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. You'll soon learn in this chapter
711 how many of the functions on symbols are really classes. This is
712 because simple symbolic arithmetic is not supported by languages like
713 C++ so in a certain way GiNaC has to implement its own arithmetic.
717 To get an idea about what kinds of symbolic composits may be built we
718 have a look at the most important classes in the class hierarchy. The
719 oval classes are atomic ones and the squared classes are containers.
720 The dashed line symbolizes a `points to' or `handles' relationship while
721 the solid lines stand for `inherits from' relationship in the class
724 @image{classhierarchy}
726 Some of the classes shown here (the ones sitting in white boxes) are
727 abstract base classes that are of no interest at all for the user. They
728 are used internally in order to avoid code duplication if two or more
729 classes derived from them share certain features. An example would be
730 @code{expairseq}, which is a container for a sequence of pairs each
731 consisting of one expression and a number (@code{numeric}). What
732 @emph{is} visible to the user are the derived classes @code{add} and
733 @code{mul}, representing sums of terms and products, respectively.
734 @xref{Internal Structures}, where these two classes are described in
737 At this point, we only summarize what kind of mathematical objects are
738 stored in the different classes in above diagram in order to give you a
742 @multitable @columnfractions .22 .78
743 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
744 @item @code{constant} @tab Constants like
751 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
752 @item @code{add} @tab Sums like @math{x+y} or @math{a+(2*b)+3}
753 @item @code{mul} @tab Products like @math{x*y} or @math{a*(x+y+z)*b*2}
754 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
759 @code{sqrt(}@math{2}@code{)}
762 @item @code{pseries} @tab Power Series, e.g. @math{x+1/6*x^3+1/120*x^5+O(x^7)}
763 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
764 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
765 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
766 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
767 @item @code{color} @tab Element of the @math{SU(3)} Lie-algebra
768 @item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
769 @item @code{idx} @tab Index of a tensor object
770 @item @code{coloridx} @tab Index of a @math{SU(3)} tensor
774 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
775 @c node-name, next, previous, up
777 @cindex @code{symbol} (class)
778 @cindex hierarchy of classes
781 Symbols are for symbolic manipulation what atoms are for chemistry. You
782 can declare objects of class @code{symbol} as any other object simply by
783 saying @code{symbol x,y;}. There is, however, a catch in here having to
784 do with the fact that C++ is a compiled language. The information about
785 the symbol's name is thrown away by the compiler but at a later stage
786 you may want to print expressions holding your symbols. In order to
787 avoid confusion GiNaC's symbols are able to know their own name. This
788 is accomplished by declaring its name for output at construction time in
789 the fashion @code{symbol x("x");}. If you declare a symbol using the
790 default constructor (i.e. without string argument) the system will deal
791 out a unique name. That name may not be suitable for printing but for
792 internal routines when no output is desired it is often enough. We'll
793 come across examples of such symbols later in this tutorial.
795 This implies that the strings passed to symbols at construction time may
796 not be used for comparing two of them. It is perfectly legitimate to
797 write @code{symbol x("x"),y("x");} but it is likely to lead into
798 trouble. Here, @code{x} and @code{y} are different symbols and
799 statements like @code{x-y} will not be simplified to zero although the
800 output @code{x-x} looks funny. Such output may also occur when there
801 are two different symbols in two scopes, for instance when you call a
802 function that declares a symbol with a name already existent in a symbol
803 in the calling function. Again, comparing them (using @code{operator==}
804 for instance) will always reveal their difference. Watch out, please.
806 @cindex @code{subs()}
807 Although symbols can be assigned expressions for internal reasons, you
808 should not do it (and we are not going to tell you how it is done). If
809 you want to replace a symbol with something else in an expression, you
810 can use the expression's @code{.subs()} method (@xref{Substituting Symbols},
811 for more information).
814 @node Numbers, Constants, Symbols, Basic Concepts
815 @c node-name, next, previous, up
817 @cindex @code{numeric} (class)
823 For storing numerical things, GiNaC uses Bruno Haible's library
824 @acronym{CLN}. The classes therein serve as foundation classes for
825 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
826 alternatively for Common Lisp Numbers. In order to find out more about
827 @acronym{CLN}'s internals the reader is refered to the documentation of
828 that library. @inforef{Introduction, , cln}, for more
829 information. Suffice to say that it is by itself build on top of another
830 library, the GNU Multiple Precision library @acronym{GMP}, which is an
831 extremely fast library for arbitrary long integers and rationals as well
832 as arbitrary precision floating point numbers. It is very commonly used
833 by several popular cryptographic applications. @acronym{CLN} extends
834 @acronym{GMP} by several useful things: First, it introduces the complex
835 number field over either reals (i.e. floating point numbers with
836 arbitrary precision) or rationals. Second, it automatically converts
837 rationals to integers if the denominator is unity and complex numbers to
838 real numbers if the imaginary part vanishes and also correctly treats
839 algebraic functions. Third it provides good implementations of
840 state-of-the-art algorithms for all trigonometric and hyperbolic
841 functions as well as for calculation of some useful constants.
843 The user can construct an object of class @code{numeric} in several
844 ways. The following example shows the four most important constructors.
845 It uses construction from C-integer, construction of fractions from two
846 integers, construction from C-float and construction from a string:
849 #include <ginac/ginac.h>
850 using namespace GiNaC;
854 numeric two(2); // exact integer 2
855 numeric r(2,3); // exact fraction 2/3
856 numeric e(2.71828); // floating point number
857 numeric p("3.1415926535897932385"); // floating point number
858 // Trott's constant in scientific notation:
859 numeric trott("1.0841015122311136151E-2");
861 cout << two*p << endl; // floating point 6.283...
865 Note that all those constructors are @emph{explicit} which means you are
866 not allowed to write @code{numeric two=2;}. This is because the basic
867 objects to be handled by GiNaC are the expressions @code{ex} and we want
868 to keep things simple and wish objects like @code{pow(x,2)} to be
869 handled the same way as @code{pow(x,a)}, which means that we need to
870 allow a general @code{ex} as base and exponent. Therefore there is an
871 implicit constructor from C-integers directly to expressions handling
872 numerics at work in most of our examples. This design really becomes
873 convenient when one declares own functions having more than one
874 parameter but it forbids using implicit constructors because that would
875 lead to compile-time ambiguities.
877 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
878 This would, however, call C's built-in operator @code{/} for integers
879 first and result in a numeric holding a plain integer 1. @strong{Never
880 use the operator @code{/} on integers} unless you know exactly what you
881 are doing! Use the constructor from two integers instead, as shown in
882 the example above. Writing @code{numeric(1)/2} may look funny but works
885 @cindex @code{Digits}
887 We have seen now the distinction between exact numbers and floating
888 point numbers. Clearly, the user should never have to worry about
889 dynamically created exact numbers, since their `exactness' always
890 determines how they ought to be handled, i.e. how `long' they are. The
891 situation is different for floating point numbers. Their accuracy is
892 controlled by one @emph{global} variable, called @code{Digits}. (For
893 those readers who know about Maple: it behaves very much like Maple's
894 @code{Digits}). All objects of class numeric that are constructed from
895 then on will be stored with a precision matching that number of decimal
899 #include <ginac/ginac.h>
900 using namespace GiNaC;
904 numeric three(3.0), one(1.0);
905 numeric x = one/three;
907 cout << "in " << Digits << " digits:" << endl;
909 cout << Pi.evalf() << endl;
921 The above example prints the following output to screen:
928 0.333333333333333333333333333333333333333333333333333333333333333333
929 3.14159265358979323846264338327950288419716939937510582097494459231
932 It should be clear that objects of class @code{numeric} should be used
933 for constructing numbers or for doing arithmetic with them. The objects
934 one deals with most of the time are the polymorphic expressions @code{ex}.
936 @subsection Tests on numbers
938 Once you have declared some numbers, assigned them to expressions and
939 done some arithmetic with them it is frequently desired to retrieve some
940 kind of information from them like asking whether that number is
941 integer, rational, real or complex. For those cases GiNaC provides
942 several useful methods. (Internally, they fall back to invocations of
943 certain CLN functions.)
945 As an example, let's construct some rational number, multiply it with
946 some multiple of its denominator and test what comes out:
949 #include <ginac/ginac.h>
950 using namespace GiNaC;
952 // some very important constants:
953 const numeric twentyone(21);
954 const numeric ten(10);
955 const numeric five(5);
959 numeric answer = twentyone;
962 cout << answer.is_integer() << endl; // false, it's 21/5
964 cout << answer.is_integer() << endl; // true, it's 42 now!
