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) 1999-2000 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{} 1999-2000 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{} 1999-2000 Johannes Gutenberg
139 University Mainz, Germany.
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 now 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.
639 * Archiving:: Storing expression libraries in files.
643 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
644 @c node-name, next, previous, up
646 @cindex expression (class @code{ex})
649 The most common class of objects a user deals with is the expression
650 @code{ex}, representing a mathematical object like a variable, number,
651 function, sum, product, etc... Expressions may be put together to form
652 new expressions, passed as arguments to functions, and so on. Here is a
653 little collection of valid expressions:
656 ex MyEx1 = 5; // simple number
657 ex MyEx2 = x + 2*y; // polynomial in x and y
658 ex MyEx3 = (x + 1)/(x - 1); // rational expression
659 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
660 ex MyEx5 = MyEx4 + 1; // similar to above
663 Expressions are handles to other more fundamental objects, that many
664 times contain other expressions thus creating a tree of expressions
665 (@xref{Internal Structures}, for particular examples). Most methods on
666 @code{ex} therefore run top-down through such an expression tree. For
667 example, the method @code{has()} scans recursively for occurrences of
668 something inside an expression. Thus, if you have declared @code{MyEx4}
669 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
670 the argument of @code{sin} and hence return @code{true}.
672 The next sections will outline the general picture of GiNaC's class
673 hierarchy and describe the classes of objects that are handled by
677 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
678 @c node-name, next, previous, up
679 @section The Class Hierarchy
681 GiNaC's class hierarchy consists of several classes representing
682 mathematical objects, all of which (except for @code{ex} and some
683 helpers) are internally derived from one abstract base class called
684 @code{basic}. You do not have to deal with objects of class
685 @code{basic}, instead you'll be dealing with symbols, numbers,
686 containers of expressions and so on. You'll soon learn in this chapter
687 how many of the functions on symbols are really classes. This is
688 because simple symbolic arithmetic is not supported by languages like
689 C++ so in a certain way GiNaC has to implement its own arithmetic.
693 To get an idea about what kinds of symbolic composits may be built we
694 have a look at the most important classes in the class hierarchy. The
695 oval classes are atomic ones and the squared classes are containers.
696 The dashed line symbolizes a `points to' or `handles' relationship while
697 the solid lines stand for `inherits from' relationship in the class
700 @image{classhierarchy}
702 Some of the classes shown here (the ones sitting in white boxes) are
703 abstract base classes that are of no interest at all for the user. They
704 are used internally in order to avoid code duplication if two or more
705 classes derived from them share certain features. An example would be
706 @code{expairseq}, which is a container for a sequence of pairs each
707 consisting of one expression and a number (@code{numeric}). What
708 @emph{is} visible to the user are the derived classes @code{add} and
709 @code{mul}, representing sums of terms and products, respectively.
710 @xref{Internal Structures}, where these two classes are described in
713 At this point, we only summarize what kind of mathematical objects are
714 stored in the different classes in above diagram in order to give you a
718 @multitable @columnfractions .22 .78
719 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
720 @item @code{constant} @tab Constants like
727 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
728 @item @code{add} @tab Sums like @math{x+y} or @math{a+(2*b)+3}
729 @item @code{mul} @tab Products like @math{x*y} or @math{a*(x+y+z)*b*2}
730 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
735 @code{sqrt(}@math{2}@code{)}
738 @item @code{pseries} @tab Power Series, e.g. @math{x+1/6*x^3+1/120*x^5+O(x^7)}
739 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
740 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
741 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
742 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
743 @item @code{color} @tab Element of the @math{SU(3)} Lie-algebra
744 @item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
745 @item @code{idx} @tab Index of a tensor object
746 @item @code{coloridx} @tab Index of a @math{SU(3)} tensor
750 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
751 @c node-name, next, previous, up
753 @cindex @code{symbol} (class)
754 @cindex hierarchy of classes
757 Symbols are for symbolic manipulation what atoms are for chemistry. You
758 can declare objects of class @code{symbol} as any other object simply by
759 saying @code{symbol x,y;}. There is, however, a catch in here having to
760 do with the fact that C++ is a compiled language. The information about
761 the symbol's name is thrown away by the compiler but at a later stage
762 you may want to print expressions holding your symbols. In order to
763 avoid confusion GiNaC's symbols are able to know their own name. This
764 is accomplished by declaring its name for output at construction time in
765 the fashion @code{symbol x("x");}. If you declare a symbol using the
766 default constructor (i.e. without string argument) the system will deal
767 out a unique name. That name may not be suitable for printing but for
768 internal routines when no output is desired it is often enough. We'll
769 come across examples of such symbols later in this tutorial.
771 This implies that the strings passed to symbols at construction time may
772 not be used for comparing two of them. It is perfectly legitimate to
773 write @code{symbol x("x"),y("x");} but it is likely to lead into
774 trouble. Here, @code{x} and @code{y} are different symbols and
775 statements like @code{x-y} will not be simplified to zero although the
776 output @code{x-x} looks funny. Such output may also occur when there
777 are two different symbols in two scopes, for instance when you call a
778 function that declares a symbol with a name already existent in a symbol
779 in the calling function. Again, comparing them (using @code{operator==}
780 for instance) will always reveal their difference. Watch out, please.
782 @cindex @code{subs()}
783 Although symbols can be assigned expressions for internal reasons, you
784 should not do it (and we are not going to tell you how it is done). If
785 you want to replace a symbol with something else in an expression, you
786 can use the expression's @code{.subs()} method.
789 @node Numbers, Constants, Symbols, Basic Concepts
790 @c node-name, next, previous, up
792 @cindex @code{numeric} (class)
798 For storing numerical things, GiNaC uses Bruno Haible's library
799 @acronym{CLN}. The classes therein serve as foundation classes for
800 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
801 alternatively for Common Lisp Numbers. In order to find out more about
802 @acronym{CLN}'s internals the reader is refered to the documentation of
803 that library. @inforef{Introduction, , cln}, for more
804 information. Suffice to say that it is by itself build on top of another
805 library, the GNU Multiple Precision library @acronym{GMP}, which is an
806 extremely fast library for arbitrary long integers and rationals as well
807 as arbitrary precision floating point numbers. It is very commonly used
808 by several popular cryptographic applications. @acronym{CLN} extends
809 @acronym{GMP} by several useful things: First, it introduces the complex
810 number field over either reals (i.e. floating point numbers with
811 arbitrary precision) or rationals. Second, it automatically converts
812 rationals to integers if the denominator is unity and complex numbers to
813 real numbers if the imaginary part vanishes and also correctly treats
814 algebraic functions. Third it provides good implementations of
815 state-of-the-art algorithms for all trigonometric and hyperbolic
816 functions as well as for calculation of some useful constants.
