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 * Methods and Functions:: 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(const symbol & x, int n)
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, n) * diff(HKer, x, n) / 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 Exact numbers are always retained as exact numbers and only evaluated as
284 floating point numbers if requested. For instance, with numeric
285 radicals is dealt pretty much as with symbols. Products of sums of them
289 > expand((1+a^(1/5)-a^(2/5))^3);
290 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
291 > expand((1+3^(1/5)-3^(2/5))^3);
293 > evalf((1+3^(1/5)-3^(2/5))^3);
294 0.33408977534118624228
297 The function @code{evalf} that was used above converts any number in
298 GiNaC's expressions into floating point numbers. This can be done to
299 arbitrary predefined accuracy:
303 0.14285714285714285714
307 0.1428571428571428571428571428571428571428571428571428571428571428571428
308 5714285714285714285714285714285714285
311 Exact numbers other than rationals that can be manipulated in GiNaC
312 include predefined constants like Archimedes' @code{Pi}. They can both
313 be used in symbolic manipulations (as an exact number) as well as in
314 numeric expressions (as an inexact number):
320 9.869604401089358619+x
324 11.869604401089358619
327 Built-in functions evaluate immediately to exact numbers if
328 this is possible. Conversions that can be safely performed are done
329 immediately; conversions that are not generally valid are not done:
340 (Note that converting the last input to @code{x} would allow one to
341 conclude that @code{42*Pi} is equal to @code{0}.)
343 Linear equation systems can be solved along with basic linear
344 algebra manipulations over symbolic expressions. In C++ GiNaC offers
345 a matrix class for this purpose but we can see what it can do using
346 @command{ginsh}'s notation of double brackets to type them in:
349 > lsolve(a+x*y==z,x);
351 > lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
353 > M = [[ [[1, 3]], [[-3, 2]] ]];
354 [[ [[1,3]], [[-3,2]] ]]
357 > charpoly(M,lambda);
361 Multivariate polynomials and rational functions may be expanded,
362 collected and normalized (i.e. converted to a ratio of two coprime
366 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
367 -3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
368 > b = x^2 + 4*x*y - y^2;
371 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
373 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)
378 You can differentiate functions and expand them as Taylor or Laurent
379 series in a very natural syntax (the second argument of @code{series} is
380 a relation defining the evaluation point, the third specifies the
383 @cindex Zeta function
387 > series(sin(x),x==0,4);
389 > series(1/tan(x),x==0,4);
390 x^(-1)-1/3*x+Order(x^2)
391 > series(tgamma(x),x==0,3);
392 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
393 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
395 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
396 -(0.90747907608088628905)*x^2+Order(x^3)
397 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
398 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
399 -Euler-1/12+Order((x-1/2*Pi)^3)
402 Here we have made use of the @command{ginsh}-command @code{"} to pop the
403 previously evaluated element from @command{ginsh}'s internal stack.
405 If you ever wanted to convert units in C or C++ and found this is
406 cumbersome, here is the solution. Symbolic types can always be used as
407 tags for different types of objects. Converting from wrong units to the
408 metric system is now easy:
416 140613.91592783185568*kg*m^(-2)
420 @node Installation, Prerequisites, What it can do for you, Top
421 @c node-name, next, previous, up
422 @chapter Installation
425 GiNaC's installation follows the spirit of most GNU software. It is
426 easily installed on your system by three steps: configuration, build,
430 * Prerequisites:: Packages upon which GiNaC depends.
431 * Configuration:: How to configure GiNaC.
432 * Building GiNaC:: How to compile GiNaC.
433 * Installing GiNaC:: How to install GiNaC on your system.
437 @node Prerequisites, Configuration, Installation, Installation
438 @c node-name, next, previous, up
439 @section Prerequisites
441 In order to install GiNaC on your system, some prerequisites need to be
442 met. First of all, you need to have a C++-compiler adhering to the
443 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
444 development so if you have a different compiler you are on your own.
445 For the configuration to succeed you need a Posix compliant shell
446 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
447 by the built process as well, since some of the source files are
448 automatically generated by Perl scripts. Last but not least, Bruno
449 Haible's library @acronym{CLN} is extensively used and needs to be
450 installed on your system. Please get it either from
451 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
452 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
453 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
454 site} (it is covered by GPL) and install it prior to trying to install
455 GiNaC. The configure script checks if it can find it and if it cannot
456 it will refuse to continue.
459 @node Configuration, Building GiNaC, Prerequisites, Installation
460 @c node-name, next, previous, up
461 @section Configuration
462 @cindex configuration
465 To configure GiNaC means to prepare the source distribution for
466 building. It is done via a shell script called @command{configure} that
467 is shipped with the sources and was originally generated by GNU
468 Autoconf. Since a configure script generated by GNU Autoconf never
469 prompts, all customization must be done either via command line
470 parameters or environment variables. It accepts a list of parameters,
471 the complete set of which can be listed by calling it with the
472 @option{--help} option. The most important ones will be shortly
473 described in what follows:
478 @option{--disable-shared}: When given, this option switches off the
479 build of a shared library, i.e. a @file{.so} file. This may be convenient
480 when developing because it considerably speeds up compilation.
483 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
484 and headers are installed. It defaults to @file{/usr/local} which means
485 that the library is installed in the directory @file{/usr/local/lib},
486 the header files in @file{/usr/local/include/ginac} and the documentation
487 (like this one) into @file{/usr/local/share/doc/GiNaC}.
490 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
491 the library installed in some other directory than
492 @file{@var{PREFIX}/lib/}.
495 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
496 to have the header files installed in some other directory than
497 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
498 @option{--includedir=/usr/include} you will end up with the header files
499 sitting in the directory @file{/usr/include/ginac/}. Note that the
500 subdirectory @file{ginac} is enforced by this process in order to
501 keep the header files separated from others. This avoids some
502 clashes and allows for an easier deinstallation of GiNaC. This ought
503 to be considered A Good Thing (tm).
506 @option{--datadir=@var{DATADIR}}: This option may be given in case you
507 want to have the documentation installed in some other directory than
508 @file{@var{PREFIX}/share/doc/GiNaC/}.
512 In addition, you may specify some environment variables.
513 @env{CXX} holds the path and the name of the C++ compiler
514 in case you want to override the default in your path. (The
515 @command{configure} script searches your path for @command{c++},
516 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
517 and @command{cc++} in that order.) It may be very useful to
518 define some compiler flags with the @env{CXXFLAGS} environment
519 variable, like optimization, debugging information and warning
520 levels. If omitted, it defaults to @option{-g -O2}.
522 The whole process is illustrated in the following two
523 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
524 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
527 Here is a simple configuration for a site-wide GiNaC library assuming
528 everything is in default paths:
531 $ export CXXFLAGS="-Wall -O2"
535 And here is a configuration for a private static GiNaC library with
536 several components sitting in custom places (site-wide @acronym{GCC} and
537 private @acronym{CLN}). The compiler is pursuaded to be picky and full
538 assertions and debugging information are switched on:
541 $ export CXX=/usr/local/gnu/bin/c++
542 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
543 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
544 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
545 $ ./configure --disable-shared --prefix=$(HOME)
549 @node Building GiNaC, Installing GiNaC, Configuration, Installation
550 @c node-name, next, previous, up
551 @section Building GiNaC
552 @cindex building GiNaC
554 After proper configuration you should just build the whole
559 at the command prompt and go for a cup of coffee. The exact time it
560 takes to compile GiNaC depends not only on the speed of your machines
561 but also on other parameters, for instance what value for @env{CXXFLAGS}
562 you entered. Optimization may be very time-consuming.
564 Just to make sure GiNaC works properly you may run a collection of
565 regression tests by typing
571 This will compile some sample programs, run them and check the output
572 for correctness. The regression tests fall in three categories. First,
573 the so called @emph{exams} are performed, simple tests where some
574 predefined input is evaluated (like a pupils' exam). Second, the
575 @emph{checks} test the coherence of results among each other with
576 possible random input. Third, some @emph{timings} are performed, which
577 benchmark some predefined problems with different sizes and display the
578 CPU time used in seconds. Each individual test should return a message
579 @samp{passed}. This is mostly intended to be a QA-check if something
580 was broken during development, not a sanity check of your system. Some
581 of the tests in sections @emph{checks} and @emph{timings} may require
582 insane amounts of memory and CPU time. Feel free to kill them if your
583 machine catches fire. Another quite important intent is to allow people
584 to fiddle around with optimization.
586 Generally, the top-level Makefile runs recursively to the
587 subdirectories. It is therfore safe to go into any subdirectory
588 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
589 @var{target} there in case something went wrong.
592 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
593 @c node-name, next, previous, up
594 @section Installing GiNaC
597 To install GiNaC on your system, simply type
603 As described in the section about configuration the files will be
604 installed in the following directories (the directories will be created
605 if they don't already exist):
610 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
611 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
612 So will @file{libginac.so} unless the configure script was
613 given the option @option{--disable-shared}. The proper symlinks
614 will be established as well.
617 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
618 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
621 All documentation (HTML and Postscript) will be stuffed into
622 @file{@var{PREFIX}/share/doc/GiNaC/} (or
623 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
627 For the sake of completeness we will list some other useful make
628 targets: @command{make clean} deletes all files generated by
629 @command{make}, i.e. all the object files. In addition @command{make
630 distclean} removes all files generated by the configuration and
631 @command{make maintainer-clean} goes one step further and deletes files
632 that may require special tools to rebuild (like the @command{libtool}
633 for instance). Finally @command{make uninstall} removes the installed
634 library, header files and documentation@footnote{Uninstallation does not
635 work after you have called @command{make distclean} since the
636 @file{Makefile} is itself generated by the configuration from
637 @file{Makefile.in} and hence deleted by @command{make distclean}. There
638 are two obvious ways out of this dilemma. First, you can run the
639 configuration again with the same @var{PREFIX} thus creating a
640 @file{Makefile} with a working @samp{uninstall} target. Second, you can
641 do it by hand since you now know where all the files went during
645 @node Basic Concepts, Expressions, Installing GiNaC, Top
646 @c node-name, next, previous, up
647 @chapter Basic Concepts
649 This chapter will describe the different fundamental objects that can be
650 handled by GiNaC. But before doing so, it is worthwhile introducing you
651 to the more commonly used class of expressions, representing a flexible
652 meta-class for storing all mathematical objects.
