i*y+sqrt(1-y^2)).
1582 If y <= 1, the realpart is 0 and the imagpart is >= 0 and <= pi/2.
1583 If y >= 1, the imagpart is pi/2 and the realpart is
1584 log(y+sqrt(y^2-1)) >= log(y) >= 0.
1587 Moreover, if z is in Range(sqrt),
1588 log(sqrt(1+z^2)+z) = 2 artanh(z/(1+sqrt(1+z^2)))
1589 (for a proof, see file src/cl_C_asinh.cc).
1592 @item cl_N acosh (const cl_N& z)
1593 @cindex @code{acosh ()}
1594 Returns @code{arcosh(z)}. This is defined as
1595 @code{arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))}.
1596 The range of the result is the half-strip in the complex domain
1597 @code{-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0},
1598 excluding the numbers with @code{realpart = 0} and @code{-pi < imagpart < 0}.
1600 Proof: sqrt((z+1)/2) and sqrt((z-1)/2)) lie in Range(sqrt), hence does
1601 their sum, hence its log has an imagpart <= pi/2 and > -pi/2.
1602 If z is in Range(sqrt), we have
1603 sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1)
1604 ==> (sqrt((z+1)/2)+sqrt((z-1)/2))^2 = (z+1)/2 + sqrt(z^2-1) + (z-1)/2
1606 ==> arcosh(z) = log(z+sqrt(z^2-1)) mod 2*pi*i
1607 and since the imagpart of both expressions is > -pi, <= pi
1608 ==> arcosh(z) = log(z+sqrt(z^2-1))
1609 To prove that the realpart of this is >= 0, write z = x+iy with x>=0,
1610 z^2-1 = u+iv with u = x^2-y^2-1, v = 2xy,
1611 sqrt(z^2-1) = p+iq with p = sqrt((sqrt(u^2+v^2)+u)/2) >= 0,
1612 q = sqrt((sqrt(u^2+v^2)-u)/2) * sign(v),
1613 then |z+sqrt(z^2-1)|^2 = |x+iy + p+iq|^2
1615 = x^2 + 2xp + p^2 + y^2 + 2yq + q^2
1616 >= x^2 + p^2 + y^2 + q^2 (since x>=0, p>=0, yq>=0)
1617 = x^2 + y^2 + sqrt(u^2+v^2)
1622 hence realpart(log(z+sqrt(z^2-1))) = log(|z+sqrt(z^2-1)|) >= 0.
1623 Equality holds only if y = 0 and u <= 0, i.e. 0 <= x < 1.
1624 In this case arcosh(z) = log(x+i*sqrt(1-x^2)) has imagpart >=0.
1625 Otherwise, -z is in Range(sqrt).
1626 If y != 0, sqrt((z+1)/2) = i^sign(y) * sqrt((-z-1)/2),
1627 sqrt((z-1)/2) = i^sign(y) * sqrt((-z+1)/2),
1628 hence arcosh(z) = sign(y)*pi/2*i + arcosh(-z),
1629 and this has realpart > 0.
1630 If y = 0 and -1<=x<=0, we still have sqrt(z+1)*sqrt(z-1) = sqrt(z^2-1),
1631 ==> arcosh(z) = log(z+sqrt(z^2-1)) = log(x+i*sqrt(1-x^2))
1632 has realpart = 0 and imagpart > 0.
1633 If y = 0 and x<=-1, however, sqrt(z+1)*sqrt(z-1) = - sqrt(z^2-1),
1634 ==> arcosh(z) = log(z-sqrt(z^2-1)) = pi*i + arcosh(-z).
1635 This has realpart >= 0 and imagpart = pi.
1638 @item cl_N atanh (const cl_N& z)
1639 @cindex @code{atanh ()}
1640 Returns @code{artanh(z)}. This is defined as
1641 @code{artanh(z) = (log(1+z)-log(1-z)) / 2} and satisfies
1642 @code{artanh(-z) = -artanh(z)}. The range of the result is
1643 the strip in the complex domain
1644 @code{-pi/2 <= imagpart(artanh(z)) <= pi/2}, excluding the numbers
1645 with @code{imagpart = -pi/2} and @code{realpart <= 0} and the numbers
1646 with @code{imagpart = pi/2} and @code{realpart >= 0}.
1648 Proof: Write z = x+iy. Examine
1649 imagpart(artanh(z)) = (atan(1+x,y) - atan(1-x,-y))/2.
1651 x > 1 ==> imagpart = -pi/2, realpart = 1/2 log((x+1)/(x-1)) > 0,
1652 x < -1 ==> imagpart = pi/2, realpart = 1/2 log((-x-1)/(-x+1)) < 0,
1653 |x| < 1 ==> imagpart = 0
1656 = (atan(1+x,y) - atan(1-x,-y))/2
1657 = ((pi/2 - atan((1+x)/y)) - (-pi/2 - atan((1-x)/-y)))/2
1658 = (pi - atan((1+x)/y) - atan((1-x)/y))/2
1659 > (pi - pi/2 - pi/2 )/2 = 0
1660 and (1+x)/y > (1-x)/y
1661 ==> atan((1+x)/y) > atan((-1+x)/y) = - atan((1-x)/y)
1662 ==> imagpart < pi/2.
1663 Hence 0 < imagpart < pi/2.
1665 By artanh(z) = -artanh(-z) and case 2, -pi/2 < imagpart < 0.
1670 @subsection Euler gamma
1671 @cindex Euler's constant
1673 Euler's constant C = 0.577@dots{} is returned by the following functions:
1676 @item cl_F eulerconst (float_format_t f)
1677 @cindex @code{eulerconst ()}
1678 Returns Euler's constant as a float of format @code{f}.
1680 @item cl_F eulerconst (const cl_F& y)
1681 Returns Euler's constant in the float format of @code{y}.
1683 @item cl_F eulerconst (void)
1684 Returns Euler's constant as a float of format @code{default_float_format}.
1687 Catalan's constant G = 0.915@dots{} is returned by the following functions:
1688 @cindex Catalan's constant
1691 @item cl_F catalanconst (float_format_t f)
1692 @cindex @code{catalanconst ()}
1693 Returns Catalan's constant as a float of format @code{f}.
1695 @item cl_F catalanconst (const cl_F& y)
1696 Returns Catalan's constant in the float format of @code{y}.
1698 @item cl_F catalanconst (void)
1699 Returns Catalan's constant as a float of format @code{default_float_format}.
1703 @subsection Riemann zeta
1704 @cindex Riemann's zeta
1706 Riemann's zeta function at an integral point @code{s>1} is returned by the
1707 following functions:
1710 @item cl_F zeta (int s, float_format_t f)
1711 @cindex @code{zeta ()}
1712 Returns Riemann's zeta function at @code{s} as a float of format @code{f}.
1714 @item cl_F zeta (int s, const cl_F& y)
1715 Returns Riemann's zeta function at @code{s} in the float format of @code{y}.
1717 @item cl_F zeta (int s)
1718 Returns Riemann's zeta function at @code{s} as a float of format
1719 @code{default_float_format}.
1723 @section Functions on integers
1725 @subsection Logical functions
1727 Integers, when viewed as in two's complement notation, can be thought as
1728 infinite bit strings where the bits' values eventually are constant.
1735 The logical operations view integers as such bit strings and operate
1736 on each of the bit positions in parallel.
1739 @item cl_I lognot (const cl_I& x)
1740 @cindex @code{lognot ()}
1741 @itemx cl_I operator ~ (const cl_I& x)
1742 @cindex @code{operator ~ ()}
1743 Logical not, like @code{~x} in C. This is the same as @code{-1-x}.
1745 @item cl_I logand (const cl_I& x, const cl_I& y)
1746 @cindex @code{logand ()}
1747 @itemx cl_I operator & (const cl_I& x, const cl_I& y)
1748 @cindex @code{operator & ()}
1749 Logical and, like @code{x & y} in C.
1751 @item cl_I logior (const cl_I& x, const cl_I& y)
1752 @cindex @code{logior ()}
1753 @itemx cl_I operator | (const cl_I& x, const cl_I& y)
1754 @cindex @code{operator | ()}
1755 Logical (inclusive) or, like @code{x | y} in C.
1757 @item cl_I logxor (const cl_I& x, const cl_I& y)
1758 @cindex @code{logxor ()}
1759 @itemx cl_I operator ^ (const cl_I& x, const cl_I& y)
1760 @cindex @code{operator ^ ()}
1761 Exclusive or, like @code{x ^ y} in C.
1763 @item cl_I logeqv (const cl_I& x, const cl_I& y)
1764 @cindex @code{logeqv ()}
1765 Bitwise equivalence, like @code{~(x ^ y)} in C.
1767 @item cl_I lognand (const cl_I& x, const cl_I& y)
1768 @cindex @code{lognand ()}
1769 Bitwise not and, like @code{~(x & y)} in C.
1771 @item cl_I lognor (const cl_I& x, const cl_I& y)
1772 @cindex @code{lognor ()}
1773 Bitwise not or, like @code{~(x | y)} in C.
1775 @item cl_I logandc1 (const cl_I& x, const cl_I& y)
1776 @cindex @code{logandc1 ()}
1777 Logical and, complementing the first argument, like @code{~x & y} in C.
1779 @item cl_I logandc2 (const cl_I& x, const cl_I& y)
1780 @cindex @code{logandc2 ()}
1781 Logical and, complementing the second argument, like @code{x & ~y} in C.
1783 @item cl_I logorc1 (const cl_I& x, const cl_I& y)
1784 @cindex @code{logorc1 ()}
1785 Logical or, complementing the first argument, like @code{~x | y} in C.
1787 @item cl_I logorc2 (const cl_I& x, const cl_I& y)
1788 @cindex @code{logorc2 ()}
1789 Logical or, complementing the second argument, like @code{x | ~y} in C.
1792 These operations are all available though the function
1794 @item cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
1795 @cindex @code{boole ()}
1797 where @code{op} must have one of the 16 values (each one stands for a function
1798 which combines two bits into one bit): @code{boole_clr}, @code{boole_set},
1799 @code{boole_1}, @code{boole_2}, @code{boole_c1}, @code{boole_c2},
1800 @code{boole_and}, @code{boole_ior}, @code{boole_xor}, @code{boole_eqv},
1801 @code{boole_nand}, @code{boole_nor}, @code{boole_andc1}, @code{boole_andc2},
1802 @code{boole_orc1}, @code{boole_orc2}.
