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) 1999-2001 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{} 1999-2001 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{} 1999-2001 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>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package Tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
216 #include <ginac/ginac.h>
218 using namespace GiNaC;
220 ex HermitePoly(const symbol & x, int n)
222 ex HKer=exp(-pow(x, 2));
223 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
224 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
231 for (int i=0; i<6; ++i)
232 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
238 When run, this will type out
244 H_3(z) == -12*z+8*z^3
245 H_4(z) == -48*z^2+16*z^4+12
246 H_5(z) == 120*z-160*z^3+32*z^5
249 This method of generating the coefficients is of course far from optimal
250 for production purposes.
252 In order to show some more examples of what GiNaC can do we will now use
253 the @command{ginsh}, a simple GiNaC interactive shell that provides a
254 convenient window into GiNaC's capabilities.
257 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
258 @c node-name, next, previous, up
259 @section What it can do for you
261 @cindex @command{ginsh}
262 After invoking @command{ginsh} one can test and experiment with GiNaC's
263 features much like in other Computer Algebra Systems except that it does
264 not provide programming constructs like loops or conditionals. For a
265 concise description of the @command{ginsh} syntax we refer to its
266 accompanied man page. Suffice to say that assignments and comparisons in
267 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
270 It can manipulate arbitrary precision integers in a very fast way.
271 Rational numbers are automatically converted to fractions of coprime
276 369988485035126972924700782451696644186473100389722973815184405301748249
278 123329495011708990974900260817232214728824366796574324605061468433916083
285 Exact numbers are always retained as exact numbers and only evaluated as
286 floating point numbers if requested. For instance, with numeric
287 radicals is dealt pretty much as with symbols. Products of sums of them
291 > expand((1+a^(1/5)-a^(2/5))^3);
292 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
293 > expand((1+3^(1/5)-3^(2/5))^3);
295 > evalf((1+3^(1/5)-3^(2/5))^3);
296 0.33408977534118624228
299 The function @code{evalf} that was used above converts any number in
300 GiNaC's expressions into floating point numbers. This can be done to
301 arbitrary predefined accuracy:
305 0.14285714285714285714
309 0.1428571428571428571428571428571428571428571428571428571428571428571428
310 5714285714285714285714285714285714285
313 Exact numbers other than rationals that can be manipulated in GiNaC
314 include predefined constants like Archimedes' @code{Pi}. They can both
315 be used in symbolic manipulations (as an exact number) as well as in
316 numeric expressions (as an inexact number):
322 9.869604401089358619+x
326 11.869604401089358619
329 Built-in functions evaluate immediately to exact numbers if
330 this is possible. Conversions that can be safely performed are done
331 immediately; conversions that are not generally valid are not done:
342 (Note that converting the last input to @code{x} would allow one to
343 conclude that @code{42*Pi} is equal to @code{0}.)
345 Linear equation systems can be solved along with basic linear
346 algebra manipulations over symbolic expressions. In C++ GiNaC offers
347 a matrix class for this purpose but we can see what it can do using
348 @command{ginsh}'s bracket notation to type them in:
351 > lsolve(a+x*y==z,x);
353 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
355 > M = [ [1, 3], [-3, 2] ];
359 > charpoly(M,lambda);
361 > A = [ [1, 1], [2, -1] ];
364 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 Multivariate polynomials and rational functions may be expanded,
370 collected and normalized (i.e. converted to a ratio of two coprime
374 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
375 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
376 > b = x^2 + 4*x*y - y^2;
379 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
381 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
383 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
388 You can differentiate functions and expand them as Taylor or Laurent
389 series in a very natural syntax (the second argument of @code{series} is
390 a relation defining the evaluation point, the third specifies the
393 @cindex Zeta function
397 > series(sin(x),x==0,4);
399 > series(1/tan(x),x==0,4);
400 x^(-1)-1/3*x+Order(x^2)
401 > series(tgamma(x),x==0,3);
402 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
403 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
405 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
406 -(0.90747907608088628905)*x^2+Order(x^3)
407 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
408 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
409 -Euler-1/12+Order((x-1/2*Pi)^3)
412 Here we have made use of the @command{ginsh}-command @code{"} to pop the
413 previously evaluated element from @command{ginsh}'s internal stack.
415 If you ever wanted to convert units in C or C++ and found this is
416 cumbersome, here is the solution. Symbolic types can always be used as
417 tags for different types of objects. Converting from wrong units to the
418 metric system is now easy:
426 140613.91592783185568*kg*m^(-2)
430 @node Installation, Prerequisites, What it can do for you, Top
431 @c node-name, next, previous, up
432 @chapter Installation
435 GiNaC's installation follows the spirit of most GNU software. It is
436 easily installed on your system by three steps: configuration, build,
440 * Prerequisites:: Packages upon which GiNaC depends.
441 * Configuration:: How to configure GiNaC.
442 * Building GiNaC:: How to compile GiNaC.
443 * Installing GiNaC:: How to install GiNaC on your system.
447 @node Prerequisites, Configuration, Installation, Installation
448 @c node-name, next, previous, up
449 @section Prerequisites
451 In order to install GiNaC on your system, some prerequisites need to be
452 met. First of all, you need to have a C++-compiler adhering to the
453 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
454 development so if you have a different compiler you are on your own.
455 For the configuration to succeed you need a Posix compliant shell
456 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
457 by the built process as well, since some of the source files are
458 automatically generated by Perl scripts. Last but not least, Bruno
459 Haible's library @acronym{CLN} is extensively used and needs to be
460 installed on your system. Please get it either from
461 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
462 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
463 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
464 site} (it is covered by GPL) and install it prior to trying to install
465 GiNaC. The configure script checks if it can find it and if it cannot
466 it will refuse to continue.
469 @node Configuration, Building GiNaC, Prerequisites, Installation
470 @c node-name, next, previous, up
471 @section Configuration
472 @cindex configuration
475 To configure GiNaC means to prepare the source distribution for
476 building. It is done via a shell script called @command{configure} that
477 is shipped with the sources and was originally generated by GNU
478 Autoconf. Since a configure script generated by GNU Autoconf never
479 prompts, all customization must be done either via command line
480 parameters or environment variables. It accepts a list of parameters,
481 the complete set of which can be listed by calling it with the
482 @option{--help} option. The most important ones will be shortly
483 described in what follows:
488 @option{--disable-shared}: When given, this option switches off the
489 build of a shared library, i.e. a @file{.so} file. This may be convenient
490 when developing because it considerably speeds up compilation.
493 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
494 and headers are installed. It defaults to @file{/usr/local} which means
495 that the library is installed in the directory @file{/usr/local/lib},
496 the header files in @file{/usr/local/include/ginac} and the documentation
497 (like this one) into @file{/usr/local/share/doc/GiNaC}.
500 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
501 the library installed in some other directory than
502 @file{@var{PREFIX}/lib/}.
505 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
506 to have the header files installed in some other directory than
507 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
508 @option{--includedir=/usr/include} you will end up with the header files
509 sitting in the directory @file{/usr/include/ginac/}. Note that the
510 subdirectory @file{ginac} is enforced by this process in order to
511 keep the header files separated from others. This avoids some
512 clashes and allows for an easier deinstallation of GiNaC. This ought
513 to be considered A Good Thing (tm).
516 @option{--datadir=@var{DATADIR}}: This option may be given in case you
517 want to have the documentation installed in some other directory than
518 @file{@var{PREFIX}/share/doc/GiNaC/}.
522 In addition, you may specify some environment variables.
523 @env{CXX} holds the path and the name of the C++ compiler
524 in case you want to override the default in your path. (The
525 @command{configure} script searches your path for @command{c++},
526 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
527 and @command{cc++} in that order.) It may be very useful to
528 define some compiler flags with the @env{CXXFLAGS} environment
529 variable, like optimization, debugging information and warning
530 levels. If omitted, it defaults to @option{-g -O2}.
532 The whole process is illustrated in the following two
533 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
534 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
537 Here is a simple configuration for a site-wide GiNaC library assuming
538 everything is in default paths:
541 $ export CXXFLAGS="-Wall -O2"
545 And here is a configuration for a private static GiNaC library with
546 several components sitting in custom places (site-wide @acronym{GCC} and
547 private @acronym{CLN}). The compiler is pursuaded to be picky and full
548 assertions and debugging information are switched on:
551 $ export CXX=/usr/local/gnu/bin/c++
552 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
553 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
554 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
555 $ ./configure --disable-shared --prefix=$(HOME)
559 @node Building GiNaC, Installing GiNaC, Configuration, Installation
560 @c node-name, next, previous, up
561 @section Building GiNaC
562 @cindex building GiNaC
564 After proper configuration you should just build the whole
569 at the command prompt and go for a cup of coffee. The exact time it
570 takes to compile GiNaC depends not only on the speed of your machines
571 but also on other parameters, for instance what value for @env{CXXFLAGS}
572 you entered. Optimization may be very time-consuming.
574 Just to make sure GiNaC works properly you may run a collection of
575 regression tests by typing
581 This will compile some sample programs, run them and check the output
582 for correctness. The regression tests fall in three categories. First,
583 the so called @emph{exams} are performed, simple tests where some
584 predefined input is evaluated (like a pupils' exam). Second, the
585 @emph{checks} test the coherence of results among each other with
586 possible random input. Third, some @emph{timings} are performed, which
587 benchmark some predefined problems with different sizes and display the
588 CPU time used in seconds. Each individual test should return a message
589 @samp{passed}. This is mostly intended to be a QA-check if something
590 was broken during development, not a sanity check of your system. Some
591 of the tests in sections @emph{checks} and @emph{timings} may require
592 insane amounts of memory and CPU time. Feel free to kill them if your
593 machine catches fire. Another quite important intent is to allow people
594 to fiddle around with optimization.
596 Generally, the top-level Makefile runs recursively to the
597 subdirectories. It is therfore safe to go into any subdirectory
598 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
599 @var{target} there in case something went wrong.
602 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
603 @c node-name, next, previous, up
604 @section Installing GiNaC
607 To install GiNaC on your system, simply type
613 As described in the section about configuration the files will be
614 installed in the following directories (the directories will be created
615 if they don't already exist):
620 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
621 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
622 So will @file{libginac.so} unless the configure script was
623 given the option @option{--disable-shared}. The proper symlinks
624 will be established as well.
627 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
628 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
631 All documentation (HTML and Postscript) will be stuffed into
632 @file{@var{PREFIX}/share/doc/GiNaC/} (or
633 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
637 For the sake of completeness we will list some other useful make
638 targets: @command{make clean} deletes all files generated by
639 @command{make}, i.e. all the object files. In addition @command{make
640 distclean} removes all files generated by the configuration and
641 @command{make maintainer-clean} goes one step further and deletes files
642 that may require special tools to rebuild (like the @command{libtool}
643 for instance). Finally @command{make uninstall} removes the installed
644 library, header files and documentation@footnote{Uninstallation does not
645 work after you have called @command{make distclean} since the
646 @file{Makefile} is itself generated by the configuration from
647 @file{Makefile.in} and hence deleted by @command{make distclean}. There
648 are two obvious ways out of this dilemma. First, you can run the
649 configuration again with the same @var{PREFIX} thus creating a
650 @file{Makefile} with a working @samp{uninstall} target. Second, you can
651 do it by hand since you now know where all the files went during
655 @node Basic Concepts, Expressions, Installing GiNaC, Top
656 @c node-name, next, previous, up
657 @chapter Basic Concepts
659 This chapter will describe the different fundamental objects that can be
660 handled by GiNaC. But before doing so, it is worthwhile introducing you
661 to the more commonly used class of expressions, representing a flexible
662 meta-class for storing all mathematical objects.
665 * Expressions:: The fundamental GiNaC class.
666 * The Class Hierarchy:: Overview of GiNaC's classes.
667 * Symbols:: Symbolic objects.
668 * Numbers:: Numerical objects.
669 * Constants:: Pre-defined constants.
670 * Fundamental containers:: The power, add and mul classes.
671 * Lists:: Lists of expressions.
672 * Mathematical functions:: Mathematical functions.
673 * Relations:: Equality, Inequality and all that.
674 * Matrices:: Matrices.
675 * Indexed objects:: Handling indexed quantities.
676 * Non-commutative objects:: Algebras with non-commutative products.
680 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
681 @c node-name, next, previous, up
683 @cindex expression (class @code{ex})
686 The most common class of objects a user deals with is the expression
687 @code{ex}, representing a mathematical object like a variable, number,
688 function, sum, product, etc@dots{} Expressions may be put together to form
689 new expressions, passed as arguments to functions, and so on. Here is a
690 little collection of valid expressions:
693 ex MyEx1 = 5; // simple number
694 ex MyEx2 = x + 2*y; // polynomial in x and y
695 ex MyEx3 = (x + 1)/(x - 1); // rational expression
696 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
697 ex MyEx5 = MyEx4 + 1; // similar to above
700 Expressions are handles to other more fundamental objects, that often
701 contain other expressions thus creating a tree of expressions
702 (@xref{Internal Structures}, for particular examples). Most methods on
703 @code{ex} therefore run top-down through such an expression tree. For
704 example, the method @code{has()} scans recursively for occurrences of
705 something inside an expression. Thus, if you have declared @code{MyEx4}
706 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
707 the argument of @code{sin} and hence return @code{true}.
709 The next sections will outline the general picture of GiNaC's class
710 hierarchy and describe the classes of objects that are handled by
714 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
715 @c node-name, next, previous, up
716 @section The Class Hierarchy
718 GiNaC's class hierarchy consists of several classes representing
719 mathematical objects, all of which (except for @code{ex} and some
720 helpers) are internally derived from one abstract base class called
721 @code{basic}. You do not have to deal with objects of class
722 @code{basic}, instead you'll be dealing with symbols, numbers,
723 containers of expressions and so on.
727 To get an idea about what kinds of symbolic composits may be built we
728 have a look at the most important classes in the class hierarchy and
729 some of the relations among the classes:
731 @image{classhierarchy}
733 The abstract classes shown here (the ones without drop-shadow) are of no
734 interest for the user. They are used internally in order to avoid code
735 duplication if two or more classes derived from them share certain
736 features. An example is @code{expairseq}, a container for a sequence of
737 pairs each consisting of one expression and a number (@code{numeric}).
738 What @emph{is} visible to the user are the derived classes @code{add}
739 and @code{mul}, representing sums and products. @xref{Internal
740 Structures}, where these two classes are described in more detail. The
741 following table shortly summarizes what kinds of mathematical objects
742 are stored in the different classes:
745 @multitable @columnfractions .22 .78
746 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
747 @item @code{constant} @tab Constants like
754 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
755 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
756 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
757 @item @code{ncmul} @tab Products of non-commutative objects
758 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
763 @code{sqrt(}@math{2}@code{)}
766 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
767 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
768 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
769 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
770 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
771 @item @code{indexed} @tab Indexed object like @math{A_ij}
772 @item @code{tensor} @tab Special tensor like the delta and metric tensors
773 @item @code{idx} @tab Index of an indexed object
774 @item @code{varidx} @tab Index with variance
775 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
776 @item @code{wildcard} @tab Wildcard for pattern matching
780 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
781 @c node-name, next, previous, up
783 @cindex @code{symbol} (class)
784 @cindex hierarchy of classes
787 Symbols are for symbolic manipulation what atoms are for chemistry. You
788 can declare objects of class @code{symbol} as any other object simply by
789 saying @code{symbol x,y;}. There is, however, a catch in here having to
790 do with the fact that C++ is a compiled language. The information about
791 the symbol's name is thrown away by the compiler but at a later stage
792 you may want to print expressions holding your symbols. In order to
793 avoid confusion GiNaC's symbols are able to know their own name. This
794 is accomplished by declaring its name for output at construction time in
795 the fashion @code{symbol x("x");}. If you declare a symbol using the
796 default constructor (i.e. without string argument) the system will deal
797 out a unique name. That name may not be suitable for printing but for
798 internal routines when no output is desired it is often enough. We'll
799 come across examples of such symbols later in this tutorial.
801 This implies that the strings passed to symbols at construction time may
802 not be used for comparing two of them. It is perfectly legitimate to
803 write @code{symbol x("x"),y("x");} but it is likely to lead into
804 trouble. Here, @code{x} and @code{y} are different symbols and
805 statements like @code{x-y} will not be simplified to zero although the
806 output @code{x-x} looks funny. Such output may also occur when there
807 are two different symbols in two scopes, for instance when you call a
808 function that declares a symbol with a name already existent in a symbol
809 in the calling function. Again, comparing them (using @code{operator==}
810 for instance) will always reveal their difference. Watch out, please.