968 Note that the variable @code{answer} is constructed here as an integer
969 by @code{numeric}'s copy constructor but in an intermediate step it
970 holds a rational number represented as integer numerator and integer
971 denominator. When multiplied by 10, the denominator becomes unity and
972 the result is automatically converted to a pure integer again.
973 Internally, the underlying @acronym{CLN} is responsible for this
974 behaviour and we refer the reader to @acronym{CLN}'s documentation.
975 Suffice to say that the same behaviour applies to complex numbers as
976 well as return values of certain functions. Complex numbers are
977 automatically converted to real numbers if the imaginary part becomes
978 zero. The full set of tests that can be applied is listed in the
982 @multitable @columnfractions .30 .70
983 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
984 @item @code{.is_zero()}
985 @tab @dots{}equal to zero
986 @item @code{.is_positive()}
987 @tab @dots{}not complex and greater than 0
988 @item @code{.is_integer()}
989 @tab @dots{}a (non-complex) integer
990 @item @code{.is_pos_integer()}
991 @tab @dots{}an integer and greater than 0
992 @item @code{.is_nonneg_integer()}
993 @tab @dots{}an integer and greater equal 0
994 @item @code{.is_even()}
995 @tab @dots{}an even integer
996 @item @code{.is_odd()}
997 @tab @dots{}an odd integer
998 @item @code{.is_prime()}
999 @tab @dots{}a prime integer (probabilistic primality test)
1000 @item @code{.is_rational()}
1001 @tab @dots{}an exact rational number (integers are rational, too)
1002 @item @code{.is_real()}
1003 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1004 @item @code{.is_cinteger()}
1005 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1006 @item @code{.is_crational()}
1007 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1012 @node Constants, Fundamental containers, Numbers, Basic Concepts
1013 @c node-name, next, previous, up
1015 @cindex @code{constant} (class)
1018 @cindex @code{Catalan}
1019 @cindex @code{Euler}
1020 @cindex @code{evalf()}
1021 Constants behave pretty much like symbols except that they return some
1022 specific number when the method @code{.evalf()} is called.
1024 The predefined known constants are:
1027 @multitable @columnfractions .14 .30 .56
1028 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1030 @tab Archimedes' constant
1031 @tab 3.14159265358979323846264338327950288
1032 @item @code{Catalan}
1033 @tab Catalan's constant
1034 @tab 0.91596559417721901505460351493238411
1036 @tab Euler's (or Euler-Mascheroni) constant
1037 @tab 0.57721566490153286060651209008240243
1042 @node Fundamental containers, Lists, Constants, Basic Concepts
1043 @c node-name, next, previous, up
1044 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1048 @cindex @code{power}
1050 Simple polynomial expressions are written down in GiNaC pretty much like
1051 in other CAS or like expressions involving numerical variables in C.
1052 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1053 been overloaded to achieve this goal. When you run the following
1054 program, the constructor for an object of type @code{mul} is
1055 automatically called to hold the product of @code{a} and @code{b} and
1056 then the constructor for an object of type @code{add} is called to hold
1057 the sum of that @code{mul} object and the number one:
1060 #include <ginac/ginac.h>
1061 using namespace GiNaC;
1065 symbol a("a"), b("b");
1071 @cindex @code{pow()}
1072 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1073 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1074 construction is necessary since we cannot safely overload the constructor
1075 @code{^} in C++ to construct a @code{power} object. If we did, it would
1076 have several counterintuitive effects:
1080 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1082 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1083 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1084 interpret this as @code{x^(a^b)}.
1086 Also, expressions involving integer exponents are very frequently used,
1087 which makes it even more dangerous to overload @code{^} since it is then
1088 hard to distinguish between the semantics as exponentiation and the one
1089 for exclusive or. (It would be embarassing to return @code{1} where one
1090 has requested @code{2^3}.)
1093 @cindex @command{ginsh}
1094 All effects are contrary to mathematical notation and differ from the
1095 way most other CAS handle exponentiation, therefore overloading @code{^}
1096 is ruled out for GiNaC's C++ part. The situation is different in
1097 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1098 that the other frequently used exponentiation operator @code{**} does
1099 not exist at all in C++).
1101 To be somewhat more precise, objects of the three classes described
1102 here, are all containers for other expressions. An object of class
1103 @code{power} is best viewed as a container with two slots, one for the
1104 basis, one for the exponent. All valid GiNaC expressions can be
1105 inserted. However, basic transformations like simplifying
1106 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1107 when this is mathematically possible. If we replace the outer exponent
1108 three in the example by some symbols @code{a}, the simplification is not
1109 safe and will not be performed, since @code{a} might be @code{1/2} and
1112 Objects of type @code{add} and @code{mul} are containers with an
1113 arbitrary number of slots for expressions to be inserted. Again, simple
1114 and safe simplifications are carried out like transforming
1115 @code{3*x+4-x} to @code{2*x+4}.
1117 The general rule is that when you construct such objects, GiNaC
1118 automatically creates them in canonical form, which might differ from
1119 the form you typed in your program. This allows for rapid comparison of
1120 expressions, since after all @code{a-a} is simply zero. Note, that the
1121 canonical form is not necessarily lexicographical ordering or in any way
1122 easily guessable. It is only guaranteed that constructing the same
1123 expression twice, either implicitly or explicitly, results in the same
1127 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1128 @c node-name, next, previous, up
1129 @section Lists of expressions
1130 @cindex @code{lst} (class)
1132 @cindex @code{nops()}
1134 @cindex @code{append()}
1135 @cindex @code{prepend()}
1137 The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
1138 These are sometimes used to supply a variable number of arguments of the same
1139 type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
1140 should have a basic understanding about them.
1142 Lists of up to 15 expressions can be directly constructed from single
1147 symbol x("x"), y("y");
1148 lst l(x, 2, y, x+y);
1149 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1153 Use the @code{nops()} method to determine the size (number of expressions) of
1154 a list and the @code{op()} method to access individual elements:
1158 cout << l.nops() << endl; // prints '4'
1159 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1163 Finally you can append or prepend an expression to a list with the
1164 @code{append()} and @code{prepend()} methods:
1168 l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
1169 l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
1174 @node Mathematical functions, Relations, Lists, Basic Concepts
1175 @c node-name, next, previous, up
1176 @section Mathematical functions
1177 @cindex @code{function} (class)
1178 @cindex trigonometric function
1179 @cindex hyperbolic function
1181 There are quite a number of useful functions hard-wired into GiNaC. For
1182 instance, all trigonometric and hyperbolic functions are implemented
1183 (@xref{Built-in Functions}, for a complete list).
1185 These functions are all objects of class @code{function}. They accept one
1186 or more expressions as arguments and return one expression. If the arguments
1187 are not numerical, the evaluation of the function may be halted, as it
1188 does in the next example:
1190 @cindex Gamma function
1191 @cindex @code{subs()}
1193 #include <ginac/ginac.h>
1194 using namespace GiNaC;
1198 symbol x("x"), y("y");
1201 cout << "tgamma(" << foo << ") -> " << tgamma(foo) << endl;
1202 ex bar = foo.subs(y==1);
1203 cout << "tgamma(" << bar << ") -> " << tgamma(bar) << endl;
1204 ex foobar = bar.subs(x==7);
1205 cout << "tgamma(" << foobar << ") -> " << tgamma(foobar) << endl;
1209 This program shows how the function returns itself twice and finally an
1210 expression that may be really useful:
1213 tgamma(x+(1/2)*y) -> tgamma(x+(1/2)*y)
1214 tgamma(x+1/2) -> tgamma(x+1/2)
1215 tgamma(15/2) -> (135135/128)*Pi^(1/2)
1218 Besides evaluation most of these functions allow differentiation, series
1219 expansion and so on. Read the next chapter in order to learn more about
1223 @node Relations, Methods and Functions, Mathematical functions, Basic Concepts
1224 @c node-name, next, previous, up
1226 @cindex @code{relational} (class)
1228 Sometimes, a relation holding between two expressions must be stored
1229 somehow. The class @code{relational} is a convenient container for such
1230 purposes. A relation is by definition a container for two @code{ex} and
1231 a relation between them that signals equality, inequality and so on.
