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 = "3.14159265358979323846"; // constructor from string
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 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
871 This would, however, call C's built-in operator @code{/} for integers
872 first and result in a numeric holding a plain integer 1. @strong{Never
873 use the operator @code{/} on integers} unless you know exactly what you
874 are doing! Use the constructor from two integers instead, as shown in
875 the example above. Writing @code{numeric(1)/2} may look funny but works
878 @cindex @code{Digits}
880 We have seen now the distinction between exact numbers and floating
881 point numbers. Clearly, the user should never have to worry about
882 dynamically created exact numbers, since their `exactness' always
883 determines how they ought to be handled, i.e. how `long' they are. The
884 situation is different for floating point numbers. Their accuracy is
885 controlled by one @emph{global} variable, called @code{Digits}. (For
886 those readers who know about Maple: it behaves very much like Maple's
887 @code{Digits}). All objects of class numeric that are constructed from
888 then on will be stored with a precision matching that number of decimal
892 #include <ginac/ginac.h>
894 using namespace GiNaC;
898 numeric three(3.0), one(1.0);
899 numeric x = one/three;
901 cout << "in " << Digits << " digits:" << endl;
903 cout << Pi.evalf() << endl;
915 The above example prints the following output to screen:
922 0.333333333333333333333333333333333333333333333333333333333333333333
923 3.14159265358979323846264338327950288419716939937510582097494459231
926 It should be clear that objects of class @code{numeric} should be used
927 for constructing numbers or for doing arithmetic with them. The objects
928 one deals with most of the time are the polymorphic expressions @code{ex}.
930 @subsection Tests on numbers
932 Once you have declared some numbers, assigned them to expressions and
933 done some arithmetic with them it is frequently desired to retrieve some
934 kind of information from them like asking whether that number is
935 integer, rational, real or complex. For those cases GiNaC provides
936 several useful methods. (Internally, they fall back to invocations of
937 certain CLN functions.)
939 As an example, let's construct some rational number, multiply it with
940 some multiple of its denominator and test what comes out:
943 #include <ginac/ginac.h>
945 using namespace GiNaC;
947 // some very important constants:
948 const numeric twentyone(21);
949 const numeric ten(10);
950 const numeric five(5);
954 numeric answer = twentyone;
957 cout << answer.is_integer() << endl; // false, it's 21/5
959 cout << answer.is_integer() << endl; // true, it's 42 now!
963 Note that the variable @code{answer} is constructed here as an integer
964 by @code{numeric}'s copy constructor but in an intermediate step it
965 holds a rational number represented as integer numerator and integer
966 denominator. When multiplied by 10, the denominator becomes unity and
967 the result is automatically converted to a pure integer again.
968 Internally, the underlying @acronym{CLN} is responsible for this
969 behaviour and we refer the reader to @acronym{CLN}'s documentation.
970 Suffice to say that the same behaviour applies to complex numbers as
971 well as return values of certain functions. Complex numbers are
972 automatically converted to real numbers if the imaginary part becomes
973 zero. The full set of tests that can be applied is listed in the
977 @multitable @columnfractions .30 .70
978 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
979 @item @code{.is_zero()}
980 @tab @dots{}equal to zero
981 @item @code{.is_positive()}
982 @tab @dots{}not complex and greater than 0
983 @item @code{.is_integer()}
984 @tab @dots{}a (non-complex) integer
985 @item @code{.is_pos_integer()}
986 @tab @dots{}an integer and greater than 0
987 @item @code{.is_nonneg_integer()}
988 @tab @dots{}an integer and greater equal 0
989 @item @code{.is_even()}
990 @tab @dots{}an even integer
991 @item @code{.is_odd()}
992 @tab @dots{}an odd integer
993 @item @code{.is_prime()}
994 @tab @dots{}a prime integer (probabilistic primality test)
995 @item @code{.is_rational()}
996 @tab @dots{}an exact rational number (integers are rational, too)
997 @item @code{.is_real()}
998 @tab @dots{}a real integer, rational or float (i.e. is not complex)
999 @item @code{.is_cinteger()}
1000 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1001 @item @code{.is_crational()}
1002 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1007 @node Constants, Fundamental containers, Numbers, Basic Concepts
1008 @c node-name, next, previous, up
1010 @cindex @code{constant} (class)
1013 @cindex @code{Catalan}
1014 @cindex @code{Euler}
1015 @cindex @code{evalf()}
1016 Constants behave pretty much like symbols except that they return some
1017 specific number when the method @code{.evalf()} is called.
1019 The predefined known constants are:
1022 @multitable @columnfractions .14 .30 .56
1023 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1025 @tab Archimedes' constant
1026 @tab 3.14159265358979323846264338327950288
1027 @item @code{Catalan}
1028 @tab Catalan's constant
1029 @tab 0.91596559417721901505460351493238411
1031 @tab Euler's (or Euler-Mascheroni) constant
1032 @tab 0.57721566490153286060651209008240243
1037 @node Fundamental containers, Lists, Constants, Basic Concepts
1038 @c node-name, next, previous, up
1039 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1043 @cindex @code{power}
1045 Simple polynomial expressions are written down in GiNaC pretty much like
1046 in other CAS or like expressions involving numerical variables in C.
1047 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1048 been overloaded to achieve this goal. When you run the following
1049 code snippet, the constructor for an object of type @code{mul} is
1050 automatically called to hold the product of @code{a} and @code{b} and
1051 then the constructor for an object of type @code{add} is called to hold
1052 the sum of that @code{mul} object and the number one:
1056 symbol a("a"), b("b");
1061 @cindex @code{pow()}
1062 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1063 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1064 construction is necessary since we cannot safely overload the constructor
1065 @code{^} in C++ to construct a @code{power} object. If we did, it would
1066 have several counterintuitive and undesired effects:
1070 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1072 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1073 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1074 interpret this as @code{x^(a^b)}.
1076 Also, expressions involving integer exponents are very frequently used,
1077 which makes it even more dangerous to overload @code{^} since it is then
1078 hard to distinguish between the semantics as exponentiation and the one
1079 for exclusive or. (It would be embarassing to return @code{1} where one
1080 has requested @code{2^3}.)
1083 @cindex @command{ginsh}
1084 All effects are contrary to mathematical notation and differ from the
1085 way most other CAS handle exponentiation, therefore overloading @code{^}
1086 is ruled out for GiNaC's C++ part. The situation is different in
1087 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1088 that the other frequently used exponentiation operator @code{**} does
1089 not exist at all in C++).
1091 To be somewhat more precise, objects of the three classes described
1092 here, are all containers for other expressions. An object of class
1093 @code{power} is best viewed as a container with two slots, one for the
1094 basis, one for the exponent. All valid GiNaC expressions can be
1095 inserted. However, basic transformations like simplifying
1096 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1097 when this is mathematically possible. If we replace the outer exponent
1098 three in the example by some symbols @code{a}, the simplification is not
1099 safe and will not be performed, since @code{a} might be @code{1/2} and
1102 Objects of type @code{add} and @code{mul} are containers with an
1103 arbitrary number of slots for expressions to be inserted. Again, simple
1104 and safe simplifications are carried out like transforming
1105 @code{3*x+4-x} to @code{2*x+4}.
