1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Important Algorithms:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistical structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999 Johannes Gutenberg University Mainz,
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of it's own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
183 using namespace GiNaC;
187 symbol x("x"), y("y");
190 for (int i=0; i<3; ++i)
191 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
193 cout << poly << endl;
198 Assuming the file is called @file{hello.cc}, on our system we can compile
199 and run it like this:
202 $ c++ hello.cc -o hello -lcln -lginac
204 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
207 (@xref{Package Tools}, for tools that help you when creating a software
208 package that uses GiNaC.)
210 @cindex Hermite polynomial
211 Next, there is a more meaningful C++ program that calls a function which
212 generates Hermite polynomials in a specified free variable.
215 #include <ginac/ginac.h>
216 using namespace GiNaC;
218 ex HermitePoly(symbol x, int deg)
220 ex HKer=exp(-pow(x,2));
221 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
222 return normal(pow(-1,deg) * diff(HKer, x, deg) / HKer);
229 for (int i=0; i<6; ++i)
230 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
236 When run, this will type out
242 H_3(z) == -12*z+8*z^3
243 H_4(z) == -48*z^2+16*z^4+12
244 H_5(z) == 120*z-160*z^3+32*z^5
247 This method of generating the coefficients is of course far from optimal
248 for production purposes.
250 In order to show some more examples of what GiNaC can do we will now use
251 the @command{ginsh}, a simple GiNaC interactive shell that provides a
252 convenient window into GiNaC's capabilities.
255 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
256 @c node-name, next, previous, up
257 @section What it can do for you
260 After invoking @command{ginsh} one can test and experiment with GiNaC's
261 features much like in other Computer Algebra Systems except that it does
262 not provide programming constructs like loops or conditionals. For a
263 concise description of the @command{ginsh} syntax we refer to its
264 accompanied man page. Suffice to say that assignments and comparisons in
265 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
268 It can manipulate arbitrary precision integers in a very fast way.
269 Rational numbers are automatically converted to fractions of coprime
274 369988485035126972924700782451696644186473100389722973815184405301748249
276 123329495011708990974900260817232214728824366796574324605061468433916083
283 All numbers occuring in GiNaC's expressions can be converted into floating
284 point numbers with the @code{evalf} method, to arbitrary accuracy:
288 0.14285714285714285714
292 0.1428571428571428571428571428571428571428571428571428571428571428571428
293 5714285714285714285714285714285714285
296 Exact numbers other than rationals that can be manipulated in GiNaC
297 include predefined constants like Archimedes' @code{Pi}. They can both
298 be used in symbolic manipulations (as an exact number) as well as in
299 numeric expressions (as an inexact number):
305 x+9.869604401089358619L0
309 11.869604401089358619L0
312 Built-in functions evaluate immediately to exact numbers if
313 this is possible. Conversions that can be safely performed are done
314 immediately; conversions that are not generally valid are not done:
325 (Note that converting the last input to @code{x} would allow one to
326 conclude that @code{42*Pi} is equal to @code{0}.)
328 Linear equation systems can be solved along with basic linear
329 algebra manipulations over symbolic expressions. In C++ GiNaC offers
330 a matrix class for this purpose but we can see what it can do using
331 @command{ginsh}'s notation of double brackets to type them in:
334 > lsolve(a+x*y==z,x);
336 lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
338 > M = [[ [[1, 3]], [[-3, 2]] ]];
339 [[ [[1,3]], [[-3,2]] ]]
342 > charpoly(M,lambda);
346 Multivariate polynomials and rational functions may be expanded,
347 collected and normalized (i.e. converted to a ratio of two coprime
351 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
352 -3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
353 > b = x^2 + 4*x*y - y^2;
356 3*y^6+x^6-24*x*y^5+43*x^2*y^4+16*x^3*y^3+17*x^4*y^2+8*x^5*y
358 3*y^6+48*x*y^4+2*x^2*y^2+x^4*(-y^2+x^2+4*x*y)+4*x^3*y*(-y^2+x^2+4*x*y)
363 You can differentiate functions and expand them as Taylor or Laurent
364 series (the third argument of @code{series} is the evaluation point, the
365 fourth defines the order):
370 > series(sin(x),x,0,4);
372 > series(1/tan(x),x,0,4);
373 x^(-1)-1/3*x+Order(x^2)
374 > series(gamma(2*sin(x)-2),x,Pi/2,6);
375 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*EulerGamma^2-1/240)*(x-1/2*Pi)^2
376 -EulerGamma-1/12+Order((x-1/2*Pi)^3)
379 If you ever wanted to convert units in C or C++ and found this
380 is cumbersome, here is the solution. Symbolic types can always be
381 used as tags for different types of objects. Converting from wrong
382 units to the metric system is therefore easy:
390 140613.91592783185568*kg*m^(-2)
394 @node Installation, Prerequisites, What it can do for you, Top
395 @c node-name, next, previous, up
396 @chapter Installation
399 GiNaC's installation follows the spirit of most GNU software. It is
400 easily installed on your system by three steps: configuration, build,
404 * Prerequisites:: Packages upon which GiNaC depends.
405 * Configuration:: How to configure GiNaC.
406 * Building GiNaC:: How to compile GiNaC.
407 * Installing GiNaC:: How to install GiNaC on your system.
411 @node Prerequisites, Configuration, Installation, Installation
412 @c node-name, next, previous, up
413 @section Prerequisites
415 In order to install GiNaC on your system, some prerequisites need
416 to be met. First of all, you need to have a C++-compiler adhering to
417 the ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
418 development so if you have a different compiler you are on your own.
419 For the configuration to succeed you need a Posix compliant shell
420 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
421 by the built process as well, since some of the source files are automatically
422 generated by Perl scripts. Last but not least, Bruno Haible's library
423 @acronym{CLN} is extensively used and needs to be installed on your system.
424 Please get it from @uref{ftp://ftp.santafe.edu/pub/gnu/} or from
425 @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP site}
426 (it is covered by GPL) and install it prior to trying to install GiNaC.
427 The configure script checks if it can find it and if it cannot
428 it will refuse to continue.
431 @node Configuration, Building GiNaC, Prerequisites, Installation
432 @c node-name, next, previous, up
433 @section Configuration
434 @cindex configuration
437 To configure GiNaC means to prepare the source distribution for
438 building. It is done via a shell script called @command{configure} that
439 is shipped with the sources and was originally generated by GNU
440 Autoconf. Since a configure script generated by GNU Autoconf never
441 prompts, all customization must be done either via command line
442 parameters or environment variables. It accepts a list of parameters,
443 the complete set of which can be listed by calling it with the
444 @option{--help} option. The most important ones will be shortly
445 described in what follows:
450 @option{--disable-shared}: When given, this option switches off the
451 build of a shared library, i.e. a @file{.so} file. This may be convenient
452 when developing because it considerably speeds up compilation.
455 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
456 and headers are installed. It defaults to @file{/usr/local} which means
457 that the library is installed in the directory @file{/usr/local/lib},
458 the header files in @file{/usr/local/include/GiNaC} and the documentation
459 (like this one) into @file{/usr/local/share/doc/GiNaC}.
462 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
463 the library installed in some other directory than
464 @file{@var{PREFIX}/lib/}.
