1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2001 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistical structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package Tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
216 #include <ginac/ginac.h>
218 using namespace GiNaC;
220 ex HermitePoly(const symbol & x, int n)
222 ex HKer=exp(-pow(x, 2));
223 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
224 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
231 for (int i=0; i<6; ++i)
232 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
238 When run, this will type out
244 H_3(z) == -12*z+8*z^3
245 H_4(z) == -48*z^2+16*z^4+12
246 H_5(z) == 120*z-160*z^3+32*z^5
249 This method of generating the coefficients is of course far from optimal
250 for production purposes.
252 In order to show some more examples of what GiNaC can do we will now use
253 the @command{ginsh}, a simple GiNaC interactive shell that provides a
254 convenient window into GiNaC's capabilities.
257 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
258 @c node-name, next, previous, up
259 @section What it can do for you
261 @cindex @command{ginsh}
262 After invoking @command{ginsh} one can test and experiment with GiNaC's
263 features much like in other Computer Algebra Systems except that it does
264 not provide programming constructs like loops or conditionals. For a
265 concise description of the @command{ginsh} syntax we refer to its
266 accompanied man page. Suffice to say that assignments and comparisons in
267 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
270 It can manipulate arbitrary precision integers in a very fast way.
271 Rational numbers are automatically converted to fractions of coprime
276 369988485035126972924700782451696644186473100389722973815184405301748249
278 123329495011708990974900260817232214728824366796574324605061468433916083
285 Exact numbers are always retained as exact numbers and only evaluated as
286 floating point numbers if requested. For instance, with numeric
287 radicals is dealt pretty much as with symbols. Products of sums of them
291 > expand((1+a^(1/5)-a^(2/5))^3);
292 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
293 > expand((1+3^(1/5)-3^(2/5))^3);
295 > evalf((1+3^(1/5)-3^(2/5))^3);
296 0.33408977534118624228
299 The function @code{evalf} that was used above converts any number in
300 GiNaC's expressions into floating point numbers. This can be done to
301 arbitrary predefined accuracy:
305 0.14285714285714285714
309 0.1428571428571428571428571428571428571428571428571428571428571428571428
310 5714285714285714285714285714285714285
313 Exact numbers other than rationals that can be manipulated in GiNaC
314 include predefined constants like Archimedes' @code{Pi}. They can both
315 be used in symbolic manipulations (as an exact number) as well as in
316 numeric expressions (as an inexact number):
322 9.869604401089358619+x
326 11.869604401089358619
329 Built-in functions evaluate immediately to exact numbers if
330 this is possible. Conversions that can be safely performed are done
331 immediately; conversions that are not generally valid are not done:
342 (Note that converting the last input to @code{x} would allow one to
343 conclude that @code{42*Pi} is equal to @code{0}.)
345 Linear equation systems can be solved along with basic linear
346 algebra manipulations over symbolic expressions. In C++ GiNaC offers
347 a matrix class for this purpose but we can see what it can do using
348 @command{ginsh}'s bracket notation to type them in:
351 > lsolve(a+x*y==z,x);
353 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
355 > M = [ [1, 3], [-3, 2] ];
359 > charpoly(M,lambda);
361 > A = [ [1, 1], [2, -1] ];
364 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 Multivariate polynomials and rational functions may be expanded,
370 collected and normalized (i.e. converted to a ratio of two coprime
374 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
375 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
376 > b = x^2 + 4*x*y - y^2;
379 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
381 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
383 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
388 You can differentiate functions and expand them as Taylor or Laurent
389 series in a very natural syntax (the second argument of @code{series} is
390 a relation defining the evaluation point, the third specifies the
393 @cindex Zeta function
397 > series(sin(x),x==0,4);
399 > series(1/tan(x),x==0,4);
400 x^(-1)-1/3*x+Order(x^2)
401 > series(tgamma(x),x==0,3);
402 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
403 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
405 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
406 -(0.90747907608088628905)*x^2+Order(x^3)
407 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
408 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
409 -Euler-1/12+Order((x-1/2*Pi)^3)
412 Here we have made use of the @command{ginsh}-command @code{"} to pop the
413 previously evaluated element from @command{ginsh}'s internal stack.
415 If you ever wanted to convert units in C or C++ and found this is
416 cumbersome, here is the solution. Symbolic types can always be used as
417 tags for different types of objects. Converting from wrong units to the
418 metric system is now easy:
426 140613.91592783185568*kg*m^(-2)
430 @node Installation, Prerequisites, What it can do for you, Top
431 @c node-name, next, previous, up
432 @chapter Installation
435 GiNaC's installation follows the spirit of most GNU software. It is
436 easily installed on your system by three steps: configuration, build,
440 * Prerequisites:: Packages upon which GiNaC depends.
441 * Configuration:: How to configure GiNaC.
442 * Building GiNaC:: How to compile GiNaC.
443 * Installing GiNaC:: How to install GiNaC on your system.
447 @node Prerequisites, Configuration, Installation, Installation
448 @c node-name, next, previous, up
449 @section Prerequisites
451 In order to install GiNaC on your system, some prerequisites need to be
452 met. First of all, you need to have a C++-compiler adhering to the
453 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
454 development so if you have a different compiler you are on your own.
455 For the configuration to succeed you need a Posix compliant shell
456 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
457 by the built process as well, since some of the source files are
458 automatically generated by Perl scripts. Last but not least, Bruno
459 Haible's library @acronym{CLN} is extensively used and needs to be
460 installed on your system. Please get it either from
461 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
462 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
463 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
464 site} (it is covered by GPL) and install it prior to trying to install
465 GiNaC. The configure script checks if it can find it and if it cannot
466 it will refuse to continue.
469 @node Configuration, Building GiNaC, Prerequisites, Installation
470 @c node-name, next, previous, up
471 @section Configuration
472 @cindex configuration
475 To configure GiNaC means to prepare the source distribution for
476 building. It is done via a shell script called @command{configure} that
477 is shipped with the sources and was originally generated by GNU
478 Autoconf. Since a configure script generated by GNU Autoconf never
479 prompts, all customization must be done either via command line
480 parameters or environment variables. It accepts a list of parameters,
481 the complete set of which can be listed by calling it with the
482 @option{--help} option. The most important ones will be shortly
483 described in what follows:
488 @option{--disable-shared}: When given, this option switches off the
489 build of a shared library, i.e. a @file{.so} file. This may be convenient
490 when developing because it considerably speeds up compilation.
493 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
494 and headers are installed. It defaults to @file{/usr/local} which means
495 that the library is installed in the directory @file{/usr/local/lib},
496 the header files in @file{/usr/local/include/ginac} and the documentation
497 (like this one) into @file{/usr/local/share/doc/GiNaC}.
500 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
501 the library installed in some other directory than
502 @file{@var{PREFIX}/lib/}.
505 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
506 to have the header files installed in some other directory than
507 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
508 @option{--includedir=/usr/include} you will end up with the header files
509 sitting in the directory @file{/usr/include/ginac/}. Note that the
510 subdirectory @file{ginac} is enforced by this process in order to
511 keep the header files separated from others. This avoids some
512 clashes and allows for an easier deinstallation of GiNaC. This ought
513 to be considered A Good Thing (tm).
516 @option{--datadir=@var{DATADIR}}: This option may be given in case you
517 want to have the documentation installed in some other directory than
518 @file{@var{PREFIX}/share/doc/GiNaC/}.
522 In addition, you may specify some environment variables.
523 @env{CXX} holds the path and the name of the C++ compiler
524 in case you want to override the default in your path. (The
525 @command{configure} script searches your path for @command{c++},
526 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
527 and @command{cc++} in that order.) It may be very useful to
528 define some compiler flags with the @env{CXXFLAGS} environment
529 variable, like optimization, debugging information and warning
530 levels. If omitted, it defaults to @option{-g -O2}.
532 The whole process is illustrated in the following two
533 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
534 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
537 Here is a simple configuration for a site-wide GiNaC library assuming
538 everything is in default paths:
541 $ export CXXFLAGS="-Wall -O2"
545 And here is a configuration for a private static GiNaC library with
546 several components sitting in custom places (site-wide @acronym{GCC} and
547 private @acronym{CLN}). The compiler is pursuaded to be picky and full
548 assertions and debugging information are switched on:
551 $ export CXX=/usr/local/gnu/bin/c++
552 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
553 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
554 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
555 $ ./configure --disable-shared --prefix=$(HOME)
559 @node Building GiNaC, Installing GiNaC, Configuration, Installation
560 @c node-name, next, previous, up
561 @section Building GiNaC
562 @cindex building GiNaC
564 After proper configuration you should just build the whole
569 at the command prompt and go for a cup of coffee. The exact time it
570 takes to compile GiNaC depends not only on the speed of your machines
571 but also on other parameters, for instance what value for @env{CXXFLAGS}
572 you entered. Optimization may be very time-consuming.
574 Just to make sure GiNaC works properly you may run a collection of
575 regression tests by typing
581 This will compile some sample programs, run them and check the output
582 for correctness. The regression tests fall in three categories. First,
583 the so called @emph{exams} are performed, simple tests where some
584 predefined input is evaluated (like a pupils' exam). Second, the
585 @emph{checks} test the coherence of results among each other with
586 possible random input. Third, some @emph{timings} are performed, which
587 benchmark some predefined problems with different sizes and display the
588 CPU time used in seconds. Each individual test should return a message
589 @samp{passed}. This is mostly intended to be a QA-check if something
590 was broken during development, not a sanity check of your system. Some
591 of the tests in sections @emph{checks} and @emph{timings} may require
592 insane amounts of memory and CPU time. Feel free to kill them if your
593 machine catches fire. Another quite important intent is to allow people
594 to fiddle around with optimization.
596 Generally, the top-level Makefile runs recursively to the
597 subdirectories. It is therfore safe to go into any subdirectory
598 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
599 @var{target} there in case something went wrong.
602 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
603 @c node-name, next, previous, up
604 @section Installing GiNaC
607 To install GiNaC on your system, simply type
613 As described in the section about configuration the files will be
614 installed in the following directories (the directories will be created
615 if they don't already exist):
620 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
621 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
622 So will @file{libginac.so} unless the configure script was
623 given the option @option{--disable-shared}. The proper symlinks
624 will be established as well.
627 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
628 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
631 All documentation (HTML and Postscript) will be stuffed into
632 @file{@var{PREFIX}/share/doc/GiNaC/} (or
633 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
637 For the sake of completeness we will list some other useful make
638 targets: @command{make clean} deletes all files generated by
639 @command{make}, i.e. all the object files. In addition @command{make
640 distclean} removes all files generated by the configuration and
641 @command{make maintainer-clean} goes one step further and deletes files
642 that may require special tools to rebuild (like the @command{libtool}
643 for instance). Finally @command{make uninstall} removes the installed
644 library, header files and documentation@footnote{Uninstallation does not
645 work after you have called @command{make distclean} since the
646 @file{Makefile} is itself generated by the configuration from
647 @file{Makefile.in} and hence deleted by @command{make distclean}. There
648 are two obvious ways out of this dilemma. First, you can run the
649 configuration again with the same @var{PREFIX} thus creating a
650 @file{Makefile} with a working @samp{uninstall} target. Second, you can
651 do it by hand since you now know where all the files went during
655 @node Basic Concepts, Expressions, Installing GiNaC, Top
656 @c node-name, next, previous, up
657 @chapter Basic Concepts
659 This chapter will describe the different fundamental objects that can be
660 handled by GiNaC. But before doing so, it is worthwhile introducing you
661 to the more commonly used class of expressions, representing a flexible
662 meta-class for storing all mathematical objects.
665 * Expressions:: The fundamental GiNaC class.
666 * The Class Hierarchy:: Overview of GiNaC's classes.
667 * Symbols:: Symbolic objects.
668 * Numbers:: Numerical objects.
669 * Constants:: Pre-defined constants.
670 * Fundamental containers:: The power, add and mul classes.
671 * Lists:: Lists of expressions.
672 * Mathematical functions:: Mathematical functions.
673 * Relations:: Equality, Inequality and all that.
674 * Matrices:: Matrices.
675 * Indexed objects:: Handling indexed quantities.
676 * Non-commutative objects:: Algebras with non-commutative products.
680 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
681 @c node-name, next, previous, up
683 @cindex expression (class @code{ex})
686 The most common class of objects a user deals with is the expression
687 @code{ex}, representing a mathematical object like a variable, number,
688 function, sum, product, etc@dots{} Expressions may be put together to form
689 new expressions, passed as arguments to functions, and so on. Here is a
690 little collection of valid expressions:
693 ex MyEx1 = 5; // simple number
694 ex MyEx2 = x + 2*y; // polynomial in x and y
695 ex MyEx3 = (x + 1)/(x - 1); // rational expression
696 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
697 ex MyEx5 = MyEx4 + 1; // similar to above
700 Expressions are handles to other more fundamental objects, that often
701 contain other expressions thus creating a tree of expressions
702 (@xref{Internal Structures}, for particular examples). Most methods on
703 @code{ex} therefore run top-down through such an expression tree. For
704 example, the method @code{has()} scans recursively for occurrences of
705 something inside an expression. Thus, if you have declared @code{MyEx4}
706 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
707 the argument of @code{sin} and hence return @code{true}.
709 The next sections will outline the general picture of GiNaC's class
710 hierarchy and describe the classes of objects that are handled by
714 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
715 @c node-name, next, previous, up
716 @section The Class Hierarchy
718 GiNaC's class hierarchy consists of several classes representing
719 mathematical objects, all of which (except for @code{ex} and some
720 helpers) are internally derived from one abstract base class called
721 @code{basic}. You do not have to deal with objects of class
722 @code{basic}, instead you'll be dealing with symbols, numbers,
723 containers of expressions and so on.
727 To get an idea about what kinds of symbolic composits may be built we
728 have a look at the most important classes in the class hierarchy and
729 some of the relations among the classes:
731 @image{classhierarchy}
733 The abstract classes shown here (the ones without drop-shadow) are of no
734 interest for the user. They are used internally in order to avoid code
735 duplication if two or more classes derived from them share certain
736 features. An example is @code{expairseq}, a container for a sequence of
737 pairs each consisting of one expression and a number (@code{numeric}).
738 What @emph{is} visible to the user are the derived classes @code{add}
739 and @code{mul}, representing sums and products. @xref{Internal
740 Structures}, where these two classes are described in more detail. The
741 following table shortly summarizes what kinds of mathematical objects
742 are stored in the different classes:
745 @multitable @columnfractions .22 .78
746 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
747 @item @code{constant} @tab Constants like
754 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
755 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
756 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
757 @item @code{ncmul} @tab Products of non-commutative objects
758 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
763 @code{sqrt(}@math{2}@code{)}
766 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
767 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
768 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
769 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
770 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
771 @item @code{indexed} @tab Indexed object like @math{A_ij}
772 @item @code{tensor} @tab Special tensor like the delta and metric tensors
773 @item @code{idx} @tab Index of an indexed object
774 @item @code{varidx} @tab Index with variance
775 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
776 @item @code{wildcard} @tab Wildcard for pattern matching
780 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
781 @c node-name, next, previous, up
783 @cindex @code{symbol} (class)
784 @cindex hierarchy of classes
787 Symbols are for symbolic manipulation what atoms are for chemistry. You
788 can declare objects of class @code{symbol} as any other object simply by
789 saying @code{symbol x,y;}. There is, however, a catch in here having to
790 do with the fact that C++ is a compiled language. The information about
791 the symbol's name is thrown away by the compiler but at a later stage
792 you may want to print expressions holding your symbols. In order to
793 avoid confusion GiNaC's symbols are able to know their own name. This
794 is accomplished by declaring its name for output at construction time in
795 the fashion @code{symbol x("x");}. If you declare a symbol using the
796 default constructor (i.e. without string argument) the system will deal
797 out a unique name. That name may not be suitable for printing but for
798 internal routines when no output is desired it is often enough. We'll
799 come across examples of such symbols later in this tutorial.
801 This implies that the strings passed to symbols at construction time may
802 not be used for comparing two of them. It is perfectly legitimate to
803 write @code{symbol x("x"),y("x");} but it is likely to lead into
804 trouble. Here, @code{x} and @code{y} are different symbols and
805 statements like @code{x-y} will not be simplified to zero although the
806 output @code{x-x} looks funny. Such output may also occur when there
807 are two different symbols in two scopes, for instance when you call a
808 function that declares a symbol with a name already existent in a symbol
809 in the calling function. Again, comparing them (using @code{operator==}
810 for instance) will always reveal their difference. Watch out, please.