1107 The general rule is that when you construct such objects, GiNaC
1108 automatically creates them in canonical form, which might differ from
1109 the form you typed in your program. This allows for rapid comparison of
1110 expressions, since after all @code{a-a} is simply zero. Note, that the
1111 canonical form is not necessarily lexicographical ordering or in any way
1112 easily guessable. It is only guaranteed that constructing the same
1113 expression twice, either implicitly or explicitly, results in the same
1117 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1118 @c node-name, next, previous, up
1119 @section Lists of expressions
1120 @cindex @code{lst} (class)
1122 @cindex @code{nops()}
1124 @cindex @code{append()}
1125 @cindex @code{prepend()}
1126 @cindex @code{remove_first()}
1127 @cindex @code{remove_last()}
1129 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1130 expressions. These are sometimes used to supply a variable number of
1131 arguments of the same type to GiNaC methods such as @code{subs()} and
1132 @code{to_rational()}, so you should have a basic understanding about them.
1134 Lists of up to 16 expressions can be directly constructed from single
1139 symbol x("x"), y("y");
1140 lst l(x, 2, y, x+y);
1141 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1145 Use the @code{nops()} method to determine the size (number of expressions) of
1146 a list and the @code{op()} method to access individual elements:
1150 cout << l.nops() << endl; // prints '4'
1151 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1155 You can append or prepend an expression to a list with the @code{append()}
1156 and @code{prepend()} methods:
1160 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1161 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1165 Finally you can remove the first or last element of a list with
1166 @code{remove_first()} and @code{remove_last()}:
1170 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1171 l.remove_last(); // l is now @{x, 2, y, x+y@}
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 (better called @emph{pseudofunctions}) are all objects
1188 of class @code{function}. They accept one or more expressions as
1189 arguments and return one expression. If the arguments are not
1190 numerical, the evaluation of the function may be halted, as it does in
1191 the next example, showing how a function returns itself twice and
1192 finally an expression that may be really useful:
1194 @cindex Gamma function
1195 @cindex @code{subs()}
1198 symbol x("x"), y("y");
1200 cout << tgamma(foo) << endl;
1201 // -> tgamma(x+(1/2)*y)
1202 ex bar = foo.subs(y==1);
1203 cout << tgamma(bar) << endl;
1205 ex foobar = bar.subs(x==7);
1206 cout << tgamma(foobar) << endl;
1207 // -> (135135/128)*Pi^(1/2)
1211 Besides evaluation most of these functions allow differentiation, series
1212 expansion and so on. Read the next chapter in order to learn more about
1215 It must be noted that these pseudofunctions are created by inline
1216 functions, where the argument list is templated. This means that
1217 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1218 @code{sin(ex(1))} and will therefore not result in a floating point
1219 numeber. Unless of course the function prototype is explicitly
1220 overridden -- which is the case for arguments of type @code{numeric}
1221 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1222 point number of class @code{numeric} you should call
1223 @code{sin(numeric(1))}. This is almost the same as calling
1224 @code{sin(1).evalf()} except that the latter will return a numeric
1225 wrapped inside an @code{ex}.
1228 @node Relations, Matrices, Mathematical functions, Basic Concepts
1229 @c node-name, next, previous, up
1231 @cindex @code{relational} (class)
1233 Sometimes, a relation holding between two expressions must be stored
1234 somehow. The class @code{relational} is a convenient container for such
1235 purposes. A relation is by definition a container for two @code{ex} and
1236 a relation between them that signals equality, inequality and so on.
1237 They are created by simply using the C++ operators @code{==}, @code{!=},
1238 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1240 @xref{Mathematical functions}, for examples where various applications
1241 of the @code{.subs()} method show how objects of class relational are
1242 used as arguments. There they provide an intuitive syntax for
1243 substitutions. They are also used as arguments to the @code{ex::series}
1244 method, where the left hand side of the relation specifies the variable
1245 to expand in and the right hand side the expansion point. They can also
1246 be used for creating systems of equations that are to be solved for
1247 unknown variables. But the most common usage of objects of this class
1248 is rather inconspicuous in statements of the form @code{if
1249 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1250 conversion from @code{relational} to @code{bool} takes place. Note,
1251 however, that @code{==} here does not perform any simplifications, hence
1252 @code{expand()} must be called explicitly.
1255 @node Matrices, Indexed objects, Relations, Basic Concepts
1256 @c node-name, next, previous, up
1258 @cindex @code{matrix} (class)
1260 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1261 matrix with @math{m} rows and @math{n} columns are accessed with two
1262 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1263 second one in the range 0@dots{}@math{n-1}.
1265 There are a couple of ways to construct matrices, with or without preset
1269 matrix::matrix(unsigned r, unsigned c);
1270 matrix::matrix(unsigned r, unsigned c, const lst & l);
1271 ex lst_to_matrix(const lst & l);
1272 ex diag_matrix(const lst & l);
1275 The first two functions are @code{matrix} constructors which create a matrix
1276 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1277 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1278 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1279 from a list of lists, each list representing a matrix row. Finally,
1280 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1281 elements. Note that the last two functions return expressions, not matrix
1284 Matrix elements can be accessed and set using the parenthesis (function call)
1288 const ex & matrix::operator()(unsigned r, unsigned c) const;
1289 ex & matrix::operator()(unsigned r, unsigned c);
1292 It is also possible to access the matrix elements in a linear fashion with
1293 the @code{op()} method. But C++-style subscripting with square brackets
1294 @samp{[]} is not available.
1296 Here are a couple of examples that all construct the same 2x2 diagonal
1301 symbol a("a"), b("b");
1309 e = matrix(2, 2, lst(a, 0, 0, b));
1311 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1313 e = diag_matrix(lst(a, b));
1320 @cindex @code{transpose()}
1321 @cindex @code{inverse()}
1322 There are three ways to do arithmetic with matrices. The first (and most
1323 efficient one) is to use the methods provided by the @code{matrix} class:
1326 matrix matrix::add(const matrix & other) const;
1327 matrix matrix::sub(const matrix & other) const;
1328 matrix matrix::mul(const matrix & other) const;
1329 matrix matrix::mul_scalar(const ex & other) const;
1330 matrix matrix::pow(const ex & expn) const;
1331 matrix matrix::transpose(void) const;
1332 matrix matrix::inverse(void) const;
1335 All of these methods return the result as a new matrix object. Here is an
1336 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1341 matrix A(2, 2, lst(1, 2, 3, 4));
1342 matrix B(2, 2, lst(-1, 0, 2, 1));
1343 matrix C(2, 2, lst(8, 4, 2, 1));
1345 matrix result = A.mul(B).sub(C.mul_scalar(2));
1346 cout << result << endl;
1347 // -> [[-13,-6],[1,2]]
1352 @cindex @code{evalm()}
1353 The second (and probably the most natural) way is to construct an expression
1354 containing matrices with the usual arithmetic operators and @code{pow()}.
1355 For efficiency reasons, expressions with sums, products and powers of
1356 matrices are not automatically evaluated in GiNaC. You have to call the
1360 ex ex::evalm() const;
1363 to obtain the result:
1370 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1371 cout << e.evalm() << endl;
1372 // -> [[-13,-6],[1,2]]
1377 The non-commutativity of the product @code{A*B} in this example is
1378 automatically recognized by GiNaC. There is no need to use a special
1379 operator here. @xref{Non-commutative objects}, for more information about
1380 dealing with non-commutative expressions.
1382 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1383 to perform the arithmetic:
1388 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1389 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1391 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1392 cout << e.simplify_indexed() << endl;
1393 // -> [[-13,-6],[1,2]].i.j
1397 Using indices is most useful when working with rectangular matrices and
1398 one-dimensional vectors because you don't have to worry about having to
1399 transpose matrices before multiplying them. @xref{Indexed objects}, for
1400 more information about using matrices with indices, and about indices in
1403 The @code{matrix} class provides a couple of additional methods for
1404 computing determinants, traces, and characteristic polynomials:
1407 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1408 ex matrix::trace(void) const;
1409 ex matrix::charpoly(const symbol & lambda) const;
1412 The @samp{algo} argument of @code{determinant()} allows to select between
1413 different algorithms for calculating the determinant. The possible values
1414 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1415 heuristic to automatically select an algorithm that is likely to give the
1416 result most quickly.