812 @cindex @code{subs()}
813 Although symbols can be assigned expressions for internal reasons, you
814 should not do it (and we are not going to tell you how it is done). If
815 you want to replace a symbol with something else in an expression, you
816 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
819 @node Numbers, Constants, Symbols, Basic Concepts
820 @c node-name, next, previous, up
822 @cindex @code{numeric} (class)
828 For storing numerical things, GiNaC uses Bruno Haible's library
829 @acronym{CLN}. The classes therein serve as foundation classes for
830 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
831 alternatively for Common Lisp Numbers. In order to find out more about
832 @acronym{CLN}'s internals the reader is refered to the documentation of
833 that library. @inforef{Introduction, , cln}, for more
834 information. Suffice to say that it is by itself build on top of another
835 library, the GNU Multiple Precision library @acronym{GMP}, which is an
836 extremely fast library for arbitrary long integers and rationals as well
837 as arbitrary precision floating point numbers. It is very commonly used
838 by several popular cryptographic applications. @acronym{CLN} extends
839 @acronym{GMP} by several useful things: First, it introduces the complex
840 number field over either reals (i.e. floating point numbers with
841 arbitrary precision) or rationals. Second, it automatically converts
842 rationals to integers if the denominator is unity and complex numbers to
843 real numbers if the imaginary part vanishes and also correctly treats
844 algebraic functions. Third it provides good implementations of
845 state-of-the-art algorithms for all trigonometric and hyperbolic
846 functions as well as for calculation of some useful constants.
848 The user can construct an object of class @code{numeric} in several
849 ways. The following example shows the four most important constructors.
850 It uses construction from C-integer, construction of fractions from two
851 integers, construction from C-float and construction from a string:
854 #include <ginac/ginac.h>
855 using namespace GiNaC;
859 numeric two(2); // exact integer 2
860 numeric r(2,3); // exact fraction 2/3
861 numeric e(2.71828); // floating point number
862 numeric p("3.1415926535897932385"); // floating point number
863 // Trott's constant in scientific notation:
864 numeric trott("1.0841015122311136151E-2");
866 std::cout << two*p << std::endl; // floating point 6.283...
870 Note that all those constructors are @emph{explicit} which means you are
871 not allowed to write @code{numeric two=2;}. This is because the basic
872 objects to be handled by GiNaC are the expressions @code{ex} and we want
873 to keep things simple and wish objects like @code{pow(x,2)} to be
874 handled the same way as @code{pow(x,a)}, which means that we need to
875 allow a general @code{ex} as base and exponent. Therefore there is an
876 implicit constructor from C-integers directly to expressions handling
877 numerics at work in most of our examples. This design really becomes
878 convenient when one declares own functions having more than one
879 parameter but it forbids using implicit constructors because that would
880 lead to compile-time ambiguities.
882 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
883 This would, however, call C's built-in operator @code{/} for integers
884 first and result in a numeric holding a plain integer 1. @strong{Never
885 use the operator @code{/} on integers} unless you know exactly what you
886 are doing! Use the constructor from two integers instead, as shown in
887 the example above. Writing @code{numeric(1)/2} may look funny but works
890 @cindex @code{Digits}
892 We have seen now the distinction between exact numbers and floating
893 point numbers. Clearly, the user should never have to worry about
894 dynamically created exact numbers, since their `exactness' always
895 determines how they ought to be handled, i.e. how `long' they are. The
896 situation is different for floating point numbers. Their accuracy is
897 controlled by one @emph{global} variable, called @code{Digits}. (For
898 those readers who know about Maple: it behaves very much like Maple's
899 @code{Digits}). All objects of class numeric that are constructed from
900 then on will be stored with a precision matching that number of decimal
904 #include <ginac/ginac.h>
906 using namespace GiNaC;
910 numeric three(3.0), one(1.0);
911 numeric x = one/three;
913 cout << "in " << Digits << " digits:" << endl;
915 cout << Pi.evalf() << endl;
927 The above example prints the following output to screen:
934 0.333333333333333333333333333333333333333333333333333333333333333333
935 3.14159265358979323846264338327950288419716939937510582097494459231
938 It should be clear that objects of class @code{numeric} should be used
939 for constructing numbers or for doing arithmetic with them. The objects
940 one deals with most of the time are the polymorphic expressions @code{ex}.
942 @subsection Tests on numbers
944 Once you have declared some numbers, assigned them to expressions and
945 done some arithmetic with them it is frequently desired to retrieve some
946 kind of information from them like asking whether that number is
947 integer, rational, real or complex. For those cases GiNaC provides
948 several useful methods. (Internally, they fall back to invocations of
949 certain CLN functions.)
951 As an example, let's construct some rational number, multiply it with
952 some multiple of its denominator and test what comes out:
955 #include <ginac/ginac.h>
957 using namespace GiNaC;
959 // some very important constants:
960 const numeric twentyone(21);
961 const numeric ten(10);
962 const numeric five(5);
966 numeric answer = twentyone;
969 cout << answer.is_integer() << endl; // false, it's 21/5
971 cout << answer.is_integer() << endl; // true, it's 42 now!
975 Note that the variable @code{answer} is constructed here as an integer
976 by @code{numeric}'s copy constructor but in an intermediate step it
977 holds a rational number represented as integer numerator and integer
978 denominator. When multiplied by 10, the denominator becomes unity and
979 the result is automatically converted to a pure integer again.
980 Internally, the underlying @acronym{CLN} is responsible for this
981 behaviour and we refer the reader to @acronym{CLN}'s documentation.
982 Suffice to say that the same behaviour applies to complex numbers as
983 well as return values of certain functions. Complex numbers are
984 automatically converted to real numbers if the imaginary part becomes
985 zero. The full set of tests that can be applied is listed in the
989 @multitable @columnfractions .30 .70
990 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
991 @item @code{.is_zero()}
992 @tab @dots{}equal to zero
993 @item @code{.is_positive()}
994 @tab @dots{}not complex and greater than 0
995 @item @code{.is_integer()}
996 @tab @dots{}a (non-complex) integer
997 @item @code{.is_pos_integer()}
998 @tab @dots{}an integer and greater than 0
999 @item @code{.is_nonneg_integer()}
1000 @tab @dots{}an integer and greater equal 0
1001 @item @code{.is_even()}
1002 @tab @dots{}an even integer
1003 @item @code{.is_odd()}
1004 @tab @dots{}an odd integer
1005 @item @code{.is_prime()}
1006 @tab @dots{}a prime integer (probabilistic primality test)
1007 @item @code{.is_rational()}
1008 @tab @dots{}an exact rational number (integers are rational, too)
1009 @item @code{.is_real()}
1010 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1011 @item @code{.is_cinteger()}
1012 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1013 @item @code{.is_crational()}
1014 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1019 @node Constants, Fundamental containers, Numbers, Basic Concepts
1020 @c node-name, next, previous, up
1022 @cindex @code{constant} (class)
1025 @cindex @code{Catalan}
1026 @cindex @code{Euler}
1027 @cindex @code{evalf()}
1028 Constants behave pretty much like symbols except that they return some
1029 specific number when the method @code{.evalf()} is called.
1031 The predefined known constants are:
1034 @multitable @columnfractions .14 .30 .56
1035 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1037 @tab Archimedes' constant
1038 @tab 3.14159265358979323846264338327950288
1039 @item @code{Catalan}
1040 @tab Catalan's constant
1041 @tab 0.91596559417721901505460351493238411
1043 @tab Euler's (or Euler-Mascheroni) constant
1044 @tab 0.57721566490153286060651209008240243
1049 @node Fundamental containers, Lists, Constants, Basic Concepts
1050 @c node-name, next, previous, up
1051 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1055 @cindex @code{power}
1057 Simple polynomial expressions are written down in GiNaC pretty much like
1058 in other CAS or like expressions involving numerical variables in C.
1059 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1060 been overloaded to achieve this goal. When you run the following
1061 code snippet, the constructor for an object of type @code{mul} is
1062 automatically called to hold the product of @code{a} and @code{b} and
1063 then the constructor for an object of type @code{add} is called to hold
1064 the sum of that @code{mul} object and the number one:
1068 symbol a("a"), b("b");
1073 @cindex @code{pow()}
1074 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1075 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1076 construction is necessary since we cannot safely overload the constructor
1077 @code{^} in C++ to construct a @code{power} object. If we did, it would
1078 have several counterintuitive and undesired effects:
1082 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1084 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1085 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1086 interpret this as @code{x^(a^b)}.
1088 Also, expressions involving integer exponents are very frequently used,
1089 which makes it even more dangerous to overload @code{^} since it is then
1090 hard to distinguish between the semantics as exponentiation and the one
1091 for exclusive or. (It would be embarassing to return @code{1} where one
1092 has requested @code{2^3}.)
1095 @cindex @command{ginsh}
1096 All effects are contrary to mathematical notation and differ from the
1097 way most other CAS handle exponentiation, therefore overloading @code{^}
1098 is ruled out for GiNaC's C++ part. The situation is different in
1099 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1100 that the other frequently used exponentiation operator @code{**} does
1101 not exist at all in C++).
1103 To be somewhat more precise, objects of the three classes described
1104 here, are all containers for other expressions. An object of class
1105 @code{power} is best viewed as a container with two slots, one for the
1106 basis, one for the exponent. All valid GiNaC expressions can be
1107 inserted. However, basic transformations like simplifying
1108 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1109 when this is mathematically possible. If we replace the outer exponent
1110 three in the example by some symbols @code{a}, the simplification is not
1111 safe and will not be performed, since @code{a} might be @code{1/2} and
1114 Objects of type @code{add} and @code{mul} are containers with an
1115 arbitrary number of slots for expressions to be inserted. Again, simple
1116 and safe simplifications are carried out like transforming
1117 @code{3*x+4-x} to @code{2*x+4}.
1119 The general rule is that when you construct such objects, GiNaC
1120 automatically creates them in canonical form, which might differ from
1121 the form you typed in your program. This allows for rapid comparison of
1122 expressions, since after all @code{a-a} is simply zero. Note, that the
1123 canonical form is not necessarily lexicographical ordering or in any way
1124 easily guessable. It is only guaranteed that constructing the same
1125 expression twice, either implicitly or explicitly, results in the same
1129 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1130 @c node-name, next, previous, up
1131 @section Lists of expressions
1132 @cindex @code{lst} (class)
1134 @cindex @code{nops()}
1136 @cindex @code{append()}
1137 @cindex @code{prepend()}
1138 @cindex @code{remove_first()}
1139 @cindex @code{remove_last()}
1141 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1142 expressions. These are sometimes used to supply a variable number of
1143 arguments of the same type to GiNaC methods such as @code{subs()} and
1144 @code{to_rational()}, so you should have a basic understanding about them.
1146 Lists of up to 16 expressions can be directly constructed from single
1151 symbol x("x"), y("y");
1152 lst l(x, 2, y, x+y);
1153 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1157 Use the @code{nops()} method to determine the size (number of expressions) of
1158 a list and the @code{op()} method to access individual elements:
1162 cout << l.nops() << endl; // prints '4'
1163 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1167 You can append or prepend an expression to a list with the @code{append()}
1168 and @code{prepend()} methods:
1172 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1173 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1177 Finally you can remove the first or last element of a list with
1178 @code{remove_first()} and @code{remove_last()}:
1182 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1183 l.remove_last(); // l is now @{x, 2, y, x+y@}
1188 @node Mathematical functions, Relations, Lists, Basic Concepts
1189 @c node-name, next, previous, up
1190 @section Mathematical functions
1191 @cindex @code{function} (class)
1192 @cindex trigonometric function
1193 @cindex hyperbolic function
1195 There are quite a number of useful functions hard-wired into GiNaC. For
1196 instance, all trigonometric and hyperbolic functions are implemented
1197 (@xref{Built-in Functions}, for a complete list).
1199 These functions are all objects of class @code{function}. They accept
1200 one or more expressions as arguments and return one expression. If the
1201 arguments are not numerical, the evaluation of the function may be
1202 halted, as it does in the next example, showing how a function returns
1203 itself twice and finally an expression that may be really useful:
1205 @cindex Gamma function
1206 @cindex @code{subs()}
1209 symbol x("x"), y("y");
1211 cout << tgamma(foo) << endl;
1212 // -> tgamma(x+(1/2)*y)
1213 ex bar = foo.subs(y==1);
1214 cout << tgamma(bar) << endl;
1216 ex foobar = bar.subs(x==7);
1217 cout << tgamma(foobar) << endl;
1218 // -> (135135/128)*Pi^(1/2)
1222 Besides evaluation most of these functions allow differentiation, series
1223 expansion and so on. Read the next chapter in order to learn more about
1227 @node Relations, Matrices, Mathematical functions, Basic Concepts
1228 @c node-name, next, previous, up
1230 @cindex @code{relational} (class)
1232 Sometimes, a relation holding between two expressions must be stored
1233 somehow. The class @code{relational} is a convenient container for such
1234 purposes. A relation is by definition a container for two @code{ex} and
1235 a relation between them that signals equality, inequality and so on.
1236 They are created by simply using the C++ operators @code{==}, @code{!=},
1237 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1239 @xref{Mathematical functions}, for examples where various applications
1240 of the @code{.subs()} method show how objects of class relational are
1241 used as arguments. There they provide an intuitive syntax for
1242 substitutions. They are also used as arguments to the @code{ex::series}
1243 method, where the left hand side of the relation specifies the variable
1244 to expand in and the right hand side the expansion point. They can also
1245 be used for creating systems of equations that are to be solved for
1246 unknown variables. But the most common usage of objects of this class
1247 is rather inconspicuous in statements of the form @code{if
1248 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1249 conversion from @code{relational} to @code{bool} takes place. Note,
1250 however, that @code{==} here does not perform any simplifications, hence
1251 @code{expand()} must be called explicitly.
1254 @node Matrices, Indexed objects, Relations, Basic Concepts
1255 @c node-name, next, previous, up
1257 @cindex @code{matrix} (class)
1259 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1260 matrix with @math{m} rows and @math{n} columns are accessed with two
1261 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1262 second one in the range 0@dots{}@math{n-1}.
1264 There are a couple of ways to construct matrices, with or without preset
1268 matrix::matrix(unsigned r, unsigned c);
1269 matrix::matrix(unsigned r, unsigned c, const lst & l);
1270 ex lst_to_matrix(const lst & l);
1271 ex diag_matrix(const lst & l);
1274 The first two functions are @code{matrix} constructors which create a matrix
1275 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1276 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1277 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1278 from a list of lists, each list representing a matrix row. Finally,
1279 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1280 elements. Note that the last two functions return expressions, not matrix
1283 Matrix elements can be accessed and set using the parenthesis (function call)
1287 const ex & matrix::operator()(unsigned r, unsigned c) const;
1288 ex & matrix::operator()(unsigned r, unsigned c);
1291 It is also possible to access the matrix elements in a linear fashion with
1292 the @code{op()} method. But C++-style subscripting with square brackets
1293 @samp{[]} is not available.
1295 Here are a couple of examples that all construct the same 2x2 diagonal
1300 symbol a("a"), b("b");
1308 e = matrix(2, 2, lst(a, 0, 0, b));
1310 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1312 e = diag_matrix(lst(a, b));
1319 @cindex @code{transpose()}
1320 @cindex @code{inverse()}
1321 There are three ways to do arithmetic with matrices. The first (and most
1322 efficient one) is to use the methods provided by the @code{matrix} class:
1325 matrix matrix::add(const matrix & other) const;
1326 matrix matrix::sub(const matrix & other) const;
1327 matrix matrix::mul(const matrix & other) const;
1328 matrix matrix::mul_scalar(const ex & other) const;
1329 matrix matrix::pow(const ex & expn) const;
1330 matrix matrix::transpose(void) const;
1331 matrix matrix::inverse(void) const;
1334 All of these methods return the result as a new matrix object. Here is an
1335 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1340 matrix A(2, 2, lst(1, 2, 3, 4));
1341 matrix B(2, 2, lst(-1, 0, 2, 1));
1342 matrix C(2, 2, lst(8, 4, 2, 1));
1344 matrix result = A.mul(B).sub(C.mul_scalar(2));
1345 cout << result << endl;
1346 // -> [[-13,-6],[1,2]]
1351 @cindex @code{evalm()}
1352 The second (and probably the most natural) way is to construct an expression
1353 containing matrices with the usual arithmetic operators and @code{pow()}.
1354 For efficiency reasons, expressions with sums, products and powers of
1355 matrices are not automatically evaluated in GiNaC. You have to call the
1359 ex ex::evalm() const;
1362 to obtain the result:
1369 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1370 cout << e.evalm() << endl;
1371 // -> [[-13,-6],[1,2]]
1376 The non-commutativity of the product @code{A*B} in this example is
1377 automatically recognized by GiNaC. There is no need to use a special
1378 operator here. @xref{Non-commutative objects}, for more information about
1379 dealing with non-commutative expressions.
1381 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1382 to perform the arithmetic:
1387 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1388 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1390 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1391 cout << e.simplify_indexed() << endl;
1392 // -> [[-13,-6],[1,2]].i.j
1396 Using indices is most useful when working with rectangular matrices and
1397 one-dimensional vectors because you don't have to worry about having to
1398 transpose matrices before multiplying them. @xref{Indexed objects}, for
1399 more information about using matrices with indices, and about indices in
1402 The @code{matrix} class provides a couple of additional methods for
1403 computing determinants, traces, and characteristic polynomials:
1406 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1407 ex matrix::trace(void) const;
1408 ex matrix::charpoly(const symbol & lambda) const;
1411 The @samp{algo} argument of @code{determinant()} allows to select between
1412 different algorithms for calculating the determinant. The possible values
1413 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1414 heuristic to automatically select an algorithm that is likely to give the
1415 result most quickly.