1803 @cindex @code{boole_clr}
1804 @cindex @code{boole_set}
1805 @cindex @code{boole_1}
1806 @cindex @code{boole_2}
1807 @cindex @code{boole_c1}
1808 @cindex @code{boole_c2}
1809 @cindex @code{boole_and}
1810 @cindex @code{boole_xor}
1811 @cindex @code{boole_eqv}
1812 @cindex @code{boole_nand}
1813 @cindex @code{boole_nor}
1814 @cindex @code{boole_andc1}
1815 @cindex @code{boole_andc2}
1816 @cindex @code{boole_orc1}
1817 @cindex @code{boole_orc2}
1820 Other functions that view integers as bit strings:
1823 @item cl_boolean logtest (const cl_I& x, const cl_I& y)
1824 @cindex @code{logtest ()}
1825 Returns true if some bit is set in both @code{x} and @code{y}, i.e. if
1826 @code{logand(x,y) != 0}.
1828 @item cl_boolean logbitp (const cl_I& n, const cl_I& x)
1829 @cindex @code{logbitp ()}
1830 Returns true if the @code{n}th bit (from the right) of @code{x} is set.
1831 Bit 0 is the least significant bit.
1833 @item uintL logcount (const cl_I& x)
1834 @cindex @code{logcount ()}
1835 Returns the number of one bits in @code{x}, if @code{x} >= 0, or
1836 the number of zero bits in @code{x}, if @code{x} < 0.
1839 The following functions operate on intervals of bits in integers.
1842 struct cl_byte @{ uintL size; uintL position; @};
1844 @cindex @code{cl_byte}
1845 represents the bit interval containing the bits
1846 @code{position}@dots{}@code{position+size-1} of an integer.
1847 The constructor @code{cl_byte(size,position)} constructs a @code{cl_byte}.
1850 @item cl_I ldb (const cl_I& n, const cl_byte& b)
1851 @cindex @code{ldb ()}
1852 extracts the bits of @code{n} described by the bit interval @code{b}
1853 and returns them as a nonnegative integer with @code{b.size} bits.
1855 @item cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
1856 @cindex @code{ldb_test ()}
1857 Returns true if some bit described by the bit interval @code{b} is set in
1860 @item cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1861 @cindex @code{dpb ()}
1862 Returns @code{n}, with the bits described by the bit interval @code{b}
1863 replaced by @code{newbyte}. Only the lowest @code{b.size} bits of
1864 @code{newbyte} are relevant.
1867 The functions @code{ldb} and @code{dpb} implicitly shift. The following
1868 functions are their counterparts without shifting:
1871 @item cl_I mask_field (const cl_I& n, const cl_byte& b)
1872 @cindex @code{mask_field ()}
1873 returns an integer with the bits described by the bit interval @code{b}
1874 copied from the corresponding bits in @code{n}, the other bits zero.
1876 @item cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
1877 @cindex @code{deposit_field ()}
1878 returns an integer where the bits described by the bit interval @code{b}
1879 come from @code{newbyte} and the other bits come from @code{n}.
1882 The following relations hold:
1886 @code{ldb (n, b) = mask_field(n, b) >> b.position},
1888 @code{dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)},
1890 @code{deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)}.
1893 The following operations on integers as bit strings are efficient shortcuts
1894 for common arithmetic operations:
1897 @item cl_boolean oddp (const cl_I& x)
1898 @cindex @code{oddp ()}
1899 Returns true if the least significant bit of @code{x} is 1. Equivalent to
1900 @code{mod(x,2) != 0}.
1902 @item cl_boolean evenp (const cl_I& x)
1903 @cindex @code{evenp ()}
1904 Returns true if the least significant bit of @code{x} is 0. Equivalent to
1905 @code{mod(x,2) == 0}.
1907 @item cl_I operator << (const cl_I& x, const cl_I& n)
1908 @cindex @code{operator << ()}
1909 Shifts @code{x} by @code{n} bits to the left. @code{n} should be >=0.
1910 Equivalent to @code{x * expt(2,n)}.
1912 @item cl_I operator >> (const cl_I& x, const cl_I& n)
1913 @cindex @code{operator >> ()}
1914 Shifts @code{x} by @code{n} bits to the right. @code{n} should be >=0.
1915 Bits shifted out to the right are thrown away.
1916 Equivalent to @code{floor(x / expt(2,n))}.
1918 @item cl_I ash (const cl_I& x, const cl_I& y)
1919 @cindex @code{ash ()}
1920 Shifts @code{x} by @code{y} bits to the left (if @code{y}>=0) or
1921 by @code{-y} bits to the right (if @code{y}<=0). In other words, this
1922 returns @code{floor(x * expt(2,y))}.
1924 @item uintL integer_length (const cl_I& x)
1925 @cindex @code{integer_length ()}
1926 Returns the number of bits (excluding the sign bit) needed to represent @code{x}
1927 in two's complement notation. This is the smallest n >= 0 such that
1928 -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1931 @item uintL ord2 (const cl_I& x)
1932 @cindex @code{ord2 ()}
1933 @code{x} must be non-zero. This function returns the number of 0 bits at the
1934 right of @code{x} in two's complement notation. This is the largest n >= 0
1935 such that 2^n divides @code{x}.
1937 @item uintL power2p (const cl_I& x)
1938 @cindex @code{power2p ()}
1939 @code{x} must be > 0. This function checks whether @code{x} is a power of 2.
1940 If @code{x} = 2^(n-1), it returns n. Else it returns 0.
1941 (See also the function @code{logp}.)
1945 @subsection Number theoretic functions
1948 @item uint32 gcd (uint32 a, uint32 b)
1949 @cindex @code{gcd ()}
1950 @itemx cl_I gcd (const cl_I& a, const cl_I& b)
1951 This function returns the greatest common divisor of @code{a} and @code{b},
1952 normalized to be >= 0.
1954 @item cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
1955 @cindex @code{xgcd ()}
1956 This function (``extended gcd'') returns the greatest common divisor @code{g} of
1957 @code{a} and @code{b} and at the same time the representation of @code{g}
1958 as an integral linear combination of @code{a} and @code{b}:
1959 @code{u} and @code{v} with @code{u*a+v*b = g}, @code{g} >= 0.
1960 @code{u} and @code{v} will be normalized to be of smallest possible absolute
1961 value, in the following sense: If @code{a} and @code{b} are non-zero, and
1962 @code{abs(a) != abs(b)}, @code{u} and @code{v} will satisfy the inequalities
1963 @code{abs(u) <= abs(b)/(2*g)}, @code{abs(v) <= abs(a)/(2*g)}.
1965 @item cl_I lcm (const cl_I& a, const cl_I& b)
1966 @cindex @code{lcm ()}
1967 This function returns the least common multiple of @code{a} and @code{b},
1968 normalized to be >= 0.
1970 @item cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
1971 @cindex @code{logp ()}
1972 @itemx cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
1973 @code{a} must be > 0. @code{b} must be >0 and != 1. If log(a,b) is
1974 rational number, this function returns true and sets *l = log(a,b), else
1979 @subsection Combinatorial functions
1982 @item cl_I factorial (uintL n)
1983 @cindex @code{factorial ()}
1984 @code{n} must be a small integer >= 0. This function returns the factorial
1985 @code{n}! = @code{1*2*@dots{}*n}.
1987 @item cl_I doublefactorial (uintL n)
1988 @cindex @code{doublefactorial ()}
1989 @code{n} must be a small integer >= 0. This function returns the
1990 doublefactorial @code{n}!! = @code{1*3*@dots{}*n} or
1991 @code{n}!! = @code{2*4*@dots{}*n}, respectively.
1993 @item cl_I binomial (uintL n, uintL k)
1994 @cindex @code{binomial ()}
1995 @code{n} and @code{k} must be small integers >= 0. This function returns the
1996 binomial coefficient
1998 ${n \choose k} = {n! \over n! (n-k)!}$
2001 (@code{n} choose @code{k}) = @code{n}! / @code{k}! @code{(n-k)}!
2003 for 0 <= k <= n, 0 else.
2007 @section Functions on floating-point numbers
2009 Recall that a floating-point number consists of a sign @code{s}, an
2010 exponent @code{e} and a mantissa @code{m}. The value of the number is
2011 @code{(-1)^s * 2^e * m}.
2014 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2015 defines the following operations.
2018 @item @var{type} scale_float (const @var{type}& x, sintL delta)
2019 @cindex @code{scale_float ()}
2020 @itemx @var{type} scale_float (const @var{type}& x, const cl_I& delta)
2021 Returns @code{x*2^delta}. This is more efficient than an explicit multiplication
2022 because it copies @code{x} and modifies the exponent.
2025 The following functions provide an abstract interface to the underlying
2026 representation of floating-point numbers.
2029 @item sintL float_exponent (const @var{type}& x)
2030 @cindex @code{float_exponent ()}
2031 Returns the exponent @code{e} of @code{x}.
2032 For @code{x = 0.0}, this is 0. For @code{x} non-zero, this is the unique
2033 integer with @code{2^(e-1) <= abs(x) < 2^e}.
2035 @item sintL float_radix (const @var{type}& x)
2036 @cindex @code{float_radix ()}
2037 Returns the base of the floating-point representation. This is always @code{2}.
2039 @item @var{type} float_sign (const @var{type}& x)
2040 @cindex @code{float_sign ()}
2041 Returns the sign @code{s} of @code{x} as a float. The value is 1 for
2042 @code{x} >= 0, -1 for @code{x} < 0.