1419 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1420 @c node-name, next, previous, up
1421 @section Indexed objects
1423 GiNaC allows you to handle expressions containing general indexed objects in
1424 arbitrary spaces. It is also able to canonicalize and simplify such
1425 expressions and perform symbolic dummy index summations. There are a number
1426 of predefined indexed objects provided, like delta and metric tensors.
1428 There are few restrictions placed on indexed objects and their indices and
1429 it is easy to construct nonsense expressions, but our intention is to
1430 provide a general framework that allows you to implement algorithms with
1431 indexed quantities, getting in the way as little as possible.
1433 @cindex @code{idx} (class)
1434 @cindex @code{indexed} (class)
1435 @subsection Indexed quantities and their indices
1437 Indexed expressions in GiNaC are constructed of two special types of objects,
1438 @dfn{index objects} and @dfn{indexed objects}.
1442 @cindex contravariant
1445 @item Index objects are of class @code{idx} or a subclass. Every index has
1446 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1447 the index lives in) which can both be arbitrary expressions but are usually
1448 a number or a simple symbol. In addition, indices of class @code{varidx} have
1449 a @dfn{variance} (they can be co- or contravariant), and indices of class
1450 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1452 @item Indexed objects are of class @code{indexed} or a subclass. They
1453 contain a @dfn{base expression} (which is the expression being indexed), and
1454 one or more indices.
1458 @strong{Note:} when printing expressions, covariant indices and indices
1459 without variance are denoted @samp{.i} while contravariant indices are
1460 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1461 value. In the following, we are going to use that notation in the text so
1462 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1463 not visible in the output.
1465 A simple example shall illustrate the concepts:
1468 #include <ginac/ginac.h>
1469 using namespace std;
1470 using namespace GiNaC;
1474 symbol i_sym("i"), j_sym("j");
1475 idx i(i_sym, 3), j(j_sym, 3);
1478 cout << indexed(A, i, j) << endl;
1483 The @code{idx} constructor takes two arguments, the index value and the
1484 index dimension. First we define two index objects, @code{i} and @code{j},
1485 both with the numeric dimension 3. The value of the index @code{i} is the
1486 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1487 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1488 construct an expression containing one indexed object, @samp{A.i.j}. It has
1489 the symbol @code{A} as its base expression and the two indices @code{i} and
1492 Note the difference between the indices @code{i} and @code{j} which are of
1493 class @code{idx}, and the index values which are the sybols @code{i_sym}
1494 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1495 or numbers but must be index objects. For example, the following is not
1496 correct and will raise an exception:
1499 symbol i("i"), j("j");
1500 e = indexed(A, i, j); // ERROR: indices must be of type idx
1503 You can have multiple indexed objects in an expression, index values can
1504 be numeric, and index dimensions symbolic:
1508 symbol B("B"), dim("dim");
1509 cout << 4 * indexed(A, i)
1510 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1515 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1516 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1517 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1518 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1519 @code{simplify_indexed()} for that, see below).
1521 In fact, base expressions, index values and index dimensions can be
1522 arbitrary expressions:
1526 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1531 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1532 get an error message from this but you will probably not be able to do
1533 anything useful with it.
1535 @cindex @code{get_value()}
1536 @cindex @code{get_dimension()}
1540 ex idx::get_value(void);
1541 ex idx::get_dimension(void);
1544 return the value and dimension of an @code{idx} object. If you have an index
1545 in an expression, such as returned by calling @code{.op()} on an indexed
1546 object, you can get a reference to the @code{idx} object with the function
1547 @code{ex_to<idx>()} on the expression.
1549 There are also the methods
1552 bool idx::is_numeric(void);
1553 bool idx::is_symbolic(void);
1554 bool idx::is_dim_numeric(void);
1555 bool idx::is_dim_symbolic(void);
1558 for checking whether the value and dimension are numeric or symbolic
1559 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1560 About Expressions}) returns information about the index value.
1562 @cindex @code{varidx} (class)
1563 If you need co- and contravariant indices, use the @code{varidx} class:
1567 symbol mu_sym("mu"), nu_sym("nu");
1568 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1569 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1571 cout << indexed(A, mu, nu) << endl;
1573 cout << indexed(A, mu_co, nu) << endl;
1575 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1580 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1581 co- or contravariant. The default is a contravariant (upper) index, but
1582 this can be overridden by supplying a third argument to the @code{varidx}
1583 constructor. The two methods
1586 bool varidx::is_covariant(void);
1587 bool varidx::is_contravariant(void);
1590 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1591 to get the object reference from an expression). There's also the very useful
1595 ex varidx::toggle_variance(void);
1598 which makes a new index with the same value and dimension but the opposite
1599 variance. By using it you only have to define the index once.
1601 @cindex @code{spinidx} (class)
1602 The @code{spinidx} class provides dotted and undotted variant indices, as
1603 used in the Weyl-van-der-Waerden spinor formalism:
1607 symbol K("K"), C_sym("C"), D_sym("D");
1608 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1609 // contravariant, undotted
1610 spinidx C_co(C_sym, 2, true); // covariant index
1611 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1612 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1614 cout << indexed(K, C, D) << endl;
1616 cout << indexed(K, C_co, D_dot) << endl;
1618 cout << indexed(K, D_co_dot, D) << endl;
1623 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1624 dotted or undotted. The default is undotted but this can be overridden by
1625 supplying a fourth argument to the @code{spinidx} constructor. The two
1629 bool spinidx::is_dotted(void);
1630 bool spinidx::is_undotted(void);
1633 allow you to check whether or not a @code{spinidx} object is dotted (use
1634 @code{ex_to<spinidx>()} to get the object reference from an expression).
1635 Finally, the two methods
1638 ex spinidx::toggle_dot(void);
1639 ex spinidx::toggle_variance_dot(void);
1642 create a new index with the same value and dimension but opposite dottedness
1643 and the same or opposite variance.
1645 @subsection Substituting indices
1647 @cindex @code{subs()}
1648 Sometimes you will want to substitute one symbolic index with another
1649 symbolic or numeric index, for example when calculating one specific element
1650 of a tensor expression. This is done with the @code{.subs()} method, as it
1651 is done for symbols (see @ref{Substituting Expressions}).
1653 You have two possibilities here. You can either substitute the whole index
1654 by another index or expression:
1658 ex e = indexed(A, mu_co);
1659 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1660 // -> A.mu becomes A~nu
1661 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1662 // -> A.mu becomes A~0
1663 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1664 // -> A.mu becomes A.0
1668 The third example shows that trying to replace an index with something that
1669 is not an index will substitute the index value instead.
1671 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1676 ex e = indexed(A, mu_co);
1677 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1678 // -> A.mu becomes A.nu
1679 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1680 // -> A.mu becomes A.0
1684 As you see, with the second method only the value of the index will get
1685 substituted. Its other properties, including its dimension, remain unchanged.
1686 If you want to change the dimension of an index you have to substitute the
1687 whole index by another one with the new dimension.
