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/}, ...) 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 * Indexed objects:: Handling indexed quantities.
675 * Non-commutative objects:: Algebras with non-commutative products.
679 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
680 @c node-name, next, previous, up
682 @cindex expression (class @code{ex})
685 The most common class of objects a user deals with is the expression
686 @code{ex}, representing a mathematical object like a variable, number,
687 function, sum, product, etc... Expressions may be put together to form
688 new expressions, passed as arguments to functions, and so on. Here is a
689 little collection of valid expressions:
692 ex MyEx1 = 5; // simple number
693 ex MyEx2 = x + 2*y; // polynomial in x and y
694 ex MyEx3 = (x + 1)/(x - 1); // rational expression
695 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
696 ex MyEx5 = MyEx4 + 1; // similar to above
699 Expressions are handles to other more fundamental objects, that often
700 contain other expressions thus creating a tree of expressions
701 (@xref{Internal Structures}, for particular examples). Most methods on
702 @code{ex} therefore run top-down through such an expression tree. For
703 example, the method @code{has()} scans recursively for occurrences of
704 something inside an expression. Thus, if you have declared @code{MyEx4}
705 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
706 the argument of @code{sin} and hence return @code{true}.
708 The next sections will outline the general picture of GiNaC's class
709 hierarchy and describe the classes of objects that are handled by
713 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
714 @c node-name, next, previous, up
715 @section The Class Hierarchy
717 GiNaC's class hierarchy consists of several classes representing
718 mathematical objects, all of which (except for @code{ex} and some
719 helpers) are internally derived from one abstract base class called
720 @code{basic}. You do not have to deal with objects of class
721 @code{basic}, instead you'll be dealing with symbols, numbers,
722 containers of expressions and so on.
726 To get an idea about what kinds of symbolic composits may be built we
727 have a look at the most important classes in the class hierarchy and
728 some of the relations among the classes:
730 @image{classhierarchy}
732 The abstract classes shown here (the ones without drop-shadow) are of no
733 interest for the user. They are used internally in order to avoid code
734 duplication if two or more classes derived from them share certain
735 features. An example is @code{expairseq}, a container for a sequence of
736 pairs each consisting of one expression and a number (@code{numeric}).
737 What @emph{is} visible to the user are the derived classes @code{add}
738 and @code{mul}, representing sums and products. @xref{Internal
739 Structures}, where these two classes are described in more detail. The
740 following table shortly summarizes what kinds of mathematical objects
741 are stored in the different classes:
744 @multitable @columnfractions .22 .78
745 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
746 @item @code{constant} @tab Constants like
753 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
754 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
755 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
756 @item @code{ncmul} @tab Products of non-commutative objects
757 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
762 @code{sqrt(}@math{2}@code{)}
765 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
766 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
767 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
768 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
769 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
770 @item @code{indexed} @tab Indexed object like @math{A_ij}
771 @item @code{tensor} @tab Special tensor like the delta and metric tensors
772 @item @code{idx} @tab Index of an indexed object
773 @item @code{varidx} @tab Index with variance
774 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
775 @item @code{wildcard} @tab Wildcard for pattern matching
779 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
780 @c node-name, next, previous, up
782 @cindex @code{symbol} (class)
783 @cindex hierarchy of classes
786 Symbols are for symbolic manipulation what atoms are for chemistry. You
787 can declare objects of class @code{symbol} as any other object simply by
788 saying @code{symbol x,y;}. There is, however, a catch in here having to
789 do with the fact that C++ is a compiled language. The information about
790 the symbol's name is thrown away by the compiler but at a later stage
791 you may want to print expressions holding your symbols. In order to
792 avoid confusion GiNaC's symbols are able to know their own name. This
793 is accomplished by declaring its name for output at construction time in
794 the fashion @code{symbol x("x");}. If you declare a symbol using the
795 default constructor (i.e. without string argument) the system will deal
796 out a unique name. That name may not be suitable for printing but for
797 internal routines when no output is desired it is often enough. We'll
798 come across examples of such symbols later in this tutorial.
800 This implies that the strings passed to symbols at construction time may
801 not be used for comparing two of them. It is perfectly legitimate to
802 write @code{symbol x("x"),y("x");} but it is likely to lead into
803 trouble. Here, @code{x} and @code{y} are different symbols and
804 statements like @code{x-y} will not be simplified to zero although the
805 output @code{x-x} looks funny. Such output may also occur when there
806 are two different symbols in two scopes, for instance when you call a
807 function that declares a symbol with a name already existent in a symbol
808 in the calling function. Again, comparing them (using @code{operator==}
809 for instance) will always reveal their difference. Watch out, please.
811 @cindex @code{subs()}
812 Although symbols can be assigned expressions for internal reasons, you
813 should not do it (and we are not going to tell you how it is done). If
814 you want to replace a symbol with something else in an expression, you
815 can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
816 for more information).
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()}
1139 The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
1140 These are sometimes used to supply a variable number of arguments of the same
1141 type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
1142 should have a basic understanding about them.
1144 Lists of up to 15 expressions can be directly constructed from single
1149 symbol x("x"), y("y");
1150 lst l(x, 2, y, x+y);
1151 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1155 Use the @code{nops()} method to determine the size (number of expressions) of
1156 a list and the @code{op()} method to access individual elements:
1160 cout << l.nops() << endl; // prints '4'
1161 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1165 Finally you can append or prepend an expression to a list with the
1166 @code{append()} and @code{prepend()} methods:
1170 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1171 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1176 @node Mathematical functions, Relations, Lists, Basic Concepts
1177 @c node-name, next, previous, up
1178 @section Mathematical functions
1179 @cindex @code{function} (class)
1180 @cindex trigonometric function
1181 @cindex hyperbolic function
1183 There are quite a number of useful functions hard-wired into GiNaC. For
1184 instance, all trigonometric and hyperbolic functions are implemented
1185 (@xref{Built-in Functions}, for a complete list).
1187 These functions are all objects of class @code{function}. They accept
1188 one or more expressions as arguments and return one expression. If the
1189 arguments are not numerical, the evaluation of the function may be
1190 halted, as it does in the next example, showing how a function returns
1191 itself twice and finally an expression that may be really useful:
1193 @cindex Gamma function
1194 @cindex @code{subs()}
1197 symbol x("x"), y("y");
1199 cout << tgamma(foo) << endl;
1200 // -> tgamma(x+(1/2)*y)
1201 ex bar = foo.subs(y==1);
1202 cout << tgamma(bar) << endl;
1204 ex foobar = bar.subs(x==7);
1205 cout << tgamma(foobar) << endl;
1206 // -> (135135/128)*Pi^(1/2)
1210 Besides evaluation most of these functions allow differentiation, series
1211 expansion and so on. Read the next chapter in order to learn more about
1215 @node Relations, Indexed objects, Mathematical functions, Basic Concepts
1216 @c node-name, next, previous, up
1218 @cindex @code{relational} (class)
1220 Sometimes, a relation holding between two expressions must be stored
1221 somehow. The class @code{relational} is a convenient container for such
1222 purposes. A relation is by definition a container for two @code{ex} and
1223 a relation between them that signals equality, inequality and so on.
1224 They are created by simply using the C++ operators @code{==}, @code{!=},
1225 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1227 @xref{Mathematical functions}, for examples where various applications
1228 of the @code{.subs()} method show how objects of class relational are
1229 used as arguments. There they provide an intuitive syntax for
1230 substitutions. They are also used as arguments to the @code{ex::series}
1231 method, where the left hand side of the relation specifies the variable
1232 to expand in and the right hand side the expansion point. They can also
1233 be used for creating systems of equations that are to be solved for
1234 unknown variables. But the most common usage of objects of this class
1235 is rather inconspicuous in statements of the form @code{if
1236 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1237 conversion from @code{relational} to @code{bool} takes place. Note,
1238 however, that @code{==} here does not perform any simplifications, hence
1239 @code{expand()} must be called explicitly.
1242 @node Indexed objects, Non-commutative objects, Relations, Basic Concepts
1243 @c node-name, next, previous, up
1244 @section Indexed objects
1246 GiNaC allows you to handle expressions containing general indexed objects in
1247 arbitrary spaces. It is also able to canonicalize and simplify such
1248 expressions and perform symbolic dummy index summations. There are a number
1249 of predefined indexed objects provided, like delta and metric tensors.
1251 There are few restrictions placed on indexed objects and their indices and
1252 it is easy to construct nonsense expressions, but our intention is to
1253 provide a general framework that allows you to implement algorithms with
1254 indexed quantities, getting in the way as little as possible.
1256 @cindex @code{idx} (class)
1257 @cindex @code{indexed} (class)
1258 @subsection Indexed quantities and their indices
1260 Indexed expressions in GiNaC are constructed of two special types of objects,
1261 @dfn{index objects} and @dfn{indexed objects}.
1265 @cindex contravariant
1268 @item Index objects are of class @code{idx} or a subclass. Every index has
1269 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1270 the index lives in) which can both be arbitrary expressions but are usually
1271 a number or a simple symbol. In addition, indices of class @code{varidx} have
1272 a @dfn{variance} (they can be co- or contravariant), and indices of class
1273 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1275 @item Indexed objects are of class @code{indexed} or a subclass. They
1276 contain a @dfn{base expression} (which is the expression being indexed), and
1277 one or more indices.
1281 @strong{Note:} when printing expressions, covariant indices and indices
1282 without variance are denoted @samp{.i} while contravariant indices are
1283 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1284 value. In the following, we are going to use that notation in the text so
1285 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1286 not visible in the output.
1288 A simple example shall illustrate the concepts:
1291 #include <ginac/ginac.h>
1292 using namespace std;
1293 using namespace GiNaC;
1297 symbol i_sym("i"), j_sym("j");
1298 idx i(i_sym, 3), j(j_sym, 3);
1301 cout << indexed(A, i, j) << endl;
1306 The @code{idx} constructor takes two arguments, the index value and the
1307 index dimension. First we define two index objects, @code{i} and @code{j},
1308 both with the numeric dimension 3. The value of the index @code{i} is the
1309 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1310 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1311 construct an expression containing one indexed object, @samp{A.i.j}. It has
1312 the symbol @code{A} as its base expression and the two indices @code{i} and
1315 Note the difference between the indices @code{i} and @code{j} which are of
1316 class @code{idx}, and the index values which are the sybols @code{i_sym}
1317 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1318 or numbers but must be index objects. For example, the following is not
1319 correct and will raise an exception:
1322 symbol i("i"), j("j");
1323 e = indexed(A, i, j); // ERROR: indices must be of type idx
1326 You can have multiple indexed objects in an expression, index values can
1327 be numeric, and index dimensions symbolic:
1331 symbol B("B"), dim("dim");
1332 cout << 4 * indexed(A, i)
1333 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1338 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1339 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1340 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1341 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1342 @code{simplify_indexed()} for that, see below).
1344 In fact, base expressions, index values and index dimensions can be
1345 arbitrary expressions:
1349 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1354 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1355 get an error message from this but you will probably not be able to do
1356 anything useful with it.
1358 @cindex @code{get_value()}
1359 @cindex @code{get_dimension()}
1363 ex idx::get_value(void);
1364 ex idx::get_dimension(void);
1367 return the value and dimension of an @code{idx} object. If you have an index
1368 in an expression, such as returned by calling @code{.op()} on an indexed
1369 object, you can get a reference to the @code{idx} object with the function
1370 @code{ex_to_idx()} on the expression.
1372 There are also the methods
1375 bool idx::is_numeric(void);
1376 bool idx::is_symbolic(void);
1377 bool idx::is_dim_numeric(void);
1378 bool idx::is_dim_symbolic(void);
1381 for checking whether the value and dimension are numeric or symbolic
1382 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1383 About Expressions}) returns information about the index value.
1385 @cindex @code{varidx} (class)
1386 If you need co- and contravariant indices, use the @code{varidx} class:
1390 symbol mu_sym("mu"), nu_sym("nu");
1391 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1392 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1394 cout << indexed(A, mu, nu) << endl;
1396 cout << indexed(A, mu_co, nu) << endl;
1398 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1403 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1404 co- or contravariant. The default is a contravariant (upper) index, but
1405 this can be overridden by supplying a third argument to the @code{varidx}
1406 constructor. The two methods
1409 bool varidx::is_covariant(void);
1410 bool varidx::is_contravariant(void);
1413 allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
1414 to get the object reference from an expression). There's also the very useful
1418 ex varidx::toggle_variance(void);
1421 which makes a new index with the same value and dimension but the opposite
1422 variance. By using it you only have to define the index once.
1424 @cindex @code{spinidx} (class)
1425 The @code{spinidx} class provides dotted and undotted variant indices, as
1426 used in the Weyl-van-der-Waerden spinor formalism:
1430 symbol K("K"), C_sym("C"), D_sym("D");
1431 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1432 // contravariant, undotted
1433 spinidx C_co(C_sym, 2, true); // covariant index
1434 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1435 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1437 cout << indexed(K, C, D) << endl;
1439 cout << indexed(K, C_co, D_dot) << endl;
1441 cout << indexed(K, D_co_dot, D) << endl;
1446 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1447 dotted or undotted. The default is undotted but this can be overridden by
1448 supplying a fourth argument to the @code{spinidx} constructor. The two
1452 bool spinidx::is_dotted(void);
1453 bool spinidx::is_undotted(void);
1456 allow you to check whether or not a @code{spinidx} object is dotted (use
1457 @code{ex_to_spinidx()} to get the object reference from an expression).
1458 Finally, the two methods
1461 ex spinidx::toggle_dot(void);
1462 ex spinidx::toggle_variance_dot(void);
1465 create a new index with the same value and dimension but opposite dottedness
1466 and the same or opposite variance.
