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 its 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
259 @cindex @command{ginsh}
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):
367 @cindex Zeta function
371 > series(sin(x),x,0,4);
373 > series(1/tan(x),x,0,4);
374 x^(-1)-1/3*x+Order(x^2)
375 > series(gamma(x),x,0,3);
376 x^(-1)-EulerGamma+(1/12*Pi^2+1/2*EulerGamma^2)*x
377 +(-1/3*zeta(3)-1/12*Pi^2*EulerGamma-1/6*EulerGamma^3)*x^2+Order(x^3)
379 x^(-1.0)-0.5772156649015328606+(0.98905599532797255544)*x
380 -(0.90747907608088628905)*x^2+Order(x^(3.0))
381 > series(gamma(2*sin(x)-2),x,Pi/2,6);
382 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*EulerGamma^2-1/240)*(x-1/2*Pi)^2
383 -EulerGamma-1/12+Order((x-1/2*Pi)^3)
386 Here we have made use of the @command{ginsh}-command @code{"} to pop the
387 previously evaluated element from @command{ginsh}'s internal stack.
389 If you ever wanted to convert units in C or C++ and found this is
390 cumbersome, here is the solution. Symbolic types can always be used as
391 tags for different types of objects. Converting from wrong units to the
392 metric system is therefore easy:
400 140613.91592783185568*kg*m^(-2)
404 @node Installation, Prerequisites, What it can do for you, Top
405 @c node-name, next, previous, up
406 @chapter Installation
409 GiNaC's installation follows the spirit of most GNU software. It is
410 easily installed on your system by three steps: configuration, build,
414 * Prerequisites:: Packages upon which GiNaC depends.
415 * Configuration:: How to configure GiNaC.
416 * Building GiNaC:: How to compile GiNaC.
417 * Installing GiNaC:: How to install GiNaC on your system.
421 @node Prerequisites, Configuration, Installation, Installation
422 @c node-name, next, previous, up
423 @section Prerequisites
425 In order to install GiNaC on your system, some prerequisites need
426 to be met. First of all, you need to have a C++-compiler adhering to
427 the ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
428 development so if you have a different compiler you are on your own.
429 For the configuration to succeed you need a Posix compliant shell
430 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
431 by the built process as well, since some of the source files are automatically
432 generated by Perl scripts. Last but not least, Bruno Haible's library
433 @acronym{CLN} is extensively used and needs to be installed on your system.
434 Please get it from @uref{ftp://ftp.santafe.edu/pub/gnu/} or from
435 @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP site}
436 (it is covered by GPL) and install it prior to trying to install GiNaC.
437 The configure script checks if it can find it and if it cannot
438 it will refuse to continue.
441 @node Configuration, Building GiNaC, Prerequisites, Installation
442 @c node-name, next, previous, up
443 @section Configuration
444 @cindex configuration
447 To configure GiNaC means to prepare the source distribution for
448 building. It is done via a shell script called @command{configure} that
449 is shipped with the sources and was originally generated by GNU
450 Autoconf. Since a configure script generated by GNU Autoconf never
451 prompts, all customization must be done either via command line
452 parameters or environment variables. It accepts a list of parameters,
453 the complete set of which can be listed by calling it with the
454 @option{--help} option. The most important ones will be shortly
455 described in what follows:
460 @option{--disable-shared}: When given, this option switches off the
461 build of a shared library, i.e. a @file{.so} file. This may be convenient
462 when developing because it considerably speeds up compilation.
465 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
466 and headers are installed. It defaults to @file{/usr/local} which means
467 that the library is installed in the directory @file{/usr/local/lib},
468 the header files in @file{/usr/local/include/ginac} and the documentation
469 (like this one) into @file{/usr/local/share/doc/GiNaC}.
472 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
473 the library installed in some other directory than
474 @file{@var{PREFIX}/lib/}.
477 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
478 to have the header files installed in some other directory than
479 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
480 @option{--includedir=/usr/include} you will end up with the header files
481 sitting in the directory @file{/usr/include/ginac/}. Note that the
482 subdirectory @file{ginac} is enforced by this process in order to
483 keep the header files separated from others. This avoids some
484 clashes and allows for an easier deinstallation of GiNaC. This ought
485 to be considered A Good Thing (tm).
488 @option{--datadir=@var{DATADIR}}: This option may be given in case you
489 want to have the documentation installed in some other directory than
490 @file{@var{PREFIX}/share/doc/GiNaC/}.
494 In addition, you may specify some environment variables.
495 @env{CXX} holds the path and the name of the C++ compiler
496 in case you want to override the default in your path. (The
497 @command{configure} script searches your path for @command{c++},
498 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
499 and @command{cc++} in that order.) It may be very useful to
500 define some compiler flags with the @env{CXXFLAGS} environment
501 variable, like optimization, debugging information and warning
502 levels. If omitted, it defaults to @option{-g -O2}.
504 The whole process is illustrated in the following two
505 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
506 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
509 Here is a simple configuration for a site-wide GiNaC library assuming
510 everything is in default paths:
513 $ export CXXFLAGS="-Wall -O2"
517 And here is a configuration for a private static GiNaC library with
518 several components sitting in custom places (site-wide @acronym{GCC} and
519 private @acronym{CLN}). The compiler is pursuaded to be picky and full
520 assertions and debugging information are switched on:
523 $ export CXX=/usr/local/gnu/bin/c++
524 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
525 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
526 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
527 $ ./configure --disable-shared --prefix=$(HOME)
531 @node Building GiNaC, Installing GiNaC, Configuration, Installation
532 @c node-name, next, previous, up
533 @section Building GiNaC
534 @cindex building GiNaC
536 After proper configuration you should just build the whole
541 at the command prompt and go for a cup of coffee. The exact time it
542 takes to compile GiNaC depends not only on the speed of your machines
543 but also on other parameters, for instance what value for @env{CXXFLAGS}
544 you entered. Optimization may be very time-consuming.
546 Just to make sure GiNaC works properly you may run a simple test
553 This will compile some sample programs, run them and compare the output
554 to reference output. Each of the checks should return a message @samp{passed}
555 together with the CPU time used for that particular test. If it does
556 not, something went wrong. This is mostly intended to be a QA-check
557 if something was broken during the development, not a sanity check
558 of your system. Another intent is to allow people to fiddle around
559 with optimization. If @acronym{CLN} was installed all right
560 this step is unlikely to return any errors.
562 Generally, the top-level Makefile runs recursively to the
563 subdirectories. It is therfore safe to go into any subdirectory
564 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
565 @var{target} there in case something went wrong.
568 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
569 @c node-name, next, previous, up
570 @section Installing GiNaC
573 To install GiNaC on your system, simply type
579 As described in the section about configuration the files will be
580 installed in the following directories (the directories will be created
581 if they don't already exist):
586 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
587 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
588 So will @file{libginac.so} unless the configure script was
589 given the option @option{--disable-shared}. The proper symlinks
590 will be established as well.