812 @cindex @code{subs()}
813 Although symbols can be assigned expressions for internal reasons, you
814 should not do it (and we are not going to tell you how it is done). If
815 you want to replace a symbol with something else in an expression, you
816 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
819 @node Numbers, Constants, Symbols, Basic Concepts
820 @c node-name, next, previous, up
822 @cindex @code{numeric} (class)
828 For storing numerical things, GiNaC uses Bruno Haible's library
829 @acronym{CLN}. The classes therein serve as foundation classes for
830 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
831 alternatively for Common Lisp Numbers. In order to find out more about
832 @acronym{CLN}'s internals the reader is refered to the documentation of
833 that library. @inforef{Introduction, , cln}, for more
834 information. Suffice to say that it is by itself build on top of another
835 library, the GNU Multiple Precision library @acronym{GMP}, which is an
836 extremely fast library for arbitrary long integers and rationals as well
837 as arbitrary precision floating point numbers. It is very commonly used
838 by several popular cryptographic applications. @acronym{CLN} extends
839 @acronym{GMP} by several useful things: First, it introduces the complex
840 number field over either reals (i.e. floating point numbers with
841 arbitrary precision) or rationals. Second, it automatically converts
842 rationals to integers if the denominator is unity and complex numbers to
843 real numbers if the imaginary part vanishes and also correctly treats
844 algebraic functions. Third it provides good implementations of
845 state-of-the-art algorithms for all trigonometric and hyperbolic
846 functions as well as for calculation of some useful constants.
848 The user can construct an object of class @code{numeric} in several
849 ways. The following example shows the four most important constructors.
850 It uses construction from C-integer, construction of fractions from two
851 integers, construction from C-float and construction from a string:
854 #include <ginac/ginac.h>
855 using namespace GiNaC;
859 numeric two(2); // exact integer 2
860 numeric r(2,3); // exact fraction 2/3
861 numeric e(2.71828); // floating point number
862 numeric p("3.1415926535897932385"); // floating point number
863 // Trott's constant in scientific notation:
864 numeric trott("1.0841015122311136151E-2");
866 std::cout << two*p << std::endl; // floating point 6.283...
870 Note that all those constructors are @emph{explicit} which means you are
871 not allowed to write @code{numeric two=2;}. This is because the basic
872 objects to be handled by GiNaC are the expressions @code{ex} and we want
873 to keep things simple and wish objects like @code{pow(x,2)} to be
874 handled the same way as @code{pow(x,a)}, which means that we need to
875 allow a general @code{ex} as base and exponent. Therefore there is an
876 implicit constructor from C-integers directly to expressions handling
877 numerics at work in most of our examples. This design really becomes
878 convenient when one declares own functions having more than one
879 parameter but it forbids using implicit constructors because that would
880 lead to compile-time ambiguities.
882 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
883 This would, however, call C's built-in operator @code{/} for integers
884 first and result in a numeric holding a plain integer 1. @strong{Never
885 use the operator @code{/} on integers} unless you know exactly what you
886 are doing! Use the constructor from two integers instead, as shown in
887 the example above. Writing @code{numeric(1)/2} may look funny but works
890 @cindex @code{Digits}
892 We have seen now the distinction between exact numbers and floating
893 point numbers. Clearly, the user should never have to worry about
894 dynamically created exact numbers, since their `exactness' always
895 determines how they ought to be handled, i.e. how `long' they are. The
896 situation is different for floating point numbers. Their accuracy is
897 controlled by one @emph{global} variable, called @code{Digits}. (For
898 those readers who know about Maple: it behaves very much like Maple's
899 @code{Digits}). All objects of class numeric that are constructed from
900 then on will be stored with a precision matching that number of decimal
904 #include <ginac/ginac.h>
906 using namespace GiNaC;
910 numeric three(3.0), one(1.0);
911 numeric x = one/three;
913 cout << "in " << Digits << " digits:" << endl;
915 cout << Pi.evalf() << endl;
927 The above example prints the following output to screen:
934 0.333333333333333333333333333333333333333333333333333333333333333333
935 3.14159265358979323846264338327950288419716939937510582097494459231
938 It should be clear that objects of class @code{numeric} should be used
939 for constructing numbers or for doing arithmetic with them. The objects
940 one deals with most of the time are the polymorphic expressions @code{ex}.
942 @subsection Tests on numbers
944 Once you have declared some numbers, assigned them to expressions and
945 done some arithmetic with them it is frequently desired to retrieve some
946 kind of information from them like asking whether that number is
947 integer, rational, real or complex. For those cases GiNaC provides
948 several useful methods. (Internally, they fall back to invocations of
949 certain CLN functions.)
951 As an example, let's construct some rational number, multiply it with
952 some multiple of its denominator and test what comes out:
955 #include <ginac/ginac.h>
957 using namespace GiNaC;
959 // some very important constants:
960 const numeric twentyone(21);
961 const numeric ten(10);
962 const numeric five(5);
966 numeric answer = twentyone;
969 cout << answer.is_integer() << endl; // false, it's 21/5
971 cout << answer.is_integer() << endl; // true, it's 42 now!
975 Note that the variable @code{answer} is constructed here as an integer
976 by @code{numeric}'s copy constructor but in an intermediate step it
977 holds a rational number represented as integer numerator and integer
978 denominator. When multiplied by 10, the denominator becomes unity and
979 the result is automatically converted to a pure integer again.
980 Internally, the underlying @acronym{CLN} is responsible for this
981 behaviour and we refer the reader to @acronym{CLN}'s documentation.
982 Suffice to say that the same behaviour applies to complex numbers as
983 well as return values of certain functions. Complex numbers are
984 automatically converted to real numbers if the imaginary part becomes
985 zero. The full set of tests that can be applied is listed in the
989 @multitable @columnfractions .30 .70
990 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
991 @item @code{.is_zero()}
992 @tab @dots{}equal to zero
993 @item @code{.is_positive()}
994 @tab @dots{}not complex and greater than 0
995 @item @code{.is_integer()}
996 @tab @dots{}a (non-complex) integer
997 @item @code{.is_pos_integer()}
998 @tab @dots{}an integer and greater than 0
999 @item @code{.is_nonneg_integer()}
1000 @tab @dots{}an integer and greater equal 0
1001 @item @code{.is_even()}
1002 @tab @dots{}an even integer
1003 @item @code{.is_odd()}
1004 @tab @dots{}an odd integer
1005 @item @code{.is_prime()}
1006 @tab @dots{}a prime integer (probabilistic primality test)
1007 @item @code{.is_rational()}
1008 @tab @dots{}an exact rational number (integers are rational, too)
1009 @item @code{.is_real()}
1010 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1011 @item @code{.is_cinteger()}
1012 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1013 @item @code{.is_crational()}
1014 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1019 @node Constants, Fundamental containers, Numbers, Basic Concepts
1020 @c node-name, next, previous, up
1022 @cindex @code{constant} (class)
1025 @cindex @code{Catalan}
1026 @cindex @code{Euler}
1027 @cindex @code{evalf()}
1028 Constants behave pretty much like symbols except that they return some
1029 specific number when the method @code{.evalf()} is called.
1031 The predefined known constants are:
1034 @multitable @columnfractions .14 .30 .56
1035 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1037 @tab Archimedes' constant
1038 @tab 3.14159265358979323846264338327950288
1039 @item @code{Catalan}
1040 @tab Catalan's constant
1041 @tab 0.91596559417721901505460351493238411
1043 @tab Euler's (or Euler-Mascheroni) constant
1044 @tab 0.57721566490153286060651209008240243
1049 @node Fundamental containers, Lists, Constants, Basic Concepts
1050 @c node-name, next, previous, up
1051 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1055 @cindex @code{power}
1057 Simple polynomial expressions are written down in GiNaC pretty much like
1058 in other CAS or like expressions involving numerical variables in C.
1059 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1060 been overloaded to achieve this goal. When you run the following
1061 code snippet, the constructor for an object of type @code{mul} is
1062 automatically called to hold the product of @code{a} and @code{b} and
1063 then the constructor for an object of type @code{add} is called to hold
1064 the sum of that @code{mul} object and the number one:
1068 symbol a("a"), b("b");
1073 @cindex @code{pow()}
1074 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1075 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1076 construction is necessary since we cannot safely overload the constructor
1077 @code{^} in C++ to construct a @code{power} object. If we did, it would
1078 have several counterintuitive and undesired effects:
1082 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1084 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1085 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1086 interpret this as @code{x^(a^b)}.
1088 Also, expressions involving integer exponents are very frequently used,
1089 which makes it even more dangerous to overload @code{^} since it is then
1090 hard to distinguish between the semantics as exponentiation and the one
1091 for exclusive or. (It would be embarassing to return @code{1} where one
1092 has requested @code{2^3}.)
1095 @cindex @command{ginsh}
1096 All effects are contrary to mathematical notation and differ from the
1097 way most other CAS handle exponentiation, therefore overloading @code{^}
1098 is ruled out for GiNaC's C++ part. The situation is different in
1099 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1100 that the other frequently used exponentiation operator @code{**} does
1101 not exist at all in C++).
1103 To be somewhat more precise, objects of the three classes described
1104 here, are all containers for other expressions. An object of class
1105 @code{power} is best viewed as a container with two slots, one for the
1106 basis, one for the exponent. All valid GiNaC expressions can be
1107 inserted. However, basic transformations like simplifying
1108 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1109 when this is mathematically possible. If we replace the outer exponent
1110 three in the example by some symbols @code{a}, the simplification is not
1111 safe and will not be performed, since @code{a} might be @code{1/2} and
1114 Objects of type @code{add} and @code{mul} are containers with an
1115 arbitrary number of slots for expressions to be inserted. Again, simple
1116 and safe simplifications are carried out like transforming
1117 @code{3*x+4-x} to @code{2*x+4}.
1119 The general rule is that when you construct such objects, GiNaC
1120 automatically creates them in canonical form, which might differ from
1121 the form you typed in your program. This allows for rapid comparison of
1122 expressions, since after all @code{a-a} is simply zero. Note, that the
1123 canonical form is not necessarily lexicographical ordering or in any way
1124 easily guessable. It is only guaranteed that constructing the same
1125 expression twice, either implicitly or explicitly, results in the same
1129 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1130 @c node-name, next, previous, up
1131 @section Lists of expressions
1132 @cindex @code{lst} (class)
1134 @cindex @code{nops()}
1136 @cindex @code{append()}
1137 @cindex @code{prepend()}
1139 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1140 expressions. These are sometimes used to supply a variable number of
1141 arguments of the same type to GiNaC methods such as @code{subs()} and
1142 @code{to_rational()}, so you should have a basic understanding about them.
1144 Lists of up to 16 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, Matrices, 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 Matrices, Indexed objects, Relations, Basic Concepts
1243 @c node-name, next, previous, up
1245 @cindex @code{matrix} (class)
1247 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1248 matrix with @math{m} rows and @math{n} columns are accessed with two
1249 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1250 second one in the range 0@dots{}@math{n-1}.
1252 There are a couple of ways to construct matrices, with or without preset
1256 matrix::matrix(unsigned r, unsigned c);
1257 matrix::matrix(unsigned r, unsigned c, const lst & l);
1258 ex lst_to_matrix(const lst & l);
1259 ex diag_matrix(const lst & l);
1262 The first two functions are @code{matrix} constructors which create a matrix
1263 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1264 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1265 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1266 from a list of lists, each list representing a matrix row. Finally,
1267 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1268 elements. Note that the last two functions return expressions, not matrix
1271 Matrix elements can be accessed and set using the parenthesis (function call)
1275 const ex & matrix::operator()(unsigned r, unsigned c) const;
1276 ex & matrix::operator()(unsigned r, unsigned c);
1279 It is also possible to access the matrix elements in a linear fashion with
1280 the @code{op()} method. But C++-style subscripting with square brackets
1281 @samp{[]} is not available.
1283 Here are a couple of examples that all construct the same 2x2 diagonal
1288 symbol a("a"), b("b");
1296 e = matrix(2, 2, lst(a, 0, 0, b));
1298 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1300 e = diag_matrix(lst(a, b));
1307 @cindex @code{transpose()}
1308 @cindex @code{inverse()}
1309 There are three ways to do arithmetic with matrices. The first (and most
1310 efficient one) is to use the methods provided by the @code{matrix} class:
1313 matrix matrix::add(const matrix & other) const;
1314 matrix matrix::sub(const matrix & other) const;
1315 matrix matrix::mul(const matrix & other) const;
1316 matrix matrix::mul_scalar(const ex & other) const;
1317 matrix matrix::pow(const ex & expn) const;
1318 matrix matrix::transpose(void) const;
1319 matrix matrix::inverse(void) const;
1322 All of these methods return the result as a new matrix object. Here is an
1323 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1328 matrix A(2, 2, lst(1, 2, 3, 4));
1329 matrix B(2, 2, lst(-1, 0, 2, 1));
1330 matrix C(2, 2, lst(8, 4, 2, 1));
1332 matrix result = A.mul(B).sub(C.mul_scalar(2));
1333 cout << result << endl;
1334 // -> [[-13,-6],[1,2]]
1339 @cindex @code{evalm()}
1340 The second (and probably the most natural) way is to construct an expression
1341 containing matrices with the usual arithmetic operators and @code{pow()}.
1342 For efficiency reasons, expressions with sums, products and powers of
1343 matrices are not automatically evaluated in GiNaC. You have to call the
1347 ex ex::evalm() const;
1350 to obtain the result:
1357 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1358 cout << e.evalm() << endl;
1359 // -> [[-13,-6],[1,2]]
1364 The non-commutativity of the product @code{A*B} in this example is
1365 automatically recognized by GiNaC. There is no need to use a special
1366 operator here. @xref{Non-commutative objects}, for more information about
1367 dealing with non-commutative expressions.
1369 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1370 to perform the arithmetic:
1375 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1376 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1378 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1379 cout << e.simplify_indexed() << endl;
1380 // -> [[-13,-6],[1,2]].i.j
1384 Using indices is most useful when working with rectangular matrices and
1385 one-dimensional vectors because you don't have to worry about having to
1386 transpose matrices before multiplying them. @xref{Indexed objects}, for
1387 more information about using matrices with indices, and about indices in
1390 The @code{matrix} class provides a couple of additional methods for
1391 computing determinants, traces, and characteristic polynomials:
1394 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1395 ex matrix::trace(void) const;
1396 ex matrix::charpoly(const symbol & lambda) const;
1399 The @samp{algo} argument of @code{determinant()} allows to select between
1400 different algorithms for calculating the determinant. The possible values
1401 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1402 heuristic to automatically select an algorithm that is likely to give the
1403 result most quickly.
1406 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1407 @c node-name, next, previous, up
1408 @section Indexed objects
1410 GiNaC allows you to handle expressions containing general indexed objects in
1411 arbitrary spaces. It is also able to canonicalize and simplify such
1412 expressions and perform symbolic dummy index summations. There are a number
1413 of predefined indexed objects provided, like delta and metric tensors.
1415 There are few restrictions placed on indexed objects and their indices and
1416 it is easy to construct nonsense expressions, but our intention is to
1417 provide a general framework that allows you to implement algorithms with
1418 indexed quantities, getting in the way as little as possible.
1420 @cindex @code{idx} (class)
1421 @cindex @code{indexed} (class)
1422 @subsection Indexed quantities and their indices
1424 Indexed expressions in GiNaC are constructed of two special types of objects,
1425 @dfn{index objects} and @dfn{indexed objects}.
1429 @cindex contravariant
1432 @item Index objects are of class @code{idx} or a subclass. Every index has
1433 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1434 the index lives in) which can both be arbitrary expressions but are usually
1435 a number or a simple symbol. In addition, indices of class @code{varidx} have
1436 a @dfn{variance} (they can be co- or contravariant), and indices of class
1437 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1439 @item Indexed objects are of class @code{indexed} or a subclass. They
1440 contain a @dfn{base expression} (which is the expression being indexed), and
1441 one or more indices.
1445 @strong{Note:} when printing expressions, covariant indices and indices
1446 without variance are denoted @samp{.i} while contravariant indices are
1447 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1448 value. In the following, we are going to use that notation in the text so
1449 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1450 not visible in the output.