1468 @subsection Substituting indices
1470 @cindex @code{subs()}
1471 Sometimes you will want to substitute one symbolic index with another
1472 symbolic or numeric index, for example when calculating one specific element
1473 of a tensor expression. This is done with the @code{.subs()} method, as it
1474 is done for symbols (see @ref{Substituting Expressions}).
1476 You have two possibilities here. You can either substitute the whole index
1477 by another index or expression:
1481 ex e = indexed(A, mu_co);
1482 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1483 // -> A.mu becomes A~nu
1484 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1485 // -> A.mu becomes A~0
1486 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1487 // -> A.mu becomes A.0
1491 The third example shows that trying to replace an index with something that
1492 is not an index will substitute the index value instead.
1494 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1499 ex e = indexed(A, mu_co);
1500 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1501 // -> A.mu becomes A.nu
1502 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1503 // -> A.mu becomes A.0
1507 As you see, with the second method only the value of the index will get
1508 substituted. Its other properties, including its dimension, remain unchanged.
1509 If you want to change the dimension of an index you have to substitute the
1510 whole index by another one with the new dimension.
1512 Finally, substituting the base expression of an indexed object works as
1517 ex e = indexed(A, mu_co);
1518 cout << e << " becomes " << e.subs(A == A+B) << endl;
1519 // -> A.mu becomes (B+A).mu
1523 @subsection Symmetries
1525 Indexed objects can be declared as being totally symmetric or antisymmetric
1526 with respect to their indices. In this case, GiNaC will automatically bring
1527 the indices into a canonical order which allows for some immediate
1532 cout << indexed(A, indexed::symmetric, i, j)
1533 + indexed(A, indexed::symmetric, j, i) << endl;
1535 cout << indexed(B, indexed::antisymmetric, i, j)
1536 + indexed(B, indexed::antisymmetric, j, j) << endl;
1538 cout << indexed(B, indexed::antisymmetric, i, j)
1539 + indexed(B, indexed::antisymmetric, j, i) << endl;
1544 @cindex @code{get_free_indices()}
1546 @subsection Dummy indices
1548 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1549 that a summation over the index range is implied. Symbolic indices which are
1550 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1551 dummy nor free indices.
1553 To be recognized as a dummy index pair, the two indices must be of the same
1554 class and dimension and their value must be the same single symbol (an index
1555 like @samp{2*n+1} is never a dummy index). If the indices are of class
1556 @code{varidx} they must also be of opposite variance; if they are of class
1557 @code{spinidx} they must be both dotted or both undotted.
1559 The method @code{.get_free_indices()} returns a vector containing the free
1560 indices of an expression. It also checks that the free indices of the terms
1561 of a sum are consistent:
1565 symbol A("A"), B("B"), C("C");
1567 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1568 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1570 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1571 cout << exprseq(e.get_free_indices()) << endl;
1573 // 'j' and 'l' are dummy indices
1575 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1576 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1578 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1579 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1580 cout << exprseq(e.get_free_indices()) << endl;
1582 // 'nu' is a dummy index, but 'sigma' is not
1584 e = indexed(A, mu, mu);
1585 cout << exprseq(e.get_free_indices()) << endl;
1587 // 'mu' is not a dummy index because it appears twice with the same
1590 e = indexed(A, mu, nu) + 42;
1591 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1592 // this will throw an exception:
1593 // "add::get_free_indices: inconsistent indices in sum"
1597 @cindex @code{simplify_indexed()}
1598 @subsection Simplifying indexed expressions
1600 In addition to the few automatic simplifications that GiNaC performs on
1601 indexed expressions (such as re-ordering the indices of symmetric tensors
1602 and calculating traces and convolutions of matrices and predefined tensors)
1606 ex ex::simplify_indexed(void);
1607 ex ex::simplify_indexed(const scalar_products & sp);
1610 that performs some more expensive operations:
1613 @item it checks the consistency of free indices in sums in the same way
1614 @code{get_free_indices()} does
1615 @item it tries to give dumy indices that appear in different terms of a sum
1616 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1617 @item it (symbolically) calculates all possible dummy index summations/contractions
1618 with the predefined tensors (this will be explained in more detail in the
1620 @item as a special case of dummy index summation, it can replace scalar products
1621 of two tensors with a user-defined value
1624 The last point is done with the help of the @code{scalar_products} class
1625 which is used to store scalar products with known values (this is not an
1626 arithmetic class, you just pass it to @code{simplify_indexed()}):
1630 symbol A("A"), B("B"), C("C"), i_sym("i");
1634 sp.add(A, B, 0); // A and B are orthogonal
1635 sp.add(A, C, 0); // A and C are orthogonal
1636 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1638 e = indexed(A + B, i) * indexed(A + C, i);
1640 // -> (B+A).i*(A+C).i
1642 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1648 The @code{scalar_products} object @code{sp} acts as a storage for the
1649 scalar products added to it with the @code{.add()} method. This method
1650 takes three arguments: the two expressions of which the scalar product is
1651 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1652 @code{simplify_indexed()} will replace all scalar products of indexed
1653 objects that have the symbols @code{A} and @code{B} as base expressions
1654 with the single value 0. The number, type and dimension of the indices
1655 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1657 @cindex @code{expand()}
1658 The example above also illustrates a feature of the @code{expand()} method:
1659 if passed the @code{expand_indexed} option it will distribute indices
1660 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1662 @cindex @code{tensor} (class)
1663 @subsection Predefined tensors
1665 Some frequently used special tensors such as the delta, epsilon and metric
1666 tensors are predefined in GiNaC. They have special properties when
1667 contracted with other tensor expressions and some of them have constant
1668 matrix representations (they will evaluate to a number when numeric
1669 indices are specified).
1671 @cindex @code{delta_tensor()}
1672 @subsubsection Delta tensor
1674 The delta tensor takes two indices, is symmetric and has the matrix
1675 representation @code{diag(1,1,1,...)}. It is constructed by the function
1676 @code{delta_tensor()}:
1680 symbol A("A"), B("B");
1682 idx i(symbol("i"), 3), j(symbol("j"), 3),
1683 k(symbol("k"), 3), l(symbol("l"), 3);
1685 ex e = indexed(A, i, j) * indexed(B, k, l)
1686 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1687 cout << e.simplify_indexed() << endl;
1690 cout << delta_tensor(i, i) << endl;
1695 @cindex @code{metric_tensor()}
1696 @subsubsection General metric tensor
1698 The function @code{metric_tensor()} creates a general symmetric metric
1699 tensor with two indices that can be used to raise/lower tensor indices. The
1700 metric tensor is denoted as @samp{g} in the output and if its indices are of
1701 mixed variance it is automatically replaced by a delta tensor:
1707 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1709 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1710 cout << e.simplify_indexed() << endl;
1713 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1714 cout << e.simplify_indexed() << endl;
1717 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1718 * metric_tensor(nu, rho);
1719 cout << e.simplify_indexed() << endl;
1722 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1723 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1724 + indexed(A, mu.toggle_variance(), rho));
1725 cout << e.simplify_indexed() << endl;
1730 @cindex @code{lorentz_g()}
1731 @subsubsection Minkowski metric tensor
1733 The Minkowski metric tensor is a special metric tensor with a constant
1734 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1735 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1736 It is created with the function @code{lorentz_g()} (although it is output as
1741 varidx mu(symbol("mu"), 4);
1743 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1744 * lorentz_g(mu, varidx(0, 4)); // negative signature
1745 cout << e.simplify_indexed() << endl;
1748 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1749 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1750 cout << e.simplify_indexed() << endl;
1755 @cindex @code{spinor_metric()}
1756 @subsubsection Spinor metric tensor
1758 The function @code{spinor_metric()} creates an antisymmetric tensor with
1759 two indices that is used to raise/lower indices of 2-component spinors.
1760 It is output as @samp{eps}:
1766 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
1767 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
1769 e = spinor_metric(A, B) * indexed(psi, B_co);
1770 cout << e.simplify_indexed() << endl;
1773 e = spinor_metric(A, B) * indexed(psi, A_co);
1774 cout << e.simplify_indexed() << endl;
1777 e = spinor_metric(A_co, B_co) * indexed(psi, B);
1778 cout << e.simplify_indexed() << endl;
1781 e = spinor_metric(A_co, B_co) * indexed(psi, A);
1782 cout << e.simplify_indexed() << endl;
1785 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
1786 cout << e.simplify_indexed() << endl;
1789 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
1790 cout << e.simplify_indexed() << endl;
1795 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
1797 @cindex @code{epsilon_tensor()}
1798 @cindex @code{lorentz_eps()}
1799 @subsubsection Epsilon tensor
1801 The epsilon tensor is totally antisymmetric, its number of indices is equal
1802 to the dimension of the index space (the indices must all be of the same
1803 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1804 defined to be 1. Its behaviour with indices that have a variance also
1805 depends on the signature of the metric. Epsilon tensors are output as
1808 There are three functions defined to create epsilon tensors in 2, 3 and 4
1812 ex epsilon_tensor(const ex & i1, const ex & i2);
1813 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1814 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1817 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1818 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1819 Minkowski space (the last @code{bool} argument specifies whether the metric
1820 has negative or positive signature, as in the case of the Minkowski metric
1823 @subsection Linear algebra
1825 The @code{matrix} class can be used with indices to do some simple linear
1826 algebra (linear combinations and products of vectors and matrices, traces
1827 and scalar products):
1831 idx i(symbol("i"), 2), j(symbol("j"), 2);
1832 symbol x("x"), y("y");
1834 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
1836 cout << indexed(A, i, i) << endl;
1839 ex e = indexed(A, i, j) * indexed(X, j);
1840 cout << e.simplify_indexed() << endl;
1841 // -> [[2*y+x],[4*y+3*x]].i
1843 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
1844 cout << e.simplify_indexed() << endl;
1845 // -> [[3*y+3*x,6*y+2*x]].j
1849 You can of course obtain the same results with the @code{matrix::add()},
1850 @code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
1851 don't have to worry about transposing matrices.
1853 Matrix indices always start at 0 and their dimension must match the number
1854 of rows/columns of the matrix. Matrices with one row or one column are
1855 vectors and can have one or two indices (it doesn't matter whether it's a
1856 row or a column vector). Other matrices must have two indices.
1858 You should be careful when using indices with variance on matrices. GiNaC
1859 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
1860 @samp{F.mu.nu} are different matrices. In this case you should use only
1861 one form for @samp{F} and explicitly multiply it with a matrix representation
1862 of the metric tensor.
1865 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
1866 @c node-name, next, previous, up
1867 @section Non-commutative objects
1869 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
1870 non-commutative objects are built-in which are mostly of use in high energy
1874 @item Clifford (Dirac) algebra (class @code{clifford})
1875 @item su(3) Lie algebra (class @code{color})
1876 @item Matrices (unindexed) (class @code{matrix})
1879 The @code{clifford} and @code{color} classes are subclasses of
1880 @code{indexed} because the elements of these algebras ususally carry
1883 Unlike most computer algebra systems, GiNaC does not primarily provide an
1884 operator (often denoted @samp{&*}) for representing inert products of
1885 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
1886 classes of objects involved, and non-commutative products are formed with
1887 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
1888 figuring out by itself which objects commute and will group the factors
1889 by their class. Consider this example:
1893 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
1894 idx a(symbol("a"), 8), b(symbol("b"), 8);
1895 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
1897 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
1901 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
1902 groups the non-commutative factors (the gammas and the su(3) generators)
1903 together while preserving the order of factors within each class (because
1904 Clifford objects commute with color objects). The resulting expression is a
1905 @emph{commutative} product with two factors that are themselves non-commutative
1906 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
1907 parentheses are placed around the non-commutative products in the output.
1909 @cindex @code{ncmul} (class)
1910 Non-commutative products are internally represented by objects of the class
1911 @code{ncmul}, as opposed to commutative products which are handled by the
1912 @code{mul} class. You will normally not have to worry about this distinction,
1915 The advantage of this approach is that you never have to worry about using
1916 (or forgetting to use) a special operator when constructing non-commutative
1917 expressions. Also, non-commutative products in GiNaC are more intelligent
1918 than in other computer algebra systems; they can, for example, automatically
1919 canonicalize themselves according to rules specified in the implementation
1920 of the non-commutative classes. The drawback is that to work with other than
1921 the built-in algebras you have to implement new classes yourself. Symbols
1922 always commute and it's not possible to construct non-commutative products
1923 using symbols to represent the algebra elements or generators. User-defined
1924 functions can, however, be specified as being non-commutative.
1926 @cindex @code{return_type()}
1927 @cindex @code{return_type_tinfo()}
1928 Information about the commutativity of an object or expression can be
1929 obtained with the two member functions
1932 unsigned ex::return_type(void) const;
1933 unsigned ex::return_type_tinfo(void) const;
1936 The @code{return_type()} function returns one of three values (defined in
1937 the header file @file{flags.h}), corresponding to three categories of
1938 expressions in GiNaC:
1941 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
1942 classes are of this kind.
1943 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
1944 certain class of non-commutative objects which can be determined with the
1945 @code{return_type_tinfo()} method. Expressions of this category commute
1946 with everything except @code{noncommutative} expressions of the same
1948 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
1949 of non-commutative objects of different classes. Expressions of this
1950 category don't commute with any other @code{noncommutative} or
1951 @code{noncommutative_composite} expressions.
1954 The value returned by the @code{return_type_tinfo()} method is valid only
1955 when the return type of the expression is @code{noncommutative}. It is a
1956 value that is unique to the class of the object and usually one of the
1957 constants in @file{tinfos.h}, or derived therefrom.
1959 Here are a couple of examples:
1962 @multitable @columnfractions 0.33 0.33 0.34
1963 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
1964 @item @code{42} @tab @code{commutative} @tab -
1965 @item @code{2*x-y} @tab @code{commutative} @tab -
1966 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1967 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1968 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
1969 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
1973 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
1974 @code{TINFO_clifford} for objects with a representation label of zero.