593 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
594 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
597 All documentation (HTML and Postscript) will be stuffed into
598 @file{@var{PREFIX}/share/doc/GiNaC/} (or
599 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
603 For the sake of completeness we will list some other useful make
604 targets: @command{make clean} deletes all files generated by
605 @command{make}, i.e. all the object files. In addition @command{make
606 distclean} removes all files generated by the configuration and
607 @command{make maintainer-clean} goes one step further and deletes files
608 that may require special tools to rebuild (like the @command{libtool}
609 for instance). Finally @command{make uninstall} removes the installed
610 library, header files and documentation@footnote{Uninstallation does not
611 work after you have called @command{make distclean} since the
612 @file{Makefile} is itself generated by the configuration from
613 @file{Makefile.in} and hence deleted by @command{make distclean}. There
614 are two obvious ways out of this dilemma. First, you can run the
615 configuration again with the same @var{PREFIX} thus creating a
616 @file{Makefile} with a working @samp{uninstall} target. Second, you can
617 do it by hand since you now know where all the files went during
621 @node Basic Concepts, Expressions, Installing GiNaC, Top
622 @c node-name, next, previous, up
623 @chapter Basic Concepts
625 This chapter will describe the different fundamental objects that can be
626 handled by GiNaC. But before doing so, it is worthwhile introducing you
627 to the more commonly used class of expressions, representing a flexible
628 meta-class for storing all mathematical objects.
631 * Expressions:: The fundamental GiNaC class.
632 * The Class Hierarchy:: Overview of GiNaC's classes.
633 * Symbols:: Symbolic objects.
634 * Numbers:: Numerical objects.
635 * Constants:: Pre-defined constants.
636 * Fundamental containers:: The power, add and mul classes.
637 * Built-in functions:: Mathematical functions.
638 * Relations:: Equality, Inequality and all that.
642 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
643 @c node-name, next, previous, up
645 @cindex expression (class @code{ex})
648 The most common class of objects a user deals with is the expression
649 @code{ex}, representing a mathematical object like a variable, number,
650 function, sum, product, etc... Expressions may be put together to form
651 new expressions, passed as arguments to functions, and so on. Here is a
652 little collection of valid expressions:
655 ex MyEx1 = 5; // simple number
656 ex MyEx2 = x + 2*y; // polynomial in x and y
657 ex MyEx3 = (x + 1)/(x - 1); // rational expression
658 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
659 ex MyEx5 = MyEx4 + 1; // similar to above
662 Expressions are handles to other more fundamental objects, that many
663 times contain other expressions thus creating a tree of expressions
664 (@xref{Internal Structures}, for particular examples). Most methods on
665 @code{ex} therefore run top-down through such an expression tree. For
666 example, the method @code{has()} scans recursively for occurrences of
667 something inside an expression. Thus, if you have declared @code{MyEx4}
668 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
669 the argument of @code{sin} and hence return @code{true}.
671 The next sections will outline the general picture of GiNaC's class
672 hierarchy and describe the classes of objects that are handled by
676 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
677 @c node-name, next, previous, up
678 @section The Class Hierarchy
680 GiNaC's class hierarchy consists of several classes representing
681 mathematical objects, all of which (except for @code{ex} and some
682 helpers) are internally derived from one abstract base class called
683 @code{basic}. You do not have to deal with objects of class
684 @code{basic}, instead you'll be dealing with symbols, numbers,
685 containers of expressions and so on. You'll soon learn in this chapter
686 how many of the functions on symbols are really classes. This is
687 because simple symbolic arithmetic is not supported by languages like
688 C++ so in a certain way GiNaC has to implement its own arithmetic.
692 To get an idea about what kinds of symbolic composits may be built we
693 have a look at the most important classes in the class hierarchy. The
694 oval classes are atomic ones and the squared classes are containers.
695 The dashed line symbolizes a `points to' or `handles' relationship while
696 the solid lines stand for `inherits from' relationship in the class
699 @image{classhierarchy}
701 Some of the classes shown here (the ones sitting in white boxes) are
702 abstract base classes that are of no interest at all for the user. They
703 are used internally in order to avoid code duplication if two or more
704 classes derived from them share certain features. An example would be
705 @code{expairseq}, which is a container for a sequence of pairs each
706 consisting of one expression and a number (@code{numeric}). What
707 @emph{is} visible to the user are the derived classes @code{add} and
708 @code{mul}, representing sums of terms and products, respectively.
709 @xref{Internal Structures}, where these two classes are described in
713 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
714 @c node-name, next, previous, up
716 @cindex Symbols (class @code{symbol})
717 @cindex hierarchy of classes
720 Symbols are for symbolic manipulation what atoms are for chemistry. You
721 can declare objects of class @code{symbol} as any other object simply by
722 saying @code{symbol x,y;}. There is, however, a catch in here having to
723 do with the fact that C++ is a compiled language. The information about
724 the symbol's name is thrown away by the compiler but at a later stage
725 you may want to print expressions holding your symbols. In order to
726 avoid confusion GiNaC's symbols are able to know their own name. This
727 is accomplished by declaring its name for output at construction time in
728 the fashion @code{symbol x("x");}. If you declare a symbol using the
729 default constructor (i.e. without string argument) the system will deal
730 out a unique name. That name may not be suitable for printing but for
731 internal routines when no output is desired it is often enough. We'll
732 come across examples of such symbols later in this tutorial.
734 This implies that the strings passed to symbols at construction time may
735 not be used for comparing two of them. It is perfectly legitimate to
736 write @code{symbol x("x"),y("x");} but it is likely to lead into
737 trouble. Here, @code{x} and @code{y} are different symbols and
738 statements like @code{x-y} will not be simplified to zero although the
739 output @code{x-x} looks funny. Such output may also occur when there
740 are two different symbols in two scopes, for instance when you call a
741 function that declares a symbol with a name already existent in a symbol
742 in the calling function. Again, comparing them (using @code{operator==}
743 for instance) will always reveal their difference. Watch out, please.
745 @cindex @code{subs()}
746 Although symbols can be assigned expressions for internal reasons, you
747 should not do it (and we are not going to tell you how it is done). If
748 you want to replace a symbol with something else in an expression, you
749 can use the expression's @code{.subs()} method.
752 @node Numbers, Constants, Symbols, Basic Concepts
753 @c node-name, next, previous, up
755 @cindex numbers (class @code{numeric})
761 For storing numerical things, GiNaC uses Bruno Haible's library
762 @acronym{CLN}. The classes therein serve as foundation classes for
763 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
764 alternatively for Common Lisp Numbers. In order to find out more about
765 @acronym{CLN}'s internals the reader is refered to the documentation of
766 that library. @inforef{Introduction, , cln}, for more
767 information. Suffice to say that it is by itself build on top of another
768 library, the GNU Multiple Precision library @acronym{GMP}, which is an
769 extremely fast library for arbitrary long integers and rationals as well
770 as arbitrary precision floating point numbers. It is very commonly used
771 by several popular cryptographic applications. @acronym{CLN} extends
772 @acronym{GMP} by several useful things: First, it introduces the complex
773 number field over either reals (i.e. floating point numbers with
774 arbitrary precision) or rationals. Second, it automatically converts
775 rationals to integers if the denominator is unity and complex numbers to
776 real numbers if the imaginary part vanishes and also correctly treats
777 algebraic functions. Third it provides good implementations of
778 state-of-the-art algorithms for all trigonometric and hyperbolic
779 functions as well as for calculation of some useful constants.