1452 A simple example shall illustrate the concepts:
1455 #include <ginac/ginac.h>
1456 using namespace std;
1457 using namespace GiNaC;
1461 symbol i_sym("i"), j_sym("j");
1462 idx i(i_sym, 3), j(j_sym, 3);
1465 cout << indexed(A, i, j) << endl;
1470 The @code{idx} constructor takes two arguments, the index value and the
1471 index dimension. First we define two index objects, @code{i} and @code{j},
1472 both with the numeric dimension 3. The value of the index @code{i} is the
1473 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1474 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1475 construct an expression containing one indexed object, @samp{A.i.j}. It has
1476 the symbol @code{A} as its base expression and the two indices @code{i} and
1479 Note the difference between the indices @code{i} and @code{j} which are of
1480 class @code{idx}, and the index values which are the sybols @code{i_sym}
1481 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1482 or numbers but must be index objects. For example, the following is not
1483 correct and will raise an exception:
1486 symbol i("i"), j("j");
1487 e = indexed(A, i, j); // ERROR: indices must be of type idx
1490 You can have multiple indexed objects in an expression, index values can
1491 be numeric, and index dimensions symbolic:
1495 symbol B("B"), dim("dim");
1496 cout << 4 * indexed(A, i)
1497 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1502 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1503 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1504 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1505 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1506 @code{simplify_indexed()} for that, see below).
1508 In fact, base expressions, index values and index dimensions can be
1509 arbitrary expressions:
1513 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1518 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1519 get an error message from this but you will probably not be able to do
1520 anything useful with it.
1522 @cindex @code{get_value()}
1523 @cindex @code{get_dimension()}
1527 ex idx::get_value(void);
1528 ex idx::get_dimension(void);
1531 return the value and dimension of an @code{idx} object. If you have an index
1532 in an expression, such as returned by calling @code{.op()} on an indexed
1533 object, you can get a reference to the @code{idx} object with the function
1534 @code{ex_to_idx()} on the expression.
1536 There are also the methods
1539 bool idx::is_numeric(void);
1540 bool idx::is_symbolic(void);
1541 bool idx::is_dim_numeric(void);
1542 bool idx::is_dim_symbolic(void);
1545 for checking whether the value and dimension are numeric or symbolic
1546 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1547 About Expressions}) returns information about the index value.
1549 @cindex @code{varidx} (class)
1550 If you need co- and contravariant indices, use the @code{varidx} class:
1554 symbol mu_sym("mu"), nu_sym("nu");
1555 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1556 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1558 cout << indexed(A, mu, nu) << endl;
1560 cout << indexed(A, mu_co, nu) << endl;
1562 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1567 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1568 co- or contravariant. The default is a contravariant (upper) index, but
1569 this can be overridden by supplying a third argument to the @code{varidx}
1570 constructor. The two methods
1573 bool varidx::is_covariant(void);
1574 bool varidx::is_contravariant(void);
1577 allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
1578 to get the object reference from an expression). There's also the very useful
1582 ex varidx::toggle_variance(void);
1585 which makes a new index with the same value and dimension but the opposite
1586 variance. By using it you only have to define the index once.
1588 @cindex @code{spinidx} (class)
1589 The @code{spinidx} class provides dotted and undotted variant indices, as
1590 used in the Weyl-van-der-Waerden spinor formalism:
1594 symbol K("K"), C_sym("C"), D_sym("D");
1595 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1596 // contravariant, undotted
1597 spinidx C_co(C_sym, 2, true); // covariant index
1598 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1599 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1601 cout << indexed(K, C, D) << endl;
1603 cout << indexed(K, C_co, D_dot) << endl;
1605 cout << indexed(K, D_co_dot, D) << endl;
1610 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1611 dotted or undotted. The default is undotted but this can be overridden by
1612 supplying a fourth argument to the @code{spinidx} constructor. The two
1616 bool spinidx::is_dotted(void);
1617 bool spinidx::is_undotted(void);
1620 allow you to check whether or not a @code{spinidx} object is dotted (use
1621 @code{ex_to_spinidx()} to get the object reference from an expression).
1622 Finally, the two methods
1625 ex spinidx::toggle_dot(void);
1626 ex spinidx::toggle_variance_dot(void);
1629 create a new index with the same value and dimension but opposite dottedness
1630 and the same or opposite variance.
1632 @subsection Substituting indices
1634 @cindex @code{subs()}
1635 Sometimes you will want to substitute one symbolic index with another
1636 symbolic or numeric index, for example when calculating one specific element
1637 of a tensor expression. This is done with the @code{.subs()} method, as it
1638 is done for symbols (see @ref{Substituting Expressions}).
1640 You have two possibilities here. You can either substitute the whole index
1641 by another index or expression:
1645 ex e = indexed(A, mu_co);
1646 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1647 // -> A.mu becomes A~nu
1648 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1649 // -> A.mu becomes A~0
1650 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1651 // -> A.mu becomes A.0
1655 The third example shows that trying to replace an index with something that
1656 is not an index will substitute the index value instead.
1658 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1663 ex e = indexed(A, mu_co);
1664 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1665 // -> A.mu becomes A.nu
1666 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1667 // -> A.mu becomes A.0
1671 As you see, with the second method only the value of the index will get
1672 substituted. Its other properties, including its dimension, remain unchanged.
1673 If you want to change the dimension of an index you have to substitute the
1674 whole index by another one with the new dimension.
1676 Finally, substituting the base expression of an indexed object works as
1681 ex e = indexed(A, mu_co);
1682 cout << e << " becomes " << e.subs(A == A+B) << endl;
1683 // -> A.mu becomes (B+A).mu
1687 @subsection Symmetries
1689 Indexed objects can be declared as being totally symmetric or antisymmetric
1690 with respect to their indices. In this case, GiNaC will automatically bring
1691 the indices into a canonical order which allows for some immediate
1696 cout << indexed(A, indexed::symmetric, i, j)
1697 + indexed(A, indexed::symmetric, j, i) << endl;
1699 cout << indexed(B, indexed::antisymmetric, i, j)
1700 + indexed(B, indexed::antisymmetric, j, j) << endl;
1702 cout << indexed(B, indexed::antisymmetric, i, j)
1703 + indexed(B, indexed::antisymmetric, j, i) << endl;
1708 @cindex @code{get_free_indices()}
1710 @subsection Dummy indices
1712 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1713 that a summation over the index range is implied. Symbolic indices which are
1714 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1715 dummy nor free indices.
1717 To be recognized as a dummy index pair, the two indices must be of the same
1718 class and dimension and their value must be the same single symbol (an index
1719 like @samp{2*n+1} is never a dummy index). If the indices are of class
1720 @code{varidx} they must also be of opposite variance; if they are of class
1721 @code{spinidx} they must be both dotted or both undotted.
1723 The method @code{.get_free_indices()} returns a vector containing the free
1724 indices of an expression. It also checks that the free indices of the terms
1725 of a sum are consistent:
1729 symbol A("A"), B("B"), C("C");
1731 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1732 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1734 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1735 cout << exprseq(e.get_free_indices()) << endl;
1737 // 'j' and 'l' are dummy indices
1739 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1740 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1742 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1743 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1744 cout << exprseq(e.get_free_indices()) << endl;
1746 // 'nu' is a dummy index, but 'sigma' is not
1748 e = indexed(A, mu, mu);
1749 cout << exprseq(e.get_free_indices()) << endl;
1751 // 'mu' is not a dummy index because it appears twice with the same
1754 e = indexed(A, mu, nu) + 42;
1755 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1756 // this will throw an exception:
1757 // "add::get_free_indices: inconsistent indices in sum"
1761 @cindex @code{simplify_indexed()}
1762 @subsection Simplifying indexed expressions
1764 In addition to the few automatic simplifications that GiNaC performs on
1765 indexed expressions (such as re-ordering the indices of symmetric tensors
1766 and calculating traces and convolutions of matrices and predefined tensors)
1770 ex ex::simplify_indexed(void);
1771 ex ex::simplify_indexed(const scalar_products & sp);
1774 that performs some more expensive operations:
1777 @item it checks the consistency of free indices in sums in the same way
1778 @code{get_free_indices()} does
1779 @item it tries to give dumy indices that appear in different terms of a sum
1780 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1781 @item it (symbolically) calculates all possible dummy index summations/contractions
1782 with the predefined tensors (this will be explained in more detail in the
1784 @item as a special case of dummy index summation, it can replace scalar products
1785 of two tensors with a user-defined value
1788 The last point is done with the help of the @code{scalar_products} class
1789 which is used to store scalar products with known values (this is not an
1790 arithmetic class, you just pass it to @code{simplify_indexed()}):
1794 symbol A("A"), B("B"), C("C"), i_sym("i");
1798 sp.add(A, B, 0); // A and B are orthogonal
1799 sp.add(A, C, 0); // A and C are orthogonal
1800 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1802 e = indexed(A + B, i) * indexed(A + C, i);
1804 // -> (B+A).i*(A+C).i
1806 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1812 The @code{scalar_products} object @code{sp} acts as a storage for the
1813 scalar products added to it with the @code{.add()} method. This method
1814 takes three arguments: the two expressions of which the scalar product is
1815 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1816 @code{simplify_indexed()} will replace all scalar products of indexed
1817 objects that have the symbols @code{A} and @code{B} as base expressions
1818 with the single value 0. The number, type and dimension of the indices
1819 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1821 @cindex @code{expand()}
1822 The example above also illustrates a feature of the @code{expand()} method:
1823 if passed the @code{expand_indexed} option it will distribute indices
1824 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1826 @cindex @code{tensor} (class)
1827 @subsection Predefined tensors
1829 Some frequently used special tensors such as the delta, epsilon and metric
1830 tensors are predefined in GiNaC. They have special properties when
1831 contracted with other tensor expressions and some of them have constant
1832 matrix representations (they will evaluate to a number when numeric
1833 indices are specified).
1835 @cindex @code{delta_tensor()}
1836 @subsubsection Delta tensor
1838 The delta tensor takes two indices, is symmetric and has the matrix
1839 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
1840 @code{delta_tensor()}:
1844 symbol A("A"), B("B");
1846 idx i(symbol("i"), 3), j(symbol("j"), 3),
1847 k(symbol("k"), 3), l(symbol("l"), 3);
1849 ex e = indexed(A, i, j) * indexed(B, k, l)
1850 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1851 cout << e.simplify_indexed() << endl;
1854 cout << delta_tensor(i, i) << endl;
1859 @cindex @code{metric_tensor()}
1860 @subsubsection General metric tensor
1862 The function @code{metric_tensor()} creates a general symmetric metric
1863 tensor with two indices that can be used to raise/lower tensor indices. The
1864 metric tensor is denoted as @samp{g} in the output and if its indices are of
1865 mixed variance it is automatically replaced by a delta tensor:
1871 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1873 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1874 cout << e.simplify_indexed() << endl;
1877 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1878 cout << e.simplify_indexed() << endl;
1881 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1882 * metric_tensor(nu, rho);
1883 cout << e.simplify_indexed() << endl;
1886 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1887 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1888 + indexed(A, mu.toggle_variance(), rho));
1889 cout << e.simplify_indexed() << endl;
1894 @cindex @code{lorentz_g()}
1895 @subsubsection Minkowski metric tensor
1897 The Minkowski metric tensor is a special metric tensor with a constant
1898 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1899 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1900 It is created with the function @code{lorentz_g()} (although it is output as
1905 varidx mu(symbol("mu"), 4);
1907 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1908 * lorentz_g(mu, varidx(0, 4)); // negative signature
1909 cout << e.simplify_indexed() << endl;
1912 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1913 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1914 cout << e.simplify_indexed() << endl;
1919 @cindex @code{spinor_metric()}
1920 @subsubsection Spinor metric tensor
1922 The function @code{spinor_metric()} creates an antisymmetric tensor with
1923 two indices that is used to raise/lower indices of 2-component spinors.
1924 It is output as @samp{eps}:
1930 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
1931 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
1933 e = spinor_metric(A, B) * indexed(psi, B_co);
1934 cout << e.simplify_indexed() << endl;
1937 e = spinor_metric(A, B) * indexed(psi, A_co);
1938 cout << e.simplify_indexed() << endl;
1941 e = spinor_metric(A_co, B_co) * indexed(psi, B);
1942 cout << e.simplify_indexed() << endl;
1945 e = spinor_metric(A_co, B_co) * indexed(psi, A);
1946 cout << e.simplify_indexed() << endl;
1949 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
1950 cout << e.simplify_indexed() << endl;
1953 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
1954 cout << e.simplify_indexed() << endl;
1959 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
1961 @cindex @code{epsilon_tensor()}
1962 @cindex @code{lorentz_eps()}
1963 @subsubsection Epsilon tensor
1965 The epsilon tensor is totally antisymmetric, its number of indices is equal
1966 to the dimension of the index space (the indices must all be of the same
1967 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1968 defined to be 1. Its behaviour with indices that have a variance also
1969 depends on the signature of the metric. Epsilon tensors are output as
1972 There are three functions defined to create epsilon tensors in 2, 3 and 4
1976 ex epsilon_tensor(const ex & i1, const ex & i2);
1977 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1978 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1981 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1982 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1983 Minkowski space (the last @code{bool} argument specifies whether the metric
1984 has negative or positive signature, as in the case of the Minkowski metric
1987 @subsection Linear algebra
1989 The @code{matrix} class can be used with indices to do some simple linear
1990 algebra (linear combinations and products of vectors and matrices, traces
1991 and scalar products):
1995 idx i(symbol("i"), 2), j(symbol("j"), 2);
1996 symbol x("x"), y("y");
1998 // A is a 2x2 matrix, X is a 2x1 vector
1999 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2001 cout << indexed(A, i, i) << endl;
2004 ex e = indexed(A, i, j) * indexed(X, j);
2005 cout << e.simplify_indexed() << endl;
2006 // -> [[2*y+x],[4*y+3*x]].i
2008 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2009 cout << e.simplify_indexed() << endl;
2010 // -> [[3*y+3*x,6*y+2*x]].j
2014 You can of course obtain the same results with the @code{matrix::add()},
2015 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2016 but with indices you don't have to worry about transposing matrices.
2018 Matrix indices always start at 0 and their dimension must match the number
2019 of rows/columns of the matrix. Matrices with one row or one column are
2020 vectors and can have one or two indices (it doesn't matter whether it's a
2021 row or a column vector). Other matrices must have two indices.
2023 You should be careful when using indices with variance on matrices. GiNaC
2024 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2025 @samp{F.mu.nu} are different matrices. In this case you should use only
2026 one form for @samp{F} and explicitly multiply it with a matrix representation
2027 of the metric tensor.
2030 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2031 @c node-name, next, previous, up
2032 @section Non-commutative objects
2034 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2035 non-commutative objects are built-in which are mostly of use in high energy
2039 @item Clifford (Dirac) algebra (class @code{clifford})
2040 @item su(3) Lie algebra (class @code{color})
2041 @item Matrices (unindexed) (class @code{matrix})
2044 The @code{clifford} and @code{color} classes are subclasses of
2045 @code{indexed} because the elements of these algebras ususally carry
2046 indices. The @code{matrix} class is described in more detail in
2049 Unlike most computer algebra systems, GiNaC does not primarily provide an
2050 operator (often denoted @samp{&*}) for representing inert products of
2051 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2052 classes of objects involved, and non-commutative products are formed with
2053 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2054 figuring out by itself which objects commute and will group the factors
2055 by their class. Consider this example:
2059 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2060 idx a(symbol("a"), 8), b(symbol("b"), 8);
2061 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2063 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2067 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2068 groups the non-commutative factors (the gammas and the su(3) generators)
2069 together while preserving the order of factors within each class (because
2070 Clifford objects commute with color objects). The resulting expression is a
2071 @emph{commutative} product with two factors that are themselves non-commutative
2072 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2073 parentheses are placed around the non-commutative products in the output.
2075 @cindex @code{ncmul} (class)
2076 Non-commutative products are internally represented by objects of the class
2077 @code{ncmul}, as opposed to commutative products which are handled by the
2078 @code{mul} class. You will normally not have to worry about this distinction,
2081 The advantage of this approach is that you never have to worry about using
2082 (or forgetting to use) a special operator when constructing non-commutative
2083 expressions. Also, non-commutative products in GiNaC are more intelligent
2084 than in other computer algebra systems; they can, for example, automatically
2085 canonicalize themselves according to rules specified in the implementation
2086 of the non-commutative classes. The drawback is that to work with other than
2087 the built-in algebras you have to implement new classes yourself. Symbols
2088 always commute and it's not possible to construct non-commutative products
2089 using symbols to represent the algebra elements or generators. User-defined
2090 functions can, however, be specified as being non-commutative.
2092 @cindex @code{return_type()}
2093 @cindex @code{return_type_tinfo()}
2094 Information about the commutativity of an object or expression can be
2095 obtained with the two member functions
2098 unsigned ex::return_type(void) const;
2099 unsigned ex::return_type_tinfo(void) const;
2102 The @code{return_type()} function returns one of three values (defined in
2103 the header file @file{flags.h}), corresponding to three categories of
2104 expressions in GiNaC:
2107 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2108 classes are of this kind.