1689 Finally, substituting the base expression of an indexed object works as
1694 ex e = indexed(A, mu_co);
1695 cout << e << " becomes " << e.subs(A == A+B) << endl;
1696 // -> A.mu becomes (B+A).mu
1700 @subsection Symmetries
1701 @cindex @code{symmetry} (class)
1702 @cindex @code{sy_none()}
1703 @cindex @code{sy_symm()}
1704 @cindex @code{sy_anti()}
1705 @cindex @code{sy_cycl()}
1707 Indexed objects can have certain symmetry properties with respect to their
1708 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1709 that is constructed with the helper functions
1712 symmetry sy_none(...);
1713 symmetry sy_symm(...);
1714 symmetry sy_anti(...);
1715 symmetry sy_cycl(...);
1718 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1719 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1720 represents a cyclic symmetry. Each of these functions accepts up to four
1721 arguments which can be either symmetry objects themselves or unsigned integer
1722 numbers that represent an index position (counting from 0). A symmetry
1723 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1724 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1727 Here are some examples of symmetry definitions:
1732 e = indexed(A, i, j);
1733 e = indexed(A, sy_none(), i, j); // equivalent
1734 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1736 // Symmetric in all three indices:
1737 e = indexed(A, sy_symm(), i, j, k);
1738 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1739 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1740 // different canonical order
1742 // Symmetric in the first two indices only:
1743 e = indexed(A, sy_symm(0, 1), i, j, k);
1744 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1746 // Antisymmetric in the first and last index only (index ranges need not
1748 e = indexed(A, sy_anti(0, 2), i, j, k);
1749 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1751 // An example of a mixed symmetry: antisymmetric in the first two and
1752 // last two indices, symmetric when swapping the first and last index
1753 // pairs (like the Riemann curvature tensor):
1754 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1756 // Cyclic symmetry in all three indices:
1757 e = indexed(A, sy_cycl(), i, j, k);
1758 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1760 // The following examples are invalid constructions that will throw
1761 // an exception at run time.
1763 // An index may not appear multiple times:
1764 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1765 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1767 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1768 // same number of indices:
1769 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1771 // And of course, you cannot specify indices which are not there:
1772 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1776 If you need to specify more than four indices, you have to use the
1777 @code{.add()} method of the @code{symmetry} class. For example, to specify
1778 full symmetry in the first six indices you would write
1779 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1781 If an indexed object has a symmetry, GiNaC will automatically bring the
1782 indices into a canonical order which allows for some immediate simplifications:
1786 cout << indexed(A, sy_symm(), i, j)
1787 + indexed(A, sy_symm(), j, i) << endl;
1789 cout << indexed(B, sy_anti(), i, j)
1790 + indexed(B, sy_anti(), j, i) << endl;
1792 cout << indexed(B, sy_anti(), i, j, k)
1793 + indexed(B, sy_anti(), j, i, k) << endl;
1798 @cindex @code{get_free_indices()}
1800 @subsection Dummy indices
1802 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1803 that a summation over the index range is implied. Symbolic indices which are
1804 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1805 dummy nor free indices.
1807 To be recognized as a dummy index pair, the two indices must be of the same
1808 class and dimension and their value must be the same single symbol (an index
1809 like @samp{2*n+1} is never a dummy index). If the indices are of class
1810 @code{varidx} they must also be of opposite variance; if they are of class
1811 @code{spinidx} they must be both dotted or both undotted.
1813 The method @code{.get_free_indices()} returns a vector containing the free
1814 indices of an expression. It also checks that the free indices of the terms
1815 of a sum are consistent:
1819 symbol A("A"), B("B"), C("C");
1821 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1822 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1824 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1825 cout << exprseq(e.get_free_indices()) << endl;
1827 // 'j' and 'l' are dummy indices
1829 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1830 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1832 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1833 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1834 cout << exprseq(e.get_free_indices()) << endl;
1836 // 'nu' is a dummy index, but 'sigma' is not
1838 e = indexed(A, mu, mu);
1839 cout << exprseq(e.get_free_indices()) << endl;
1841 // 'mu' is not a dummy index because it appears twice with the same
1844 e = indexed(A, mu, nu) + 42;
1845 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1846 // this will throw an exception:
1847 // "add::get_free_indices: inconsistent indices in sum"
1851 @cindex @code{simplify_indexed()}
1852 @subsection Simplifying indexed expressions
1854 In addition to the few automatic simplifications that GiNaC performs on
1855 indexed expressions (such as re-ordering the indices of symmetric tensors
1856 and calculating traces and convolutions of matrices and predefined tensors)
1860 ex ex::simplify_indexed(void);
1861 ex ex::simplify_indexed(const scalar_products & sp);
1864 that performs some more expensive operations:
1867 @item it checks the consistency of free indices in sums in the same way
1868 @code{get_free_indices()} does
1869 @item it tries to give dumy indices that appear in different terms of a sum
1870 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1871 @item it (symbolically) calculates all possible dummy index summations/contractions
1872 with the predefined tensors (this will be explained in more detail in the
1874 @item it detects contractions that vanish for symmetry reasons, for example
1875 the contraction of a symmetric and a totally antisymmetric tensor
1876 @item as a special case of dummy index summation, it can replace scalar products
1877 of two tensors with a user-defined value
1880 The last point is done with the help of the @code{scalar_products} class
1881 which is used to store scalar products with known values (this is not an
1882 arithmetic class, you just pass it to @code{simplify_indexed()}):
1886 symbol A("A"), B("B"), C("C"), i_sym("i");
1890 sp.add(A, B, 0); // A and B are orthogonal
1891 sp.add(A, C, 0); // A and C are orthogonal
1892 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1894 e = indexed(A + B, i) * indexed(A + C, i);
1896 // -> (B+A).i*(A+C).i
1898 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1904 The @code{scalar_products} object @code{sp} acts as a storage for the
1905 scalar products added to it with the @code{.add()} method. This method
1906 takes three arguments: the two expressions of which the scalar product is
1907 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1908 @code{simplify_indexed()} will replace all scalar products of indexed
1909 objects that have the symbols @code{A} and @code{B} as base expressions
1910 with the single value 0. The number, type and dimension of the indices
1911 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1913 @cindex @code{expand()}
1914 The example above also illustrates a feature of the @code{expand()} method:
1915 if passed the @code{expand_indexed} option it will distribute indices
1916 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1918 @cindex @code{tensor} (class)
1919 @subsection Predefined tensors
1921 Some frequently used special tensors such as the delta, epsilon and metric
1922 tensors are predefined in GiNaC. They have special properties when
1923 contracted with other tensor expressions and some of them have constant
1924 matrix representations (they will evaluate to a number when numeric
1925 indices are specified).
1927 @cindex @code{delta_tensor()}
1928 @subsubsection Delta tensor
1930 The delta tensor takes two indices, is symmetric and has the matrix
1931 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
1932 @code{delta_tensor()}:
1936 symbol A("A"), B("B");
1938 idx i(symbol("i"), 3), j(symbol("j"), 3),
1939 k(symbol("k"), 3), l(symbol("l"), 3);
1941 ex e = indexed(A, i, j) * indexed(B, k, l)
1942 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1943 cout << e.simplify_indexed() << endl;
1946 cout << delta_tensor(i, i) << endl;
1951 @cindex @code{metric_tensor()}
1952 @subsubsection General metric tensor
1954 The function @code{metric_tensor()} creates a general symmetric metric
1955 tensor with two indices that can be used to raise/lower tensor indices. The
1956 metric tensor is denoted as @samp{g} in the output and if its indices are of
1957 mixed variance it is automatically replaced by a delta tensor:
1963 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1965 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1966 cout << e.simplify_indexed() << endl;
1969 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1970 cout << e.simplify_indexed() << endl;
1973 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1974 * metric_tensor(nu, rho);
1975 cout << e.simplify_indexed() << endl;
1978 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1979 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1980 + indexed(A, mu.toggle_variance(), rho));
1981 cout << e.simplify_indexed() << endl;
1986 @cindex @code{lorentz_g()}
1987 @subsubsection Minkowski metric tensor
1989 The Minkowski metric tensor is a special metric tensor with a constant
1990 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1991 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1992 It is created with the function @code{lorentz_g()} (although it is output as
1997 varidx mu(symbol("mu"), 4);
1999 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2000 * lorentz_g(mu, varidx(0, 4)); // negative signature
2001 cout << e.simplify_indexed() << endl;
2004 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2005 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2006 cout << e.simplify_indexed() << endl;
2011 @cindex @code{spinor_metric()}
2012 @subsubsection Spinor metric tensor
2014 The function @code{spinor_metric()} creates an antisymmetric tensor with
2015 two indices that is used to raise/lower indices of 2-component spinors.