781 The user can construct an object of class @code{numeric} in several
782 ways. The following example shows the four most important constructors.
783 It uses construction from C-integer, construction of fractions from two
784 integers, construction from C-float and construction from a string:
787 #include <ginac/ginac.h>
788 using namespace GiNaC;
792 numeric two(2); // exact integer 2
793 numeric r(2,3); // exact fraction 2/3
794 numeric e(2.71828); // floating point number
795 numeric p("3.1415926535897932385"); // floating point number
797 cout << two*p << endl; // floating point 6.283...
802 Note that all those constructors are @emph{explicit} which means you are
803 not allowed to write @code{numeric two=2;}. This is because the basic
804 objects to be handled by GiNaC are the expressions @code{ex} and we want
805 to keep things simple and wish objects like @code{pow(x,2)} to be
806 handled the same way as @code{pow(x,a)}, which means that we need to
807 allow a general @code{ex} as base and exponent. Therefore there is an
808 implicit constructor from C-integers directly to expressions handling
809 numerics at work in most of our examples. This design really becomes
810 convenient when one declares own functions having more than one
811 parameter but it forbids using implicit constructors because that would
812 lead to compile-time ambiguities.
814 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
815 This would, however, call C's built-in operator @code{/} for integers
816 first and result in a numeric holding a plain integer 1. @strong{Never
817 use the operator @code{/} on integers} unless you know exactly what you
818 are doing! Use the constructor from two integers instead, as shown in
819 the example above. Writing @code{numeric(1)/2} may look funny but works
822 @cindex @code{Digits}
824 We have seen now the distinction between exact numbers and floating
825 point numbers. Clearly, the user should never have to worry about
826 dynamically created exact numbers, since their `exactness' always
827 determines how they ought to be handled, i.e. how `long' they are. The
828 situation is different for floating point numbers. Their accuracy is
829 controlled by one @emph{global} variable, called @code{Digits}. (For
830 those readers who know about Maple: it behaves very much like Maple's
831 @code{Digits}). All objects of class numeric that are constructed from
832 then on will be stored with a precision matching that number of decimal
836 #include <ginac/ginac.h>
837 using namespace GiNaC;
841 numeric three(3.0), one(1.0);
842 numeric x = one/three;
844 cout << "in " << Digits << " digits:" << endl;
846 cout << Pi.evalf() << endl;
858 The above example prints the following output to screen:
865 0.333333333333333333333333333333333333333333333333333333333333333333
866 3.14159265358979323846264338327950288419716939937510582097494459231
869 It should be clear that objects of class @code{numeric} should be used
870 for constructing numbers or for doing arithmetic with them. The objects
871 one deals with most of the time are the polymorphic expressions @code{ex}.
873 @subsection Tests on numbers
875 Once you have declared some numbers, assigned them to expressions and
876 done some arithmetic with them it is frequently desired to retrieve some
877 kind of information from them like asking whether that number is
878 integer, rational, real or complex. For those cases GiNaC provides
879 several useful methods. (Internally, they fall back to invocations of
880 certain CLN functions.)
882 As an example, let's construct some rational number, multiply it with
883 some multiple of its denominator and test what comes out:
886 #include <ginac/ginac.h>
887 using namespace GiNaC;
889 // some very important constants:
890 const numeric twentyone(21);
891 const numeric ten(10);
892 const numeric five(5);
896 numeric answer = twentyone;
899 cout << answer.is_integer() << endl; // false, it's 21/5
901 cout << answer.is_integer() << endl; // true, it's 42 now!
906 Note that the variable @code{answer} is constructed here as an integer
907 by @code{numeric}'s copy constructor but in an intermediate step it
908 holds a rational number represented as integer numerator and integer
909 denominator. When multiplied by 10, the denominator becomes unity and
910 the result is automatically converted to a pure integer again.
911 Internally, the underlying @acronym{CLN} is responsible for this
912 behaviour and we refer the reader to @acronym{CLN}'s documentation.
913 Suffice to say that the same behaviour applies to complex numbers as
914 well as return values of certain functions. Complex numbers are
915 automatically converted to real numbers if the imaginary part becomes
916 zero. The full set of tests that can be applied is listed in the
920 @multitable @columnfractions .30 .70
921 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
922 @item @code{.is_zero()}
923 @tab @dots{}equal to zero
924 @item @code{.is_positive()}
925 @tab @dots{}not complex and greater than 0
926 @item @code{.is_integer()}
927 @tab @dots{}a (non-complex) integer
928 @item @code{.is_pos_integer()}
929 @tab @dots{}an integer and greater than 0
930 @item @code{.is_nonneg_integer()}
931 @tab @dots{}an integer and greater equal 0
932 @item @code{.is_even()}
933 @tab @dots{}an even integer
934 @item @code{.is_odd()}
935 @tab @dots{}an odd integer
936 @item @code{.is_prime()}
937 @tab @dots{}a prime integer (probabilistic primality test)
938 @item @code{.is_rational()}
939 @tab @dots{}an exact rational number (integers are rational, too)
940 @item @code{.is_real()}
941 @tab @dots{}a real integer, rational or float (i.e. is not complex)
942 @item @code{.is_cinteger()}
943 @tab @dots{}a (complex) integer, such as @math{2-3*I}
944 @item @code{.is_crational()}
945 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
950 @node Constants, Fundamental containers, Numbers, Basic Concepts
951 @c node-name, next, previous, up
953 @cindex constants (class @code{constant})
956 @cindex @code{Catalan}
957 @cindex @code{EulerGamma}
958 @cindex @code{evalf()}
959 Constants behave pretty much like symbols except that they return some
960 specific number when the method @code{.evalf()} is called.
962 The predefined known constants are:
965 @multitable @columnfractions .14 .30 .56
966 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
968 @tab Archimedes' constant
969 @tab 3.14159265358979323846264338327950288
971 @tab Catalan's constant
972 @tab 0.91596559417721901505460351493238411
973 @item @code{EulerGamma}
974 @tab Euler's (or Euler-Mascheroni) constant
975 @tab 0.57721566490153286060651209008240243
980 @node Fundamental containers, Built-in functions, Constants, Basic Concepts
981 @c node-name, next, previous, up
982 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
988 Simple polynomial expressions are written down in GiNaC pretty much like
989 in other CAS or like expressions involving numerical variables in C.
990 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
991 been overloaded to achieve this goal. When you run the following
992 program, the constructor for an object of type @code{mul} is
993 automatically called to hold the product of @code{a} and @code{b} and
994 then the constructor for an object of type @code{add} is called to hold
995 the sum of that @code{mul} object and the number one:
998 #include <ginac/ginac.h>
999 using namespace GiNaC;
1003 symbol a("a"), b("b");
1009 @cindex @code{pow()}
1010 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1011 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1012 construction is necessary since we cannot safely overload the constructor
1013 @code{^} in C++ to construct a @code{power} object. If we did, it would
1014 have several counterintuitive effects:
1018 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1020 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1021 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1022 interpret this as @code{x^(a^b)}.