2109 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2110 certain class of non-commutative objects which can be determined with the
2111 @code{return_type_tinfo()} method. Expressions of this category commute
2112 with everything except @code{noncommutative} expressions of the same
2114 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2115 of non-commutative objects of different classes. Expressions of this
2116 category don't commute with any other @code{noncommutative} or
2117 @code{noncommutative_composite} expressions.
2120 The value returned by the @code{return_type_tinfo()} method is valid only
2121 when the return type of the expression is @code{noncommutative}. It is a
2122 value that is unique to the class of the object and usually one of the
2123 constants in @file{tinfos.h}, or derived therefrom.
2125 Here are a couple of examples:
2128 @multitable @columnfractions 0.33 0.33 0.34
2129 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2130 @item @code{42} @tab @code{commutative} @tab -
2131 @item @code{2*x-y} @tab @code{commutative} @tab -
2132 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2133 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2134 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2135 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2139 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2140 @code{TINFO_clifford} for objects with a representation label of zero.
2141 Other representation labels yield a different @code{return_type_tinfo()},
2142 but it's the same for any two objects with the same label. This is also true
2145 A last note: With the exception of matrices, positive integer powers of
2146 non-commutative objects are automatically expanded in GiNaC. For example,
2147 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2148 non-commutative expressions).
2151 @cindex @code{clifford} (class)
2152 @subsection Clifford algebra
2154 @cindex @code{dirac_gamma()}
2155 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2156 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2157 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2158 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2161 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2164 which takes two arguments: the index and a @dfn{representation label} in the
2165 range 0 to 255 which is used to distinguish elements of different Clifford
2166 algebras (this is also called a @dfn{spin line index}). Gammas with different
2167 labels commute with each other. The dimension of the index can be 4 or (in
2168 the framework of dimensional regularization) any symbolic value. Spinor
2169 indices on Dirac gammas are not supported in GiNaC.
2171 @cindex @code{dirac_ONE()}
2172 The unity element of a Clifford algebra is constructed by
2175 ex dirac_ONE(unsigned char rl = 0);
2178 @cindex @code{dirac_gamma5()}
2179 and there's a special element @samp{gamma5} that commutes with all other
2180 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2184 ex dirac_gamma5(unsigned char rl = 0);
2187 @cindex @code{dirac_gamma6()}
2188 @cindex @code{dirac_gamma7()}
2189 The two additional functions
2192 ex dirac_gamma6(unsigned char rl = 0);
2193 ex dirac_gamma7(unsigned char rl = 0);
2196 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2199 @cindex @code{dirac_slash()}
2200 Finally, the function
2203 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2206 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2207 whose dimension is given by the @code{dim} argument.
2209 In products of dirac gammas, superfluous unity elements are automatically
2210 removed, squares are replaced by their values and @samp{gamma5} is
2211 anticommuted to the front. The @code{simplify_indexed()} function performs
2212 contractions in gamma strings, for example
2217 symbol a("a"), b("b"), D("D");
2218 varidx mu(symbol("mu"), D);
2219 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2220 * dirac_gamma(mu.toggle_variance());
2222 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2223 e = e.simplify_indexed();
2225 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2226 cout << e.subs(D == 4) << endl;
2227 // -> -2*gamma~symbol10*a.symbol10
2228 // [ == -2 * dirac_slash(a, D) ]
2233 @cindex @code{dirac_trace()}
2234 To calculate the trace of an expression containing strings of Dirac gammas
2235 you use the function
2238 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2241 This function takes the trace of all gammas with the specified representation
2242 label; gammas with other labels are left standing. The last argument to
2243 @code{dirac_trace()} is the value to be returned for the trace of the unity
2244 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2245 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2246 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2247 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2248 This @samp{gamma5} scheme is described in greater detail in
2249 @cite{The Role of gamma5 in Dimensional Regularization}.
2251 The value of the trace itself is also usually different in 4 and in
2252 @math{D != 4} dimensions:
2257 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2258 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2259 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2260 cout << dirac_trace(e).simplify_indexed() << endl;
2267 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2268 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2269 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2270 cout << dirac_trace(e).simplify_indexed() << endl;
2271 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2275 Here is an example for using @code{dirac_trace()} to compute a value that
2276 appears in the calculation of the one-loop vacuum polarization amplitude in
2281 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2282 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2285 sp.add(l, l, pow(l, 2));
2286 sp.add(l, q, ldotq);
2288 ex e = dirac_gamma(mu) *
2289 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2290 dirac_gamma(mu.toggle_variance()) *
2291 (dirac_slash(l, D) + m * dirac_ONE());
2292 e = dirac_trace(e).simplify_indexed(sp);
2293 e = e.collect(lst(l, ldotq, m), true);
2295 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2299 The @code{canonicalize_clifford()} function reorders all gamma products that
2300 appear in an expression to a canonical (but not necessarily simple) form.
2301 You can use this to compare two expressions or for further simplifications:
2305 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2306 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2308 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2310 e = canonicalize_clifford(e);
2317 @cindex @code{color} (class)
2318 @subsection Color algebra
2320 @cindex @code{color_T()}
2321 For computations in quantum chromodynamics, GiNaC implements the base elements
2322 and structure constants of the su(3) Lie algebra (color algebra). The base
2323 elements @math{T_a} are constructed by the function
2326 ex color_T(const ex & a, unsigned char rl = 0);
2329 which takes two arguments: the index and a @dfn{representation label} in the
2330 range 0 to 255 which is used to distinguish elements of different color
2331 algebras. Objects with different labels commute with each other. The
2332 dimension of the index must be exactly 8 and it should be of class @code{idx},
2335 @cindex @code{color_ONE()}
2336 The unity element of a color algebra is constructed by
2339 ex color_ONE(unsigned char rl = 0);
2342 @cindex @code{color_d()}
2343 @cindex @code{color_f()}
2347 ex color_d(const ex & a, const ex & b, const ex & c);
2348 ex color_f(const ex & a, const ex & b, const ex & c);
2351 create the symmetric and antisymmetric structure constants @math{d_abc} and
2352 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2353 and @math{[T_a, T_b] = i f_abc T_c}.
2355 @cindex @code{color_h()}
2356 There's an additional function
2359 ex color_h(const ex & a, const ex & b, const ex & c);
2362 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2364 The function @code{simplify_indexed()} performs some simplifications on
2365 expressions containing color objects:
2370 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2371 k(symbol("k"), 8), l(symbol("l"), 8);
2373 e = color_d(a, b, l) * color_f(a, b, k);
2374 cout << e.simplify_indexed() << endl;
2377 e = color_d(a, b, l) * color_d(a, b, k);
2378 cout << e.simplify_indexed() << endl;
2381 e = color_f(l, a, b) * color_f(a, b, k);
2382 cout << e.simplify_indexed() << endl;
2385 e = color_h(a, b, c) * color_h(a, b, c);
2386 cout << e.simplify_indexed() << endl;
2389 e = color_h(a, b, c) * color_T(b) * color_T(c);
2390 cout << e.simplify_indexed() << endl;
2393 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2394 cout << e.simplify_indexed() << endl;
2397 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2398 cout << e.simplify_indexed() << endl;
2399 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2403 @cindex @code{color_trace()}
2404 To calculate the trace of an expression containing color objects you use the
2408 ex color_trace(const ex & e, unsigned char rl = 0);
2411 This function takes the trace of all color @samp{T} objects with the
2412 specified representation label; @samp{T}s with other labels are left
2413 standing. For example:
2417 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2419 // -> -I*f.a.c.b+d.a.c.b
2424 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2425 @c node-name, next, previous, up
2426 @chapter Methods and Functions
2429 In this chapter the most important algorithms provided by GiNaC will be
2430 described. Some of them are implemented as functions on expressions,
2431 others are implemented as methods provided by expression objects. If
2432 they are methods, there exists a wrapper function around it, so you can
2433 alternatively call it in a functional way as shown in the simple
2438 cout << "As method: " << sin(1).evalf() << endl;
2439 cout << "As function: " << evalf(sin(1)) << endl;
2443 @cindex @code{subs()}
2444 The general rule is that wherever methods accept one or more parameters
2445 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2446 wrapper accepts is the same but preceded by the object to act on
2447 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2448 most natural one in an OO model but it may lead to confusion for MapleV
2449 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2450 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2451 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2452 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2453 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2454 here. Also, users of MuPAD will in most cases feel more comfortable
2455 with GiNaC's convention. All function wrappers are implemented
2456 as simple inline functions which just call the corresponding method and
2457 are only provided for users uncomfortable with OO who are dead set to
2458 avoid method invocations. Generally, nested function wrappers are much
2459 harder to read than a sequence of methods and should therefore be
2460 avoided if possible. On the other hand, not everything in GiNaC is a
2461 method on class @code{ex} and sometimes calling a function cannot be
2465 * Information About Expressions::
2466 * Substituting Expressions::
2467 * Pattern Matching and Advanced Substitutions::
2468 * Polynomial Arithmetic:: Working with polynomials.
2469 * Rational Expressions:: Working with rational functions.
2470 * Symbolic Differentiation::
2471 * Series Expansion:: Taylor and Laurent expansion.
2473 * Built-in Functions:: List of predefined mathematical functions.
2474 * Input/Output:: Input and output of expressions.
2478 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2479 @c node-name, next, previous, up
2480 @section Getting information about expressions
2482 @subsection Checking expression types
2483 @cindex @code{is_ex_of_type()}
2484 @cindex @code{ex_to_numeric()}
2485 @cindex @code{ex_to_@dots{}}
2486 @cindex @code{Converting ex to other classes}
2487 @cindex @code{info()}
2488 @cindex @code{return_type()}
2489 @cindex @code{return_type_tinfo()}
2491 Sometimes it's useful to check whether a given expression is a plain number,
2492 a sum, a polynomial with integer coefficients, or of some other specific type.
2493 GiNaC provides a couple of functions for this (the first one is actually a macro):
2496 bool is_ex_of_type(const ex & e, TYPENAME t);
2497 bool ex::info(unsigned flag);
2498 unsigned ex::return_type(void) const;
2499 unsigned ex::return_type_tinfo(void) const;
2502 When the test made by @code{is_ex_of_type()} returns true, it is safe to
2503 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
2504 one of the class names (@xref{The Class Hierarchy}, for a list of all
2505 classes). For example, assuming @code{e} is an @code{ex}:
2510 if (is_ex_of_type(e, numeric))
2511 numeric n = ex_to_numeric(e);
2516 @code{is_ex_of_type()} allows you to check whether the top-level object of
2517 an expression @samp{e} is an instance of the GiNaC class @samp{t}
2518 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2519 e.g., for checking whether an expression is a number, a sum, or a product:
2526 is_ex_of_type(e1, numeric); // true
2527 is_ex_of_type(e2, numeric); // false
2528 is_ex_of_type(e1, add); // false
2529 is_ex_of_type(e2, add); // true
2530 is_ex_of_type(e1, mul); // false
2531 is_ex_of_type(e2, mul); // false
2535 The @code{info()} method is used for checking certain attributes of
2536 expressions. The possible values for the @code{flag} argument are defined
2537 in @file{ginac/flags.h}, the most important being explained in the following
2541 @multitable @columnfractions .30 .70
2542 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2543 @item @code{numeric}
2544 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
2546 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2547 @item @code{rational}
2548 @tab @dots{}an exact rational number (integers are rational, too)
2549 @item @code{integer}
2550 @tab @dots{}a (non-complex) integer
2551 @item @code{crational}
2552 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2553 @item @code{cinteger}
2554 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2555 @item @code{positive}
2556 @tab @dots{}not complex and greater than 0
2557 @item @code{negative}
2558 @tab @dots{}not complex and less than 0
2559 @item @code{nonnegative}
2560 @tab @dots{}not complex and greater than or equal to 0
2562 @tab @dots{}an integer greater than 0
2564 @tab @dots{}an integer less than 0
2565 @item @code{nonnegint}
2566 @tab @dots{}an integer greater than or equal to 0
2568 @tab @dots{}an even integer
2570 @tab @dots{}an odd integer
2572 @tab @dots{}a prime integer (probabilistic primality test)
2573 @item @code{relation}
2574 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
2575 @item @code{relation_equal}
2576 @tab @dots{}a @code{==} relation
2577 @item @code{relation_not_equal}
2578 @tab @dots{}a @code{!=} relation
2579 @item @code{relation_less}
2580 @tab @dots{}a @code{<} relation
2581 @item @code{relation_less_or_equal}
2582 @tab @dots{}a @code{<=} relation
2583 @item @code{relation_greater}
2584 @tab @dots{}a @code{>} relation
2585 @item @code{relation_greater_or_equal}
2586 @tab @dots{}a @code{>=} relation
2588 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
2590 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
2591 @item @code{polynomial}
2592 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2593 @item @code{integer_polynomial}
2594 @tab @dots{}a polynomial with (non-complex) integer coefficients
2595 @item @code{cinteger_polynomial}
2596 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2597 @item @code{rational_polynomial}
2598 @tab @dots{}a polynomial with (non-complex) rational coefficients
2599 @item @code{crational_polynomial}
2600 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2601 @item @code{rational_function}
2602 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2603 @item @code{algebraic}
2604 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2608 To determine whether an expression is commutative or non-commutative and if
2609 so, with which other expressions it would commute, you use the methods
2610 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2611 for an explanation of these.
2614 @subsection Accessing subexpressions
2615 @cindex @code{nops()}
2618 @cindex @code{relational} (class)
2620 GiNaC provides the two methods
2623 unsigned ex::nops();
2624 ex ex::op(unsigned i);
2627 for accessing the subexpressions in the container-like GiNaC classes like
2628 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2629 determines the number of subexpressions (@samp{operands}) contained, while
2630 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2631 In the case of a @code{power} object, @code{op(0)} will return the basis
2632 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2633 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2635 The left-hand and right-hand side expressions of objects of class
2636 @code{relational} (and only of these) can also be accessed with the methods
2644 @subsection Comparing expressions
2645 @cindex @code{is_equal()}
2646 @cindex @code{is_zero()}
2648 Expressions can be compared with the usual C++ relational operators like
2649 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2650 the result is usually not determinable and the result will be @code{false},
2651 except in the case of the @code{!=} operator. You should also be aware that
2652 GiNaC will only do the most trivial test for equality (subtracting both
2653 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2656 Actually, if you construct an expression like @code{a == b}, this will be
2657 represented by an object of the @code{relational} class (@pxref{Relations})
2658 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2660 There are also two methods
2663 bool ex::is_equal(const ex & other);
2667 for checking whether one expression is equal to another, or equal to zero,
2670 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2671 GiNaC header files. This method is however only to be used internally by
2672 GiNaC to establish a canonical sort order for terms, and using it to compare
2673 expressions will give very surprising results.
2676 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2677 @c node-name, next, previous, up
2678 @section Substituting expressions
2679 @cindex @code{subs()}
2681 Algebraic objects inside expressions can be replaced with arbitrary
2682 expressions via the @code{.subs()} method:
2685 ex ex::subs(const ex & e);
2686 ex ex::subs(const lst & syms, const lst & repls);
2689 In the first form, @code{subs()} accepts a relational of the form
2690 @samp{object == expression} or a @code{lst} of such relationals:
2694 symbol x("x"), y("y");
2696 ex e1 = 2*x^2-4*x+3;
2697 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2701 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2706 If you specify multiple substitutions, they are performed in parallel, so e.g.
2707 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2709 The second form of @code{subs()} takes two lists, one for the objects to be
2710 replaced and one for the expressions to be substituted (both lists must
2711 contain the same number of elements). Using this form, you would write
2712 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2714 @code{subs()} performs syntactic substitution of any complete algebraic
2715 object; it does not try to match sub-expressions as is demonstrated by the
2720 symbol x("x"), y("y"), z("z");
2722 ex e1 = pow(x+y, 2);
2723 cout << e1.subs(x+y == 4) << endl;
2726 ex e2 = sin(x)*sin(y)*cos(x);
2727 cout << e2.subs(sin(x) == cos(x)) << endl;
2728 // -> cos(x)^2*sin(y)
2731 cout << e3.subs(x+y == 4) << endl;
2733 // (and not 4+z as one might expect)
2737 A more powerful form of substitution using wildcards is described in the
2741 @node Pattern Matching and Advanced Substitutions, Polynomial Arithmetic, Substituting Expressions, Methods and Functions
2742 @c node-name, next, previous, up
2743 @section Pattern matching and advanced substitutions
2744 @cindex @code{wildcard} (class)
2745 @cindex Pattern matching
2747 GiNaC allows the use of patterns for checking whether an expression is of a
2748 certain form or contains subexpressions of a certain form, and for
2749 substituting expressions in a more general way.