2016 It is output as @samp{eps}:
2022 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2023 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2025 e = spinor_metric(A, B) * indexed(psi, B_co);
2026 cout << e.simplify_indexed() << endl;
2029 e = spinor_metric(A, B) * indexed(psi, A_co);
2030 cout << e.simplify_indexed() << endl;
2033 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2034 cout << e.simplify_indexed() << endl;
2037 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2038 cout << e.simplify_indexed() << endl;
2041 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2042 cout << e.simplify_indexed() << endl;
2045 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2046 cout << e.simplify_indexed() << endl;
2051 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2053 @cindex @code{epsilon_tensor()}
2054 @cindex @code{lorentz_eps()}
2055 @subsubsection Epsilon tensor
2057 The epsilon tensor is totally antisymmetric, its number of indices is equal
2058 to the dimension of the index space (the indices must all be of the same
2059 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2060 defined to be 1. Its behaviour with indices that have a variance also
2061 depends on the signature of the metric. Epsilon tensors are output as
2064 There are three functions defined to create epsilon tensors in 2, 3 and 4
2068 ex epsilon_tensor(const ex & i1, const ex & i2);
2069 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2070 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2073 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2074 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2075 Minkowski space (the last @code{bool} argument specifies whether the metric
2076 has negative or positive signature, as in the case of the Minkowski metric
2081 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2082 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2083 e = lorentz_eps(mu, nu, rho, sig) *
2084 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2085 cout << simplify_indexed(e) << endl;
2086 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2088 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2089 symbol A("A"), B("B");
2090 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2091 cout << simplify_indexed(e) << endl;
2092 // -> -B.k*A.j*eps.i.k.j
2093 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2094 cout << simplify_indexed(e) << endl;
2099 @subsection Linear algebra
2101 The @code{matrix} class can be used with indices to do some simple linear
2102 algebra (linear combinations and products of vectors and matrices, traces
2103 and scalar products):
2107 idx i(symbol("i"), 2), j(symbol("j"), 2);
2108 symbol x("x"), y("y");
2110 // A is a 2x2 matrix, X is a 2x1 vector
2111 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2113 cout << indexed(A, i, i) << endl;
2116 ex e = indexed(A, i, j) * indexed(X, j);
2117 cout << e.simplify_indexed() << endl;
2118 // -> [[2*y+x],[4*y+3*x]].i
2120 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2121 cout << e.simplify_indexed() << endl;
2122 // -> [[3*y+3*x,6*y+2*x]].j
2126 You can of course obtain the same results with the @code{matrix::add()},
2127 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2128 but with indices you don't have to worry about transposing matrices.
2130 Matrix indices always start at 0 and their dimension must match the number
2131 of rows/columns of the matrix. Matrices with one row or one column are
2132 vectors and can have one or two indices (it doesn't matter whether it's a
2133 row or a column vector). Other matrices must have two indices.
2135 You should be careful when using indices with variance on matrices. GiNaC
2136 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2137 @samp{F.mu.nu} are different matrices. In this case you should use only
2138 one form for @samp{F} and explicitly multiply it with a matrix representation
2139 of the metric tensor.
2142 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2143 @c node-name, next, previous, up
2144 @section Non-commutative objects
2146 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2147 non-commutative objects are built-in which are mostly of use in high energy
2151 @item Clifford (Dirac) algebra (class @code{clifford})
2152 @item su(3) Lie algebra (class @code{color})
2153 @item Matrices (unindexed) (class @code{matrix})
2156 The @code{clifford} and @code{color} classes are subclasses of
2157 @code{indexed} because the elements of these algebras ususally carry
2158 indices. The @code{matrix} class is described in more detail in
2161 Unlike most computer algebra systems, GiNaC does not primarily provide an
2162 operator (often denoted @samp{&*}) for representing inert products of
2163 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2164 classes of objects involved, and non-commutative products are formed with
2165 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2166 figuring out by itself which objects commute and will group the factors
2167 by their class. Consider this example:
2171 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2172 idx a(symbol("a"), 8), b(symbol("b"), 8);
2173 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2175 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2179 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2180 groups the non-commutative factors (the gammas and the su(3) generators)
2181 together while preserving the order of factors within each class (because
2182 Clifford objects commute with color objects). The resulting expression is a
2183 @emph{commutative} product with two factors that are themselves non-commutative
2184 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2185 parentheses are placed around the non-commutative products in the output.
2187 @cindex @code{ncmul} (class)
2188 Non-commutative products are internally represented by objects of the class
2189 @code{ncmul}, as opposed to commutative products which are handled by the
2190 @code{mul} class. You will normally not have to worry about this distinction,
2193 The advantage of this approach is that you never have to worry about using
2194 (or forgetting to use) a special operator when constructing non-commutative
2195 expressions. Also, non-commutative products in GiNaC are more intelligent
2196 than in other computer algebra systems; they can, for example, automatically
2197 canonicalize themselves according to rules specified in the implementation
2198 of the non-commutative classes. The drawback is that to work with other than
2199 the built-in algebras you have to implement new classes yourself. Symbols
2200 always commute and it's not possible to construct non-commutative products
2201 using symbols to represent the algebra elements or generators. User-defined
2202 functions can, however, be specified as being non-commutative.
2204 @cindex @code{return_type()}
2205 @cindex @code{return_type_tinfo()}
2206 Information about the commutativity of an object or expression can be
2207 obtained with the two member functions
2210 unsigned ex::return_type(void) const;
2211 unsigned ex::return_type_tinfo(void) const;
2214 The @code{return_type()} function returns one of three values (defined in
2215 the header file @file{flags.h}), corresponding to three categories of
2216 expressions in GiNaC:
2219 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2220 classes are of this kind.
2221 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2222 certain class of non-commutative objects which can be determined with the
2223 @code{return_type_tinfo()} method. Expressions of this category commute
2224 with everything except @code{noncommutative} expressions of the same
2226 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2227 of non-commutative objects of different classes. Expressions of this
2228 category don't commute with any other @code{noncommutative} or
2229 @code{noncommutative_composite} expressions.
2232 The value returned by the @code{return_type_tinfo()} method is valid only
2233 when the return type of the expression is @code{noncommutative}. It is a
2234 value that is unique to the class of the object and usually one of the
2235 constants in @file{tinfos.h}, or derived therefrom.
2237 Here are a couple of examples:
2240 @multitable @columnfractions 0.33 0.33 0.34
2241 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2242 @item @code{42} @tab @code{commutative} @tab -
2243 @item @code{2*x-y} @tab @code{commutative} @tab -
2244 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2245 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2246 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2247 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2251 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2252 @code{TINFO_clifford} for objects with a representation label of zero.