467 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
468 to have the header files installed in some other directory than
469 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
470 @option{--includedir=/usr/include} you will end up with the header files
471 sitting in the directory @file{/usr/include/ginac/}. Note that the
472 subdirectory @file{GiNaC} is enforced by this process in order to
473 keep the header files separated from others. This avoids some
474 clashes and allows for an easier deinstallation of GiNaC. This ought
475 to be considered A Good Thing (tm).
478 @option{--datadir=@var{DATADIR}}: This option may be given in case you
479 want to have the documentation installed in some other directory than
480 @file{@var{PREFIX}/share/doc/GiNaC/}.
484 In addition, you may specify some environment variables.
485 @env{CXX} holds the path and the name of the C++ compiler
486 in case you want to override the default in your path. (The
487 @command{configure} script searches your path for @command{c++},
488 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
489 and @command{cc++} in that order.) It may be very useful to
490 define some compiler flags with the @env{CXXFLAGS} environment
491 variable, like optimization, debugging information and warning
492 levels. If omitted, it defaults to @option{-g -O2}.
494 The whole process is illustrated in the following two
495 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
496 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
499 Here is a simple configuration for a site-wide GiNaC library assuming
500 everything is in default paths:
503 $ export CXXFLAGS="-Wall -O2"
507 And here is a configuration for a private static GiNaC library with
508 several components sitting in custom places (site-wide @acronym{GCC} and
509 private @acronym{CLN}). The compiler is pursuaded to be picky and full
510 assertions and debugging are switched on:
513 $ export CXX=/usr/local/gnu/bin/c++
514 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
515 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
516 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
517 $ ./configure --disable-shared --prefix=$(HOME)
521 @node Building GiNaC, Installing GiNaC, Configuration, Installation
522 @c node-name, next, previous, up
523 @section Building GiNaC
524 @cindex building GiNaC
526 After proper configuration you should just build the whole
531 at the command prompt and go for a cup of coffee. The exact time it
532 takes to compile GiNaC depends not only on the speed of your machines
533 but also on other parameters, for instance what value for @env{CXXFLAGS}
534 you entered. Optimization may be very time-consuming.
536 Just to make sure GiNaC works properly you may run a simple test
543 This will compile some sample programs, run them and compare the output
544 to reference output. Each of the checks should return a message @samp{passed}
545 together with the CPU time used for that particular test. If it does
546 not, something went wrong. This is mostly intended to be a QA-check
547 if something was broken during the development, not a sanity check
548 of your system. Another intent is to allow people to fiddle around
549 with optimization. If @acronym{CLN} was installed all right
550 this step is unlikely to return any errors.
552 Generally, the top-level Makefile runs recursively to the
553 subdirectories. It is therfore safe to go into any subdirectory
554 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
555 @var{target} there in case something went wrong.
558 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
559 @c node-name, next, previous, up
560 @section Installing GiNaC
563 To install GiNaC on your system, simply type
569 As described in the section about configuration the files will be
570 installed in the following directories (the directories will be created
571 if they don't already exist):
576 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
577 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
578 So will @file{libginac.so} unless the configure script was
579 given the option @option{--disable-shared}. The proper symlinks
580 will be established as well.
583 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
584 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
587 All documentation (HTML and Postscript) will be stuffed into
588 @file{@var{PREFIX}/share/doc/GiNaC/} (or
589 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
593 For the sake of completeness we will list some other useful make
594 targets: @command{make clean} deletes all files generated by
595 @command{make}, i.e. all the object files. In addition @command{make
596 distclean} removes all files generated by the configuration and
597 @command{make maintainer-clean} goes one step further and deletes files
598 that may require special tools to rebuild (like the @command{libtool}
599 for instance). Finally @command{make uninstall} removes the installed
600 library, header files and documentation@footnote{Uninstallation does not
601 work after you have called @command{make distclean} since the
602 @file{Makefile} is itself generated by the configuration from
603 @file{Makefile.in} and hence deleted by @command{make distclean}. There
604 are two obvious ways out of this dilemma. First, you can run the
605 configuration again with the same @var{PREFIX} thus creating a
606 @file{Makefile} with a working @samp{uninstall} target. Second, you can
607 do it by hand since you now know where all the files went during
611 @node Basic Concepts, Expressions, Installing GiNaC, Top
612 @c node-name, next, previous, up
613 @chapter Basic Concepts
615 This chapter will describe the different fundamental objects that can be
616 handled by GiNaC. But before doing so, it is worthwhile introducing you
617 to the more commonly used class of expressions, representing a flexible
618 meta-class for storing all mathematical objects.
621 * Expressions:: The fundamental GiNaC class.
622 * The Class Hierarchy:: Overview of GiNaC's classes.
623 * Symbols:: Symbolic objects.
624 * Numbers:: Numerical objects.
625 * Constants:: Pre-defined constants.
626 * Fundamental operations:: The power, add and mul classes.
627 * Built-in functions:: Mathematical functions.
631 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
632 @c node-name, next, previous, up
634 @cindex expression (class @code{ex})
637 The most common class of objects a user deals with is the expression
638 @code{ex}, representing a mathematical object like a variable, number,
639 function, sum, product, etc... Expressions may be put together to form
640 new expressions, passed as arguments to functions, and so on. Here is a
641 little collection of valid expressions:
644 ex MyEx1 = 5; // simple number
645 ex MyEx2 = x + 2*y; // polynomial in x and y
646 ex MyEx3 = (x + 1)/(x - 1); // rational expression
647 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
648 ex MyEx5 = MyEx4 + 1; // similar to above
651 Expressions are handles to other more fundamental objects, that many
652 times contain other expressions thus creating a tree of expressions
653 (@xref{Internal Structures}, for particular examples). Most methods on
654 @code{ex} therefore run top-down through such an expression tree. For
655 example, the method @code{has()} scans recursively for occurrences of
656 something inside an expression. Thus, if you have declared @code{MyEx4}
657 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
658 the argument of @code{sin} and hence return @code{true}.
660 The next sections will outline the general picture of GiNaC's class
661 hierarchy and describe the classes of objects that are handled by
665 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
666 @c node-name, next, previous, up
667 @section The Class Hierarchy
669 GiNaC's class hierarchy consists of several classes representing
670 mathematical objects, all of which (except for @code{ex} and some
671 helpers) are internally derived from one abstract base class called
672 @code{basic}. You do not have to deal with objects of class
673 @code{basic}, instead you'll be dealing with symbols and functions of
674 symbols. You'll soon learn in this chapter how many of the functions on
675 symbols are really classes. This is because simple symbolic arithmetic
676 is not supported by languages like C++ so in a certain way GiNaC has to
677 implement its own arithmetic.
679 To give an idea about what kinds of symbolic composits may be built we
680 have a look at the most important classes in the class hierarchy. The
681 oval classes are atomic ones and the squared classes are containers.
682 The dashed line symbolizes a "points to" or "handles" relationship while
683 the solid lines stand for "inherits from" relationship in the class
686 @image{classhierarchy}
688 Some of the classes shown here (the ones sitting in white boxes) are
689 abstract base classes that are of no interest at all for the user. They
690 are used internally in order to avoid code duplication if two or more
691 classes derived from them share certain features. An example would be
692 @code{expairseq}, which is a container for a sequence of pairs each
693 consisting of one expression and a number (@code{numeric}). What
694 @emph{is} visible to the user are the derived classes @code{add} and
695 @code{mul}, representing sums of terms and products, respectively.
696 We'll come back later to some more details about these two classes and
697 motivate the use of pairs in sums and products here.