818 The user can construct an object of class @code{numeric} in several
819 ways. The following example shows the four most important constructors.
820 It uses construction from C-integer, construction of fractions from two
821 integers, construction from C-float and construction from a string:
824 #include <ginac/ginac.h>
825 using namespace GiNaC;
829 numeric two(2); // exact integer 2
830 numeric r(2,3); // exact fraction 2/3
831 numeric e(2.71828); // floating point number
832 numeric p("3.1415926535897932385"); // floating point number
834 cout << two*p << endl; // floating point 6.283...
839 Note that all those constructors are @emph{explicit} which means you are
840 not allowed to write @code{numeric two=2;}. This is because the basic
841 objects to be handled by GiNaC are the expressions @code{ex} and we want
842 to keep things simple and wish objects like @code{pow(x,2)} to be
843 handled the same way as @code{pow(x,a)}, which means that we need to
844 allow a general @code{ex} as base and exponent. Therefore there is an
845 implicit constructor from C-integers directly to expressions handling
846 numerics at work in most of our examples. This design really becomes
847 convenient when one declares own functions having more than one
848 parameter but it forbids using implicit constructors because that would
849 lead to compile-time ambiguities.
851 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
852 This would, however, call C's built-in operator @code{/} for integers
853 first and result in a numeric holding a plain integer 1. @strong{Never
854 use the operator @code{/} on integers} unless you know exactly what you
855 are doing! Use the constructor from two integers instead, as shown in
856 the example above. Writing @code{numeric(1)/2} may look funny but works
859 @cindex @code{Digits}
861 We have seen now the distinction between exact numbers and floating
862 point numbers. Clearly, the user should never have to worry about
863 dynamically created exact numbers, since their `exactness' always
864 determines how they ought to be handled, i.e. how `long' they are. The
865 situation is different for floating point numbers. Their accuracy is
866 controlled by one @emph{global} variable, called @code{Digits}. (For
867 those readers who know about Maple: it behaves very much like Maple's
868 @code{Digits}). All objects of class numeric that are constructed from
869 then on will be stored with a precision matching that number of decimal
873 #include <ginac/ginac.h>
874 using namespace GiNaC;
878 numeric three(3.0), one(1.0);
879 numeric x = one/three;
881 cout << "in " << Digits << " digits:" << endl;
883 cout << Pi.evalf() << endl;
895 The above example prints the following output to screen:
902 0.333333333333333333333333333333333333333333333333333333333333333333
903 3.14159265358979323846264338327950288419716939937510582097494459231
906 It should be clear that objects of class @code{numeric} should be used
907 for constructing numbers or for doing arithmetic with them. The objects
908 one deals with most of the time are the polymorphic expressions @code{ex}.
910 @subsection Tests on numbers
912 Once you have declared some numbers, assigned them to expressions and
913 done some arithmetic with them it is frequently desired to retrieve some
914 kind of information from them like asking whether that number is
915 integer, rational, real or complex. For those cases GiNaC provides
916 several useful methods. (Internally, they fall back to invocations of
917 certain CLN functions.)
919 As an example, let's construct some rational number, multiply it with
920 some multiple of its denominator and test what comes out:
923 #include <ginac/ginac.h>
924 using namespace GiNaC;
926 // some very important constants:
927 const numeric twentyone(21);
928 const numeric ten(10);
929 const numeric five(5);
933 numeric answer = twentyone;
936 cout << answer.is_integer() << endl; // false, it's 21/5
938 cout << answer.is_integer() << endl; // true, it's 42 now!
943 Note that the variable @code{answer} is constructed here as an integer
944 by @code{numeric}'s copy constructor but in an intermediate step it
945 holds a rational number represented as integer numerator and integer
946 denominator. When multiplied by 10, the denominator becomes unity and
947 the result is automatically converted to a pure integer again.
948 Internally, the underlying @acronym{CLN} is responsible for this
949 behaviour and we refer the reader to @acronym{CLN}'s documentation.
950 Suffice to say that the same behaviour applies to complex numbers as
951 well as return values of certain functions. Complex numbers are
952 automatically converted to real numbers if the imaginary part becomes
953 zero. The full set of tests that can be applied is listed in the
957 @multitable @columnfractions .30 .70
958 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
959 @item @code{.is_zero()}
960 @tab @dots{}equal to zero
961 @item @code{.is_positive()}
962 @tab @dots{}not complex and greater than 0
963 @item @code{.is_integer()}
964 @tab @dots{}a (non-complex) integer
965 @item @code{.is_pos_integer()}
966 @tab @dots{}an integer and greater than 0
967 @item @code{.is_nonneg_integer()}
968 @tab @dots{}an integer and greater equal 0
969 @item @code{.is_even()}
970 @tab @dots{}an even integer
971 @item @code{.is_odd()}
972 @tab @dots{}an odd integer
973 @item @code{.is_prime()}
974 @tab @dots{}a prime integer (probabilistic primality test)
975 @item @code{.is_rational()}
976 @tab @dots{}an exact rational number (integers are rational, too)
977 @item @code{.is_real()}
978 @tab @dots{}a real integer, rational or float (i.e. is not complex)
979 @item @code{.is_cinteger()}
980 @tab @dots{}a (complex) integer, such as @math{2-3*I}
981 @item @code{.is_crational()}
982 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
987 @node Constants, Fundamental containers, Numbers, Basic Concepts
988 @c node-name, next, previous, up
990 @cindex @code{constant} (class)
993 @cindex @code{Catalan}
994 @cindex @code{EulerGamma}
995 @cindex @code{evalf()}
996 Constants behave pretty much like symbols except that they return some
997 specific number when the method @code{.evalf()} is called.
999 The predefined known constants are:
1002 @multitable @columnfractions .14 .30 .56
1003 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1005 @tab Archimedes' constant
1006 @tab 3.14159265358979323846264338327950288
1007 @item @code{Catalan}
1008 @tab Catalan's constant
1009 @tab 0.91596559417721901505460351493238411
1010 @item @code{EulerGamma}
1011 @tab Euler's (or Euler-Mascheroni) constant
1012 @tab 0.57721566490153286060651209008240243
1017 @node Fundamental containers, Built-in functions, Constants, Basic Concepts
1018 @c node-name, next, previous, up
1019 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1023 @cindex @code{power}
1025 Simple polynomial expressions are written down in GiNaC pretty much like
1026 in other CAS or like expressions involving numerical variables in C.