1232 They are created by simply using the C++ operators @code{==}, @code{!=},
1233 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1235 @xref{Mathematical functions}, for examples where various applications
1236 of the @code{.subs()} method show how objects of class relational are
1237 used as arguments. There they provide an intuitive syntax for
1238 substitutions. They are also used as arguments to the @code{ex::series}
1239 method, where the left hand side of the relation specifies the variable
1240 to expand in and the right hand side the expansion point. They can also
1241 be used for creating systems of equations that are to be solved for
1242 unknown variables. But the most common usage of objects of this class
1243 is rather inconspicuous in statements of the form @code{if
1244 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1245 conversion from @code{relational} to @code{bool} takes place. Note,
1246 however, that @code{==} here does not perform any simplifications, hence
1247 @code{expand()} must be called explicitly.
1250 @node Methods and Functions, Information About Expressions, Relations, Top
1251 @c node-name, next, previous, up
1252 @chapter Methods and Functions
1255 In this chapter the most important algorithms provided by GiNaC will be
1256 described. Some of them are implemented as functions on expressions,
1257 others are implemented as methods provided by expression objects. If
1258 they are methods, there exists a wrapper function around it, so you can
1259 alternatively call it in a functional way as shown in the simple
1263 #include <ginac/ginac.h>
1264 using namespace GiNaC;
1268 ex x = numeric(1.0);
1270 cout << "As method: " << sin(x).evalf() << endl;
1271 cout << "As function: " << evalf(sin(x)) << endl;
1275 @cindex @code{subs()}
1276 The general rule is that wherever methods accept one or more parameters
1277 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1278 wrapper accepts is the same but preceded by the object to act on
1279 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1280 most natural one in an OO model but it may lead to confusion for MapleV
1281 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1282 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1283 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1284 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1285 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1286 here. Also, users of MuPAD will in most cases feel more comfortable
1287 with GiNaC's convention. All function wrappers are implemented
1288 as simple inline functions which just call the corresponding method and
1289 are only provided for users uncomfortable with OO who are dead set to
1290 avoid method invocations. Generally, nested function wrappers are much
1291 harder to read than a sequence of methods and should therefore be
1292 avoided if possible. On the other hand, not everything in GiNaC is a
1293 method on class @code{ex} and sometimes calling a function cannot be
1297 * Information About Expressions::
1298 * Substituting Symbols::
1299 * Polynomial Arithmetic:: Working with polynomials.
1300 * Rational Expressions:: Working with rational functions.
1301 * Symbolic Differentiation::
1302 * Series Expansion:: Taylor and Laurent expansion.
1303 * Built-in Functions:: List of predefined mathematical functions.
1304 * Input/Output:: Input and output of expressions.
1308 @node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions
1309 @c node-name, next, previous, up
1310 @section Getting information about expressions
1312 @subsection Checking expression types
1313 @cindex @code{is_ex_of_type()}
1314 @cindex @code{info()}
1316 Sometimes it's useful to check whether a given expression is a plain number,
1317 a sum, a polynomial with integer coefficients, or of some other specific type.
1318 GiNaC provides two functions for this (the first one is actually a macro):
1321 bool is_ex_of_type(const ex & e, TYPENAME t);
1322 bool ex::info(unsigned flag);
1325 @code{is_ex_of_type()} allows you to check whether the top-level object of
1326 an expression @samp{e} is an instance of the GiNaC class @samp{t}
1327 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
1328 e.g., for checking whether an expression is a number, a sum, or a product:
1335 is_ex_of_type(e1, numeric); // true
1336 is_ex_of_type(e2, numeric); // false
1337 is_ex_of_type(e1, add); // false
1338 is_ex_of_type(e2, add); // true
1339 is_ex_of_type(e1, mul); // false
1340 is_ex_of_type(e2, mul); // false
1344 The @code{info()} method is used for checking certain attributes of
1345 expressions. The possible values for the @code{flag} argument are defined
1346 in @file{ginac/flags.h}, the most important being explained in the following
1350 @multitable @columnfractions .30 .70
1351 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
1352 @item @code{numeric}
1353 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
1355 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1356 @item @code{rational}
1357 @tab @dots{}an exact rational number (integers are rational, too)
1358 @item @code{integer}
1359 @tab @dots{}a (non-complex) integer
1360 @item @code{crational}
1361 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1362 @item @code{cinteger}
1363 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1364 @item @code{positive}
1365 @tab @dots{}not complex and greater than 0
1366 @item @code{negative}
1367 @tab @dots{}not complex and less than 0
1368 @item @code{nonnegative}
1369 @tab @dots{}not complex and greater than or equal to 0
1371 @tab @dots{}an integer greater than 0
1373 @tab @dots{}an integer less than 0
1374 @item @code{nonnegint}
1375 @tab @dots{}an integer greater than or equal to 0
1377 @tab @dots{}an even integer
1379 @tab @dots{}an odd integer
1381 @tab @dots{}a prime integer (probabilistic primality test)
1382 @item @code{relation}
1383 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
1384 @item @code{relation_equal}
1385 @tab @dots{}a @code{==} relation
1386 @item @code{relation_not_equal}
1387 @tab @dots{}a @code{!=} relation
1388 @item @code{relation_less}
1389 @tab @dots{}a @code{<} relation
1390 @item @code{relation_less_or_equal}
1391 @tab @dots{}a @code{<=} relation
1392 @item @code{relation_greater}
1393 @tab @dots{}a @code{>} relation
1394 @item @code{relation_greater_or_equal}
1395 @tab @dots{}a @code{>=} relation
1397 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
1399 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
1400 @item @code{polynomial}
1401 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
1402 @item @code{integer_polynomial}
1403 @tab @dots{}a polynomial with (non-complex) integer coefficients
1404 @item @code{cinteger_polynomial}
1405 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
1406 @item @code{rational_polynomial}
1407 @tab @dots{}a polynomial with (non-complex) rational coefficients
1408 @item @code{crational_polynomial}
1409 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
1410 @item @code{rational_function}
1411 @tab @dots{}a rational function
1416 @subsection Accessing subexpressions
1417 @cindex @code{nops()}
1419 @cindex @code{has()}
1421 @cindex @code{relational} (class)
1423 GiNaC provides the two methods
1426 unsigned ex::nops();
1427 ex ex::op(unsigned i);
1430 for accessing the subexpressions in the container-like GiNaC classes like
1431 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
1432 determines the number of subexpressions (@samp{operands}) contained, while
1433 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
1434 In the case of a @code{power} object, @code{op(0)} will return the basis
1435 and @code{op(1)} the exponent.
1437 The left-hand and right-hand side expressions of objects of class
1438 @code{relational} (and only of these) can also be accessed with the methods
1448 bool ex::has(const ex & other);
1451 checks whether an expression contains the given subexpression @code{other}.
1452 This only works reliably if @code{other} is of an atomic class such as a
1453 @code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
1454 @code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
1457 @subsection Comparing expressions
1458 @cindex @code{is_equal()}
1459 @cindex @code{is_zero()}
1461 Expressions can be compared with the usual C++ relational operators like
1462 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
1463 the result is usually not determinable and the result will be @code{false},
1464 except in the case of the @code{!=} operator. You should also be aware that
1465 GiNaC will only do the most trivial test for equality (subtracting both
1466 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
1469 Actually, if you construct an expression like @code{a == b}, this will be
1470 represented by an object of the @code{relational} class (@xref{Relations}.)
1471 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
1473 There are also two methods
1476 bool ex::is_equal(const ex & other);
1480 for checking whether one expression is equal to another, or equal to zero,
1483 @strong{Warning:} You will also find an @code{ex::compare()} method in the
1484 GiNaC header files. This method is however only to be used internally by
1485 GiNaC to establish a canonical sort order for terms, and using it to compare
1486 expressions will give very surprising results.
1489 @node Substituting Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions
1490 @c node-name, next, previous, up
1491 @section Substituting symbols
1492 @cindex @code{subs()}
1494 Symbols can be replaced with expressions via the @code{.subs()} method:
1497 ex ex::subs(const ex & e);
1498 ex ex::subs(const lst & syms, const lst & repls);
1501 In the first form, @code{subs()} accepts a relational of the form
1502 @samp{symbol == expression} or a @code{lst} of such relationals. E.g.