700 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
701 @c node-name, next, previous, up
703 @cindex Symbols (class @code{symbol})
705 Symbols are for symbolic manipulation what atoms are for chemistry. You
706 can declare objects of class @code{symbol} as any other object simply by
707 saying @code{symbol x,y;}. There is, however, a catch in here having to
708 do with the fact that C++ is a compiled language. The information about
709 the symbol's name is thrown away by the compiler but at a later stage
710 you may want to print expressions holding your symbols. In order to
711 avoid confusion GiNaC's symbols are able to know their own name. This
712 is accomplished by declaring its name for output at construction time in
713 the fashion @code{symbol x("x");}. If you declare a symbol using the
714 default constructor (i.e. without string argument) the system will deal
715 out a unique name. That name may not be suitable for printing but for
716 internal routines when no output is desired it is often enough. We'll
717 come across examples of such symbols later in this tutorial.
719 This implies that the strings passed to symbols at construction time may
720 not be used for comparing two of them. It is perfectly legitimate to
721 write @code{symbol x("x"),y("x");} but it is likely to lead into
722 trouble. Here, @code{x} and @code{y} are different symbols and
723 statements like @code{x-y} will not be simplified to zero although the
724 output @code{x-x} looks funny. Such output may also occur when there
725 are two different symbols in two scopes, for instance when you call a
726 function that declares a symbol with a name already existent in a symbol
727 in the calling function. Again, comparing them (using @code{operator==}
728 for instance) will always reveal their difference. Watch out, please.
730 Although symbols can be assigned expressions for internal reasons, you
731 should not do it (and we are not going to tell you how it is done). If
732 you want to replace a symbol with something else in an expression, you
733 can use the expression's @code{.subs()} method.
736 @node Numbers, Constants, Symbols, Basic Concepts
737 @c node-name, next, previous, up
739 @cindex numbers (class @code{numeric})
742 For storing numerical things, GiNaC uses Bruno Haible's library
743 @acronym{CLN}. The classes therein serve as foundation classes for
744 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
745 alternatively for Common Lisp Numbers. In order to find out more about
746 @acronym{CLN}'s internals the reader is refered to the documentation of
747 that library. @inforef{Introduction, , cln}, for more
748 information. Suffice to say that it is by itself build on top of another
749 library, the GNU Multiple Precision library @acronym{GMP}, which is an
750 extremely fast library for arbitrary long integers and rationals as well
751 as arbitrary precision floating point numbers. It is very commonly used
752 by several popular cryptographic applications. @acronym{CLN} extends
753 @acronym{GMP} by several useful things: First, it introduces the complex
754 number field over either reals (i.e. floating point numbers with
755 arbitrary precision) or rationals. Second, it automatically converts
756 rationals to integers if the denominator is unity and complex numbers to
757 real numbers if the imaginary part vanishes and also correctly treats
758 algebraic functions. Third it provides good implementations of
759 state-of-the-art algorithms for all trigonometric and hyperbolic
760 functions as well as for calculation of some useful constants.
762 The user can construct an object of class @code{numeric} in several
763 ways. The following example shows the four most important constructors.
764 It uses construction from C-integer, construction of fractions from two
765 integers, construction from C-float and construction from a string:
768 #include <ginac/ginac.h>
769 using namespace GiNaC;
773 numeric two(2); // exact integer 2
774 numeric r(2,3); // exact fraction 2/3
775 numeric e(2.71828); // floating point number
776 numeric p("3.1415926535897932385"); // floating point number
778 cout << two*p << endl; // floating point 6.283...
783 Note that all those constructors are @emph{explicit} which means you are
784 not allowed to write @code{numeric two=2;}. This is because the basic
785 objects to be handled by GiNaC are the expressions @code{ex} and we want
786 to keep things simple and wish objects like @code{pow(x,2)} to be
787 handled the same way as @code{pow(x,a)}, which means that we need to
788 allow a general @code{ex} as base and exponent. Therefore there is an
789 implicit constructor from C-integers directly to expressions handling
790 numerics at work in most of our examples. This design really becomes
791 convenient when one declares own functions having more than one
792 parameter but it forbids using implicit constructors because that would
795 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
796 This would, however, call C's built-in operator @code{/} for integers
797 first and result in a numeric holding a plain integer 1. @strong{Never
798 use @code{/} on integers!} Use the constructor from two integers
799 instead, as shown in the example above. Writing @code{numeric(1)/2} may
800 look funny but works also.
802 @cindex @code{Digits}
804 We have seen now the distinction between exact numbers and floating
805 point numbers. Clearly, the user should never have to worry about
806 dynamically created exact numbers, since their "exactness" always
807 determines how they ought to be handled, i.e. how "long" they are. The
808 situation is different for floating point numbers. Their accuracy is
809 controlled by one @emph{global} variable, called @code{Digits}. (For
810 those readers who know about Maple: it behaves very much like Maple's
811 @code{Digits}). All objects of class numeric that are constructed from
812 then on will be stored with a precision matching that number of decimal
816 #include <ginac/ginac.h>
817 using namespace GiNaC;
821 numeric three(3.0), one(1.0);
822 numeric x = one/three;
824 cout << "in " << Digits << " digits:" << endl;
826 cout << Pi.evalf() << endl;
838 The above example prints the following output to screen:
845 0.333333333333333333333333333333333333333333333333333333333333333333
846 3.14159265358979323846264338327950288419716939937510582097494459231
849 It should be clear that objects of class @code{numeric} should be used
850 for constructing numbers or for doing arithmetic with them. The objects
851 one deals with most of the time are the polymorphic expressions @code{ex}.
853 @subsection Tests on numbers
855 Once you have declared some numbers, assigned them to expressions and
856 done some arithmetic with them it is frequently desired to retrieve some
857 kind of information from them like asking whether that number is
858 integer, rational, real or complex. For those cases GiNaC provides
859 several useful methods. (Internally, they fall back to invocations of
860 certain CLN functions.)
862 As an example, let's construct some rational number, multiply it with
863 some multiple of its denominator and test what comes out:
866 #include <ginac/ginac.h>
867 using namespace GiNaC;
869 // some very important constants:
870 const numeric twentyone(21);
871 const numeric ten(10);
872 const numeric five(5);
876 numeric answer = twentyone;
879 cout << answer.is_integer() << endl; // false, it's 21/5
881 cout << answer.is_integer() << endl; // true, it's 42 now!
886 Note that the variable @code{answer} is constructed here as an integer
887 by @code{numeric}'s copy constructor but in an intermediate step it
888 holds a rational number represented as integer numerator and integer
889 denominator. When multiplied by 10, the denominator becomes unity and
890 the result is automatically converted to a pure integer again.
891 Internally, the underlying @acronym{CLN} is responsible for this
892 behaviour and we refer the reader to @acronym{CLN}'s documentation.