655 * Expressions:: The fundamental GiNaC class.
656 * The Class Hierarchy:: Overview of GiNaC's classes.
657 * Symbols:: Symbolic objects.
658 * Numbers:: Numerical objects.
659 * Constants:: Pre-defined constants.
660 * Fundamental containers:: The power, add and mul classes.
661 * Lists:: Lists of expressions.
662 * Mathematical functions:: Mathematical functions.
663 * Relations:: Equality, Inequality and all that.
667 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
668 @c node-name, next, previous, up
670 @cindex expression (class @code{ex})
673 The most common class of objects a user deals with is the expression
674 @code{ex}, representing a mathematical object like a variable, number,
675 function, sum, product, etc... Expressions may be put together to form
676 new expressions, passed as arguments to functions, and so on. Here is a
677 little collection of valid expressions:
680 ex MyEx1 = 5; // simple number
681 ex MyEx2 = x + 2*y; // polynomial in x and y
682 ex MyEx3 = (x + 1)/(x - 1); // rational expression
683 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
684 ex MyEx5 = MyEx4 + 1; // similar to above
687 Expressions are handles to other more fundamental objects, that often
688 contain other expressions thus creating a tree of expressions
689 (@xref{Internal Structures}, for particular examples). Most methods on
690 @code{ex} therefore run top-down through such an expression tree. For
691 example, the method @code{has()} scans recursively for occurrences of
692 something inside an expression. Thus, if you have declared @code{MyEx4}
693 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
694 the argument of @code{sin} and hence return @code{true}.
696 The next sections will outline the general picture of GiNaC's class
697 hierarchy and describe the classes of objects that are handled by
701 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
702 @c node-name, next, previous, up
703 @section The Class Hierarchy
705 GiNaC's class hierarchy consists of several classes representing
706 mathematical objects, all of which (except for @code{ex} and some
707 helpers) are internally derived from one abstract base class called
708 @code{basic}. You do not have to deal with objects of class
709 @code{basic}, instead you'll be dealing with symbols, numbers,
710 containers of expressions and so on.
714 To get an idea about what kinds of symbolic composits may be built we
715 have a look at the most important classes in the class hierarchy and
716 some of the relations among the classes:
718 @image{classhierarchy}
720 The abstract classes shown here (the ones without drop-shadow) are of no
721 interest for the user. They are used internally in order to avoid code
722 duplication if two or more classes derived from them share certain
723 features. An example is @code{expairseq}, a container for a sequence of
724 pairs each consisting of one expression and a number (@code{numeric}).
725 What @emph{is} visible to the user are the derived classes @code{add}
726 and @code{mul}, representing sums and products. @xref{Internal
727 Structures}, where these two classes are described in more detail. The
728 following table shortly summarizes what kinds of mathematical objects
729 are stored in the different classes:
732 @multitable @columnfractions .22 .78
733 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
734 @item @code{constant} @tab Constants like
741 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
742 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
743 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
744 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
749 @code{sqrt(}@math{2}@code{)}
752 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
753 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
754 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
755 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
756 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
757 @item @code{color}, @code{coloridx} @tab Element and index of the @math{SU(3)} Lie-algebra
758 @item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
759 @item @code{idx} @tab Index of a general tensor object
763 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
764 @c node-name, next, previous, up
766 @cindex @code{symbol} (class)
767 @cindex hierarchy of classes
770 Symbols are for symbolic manipulation what atoms are for chemistry. You
771 can declare objects of class @code{symbol} as any other object simply by
772 saying @code{symbol x,y;}. There is, however, a catch in here having to
773 do with the fact that C++ is a compiled language. The information about
774 the symbol's name is thrown away by the compiler but at a later stage
775 you may want to print expressions holding your symbols. In order to
776 avoid confusion GiNaC's symbols are able to know their own name. This
777 is accomplished by declaring its name for output at construction time in
778 the fashion @code{symbol x("x");}. If you declare a symbol using the
779 default constructor (i.e. without string argument) the system will deal
780 out a unique name. That name may not be suitable for printing but for
781 internal routines when no output is desired it is often enough. We'll
782 come across examples of such symbols later in this tutorial.
784 This implies that the strings passed to symbols at construction time may
785 not be used for comparing two of them. It is perfectly legitimate to
786 write @code{symbol x("x"),y("x");} but it is likely to lead into
787 trouble. Here, @code{x} and @code{y} are different symbols and
788 statements like @code{x-y} will not be simplified to zero although the
789 output @code{x-x} looks funny. Such output may also occur when there
790 are two different symbols in two scopes, for instance when you call a
791 function that declares a symbol with a name already existent in a symbol
792 in the calling function. Again, comparing them (using @code{operator==}
793 for instance) will always reveal their difference. Watch out, please.
795 @cindex @code{subs()}
796 Although symbols can be assigned expressions for internal reasons, you
797 should not do it (and we are not going to tell you how it is done). If
798 you want to replace a symbol with something else in an expression, you
799 can use the expression's @code{.subs()} method (@xref{Substituting Symbols},
800 for more information).
803 @node Numbers, Constants, Symbols, Basic Concepts
804 @c node-name, next, previous, up
806 @cindex @code{numeric} (class)
812 For storing numerical things, GiNaC uses Bruno Haible's library
813 @acronym{CLN}. The classes therein serve as foundation classes for
814 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
815 alternatively for Common Lisp Numbers. In order to find out more about
816 @acronym{CLN}'s internals the reader is refered to the documentation of
817 that library. @inforef{Introduction, , cln}, for more
818 information. Suffice to say that it is by itself build on top of another
819 library, the GNU Multiple Precision library @acronym{GMP}, which is an
820 extremely fast library for arbitrary long integers and rationals as well
821 as arbitrary precision floating point numbers. It is very commonly used
822 by several popular cryptographic applications. @acronym{CLN} extends
823 @acronym{GMP} by several useful things: First, it introduces the complex
824 number field over either reals (i.e. floating point numbers with
825 arbitrary precision) or rationals. Second, it automatically converts
826 rationals to integers if the denominator is unity and complex numbers to
827 real numbers if the imaginary part vanishes and also correctly treats
828 algebraic functions. Third it provides good implementations of
829 state-of-the-art algorithms for all trigonometric and hyperbolic
830 functions as well as for calculation of some useful constants.
832 The user can construct an object of class @code{numeric} in several
833 ways. The following example shows the four most important constructors.
834 It uses construction from C-integer, construction of fractions from two
835 integers, construction from C-float and construction from a string:
838 #include <ginac/ginac.h>
839 using namespace GiNaC;
843 numeric two(2); // exact integer 2
844 numeric r(2,3); // exact fraction 2/3
845 numeric e(2.71828); // floating point number
846 numeric p("3.1415926535897932385"); // floating point number
847 // Trott's constant in scientific notation:
848 numeric trott("1.0841015122311136151E-2");
850 cout << two*p << endl; // floating point 6.283...
854 Note that all those constructors are @emph{explicit} which means you are
855 not allowed to write @code{numeric two=2;}. This is because the basic
856 objects to be handled by GiNaC are the expressions @code{ex} and we want
857 to keep things simple and wish objects like @code{pow(x,2)} to be
858 handled the same way as @code{pow(x,a)}, which means that we need to
859 allow a general @code{ex} as base and exponent. Therefore there is an
860 implicit constructor from C-integers directly to expressions handling
861 numerics at work in most of our examples. This design really becomes
862 convenient when one declares own functions having more than one
863 parameter but it forbids using implicit constructors because that would
864 lead to compile-time ambiguities.
866 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
867 This would, however, call C's built-in operator @code{/} for integers
868 first and result in a numeric holding a plain integer 1. @strong{Never
869 use the operator @code{/} on integers} unless you know exactly what you
870 are doing! Use the constructor from two integers instead, as shown in
871 the example above. Writing @code{numeric(1)/2} may look funny but works
874 @cindex @code{Digits}
876 We have seen now the distinction between exact numbers and floating
877 point numbers. Clearly, the user should never have to worry about
878 dynamically created exact numbers, since their `exactness' always
879 determines how they ought to be handled, i.e. how `long' they are. The
880 situation is different for floating point numbers. Their accuracy is
881 controlled by one @emph{global} variable, called @code{Digits}. (For
882 those readers who know about Maple: it behaves very much like Maple's
883 @code{Digits}). All objects of class numeric that are constructed from
884 then on will be stored with a precision matching that number of decimal
888 #include <ginac/ginac.h>
889 using namespace GiNaC;
893 numeric three(3.0), one(1.0);
894 numeric x = one/three;
896 cout << "in " << Digits << " digits:" << endl;
898 cout << Pi.evalf() << endl;
910 The above example prints the following output to screen:
917 0.333333333333333333333333333333333333333333333333333333333333333333
918 3.14159265358979323846264338327950288419716939937510582097494459231
921 It should be clear that objects of class @code{numeric} should be used
922 for constructing numbers or for doing arithmetic with them. The objects
923 one deals with most of the time are the polymorphic expressions @code{ex}.
925 @subsection Tests on numbers
927 Once you have declared some numbers, assigned them to expressions and
928 done some arithmetic with them it is frequently desired to retrieve some
929 kind of information from them like asking whether that number is
930 integer, rational, real or complex. For those cases GiNaC provides
931 several useful methods. (Internally, they fall back to invocations of
932 certain CLN functions.)