2253 Other representation labels yield a different @code{return_type_tinfo()},
2254 but it's the same for any two objects with the same label. This is also true
2257 A last note: With the exception of matrices, positive integer powers of
2258 non-commutative objects are automatically expanded in GiNaC. For example,
2259 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2260 non-commutative expressions).
2263 @cindex @code{clifford} (class)
2264 @subsection Clifford algebra
2266 @cindex @code{dirac_gamma()}
2267 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2268 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2269 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2270 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2273 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2276 which takes two arguments: the index and a @dfn{representation label} in the
2277 range 0 to 255 which is used to distinguish elements of different Clifford
2278 algebras (this is also called a @dfn{spin line index}). Gammas with different
2279 labels commute with each other. The dimension of the index can be 4 or (in
2280 the framework of dimensional regularization) any symbolic value. Spinor
2281 indices on Dirac gammas are not supported in GiNaC.
2283 @cindex @code{dirac_ONE()}
2284 The unity element of a Clifford algebra is constructed by
2287 ex dirac_ONE(unsigned char rl = 0);
2290 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2291 multiples of the unity element, even though it's customary to omit it.
2292 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2293 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2294 GiNaC may produce incorrect results.
2296 @cindex @code{dirac_gamma5()}
2297 There's a special element @samp{gamma5} that commutes with all other
2298 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2302 ex dirac_gamma5(unsigned char rl = 0);
2305 @cindex @code{dirac_gamma6()}
2306 @cindex @code{dirac_gamma7()}
2307 The two additional functions
2310 ex dirac_gamma6(unsigned char rl = 0);
2311 ex dirac_gamma7(unsigned char rl = 0);
2314 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2317 @cindex @code{dirac_slash()}
2318 Finally, the function
2321 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2324 creates a term that represents a contraction of @samp{e} with the Dirac
2325 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2326 with a unique index whose dimension is given by the @code{dim} argument).
2327 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2329 In products of dirac gammas, superfluous unity elements are automatically
2330 removed, squares are replaced by their values and @samp{gamma5} is
2331 anticommuted to the front. The @code{simplify_indexed()} function performs
2332 contractions in gamma strings, for example
2337 symbol a("a"), b("b"), D("D");
2338 varidx mu(symbol("mu"), D);
2339 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2340 * dirac_gamma(mu.toggle_variance());
2342 // -> gamma~mu*a\*gamma.mu
2343 e = e.simplify_indexed();
2346 cout << e.subs(D == 4) << endl;
2352 @cindex @code{dirac_trace()}
2353 To calculate the trace of an expression containing strings of Dirac gammas
2354 you use the function
2357 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2360 This function takes the trace of all gammas with the specified representation
2361 label; gammas with other labels are left standing. The last argument to
2362 @code{dirac_trace()} is the value to be returned for the trace of the unity
2363 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2364 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2365 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2366 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2367 This @samp{gamma5} scheme is described in greater detail in
2368 @cite{The Role of gamma5 in Dimensional Regularization}.
2370 The value of the trace itself is also usually different in 4 and in
2371 @math{D != 4} dimensions:
2376 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2377 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2378 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2379 cout << dirac_trace(e).simplify_indexed() << endl;
2386 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2387 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2388 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2389 cout << dirac_trace(e).simplify_indexed() << endl;
2390 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2394 Here is an example for using @code{dirac_trace()} to compute a value that
2395 appears in the calculation of the one-loop vacuum polarization amplitude in
2400 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2401 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2404 sp.add(l, l, pow(l, 2));
2405 sp.add(l, q, ldotq);
2407 ex e = dirac_gamma(mu) *
2408 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2409 dirac_gamma(mu.toggle_variance()) *
2410 (dirac_slash(l, D) + m * dirac_ONE());
2411 e = dirac_trace(e).simplify_indexed(sp);
2412 e = e.collect(lst(l, ldotq, m));
2414 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2418 The @code{canonicalize_clifford()} function reorders all gamma products that
2419 appear in an expression to a canonical (but not necessarily simple) form.
2420 You can use this to compare two expressions or for further simplifications:
2424 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2425 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2427 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2429 e = canonicalize_clifford(e);
2436 @cindex @code{color} (class)
2437 @subsection Color algebra
2439 @cindex @code{color_T()}
2440 For computations in quantum chromodynamics, GiNaC implements the base elements
2441 and structure constants of the su(3) Lie algebra (color algebra). The base
2442 elements @math{T_a} are constructed by the function
2445 ex color_T(const ex & a, unsigned char rl = 0);
2448 which takes two arguments: the index and a @dfn{representation label} in the
2449 range 0 to 255 which is used to distinguish elements of different color
2450 algebras. Objects with different labels commute with each other. The
2451 dimension of the index must be exactly 8 and it should be of class @code{idx},
2454 @cindex @code{color_ONE()}
2455 The unity element of a color algebra is constructed by
2458 ex color_ONE(unsigned char rl = 0);
2461 @strong{Note:} You must always use @code{color_ONE()} when referring to
2462 multiples of the unity element, even though it's customary to omit it.
2463 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2464 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2465 GiNaC may produce incorrect results.
2467 @cindex @code{color_d()}
2468 @cindex @code{color_f()}
2472 ex color_d(const ex & a, const ex & b, const ex & c);
2473 ex color_f(const ex & a, const ex & b, const ex & c);
2476 create the symmetric and antisymmetric structure constants @math{d_abc} and
2477 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2478 and @math{[T_a, T_b] = i f_abc T_c}.
2480 @cindex @code{color_h()}
2481 There's an additional function
2484 ex color_h(const ex & a, const ex & b, const ex & c);
2487 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2489 The function @code{simplify_indexed()} performs some simplifications on
2490 expressions containing color objects:
2495 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2496 k(symbol("k"), 8), l(symbol("l"), 8);
2498 e = color_d(a, b, l) * color_f(a, b, k);
2499 cout << e.simplify_indexed() << endl;
2502 e = color_d(a, b, l) * color_d(a, b, k);
2503 cout << e.simplify_indexed() << endl;
2506 e = color_f(l, a, b) * color_f(a, b, k);
2507 cout << e.simplify_indexed() << endl;
2510 e = color_h(a, b, c) * color_h(a, b, c);
2511 cout << e.simplify_indexed() << endl;
2514 e = color_h(a, b, c) * color_T(b) * color_T(c);
2515 cout << e.simplify_indexed() << endl;
2518 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2519 cout << e.simplify_indexed() << endl;
2522 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2523 cout << e.simplify_indexed() << endl;
2524 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2528 @cindex @code{color_trace()}
2529 To calculate the trace of an expression containing color objects you use the
2533 ex color_trace(const ex & e, unsigned char rl = 0);
2536 This function takes the trace of all color @samp{T} objects with the
2537 specified representation label; @samp{T}s with other labels are left
2538 standing. For example:
2542 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2544 // -> -I*f.a.c.b+d.a.c.b
2549 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2550 @c node-name, next, previous, up
2551 @chapter Methods and Functions
2554 In this chapter the most important algorithms provided by GiNaC will be
2555 described. Some of them are implemented as functions on expressions,
2556 others are implemented as methods provided by expression objects. If
2557 they are methods, there exists a wrapper function around it, so you can
2558 alternatively call it in a functional way as shown in the simple
2563 cout << "As method: " << sin(1).evalf() << endl;
2564 cout << "As function: " << evalf(sin(1)) << endl;
2568 @cindex @code{subs()}
2569 The general rule is that wherever methods accept one or more parameters
2570 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2571 wrapper accepts is the same but preceded by the object to act on
2572 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2573 most natural one in an OO model but it may lead to confusion for MapleV
2574 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2575 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2576 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2577 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2578 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2579 here. Also, users of MuPAD will in most cases feel more comfortable
2580 with GiNaC's convention. All function wrappers are implemented
2581 as simple inline functions which just call the corresponding method and
2582 are only provided for users uncomfortable with OO who are dead set to
2583 avoid method invocations. Generally, nested function wrappers are much
2584 harder to read than a sequence of methods and should therefore be
2585 avoided if possible. On the other hand, not everything in GiNaC is a
2586 method on class @code{ex} and sometimes calling a function cannot be
2590 * Information About Expressions::
2591 * Substituting Expressions::
2592 * Pattern Matching and Advanced Substitutions::
2593 * Applying a Function on Subexpressions::
2594 * Polynomial Arithmetic:: Working with polynomials.