893 Suffice to say that the same behaviour applies to complex numbers as
894 well as return values of certain functions. Complex numbers are
895 automatically converted to real numbers if the imaginary part becomes
896 zero. The full set of tests that can be applied is listed in the
900 @multitable @columnfractions .33 .67
901 @item Method @tab Returns true if@dots{}
902 @item @code{.is_zero()}
903 @tab object is equal to zero
904 @item @code{.is_positive()}
905 @tab object is not complex and greater than 0
906 @item @code{.is_integer()}
907 @tab object is a (non-complex) integer
908 @item @code{.is_pos_integer()}
909 @tab object is an integer and greater than 0
910 @item @code{.is_nonneg_integer()}
911 @tab object is an integer and greater equal 0
912 @item @code{.is_even()}
913 @tab object is an even integer
914 @item @code{.is_odd()}
915 @tab object is an odd integer
916 @item @code{.is_prime()}
917 @tab object is a prime integer (probabilistic primality test)
918 @item @code{.is_rational()}
919 @tab object is an exact rational number (integers are rational, too)
920 @item @code{.is_real()}
921 @tab object is a real integer, rational or float (i.e. is not complex)
922 @item @code{.is_cinteger()}
923 @tab object is a (complex) integer, such as @math{2-3*I}
924 @item @code{.is_crational()}
925 @tab object is an exact (complex) rational number (such as @math{2/3+7/2*I})
930 @node Constants, Fundamental operations, Numbers, Basic Concepts
931 @c node-name, next, previous, up
933 @cindex constants (class @code{constant})
934 @cindex @code{evalf()}
936 Constants behave pretty much like symbols except that they return some
937 specific number when the method @code{.evalf()} is called.
939 The predefined known constants are:
942 @multitable @columnfractions .14 .29 .57
943 @item Name @tab Common Name @tab Numerical Value (35 digits)
945 @tab Archimedes' constant
946 @tab 3.14159265358979323846264338327950288
948 @tab Catalan's constant
949 @tab 0.91596559417721901505460351493238411
950 @item @code{EulerGamma}
951 @tab Euler's (or Euler-Mascheroni) constant
952 @tab 0.57721566490153286060651209008240243
957 @node Fundamental operations, Built-in functions, Constants, Basic Concepts
958 @c node-name, next, previous, up
959 @section Fundamental operations: the @code{power}, @code{add} and @code{mul} classes
965 Simple polynomial expressions are written down in GiNaC pretty much like
966 in other CAS or like expressions involving numerical variables in C.
967 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
968 been overloaded to achieve this goal. When you run the following
969 program, the constructor for an object of type @code{mul} is
970 automatically called to hold the product of @code{a} and @code{b} and
971 then the constructor for an object of type @code{add} is called to hold
972 the sum of that @code{mul} object and the number one:
975 #include <ginac/ginac.h>
976 using namespace GiNaC;
980 symbol a("a"), b("b");
986 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
987 statement @code{pow(x,2);} to represent @code{x} squared. This direct
988 construction is necessary since we cannot safely overload the constructor
989 @code{^} in C++ to construct a @code{power} object. If we did, it would
990 have several counterintuitive effects:
994 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
996 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
997 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
998 interpret this as @code{x^(a^b)}.
1000 Also, expressions involving integer exponents are very frequently used,
1001 which makes it even more dangerous to overload @code{^} since it is then
1002 hard to distinguish between the semantics as exponentiation and the one
1003 for exclusive or. (It would be embarassing to return @code{1} where one
1004 has requested @code{2^3}.)
1007 @cindex @code{ginsh}
1008 All effects are contrary to mathematical notation and differ from the
1009 way most other CAS handle exponentiation, therefore overloading @code{^}
1010 is ruled out for GiNaC's C++ part. The situation is different in
1011 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1012 that the other frequently used exponentiation operator @code{**} does
1013 not exist at all in C++).
1015 To be somewhat more precise, objects of the three classes described
1016 here, are all containers for other expressions. An object of class
1017 @code{power} is best viewed as a container with two slots, one for the
1018 basis, one for the exponent. All valid GiNaC expressions can be
1019 inserted. However, basic transformations like simplifying
1020 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1021 when this is mathematically possible. If we replace the outer exponent
1022 three in the example by some symbols @code{a}, the simplification is not
1023 safe and will not be performed, since @code{a} might be @code{1/2} and
1026 Objects of type @code{add} and @code{mul} are containers with an
1027 arbitrary number of slots for expressions to be inserted. Again, simple
1028 and safe simplifications are carried out like transforming
1029 @code{3*x+4-x} to @code{2*x+4}.
1031 The general rule is that when you construct such objects, GiNaC
1032 automatically creates them in canonical form, which might differ from
1033 the form you typed in your program. This allows for rapid comparison of
1034 expressions, since after all @code{a-a} is simply zero. Note, that the
1035 canonical form is not necessarily lexicographical ordering or in any way
1036 easily guessable. It is only guaranteed that constructing the same
1037 expression twice, either implicitly or explicitly, results in the same
1041 @node Built-in functions, Important Algorithms, Fundamental operations, Basic Concepts
1042 @c node-name, next, previous, up
1043 @section Built-in functions
1045 There are quite a number of useful functions built into GiNaC. They are
1046 all objects of class @code{function}. They accept one or more
1047 expressions as arguments and return one expression. If the arguments
1048 are not numerical, the evaluation of the function may be halted, as it
1049 does in the next example:
1052 #include <ginac/ginac.h>
1053 using namespace GiNaC;
1057 symbol x("x"), y("y");
1060 cout << "gamma(" << foo << ") -> " << gamma(foo) << endl;
1061 ex bar = foo.subs(y==1);
1062 cout << "gamma(" << bar << ") -> " << gamma(bar) << endl;
1063 ex foobar = bar.subs(x==7);
1064 cout << "gamma(" << foobar << ") -> " << gamma(foobar) << endl;
1069 This program shows how the function returns itself twice and finally an
1070 expression that may be really useful:
1073 gamma(x+(1/2)*y) -> gamma(x+(1/2)*y)
1074 gamma(x+1/2) -> gamma(x+1/2)
1075 gamma(15/2) -> (135135/128)*Pi^(1/2)
1078 Most of these functions can be differentiated, series expanded and so
1079 on. Read the next chapter in order to learn more about this.
1082 @node Important Algorithms, Polynomial Expansion, Built-in functions, Top
1083 @c node-name, next, previous, up
1084 @chapter Important Algorithms
1087 In this chapter the most important algorithms provided by GiNaC will be
1088 described. Some of them are implemented as functions on expressions,
1089 others are implemented as methods provided by expression objects. If
1090 they are methods, there exists a wrapper function around it, so you can
1091 alternatively call it in a functional way as shown in the simple
1095 #include <ginac/ginac.h>
1096 using namespace GiNaC;
1100 ex x = numeric(1.0);
1102 cout << "As method: " << sin(x).evalf() << endl;
1103 cout << "As function: " << evalf(sin(x)) << endl;
1108 The general rule is that wherever methods accept one or more parameters
1109 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1110 wrapper accepts is the same but preceded by the object to act on
1111 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1112 most natural one in an OO model but it may lead to confusion for MapleV
1113 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1114 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1115 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1116 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1117 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1118 here. Also, users of MuPAD will in most cases feel more comfortable
1119 with GiNaC's convention. All function wrappers are always implemented
1120 as simple inline functions which just call the corresponding method and
1121 are only provided for users uncomfortable with OO who are dead set to
1122 avoid method invocations. Generally, a chain of function wrappers is
1123 much harder to read than a chain of methods and should therefore be
1124 avoided if possible. On the other hand, not everything in GiNaC is a
1125 method on class @code{ex} and sometimes calling a function cannot be
1129 * Polynomial Expansion::
1130 * Collecting expressions::
1131 * Polynomial Arithmetic::
1132 * Symbolic Differentiation::
1133 * Series Expansion::
1137 @node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
1138 @c node-name, next, previous, up
1139 @section Polynomial Expansion
1140 @cindex @code{expand()}
1142 A polynomial in one or more variables has many equivalent
1143 representations. Some useful ones serve a specific purpose. Consider
1144 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1145 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1146 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1147 representations are the recursive ones where one collects for exponents
1148 in one of the three variable. Since the factors are themselves
1149 polynomials in the remaining two variables the procedure can be
1150 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1151 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1154 To bring an expression into expanded form, its method @code{.expand()}
1155 may be called. In our example above, this corresponds to @math{4*x*y +
1156 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1157 GiNaC is not easily guessable you should be prepared to see different
1158 orderings of terms in such sums!