934 As an example, let's construct some rational number, multiply it with
935 some multiple of its denominator and test what comes out:
938 #include <ginac/ginac.h>
939 using namespace GiNaC;
941 // some very important constants:
942 const numeric twentyone(21);
943 const numeric ten(10);
944 const numeric five(5);
948 numeric answer = twentyone;
951 cout << answer.is_integer() << endl; // false, it's 21/5
953 cout << answer.is_integer() << endl; // true, it's 42 now!
957 Note that the variable @code{answer} is constructed here as an integer
958 by @code{numeric}'s copy constructor but in an intermediate step it
959 holds a rational number represented as integer numerator and integer
960 denominator. When multiplied by 10, the denominator becomes unity and
961 the result is automatically converted to a pure integer again.
962 Internally, the underlying @acronym{CLN} is responsible for this
963 behaviour and we refer the reader to @acronym{CLN}'s documentation.
964 Suffice to say that the same behaviour applies to complex numbers as
965 well as return values of certain functions. Complex numbers are
966 automatically converted to real numbers if the imaginary part becomes
967 zero. The full set of tests that can be applied is listed in the
971 @multitable @columnfractions .30 .70
972 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
973 @item @code{.is_zero()}
974 @tab @dots{}equal to zero
975 @item @code{.is_positive()}
976 @tab @dots{}not complex and greater than 0
977 @item @code{.is_integer()}
978 @tab @dots{}a (non-complex) integer
979 @item @code{.is_pos_integer()}
980 @tab @dots{}an integer and greater than 0
981 @item @code{.is_nonneg_integer()}
982 @tab @dots{}an integer and greater equal 0
983 @item @code{.is_even()}
984 @tab @dots{}an even integer
985 @item @code{.is_odd()}
986 @tab @dots{}an odd integer
987 @item @code{.is_prime()}
988 @tab @dots{}a prime integer (probabilistic primality test)
989 @item @code{.is_rational()}
990 @tab @dots{}an exact rational number (integers are rational, too)
991 @item @code{.is_real()}
992 @tab @dots{}a real integer, rational or float (i.e. is not complex)
993 @item @code{.is_cinteger()}
994 @tab @dots{}a (complex) integer (such as @math{2-3*I})
995 @item @code{.is_crational()}
996 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1001 @node Constants, Fundamental containers, Numbers, Basic Concepts
1002 @c node-name, next, previous, up
1004 @cindex @code{constant} (class)
1007 @cindex @code{Catalan}
1008 @cindex @code{Euler}
1009 @cindex @code{evalf()}
1010 Constants behave pretty much like symbols except that they return some
1011 specific number when the method @code{.evalf()} is called.
1013 The predefined known constants are:
1016 @multitable @columnfractions .14 .30 .56
1017 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1019 @tab Archimedes' constant
1020 @tab 3.14159265358979323846264338327950288
1021 @item @code{Catalan}
1022 @tab Catalan's constant
1023 @tab 0.91596559417721901505460351493238411
1025 @tab Euler's (or Euler-Mascheroni) constant
1026 @tab 0.57721566490153286060651209008240243
1031 @node Fundamental containers, Lists, Constants, Basic Concepts
1032 @c node-name, next, previous, up
1033 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1037 @cindex @code{power}
1039 Simple polynomial expressions are written down in GiNaC pretty much like
1040 in other CAS or like expressions involving numerical variables in C.
1041 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1042 been overloaded to achieve this goal. When you run the following
1043 program, the constructor for an object of type @code{mul} is
1044 automatically called to hold the product of @code{a} and @code{b} and
1045 then the constructor for an object of type @code{add} is called to hold
1046 the sum of that @code{mul} object and the number one:
1049 #include <ginac/ginac.h>
1050 using namespace GiNaC;
1054 symbol a("a"), b("b");
1060 @cindex @code{pow()}
1061 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1062 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1063 construction is necessary since we cannot safely overload the constructor
1064 @code{^} in C++ to construct a @code{power} object. If we did, it would
1065 have several counterintuitive effects:
1069 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1071 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1072 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1073 interpret this as @code{x^(a^b)}.
1075 Also, expressions involving integer exponents are very frequently used,
1076 which makes it even more dangerous to overload @code{^} since it is then
1077 hard to distinguish between the semantics as exponentiation and the one
1078 for exclusive or. (It would be embarassing to return @code{1} where one
1079 has requested @code{2^3}.)
1082 @cindex @command{ginsh}
1083 All effects are contrary to mathematical notation and differ from the
1084 way most other CAS handle exponentiation, therefore overloading @code{^}
1085 is ruled out for GiNaC's C++ part. The situation is different in
1086 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1087 that the other frequently used exponentiation operator @code{**} does
1088 not exist at all in C++).
1090 To be somewhat more precise, objects of the three classes described
1091 here, are all containers for other expressions. An object of class
1092 @code{power} is best viewed as a container with two slots, one for the
1093 basis, one for the exponent. All valid GiNaC expressions can be
1094 inserted. However, basic transformations like simplifying
1095 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1096 when this is mathematically possible. If we replace the outer exponent
1097 three in the example by some symbols @code{a}, the simplification is not
1098 safe and will not be performed, since @code{a} might be @code{1/2} and
1101 Objects of type @code{add} and @code{mul} are containers with an
1102 arbitrary number of slots for expressions to be inserted. Again, simple
1103 and safe simplifications are carried out like transforming
1104 @code{3*x+4-x} to @code{2*x+4}.
1106 The general rule is that when you construct such objects, GiNaC
1107 automatically creates them in canonical form, which might differ from
1108 the form you typed in your program. This allows for rapid comparison of
1109 expressions, since after all @code{a-a} is simply zero. Note, that the
1110 canonical form is not necessarily lexicographical ordering or in any way
1111 easily guessable. It is only guaranteed that constructing the same
1112 expression twice, either implicitly or explicitly, results in the same
1116 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1117 @c node-name, next, previous, up
1118 @section Lists of expressions
1119 @cindex @code{lst} (class)
1121 @cindex @code{nops()}
1123 @cindex @code{append()}
1124 @cindex @code{prepend()}
1126 The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
1127 These are sometimes used to supply a variable number of arguments of the same
1128 type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
1129 should have a basic understanding about them.
1131 Lists of up to 15 expressions can be directly constructed from single
1136 symbol x("x"), y("y");
1137 lst l(x, 2, y, x+y);
1138 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1142 Use the @code{nops()} method to determine the size (number of expressions) of
1143 a list and the @code{op()} method to access individual elements:
1147 cout << l.nops() << endl; // prints '4'
1148 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1152 Finally you can append or prepend an expression to a list with the
1153 @code{append()} and @code{prepend()} methods:
1157 l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
1158 l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
1163 @node Mathematical functions, Relations, Lists, Basic Concepts
1164 @c node-name, next, previous, up
1165 @section Mathematical functions
1166 @cindex @code{function} (class)
1167 @cindex trigonometric function
1168 @cindex hyperbolic function
1170 There are quite a number of useful functions hard-wired into GiNaC. For
1171 instance, all trigonometric and hyperbolic functions are implemented
1172 (@xref{Built-in Functions}, for a complete list).
1174 These functions are all objects of class @code{function}. They accept one
1175 or more expressions as arguments and return one expression. If the arguments
1176 are not numerical, the evaluation of the function may be halted, as it
1177 does in the next example:
1179 @cindex Gamma function
1180 @cindex @code{subs()}
1182 #include <ginac/ginac.h>
1183 using namespace GiNaC;
1187 symbol x("x"), y("y");
1190 cout << "tgamma(" << foo << ") -> " << tgamma(foo) << endl;
1191 ex bar = foo.subs(y==1);
1192 cout << "tgamma(" << bar << ") -> " << tgamma(bar) << endl;
1193 ex foobar = bar.subs(x==7);
1194 cout << "tgamma(" << foobar << ") -> " << tgamma(foobar) << endl;
1198 This program shows how the function returns itself twice and finally an
1199 expression that may be really useful:
1202 tgamma(x+(1/2)*y) -> tgamma(x+(1/2)*y)
1203 tgamma(x+1/2) -> tgamma(x+1/2)
1204 tgamma(15/2) -> (135135/128)*Pi^(1/2)
1207 Besides evaluation most of these functions allow differentiation, series
1208 expansion and so on. Read the next chapter in order to learn more about
1212 @node Relations, Methods and Functions, Mathematical functions, Basic Concepts
1213 @c node-name, next, previous, up
1215 @cindex @code{relational} (class)
1217 Sometimes, a relation holding between two expressions must be stored
1218 somehow. The class @code{relational} is a convenient container for such
1219 purposes. A relation is by definition a container for two @code{ex} and
1220 a relation between them that signals equality, inequality and so on.
1221 They are created by simply using the C++ operators @code{==}, @code{!=},
1222 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1224 @xref{Mathematical functions}, for examples where various applications
1225 of the @code{.subs()} method show how objects of class relational are
1226 used as arguments. There they provide an intuitive syntax for
1227 substitutions. They are also used as arguments to the @code{ex::series}
1228 method, where the left hand side of the relation specifies the variable
1229 to expand in and the right hand side the expansion point. They can also
1230 be used for creating systems of equations that are to be solved for
1231 unknown variables. But the most common usage of objects of this class
1232 is rather inconspicuous in statements of the form @code{if
1233 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1234 conversion from @code{relational} to @code{bool} takes place. Note,
1235 however, that @code{==} here does not perform any simplifications, hence
1236 @code{expand()} must be called explicitly.