1418 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1419 @c node-name, next, previous, up
1420 @section Indexed objects
1422 GiNaC allows you to handle expressions containing general indexed objects in
1423 arbitrary spaces. It is also able to canonicalize and simplify such
1424 expressions and perform symbolic dummy index summations. There are a number
1425 of predefined indexed objects provided, like delta and metric tensors.
1427 There are few restrictions placed on indexed objects and their indices and
1428 it is easy to construct nonsense expressions, but our intention is to
1429 provide a general framework that allows you to implement algorithms with
1430 indexed quantities, getting in the way as little as possible.
1432 @cindex @code{idx} (class)
1433 @cindex @code{indexed} (class)
1434 @subsection Indexed quantities and their indices
1436 Indexed expressions in GiNaC are constructed of two special types of objects,
1437 @dfn{index objects} and @dfn{indexed objects}.
1441 @cindex contravariant
1444 @item Index objects are of class @code{idx} or a subclass. Every index has
1445 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1446 the index lives in) which can both be arbitrary expressions but are usually
1447 a number or a simple symbol. In addition, indices of class @code{varidx} have
1448 a @dfn{variance} (they can be co- or contravariant), and indices of class
1449 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1451 @item Indexed objects are of class @code{indexed} or a subclass. They
1452 contain a @dfn{base expression} (which is the expression being indexed), and
1453 one or more indices.
1457 @strong{Note:} when printing expressions, covariant indices and indices
1458 without variance are denoted @samp{.i} while contravariant indices are
1459 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1460 value. In the following, we are going to use that notation in the text so
1461 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1462 not visible in the output.
1464 A simple example shall illustrate the concepts:
1467 #include <ginac/ginac.h>
1468 using namespace std;
1469 using namespace GiNaC;
1473 symbol i_sym("i"), j_sym("j");
1474 idx i(i_sym, 3), j(j_sym, 3);
1477 cout << indexed(A, i, j) << endl;
1482 The @code{idx} constructor takes two arguments, the index value and the
1483 index dimension. First we define two index objects, @code{i} and @code{j},
1484 both with the numeric dimension 3. The value of the index @code{i} is the
1485 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1486 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1487 construct an expression containing one indexed object, @samp{A.i.j}. It has
1488 the symbol @code{A} as its base expression and the two indices @code{i} and
1491 Note the difference between the indices @code{i} and @code{j} which are of
1492 class @code{idx}, and the index values which are the sybols @code{i_sym}
1493 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1494 or numbers but must be index objects. For example, the following is not
1495 correct and will raise an exception:
1498 symbol i("i"), j("j");
1499 e = indexed(A, i, j); // ERROR: indices must be of type idx
1502 You can have multiple indexed objects in an expression, index values can
1503 be numeric, and index dimensions symbolic:
1507 symbol B("B"), dim("dim");
1508 cout << 4 * indexed(A, i)
1509 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1514 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1515 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1516 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1517 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1518 @code{simplify_indexed()} for that, see below).
1520 In fact, base expressions, index values and index dimensions can be
1521 arbitrary expressions:
1525 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1530 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1531 get an error message from this but you will probably not be able to do
1532 anything useful with it.
1534 @cindex @code{get_value()}
1535 @cindex @code{get_dimension()}
1539 ex idx::get_value(void);
1540 ex idx::get_dimension(void);
1543 return the value and dimension of an @code{idx} object. If you have an index
1544 in an expression, such as returned by calling @code{.op()} on an indexed
1545 object, you can get a reference to the @code{idx} object with the function
1546 @code{ex_to<idx>()} on the expression.
1548 There are also the methods
1551 bool idx::is_numeric(void);
1552 bool idx::is_symbolic(void);
1553 bool idx::is_dim_numeric(void);
1554 bool idx::is_dim_symbolic(void);
1557 for checking whether the value and dimension are numeric or symbolic
1558 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1559 About Expressions}) returns information about the index value.
1561 @cindex @code{varidx} (class)
1562 If you need co- and contravariant indices, use the @code{varidx} class:
1566 symbol mu_sym("mu"), nu_sym("nu");
1567 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1568 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1570 cout << indexed(A, mu, nu) << endl;
1572 cout << indexed(A, mu_co, nu) << endl;
1574 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1579 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1580 co- or contravariant. The default is a contravariant (upper) index, but
1581 this can be overridden by supplying a third argument to the @code{varidx}
1582 constructor. The two methods
1585 bool varidx::is_covariant(void);
1586 bool varidx::is_contravariant(void);
1589 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1590 to get the object reference from an expression). There's also the very useful
1594 ex varidx::toggle_variance(void);
1597 which makes a new index with the same value and dimension but the opposite
1598 variance. By using it you only have to define the index once.
1600 @cindex @code{spinidx} (class)
1601 The @code{spinidx} class provides dotted and undotted variant indices, as
1602 used in the Weyl-van-der-Waerden spinor formalism:
1606 symbol K("K"), C_sym("C"), D_sym("D");
1607 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1608 // contravariant, undotted
1609 spinidx C_co(C_sym, 2, true); // covariant index
1610 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1611 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1613 cout << indexed(K, C, D) << endl;
1615 cout << indexed(K, C_co, D_dot) << endl;
1617 cout << indexed(K, D_co_dot, D) << endl;
1622 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1623 dotted or undotted. The default is undotted but this can be overridden by
1624 supplying a fourth argument to the @code{spinidx} constructor. The two
1628 bool spinidx::is_dotted(void);
1629 bool spinidx::is_undotted(void);
1632 allow you to check whether or not a @code{spinidx} object is dotted (use
1633 @code{ex_to<spinidx>()} to get the object reference from an expression).
1634 Finally, the two methods
1637 ex spinidx::toggle_dot(void);
1638 ex spinidx::toggle_variance_dot(void);
1641 create a new index with the same value and dimension but opposite dottedness
1642 and the same or opposite variance.
1644 @subsection Substituting indices
1646 @cindex @code{subs()}
1647 Sometimes you will want to substitute one symbolic index with another
1648 symbolic or numeric index, for example when calculating one specific element
1649 of a tensor expression. This is done with the @code{.subs()} method, as it
1650 is done for symbols (see @ref{Substituting Expressions}).
1652 You have two possibilities here. You can either substitute the whole index
1653 by another index or expression:
1657 ex e = indexed(A, mu_co);
1658 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1659 // -> A.mu becomes A~nu
1660 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1661 // -> A.mu becomes A~0
1662 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1663 // -> A.mu becomes A.0
1667 The third example shows that trying to replace an index with something that
1668 is not an index will substitute the index value instead.
1670 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1675 ex e = indexed(A, mu_co);
1676 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1677 // -> A.mu becomes A.nu
1678 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1679 // -> A.mu becomes A.0
1683 As you see, with the second method only the value of the index will get
1684 substituted. Its other properties, including its dimension, remain unchanged.
1685 If you want to change the dimension of an index you have to substitute the
1686 whole index by another one with the new dimension.
1688 Finally, substituting the base expression of an indexed object works as
1693 ex e = indexed(A, mu_co);
1694 cout << e << " becomes " << e.subs(A == A+B) << endl;
1695 // -> A.mu becomes (B+A).mu
1699 @subsection Symmetries
1701 Indexed objects can be declared as being totally symmetric or antisymmetric
1702 with respect to their indices. In this case, GiNaC will automatically bring
1703 the indices into a canonical order which allows for some immediate
1708 cout << indexed(A, indexed::symmetric, i, j)
1709 + indexed(A, indexed::symmetric, j, i) << endl;
1711 cout << indexed(B, indexed::antisymmetric, i, j)
1712 + indexed(B, indexed::antisymmetric, j, j) << endl;
1714 cout << indexed(B, indexed::antisymmetric, i, j)
1715 + indexed(B, indexed::antisymmetric, j, i) << endl;
1720 @cindex @code{get_free_indices()}
1722 @subsection Dummy indices
1724 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1725 that a summation over the index range is implied. Symbolic indices which are
1726 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1727 dummy nor free indices.
1729 To be recognized as a dummy index pair, the two indices must be of the same
1730 class and dimension and their value must be the same single symbol (an index
1731 like @samp{2*n+1} is never a dummy index). If the indices are of class
1732 @code{varidx} they must also be of opposite variance; if they are of class
1733 @code{spinidx} they must be both dotted or both undotted.
1735 The method @code{.get_free_indices()} returns a vector containing the free
1736 indices of an expression. It also checks that the free indices of the terms
1737 of a sum are consistent:
1741 symbol A("A"), B("B"), C("C");
1743 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1744 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1746 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1747 cout << exprseq(e.get_free_indices()) << endl;
1749 // 'j' and 'l' are dummy indices
1751 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1752 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1754 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1755 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1756 cout << exprseq(e.get_free_indices()) << endl;
1758 // 'nu' is a dummy index, but 'sigma' is not
1760 e = indexed(A, mu, mu);
1761 cout << exprseq(e.get_free_indices()) << endl;
1763 // 'mu' is not a dummy index because it appears twice with the same
1766 e = indexed(A, mu, nu) + 42;
1767 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1768 // this will throw an exception:
1769 // "add::get_free_indices: inconsistent indices in sum"
1773 @cindex @code{simplify_indexed()}
1774 @subsection Simplifying indexed expressions
1776 In addition to the few automatic simplifications that GiNaC performs on
1777 indexed expressions (such as re-ordering the indices of symmetric tensors
1778 and calculating traces and convolutions of matrices and predefined tensors)
1782 ex ex::simplify_indexed(void);
1783 ex ex::simplify_indexed(const scalar_products & sp);
1786 that performs some more expensive operations:
1789 @item it checks the consistency of free indices in sums in the same way
1790 @code{get_free_indices()} does
1791 @item it tries to give dumy indices that appear in different terms of a sum
1792 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1793 @item it (symbolically) calculates all possible dummy index summations/contractions
1794 with the predefined tensors (this will be explained in more detail in the
1796 @item as a special case of dummy index summation, it can replace scalar products
1797 of two tensors with a user-defined value
1800 The last point is done with the help of the @code{scalar_products} class
1801 which is used to store scalar products with known values (this is not an
1802 arithmetic class, you just pass it to @code{simplify_indexed()}):
1806 symbol A("A"), B("B"), C("C"), i_sym("i");
1810 sp.add(A, B, 0); // A and B are orthogonal
1811 sp.add(A, C, 0); // A and C are orthogonal
1812 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1814 e = indexed(A + B, i) * indexed(A + C, i);
1816 // -> (B+A).i*(A+C).i
1818 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1824 The @code{scalar_products} object @code{sp} acts as a storage for the
1825 scalar products added to it with the @code{.add()} method. This method
1826 takes three arguments: the two expressions of which the scalar product is
1827 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1828 @code{simplify_indexed()} will replace all scalar products of indexed
1829 objects that have the symbols @code{A} and @code{B} as base expressions
1830 with the single value 0. The number, type and dimension of the indices
1831 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1833 @cindex @code{expand()}
1834 The example above also illustrates a feature of the @code{expand()} method:
1835 if passed the @code{expand_indexed} option it will distribute indices
1836 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1838 @cindex @code{tensor} (class)
1839 @subsection Predefined tensors
1841 Some frequently used special tensors such as the delta, epsilon and metric
1842 tensors are predefined in GiNaC. They have special properties when
1843 contracted with other tensor expressions and some of them have constant
1844 matrix representations (they will evaluate to a number when numeric
1845 indices are specified).
1847 @cindex @code{delta_tensor()}
1848 @subsubsection Delta tensor
1850 The delta tensor takes two indices, is symmetric and has the matrix
1851 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
1852 @code{delta_tensor()}:
1856 symbol A("A"), B("B");
1858 idx i(symbol("i"), 3), j(symbol("j"), 3),
1859 k(symbol("k"), 3), l(symbol("l"), 3);
1861 ex e = indexed(A, i, j) * indexed(B, k, l)
1862 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1863 cout << e.simplify_indexed() << endl;
1866 cout << delta_tensor(i, i) << endl;
1871 @cindex @code{metric_tensor()}
1872 @subsubsection General metric tensor
1874 The function @code{metric_tensor()} creates a general symmetric metric
1875 tensor with two indices that can be used to raise/lower tensor indices. The
1876 metric tensor is denoted as @samp{g} in the output and if its indices are of
1877 mixed variance it is automatically replaced by a delta tensor:
1883 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1885 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1886 cout << e.simplify_indexed() << endl;
1889 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1890 cout << e.simplify_indexed() << endl;
1893 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1894 * metric_tensor(nu, rho);
1895 cout << e.simplify_indexed() << endl;
1898 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1899 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1900 + indexed(A, mu.toggle_variance(), rho));
1901 cout << e.simplify_indexed() << endl;
1906 @cindex @code{lorentz_g()}
1907 @subsubsection Minkowski metric tensor
1909 The Minkowski metric tensor is a special metric tensor with a constant
1910 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1911 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1912 It is created with the function @code{lorentz_g()} (although it is output as
1917 varidx mu(symbol("mu"), 4);
1919 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1920 * lorentz_g(mu, varidx(0, 4)); // negative signature
1921 cout << e.simplify_indexed() << endl;
1924 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1925 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1926 cout << e.simplify_indexed() << endl;
1931 @cindex @code{spinor_metric()}
1932 @subsubsection Spinor metric tensor
1934 The function @code{spinor_metric()} creates an antisymmetric tensor with
1935 two indices that is used to raise/lower indices of 2-component spinors.
1936 It is output as @samp{eps}:
1942 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
1943 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
1945 e = spinor_metric(A, B) * indexed(psi, B_co);
1946 cout << e.simplify_indexed() << endl;
1949 e = spinor_metric(A, B) * indexed(psi, A_co);
1950 cout << e.simplify_indexed() << endl;
1953 e = spinor_metric(A_co, B_co) * indexed(psi, B);
1954 cout << e.simplify_indexed() << endl;
1957 e = spinor_metric(A_co, B_co) * indexed(psi, A);
1958 cout << e.simplify_indexed() << endl;
1961 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
1962 cout << e.simplify_indexed() << endl;
1965 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
1966 cout << e.simplify_indexed() << endl;
1971 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
1973 @cindex @code{epsilon_tensor()}
1974 @cindex @code{lorentz_eps()}
1975 @subsubsection Epsilon tensor
1977 The epsilon tensor is totally antisymmetric, its number of indices is equal
1978 to the dimension of the index space (the indices must all be of the same
1979 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1980 defined to be 1. Its behaviour with indices that have a variance also
1981 depends on the signature of the metric. Epsilon tensors are output as
1984 There are three functions defined to create epsilon tensors in 2, 3 and 4
1988 ex epsilon_tensor(const ex & i1, const ex & i2);
1989 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1990 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1993 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1994 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1995 Minkowski space (the last @code{bool} argument specifies whether the metric
1996 has negative or positive signature, as in the case of the Minkowski metric
1999 @subsection Linear algebra
2001 The @code{matrix} class can be used with indices to do some simple linear
2002 algebra (linear combinations and products of vectors and matrices, traces
2003 and scalar products):
2007 idx i(symbol("i"), 2), j(symbol("j"), 2);
2008 symbol x("x"), y("y");
2010 // A is a 2x2 matrix, X is a 2x1 vector
2011 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2013 cout << indexed(A, i, i) << endl;
2016 ex e = indexed(A, i, j) * indexed(X, j);
2017 cout << e.simplify_indexed() << endl;
2018 // -> [[2*y+x],[4*y+3*x]].i
2020 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2021 cout << e.simplify_indexed() << endl;
2022 // -> [[3*y+3*x,6*y+2*x]].j
2026 You can of course obtain the same results with the @code{matrix::add()},
2027 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2028 but with indices you don't have to worry about transposing matrices.
2030 Matrix indices always start at 0 and their dimension must match the number
2031 of rows/columns of the matrix. Matrices with one row or one column are
2032 vectors and can have one or two indices (it doesn't matter whether it's a
2033 row or a column vector). Other matrices must have two indices.
2035 You should be careful when using indices with variance on matrices. GiNaC
2036 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2037 @samp{F.mu.nu} are different matrices. In this case you should use only
2038 one form for @samp{F} and explicitly multiply it with a matrix representation
2039 of the metric tensor.