1027 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1028 been overloaded to achieve this goal. When you run the following
1029 program, the constructor for an object of type @code{mul} is
1030 automatically called to hold the product of @code{a} and @code{b} and
1031 then the constructor for an object of type @code{add} is called to hold
1032 the sum of that @code{mul} object and the number one:
1035 #include <ginac/ginac.h>
1036 using namespace GiNaC;
1040 symbol a("a"), b("b");
1046 @cindex @code{pow()}
1047 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1048 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1049 construction is necessary since we cannot safely overload the constructor
1050 @code{^} in C++ to construct a @code{power} object. If we did, it would
1051 have several counterintuitive effects:
1055 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1057 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1058 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1059 interpret this as @code{x^(a^b)}.
1061 Also, expressions involving integer exponents are very frequently used,
1062 which makes it even more dangerous to overload @code{^} since it is then
1063 hard to distinguish between the semantics as exponentiation and the one
1064 for exclusive or. (It would be embarassing to return @code{1} where one
1065 has requested @code{2^3}.)
1068 @cindex @command{ginsh}
1069 All effects are contrary to mathematical notation and differ from the
1070 way most other CAS handle exponentiation, therefore overloading @code{^}
1071 is ruled out for GiNaC's C++ part. The situation is different in
1072 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1073 that the other frequently used exponentiation operator @code{**} does
1074 not exist at all in C++).
1076 To be somewhat more precise, objects of the three classes described
1077 here, are all containers for other expressions. An object of class
1078 @code{power} is best viewed as a container with two slots, one for the
1079 basis, one for the exponent. All valid GiNaC expressions can be
1080 inserted. However, basic transformations like simplifying
1081 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1082 when this is mathematically possible. If we replace the outer exponent
1083 three in the example by some symbols @code{a}, the simplification is not
1084 safe and will not be performed, since @code{a} might be @code{1/2} and
1087 Objects of type @code{add} and @code{mul} are containers with an
1088 arbitrary number of slots for expressions to be inserted. Again, simple
1089 and safe simplifications are carried out like transforming
1090 @code{3*x+4-x} to @code{2*x+4}.
1092 The general rule is that when you construct such objects, GiNaC
1093 automatically creates them in canonical form, which might differ from
1094 the form you typed in your program. This allows for rapid comparison of
1095 expressions, since after all @code{a-a} is simply zero. Note, that the
1096 canonical form is not necessarily lexicographical ordering or in any way
1097 easily guessable. It is only guaranteed that constructing the same
1098 expression twice, either implicitly or explicitly, results in the same
1102 @node Built-in functions, Relations, Fundamental containers, Basic Concepts
1103 @c node-name, next, previous, up
1104 @section Built-in functions
1105 @cindex @code{function} (class)
1106 @cindex trigonometric function
1107 @cindex hyperbolic function
1109 There are quite a number of useful functions hard-wired into GiNaC. For
1110 instance, all trigonometric and hyperbolic functions are implemented.
1111 They are all objects of class @code{function}. They accept one or more
1112 expressions as arguments and return one expression. If the arguments
1113 are not numerical, the evaluation of the function may be halted, as it
1114 does in the next example:
1116 @cindex Gamma function
1117 @cindex @code{subs()}
1119 #include <ginac/ginac.h>
1120 using namespace GiNaC;
1124 symbol x("x"), y("y");
1127 cout << "gamma(" << foo << ") -> " << gamma(foo) << endl;
1128 ex bar = foo.subs(y==1);
1129 cout << "gamma(" << bar << ") -> " << gamma(bar) << endl;
1130 ex foobar = bar.subs(x==7);
1131 cout << "gamma(" << foobar << ") -> " << gamma(foobar) << endl;
1136 This program shows how the function returns itself twice and finally an
1137 expression that may be really useful:
1140 gamma(x+(1/2)*y) -> gamma(x+(1/2)*y)
1141 gamma(x+1/2) -> gamma(x+1/2)
1142 gamma(15/2) -> (135135/128)*Pi^(1/2)
1146 For functions that have a branch cut in the complex plane GiNaC follows
1147 the conventions for C++ as defined in the ANSI standard. In particular:
1148 the natural logarithm (@code{log}) and the square root (@code{sqrt})
1149 both have their branch cuts running along the negative real axis where
1150 the points on the axis itself belong to the upper part.
1152 Besides evaluation most of these functions allow differentiation, series
1153 expansion and so on. Read the next chapter in order to learn more about
1157 @node Relations, Archiving, Built-in functions, Basic Concepts
1158 @c node-name, next, previous, up
1160 @cindex @code{relational} (class)
1162 Sometimes, a relation holding between two expressions must be stored
1163 somehow. The class @code{relational} is a convenient container for such
1164 purposes. A relation is by definition a container for two @code{ex} and
1165 a relation between them that signals equality, inequality and so on.
1166 They are created by simply using the C++ operators @code{==}, @code{!=},
1167 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1169 @xref{Built-in functions}, for examples where various applications of
1170 the @code{.subs()} method show how objects of class relational are used
1171 as arguments. There they provide an intuitive syntax for substitutions.
1174 @node Archiving, Important Algorithms, Relations, Basic Concepts
1175 @c node-name, next, previous, up
1176 @section Archiving Expressions
1178 @cindex @code{archive} (class)
1180 GiNaC allows creating @dfn{archives} of expressions which can be stored
1181 to or retrieved from files. To create an archive, you declare an object
1182 of class @code{archive} and archive expressions in it, giving each
1183 expressions a unique name:
1186 #include <ginac/ginac.h>
1188 using namespace GiNaC;
1192 symbol x("x"), y("y"), z("z");
1194 ex foo = sin(x + 2*y) + 3*z + 41;
1198 a.archive_ex(foo, "foo");
1199 a.archive_ex(bar, "the second one");
1203 The archive can then be written to a file:
1207 ofstream out("foobar.gar");
1213 The file @file{foobar.gar} contains all information that is needed to
1214 reconstruct the expressions @code{foo} and @code{bar}.