1239 @node Methods and Functions, Information About Expressions, Relations, Top
1240 @c node-name, next, previous, up
1241 @chapter Methods and Functions
1244 In this chapter the most important algorithms provided by GiNaC will be
1245 described. Some of them are implemented as functions on expressions,
1246 others are implemented as methods provided by expression objects. If
1247 they are methods, there exists a wrapper function around it, so you can
1248 alternatively call it in a functional way as shown in the simple
1252 #include <ginac/ginac.h>
1253 using namespace GiNaC;
1257 ex x = numeric(1.0);
1259 cout << "As method: " << sin(x).evalf() << endl;
1260 cout << "As function: " << evalf(sin(x)) << endl;
1264 @cindex @code{subs()}
1265 The general rule is that wherever methods accept one or more parameters
1266 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1267 wrapper accepts is the same but preceded by the object to act on
1268 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1269 most natural one in an OO model but it may lead to confusion for MapleV
1270 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1271 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1272 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1273 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1274 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1275 here. Also, users of MuPAD will in most cases feel more comfortable
1276 with GiNaC's convention. All function wrappers are implemented
1277 as simple inline functions which just call the corresponding method and
1278 are only provided for users uncomfortable with OO who are dead set to
1279 avoid method invocations. Generally, nested function wrappers are much
1280 harder to read than a sequence of methods and should therefore be
1281 avoided if possible. On the other hand, not everything in GiNaC is a
1282 method on class @code{ex} and sometimes calling a function cannot be
1286 * Information About Expressions::
1287 * Substituting Symbols::
1288 * Polynomial Arithmetic:: Working with polynomials.
1289 * Rational Expressions:: Working with rational functions.
1290 * Symbolic Differentiation::
1291 * Series Expansion:: Taylor and Laurent expansion.
1292 * Built-in Functions:: List of predefined mathematical functions.
1293 * Input/Output:: Input and output of expressions.
1297 @node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions
1298 @c node-name, next, previous, up
1299 @section Getting information about expressions
1301 @subsection Checking expression types
1302 @cindex @code{is_ex_of_type()}
1303 @cindex @code{info()}
1305 Sometimes it's useful to check whether a given expression is a plain number,
1306 a sum, a polynomial with integer coefficients, or of some other specific type.
1307 GiNaC provides two functions for this (the first one is actually a macro):
1310 bool is_ex_of_type(const ex & e, TYPENAME t);
1311 bool ex::info(unsigned flag);
1314 @code{is_ex_of_type()} allows you to check whether the top-level object of
1315 an expression @samp{e} is an instance of the GiNaC class @samp{t}
1316 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
1317 e.g., for checking whether an expression is a number, a sum, or a product:
1324 is_ex_of_type(e1, numeric); // true
1325 is_ex_of_type(e2, numeric); // false
1326 is_ex_of_type(e1, add); // false
1327 is_ex_of_type(e2, add); // true
1328 is_ex_of_type(e1, mul); // false
1329 is_ex_of_type(e2, mul); // false
1333 The @code{info()} method is used for checking certain attributes of
1334 expressions. The possible values for the @code{flag} argument are defined
1335 in @file{ginac/flags.h}, the most important being explained in the following
1339 @multitable @columnfractions .30 .70
1340 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
1341 @item @code{numeric}
1342 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
1344 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1345 @item @code{rational}
1346 @tab @dots{}an exact rational number (integers are rational, too)
1347 @item @code{integer}
1348 @tab @dots{}a (non-complex) integer
1349 @item @code{crational}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1351 @item @code{cinteger}
1352 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1353 @item @code{positive}
1354 @tab @dots{}not complex and greater than 0
1355 @item @code{negative}
1356 @tab @dots{}not complex and less than 0
1357 @item @code{nonnegative}
1358 @tab @dots{}not complex and greater than or equal to 0
1360 @tab @dots{}an integer greater than 0
1362 @tab @dots{}an integer less than 0
1363 @item @code{nonnegint}
1364 @tab @dots{}an integer greater than or equal to 0
1366 @tab @dots{}an even integer
1368 @tab @dots{}an odd integer
1370 @tab @dots{}a prime integer (probabilistic primality test)
1371 @item @code{relation}
1372 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
1373 @item @code{relation_equal}
1374 @tab @dots{}a @code{==} relation
1375 @item @code{relation_not_equal}
1376 @tab @dots{}a @code{!=} relation
1377 @item @code{relation_less}
1378 @tab @dots{}a @code{<} relation
1379 @item @code{relation_less_or_equal}
1380 @tab @dots{}a @code{<=} relation
1381 @item @code{relation_greater}
1382 @tab @dots{}a @code{>} relation
1383 @item @code{relation_greater_or_equal}
1384 @tab @dots{}a @code{>=} relation
1386 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
1388 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
1389 @item @code{polynomial}
1390 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
1391 @item @code{integer_polynomial}
1392 @tab @dots{}a polynomial with (non-complex) integer coefficients
1393 @item @code{cinteger_polynomial}
1394 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
1395 @item @code{rational_polynomial}
1396 @tab @dots{}a polynomial with (non-complex) rational coefficients
1397 @item @code{crational_polynomial}
1398 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
1399 @item @code{rational_function}
1400 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
1401 @item @code{algebraic}
1402 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
1407 @subsection Accessing subexpressions
1408 @cindex @code{nops()}
1410 @cindex @code{has()}
1412 @cindex @code{relational} (class)
1414 GiNaC provides the two methods
1417 unsigned ex::nops();
1418 ex ex::op(unsigned i);
1421 for accessing the subexpressions in the container-like GiNaC classes like
1422 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
1423 determines the number of subexpressions (@samp{operands}) contained, while
1424 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
1425 In the case of a @code{power} object, @code{op(0)} will return the basis
1426 and @code{op(1)} the exponent.
1428 The left-hand and right-hand side expressions of objects of class
1429 @code{relational} (and only of these) can also be accessed with the methods
1439 bool ex::has(const ex & other);
1442 checks whether an expression contains the given subexpression @code{other}.
1443 This only works reliably if @code{other} is of an atomic class such as a
1444 @code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
1445 @code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
1448 @subsection Comparing expressions
1449 @cindex @code{is_equal()}
1450 @cindex @code{is_zero()}
1452 Expressions can be compared with the usual C++ relational operators like
1453 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
1454 the result is usually not determinable and the result will be @code{false},
1455 except in the case of the @code{!=} operator. You should also be aware that
1456 GiNaC will only do the most trivial test for equality (subtracting both
1457 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
1460 Actually, if you construct an expression like @code{a == b}, this will be
1461 represented by an object of the @code{relational} class (@xref{Relations}.)
1462 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
1464 There are also two methods
1467 bool ex::is_equal(const ex & other);
1471 for checking whether one expression is equal to another, or equal to zero,
1474 @strong{Warning:} You will also find an @code{ex::compare()} method in the
1475 GiNaC header files. This method is however only to be used internally by
1476 GiNaC to establish a canonical sort order for terms, and using it to compare
1477 expressions will give very surprising results.
1480 @node Substituting Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions
1481 @c node-name, next, previous, up
1482 @section Substituting symbols
1483 @cindex @code{subs()}
1485 Symbols can be replaced with expressions via the @code{.subs()} method:
1488 ex ex::subs(const ex & e);
1489 ex ex::subs(const lst & syms, const lst & repls);
1492 In the first form, @code{subs()} accepts a relational of the form
1493 @samp{symbol == expression} or a @code{lst} of such relationals. E.g.
1497 symbol x("x"), y("y");
1498 ex e1 = 2*x^2-4*x+3;
1499 cout << "e1(7) = " << e1.subs(x == 7) << endl;
1501 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
1505 will print @samp{73} and @samp{-10}, respectively.
1507 If you specify multiple substitutions, they are performed in parallel, so e.g.
1508 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
1510 The second form of @code{subs()} takes two lists, one for the symbols and
1511 one for the expressions to be substituted (both lists must contain the same
1512 number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))}
1513 to exchange @samp{x} and @samp{y}.
1516 @node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, Methods and Functions
1517 @c node-name, next, previous, up
1518 @section Polynomial arithmetic
1520 @subsection Expanding and collecting
1521 @cindex @code{expand()}
1522 @cindex @code{collect()}
1524 A polynomial in one or more variables has many equivalent
1525 representations. Some useful ones serve a specific purpose. Consider
1526 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1527 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1528 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1529 representations are the recursive ones where one collects for exponents
1530 in one of the three variable. Since the factors are themselves
1531 polynomials in the remaining two variables the procedure can be
1532 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1533 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1536 To bring an expression into expanded form, its method
1542 may be called. In our example above, this corresponds to @math{4*x*y +
1543 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1544 GiNaC is not easily guessable you should be prepared to see different
1545 orderings of terms in such sums!
1547 Another useful representation of multivariate polynomials is as a
1548 univariate polynomial in one of the variables with the coefficients
1549 being polynomials in the remaining variables. The method
1550 @code{collect()} accomplishes this task:
1553 ex ex::collect(const symbol & s);
1556 Note that the original polynomial needs to be in expanded form in order
1557 to be able to find the coefficients properly.
1559 @subsection Degree and coefficients
1560 @cindex @code{degree()}
1561 @cindex @code{ldegree()}
1562 @cindex @code{coeff()}
1564 The degree and low degree of a polynomial can be obtained using the two
1568 int ex::degree(const symbol & s);
1569 int ex::ldegree(const symbol & s);
1572 which also work reliably on non-expanded input polynomials (they even work
1573 on rational functions, returning the asymptotic degree). To extract
1574 a coefficient with a certain power from an expanded polynomial you use
1577 ex ex::coeff(const symbol & s, int n);
1580 You can also obtain the leading and trailing coefficients with the methods
1583 ex ex::lcoeff(const symbol & s);
1584 ex ex::tcoeff(const symbol & s);
1587 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
1590 An application is illustrated in the next example, where a multivariate
1591 polynomial is analyzed:
1594 #include <ginac/ginac.h>
1595 using namespace GiNaC;
1599 symbol x("x"), y("y");
1600 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1601 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1602 ex Poly = PolyInp.expand();
1604 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1605 cout << "The x^" << i << "-coefficient is "
1606 << Poly.coeff(x,i) << endl;
1608 cout << "As polynomial in y: "
1609 << Poly.collect(y) << endl;
1613 When run, it returns an output in the following fashion:
1616 The x^0-coefficient is y^2+11*y
1617 The x^1-coefficient is 5*y^2-2*y
1618 The x^2-coefficient is -1
1619 The x^3-coefficient is 4*y
1620 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1623 As always, the exact output may vary between different versions of GiNaC
1624 or even from run to run since the internal canonical ordering is not
1625 within the user's sphere of influence.