2042 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2043 @c node-name, next, previous, up
2044 @section Non-commutative objects
2046 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2047 non-commutative objects are built-in which are mostly of use in high energy
2051 @item Clifford (Dirac) algebra (class @code{clifford})
2052 @item su(3) Lie algebra (class @code{color})
2053 @item Matrices (unindexed) (class @code{matrix})
2056 The @code{clifford} and @code{color} classes are subclasses of
2057 @code{indexed} because the elements of these algebras ususally carry
2058 indices. The @code{matrix} class is described in more detail in
2061 Unlike most computer algebra systems, GiNaC does not primarily provide an
2062 operator (often denoted @samp{&*}) for representing inert products of
2063 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2064 classes of objects involved, and non-commutative products are formed with
2065 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2066 figuring out by itself which objects commute and will group the factors
2067 by their class. Consider this example:
2071 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2072 idx a(symbol("a"), 8), b(symbol("b"), 8);
2073 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2075 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2079 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2080 groups the non-commutative factors (the gammas and the su(3) generators)
2081 together while preserving the order of factors within each class (because
2082 Clifford objects commute with color objects). The resulting expression is a
2083 @emph{commutative} product with two factors that are themselves non-commutative
2084 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2085 parentheses are placed around the non-commutative products in the output.
2087 @cindex @code{ncmul} (class)
2088 Non-commutative products are internally represented by objects of the class
2089 @code{ncmul}, as opposed to commutative products which are handled by the
2090 @code{mul} class. You will normally not have to worry about this distinction,
2093 The advantage of this approach is that you never have to worry about using
2094 (or forgetting to use) a special operator when constructing non-commutative
2095 expressions. Also, non-commutative products in GiNaC are more intelligent
2096 than in other computer algebra systems; they can, for example, automatically
2097 canonicalize themselves according to rules specified in the implementation
2098 of the non-commutative classes. The drawback is that to work with other than
2099 the built-in algebras you have to implement new classes yourself. Symbols
2100 always commute and it's not possible to construct non-commutative products
2101 using symbols to represent the algebra elements or generators. User-defined
2102 functions can, however, be specified as being non-commutative.
2104 @cindex @code{return_type()}
2105 @cindex @code{return_type_tinfo()}
2106 Information about the commutativity of an object or expression can be
2107 obtained with the two member functions
2110 unsigned ex::return_type(void) const;
2111 unsigned ex::return_type_tinfo(void) const;
2114 The @code{return_type()} function returns one of three values (defined in
2115 the header file @file{flags.h}), corresponding to three categories of
2116 expressions in GiNaC:
2119 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2120 classes are of this kind.
2121 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2122 certain class of non-commutative objects which can be determined with the
2123 @code{return_type_tinfo()} method. Expressions of this category commute
2124 with everything except @code{noncommutative} expressions of the same
2126 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2127 of non-commutative objects of different classes. Expressions of this
2128 category don't commute with any other @code{noncommutative} or
2129 @code{noncommutative_composite} expressions.
2132 The value returned by the @code{return_type_tinfo()} method is valid only
2133 when the return type of the expression is @code{noncommutative}. It is a
2134 value that is unique to the class of the object and usually one of the
2135 constants in @file{tinfos.h}, or derived therefrom.
2137 Here are a couple of examples:
2140 @multitable @columnfractions 0.33 0.33 0.34
2141 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2142 @item @code{42} @tab @code{commutative} @tab -
2143 @item @code{2*x-y} @tab @code{commutative} @tab -
2144 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2145 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2146 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2147 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2151 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2152 @code{TINFO_clifford} for objects with a representation label of zero.
2153 Other representation labels yield a different @code{return_type_tinfo()},
2154 but it's the same for any two objects with the same label. This is also true
2157 A last note: With the exception of matrices, positive integer powers of
2158 non-commutative objects are automatically expanded in GiNaC. For example,
2159 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2160 non-commutative expressions).
2163 @cindex @code{clifford} (class)
2164 @subsection Clifford algebra
2166 @cindex @code{dirac_gamma()}
2167 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2168 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2169 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2170 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2173 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2176 which takes two arguments: the index and a @dfn{representation label} in the
2177 range 0 to 255 which is used to distinguish elements of different Clifford
2178 algebras (this is also called a @dfn{spin line index}). Gammas with different
2179 labels commute with each other. The dimension of the index can be 4 or (in
2180 the framework of dimensional regularization) any symbolic value. Spinor
2181 indices on Dirac gammas are not supported in GiNaC.
2183 @cindex @code{dirac_ONE()}
2184 The unity element of a Clifford algebra is constructed by
2187 ex dirac_ONE(unsigned char rl = 0);
2190 @cindex @code{dirac_gamma5()}
2191 and there's a special element @samp{gamma5} that commutes with all other
2192 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2196 ex dirac_gamma5(unsigned char rl = 0);
2199 @cindex @code{dirac_gamma6()}
2200 @cindex @code{dirac_gamma7()}
2201 The two additional functions
2204 ex dirac_gamma6(unsigned char rl = 0);
2205 ex dirac_gamma7(unsigned char rl = 0);
2208 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2211 @cindex @code{dirac_slash()}
2212 Finally, the function
2215 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2218 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2219 whose dimension is given by the @code{dim} argument.
2221 In products of dirac gammas, superfluous unity elements are automatically
2222 removed, squares are replaced by their values and @samp{gamma5} is
2223 anticommuted to the front. The @code{simplify_indexed()} function performs
2224 contractions in gamma strings, for example
2229 symbol a("a"), b("b"), D("D");
2230 varidx mu(symbol("mu"), D);
2231 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2232 * dirac_gamma(mu.toggle_variance());
2234 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2235 e = e.simplify_indexed();
2237 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2238 cout << e.subs(D == 4) << endl;
2239 // -> -2*gamma~symbol10*a.symbol10
2240 // [ == -2 * dirac_slash(a, D) ]
2245 @cindex @code{dirac_trace()}
2246 To calculate the trace of an expression containing strings of Dirac gammas
2247 you use the function
2250 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2253 This function takes the trace of all gammas with the specified representation
2254 label; gammas with other labels are left standing. The last argument to
2255 @code{dirac_trace()} is the value to be returned for the trace of the unity
2256 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2257 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2258 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2259 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2260 This @samp{gamma5} scheme is described in greater detail in
2261 @cite{The Role of gamma5 in Dimensional Regularization}.
2263 The value of the trace itself is also usually different in 4 and in
2264 @math{D != 4} dimensions:
2269 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2270 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2271 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2272 cout << dirac_trace(e).simplify_indexed() << endl;
2279 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2280 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2281 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2282 cout << dirac_trace(e).simplify_indexed() << endl;
2283 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2287 Here is an example for using @code{dirac_trace()} to compute a value that
2288 appears in the calculation of the one-loop vacuum polarization amplitude in
2293 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2294 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2297 sp.add(l, l, pow(l, 2));
2298 sp.add(l, q, ldotq);
2300 ex e = dirac_gamma(mu) *
2301 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2302 dirac_gamma(mu.toggle_variance()) *
2303 (dirac_slash(l, D) + m * dirac_ONE());
2304 e = dirac_trace(e).simplify_indexed(sp);
2305 e = e.collect(lst(l, ldotq, m), true);
2307 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2311 The @code{canonicalize_clifford()} function reorders all gamma products that
2312 appear in an expression to a canonical (but not necessarily simple) form.
2313 You can use this to compare two expressions or for further simplifications:
2317 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2318 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2320 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2322 e = canonicalize_clifford(e);
2329 @cindex @code{color} (class)
2330 @subsection Color algebra
2332 @cindex @code{color_T()}
2333 For computations in quantum chromodynamics, GiNaC implements the base elements
2334 and structure constants of the su(3) Lie algebra (color algebra). The base
2335 elements @math{T_a} are constructed by the function
2338 ex color_T(const ex & a, unsigned char rl = 0);
2341 which takes two arguments: the index and a @dfn{representation label} in the
2342 range 0 to 255 which is used to distinguish elements of different color
2343 algebras. Objects with different labels commute with each other. The
2344 dimension of the index must be exactly 8 and it should be of class @code{idx},
2347 @cindex @code{color_ONE()}
2348 The unity element of a color algebra is constructed by
2351 ex color_ONE(unsigned char rl = 0);
2354 @cindex @code{color_d()}
2355 @cindex @code{color_f()}
2359 ex color_d(const ex & a, const ex & b, const ex & c);
2360 ex color_f(const ex & a, const ex & b, const ex & c);
2363 create the symmetric and antisymmetric structure constants @math{d_abc} and
2364 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2365 and @math{[T_a, T_b] = i f_abc T_c}.
2367 @cindex @code{color_h()}
2368 There's an additional function
2371 ex color_h(const ex & a, const ex & b, const ex & c);
2374 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2376 The function @code{simplify_indexed()} performs some simplifications on
2377 expressions containing color objects:
2382 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2383 k(symbol("k"), 8), l(symbol("l"), 8);
2385 e = color_d(a, b, l) * color_f(a, b, k);
2386 cout << e.simplify_indexed() << endl;
2389 e = color_d(a, b, l) * color_d(a, b, k);
2390 cout << e.simplify_indexed() << endl;
2393 e = color_f(l, a, b) * color_f(a, b, k);
2394 cout << e.simplify_indexed() << endl;
2397 e = color_h(a, b, c) * color_h(a, b, c);
2398 cout << e.simplify_indexed() << endl;
2401 e = color_h(a, b, c) * color_T(b) * color_T(c);
2402 cout << e.simplify_indexed() << endl;
2405 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2406 cout << e.simplify_indexed() << endl;
2409 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2410 cout << e.simplify_indexed() << endl;
2411 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2415 @cindex @code{color_trace()}
2416 To calculate the trace of an expression containing color objects you use the
2420 ex color_trace(const ex & e, unsigned char rl = 0);
2423 This function takes the trace of all color @samp{T} objects with the
2424 specified representation label; @samp{T}s with other labels are left
2425 standing. For example:
2429 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2431 // -> -I*f.a.c.b+d.a.c.b
2436 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2437 @c node-name, next, previous, up
2438 @chapter Methods and Functions
2441 In this chapter the most important algorithms provided by GiNaC will be
2442 described. Some of them are implemented as functions on expressions,
2443 others are implemented as methods provided by expression objects. If
2444 they are methods, there exists a wrapper function around it, so you can
2445 alternatively call it in a functional way as shown in the simple
2450 cout << "As method: " << sin(1).evalf() << endl;
2451 cout << "As function: " << evalf(sin(1)) << endl;
2455 @cindex @code{subs()}
2456 The general rule is that wherever methods accept one or more parameters
2457 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2458 wrapper accepts is the same but preceded by the object to act on
2459 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2460 most natural one in an OO model but it may lead to confusion for MapleV
2461 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2462 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2463 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2464 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2465 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2466 here. Also, users of MuPAD will in most cases feel more comfortable
2467 with GiNaC's convention. All function wrappers are implemented
2468 as simple inline functions which just call the corresponding method and
2469 are only provided for users uncomfortable with OO who are dead set to
2470 avoid method invocations. Generally, nested function wrappers are much
2471 harder to read than a sequence of methods and should therefore be
2472 avoided if possible. On the other hand, not everything in GiNaC is a
2473 method on class @code{ex} and sometimes calling a function cannot be
2477 * Information About Expressions::
2478 * Substituting Expressions::
2479 * Pattern Matching and Advanced Substitutions::
2480 * Polynomial Arithmetic:: Working with polynomials.
2481 * Rational Expressions:: Working with rational functions.
2482 * Symbolic Differentiation::
2483 * Series Expansion:: Taylor and Laurent expansion.
2485 * Built-in Functions:: List of predefined mathematical functions.
2486 * Input/Output:: Input and output of expressions.
2490 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2491 @c node-name, next, previous, up
2492 @section Getting information about expressions
2494 @subsection Checking expression types
2495 @cindex @code{is_a<@dots{}>()}
2496 @cindex @code{is_exactly_a<@dots{}>()}
2497 @cindex @code{ex_to<@dots{}>()}
2498 @cindex Converting @code{ex} to other classes
2499 @cindex @code{info()}
2500 @cindex @code{return_type()}
2501 @cindex @code{return_type_tinfo()}
2503 Sometimes it's useful to check whether a given expression is a plain number,
2504 a sum, a polynomial with integer coefficients, or of some other specific type.
2505 GiNaC provides a couple of functions for this:
2508 bool is_a<T>(const ex & e);
2509 bool is_exactly_a<T>(const ex & e);
2510 bool ex::info(unsigned flag);
2511 unsigned ex::return_type(void) const;
2512 unsigned ex::return_type_tinfo(void) const;
2515 When the test made by @code{is_a<T>()} returns true, it is safe to call
2516 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2517 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2518 example, assuming @code{e} is an @code{ex}:
2523 if (is_a<numeric>(e))
2524 numeric n = ex_to<numeric>(e);
2529 @code{is_a<T>(e)} allows you to check whether the top-level object of
2530 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2531 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2532 e.g., for checking whether an expression is a number, a sum, or a product:
2539 is_a<numeric>(e1); // true
2540 is_a<numeric>(e2); // false
2541 is_a<add>(e1); // false
2542 is_a<add>(e2); // true
2543 is_a<mul>(e1); // false
2544 is_a<mul>(e2); // false
2548 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2549 top-level object of an expression @samp{e} is an instance of the GiNaC
2550 class @samp{T}, not including parent classes.
2552 The @code{info()} method is used for checking certain attributes of
2553 expressions. The possible values for the @code{flag} argument are defined
2554 in @file{ginac/flags.h}, the most important being explained in the following
2558 @multitable @columnfractions .30 .70
2559 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2560 @item @code{numeric}
2561 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2563 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2564 @item @code{rational}
2565 @tab @dots{}an exact rational number (integers are rational, too)
2566 @item @code{integer}
2567 @tab @dots{}a (non-complex) integer
2568 @item @code{crational}
2569 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2570 @item @code{cinteger}
2571 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2572 @item @code{positive}
2573 @tab @dots{}not complex and greater than 0
2574 @item @code{negative}
2575 @tab @dots{}not complex and less than 0
2576 @item @code{nonnegative}
2577 @tab @dots{}not complex and greater than or equal to 0
2579 @tab @dots{}an integer greater than 0
2581 @tab @dots{}an integer less than 0
2582 @item @code{nonnegint}
2583 @tab @dots{}an integer greater than or equal to 0
2585 @tab @dots{}an even integer
2587 @tab @dots{}an odd integer
2589 @tab @dots{}a prime integer (probabilistic primality test)
2590 @item @code{relation}
2591 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2592 @item @code{relation_equal}
2593 @tab @dots{}a @code{==} relation
2594 @item @code{relation_not_equal}
2595 @tab @dots{}a @code{!=} relation
2596 @item @code{relation_less}
2597 @tab @dots{}a @code{<} relation
2598 @item @code{relation_less_or_equal}
2599 @tab @dots{}a @code{<=} relation
2600 @item @code{relation_greater}
2601 @tab @dots{}a @code{>} relation
2602 @item @code{relation_greater_or_equal}
2603 @tab @dots{}a @code{>=} relation
2605 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2607 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2608 @item @code{polynomial}
2609 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2610 @item @code{integer_polynomial}
2611 @tab @dots{}a polynomial with (non-complex) integer coefficients
2612 @item @code{cinteger_polynomial}
2613 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2614 @item @code{rational_polynomial}
2615 @tab @dots{}a polynomial with (non-complex) rational coefficients
2616 @item @code{crational_polynomial}
2617 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2618 @item @code{rational_function}
2619 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2620 @item @code{algebraic}
2621 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2625 To determine whether an expression is commutative or non-commutative and if
2626 so, with which other expressions it would commute, you use the methods
2627 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2628 for an explanation of these.
2631 @subsection Accessing subexpressions
2632 @cindex @code{nops()}
2635 @cindex @code{relational} (class)
2637 GiNaC provides the two methods
2640 unsigned ex::nops();
2641 ex ex::op(unsigned i);
2644 for accessing the subexpressions in the container-like GiNaC classes like
2645 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2646 determines the number of subexpressions (@samp{operands}) contained, while
2647 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2648 In the case of a @code{power} object, @code{op(0)} will return the basis
2649 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2650 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2652 The left-hand and right-hand side expressions of objects of class
2653 @code{relational} (and only of these) can also be accessed with the methods
2661 @subsection Comparing expressions
2662 @cindex @code{is_equal()}
2663 @cindex @code{is_zero()}
2665 Expressions can be compared with the usual C++ relational operators like
2666 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2667 the result is usually not determinable and the result will be @code{false},
2668 except in the case of the @code{!=} operator. You should also be aware that
2669 GiNaC will only do the most trivial test for equality (subtracting both
2670 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2673 Actually, if you construct an expression like @code{a == b}, this will be
2674 represented by an object of the @code{relational} class (@pxref{Relations})
2675 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2677 There are also two methods
2680 bool ex::is_equal(const ex & other);
2684 for checking whether one expression is equal to another, or equal to zero,
2687 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2688 GiNaC header files. This method is however only to be used internally by
2689 GiNaC to establish a canonical sort order for terms, and using it to compare
2690 expressions will give very surprising results.