1161 @node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
1162 @c node-name, next, previous, up
1163 @section Collecting expressions
1164 @cindex @code{collect()}
1165 @cindex @code{coeff()}
1167 Another useful representation of multivariate polynomials is as a
1168 univariate polynomial in one of the variables with the coefficients
1169 being polynomials in the remaining variables. The method
1170 @code{collect()} accomplishes this task. Here is its declaration:
1173 ex ex::collect(symbol const & s);
1176 Note that the original polynomial needs to be in expanded form in order
1177 to be able to find the coefficients properly. The range of occuring
1178 coefficients can be checked using the two methods
1181 int ex::degree(symbol const & s);
1182 int ex::ldegree(symbol const & s);
1185 where @code{degree()} returns the highest coefficient and
1186 @code{ldegree()} the lowest one. (These two methods work also reliably
1187 on non-expanded input polynomials). An application is illustrated in
1188 the next example, where a multivariate polynomial is analysed:
1191 #include <ginac/ginac.h>
1192 using namespace GiNaC;
1196 symbol x("x"), y("y");
1197 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1198 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1199 ex Poly = PolyInp.expand();
1201 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1202 cout << "The x^" << i << "-coefficient is "
1203 << Poly.coeff(x,i) << endl;
1205 cout << "As polynomial in y: "
1206 << Poly.collect(y) << endl;
1211 When run, it returns an output in the following fashion:
1214 The x^0-coefficient is y^2+11*y
1215 The x^1-coefficient is 5*y^2-2*y
1216 The x^2-coefficient is -1
1217 The x^3-coefficient is 4*y
1218 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1221 As always, the exact output may vary between different versions of GiNaC
1222 or even from run to run since the internal canonical ordering is not
1223 within the user's sphere of influence.
1226 @node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
1227 @c node-name, next, previous, up
1228 @section Polynomial Arithmetic
1230 @subsection GCD and LCM
1234 The functions for polynomial greatest common divisor and least common
1235 multiple have the synopsis:
1238 ex gcd(const ex & a, const ex & b);
1239 ex lcm(const ex & a, const ex & b);
1242 The functions @code{gcd()} and @code{lcm()} accept two expressions
1243 @code{a} and @code{b} as arguments and return a new expression, their
1244 greatest common divisor or least common multiple, respectively. If the
1245 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1246 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1249 #include <ginac/ginac.h>
1250 using namespace GiNaC;
1254 symbol x("x"), y("y"), z("z");
1255 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1256 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1258 ex P_gcd = gcd(P_a, P_b);
1260 ex P_lcm = lcm(P_a, P_b);
1261 // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
1266 @subsection The @code{normal} method
1267 @cindex @code{normal()}
1268 @cindex temporary replacement
1270 While in common symbolic code @code{gcd()} and @code{lcm()} are not too
1271 heavily used, simplification is called for frequently. Therefore
1272 @code{.normal()}, which provides some basic form of simplification, has
1273 become a method of class @code{ex}, just like @code{.expand()}. It
1274 converts a rational function into an equivalent rational function where
1275 numerator and denominator are coprime. This means, it finds the GCD of
1276 numerator and denominator and cancels it. If it encounters some object
1277 which does not belong to the domain of rationals (a function for
1278 instance), that object is replaced by a temporary symbol. This means
1279 that both expressions @code{t1} and @code{t2} are indeed simplified in
1280 this little program:
1283 #include <ginac/ginac.h>
1284 using namespace GiNaC;
1289 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1290 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1291 cout << "t1 is " << t1.normal() << endl;
1292 cout << "t2 is " << t2.normal() << endl;
1297 Of course this works for multivariate polynomials too, so the ratio of
1298 the sample-polynomials from the section about GCD and LCM above would be
1299 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1302 @node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
1303 @c node-name, next, previous, up
1304 @section Symbolic Differentiation
1305 @cindex differentiation
1307 @cindex product rule
1309 GiNaC's objects know how to differentiate themselves. Thus, a
1310 polynomial (class @code{add}) knows that its derivative is the sum of
1311 the derivatives of all the monomials:
1314 #include <ginac/ginac.h>
1315 using namespace GiNaC;
1319 symbol x("x"), y("y"), z("z");
1320 ex P = pow(x, 5) + pow(x, 2) + y;
1322 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1323 cout << P.diff(y) << endl; // 1
1324 cout << P.diff(z) << endl; // 0
1329 If a second integer parameter @var{n} is given, the @code{diff} method
1330 returns the @var{n}th derivative.
1332 If @emph{every} object and every function is told what its derivative
1333 is, all derivatives of composed objects can be calculated using the
1334 chain rule and the product rule. Consider, for instance the expression
1335 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1336 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1337 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1338 out that the composition is the generating function for Euler Numbers,
1339 i.e. the so called @var{n}th Euler number is the coefficient of
1340 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1341 identity to code a function that generates Euler numbers in just three
1344 @cindex Euler numbers
1346 #include <ginac/ginac.h>
1347 using namespace GiNaC;
1349 ex EulerNumber(unsigned n)
1352 ex generator = pow(cosh(x),-1);
1353 return generator.diff(x,n).subs(x==0);
1358 for (unsigned i=0; i<11; i+=2)
1359 cout << EulerNumber(i) << endl;
1364 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1365 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1366 @code{i} by two since all odd Euler numbers vanish anyways.