2751 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2752 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2753 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2754 an unsigned integer number to allow having multiple different wildcards in a
2755 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2756 are specified in @command{ginsh}. In C++ code, wildcard objects are created
2760 ex wild(unsigned label = 0);
2763 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2766 Some examples for patterns:
2768 @multitable @columnfractions .5 .5
2769 @item @strong{Constructed as} @tab @strong{Output as}
2770 @item @code{wild()} @tab @samp{$0}
2771 @item @code{pow(x,wild())} @tab @samp{x^$0}
2772 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2773 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2779 @item Wildcards behave like symbols and are subject to the same algebraic
2780 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2781 @item As shown in the last example, to use wildcards for indices you have to
2782 use them as the value of an @code{idx} object. This is because indices must
2783 always be of class @code{idx} (or a subclass).
2784 @item Wildcards only represent expressions or subexpressions. It is not
2785 possible to use them as placeholders for other properties like index
2786 dimension or variance, representation labels, symmetry of indexed objects
2788 @item Because wildcards are commutative, it is not possible to use wildcards
2789 as part of noncommutative products.
2790 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2791 are also valid patterns.
2794 @cindex @code{match()}
2795 The most basic application of patterns is to check whether an expression
2796 matches a given pattern. This is done by the function
2799 bool ex::match(const ex & pattern);
2800 bool ex::match(const ex & pattern, lst & repls);
2803 This function returns @code{true} when the expression matches the pattern
2804 and @code{false} if it doesn't. If used in the second form, the actual
2805 subexpressions matched by the wildcards get returned in the @code{repls}
2806 object as a list of relations of the form @samp{wildcard == expression}.
2807 If @code{match()} returns false, the state of @code{repls} is undefined.
2808 For reproducible results, the list should be empty when passed to
2809 @code{match()}, but it is also possible to find similarities in multiple
2810 expressions by passing in the result of a previous match.
2812 The matching algorithm works as follows:
2815 @item A single wildcard matches any expression. If one wildcard appears
2816 multiple times in a pattern, it must match the same expression in all
2817 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2818 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2819 @item If the expression is not of the same class as the pattern, the match
2820 fails (i.e. a sum only matches a sum, a function only matches a function,
2822 @item If the pattern is a function, it only matches the same function
2823 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2824 @item Except for sums and products, the match fails if the number of
2825 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2827 @item If there are no subexpressions, the expressions and the pattern must
2828 be equal (in the sense of @code{is_equal()}).
2829 @item Except for sums and products, each subexpression (@code{op()}) must
2830 match the corresponding subexpression of the pattern.
2833 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2834 account for their commutativity and associativity:
2837 @item If the pattern contains a term or factor that is a single wildcard,
2838 this one is used as the @dfn{global wildcard}. If there is more than one
2839 such wildcard, one of them is chosen as the global wildcard in a random
2841 @item Every term/factor of the pattern, except the global wildcard, is
2842 matched against every term of the expression in sequence. If no match is
2843 found, the whole match fails. Terms that did match are not considered in
2845 @item If there are no unmatched terms left, the match succeeds. Otherwise
2846 the match fails unless there is a global wildcard in the pattern, in
2847 which case this wildcard matches the remaining terms.
2850 In general, having more than one single wildcard as a term of a sum or a
2851 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2854 Here are some examples in @command{ginsh} to demonstrate how it works (the
2855 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2856 match fails, and the list of wildcard replacements otherwise):
2859 > match((x+y)^a,(x+y)^a);
2861 > match((x+y)^a,(x+y)^b);
2863 > match((x+y)^a,$1^$2);
2865 > match((x+y)^a,$1^$1);
2867 > match((x+y)^(x+y),$1^$1);
2869 > match((x+y)^(x+y),$1^$2);
2871 > match((a+b)*(a+c),($1+b)*($1+c));
2873 > match((a+b)*(a+c),(a+$1)*(a+$2));
2875 (Unpredictable. The result might also be [$1==c,$2==b].)
2876 > match((a+b)*(a+c),($1+$2)*($1+$3));
2877 (The result is undefined. Due to the sequential nature of the algorithm
2878 and the re-ordering of terms in GiNaC, the match for the first factor
2879 may be @{$1==a,$2==b@} in which case the match for the second factor
2880 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
2882 > match(a*(x+y)+a*z+b,a*$1+$2);
2883 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
2884 @{$1=x+y,$2=a*z+b@}.)
2885 > match(a+b+c+d+e+f,c);
2887 > match(a+b+c+d+e+f,c+$0);
2889 > match(a+b+c+d+e+f,c+e+$0);
2891 > match(a+b,a+b+$0);
2893 > match(a*b^2,a^$1*b^$2);
2895 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
2897 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
2899 > match(atan2(y,x^2),atan2(y,$0));
2903 @cindex @code{has()}
2904 A more general way to look for patterns in expressions is provided by the
2908 bool ex::has(const ex & pattern);
2911 This function checks whether a pattern is matched by an expression itself or
2912 by any of its subexpressions.
2914 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
2915 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
2918 > has(x*sin(x+y+2*a),y);
2920 > has(x*sin(x+y+2*a),x+y);
2922 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
2923 has the subexpressions "x", "y" and "2*a".)
2924 > has(x*sin(x+y+2*a),x+y+$1);
2926 (But this is possible.)
2927 > has(x*sin(2*(x+y)+2*a),x+y);
2929 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
2930 which "x+y" is not a subexpression.)
2933 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
2935 > has(4*x^2-x+3,$1*x);
2937 > has(4*x^2+x+3,$1*x);
2939 (Another possible pitfall. The first expression matches because the term
2940 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
2941 contains a linear term you should use the coeff() function instead.)
2944 @cindex @code{subs()}
2945 Probably the most useful application of patterns is to use them for
2946 substituting expressions with the @code{subs()} method. Wildcards can be
2947 used in the search patterns as well as in the replacement expressions, where
2948 they get replaced by the expressions matched by them. @code{subs()} doesn't
2949 know anything about algebra; it performs purely syntactic substitutions.
2954 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
2956 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
2958 > subs((a+b+c)^2,a+b=x);
2960 > subs((a+b+c)^2,a+b+$1==x+$1);
2962 > subs(a+2*b,a+b=x);
2964 > subs(4*x^3-2*x^2+5*x-1,x==a);
2966 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
2968 > subs(sin(1+sin(x)),sin($1)==cos($1));
2970 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
2974 The last example would be written in C++ in this way:
2978 symbol a("a"), b("b"), x("x"), y("y");
2979 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
2980 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
2981 cout << e.expand() << endl;
2987 @node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
2988 @c node-name, next, previous, up
2989 @section Polynomial arithmetic
2991 @subsection Expanding and collecting
2992 @cindex @code{expand()}
2993 @cindex @code{collect()}
2995 A polynomial in one or more variables has many equivalent
2996 representations. Some useful ones serve a specific purpose. Consider
2997 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2998 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2999 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3000 representations are the recursive ones where one collects for exponents
3001 in one of the three variable. Since the factors are themselves
3002 polynomials in the remaining two variables the procedure can be
3003 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
3004 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3007 To bring an expression into expanded form, its method
3013 may be called. In our example above, this corresponds to @math{4*x*y +
3014 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3015 GiNaC is not easily guessable you should be prepared to see different
3016 orderings of terms in such sums!
3018 Another useful representation of multivariate polynomials is as a
3019 univariate polynomial in one of the variables with the coefficients
3020 being polynomials in the remaining variables. The method
3021 @code{collect()} accomplishes this task:
3024 ex ex::collect(const ex & s, bool distributed = false);
3027 The first argument to @code{collect()} can also be a list of objects in which
3028 case the result is either a recursively collected polynomial, or a polynomial
3029 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3030 by the @code{distributed} flag.
3032 Note that the original polynomial needs to be in expanded form in order
3033 for @code{collect()} to be able to find the coefficients properly.
3035 @subsection Degree and coefficients
3036 @cindex @code{degree()}
3037 @cindex @code{ldegree()}
3038 @cindex @code{coeff()}
3040 The degree and low degree of a polynomial can be obtained using the two
3044 int ex::degree(const ex & s);
3045 int ex::ldegree(const ex & s);
3048 which also work reliably on non-expanded input polynomials (they even work
3049 on rational functions, returning the asymptotic degree). To extract
3050 a coefficient with a certain power from an expanded polynomial you use
3053 ex ex::coeff(const ex & s, int n);
3056 You can also obtain the leading and trailing coefficients with the methods
3059 ex ex::lcoeff(const ex & s);
3060 ex ex::tcoeff(const ex & s);
3063 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3066 An application is illustrated in the next example, where a multivariate
3067 polynomial is analyzed:
3070 #include <ginac/ginac.h>
3071 using namespace std;
3072 using namespace GiNaC;
3076 symbol x("x"), y("y");
3077 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3078 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3079 ex Poly = PolyInp.expand();
3081 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3082 cout << "The x^" << i << "-coefficient is "
3083 << Poly.coeff(x,i) << endl;
3085 cout << "As polynomial in y: "
3086 << Poly.collect(y) << endl;
3090 When run, it returns an output in the following fashion:
3093 The x^0-coefficient is y^2+11*y
3094 The x^1-coefficient is 5*y^2-2*y
3095 The x^2-coefficient is -1
3096 The x^3-coefficient is 4*y
3097 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3100 As always, the exact output may vary between different versions of GiNaC
3101 or even from run to run since the internal canonical ordering is not
3102 within the user's sphere of influence.
3104 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3105 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3106 with non-polynomial expressions as they not only work with symbols but with
3107 constants, functions and indexed objects as well:
3111 symbol a("a"), b("b"), c("c");
3112 idx i(symbol("i"), 3);
3114 ex e = pow(sin(x) - cos(x), 4);
3115 cout << e.degree(cos(x)) << endl;
3117 cout << e.expand().coeff(sin(x), 3) << endl;
3120 e = indexed(a+b, i) * indexed(b+c, i);
3121 e = e.expand(expand_options::expand_indexed);
3122 cout << e.collect(indexed(b, i)) << endl;
3123 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3128 @subsection Polynomial division
3129 @cindex polynomial division
3132 @cindex pseudo-remainder
3133 @cindex @code{quo()}
3134 @cindex @code{rem()}
3135 @cindex @code{prem()}
3136 @cindex @code{divide()}
3141 ex quo(const ex & a, const ex & b, const symbol & x);
3142 ex rem(const ex & a, const ex & b, const symbol & x);
3145 compute the quotient and remainder of univariate polynomials in the variable
3146 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3148 The additional function
3151 ex prem(const ex & a, const ex & b, const symbol & x);
3154 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3155 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3157 Exact division of multivariate polynomials is performed by the function
3160 bool divide(const ex & a, const ex & b, ex & q);
3163 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3164 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3165 in which case the value of @code{q} is undefined.
3168 @subsection Unit, content and primitive part
3169 @cindex @code{unit()}
3170 @cindex @code{content()}
3171 @cindex @code{primpart()}
3176 ex ex::unit(const symbol & x);
3177 ex ex::content(const symbol & x);
3178 ex ex::primpart(const symbol & x);
3181 return the unit part, content part, and primitive polynomial of a multivariate
3182 polynomial with respect to the variable @samp{x} (the unit part being the sign
3183 of the leading coefficient, the content part being the GCD of the coefficients,
3184 and the primitive polynomial being the input polynomial divided by the unit and
3185 content parts). The product of unit, content, and primitive part is the
3186 original polynomial.
3189 @subsection GCD and LCM
3192 @cindex @code{gcd()}
3193 @cindex @code{lcm()}
3195 The functions for polynomial greatest common divisor and least common
3196 multiple have the synopsis
3199 ex gcd(const ex & a, const ex & b);
3200 ex lcm(const ex & a, const ex & b);
3203 The functions @code{gcd()} and @code{lcm()} accept two expressions
3204 @code{a} and @code{b} as arguments and return a new expression, their
3205 greatest common divisor or least common multiple, respectively. If the
3206 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3207 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3210 #include <ginac/ginac.h>
3211 using namespace GiNaC;
3215 symbol x("x"), y("y"), z("z");
3216 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3217 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3219 ex P_gcd = gcd(P_a, P_b);
3221 ex P_lcm = lcm(P_a, P_b);
3222 // 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
3227 @subsection Square-free decomposition
3228 @cindex square-free decomposition
3229 @cindex factorization
3230 @cindex @code{sqrfree()}
3232 GiNaC still lacks proper factorization support. Some form of
3233 factorization is, however, easily implemented by noting that factors
3234 appearing in a polynomial with power two or more also appear in the
3235 derivative and hence can easily be found by computing the GCD of the
3236 original polynomial and its derivatives. Any system has an interface
3237 for this so called square-free factorization. So we provide one, too:
3239 ex sqrfree(const ex & a, const lst & l = lst());
3241 Here is an example that by the way illustrates how the result may depend
3242 on the order of differentiation:
3245 symbol x("x"), y("y");
3246 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3248 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3249 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3251 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3252 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3254 cout << sqrfree(BiVarPol) << endl;
3255 // -> depending on luck, any of the above
3260 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3261 @c node-name, next, previous, up
3262 @section Rational expressions
3264 @subsection The @code{normal} method
3265 @cindex @code{normal()}
3266 @cindex simplification
3267 @cindex temporary replacement
3269 Some basic form of simplification of expressions is called for frequently.
3270 GiNaC provides the method @code{.normal()}, which converts a rational function
3271 into an equivalent rational function of the form @samp{numerator/denominator}
3272 where numerator and denominator are coprime. If the input expression is already
3273 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3274 otherwise it performs fraction addition and multiplication.
3276 @code{.normal()} can also be used on expressions which are not rational functions
3277 as it will replace all non-rational objects (like functions or non-integer
3278 powers) by temporary symbols to bring the expression to the domain of rational
3279 functions before performing the normalization, and re-substituting these
3280 symbols afterwards. This algorithm is also available as a separate method
3281 @code{.to_rational()}, described below.
3283 This means that both expressions @code{t1} and @code{t2} are indeed
3284 simplified in this little program:
3287 #include <ginac/ginac.h>
3288 using namespace GiNaC;
3293 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3294 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3295 std::cout << "t1 is " << t1.normal() << std::endl;
3296 std::cout << "t2 is " << t2.normal() << std::endl;
3300 Of course this works for multivariate polynomials too, so the ratio of
3301 the sample-polynomials from the section about GCD and LCM above would be
3302 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3305 @subsection Numerator and denominator
3308 @cindex @code{numer()}
3309 @cindex @code{denom()}
3310 @cindex @code{numer_denom()}
3312 The numerator and denominator of an expression can be obtained with
3317 ex ex::numer_denom();
3320 These functions will first normalize the expression as described above and
3321 then return the numerator, denominator, or both as a list, respectively.
3322 If you need both numerator and denominator, calling @code{numer_denom()} is
3323 faster than using @code{numer()} and @code{denom()} separately.
3326 @subsection Converting to a rational expression
3327 @cindex @code{to_rational()}
3329 Some of the methods described so far only work on polynomials or rational
3330 functions. GiNaC provides a way to extend the domain of these functions to
3331 general expressions by using the temporary replacement algorithm described
3332 above. You do this by calling
3335 ex ex::to_rational(lst &l);
3338 on the expression to be converted. The supplied @code{lst} will be filled
3339 with the generated temporary symbols and their replacement expressions in
3340 a format that can be used directly for the @code{subs()} method. It can also
3341 already contain a list of replacements from an earlier application of
3342 @code{.to_rational()}, so it's possible to use it on multiple expressions
3343 and get consistent results.
3350 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3351 ex b = sin(x) + cos(x);
3354 divide(a.to_rational(l), b.to_rational(l), q);
3355 cout << q.subs(l) << endl;
3359 will print @samp{sin(x)-cos(x)}.
3362 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3363 @c node-name, next, previous, up
3364 @section Symbolic differentiation
3365 @cindex differentiation
3366 @cindex @code{diff()}
3368 @cindex product rule
3370 GiNaC's objects know how to differentiate themselves. Thus, a
3371 polynomial (class @code{add}) knows that its derivative is the sum of
3372 the derivatives of all the monomials:
3375 #include <ginac/ginac.h>
3376 using namespace GiNaC;
3380 symbol x("x"), y("y"), z("z");
3381 ex P = pow(x, 5) + pow(x, 2) + y;
3383 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3384 cout << P.diff(y) << endl; // 1
3385 cout << P.diff(z) << endl; // 0
3389 If a second integer parameter @var{n} is given, the @code{diff} method
3390 returns the @var{n}th derivative.