1216 @cindex @command{viewgar}
1217 The tool @command{viewgar} that comes with GiNaC can be used to view
1218 the contents of GiNaC archive files:
1221 $ viewgar foobar.gar
1222 foo = 41+sin(x+2*y)+3*z
1223 the second one = 42+sin(x+2*y)+3*z
1226 The point of writing archive files is of course that they can later be
1232 ifstream in("foobar.gar");
1237 And the stored expressions can be retrieved by their name:
1242 syms.append(x); syms.append(y);
1244 ex ex1 = a2.unarchive_ex(syms, "foo");
1245 ex ex2 = a2.unarchive_ex(syms, "the second one");
1247 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
1248 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
1249 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
1254 Note that you have to supply a list of the symbols which are to be inserted
1255 in the expressions. Symbols in archives are stored by their name only and
1256 if you don't specify which symbols you have, unarchiving the expression will
1257 create new symbols with that name. E.g. if you hadn't included @code{x} in
1258 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
1259 have had no effect because the @code{x} in @code{ex1} would have been a
1260 different symbol than the @code{x} which was defined at the beginning of
1261 the program, altough both would appear as @samp{x} when printed.
1265 @node Important Algorithms, Polynomial Expansion, Archiving, Top
1266 @c node-name, next, previous, up
1267 @chapter Important Algorithms
1270 In this chapter the most important algorithms provided by GiNaC will be
1271 described. Some of them are implemented as functions on expressions,
1272 others are implemented as methods provided by expression objects. If
1273 they are methods, there exists a wrapper function around it, so you can
1274 alternatively call it in a functional way as shown in the simple
1278 #include <ginac/ginac.h>
1279 using namespace GiNaC;
1283 ex x = numeric(1.0);
1285 cout << "As method: " << sin(x).evalf() << endl;
1286 cout << "As function: " << evalf(sin(x)) << endl;
1291 @cindex @code{subs()}
1292 The general rule is that wherever methods accept one or more parameters
1293 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1294 wrapper accepts is the same but preceded by the object to act on
1295 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1296 most natural one in an OO model but it may lead to confusion for MapleV
1297 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1298 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1299 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1300 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1301 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1302 here. Also, users of MuPAD will in most cases feel more comfortable
1303 with GiNaC's convention. All function wrappers are always implemented
1304 as simple inline functions which just call the corresponding method and
1305 are only provided for users uncomfortable with OO who are dead set to
1306 avoid method invocations. Generally, nested function wrappers are much
1307 harder to read than a sequence of methods and should therefore be
1308 avoided if possible. On the other hand, not everything in GiNaC is a
1309 method on class @code{ex} and sometimes calling a function cannot be
1313 * Polynomial Expansion::
1314 * Collecting expressions::
1315 * Polynomial Arithmetic::
1316 * Symbolic Differentiation::
1317 * Series Expansion::
1321 @node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
1322 @c node-name, next, previous, up
1323 @section Polynomial Expansion
1324 @cindex @code{expand()}
1326 A polynomial in one or more variables has many equivalent
1327 representations. Some useful ones serve a specific purpose. Consider
1328 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1329 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1330 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1331 representations are the recursive ones where one collects for exponents
1332 in one of the three variable. Since the factors are themselves
1333 polynomials in the remaining two variables the procedure can be
1334 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1335 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1338 To bring an expression into expanded form, its method @code{.expand()}
1339 may be called. In our example above, this corresponds to @math{4*x*y +
1340 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1341 GiNaC is not easily guessable you should be prepared to see different
1342 orderings of terms in such sums!
1345 @node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
1346 @c node-name, next, previous, up
1347 @section Collecting expressions
1348 @cindex @code{collect()}
1349 @cindex @code{coeff()}
1351 Another useful representation of multivariate polynomials is as a
1352 univariate polynomial in one of the variables with the coefficients
1353 being polynomials in the remaining variables. The method
1354 @code{collect()} accomplishes this task. Here is its declaration:
1357 ex ex::collect(const symbol & s);
1360 Note that the original polynomial needs to be in expanded form in order
1361 to be able to find the coefficients properly. The range of occuring
1362 coefficients can be checked using the two methods
1364 @cindex @code{degree()}
1365 @cindex @code{ldegree()}
1367 int ex::degree(const symbol & s);
1368 int ex::ldegree(const symbol & s);
1371 where @code{degree()} returns the highest coefficient and
1372 @code{ldegree()} the lowest one. (These two methods work also reliably
1373 on non-expanded input polynomials). An application is illustrated in
1374 the next example, where a multivariate polynomial is analyzed:
1377 #include <ginac/ginac.h>
1378 using namespace GiNaC;
1382 symbol x("x"), y("y");
1383 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1384 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1385 ex Poly = PolyInp.expand();
1387 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1388 cout << "The x^" << i << "-coefficient is "
1389 << Poly.coeff(x,i) << endl;
1391 cout << "As polynomial in y: "
1392 << Poly.collect(y) << endl;
1397 When run, it returns an output in the following fashion:
1400 The x^0-coefficient is y^2+11*y
1401 The x^1-coefficient is 5*y^2-2*y
1402 The x^2-coefficient is -1
1403 The x^3-coefficient is 4*y
1404 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1407 As always, the exact output may vary between different versions of GiNaC
1408 or even from run to run since the internal canonical ordering is not
1409 within the user's sphere of influence.
1412 @node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
1413 @c node-name, next, previous, up
1414 @section Polynomial Arithmetic
1416 @subsection GCD and LCM
1420 The functions for polynomial greatest common divisor and least common
1421 multiple have the synopsis:
1424 ex gcd(const ex & a, const ex & b);
1425 ex lcm(const ex & a, const ex & b);
1428 The functions @code{gcd()} and @code{lcm()} accept two expressions
1429 @code{a} and @code{b} as arguments and return a new expression, their
1430 greatest common divisor or least common multiple, respectively. If the
1431 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1432 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1435 #include <ginac/ginac.h>
1436 using namespace GiNaC;
1440 symbol x("x"), y("y"), z("z");
1441 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1442 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1444 ex P_gcd = gcd(P_a, P_b);
1446 ex P_lcm = lcm(P_a, P_b);
1447 // 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
1452 @subsection The @code{normal} method
1453 @cindex @code{normal()}
1454 @cindex temporary replacement
1456 While in common symbolic code @code{gcd()} and @code{lcm()} are not too
1457 heavily used, simplification is called for frequently. Therefore
1458 @code{.normal()}, which provides some basic form of simplification, has
1459 become a method of class @code{ex}, just like @code{.expand()}. It
1460 converts a rational function into an equivalent rational function where
1461 numerator and denominator are coprime. This means, it finds the GCD of
1462 numerator and denominator and cancels it. If it encounters some object
1463 which does not belong to the domain of rationals (a function for
1464 instance), that object is replaced by a temporary symbol. This means
1465 that both expressions @code{t1} and @code{t2} are indeed simplified in
1466 this little program:
1469 #include <ginac/ginac.h>
1470 using namespace GiNaC;
1475 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1476 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1477 cout << "t1 is " << t1.normal() << endl;
1478 cout << "t2 is " << t2.normal() << endl;
1483 Of course this works for multivariate polynomials too, so the ratio of
1484 the sample-polynomials from the section about GCD and LCM above would be
1485 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1488 @node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
1489 @c node-name, next, previous, up
1490 @section Symbolic Differentiation
1491 @cindex differentiation
1492 @cindex @code{diff()}
1494 @cindex product rule
1496 GiNaC's objects know how to differentiate themselves. Thus, a
1497 polynomial (class @code{add}) knows that its derivative is the sum of
1498 the derivatives of all the monomials:
1501 #include <ginac/ginac.h>
1502 using namespace GiNaC;
1506 symbol x("x"), y("y"), z("z");
1507 ex P = pow(x, 5) + pow(x, 2) + y;
1509 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1510 cout << P.diff(y) << endl; // 1
1511 cout << P.diff(z) << endl; // 0
1516 If a second integer parameter @var{n} is given, the @code{diff} method
1517 returns the @var{n}th derivative.