1506 symbol x("x"), y("y");
1507 ex e1 = 2*x^2-4*x+3;
1508 cout << "e1(7) = " << e1.subs(x == 7) << endl;
1510 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
1514 will print @samp{73} and @samp{-10}, respectively.
1516 If you specify multiple substitutions, they are performed in parallel, so e.g.
1517 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
1519 The second form of @code{subs()} takes two lists, one for the symbols and
1520 one for the expressions to be substituted (both lists must contain the same
1521 number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))}
1522 to exchange @samp{x} and @samp{y}.
1525 @node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, Methods and Functions
1526 @c node-name, next, previous, up
1527 @section Polynomial arithmetic
1529 @subsection Expanding and collecting
1530 @cindex @code{expand()}
1531 @cindex @code{collect()}
1533 A polynomial in one or more variables has many equivalent
1534 representations. Some useful ones serve a specific purpose. Consider
1535 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1536 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1537 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1538 representations are the recursive ones where one collects for exponents
1539 in one of the three variable. Since the factors are themselves
1540 polynomials in the remaining two variables the procedure can be
1541 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1542 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1545 To bring an expression into expanded form, its method
1551 may be called. In our example above, this corresponds to @math{4*x*y +
1552 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1553 GiNaC is not easily guessable you should be prepared to see different
1554 orderings of terms in such sums!
1556 Another useful representation of multivariate polynomials is as a
1557 univariate polynomial in one of the variables with the coefficients
1558 being polynomials in the remaining variables. The method
1559 @code{collect()} accomplishes this task:
1562 ex ex::collect(const symbol & s);
1565 Note that the original polynomial needs to be in expanded form in order
1566 to be able to find the coefficients properly.
1568 @subsection Degree and coefficients
1569 @cindex @code{degree()}
1570 @cindex @code{ldegree()}
1571 @cindex @code{coeff()}
1573 The degree and low degree of a polynomial can be obtained using the two
1577 int ex::degree(const symbol & s);
1578 int ex::ldegree(const symbol & s);
1581 which also work reliably on non-expanded input polynomials (they even work
1582 on rational functions, returning the asymptotic degree). To extract
1583 a coefficient with a certain power from an expanded polynomial you use
1586 ex ex::coeff(const symbol & s, int n);
1589 You can also obtain the leading and trailing coefficients with the methods
1592 ex ex::lcoeff(const symbol & s);
1593 ex ex::tcoeff(const symbol & s);
1596 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
1599 An application is illustrated in the next example, where a multivariate
1600 polynomial is analyzed:
1603 #include <ginac/ginac.h>
1604 using namespace GiNaC;
1608 symbol x("x"), y("y");
1609 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1610 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1611 ex Poly = PolyInp.expand();
1613 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1614 cout << "The x^" << i << "-coefficient is "
1615 << Poly.coeff(x,i) << endl;
1617 cout << "As polynomial in y: "
1618 << Poly.collect(y) << endl;
1622 When run, it returns an output in the following fashion:
1625 The x^0-coefficient is y^2+11*y
1626 The x^1-coefficient is 5*y^2-2*y
1627 The x^2-coefficient is -1
1628 The x^3-coefficient is 4*y
1629 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1632 As always, the exact output may vary between different versions of GiNaC
1633 or even from run to run since the internal canonical ordering is not
1634 within the user's sphere of influence.
1637 @subsection Polynomial division
1638 @cindex polynomial division
1641 @cindex pseudo-remainder
1642 @cindex @code{quo()}
1643 @cindex @code{rem()}
1644 @cindex @code{prem()}
1645 @cindex @code{divide()}
1650 ex quo(const ex & a, const ex & b, const symbol & x);
1651 ex rem(const ex & a, const ex & b, const symbol & x);
1654 compute the quotient and remainder of univariate polynomials in the variable
1655 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
1657 The additional function
1660 ex prem(const ex & a, const ex & b, const symbol & x);
1663 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
1664 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
1666 Exact division of multivariate polynomials is performed by the function
1669 bool divide(const ex & a, const ex & b, ex & q);
1672 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
1673 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
1674 in which case the value of @code{q} is undefined.
1677 @subsection Unit, content and primitive part
1678 @cindex @code{unit()}
1679 @cindex @code{content()}
1680 @cindex @code{primpart()}
1685 ex ex::unit(const symbol & x);
1686 ex ex::content(const symbol & x);
1687 ex ex::primpart(const symbol & x);
1690 return the unit part, content part, and primitive polynomial of a multivariate
1691 polynomial with respect to the variable @samp{x} (the unit part being the sign
1692 of the leading coefficient, the content part being the GCD of the coefficients,
1693 and the primitive polynomial being the input polynomial divided by the unit and
1694 content parts). The product of unit, content, and primitive part is the
1695 original polynomial.
1698 @subsection GCD and LCM
1701 @cindex @code{gcd()}
1702 @cindex @code{lcm()}
1704 The functions for polynomial greatest common divisor and least common
1705 multiple have the synopsis
1708 ex gcd(const ex & a, const ex & b);
1709 ex lcm(const ex & a, const ex & b);
1712 The functions @code{gcd()} and @code{lcm()} accept two expressions
1713 @code{a} and @code{b} as arguments and return a new expression, their
1714 greatest common divisor or least common multiple, respectively. If the
1715 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1716 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1719 #include <ginac/ginac.h>
1720 using namespace GiNaC;
1724 symbol x("x"), y("y"), z("z");
1725 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1726 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1728 ex P_gcd = gcd(P_a, P_b);
1730 ex P_lcm = lcm(P_a, P_b);
1731 // 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
1736 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
1737 @c node-name, next, previous, up
1738 @section Rational expressions
1740 @subsection The @code{normal} method
1741 @cindex @code{normal()}
1742 @cindex simplification
1743 @cindex temporary replacement
1745 Some basic from of simplification of expressions is called for frequently.
1746 GiNaC provides the method @code{.normal()}, which converts a rational function
1747 into an equivalent rational function of the form @samp{numerator/denominator}
1748 where numerator and denominator are coprime. If the input expression is already
1749 a fraction, it just finds the GCD of numerator and denominator and cancels it,
1750 otherwise it performs fraction addition and multiplication.
1752 @code{.normal()} can also be used on expressions which are not rational functions
1753 as it will replace all non-rational objects (like functions or non-integer
1754 powers) by temporary symbols to bring the expression to the domain of rational
1755 functions before performing the normalization, and re-substituting these
1756 symbols afterwards. This algorithm is also available as a separate method
1757 @code{.to_rational()}, described below.
1759 This means that both expressions @code{t1} and @code{t2} are indeed
1760 simplified in this little program:
1763 #include <ginac/ginac.h>
1764 using namespace GiNaC;
1769 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1770 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1771 cout << "t1 is " << t1.normal() << endl;
1772 cout << "t2 is " << t2.normal() << endl;
1776 Of course this works for multivariate polynomials too, so the ratio of
1777 the sample-polynomials from the section about GCD and LCM above would be
1778 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1781 @subsection Numerator and denominator
1784 @cindex @code{numer()}
1785 @cindex @code{denom()}
1787 The numerator and denominator of an expression can be obtained with
1794 These functions will first normalize the expression as described above and
1795 then return the numerator or denominator, respectively.
1798 @subsection Converting to a rational expression
1799 @cindex @code{to_rational()}
1801 Some of the methods described so far only work on polynomials or rational
1802 functions. GiNaC provides a way to extend the domain of these functions to
1803 general expressions by using the temporary replacement algorithm described
1804 above. You do this by calling
1807 ex ex::to_rational(lst &l);
1810 on the expression to be converted. The supplied @code{lst} will be filled
1811 with the generated temporary symbols and their replacement expressions in
1812 a format that can be used directly for the @code{subs()} method. It can also
1813 already contain a list of replacements from an earlier application of
1814 @code{.to_rational()}, so it's possible to use it on multiple expressions
1815 and get consistent results.
1822 ex a = pow(sin(x), 2) - pow(cos(x), 2);
1823 ex b = sin(x) + cos(x);
1826 divide(a.to_rational(l), b.to_rational(l), q);
1827 cout << q.subs(l) << endl;
1831 will print @samp{sin(x)-cos(x)}.