2693 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2694 @c node-name, next, previous, up
2695 @section Substituting expressions
2696 @cindex @code{subs()}
2698 Algebraic objects inside expressions can be replaced with arbitrary
2699 expressions via the @code{.subs()} method:
2702 ex ex::subs(const ex & e);
2703 ex ex::subs(const lst & syms, const lst & repls);
2706 In the first form, @code{subs()} accepts a relational of the form
2707 @samp{object == expression} or a @code{lst} of such relationals:
2711 symbol x("x"), y("y");
2713 ex e1 = 2*x^2-4*x+3;
2714 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2718 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2723 If you specify multiple substitutions, they are performed in parallel, so e.g.
2724 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2726 The second form of @code{subs()} takes two lists, one for the objects to be
2727 replaced and one for the expressions to be substituted (both lists must
2728 contain the same number of elements). Using this form, you would write
2729 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2731 @code{subs()} performs syntactic substitution of any complete algebraic
2732 object; it does not try to match sub-expressions as is demonstrated by the
2737 symbol x("x"), y("y"), z("z");
2739 ex e1 = pow(x+y, 2);
2740 cout << e1.subs(x+y == 4) << endl;
2743 ex e2 = sin(x)*sin(y)*cos(x);
2744 cout << e2.subs(sin(x) == cos(x)) << endl;
2745 // -> cos(x)^2*sin(y)
2748 cout << e3.subs(x+y == 4) << endl;
2750 // (and not 4+z as one might expect)
2754 A more powerful form of substitution using wildcards is described in the
2758 @node Pattern Matching and Advanced Substitutions, Polynomial Arithmetic, Substituting Expressions, Methods and Functions
2759 @c node-name, next, previous, up
2760 @section Pattern matching and advanced substitutions
2761 @cindex @code{wildcard} (class)
2762 @cindex Pattern matching
2764 GiNaC allows the use of patterns for checking whether an expression is of a
2765 certain form or contains subexpressions of a certain form, and for
2766 substituting expressions in a more general way.
2768 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2769 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2770 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2771 an unsigned integer number to allow having multiple different wildcards in a
2772 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2773 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2777 ex wild(unsigned label = 0);
2780 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2783 Some examples for patterns:
2785 @multitable @columnfractions .5 .5
2786 @item @strong{Constructed as} @tab @strong{Output as}
2787 @item @code{wild()} @tab @samp{$0}
2788 @item @code{pow(x,wild())} @tab @samp{x^$0}
2789 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2790 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2796 @item Wildcards behave like symbols and are subject to the same algebraic
2797 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2798 @item As shown in the last example, to use wildcards for indices you have to
2799 use them as the value of an @code{idx} object. This is because indices must
2800 always be of class @code{idx} (or a subclass).
2801 @item Wildcards only represent expressions or subexpressions. It is not
2802 possible to use them as placeholders for other properties like index
2803 dimension or variance, representation labels, symmetry of indexed objects
2805 @item Because wildcards are commutative, it is not possible to use wildcards
2806 as part of noncommutative products.
2807 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2808 are also valid patterns.
2811 @cindex @code{match()}
2812 The most basic application of patterns is to check whether an expression
2813 matches a given pattern. This is done by the function
2816 bool ex::match(const ex & pattern);
2817 bool ex::match(const ex & pattern, lst & repls);
2820 This function returns @code{true} when the expression matches the pattern
2821 and @code{false} if it doesn't. If used in the second form, the actual
2822 subexpressions matched by the wildcards get returned in the @code{repls}
2823 object as a list of relations of the form @samp{wildcard == expression}.
2824 If @code{match()} returns false, the state of @code{repls} is undefined.
2825 For reproducible results, the list should be empty when passed to
2826 @code{match()}, but it is also possible to find similarities in multiple
2827 expressions by passing in the result of a previous match.
2829 The matching algorithm works as follows:
2832 @item A single wildcard matches any expression. If one wildcard appears
2833 multiple times in a pattern, it must match the same expression in all
2834 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2835 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2836 @item If the expression is not of the same class as the pattern, the match
2837 fails (i.e. a sum only matches a sum, a function only matches a function,
2839 @item If the pattern is a function, it only matches the same function
2840 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2841 @item Except for sums and products, the match fails if the number of
2842 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2844 @item If there are no subexpressions, the expressions and the pattern must
2845 be equal (in the sense of @code{is_equal()}).
2846 @item Except for sums and products, each subexpression (@code{op()}) must
2847 match the corresponding subexpression of the pattern.
2850 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2851 account for their commutativity and associativity:
2854 @item If the pattern contains a term or factor that is a single wildcard,
2855 this one is used as the @dfn{global wildcard}. If there is more than one
2856 such wildcard, one of them is chosen as the global wildcard in a random
2858 @item Every term/factor of the pattern, except the global wildcard, is
2859 matched against every term of the expression in sequence. If no match is
2860 found, the whole match fails. Terms that did match are not considered in
2862 @item If there are no unmatched terms left, the match succeeds. Otherwise
2863 the match fails unless there is a global wildcard in the pattern, in
2864 which case this wildcard matches the remaining terms.
2867 In general, having more than one single wildcard as a term of a sum or a
2868 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2871 Here are some examples in @command{ginsh} to demonstrate how it works (the
2872 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2873 match fails, and the list of wildcard replacements otherwise):
2876 > match((x+y)^a,(x+y)^a);
2878 > match((x+y)^a,(x+y)^b);
2880 > match((x+y)^a,$1^$2);
2882 > match((x+y)^a,$1^$1);
2884 > match((x+y)^(x+y),$1^$1);
2886 > match((x+y)^(x+y),$1^$2);
2888 > match((a+b)*(a+c),($1+b)*($1+c));
2890 > match((a+b)*(a+c),(a+$1)*(a+$2));
2892 (Unpredictable. The result might also be [$1==c,$2==b].)
2893 > match((a+b)*(a+c),($1+$2)*($1+$3));
2894 (The result is undefined. Due to the sequential nature of the algorithm
2895 and the re-ordering of terms in GiNaC, the match for the first factor
2896 may be @{$1==a,$2==b@} in which case the match for the second factor
2897 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
2899 > match(a*(x+y)+a*z+b,a*$1+$2);
2900 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
2901 @{$1=x+y,$2=a*z+b@}.)
2902 > match(a+b+c+d+e+f,c);
2904 > match(a+b+c+d+e+f,c+$0);
2906 > match(a+b+c+d+e+f,c+e+$0);
2908 > match(a+b,a+b+$0);
2910 > match(a*b^2,a^$1*b^$2);
2912 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
2913 even though a==a^1.)
2914 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
2916 > match(atan2(y,x^2),atan2(y,$0));
2920 @cindex @code{has()}
2921 A more general way to look for patterns in expressions is provided by the
2925 bool ex::has(const ex & pattern);
2928 This function checks whether a pattern is matched by an expression itself or
2929 by any of its subexpressions.
2931 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
2932 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
2935 > has(x*sin(x+y+2*a),y);
2937 > has(x*sin(x+y+2*a),x+y);
2939 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
2940 has the subexpressions "x", "y" and "2*a".)
2941 > has(x*sin(x+y+2*a),x+y+$1);
2943 (But this is possible.)
2944 > has(x*sin(2*(x+y)+2*a),x+y);
2946 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
2947 which "x+y" is not a subexpression.)
2950 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
2952 > has(4*x^2-x+3,$1*x);
2954 > has(4*x^2+x+3,$1*x);
2956 (Another possible pitfall. The first expression matches because the term
2957 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
2958 contains a linear term you should use the coeff() function instead.)
2961 @cindex @code{subs()}
2962 Probably the most useful application of patterns is to use them for
2963 substituting expressions with the @code{subs()} method. Wildcards can be
2964 used in the search patterns as well as in the replacement expressions, where
2965 they get replaced by the expressions matched by them. @code{subs()} doesn't
2966 know anything about algebra; it performs purely syntactic substitutions.
2971 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
2973 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
2975 > subs((a+b+c)^2,a+b=x);
2977 > subs((a+b+c)^2,a+b+$1==x+$1);
2979 > subs(a+2*b,a+b=x);
2981 > subs(4*x^3-2*x^2+5*x-1,x==a);
2983 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
2985 > subs(sin(1+sin(x)),sin($1)==cos($1));
2987 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
2991 The last example would be written in C++ in this way:
2995 symbol a("a"), b("b"), x("x"), y("y");
2996 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
2997 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
2998 cout << e.expand() << endl;
3004 @node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
3005 @c node-name, next, previous, up
3006 @section Polynomial arithmetic
3008 @subsection Expanding and collecting
3009 @cindex @code{expand()}
3010 @cindex @code{collect()}
3012 A polynomial in one or more variables has many equivalent
3013 representations. Some useful ones serve a specific purpose. Consider
3014 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3015 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3016 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3017 representations are the recursive ones where one collects for exponents
3018 in one of the three variable. Since the factors are themselves
3019 polynomials in the remaining two variables the procedure can be
3020 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
3021 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3024 To bring an expression into expanded form, its method
3030 may be called. In our example above, this corresponds to @math{4*x*y +
3031 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3032 GiNaC is not easily guessable you should be prepared to see different
3033 orderings of terms in such sums!
3035 Another useful representation of multivariate polynomials is as a
3036 univariate polynomial in one of the variables with the coefficients
3037 being polynomials in the remaining variables. The method
3038 @code{collect()} accomplishes this task:
3041 ex ex::collect(const ex & s, bool distributed = false);
3044 The first argument to @code{collect()} can also be a list of objects in which
3045 case the result is either a recursively collected polynomial, or a polynomial
3046 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3047 by the @code{distributed} flag.
3049 Note that the original polynomial needs to be in expanded form in order
3050 for @code{collect()} to be able to find the coefficients properly.
3052 @subsection Degree and coefficients
3053 @cindex @code{degree()}
3054 @cindex @code{ldegree()}
3055 @cindex @code{coeff()}
3057 The degree and low degree of a polynomial can be obtained using the two
3061 int ex::degree(const ex & s);
3062 int ex::ldegree(const ex & s);
3065 which also work reliably on non-expanded input polynomials (they even work
3066 on rational functions, returning the asymptotic degree). To extract
3067 a coefficient with a certain power from an expanded polynomial you use
3070 ex ex::coeff(const ex & s, int n);
3073 You can also obtain the leading and trailing coefficients with the methods
3076 ex ex::lcoeff(const ex & s);
3077 ex ex::tcoeff(const ex & s);
3080 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3083 An application is illustrated in the next example, where a multivariate
3084 polynomial is analyzed:
3087 #include <ginac/ginac.h>
3088 using namespace std;
3089 using namespace GiNaC;
3093 symbol x("x"), y("y");
3094 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3095 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3096 ex Poly = PolyInp.expand();
3098 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3099 cout << "The x^" << i << "-coefficient is "
3100 << Poly.coeff(x,i) << endl;
3102 cout << "As polynomial in y: "
3103 << Poly.collect(y) << endl;
3107 When run, it returns an output in the following fashion:
3110 The x^0-coefficient is y^2+11*y
3111 The x^1-coefficient is 5*y^2-2*y
3112 The x^2-coefficient is -1
3113 The x^3-coefficient is 4*y
3114 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3117 As always, the exact output may vary between different versions of GiNaC
3118 or even from run to run since the internal canonical ordering is not
3119 within the user's sphere of influence.
3121 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3122 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3123 with non-polynomial expressions as they not only work with symbols but with
3124 constants, functions and indexed objects as well:
3128 symbol a("a"), b("b"), c("c");
3129 idx i(symbol("i"), 3);
3131 ex e = pow(sin(x) - cos(x), 4);
3132 cout << e.degree(cos(x)) << endl;
3134 cout << e.expand().coeff(sin(x), 3) << endl;
3137 e = indexed(a+b, i) * indexed(b+c, i);
3138 e = e.expand(expand_options::expand_indexed);
3139 cout << e.collect(indexed(b, i)) << endl;
3140 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3145 @subsection Polynomial division
3146 @cindex polynomial division
3149 @cindex pseudo-remainder
3150 @cindex @code{quo()}
3151 @cindex @code{rem()}
3152 @cindex @code{prem()}
3153 @cindex @code{divide()}
3158 ex quo(const ex & a, const ex & b, const symbol & x);
3159 ex rem(const ex & a, const ex & b, const symbol & x);
3162 compute the quotient and remainder of univariate polynomials in the variable
3163 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3165 The additional function
3168 ex prem(const ex & a, const ex & b, const symbol & x);
3171 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3172 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3174 Exact division of multivariate polynomials is performed by the function
3177 bool divide(const ex & a, const ex & b, ex & q);
3180 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3181 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3182 in which case the value of @code{q} is undefined.
3185 @subsection Unit, content and primitive part
3186 @cindex @code{unit()}
3187 @cindex @code{content()}
3188 @cindex @code{primpart()}
3193 ex ex::unit(const symbol & x);
3194 ex ex::content(const symbol & x);
3195 ex ex::primpart(const symbol & x);
3198 return the unit part, content part, and primitive polynomial of a multivariate
3199 polynomial with respect to the variable @samp{x} (the unit part being the sign
3200 of the leading coefficient, the content part being the GCD of the coefficients,
3201 and the primitive polynomial being the input polynomial divided by the unit and
3202 content parts). The product of unit, content, and primitive part is the
3203 original polynomial.
3206 @subsection GCD and LCM
3209 @cindex @code{gcd()}
3210 @cindex @code{lcm()}
3212 The functions for polynomial greatest common divisor and least common
3213 multiple have the synopsis
3216 ex gcd(const ex & a, const ex & b);
3217 ex lcm(const ex & a, const ex & b);
3220 The functions @code{gcd()} and @code{lcm()} accept two expressions
3221 @code{a} and @code{b} as arguments and return a new expression, their
3222 greatest common divisor or least common multiple, respectively. If the
3223 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3224 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3227 #include <ginac/ginac.h>
3228 using namespace GiNaC;
3232 symbol x("x"), y("y"), z("z");
3233 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3234 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3236 ex P_gcd = gcd(P_a, P_b);
3238 ex P_lcm = lcm(P_a, P_b);
3239 // 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
3244 @subsection Square-free decomposition
3245 @cindex square-free decomposition
3246 @cindex factorization
3247 @cindex @code{sqrfree()}
3249 GiNaC still lacks proper factorization support. Some form of
3250 factorization is, however, easily implemented by noting that factors
3251 appearing in a polynomial with power two or more also appear in the
3252 derivative and hence can easily be found by computing the GCD of the
3253 original polynomial and its derivatives. Any system has an interface
3254 for this so called square-free factorization. So we provide one, too:
3256 ex sqrfree(const ex & a, const lst & l = lst());
3258 Here is an example that by the way illustrates how the result may depend
3259 on the order of differentiation:
3262 symbol x("x"), y("y");
3263 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3265 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3266 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3268 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3269 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3271 cout << sqrfree(BiVarPol) << endl;
3272 // -> depending on luck, any of the above
3277 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3278 @c node-name, next, previous, up
3279 @section Rational expressions
3281 @subsection The @code{normal} method
3282 @cindex @code{normal()}
3283 @cindex simplification
3284 @cindex temporary replacement
3286 Some basic form of simplification of expressions is called for frequently.
3287 GiNaC provides the method @code{.normal()}, which converts a rational function
3288 into an equivalent rational function of the form @samp{numerator/denominator}
3289 where numerator and denominator are coprime. If the input expression is already
3290 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3291 otherwise it performs fraction addition and multiplication.
3293 @code{.normal()} can also be used on expressions which are not rational functions
3294 as it will replace all non-rational objects (like functions or non-integer
3295 powers) by temporary symbols to bring the expression to the domain of rational
3296 functions before performing the normalization, and re-substituting these
3297 symbols afterwards. This algorithm is also available as a separate method
3298 @code{.to_rational()}, described below.
3300 This means that both expressions @code{t1} and @code{t2} are indeed
3301 simplified in this little program:
3304 #include <ginac/ginac.h>
3305 using namespace GiNaC;
3310 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3311 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3312 std::cout << "t1 is " << t1.normal() << std::endl;
3313 std::cout << "t2 is " << t2.normal() << std::endl;
3317 Of course this works for multivariate polynomials too, so the ratio of
3318 the sample-polynomials from the section about GCD and LCM above would be
3319 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3322 @subsection Numerator and denominator
3325 @cindex @code{numer()}
3326 @cindex @code{denom()}
3327 @cindex @code{numer_denom()}
3329 The numerator and denominator of an expression can be obtained with
3334 ex ex::numer_denom();
3337 These functions will first normalize the expression as described above and
3338 then return the numerator, denominator, or both as a list, respectively.
3339 If you need both numerator and denominator, calling @code{numer_denom()} is
3340 faster than using @code{numer()} and @code{denom()} separately.
3343 @subsection Converting to a rational expression
3344 @cindex @code{to_rational()}
3346 Some of the methods described so far only work on polynomials or rational
3347 functions. GiNaC provides a way to extend the domain of these functions to
3348 general expressions by using the temporary replacement algorithm described
3349 above. You do this by calling
3352 ex ex::to_rational(lst &l);
3355 on the expression to be converted. The supplied @code{lst} will be filled
3356 with the generated temporary symbols and their replacement expressions in
3357 a format that can be used directly for the @code{subs()} method. It can also
3358 already contain a list of replacements from an earlier application of
3359 @code{.to_rational()}, so it's possible to use it on multiple expressions
3360 and get consistent results.