1519 If @emph{every} object and every function is told what its derivative
1520 is, all derivatives of composed objects can be calculated using the
1521 chain rule and the product rule. Consider, for instance the expression
1522 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1523 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1524 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1525 out that the composition is the generating function for Euler Numbers,
1526 i.e. the so called @var{n}th Euler number is the coefficient of
1527 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1528 identity to code a function that generates Euler numbers in just three
1531 @cindex Euler numbers
1533 #include <ginac/ginac.h>
1534 using namespace GiNaC;
1536 ex EulerNumber(unsigned n)
1539 ex generator = pow(cosh(x),-1);
1540 return generator.diff(x,n).subs(x==0);
1545 for (unsigned i=0; i<11; i+=2)
1546 cout << EulerNumber(i) << endl;
1551 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1552 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1553 @code{i} by two since all odd Euler numbers vanish anyways.
1556 @node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
1557 @c node-name, next, previous, up
1558 @section Series Expansion
1559 @cindex @code{series()}
1560 @cindex Taylor expansion
1561 @cindex Laurent expansion
1562 @cindex @code{pseries} (class)
1564 Expressions know how to expand themselves as a Taylor series or (more
1565 generally) a Laurent series. As in most conventional Computer Algebra
1566 Systems, no distinction is made between those two. There is a class of
1567 its own for storing such series (@code{class pseries}) and a built-in
1568 function (called @code{Order}) for storing the order term of the series.
1569 As a consequence, if you want to work with series, i.e. multiply two
1570 series, you need to call the method @code{ex::series} again to convert
1571 it to a series object with the usual structure (expansion plus order
1572 term). A sample application from special relativity could read:
1575 #include <ginac/ginac.h>
1576 using namespace GiNaC;
1580 symbol v("v"), c("c");
1582 ex gamma = 1/sqrt(1 - pow(v/c,2));
1583 ex mass_nonrel = gamma.series(v, 0, 10);
1585 cout << "the relativistic mass increase with v is " << endl
1586 << mass_nonrel << endl;
1588 cout << "the inverse square of this series is " << endl
1589 << pow(mass_nonrel,-2).series(v, 0, 10) << endl;
1595 Only calling the series method makes the last output simplify to
1596 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
1597 series raised to the power @math{-2}.
1599 @cindex M@'echain's formula
1600 As another instructive application, let us calculate the numerical
1601 value of Archimedes' constant
1605 (for which there already exists the built-in constant @code{Pi})
1606 using M@'echain's amazing formula
1608 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1611 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1613 We may expand the arcus tangent around @code{0} and insert the fractions
1614 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1615 carries an order term with it and the question arises what the system is
1616 supposed to do when the fractions are plugged into that order term. The
1617 solution is to use the function @code{series_to_poly()} to simply strip
1621 #include <ginac/ginac.h>
1622 using namespace GiNaC;
1624 ex mechain_pi(int degr)
1627 ex pi_expansion = series_to_poly(atan(x).series(x,0,degr));
1628 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1629 -4*pi_expansion.subs(x==numeric(1,239));
1636 for (int i=2; i<12; i+=2) @{
1637 pi_frac = mechain_pi(i);
1638 cout << i << ":\t" << pi_frac << endl
1639 << "\t" << pi_frac.evalf() << endl;
1645 When you run this program, it will type out:
1649 3.1832635983263598326
1650 4: 5359397032/1706489875
1651 3.1405970293260603143
1652 6: 38279241713339684/12184551018734375
1653 3.141621029325034425
1654 8: 76528487109180192540976/24359780855939418203125
1655 3.141591772182177295
1656 10: 327853873402258685803048818236/104359128170408663038552734375
1657 3.1415926824043995174
1661 @node Extending GiNaC, What does not belong into GiNaC, Series Expansion, Top
1662 @c node-name, next, previous, up
1663 @chapter Extending GiNaC
1665 By reading so far you should have gotten a fairly good understanding of
1666 GiNaC's design-patterns. From here on you should start reading the
1667 sources. All we can do now is issue some recommendations how to tackle
1668 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
1669 develop some useful extension please don't hesitate to contact the GiNaC
1670 authors---they will happily incorporate them into future versions.
1673 * What does not belong into GiNaC:: What to avoid.
1674 * Symbolic functions:: Implementing symbolic functions.
1678 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
1679 @c node-name, next, previous, up
1680 @section What doesn't belong into GiNaC
1682 @cindex @command{ginsh}
1683 First of all, GiNaC's name must be read literally. It is designed to be
1684 a library for use within C++. The tiny @command{ginsh} accompanying
1685 GiNaC makes this even more clear: it doesn't even attempt to provide a
1686 language. There are no loops or conditional expressions in
1687 @command{ginsh}, it is merely a window into the library for the
1688 programmer to test stuff (or to show off). Still, the design of a
1689 complete CAS with a language of its own, graphical capabilites and all
1690 this on top of GiNaC is possible and is without doubt a nice project for
1693 There are many built-in functions in GiNaC that do not know how to
1694 evaluate themselves numerically to a precision declared at runtime
1695 (using @code{Digits}). Some may be evaluated at certain points, but not
1696 generally. This ought to be fixed. However, doing numerical
1697 computations with GiNaC's quite abstract classes is doomed to be
1698 inefficient. For this purpose, the underlying foundation classes
1699 provided by @acronym{CLN} are much better suited.