2595 * Rational Expressions:: Working with rational functions.
2596 * Symbolic Differentiation::
2597 * Series Expansion:: Taylor and Laurent expansion.
2599 * Built-in Functions:: List of predefined mathematical functions.
2600 * Input/Output:: Input and output of expressions.
2604 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2605 @c node-name, next, previous, up
2606 @section Getting information about expressions
2608 @subsection Checking expression types
2609 @cindex @code{is_a<@dots{}>()}
2610 @cindex @code{is_exactly_a<@dots{}>()}
2611 @cindex @code{ex_to<@dots{}>()}
2612 @cindex Converting @code{ex} to other classes
2613 @cindex @code{info()}
2614 @cindex @code{return_type()}
2615 @cindex @code{return_type_tinfo()}
2617 Sometimes it's useful to check whether a given expression is a plain number,
2618 a sum, a polynomial with integer coefficients, or of some other specific type.
2619 GiNaC provides a couple of functions for this:
2622 bool is_a<T>(const ex & e);
2623 bool is_exactly_a<T>(const ex & e);
2624 bool ex::info(unsigned flag);
2625 unsigned ex::return_type(void) const;
2626 unsigned ex::return_type_tinfo(void) const;
2629 When the test made by @code{is_a<T>()} returns true, it is safe to call
2630 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2631 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2632 example, assuming @code{e} is an @code{ex}:
2637 if (is_a<numeric>(e))
2638 numeric n = ex_to<numeric>(e);
2643 @code{is_a<T>(e)} allows you to check whether the top-level object of
2644 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2645 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2646 e.g., for checking whether an expression is a number, a sum, or a product:
2653 is_a<numeric>(e1); // true
2654 is_a<numeric>(e2); // false
2655 is_a<add>(e1); // false
2656 is_a<add>(e2); // true
2657 is_a<mul>(e1); // false
2658 is_a<mul>(e2); // false
2662 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2663 top-level object of an expression @samp{e} is an instance of the GiNaC
2664 class @samp{T}, not including parent classes.
2666 The @code{info()} method is used for checking certain attributes of
2667 expressions. The possible values for the @code{flag} argument are defined
2668 in @file{ginac/flags.h}, the most important being explained in the following
2672 @multitable @columnfractions .30 .70
2673 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2674 @item @code{numeric}
2675 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2677 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2678 @item @code{rational}
2679 @tab @dots{}an exact rational number (integers are rational, too)
2680 @item @code{integer}
2681 @tab @dots{}a (non-complex) integer
2682 @item @code{crational}
2683 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2684 @item @code{cinteger}
2685 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2686 @item @code{positive}
2687 @tab @dots{}not complex and greater than 0
2688 @item @code{negative}
2689 @tab @dots{}not complex and less than 0
2690 @item @code{nonnegative}
2691 @tab @dots{}not complex and greater than or equal to 0
2693 @tab @dots{}an integer greater than 0
2695 @tab @dots{}an integer less than 0
2696 @item @code{nonnegint}
2697 @tab @dots{}an integer greater than or equal to 0
2699 @tab @dots{}an even integer
2701 @tab @dots{}an odd integer
2703 @tab @dots{}a prime integer (probabilistic primality test)
2704 @item @code{relation}
2705 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2706 @item @code{relation_equal}
2707 @tab @dots{}a @code{==} relation
2708 @item @code{relation_not_equal}
2709 @tab @dots{}a @code{!=} relation
2710 @item @code{relation_less}
2711 @tab @dots{}a @code{<} relation
2712 @item @code{relation_less_or_equal}
2713 @tab @dots{}a @code{<=} relation
2714 @item @code{relation_greater}
2715 @tab @dots{}a @code{>} relation
2716 @item @code{relation_greater_or_equal}
2717 @tab @dots{}a @code{>=} relation
2719 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2721 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2722 @item @code{polynomial}
2723 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2724 @item @code{integer_polynomial}
2725 @tab @dots{}a polynomial with (non-complex) integer coefficients
2726 @item @code{cinteger_polynomial}
2727 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2728 @item @code{rational_polynomial}
2729 @tab @dots{}a polynomial with (non-complex) rational coefficients
2730 @item @code{crational_polynomial}
2731 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2732 @item @code{rational_function}
2733 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2734 @item @code{algebraic}
2735 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2739 To determine whether an expression is commutative or non-commutative and if
2740 so, with which other expressions it would commute, you use the methods
2741 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2742 for an explanation of these.
2745 @subsection Accessing subexpressions
2746 @cindex @code{nops()}
2749 @cindex @code{relational} (class)
2751 GiNaC provides the two methods
2754 unsigned ex::nops();
2755 ex ex::op(unsigned i);
2758 for accessing the subexpressions in the container-like GiNaC classes like
2759 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2760 determines the number of subexpressions (@samp{operands}) contained, while
2761 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2762 In the case of a @code{power} object, @code{op(0)} will return the basis
2763 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2764 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2766 The left-hand and right-hand side expressions of objects of class
2767 @code{relational} (and only of these) can also be accessed with the methods
2775 @subsection Comparing expressions
2776 @cindex @code{is_equal()}
2777 @cindex @code{is_zero()}
2779 Expressions can be compared with the usual C++ relational operators like
2780 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2781 the result is usually not determinable and the result will be @code{false},
2782 except in the case of the @code{!=} operator. You should also be aware that
2783 GiNaC will only do the most trivial test for equality (subtracting both
2784 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2787 Actually, if you construct an expression like @code{a == b}, this will be
2788 represented by an object of the @code{relational} class (@pxref{Relations})
2789 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2791 There are also two methods
2794 bool ex::is_equal(const ex & other);
2798 for checking whether one expression is equal to another, or equal to zero,
2801 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2802 GiNaC header files. This method is however only to be used internally by
2803 GiNaC to establish a canonical sort order for terms, and using it to compare
2804 expressions will give very surprising results.
2807 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2808 @c node-name, next, previous, up
2809 @section Substituting expressions
2810 @cindex @code{subs()}
2812 Algebraic objects inside expressions can be replaced with arbitrary
2813 expressions via the @code{.subs()} method:
2816 ex ex::subs(const ex & e);
2817 ex ex::subs(const lst & syms, const lst & repls);
2820 In the first form, @code{subs()} accepts a relational of the form
2821 @samp{object == expression} or a @code{lst} of such relationals:
2825 symbol x("x"), y("y");
2827 ex e1 = 2*x^2-4*x+3;
2828 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2832 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2837 If you specify multiple substitutions, they are performed in parallel, so e.g.