2044 @item uintL float_digits (const @var{type}& x)
2045 @cindex @code{float_digits ()}
2046 Returns the number of mantissa bits in the floating-point representation
2047 of @code{x}, including the hidden bit. The value only depends on the type
2048 of @code{x}, not on its value.
2050 @item uintL float_precision (const @var{type}& x)
2051 @cindex @code{float_precision ()}
2052 Returns the number of significant mantissa bits in the floating-point
2053 representation of @code{x}. Since denormalized numbers are not supported,
2054 this is the same as @code{float_digits(x)} if @code{x} is non-zero, and
2058 The complete internal representation of a float is encoded in the type
2059 @cindex @code{decoded_float}
2060 @cindex @code{decoded_sfloat}
2061 @cindex @code{decoded_ffloat}
2062 @cindex @code{decoded_dfloat}
2063 @cindex @code{decoded_lfloat}
2064 @code{decoded_float} (or @code{decoded_sfloat}, @code{decoded_ffloat},
2065 @code{decoded_dfloat}, @code{decoded_lfloat}, respectively), defined by
2067 struct decoded_@var{type}float @{
2068 @var{type} mantissa; cl_I exponent; @var{type} sign;
2072 and returned by the function
2075 @item decoded_@var{type}float decode_float (const @var{type}& x)
2076 @cindex @code{decode_float ()}
2077 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2078 @code{x = (-1)^s * 2^e * m} and @code{0.5 <= m < 1.0}. For @code{x} = 0,
2079 it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2080 @code{e} is the same as returned by the function @code{float_exponent}.
2083 A complete decoding in terms of integers is provided as type
2084 @cindex @code{cl_idecoded_float}
2086 struct cl_idecoded_float @{
2087 cl_I mantissa; cl_I exponent; cl_I sign;
2090 by the following function:
2093 @item cl_idecoded_float integer_decode_float (const @var{type}& x)
2094 @cindex @code{integer_decode_float ()}
2095 For @code{x} non-zero, this returns @code{(-1)^s}, @code{e}, @code{m} with
2096 @code{x = (-1)^s * 2^e * m} and @code{m} an integer with @code{float_digits(x)}
2097 bits. For @code{x} = 0, it returns @code{(-1)^s}=1, @code{e}=0, @code{m}=0.
2098 WARNING: The exponent @code{e} is not the same as the one returned by
2099 the functions @code{decode_float} and @code{float_exponent}.
2102 Some other function, implemented only for class @code{cl_F}:
2105 @item cl_F float_sign (const cl_F& x, const cl_F& y)
2106 @cindex @code{float_sign ()}
2107 This returns a floating point number whose precision and absolute value
2108 is that of @code{y} and whose sign is that of @code{x}. If @code{x} is
2109 zero, it is treated as positive. Same for @code{y}.
2113 @section Conversion functions
2116 @subsection Conversion to floating-point numbers
2118 The type @code{float_format_t} describes a floating-point format.
2119 @cindex @code{float_format_t}
2122 @item float_format_t float_format (uintL n)
2123 @cindex @code{float_format ()}
2124 Returns the smallest float format which guarantees at least @code{n}
2125 decimal digits in the mantissa (after the decimal point).
2127 @item float_format_t float_format (const cl_F& x)
2128 Returns the floating point format of @code{x}.
2130 @item float_format_t default_float_format
2131 @cindex @code{default_float_format}
2132 Global variable: the default float format used when converting rational numbers
2136 To convert a real number to a float, each of the types
2137 @code{cl_R}, @code{cl_F}, @code{cl_I}, @code{cl_RA},
2138 @code{int}, @code{unsigned int}, @code{float}, @code{double}
2139 defines the following operations:
2142 @item cl_F cl_float (const @var{type}&x, float_format_t f)
2143 @cindex @code{cl_float ()}
2144 Returns @code{x} as a float of format @code{f}.
2145 @item cl_F cl_float (const @var{type}&x, const cl_F& y)
2146 Returns @code{x} in the float format of @code{y}.
2147 @item cl_F cl_float (const @var{type}&x)
2148 Returns @code{x} as a float of format @code{default_float_format} if
2149 it is an exact number, or @code{x} itself if it is already a float.
2152 Of course, converting a number to a float can lose precision.
2154 Every floating-point format has some characteristic numbers:
2157 @item cl_F most_positive_float (float_format_t f)
2158 @cindex @code{most_positive_float ()}
2159 Returns the largest (most positive) floating point number in float format @code{f}.
2161 @item cl_F most_negative_float (float_format_t f)
2162 @cindex @code{most_negative_float ()}
2163 Returns the smallest (most negative) floating point number in float format @code{f}.
2165 @item cl_F least_positive_float (float_format_t f)
2166 @cindex @code{least_positive_float ()}
2167 Returns the least positive floating point number (i.e. > 0 but closest to 0)
2168 in float format @code{f}.
2170 @item cl_F least_negative_float (float_format_t f)
2171 @cindex @code{least_negative_float ()}
2172 Returns the least negative floating point number (i.e. < 0 but closest to 0)
2173 in float format @code{f}.
2175 @item cl_F float_epsilon (float_format_t f)
2176 @cindex @code{float_epsilon ()}
2177 Returns the smallest floating point number e > 0 such that @code{1+e != 1}.
2179 @item cl_F float_negative_epsilon (float_format_t f)
2180 @cindex @code{float_negative_epsilon ()}
2181 Returns the smallest floating point number e > 0 such that @code{1-e != 1}.
2185 @subsection Conversion to rational numbers
2187 Each of the classes @code{cl_R}, @code{cl_RA}, @code{cl_F}
2188 defines the following operation:
2191 @item cl_RA rational (const @var{type}& x)
2192 @cindex @code{rational ()}
2193 Returns the value of @code{x} as an exact number. If @code{x} is already
2194 an exact number, this is @code{x}. If @code{x} is a floating-point number,
2195 the value is a rational number whose denominator is a power of 2.
2198 In order to convert back, say, @code{(cl_F)(cl_R)"1/3"} to @code{1/3}, there is
2202 @item cl_RA rationalize (const cl_R& x)
2203 @cindex @code{rationalize ()}
2204 If @code{x} is a floating-point number, it actually represents an interval
2205 of real numbers, and this function returns the rational number with
2206 smallest denominator (and smallest numerator, in magnitude)
2207 which lies in this interval.
2208 If @code{x} is already an exact number, this function returns @code{x}.
2211 If @code{x} is any float, one has
2215 @code{cl_float(rational(x),x) = x}
2217 @code{cl_float(rationalize(x),x) = x}
2221 @section Random number generators
2224 A random generator is a machine which produces (pseudo-)random numbers.
2225 The include file @code{<cln/random.h>} defines a class @code{random_state}
2226 which contains the state of a random generator. If you make a copy
2227 of the random number generator, the original one and the copy will produce
2228 the same sequence of random numbers.
2230 The following functions return (pseudo-)random numbers in different formats.
2231 Calling one of these modifies the state of the random number generator in
2232 a complicated but deterministic way.
2235 @cindex @code{random_state}
2236 @cindex @code{default_random_state}
2238 random_state default_random_state
2240 contains a default random number generator. It is used when the functions
2241 below are called without @code{random_state} argument.
2244 @item uint32 random32 (random_state& randomstate)
2245 @itemx uint32 random32 ()
2246 @cindex @code{random32 ()}
2247 Returns a random unsigned 32-bit number. All bits are equally random.
2249 @item cl_I random_I (random_state& randomstate, const cl_I& n)
2250 @itemx cl_I random_I (const cl_I& n)
2251 @cindex @code{random_I ()}
2252 @code{n} must be an integer > 0. This function returns a random integer @code{x}
2253 in the range @code{0 <= x < n}.
2255 @item cl_F random_F (random_state& randomstate, const cl_F& n)
2256 @itemx cl_F random_F (const cl_F& n)
2257 @cindex @code{random_F ()}
2258 @code{n} must be a float > 0. This function returns a random floating-point
2259 number of the same format as @code{n} in the range @code{0 <= x < n}.
2261 @item cl_R random_R (random_state& randomstate, const cl_R& n)
2262 @itemx cl_R random_R (const cl_R& n)
2263 @cindex @code{random_R ()}
2264 Behaves like @code{random_I} if @code{n} is an integer and like @code{random_F}
2265 if @code{n} is a float.
2269 @section Obfuscating operators
2270 @cindex modifying operators
2272 The modifying C/C++ operators @code{+=}, @code{-=}, @code{*=}, @code{/=},
2273 @code{&=}, @code{|=}, @code{^=}, @code{<<=}, @code{>>=}
2274 are not available by default because their
2275 use tends to make programs unreadable. It is trivial to get away without
2276 them. However, if you feel that you absolutely need these operators
2277 to get happy, then add
2279 #define WANT_OBFUSCATING_OPERATORS
2281 @cindex @code{WANT_OBFUSCATING_OPERATORS}
2282 to the beginning of your source files, before the inclusion of any CLN
2283 include files. This flag will enable the following operators:
2285 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA},
2286 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2289 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2290 @cindex @code{operator += ()}
2291 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2292 @cindex @code{operator -= ()}
2293 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2294 @cindex @code{operator *= ()}
2295 @itemx @var{type}& operator /= (@var{type}&, const @var{type}&)
2296 @cindex @code{operator /= ()}
2299 For the class @code{cl_I}:
2302 @item @var{type}& operator += (@var{type}&, const @var{type}&)
2303 @itemx @var{type}& operator -= (@var{type}&, const @var{type}&)
2304 @itemx @var{type}& operator *= (@var{type}&, const @var{type}&)
2305 @itemx @var{type}& operator &= (@var{type}&, const @var{type}&)
2306 @cindex @code{operator &= ()}
2307 @itemx @var{type}& operator |= (@var{type}&, const @var{type}&)
2308 @cindex @code{operator |= ()}
2309 @itemx @var{type}& operator ^= (@var{type}&, const @var{type}&)
2310 @cindex @code{operator ^= ()}
2311 @itemx @var{type}& operator <<= (@var{type}&, const @var{type}&)
2312 @cindex @code{operator <<= ()}
2313 @itemx @var{type}& operator >>= (@var{type}&, const @var{type}&)
2314 @cindex @code{operator >>= ()}
2317 For the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2318 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}:
2321 @item @var{type}& operator ++ (@var{type}& x)
2322 @cindex @code{operator ++ ()}
2323 The prefix operator @code{++x}.
2325 @item void operator ++ (@var{type}& x, int)
2326 The postfix operator @code{x++}.
2328 @item @var{type}& operator -- (@var{type}& x)
2329 @cindex @code{operator -- ()}
2330 The prefix operator @code{--x}.
2332 @item void operator -- (@var{type}& x, int)
2333 The postfix operator @code{x--}.
2336 Note that by using these obfuscating operators, you wouldn't gain efficiency:
2337 In CLN @samp{x += y;} is exactly the same as @samp{x = x+y;}, not more
2341 @chapter Input/Output
2342 @cindex Input/Output
2344 @section Internal and printed representation
2345 @cindex representation
2347 All computations deal with the internal representations of the numbers.
2349 Every number has an external representation as a sequence of ASCII characters.
2350 Several external representations may denote the same number, for example,
2351 "20.0" and "20.000".