1975 Other representation labels yield a different @code{return_type_tinfo()},
1976 but it's the same for any two objects with the same label. This is also true
1979 A last note: With the exception of matrices, positive integer powers of
1980 non-commutative objects are automatically expanded in GiNaC. For example,
1981 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
1982 non-commutative expressions).
1985 @cindex @code{clifford} (class)
1986 @subsection Clifford algebra
1988 @cindex @code{dirac_gamma()}
1989 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
1990 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
1991 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
1992 is the Minkowski metric tensor. Dirac gammas are constructed by the function
1995 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
1998 which takes two arguments: the index and a @dfn{representation label} in the
1999 range 0 to 255 which is used to distinguish elements of different Clifford
2000 algebras (this is also called a @dfn{spin line index}). Gammas with different
2001 labels commute with each other. The dimension of the index can be 4 or (in
2002 the framework of dimensional regularization) any symbolic value. Spinor
2003 indices on Dirac gammas are not supported in GiNaC.
2005 @cindex @code{dirac_ONE()}
2006 The unity element of a Clifford algebra is constructed by
2009 ex dirac_ONE(unsigned char rl = 0);
2012 @cindex @code{dirac_gamma5()}
2013 and there's a special element @samp{gamma5} that commutes with all other
2014 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2018 ex dirac_gamma5(unsigned char rl = 0);
2021 @cindex @code{dirac_gamma6()}
2022 @cindex @code{dirac_gamma7()}
2023 The two additional functions
2026 ex dirac_gamma6(unsigned char rl = 0);
2027 ex dirac_gamma7(unsigned char rl = 0);
2030 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2033 @cindex @code{dirac_slash()}
2034 Finally, the function
2037 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2040 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2041 whose dimension is given by the @code{dim} argument.
2043 In products of dirac gammas, superfluous unity elements are automatically
2044 removed, squares are replaced by their values and @samp{gamma5} is
2045 anticommuted to the front. The @code{simplify_indexed()} function performs
2046 contractions in gamma strings, for example
2051 symbol a("a"), b("b"), D("D");
2052 varidx mu(symbol("mu"), D);
2053 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2054 * dirac_gamma(mu.toggle_variance());
2056 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2057 e = e.simplify_indexed();
2059 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2060 cout << e.subs(D == 4) << endl;
2061 // -> -2*gamma~symbol10*a.symbol10
2062 // [ == -2 * dirac_slash(a, D) ]
2067 @cindex @code{dirac_trace()}
2068 To calculate the trace of an expression containing strings of Dirac gammas
2069 you use the function
2072 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2075 This function takes the trace of all gammas with the specified representation
2076 label; gammas with other labels are left standing. The last argument to
2077 @code{dirac_trace()} is the value to be returned for the trace of the unity
2078 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2079 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2080 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2081 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2082 This @samp{gamma5} scheme is described in greater detail in
2083 @cite{The Role of gamma5 in Dimensional Regularization}.
2085 The value of the trace itself is also usually different in 4 and in
2086 @math{D != 4} dimensions:
2091 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2092 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2093 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2094 cout << dirac_trace(e).simplify_indexed() << endl;
2101 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2102 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2103 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2104 cout << dirac_trace(e).simplify_indexed() << endl;
2105 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2109 Here is an example for using @code{dirac_trace()} to compute a value that
2110 appears in the calculation of the one-loop vacuum polarization amplitude in
2115 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2116 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2119 sp.add(l, l, pow(l, 2));
2120 sp.add(l, q, ldotq);
2122 ex e = dirac_gamma(mu) *
2123 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2124 dirac_gamma(mu.toggle_variance()) *
2125 (dirac_slash(l, D) + m * dirac_ONE());
2126 e = dirac_trace(e).simplify_indexed(sp);
2127 e = e.collect(lst(l, ldotq, m), true);
2129 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2133 The @code{canonicalize_clifford()} function reorders all gamma products that
2134 appear in an expression to a canonical (but not necessarily simple) form.
2135 You can use this to compare two expressions or for further simplifications:
2139 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2140 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2142 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2144 e = canonicalize_clifford(e);
2151 @cindex @code{color} (class)
2152 @subsection Color algebra
2154 @cindex @code{color_T()}
2155 For computations in quantum chromodynamics, GiNaC implements the base elements
2156 and structure constants of the su(3) Lie algebra (color algebra). The base
2157 elements @math{T_a} are constructed by the function
2160 ex color_T(const ex & a, unsigned char rl = 0);
2163 which takes two arguments: the index and a @dfn{representation label} in the
2164 range 0 to 255 which is used to distinguish elements of different color
2165 algebras. Objects with different labels commute with each other. The
2166 dimension of the index must be exactly 8 and it should be of class @code{idx},
2169 @cindex @code{color_ONE()}
2170 The unity element of a color algebra is constructed by
2173 ex color_ONE(unsigned char rl = 0);
2176 @cindex @code{color_d()}
2177 @cindex @code{color_f()}
2181 ex color_d(const ex & a, const ex & b, const ex & c);
2182 ex color_f(const ex & a, const ex & b, const ex & c);
2185 create the symmetric and antisymmetric structure constants @math{d_abc} and
2186 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2187 and @math{[T_a, T_b] = i f_abc T_c}.
2189 @cindex @code{color_h()}
2190 There's an additional function
2193 ex color_h(const ex & a, const ex & b, const ex & c);
2196 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2198 The function @code{simplify_indexed()} performs some simplifications on
2199 expressions containing color objects:
2204 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2205 k(symbol("k"), 8), l(symbol("l"), 8);
2207 e = color_d(a, b, l) * color_f(a, b, k);
2208 cout << e.simplify_indexed() << endl;
2211 e = color_d(a, b, l) * color_d(a, b, k);
2212 cout << e.simplify_indexed() << endl;
2215 e = color_f(l, a, b) * color_f(a, b, k);
2216 cout << e.simplify_indexed() << endl;
2219 e = color_h(a, b, c) * color_h(a, b, c);
2220 cout << e.simplify_indexed() << endl;
2223 e = color_h(a, b, c) * color_T(b) * color_T(c);
2224 cout << e.simplify_indexed() << endl;
2227 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2228 cout << e.simplify_indexed() << endl;
2231 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2232 cout << e.simplify_indexed() << endl;
2233 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2237 @cindex @code{color_trace()}
2238 To calculate the trace of an expression containing color objects you use the
2242 ex color_trace(const ex & e, unsigned char rl = 0);
2245 This function takes the trace of all color @samp{T} objects with the
2246 specified representation label; @samp{T}s with other labels are left
2247 standing. For example:
2251 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2253 // -> -I*f.a.c.b+d.a.c.b
2258 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2259 @c node-name, next, previous, up
2260 @chapter Methods and Functions
2263 In this chapter the most important algorithms provided by GiNaC will be
2264 described. Some of them are implemented as functions on expressions,
2265 others are implemented as methods provided by expression objects. If
2266 they are methods, there exists a wrapper function around it, so you can
2267 alternatively call it in a functional way as shown in the simple
2272 cout << "As method: " << sin(1).evalf() << endl;
2273 cout << "As function: " << evalf(sin(1)) << endl;
2277 @cindex @code{subs()}
2278 The general rule is that wherever methods accept one or more parameters
2279 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2280 wrapper accepts is the same but preceded by the object to act on
2281 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2282 most natural one in an OO model but it may lead to confusion for MapleV
2283 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2284 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2285 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2286 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2287 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2288 here. Also, users of MuPAD will in most cases feel more comfortable
2289 with GiNaC's convention. All function wrappers are implemented
2290 as simple inline functions which just call the corresponding method and
2291 are only provided for users uncomfortable with OO who are dead set to
2292 avoid method invocations. Generally, nested function wrappers are much
2293 harder to read than a sequence of methods and should therefore be
2294 avoided if possible. On the other hand, not everything in GiNaC is a
2295 method on class @code{ex} and sometimes calling a function cannot be
2299 * Information About Expressions::
2300 * Substituting Expressions::
2301 * Pattern Matching and Advanced Substitutions::
2302 * Polynomial Arithmetic:: Working with polynomials.
2303 * Rational Expressions:: Working with rational functions.
2304 * Symbolic Differentiation::
2305 * Series Expansion:: Taylor and Laurent expansion.
2307 * Built-in Functions:: List of predefined mathematical functions.
2308 * Input/Output:: Input and output of expressions.
2312 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2313 @c node-name, next, previous, up
2314 @section Getting information about expressions
2316 @subsection Checking expression types
2317 @cindex @code{is_ex_of_type()}
2318 @cindex @code{ex_to_numeric()}
2319 @cindex @code{ex_to_@dots{}}
2320 @cindex @code{Converting ex to other classes}
2321 @cindex @code{info()}
2322 @cindex @code{return_type()}
2323 @cindex @code{return_type_tinfo()}
2325 Sometimes it's useful to check whether a given expression is a plain number,
2326 a sum, a polynomial with integer coefficients, or of some other specific type.
2327 GiNaC provides a couple of functions for this (the first one is actually a macro):
2330 bool is_ex_of_type(const ex & e, TYPENAME t);
2331 bool ex::info(unsigned flag);
2332 unsigned ex::return_type(void) const;
2333 unsigned ex::return_type_tinfo(void) const;
2336 When the test made by @code{is_ex_of_type()} returns true, it is safe to
2337 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
2338 one of the class names (@xref{The Class Hierarchy}, for a list of all
2339 classes). For example, assuming @code{e} is an @code{ex}:
2344 if (is_ex_of_type(e, numeric))
2345 numeric n = ex_to_numeric(e);
2350 @code{is_ex_of_type()} allows you to check whether the top-level object of
2351 an expression @samp{e} is an instance of the GiNaC class @samp{t}
2352 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2353 e.g., for checking whether an expression is a number, a sum, or a product:
2360 is_ex_of_type(e1, numeric); // true
2361 is_ex_of_type(e2, numeric); // false
2362 is_ex_of_type(e1, add); // false
2363 is_ex_of_type(e2, add); // true
2364 is_ex_of_type(e1, mul); // false
2365 is_ex_of_type(e2, mul); // false
2369 The @code{info()} method is used for checking certain attributes of
2370 expressions. The possible values for the @code{flag} argument are defined
2371 in @file{ginac/flags.h}, the most important being explained in the following
2375 @multitable @columnfractions .30 .70
2376 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2377 @item @code{numeric}
2378 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
2380 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2381 @item @code{rational}
2382 @tab @dots{}an exact rational number (integers are rational, too)
2383 @item @code{integer}
2384 @tab @dots{}a (non-complex) integer
2385 @item @code{crational}
2386 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2387 @item @code{cinteger}
2388 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2389 @item @code{positive}
2390 @tab @dots{}not complex and greater than 0
2391 @item @code{negative}
2392 @tab @dots{}not complex and less than 0
2393 @item @code{nonnegative}
2394 @tab @dots{}not complex and greater than or equal to 0
2396 @tab @dots{}an integer greater than 0
2398 @tab @dots{}an integer less than 0
2399 @item @code{nonnegint}
2400 @tab @dots{}an integer greater than or equal to 0
2402 @tab @dots{}an even integer
2404 @tab @dots{}an odd integer
2406 @tab @dots{}a prime integer (probabilistic primality test)
2407 @item @code{relation}
2408 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
2409 @item @code{relation_equal}
2410 @tab @dots{}a @code{==} relation
2411 @item @code{relation_not_equal}
2412 @tab @dots{}a @code{!=} relation
2413 @item @code{relation_less}
2414 @tab @dots{}a @code{<} relation
2415 @item @code{relation_less_or_equal}
2416 @tab @dots{}a @code{<=} relation
2417 @item @code{relation_greater}
2418 @tab @dots{}a @code{>} relation
2419 @item @code{relation_greater_or_equal}
2420 @tab @dots{}a @code{>=} relation
2422 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
2424 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
2425 @item @code{polynomial}
2426 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2427 @item @code{integer_polynomial}
2428 @tab @dots{}a polynomial with (non-complex) integer coefficients
2429 @item @code{cinteger_polynomial}
2430 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2431 @item @code{rational_polynomial}
2432 @tab @dots{}a polynomial with (non-complex) rational coefficients
2433 @item @code{crational_polynomial}
2434 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2435 @item @code{rational_function}
2436 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2437 @item @code{algebraic}
2438 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2442 To determine whether an expression is commutative or non-commutative and if
2443 so, with which other expressions it would commute, you use the methods
2444 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2445 for an explanation of these.
2448 @subsection Accessing subexpressions
2449 @cindex @code{nops()}
2452 @cindex @code{relational} (class)
2454 GiNaC provides the two methods
2457 unsigned ex::nops();
2458 ex ex::op(unsigned i);
2461 for accessing the subexpressions in the container-like GiNaC classes like
2462 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2463 determines the number of subexpressions (@samp{operands}) contained, while
2464 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2465 In the case of a @code{power} object, @code{op(0)} will return the basis
2466 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2467 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2469 The left-hand and right-hand side expressions of objects of class
2470 @code{relational} (and only of these) can also be accessed with the methods
2478 @subsection Comparing expressions
2479 @cindex @code{is_equal()}
2480 @cindex @code{is_zero()}
2482 Expressions can be compared with the usual C++ relational operators like
2483 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2484 the result is usually not determinable and the result will be @code{false},
2485 except in the case of the @code{!=} operator. You should also be aware that
2486 GiNaC will only do the most trivial test for equality (subtracting both
2487 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2490 Actually, if you construct an expression like @code{a == b}, this will be
2491 represented by an object of the @code{relational} class (@xref{Relations}.)
2492 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2494 There are also two methods
2497 bool ex::is_equal(const ex & other);
2501 for checking whether one expression is equal to another, or equal to zero,
2504 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2505 GiNaC header files. This method is however only to be used internally by
2506 GiNaC to establish a canonical sort order for terms, and using it to compare
2507 expressions will give very surprising results.