1628 @subsection Polynomial division
1629 @cindex polynomial division
1632 @cindex pseudo-remainder
1633 @cindex @code{quo()}
1634 @cindex @code{rem()}
1635 @cindex @code{prem()}
1636 @cindex @code{divide()}
1641 ex quo(const ex & a, const ex & b, const symbol & x);
1642 ex rem(const ex & a, const ex & b, const symbol & x);
1645 compute the quotient and remainder of univariate polynomials in the variable
1646 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
1648 The additional function
1651 ex prem(const ex & a, const ex & b, const symbol & x);
1654 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
1655 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
1657 Exact division of multivariate polynomials is performed by the function
1660 bool divide(const ex & a, const ex & b, ex & q);
1663 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
1664 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
1665 in which case the value of @code{q} is undefined.
1668 @subsection Unit, content and primitive part
1669 @cindex @code{unit()}
1670 @cindex @code{content()}
1671 @cindex @code{primpart()}
1676 ex ex::unit(const symbol & x);
1677 ex ex::content(const symbol & x);
1678 ex ex::primpart(const symbol & x);
1681 return the unit part, content part, and primitive polynomial of a multivariate
1682 polynomial with respect to the variable @samp{x} (the unit part being the sign
1683 of the leading coefficient, the content part being the GCD of the coefficients,
1684 and the primitive polynomial being the input polynomial divided by the unit and
1685 content parts). The product of unit, content, and primitive part is the
1686 original polynomial.
1689 @subsection GCD and LCM
1692 @cindex @code{gcd()}
1693 @cindex @code{lcm()}
1695 The functions for polynomial greatest common divisor and least common
1696 multiple have the synopsis
1699 ex gcd(const ex & a, const ex & b);
1700 ex lcm(const ex & a, const ex & b);
1703 The functions @code{gcd()} and @code{lcm()} accept two expressions
1704 @code{a} and @code{b} as arguments and return a new expression, their
1705 greatest common divisor or least common multiple, respectively. If the
1706 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1707 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1710 #include <ginac/ginac.h>
1711 using namespace GiNaC;
1715 symbol x("x"), y("y"), z("z");
1716 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1717 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1719 ex P_gcd = gcd(P_a, P_b);
1721 ex P_lcm = lcm(P_a, P_b);
1722 // 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
1727 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
1728 @c node-name, next, previous, up
1729 @section Rational expressions
1731 @subsection The @code{normal} method
1732 @cindex @code{normal()}
1733 @cindex simplification
1734 @cindex temporary replacement
1736 Some basic from of simplification of expressions is called for frequently.
1737 GiNaC provides the method @code{.normal()}, which converts a rational function
1738 into an equivalent rational function of the form @samp{numerator/denominator}
1739 where numerator and denominator are coprime. If the input expression is already
1740 a fraction, it just finds the GCD of numerator and denominator and cancels it,
1741 otherwise it performs fraction addition and multiplication.
1743 @code{.normal()} can also be used on expressions which are not rational functions
1744 as it will replace all non-rational objects (like functions or non-integer
1745 powers) by temporary symbols to bring the expression to the domain of rational
1746 functions before performing the normalization, and re-substituting these
1747 symbols afterwards. This algorithm is also available as a separate method
1748 @code{.to_rational()}, described below.
1750 This means that both expressions @code{t1} and @code{t2} are indeed
1751 simplified in this little program:
1754 #include <ginac/ginac.h>
1755 using namespace GiNaC;
1760 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1761 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1762 cout << "t1 is " << t1.normal() << endl;
1763 cout << "t2 is " << t2.normal() << endl;
1767 Of course this works for multivariate polynomials too, so the ratio of
1768 the sample-polynomials from the section about GCD and LCM above would be
1769 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1772 @subsection Numerator and denominator
1775 @cindex @code{numer()}
1776 @cindex @code{denom()}
1778 The numerator and denominator of an expression can be obtained with
1785 These functions will first normalize the expression as described above and
1786 then return the numerator or denominator, respectively.
1789 @subsection Converting to a rational expression
1790 @cindex @code{to_rational()}
1792 Some of the methods described so far only work on polynomials or rational
1793 functions. GiNaC provides a way to extend the domain of these functions to
1794 general expressions by using the temporary replacement algorithm described
1795 above. You do this by calling
1798 ex ex::to_rational(lst &l);
1801 on the expression to be converted. The supplied @code{lst} will be filled
1802 with the generated temporary symbols and their replacement expressions in
1803 a format that can be used directly for the @code{subs()} method. It can also
1804 already contain a list of replacements from an earlier application of
1805 @code{.to_rational()}, so it's possible to use it on multiple expressions
1806 and get consistent results.
1813 ex a = pow(sin(x), 2) - pow(cos(x), 2);
1814 ex b = sin(x) + cos(x);
1817 divide(a.to_rational(l), b.to_rational(l), q);
1818 cout << q.subs(l) << endl;
1822 will print @samp{sin(x)-cos(x)}.
1825 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
1826 @c node-name, next, previous, up
1827 @section Symbolic differentiation
1828 @cindex differentiation
1829 @cindex @code{diff()}
1831 @cindex product rule
1833 GiNaC's objects know how to differentiate themselves. Thus, a
1834 polynomial (class @code{add}) knows that its derivative is the sum of
1835 the derivatives of all the monomials:
1838 #include <ginac/ginac.h>
1839 using namespace GiNaC;
1843 symbol x("x"), y("y"), z("z");
1844 ex P = pow(x, 5) + pow(x, 2) + y;
1846 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1847 cout << P.diff(y) << endl; // 1
1848 cout << P.diff(z) << endl; // 0
1852 If a second integer parameter @var{n} is given, the @code{diff} method
1853 returns the @var{n}th derivative.
1855 If @emph{every} object and every function is told what its derivative
1856 is, all derivatives of composed objects can be calculated using the
1857 chain rule and the product rule. Consider, for instance the expression
1858 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1859 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1860 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1861 out that the composition is the generating function for Euler Numbers,
1862 i.e. the so called @var{n}th Euler number is the coefficient of
1863 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1864 identity to code a function that generates Euler numbers in just three
1867 @cindex Euler numbers
1869 #include <ginac/ginac.h>
1870 using namespace GiNaC;
1872 ex EulerNumber(unsigned n)
1875 const ex generator = pow(cosh(x),-1);
1876 return generator.diff(x,n).subs(x==0);
1881 for (unsigned i=0; i<11; i+=2)
1882 cout << EulerNumber(i) << endl;
1887 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1888 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1889 @code{i} by two since all odd Euler numbers vanish anyways.
1892 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
1893 @c node-name, next, previous, up
1894 @section Series expansion
1895 @cindex @code{series()}
1896 @cindex Taylor expansion
1897 @cindex Laurent expansion
1898 @cindex @code{pseries} (class)
1900 Expressions know how to expand themselves as a Taylor series or (more
1901 generally) a Laurent series. As in most conventional Computer Algebra
1902 Systems, no distinction is made between those two. There is a class of
1903 its own for storing such series (@code{class pseries}) and a built-in
1904 function (called @code{Order}) for storing the order term of the series.
1905 As a consequence, if you want to work with series, i.e. multiply two
1906 series, you need to call the method @code{ex::series} again to convert
1907 it to a series object with the usual structure (expansion plus order
1908 term). A sample application from special relativity could read:
1911 #include <ginac/ginac.h>
1912 using namespace GiNaC;
1916 symbol v("v"), c("c");
1918 ex gamma = 1/sqrt(1 - pow(v/c,2));
1919 ex mass_nonrel = gamma.series(v==0, 10);
1921 cout << "the relativistic mass increase with v is " << endl
1922 << mass_nonrel << endl;
1924 cout << "the inverse square of this series is " << endl
1925 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
1929 Only calling the series method makes the last output simplify to
1930 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
1931 series raised to the power @math{-2}.