1024 Also, expressions involving integer exponents are very frequently used,
1025 which makes it even more dangerous to overload @code{^} since it is then
1026 hard to distinguish between the semantics as exponentiation and the one
1027 for exclusive or. (It would be embarassing to return @code{1} where one
1028 has requested @code{2^3}.)
1031 @cindex @command{ginsh}
1032 All effects are contrary to mathematical notation and differ from the
1033 way most other CAS handle exponentiation, therefore overloading @code{^}
1034 is ruled out for GiNaC's C++ part. The situation is different in
1035 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1036 that the other frequently used exponentiation operator @code{**} does
1037 not exist at all in C++).
1039 To be somewhat more precise, objects of the three classes described
1040 here, are all containers for other expressions. An object of class
1041 @code{power} is best viewed as a container with two slots, one for the
1042 basis, one for the exponent. All valid GiNaC expressions can be
1043 inserted. However, basic transformations like simplifying
1044 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1045 when this is mathematically possible. If we replace the outer exponent
1046 three in the example by some symbols @code{a}, the simplification is not
1047 safe and will not be performed, since @code{a} might be @code{1/2} and
1050 Objects of type @code{add} and @code{mul} are containers with an
1051 arbitrary number of slots for expressions to be inserted. Again, simple
1052 and safe simplifications are carried out like transforming
1053 @code{3*x+4-x} to @code{2*x+4}.
1055 The general rule is that when you construct such objects, GiNaC
1056 automatically creates them in canonical form, which might differ from
1057 the form you typed in your program. This allows for rapid comparison of
1058 expressions, since after all @code{a-a} is simply zero. Note, that the
1059 canonical form is not necessarily lexicographical ordering or in any way
1060 easily guessable. It is only guaranteed that constructing the same
1061 expression twice, either implicitly or explicitly, results in the same
1065 @node Built-in functions, Relations, Fundamental containers, Basic Concepts
1066 @c node-name, next, previous, up
1067 @section Built-in functions
1068 @cindex functions (class @code{function})
1069 @cindex trigonometric function
1070 @cindex hyperbolic function
1072 There are quite a number of useful functions hard-wired into GiNaC. For
1073 instance, all trigonometric and hyperbolic functions are implemented.
1074 They are all objects of class @code{function}. They accept one or more
1075 expressions as arguments and return one expression. If the arguments
1076 are not numerical, the evaluation of the function may be halted, as it
1077 does in the next example:
1079 @cindex Gamma function
1080 @cindex @code{subs()}
1082 #include <ginac/ginac.h>
1083 using namespace GiNaC;
1087 symbol x("x"), y("y");
1090 cout << "gamma(" << foo << ") -> " << gamma(foo) << endl;
1091 ex bar = foo.subs(y==1);
1092 cout << "gamma(" << bar << ") -> " << gamma(bar) << endl;
1093 ex foobar = bar.subs(x==7);
1094 cout << "gamma(" << foobar << ") -> " << gamma(foobar) << endl;
1099 This program shows how the function returns itself twice and finally an
1100 expression that may be really useful:
1103 gamma(x+(1/2)*y) -> gamma(x+(1/2)*y)
1104 gamma(x+1/2) -> gamma(x+1/2)
1105 gamma(15/2) -> (135135/128)*Pi^(1/2)
1109 For functions that have a branch cut in the complex plane GiNaC follows
1110 the conventions for C++ as defined in the ANSI standard. In particular:
1111 the natural logarithm (@code{log}) and the square root (@code{sqrt})
1112 both have their branch cuts running along the negative real axis where
1113 the points on the axis itself belong to the upper part.
1115 Besides evaluation most of these functions allow differentiation, series
1116 expansion and so on. Read the next chapter in order to learn more about
1120 @node Relations, Important Algorithms, Built-in functions, Basic Concepts
1121 @c node-name, next, previous, up
1123 @cindex relations (class @code{relational})
1125 Sometimes, a relation holding between two expressions must be stored
1126 somehow. The class @code{relational} is a convenient container for such
1127 purposes. A relation is by definition a container for two @code{ex} and
1128 a relation between them that signals equality, inequality and so on.
1129 They are created by simply using the C++ operators @code{==}, @code{!=},
1130 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1132 @xref{Built-in functions}, for examples where various applications of
1133 the @code{.subs()} method show how objects of class relational are used
1134 as arguments. There they provide an intuitive syntax for substitutions.
1137 @node Important Algorithms, Polynomial Expansion, Relations, Top
1138 @c node-name, next, previous, up
1139 @chapter Important Algorithms
1142 In this chapter the most important algorithms provided by GiNaC will be
1143 described. Some of them are implemented as functions on expressions,
1144 others are implemented as methods provided by expression objects. If
1145 they are methods, there exists a wrapper function around it, so you can
1146 alternatively call it in a functional way as shown in the simple
1150 #include <ginac/ginac.h>
1151 using namespace GiNaC;
1155 ex x = numeric(1.0);
1157 cout << "As method: " << sin(x).evalf() << endl;
1158 cout << "As function: " << evalf(sin(x)) << endl;
1163 @cindex @code{subs()}
1164 The general rule is that wherever methods accept one or more parameters
1165 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1166 wrapper accepts is the same but preceded by the object to act on
1167 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1168 most natural one in an OO model but it may lead to confusion for MapleV
1169 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1170 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1171 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1172 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1173 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1174 here. Also, users of MuPAD will in most cases feel more comfortable
1175 with GiNaC's convention. All function wrappers are always implemented
1176 as simple inline functions which just call the corresponding method and
1177 are only provided for users uncomfortable with OO who are dead set to
1178 avoid method invocations. Generally, nested function wrappers are much
1179 harder to read than a sequence of methods and should therefore be
1180 avoided if possible. On the other hand, not everything in GiNaC is a
1181 method on class @code{ex} and sometimes calling a function cannot be
1185 * Polynomial Expansion::
1186 * Collecting expressions::
1187 * Polynomial Arithmetic::
1188 * Symbolic Differentiation::
1189 * Series Expansion::
1193 @node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
1194 @c node-name, next, previous, up
1195 @section Polynomial Expansion
1196 @cindex @code{expand()}
1198 A polynomial in one or more variables has many equivalent
1199 representations. Some useful ones serve a specific purpose. Consider
1200 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1201 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1202 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1203 representations are the recursive ones where one collects for exponents
1204 in one of the three variable. Since the factors are themselves
1205 polynomials in the remaining two variables the procedure can be
1206 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1207 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1210 To bring an expression into expanded form, its method @code{.expand()}
1211 may be called. In our example above, this corresponds to @math{4*x*y +
1212 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1213 GiNaC is not easily guessable you should be prepared to see different
1214 orderings of terms in such sums!