1702 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
1703 @c node-name, next, previous, up
1704 @section Symbolic functions
1706 The easiest and most instructive way to start with is probably to
1707 implement your own function. Objects of class @code{function} are
1708 inserted into the system via a kind of `registry'. They get a serial
1709 number that is used internally to identify them but you usually need not
1710 worry about this. What you have to care for are functions that are
1711 called when the user invokes certain methods. These are usual
1712 C++-functions accepting a number of @code{ex} as arguments and returning
1713 one @code{ex}. As an example, if we have a look at a simplified
1714 implementation of the cosine trigonometric function, we first need a
1715 function that is called when one wishes to @code{eval} it. It could
1716 look something like this:
1719 static ex cos_eval_method(const ex & x)
1721 // if (!x%(2*Pi)) return 1
1722 // if (!x%Pi) return -1
1723 // if (!x%Pi/2) return 0
1724 // care for other cases...
1725 return cos(x).hold();
1729 @cindex @code{hold()}
1731 The last line returns @code{cos(x)} if we don't know what else to do and
1732 stops a potential recursive evaluation by saying @code{.hold()}. We
1733 should also implement a method for numerical evaluation and since we are
1734 lazy we sweep the problem under the rug by calling someone else's
1735 function that does so, in this case the one in class @code{numeric}:
1738 static ex cos_evalf_method(const ex & x)
1740 return sin(ex_to_numeric(x));
1744 Differentiation will surely turn up and so we need to tell
1745 @code{sin} how to differentiate itself:
1748 static ex cos_diff_method(const ex & x, unsigned diff_param)
1754 @cindex product rule
1755 The second parameter is obligatory but uninteresting at this point. It
1756 specifies which parameter to differentiate in a partial derivative in
1757 case the function has more than one parameter and its main application
1758 is for correct handling of the chain rule. For Taylor expansion, it is
1759 enough to know how to differentiate. But if the function you want to
1760 implement does have a pole somewhere in the complex plane, you need to
1761 write another method for Laurent expansion around that point.
1763 Now that all the ingrediences for @code{cos} have been set up, we need
1764 to tell the system about it. This is done by a macro and we are not
1765 going to descibe how it expands, please consult your preprocessor if you
1769 REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);
1772 The first argument is the function's name, the second, third and fourth
1773 bind the corresponding methods to this objects and the fifth is a slot
1774 for inserting a method for series expansion. (If set to @code{NULL} it
1775 defaults to simple Taylor expansion, which is correct if there are no
1776 poles involved. The way GiNaC handles poles in case there are any is
1777 best understood by studying one of the examples, like the Gamma function
1778 for instance. In essence the function first checks if there is a pole
1779 at the evaluation point and falls back to Taylor expansion if there
1780 isn't. Then, the pole is regularized by some suitable transformation.)
1781 Also, the new function needs to be declared somewhere. This may also be
1782 done by a convenient preprocessor macro:
1785 DECLARE_FUNCTION_1P(cos)
1788 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
1789 implementation of @code{cos} is very incomplete and lacks several safety
1790 mechanisms. Please, have a look at the real implementation in GiNaC.
1791 (By the way: in case you are worrying about all the macros above we can
1792 assure you that functions are GiNaC's most macro-intense classes. We
1793 have done our best to avoid them where we can.)
1795 That's it. May the source be with you!
1798 @node A Comparison With Other CAS, Advantages, Symbolic functions, Top
1799 @c node-name, next, previous, up
1800 @chapter A Comparison With Other CAS
1803 This chapter will give you some information on how GiNaC compares to
1804 other, traditional Computer Algebra Systems, like @emph{Maple},
1805 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
1806 disadvantages over these systems.
1809 * Advantages:: Stengths of the GiNaC approach.
1810 * Disadvantages:: Weaknesses of the GiNaC approach.
1811 * Why C++?:: Attractiveness of C++.
1814 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
1815 @c node-name, next, previous, up
1818 GiNaC has several advantages over traditional Computer
1819 Algebra Systems, like
1824 familiar language: all common CAS implement their own proprietary
1825 grammar which you have to learn first (and maybe learn again when your
1826 vendor decides to `enhance' it). With GiNaC you can write your program
1827 in common C++, which is standardized.
1831 structured data types: you can build up structured data types using
1832 @code{struct}s or @code{class}es together with STL features instead of
1833 using unnamed lists of lists of lists.
1836 strongly typed: in CAS, you usually have only one kind of variables
1837 which can hold contents of an arbitrary type. This 4GL like feature is
1838 nice for novice programmers, but dangerous.
1841 development tools: powerful development tools exist for C++, like fancy
1842 editors (e.g. with automatic indentation and syntax highlighting),
1843 debuggers, visualization tools, documentation tools...
1846 modularization: C++ programs can easily be split into modules by
1847 separating interface and implementation.
1850 price: GiNaC is distributed under the GNU Public License which means
1851 that it is free and available with source code. And there are excellent
1852 C++-compilers for free, too.
1855 extendable: you can add your own classes to GiNaC, thus extending it on
1856 a very low level. Compare this to a traditional CAS that you can
1857 usually only extend on a high level by writing in the language defined
1858 by the parser. In particular, it turns out to be almost impossible to
1859 fix bugs in a traditional system.
1862 seemless integration: it is somewhere between difficult and impossible
1863 to call CAS functions from within a program written in C++ or any other
1864 programming language and vice versa. With GiNaC, your symbolic routines
1865 are part of your program. You can easily call third party libraries,
1866 e.g. for numerical evaluation or graphical interaction. All other
1867 approaches are much more cumbersome: they range from simply ignoring the
1868 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
1869 system (i.e. @emph{Yacas}).
1872 efficiency: often large parts of a program do not need symbolic
1873 calculations at all. Why use large integers for loop variables or
1874 arbitrary precision arithmetics where double accuracy is sufficient?
1875 For pure symbolic applications, GiNaC is comparable in speed with other
1881 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
1882 @c node-name, next, previous, up
1883 @section Disadvantages
1885 Of course it also has some disadvantages:
1890 not interactive: GiNaC programs have to be written in an editor,
1891 compiled and executed. You cannot play with expressions interactively.
1892 However, such an extension is not inherently forbidden by design. In
1893 fact, two interactive interfaces are possible: First, a shell that
1894 exposes GiNaC's types to a command line can readily be written (the tiny
1895 @command{ginsh} that is part of the distribution being an example) and
1896 second, as a more consistent approach we are working on an integration
1897 with the @acronym{CINT} C++ interpreter.