2510 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2511 @c node-name, next, previous, up
2512 @section Substituting expressions
2513 @cindex @code{subs()}
2515 Algebraic objects inside expressions can be replaced with arbitrary
2516 expressions via the @code{.subs()} method:
2519 ex ex::subs(const ex & e);
2520 ex ex::subs(const lst & syms, const lst & repls);
2523 In the first form, @code{subs()} accepts a relational of the form
2524 @samp{object == expression} or a @code{lst} of such relationals:
2528 symbol x("x"), y("y");
2530 ex e1 = 2*x^2-4*x+3;
2531 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2535 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2540 If you specify multiple substitutions, they are performed in parallel, so e.g.
2541 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2543 The second form of @code{subs()} takes two lists, one for the objects to be
2544 replaced and one for the expressions to be substituted (both lists must
2545 contain the same number of elements). Using this form, you would write
2546 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2548 @code{subs()} performs syntactic substitution of any complete algebraic
2549 object; it does not try to match sub-expressions as is demonstrated by the
2554 symbol x("x"), y("y"), z("z");
2556 ex e1 = pow(x+y, 2);
2557 cout << e1.subs(x+y == 4) << endl;
2560 ex e2 = sin(x)*sin(y)*cos(x);
2561 cout << e2.subs(sin(x) == cos(x)) << endl;
2562 // -> cos(x)^2*sin(y)
2565 cout << e3.subs(x+y == 4) << endl;
2567 // (and not 4+z as one might expect)
2571 A more powerful form of substitution using wildcards is described in the
2575 @node Pattern Matching and Advanced Substitutions, Polynomial Arithmetic, Substituting Expressions, Methods and Functions
2576 @c node-name, next, previous, up
2577 @section Pattern matching and advanced substitutions
2579 GiNaC allows the use of patterns for checking whether an expression is of a
2580 certain form or contains subexpressions of a certain form, and for
2581 substituting expressions in a more general way.
2583 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2584 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2585 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2586 an unsigned integer number to allow having multiple different wildcards in a
2587 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2588 are specified in @command{ginsh}. In C++ code, wildcard objects are created
2592 ex wild(unsigned label = 0);
2595 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2598 Some examples for patterns:
2600 @multitable @columnfractions .5 .5
2601 @item @strong{Constructed as} @tab @strong{Output as}
2602 @item @code{wild()} @tab @samp{$0}
2603 @item @code{pow(x,wild())} @tab @samp{x^$0}
2604 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2605 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2611 @item Wildcards behave like symbols and are subject to the same algebraic
2612 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2613 @item As shown in the last example, to use wildcards for indices you have to
2614 use them as the value of an @code{idx} object. This is because indices must
2615 always be of class @code{idx} (or a subclass).
2616 @item Wildcards only represent expressions or subexpressions. It is not
2617 possible to use them as placeholders for other properties like index
2618 dimension or variance, representation labels, symmetry of indexed objects
2620 @item Because wildcards are commutative, it is not possible to use wildcards
2621 as part of noncommutative products.
2622 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2623 are also valid patterns.
2626 @cindex @code{match()}
2627 The most basic application of patterns is to check whether an expression
2628 matches a given pattern. This is done by the function
2631 bool ex::match(const ex & pattern);
2632 bool ex::match(const ex & pattern, lst & repls);
2635 This function returns @code{true} when the expression matches the pattern
2636 and @code{false} if it doesn't. If used in the second form, the actual
2637 subexpressions matched by the wildcards get returned in the @code{repls}
2638 object as a list of relations of the form @samp{wildcard == expression}.
2639 If @code{match()} returns false, the state of @code{repls} is undefined.
2640 For reproducible results, the list should be empty when passed to
2641 @code{match()}, but it is also possible to find similarities in multiple
2642 expressions by passing in the result of a previous match.
2644 The matching algorithm works as follows:
2647 @item A single wildcard matches any expression. If one wildcard appears
2648 multiple times in a pattern, it must match the same expression in all
2649 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2650 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2651 @item If the expression is not of the same class as the pattern, the match
2652 fails (i.e. a sum only matches a sum, a function only matches a function,
2654 @item If the pattern is a function, it only matches the same function
2655 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2656 @item Except for sums and products, the match fails if the number of
2657 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2659 @item If there are no subexpressions, the expressions and the pattern must
2660 be equal (in the sense of @code{is_equal()}).
2661 @item Except for sums and products, each subexpression (@code{op()}) must
2662 match the corresponding subexpression of the pattern.
2665 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2666 account for their commutativity and associativity:
2669 @item If the pattern contains a term or factor that is a single wildcard,
2670 this one is used as the @dfn{global wildcard}. If there is more than one
2671 such wildcard, one of them is chosen as the global wildcard in a random
2673 @item Every term/factor of the pattern, except the global wildcard, is
2674 matched against every term of the expression in sequence. If no match is
2675 found, the whole match fails. Terms that did match are not considered in
2677 @item If there are no unmatched terms left, the match succeeds. Otherwise
2678 the match fails unless there is a global wildcard in the pattern, in
2679 which case this wildcard matches the remaining terms.
2682 In general, having more than one single wildcard as a term of a sum or a
2683 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2686 Here are some examples in @command{ginsh} to demonstrate how it works (the
2687 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2688 match fails, and the list of wildcard replacements otherwise):
2691 > match((x+y)^a,(x+y)^a);
2693 > match((x+y)^a,(x+y)^b);
2695 > match((x+y)^a,$1^$2);
2697 > match((x+y)^a,$1^$1);
2699 > match((x+y)^(x+y),$1^$1);
2701 > match((x+y)^(x+y),$1^$2);
2703 > match((a+b)*(a+c),($1+b)*($1+c));
2705 > match((a+b)*(a+c),(a+$1)*(a+$2));
2707 (Unpredictable. The result might also be [$1==c,$2==b].)
2708 > match((a+b)*(a+c),($1+$2)*($1+$3));
2709 (The result is undefined. Due to the sequential nature of the algorithm
2710 and the re-ordering of terms in GiNaC, the match for the first factor
2711 may be @{$1==a,$2==b@} in which case the match for the second factor
2712 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
2714 > match(a*(x+y)+a*z+b,a*$1+$2);
2715 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
2716 @{$1=x+y,$2=a*z+b@}.)
2717 > match(a+b+c+d+e+f,c);
2719 > match(a+b+c+d+e+f,c+$0);
2721 > match(a+b+c+d+e+f,c+e+$0);
2723 > match(a+b,a+b+$0);
2725 > match(a*b^2,a^$1*b^$2);
2727 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
2729 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
2731 > match(atan2(y,x^2),atan2(y,$0));
2735 @cindex @code{has()}
2736 A more general way to look for patterns in expressions is provided by the
2740 bool ex::has(const ex & pattern);
2743 This function checks whether a pattern is matched by an expression itself or
2744 by any of its subexpressions.
2746 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
2747 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
2750 > has(x*sin(x+y+2*a),y);
2752 > has(x*sin(x+y+2*a),x+y);
2754 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
2755 has the subexpressions "x", "y" and "2*a".)
2756 > has(x*sin(x+y+2*a),x+y+$1);
2758 (But this is possible.)
2759 > has(x*sin(2*(x+y)+2*a),x+y);
2761 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
2762 which "x+y" is not a subexpression.)
2765 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
2767 > has(4*x^2-x+3,$1*x);
2769 > has(4*x^2+x+3,$1*x);
2771 (Another possible pitfall. The first expression matches because the term
2772 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
2773 contains a linear term you should use the coeff() function instead.)
2776 @cindex @code{subs()}
2777 Probably the most useful application of patterns is to use them for
2778 substituting expressions with the @code{subs()} method. Wildcards can be
2779 used in the search patterns as well as in the replacement expressions, where
2780 they get replaced by the expressions matched by them. @code{subs()} doesn't
2781 know anything about algebra; it performs purely syntactic substitutions.
2786 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
2788 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
2790 > subs((a+b+c)^2,a+b=x);
2792 > subs((a+b+c)^2,a+b+$1==x+$1);
2794 > subs(a+2*b,a+b=x);
2796 > subs(4*x^3-2*x^2+5*x-1,x==a);
2798 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
2800 > subs(sin(1+sin(x)),sin($1)==cos($1));
2802 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
2806 The last example would be written in C++ in this way:
2810 symbol a("a"), b("b"), x("x"), y("y");
2811 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
2812 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
2813 cout << e.expand() << endl;
2819 @node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
2820 @c node-name, next, previous, up
2821 @section Polynomial arithmetic
2823 @subsection Expanding and collecting
2824 @cindex @code{expand()}
2825 @cindex @code{collect()}
2827 A polynomial in one or more variables has many equivalent
2828 representations. Some useful ones serve a specific purpose. Consider
2829 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2830 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2831 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
2832 representations are the recursive ones where one collects for exponents
2833 in one of the three variable. Since the factors are themselves
2834 polynomials in the remaining two variables the procedure can be
2835 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
2836 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
2839 To bring an expression into expanded form, its method
2845 may be called. In our example above, this corresponds to @math{4*x*y +
2846 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
2847 GiNaC is not easily guessable you should be prepared to see different
2848 orderings of terms in such sums!
2850 Another useful representation of multivariate polynomials is as a
2851 univariate polynomial in one of the variables with the coefficients
2852 being polynomials in the remaining variables. The method
2853 @code{collect()} accomplishes this task:
2856 ex ex::collect(const ex & s, bool distributed = false);
2859 The first argument to @code{collect()} can also be a list of objects in which
2860 case the result is either a recursively collected polynomial, or a polynomial
2861 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
2862 by the @code{distributed} flag.
2864 Note that the original polynomial needs to be in expanded form in order
2865 for @code{collect()} to be able to find the coefficients properly.
2867 @subsection Degree and coefficients
2868 @cindex @code{degree()}
2869 @cindex @code{ldegree()}
2870 @cindex @code{coeff()}
2872 The degree and low degree of a polynomial can be obtained using the two
2876 int ex::degree(const ex & s);
2877 int ex::ldegree(const ex & s);
2880 which also work reliably on non-expanded input polynomials (they even work
2881 on rational functions, returning the asymptotic degree). To extract
2882 a coefficient with a certain power from an expanded polynomial you use
2885 ex ex::coeff(const ex & s, int n);
2888 You can also obtain the leading and trailing coefficients with the methods
2891 ex ex::lcoeff(const ex & s);
2892 ex ex::tcoeff(const ex & s);
2895 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
2898 An application is illustrated in the next example, where a multivariate
2899 polynomial is analyzed:
2902 #include <ginac/ginac.h>
2903 using namespace std;
2904 using namespace GiNaC;
2908 symbol x("x"), y("y");
2909 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
2910 - pow(x+y,2) + 2*pow(y+2,2) - 8;
2911 ex Poly = PolyInp.expand();
2913 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
2914 cout << "The x^" << i << "-coefficient is "
2915 << Poly.coeff(x,i) << endl;
2917 cout << "As polynomial in y: "
2918 << Poly.collect(y) << endl;
2922 When run, it returns an output in the following fashion:
2925 The x^0-coefficient is y^2+11*y
2926 The x^1-coefficient is 5*y^2-2*y
2927 The x^2-coefficient is -1
2928 The x^3-coefficient is 4*y
2929 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
2932 As always, the exact output may vary between different versions of GiNaC
2933 or even from run to run since the internal canonical ordering is not
2934 within the user's sphere of influence.
2936 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
2937 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
2938 with non-polynomial expressions as they not only work with symbols but with
2939 constants, functions and indexed objects as well:
2943 symbol a("a"), b("b"), c("c");
2944 idx i(symbol("i"), 3);
2946 ex e = pow(sin(x) - cos(x), 4);
2947 cout << e.degree(cos(x)) << endl;
2949 cout << e.expand().coeff(sin(x), 3) << endl;
2952 e = indexed(a+b, i) * indexed(b+c, i);
2953 e = e.expand(expand_options::expand_indexed);
2954 cout << e.collect(indexed(b, i)) << endl;
2955 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
2960 @subsection Polynomial division
2961 @cindex polynomial division
2964 @cindex pseudo-remainder
2965 @cindex @code{quo()}
2966 @cindex @code{rem()}
2967 @cindex @code{prem()}
2968 @cindex @code{divide()}
2973 ex quo(const ex & a, const ex & b, const symbol & x);
2974 ex rem(const ex & a, const ex & b, const symbol & x);
2977 compute the quotient and remainder of univariate polynomials in the variable
2978 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
2980 The additional function
2983 ex prem(const ex & a, const ex & b, const symbol & x);
2986 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
2987 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
2989 Exact division of multivariate polynomials is performed by the function
2992 bool divide(const ex & a, const ex & b, ex & q);
2995 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
2996 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
2997 in which case the value of @code{q} is undefined.
3000 @subsection Unit, content and primitive part
3001 @cindex @code{unit()}
3002 @cindex @code{content()}
3003 @cindex @code{primpart()}
3008 ex ex::unit(const symbol & x);
3009 ex ex::content(const symbol & x);
3010 ex ex::primpart(const symbol & x);
3013 return the unit part, content part, and primitive polynomial of a multivariate
3014 polynomial with respect to the variable @samp{x} (the unit part being the sign
3015 of the leading coefficient, the content part being the GCD of the coefficients,
3016 and the primitive polynomial being the input polynomial divided by the unit and
3017 content parts). The product of unit, content, and primitive part is the
3018 original polynomial.