3367 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3368 ex b = sin(x) + cos(x);
3371 divide(a.to_rational(l), b.to_rational(l), q);
3372 cout << q.subs(l) << endl;
3376 will print @samp{sin(x)-cos(x)}.
3379 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3380 @c node-name, next, previous, up
3381 @section Symbolic differentiation
3382 @cindex differentiation
3383 @cindex @code{diff()}
3385 @cindex product rule
3387 GiNaC's objects know how to differentiate themselves. Thus, a
3388 polynomial (class @code{add}) knows that its derivative is the sum of
3389 the derivatives of all the monomials:
3392 #include <ginac/ginac.h>
3393 using namespace GiNaC;
3397 symbol x("x"), y("y"), z("z");
3398 ex P = pow(x, 5) + pow(x, 2) + y;
3400 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3401 cout << P.diff(y) << endl; // 1
3402 cout << P.diff(z) << endl; // 0
3406 If a second integer parameter @var{n} is given, the @code{diff} method
3407 returns the @var{n}th derivative.
3409 If @emph{every} object and every function is told what its derivative
3410 is, all derivatives of composed objects can be calculated using the
3411 chain rule and the product rule. Consider, for instance the expression
3412 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3413 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3414 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3415 out that the composition is the generating function for Euler Numbers,
3416 i.e. the so called @var{n}th Euler number is the coefficient of
3417 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3418 identity to code a function that generates Euler numbers in just three
3421 @cindex Euler numbers
3423 #include <ginac/ginac.h>
3424 using namespace GiNaC;
3426 ex EulerNumber(unsigned n)
3429 const ex generator = pow(cosh(x),-1);
3430 return generator.diff(x,n).subs(x==0);
3435 for (unsigned i=0; i<11; i+=2)
3436 std::cout << EulerNumber(i) << std::endl;
3441 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3442 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3443 @code{i} by two since all odd Euler numbers vanish anyways.
3446 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3447 @c node-name, next, previous, up
3448 @section Series expansion
3449 @cindex @code{series()}
3450 @cindex Taylor expansion
3451 @cindex Laurent expansion
3452 @cindex @code{pseries} (class)
3454 Expressions know how to expand themselves as a Taylor series or (more
3455 generally) a Laurent series. As in most conventional Computer Algebra
3456 Systems, no distinction is made between those two. There is a class of
3457 its own for storing such series (@code{class pseries}) and a built-in
3458 function (called @code{Order}) for storing the order term of the series.
3459 As a consequence, if you want to work with series, i.e. multiply two
3460 series, you need to call the method @code{ex::series} again to convert
3461 it to a series object with the usual structure (expansion plus order
3462 term). A sample application from special relativity could read:
3465 #include <ginac/ginac.h>
3466 using namespace std;
3467 using namespace GiNaC;
3471 symbol v("v"), c("c");
3473 ex gamma = 1/sqrt(1 - pow(v/c,2));
3474 ex mass_nonrel = gamma.series(v==0, 10);
3476 cout << "the relativistic mass increase with v is " << endl
3477 << mass_nonrel << endl;
3479 cout << "the inverse square of this series is " << endl
3480 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3484 Only calling the series method makes the last output simplify to
3485 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3486 series raised to the power @math{-2}.
3488 @cindex M@'echain's formula
3489 As another instructive application, let us calculate the numerical
3490 value of Archimedes' constant
3494 (for which there already exists the built-in constant @code{Pi})
3495 using M@'echain's amazing formula
3497 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3500 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3502 We may expand the arcus tangent around @code{0} and insert the fractions
3503 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3504 carries an order term with it and the question arises what the system is
3505 supposed to do when the fractions are plugged into that order term. The
3506 solution is to use the function @code{series_to_poly()} to simply strip
3510 #include <ginac/ginac.h>
3511 using namespace GiNaC;
3513 ex mechain_pi(int degr)
3516 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3517 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3518 -4*pi_expansion.subs(x==numeric(1,239));
3524 using std::cout; // just for fun, another way of...
3525 using std::endl; // ...dealing with this namespace std.
3527 for (int i=2; i<12; i+=2) @{
3528 pi_frac = mechain_pi(i);
3529 cout << i << ":\t" << pi_frac << endl
3530 << "\t" << pi_frac.evalf() << endl;
3536 Note how we just called @code{.series(x,degr)} instead of
3537 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3538 method @code{series()}: if the first argument is a symbol the expression
3539 is expanded in that symbol around point @code{0}. When you run this
3540 program, it will type out:
3544 3.1832635983263598326
3545 4: 5359397032/1706489875
3546 3.1405970293260603143
3547 6: 38279241713339684/12184551018734375
3548 3.141621029325034425
3549 8: 76528487109180192540976/24359780855939418203125
3550 3.141591772182177295
3551 10: 327853873402258685803048818236/104359128170408663038552734375
3552 3.1415926824043995174
3556 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3557 @c node-name, next, previous, up
3558 @section Symmetrization
3559 @cindex @code{symmetrize()}
3560 @cindex @code{antisymmetrize()}
3561 @cindex @code{symmetrize_cyclic()}
3566 ex ex::symmetrize(const lst & l);
3567 ex ex::antisymmetrize(const lst & l);
3568 ex ex::symmetrize_cyclic(const lst & l);
3571 symmetrize an expression by returning the sum over all symmetric,
3572 antisymmetric or cyclic permutations of the specified list of objects,
3573 weighted by the number of permutations.
3575 The three additional methods
3578 ex ex::symmetrize();
3579 ex ex::antisymmetrize();
3580 ex ex::symmetrize_cyclic();
3583 symmetrize or antisymmetrize an expression over its free indices.
3585 Symmetrization is most useful with indexed expressions but can be used with
3586 almost any kind of object (anything that is @code{subs()}able):
3590 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3591 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3593 cout << indexed(A, i, j).symmetrize() << endl;
3594 // -> 1/2*A.j.i+1/2*A.i.j
3595 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3596 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3597 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3598 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
3603 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3604 @c node-name, next, previous, up
3605 @section Predefined mathematical functions
3607 GiNaC contains the following predefined mathematical functions:
3610 @multitable @columnfractions .30 .70
3611 @item @strong{Name} @tab @strong{Function}
3614 @item @code{csgn(x)}
3616 @item @code{sqrt(x)}
3617 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3624 @item @code{asin(x)}
3626 @item @code{acos(x)}
3628 @item @code{atan(x)}
3629 @tab inverse tangent
3630 @item @code{atan2(y, x)}
3631 @tab inverse tangent with two arguments
3632 @item @code{sinh(x)}
3633 @tab hyperbolic sine
3634 @item @code{cosh(x)}
3635 @tab hyperbolic cosine
3636 @item @code{tanh(x)}
3637 @tab hyperbolic tangent
3638 @item @code{asinh(x)}
3639 @tab inverse hyperbolic sine
3640 @item @code{acosh(x)}
3641 @tab inverse hyperbolic cosine
3642 @item @code{atanh(x)}
3643 @tab inverse hyperbolic tangent
3645 @tab exponential function
3647 @tab natural logarithm
3650 @item @code{zeta(x)}
3651 @tab Riemann's zeta function
3652 @item @code{zeta(n, x)}
3653 @tab derivatives of Riemann's zeta function
3654 @item @code{tgamma(x)}
3656 @item @code{lgamma(x)}
3657 @tab logarithm of Gamma function
3658 @item @code{beta(x, y)}
3659 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3661 @tab psi (digamma) function
3662 @item @code{psi(n, x)}
3663 @tab derivatives of psi function (polygamma functions)
3664 @item @code{factorial(n)}
3665 @tab factorial function
3666 @item @code{binomial(n, m)}
3667 @tab binomial coefficients
3668 @item @code{Order(x)}
3669 @tab order term function in truncated power series
3670 @item @code{Derivative(x, l)}
3671 @tab inert partial differentiation operator (used internally)
3676 For functions that have a branch cut in the complex plane GiNaC follows
3677 the conventions for C++ as defined in the ANSI standard as far as
3678 possible. In particular: the natural logarithm (@code{log}) and the
3679 square root (@code{sqrt}) both have their branch cuts running along the
3680 negative real axis where the points on the axis itself belong to the
3681 upper part (i.e. continuous with quadrant II). The inverse
3682 trigonometric and hyperbolic functions are not defined for complex
3683 arguments by the C++ standard, however. In GiNaC we follow the
3684 conventions used by CLN, which in turn follow the carefully designed
3685 definitions in the Common Lisp standard. It should be noted that this
3686 convention is identical to the one used by the C99 standard and by most
3687 serious CAS. It is to be expected that future revisions of the C++
3688 standard incorporate these functions in the complex domain in a manner
3689 compatible with C99.
3692 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3693 @c node-name, next, previous, up
3694 @section Input and output of expressions
3697 @subsection Expression output
3699 @cindex output of expressions
3701 The easiest way to print an expression is to write it to a stream:
3706 ex e = 4.5+pow(x,2)*3/2;
3707 cout << e << endl; // prints '(4.5)+3/2*x^2'
3711 The output format is identical to the @command{ginsh} input syntax and
3712 to that used by most computer algebra systems, but not directly pastable
3713 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3714 is printed as @samp{x^2}).
3716 It is possible to print expressions in a number of different formats with
3720 void ex::print(const print_context & c, unsigned level = 0);
3723 @cindex @code{print_context} (class)
3724 The type of @code{print_context} object passed in determines the format
3725 of the output. The possible types are defined in @file{ginac/print.h}.
3726 All constructors of @code{print_context} and derived classes take an
3727 @code{ostream &} as their first argument.
3729 To print an expression in a way that can be directly used in a C or C++
3730 program, you pass a @code{print_csrc} object like this:
3734 cout << "float f = ";
3735 e.print(print_csrc_float(cout));
3738 cout << "double d = ";
3739 e.print(print_csrc_double(cout));
3742 cout << "cl_N n = ";
3743 e.print(print_csrc_cl_N(cout));
3748 The three possible types mostly affect the way in which floating point
3749 numbers are written.
3751 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3754 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3755 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3756 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3759 The @code{print_context} type @code{print_tree} provides a dump of the
3760 internal structure of an expression for debugging purposes:
3764 e.print(print_tree(cout));
3771 add, hash=0x0, flags=0x3, nops=2
3772 power, hash=0x9, flags=0x3, nops=2
3773 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
3774 2 (numeric), hash=0x80000042, flags=0xf
3775 3/2 (numeric), hash=0x80000061, flags=0xf
3778 4.5L0 (numeric), hash=0x8000004b, flags=0xf
3782 This kind of output is also available in @command{ginsh} as the @code{print()}
3785 Another useful output format is for LaTeX parsing in mathematical mode.
3786 It is rather similar to the default @code{print_context} but provides
3787 some braces needed by LaTeX for delimiting boxes and also converts some
3788 common objects to conventional LaTeX names. It is possible to give symbols
3789 a special name for LaTeX output by supplying it as a second argument to
3790 the @code{symbol} constructor.
3792 For example, the code snippet
3797 ex foo = lgamma(x).series(x==0,3);
3798 foo.print(print_latex(std::cout));
3804 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
3807 If you need any fancy special output format, e.g. for interfacing GiNaC
3808 with other algebra systems or for producing code for different
3809 programming languages, you can always traverse the expression tree yourself:
3812 static void my_print(const ex & e)
3814 if (is_a<function>(e))
3815 cout << ex_to<function>(e).get_name();
3817 cout << e.bp->class_name();
3819 unsigned n = e.nops();
3821 for (unsigned i=0; i<n; i++) @{
3833 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
3841 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
3842 symbol(y))),numeric(-2)))
3845 If you need an output format that makes it possible to accurately
3846 reconstruct an expression by feeding the output to a suitable parser or
3847 object factory, you should consider storing the expression in an
3848 @code{archive} object and reading the object properties from there.
3849 See the section on archiving for more information.
3852 @subsection Expression input
3853 @cindex input of expressions
3855 GiNaC provides no way to directly read an expression from a stream because
3856 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
3857 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
3858 @code{y} you defined in your program and there is no way to specify the
3859 desired symbols to the @code{>>} stream input operator.
3861 Instead, GiNaC lets you construct an expression from a string, specifying the
3862 list of symbols to be used:
3866 symbol x("x"), y("y");
3867 ex e("2*x+sin(y)", lst(x, y));
3871 The input syntax is the same as that used by @command{ginsh} and the stream
3872 output operator @code{<<}. The symbols in the string are matched by name to
3873 the symbols in the list and if GiNaC encounters a symbol not specified in
3874 the list it will throw an exception.
3876 With this constructor, it's also easy to implement interactive GiNaC programs:
3881 #include <stdexcept>
3882 #include <ginac/ginac.h>
3883 using namespace std;
3884 using namespace GiNaC;
3891 cout << "Enter an expression containing 'x': ";
3896 cout << "The derivative of " << e << " with respect to x is ";
3897 cout << e.diff(x) << ".\n";
3898 @} catch (exception &p) @{
3899 cerr << p.what() << endl;
3905 @subsection Archiving
3906 @cindex @code{archive} (class)
3909 GiNaC allows creating @dfn{archives} of expressions which can be stored
3910 to or retrieved from files. To create an archive, you declare an object
3911 of class @code{archive} and archive expressions in it, giving each
3912 expression a unique name:
3916 using namespace std;
3917 #include <ginac/ginac.h>
3918 using namespace GiNaC;
3922 symbol x("x"), y("y"), z("z");
3924 ex foo = sin(x + 2*y) + 3*z + 41;
3928 a.archive_ex(foo, "foo");
3929 a.archive_ex(bar, "the second one");
3933 The archive can then be written to a file:
3937 ofstream out("foobar.gar");
3943 The file @file{foobar.gar} contains all information that is needed to
3944 reconstruct the expressions @code{foo} and @code{bar}.
3946 @cindex @command{viewgar}
3947 The tool @command{viewgar} that comes with GiNaC can be used to view
3948 the contents of GiNaC archive files:
3951 $ viewgar foobar.gar
3952 foo = 41+sin(x+2*y)+3*z
3953 the second one = 42+sin(x+2*y)+3*z
3956 The point of writing archive files is of course that they can later be
3962 ifstream in("foobar.gar");
3967 And the stored expressions can be retrieved by their name:
3973 ex ex1 = a2.unarchive_ex(syms, "foo");
3974 ex ex2 = a2.unarchive_ex(syms, "the second one");
3976 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
3977 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
3978 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
3982 Note that you have to supply a list of the symbols which are to be inserted
3983 in the expressions. Symbols in archives are stored by their name only and
3984 if you don't specify which symbols you have, unarchiving the expression will
3985 create new symbols with that name. E.g. if you hadn't included @code{x} in
3986 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
3987 have had no effect because the @code{x} in @code{ex1} would have been a
3988 different symbol than the @code{x} which was defined at the beginning of
3989 the program, altough both would appear as @samp{x} when printed.
3991 You can also use the information stored in an @code{archive} object to
3992 output expressions in a format suitable for exact reconstruction. The
3993 @code{archive} and @code{archive_node} classes have a couple of member
3994 functions that let you access the stored properties:
3997 static void my_print2(const archive_node & n)
4000 n.find_string("class", class_name);
4001 cout << class_name << "(";
4003 archive_node::propinfovector p;
4004 n.get_properties(p);
4006 unsigned num = p.size();
4007 for (unsigned i=0; i<num; i++) @{
4008 const string &name = p[i].name;
4009 if (name == "class")
4011 cout << name << "=";
4013 unsigned count = p[i].count;
4017 for (unsigned j=0; j<count; j++) @{
4018 switch (p[i].type) @{
4019 case archive_node::PTYPE_BOOL: @{
4021 n.find_bool(name, x);
4022 cout << (x ? "true" : "false");
4025 case archive_node::PTYPE_UNSIGNED: @{
4027 n.find_unsigned(name, x);
4031 case archive_node::PTYPE_STRING: @{
4033 n.find_string(name, x);
4034 cout << '\"' << x << '\"';
4037 case archive_node::PTYPE_NODE: @{
4038 const archive_node &x = n.find_ex_node(name, j);
4060 ex e = pow(2, x) - y;
4062 my_print2(ar.get_top_node(0)); cout << endl;
4070 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4071 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4072 overall_coeff=numeric(number="0"))
4075 Be warned, however, that the set of properties and their meaning for each
4076 class may change between GiNaC versions.
4079 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4080 @c node-name, next, previous, up
4081 @chapter Extending GiNaC
4083 By reading so far you should have gotten a fairly good understanding of
4084 GiNaC's design-patterns. From here on you should start reading the
4085 sources. All we can do now is issue some recommendations how to tackle
4086 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4087 develop some useful extension please don't hesitate to contact the GiNaC
4088 authors---they will happily incorporate them into future versions.
4091 * What does not belong into GiNaC:: What to avoid.
4092 * Symbolic functions:: Implementing symbolic functions.
4093 * Adding classes:: Defining new algebraic classes.