2353 Converting an internal to an external representation is called ``printing'',
2355 converting an external to an internal representation is called ``reading''.
2357 In CLN, it is always true that conversion of an internal to an external
2358 representation and then back to an internal representation will yield the
2359 same internal representation. Symbolically: @code{read(print(x)) == x}.
2360 This is called ``print-read consistency''.
2362 Different types of numbers have different external representations (case
2367 External representation: @var{sign}@{@var{digit}@}+. The reader also accepts the
2368 Common Lisp syntaxes @var{sign}@{@var{digit}@}+@code{.} with a trailing dot
2369 for decimal integers
2370 and the @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes.
2372 @item Rational numbers
2373 External representation: @var{sign}@{@var{digit}@}+@code{/}@{@var{digit}@}+.
2374 The @code{#@var{n}R}, @code{#b}, @code{#o}, @code{#x} prefixes are allowed
2377 @item Floating-point numbers
2378 External representation: @var{sign}@{@var{digit}@}*@var{exponent} or
2379 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}*@var{exponent} or
2380 @var{sign}@{@var{digit}@}*@code{.}@{@var{digit}@}+. A precision specifier
2381 of the form _@var{prec} may be appended. There must be at least
2382 one digit in the non-exponent part. The exponent has the syntax
2383 @var{expmarker} @var{expsign} @{@var{digit}@}+.
2384 The exponent marker is
2388 @samp{s} for short-floats,
2390 @samp{f} for single-floats,
2392 @samp{d} for double-floats,
2394 @samp{L} for long-floats,
2397 or @samp{e}, which denotes a default float format. The precision specifying
2398 suffix has the syntax _@var{prec} where @var{prec} denotes the number of
2399 valid mantissa digits (in decimal, excluding leading zeroes), cf. also
2400 function @samp{float_format}.
2402 @item Complex numbers
2403 External representation:
2406 In algebraic notation: @code{@var{realpart}+@var{imagpart}i}. Of course,
2407 if @var{imagpart} is negative, its printed representation begins with
2408 a @samp{-}, and the @samp{+} between @var{realpart} and @var{imagpart}
2409 may be omitted. Note that this notation cannot be used when the @var{imagpart}
2410 is rational and the rational number's base is >18, because the @samp{i}
2411 is then read as a digit.
2413 In Common Lisp notation: @code{#C(@var{realpart} @var{imagpart})}.
2418 @section Input functions
2420 Including @code{<cln/io.h>} defines a number of simple input functions
2421 that read from @code{std::istream&}:
2424 @item int freadchar (std::istream& stream)
2425 Reads a character from @code{stream}. Returns @code{cl_EOF} (not a @samp{char}!)
2426 if the end of stream was encountered or an error occurred.
2428 @item int funreadchar (std::istream& stream, int c)
2429 Puts back @code{c} onto @code{stream}. @code{c} must be the result of the
2430 last @code{freadchar} operation on @code{stream}.
2433 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2434 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2435 defines, in @code{<cln/@var{type}_io.h>}, the following input function:
2438 @item std::istream& operator>> (std::istream& stream, @var{type}& result)
2439 Reads a number from @code{stream} and stores it in the @code{result}.
2442 The most flexible input functions, defined in @code{<cln/@var{type}_io.h>},
2446 @item cl_N read_complex (std::istream& stream, const cl_read_flags& flags)
2447 @itemx cl_R read_real (std::istream& stream, const cl_read_flags& flags)
2448 @itemx cl_F read_float (std::istream& stream, const cl_read_flags& flags)
2449 @itemx cl_RA read_rational (std::istream& stream, const cl_read_flags& flags)
2450 @itemx cl_I read_integer (std::istream& stream, const cl_read_flags& flags)
2451 Reads a number from @code{stream}. The @code{flags} are parameters which
2452 affect the input syntax. Whitespace before the number is silently skipped.
2454 @item cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2455 @itemx cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2456 @itemx cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2457 @itemx cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2458 @itemx cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)
2459 Reads a number from a string in memory. The @code{flags} are parameters which
2460 affect the input syntax. The string starts at @code{string} and ends at
2461 @code{string_limit} (exclusive limit). @code{string_limit} may also be
2462 @code{NULL}, denoting the entire string, i.e. equivalent to
2463 @code{string_limit = string + strlen(string)}. If @code{end_of_parse} is
2464 @code{NULL}, the string in memory must contain exactly one number and nothing
2465 more, else a fatal error will be signalled. If @code{end_of_parse}
2466 is not @code{NULL}, @code{*end_of_parse} will be assigned a pointer past
2467 the last parsed character (i.e. @code{string_limit} if nothing came after
2468 the number). Whitespace is not allowed.
2471 The structure @code{cl_read_flags} contains the following fields:
2474 @item cl_read_syntax_t syntax
2475 The possible results of the read operation. Possible values are
2476 @code{syntax_number}, @code{syntax_real}, @code{syntax_rational},
2477 @code{syntax_integer}, @code{syntax_float}, @code{syntax_sfloat},
2478 @code{syntax_ffloat}, @code{syntax_dfloat}, @code{syntax_lfloat}.
2480 @item cl_read_lsyntax_t lsyntax
2481 Specifies the language-dependent syntax variant for the read operation.
2485 @item lsyntax_standard
2486 accept standard algebraic notation only, no complex numbers,
2487 @item lsyntax_algebraic
2488 accept the algebraic notation @code{@var{x}+@var{y}i} for complex numbers,
2489 @item lsyntax_commonlisp
2490 accept the @code{#b}, @code{#o}, @code{#x} syntaxes for binary, octal,
2491 hexadecimal numbers,
2492 @code{#@var{base}R} for rational numbers in a given base,
2493 @code{#c(@var{realpart} @var{imagpart})} for complex numbers,
2495 accept all of these extensions.
2498 @item unsigned int rational_base
2499 The base in which rational numbers are read.
2501 @item float_format_t float_flags.default_float_format
2502 The float format used when reading floats with exponent marker @samp{e}.
2504 @item float_format_t float_flags.default_lfloat_format
2505 The float format used when reading floats with exponent marker @samp{l}.
2507 @item cl_boolean float_flags.mantissa_dependent_float_format
2508 When this flag is true, floats specified with more digits than corresponding
2509 to the exponent marker they contain, but without @var{_nnn} suffix, will get a
2510 precision corresponding to their number of significant digits.
2514 @section Output functions
2516 Including @code{<cln/io.h>} defines a number of simple output functions
2517 that write to @code{std::ostream&}:
2520 @item void fprintchar (std::ostream& stream, char c)
2521 Prints the character @code{x} literally on the @code{stream}.
2523 @item void fprint (std::ostream& stream, const char * string)
2524 Prints the @code{string} literally on the @code{stream}.
2526 @item void fprintdecimal (std::ostream& stream, int x)
2527 @itemx void fprintdecimal (std::ostream& stream, const cl_I& x)
2528 Prints the integer @code{x} in decimal on the @code{stream}.
2530 @item void fprintbinary (std::ostream& stream, const cl_I& x)
2531 Prints the integer @code{x} in binary (base 2, without prefix)
2532 on the @code{stream}.
2534 @item void fprintoctal (std::ostream& stream, const cl_I& x)
2535 Prints the integer @code{x} in octal (base 8, without prefix)
2536 on the @code{stream}.
2538 @item void fprinthexadecimal (std::ostream& stream, const cl_I& x)
2539 Prints the integer @code{x} in hexadecimal (base 16, without prefix)
2540 on the @code{stream}.
2543 Each of the classes @code{cl_N}, @code{cl_R}, @code{cl_RA}, @code{cl_I},
2544 @code{cl_F}, @code{cl_SF}, @code{cl_FF}, @code{cl_DF}, @code{cl_LF}
2545 defines, in @code{<cln/@var{type}_io.h>}, the following output functions:
2548 @item void fprint (std::ostream& stream, const @var{type}& x)
2549 @itemx std::ostream& operator<< (std::ostream& stream, const @var{type}& x)
2550 Prints the number @code{x} on the @code{stream}. The output may depend
2551 on the global printer settings in the variable @code{default_print_flags}.
2552 The @code{ostream} flags and settings (flags, width and locale) are
2556 The most flexible output function, defined in @code{<cln/@var{type}_io.h>},
2559 void print_complex (std::ostream& stream, const cl_print_flags& flags,
2561 void print_real (std::ostream& stream, const cl_print_flags& flags,
2563 void print_float (std::ostream& stream, const cl_print_flags& flags,
2565 void print_rational (std::ostream& stream, const cl_print_flags& flags,
2567 void print_integer (std::ostream& stream, const cl_print_flags& flags,
2570 Prints the number @code{x} on the @code{stream}. The @code{flags} are
2571 parameters which affect the output.