3021 @subsection GCD and LCM
3024 @cindex @code{gcd()}
3025 @cindex @code{lcm()}
3027 The functions for polynomial greatest common divisor and least common
3028 multiple have the synopsis
3031 ex gcd(const ex & a, const ex & b);
3032 ex lcm(const ex & a, const ex & b);
3035 The functions @code{gcd()} and @code{lcm()} accept two expressions
3036 @code{a} and @code{b} as arguments and return a new expression, their
3037 greatest common divisor or least common multiple, respectively. If the
3038 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3039 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3042 #include <ginac/ginac.h>
3043 using namespace GiNaC;
3047 symbol x("x"), y("y"), z("z");
3048 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3049 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3051 ex P_gcd = gcd(P_a, P_b);
3053 ex P_lcm = lcm(P_a, P_b);
3054 // 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
3059 @subsection Square-free decomposition
3060 @cindex square-free decomposition
3061 @cindex factorization
3062 @cindex @code{sqrfree()}
3064 GiNaC still lacks proper factorization support. Some form of
3065 factorization is, however, easily implemented by noting that factors
3066 appearing in a polynomial with power two or more also appear in the
3067 derivative and hence can easily be found by computing the GCD of the
3068 original polynomial and its derivatives. Any system has an interface
3069 for this so called square-free factorization. So we provide one, too:
3071 ex sqrfree(const ex & a, const lst & l = lst());
3073 Here is an example that by the way illustrates how the result may depend
3074 on the order of differentiation:
3077 symbol x("x"), y("y");
3078 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3080 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3081 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3083 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3084 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3086 cout << sqrfree(BiVarPol) << endl;
3087 // -> depending on luck, any of the above
3092 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3093 @c node-name, next, previous, up
3094 @section Rational expressions
3096 @subsection The @code{normal} method
3097 @cindex @code{normal()}
3098 @cindex simplification
3099 @cindex temporary replacement
3101 Some basic form of simplification of expressions is called for frequently.
3102 GiNaC provides the method @code{.normal()}, which converts a rational function
3103 into an equivalent rational function of the form @samp{numerator/denominator}
3104 where numerator and denominator are coprime. If the input expression is already
3105 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3106 otherwise it performs fraction addition and multiplication.
3108 @code{.normal()} can also be used on expressions which are not rational functions
3109 as it will replace all non-rational objects (like functions or non-integer
3110 powers) by temporary symbols to bring the expression to the domain of rational
3111 functions before performing the normalization, and re-substituting these
3112 symbols afterwards. This algorithm is also available as a separate method
3113 @code{.to_rational()}, described below.
3115 This means that both expressions @code{t1} and @code{t2} are indeed
3116 simplified in this little program:
3119 #include <ginac/ginac.h>
3120 using namespace GiNaC;
3125 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3126 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3127 std::cout << "t1 is " << t1.normal() << std::endl;
3128 std::cout << "t2 is " << t2.normal() << std::endl;
3132 Of course this works for multivariate polynomials too, so the ratio of
3133 the sample-polynomials from the section about GCD and LCM above would be
3134 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3137 @subsection Numerator and denominator
3140 @cindex @code{numer()}
3141 @cindex @code{denom()}
3142 @cindex @code{numer_denom()}
3144 The numerator and denominator of an expression can be obtained with
3149 ex ex::numer_denom();
3152 These functions will first normalize the expression as described above and
3153 then return the numerator, denominator, or both as a list, respectively.
3154 If you need both numerator and denominator, calling @code{numer_denom()} is
3155 faster than using @code{numer()} and @code{denom()} separately.
3158 @subsection Converting to a rational expression
3159 @cindex @code{to_rational()}
3161 Some of the methods described so far only work on polynomials or rational
3162 functions. GiNaC provides a way to extend the domain of these functions to
3163 general expressions by using the temporary replacement algorithm described
3164 above. You do this by calling
3167 ex ex::to_rational(lst &l);
3170 on the expression to be converted. The supplied @code{lst} will be filled
3171 with the generated temporary symbols and their replacement expressions in
3172 a format that can be used directly for the @code{subs()} method. It can also
3173 already contain a list of replacements from an earlier application of
3174 @code{.to_rational()}, so it's possible to use it on multiple expressions
3175 and get consistent results.
3182 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3183 ex b = sin(x) + cos(x);
3186 divide(a.to_rational(l), b.to_rational(l), q);
3187 cout << q.subs(l) << endl;
3191 will print @samp{sin(x)-cos(x)}.
3194 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3195 @c node-name, next, previous, up
3196 @section Symbolic differentiation
3197 @cindex differentiation
3198 @cindex @code{diff()}
3200 @cindex product rule
3202 GiNaC's objects know how to differentiate themselves. Thus, a
3203 polynomial (class @code{add}) knows that its derivative is the sum of
3204 the derivatives of all the monomials:
3207 #include <ginac/ginac.h>
3208 using namespace GiNaC;
3212 symbol x("x"), y("y"), z("z");
3213 ex P = pow(x, 5) + pow(x, 2) + y;
3215 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3216 cout << P.diff(y) << endl; // 1
3217 cout << P.diff(z) << endl; // 0
3221 If a second integer parameter @var{n} is given, the @code{diff} method
3222 returns the @var{n}th derivative.
3224 If @emph{every} object and every function is told what its derivative
3225 is, all derivatives of composed objects can be calculated using the
3226 chain rule and the product rule. Consider, for instance the expression
3227 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3228 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3229 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3230 out that the composition is the generating function for Euler Numbers,
3231 i.e. the so called @var{n}th Euler number is the coefficient of
3232 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3233 identity to code a function that generates Euler numbers in just three
3236 @cindex Euler numbers
3238 #include <ginac/ginac.h>
3239 using namespace GiNaC;
3241 ex EulerNumber(unsigned n)
3244 const ex generator = pow(cosh(x),-1);
3245 return generator.diff(x,n).subs(x==0);
3250 for (unsigned i=0; i<11; i+=2)
3251 std::cout << EulerNumber(i) << std::endl;
3256 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3257 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3258 @code{i} by two since all odd Euler numbers vanish anyways.
3261 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3262 @c node-name, next, previous, up
3263 @section Series expansion
3264 @cindex @code{series()}
3265 @cindex Taylor expansion
3266 @cindex Laurent expansion
3267 @cindex @code{pseries} (class)
3269 Expressions know how to expand themselves as a Taylor series or (more
3270 generally) a Laurent series. As in most conventional Computer Algebra
3271 Systems, no distinction is made between those two. There is a class of
3272 its own for storing such series (@code{class pseries}) and a built-in
3273 function (called @code{Order}) for storing the order term of the series.
3274 As a consequence, if you want to work with series, i.e. multiply two
3275 series, you need to call the method @code{ex::series} again to convert
3276 it to a series object with the usual structure (expansion plus order
3277 term). A sample application from special relativity could read:
3280 #include <ginac/ginac.h>
3281 using namespace std;
3282 using namespace GiNaC;
3286 symbol v("v"), c("c");
3288 ex gamma = 1/sqrt(1 - pow(v/c,2));
3289 ex mass_nonrel = gamma.series(v==0, 10);
3291 cout << "the relativistic mass increase with v is " << endl
3292 << mass_nonrel << endl;
3294 cout << "the inverse square of this series is " << endl
3295 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3299 Only calling the series method makes the last output simplify to
3300 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3301 series raised to the power @math{-2}.
3303 @cindex M@'echain's formula
3304 As another instructive application, let us calculate the numerical
3305 value of Archimedes' constant
3309 (for which there already exists the built-in constant @code{Pi})
3310 using M@'echain's amazing formula
3312 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3315 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3317 We may expand the arcus tangent around @code{0} and insert the fractions
3318 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3319 carries an order term with it and the question arises what the system is
3320 supposed to do when the fractions are plugged into that order term. The
3321 solution is to use the function @code{series_to_poly()} to simply strip
3325 #include <ginac/ginac.h>
3326 using namespace GiNaC;
3328 ex mechain_pi(int degr)
3331 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3332 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3333 -4*pi_expansion.subs(x==numeric(1,239));
3339 using std::cout; // just for fun, another way of...
3340 using std::endl; // ...dealing with this namespace std.
3342 for (int i=2; i<12; i+=2) @{
3343 pi_frac = mechain_pi(i);
3344 cout << i << ":\t" << pi_frac << endl
3345 << "\t" << pi_frac.evalf() << endl;
3351 Note how we just called @code{.series(x,degr)} instead of
3352 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3353 method @code{series()}: if the first argument is a symbol the expression
3354 is expanded in that symbol around point @code{0}. When you run this
3355 program, it will type out:
3359 3.1832635983263598326
3360 4: 5359397032/1706489875
3361 3.1405970293260603143
3362 6: 38279241713339684/12184551018734375
3363 3.141621029325034425
3364 8: 76528487109180192540976/24359780855939418203125
3365 3.141591772182177295
3366 10: 327853873402258685803048818236/104359128170408663038552734375
3367 3.1415926824043995174
3371 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3372 @c node-name, next, previous, up
3373 @section Symmetrization
3374 @cindex @code{symmetrize()}
3375 @cindex @code{antisymmetrize()}
3380 ex ex::symmetrize(const lst & l);
3381 ex ex::antisymmetrize(const lst & l);
3384 symmetrize an expression by returning the symmetric or antisymmetric sum
3385 over all permutations of the specified list of objects, weighted by the
3386 number of permutations.
3388 The two additional methods
3391 ex ex::symmetrize();
3392 ex ex::antisymmetrize();
3395 symmetrize or antisymmetrize an expression over its free indices.
3397 Symmetrization is most useful with indexed expressions but can be used with
3398 almost any kind of object (anything that is @code{subs()}able):
3402 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3403 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3405 cout << indexed(A, i, j).symmetrize() << endl;
3406 // -> 1/2*A.j.i+1/2*A.i.j
3407 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3408 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3409 cout << lst(a, b, c).symmetrize(lst(a, b, c)) << endl;
3410 // -> 1/6*@{a,b,c@}+1/6*@{c,a,b@}+1/6*@{b,a,c@}+1/6*@{c,b,a@}+1/6*@{b,c,a@}+1/6*@{a,c,b@}
3415 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3416 @c node-name, next, previous, up
3417 @section Predefined mathematical functions
3419 GiNaC contains the following predefined mathematical functions:
3422 @multitable @columnfractions .30 .70
3423 @item @strong{Name} @tab @strong{Function}
3426 @item @code{csgn(x)}
3428 @item @code{sqrt(x)}
3429 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3436 @item @code{asin(x)}
3438 @item @code{acos(x)}
3440 @item @code{atan(x)}
3441 @tab inverse tangent
3442 @item @code{atan2(y, x)}
3443 @tab inverse tangent with two arguments
3444 @item @code{sinh(x)}
3445 @tab hyperbolic sine
3446 @item @code{cosh(x)}
3447 @tab hyperbolic cosine
3448 @item @code{tanh(x)}
3449 @tab hyperbolic tangent
3450 @item @code{asinh(x)}
3451 @tab inverse hyperbolic sine
3452 @item @code{acosh(x)}
3453 @tab inverse hyperbolic cosine
3454 @item @code{atanh(x)}
3455 @tab inverse hyperbolic tangent
3457 @tab exponential function
3459 @tab natural logarithm
3462 @item @code{zeta(x)}
3463 @tab Riemann's zeta function
3464 @item @code{zeta(n, x)}
3465 @tab derivatives of Riemann's zeta function
3466 @item @code{tgamma(x)}
3468 @item @code{lgamma(x)}
3469 @tab logarithm of Gamma function
3470 @item @code{beta(x, y)}
3471 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3473 @tab psi (digamma) function
3474 @item @code{psi(n, x)}
3475 @tab derivatives of psi function (polygamma functions)
3476 @item @code{factorial(n)}
3477 @tab factorial function
3478 @item @code{binomial(n, m)}
3479 @tab binomial coefficients
3480 @item @code{Order(x)}
3481 @tab order term function in truncated power series
3482 @item @code{Derivative(x, l)}
3483 @tab inert partial differentiation operator (used internally)
3488 For functions that have a branch cut in the complex plane GiNaC follows
3489 the conventions for C++ as defined in the ANSI standard as far as
3490 possible. In particular: the natural logarithm (@code{log}) and the
3491 square root (@code{sqrt}) both have their branch cuts running along the
3492 negative real axis where the points on the axis itself belong to the
3493 upper part (i.e. continuous with quadrant II). The inverse
3494 trigonometric and hyperbolic functions are not defined for complex
3495 arguments by the C++ standard, however. In GiNaC we follow the
3496 conventions used by CLN, which in turn follow the carefully designed
3497 definitions in the Common Lisp standard. It should be noted that this
3498 convention is identical to the one used by the C99 standard and by most
3499 serious CAS. It is to be expected that future revisions of the C++
3500 standard incorporate these functions in the complex domain in a manner
3501 compatible with C99.
3504 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3505 @c node-name, next, previous, up
3506 @section Input and output of expressions
3509 @subsection Expression output
3511 @cindex output of expressions
3513 The easiest way to print an expression is to write it to a stream:
3518 ex e = 4.5+pow(x,2)*3/2;
3519 cout << e << endl; // prints '(4.5)+3/2*x^2'
3523 The output format is identical to the @command{ginsh} input syntax and
3524 to that used by most computer algebra systems, but not directly pastable
3525 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3526 is printed as @samp{x^2}).
3528 It is possible to print expressions in a number of different formats with
3532 void ex::print(const print_context & c, unsigned level = 0);
3535 @cindex @code{print_context} (class)
3536 The type of @code{print_context} object passed in determines the format
3537 of the output. The possible types are defined in @file{ginac/print.h}.
3538 All constructors of @code{print_context} and derived classes take an
3539 @code{ostream &} as their first argument.
3541 To print an expression in a way that can be directly used in a C or C++
3542 program, you pass a @code{print_csrc} object like this:
3546 cout << "float f = ";
3547 e.print(print_csrc_float(cout));
3550 cout << "double d = ";
3551 e.print(print_csrc_double(cout));
3554 cout << "cl_N n = ";
3555 e.print(print_csrc_cl_N(cout));
3560 The three possible types mostly affect the way in which floating point
3561 numbers are written.