3392 If @emph{every} object and every function is told what its derivative
3393 is, all derivatives of composed objects can be calculated using the
3394 chain rule and the product rule. Consider, for instance the expression
3395 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3396 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3397 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3398 out that the composition is the generating function for Euler Numbers,
3399 i.e. the so called @var{n}th Euler number is the coefficient of
3400 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3401 identity to code a function that generates Euler numbers in just three
3404 @cindex Euler numbers
3406 #include <ginac/ginac.h>
3407 using namespace GiNaC;
3409 ex EulerNumber(unsigned n)
3412 const ex generator = pow(cosh(x),-1);
3413 return generator.diff(x,n).subs(x==0);
3418 for (unsigned i=0; i<11; i+=2)
3419 std::cout << EulerNumber(i) << std::endl;
3424 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3425 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3426 @code{i} by two since all odd Euler numbers vanish anyways.
3429 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3430 @c node-name, next, previous, up
3431 @section Series expansion
3432 @cindex @code{series()}
3433 @cindex Taylor expansion
3434 @cindex Laurent expansion
3435 @cindex @code{pseries} (class)
3437 Expressions know how to expand themselves as a Taylor series or (more
3438 generally) a Laurent series. As in most conventional Computer Algebra
3439 Systems, no distinction is made between those two. There is a class of
3440 its own for storing such series (@code{class pseries}) and a built-in
3441 function (called @code{Order}) for storing the order term of the series.
3442 As a consequence, if you want to work with series, i.e. multiply two
3443 series, you need to call the method @code{ex::series} again to convert
3444 it to a series object with the usual structure (expansion plus order
3445 term). A sample application from special relativity could read:
3448 #include <ginac/ginac.h>
3449 using namespace std;
3450 using namespace GiNaC;
3454 symbol v("v"), c("c");
3456 ex gamma = 1/sqrt(1 - pow(v/c,2));
3457 ex mass_nonrel = gamma.series(v==0, 10);
3459 cout << "the relativistic mass increase with v is " << endl
3460 << mass_nonrel << endl;
3462 cout << "the inverse square of this series is " << endl
3463 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3467 Only calling the series method makes the last output simplify to
3468 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3469 series raised to the power @math{-2}.
3471 @cindex M@'echain's formula
3472 As another instructive application, let us calculate the numerical
3473 value of Archimedes' constant
3477 (for which there already exists the built-in constant @code{Pi})
3478 using M@'echain's amazing formula
3480 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3483 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3485 We may expand the arcus tangent around @code{0} and insert the fractions
3486 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3487 carries an order term with it and the question arises what the system is
3488 supposed to do when the fractions are plugged into that order term. The
3489 solution is to use the function @code{series_to_poly()} to simply strip
3493 #include <ginac/ginac.h>
3494 using namespace GiNaC;
3496 ex mechain_pi(int degr)
3499 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3500 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3501 -4*pi_expansion.subs(x==numeric(1,239));
3507 using std::cout; // just for fun, another way of...
3508 using std::endl; // ...dealing with this namespace std.
3510 for (int i=2; i<12; i+=2) @{
3511 pi_frac = mechain_pi(i);
3512 cout << i << ":\t" << pi_frac << endl
3513 << "\t" << pi_frac.evalf() << endl;
3519 Note how we just called @code{.series(x,degr)} instead of
3520 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3521 method @code{series()}: if the first argument is a symbol the expression
3522 is expanded in that symbol around point @code{0}. When you run this
3523 program, it will type out:
3527 3.1832635983263598326
3528 4: 5359397032/1706489875
3529 3.1405970293260603143
3530 6: 38279241713339684/12184551018734375
3531 3.141621029325034425
3532 8: 76528487109180192540976/24359780855939418203125
3533 3.141591772182177295
3534 10: 327853873402258685803048818236/104359128170408663038552734375
3535 3.1415926824043995174
3539 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3540 @c node-name, next, previous, up
3541 @section Symmetrization
3542 @cindex @code{symmetrize()}
3543 @cindex @code{antisymmetrize()}
3548 ex ex::symmetrize(const lst & l);
3549 ex ex::antisymmetrize(const lst & l);
3552 symmetrize an expression by returning the symmetric or antisymmetric sum
3553 over all permutations of the specified list of objects, weighted by the
3554 number of permutations.
3556 The two additional methods
3559 ex ex::symmetrize();
3560 ex ex::antisymmetrize();
3563 symmetrize or antisymmetrize an expression over its free indices.
3565 Symmetrization is most useful with indexed expressions but can be used with
3566 almost any kind of object (anything that is @code{subs()}able):
3570 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3571 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3573 cout << indexed(A, i, j).symmetrize() << endl;
3574 // -> 1/2*A.j.i+1/2*A.i.j
3575 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3576 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3577 cout << lst(a, b, c).symmetrize(lst(a, b, c)) << endl;
3578 // -> 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@}
3583 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3584 @c node-name, next, previous, up
3585 @section Predefined mathematical functions
3587 GiNaC contains the following predefined mathematical functions:
3590 @multitable @columnfractions .30 .70
3591 @item @strong{Name} @tab @strong{Function}
3594 @item @code{csgn(x)}
3596 @item @code{sqrt(x)}
3597 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3604 @item @code{asin(x)}
3606 @item @code{acos(x)}
3608 @item @code{atan(x)}
3609 @tab inverse tangent
3610 @item @code{atan2(y, x)}
3611 @tab inverse tangent with two arguments
3612 @item @code{sinh(x)}
3613 @tab hyperbolic sine
3614 @item @code{cosh(x)}
3615 @tab hyperbolic cosine
3616 @item @code{tanh(x)}
3617 @tab hyperbolic tangent
3618 @item @code{asinh(x)}
3619 @tab inverse hyperbolic sine
3620 @item @code{acosh(x)}
3621 @tab inverse hyperbolic cosine
3622 @item @code{atanh(x)}
3623 @tab inverse hyperbolic tangent
3625 @tab exponential function
3627 @tab natural logarithm
3630 @item @code{zeta(x)}
3631 @tab Riemann's zeta function
3632 @item @code{zeta(n, x)}
3633 @tab derivatives of Riemann's zeta function
3634 @item @code{tgamma(x)}
3636 @item @code{lgamma(x)}
3637 @tab logarithm of Gamma function
3638 @item @code{beta(x, y)}
3639 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3641 @tab psi (digamma) function
3642 @item @code{psi(n, x)}
3643 @tab derivatives of psi function (polygamma functions)
3644 @item @code{factorial(n)}
3645 @tab factorial function
3646 @item @code{binomial(n, m)}
3647 @tab binomial coefficients
3648 @item @code{Order(x)}
3649 @tab order term function in truncated power series
3650 @item @code{Derivative(x, l)}
3651 @tab inert partial differentiation operator (used internally)
3656 For functions that have a branch cut in the complex plane GiNaC follows
3657 the conventions for C++ as defined in the ANSI standard as far as
3658 possible. In particular: the natural logarithm (@code{log}) and the
3659 square root (@code{sqrt}) both have their branch cuts running along the
3660 negative real axis where the points on the axis itself belong to the
3661 upper part (i.e. continuous with quadrant II). The inverse
3662 trigonometric and hyperbolic functions are not defined for complex
3663 arguments by the C++ standard, however. In GiNaC we follow the
3664 conventions used by CLN, which in turn follow the carefully designed
3665 definitions in the Common Lisp standard. It should be noted that this
3666 convention is identical to the one used by the C99 standard and by most
3667 serious CAS. It is to be expected that future revisions of the C++
3668 standard incorporate these functions in the complex domain in a manner
3669 compatible with C99.
3672 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3673 @c node-name, next, previous, up
3674 @section Input and output of expressions
3677 @subsection Expression output
3679 @cindex output of expressions
3681 The easiest way to print an expression is to write it to a stream:
3686 ex e = 4.5+pow(x,2)*3/2;
3687 cout << e << endl; // prints '(4.5)+3/2*x^2'
3691 The output format is identical to the @command{ginsh} input syntax and
3692 to that used by most computer algebra systems, but not directly pastable
3693 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3694 is printed as @samp{x^2}).
3696 It is possible to print expressions in a number of different formats with
3700 void ex::print(const print_context & c, unsigned level = 0);
3703 @cindex @code{print_context} (class)
3704 The type of @code{print_context} object passed in determines the format
3705 of the output. The possible types are defined in @file{ginac/print.h}.
3706 All constructors of @code{print_context} and derived classes take an
3707 @code{ostream &} as their first argument.
3709 To print an expression in a way that can be directly used in a C or C++
3710 program, you pass a @code{print_csrc} object like this:
3714 cout << "float f = ";
3715 e.print(print_csrc_float(cout));
3718 cout << "double d = ";
3719 e.print(print_csrc_double(cout));
3722 cout << "cl_N n = ";
3723 e.print(print_csrc_cl_N(cout));
3728 The three possible types mostly affect the way in which floating point
3729 numbers are written.
3731 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3734 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3735 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3736 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3739 The @code{print_context} type @code{print_tree} provides a dump of the
3740 internal structure of an expression for debugging purposes:
3744 e.print(print_tree(cout));
3751 add, hash=0x0, flags=0x3, nops=2
3752 power, hash=0x9, flags=0x3, nops=2
3753 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
3754 2 (numeric), hash=0x80000042, flags=0xf
3755 3/2 (numeric), hash=0x80000061, flags=0xf
3758 4.5L0 (numeric), hash=0x8000004b, flags=0xf
3762 This kind of output is also available in @command{ginsh} as the @code{print()}
3765 Another useful output format is for LaTeX parsing in mathematical mode.
3766 It is rather similar to the default @code{print_context} but provides
3767 some braces needed by LaTeX for delimiting boxes and also converts some
3768 common objects to conventional LaTeX names. It is possible to give symbols
3769 a special name for LaTeX output by supplying it as a second argument to
3770 the @code{symbol} constructor.
3772 For example, the code snippet
3777 ex foo = lgamma(x).series(x==0,3);
3778 foo.print(print_latex(std::cout));
3784 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
3787 If you need any fancy special output format, e.g. for interfacing GiNaC
3788 with other algebra systems or for producing code for different
3789 programming languages, you can always traverse the expression tree yourself:
3792 static void my_print(const ex & e)
3794 if (is_ex_of_type(e, function))
3795 cout << ex_to_function(e).get_name();
3797 cout << e.bp->class_name();
3799 unsigned n = e.nops();
3801 for (unsigned i=0; i<n; i++) @{
3813 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
3821 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
3822 symbol(y))),numeric(-2)))
3825 If you need an output format that makes it possible to accurately
3826 reconstruct an expression by feeding the output to a suitable parser or
3827 object factory, you should consider storing the expression in an
3828 @code{archive} object and reading the object properties from there.
3829 See the section on archiving for more information.
3832 @subsection Expression input
3833 @cindex input of expressions
3835 GiNaC provides no way to directly read an expression from a stream because
3836 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
3837 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
3838 @code{y} you defined in your program and there is no way to specify the
3839 desired symbols to the @code{>>} stream input operator.
3841 Instead, GiNaC lets you construct an expression from a string, specifying the
3842 list of symbols to be used:
3846 symbol x("x"), y("y");
3847 ex e("2*x+sin(y)", lst(x, y));
3851 The input syntax is the same as that used by @command{ginsh} and the stream
3852 output operator @code{<<}. The symbols in the string are matched by name to
3853 the symbols in the list and if GiNaC encounters a symbol not specified in
3854 the list it will throw an exception.
3856 With this constructor, it's also easy to implement interactive GiNaC programs:
3861 #include <stdexcept>
3862 #include <ginac/ginac.h>
3863 using namespace std;
3864 using namespace GiNaC;
3871 cout << "Enter an expression containing 'x': ";
3876 cout << "The derivative of " << e << " with respect to x is ";
3877 cout << e.diff(x) << ".\n";
3878 @} catch (exception &p) @{
3879 cerr << p.what() << endl;
3885 @subsection Archiving
3886 @cindex @code{archive} (class)
3889 GiNaC allows creating @dfn{archives} of expressions which can be stored
3890 to or retrieved from files. To create an archive, you declare an object
3891 of class @code{archive} and archive expressions in it, giving each
3892 expression a unique name:
3896 using namespace std;
3897 #include <ginac/ginac.h>
3898 using namespace GiNaC;
3902 symbol x("x"), y("y"), z("z");
3904 ex foo = sin(x + 2*y) + 3*z + 41;
3908 a.archive_ex(foo, "foo");
3909 a.archive_ex(bar, "the second one");
3913 The archive can then be written to a file:
3917 ofstream out("foobar.gar");
3923 The file @file{foobar.gar} contains all information that is needed to
3924 reconstruct the expressions @code{foo} and @code{bar}.
3926 @cindex @command{viewgar}
3927 The tool @command{viewgar} that comes with GiNaC can be used to view
3928 the contents of GiNaC archive files:
3931 $ viewgar foobar.gar
3932 foo = 41+sin(x+2*y)+3*z
3933 the second one = 42+sin(x+2*y)+3*z
3936 The point of writing archive files is of course that they can later be
3942 ifstream in("foobar.gar");
3947 And the stored expressions can be retrieved by their name:
3953 ex ex1 = a2.unarchive_ex(syms, "foo");
3954 ex ex2 = a2.unarchive_ex(syms, "the second one");
3956 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
3957 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
3958 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
3962 Note that you have to supply a list of the symbols which are to be inserted
3963 in the expressions. Symbols in archives are stored by their name only and
3964 if you don't specify which symbols you have, unarchiving the expression will
3965 create new symbols with that name. E.g. if you hadn't included @code{x} in
3966 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
3967 have had no effect because the @code{x} in @code{ex1} would have been a
3968 different symbol than the @code{x} which was defined at the beginning of
3969 the program, altough both would appear as @samp{x} when printed.
3971 You can also use the information stored in an @code{archive} object to
3972 output expressions in a format suitable for exact reconstruction. The
3973 @code{archive} and @code{archive_node} classes have a couple of member
3974 functions that let you access the stored properties:
3977 static void my_print2(const archive_node & n)
3980 n.find_string("class", class_name);
3981 cout << class_name << "(";
3983 archive_node::propinfovector p;
3984 n.get_properties(p);
3986 unsigned num = p.size();
3987 for (unsigned i=0; i<num; i++) @{
3988 const string &name = p[i].name;
3989 if (name == "class")
3991 cout << name << "=";
3993 unsigned count = p[i].count;
3997 for (unsigned j=0; j<count; j++) @{
3998 switch (p[i].type) @{
3999 case archive_node::PTYPE_BOOL: @{
4001 n.find_bool(name, x);
4002 cout << (x ? "true" : "false");
4005 case archive_node::PTYPE_UNSIGNED: @{
4007 n.find_unsigned(name, x);
4011 case archive_node::PTYPE_STRING: @{
4013 n.find_string(name, x);
4014 cout << '\"' << x << '\"';
4017 case archive_node::PTYPE_NODE: @{
4018 const archive_node &x = n.find_ex_node(name, j);
4040 ex e = pow(2, x) - y;
4042 my_print2(ar.get_top_node(0)); cout << endl;
4050 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4051 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4052 overall_coeff=numeric(number="0"))
4055 Be warned, however, that the set of properties and their meaning for each
4056 class may change between GiNaC versions.
4059 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4060 @c node-name, next, previous, up
4061 @chapter Extending GiNaC
4063 By reading so far you should have gotten a fairly good understanding of
4064 GiNaC's design-patterns. From here on you should start reading the
4065 sources. All we can do now is issue some recommendations how to tackle
4066 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4067 develop some useful extension please don't hesitate to contact the GiNaC
4068 authors---they will happily incorporate them into future versions.
4071 * What does not belong into GiNaC:: What to avoid.
4072 * Symbolic functions:: Implementing symbolic functions.
4073 * Adding classes:: Defining new algebraic classes.
4077 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4078 @c node-name, next, previous, up
4079 @section What doesn't belong into GiNaC
4081 @cindex @command{ginsh}
4082 First of all, GiNaC's name must be read literally. It is designed to be
4083 a library for use within C++. The tiny @command{ginsh} accompanying
4084 GiNaC makes this even more clear: it doesn't even attempt to provide a
4085 language. There are no loops or conditional expressions in
4086 @command{ginsh}, it is merely a window into the library for the
4087 programmer to test stuff (or to show off). Still, the design of a
4088 complete CAS with a language of its own, graphical capabilites and all
4089 this on top of GiNaC is possible and is without doubt a nice project for
4092 There are many built-in functions in GiNaC that do not know how to
4093 evaluate themselves numerically to a precision declared at runtime
4094 (using @code{Digits}). Some may be evaluated at certain points, but not
4095 generally. This ought to be fixed. However, doing numerical
4096 computations with GiNaC's quite abstract classes is doomed to be
4097 inefficient. For this purpose, the underlying foundation classes
4098 provided by @acronym{CLN} are much better suited.