2573 The structure type @code{cl_print_flags} contains the following fields:
2576 @item unsigned int rational_base
2577 The base in which rational numbers are printed. Default is @code{10}.
2579 @item cl_boolean rational_readably
2580 If this flag is true, rational numbers are printed with radix specifiers in
2581 Common Lisp syntax (@code{#@var{n}R} or @code{#b} or @code{#o} or @code{#x}
2582 prefixes, trailing dot). Default is false.
2584 @item cl_boolean float_readably
2585 If this flag is true, type specific exponent markers have precedence over 'E'.
2588 @item float_format_t default_float_format
2589 Floating point numbers of this format will be printed using the 'E' exponent
2590 marker. Default is @code{float_format_ffloat}.
2592 @item cl_boolean complex_readably
2593 If this flag is true, complex numbers will be printed using the Common Lisp
2594 syntax @code{#C(@var{realpart} @var{imagpart})}. Default is false.
2596 @item cl_string univpoly_varname
2597 Univariate polynomials with no explicit indeterminate name will be printed
2598 using this variable name. Default is @code{"x"}.
2601 The global variable @code{default_print_flags} contains the default values,
2602 used by the function @code{fprint}.
2607 CLN has a class of abstract rings.
2615 Rings can be compared for equality:
2618 @item bool operator== (const cl_ring&, const cl_ring&)
2619 @itemx bool operator!= (const cl_ring&, const cl_ring&)
2620 These compare two rings for equality.
2623 Given a ring @code{R}, the following members can be used.
2626 @item void R->fprint (std::ostream& stream, const cl_ring_element& x)
2627 @cindex @code{fprint ()}
2628 @itemx cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)
2629 @cindex @code{equal ()}
2630 @itemx cl_ring_element R->zero ()
2631 @cindex @code{zero ()}
2632 @itemx cl_boolean R->zerop (const cl_ring_element& x)
2633 @cindex @code{zerop ()}
2634 @itemx cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)
2635 @cindex @code{plus ()}
2636 @itemx cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)
2637 @cindex @code{minus ()}
2638 @itemx cl_ring_element R->uminus (const cl_ring_element& x)
2639 @cindex @code{uminus ()}
2640 @itemx cl_ring_element R->one ()
2641 @cindex @code{one ()}
2642 @itemx cl_ring_element R->canonhom (const cl_I& x)
2643 @cindex @code{canonhom ()}
2644 @itemx cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)
2645 @cindex @code{mul ()}
2646 @itemx cl_ring_element R->square (const cl_ring_element& x)
2647 @cindex @code{square ()}
2648 @itemx cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)
2649 @cindex @code{expt_pos ()}
2652 The following rings are built-in.
2655 @item cl_null_ring cl_0_ring
2656 The null ring, containing only zero.
2658 @item cl_complex_ring cl_C_ring
2659 The ring of complex numbers. This corresponds to the type @code{cl_N}.
2661 @item cl_real_ring cl_R_ring
2662 The ring of real numbers. This corresponds to the type @code{cl_R}.
2664 @item cl_rational_ring cl_RA_ring
2665 The ring of rational numbers. This corresponds to the type @code{cl_RA}.
2667 @item cl_integer_ring cl_I_ring
2668 The ring of integers. This corresponds to the type @code{cl_I}.
2671 Type tests can be performed for any of @code{cl_C_ring}, @code{cl_R_ring},
2672 @code{cl_RA_ring}, @code{cl_I_ring}:
2675 @item cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)
2676 @cindex @code{instanceof ()}
2677 Tests whether the given number is an element of the number ring R.
2681 @chapter Modular integers
2682 @cindex modular integer
2684 @section Modular integer rings
2687 CLN implements modular integers, i.e. integers modulo a fixed integer N.
2688 The modulus is explicitly part of every modular integer. CLN doesn't
2689 allow you to (accidentally) mix elements of different modular rings,
2690 e.g. @code{(3 mod 4) + (2 mod 5)} will result in a runtime error.
2691 (Ideally one would imagine a generic data type @code{cl_MI(N)}, but C++
2692 doesn't have generic types. So one has to live with runtime checks.)
2694 The class of modular integer rings is
2702 Modular integer ring
2706 @cindex @code{cl_modint_ring}
2708 and the class of all modular integers (elements of modular integer rings) is
2716 Modular integer rings are constructed using the function
2719 @item cl_modint_ring find_modint_ring (const cl_I& N)
2720 @cindex @code{find_modint_ring ()}
2721 This function returns the modular ring @samp{Z/NZ}. It takes care
2722 of finding out about special cases of @code{N}, like powers of two
2723 and odd numbers for which Montgomery multiplication will be a win,
2724 @cindex Montgomery multiplication
2725 and precomputes any necessary auxiliary data for computing modulo @code{N}.
2726 There is a cache table of rings, indexed by @code{N} (or, more precisely,
2727 by @code{abs(N)}). This ensures that the precomputation costs are reduced
2731 Modular integer rings can be compared for equality:
2734 @item bool operator== (const cl_modint_ring&, const cl_modint_ring&)
2735 @cindex @code{operator == ()}
2736 @itemx bool operator!= (const cl_modint_ring&, const cl_modint_ring&)
2737 @cindex @code{operator != ()}
2738 These compare two modular integer rings for equality. Two different calls
2739 to @code{find_modint_ring} with the same argument necessarily return the
2740 same ring because it is memoized in the cache table.
2743 @section Functions on modular integers
2745 Given a modular integer ring @code{R}, the following members can be used.
2748 @item cl_I R->modulus
2749 @cindex @code{modulus}
2750 This is the ring's modulus, normalized to be nonnegative: @code{abs(N)}.
2752 @item cl_MI R->zero()
2753 @cindex @code{zero ()}
2754 This returns @code{0 mod N}.
2756 @item cl_MI R->one()
2757 @cindex @code{one ()}
2758 This returns @code{1 mod N}.
2760 @item cl_MI R->canonhom (const cl_I& x)
2761 @cindex @code{canonhom ()}
2762 This returns @code{x mod N}.
2764 @item cl_I R->retract (const cl_MI& x)
2765 @cindex @code{retract ()}
2766 This is a partial inverse function to @code{R->canonhom}. It returns the
2767 standard representative (@code{>=0}, @code{<N}) of @code{x}.
2769 @item cl_MI R->random(random_state& randomstate)
2770 @itemx cl_MI R->random()
2771 @cindex @code{random ()}
2772 This returns a random integer modulo @code{N}.
2775 The following operations are defined on modular integers.
2778 @item cl_modint_ring x.ring ()
2779 @cindex @code{ring ()}
2780 Returns the ring to which the modular integer @code{x} belongs.
2782 @item cl_MI operator+ (const cl_MI&, const cl_MI&)
2783 @cindex @code{operator + ()}
2784 Returns the sum of two modular integers. One of the arguments may also
2787 @item cl_MI operator- (const cl_MI&, const cl_MI&)
2788 @cindex @code{operator - ()}
2789 Returns the difference of two modular integers. One of the arguments may also
2792 @item cl_MI operator- (const cl_MI&)
2793 Returns the negative of a modular integer.
2795 @item cl_MI operator* (const cl_MI&, const cl_MI&)
2796 @cindex @code{operator * ()}
2797 Returns the product of two modular integers. One of the arguments may also
2800 @item cl_MI square (const cl_MI&)
2801 @cindex @code{square ()}
2802 Returns the square of a modular integer.
2804 @item cl_MI recip (const cl_MI& x)
2805 @cindex @code{recip ()}
2806 Returns the reciprocal @code{x^-1} of a modular integer @code{x}. @code{x}
2807 must be coprime to the modulus, otherwise an error message is issued.
2809 @item cl_MI div (const cl_MI& x, const cl_MI& y)
2810 @cindex @code{div ()}
2811 Returns the quotient @code{x*y^-1} of two modular integers @code{x}, @code{y}.
2812 @code{y} must be coprime to the modulus, otherwise an error message is issued.
2814 @item cl_MI expt_pos (const cl_MI& x, const cl_I& y)
2815 @cindex @code{expt_pos ()}
2816 @code{y} must be > 0. Returns @code{x^y}.
2818 @item cl_MI expt (const cl_MI& x, const cl_I& y)
2819 @cindex @code{expt ()}
2820 Returns @code{x^y}. If @code{y} is negative, @code{x} must be coprime to the
2821 modulus, else an error message is issued.
2823 @item cl_MI operator<< (const cl_MI& x, const cl_I& y)
2824 @cindex @code{operator << ()}
2825 Returns @code{x*2^y}.
2827 @item cl_MI operator>> (const cl_MI& x, const cl_I& y)
2828 @cindex @code{operator >> ()}
2829 Returns @code{x*2^-y}. When @code{y} is positive, the modulus must be odd,
2830 or an error message is issued.
2832 @item bool operator== (const cl_MI&, const cl_MI&)
2833 @cindex @code{operator == ()}
2834 @itemx bool operator!= (const cl_MI&, const cl_MI&)
2835 @cindex @code{operator != ()}
2836 Compares two modular integers, belonging to the same modular integer ring,
2839 @item cl_boolean zerop (const cl_MI& x)
2840 @cindex @code{zerop ()}
2841 Returns true if @code{x} is @code{0 mod N}.
2844 The following output functions are defined (see also the chapter on
2848 @item void fprint (std::ostream& stream, const cl_MI& x)
2849 @cindex @code{fprint ()}
2850 @itemx std::ostream& operator<< (std::ostream& stream, const cl_MI& x)
2851 @cindex @code{operator << ()}
2852 Prints the modular integer @code{x} on the @code{stream}. The output may depend
2853 on the global printer settings in the variable @code{default_print_flags}.