3563 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3566 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3567 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3568 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3571 The @code{print_context} type @code{print_tree} provides a dump of the
3572 internal structure of an expression for debugging purposes:
3576 e.print(print_tree(cout));
3583 add, hash=0x0, flags=0x3, nops=2
3584 power, hash=0x9, flags=0x3, nops=2
3585 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
3586 2 (numeric), hash=0x80000042, flags=0xf
3587 3/2 (numeric), hash=0x80000061, flags=0xf
3590 4.5L0 (numeric), hash=0x8000004b, flags=0xf
3594 This kind of output is also available in @command{ginsh} as the @code{print()}
3597 Another useful output format is for LaTeX parsing in mathematical mode.
3598 It is rather similar to the default @code{print_context} but provides
3599 some braces needed by LaTeX for delimiting boxes and also converts some
3600 common objects to conventional LaTeX names. It is possible to give symbols
3601 a special name for LaTeX output by supplying it as a second argument to
3602 the @code{symbol} constructor.
3604 For example, the code snippet
3609 ex foo = lgamma(x).series(x==0,3);
3610 foo.print(print_latex(std::cout));
3616 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
3619 If you need any fancy special output format, e.g. for interfacing GiNaC
3620 with other algebra systems or for producing code for different
3621 programming languages, you can always traverse the expression tree yourself:
3624 static void my_print(const ex & e)
3626 if (is_ex_of_type(e, function))
3627 cout << ex_to_function(e).get_name();
3629 cout << e.bp->class_name();
3631 unsigned n = e.nops();
3633 for (unsigned i=0; i<n; i++) @{
3645 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
3653 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
3654 symbol(y))),numeric(-2)))
3657 If you need an output format that makes it possible to accurately
3658 reconstruct an expression by feeding the output to a suitable parser or
3659 object factory, you should consider storing the expression in an
3660 @code{archive} object and reading the object properties from there.
3661 See the section on archiving for more information.
3664 @subsection Expression input
3665 @cindex input of expressions
3667 GiNaC provides no way to directly read an expression from a stream because
3668 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
3669 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
3670 @code{y} you defined in your program and there is no way to specify the
3671 desired symbols to the @code{>>} stream input operator.
3673 Instead, GiNaC lets you construct an expression from a string, specifying the
3674 list of symbols to be used:
3678 symbol x("x"), y("y");
3679 ex e("2*x+sin(y)", lst(x, y));
3683 The input syntax is the same as that used by @command{ginsh} and the stream
3684 output operator @code{<<}. The symbols in the string are matched by name to
3685 the symbols in the list and if GiNaC encounters a symbol not specified in
3686 the list it will throw an exception.
3688 With this constructor, it's also easy to implement interactive GiNaC programs:
3693 #include <stdexcept>
3694 #include <ginac/ginac.h>
3695 using namespace std;
3696 using namespace GiNaC;
3703 cout << "Enter an expression containing 'x': ";
3708 cout << "The derivative of " << e << " with respect to x is ";
3709 cout << e.diff(x) << ".\n";
3710 @} catch (exception &p) @{
3711 cerr << p.what() << endl;
3717 @subsection Archiving
3718 @cindex @code{archive} (class)
3721 GiNaC allows creating @dfn{archives} of expressions which can be stored
3722 to or retrieved from files. To create an archive, you declare an object
3723 of class @code{archive} and archive expressions in it, giving each
3724 expression a unique name:
3728 using namespace std;
3729 #include <ginac/ginac.h>
3730 using namespace GiNaC;
3734 symbol x("x"), y("y"), z("z");
3736 ex foo = sin(x + 2*y) + 3*z + 41;
3740 a.archive_ex(foo, "foo");
3741 a.archive_ex(bar, "the second one");
3745 The archive can then be written to a file:
3749 ofstream out("foobar.gar");
3755 The file @file{foobar.gar} contains all information that is needed to
3756 reconstruct the expressions @code{foo} and @code{bar}.
3758 @cindex @command{viewgar}
3759 The tool @command{viewgar} that comes with GiNaC can be used to view
3760 the contents of GiNaC archive files:
3763 $ viewgar foobar.gar
3764 foo = 41+sin(x+2*y)+3*z
3765 the second one = 42+sin(x+2*y)+3*z
3768 The point of writing archive files is of course that they can later be
3774 ifstream in("foobar.gar");
3779 And the stored expressions can be retrieved by their name:
3785 ex ex1 = a2.unarchive_ex(syms, "foo");
3786 ex ex2 = a2.unarchive_ex(syms, "the second one");
3788 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
3789 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
3790 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
3794 Note that you have to supply a list of the symbols which are to be inserted
3795 in the expressions. Symbols in archives are stored by their name only and
3796 if you don't specify which symbols you have, unarchiving the expression will
3797 create new symbols with that name. E.g. if you hadn't included @code{x} in
3798 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
3799 have had no effect because the @code{x} in @code{ex1} would have been a
3800 different symbol than the @code{x} which was defined at the beginning of
3801 the program, altough both would appear as @samp{x} when printed.
3803 You can also use the information stored in an @code{archive} object to
3804 output expressions in a format suitable for exact reconstruction. The
3805 @code{archive} and @code{archive_node} classes have a couple of member
3806 functions that let you access the stored properties:
3809 static void my_print2(const archive_node & n)
3812 n.find_string("class", class_name);
3813 cout << class_name << "(";
3815 archive_node::propinfovector p;
3816 n.get_properties(p);
3818 unsigned num = p.size();
3819 for (unsigned i=0; i<num; i++) @{
3820 const string &name = p[i].name;
3821 if (name == "class")
3823 cout << name << "=";
3825 unsigned count = p[i].count;
3829 for (unsigned j=0; j<count; j++) @{
3830 switch (p[i].type) @{
3831 case archive_node::PTYPE_BOOL: @{
3833 n.find_bool(name, x);
3834 cout << (x ? "true" : "false");
3837 case archive_node::PTYPE_UNSIGNED: @{
3839 n.find_unsigned(name, x);
3843 case archive_node::PTYPE_STRING: @{
3845 n.find_string(name, x);
3846 cout << '\"' << x << '\"';
3849 case archive_node::PTYPE_NODE: @{
3850 const archive_node &x = n.find_ex_node(name, j);
3872 ex e = pow(2, x) - y;
3874 my_print2(ar.get_top_node(0)); cout << endl;
3882 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
3883 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
3884 overall_coeff=numeric(number="0"))
3887 Be warned, however, that the set of properties and their meaning for each
3888 class may change between GiNaC versions.
3891 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
3892 @c node-name, next, previous, up
3893 @chapter Extending GiNaC
3895 By reading so far you should have gotten a fairly good understanding of
3896 GiNaC's design-patterns. From here on you should start reading the
3897 sources. All we can do now is issue some recommendations how to tackle
3898 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
3899 develop some useful extension please don't hesitate to contact the GiNaC
3900 authors---they will happily incorporate them into future versions.
3903 * What does not belong into GiNaC:: What to avoid.
3904 * Symbolic functions:: Implementing symbolic functions.
3905 * Adding classes:: Defining new algebraic classes.
3909 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
3910 @c node-name, next, previous, up
3911 @section What doesn't belong into GiNaC
3913 @cindex @command{ginsh}
3914 First of all, GiNaC's name must be read literally. It is designed to be
3915 a library for use within C++. The tiny @command{ginsh} accompanying
3916 GiNaC makes this even more clear: it doesn't even attempt to provide a
3917 language. There are no loops or conditional expressions in
3918 @command{ginsh}, it is merely a window into the library for the
3919 programmer to test stuff (or to show off). Still, the design of a
3920 complete CAS with a language of its own, graphical capabilites and all
3921 this on top of GiNaC is possible and is without doubt a nice project for
3924 There are many built-in functions in GiNaC that do not know how to
3925 evaluate themselves numerically to a precision declared at runtime
3926 (using @code{Digits}). Some may be evaluated at certain points, but not
3927 generally. This ought to be fixed. However, doing numerical
3928 computations with GiNaC's quite abstract classes is doomed to be
3929 inefficient. For this purpose, the underlying foundation classes
3930 provided by @acronym{CLN} are much better suited.
3933 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
3934 @c node-name, next, previous, up
3935 @section Symbolic functions
3937 The easiest and most instructive way to start with is probably to
3938 implement your own function. GiNaC's functions are objects of class
3939 @code{function}. The preprocessor is then used to convert the function
3940 names to objects with a corresponding serial number that is used
3941 internally to identify them. You usually need not worry about this
3942 number. New functions may be inserted into the system via a kind of
3943 `registry'. It is your responsibility to care for some functions that
3944 are called when the user invokes certain methods. These are usual
3945 C++-functions accepting a number of @code{ex} as arguments and returning
3946 one @code{ex}. As an example, if we have a look at a simplified
3947 implementation of the cosine trigonometric function, we first need a
3948 function that is called when one wishes to @code{eval} it. It could
3949 look something like this:
3952 static ex cos_eval_method(const ex & x)
3954 // if (!x%(2*Pi)) return 1
3955 // if (!x%Pi) return -1
3956 // if (!x%Pi/2) return 0
3957 // care for other cases...
3958 return cos(x).hold();
3962 @cindex @code{hold()}
3964 The last line returns @code{cos(x)} if we don't know what else to do and
3965 stops a potential recursive evaluation by saying @code{.hold()}, which
3966 sets a flag to the expression signaling that it has been evaluated. We
3967 should also implement a method for numerical evaluation and since we are
3968 lazy we sweep the problem under the rug by calling someone else's
3969 function that does so, in this case the one in class @code{numeric}:
3972 static ex cos_evalf(const ex & x)
3974 return cos(ex_to_numeric(x));
3978 Differentiation will surely turn up and so we need to tell @code{cos}
3979 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
3980 instance are then handled automatically by @code{basic::diff} and
3984 static ex cos_deriv(const ex & x, unsigned diff_param)
3990 @cindex product rule
3991 The second parameter is obligatory but uninteresting at this point. It
3992 specifies which parameter to differentiate in a partial derivative in
3993 case the function has more than one parameter and its main application
3994 is for correct handling of the chain rule. For Taylor expansion, it is
3995 enough to know how to differentiate. But if the function you want to
3996 implement does have a pole somewhere in the complex plane, you need to
3997 write another method for Laurent expansion around that point.
3999 Now that all the ingredients for @code{cos} have been set up, we need
4000 to tell the system about it. This is done by a macro and we are not
4001 going to descibe how it expands, please consult your preprocessor if you
4005 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4006 evalf_func(cos_evalf).
4007 derivative_func(cos_deriv));
4010 The first argument is the function's name used for calling it and for
4011 output. The second binds the corresponding methods as options to this
4012 object. Options are separated by a dot and can be given in an arbitrary
4013 order. GiNaC functions understand several more options which are always
4014 specified as @code{.option(params)}, for example a method for series
4015 expansion @code{.series_func(cos_series)}. Again, if no series
4016 expansion method is given, GiNaC defaults to simple Taylor expansion,
4017 which is correct if there are no poles involved as is the case for the
4018 @code{cos} function. The way GiNaC handles poles in case there are any
4019 is best understood by studying one of the examples, like the Gamma
4020 (@code{tgamma}) function for instance. (In essence the function first
4021 checks if there is a pole at the evaluation point and falls back to
4022 Taylor expansion if there isn't. Then, the pole is regularized by some
4023 suitable transformation.) Also, the new function needs to be declared
4024 somewhere. This may also be done by a convenient preprocessor macro:
4027 DECLARE_FUNCTION_1P(cos)
4030 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4031 implementation of @code{cos} is very incomplete and lacks several safety
4032 mechanisms. Please, have a look at the real implementation in GiNaC.
4033 (By the way: in case you are worrying about all the macros above we can
4034 assure you that functions are GiNaC's most macro-intense classes. We
4035 have done our best to avoid macros where we can.)
4038 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4039 @c node-name, next, previous, up
4040 @section Adding classes
4042 If you are doing some very specialized things with GiNaC you may find that
4043 you have to implement your own algebraic classes to fit your needs. This
4044 section will explain how to do this by giving the example of a simple
4045 'string' class. After reading this section you will know how to properly
4046 declare a GiNaC class and what the minimum required member functions are
4047 that you have to implement. We only cover the implementation of a 'leaf'
4048 class here (i.e. one that doesn't contain subexpressions). Creating a
4049 container class like, for example, a class representing tensor products is
4050 more involved but this section should give you enough information so you can
4051 consult the source to GiNaC's predefined classes if you want to implement
4052 something more complicated.
4054 @subsection GiNaC's run-time type information system
4056 @cindex hierarchy of classes
4058 All algebraic classes (that is, all classes that can appear in expressions)
4059 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4060 @code{basic *} (which is essentially what an @code{ex} is) represents a
4061 generic pointer to an algebraic class. Occasionally it is necessary to find
4062 out what the class of an object pointed to by a @code{basic *} really is.
4063 Also, for the unarchiving of expressions it must be possible to find the
4064 @code{unarchive()} function of a class given the class name (as a string). A
4065 system that provides this kind of information is called a run-time type
4066 information (RTTI) system. The C++ language provides such a thing (see the
4067 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4068 implements its own, simpler RTTI.
4070 The RTTI in GiNaC is based on two mechanisms:
4075 The @code{basic} class declares a member variable @code{tinfo_key} which
4076 holds an unsigned integer that identifies the object's class. These numbers
4077 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4078 classes. They all start with @code{TINFO_}.
4081 By means of some clever tricks with static members, GiNaC maintains a list
4082 of information for all classes derived from @code{basic}. The information
4083 available includes the class names, the @code{tinfo_key}s, and pointers
4084 to the unarchiving functions. This class registry is defined in the
4085 @file{registrar.h} header file.