1834 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
1835 @c node-name, next, previous, up
1836 @section Symbolic differentiation
1837 @cindex differentiation
1838 @cindex @code{diff()}
1840 @cindex product rule
1842 GiNaC's objects know how to differentiate themselves. Thus, a
1843 polynomial (class @code{add}) knows that its derivative is the sum of
1844 the derivatives of all the monomials:
1847 #include <ginac/ginac.h>
1848 using namespace GiNaC;
1852 symbol x("x"), y("y"), z("z");
1853 ex P = pow(x, 5) + pow(x, 2) + y;
1855 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1856 cout << P.diff(y) << endl; // 1
1857 cout << P.diff(z) << endl; // 0
1861 If a second integer parameter @var{n} is given, the @code{diff} method
1862 returns the @var{n}th derivative.
1864 If @emph{every} object and every function is told what its derivative
1865 is, all derivatives of composed objects can be calculated using the
1866 chain rule and the product rule. Consider, for instance the expression
1867 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1868 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1869 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1870 out that the composition is the generating function for Euler Numbers,
1871 i.e. the so called @var{n}th Euler number is the coefficient of
1872 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1873 identity to code a function that generates Euler numbers in just three
1876 @cindex Euler numbers
1878 #include <ginac/ginac.h>
1879 using namespace GiNaC;
1881 ex EulerNumber(unsigned n)
1884 const ex generator = pow(cosh(x),-1);
1885 return generator.diff(x,n).subs(x==0);
1890 for (unsigned i=0; i<11; i+=2)
1891 cout << EulerNumber(i) << endl;
1896 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1897 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1898 @code{i} by two since all odd Euler numbers vanish anyways.
1901 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
1902 @c node-name, next, previous, up
1903 @section Series expansion
1904 @cindex @code{series()}
1905 @cindex Taylor expansion
1906 @cindex Laurent expansion
1907 @cindex @code{pseries} (class)
1909 Expressions know how to expand themselves as a Taylor series or (more
1910 generally) a Laurent series. As in most conventional Computer Algebra
1911 Systems, no distinction is made between those two. There is a class of
1912 its own for storing such series (@code{class pseries}) and a built-in
1913 function (called @code{Order}) for storing the order term of the series.
1914 As a consequence, if you want to work with series, i.e. multiply two
1915 series, you need to call the method @code{ex::series} again to convert
1916 it to a series object with the usual structure (expansion plus order
1917 term). A sample application from special relativity could read:
1920 #include <ginac/ginac.h>
1921 using namespace GiNaC;
1925 symbol v("v"), c("c");
1927 ex gamma = 1/sqrt(1 - pow(v/c,2));
1928 ex mass_nonrel = gamma.series(v==0, 10);
1930 cout << "the relativistic mass increase with v is " << endl
1931 << mass_nonrel << endl;
1933 cout << "the inverse square of this series is " << endl
1934 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
1938 Only calling the series method makes the last output simplify to
1939 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
1940 series raised to the power @math{-2}.
1942 @cindex M@'echain's formula
1943 As another instructive application, let us calculate the numerical
1944 value of Archimedes' constant
1948 (for which there already exists the built-in constant @code{Pi})
1949 using M@'echain's amazing formula
1951 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1954 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1956 We may expand the arcus tangent around @code{0} and insert the fractions
1957 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1958 carries an order term with it and the question arises what the system is
1959 supposed to do when the fractions are plugged into that order term. The
1960 solution is to use the function @code{series_to_poly()} to simply strip
1964 #include <ginac/ginac.h>
1965 using namespace GiNaC;
1967 ex mechain_pi(int degr)
1970 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
1971 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1972 -4*pi_expansion.subs(x==numeric(1,239));
1979 for (int i=2; i<12; i+=2) @{
1980 pi_frac = mechain_pi(i);
1981 cout << i << ":\t" << pi_frac << endl
1982 << "\t" << pi_frac.evalf() << endl;
1988 Note how we just called @code{.series(x,degr)} instead of
1989 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
1990 method @code{series()}: if the first argument is a symbol the expression
1991 is expanded in that symbol around point @code{0}. When you run this
1992 program, it will type out:
1996 3.1832635983263598326
1997 4: 5359397032/1706489875
1998 3.1405970293260603143
1999 6: 38279241713339684/12184551018734375
2000 3.141621029325034425
2001 8: 76528487109180192540976/24359780855939418203125
2002 3.141591772182177295
2003 10: 327853873402258685803048818236/104359128170408663038552734375
2004 3.1415926824043995174
2008 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
2009 @c node-name, next, previous, up
2010 @section Predefined mathematical functions
2012 GiNaC contains the following predefined mathematical functions:
2015 @multitable @columnfractions .30 .70
2016 @item @strong{Name} @tab @strong{Function}
2019 @item @code{csgn(x)}
2021 @item @code{sqrt(x)}
2022 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
2029 @item @code{asin(x)}
2031 @item @code{acos(x)}
2033 @item @code{atan(x)}
2034 @tab inverse tangent
2035 @item @code{atan2(y, x)}
2036 @tab inverse tangent with two arguments
2037 @item @code{sinh(x)}
2038 @tab hyperbolic sine
2039 @item @code{cosh(x)}
2040 @tab hyperbolic cosine
2041 @item @code{tanh(x)}
2042 @tab hyperbolic tangent
2043 @item @code{asinh(x)}
2044 @tab inverse hyperbolic sine
2045 @item @code{acosh(x)}
2046 @tab inverse hyperbolic cosine
2047 @item @code{atanh(x)}
2048 @tab inverse hyperbolic tangent
2050 @tab exponential function
2052 @tab natural logarithm
2053 @item @code{zeta(x)}
2054 @tab Riemann's zeta function
2055 @item @code{zeta(n, x)}
2056 @tab derivatives of Riemann's zeta function
2057 @item @code{tgamma(x)}
2059 @item @code{lgamma(x)}
2060 @tab logarithm of Gamma function
2061 @item @code{beta(x, y)}
2062 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
2064 @tab psi (digamma) function
2065 @item @code{psi(n, x)}
2066 @tab derivatives of psi function (polygamma functions)
2067 @item @code{factorial(n)}
2068 @tab factorial function
2069 @item @code{binomial(n, m)}
2070 @tab binomial coefficients
2071 @item @code{Order(x)}
2072 @tab order term function in truncated power series
2073 @item @code{Derivative(x, l)}
2074 @tab inert partial differentiation operator (used internally)
2079 For functions that have a branch cut in the complex plane GiNaC follows
2080 the conventions for C++ as defined in the ANSI standard. In particular:
2081 the natural logarithm (@code{log}) and the square root (@code{sqrt})
2082 both have their branch cuts running along the negative real axis where
2083 the points on the axis itself belong to the upper part.
2086 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
2087 @c node-name, next, previous, up
2088 @section Input and output of expressions
2091 @subsection Expression output
2093 @cindex output of expressions
2095 The easiest way to print an expression is to write it to a stream:
2100 ex e = 4.5+pow(x,2)*3/2;
2101 cout << e << endl; // prints '4.5+3/2*x^2'
2105 The output format is identical to the @command{ginsh} input syntax and
2106 to that used by most computer algebra systems, but not directly pastable
2107 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
2108 is printed as @samp{x^2}).
2110 To print an expression in a way that can be directly used in a C or C++
2111 program, you use the method
2114 void ex::printcsrc(ostream & os, unsigned type, const char *name);
2117 This outputs a line in the form of a variable definition @code{<type> <name> = <expression>}.
2118 The possible types are defined in @file{ginac/flags.h} (@code{csrc_types})
2119 and mostly affect the way in which floating point numbers are written:
2123 e.printcsrc(cout, csrc_types::ctype_float, "f");
2124 e.printcsrc(cout, csrc_types::ctype_double, "d");
2125 e.printcsrc(cout, csrc_types::ctype_cl_N, "n");
2129 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
2132 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2133 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2134 cl_N n = (cl_F("3.0")/cl_F("2.0"))*(x*x)+cl_F("4.5");
2137 Finally, there are the two methods @code{printraw()} and @code{printtree()} intended for GiNaC
2138 developers, that provide a dump of the internal structure of an expression for
2143 e.printraw(cout); cout << endl << endl;
2151 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))
2153 type=Q25GiNaC3add, hash=0 (0x0), flags=3, nops=2
2154 power: hash=2 (0x2), flags=3
2155 x (symbol): serial=1, hash=150875740 (0x8fe2e5c), flags=11
2156 2 (numeric): hash=2147483714 (0x80000042), flags=11
2157 3/2 (numeric): hash=2147483745 (0x80000061), flags=11
2160 4.5L0 (numeric): hash=2147483723 (0x8000004b), flags=11
2164 The @code{printtree()} method is also available in @command{ginsh} as the
2165 @code{print()} function.