2857 @chapter Symbolic data types
2858 @cindex symbolic type
2860 CLN implements two symbolic (non-numeric) data types: strings and symbols.
2864 @cindex @code{cl_string}
2874 implements immutable strings.
2876 Strings are constructed through the following constructors:
2879 @item cl_string (const char * s)
2880 Returns an immutable copy of the (zero-terminated) C string @code{s}.
2882 @item cl_string (const char * ptr, unsigned long len)
2883 Returns an immutable copy of the @code{len} characters at
2884 @code{ptr[0]}, @dots{}, @code{ptr[len-1]}. NUL characters are allowed.
2887 The following functions are available on strings:
2891 Assignment from @code{cl_string} and @code{const char *}.
2894 @cindex @code{length ()}
2896 @cindex @code{strlen ()}
2897 Returns the length of the string @code{s}.
2900 @cindex @code{operator [] ()}
2901 Returns the @code{i}th character of the string @code{s}.
2902 @code{i} must be in the range @code{0 <= i < s.length()}.
2904 @item bool equal (const cl_string& s1, const cl_string& s2)
2905 @cindex @code{equal ()}
2906 Compares two strings for equality. One of the arguments may also be a
2907 plain @code{const char *}.
2912 @cindex @code{cl_symbol}
2914 Symbols are uniquified strings: all symbols with the same name are shared.
2915 This means that comparison of two symbols is fast (effectively just a pointer
2916 comparison), whereas comparison of two strings must in the worst case walk
2917 both strings until their end.
2918 Symbols are used, for example, as tags for properties, as names of variables
2919 in polynomial rings, etc.
2921 Symbols are constructed through the following constructor:
2924 @item cl_symbol (const cl_string& s)
2925 Looks up or creates a new symbol with a given name.
2928 The following operations are available on symbols:
2931 @item cl_string (const cl_symbol& sym)
2932 Conversion to @code{cl_string}: Returns the string which names the symbol
2935 @item bool equal (const cl_symbol& sym1, const cl_symbol& sym2)
2936 @cindex @code{equal ()}
2937 Compares two symbols for equality. This is very fast.
2941 @chapter Univariate polynomials
2943 @cindex univariate polynomial
2945 @section Univariate polynomial rings
2947 CLN implements univariate polynomials (polynomials in one variable) over an
2948 arbitrary ring. The indeterminate variable may be either unnamed (and will be
2949 printed according to @code{default_print_flags.univpoly_varname}, which
2950 defaults to @samp{x}) or carry a given name. The base ring and the
2951 indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
2952 (accidentally) mix elements of different polynomial rings, e.g.
2953 @code{(a^2+1) * (b^3-1)} will result in a runtime error. (Ideally this should
2954 return a multivariate polynomial, but they are not yet implemented in CLN.)
2956 The classes of univariate polynomial rings are
2964 Univariate polynomial ring
2968 +----------------+-------------------+
2970 Complex polynomial ring | Modular integer polynomial ring
2971 cl_univpoly_complex_ring | cl_univpoly_modint_ring
2972 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
2976 Real polynomial ring |
2977 cl_univpoly_real_ring |
2978 <cln/univpoly_real.h> |
2982 Rational polynomial ring |
2983 cl_univpoly_rational_ring |
2984 <cln/univpoly_rational.h> |
2988 Integer polynomial ring
2989 cl_univpoly_integer_ring
2990 <cln/univpoly_integer.h>
2993 and the corresponding classes of univariate polynomials are
2996 Univariate polynomial
3000 +----------------+-------------------+
3002 Complex polynomial | Modular integer polynomial
3004 <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
3010 <cln/univpoly_real.h> |
3014 Rational polynomial |
3016 <cln/univpoly_rational.h> |
3022 <cln/univpoly_integer.h>
3025 Univariate polynomial rings are constructed using the functions
3028 @item cl_univpoly_ring find_univpoly_ring (const cl_ring& R)
3029 @itemx cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)
3030 This function returns the polynomial ring @samp{R[X]}, unnamed or named.
3031 @code{R} may be an arbitrary ring. This function takes care of finding out
3032 about special cases of @code{R}, such as the rings of complex numbers,
3033 real numbers, rational numbers, integers, or modular integer rings.
3034 There is a cache table of rings, indexed by @code{R} and @code{varname}.
3035 This ensures that two calls of this function with the same arguments will
3036 return the same polynomial ring.
3038 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)
3039 @cindex @code{find_univpoly_ring ()}
3040 @itemx cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)
3041 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)
3042 @itemx cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)
3043 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)
3044 @itemx cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)
3045 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)
3046 @itemx cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)
3047 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)
3048 @itemx cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)
3049 These functions are equivalent to the general @code{find_univpoly_ring},
3050 only the return type is more specific, according to the base ring's type.
3053 @section Functions on univariate polynomials
3055 Given a univariate polynomial ring @code{R}, the following members can be used.
3058 @item cl_ring R->basering()
3059 @cindex @code{basering ()}
3060 This returns the base ring, as passed to @samp{find_univpoly_ring}.
3062 @item cl_UP R->zero()
3063 @cindex @code{zero ()}
3064 This returns @code{0 in R}, a polynomial of degree -1.
3066 @item cl_UP R->one()
3067 @cindex @code{one ()}
3068 This returns @code{1 in R}, a polynomial of degree <= 0.
3070 @item cl_UP R->canonhom (const cl_I& x)
3071 @cindex @code{canonhom ()}
3072 This returns @code{x in R}, a polynomial of degree <= 0.
3074 @item cl_UP R->monomial (const cl_ring_element& x, uintL e)
3075 @cindex @code{monomial ()}
3076 This returns a sparse polynomial: @code{x * X^e}, where @code{X} is the
3079 @item cl_UP R->create (sintL degree)
3080 @cindex @code{create ()}
3081 Creates a new polynomial with a given degree. The zero polynomial has degree
3082 @code{-1}. After creating the polynomial, you should put in the coefficients,
3083 using the @code{set_coeff} member function, and then call the @code{finalize}
3087 The following are the only destructive operations on univariate polynomials.
3090 @item void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)
3091 @cindex @code{set_coeff ()}
3092 This changes the coefficient of @code{X^index} in @code{x} to be @code{y}.
3093 After changing a polynomial and before applying any "normal" operation on it,
3094 you should call its @code{finalize} member function.
3096 @item void finalize (cl_UP& x)
3097 @cindex @code{finalize ()}
3098 This function marks the endpoint of destructive modifications of a polynomial.
3099 It normalizes the internal representation so that subsequent computations have
3100 less overhead. Doing normal computations on unnormalized polynomials may
3101 produce wrong results or crash the program.
3104 The following operations are defined on univariate polynomials.
3107 @item cl_univpoly_ring x.ring ()
3108 @cindex @code{ring ()}
3109 Returns the ring to which the univariate polynomial @code{x} belongs.
3111 @item cl_UP operator+ (const cl_UP&, const cl_UP&)
3112 @cindex @code{operator + ()}
3113 Returns the sum of two univariate polynomials.
3115 @item cl_UP operator- (const cl_UP&, const cl_UP&)
3116 @cindex @code{operator - ()}
3117 Returns the difference of two univariate polynomials.
3119 @item cl_UP operator- (const cl_UP&)
3120 Returns the negative of a univariate polynomial.
3122 @item cl_UP operator* (const cl_UP&, const cl_UP&)
3123 @cindex @code{operator * ()}
3124 Returns the product of two univariate polynomials. One of the arguments may
3125 also be a plain integer or an element of the base ring.
3127 @item cl_UP square (const cl_UP&)
3128 @cindex @code{square ()}
3129 Returns the square of a univariate polynomial.
3131 @item cl_UP expt_pos (const cl_UP& x, const cl_I& y)
3132 @cindex @code{expt_pos ()}
3133 @code{y} must be > 0. Returns @code{x^y}.
3135 @item bool operator== (const cl_UP&, const cl_UP&)
3136 @cindex @code{operator == ()}
3137 @itemx bool operator!= (const cl_UP&, const cl_UP&)
3138 @cindex @code{operator != ()}
3139 Compares two univariate polynomials, belonging to the same univariate
3140 polynomial ring, for equality.
3142 @item cl_boolean zerop (const cl_UP& x)
3143 @cindex @code{zerop ()}
3144 Returns true if @code{x} is @code{0 in R}.
3146 @item sintL degree (const cl_UP& x)
3147 @cindex @code{degree ()}
3148 Returns the degree of the polynomial. The zero polynomial has degree @code{-1}.
3150 @item cl_ring_element coeff (const cl_UP& x, uintL index)
3151 @cindex @code{coeff ()}
3152 Returns the coefficient of @code{X^index} in the polynomial @code{x}.
3154 @item cl_ring_element x (const cl_ring_element& y)
3155 @cindex @code{operator () ()}
3156 Evaluation: If @code{x} is a polynomial and @code{y} belongs to the base ring,
3157 then @samp{x(y)} returns the value of the substitution of @code{y} into
3160 @item cl_UP deriv (const cl_UP& x)
3161 @cindex @code{deriv ()}
3162 Returns the derivative of the polynomial @code{x} with respect to the
3163 indeterminate @code{X}.
3166 The following output functions are defined (see also the chapter on
3170 @item void fprint (std::ostream& stream, const cl_UP& x)
3171 @cindex @code{fprint ()}
3172 @itemx std::ostream& operator<< (std::ostream& stream, const cl_UP& x)
3173 @cindex @code{operator << ()}
3174 Prints the univariate polynomial @code{x} on the @code{stream}. The output may
3175 depend on the global printer settings in the variable
3176 @code{default_print_flags}.