1217 @node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
1218 @c node-name, next, previous, up
1219 @section Collecting expressions
1220 @cindex @code{collect()}
1221 @cindex @code{coeff()}
1223 Another useful representation of multivariate polynomials is as a
1224 univariate polynomial in one of the variables with the coefficients
1225 being polynomials in the remaining variables. The method
1226 @code{collect()} accomplishes this task. Here is its declaration:
1229 ex ex::collect(symbol const & s);
1232 Note that the original polynomial needs to be in expanded form in order
1233 to be able to find the coefficients properly. The range of occuring
1234 coefficients can be checked using the two methods
1236 @cindex @code{degree()}
1237 @cindex @code{ldegree()}
1239 int ex::degree(symbol const & s);
1240 int ex::ldegree(symbol const & s);
1243 where @code{degree()} returns the highest coefficient and
1244 @code{ldegree()} the lowest one. (These two methods work also reliably
1245 on non-expanded input polynomials). An application is illustrated in
1246 the next example, where a multivariate polynomial is analyzed:
1249 #include <ginac/ginac.h>
1250 using namespace GiNaC;
1254 symbol x("x"), y("y");
1255 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1256 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1257 ex Poly = PolyInp.expand();
1259 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1260 cout << "The x^" << i << "-coefficient is "
1261 << Poly.coeff(x,i) << endl;
1263 cout << "As polynomial in y: "
1264 << Poly.collect(y) << endl;
1269 When run, it returns an output in the following fashion:
1272 The x^0-coefficient is y^2+11*y
1273 The x^1-coefficient is 5*y^2-2*y
1274 The x^2-coefficient is -1
1275 The x^3-coefficient is 4*y
1276 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1279 As always, the exact output may vary between different versions of GiNaC
1280 or even from run to run since the internal canonical ordering is not
1281 within the user's sphere of influence.
1284 @node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
1285 @c node-name, next, previous, up
1286 @section Polynomial Arithmetic
1288 @subsection GCD and LCM
1292 The functions for polynomial greatest common divisor and least common
1293 multiple have the synopsis:
1296 ex gcd(const ex & a, const ex & b);
1297 ex lcm(const ex & a, const ex & b);
1300 The functions @code{gcd()} and @code{lcm()} accept two expressions
1301 @code{a} and @code{b} as arguments and return a new expression, their
1302 greatest common divisor or least common multiple, respectively. If the
1303 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1304 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1307 #include <ginac/ginac.h>
1308 using namespace GiNaC;
1312 symbol x("x"), y("y"), z("z");
1313 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1314 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1316 ex P_gcd = gcd(P_a, P_b);
1318 ex P_lcm = lcm(P_a, P_b);
1319 // 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
1324 @subsection The @code{normal} method
1325 @cindex @code{normal()}
1326 @cindex temporary replacement
1328 While in common symbolic code @code{gcd()} and @code{lcm()} are not too
1329 heavily used, simplification is called for frequently. Therefore
1330 @code{.normal()}, which provides some basic form of simplification, has
1331 become a method of class @code{ex}, just like @code{.expand()}. It
1332 converts a rational function into an equivalent rational function where
1333 numerator and denominator are coprime. This means, it finds the GCD of
1334 numerator and denominator and cancels it. If it encounters some object
1335 which does not belong to the domain of rationals (a function for
1336 instance), that object is replaced by a temporary symbol. This means
1337 that both expressions @code{t1} and @code{t2} are indeed simplified in
1338 this little program:
1341 #include <ginac/ginac.h>
1342 using namespace GiNaC;
1347 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1348 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1349 cout << "t1 is " << t1.normal() << endl;
1350 cout << "t2 is " << t2.normal() << endl;
1355 Of course this works for multivariate polynomials too, so the ratio of
1356 the sample-polynomials from the section about GCD and LCM above would be
1357 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1360 @node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
1361 @c node-name, next, previous, up
1362 @section Symbolic Differentiation
1363 @cindex differentiation
1364 @cindex @code{diff()}
1366 @cindex product rule
1368 GiNaC's objects know how to differentiate themselves. Thus, a
1369 polynomial (class @code{add}) knows that its derivative is the sum of
1370 the derivatives of all the monomials:
1373 #include <ginac/ginac.h>
1374 using namespace GiNaC;
1378 symbol x("x"), y("y"), z("z");
1379 ex P = pow(x, 5) + pow(x, 2) + y;
1381 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1382 cout << P.diff(y) << endl; // 1
1383 cout << P.diff(z) << endl; // 0
1388 If a second integer parameter @var{n} is given, the @code{diff} method
1389 returns the @var{n}th derivative.
1391 If @emph{every} object and every function is told what its derivative
1392 is, all derivatives of composed objects can be calculated using the
1393 chain rule and the product rule. Consider, for instance the expression
1394 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1395 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1396 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1397 out that the composition is the generating function for Euler Numbers,
1398 i.e. the so called @var{n}th Euler number is the coefficient of
1399 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1400 identity to code a function that generates Euler numbers in just three
1403 @cindex Euler numbers
1405 #include <ginac/ginac.h>
1406 using namespace GiNaC;
1408 ex EulerNumber(unsigned n)
1411 ex generator = pow(cosh(x),-1);
1412 return generator.diff(x,n).subs(x==0);
1417 for (unsigned i=0; i<11; i+=2)
1418 cout << EulerNumber(i) << endl;
1423 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1424 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1425 @code{i} by two since all odd Euler numbers vanish anyways.
1428 @node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
1429 @c node-name, next, previous, up
1430 @section Series Expansion
1431 @cindex series expansion
1432 @cindex Taylor expansion
1433 @cindex Laurent expansion
1435 Expressions know how to expand themselves as a Taylor series or (more
1436 generally) a Laurent series. Similar to most conventional Computer
1437 Algebra Systems, no distinction is made between those two. There is a
1438 class of its own for storing such series as well as a class for storing
1439 the order of the series. A sample program could read:
1442 #include <ginac/ginac.h>
1443 using namespace GiNaC;
1449 ex MyExpr1 = sin(x);
1450 ex MyExpr2 = 1/(x - pow(x, 2) - pow(x, 3));
1451 ex MyTailor, MySeries;
1453 MyTailor = MyExpr1.series(x, point, 5);
1454 cout << MyExpr1 << " == " << MyTailor
1455 << " for small " << x << endl;
1456 MySeries = MyExpr2.series(x, point, 7);
1457 cout << MyExpr2 << " == " << MySeries
1458 << " for small " << x << endl;
1463 @cindex M@'echain's formula
1464 As an instructive application, let us calculate the numerical value of
1465 Archimedes' constant
1469 (for which there already exists the built-in constant @code{Pi})
1470 using M@'echain's amazing formula
1472 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1475 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1477 We may expand the arcus tangent around @code{0} and insert the fractions
1478 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1479 carries an order term with it and the question arises what the system is
1480 supposed to do when the fractions are plugged into that order term. The
1481 solution is to use the function @code{series_to_poly()} to simply strip
1485 #include <ginac/ginac.h>
1486 using namespace GiNaC;
1488 ex mechain_pi(int degr)
1491 ex pi_expansion = series_to_poly(atan(x).series(x,0,degr));
1492 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1493 -4*pi_expansion.subs(x==numeric(1,239));
1500 for (int i=2; i<12; i+=2) @{
1501 pi_frac = mechain_pi(i);
1502 cout << i << ":\t" << pi_frac << endl
1503 << "\t" << pi_frac.evalf() << endl;
1509 When you run this program, it will type out:
1513 3.1832635983263598326
1514 4: 5359397032/1706489875
1515 3.1405970293260603143
1516 6: 38279241713339684/12184551018734375
1517 3.141621029325034425
1518 8: 76528487109180192540976/24359780855939418203125
1519 3.141591772182177295
1520 10: 327853873402258685803048818236/104359128170408663038552734375
1521 3.1415926824043995174
1525 @node Extending GiNaC, What does not belong into GiNaC, Series Expansion, Top
1526 @c node-name, next, previous, up
1527 @chapter Extending GiNaC
1529 By reading so far you should have gotten a fairly good understanding of
1530 GiNaC's design-patterns. From here on you should start reading the
1531 sources. All we can do now is issue some recommendations how to tackle
1532 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
1533 develop some useful extension please don't hesitate to contact the GiNaC
1534 authors---they will happily incorporate them into future versions.