1900 advanced features: GiNaC cannot compete with a program like
1901 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
1902 which grows since 1981 by the work of dozens of programmers, with
1903 respect to mathematical features. Integration, factorization,
1904 non-trivial simplifications, limits etc. are missing in GiNaC (and are
1905 not planned for the near future).
1908 portability: While the GiNaC library itself is designed to avoid any
1909 platform dependent features (it should compile on any ANSI compliant C++
1910 compiler), the currently used version of the CLN library (fast large
1911 integer and arbitrary precision arithmetics) can be compiled only on
1912 systems with a recently new C++ compiler from the GNU Compiler
1913 Collection (@acronym{GCC}).@footnote{This is because CLN uses
1914 PROVIDE/REQUIRE like macros to let the compiler gather all static
1915 initializations, which works for GNU C++ only.} GiNaC uses recent
1916 language features like explicit constructors, mutable members, RTTI,
1917 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
1918 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
1919 ANSI compliant, support all needed features.
1924 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
1925 @c node-name, next, previous, up
1928 Why did we choose to implement GiNaC in C++ instead of Java or any other
1929 language? C++ is not perfect: type checking is not strict (casting is
1930 possible), separation between interface and implementation is not
1931 complete, object oriented design is not enforced. The main reason is
1932 the often scolded feature of operator overloading in C++. While it may
1933 be true that operating on classes with a @code{+} operator is rarely
1934 meaningful, it is perfectly suited for algebraic expressions. Writing
1935 @math{3x+5y} as @code{3*x+5*y} instead of
1936 @code{x.times(3).plus(y.times(5))} looks much more natural.
1937 Furthermore, the main developers are more familiar with C++ than with
1938 any other programming language.
1941 @node Internal Structures, Expressions are reference counted, Why C++? , Top
1942 @c node-name, next, previous, up
1943 @appendix Internal Structures
1946 * Expressions are reference counted::
1947 * Internal representation of products and sums::
1950 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
1951 @c node-name, next, previous, up
1952 @appendixsection Expressions are reference counted
1954 @cindex reference counting
1955 @cindex copy-on-write
1956 @cindex garbage collection
1957 An expression is extremely light-weight since internally it works like a
1958 handle to the actual representation and really holds nothing more than a
1959 pointer to some other object. What this means in practice is that
1960 whenever you create two @code{ex} and set the second equal to the first
1961 no copying process is involved. Instead, the copying takes place as soon
1962 as you try to change the second. Consider the simple sequence of code:
1965 #include <ginac/ginac.h>
1966 using namespace GiNaC;
1970 symbol x("x"), y("y"), z("z");
1973 e1 = sin(x + 2*y) + 3*z + 41;
1974 e2 = e1; // e2 points to same object as e1
1975 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
1976 e2 += 1; // e2 is copied into a new object
1977 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
1982 The line @code{e2 = e1;} creates a second expression pointing to the
1983 object held already by @code{e1}. The time involved for this operation
1984 is therefore constant, no matter how large @code{e1} was. Actual
1985 copying, however, must take place in the line @code{e2 += 1;} because
1986 @code{e1} and @code{e2} are not handles for the same object any more.
1987 This concept is called @dfn{copy-on-write semantics}. It increases
1988 performance considerably whenever one object occurs multiple times and
1989 represents a simple garbage collection scheme because when an @code{ex}
1990 runs out of scope its destructor checks whether other expressions handle
1991 the object it points to too and deletes the object from memory if that
1992 turns out not to be the case. A slightly less trivial example of
1993 differentiation using the chain-rule should make clear how powerful this
1997 #include <ginac/ginac.h>
1998 using namespace GiNaC;
2002 symbol x("x"), y("y");
2006 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
2007 cout << e1 << endl // prints x+3*y
2008 << e2 << endl // prints (x+3*y)^3
2009 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
2014 Here, @code{e1} will actually be referenced three times while @code{e2}
2015 will be referenced two times. When the power of an expression is built,
2016 that expression needs not be copied. Likewise, since the derivative of
2017 a power of an expression can be easily expressed in terms of that
2018 expression, no copying of @code{e1} is involved when @code{e3} is
2019 constructed. So, when @code{e3} is constructed it will print as
2020 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
2021 holds a reference to @code{e2} and the factor in front is just
2024 As a user of GiNaC, you cannot see this mechanism of copy-on-write
2025 semantics. When you insert an expression into a second expression, the
2026 result behaves exactly as if the contents of the first expression were
2027 inserted. But it may be useful to remember that this is not what
2028 happens. Knowing this will enable you to write much more efficient
2029 code. If you still have an uncertain feeling with copy-on-write
2030 semantics, we recommend you have a look at the
2031 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
2032 Marshall Cline. Chapter 16 covers this issue and presents an
2033 implementation which is pretty close to the one in GiNaC.
2036 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
2037 @c node-name, next, previous, up
2038 @appendixsection Internal representation of products and sums
2040 @cindex representation
2043 @cindex @code{power}
2044 Although it should be completely transparent for the user of
2045 GiNaC a short discussion of this topic helps to understand the sources
2046 and also explain performance to a large degree. Consider the
2047 unexpanded symbolic expression
2049 $2d^3 \left( 4a + 5b - 3 \right)$
2052 @math{2*d^3*(4*a+5*b-3)}
2054 which could naively be represented by a tree of linear containers for
2055 addition and multiplication, one container for exponentiation with base
2056 and exponent and some atomic leaves of symbols and numbers in this
2061 @cindex pair-wise representation
2062 However, doing so results in a rather deeply nested tree which will
2063 quickly become inefficient to manipulate. We can improve on this by
2064 representing the sum as a sequence of terms, each one being a pair of a
2065 purely numeric multiplicative coefficient and its rest. In the same
2066 spirit we can store the multiplication as a sequence of terms, each
2067 having a numeric exponent and a possibly complicated base, the tree
2068 becomes much more flat:
2072 The number @code{3} above the symbol @code{d} shows that @code{mul}
2073 objects are treated similarly where the coefficients are interpreted as
2074 @emph{exponents} now. Addition of sums of terms or multiplication of
2075 products with numerical exponents can be coded to be very efficient with
2076 such a pair-wise representation. Internally, this handling is performed
2077 by most CAS in this way. It typically speeds up manipulations by an
2078 order of magnitude. The overall multiplicative factor @code{2} and the
2079 additive term @code{-3} look somewhat out of place in this
2080 representation, however, since they are still carrying a trivial
2081 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
2082 this is avoided by adding a field that carries an overall numeric
2083 coefficient. This results in the realistic picture of internal
2086 $2d^3 \left( 4a + 5b - 3 \right)$:
2089 @math{2*d^3*(4*a+5*b-3)}:
2095 This also allows for a better handling of numeric radicals, since
2096 @code{sqrt(2)} can now be carried along calculations. Now it should be
2097 clear, why both classes @code{add} and @code{mul} are derived from the
2098 same abstract class: the data representation is the same, only the
2099 semantics differs. In the class hierarchy, methods for polynomial
2100 expansion and the like are reimplemented for @code{add} and @code{mul},
2101 but the data structure is inherited from @code{expairseq}.