4089 The disadvantage of this proprietary RTTI implementation is that there's
4090 a little more to do when implementing new classes (C++'s RTTI works more
4091 or less automatic) but don't worry, most of the work is simplified by
4094 @subsection A minimalistic example
4096 Now we will start implementing a new class @code{mystring} that allows
4097 placing character strings in algebraic expressions (this is not very useful,
4098 but it's just an example). This class will be a direct subclass of
4099 @code{basic}. You can use this sample implementation as a starting point
4100 for your own classes.
4102 The code snippets given here assume that you have included some header files
4108 #include <stdexcept>
4109 using namespace std;
4111 #include <ginac/ginac.h>
4112 using namespace GiNaC;
4115 The first thing we have to do is to define a @code{tinfo_key} for our new
4116 class. This can be any arbitrary unsigned number that is not already taken
4117 by one of the existing classes but it's better to come up with something
4118 that is unlikely to clash with keys that might be added in the future. The
4119 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4120 which is not a requirement but we are going to stick with this scheme:
4123 const unsigned TINFO_mystring = 0x42420001U;
4126 Now we can write down the class declaration. The class stores a C++
4127 @code{string} and the user shall be able to construct a @code{mystring}
4128 object from a C or C++ string:
4131 class mystring : public basic
4133 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4136 mystring(const string &s);
4137 mystring(const char *s);
4143 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4146 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4147 macros are defined in @file{registrar.h}. They take the name of the class
4148 and its direct superclass as arguments and insert all required declarations
4149 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4150 the first line after the opening brace of the class definition. The
4151 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4152 source (at global scope, of course, not inside a function).
4154 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4155 declarations of the default and copy constructor, the destructor, the
4156 assignment operator and a couple of other functions that are required. It
4157 also defines a type @code{inherited} which refers to the superclass so you
4158 don't have to modify your code every time you shuffle around the class
4159 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4160 constructor, the destructor and the assignment operator.
4162 Now there are nine member functions we have to implement to get a working
4168 @code{mystring()}, the default constructor.
4171 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4172 assignment operator to free dynamically allocated members. The @code{call_parent}
4173 specifies whether the @code{destroy()} function of the superclass is to be
4177 @code{void copy(const mystring &other)}, which is used in the copy constructor
4178 and assignment operator to copy the member variables over from another
4179 object of the same class.
4182 @code{void archive(archive_node &n)}, the archiving function. This stores all
4183 information needed to reconstruct an object of this class inside an
4184 @code{archive_node}.
4187 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4188 constructor. This constructs an instance of the class from the information
4189 found in an @code{archive_node}.
4192 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4193 unarchiving function. It constructs a new instance by calling the unarchiving
4197 @code{int compare_same_type(const basic &other)}, which is used internally
4198 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4199 -1, depending on the relative order of this object and the @code{other}
4200 object. If it returns 0, the objects are considered equal.
4201 @strong{Note:} This has nothing to do with the (numeric) ordering
4202 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4203 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4204 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4205 must provide a @code{compare_same_type()} function, even those representing
4206 objects for which no reasonable algebraic ordering relationship can be
4210 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4211 which are the two constructors we declared.
4215 Let's proceed step-by-step. The default constructor looks like this:
4218 mystring::mystring() : inherited(TINFO_mystring)
4220 // dynamically allocate resources here if required
4224 The golden rule is that in all constructors you have to set the
4225 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4226 it will be set by the constructor of the superclass and all hell will break
4227 loose in the RTTI. For your convenience, the @code{basic} class provides
4228 a constructor that takes a @code{tinfo_key} value, which we are using here
4229 (remember that in our case @code{inherited = basic}). If the superclass
4230 didn't have such a constructor, we would have to set the @code{tinfo_key}
4231 to the right value manually.
4233 In the default constructor you should set all other member variables to
4234 reasonable default values (we don't need that here since our @code{str}
4235 member gets set to an empty string automatically). The constructor(s) are of
4236 course also the right place to allocate any dynamic resources you require.
4238 Next, the @code{destroy()} function:
4241 void mystring::destroy(bool call_parent)
4243 // free dynamically allocated resources here if required
4245 inherited::destroy(call_parent);
4249 This function is where we free all dynamically allocated resources. We don't
4250 have any so we're not doing anything here, but if we had, for example, used
4251 a C-style @code{char *} to store our string, this would be the place to
4252 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4253 to call the @code{destroy()} function of the superclass after we're done
4254 (to mimic C++'s automatic invocation of superclass destructors where
4255 @code{destroy()} is called from outside a destructor).
4257 The @code{copy()} function just copies over the member variables from
4261 void mystring::copy(const mystring &other)
4263 inherited::copy(other);
4268 We can simply overwrite the member variables here. There's no need to worry
4269 about dynamically allocated storage. The assignment operator (which is
4270 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4271 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4272 explicitly call the @code{copy()} function of the superclass here so
4273 all the member variables will get copied.
4275 Next are the three functions for archiving. You have to implement them even
4276 if you don't plan to use archives, but the minimum required implementation
4277 is really simple. First, the archiving function:
4280 void mystring::archive(archive_node &n) const
4282 inherited::archive(n);
4283 n.add_string("string", str);
4287 The only thing that is really required is calling the @code{archive()}
4288 function of the superclass. Optionally, you can store all information you
4289 deem necessary for representing the object into the passed
4290 @code{archive_node}. We are just storing our string here. For more
4291 information on how the archiving works, consult the @file{archive.h} header
4294 The unarchiving constructor is basically the inverse of the archiving
4298 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4300 n.find_string("string", str);
4304 If you don't need archiving, just leave this function empty (but you must
4305 invoke the unarchiving constructor of the superclass). Note that we don't
4306 have to set the @code{tinfo_key} here because it is done automatically
4307 by the unarchiving constructor of the @code{basic} class.
4309 Finally, the unarchiving function:
4312 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4314 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4318 You don't have to understand how exactly this works. Just copy these four
4319 lines into your code literally (replacing the class name, of course). It
4320 calls the unarchiving constructor of the class and unless you are doing
4321 something very special (like matching @code{archive_node}s to global
4322 objects) you don't need a different implementation. For those who are
4323 interested: setting the @code{dynallocated} flag puts the object under
4324 the control of GiNaC's garbage collection. It will get deleted automatically
4325 once it is no longer referenced.
4327 Our @code{compare_same_type()} function uses a provided function to compare
4331 int mystring::compare_same_type(const basic &other) const
4333 const mystring &o = static_cast<const mystring &>(other);
4334 int cmpval = str.compare(o.str);
4337 else if (cmpval < 0)
4344 Although this function takes a @code{basic &}, it will always be a reference
4345 to an object of exactly the same class (objects of different classes are not
4346 comparable), so the cast is safe. If this function returns 0, the two objects
4347 are considered equal (in the sense that @math{A-B=0}), so you should compare
4348 all relevant member variables.
4350 Now the only thing missing is our two new constructors:
4353 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4355 // dynamically allocate resources here if required
4358 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4360 // dynamically allocate resources here if required
4364 No surprises here. We set the @code{str} member from the argument and
4365 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4367 That's it! We now have a minimal working GiNaC class that can store
4368 strings in algebraic expressions. Let's confirm that the RTTI works:
4371 ex e = mystring("Hello, world!");
4372 cout << is_ex_of_type(e, mystring) << endl;
4375 cout << e.bp->class_name() << endl;
4379 Obviously it does. Let's see what the expression @code{e} looks like:
4383 // -> [mystring object]
4386 Hm, not exactly what we expect, but of course the @code{mystring} class
4387 doesn't yet know how to print itself. This is done in the @code{print()}
4388 member function. Let's say that we wanted to print the string surrounded
4392 class mystring : public basic
4396 void print(const print_context &c, unsigned level = 0) const;
4400 void mystring::print(const print_context &c, unsigned level) const
4402 // print_context::s is a reference to an ostream
4403 c.s << '\"' << str << '\"';
4407 The @code{level} argument is only required for container classes to
4408 correctly parenthesize the output. Let's try again to print the expression:
4412 // -> "Hello, world!"
4415 Much better. The @code{mystring} class can be used in arbitrary expressions:
4418 e += mystring("GiNaC rulez");
4420 // -> "GiNaC rulez"+"Hello, world!"
4423 (note that GiNaC's automatic term reordering is in effect here), or even
4426 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4428 // -> "One string"^(2*sin(-"Another string"+Pi))
4431 Whether this makes sense is debatable but remember that this is only an
4432 example. At least it allows you to implement your own symbolic algorithms
4435 Note that GiNaC's algebraic rules remain unchanged:
4438 e = mystring("Wow") * mystring("Wow");
4442 e = pow(mystring("First")-mystring("Second"), 2);
4443 cout << e.expand() << endl;
4444 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4447 There's no way to, for example, make GiNaC's @code{add} class perform string
4448 concatenation. You would have to implement this yourself.
4450 @subsection Automatic evaluation
4452 @cindex @code{hold()}
4454 When dealing with objects that are just a little more complicated than the
4455 simple string objects we have implemented, chances are that you will want to
4456 have some automatic simplifications or canonicalizations performed on them.
4457 This is done in the evaluation member function @code{eval()}. Let's say that
4458 we wanted all strings automatically converted to lowercase with
4459 non-alphabetic characters stripped, and empty strings removed:
4462 class mystring : public basic
4466 ex eval(int level = 0) const;
4470 ex mystring::eval(int level) const
4473 for (int i=0; i<str.length(); i++) @{
4475 if (c >= 'A' && c <= 'Z')
4476 new_str += tolower(c);
4477 else if (c >= 'a' && c <= 'z')
4481 if (new_str.length() == 0)
4484 return mystring(new_str).hold();
4488 The @code{level} argument is used to limit the recursion depth of the
4489 evaluation. We don't have any subexpressions in the @code{mystring} class
4490 so we are not concerned with this. If we had, we would call the @code{eval()}
4491 functions of the subexpressions with @code{level - 1} as the argument if
4492 @code{level != 1}. The @code{hold()} member function sets a flag in the
4493 object that prevents further evaluation. Otherwise we might end up in an
4494 endless loop. When you want to return the object unmodified, use
4495 @code{return this->hold();}.
4497 Let's confirm that it works:
4500 ex e = mystring("Hello, world!") + mystring("!?#");
4504 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4509 @subsection Other member functions
4511 We have implemented only a small set of member functions to make the class
4512 work in the GiNaC framework. For a real algebraic class, there are probably
4513 some more functions that you will want to re-implement, such as
4514 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4515 or the header file of the class you want to make a subclass of to see
4516 what's there. One member function that you will most likely want to
4517 implement for terminal classes like the described string class is
4518 @code{calcchash()} that returns an @code{unsigned} hash value for the object
4519 which will allow GiNaC to compare and canonicalize expressions much more
4522 You can, of course, also add your own new member functions. In this case you
4523 will probably want to define a little helper function like
4526 inline const mystring &ex_to_mystring(const ex &e)
4528 return static_cast<const mystring &>(*e.bp);
4532 that let's you get at the object inside an expression (after you have
4533 verified that the type is correct) so you can call member functions that are
4534 specific to the class.
4536 That's it. May the source be with you!
4539 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4540 @c node-name, next, previous, up
4541 @chapter A Comparison With Other CAS
4544 This chapter will give you some information on how GiNaC compares to
4545 other, traditional Computer Algebra Systems, like @emph{Maple},
4546 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4547 disadvantages over these systems.
4550 * Advantages:: Stengths of the GiNaC approach.
4551 * Disadvantages:: Weaknesses of the GiNaC approach.
4552 * Why C++?:: Attractiveness of C++.
4555 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4556 @c node-name, next, previous, up
4559 GiNaC has several advantages over traditional Computer
4560 Algebra Systems, like
4565 familiar language: all common CAS implement their own proprietary
4566 grammar which you have to learn first (and maybe learn again when your
4567 vendor decides to `enhance' it). With GiNaC you can write your program
4568 in common C++, which is standardized.
4572 structured data types: you can build up structured data types using
4573 @code{struct}s or @code{class}es together with STL features instead of
4574 using unnamed lists of lists of lists.
4577 strongly typed: in CAS, you usually have only one kind of variables
4578 which can hold contents of an arbitrary type. This 4GL like feature is
4579 nice for novice programmers, but dangerous.
4582 development tools: powerful development tools exist for C++, like fancy
4583 editors (e.g. with automatic indentation and syntax highlighting),
4584 debuggers, visualization tools, documentation generators...
4587 modularization: C++ programs can easily be split into modules by
4588 separating interface and implementation.
4591 price: GiNaC is distributed under the GNU Public License which means
4592 that it is free and available with source code. And there are excellent
4593 C++-compilers for free, too.
4596 extendable: you can add your own classes to GiNaC, thus extending it on
4597 a very low level. Compare this to a traditional CAS that you can
4598 usually only extend on a high level by writing in the language defined
4599 by the parser. In particular, it turns out to be almost impossible to
4600 fix bugs in a traditional system.
4603 multiple interfaces: Though real GiNaC programs have to be written in
4604 some editor, then be compiled, linked and executed, there are more ways
4605 to work with the GiNaC engine. Many people want to play with
4606 expressions interactively, as in traditional CASs. Currently, two such
4607 windows into GiNaC have been implemented and many more are possible: the
4608 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
4609 types to a command line and second, as a more consistent approach, an
4610 interactive interface to the @acronym{Cint} C++ interpreter has been put
4611 together (called @acronym{GiNaC-cint}) that allows an interactive
4612 scripting interface consistent with the C++ language.
4615 seemless integration: it is somewhere between difficult and impossible
4616 to call CAS functions from within a program written in C++ or any other
4617 programming language and vice versa. With GiNaC, your symbolic routines
4618 are part of your program. You can easily call third party libraries,
4619 e.g. for numerical evaluation or graphical interaction. All other
4620 approaches are much more cumbersome: they range from simply ignoring the
4621 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
4622 system (i.e. @emph{Yacas}).