3179 @section Special polynomials
3181 The following functions return special polynomials.
3184 @item cl_UP_I tschebychev (sintL n)
3185 @cindex @code{tschebychev ()}
3186 @cindex Chebyshev polynomial
3187 Returns the n-th Chebyshev polynomial (n >= 0).
3189 @item cl_UP_I hermite (sintL n)
3190 @cindex @code{hermite ()}
3191 @cindex Hermite polynomial
3192 Returns the n-th Hermite polynomial (n >= 0).
3194 @item cl_UP_RA legendre (sintL n)
3195 @cindex @code{legendre ()}
3196 @cindex Legende polynomial
3197 Returns the n-th Legendre polynomial (n >= 0).
3199 @item cl_UP_I laguerre (sintL n)
3200 @cindex @code{laguerre ()}
3201 @cindex Laguerre polynomial
3202 Returns the n-th Laguerre polynomial (n >= 0).
3205 Information how to derive the differential equation satisfied by each
3206 of these polynomials from their definition can be found in the
3207 @code{doc/polynomial/} directory.
3215 Using C++ as an implementation language provides
3219 Efficiency: It compiles to machine code.
3223 Portability: It runs on all platforms supporting a C++ compiler. Because
3224 of the availability of GNU C++, this includes all currently used 32-bit and
3225 64-bit platforms, independently of the quality of the vendor's C++ compiler.
3228 Type safety: The C++ compilers knows about the number types and complains if,
3229 for example, you try to assign a float to an integer variable. However,
3230 a drawback is that C++ doesn't know about generic types, hence a restriction
3231 like that @code{operator+ (const cl_MI&, const cl_MI&)} requires that both
3232 arguments belong to the same modular ring cannot be expressed as a compile-time
3236 Algebraic syntax: The elementary operations @code{+}, @code{-}, @code{*},
3237 @code{=}, @code{==}, ... can be used in infix notation, which is more
3238 convenient than Lisp notation @samp{(+ x y)} or C notation @samp{add(x,y,&z)}.
3241 With these language features, there is no need for two separate languages,
3242 one for the implementation of the library and one in which the library's users
3243 can program. This means that a prototype implementation of an algorithm
3244 can be integrated into the library immediately after it has been tested and
3245 debugged. No need to rewrite it in a low-level language after having prototyped
3246 in a high-level language.
3249 @section Memory efficiency
3251 In order to save memory allocations, CLN implements:
3255 Object sharing: An operation like @code{x+0} returns @code{x} without copying
3258 @cindex garbage collection
3259 @cindex reference counting
3260 Garbage collection: A reference counting mechanism makes sure that any
3261 number object's storage is freed immediately when the last reference to the
3264 @cindex immediate numbers
3265 Small integers are represented as immediate values instead of pointers
3266 to heap allocated storage. This means that integers @code{> -2^29},
3267 @code{< 2^29} don't consume heap memory, unless they were explicitly allocated
3272 @section Speed efficiency
3274 Speed efficiency is obtained by the combination of the following tricks
3279 Small integers, being represented as immediate values, don't require
3280 memory access, just a couple of instructions for each elementary operation.
3282 The kernel of CLN has been written in assembly language for some CPUs
3283 (@code{i386}, @code{m68k}, @code{sparc}, @code{mips}, @code{arm}).
3285 On all CPUs, CLN may be configured to use the superefficient low-level
3286 routines from GNU GMP version 3.
3288 For large numbers, CLN uses, instead of the standard @code{O(N^2)}
3289 algorithm, the Karatsuba multiplication, which is an
3300 For very large numbers (more than 12000 decimal digits), CLN uses
3302 Sch{@"o}nhage-Strassen
3303 @cindex Sch{@"o}nhage-Strassen multiplication
3307 @cindex Schönhage-Strassen multiplication
3309 multiplication, which is an asymptotically optimal multiplication
3312 These fast multiplication algorithms also give improvements in the speed
3313 of division and radix conversion.
3317 @section Garbage collection
3318 @cindex garbage collection
3320 All the number classes are reference count classes: They only contain a pointer
3321 to an object in the heap. Upon construction, assignment and destruction of
3322 number objects, only the objects' reference count are manipulated.
3324 Memory occupied by number objects are automatically reclaimed as soon as
3325 their reference count drops to zero.
3327 For number rings, another strategy is implemented: There is a cache of,
3328 for example, the modular integer rings. A modular integer ring is destroyed
3329 only if its reference count dropped to zero and the cache is about to be
3330 resized. The effect of this strategy is that recently used rings remain
3331 cached, whereas undue memory consumption through cached rings is avoided.
3334 @chapter Using the library
3336 For the following discussion, we will assume that you have installed
3337 the CLN source in @code{$CLN_DIR} and built it in @code{$CLN_TARGETDIR}.
3338 For example, for me it's @code{CLN_DIR="$HOME/cln"} and
3339 @code{CLN_TARGETDIR="$HOME/cln/linuxelf"}. You might define these as
3340 environment variables, or directly substitute the appropriate values.
3343 @section Compiler options
3344 @cindex compiler options
3346 Until you have installed CLN in a public place, the following options are
3349 When you compile CLN application code, add the flags
3351 -I$CLN_DIR/include -I$CLN_TARGETDIR/include
3353 to the C++ compiler's command line (@code{make} variable CFLAGS or CXXFLAGS).
3354 When you link CLN application code to form an executable, add the flags
3356 $CLN_TARGETDIR/src/libcln.a
3358 to the C/C++ compiler's command line (@code{make} variable LIBS).
3360 If you did a @code{make install}, the include files are installed in a
3361 public directory (normally @code{/usr/local/include}), hence you don't
3362 need special flags for compiling. The library has been installed to a
3363 public directory as well (normally @code{/usr/local/lib}), hence when
3364 linking a CLN application it is sufficient to give the flag @code{-lcln}.
3366 Since CLN version 1.1, there are two tools to make the creation of
3367 software packages that use CLN easier:
3370 @cindex @code{cln-config}
3371 @code{cln-config} is a shell script that you can use to determine the
3372 compiler and linker command line options required to compile and link a
3373 program with CLN. Start it with @code{--help} to learn about its options
3374 or consult the manpage that comes with it.
3376 @cindex @code{AC_PATH_CLN}
3377 @code{AC_PATH_CLN} is for packages configured using GNU automake.
3380 @code{AC_PATH_CLN([@var{MIN-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])}
3382 This macro determines the location of CLN using @code{cln-config}, which
3383 is either found in the user's path, or from the environment variable
3384 @code{CLN_CONFIG}. It tests the installed libraries to make sure that
3385 their version is not earlier than @var{MIN-VERSION} (a default version
3386 will be used if not specified). If the required version was found, sets
3387 the @env{CLN_CPPFLAGS} and the @env{CLN_LIBS} variables. This
3388 macro is in the file @file{cln.m4} which is installed in
3389 @file{$datadir/aclocal}. Note that if automake was installed with a
3390 different @samp{--prefix} than CLN, you will either have to manually
3391 move @file{cln.m4} to automake's @file{$datadir/aclocal}, or give
3392 aclocal the @samp{-I} option when running it. Here is a possible example
3393 to be included in your package's @file{configure.in}:
3395 AC_PATH_CLN(1.1.0, [
3396 LIBS="$LIBS $CLN_LIBS"
3397 CPPFLAGS="$CPPFLAGS $CLN_CPPFLAGS"
3398 ], AC_MSG_ERROR([No suitable installed version of CLN could be found.]))
3403 @section Compatibility to old CLN versions
3405 @cindex compatibility
3407 As of CLN version 1.1 all non-macro identifiers were hidden in namespace
3408 @code{cln} in order to avoid potential name clashes with other C++
3409 libraries. If you have an old application, you will have to manually
3410 port it to the new scheme. The following principles will help during
3414 All headers are now in a separate subdirectory. Instead of including
3415 @code{cl_}@var{something}@code{.h}, include
3416 @code{cln/}@var{something}@code{.h} now.
3418 All public identifiers (typenames and functions) have lost their
3419 @code{cl_} prefix. Exceptions are all the typenames of number types,
3420 (cl_N, cl_I, cl_MI, @dots{}), rings, symbolic types (cl_string,
3421 cl_symbol) and polynomials (cl_UP_@var{type}). (This is because their
3422 names would not be mnemonic enough once the namespace @code{cln} is
3423 imported. Even in a namespace we favor @code{cl_N} over @code{N}.)
3425 All public @emph{functions} that had by a @code{cl_} in their name still
3426 carry that @code{cl_} if it is intrinsic part of a typename (as in
3427 @code{cl_I_to_int ()}).
3429 When developing other libraries, please keep in mind not to import the
3430 namespace @code{cln} in one of your public header files by saying
3431 @code{using namespace cln;}. This would propagate to other applications
3432 and can cause name clashes there.
3435 @section Include files
3436 @cindex include files
3437 @cindex header files
3439 Here is a summary of the include files and their contents.
3442 @item <cln/object.h>
3443 General definitions, reference counting, garbage collection.
3444 @item <cln/number.h>
3445 The class cl_number.
3446 @item <cln/complex.h>
3447 Functions for class cl_N, the complex numbers.
3449 Functions for class cl_R, the real numbers.
3451 Functions for class cl_F, the floats.
3452 @item <cln/sfloat.h>
3453 Functions for class cl_SF, the short-floats.
3454 @item <cln/ffloat.h>
3455 Functions for class cl_FF, the single-floats.
3456 @item <cln/dfloat.h>
3457 Functions for class cl_DF, the double-floats.
3458 @item <cln/lfloat.h>
3459 Functions for class cl_LF, the long-floats.
3460 @item <cln/rational.h>
3461 Functions for class cl_RA, the rational numbers.
3462 @item <cln/integer.h>
3463 Functions for class cl_I, the integers.
3466 @item <cln/complex_io.h>
3467 Input/Output for class cl_N, the complex numbers.
3468 @item <cln/real_io.h>
3469 Input/Output for class cl_R, the real numbers.
3470 @item <cln/float_io.h>
3471 Input/Output for class cl_F, the floats.
3472 @item <cln/sfloat_io.h>
3473 Input/Output for class cl_SF, the short-floats.
3474 @item <cln/ffloat_io.h>
3475 Input/Output for class cl_FF, the single-floats.
3476 @item <cln/dfloat_io.h>
3477 Input/Output for class cl_DF, the double-floats.
3478 @item <cln/lfloat_io.h>
3479 Input/Output for class cl_LF, the long-floats.
3480 @item <cln/rational_io.h>
3481 Input/Output for class cl_RA, the rational numbers.
3482 @item <cln/integer_io.h>
3483 Input/Output for class cl_I, the integers.
3485 Flags for customizing input operations.
3486 @item <cln/output.h>
3487 Flags for customizing output operations.
3488 @item <cln/malloc.h>
3489 @code{malloc_hook}, @code{free_hook}.