1369 @node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
1370 @c node-name, next, previous, up
1371 @section Series Expansion
1372 @cindex series expansion
1373 @cindex Taylor expansion
1374 @cindex Laurent expansion
1376 Expressions know how to expand themselves as a Taylor series or (more
1377 generally) a Laurent series. Similar to most conventional Computer
1378 Algebra Systems, no distinction is made between those two. There is a
1379 class of its own for storing such series as well as a class for storing
1380 the order of the series. A sample program could read:
1383 #include <ginac/ginac.h>
1384 using namespace GiNaC;
1390 ex MyExpr1 = sin(x);
1391 ex MyExpr2 = 1/(x - pow(x, 2) - pow(x, 3));
1392 ex MyTailor, MySeries;
1394 MyTailor = MyExpr1.series(x, point, 5);
1395 cout << MyExpr1 << " == " << MyTailor
1396 << " for small " << x << endl;
1397 MySeries = MyExpr2.series(x, point, 7);
1398 cout << MyExpr2 << " == " << MySeries
1399 << " for small " << x << endl;
1404 @cindex M@'echain's formula
1405 As an instructive application, let us calculate the numerical value of
1406 Archimedes' constant
1410 (for which there already exists the built-in constant @code{Pi})
1411 using M@'echain's amazing formula
1413 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1416 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1418 We may expand the arcus tangent around @code{0} and insert the fractions
1419 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1420 carries an order term with it and the question arises what the system is
1421 supposed to do when the fractions are plugged into that order term. The
1422 solution is to use the function @code{series_to_poly()} to simply strip
1426 #include <ginac/ginac.h>
1427 using namespace GiNaC;
1429 ex mechain_pi(int degr)
1432 ex pi_expansion = series_to_poly(atan(x).series(x,0,degr));
1433 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1434 -4*pi_expansion.subs(x==numeric(1,239));
1441 for (int i=2; i<12; i+=2) @{
1442 pi_frac = mechain_pi(i);
1443 cout << i << ":\t" << pi_frac << endl
1444 << "\t" << pi_frac.evalf() << endl;
1450 When you run this program, it will type out:
1454 3.1832635983263598326
1455 4: 5359397032/1706489875
1456 3.1405970293260603143
1457 6: 38279241713339684/12184551018734375
1458 3.141621029325034425
1459 8: 76528487109180192540976/24359780855939418203125
1460 3.141591772182177295
1461 10: 327853873402258685803048818236/104359128170408663038552734375
1462 3.1415926824043995174
1466 @node Extending GiNaC, What does not belong into GiNaC, Series Expansion, Top
1467 @c node-name, next, previous, up
1468 @chapter Extending GiNaC
1470 By reading so far you should have gotten a fairly good understanding of
1471 GiNaC's design-patterns. From here on you should start reading the
1472 sources. All we can do now is issue some recommendations how to tackle
1473 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
1474 develop some useful extension please don't hesitate to contact the GiNaC
1475 authors---they will happily incorporate them into future versions.
1478 * What does not belong into GiNaC:: What to avoid.
1479 * Symbolic functions:: Implementing symbolic functions.
1483 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
1484 @c node-name, next, previous, up
1485 @section What doesn't belong into GiNaC
1487 @cindex @code{ginsh}
1488 First of all, GiNaC's name must be read literally. It is designed to be
1489 a library for use within C++. The tiny @command{ginsh} accompanying
1490 GiNaC makes this even more clear: it doesn't even attempt to provide a
1491 language. There are no loops or conditional expressions in
1492 @command{ginsh}, it is merely a window into the library for the
1493 programmer to test stuff (or to show off). Still, the design of a
1494 complete CAS with a language of its own, graphical capabilites and all
1495 this on top of GiNaC is possible and is without doubt a nice project for
1498 There are many built-in functions in GiNaC that do not know how to
1499 evaluate themselves numerically to a precision declared at runtime
1500 (using @code{Digits}). Some may be evaluated at certain points, but not
1501 generally. This ought to be fixed. However, doing numerical
1502 computations with GiNaC's quite abstract classes is doomed to be
1503 inefficient. For this purpose, the underlying bignum-package
1504 @acronym{CLN} is much better suited.
1507 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
1508 @c node-name, next, previous, up
1509 @section Symbolic functions
1511 The easiest and most instructive way to start with is probably to
1512 implement your own function. Objects of class @code{function} are
1513 inserted into the system via a kind of "registry". They get a serial
1514 number that is used internally to identify them but you usually need not
1515 worry about this. What you have to care for are functions that are
1516 called when the user invokes certain methods. These are usual
1517 C++-functions accepting a number of @code{ex} as arguments and returning
1518 one @code{ex}. As an example, if we have a look at a simplified
1519 implementation of the cosine trigonometric function, we first need a
1520 function that is called when one wishes to @code{eval} it. It could
1521 look something like this:
1524 static ex cos_eval_method(ex const & x)
1526 // if x%2*Pi return 1
1527 // if x%Pi return -1
1528 // if x%Pi/2 return 0
1529 // care for other cases...
1530 return cos(x).hold();
1534 The last line returns @code{cos(x)} if we don't know what else to do and
1535 stops a potential recursive evaluation by saying @code{.hold()}. We
1536 should also implement a method for numerical evaluation and since we are
1537 lazy we sweep the problem under the rug by calling someone else's
1538 function that does so, in this case the one in class @code{numeric}:
1541 static ex cos_evalf_method(ex const & x)
1543 return sin(ex_to_numeric(x));
1547 Differentiation will surely turn up and so we need to tell
1548 @code{sin} how to differentiate itself:
1551 static ex cos_diff_method(ex const & x, unsigned diff_param)
1557 @cindex product rule
1558 The second parameter is obligatory but uninteresting at this point. It
1559 specifies which parameter to differentiate in a partial derivative in
1560 case the function has more than one parameter and its main application
1561 is for correct handling of the chain rule. For Taylor expansion, it is
1562 enough to know how to differentiate. But if the function you want to
1563 implement does have a pole somewhere in the complex plane, you need to
1564 write another method for Laurent expansion around that point.
1566 Now that all the ingrediences for @code{cos} have been set up, we need
1567 to tell the system about it. This is done by a macro and we are not
1568 going to descibe how it expands, please consult your preprocessor if you
1572 REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);
1575 The first argument is the function's name, the second, third and fourth
1576 bind the corresponding methods to this objects and the fifth is a slot
1577 for inserting a method for series expansion. (If set to @code{NULL} it
1578 defaults to simple Taylor expansion, which is correct if there are no
1579 poles involved. The way GiNaC handles poles in case there are any is
1580 best understood by studying one of the examples, like the Gamma function
1581 for instance. In essence the function first checks if there is a pole
1582 at the evaluation point and falls back to Taylor expansion if there
1583 isn't. Then, the pole is regularized by some suitable transformation.)
1584 Also, the new function needs to be declared somewhere. This may also be
1585 done by a convenient preprocessor macro:
1588 DECLARE_FUNCTION_1P(cos)
1591 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
1592 implementation of @code{cos} is very incomplete and lacks several safety
1593 mechanisms. Please, have a look at the real implementation in GiNaC.
1594 (By the way: in case you are worrying about all the macros above we can
1595 assure you that functions are GiNaC's most macro-intense classes. We
1596 have done our best to avoid them where we can.)
1598 That's it. May the source be with you!
1601 @node A Comparison With Other CAS, Internal Structures, Symbolic functions, Top
1602 @c node-name, next, previous, up
1603 @chapter A Comparison With Other CAS
1606 This chapter will give you some information on how GiNaC compares to
1607 other, traditional Computer Algebra Systems, like @emph{Maple},
1608 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
1609 disadvantages over these systems.
1614 GiNaC has several advantages over traditional Computer
1615 Algebra Systems, like
1620 familiar language: all common CAS implement their own proprietary
1621 grammar which you have to learn first (and maybe learn again when your
1622 vendor chooses to "enhance" it). With GiNaC you can write your program
1623 in common C++, which is standardized.
1626 structured data types: you can build up structured data types using
1627 @code{struct}s or @code{class}es together with STL features instead of
1628 using unnamed lists of lists of lists.
1631 strongly typed: in CAS, you usually have only one kind of variables
1632 which can hold contents of an arbitrary type. This 4GL like feature is
1633 nice for novice programmers, but dangerous.
1636 development tools: powerful development tools exist for C++, like fancy
1637 editors (e.g. with automatic indentation and syntax highlighting),
1638 debuggers, visualization tools, documentation tools...
1641 modularization: C++ programs can easily be split into modules by
1642 separating interface and implementation.