1537 * What does not belong into GiNaC:: What to avoid.
1538 * Symbolic functions:: Implementing symbolic functions.
1542 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
1543 @c node-name, next, previous, up
1544 @section What doesn't belong into GiNaC
1546 @cindex @command{ginsh}
1547 First of all, GiNaC's name must be read literally. It is designed to be
1548 a library for use within C++. The tiny @command{ginsh} accompanying
1549 GiNaC makes this even more clear: it doesn't even attempt to provide a
1550 language. There are no loops or conditional expressions in
1551 @command{ginsh}, it is merely a window into the library for the
1552 programmer to test stuff (or to show off). Still, the design of a
1553 complete CAS with a language of its own, graphical capabilites and all
1554 this on top of GiNaC is possible and is without doubt a nice project for
1557 There are many built-in functions in GiNaC that do not know how to
1558 evaluate themselves numerically to a precision declared at runtime
1559 (using @code{Digits}). Some may be evaluated at certain points, but not
1560 generally. This ought to be fixed. However, doing numerical
1561 computations with GiNaC's quite abstract classes is doomed to be
1562 inefficient. For this purpose, the underlying foundation classes
1563 provided by @acronym{CLN} are much better suited.
1566 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
1567 @c node-name, next, previous, up
1568 @section Symbolic functions
1570 The easiest and most instructive way to start with is probably to
1571 implement your own function. Objects of class @code{function} are
1572 inserted into the system via a kind of `registry'. They get a serial
1573 number that is used internally to identify them but you usually need not
1574 worry about this. What you have to care for are functions that are
1575 called when the user invokes certain methods. These are usual
1576 C++-functions accepting a number of @code{ex} as arguments and returning
1577 one @code{ex}. As an example, if we have a look at a simplified
1578 implementation of the cosine trigonometric function, we first need a
1579 function that is called when one wishes to @code{eval} it. It could
1580 look something like this:
1583 static ex cos_eval_method(ex const & x)
1585 // if (!x%(2*Pi)) return 1
1586 // if (!x%Pi) return -1
1587 // if (!x%Pi/2) return 0
1588 // care for other cases...
1589 return cos(x).hold();
1593 @cindex @code{hold()}
1595 The last line returns @code{cos(x)} if we don't know what else to do and
1596 stops a potential recursive evaluation by saying @code{.hold()}. We
1597 should also implement a method for numerical evaluation and since we are
1598 lazy we sweep the problem under the rug by calling someone else's
1599 function that does so, in this case the one in class @code{numeric}:
1602 static ex cos_evalf_method(ex const & x)
1604 return sin(ex_to_numeric(x));
1608 Differentiation will surely turn up and so we need to tell
1609 @code{sin} how to differentiate itself:
1612 static ex cos_diff_method(ex const & x, unsigned diff_param)
1618 @cindex product rule
1619 The second parameter is obligatory but uninteresting at this point. It
1620 specifies which parameter to differentiate in a partial derivative in
1621 case the function has more than one parameter and its main application
1622 is for correct handling of the chain rule. For Taylor expansion, it is
1623 enough to know how to differentiate. But if the function you want to
1624 implement does have a pole somewhere in the complex plane, you need to
1625 write another method for Laurent expansion around that point.
1627 Now that all the ingrediences for @code{cos} have been set up, we need
1628 to tell the system about it. This is done by a macro and we are not
1629 going to descibe how it expands, please consult your preprocessor if you
1633 REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);
1636 The first argument is the function's name, the second, third and fourth
1637 bind the corresponding methods to this objects and the fifth is a slot
1638 for inserting a method for series expansion. (If set to @code{NULL} it
1639 defaults to simple Taylor expansion, which is correct if there are no
1640 poles involved. The way GiNaC handles poles in case there are any is
1641 best understood by studying one of the examples, like the Gamma function
1642 for instance. In essence the function first checks if there is a pole
1643 at the evaluation point and falls back to Taylor expansion if there
1644 isn't. Then, the pole is regularized by some suitable transformation.)
1645 Also, the new function needs to be declared somewhere. This may also be
1646 done by a convenient preprocessor macro:
1649 DECLARE_FUNCTION_1P(cos)
1652 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
1653 implementation of @code{cos} is very incomplete and lacks several safety
1654 mechanisms. Please, have a look at the real implementation in GiNaC.
1655 (By the way: in case you are worrying about all the macros above we can
1656 assure you that functions are GiNaC's most macro-intense classes. We
1657 have done our best to avoid them where we can.)
1659 That's it. May the source be with you!
1662 @node A Comparison With Other CAS, Advantages, Symbolic functions, Top
1663 @c node-name, next, previous, up
1664 @chapter A Comparison With Other CAS
1667 This chapter will give you some information on how GiNaC compares to
1668 other, traditional Computer Algebra Systems, like @emph{Maple},
1669 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
1670 disadvantages over these systems.
1673 * Advantages:: Stengths of the GiNaC approach.
1674 * Disadvantages:: Weaknesses of the GiNaC approach.
1675 * Why C++?:: Attractiveness of C++.
1678 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
1679 @c node-name, next, previous, up
1682 GiNaC has several advantages over traditional Computer
1683 Algebra Systems, like
1688 familiar language: all common CAS implement their own proprietary
1689 grammar which you have to learn first (and maybe learn again when your
1690 vendor decides to `enhance' it). With GiNaC you can write your program
1691 in common C++, which is standardized.
1695 structured data types: you can build up structured data types using
1696 @code{struct}s or @code{class}es together with STL features instead of
1697 using unnamed lists of lists of lists.
1700 strongly typed: in CAS, you usually have only one kind of variables
1701 which can hold contents of an arbitrary type. This 4GL like feature is
1702 nice for novice programmers, but dangerous.
1705 development tools: powerful development tools exist for C++, like fancy
1706 editors (e.g. with automatic indentation and syntax highlighting),
1707 debuggers, visualization tools, documentation tools...
1710 modularization: C++ programs can easily be split into modules by
1711 separating interface and implementation.
1714 price: GiNaC is distributed under the GNU Public License which means
1715 that it is free and available with source code. And there are excellent
1716 C++-compilers for free, too.
1719 extendable: you can add your own classes to GiNaC, thus extending it on
1720 a very low level. Compare this to a traditional CAS that you can
1721 usually only extend on a high level by writing in the language defined
1722 by the parser. In particular, it turns out to be almost impossible to
1723 fix bugs in a traditional system.