4625 efficiency: often large parts of a program do not need symbolic
4626 calculations at all. Why use large integers for loop variables or
4627 arbitrary precision arithmetics where @code{int} and @code{double} are
4628 sufficient? For pure symbolic applications, GiNaC is comparable in
4629 speed with other CAS.
4634 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
4635 @c node-name, next, previous, up
4636 @section Disadvantages
4638 Of course it also has some disadvantages:
4643 advanced features: GiNaC cannot compete with a program like
4644 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
4645 which grows since 1981 by the work of dozens of programmers, with
4646 respect to mathematical features. Integration, factorization,
4647 non-trivial simplifications, limits etc. are missing in GiNaC (and are
4648 not planned for the near future).
4651 portability: While the GiNaC library itself is designed to avoid any
4652 platform dependent features (it should compile on any ANSI compliant C++
4653 compiler), the currently used version of the CLN library (fast large
4654 integer and arbitrary precision arithmetics) can be compiled only on
4655 systems with a recently new C++ compiler from the GNU Compiler
4656 Collection (@acronym{GCC}).@footnote{This is because CLN uses
4657 PROVIDE/REQUIRE like macros to let the compiler gather all static
4658 initializations, which works for GNU C++ only.} GiNaC uses recent
4659 language features like explicit constructors, mutable members, RTTI,
4660 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
4661 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
4662 ANSI compliant, support all needed features.
4667 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
4668 @c node-name, next, previous, up
4671 Why did we choose to implement GiNaC in C++ instead of Java or any other
4672 language? C++ is not perfect: type checking is not strict (casting is
4673 possible), separation between interface and implementation is not
4674 complete, object oriented design is not enforced. The main reason is
4675 the often scolded feature of operator overloading in C++. While it may
4676 be true that operating on classes with a @code{+} operator is rarely
4677 meaningful, it is perfectly suited for algebraic expressions. Writing
4678 @math{3x+5y} as @code{3*x+5*y} instead of
4679 @code{x.times(3).plus(y.times(5))} looks much more natural.
4680 Furthermore, the main developers are more familiar with C++ than with
4681 any other programming language.
4684 @node Internal Structures, Expressions are reference counted, Why C++? , Top
4685 @c node-name, next, previous, up
4686 @appendix Internal Structures
4689 * Expressions are reference counted::
4690 * Internal representation of products and sums::
4693 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
4694 @c node-name, next, previous, up
4695 @appendixsection Expressions are reference counted
4697 @cindex reference counting
4698 @cindex copy-on-write
4699 @cindex garbage collection
4700 An expression is extremely light-weight since internally it works like a
4701 handle to the actual representation and really holds nothing more than a
4702 pointer to some other object. What this means in practice is that
4703 whenever you create two @code{ex} and set the second equal to the first
4704 no copying process is involved. Instead, the copying takes place as soon
4705 as you try to change the second. Consider the simple sequence of code:
4708 #include <ginac/ginac.h>
4709 using namespace std;
4710 using namespace GiNaC;
4714 symbol x("x"), y("y"), z("z");
4717 e1 = sin(x + 2*y) + 3*z + 41;
4718 e2 = e1; // e2 points to same object as e1
4719 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
4720 e2 += 1; // e2 is copied into a new object
4721 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
4725 The line @code{e2 = e1;} creates a second expression pointing to the
4726 object held already by @code{e1}. The time involved for this operation
4727 is therefore constant, no matter how large @code{e1} was. Actual
4728 copying, however, must take place in the line @code{e2 += 1;} because
4729 @code{e1} and @code{e2} are not handles for the same object any more.
4730 This concept is called @dfn{copy-on-write semantics}. It increases
4731 performance considerably whenever one object occurs multiple times and
4732 represents a simple garbage collection scheme because when an @code{ex}
4733 runs out of scope its destructor checks whether other expressions handle
4734 the object it points to too and deletes the object from memory if that
4735 turns out not to be the case. A slightly less trivial example of
4736 differentiation using the chain-rule should make clear how powerful this
4740 #include <ginac/ginac.h>
4741 using namespace std;
4742 using namespace GiNaC;
4746 symbol x("x"), y("y");
4750 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
4751 cout << e1 << endl // prints x+3*y
4752 << e2 << endl // prints (x+3*y)^3
4753 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
4757 Here, @code{e1} will actually be referenced three times while @code{e2}
4758 will be referenced two times. When the power of an expression is built,
4759 that expression needs not be copied. Likewise, since the derivative of
4760 a power of an expression can be easily expressed in terms of that
4761 expression, no copying of @code{e1} is involved when @code{e3} is
4762 constructed. So, when @code{e3} is constructed it will print as
4763 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
4764 holds a reference to @code{e2} and the factor in front is just
4767 As a user of GiNaC, you cannot see this mechanism of copy-on-write
4768 semantics. When you insert an expression into a second expression, the
4769 result behaves exactly as if the contents of the first expression were
4770 inserted. But it may be useful to remember that this is not what
4771 happens. Knowing this will enable you to write much more efficient
4772 code. If you still have an uncertain feeling with copy-on-write
4773 semantics, we recommend you have a look at the
4774 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
4775 Marshall Cline. Chapter 16 covers this issue and presents an
4776 implementation which is pretty close to the one in GiNaC.
4779 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
4780 @c node-name, next, previous, up
4781 @appendixsection Internal representation of products and sums
4783 @cindex representation
4786 @cindex @code{power}
4787 Although it should be completely transparent for the user of
4788 GiNaC a short discussion of this topic helps to understand the sources
4789 and also explain performance to a large degree. Consider the
4790 unexpanded symbolic expression
4792 $2d^3 \left( 4a + 5b - 3 \right)$
4795 @math{2*d^3*(4*a+5*b-3)}
4797 which could naively be represented by a tree of linear containers for
4798 addition and multiplication, one container for exponentiation with base
4799 and exponent and some atomic leaves of symbols and numbers in this
4804 @cindex pair-wise representation
4805 However, doing so results in a rather deeply nested tree which will
4806 quickly become inefficient to manipulate. We can improve on this by
4807 representing the sum as a sequence of terms, each one being a pair of a
4808 purely numeric multiplicative coefficient and its rest. In the same
4809 spirit we can store the multiplication as a sequence of terms, each
4810 having a numeric exponent and a possibly complicated base, the tree
4811 becomes much more flat:
4815 The number @code{3} above the symbol @code{d} shows that @code{mul}
4816 objects are treated similarly where the coefficients are interpreted as
4817 @emph{exponents} now. Addition of sums of terms or multiplication of
4818 products with numerical exponents can be coded to be very efficient with
4819 such a pair-wise representation. Internally, this handling is performed
4820 by most CAS in this way. It typically speeds up manipulations by an
4821 order of magnitude. The overall multiplicative factor @code{2} and the
4822 additive term @code{-3} look somewhat out of place in this
4823 representation, however, since they are still carrying a trivial
4824 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
4825 this is avoided by adding a field that carries an overall numeric
4826 coefficient. This results in the realistic picture of internal
4829 $2d^3 \left( 4a + 5b - 3 \right)$:
4832 @math{2*d^3*(4*a+5*b-3)}:
4838 This also allows for a better handling of numeric radicals, since
4839 @code{sqrt(2)} can now be carried along calculations. Now it should be
4840 clear, why both classes @code{add} and @code{mul} are derived from the
4841 same abstract class: the data representation is the same, only the
4842 semantics differs. In the class hierarchy, methods for polynomial
4843 expansion and the like are reimplemented for @code{add} and @code{mul},
4844 but the data structure is inherited from @code{expairseq}.
4847 @node Package Tools, ginac-config, Internal representation of products and sums, Top
4848 @c node-name, next, previous, up
4849 @appendix Package Tools
4851 If you are creating a software package that uses the GiNaC library,
4852 setting the correct command line options for the compiler and linker
4853 can be difficult. GiNaC includes two tools to make this process easier.
4856 * ginac-config:: A shell script to detect compiler and linker flags.
4857 * AM_PATH_GINAC:: Macro for GNU automake.
4861 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
4862 @c node-name, next, previous, up
4863 @section @command{ginac-config}
4864 @cindex ginac-config
4866 @command{ginac-config} is a shell script that you can use to determine
4867 the compiler and linker command line options required to compile and
4868 link a program with the GiNaC library.
4870 @command{ginac-config} takes the following flags:
4874 Prints out the version of GiNaC installed.
4876 Prints '-I' flags pointing to the installed header files.
4878 Prints out the linker flags necessary to link a program against GiNaC.
4879 @item --prefix[=@var{PREFIX}]
4880 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
4881 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
4882 Otherwise, prints out the configured value of @env{$prefix}.
4883 @item --exec-prefix[=@var{PREFIX}]
4884 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
4885 Otherwise, prints out the configured value of @env{$exec_prefix}.
4888 Typically, @command{ginac-config} will be used within a configure
4889 script, as described below. It, however, can also be used directly from
4890 the command line using backquotes to compile a simple program. For
4894 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
4897 This command line might expand to (for example):
4900 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
4901 -lginac -lcln -lstdc++
4904 Not only is the form using @command{ginac-config} easier to type, it will
4905 work on any system, no matter how GiNaC was configured.
4908 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
4909 @c node-name, next, previous, up
4910 @section @samp{AM_PATH_GINAC}
4911 @cindex AM_PATH_GINAC
4913 For packages configured using GNU automake, GiNaC also provides
4914 a macro to automate the process of checking for GiNaC.
4917 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
4925 Determines the location of GiNaC using @command{ginac-config}, which is
4926 either found in the user's path, or from the environment variable
4927 @env{GINACLIB_CONFIG}.
4930 Tests the installed libraries to make sure that their version
4931 is later than @var{MINIMUM-VERSION}. (A default version will be used
4935 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
4936 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
4937 variable to the output of @command{ginac-config --libs}, and calls
4938 @samp{AC_SUBST()} for these variables so they can be used in generated
4939 makefiles, and then executes @var{ACTION-IF-FOUND}.
4942 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
4943 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
4947 This macro is in file @file{ginac.m4} which is installed in
4948 @file{$datadir/aclocal}. Note that if automake was installed with a
4949 different @samp{--prefix} than GiNaC, you will either have to manually
4950 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
4951 aclocal the @samp{-I} option when running it.
4954 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
4955 * Example package:: Example of a package using AM_PATH_GINAC.
4959 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
4960 @c node-name, next, previous, up
4961 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
4963 Simply make sure that @command{ginac-config} is in your path, and run
4964 the configure script.
4971 The directory where the GiNaC libraries are installed needs
4972 to be found by your system's dynamic linker.
4974 This is generally done by
4977 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
4983 setting the environment variable @env{LD_LIBRARY_PATH},
4986 or, as a last resort,
4989 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
4990 running configure, for instance:
4993 LDFLAGS=-R/home/cbauer/lib ./configure
4998 You can also specify a @command{ginac-config} not in your path by
4999 setting the @env{GINACLIB_CONFIG} environment variable to the
5000 name of the executable
5003 If you move the GiNaC package from its installed location,
5004 you will either need to modify @command{ginac-config} script
5005 manually to point to the new location or rebuild GiNaC.
5016 --with-ginac-prefix=@var{PREFIX}
5017 --with-ginac-exec-prefix=@var{PREFIX}
5020 are provided to override the prefix and exec-prefix that were stored
5021 in the @command{ginac-config} shell script by GiNaC's configure. You are
5022 generally better off configuring GiNaC with the right path to begin with.
5026 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5027 @c node-name, next, previous, up
5028 @subsection Example of a package using @samp{AM_PATH_GINAC}
5030 The following shows how to build a simple package using automake
5031 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5034 #include <ginac/ginac.h>
5038 GiNaC::symbol x("x");
5039 GiNaC::ex a = GiNaC::sin(x);
5040 std::cout << "Derivative of " << a
5041 << " is " << a.diff(x) << std::endl;
5046 You should first read the introductory portions of the automake
5047 Manual, if you are not already familiar with it.
5049 Two files are needed, @file{configure.in}, which is used to build the
5053 dnl Process this file with autoconf to produce a configure script.
5055 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5061 AM_PATH_GINAC(0.7.0, [
5062 LIBS="$LIBS $GINACLIB_LIBS"
5063 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5064 ], AC_MSG_ERROR([need to have GiNaC installed]))
5069 The only command in this which is not standard for automake
5070 is the @samp{AM_PATH_GINAC} macro.
5072 That command does the following: If a GiNaC version greater or equal
5073 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5074 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5075 the error message `need to have GiNaC installed'
5077 And the @file{Makefile.am}, which will be used to build the Makefile.
5080 ## Process this file with automake to produce Makefile.in
5081 bin_PROGRAMS = simple
5082 simple_SOURCES = simple.cpp
5085 This @file{Makefile.am}, says that we are building a single executable,
5086 from a single sourcefile @file{simple.cpp}. Since every program
5087 we are building uses GiNaC we simply added the GiNaC options
5088 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5089 want to specify them on a per-program basis: for instance by
5093 simple_LDADD = $(GINACLIB_LIBS)
5094 INCLUDES = $(GINACLIB_CPPFLAGS)
5097 to the @file{Makefile.am}.
5099 To try this example out, create a new directory and add the three
5102 Now execute the following commands:
5105 $ automake --add-missing
5110 You now have a package that can be built in the normal fashion
5119 @node Bibliography, Concept Index, Example package, Top
5120 @c node-name, next, previous, up
5121 @appendix Bibliography
5126 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5129 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5132 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5135 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5138 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5139 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5142 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5143 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5144 Academic Press, London
5147 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5152 @node Concept Index, , Bibliography, Top
5153 @c node-name, next, previous, up
5154 @unnumbered Concept Index