1645 price: GiNaC is distributed under the GNU Public License which means
1646 that it is free and available with source code. And there are excellent
1647 C++-compilers for free, too.
1650 extendable: you can add your own classes to GiNaC, thus extending it on
1651 a very low level. Compare this to a traditional CAS that you can
1652 usually only extend on a high level by writing in the language defined
1653 by the parser. In particular, it turns out to be almost impossible to
1654 fix bugs in a traditional system.
1657 seemless integration: it is somewhere between difficult and impossible
1658 to call CAS functions from within a program written in C++ or any other
1659 programming language and vice versa. With GiNaC, your symbolic routines
1660 are part of your program. You can easily call third party libraries,
1661 e.g. for numerical evaluation or graphical interaction. All other
1662 approaches are much more cumbersome: they range from simply ignoring the
1663 problem (i.e. @emph{Maple}) to providing a method for "embedding" the
1664 system (i.e. @emph{Yacas}).
1667 efficiency: often large parts of a program do not need symbolic
1668 calculations at all. Why use large integers for loop variables or
1669 arbitrary precision arithmetics where double accuracy is sufficient?
1670 For pure symbolic applications, GiNaC is comparable in speed with other
1676 @heading Disadvantages
1678 Of course it also has some disadvantages:
1683 not interactive: GiNaC programs have to be written in an editor,
1684 compiled and executed. You cannot play with expressions interactively.
1685 However, such an extension is not inherently forbidden by design. In
1686 fact, two interactive interfaces are possible: First, a simple shell
1687 that exposes GiNaC's types to a command line can readily be written (and
1688 has been written) and second, as a more consistent approach we plan an
1689 integration with the @acronym{CINT} C++ interpreter.
1692 advanced features: GiNaC cannot compete with a program like
1693 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
1694 which grows since 1981 by the work of dozens of programmers, with
1695 respect to mathematical features. Integration, factorization,
1696 non-trivial simplifications, limits etc. are missing in GiNaC (and are
1697 not planned for the near future).
1700 portability: While the GiNaC library itself is designed to avoid any
1701 platform dependent features (it should compile on any ANSI compliant C++
1702 compiler), the currently used version of the CLN library (fast large
1703 integer and arbitrary precision arithmetics) can be compiled only on
1704 systems with a recently new C++ compiler from the GNU Compiler
1705 Collection (@acronym{GCC}). GiNaC uses recent language features like
1706 explicit constructors, mutable members, RTTI, @code{dynamic_cast}s and
1707 STL, so ANSI compliance is meant literally. Recent @acronym{GCC}
1708 versions starting at 2.95, although itself not yet ANSI compliant,
1709 support all needed features.
1716 Why did we choose to implement GiNaC in C++ instead of Java or any other
1717 language? C++ is not perfect: type checking is not strict (casting is
1718 possible), separation between interface and implementation is not
1719 complete, object oriented design is not enforced. The main reason is
1720 the often scolded feature of operator overloading in C++. While it may
1721 be true that operating on classes with a @code{+} operator is rarely
1722 meaningful, it is perfectly suited for algebraic expressions. Writing
1723 @math{3x+5y} as @code{3*x+5*y} instead of
1724 @code{x.times(3).plus(y.times(5))} looks much more natural. Furthermore,
1725 the main developers are more familiar with C++ than with any other
1726 programming language.
1729 @node Internal Structures, Expressions are reference counted, A Comparison With Other CAS, Top
1730 @c node-name, next, previous, up
1731 @appendix Internal Structures
1734 * Expressions are reference counted::
1735 * Internal representation of products and sums::
1738 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
1739 @c node-name, next, previous, up
1740 @appendixsection Expressions are reference counted
1742 @cindex reference counting
1743 @cindex copy-on-write
1744 @cindex garbage collection
1745 An expression is extremely light-weight since internally it works like a
1746 handle to the actual representation and really holds nothing more than a
1747 pointer to some other object. What this means in practice is that
1748 whenever you create two @code{ex} and set the second equal to the first
1749 no copying process is involved. Instead, the copying takes place as soon
1750 as you try to change the second. Consider the simple sequence of code:
1753 #include <ginac/ginac.h>
1754 using namespace GiNaC;
1758 symbol x("x"), y("y"), z("z");
1761 e1 = sin(x + 2*y) + 3*z + 41;
1762 e2 = e1; // e2 points to same object as e1
1763 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
1764 e2 += 1; // e2 is copied into a new object
1765 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
1770 The line @code{e2 = e1;} creates a second expression pointing to the
1771 object held already by @code{e1}. The time involved for this operation
1772 is therefore constant, no matter how large @code{e1} was. Actual
1773 copying, however, must take place in the line @code{e2 += 1;} because
1774 @code{e1} and @code{e2} are not handles for the same object any more.
1775 This concept is called @dfn{copy-on-write semantics}. It increases
1776 performance considerably whenever one object occurs multiple times and
1777 represents a simple garbage collection scheme because when an @code{ex}
1778 runs out of scope its destructor checks whether other expressions handle
1779 the object it points to too and deletes the object from memory if that
1780 turns out not to be the case. A slightly less trivial example of
1781 differentiation using the chain-rule should make clear how powerful this
1785 #include <ginac/ginac.h>
1786 using namespace GiNaC;
1790 symbol x("x"), y("y");
1794 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
1795 cout << e1 << endl // prints x+3*y
1796 << e2 << endl // prints (x+3*y)^3
1797 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
1802 Here, @code{e1} will actually be referenced three times while @code{e2}
1803 will be referenced two times. When the power of an expression is built,
1804 that expression needs not be copied. Likewise, since the derivative of
1805 a power of an expression can be easily expressed in terms of that
1806 expression, no copying of @code{e1} is involved when @code{e3} is
1807 constructed. So, when @code{e3} is constructed it will print as
1808 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
1809 holds a reference to @code{e2} and the factor in front is just
1812 As a user of GiNaC, you cannot see this mechanism of copy-on-write
1813 semantics. When you insert an expression into a second expression, the
1814 result behaves exactly as if the contents of the first expression were
1815 inserted. But it may be useful to remember that this is not what
1816 happens. Knowing this will enable you to write much more efficient
1817 code. If you still have an uncertain feeling with copy-on-write
1818 semantics, we recommend you have a look at the
1819 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
1820 Marshall Cline. Chapter 16 covers this issue and presents an
1821 implementation which is pretty close to the one in GiNaC.