4101 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4102 @c node-name, next, previous, up
4103 @section Symbolic functions
4105 The easiest and most instructive way to start with is probably to
4106 implement your own function. GiNaC's functions are objects of class
4107 @code{function}. The preprocessor is then used to convert the function
4108 names to objects with a corresponding serial number that is used
4109 internally to identify them. You usually need not worry about this
4110 number. New functions may be inserted into the system via a kind of
4111 `registry'. It is your responsibility to care for some functions that
4112 are called when the user invokes certain methods. These are usual
4113 C++-functions accepting a number of @code{ex} as arguments and returning
4114 one @code{ex}. As an example, if we have a look at a simplified
4115 implementation of the cosine trigonometric function, we first need a
4116 function that is called when one wishes to @code{eval} it. It could
4117 look something like this:
4120 static ex cos_eval_method(const ex & x)
4122 // if (!x%(2*Pi)) return 1
4123 // if (!x%Pi) return -1
4124 // if (!x%Pi/2) return 0
4125 // care for other cases...
4126 return cos(x).hold();
4130 @cindex @code{hold()}
4132 The last line returns @code{cos(x)} if we don't know what else to do and
4133 stops a potential recursive evaluation by saying @code{.hold()}, which
4134 sets a flag to the expression signaling that it has been evaluated. We
4135 should also implement a method for numerical evaluation and since we are
4136 lazy we sweep the problem under the rug by calling someone else's
4137 function that does so, in this case the one in class @code{numeric}:
4140 static ex cos_evalf(const ex & x)
4142 return cos(ex_to_numeric(x));
4146 Differentiation will surely turn up and so we need to tell @code{cos}
4147 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4148 instance are then handled automatically by @code{basic::diff} and
4152 static ex cos_deriv(const ex & x, unsigned diff_param)
4158 @cindex product rule
4159 The second parameter is obligatory but uninteresting at this point. It
4160 specifies which parameter to differentiate in a partial derivative in
4161 case the function has more than one parameter and its main application
4162 is for correct handling of the chain rule. For Taylor expansion, it is
4163 enough to know how to differentiate. But if the function you want to
4164 implement does have a pole somewhere in the complex plane, you need to
4165 write another method for Laurent expansion around that point.
4167 Now that all the ingredients for @code{cos} have been set up, we need
4168 to tell the system about it. This is done by a macro and we are not
4169 going to descibe how it expands, please consult your preprocessor if you
4173 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4174 evalf_func(cos_evalf).
4175 derivative_func(cos_deriv));
4178 The first argument is the function's name used for calling it and for
4179 output. The second binds the corresponding methods as options to this
4180 object. Options are separated by a dot and can be given in an arbitrary
4181 order. GiNaC functions understand several more options which are always
4182 specified as @code{.option(params)}, for example a method for series
4183 expansion @code{.series_func(cos_series)}. Again, if no series
4184 expansion method is given, GiNaC defaults to simple Taylor expansion,
4185 which is correct if there are no poles involved as is the case for the
4186 @code{cos} function. The way GiNaC handles poles in case there are any
4187 is best understood by studying one of the examples, like the Gamma
4188 (@code{tgamma}) function for instance. (In essence the function first
4189 checks if there is a pole at the evaluation point and falls back to
4190 Taylor expansion if there isn't. Then, the pole is regularized by some
4191 suitable transformation.) Also, the new function needs to be declared
4192 somewhere. This may also be done by a convenient preprocessor macro:
4195 DECLARE_FUNCTION_1P(cos)
4198 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4199 implementation of @code{cos} is very incomplete and lacks several safety
4200 mechanisms. Please, have a look at the real implementation in GiNaC.
4201 (By the way: in case you are worrying about all the macros above we can
4202 assure you that functions are GiNaC's most macro-intense classes. We
4203 have done our best to avoid macros where we can.)
4206 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4207 @c node-name, next, previous, up
4208 @section Adding classes
4210 If you are doing some very specialized things with GiNaC you may find that
4211 you have to implement your own algebraic classes to fit your needs. This
4212 section will explain how to do this by giving the example of a simple
4213 'string' class. After reading this section you will know how to properly
4214 declare a GiNaC class and what the minimum required member functions are
4215 that you have to implement. We only cover the implementation of a 'leaf'
4216 class here (i.e. one that doesn't contain subexpressions). Creating a
4217 container class like, for example, a class representing tensor products is
4218 more involved but this section should give you enough information so you can
4219 consult the source to GiNaC's predefined classes if you want to implement
4220 something more complicated.
4222 @subsection GiNaC's run-time type information system
4224 @cindex hierarchy of classes
4226 All algebraic classes (that is, all classes that can appear in expressions)
4227 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4228 @code{basic *} (which is essentially what an @code{ex} is) represents a
4229 generic pointer to an algebraic class. Occasionally it is necessary to find
4230 out what the class of an object pointed to by a @code{basic *} really is.
4231 Also, for the unarchiving of expressions it must be possible to find the
4232 @code{unarchive()} function of a class given the class name (as a string). A
4233 system that provides this kind of information is called a run-time type
4234 information (RTTI) system. The C++ language provides such a thing (see the
4235 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4236 implements its own, simpler RTTI.
4238 The RTTI in GiNaC is based on two mechanisms:
4243 The @code{basic} class declares a member variable @code{tinfo_key} which
4244 holds an unsigned integer that identifies the object's class. These numbers
4245 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4246 classes. They all start with @code{TINFO_}.
4249 By means of some clever tricks with static members, GiNaC maintains a list
4250 of information for all classes derived from @code{basic}. The information
4251 available includes the class names, the @code{tinfo_key}s, and pointers
4252 to the unarchiving functions. This class registry is defined in the
4253 @file{registrar.h} header file.
4257 The disadvantage of this proprietary RTTI implementation is that there's
4258 a little more to do when implementing new classes (C++'s RTTI works more
4259 or less automatic) but don't worry, most of the work is simplified by
4262 @subsection A minimalistic example
4264 Now we will start implementing a new class @code{mystring} that allows
4265 placing character strings in algebraic expressions (this is not very useful,
4266 but it's just an example). This class will be a direct subclass of
4267 @code{basic}. You can use this sample implementation as a starting point
4268 for your own classes.
4270 The code snippets given here assume that you have included some header files
4276 #include <stdexcept>
4277 using namespace std;
4279 #include <ginac/ginac.h>
4280 using namespace GiNaC;
4283 The first thing we have to do is to define a @code{tinfo_key} for our new
4284 class. This can be any arbitrary unsigned number that is not already taken
4285 by one of the existing classes but it's better to come up with something
4286 that is unlikely to clash with keys that might be added in the future. The
4287 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4288 which is not a requirement but we are going to stick with this scheme:
4291 const unsigned TINFO_mystring = 0x42420001U;
4294 Now we can write down the class declaration. The class stores a C++
4295 @code{string} and the user shall be able to construct a @code{mystring}
4296 object from a C or C++ string:
4299 class mystring : public basic
4301 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4304 mystring(const string &s);
4305 mystring(const char *s);
4311 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4314 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4315 macros are defined in @file{registrar.h}. They take the name of the class
4316 and its direct superclass as arguments and insert all required declarations
4317 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4318 the first line after the opening brace of the class definition. The
4319 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4320 source (at global scope, of course, not inside a function).
4322 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4323 declarations of the default and copy constructor, the destructor, the
4324 assignment operator and a couple of other functions that are required. It
4325 also defines a type @code{inherited} which refers to the superclass so you
4326 don't have to modify your code every time you shuffle around the class
4327 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4328 constructor, the destructor and the assignment operator.
4330 Now there are nine member functions we have to implement to get a working
4336 @code{mystring()}, the default constructor.
4339 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4340 assignment operator to free dynamically allocated members. The @code{call_parent}
4341 specifies whether the @code{destroy()} function of the superclass is to be
4345 @code{void copy(const mystring &other)}, which is used in the copy constructor
4346 and assignment operator to copy the member variables over from another
4347 object of the same class.
4350 @code{void archive(archive_node &n)}, the archiving function. This stores all
4351 information needed to reconstruct an object of this class inside an
4352 @code{archive_node}.
4355 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4356 constructor. This constructs an instance of the class from the information
4357 found in an @code{archive_node}.
4360 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4361 unarchiving function. It constructs a new instance by calling the unarchiving
4365 @code{int compare_same_type(const basic &other)}, which is used internally
4366 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4367 -1, depending on the relative order of this object and the @code{other}
4368 object. If it returns 0, the objects are considered equal.
4369 @strong{Note:} This has nothing to do with the (numeric) ordering
4370 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4371 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4372 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4373 must provide a @code{compare_same_type()} function, even those representing
4374 objects for which no reasonable algebraic ordering relationship can be
4378 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4379 which are the two constructors we declared.
4383 Let's proceed step-by-step. The default constructor looks like this:
4386 mystring::mystring() : inherited(TINFO_mystring)
4388 // dynamically allocate resources here if required
4392 The golden rule is that in all constructors you have to set the
4393 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4394 it will be set by the constructor of the superclass and all hell will break
4395 loose in the RTTI. For your convenience, the @code{basic} class provides
4396 a constructor that takes a @code{tinfo_key} value, which we are using here
4397 (remember that in our case @code{inherited = basic}). If the superclass
4398 didn't have such a constructor, we would have to set the @code{tinfo_key}
4399 to the right value manually.
4401 In the default constructor you should set all other member variables to
4402 reasonable default values (we don't need that here since our @code{str}
4403 member gets set to an empty string automatically). The constructor(s) are of
4404 course also the right place to allocate any dynamic resources you require.
4406 Next, the @code{destroy()} function:
4409 void mystring::destroy(bool call_parent)
4411 // free dynamically allocated resources here if required
4413 inherited::destroy(call_parent);
4417 This function is where we free all dynamically allocated resources. We don't
4418 have any so we're not doing anything here, but if we had, for example, used
4419 a C-style @code{char *} to store our string, this would be the place to
4420 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4421 to call the @code{destroy()} function of the superclass after we're done
4422 (to mimic C++'s automatic invocation of superclass destructors where
4423 @code{destroy()} is called from outside a destructor).
4425 The @code{copy()} function just copies over the member variables from
4429 void mystring::copy(const mystring &other)
4431 inherited::copy(other);
4436 We can simply overwrite the member variables here. There's no need to worry
4437 about dynamically allocated storage. The assignment operator (which is
4438 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4439 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4440 explicitly call the @code{copy()} function of the superclass here so
4441 all the member variables will get copied.
4443 Next are the three functions for archiving. You have to implement them even
4444 if you don't plan to use archives, but the minimum required implementation
4445 is really simple. First, the archiving function:
4448 void mystring::archive(archive_node &n) const
4450 inherited::archive(n);
4451 n.add_string("string", str);
4455 The only thing that is really required is calling the @code{archive()}
4456 function of the superclass. Optionally, you can store all information you
4457 deem necessary for representing the object into the passed
4458 @code{archive_node}. We are just storing our string here. For more
4459 information on how the archiving works, consult the @file{archive.h} header
4462 The unarchiving constructor is basically the inverse of the archiving
4466 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4468 n.find_string("string", str);
4472 If you don't need archiving, just leave this function empty (but you must
4473 invoke the unarchiving constructor of the superclass). Note that we don't
4474 have to set the @code{tinfo_key} here because it is done automatically
4475 by the unarchiving constructor of the @code{basic} class.
4477 Finally, the unarchiving function:
4480 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4482 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4486 You don't have to understand how exactly this works. Just copy these four
4487 lines into your code literally (replacing the class name, of course). It
4488 calls the unarchiving constructor of the class and unless you are doing
4489 something very special (like matching @code{archive_node}s to global
4490 objects) you don't need a different implementation. For those who are
4491 interested: setting the @code{dynallocated} flag puts the object under
4492 the control of GiNaC's garbage collection. It will get deleted automatically
4493 once it is no longer referenced.
4495 Our @code{compare_same_type()} function uses a provided function to compare
4499 int mystring::compare_same_type(const basic &other) const
4501 const mystring &o = static_cast<const mystring &>(other);
4502 int cmpval = str.compare(o.str);
4505 else if (cmpval < 0)
4512 Although this function takes a @code{basic &}, it will always be a reference
4513 to an object of exactly the same class (objects of different classes are not
4514 comparable), so the cast is safe. If this function returns 0, the two objects
4515 are considered equal (in the sense that @math{A-B=0}), so you should compare
4516 all relevant member variables.
4518 Now the only thing missing is our two new constructors:
4521 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4523 // dynamically allocate resources here if required
4526 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4528 // dynamically allocate resources here if required
4532 No surprises here. We set the @code{str} member from the argument and
4533 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4535 That's it! We now have a minimal working GiNaC class that can store
4536 strings in algebraic expressions. Let's confirm that the RTTI works:
4539 ex e = mystring("Hello, world!");
4540 cout << is_ex_of_type(e, mystring) << endl;
4543 cout << e.bp->class_name() << endl;
4547 Obviously it does. Let's see what the expression @code{e} looks like:
4551 // -> [mystring object]
4554 Hm, not exactly what we expect, but of course the @code{mystring} class
4555 doesn't yet know how to print itself. This is done in the @code{print()}
4556 member function. Let's say that we wanted to print the string surrounded
4560 class mystring : public basic
4564 void print(const print_context &c, unsigned level = 0) const;
4568 void mystring::print(const print_context &c, unsigned level) const
4570 // print_context::s is a reference to an ostream
4571 c.s << '\"' << str << '\"';
4575 The @code{level} argument is only required for container classes to
4576 correctly parenthesize the output. Let's try again to print the expression:
4580 // -> "Hello, world!"
4583 Much better. The @code{mystring} class can be used in arbitrary expressions:
4586 e += mystring("GiNaC rulez");
4588 // -> "GiNaC rulez"+"Hello, world!"
4591 (note that GiNaC's automatic term reordering is in effect here), or even
4594 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4596 // -> "One string"^(2*sin(-"Another string"+Pi))
4599 Whether this makes sense is debatable but remember that this is only an
4600 example. At least it allows you to implement your own symbolic algorithms
4603 Note that GiNaC's algebraic rules remain unchanged:
4606 e = mystring("Wow") * mystring("Wow");
4610 e = pow(mystring("First")-mystring("Second"), 2);
4611 cout << e.expand() << endl;
4612 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4615 There's no way to, for example, make GiNaC's @code{add} class perform string
4616 concatenation. You would have to implement this yourself.
4618 @subsection Automatic evaluation
4620 @cindex @code{hold()}
4622 When dealing with objects that are just a little more complicated than the
4623 simple string objects we have implemented, chances are that you will want to
4624 have some automatic simplifications or canonicalizations performed on them.
4625 This is done in the evaluation member function @code{eval()}. Let's say that
4626 we wanted all strings automatically converted to lowercase with
4627 non-alphabetic characters stripped, and empty strings removed:
4630 class mystring : public basic
4634 ex eval(int level = 0) const;
4638 ex mystring::eval(int level) const
4641 for (int i=0; i<str.length(); i++) @{
4643 if (c >= 'A' && c <= 'Z')
4644 new_str += tolower(c);
4645 else if (c >= 'a' && c <= 'z')
4649 if (new_str.length() == 0)
4652 return mystring(new_str).hold();
4656 The @code{level} argument is used to limit the recursion depth of the
4657 evaluation. We don't have any subexpressions in the @code{mystring} class
4658 so we are not concerned with this. If we had, we would call the @code{eval()}
4659 functions of the subexpressions with @code{level - 1} as the argument if
4660 @code{level != 1}. The @code{hold()} member function sets a flag in the
4661 object that prevents further evaluation. Otherwise we might end up in an
4662 endless loop. When you want to return the object unmodified, use
4663 @code{return this->hold();}.
4665 Let's confirm that it works:
4668 ex e = mystring("Hello, world!") + mystring("!?#");
4672 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4677 @subsection Other member functions
4679 We have implemented only a small set of member functions to make the class
4680 work in the GiNaC framework. For a real algebraic class, there are probably
4681 some more functions that you will want to re-implement, such as
4682 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4683 or the header file of the class you want to make a subclass of to see
4684 what's there. One member function that you will most likely want to
4685 implement for terminal classes like the described string class is
4686 @code{calcchash()} that returns an @code{unsigned} hash value for the object
4687 which will allow GiNaC to compare and canonicalize expressions much more
4690 You can, of course, also add your own new member functions. In this case you
4691 will probably want to define a little helper function like
4694 inline const mystring &ex_to_mystring(const ex &e)
4696 return static_cast<const mystring &>(*e.bp);
4700 that let's you get at the object inside an expression (after you have
4701 verified that the type is correct) so you can call member functions that are
4702 specific to the class.