4097 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4098 @c node-name, next, previous, up
4099 @section What doesn't belong into GiNaC
4101 @cindex @command{ginsh}
4102 First of all, GiNaC's name must be read literally. It is designed to be
4103 a library for use within C++. The tiny @command{ginsh} accompanying
4104 GiNaC makes this even more clear: it doesn't even attempt to provide a
4105 language. There are no loops or conditional expressions in
4106 @command{ginsh}, it is merely a window into the library for the
4107 programmer to test stuff (or to show off). Still, the design of a
4108 complete CAS with a language of its own, graphical capabilites and all
4109 this on top of GiNaC is possible and is without doubt a nice project for
4112 There are many built-in functions in GiNaC that do not know how to
4113 evaluate themselves numerically to a precision declared at runtime
4114 (using @code{Digits}). Some may be evaluated at certain points, but not
4115 generally. This ought to be fixed. However, doing numerical
4116 computations with GiNaC's quite abstract classes is doomed to be
4117 inefficient. For this purpose, the underlying foundation classes
4118 provided by @acronym{CLN} are much better suited.
4121 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4122 @c node-name, next, previous, up
4123 @section Symbolic functions
4125 The easiest and most instructive way to start with is probably to
4126 implement your own function. GiNaC's functions are objects of class
4127 @code{function}. The preprocessor is then used to convert the function
4128 names to objects with a corresponding serial number that is used
4129 internally to identify them. You usually need not worry about this
4130 number. New functions may be inserted into the system via a kind of
4131 `registry'. It is your responsibility to care for some functions that
4132 are called when the user invokes certain methods. These are usual
4133 C++-functions accepting a number of @code{ex} as arguments and returning
4134 one @code{ex}. As an example, if we have a look at a simplified
4135 implementation of the cosine trigonometric function, we first need a
4136 function that is called when one wishes to @code{eval} it. It could
4137 look something like this:
4140 static ex cos_eval_method(const ex & x)
4142 // if (!x%(2*Pi)) return 1
4143 // if (!x%Pi) return -1
4144 // if (!x%Pi/2) return 0
4145 // care for other cases...
4146 return cos(x).hold();
4150 @cindex @code{hold()}
4152 The last line returns @code{cos(x)} if we don't know what else to do and
4153 stops a potential recursive evaluation by saying @code{.hold()}, which
4154 sets a flag to the expression signaling that it has been evaluated. We
4155 should also implement a method for numerical evaluation and since we are
4156 lazy we sweep the problem under the rug by calling someone else's
4157 function that does so, in this case the one in class @code{numeric}:
4160 static ex cos_evalf(const ex & x)
4162 return cos(ex_to<numeric>(x));
4166 Differentiation will surely turn up and so we need to tell @code{cos}
4167 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4168 instance are then handled automatically by @code{basic::diff} and
4172 static ex cos_deriv(const ex & x, unsigned diff_param)
4178 @cindex product rule
4179 The second parameter is obligatory but uninteresting at this point. It
4180 specifies which parameter to differentiate in a partial derivative in
4181 case the function has more than one parameter and its main application
4182 is for correct handling of the chain rule. For Taylor expansion, it is
4183 enough to know how to differentiate. But if the function you want to
4184 implement does have a pole somewhere in the complex plane, you need to
4185 write another method for Laurent expansion around that point.
4187 Now that all the ingredients for @code{cos} have been set up, we need
4188 to tell the system about it. This is done by a macro and we are not
4189 going to descibe how it expands, please consult your preprocessor if you
4193 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4194 evalf_func(cos_evalf).
4195 derivative_func(cos_deriv));
4198 The first argument is the function's name used for calling it and for
4199 output. The second binds the corresponding methods as options to this
4200 object. Options are separated by a dot and can be given in an arbitrary
4201 order. GiNaC functions understand several more options which are always
4202 specified as @code{.option(params)}, for example a method for series
4203 expansion @code{.series_func(cos_series)}. Again, if no series
4204 expansion method is given, GiNaC defaults to simple Taylor expansion,
4205 which is correct if there are no poles involved as is the case for the
4206 @code{cos} function. The way GiNaC handles poles in case there are any
4207 is best understood by studying one of the examples, like the Gamma
4208 (@code{tgamma}) function for instance. (In essence the function first
4209 checks if there is a pole at the evaluation point and falls back to
4210 Taylor expansion if there isn't. Then, the pole is regularized by some
4211 suitable transformation.) Also, the new function needs to be declared
4212 somewhere. This may also be done by a convenient preprocessor macro:
4215 DECLARE_FUNCTION_1P(cos)
4218 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4219 implementation of @code{cos} is very incomplete and lacks several safety
4220 mechanisms. Please, have a look at the real implementation in GiNaC.
4221 (By the way: in case you are worrying about all the macros above we can
4222 assure you that functions are GiNaC's most macro-intense classes. We
4223 have done our best to avoid macros where we can.)
4226 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4227 @c node-name, next, previous, up
4228 @section Adding classes
4230 If you are doing some very specialized things with GiNaC you may find that
4231 you have to implement your own algebraic classes to fit your needs. This
4232 section will explain how to do this by giving the example of a simple
4233 'string' class. After reading this section you will know how to properly
4234 declare a GiNaC class and what the minimum required member functions are
4235 that you have to implement. We only cover the implementation of a 'leaf'
4236 class here (i.e. one that doesn't contain subexpressions). Creating a
4237 container class like, for example, a class representing tensor products is
4238 more involved but this section should give you enough information so you can
4239 consult the source to GiNaC's predefined classes if you want to implement
4240 something more complicated.
4242 @subsection GiNaC's run-time type information system
4244 @cindex hierarchy of classes
4246 All algebraic classes (that is, all classes that can appear in expressions)
4247 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4248 @code{basic *} (which is essentially what an @code{ex} is) represents a
4249 generic pointer to an algebraic class. Occasionally it is necessary to find
4250 out what the class of an object pointed to by a @code{basic *} really is.
4251 Also, for the unarchiving of expressions it must be possible to find the
4252 @code{unarchive()} function of a class given the class name (as a string). A
4253 system that provides this kind of information is called a run-time type
4254 information (RTTI) system. The C++ language provides such a thing (see the
4255 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4256 implements its own, simpler RTTI.
4258 The RTTI in GiNaC is based on two mechanisms:
4263 The @code{basic} class declares a member variable @code{tinfo_key} which
4264 holds an unsigned integer that identifies the object's class. These numbers
4265 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4266 classes. They all start with @code{TINFO_}.
4269 By means of some clever tricks with static members, GiNaC maintains a list
4270 of information for all classes derived from @code{basic}. The information
4271 available includes the class names, the @code{tinfo_key}s, and pointers
4272 to the unarchiving functions. This class registry is defined in the
4273 @file{registrar.h} header file.
4277 The disadvantage of this proprietary RTTI implementation is that there's
4278 a little more to do when implementing new classes (C++'s RTTI works more
4279 or less automatic) but don't worry, most of the work is simplified by
4282 @subsection A minimalistic example
4284 Now we will start implementing a new class @code{mystring} that allows
4285 placing character strings in algebraic expressions (this is not very useful,
4286 but it's just an example). This class will be a direct subclass of
4287 @code{basic}. You can use this sample implementation as a starting point
4288 for your own classes.
4290 The code snippets given here assume that you have included some header files
4296 #include <stdexcept>
4297 using namespace std;
4299 #include <ginac/ginac.h>
4300 using namespace GiNaC;
4303 The first thing we have to do is to define a @code{tinfo_key} for our new
4304 class. This can be any arbitrary unsigned number that is not already taken
4305 by one of the existing classes but it's better to come up with something
4306 that is unlikely to clash with keys that might be added in the future. The
4307 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4308 which is not a requirement but we are going to stick with this scheme:
4311 const unsigned TINFO_mystring = 0x42420001U;
4314 Now we can write down the class declaration. The class stores a C++
4315 @code{string} and the user shall be able to construct a @code{mystring}
4316 object from a C or C++ string:
4319 class mystring : public basic
4321 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4324 mystring(const string &s);
4325 mystring(const char *s);
4331 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4334 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4335 macros are defined in @file{registrar.h}. They take the name of the class
4336 and its direct superclass as arguments and insert all required declarations
4337 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4338 the first line after the opening brace of the class definition. The
4339 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4340 source (at global scope, of course, not inside a function).
4342 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4343 declarations of the default and copy constructor, the destructor, the
4344 assignment operator and a couple of other functions that are required. It
4345 also defines a type @code{inherited} which refers to the superclass so you
4346 don't have to modify your code every time you shuffle around the class
4347 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4348 constructor, the destructor and the assignment operator.
4350 Now there are nine member functions we have to implement to get a working
4356 @code{mystring()}, the default constructor.
4359 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4360 assignment operator to free dynamically allocated members. The @code{call_parent}
4361 specifies whether the @code{destroy()} function of the superclass is to be
4365 @code{void copy(const mystring &other)}, which is used in the copy constructor
4366 and assignment operator to copy the member variables over from another
4367 object of the same class.
4370 @code{void archive(archive_node &n)}, the archiving function. This stores all
4371 information needed to reconstruct an object of this class inside an
4372 @code{archive_node}.
4375 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4376 constructor. This constructs an instance of the class from the information
4377 found in an @code{archive_node}.
4380 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4381 unarchiving function. It constructs a new instance by calling the unarchiving
4385 @code{int compare_same_type(const basic &other)}, which is used internally
4386 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4387 -1, depending on the relative order of this object and the @code{other}
4388 object. If it returns 0, the objects are considered equal.
4389 @strong{Note:} This has nothing to do with the (numeric) ordering
4390 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4391 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4392 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4393 must provide a @code{compare_same_type()} function, even those representing
4394 objects for which no reasonable algebraic ordering relationship can be
4398 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4399 which are the two constructors we declared.
4403 Let's proceed step-by-step. The default constructor looks like this:
4406 mystring::mystring() : inherited(TINFO_mystring)
4408 // dynamically allocate resources here if required
4412 The golden rule is that in all constructors you have to set the
4413 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4414 it will be set by the constructor of the superclass and all hell will break
4415 loose in the RTTI. For your convenience, the @code{basic} class provides
4416 a constructor that takes a @code{tinfo_key} value, which we are using here
4417 (remember that in our case @code{inherited = basic}). If the superclass
4418 didn't have such a constructor, we would have to set the @code{tinfo_key}
4419 to the right value manually.
4421 In the default constructor you should set all other member variables to
4422 reasonable default values (we don't need that here since our @code{str}
4423 member gets set to an empty string automatically). The constructor(s) are of
4424 course also the right place to allocate any dynamic resources you require.
4426 Next, the @code{destroy()} function:
4429 void mystring::destroy(bool call_parent)
4431 // free dynamically allocated resources here if required
4433 inherited::destroy(call_parent);
4437 This function is where we free all dynamically allocated resources. We don't
4438 have any so we're not doing anything here, but if we had, for example, used
4439 a C-style @code{char *} to store our string, this would be the place to
4440 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4441 to call the @code{destroy()} function of the superclass after we're done
4442 (to mimic C++'s automatic invocation of superclass destructors where
4443 @code{destroy()} is called from outside a destructor).
4445 The @code{copy()} function just copies over the member variables from
4449 void mystring::copy(const mystring &other)
4451 inherited::copy(other);
4456 We can simply overwrite the member variables here. There's no need to worry
4457 about dynamically allocated storage. The assignment operator (which is
4458 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4459 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4460 explicitly call the @code{copy()} function of the superclass here so
4461 all the member variables will get copied.
4463 Next are the three functions for archiving. You have to implement them even
4464 if you don't plan to use archives, but the minimum required implementation
4465 is really simple. First, the archiving function:
4468 void mystring::archive(archive_node &n) const
4470 inherited::archive(n);
4471 n.add_string("string", str);
4475 The only thing that is really required is calling the @code{archive()}
4476 function of the superclass. Optionally, you can store all information you
4477 deem necessary for representing the object into the passed
4478 @code{archive_node}. We are just storing our string here. For more
4479 information on how the archiving works, consult the @file{archive.h} header
4482 The unarchiving constructor is basically the inverse of the archiving
4486 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4488 n.find_string("string", str);
4492 If you don't need archiving, just leave this function empty (but you must
4493 invoke the unarchiving constructor of the superclass). Note that we don't
4494 have to set the @code{tinfo_key} here because it is done automatically
4495 by the unarchiving constructor of the @code{basic} class.
4497 Finally, the unarchiving function:
4500 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4502 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4506 You don't have to understand how exactly this works. Just copy these four
4507 lines into your code literally (replacing the class name, of course). It
4508 calls the unarchiving constructor of the class and unless you are doing
4509 something very special (like matching @code{archive_node}s to global
4510 objects) you don't need a different implementation. For those who are
4511 interested: setting the @code{dynallocated} flag puts the object under
4512 the control of GiNaC's garbage collection. It will get deleted automatically
4513 once it is no longer referenced.
4515 Our @code{compare_same_type()} function uses a provided function to compare
4519 int mystring::compare_same_type(const basic &other) const
4521 const mystring &o = static_cast<const mystring &>(other);
4522 int cmpval = str.compare(o.str);
4525 else if (cmpval < 0)
4532 Although this function takes a @code{basic &}, it will always be a reference
4533 to an object of exactly the same class (objects of different classes are not
4534 comparable), so the cast is safe. If this function returns 0, the two objects
4535 are considered equal (in the sense that @math{A-B=0}), so you should compare
4536 all relevant member variables.
4538 Now the only thing missing is our two new constructors:
4541 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4543 // dynamically allocate resources here if required
4546 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4548 // dynamically allocate resources here if required
4552 No surprises here. We set the @code{str} member from the argument and
4553 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4555 That's it! We now have a minimal working GiNaC class that can store
4556 strings in algebraic expressions. Let's confirm that the RTTI works:
4559 ex e = mystring("Hello, world!");
4560 cout << is_a<mystring>(e) << endl;
4563 cout << e.bp->class_name() << endl;
4567 Obviously it does. Let's see what the expression @code{e} looks like:
4571 // -> [mystring object]
4574 Hm, not exactly what we expect, but of course the @code{mystring} class
4575 doesn't yet know how to print itself. This is done in the @code{print()}
4576 member function. Let's say that we wanted to print the string surrounded
4580 class mystring : public basic
4584 void print(const print_context &c, unsigned level = 0) const;
4588 void mystring::print(const print_context &c, unsigned level) const
4590 // print_context::s is a reference to an ostream
4591 c.s << '\"' << str << '\"';
4595 The @code{level} argument is only required for container classes to
4596 correctly parenthesize the output. Let's try again to print the expression:
4600 // -> "Hello, world!"
4603 Much better. The @code{mystring} class can be used in arbitrary expressions:
4606 e += mystring("GiNaC rulez");
4608 // -> "GiNaC rulez"+"Hello, world!"
4611 (GiNaC's automatic term reordering is in effect here), or even
4614 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4616 // -> "One string"^(2*sin(-"Another string"+Pi))
4619 Whether this makes sense is debatable but remember that this is only an
4620 example. At least it allows you to implement your own symbolic algorithms
4623 Note that GiNaC's algebraic rules remain unchanged:
4626 e = mystring("Wow") * mystring("Wow");
4630 e = pow(mystring("First")-mystring("Second"), 2);
4631 cout << e.expand() << endl;
4632 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4635 There's no way to, for example, make GiNaC's @code{add} class perform string
4636 concatenation. You would have to implement this yourself.
4638 @subsection Automatic evaluation
4640 @cindex @code{hold()}
4642 When dealing with objects that are just a little more complicated than the
4643 simple string objects we have implemented, chances are that you will want to
4644 have some automatic simplifications or canonicalizations performed on them.
4645 This is done in the evaluation member function @code{eval()}. Let's say that
4646 we wanted all strings automatically converted to lowercase with
4647 non-alphabetic characters stripped, and empty strings removed:
4650 class mystring : public basic
4654 ex eval(int level = 0) const;
4658 ex mystring::eval(int level) const
4661 for (int i=0; i<str.length(); i++) @{
4663 if (c >= 'A' && c <= 'Z')
4664 new_str += tolower(c);
4665 else if (c >= 'a' && c <= 'z')
4669 if (new_str.length() == 0)
4672 return mystring(new_str).hold();
4676 The @code{level} argument is used to limit the recursion depth of the
4677 evaluation. We don't have any subexpressions in the @code{mystring} class
4678 so we are not concerned with this. If we had, we would call the @code{eval()}
4679 functions of the subexpressions with @code{level - 1} as the argument if
4680 @code{level != 1}. The @code{hold()} member function sets a flag in the
4681 object that prevents further evaluation. Otherwise we might end up in an
4682 endless loop. When you want to return the object unmodified, use
4683 @code{return this->hold();}.