1824 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
1825 @c node-name, next, previous, up
1826 @appendixsection Internal representation of products and sums
1828 @cindex representation
1831 @cindex @code{power}
1832 Although it should be completely transparent for the user of
1833 GiNaC a short discussion of this topic helps to understand the sources
1834 and also explain performance to a large degree. Consider the
1835 unexpanded symbolic expression
1837 $2d^3 \left( 4a + 5b - 3 \right)$
1840 @math{2*d^3*(4*a+5*b-3)}
1842 which could naively be represented by a tree of linear containers for
1843 addition and multiplication, one container for exponentiation with base
1844 and exponent and some atomic leaves of symbols and numbers in this
1849 @cindex pair-wise representation
1850 However, doing so results in a rather deeply nested tree which will
1851 quickly become inefficient to manipulate. We can improve on this by
1852 representing the sum instead as a sequence of terms, each one being a
1853 pair of a purely numeric multiplicative coefficient and its rest. In
1854 the same spirit we can store the multiplication as a sequence of terms,
1855 each having a numeric exponent and a possibly complicated base, the tree
1856 becomes much more flat:
1860 The number @code{3} above the symbol @code{d} shows that @code{mul}
1861 objects are treated similarly where the coefficients are interpreted as
1862 @emph{exponents} now. Addition of sums of terms or multiplication of
1863 products with numerical exponents can be coded to be very efficient with
1864 such a pair-wise representation. Internally, this handling is performed
1865 by most CAS in this way. It typically speeds up manipulations by an
1866 order of magnitude. The overall multiplicative factor @code{2} and the
1867 additive term @code{-3} look somewhat out of place in this
1868 representation, however, since they are still carrying a trivial
1869 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
1870 this is avoided by adding a field that carries an overall numeric
1871 coefficient. This results in the realistic picture of internal
1874 $2d^3 \left( 4a + 5b - 3 \right)$
1877 @math{2*d^3*(4*a+5*b-3)}
1883 This also allows for a better handling of numeric radicals, since
1884 @code{sqrt(2)} can now be carried along calculations. Now it should be
1885 clear, why both classes @code{add} and @code{mul} are derived from the
1886 same abstract class: the data representation is the same, only the
1887 semantics differs. In the class hierarchy, methods for polynomial
1888 expansion and the like are reimplemented for @code{add} and @code{mul},
1889 but the data structure is inherited from @code{expairseq}.
1892 @node Package Tools, ginac-config, Internal representation of products and sums, Top
1893 @c node-name, next, previous, up
1894 @appendix Package Tools
1896 If you are creating a software package that uses the GiNaC library,
1897 setting the correct command line options for the compiler and linker
1898 can be difficult. GiNaC includes two tools to make this process easier.
1901 * ginac-config:: A shell script to detect compiler and linker flags.
1902 * AM_PATH_GINAC:: Macro for GNU automake.
1906 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
1907 @c node-name, next, previous, up
1908 @section @command{ginac-config}
1909 @cindex ginac-config
1911 @command{ginac-config} is a shell script that you can use to determine
1912 the compiler and linker command line options required to compile and
1913 link a program with the GiNaC library.
1915 @command{ginac-config} takes the following flags:
1919 Prints out the version of GiNaC installed.
1921 Prints '-I' flags pointing to the installed header files.
1923 Prints out the linker flags necessary to link a program against GiNaC.
1924 @item --prefix[=@var{PREFIX}]
1925 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
1926 (And of exec-prefix, unless --exec-prefix is also specified)
1927 Otherwise, prints out the configured value of @env{$prefix}.
1928 @item --exec-prefix[=@var{PREFIX}]
1929 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
1930 Otherwise, prints out the configured value of @env{$exec_prefix}.
1933 Typically, @command{ginac-config} will be used within a configure script,
1934 as described below. It, however, can also be used directly
1935 from the command line to compile a simple program. For example:
1938 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
1941 This command line might expand to (for example):
1944 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
1945 -lginac -lcln -lstdc++
1948 Not only is the form using @command{ginac-config} easier to type, it will
1949 work on any system, no matter how GiNaC was configured.
1952 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
1953 @c node-name, next, previous, up
1954 @section @samp{AM_PATH_GINAC}
1955 @cindex AM_PATH_GINAC
1957 For packages configured using GNU automake, GiNaC also provides
1958 a macro to automate the process of checking for GiNaC.
1961 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
1969 Determines the location of GiNaC using @command{ginac-config}, which is
1970 either found in the user's path, or from the environment variable
1971 @env{GINACLIB_CONFIG}.
1974 Tests the installed libraries to make sure that there version
1975 is later than @var{MINIMUM-VERSION}. (A default version will be used
1979 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
1980 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
1981 variable to the output of @command{ginac-config --libs}, and calls
1982 @samp{AC_SUBST()} for these variables so they can be used in generated
1983 makefiles, and then executes @var{ACTION-IF-FOUND}.
1986 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
1987 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
1991 This macro is in file @file{ginac.m4} which is installed in
1992 @file{$datadir/aclocal}. Note that if automake was installed with a
1993 different @samp{--prefix} than GiNaC, you will either have to manually
1994 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
1995 aclocal the @samp{-I} option when running it.
1998 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
1999 * Example package:: Example of a package using AM_PATH_GINAC.
2003 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2004 @c node-name, next, previous, up
2005 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2007 Simply make sure that @command{ginac-config} is in your path, and run
2008 the configure script.
2015 The directory where the GiNaC libraries are installed needs
2016 to be found by your system's dynamic linker.
2018 This is generally done by
2021 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2027 setting the environment variable @env{LD_LIBRARY_PATH},
2030 or, as a last resort,
2033 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2034 running configure, for instance:
2037 LDFLAGS=-R/home/cbauer/lib ./configure
2042 You can also specify a @command{ginac-config} not in your path by
2043 setting the @env{GINACLIB_CONFIG} environment variable to the
2044 name of the executable
2047 If you move the GiNaC package from its installed location,
2048 you will need either need to modify @command{ginac-config} script
2049 manually to point to the new location or rebuild GiNaC.
2060 --with-ginac-prefix=@var{PREFIX}
2061 --with-ginac-exec-prefix=@var{PREFIX}
2064 are provided to override the prefix and exec-prefix that were stored
2065 in the @command{ginac-config} shell script by GiNaC's configure. You are
2066 generally better off configuring GiNaC with the right path to begin with.
2070 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2071 @c node-name, next, previous, up
2072 @subsection Example of a package using @samp{AM_PATH_GINAC}
2074 The following shows how to build a simple package using automake
2075 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2078 #include <ginac/ginac.h>
2079 using namespace GiNaC;
2085 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2090 You should first read the introductory portions of the automake
2091 Manual, if you are not already familiar with it.
2093 Two files are needed, @file{configure.in}, which is used to build the
2097 dnl Process this file with autoconf to produce a configure script.
2099 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2105 AM_PATH_GINAC(0.4.0, [
2106 LIBS="$LIBS $GINACLIB_LIBS"
2107 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2108 ], AC_MSG_ERROR([need to have GiNaC installed]))
2113 The only command in this which is not standard for automake
2114 is the @samp{AM_PATH_GINAC} macro.
2116 That command does the following:
2119 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2120 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2121 with the error message "need to have GiNaC installed"
2124 And the @file{Makefile.am}, which will be used to build the Makefile.
2127 ## Process this file with automake to produce Makefile.in
2128 bin_PROGRAMS = simple
2129 simple_SOURCES = simple.cpp
2132 This @file{Makefile.am}, says that we are building a single executable,
2133 from a single sourcefile @file{simple.cpp}. Since every program
2134 we are building uses GiNaC we simply added the GiNaC options
2135 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
2136 want to specify them on a per-program basis: for instance by
2140 simple_LDADD = $(GINACLIB_LIBS)
2141 INCLUDES = $(GINACLIB_CPPFLAGS)
2144 to the @file{Makefile.am}.
2146 To try this example out, create a new directory and add the three
2149 Now execute the following commands:
2152 $ automake --add-missing
2157 You now have a package that can be built in the normal fashion
2166 @node Bibliography, Concept Index, Example package, Top
2167 @c node-name, next, previous, up
2168 @appendix Bibliography
2173 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
2176 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
2179 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
2182 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
2185 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
2186 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
2189 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
2190 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
2191 Academic Press, London
2196 @node Concept Index, , Bibliography, Top
2197 @c node-name, next, previous, up
2198 @unnumbered Concept Index