1933 @cindex M@'echain's formula
1934 As another instructive application, let us calculate the numerical
1935 value of Archimedes' constant
1939 (for which there already exists the built-in constant @code{Pi})
1940 using M@'echain's amazing formula
1942 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1945 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1947 We may expand the arcus tangent around @code{0} and insert the fractions
1948 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1949 carries an order term with it and the question arises what the system is
1950 supposed to do when the fractions are plugged into that order term. The
1951 solution is to use the function @code{series_to_poly()} to simply strip
1955 #include <ginac/ginac.h>
1956 using namespace GiNaC;
1958 ex mechain_pi(int degr)
1961 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
1962 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1963 -4*pi_expansion.subs(x==numeric(1,239));
1970 for (int i=2; i<12; i+=2) @{
1971 pi_frac = mechain_pi(i);
1972 cout << i << ":\t" << pi_frac << endl
1973 << "\t" << pi_frac.evalf() << endl;
1979 Note how we just called @code{.series(x,degr)} instead of
1980 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
1981 method @code{series()}: if the first argument is a symbol the expression
1982 is expanded in that symbol around point @code{0}. When you run this
1983 program, it will type out:
1987 3.1832635983263598326
1988 4: 5359397032/1706489875
1989 3.1405970293260603143
1990 6: 38279241713339684/12184551018734375
1991 3.141621029325034425
1992 8: 76528487109180192540976/24359780855939418203125
1993 3.141591772182177295
1994 10: 327853873402258685803048818236/104359128170408663038552734375
1995 3.1415926824043995174
1999 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
2000 @c node-name, next, previous, up
2001 @section Predefined mathematical functions
2003 GiNaC contains the following predefined mathematical functions:
2006 @multitable @columnfractions .30 .70
2007 @item @strong{Name} @tab @strong{Function}
2010 @item @code{csgn(x)}
2012 @item @code{sqrt(x)}
2013 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
2020 @item @code{asin(x)}
2022 @item @code{acos(x)}
2024 @item @code{atan(x)}
2025 @tab inverse tangent
2026 @item @code{atan2(y, x)}
2027 @tab inverse tangent with two arguments
2028 @item @code{sinh(x)}
2029 @tab hyperbolic sine
2030 @item @code{cosh(x)}
2031 @tab hyperbolic cosine
2032 @item @code{tanh(x)}
2033 @tab hyperbolic tangent
2034 @item @code{asinh(x)}
2035 @tab inverse hyperbolic sine
2036 @item @code{acosh(x)}
2037 @tab inverse hyperbolic cosine
2038 @item @code{atanh(x)}
2039 @tab inverse hyperbolic tangent
2041 @tab exponential function
2043 @tab natural logarithm
2044 @item @code{zeta(x)}
2045 @tab Riemann's zeta function
2046 @item @code{zeta(n, x)}
2047 @tab derivatives of Riemann's zeta function
2048 @item @code{tgamma(x)}
2050 @item @code{lgamma(x)}
2051 @tab logarithm of Gamma function
2052 @item @code{beta(x, y)}
2053 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
2055 @tab psi (digamma) function
2056 @item @code{psi(n, x)}
2057 @tab derivatives of psi function (polygamma functions)
2058 @item @code{factorial(n)}
2059 @tab factorial function
2060 @item @code{binomial(n, m)}
2061 @tab binomial coefficients
2062 @item @code{Order(x)}
2063 @tab order term function in truncated power series
2064 @item @code{Derivative(x, l)}
2065 @tab inert partial differentiation operator (used internally)
2070 For functions that have a branch cut in the complex plane GiNaC follows
2071 the conventions for C++ as defined in the ANSI standard. In particular:
2072 the natural logarithm (@code{log}) and the square root (@code{sqrt})
2073 both have their branch cuts running along the negative real axis where
2074 the points on the axis itself belong to the upper part.
2077 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
2078 @c node-name, next, previous, up
2079 @section Input and output of expressions
2082 @subsection Expression output
2084 @cindex output of expressions
2086 The easiest way to print an expression is to write it to a stream:
2091 ex e = 4.5+pow(x,2)*3/2;
2092 cout << e << endl; // prints '4.5+3/2*x^2'
2096 The output format is identical to the @command{ginsh} input syntax and
2097 to that used by most computer algebra systems, but not directly pastable
2098 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
2099 is printed as @samp{x^2}).
2101 To print an expression in a way that can be directly used in a C or C++
2102 program, you use the method
2105 void ex::printcsrc(ostream & os, unsigned type, const char *name);
2108 This outputs a line in the form of a variable definition @code{<type> <name> = <expression>}.
2109 The possible types are defined in @file{ginac/flags.h} (@code{csrc_types})
2110 and mostly affect the way in which floating point numbers are written:
2114 e.printcsrc(cout, csrc_types::ctype_float, "f");
2115 e.printcsrc(cout, csrc_types::ctype_double, "d");
2116 e.printcsrc(cout, csrc_types::ctype_cl_N, "n");
2120 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
2123 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2124 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2125 cl_N n = (cl_F("3.0")/cl_F("2.0"))*(x*x)+cl_F("4.5");
2128 Finally, there are the two methods @code{printraw()} and @code{printtree()} intended for GiNaC
2129 developers, that provide a dump of the internal structure of an expression for
2134 e.printraw(cout); cout << endl << endl;
2142 ex(+((power(ex(symbol(name=x,serial=1,hash=150875740,flags=11)),ex(numeric(2)),hash=2,flags=3),numeric(3/2)),,hash=0,flags=3))
2144 type=Q25GiNaC3add, hash=0 (0x0), flags=3, nops=2
2145 power: hash=2 (0x2), flags=3
2146 x (symbol): serial=1, hash=150875740 (0x8fe2e5c), flags=11
2147 2 (numeric): hash=2147483714 (0x80000042), flags=11
2148 3/2 (numeric): hash=2147483745 (0x80000061), flags=11
2151 4.5L0 (numeric): hash=2147483723 (0x8000004b), flags=11
2155 The @code{printtree()} method is also available in @command{ginsh} as the
2156 @code{print()} function.
2159 @subsection Expression input
2160 @cindex input of expressions
2162 GiNaC provides no way to directly read an expression from a stream because
2163 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
2164 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
2165 @code{y} you defined in your program and there is no way to specify the
2166 desired symbols to the @code{>>} stream input operator.
2168 Instead, GiNaC lets you construct an expression from a string, specifying the
2169 list of symbols to be used:
2173 symbol x("x"), y("y");
2174 ex e("2*x+sin(y)", lst(x, y));
2178 The input syntax is the same as that used by @command{ginsh} and the stream
2179 output operator @code{<<}. The symbols in the string are matched by name to
2180 the symbols in the list and if GiNaC encounters a symbol not specified in
2181 the list it will throw an exception.
2183 With this constructor, it's also easy to implement interactive GiNaC programs:
2188 #include <stdexcept>
2189 #include <ginac/ginac.h>
2190 using namespace GiNaC;
2197 cout << "Enter an expression containing 'x': ";
2202 cout << "The derivative of " << e << " with respect to x is ";
2203 cout << e.diff(x) << ".\n";
2204 @} catch (exception &p) @{
2205 cerr << p.what() << endl;
2211 @subsection Archiving
2212 @cindex @code{archive} (class)
2215 GiNaC allows creating @dfn{archives} of expressions which can be stored
2216 to or retrieved from files. To create an archive, you declare an object
2217 of class @code{archive} and archive expressions in it, giving each
2218 expression a unique name:
2221 #include <ginac/ginac.h>
2223 using namespace GiNaC;
2227 symbol x("x"), y("y"), z("z");
2229 ex foo = sin(x + 2*y) + 3*z + 41;
2233 a.archive_ex(foo, "foo");
2234 a.archive_ex(bar, "the second one");
2238 The archive can then be written to a file:
2242 ofstream out("foobar.gar");
2248 The file @file{foobar.gar} contains all information that is needed to
2249 reconstruct the expressions @code{foo} and @code{bar}.
2251 @cindex @command{viewgar}
2252 The tool @command{viewgar} that comes with GiNaC can be used to view
2253 the contents of GiNaC archive files:
2256 $ viewgar foobar.gar
2257 foo = 41+sin(x+2*y)+3*z
2258 the second one = 42+sin(x+2*y)+3*z
2261 The point of writing archive files is of course that they can later be
2267 ifstream in("foobar.gar");
2272 And the stored expressions can be retrieved by their name:
2278 ex ex1 = a2.unarchive_ex(syms, "foo");
2279 ex ex2 = a2.unarchive_ex(syms, "the second one");
2281 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
2282 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
2283 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
2287 Note that you have to supply a list of the symbols which are to be inserted
2288 in the expressions. Symbols in archives are stored by their name only and
2289 if you don't specify which symbols you have, unarchiving the expression will
2290 create new symbols with that name. E.g. if you hadn't included @code{x} in
2291 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
2292 have had no effect because the @code{x} in @code{ex1} would have been a
2293 different symbol than the @code{x} which was defined at the beginning of
2294 the program, altough both would appear as @samp{x} when printed.
2298 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
2299 @c node-name, next, previous, up
2300 @chapter Extending GiNaC
2302 By reading so far you should have gotten a fairly good understanding of
2303 GiNaC's design-patterns. From here on you should start reading the
2304 sources. All we can do now is issue some recommendations how to tackle
2305 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
2306 develop some useful extension please don't hesitate to contact the GiNaC
2307 authors---they will happily incorporate them into future versions.
2310 * What does not belong into GiNaC:: What to avoid.
2311 * Symbolic functions:: Implementing symbolic functions.
2315 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
2316 @c node-name, next, previous, up
2317 @section What doesn't belong into GiNaC
2319 @cindex @command{ginsh}
2320 First of all, GiNaC's name must be read literally. It is designed to be
2321 a library for use within C++. The tiny @command{ginsh} accompanying
2322 GiNaC makes this even more clear: it doesn't even attempt to provide a
2323 language. There are no loops or conditional expressions in
2324 @command{ginsh}, it is merely a window into the library for the
2325 programmer to test stuff (or to show off). Still, the design of a
2326 complete CAS with a language of its own, graphical capabilites and all
2327 this on top of GiNaC is possible and is without doubt a nice project for
2330 There are many built-in functions in GiNaC that do not know how to
2331 evaluate themselves numerically to a precision declared at runtime
2332 (using @code{Digits}). Some may be evaluated at certain points, but not
2333 generally. This ought to be fixed. However, doing numerical
2334 computations with GiNaC's quite abstract classes is doomed to be
2335 inefficient. For this purpose, the underlying foundation classes
2336 provided by @acronym{CLN} are much better suited.
2339 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
2340 @c node-name, next, previous, up
2341 @section Symbolic functions
2343 The easiest and most instructive way to start with is probably to
2344 implement your own function. GiNaC's functions are objects of class
2345 @code{function}. The preprocessor is then used to convert the function
2346 names to objects with a corresponding serial number that is used
2347 internally to identify them. You usually need not worry about this
2348 number. New functions may be inserted into the system via a kind of
2349 `registry'. It is your responsibility to care for some functions that
2350 are called when the user invokes certain methods. These are usual
2351 C++-functions accepting a number of @code{ex} as arguments and returning
2352 one @code{ex}. As an example, if we have a look at a simplified
2353 implementation of the cosine trigonometric function, we first need a
2354 function that is called when one wishes to @code{eval} it. It could
2355 look something like this:
2358 static ex cos_eval_method(const ex & x)
2360 // if (!x%(2*Pi)) return 1
2361 // if (!x%Pi) return -1
2362 // if (!x%Pi/2) return 0
2363 // care for other cases...