2104 @node Package Tools, ginac-config, Internal representation of products and sums, Top
2105 @c node-name, next, previous, up
2106 @appendix Package Tools
2108 If you are creating a software package that uses the GiNaC library,
2109 setting the correct command line options for the compiler and linker
2110 can be difficult. GiNaC includes two tools to make this process easier.
2113 * ginac-config:: A shell script to detect compiler and linker flags.
2114 * AM_PATH_GINAC:: Macro for GNU automake.
2118 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
2119 @c node-name, next, previous, up
2120 @section @command{ginac-config}
2121 @cindex ginac-config
2123 @command{ginac-config} is a shell script that you can use to determine
2124 the compiler and linker command line options required to compile and
2125 link a program with the GiNaC library.
2127 @command{ginac-config} takes the following flags:
2131 Prints out the version of GiNaC installed.
2133 Prints '-I' flags pointing to the installed header files.
2135 Prints out the linker flags necessary to link a program against GiNaC.
2136 @item --prefix[=@var{PREFIX}]
2137 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
2138 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
2139 Otherwise, prints out the configured value of @env{$prefix}.
2140 @item --exec-prefix[=@var{PREFIX}]
2141 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
2142 Otherwise, prints out the configured value of @env{$exec_prefix}.
2145 Typically, @command{ginac-config} will be used within a configure
2146 script, as described below. It, however, can also be used directly from
2147 the command line using backquotes to compile a simple program. For
2151 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
2154 This command line might expand to (for example):
2157 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
2158 -lginac -lcln -lstdc++
2161 Not only is the form using @command{ginac-config} easier to type, it will
2162 work on any system, no matter how GiNaC was configured.
2165 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
2166 @c node-name, next, previous, up
2167 @section @samp{AM_PATH_GINAC}
2168 @cindex AM_PATH_GINAC
2170 For packages configured using GNU automake, GiNaC also provides
2171 a macro to automate the process of checking for GiNaC.
2174 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
2182 Determines the location of GiNaC using @command{ginac-config}, which is
2183 either found in the user's path, or from the environment variable
2184 @env{GINACLIB_CONFIG}.
2187 Tests the installed libraries to make sure that their version
2188 is later than @var{MINIMUM-VERSION}. (A default version will be used
2192 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
2193 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
2194 variable to the output of @command{ginac-config --libs}, and calls
2195 @samp{AC_SUBST()} for these variables so they can be used in generated
2196 makefiles, and then executes @var{ACTION-IF-FOUND}.
2199 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
2200 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
2204 This macro is in file @file{ginac.m4} which is installed in
2205 @file{$datadir/aclocal}. Note that if automake was installed with a
2206 different @samp{--prefix} than GiNaC, you will either have to manually
2207 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
2208 aclocal the @samp{-I} option when running it.
2211 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
2212 * Example package:: Example of a package using AM_PATH_GINAC.
2216 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2217 @c node-name, next, previous, up
2218 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2220 Simply make sure that @command{ginac-config} is in your path, and run
2221 the configure script.
2228 The directory where the GiNaC libraries are installed needs
2229 to be found by your system's dynamic linker.
2231 This is generally done by
2234 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2240 setting the environment variable @env{LD_LIBRARY_PATH},
2243 or, as a last resort,
2246 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2247 running configure, for instance:
2250 LDFLAGS=-R/home/cbauer/lib ./configure
2255 You can also specify a @command{ginac-config} not in your path by
2256 setting the @env{GINACLIB_CONFIG} environment variable to the
2257 name of the executable
2260 If you move the GiNaC package from its installed location,
2261 you will need either need to modify @command{ginac-config} script
2262 manually to point to the new location or rebuild GiNaC.
2273 --with-ginac-prefix=@var{PREFIX}
2274 --with-ginac-exec-prefix=@var{PREFIX}
2277 are provided to override the prefix and exec-prefix that were stored
2278 in the @command{ginac-config} shell script by GiNaC's configure. You are
2279 generally better off configuring GiNaC with the right path to begin with.
2283 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2284 @c node-name, next, previous, up
2285 @subsection Example of a package using @samp{AM_PATH_GINAC}
2287 The following shows how to build a simple package using automake
2288 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2291 #include <ginac/ginac.h>
2292 using namespace GiNaC;
2298 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2303 You should first read the introductory portions of the automake
2304 Manual, if you are not already familiar with it.
2306 Two files are needed, @file{configure.in}, which is used to build the
2310 dnl Process this file with autoconf to produce a configure script.
2312 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2318 AM_PATH_GINAC(0.4.0, [
2319 LIBS="$LIBS $GINACLIB_LIBS"
2320 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2321 ], AC_MSG_ERROR([need to have GiNaC installed]))
2326 The only command in this which is not standard for automake
2327 is the @samp{AM_PATH_GINAC} macro.
2329 That command does the following:
2332 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2333 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2334 with the error message `need to have GiNaC installed'
2337 And the @file{Makefile.am}, which will be used to build the Makefile.
2340 ## Process this file with automake to produce Makefile.in
2341 bin_PROGRAMS = simple
2342 simple_SOURCES = simple.cpp
2345 This @file{Makefile.am}, says that we are building a single executable,
2346 from a single sourcefile @file{simple.cpp}. Since every program
2347 we are building uses GiNaC we simply added the GiNaC options
2348 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
2349 want to specify them on a per-program basis: for instance by
2353 simple_LDADD = $(GINACLIB_LIBS)
2354 INCLUDES = $(GINACLIB_CPPFLAGS)
2357 to the @file{Makefile.am}.
2359 To try this example out, create a new directory and add the three
2362 Now execute the following commands:
2365 $ automake --add-missing
2370 You now have a package that can be built in the normal fashion
2379 @node Bibliography, Concept Index, Example package, Top
2380 @c node-name, next, previous, up
2381 @appendix Bibliography
2386 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
2389 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
2392 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
2395 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
2398 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
2399 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
2402 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
2403 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
2404 Academic Press, London
2409 @node Concept Index, , Bibliography, Top
2410 @c node-name, next, previous, up
2411 @unnumbered Concept Index