1726 seemless integration: it is somewhere between difficult and impossible
1727 to call CAS functions from within a program written in C++ or any other
1728 programming language and vice versa. With GiNaC, your symbolic routines
1729 are part of your program. You can easily call third party libraries,
1730 e.g. for numerical evaluation or graphical interaction. All other
1731 approaches are much more cumbersome: they range from simply ignoring the
1732 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
1733 system (i.e. @emph{Yacas}).
1736 efficiency: often large parts of a program do not need symbolic
1737 calculations at all. Why use large integers for loop variables or
1738 arbitrary precision arithmetics where double accuracy is sufficient?
1739 For pure symbolic applications, GiNaC is comparable in speed with other
1745 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
1746 @c node-name, next, previous, up
1747 @section Disadvantages
1749 Of course it also has some disadvantages:
1754 not interactive: GiNaC programs have to be written in an editor,
1755 compiled and executed. You cannot play with expressions interactively.
1756 However, such an extension is not inherently forbidden by design. In
1757 fact, two interactive interfaces are possible: First, a shell that
1758 exposes GiNaC's types to a command line can readily be written (the tiny
1759 @command{ginsh} that is part of the distribution being an example) and
1760 second, as a more consistent approach we are working on an integration
1761 with the @acronym{CINT} C++ interpreter.
1764 advanced features: GiNaC cannot compete with a program like
1765 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
1766 which grows since 1981 by the work of dozens of programmers, with
1767 respect to mathematical features. Integration, factorization,
1768 non-trivial simplifications, limits etc. are missing in GiNaC (and are
1769 not planned for the near future).
1772 portability: While the GiNaC library itself is designed to avoid any
1773 platform dependent features (it should compile on any ANSI compliant C++
1774 compiler), the currently used version of the CLN library (fast large
1775 integer and arbitrary precision arithmetics) can be compiled only on
1776 systems with a recently new C++ compiler from the GNU Compiler
1777 Collection (@acronym{GCC}).@footnote{This is because CLN uses
1778 PROVIDE/REQUIRE like macros to let the compiler gather all static
1779 initializations, which works for GNU C++ only.} GiNaC uses recent
1780 language features like explicit constructors, mutable members, RTTI,
1781 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
1782 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
1783 ANSI compliant, support all needed features.
1788 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
1789 @c node-name, next, previous, up
1792 Why did we choose to implement GiNaC in C++ instead of Java or any other
1793 language? C++ is not perfect: type checking is not strict (casting is
1794 possible), separation between interface and implementation is not
1795 complete, object oriented design is not enforced. The main reason is
1796 the often scolded feature of operator overloading in C++. While it may
1797 be true that operating on classes with a @code{+} operator is rarely
1798 meaningful, it is perfectly suited for algebraic expressions. Writing
1799 @math{3x+5y} as @code{3*x+5*y} instead of
1800 @code{x.times(3).plus(y.times(5))} looks much more natural.
1801 Furthermore, the main developers are more familiar with C++ than with
1802 any other programming language.
1805 @node Internal Structures, Expressions are reference counted, Why C++? , Top
1806 @c node-name, next, previous, up
1807 @appendix Internal Structures
1810 * Expressions are reference counted::
1811 * Internal representation of products and sums::
1814 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
1815 @c node-name, next, previous, up
1816 @appendixsection Expressions are reference counted
1818 @cindex reference counting
1819 @cindex copy-on-write
1820 @cindex garbage collection
1821 An expression is extremely light-weight since internally it works like a
1822 handle to the actual representation and really holds nothing more than a
1823 pointer to some other object. What this means in practice is that
1824 whenever you create two @code{ex} and set the second equal to the first
1825 no copying process is involved. Instead, the copying takes place as soon
1826 as you try to change the second. Consider the simple sequence of code:
1829 #include <ginac/ginac.h>
1830 using namespace GiNaC;
1834 symbol x("x"), y("y"), z("z");
1837 e1 = sin(x + 2*y) + 3*z + 41;
1838 e2 = e1; // e2 points to same object as e1
1839 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
1840 e2 += 1; // e2 is copied into a new object
1841 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
1846 The line @code{e2 = e1;} creates a second expression pointing to the
1847 object held already by @code{e1}. The time involved for this operation
1848 is therefore constant, no matter how large @code{e1} was. Actual
1849 copying, however, must take place in the line @code{e2 += 1;} because
1850 @code{e1} and @code{e2} are not handles for the same object any more.
1851 This concept is called @dfn{copy-on-write semantics}. It increases
1852 performance considerably whenever one object occurs multiple times and
1853 represents a simple garbage collection scheme because when an @code{ex}
1854 runs out of scope its destructor checks whether other expressions handle
1855 the object it points to too and deletes the object from memory if that
1856 turns out not to be the case. A slightly less trivial example of
1857 differentiation using the chain-rule should make clear how powerful this
1861 #include <ginac/ginac.h>
1862 using namespace GiNaC;
1866 symbol x("x"), y("y");
1870 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
1871 cout << e1 << endl // prints x+3*y
1872 << e2 << endl // prints (x+3*y)^3
1873 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
1878 Here, @code{e1} will actually be referenced three times while @code{e2}
1879 will be referenced two times. When the power of an expression is built,
1880 that expression needs not be copied. Likewise, since the derivative of
1881 a power of an expression can be easily expressed in terms of that
1882 expression, no copying of @code{e1} is involved when @code{e3} is
1883 constructed. So, when @code{e3} is constructed it will print as
1884 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
1885 holds a reference to @code{e2} and the factor in front is just
1888 As a user of GiNaC, you cannot see this mechanism of copy-on-write
1889 semantics. When you insert an expression into a second expression, the
1890 result behaves exactly as if the contents of the first expression were
1891 inserted. But it may be useful to remember that this is not what
1892 happens. Knowing this will enable you to write much more efficient
1893 code. If you still have an uncertain feeling with copy-on-write
1894 semantics, we recommend you have a look at the
1895 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
1896 Marshall Cline. Chapter 16 covers this issue and presents an
1897 implementation which is pretty close to the one in GiNaC.
1900 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
1901 @c node-name, next, previous, up
1902 @appendixsection Internal representation of products and sums
1904 @cindex representation
1907 @cindex @code{power}
1908 Although it should be completely transparent for the user of
1909 GiNaC a short discussion of this topic helps to understand the sources
1910 and also explain performance to a large degree. Consider the
1911 unexpanded symbolic expression
1913 $2d^3 \left( 4a + 5b - 3 \right)$
1916 @math{2*d^3*(4*a+5*b-3)}
1918 which could naively be represented by a tree of linear containers for
1919 addition and multiplication, one container for exponentiation with base
1920 and exponent and some atomic leaves of symbols and numbers in this
1925 @cindex pair-wise representation
1926 However, doing so results in a rather deeply nested tree which will
1927 quickly become inefficient to manipulate. We can improve on this by
1928 representing the sum as a sequence of terms, each one being a pair of a
1929 purely numeric multiplicative coefficient and its rest. In the same
1930 spirit we can store the multiplication as a sequence of terms, each
1931 having a numeric exponent and a possibly complicated base, the tree
1932 becomes much more flat:
1936 The number @code{3} above the symbol @code{d} shows that @code{mul}
1937 objects are treated similarly where the coefficients are interpreted as
1938 @emph{exponents} now. Addition of sums of terms or multiplication of
1939 products with numerical exponents can be coded to be very efficient with
1940 such a pair-wise representation. Internally, this handling is performed
1941 by most CAS in this way. It typically speeds up manipulations by an
1942 order of magnitude. The overall multiplicative factor @code{2} and the
1943 additive term @code{-3} look somewhat out of place in this
1944 representation, however, since they are still carrying a trivial
1945 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
1946 this is avoided by adding a field that carries an overall numeric
1947 coefficient. This results in the realistic picture of internal
1950 $2d^3 \left( 4a + 5b - 3 \right)$:
1953 @math{2*d^3*(4*a+5*b-3)}:
1959 This also allows for a better handling of numeric radicals, since
1960 @code{sqrt(2)} can now be carried along calculations. Now it should be
1961 clear, why both classes @code{add} and @code{mul} are derived from the
1962 same abstract class: the data representation is the same, only the
1963 semantics differs. In the class hierarchy, methods for polynomial
1964 expansion and the like are reimplemented for @code{add} and @code{mul},
1965 but the data structure is inherited from @code{expairseq}.