2364 return cos(x).hold();
2368 @cindex @code{hold()}
2370 The last line returns @code{cos(x)} if we don't know what else to do and
2371 stops a potential recursive evaluation by saying @code{.hold()}, which
2372 sets a flag to the expression signaling that it has been evaluated. We
2373 should also implement a method for numerical evaluation and since we are
2374 lazy we sweep the problem under the rug by calling someone else's
2375 function that does so, in this case the one in class @code{numeric}:
2378 static ex cos_evalf(const ex & x)
2380 return cos(ex_to_numeric(x));
2384 Differentiation will surely turn up and so we need to tell @code{cos}
2385 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
2386 instance are then handled automatically by @code{basic::diff} and
2390 static ex cos_deriv(const ex & x, unsigned diff_param)
2396 @cindex product rule
2397 The second parameter is obligatory but uninteresting at this point. It
2398 specifies which parameter to differentiate in a partial derivative in
2399 case the function has more than one parameter and its main application
2400 is for correct handling of the chain rule. For Taylor expansion, it is
2401 enough to know how to differentiate. But if the function you want to
2402 implement does have a pole somewhere in the complex plane, you need to
2403 write another method for Laurent expansion around that point.
2405 Now that all the ingrediences for @code{cos} have been set up, we need
2406 to tell the system about it. This is done by a macro and we are not
2407 going to descibe how it expands, please consult your preprocessor if you
2411 REGISTER_FUNCTION(cos, eval_func(cos_eval).
2412 evalf_func(cos_evalf).
2413 derivative_func(cos_deriv));
2416 The first argument is the function's name used for calling it and for
2417 output. The second binds the corresponding methods as options to this
2418 object. Options are separated by a dot and can be given in an arbitrary
2419 order. GiNaC functions understand several more options which are always
2420 specified as @code{.option(params)}, for example a method for series
2421 expansion @code{.series_func(cos_series)}. Again, if no series
2422 expansion method is given, GiNaC defaults to simple Taylor expansion,
2423 which is correct if there are no poles involved as is the case for the
2424 @code{cos} function. The way GiNaC handles poles in case there are any
2425 is best understood by studying one of the examples, like the Gamma
2426 (@code{tgamma}) function for instance. (In essence the function first
2427 checks if there is a pole at the evaluation point and falls back to
2428 Taylor expansion if there isn't. Then, the pole is regularized by some
2429 suitable transformation.) Also, the new function needs to be declared
2430 somewhere. This may also be done by a convenient preprocessor macro:
2433 DECLARE_FUNCTION_1P(cos)
2436 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
2437 implementation of @code{cos} is very incomplete and lacks several safety
2438 mechanisms. Please, have a look at the real implementation in GiNaC.
2439 (By the way: in case you are worrying about all the macros above we can
2440 assure you that functions are GiNaC's most macro-intense classes. We
2441 have done our best to avoid macros where we can.)
2443 That's it. May the source be with you!
2446 @node A Comparison With Other CAS, Advantages, Symbolic functions, Top
2447 @c node-name, next, previous, up
2448 @chapter A Comparison With Other CAS
2451 This chapter will give you some information on how GiNaC compares to
2452 other, traditional Computer Algebra Systems, like @emph{Maple},
2453 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
2454 disadvantages over these systems.
2457 * Advantages:: Stengths of the GiNaC approach.
2458 * Disadvantages:: Weaknesses of the GiNaC approach.
2459 * Why C++?:: Attractiveness of C++.
2462 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
2463 @c node-name, next, previous, up
2466 GiNaC has several advantages over traditional Computer
2467 Algebra Systems, like
2472 familiar language: all common CAS implement their own proprietary
2473 grammar which you have to learn first (and maybe learn again when your
2474 vendor decides to `enhance' it). With GiNaC you can write your program
2475 in common C++, which is standardized.
2479 structured data types: you can build up structured data types using
2480 @code{struct}s or @code{class}es together with STL features instead of
2481 using unnamed lists of lists of lists.
2484 strongly typed: in CAS, you usually have only one kind of variables
2485 which can hold contents of an arbitrary type. This 4GL like feature is
2486 nice for novice programmers, but dangerous.
2489 development tools: powerful development tools exist for C++, like fancy
2490 editors (e.g. with automatic indentation and syntax highlighting),
2491 debuggers, visualization tools, documentation generators...
2494 modularization: C++ programs can easily be split into modules by
2495 separating interface and implementation.
2498 price: GiNaC is distributed under the GNU Public License which means
2499 that it is free and available with source code. And there are excellent
2500 C++-compilers for free, too.
2503 extendable: you can add your own classes to GiNaC, thus extending it on
2504 a very low level. Compare this to a traditional CAS that you can
2505 usually only extend on a high level by writing in the language defined
2506 by the parser. In particular, it turns out to be almost impossible to
2507 fix bugs in a traditional system.
2510 multiple interfaces: Though real GiNaC programs have to be written in
2511 some editor, then be compiled, linked and executed, there are more ways
2512 to work with the GiNaC engine. Many people want to play with
2513 expressions interactively, as in traditional CASs. Currently, two such
2514 windows into GiNaC have been implemented and many more are possible: the
2515 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
2516 types to a command line and second, as a more consistent approach, an
2517 interactive interface to the @acronym{Cint} C++ interpreter has been put
2518 together (called @acronym{GiNaC-cint}) that allows an interactive
2519 scripting interface consistent with the C++ language.
2522 seemless integration: it is somewhere between difficult and impossible
2523 to call CAS functions from within a program written in C++ or any other
2524 programming language and vice versa. With GiNaC, your symbolic routines
2525 are part of your program. You can easily call third party libraries,
2526 e.g. for numerical evaluation or graphical interaction. All other
2527 approaches are much more cumbersome: they range from simply ignoring the
2528 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
2529 system (i.e. @emph{Yacas}).
2532 efficiency: often large parts of a program do not need symbolic
2533 calculations at all. Why use large integers for loop variables or
2534 arbitrary precision arithmetics where @code{int} and @code{double} are
2535 sufficient? For pure symbolic applications, GiNaC is comparable in
2536 speed with other CAS.
2541 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
2542 @c node-name, next, previous, up
2543 @section Disadvantages
2545 Of course it also has some disadvantages:
2550 advanced features: GiNaC cannot compete with a program like
2551 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
2552 which grows since 1981 by the work of dozens of programmers, with
2553 respect to mathematical features. Integration, factorization,
2554 non-trivial simplifications, limits etc. are missing in GiNaC (and are
2555 not planned for the near future).
2558 portability: While the GiNaC library itself is designed to avoid any
2559 platform dependent features (it should compile on any ANSI compliant C++
2560 compiler), the currently used version of the CLN library (fast large
2561 integer and arbitrary precision arithmetics) can be compiled only on
2562 systems with a recently new C++ compiler from the GNU Compiler
2563 Collection (@acronym{GCC}).@footnote{This is because CLN uses
2564 PROVIDE/REQUIRE like macros to let the compiler gather all static
2565 initializations, which works for GNU C++ only.} GiNaC uses recent
2566 language features like explicit constructors, mutable members, RTTI,
2567 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
2568 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
2569 ANSI compliant, support all needed features.
2574 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
2575 @c node-name, next, previous, up
2578 Why did we choose to implement GiNaC in C++ instead of Java or any other
2579 language? C++ is not perfect: type checking is not strict (casting is
2580 possible), separation between interface and implementation is not
2581 complete, object oriented design is not enforced. The main reason is
2582 the often scolded feature of operator overloading in C++. While it may
2583 be true that operating on classes with a @code{+} operator is rarely
2584 meaningful, it is perfectly suited for algebraic expressions. Writing
2585 @math{3x+5y} as @code{3*x+5*y} instead of
2586 @code{x.times(3).plus(y.times(5))} looks much more natural.
2587 Furthermore, the main developers are more familiar with C++ than with
2588 any other programming language.
2591 @node Internal Structures, Expressions are reference counted, Why C++? , Top
2592 @c node-name, next, previous, up
2593 @appendix Internal Structures
2596 * Expressions are reference counted::
2597 * Internal representation of products and sums::
2600 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
2601 @c node-name, next, previous, up
2602 @appendixsection Expressions are reference counted
2604 @cindex reference counting
2605 @cindex copy-on-write
2606 @cindex garbage collection
2607 An expression is extremely light-weight since internally it works like a
2608 handle to the actual representation and really holds nothing more than a
2609 pointer to some other object. What this means in practice is that
2610 whenever you create two @code{ex} and set the second equal to the first
2611 no copying process is involved. Instead, the copying takes place as soon
2612 as you try to change the second. Consider the simple sequence of code:
2615 #include <ginac/ginac.h>
2616 using namespace GiNaC;
2620 symbol x("x"), y("y"), z("z");
2623 e1 = sin(x + 2*y) + 3*z + 41;
2624 e2 = e1; // e2 points to same object as e1
2625 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
2626 e2 += 1; // e2 is copied into a new object
2627 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
2631 The line @code{e2 = e1;} creates a second expression pointing to the
2632 object held already by @code{e1}. The time involved for this operation
2633 is therefore constant, no matter how large @code{e1} was. Actual
2634 copying, however, must take place in the line @code{e2 += 1;} because
2635 @code{e1} and @code{e2} are not handles for the same object any more.