2168 @subsection Expression input
2169 @cindex input of expressions
2171 GiNaC provides no way to directly read an expression from a stream because
2172 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
2173 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
2174 @code{y} you defined in your program and there is no way to specify the
2175 desired symbols to the @code{>>} stream input operator.
2177 Instead, GiNaC lets you construct an expression from a string, specifying the
2178 list of symbols to be used:
2182 symbol x("x"), y("y");
2183 ex e("2*x+sin(y)", lst(x, y));
2187 The input syntax is the same as that used by @command{ginsh} and the stream
2188 output operator @code{<<}. The symbols in the string are matched by name to
2189 the symbols in the list and if GiNaC encounters a symbol not specified in
2190 the list it will throw an exception.
2192 With this constructor, it's also easy to implement interactive GiNaC programs:
2197 #include <stdexcept>
2198 #include <ginac/ginac.h>
2199 using namespace GiNaC;
2206 cout << "Enter an expression containing 'x': ";
2211 cout << "The derivative of " << e << " with respect to x is ";
2212 cout << e.diff(x) << ".\n";
2213 @} catch (exception &p) @{
2214 cerr << p.what() << endl;
2220 @subsection Archiving
2221 @cindex @code{archive} (class)
2224 GiNaC allows creating @dfn{archives} of expressions which can be stored
2225 to or retrieved from files. To create an archive, you declare an object
2226 of class @code{archive} and archive expressions in it, giving each
2227 expression a unique name:
2230 #include <ginac/ginac.h>
2232 using namespace GiNaC;
2236 symbol x("x"), y("y"), z("z");
2238 ex foo = sin(x + 2*y) + 3*z + 41;
2242 a.archive_ex(foo, "foo");
2243 a.archive_ex(bar, "the second one");
2247 The archive can then be written to a file:
2251 ofstream out("foobar.gar");
2257 The file @file{foobar.gar} contains all information that is needed to
2258 reconstruct the expressions @code{foo} and @code{bar}.
2260 @cindex @command{viewgar}
2261 The tool @command{viewgar} that comes with GiNaC can be used to view
2262 the contents of GiNaC archive files:
2265 $ viewgar foobar.gar
2266 foo = 41+sin(x+2*y)+3*z
2267 the second one = 42+sin(x+2*y)+3*z
2270 The point of writing archive files is of course that they can later be
2276 ifstream in("foobar.gar");
2281 And the stored expressions can be retrieved by their name:
2287 ex ex1 = a2.unarchive_ex(syms, "foo");
2288 ex ex2 = a2.unarchive_ex(syms, "the second one");
2290 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
2291 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
2292 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
2296 Note that you have to supply a list of the symbols which are to be inserted
2297 in the expressions. Symbols in archives are stored by their name only and
2298 if you don't specify which symbols you have, unarchiving the expression will
2299 create new symbols with that name. E.g. if you hadn't included @code{x} in
2300 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
2301 have had no effect because the @code{x} in @code{ex1} would have been a
2302 different symbol than the @code{x} which was defined at the beginning of
2303 the program, altough both would appear as @samp{x} when printed.
2307 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
2308 @c node-name, next, previous, up
2309 @chapter Extending GiNaC
2311 By reading so far you should have gotten a fairly good understanding of
2312 GiNaC's design-patterns. From here on you should start reading the
2313 sources. All we can do now is issue some recommendations how to tackle
2314 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
2315 develop some useful extension please don't hesitate to contact the GiNaC
2316 authors---they will happily incorporate them into future versions.
2319 * What does not belong into GiNaC:: What to avoid.
2320 * Symbolic functions:: Implementing symbolic functions.
2324 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
2325 @c node-name, next, previous, up
2326 @section What doesn't belong into GiNaC
2328 @cindex @command{ginsh}
2329 First of all, GiNaC's name must be read literally. It is designed to be
2330 a library for use within C++. The tiny @command{ginsh} accompanying
2331 GiNaC makes this even more clear: it doesn't even attempt to provide a
2332 language. There are no loops or conditional expressions in
2333 @command{ginsh}, it is merely a window into the library for the
2334 programmer to test stuff (or to show off). Still, the design of a
2335 complete CAS with a language of its own, graphical capabilites and all
2336 this on top of GiNaC is possible and is without doubt a nice project for
2339 There are many built-in functions in GiNaC that do not know how to
2340 evaluate themselves numerically to a precision declared at runtime
2341 (using @code{Digits}). Some may be evaluated at certain points, but not
2342 generally. This ought to be fixed. However, doing numerical
2343 computations with GiNaC's quite abstract classes is doomed to be
2344 inefficient. For this purpose, the underlying foundation classes
2345 provided by @acronym{CLN} are much better suited.
2348 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
2349 @c node-name, next, previous, up
2350 @section Symbolic functions
2352 The easiest and most instructive way to start with is probably to
2353 implement your own function. Objects of class @code{function} are
2354 inserted into the system via a kind of `registry'. They get a serial
2355 number that is used internally to identify them but you usually need not
2356 worry about this. What you have to care for are functions that are
2357 called when the user invokes certain methods. These are usual
2358 C++-functions accepting a number of @code{ex} as arguments and returning
2359 one @code{ex}. As an example, if we have a look at a simplified
2360 implementation of the cosine trigonometric function, we first need a
2361 function that is called when one wishes to @code{eval} it. It could
2362 look something like this:
2365 static ex cos_eval_method(const ex & x)
2367 // if (!x%(2*Pi)) return 1
2368 // if (!x%Pi) return -1
2369 // if (!x%Pi/2) return 0
2370 // care for other cases...
2371 return cos(x).hold();
2375 @cindex @code{hold()}
2377 The last line returns @code{cos(x)} if we don't know what else to do and
2378 stops a potential recursive evaluation by saying @code{.hold()}, which
2379 sets a flag to the expression signaling that it has been evaluated. We
2380 should also implement a method for numerical evaluation and since we are
2381 lazy we sweep the problem under the rug by calling someone else's
2382 function that does so, in this case the one in class @code{numeric}:
2385 static ex cos_evalf(const ex & x)
2387 return cos(ex_to_numeric(x));
2391 Differentiation will surely turn up and so we need to tell @code{cos}
2392 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
2393 instance are then handled automatically by @code{basic::diff} and
2397 static ex cos_deriv(const ex & x, unsigned diff_param)
2403 @cindex product rule
2404 The second parameter is obligatory but uninteresting at this point. It
2405 specifies which parameter to differentiate in a partial derivative in
2406 case the function has more than one parameter and its main application
2407 is for correct handling of the chain rule. For Taylor expansion, it is
2408 enough to know how to differentiate. But if the function you want to
2409 implement does have a pole somewhere in the complex plane, you need to
2410 write another method for Laurent expansion around that point.
2412 Now that all the ingrediences for @code{cos} have been set up, we need
2413 to tell the system about it. This is done by a macro and we are not
2414 going to descibe how it expands, please consult your preprocessor if you
2418 REGISTER_FUNCTION(cos, eval_func(cos_eval).
2419 evalf_func(cos_evalf).
2420 derivative_func(cos_deriv));
2423 The first argument is the function's name used for calling it and for
2424 output. The second binds the corresponding methods as options to this
2425 object. Options are separated by a dot and can be given in an arbitrary
2426 order. GiNaC functions understand several more options which are always
2427 specified as @code{.option(params)}, for example a method for series
2428 expansion @code{.series_func(cos_series)}. Again, if no series
2429 expansion method is given, GiNaC defaults to simple Taylor expansion,
2430 which is correct if there are no poles involved as is the case for the
2431 @code{cos} function. The way GiNaC handles poles in case there are any
2432 is best understood by studying one of the examples, like the Gamma
2433 (@code{tgamma}) function for instance. (In essence the function first
2434 checks if there is a pole at the evaluation point and falls back to
2435 Taylor expansion if there isn't. Then, the pole is regularized by some
2436 suitable transformation.) Also, the new function needs to be declared
2437 somewhere. This may also be done by a convenient preprocessor macro:
2440 DECLARE_FUNCTION_1P(cos)
2443 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
2444 implementation of @code{cos} is very incomplete and lacks several safety
2445 mechanisms. Please, have a look at the real implementation in GiNaC.