4685 Let's confirm that it works:
4688 ex e = mystring("Hello, world!") + mystring("!?#");
4692 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4697 @subsection Other member functions
4699 We have implemented only a small set of member functions to make the class
4700 work in the GiNaC framework. For a real algebraic class, there are probably
4701 some more functions that you will want to re-implement, such as
4702 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4703 or the header file of the class you want to make a subclass of to see
4704 what's there. One member function that you will most likely want to
4705 implement for terminal classes like the described string class is
4706 @code{calcchash()} that returns an @code{unsigned} hash value for the object
4707 which will allow GiNaC to compare and canonicalize expressions much more
4710 You can, of course, also add your own new member functions. Remember,
4711 that the RTTI may be used to get information about what kinds of objects
4712 you are dealing with (the position in the class hierarchy) and that you
4713 can always extract the bare object from an @code{ex} by stripping the
4714 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
4715 should become a need.
4717 That's it. May the source be with you!
4720 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4721 @c node-name, next, previous, up
4722 @chapter A Comparison With Other CAS
4725 This chapter will give you some information on how GiNaC compares to
4726 other, traditional Computer Algebra Systems, like @emph{Maple},
4727 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4728 disadvantages over these systems.
4731 * Advantages:: Stengths of the GiNaC approach.
4732 * Disadvantages:: Weaknesses of the GiNaC approach.
4733 * Why C++?:: Attractiveness of C++.
4736 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4737 @c node-name, next, previous, up
4740 GiNaC has several advantages over traditional Computer
4741 Algebra Systems, like
4746 familiar language: all common CAS implement their own proprietary
4747 grammar which you have to learn first (and maybe learn again when your
4748 vendor decides to `enhance' it). With GiNaC you can write your program
4749 in common C++, which is standardized.
4753 structured data types: you can build up structured data types using
4754 @code{struct}s or @code{class}es together with STL features instead of
4755 using unnamed lists of lists of lists.
4758 strongly typed: in CAS, you usually have only one kind of variables
4759 which can hold contents of an arbitrary type. This 4GL like feature is
4760 nice for novice programmers, but dangerous.
4763 development tools: powerful development tools exist for C++, like fancy
4764 editors (e.g. with automatic indentation and syntax highlighting),
4765 debuggers, visualization tools, documentation generators@dots{}
4768 modularization: C++ programs can easily be split into modules by
4769 separating interface and implementation.
4772 price: GiNaC is distributed under the GNU Public License which means
4773 that it is free and available with source code. And there are excellent
4774 C++-compilers for free, too.
4777 extendable: you can add your own classes to GiNaC, thus extending it on
4778 a very low level. Compare this to a traditional CAS that you can
4779 usually only extend on a high level by writing in the language defined
4780 by the parser. In particular, it turns out to be almost impossible to
4781 fix bugs in a traditional system.
4784 multiple interfaces: Though real GiNaC programs have to be written in
4785 some editor, then be compiled, linked and executed, there are more ways
4786 to work with the GiNaC engine. Many people want to play with
4787 expressions interactively, as in traditional CASs. Currently, two such
4788 windows into GiNaC have been implemented and many more are possible: the
4789 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
4790 types to a command line and second, as a more consistent approach, an
4791 interactive interface to the @acronym{Cint} C++ interpreter has been put
4792 together (called @acronym{GiNaC-cint}) that allows an interactive
4793 scripting interface consistent with the C++ language.
4796 seemless integration: it is somewhere between difficult and impossible
4797 to call CAS functions from within a program written in C++ or any other
4798 programming language and vice versa. With GiNaC, your symbolic routines
4799 are part of your program. You can easily call third party libraries,
4800 e.g. for numerical evaluation or graphical interaction. All other
4801 approaches are much more cumbersome: they range from simply ignoring the
4802 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
4803 system (i.e. @emph{Yacas}).
4806 efficiency: often large parts of a program do not need symbolic
4807 calculations at all. Why use large integers for loop variables or
4808 arbitrary precision arithmetics where @code{int} and @code{double} are
4809 sufficient? For pure symbolic applications, GiNaC is comparable in
4810 speed with other CAS.
4815 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
4816 @c node-name, next, previous, up
4817 @section Disadvantages
4819 Of course it also has some disadvantages:
4824 advanced features: GiNaC cannot compete with a program like
4825 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
4826 which grows since 1981 by the work of dozens of programmers, with
4827 respect to mathematical features. Integration, factorization,
4828 non-trivial simplifications, limits etc. are missing in GiNaC (and are
4829 not planned for the near future).
4832 portability: While the GiNaC library itself is designed to avoid any
4833 platform dependent features (it should compile on any ANSI compliant C++
4834 compiler), the currently used version of the CLN library (fast large
4835 integer and arbitrary precision arithmetics) can be compiled only on
4836 systems with a recently new C++ compiler from the GNU Compiler
4837 Collection (@acronym{GCC}).@footnote{This is because CLN uses
4838 PROVIDE/REQUIRE like macros to let the compiler gather all static
4839 initializations, which works for GNU C++ only.} GiNaC uses recent
4840 language features like explicit constructors, mutable members, RTTI,
4841 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
4842 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
4843 ANSI compliant, support all needed features.
4848 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
4849 @c node-name, next, previous, up
4852 Why did we choose to implement GiNaC in C++ instead of Java or any other
4853 language? C++ is not perfect: type checking is not strict (casting is
4854 possible), separation between interface and implementation is not
4855 complete, object oriented design is not enforced. The main reason is
4856 the often scolded feature of operator overloading in C++. While it may
4857 be true that operating on classes with a @code{+} operator is rarely
4858 meaningful, it is perfectly suited for algebraic expressions. Writing
4859 @math{3x+5y} as @code{3*x+5*y} instead of
4860 @code{x.times(3).plus(y.times(5))} looks much more natural.
4861 Furthermore, the main developers are more familiar with C++ than with
4862 any other programming language.
4865 @node Internal Structures, Expressions are reference counted, Why C++? , Top
4866 @c node-name, next, previous, up
4867 @appendix Internal Structures
4870 * Expressions are reference counted::
4871 * Internal representation of products and sums::
4874 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
4875 @c node-name, next, previous, up
4876 @appendixsection Expressions are reference counted
4878 @cindex reference counting
4879 @cindex copy-on-write
4880 @cindex garbage collection
4881 An expression is extremely light-weight since internally it works like a
4882 handle to the actual representation and really holds nothing more than a
4883 pointer to some other object. What this means in practice is that
4884 whenever you create two @code{ex} and set the second equal to the first
4885 no copying process is involved. Instead, the copying takes place as soon
4886 as you try to change the second. Consider the simple sequence of code:
4889 #include <ginac/ginac.h>
4890 using namespace std;
4891 using namespace GiNaC;
4895 symbol x("x"), y("y"), z("z");
4898 e1 = sin(x + 2*y) + 3*z + 41;
4899 e2 = e1; // e2 points to same object as e1
4900 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
4901 e2 += 1; // e2 is copied into a new object
4902 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
4906 The line @code{e2 = e1;} creates a second expression pointing to the
4907 object held already by @code{e1}. The time involved for this operation
4908 is therefore constant, no matter how large @code{e1} was. Actual
4909 copying, however, must take place in the line @code{e2 += 1;} because
4910 @code{e1} and @code{e2} are not handles for the same object any more.
4911 This concept is called @dfn{copy-on-write semantics}. It increases
4912 performance considerably whenever one object occurs multiple times and
4913 represents a simple garbage collection scheme because when an @code{ex}
4914 runs out of scope its destructor checks whether other expressions handle
4915 the object it points to too and deletes the object from memory if that
4916 turns out not to be the case. A slightly less trivial example of
4917 differentiation using the chain-rule should make clear how powerful this
4921 #include <ginac/ginac.h>
4922 using namespace std;
4923 using namespace GiNaC;
4927 symbol x("x"), y("y");
4931 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
4932 cout << e1 << endl // prints x+3*y
4933 << e2 << endl // prints (x+3*y)^3
4934 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
4938 Here, @code{e1} will actually be referenced three times while @code{e2}
4939 will be referenced two times. When the power of an expression is built,
4940 that expression needs not be copied. Likewise, since the derivative of
4941 a power of an expression can be easily expressed in terms of that
4942 expression, no copying of @code{e1} is involved when @code{e3} is
4943 constructed. So, when @code{e3} is constructed it will print as
4944 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
4945 holds a reference to @code{e2} and the factor in front is just
4948 As a user of GiNaC, you cannot see this mechanism of copy-on-write
4949 semantics. When you insert an expression into a second expression, the
4950 result behaves exactly as if the contents of the first expression were
4951 inserted. But it may be useful to remember that this is not what
4952 happens. Knowing this will enable you to write much more efficient
4953 code. If you still have an uncertain feeling with copy-on-write
4954 semantics, we recommend you have a look at the
4955 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
4956 Marshall Cline. Chapter 16 covers this issue and presents an
4957 implementation which is pretty close to the one in GiNaC.
4960 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
4961 @c node-name, next, previous, up
4962 @appendixsection Internal representation of products and sums
4964 @cindex representation
4967 @cindex @code{power}
4968 Although it should be completely transparent for the user of
4969 GiNaC a short discussion of this topic helps to understand the sources
4970 and also explain performance to a large degree. Consider the
4971 unexpanded symbolic expression
4973 $2d^3 \left( 4a + 5b - 3 \right)$
4976 @math{2*d^3*(4*a+5*b-3)}
4978 which could naively be represented by a tree of linear containers for
4979 addition and multiplication, one container for exponentiation with base
4980 and exponent and some atomic leaves of symbols and numbers in this
4985 @cindex pair-wise representation
4986 However, doing so results in a rather deeply nested tree which will
4987 quickly become inefficient to manipulate. We can improve on this by
4988 representing the sum as a sequence of terms, each one being a pair of a
4989 purely numeric multiplicative coefficient and its rest. In the same
4990 spirit we can store the multiplication as a sequence of terms, each
4991 having a numeric exponent and a possibly complicated base, the tree
4992 becomes much more flat:
4996 The number @code{3} above the symbol @code{d} shows that @code{mul}
4997 objects are treated similarly where the coefficients are interpreted as
4998 @emph{exponents} now. Addition of sums of terms or multiplication of
4999 products with numerical exponents can be coded to be very efficient with
5000 such a pair-wise representation. Internally, this handling is performed
5001 by most CAS in this way. It typically speeds up manipulations by an
5002 order of magnitude. The overall multiplicative factor @code{2} and the
5003 additive term @code{-3} look somewhat out of place in this
5004 representation, however, since they are still carrying a trivial
5005 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5006 this is avoided by adding a field that carries an overall numeric
5007 coefficient. This results in the realistic picture of internal
5010 $2d^3 \left( 4a + 5b - 3 \right)$:
5013 @math{2*d^3*(4*a+5*b-3)}:
5019 This also allows for a better handling of numeric radicals, since
5020 @code{sqrt(2)} can now be carried along calculations. Now it should be
5021 clear, why both classes @code{add} and @code{mul} are derived from the
5022 same abstract class: the data representation is the same, only the
5023 semantics differs. In the class hierarchy, methods for polynomial
5024 expansion and the like are reimplemented for @code{add} and @code{mul},
5025 but the data structure is inherited from @code{expairseq}.
5028 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5029 @c node-name, next, previous, up
5030 @appendix Package Tools
5032 If you are creating a software package that uses the GiNaC library,
5033 setting the correct command line options for the compiler and linker
5034 can be difficult. GiNaC includes two tools to make this process easier.
5037 * ginac-config:: A shell script to detect compiler and linker flags.
5038 * AM_PATH_GINAC:: Macro for GNU automake.
5042 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5043 @c node-name, next, previous, up
5044 @section @command{ginac-config}
5045 @cindex ginac-config
5047 @command{ginac-config} is a shell script that you can use to determine
5048 the compiler and linker command line options required to compile and
5049 link a program with the GiNaC library.
5051 @command{ginac-config} takes the following flags:
5055 Prints out the version of GiNaC installed.
5057 Prints '-I' flags pointing to the installed header files.
5059 Prints out the linker flags necessary to link a program against GiNaC.
5060 @item --prefix[=@var{PREFIX}]
5061 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5062 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5063 Otherwise, prints out the configured value of @env{$prefix}.
5064 @item --exec-prefix[=@var{PREFIX}]
5065 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5066 Otherwise, prints out the configured value of @env{$exec_prefix}.
5069 Typically, @command{ginac-config} will be used within a configure
5070 script, as described below. It, however, can also be used directly from
5071 the command line using backquotes to compile a simple program. For
5075 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5078 This command line might expand to (for example):
5081 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5082 -lginac -lcln -lstdc++
5085 Not only is the form using @command{ginac-config} easier to type, it will
5086 work on any system, no matter how GiNaC was configured.
5089 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5090 @c node-name, next, previous, up
5091 @section @samp{AM_PATH_GINAC}
5092 @cindex AM_PATH_GINAC
5094 For packages configured using GNU automake, GiNaC also provides
5095 a macro to automate the process of checking for GiNaC.
5098 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5106 Determines the location of GiNaC using @command{ginac-config}, which is
5107 either found in the user's path, or from the environment variable
5108 @env{GINACLIB_CONFIG}.
5111 Tests the installed libraries to make sure that their version
5112 is later than @var{MINIMUM-VERSION}. (A default version will be used
5116 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5117 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5118 variable to the output of @command{ginac-config --libs}, and calls
5119 @samp{AC_SUBST()} for these variables so they can be used in generated
5120 makefiles, and then executes @var{ACTION-IF-FOUND}.
5123 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5124 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5128 This macro is in file @file{ginac.m4} which is installed in
5129 @file{$datadir/aclocal}. Note that if automake was installed with a
5130 different @samp{--prefix} than GiNaC, you will either have to manually
5131 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5132 aclocal the @samp{-I} option when running it.
5135 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5136 * Example package:: Example of a package using AM_PATH_GINAC.
5140 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5141 @c node-name, next, previous, up
5142 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5144 Simply make sure that @command{ginac-config} is in your path, and run
5145 the configure script.
5152 The directory where the GiNaC libraries are installed needs
5153 to be found by your system's dynamic linker.
5155 This is generally done by
5158 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5164 setting the environment variable @env{LD_LIBRARY_PATH},
5167 or, as a last resort,
5170 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5171 running configure, for instance:
5174 LDFLAGS=-R/home/cbauer/lib ./configure
5179 You can also specify a @command{ginac-config} not in your path by
5180 setting the @env{GINACLIB_CONFIG} environment variable to the
5181 name of the executable
5184 If you move the GiNaC package from its installed location,
5185 you will either need to modify @command{ginac-config} script
5186 manually to point to the new location or rebuild GiNaC.
5197 --with-ginac-prefix=@var{PREFIX}
5198 --with-ginac-exec-prefix=@var{PREFIX}
5201 are provided to override the prefix and exec-prefix that were stored
5202 in the @command{ginac-config} shell script by GiNaC's configure. You are
5203 generally better off configuring GiNaC with the right path to begin with.
5207 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5208 @c node-name, next, previous, up
5209 @subsection Example of a package using @samp{AM_PATH_GINAC}
5211 The following shows how to build a simple package using automake
5212 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5215 #include <ginac/ginac.h>
5219 GiNaC::symbol x("x");
5220 GiNaC::ex a = GiNaC::sin(x);
5221 std::cout << "Derivative of " << a
5222 << " is " << a.diff(x) << std::endl;
5227 You should first read the introductory portions of the automake
5228 Manual, if you are not already familiar with it.
5230 Two files are needed, @file{configure.in}, which is used to build the
5234 dnl Process this file with autoconf to produce a configure script.
5236 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5242 AM_PATH_GINAC(0.7.0, [
5243 LIBS="$LIBS $GINACLIB_LIBS"
5244 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5245 ], AC_MSG_ERROR([need to have GiNaC installed]))
5250 The only command in this which is not standard for automake
5251 is the @samp{AM_PATH_GINAC} macro.
5253 That command does the following: If a GiNaC version greater or equal
5254 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5255 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5256 the error message `need to have GiNaC installed'
5258 And the @file{Makefile.am}, which will be used to build the Makefile.
5261 ## Process this file with automake to produce Makefile.in
5262 bin_PROGRAMS = simple
5263 simple_SOURCES = simple.cpp
5266 This @file{Makefile.am}, says that we are building a single executable,
5267 from a single sourcefile @file{simple.cpp}. Since every program
5268 we are building uses GiNaC we simply added the GiNaC options
5269 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5270 want to specify them on a per-program basis: for instance by
5274 simple_LDADD = $(GINACLIB_LIBS)
5275 INCLUDES = $(GINACLIB_CPPFLAGS)
5278 to the @file{Makefile.am}.
5280 To try this example out, create a new directory and add the three
5283 Now execute the following commands:
5286 $ automake --add-missing
5291 You now have a package that can be built in the normal fashion
5300 @node Bibliography, Concept Index, Example package, Top
5301 @c node-name, next, previous, up
5302 @appendix Bibliography
5307 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5310 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5313 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5316 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5319 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5320 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5323 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5324 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5325 Academic Press, London
5328 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5333 @node Concept Index, , Bibliography, Top
5334 @c node-name, next, previous, up
5335 @unnumbered Concept Index