3492 @item <cln/condition.h>
3493 Conditions/exceptions.
3494 @item <cln/string.h>
3496 @item <cln/symbol.h>
3498 @item <cln/proplist.h>
3502 @item <cln/null_ring.h>
3504 @item <cln/complex_ring.h>
3505 The ring of complex numbers.
3506 @item <cln/real_ring.h>
3507 The ring of real numbers.
3508 @item <cln/rational_ring.h>
3509 The ring of rational numbers.
3510 @item <cln/integer_ring.h>
3511 The ring of integers.
3512 @item <cln/numtheory.h>
3513 Number threory functions.
3514 @item <cln/modinteger.h>
3520 @item <cln/GV_number.h>
3521 General vectors over cl_number.
3522 @item <cln/GV_complex.h>
3523 General vectors over cl_N.
3524 @item <cln/GV_real.h>
3525 General vectors over cl_R.
3526 @item <cln/GV_rational.h>
3527 General vectors over cl_RA.
3528 @item <cln/GV_integer.h>
3529 General vectors over cl_I.
3530 @item <cln/GV_modinteger.h>
3531 General vectors of modular integers.
3534 @item <cln/SV_number.h>
3535 Simple vectors over cl_number.
3536 @item <cln/SV_complex.h>
3537 Simple vectors over cl_N.
3538 @item <cln/SV_real.h>
3539 Simple vectors over cl_R.
3540 @item <cln/SV_rational.h>
3541 Simple vectors over cl_RA.
3542 @item <cln/SV_integer.h>
3543 Simple vectors over cl_I.
3544 @item <cln/SV_ringelt.h>
3545 Simple vectors of general ring elements.
3546 @item <cln/univpoly.h>
3547 Univariate polynomials.
3548 @item <cln/univpoly_integer.h>
3549 Univariate polynomials over the integers.
3550 @item <cln/univpoly_rational.h>
3551 Univariate polynomials over the rational numbers.
3552 @item <cln/univpoly_real.h>
3553 Univariate polynomials over the real numbers.
3554 @item <cln/univpoly_complex.h>
3555 Univariate polynomials over the complex numbers.
3556 @item <cln/univpoly_modint.h>
3557 Univariate polynomials over modular integer rings.
3558 @item <cln/timing.h>
3561 Includes all of the above.
3567 A function which computes the nth Fibonacci number can be written as follows.
3568 @cindex Fibonacci number
3571 #include <cln/integer.h>
3572 #include <cln/real.h>
3573 using namespace cln;
3575 // Returns F_n, computed as the nearest integer to
3576 // ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
3577 const cl_I fibonacci (int n)
3579 // Need a precision of ((1+sqrt(5))/2)^-n.
3580 float_format_t prec = float_format((int)(0.208987641*n+5));
3581 cl_R sqrt5 = sqrt(cl_float(5,prec));
3582 cl_R phi = (1+sqrt5)/2;
3583 return round1( expt(phi,n)/sqrt5 );
3587 Let's explain what is going on in detail.
3589 The include file @code{<cln/integer.h>} is necessary because the type
3590 @code{cl_I} is used in the function, and the include file @code{<cln/real.h>}
3591 is needed for the type @code{cl_R} and the floating point number functions.
3592 The order of the include files does not matter. In order not to write
3593 out @code{cln::}@var{foo} in this simple example we can safely import
3594 the whole namespace @code{cln}.
3596 Then comes the function declaration. The argument is an @code{int}, the
3597 result an integer. The return type is defined as @samp{const cl_I}, not
3598 simply @samp{cl_I}, because that allows the compiler to detect typos like
3599 @samp{fibonacci(n) = 100}. It would be possible to declare the return
3600 type as @code{const cl_R} (real number) or even @code{const cl_N} (complex
3601 number). We use the most specialized possible return type because functions
3602 which call @samp{fibonacci} will be able to profit from the compiler's type
3603 analysis: Adding two integers is slightly more efficient than adding the
3604 same objects declared as complex numbers, because it needs less type
3605 dispatch. Also, when linking to CLN as a non-shared library, this minimizes
3606 the size of the resulting executable program.
3608 The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
3609 integer. In order to get a correct result, the absolute error should be less
3610 than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
3611 To this end, the first line computes a floating point precision for sqrt(5)
3614 Then sqrt(5) is computed by first converting the integer 5 to a floating point
3615 number and than taking the square root. The converse, first taking the square
3616 root of 5, and then converting to the desired precision, would not work in
3617 CLN: The square root would be computed to a default precision (normally
3618 single-float precision), and the following conversion could not help about
3619 the lacking accuracy. This is because CLN is not a symbolic computer algebra
3620 system and does not represent sqrt(5) in a non-numeric way.
3622 The type @code{cl_R} for sqrt5 and, in the following line, phi is the only
3623 possible choice. You cannot write @code{cl_F} because the C++ compiler can
3624 only infer that @code{cl_float(5,prec)} is a real number. You cannot write
3625 @code{cl_N} because a @samp{round1} does not exist for general complex
3628 When the function returns, all the local variables in the function are
3629 automatically reclaimed (garbage collected). Only the result survives and
3630 gets passed to the caller.
3632 The file @code{fibonacci.cc} in the subdirectory @code{examples}
3633 contains this implementation together with an even faster algorithm.
3635 @section Debugging support
3638 When debugging a CLN application with GNU @code{gdb}, two facilities are
3639 available from the library:
3642 @item The library does type checks, range checks, consistency checks at
3643 many places. When one of these fails, the function @code{cl_abort()} is
3644 called. Its default implementation is to perform an @code{exit(1)}, so
3645 you won't have a core dump. But for debugging, it is best to set a
3646 breakpoint at this function:
3648 (gdb) break cl_abort
3650 When this breakpoint is hit, look at the stack's backtrace:
3655 @item The debugger's normal @code{print} command doesn't know about
3656 CLN's types and therefore prints mostly useless hexadecimal addresses.
3657 CLN offers a function @code{cl_print}, callable from the debugger,
3658 for printing number objects. In order to get this function, you have
3659 to define the macro @samp{CL_DEBUG} and then include all the header files
3660 for which you want @code{cl_print} debugging support. For example:
3661 @cindex @code{CL_DEBUG}
3664 #include <cln/string.h>
3666 Now, if you have in your program a variable @code{cl_string s}, and
3667 inspect it under @code{gdb}, the output may look like this:
3670 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3671 word = 134568800@}@}, @}
3672 (gdb) call cl_print(s)
3676 Note that the output of @code{cl_print} goes to the program's error output,
3677 not to gdb's standard output.
3679 Note, however, that the above facility does not work with all CLN types,
3680 only with number objects and similar. Therefore CLN offers a member function
3681 @code{debug_print()} on all CLN types. The same macro @samp{CL_DEBUG}
3682 is needed for this member function to be implemented. Under @code{gdb},
3683 you call it like this:
3684 @cindex @code{debug_print ()}
3687 $7 = @{<cl_gcpointer> = @{ = @{pointer = 0x8055b60, heappointer = 0x8055b60,
3688 word = 134568800@}@}, @}
3689 (gdb) call s.debug_print()
3692 >call ($1).debug_print()
3697 Unfortunately, this feature does not seem to work under all circumstances.
3701 @chapter Customizing
3704 @section Error handling
3706 When a fatal error occurs, an error message is output to the standard error
3707 output stream, and the function @code{cl_abort} is called. The default
3708 version of this function (provided in the library) terminates the application.
3709 To catch such a fatal error, you need to define the function @code{cl_abort}
3710 yourself, with the prototype
3712 #include <cln/abort.h>
3713 void cl_abort (void);
3715 @cindex @code{cl_abort ()}
3716 This function must not return control to its caller.
3719 @section Floating-point underflow
3722 Floating point underflow denotes the situation when a floating-point number
3723 is to be created which is so close to @code{0} that its exponent is too
3724 low to be represented internally. By default, this causes a fatal error.
3725 If you set the global variable
3727 cl_boolean cl_inhibit_floating_point_underflow
3729 to @code{cl_true}, the error will be inhibited, and a floating-point zero
3730 will be generated instead. The default value of
3731 @code{cl_inhibit_floating_point_underflow} is @code{cl_false}.
3734 @section Customizing I/O
3736 The output of the function @code{fprint} may be customized by changing the
3737 value of the global variable @code{default_print_flags}.
3738 @cindex @code{default_print_flags}
3741 @section Customizing the memory allocator
3743 Every memory allocation of CLN is done through the function pointer
3744 @code{malloc_hook}. Freeing of this memory is done through the function
3745 pointer @code{free_hook}. The default versions of these functions,
3746 provided in the library, call @code{malloc} and @code{free} and check
3747 the @code{malloc} result against @code{NULL}.
3748 If you want to provide another memory allocator, you need to define
3749 the variables @code{malloc_hook} and @code{free_hook} yourself,
3752 #include <cln/malloc.h>
3754 void* (*malloc_hook) (size_t size) = @dots{};
3755 void (*free_hook) (void* ptr) = @dots{};
3758 @cindex @code{malloc_hook ()}
3759 @cindex @code{free_hook ()}
3760 The @code{cl_malloc_hook} function must not return a @code{NULL} pointer.
3762 It is not possible to change the memory allocator at runtime, because
3763 it is already called at program startup by the constructors of some
3776 @c Table of contents