1968 @node Package Tools, ginac-config, Internal representation of products and sums, Top
1969 @c node-name, next, previous, up
1970 @appendix Package Tools
1972 If you are creating a software package that uses the GiNaC library,
1973 setting the correct command line options for the compiler and linker
1974 can be difficult. GiNaC includes two tools to make this process easier.
1977 * ginac-config:: A shell script to detect compiler and linker flags.
1978 * AM_PATH_GINAC:: Macro for GNU automake.
1982 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
1983 @c node-name, next, previous, up
1984 @section @command{ginac-config}
1985 @cindex ginac-config
1987 @command{ginac-config} is a shell script that you can use to determine
1988 the compiler and linker command line options required to compile and
1989 link a program with the GiNaC library.
1991 @command{ginac-config} takes the following flags:
1995 Prints out the version of GiNaC installed.
1997 Prints '-I' flags pointing to the installed header files.
1999 Prints out the linker flags necessary to link a program against GiNaC.
2000 @item --prefix[=@var{PREFIX}]
2001 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
2002 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
2003 Otherwise, prints out the configured value of @env{$prefix}.
2004 @item --exec-prefix[=@var{PREFIX}]
2005 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
2006 Otherwise, prints out the configured value of @env{$exec_prefix}.
2009 Typically, @command{ginac-config} will be used within a configure
2010 script, as described below. It, however, can also be used directly from
2011 the command line using backquotes to compile a simple program. For
2015 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
2018 This command line might expand to (for example):
2021 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
2022 -lginac -lcln -lstdc++
2025 Not only is the form using @command{ginac-config} easier to type, it will
2026 work on any system, no matter how GiNaC was configured.
2029 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
2030 @c node-name, next, previous, up
2031 @section @samp{AM_PATH_GINAC}
2032 @cindex AM_PATH_GINAC
2034 For packages configured using GNU automake, GiNaC also provides
2035 a macro to automate the process of checking for GiNaC.
2038 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
2046 Determines the location of GiNaC using @command{ginac-config}, which is
2047 either found in the user's path, or from the environment variable
2048 @env{GINACLIB_CONFIG}.
2051 Tests the installed libraries to make sure that their version
2052 is later than @var{MINIMUM-VERSION}. (A default version will be used
2056 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
2057 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
2058 variable to the output of @command{ginac-config --libs}, and calls
2059 @samp{AC_SUBST()} for these variables so they can be used in generated
2060 makefiles, and then executes @var{ACTION-IF-FOUND}.
2063 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
2064 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
2068 This macro is in file @file{ginac.m4} which is installed in
2069 @file{$datadir/aclocal}. Note that if automake was installed with a
2070 different @samp{--prefix} than GiNaC, you will either have to manually
2071 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
2072 aclocal the @samp{-I} option when running it.
2075 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
2076 * Example package:: Example of a package using AM_PATH_GINAC.
2080 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2081 @c node-name, next, previous, up
2082 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2084 Simply make sure that @command{ginac-config} is in your path, and run
2085 the configure script.
2092 The directory where the GiNaC libraries are installed needs
2093 to be found by your system's dynamic linker.
2095 This is generally done by
2098 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2104 setting the environment variable @env{LD_LIBRARY_PATH},
2107 or, as a last resort,
2110 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2111 running configure, for instance:
2114 LDFLAGS=-R/home/cbauer/lib ./configure
2119 You can also specify a @command{ginac-config} not in your path by
2120 setting the @env{GINACLIB_CONFIG} environment variable to the
2121 name of the executable
2124 If you move the GiNaC package from its installed location,
2125 you will need either need to modify @command{ginac-config} script
2126 manually to point to the new location or rebuild GiNaC.
2137 --with-ginac-prefix=@var{PREFIX}
2138 --with-ginac-exec-prefix=@var{PREFIX}
2141 are provided to override the prefix and exec-prefix that were stored
2142 in the @command{ginac-config} shell script by GiNaC's configure. You are
2143 generally better off configuring GiNaC with the right path to begin with.
2147 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2148 @c node-name, next, previous, up
2149 @subsection Example of a package using @samp{AM_PATH_GINAC}
2151 The following shows how to build a simple package using automake
2152 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2155 #include <ginac/ginac.h>
2156 using namespace GiNaC;
2162 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2167 You should first read the introductory portions of the automake
2168 Manual, if you are not already familiar with it.
2170 Two files are needed, @file{configure.in}, which is used to build the
2174 dnl Process this file with autoconf to produce a configure script.
2176 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2182 AM_PATH_GINAC(0.4.0, [
2183 LIBS="$LIBS $GINACLIB_LIBS"
2184 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2185 ], AC_MSG_ERROR([need to have GiNaC installed]))
2190 The only command in this which is not standard for automake
2191 is the @samp{AM_PATH_GINAC} macro.
2193 That command does the following:
2196 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2197 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2198 with the error message `need to have GiNaC installed'
2201 And the @file{Makefile.am}, which will be used to build the Makefile.
2204 ## Process this file with automake to produce Makefile.in
2205 bin_PROGRAMS = simple
2206 simple_SOURCES = simple.cpp
2209 This @file{Makefile.am}, says that we are building a single executable,
2210 from a single sourcefile @file{simple.cpp}. Since every program
2211 we are building uses GiNaC we simply added the GiNaC options
2212 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
2213 want to specify them on a per-program basis: for instance by
2217 simple_LDADD = $(GINACLIB_LIBS)
2218 INCLUDES = $(GINACLIB_CPPFLAGS)
2221 to the @file{Makefile.am}.
2223 To try this example out, create a new directory and add the three
2226 Now execute the following commands:
2229 $ automake --add-missing
2234 You now have a package that can be built in the normal fashion
2243 @node Bibliography, Concept Index, Example package, Top
2244 @c node-name, next, previous, up
2245 @appendix Bibliography
2250 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
2253 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
2256 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
2259 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
2262 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
2263 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
2266 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
2267 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
2268 Academic Press, London
2273 @node Concept Index, , Bibliography, Top
2274 @c node-name, next, previous, up
2275 @unnumbered Concept Index