4704 That's it. May the source be with you!
4707 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4708 @c node-name, next, previous, up
4709 @chapter A Comparison With Other CAS
4712 This chapter will give you some information on how GiNaC compares to
4713 other, traditional Computer Algebra Systems, like @emph{Maple},
4714 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4715 disadvantages over these systems.
4718 * Advantages:: Stengths of the GiNaC approach.
4719 * Disadvantages:: Weaknesses of the GiNaC approach.
4720 * Why C++?:: Attractiveness of C++.
4723 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4724 @c node-name, next, previous, up
4727 GiNaC has several advantages over traditional Computer
4728 Algebra Systems, like
4733 familiar language: all common CAS implement their own proprietary
4734 grammar which you have to learn first (and maybe learn again when your
4735 vendor decides to `enhance' it). With GiNaC you can write your program
4736 in common C++, which is standardized.
4740 structured data types: you can build up structured data types using
4741 @code{struct}s or @code{class}es together with STL features instead of
4742 using unnamed lists of lists of lists.
4745 strongly typed: in CAS, you usually have only one kind of variables
4746 which can hold contents of an arbitrary type. This 4GL like feature is
4747 nice for novice programmers, but dangerous.
4750 development tools: powerful development tools exist for C++, like fancy
4751 editors (e.g. with automatic indentation and syntax highlighting),
4752 debuggers, visualization tools, documentation generators@dots{}
4755 modularization: C++ programs can easily be split into modules by
4756 separating interface and implementation.
4759 price: GiNaC is distributed under the GNU Public License which means
4760 that it is free and available with source code. And there are excellent
4761 C++-compilers for free, too.
4764 extendable: you can add your own classes to GiNaC, thus extending it on
4765 a very low level. Compare this to a traditional CAS that you can
4766 usually only extend on a high level by writing in the language defined
4767 by the parser. In particular, it turns out to be almost impossible to
4768 fix bugs in a traditional system.
4771 multiple interfaces: Though real GiNaC programs have to be written in
4772 some editor, then be compiled, linked and executed, there are more ways
4773 to work with the GiNaC engine. Many people want to play with
4774 expressions interactively, as in traditional CASs. Currently, two such
4775 windows into GiNaC have been implemented and many more are possible: the
4776 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
4777 types to a command line and second, as a more consistent approach, an
4778 interactive interface to the @acronym{Cint} C++ interpreter has been put
4779 together (called @acronym{GiNaC-cint}) that allows an interactive
4780 scripting interface consistent with the C++ language.
4783 seemless integration: it is somewhere between difficult and impossible
4784 to call CAS functions from within a program written in C++ or any other
4785 programming language and vice versa. With GiNaC, your symbolic routines
4786 are part of your program. You can easily call third party libraries,
4787 e.g. for numerical evaluation or graphical interaction. All other
4788 approaches are much more cumbersome: they range from simply ignoring the
4789 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
4790 system (i.e. @emph{Yacas}).
4793 efficiency: often large parts of a program do not need symbolic
4794 calculations at all. Why use large integers for loop variables or
4795 arbitrary precision arithmetics where @code{int} and @code{double} are
4796 sufficient? For pure symbolic applications, GiNaC is comparable in
4797 speed with other CAS.
4802 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
4803 @c node-name, next, previous, up
4804 @section Disadvantages
4806 Of course it also has some disadvantages:
4811 advanced features: GiNaC cannot compete with a program like
4812 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
4813 which grows since 1981 by the work of dozens of programmers, with
4814 respect to mathematical features. Integration, factorization,
4815 non-trivial simplifications, limits etc. are missing in GiNaC (and are
4816 not planned for the near future).
4819 portability: While the GiNaC library itself is designed to avoid any
4820 platform dependent features (it should compile on any ANSI compliant C++
4821 compiler), the currently used version of the CLN library (fast large
4822 integer and arbitrary precision arithmetics) can be compiled only on
4823 systems with a recently new C++ compiler from the GNU Compiler
4824 Collection (@acronym{GCC}).@footnote{This is because CLN uses
4825 PROVIDE/REQUIRE like macros to let the compiler gather all static
4826 initializations, which works for GNU C++ only.} GiNaC uses recent
4827 language features like explicit constructors, mutable members, RTTI,
4828 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
4829 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
4830 ANSI compliant, support all needed features.
4835 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
4836 @c node-name, next, previous, up
4839 Why did we choose to implement GiNaC in C++ instead of Java or any other
4840 language? C++ is not perfect: type checking is not strict (casting is
4841 possible), separation between interface and implementation is not
4842 complete, object oriented design is not enforced. The main reason is
4843 the often scolded feature of operator overloading in C++. While it may
4844 be true that operating on classes with a @code{+} operator is rarely
4845 meaningful, it is perfectly suited for algebraic expressions. Writing
4846 @math{3x+5y} as @code{3*x+5*y} instead of
4847 @code{x.times(3).plus(y.times(5))} looks much more natural.
4848 Furthermore, the main developers are more familiar with C++ than with
4849 any other programming language.
4852 @node Internal Structures, Expressions are reference counted, Why C++? , Top
4853 @c node-name, next, previous, up
4854 @appendix Internal Structures
4857 * Expressions are reference counted::
4858 * Internal representation of products and sums::
4861 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
4862 @c node-name, next, previous, up
4863 @appendixsection Expressions are reference counted
4865 @cindex reference counting
4866 @cindex copy-on-write
4867 @cindex garbage collection
4868 An expression is extremely light-weight since internally it works like a
4869 handle to the actual representation and really holds nothing more than a
4870 pointer to some other object. What this means in practice is that
4871 whenever you create two @code{ex} and set the second equal to the first
4872 no copying process is involved. Instead, the copying takes place as soon
4873 as you try to change the second. Consider the simple sequence of code:
4876 #include <ginac/ginac.h>
4877 using namespace std;
4878 using namespace GiNaC;
4882 symbol x("x"), y("y"), z("z");
4885 e1 = sin(x + 2*y) + 3*z + 41;
4886 e2 = e1; // e2 points to same object as e1
4887 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
4888 e2 += 1; // e2 is copied into a new object
4889 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
4893 The line @code{e2 = e1;} creates a second expression pointing to the
4894 object held already by @code{e1}. The time involved for this operation
4895 is therefore constant, no matter how large @code{e1} was. Actual
4896 copying, however, must take place in the line @code{e2 += 1;} because
4897 @code{e1} and @code{e2} are not handles for the same object any more.
4898 This concept is called @dfn{copy-on-write semantics}. It increases
4899 performance considerably whenever one object occurs multiple times and
4900 represents a simple garbage collection scheme because when an @code{ex}
4901 runs out of scope its destructor checks whether other expressions handle
4902 the object it points to too and deletes the object from memory if that
4903 turns out not to be the case. A slightly less trivial example of
4904 differentiation using the chain-rule should make clear how powerful this
4908 #include <ginac/ginac.h>
4909 using namespace std;
4910 using namespace GiNaC;
4914 symbol x("x"), y("y");
4918 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
4919 cout << e1 << endl // prints x+3*y
4920 << e2 << endl // prints (x+3*y)^3
4921 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
4925 Here, @code{e1} will actually be referenced three times while @code{e2}
4926 will be referenced two times. When the power of an expression is built,
4927 that expression needs not be copied. Likewise, since the derivative of
4928 a power of an expression can be easily expressed in terms of that
4929 expression, no copying of @code{e1} is involved when @code{e3} is
4930 constructed. So, when @code{e3} is constructed it will print as
4931 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
4932 holds a reference to @code{e2} and the factor in front is just
4935 As a user of GiNaC, you cannot see this mechanism of copy-on-write
4936 semantics. When you insert an expression into a second expression, the
4937 result behaves exactly as if the contents of the first expression were
4938 inserted. But it may be useful to remember that this is not what
4939 happens. Knowing this will enable you to write much more efficient
4940 code. If you still have an uncertain feeling with copy-on-write
4941 semantics, we recommend you have a look at the
4942 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
4943 Marshall Cline. Chapter 16 covers this issue and presents an
4944 implementation which is pretty close to the one in GiNaC.
4947 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
4948 @c node-name, next, previous, up
4949 @appendixsection Internal representation of products and sums
4951 @cindex representation
4954 @cindex @code{power}
4955 Although it should be completely transparent for the user of
4956 GiNaC a short discussion of this topic helps to understand the sources
4957 and also explain performance to a large degree. Consider the
4958 unexpanded symbolic expression
4960 $2d^3 \left( 4a + 5b - 3 \right)$
4963 @math{2*d^3*(4*a+5*b-3)}
4965 which could naively be represented by a tree of linear containers for
4966 addition and multiplication, one container for exponentiation with base
4967 and exponent and some atomic leaves of symbols and numbers in this
4972 @cindex pair-wise representation
4973 However, doing so results in a rather deeply nested tree which will
4974 quickly become inefficient to manipulate. We can improve on this by
4975 representing the sum as a sequence of terms, each one being a pair of a
4976 purely numeric multiplicative coefficient and its rest. In the same
4977 spirit we can store the multiplication as a sequence of terms, each
4978 having a numeric exponent and a possibly complicated base, the tree
4979 becomes much more flat:
4983 The number @code{3} above the symbol @code{d} shows that @code{mul}
4984 objects are treated similarly where the coefficients are interpreted as
4985 @emph{exponents} now. Addition of sums of terms or multiplication of
4986 products with numerical exponents can be coded to be very efficient with
4987 such a pair-wise representation. Internally, this handling is performed
4988 by most CAS in this way. It typically speeds up manipulations by an
4989 order of magnitude. The overall multiplicative factor @code{2} and the
4990 additive term @code{-3} look somewhat out of place in this
4991 representation, however, since they are still carrying a trivial
4992 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
4993 this is avoided by adding a field that carries an overall numeric
4994 coefficient. This results in the realistic picture of internal
4997 $2d^3 \left( 4a + 5b - 3 \right)$:
5000 @math{2*d^3*(4*a+5*b-3)}:
5006 This also allows for a better handling of numeric radicals, since
5007 @code{sqrt(2)} can now be carried along calculations. Now it should be
5008 clear, why both classes @code{add} and @code{mul} are derived from the
5009 same abstract class: the data representation is the same, only the
5010 semantics differs. In the class hierarchy, methods for polynomial
5011 expansion and the like are reimplemented for @code{add} and @code{mul},
5012 but the data structure is inherited from @code{expairseq}.
5015 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5016 @c node-name, next, previous, up
5017 @appendix Package Tools
5019 If you are creating a software package that uses the GiNaC library,
5020 setting the correct command line options for the compiler and linker
5021 can be difficult. GiNaC includes two tools to make this process easier.
5024 * ginac-config:: A shell script to detect compiler and linker flags.
5025 * AM_PATH_GINAC:: Macro for GNU automake.
5029 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5030 @c node-name, next, previous, up
5031 @section @command{ginac-config}
5032 @cindex ginac-config
5034 @command{ginac-config} is a shell script that you can use to determine
5035 the compiler and linker command line options required to compile and
5036 link a program with the GiNaC library.
5038 @command{ginac-config} takes the following flags:
5042 Prints out the version of GiNaC installed.
5044 Prints '-I' flags pointing to the installed header files.
5046 Prints out the linker flags necessary to link a program against GiNaC.
5047 @item --prefix[=@var{PREFIX}]
5048 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5049 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5050 Otherwise, prints out the configured value of @env{$prefix}.
5051 @item --exec-prefix[=@var{PREFIX}]
5052 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5053 Otherwise, prints out the configured value of @env{$exec_prefix}.
5056 Typically, @command{ginac-config} will be used within a configure
5057 script, as described below. It, however, can also be used directly from
5058 the command line using backquotes to compile a simple program. For
5062 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5065 This command line might expand to (for example):
5068 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5069 -lginac -lcln -lstdc++
5072 Not only is the form using @command{ginac-config} easier to type, it will
5073 work on any system, no matter how GiNaC was configured.
5076 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5077 @c node-name, next, previous, up
5078 @section @samp{AM_PATH_GINAC}
5079 @cindex AM_PATH_GINAC
5081 For packages configured using GNU automake, GiNaC also provides
5082 a macro to automate the process of checking for GiNaC.
5085 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5093 Determines the location of GiNaC using @command{ginac-config}, which is
5094 either found in the user's path, or from the environment variable
5095 @env{GINACLIB_CONFIG}.
5098 Tests the installed libraries to make sure that their version
5099 is later than @var{MINIMUM-VERSION}. (A default version will be used
5103 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5104 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5105 variable to the output of @command{ginac-config --libs}, and calls
5106 @samp{AC_SUBST()} for these variables so they can be used in generated
5107 makefiles, and then executes @var{ACTION-IF-FOUND}.
5110 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5111 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5115 This macro is in file @file{ginac.m4} which is installed in
5116 @file{$datadir/aclocal}. Note that if automake was installed with a
5117 different @samp{--prefix} than GiNaC, you will either have to manually
5118 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5119 aclocal the @samp{-I} option when running it.
5122 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5123 * Example package:: Example of a package using AM_PATH_GINAC.
5127 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5128 @c node-name, next, previous, up
5129 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5131 Simply make sure that @command{ginac-config} is in your path, and run
5132 the configure script.
5139 The directory where the GiNaC libraries are installed needs
5140 to be found by your system's dynamic linker.
5142 This is generally done by
5145 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5151 setting the environment variable @env{LD_LIBRARY_PATH},
5154 or, as a last resort,
5157 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5158 running configure, for instance:
5161 LDFLAGS=-R/home/cbauer/lib ./configure
5166 You can also specify a @command{ginac-config} not in your path by
5167 setting the @env{GINACLIB_CONFIG} environment variable to the
5168 name of the executable
5171 If you move the GiNaC package from its installed location,
5172 you will either need to modify @command{ginac-config} script
5173 manually to point to the new location or rebuild GiNaC.
5184 --with-ginac-prefix=@var{PREFIX}
5185 --with-ginac-exec-prefix=@var{PREFIX}
5188 are provided to override the prefix and exec-prefix that were stored
5189 in the @command{ginac-config} shell script by GiNaC's configure. You are
5190 generally better off configuring GiNaC with the right path to begin with.
5194 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5195 @c node-name, next, previous, up
5196 @subsection Example of a package using @samp{AM_PATH_GINAC}
5198 The following shows how to build a simple package using automake
5199 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5202 #include <ginac/ginac.h>
5206 GiNaC::symbol x("x");
5207 GiNaC::ex a = GiNaC::sin(x);
5208 std::cout << "Derivative of " << a
5209 << " is " << a.diff(x) << std::endl;
5214 You should first read the introductory portions of the automake
5215 Manual, if you are not already familiar with it.
5217 Two files are needed, @file{configure.in}, which is used to build the
5221 dnl Process this file with autoconf to produce a configure script.
5223 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5229 AM_PATH_GINAC(0.7.0, [
5230 LIBS="$LIBS $GINACLIB_LIBS"
5231 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5232 ], AC_MSG_ERROR([need to have GiNaC installed]))
5237 The only command in this which is not standard for automake
5238 is the @samp{AM_PATH_GINAC} macro.
5240 That command does the following: If a GiNaC version greater or equal
5241 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5242 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5243 the error message `need to have GiNaC installed'
5245 And the @file{Makefile.am}, which will be used to build the Makefile.
5248 ## Process this file with automake to produce Makefile.in
5249 bin_PROGRAMS = simple
5250 simple_SOURCES = simple.cpp
5253 This @file{Makefile.am}, says that we are building a single executable,
5254 from a single sourcefile @file{simple.cpp}. Since every program
5255 we are building uses GiNaC we simply added the GiNaC options
5256 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5257 want to specify them on a per-program basis: for instance by
5261 simple_LDADD = $(GINACLIB_LIBS)
5262 INCLUDES = $(GINACLIB_CPPFLAGS)
5265 to the @file{Makefile.am}.
5267 To try this example out, create a new directory and add the three
5270 Now execute the following commands:
5273 $ automake --add-missing
5278 You now have a package that can be built in the normal fashion
5287 @node Bibliography, Concept Index, Example package, Top
5288 @c node-name, next, previous, up
5289 @appendix Bibliography
5294 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5297 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5300 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5303 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5306 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5307 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5310 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5311 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5312 Academic Press, London
5315 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5320 @node Concept Index, , Bibliography, Top
5321 @c node-name, next, previous, up
5322 @unnumbered Concept Index