2446 (By the way: in case you are worrying about all the macros above we can
2447 assure you that functions are GiNaC's most macro-intense classes. We
2448 have done our best to avoid macros where we can.)
2450 That's it. May the source be with you!
2453 @node A Comparison With Other CAS, Advantages, Symbolic functions, Top
2454 @c node-name, next, previous, up
2455 @chapter A Comparison With Other CAS
2458 This chapter will give you some information on how GiNaC compares to
2459 other, traditional Computer Algebra Systems, like @emph{Maple},
2460 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
2461 disadvantages over these systems.
2464 * Advantages:: Stengths of the GiNaC approach.
2465 * Disadvantages:: Weaknesses of the GiNaC approach.
2466 * Why C++?:: Attractiveness of C++.
2469 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
2470 @c node-name, next, previous, up
2473 GiNaC has several advantages over traditional Computer
2474 Algebra Systems, like
2479 familiar language: all common CAS implement their own proprietary
2480 grammar which you have to learn first (and maybe learn again when your
2481 vendor decides to `enhance' it). With GiNaC you can write your program
2482 in common C++, which is standardized.
2486 structured data types: you can build up structured data types using
2487 @code{struct}s or @code{class}es together with STL features instead of
2488 using unnamed lists of lists of lists.
2491 strongly typed: in CAS, you usually have only one kind of variables
2492 which can hold contents of an arbitrary type. This 4GL like feature is
2493 nice for novice programmers, but dangerous.
2496 development tools: powerful development tools exist for C++, like fancy
2497 editors (e.g. with automatic indentation and syntax highlighting),
2498 debuggers, visualization tools, documentation generators...
2501 modularization: C++ programs can easily be split into modules by
2502 separating interface and implementation.
2505 price: GiNaC is distributed under the GNU Public License which means
2506 that it is free and available with source code. And there are excellent
2507 C++-compilers for free, too.
2510 extendable: you can add your own classes to GiNaC, thus extending it on
2511 a very low level. Compare this to a traditional CAS that you can
2512 usually only extend on a high level by writing in the language defined
2513 by the parser. In particular, it turns out to be almost impossible to
2514 fix bugs in a traditional system.
2517 multiple interfaces: Though real GiNaC programs have to be written in
2518 some editor, then be compiled, linked and executed, there are more ways
2519 to work with the GiNaC engine. Many people want to play with
2520 expressions interactively, as in traditional CASs. Currently, two such
2521 windows into GiNaC have been implemented and many more are possible: the
2522 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
2523 types to a command line and second, as a more consistent approach, an
2524 interactive interface to the @acronym{Cint} C++ interpreter has been put
2525 together (called @acronym{GiNaC-cint}) that allows an interactive
2526 scripting interface consistent with the C++ language.
2529 seemless integration: it is somewhere between difficult and impossible
2530 to call CAS functions from within a program written in C++ or any other
2531 programming language and vice versa. With GiNaC, your symbolic routines
2532 are part of your program. You can easily call third party libraries,
2533 e.g. for numerical evaluation or graphical interaction. All other
2534 approaches are much more cumbersome: they range from simply ignoring the
2535 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
2536 system (i.e. @emph{Yacas}).
2539 efficiency: often large parts of a program do not need symbolic
2540 calculations at all. Why use large integers for loop variables or
2541 arbitrary precision arithmetics where @code{int} and @code{double} are
2542 sufficient? For pure symbolic applications, GiNaC is comparable in
2543 speed with other CAS.
2548 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
2549 @c node-name, next, previous, up
2550 @section Disadvantages
2552 Of course it also has some disadvantages:
2557 advanced features: GiNaC cannot compete with a program like
2558 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
2559 which grows since 1981 by the work of dozens of programmers, with
2560 respect to mathematical features. Integration, factorization,
2561 non-trivial simplifications, limits etc. are missing in GiNaC (and are
2562 not planned for the near future).
2565 portability: While the GiNaC library itself is designed to avoid any
2566 platform dependent features (it should compile on any ANSI compliant C++
2567 compiler), the currently used version of the CLN library (fast large
2568 integer and arbitrary precision arithmetics) can be compiled only on
2569 systems with a recently new C++ compiler from the GNU Compiler
2570 Collection (@acronym{GCC}).@footnote{This is because CLN uses
2571 PROVIDE/REQUIRE like macros to let the compiler gather all static
2572 initializations, which works for GNU C++ only.} GiNaC uses recent
2573 language features like explicit constructors, mutable members, RTTI,
2574 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
2575 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
2576 ANSI compliant, support all needed features.
2581 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
2582 @c node-name, next, previous, up
2585 Why did we choose to implement GiNaC in C++ instead of Java or any other
2586 language? C++ is not perfect: type checking is not strict (casting is
2587 possible), separation between interface and implementation is not
2588 complete, object oriented design is not enforced. The main reason is
2589 the often scolded feature of operator overloading in C++. While it may
2590 be true that operating on classes with a @code{+} operator is rarely
2591 meaningful, it is perfectly suited for algebraic expressions. Writing
2592 @math{3x+5y} as @code{3*x+5*y} instead of
2593 @code{x.times(3).plus(y.times(5))} looks much more natural.
2594 Furthermore, the main developers are more familiar with C++ than with
2595 any other programming language.
2598 @node Internal Structures, Expressions are reference counted, Why C++? , Top
2599 @c node-name, next, previous, up
2600 @appendix Internal Structures
2603 * Expressions are reference counted::
2604 * Internal representation of products and sums::
2607 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
2608 @c node-name, next, previous, up
2609 @appendixsection Expressions are reference counted
2611 @cindex reference counting
2612 @cindex copy-on-write
2613 @cindex garbage collection
2614 An expression is extremely light-weight since internally it works like a
2615 handle to the actual representation and really holds nothing more than a
2616 pointer to some other object. What this means in practice is that
2617 whenever you create two @code{ex} and set the second equal to the first
2618 no copying process is involved. Instead, the copying takes place as soon
2619 as you try to change the second. Consider the simple sequence of code:
2622 #include <ginac/ginac.h>
2623 using namespace GiNaC;
2627 symbol x("x"), y("y"), z("z");
2630 e1 = sin(x + 2*y) + 3*z + 41;
2631 e2 = e1; // e2 points to same object as e1
2632 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
2633 e2 += 1; // e2 is copied into a new object
2634 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
2638 The line @code{e2 = e1;} creates a second expression pointing to the
2639 object held already by @code{e1}. The time involved for this operation
2640 is therefore constant, no matter how large @code{e1} was. Actual
2641 copying, however, must take place in the line @code{e2 += 1;} because
2642 @code{e1} and @code{e2} are not handles for the same object any more.
2643 This concept is called @dfn{copy-on-write semantics}. It increases
2644 performance considerably whenever one object occurs multiple times and
2645 represents a simple garbage collection scheme because when an @code{ex}
2646 runs out of scope its destructor checks whether other expressions handle
2647 the object it points to too and deletes the object from memory if that
2648 turns out not to be the case. A slightly less trivial example of
2649 differentiation using the chain-rule should make clear how powerful this
2653 #include <ginac/ginac.h>
2654 using namespace GiNaC;
2658 symbol x("x"), y("y");
2662 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
2663 cout << e1 << endl // prints x+3*y
2664 << e2 << endl // prints (x+3*y)^3
2665 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
2669 Here, @code{e1} will actually be referenced three times while @code{e2}
2670 will be referenced two times. When the power of an expression is built,
2671 that expression needs not be copied. Likewise, since the derivative of
2672 a power of an expression can be easily expressed in terms of that
2673 expression, no copying of @code{e1} is involved when @code{e3} is
2674 constructed. So, when @code{e3} is constructed it will print as
2675 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
2676 holds a reference to @code{e2} and the factor in front is just
2679 As a user of GiNaC, you cannot see this mechanism of copy-on-write
2680 semantics. When you insert an expression into a second expression, the
2681 result behaves exactly as if the contents of the first expression were
2682 inserted. But it may be useful to remember that this is not what
2683 happens. Knowing this will enable you to write much more efficient
2684 code. If you still have an uncertain feeling with copy-on-write
2685 semantics, we recommend you have a look at the
2686 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
2687 Marshall Cline. Chapter 16 covers this issue and presents an
2688 implementation which is pretty close to the one in GiNaC.