2636 This concept is called @dfn{copy-on-write semantics}. It increases
2637 performance considerably whenever one object occurs multiple times and
2638 represents a simple garbage collection scheme because when an @code{ex}
2639 runs out of scope its destructor checks whether other expressions handle
2640 the object it points to too and deletes the object from memory if that
2641 turns out not to be the case. A slightly less trivial example of
2642 differentiation using the chain-rule should make clear how powerful this
2646 #include <ginac/ginac.h>
2647 using namespace GiNaC;
2651 symbol x("x"), y("y");
2655 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
2656 cout << e1 << endl // prints x+3*y
2657 << e2 << endl // prints (x+3*y)^3
2658 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
2662 Here, @code{e1} will actually be referenced three times while @code{e2}
2663 will be referenced two times. When the power of an expression is built,
2664 that expression needs not be copied. Likewise, since the derivative of
2665 a power of an expression can be easily expressed in terms of that
2666 expression, no copying of @code{e1} is involved when @code{e3} is
2667 constructed. So, when @code{e3} is constructed it will print as
2668 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
2669 holds a reference to @code{e2} and the factor in front is just
2672 As a user of GiNaC, you cannot see this mechanism of copy-on-write
2673 semantics. When you insert an expression into a second expression, the
2674 result behaves exactly as if the contents of the first expression were
2675 inserted. But it may be useful to remember that this is not what
2676 happens. Knowing this will enable you to write much more efficient
2677 code. If you still have an uncertain feeling with copy-on-write
2678 semantics, we recommend you have a look at the
2679 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
2680 Marshall Cline. Chapter 16 covers this issue and presents an
2681 implementation which is pretty close to the one in GiNaC.
2684 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
2685 @c node-name, next, previous, up
2686 @appendixsection Internal representation of products and sums
2688 @cindex representation
2691 @cindex @code{power}
2692 Although it should be completely transparent for the user of
2693 GiNaC a short discussion of this topic helps to understand the sources
2694 and also explain performance to a large degree. Consider the
2695 unexpanded symbolic expression
2697 $2d^3 \left( 4a + 5b - 3 \right)$
2700 @math{2*d^3*(4*a+5*b-3)}
2702 which could naively be represented by a tree of linear containers for
2703 addition and multiplication, one container for exponentiation with base
2704 and exponent and some atomic leaves of symbols and numbers in this
2709 @cindex pair-wise representation
2710 However, doing so results in a rather deeply nested tree which will
2711 quickly become inefficient to manipulate. We can improve on this by
2712 representing the sum as a sequence of terms, each one being a pair of a
2713 purely numeric multiplicative coefficient and its rest. In the same
2714 spirit we can store the multiplication as a sequence of terms, each
2715 having a numeric exponent and a possibly complicated base, the tree
2716 becomes much more flat:
2720 The number @code{3} above the symbol @code{d} shows that @code{mul}
2721 objects are treated similarly where the coefficients are interpreted as
2722 @emph{exponents} now. Addition of sums of terms or multiplication of
2723 products with numerical exponents can be coded to be very efficient with
2724 such a pair-wise representation. Internally, this handling is performed
2725 by most CAS in this way. It typically speeds up manipulations by an
2726 order of magnitude. The overall multiplicative factor @code{2} and the
2727 additive term @code{-3} look somewhat out of place in this
2728 representation, however, since they are still carrying a trivial
2729 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
2730 this is avoided by adding a field that carries an overall numeric
2731 coefficient. This results in the realistic picture of internal
2734 $2d^3 \left( 4a + 5b - 3 \right)$:
2737 @math{2*d^3*(4*a+5*b-3)}:
2743 This also allows for a better handling of numeric radicals, since
2744 @code{sqrt(2)} can now be carried along calculations. Now it should be
2745 clear, why both classes @code{add} and @code{mul} are derived from the
2746 same abstract class: the data representation is the same, only the
2747 semantics differs. In the class hierarchy, methods for polynomial
2748 expansion and the like are reimplemented for @code{add} and @code{mul},
2749 but the data structure is inherited from @code{expairseq}.
2752 @node Package Tools, ginac-config, Internal representation of products and sums, Top
2753 @c node-name, next, previous, up
2754 @appendix Package Tools
2756 If you are creating a software package that uses the GiNaC library,
2757 setting the correct command line options for the compiler and linker
2758 can be difficult. GiNaC includes two tools to make this process easier.
2761 * ginac-config:: A shell script to detect compiler and linker flags.
2762 * AM_PATH_GINAC:: Macro for GNU automake.
2766 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
2767 @c node-name, next, previous, up
2768 @section @command{ginac-config}
2769 @cindex ginac-config
2771 @command{ginac-config} is a shell script that you can use to determine
2772 the compiler and linker command line options required to compile and
2773 link a program with the GiNaC library.
2775 @command{ginac-config} takes the following flags:
2779 Prints out the version of GiNaC installed.
2781 Prints '-I' flags pointing to the installed header files.
2783 Prints out the linker flags necessary to link a program against GiNaC.
2784 @item --prefix[=@var{PREFIX}]
2785 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
2786 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
2787 Otherwise, prints out the configured value of @env{$prefix}.
2788 @item --exec-prefix[=@var{PREFIX}]
2789 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
2790 Otherwise, prints out the configured value of @env{$exec_prefix}.
2793 Typically, @command{ginac-config} will be used within a configure
2794 script, as described below. It, however, can also be used directly from
2795 the command line using backquotes to compile a simple program. For
2799 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
2802 This command line might expand to (for example):
2805 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
2806 -lginac -lcln -lstdc++
2809 Not only is the form using @command{ginac-config} easier to type, it will
2810 work on any system, no matter how GiNaC was configured.
2813 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
2814 @c node-name, next, previous, up
2815 @section @samp{AM_PATH_GINAC}
2816 @cindex AM_PATH_GINAC
2818 For packages configured using GNU automake, GiNaC also provides
2819 a macro to automate the process of checking for GiNaC.
2822 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
2830 Determines the location of GiNaC using @command{ginac-config}, which is
2831 either found in the user's path, or from the environment variable
2832 @env{GINACLIB_CONFIG}.
2835 Tests the installed libraries to make sure that their version
2836 is later than @var{MINIMUM-VERSION}. (A default version will be used
2840 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
2841 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
2842 variable to the output of @command{ginac-config --libs}, and calls
2843 @samp{AC_SUBST()} for these variables so they can be used in generated
2844 makefiles, and then executes @var{ACTION-IF-FOUND}.
2847 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
2848 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
2852 This macro is in file @file{ginac.m4} which is installed in
2853 @file{$datadir/aclocal}. Note that if automake was installed with a
2854 different @samp{--prefix} than GiNaC, you will either have to manually
2855 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
2856 aclocal the @samp{-I} option when running it.
2859 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
2860 * Example package:: Example of a package using AM_PATH_GINAC.
2864 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2865 @c node-name, next, previous, up
2866 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2868 Simply make sure that @command{ginac-config} is in your path, and run
2869 the configure script.
2876 The directory where the GiNaC libraries are installed needs
2877 to be found by your system's dynamic linker.
2879 This is generally done by
2882 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2888 setting the environment variable @env{LD_LIBRARY_PATH},
2891 or, as a last resort,
2894 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2895 running configure, for instance:
2898 LDFLAGS=-R/home/cbauer/lib ./configure
2903 You can also specify a @command{ginac-config} not in your path by
2904 setting the @env{GINACLIB_CONFIG} environment variable to the
2905 name of the executable
2908 If you move the GiNaC package from its installed location,
2909 you will either need to modify @command{ginac-config} script
2910 manually to point to the new location or rebuild GiNaC.
2921 --with-ginac-prefix=@var{PREFIX}
2922 --with-ginac-exec-prefix=@var{PREFIX}
2925 are provided to override the prefix and exec-prefix that were stored
2926 in the @command{ginac-config} shell script by GiNaC's configure. You are
2927 generally better off configuring GiNaC with the right path to begin with.
2931 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2932 @c node-name, next, previous, up
2933 @subsection Example of a package using @samp{AM_PATH_GINAC}
2935 The following shows how to build a simple package using automake
2936 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2939 #include <ginac/ginac.h>
2940 using namespace GiNaC;
2946 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2951 You should first read the introductory portions of the automake
2952 Manual, if you are not already familiar with it.
2954 Two files are needed, @file{configure.in}, which is used to build the
2958 dnl Process this file with autoconf to produce a configure script.
2960 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2966 AM_PATH_GINAC(0.4.0, [
2967 LIBS="$LIBS $GINACLIB_LIBS"
2968 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2969 ], AC_MSG_ERROR([need to have GiNaC installed]))
2974 The only command in this which is not standard for automake
2975 is the @samp{AM_PATH_GINAC} macro.
2977 That command does the following:
2980 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2981 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2982 with the error message `need to have GiNaC installed'
2985 And the @file{Makefile.am}, which will be used to build the Makefile.
2988 ## Process this file with automake to produce Makefile.in
2989 bin_PROGRAMS = simple
2990 simple_SOURCES = simple.cpp
2993 This @file{Makefile.am}, says that we are building a single executable,
2994 from a single sourcefile @file{simple.cpp}. Since every program
2995 we are building uses GiNaC we simply added the GiNaC options
2996 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
2997 want to specify them on a per-program basis: for instance by
3001 simple_LDADD = $(GINACLIB_LIBS)
3002 INCLUDES = $(GINACLIB_CPPFLAGS)
3005 to the @file{Makefile.am}.
3007 To try this example out, create a new directory and add the three
3010 Now execute the following commands:
3013 $ automake --add-missing
3018 You now have a package that can be built in the normal fashion
3027 @node Bibliography, Concept Index, Example package, Top
3028 @c node-name, next, previous, up
3029 @appendix Bibliography
3034 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
3037 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
3040 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
3043 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
3046 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
3047 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
3050 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
3051 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
3052 Academic Press, London
3057 @node Concept Index, , Bibliography, Top
3058 @c node-name, next, previous, up
3059 @unnumbered Concept Index