2691 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
2692 @c node-name, next, previous, up
2693 @appendixsection Internal representation of products and sums
2695 @cindex representation
2698 @cindex @code{power}
2699 Although it should be completely transparent for the user of
2700 GiNaC a short discussion of this topic helps to understand the sources
2701 and also explain performance to a large degree. Consider the
2702 unexpanded symbolic expression
2704 $2d^3 \left( 4a + 5b - 3 \right)$
2707 @math{2*d^3*(4*a+5*b-3)}
2709 which could naively be represented by a tree of linear containers for
2710 addition and multiplication, one container for exponentiation with base
2711 and exponent and some atomic leaves of symbols and numbers in this
2716 @cindex pair-wise representation
2717 However, doing so results in a rather deeply nested tree which will
2718 quickly become inefficient to manipulate. We can improve on this by
2719 representing the sum as a sequence of terms, each one being a pair of a
2720 purely numeric multiplicative coefficient and its rest. In the same
2721 spirit we can store the multiplication as a sequence of terms, each
2722 having a numeric exponent and a possibly complicated base, the tree
2723 becomes much more flat:
2727 The number @code{3} above the symbol @code{d} shows that @code{mul}
2728 objects are treated similarly where the coefficients are interpreted as
2729 @emph{exponents} now. Addition of sums of terms or multiplication of
2730 products with numerical exponents can be coded to be very efficient with
2731 such a pair-wise representation. Internally, this handling is performed
2732 by most CAS in this way. It typically speeds up manipulations by an
2733 order of magnitude. The overall multiplicative factor @code{2} and the
2734 additive term @code{-3} look somewhat out of place in this
2735 representation, however, since they are still carrying a trivial
2736 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
2737 this is avoided by adding a field that carries an overall numeric
2738 coefficient. This results in the realistic picture of internal
2741 $2d^3 \left( 4a + 5b - 3 \right)$:
2744 @math{2*d^3*(4*a+5*b-3)}:
2750 This also allows for a better handling of numeric radicals, since
2751 @code{sqrt(2)} can now be carried along calculations. Now it should be
2752 clear, why both classes @code{add} and @code{mul} are derived from the
2753 same abstract class: the data representation is the same, only the
2754 semantics differs. In the class hierarchy, methods for polynomial
2755 expansion and the like are reimplemented for @code{add} and @code{mul},
2756 but the data structure is inherited from @code{expairseq}.
2759 @node Package Tools, ginac-config, Internal representation of products and sums, Top
2760 @c node-name, next, previous, up
2761 @appendix Package Tools
2763 If you are creating a software package that uses the GiNaC library,
2764 setting the correct command line options for the compiler and linker
2765 can be difficult. GiNaC includes two tools to make this process easier.
2768 * ginac-config:: A shell script to detect compiler and linker flags.
2769 * AM_PATH_GINAC:: Macro for GNU automake.
2773 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
2774 @c node-name, next, previous, up
2775 @section @command{ginac-config}
2776 @cindex ginac-config
2778 @command{ginac-config} is a shell script that you can use to determine
2779 the compiler and linker command line options required to compile and
2780 link a program with the GiNaC library.
2782 @command{ginac-config} takes the following flags:
2786 Prints out the version of GiNaC installed.
2788 Prints '-I' flags pointing to the installed header files.
2790 Prints out the linker flags necessary to link a program against GiNaC.
2791 @item --prefix[=@var{PREFIX}]
2792 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
2793 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
2794 Otherwise, prints out the configured value of @env{$prefix}.
2795 @item --exec-prefix[=@var{PREFIX}]
2796 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
2797 Otherwise, prints out the configured value of @env{$exec_prefix}.
2800 Typically, @command{ginac-config} will be used within a configure
2801 script, as described below. It, however, can also be used directly from
2802 the command line using backquotes to compile a simple program. For
2806 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
2809 This command line might expand to (for example):
2812 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
2813 -lginac -lcln -lstdc++
2816 Not only is the form using @command{ginac-config} easier to type, it will
2817 work on any system, no matter how GiNaC was configured.
2820 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
2821 @c node-name, next, previous, up
2822 @section @samp{AM_PATH_GINAC}
2823 @cindex AM_PATH_GINAC
2825 For packages configured using GNU automake, GiNaC also provides
2826 a macro to automate the process of checking for GiNaC.
2829 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
2837 Determines the location of GiNaC using @command{ginac-config}, which is
2838 either found in the user's path, or from the environment variable
2839 @env{GINACLIB_CONFIG}.
2842 Tests the installed libraries to make sure that their version
2843 is later than @var{MINIMUM-VERSION}. (A default version will be used
2847 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
2848 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
2849 variable to the output of @command{ginac-config --libs}, and calls
2850 @samp{AC_SUBST()} for these variables so they can be used in generated
2851 makefiles, and then executes @var{ACTION-IF-FOUND}.
2854 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
2855 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
2859 This macro is in file @file{ginac.m4} which is installed in
2860 @file{$datadir/aclocal}. Note that if automake was installed with a
2861 different @samp{--prefix} than GiNaC, you will either have to manually
2862 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
2863 aclocal the @samp{-I} option when running it.
2866 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
2867 * Example package:: Example of a package using AM_PATH_GINAC.
2871 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2872 @c node-name, next, previous, up
2873 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2875 Simply make sure that @command{ginac-config} is in your path, and run
2876 the configure script.
2883 The directory where the GiNaC libraries are installed needs
2884 to be found by your system's dynamic linker.
2886 This is generally done by
2889 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2895 setting the environment variable @env{LD_LIBRARY_PATH},
2898 or, as a last resort,
2901 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2902 running configure, for instance:
2905 LDFLAGS=-R/home/cbauer/lib ./configure
2910 You can also specify a @command{ginac-config} not in your path by
2911 setting the @env{GINACLIB_CONFIG} environment variable to the
2912 name of the executable
2915 If you move the GiNaC package from its installed location,
2916 you will either need to modify @command{ginac-config} script
2917 manually to point to the new location or rebuild GiNaC.
2928 --with-ginac-prefix=@var{PREFIX}
2929 --with-ginac-exec-prefix=@var{PREFIX}
2932 are provided to override the prefix and exec-prefix that were stored
2933 in the @command{ginac-config} shell script by GiNaC's configure. You are
2934 generally better off configuring GiNaC with the right path to begin with.
2938 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2939 @c node-name, next, previous, up
2940 @subsection Example of a package using @samp{AM_PATH_GINAC}
2942 The following shows how to build a simple package using automake
2943 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2946 #include <ginac/ginac.h>
2947 using namespace GiNaC;
2953 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2958 You should first read the introductory portions of the automake
2959 Manual, if you are not already familiar with it.
2961 Two files are needed, @file{configure.in}, which is used to build the
2965 dnl Process this file with autoconf to produce a configure script.
2967 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2973 AM_PATH_GINAC(0.4.0, [
2974 LIBS="$LIBS $GINACLIB_LIBS"
2975 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2976 ], AC_MSG_ERROR([need to have GiNaC installed]))
2981 The only command in this which is not standard for automake
2982 is the @samp{AM_PATH_GINAC} macro.
2984 That command does the following:
2987 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2988 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2989 with the error message `need to have GiNaC installed'
2992 And the @file{Makefile.am}, which will be used to build the Makefile.
2995 ## Process this file with automake to produce Makefile.in
2996 bin_PROGRAMS = simple
2997 simple_SOURCES = simple.cpp
3000 This @file{Makefile.am}, says that we are building a single executable,
3001 from a single sourcefile @file{simple.cpp}. Since every program
3002 we are building uses GiNaC we simply added the GiNaC options
3003 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
3004 want to specify them on a per-program basis: for instance by
3008 simple_LDADD = $(GINACLIB_LIBS)
3009 INCLUDES = $(GINACLIB_CPPFLAGS)
3012 to the @file{Makefile.am}.
3014 To try this example out, create a new directory and add the three
3017 Now execute the following commands:
3020 $ automake --add-missing
3025 You now have a package that can be built in the normal fashion
3034 @node Bibliography, Concept Index, Example package, Top
3035 @c node-name, next, previous, up
3036 @appendix Bibliography
3041 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
3044 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
3047 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
3050 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
3053 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
3054 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
3057 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
3058 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
3059 Academic Press, London
3064 @node Concept Index, , Bibliography, Top
3065 @c node-name, next, previous, up
3066 @unnumbered Concept Index