2838 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2840 The second form of @code{subs()} takes two lists, one for the objects to be
2841 replaced and one for the expressions to be substituted (both lists must
2842 contain the same number of elements). Using this form, you would write
2843 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2845 @code{subs()} performs syntactic substitution of any complete algebraic
2846 object; it does not try to match sub-expressions as is demonstrated by the
2851 symbol x("x"), y("y"), z("z");
2853 ex e1 = pow(x+y, 2);
2854 cout << e1.subs(x+y == 4) << endl;
2857 ex e2 = sin(x)*sin(y)*cos(x);
2858 cout << e2.subs(sin(x) == cos(x)) << endl;
2859 // -> cos(x)^2*sin(y)
2862 cout << e3.subs(x+y == 4) << endl;
2864 // (and not 4+z as one might expect)
2868 A more powerful form of substitution using wildcards is described in the
2872 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2873 @c node-name, next, previous, up
2874 @section Pattern matching and advanced substitutions
2875 @cindex @code{wildcard} (class)
2876 @cindex Pattern matching
2878 GiNaC allows the use of patterns for checking whether an expression is of a
2879 certain form or contains subexpressions of a certain form, and for
2880 substituting expressions in a more general way.
2882 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2883 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2884 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2885 an unsigned integer number to allow having multiple different wildcards in a
2886 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2887 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2891 ex wild(unsigned label = 0);
2894 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2897 Some examples for patterns:
2899 @multitable @columnfractions .5 .5
2900 @item @strong{Constructed as} @tab @strong{Output as}
2901 @item @code{wild()} @tab @samp{$0}
2902 @item @code{pow(x,wild())} @tab @samp{x^$0}
2903 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2904 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2910 @item Wildcards behave like symbols and are subject to the same algebraic
2911 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2912 @item As shown in the last example, to use wildcards for indices you have to
2913 use them as the value of an @code{idx} object. This is because indices must
2914 always be of class @code{idx} (or a subclass).
2915 @item Wildcards only represent expressions or subexpressions. It is not
2916 possible to use them as placeholders for other properties like index
2917 dimension or variance, representation labels, symmetry of indexed objects
2919 @item Because wildcards are commutative, it is not possible to use wildcards
2920 as part of noncommutative products.
2921 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2922 are also valid patterns.
2925 @cindex @code{match()}
2926 The most basic application of patterns is to check whether an expression
2927 matches a given pattern. This is done by the function
2930 bool ex::match(const ex & pattern);
2931 bool ex::match(const ex & pattern, lst & repls);
2934 This function returns @code{true} when the expression matches the pattern
2935 and @code{false} if it doesn't. If used in the second form, the actual
2936 subexpressions matched by the wildcards get returned in the @code{repls}
2937 object as a list of relations of the form @samp{wildcard == expression}.
2938 If @code{match()} returns false, the state of @code{repls} is undefined.
2939 For reproducible results, the list should be empty when passed to
2940 @code{match()}, but it is also possible to find similarities in multiple
2941 expressions by passing in the result of a previous match.
2943 The matching algorithm works as follows:
2946 @item A single wildcard matches any expression. If one wildcard appears
2947 multiple times in a pattern, it must match the same expression in all
2948 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2949 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2950 @item If the expression is not of the same class as the pattern, the match
2951 fails (i.e. a sum only matches a sum, a function only matches a function,
2953 @item If the pattern is a function, it only matches the same function
2954 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2955 @item Except for sums and products, the match fails if the number of
2956 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2958 @item If there are no subexpressions, the expressions and the pattern must
2959 be equal (in the sense of @code{is_equal()}).
2960 @item Except for sums and products, each subexpression (@code{op()}) must
2961 match the corresponding subexpression of the pattern.
2964 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2965 account for their commutativity and associativity:
2968 @item If the pattern contains a term or factor that is a single wildcard,
2969 this one is used as the @dfn{global wildcard}. If there is more than one
2970 such wildcard, one of them is chosen as the global wildcard in a random
2972 @item Every term/factor of the pattern, except the global wildcard, is
2973 matched against every term of the expression in sequence. If no match is
2974 found, the whole match fails. Terms that did match are not considered in
2976 @item If there are no unmatched terms left, the match succeeds. Otherwise
2977 the match fails unless there is a global wildcard in the pattern, in
2978 which case this wildcard matches the remaining terms.
2981 In general, having more than one single wildcard as a term of a sum or a
2982 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2985 Here are some examples in @command{ginsh} to demonstrate how it works (the
2986 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2987 match fails, and the list of wildcard replacements otherwise):
2990 > match((x+y)^a,(x+y)^a);
2992 > match((x+y)^a,(x+y)^b);
2994 > match((x+y)^a,$1^$2);
2996 > match((x+y)^a,$1^$1);
2998 > match((x+y)^(x+y),$1^$1);
3000 > match((x+y)^(x+y),$1^$2);
3002 > match((a+b)*(a+c),($1+b)*($1+c));
3004 > match((a+b)*(a+c),(a+$1)*(a+$2));
3006 (Unpredictable. The result might also be [$1==c,$2==b].)
3007 > match((a+b)*(a+c),($1+$2)*($1+$3));
3008 (The result is undefined. Due to the sequential nature of the algorithm
3009 and the re-ordering of terms in GiNaC, the match for the first factor
3010 may be @{$1==a,$2==b@} in which case the match for the second factor
3011 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3013 > match(a*(x+y)+a*z+b,a*$1+$2);
3014 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3015 @{$1=x+y,$2=a*z+b@}.)
3016 > match(a+b+c+d+e+f,c);
3018 > match(a+b+c+d+e+f,c+$0);
3020 > match(a+b+c+d+e+f,c+e+$0);
3022 > match(a+b,a+b+$0);
3024 > match(a*b^2,a^$1*b^$2);
3026 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3027 even though a==a^1.)
3028 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3030 > match(atan2(y,x^2),atan2(y,$0));
3034 @cindex @code{has()}
3035 A more general way to look for patterns in expressions is provided by the
3039 bool ex::has(const ex & pattern);
3042 This function checks whether a pattern is matched by an expression itself or
3043 by any of its subexpressions.
3045 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3046 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3049 > has(x*sin(x+y+2*a),y);
3051 > has(x*sin(x+y+2*a),x+y);
3053 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3054 has the subexpressions "x", "y" and "2*a".)
3055 > has(x*sin(x+y+2*a),x+y+$1);
3057 (But this is possible.)
3058 > has(x*sin(2*(x+y)+2*a),x+y);
3060 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3061 which "x+y" is not a subexpression.)
3064 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3066 > has(4*x^2-x+3,$1*x);
3068 > has(4*x^2+x+3,$1*x);
3070 (Another possible pitfall. The first expression matches because the term
3071 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3072 contains a linear term you should use the coeff() function instead.)
3075 @cindex @code{find()}
3079 bool ex::find(const ex & pattern, lst & found);
3082 works a bit like @code{has()} but it doesn't stop upon finding the first
3083 match. Instead, it appends all found matches to the specified list. If there
3084 are multiple occurrences of the same expression, it is entered only once to
3085 the list. @code{find()} returns false if no matches were found (in
3086 @command{ginsh}, it returns an empty list):
3089 > find(1+x+x^2+x^3,x);
3091 > find(1+x+x^2+x^3,y);
3093 > find(1+x+x^2+x^3,x^$1);
3095 (Note the absence of "x".)
3096 > expand((sin(x)+sin(y))*(a+b));
3097 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3102 @cindex @code{subs()}
3103 Probably the most useful application of patterns is to use them for
3104 substituting expressions with the @code{subs()} method. Wildcards can be
3105 used in the search patterns as well as in the replacement expressions, where
3106 they get replaced by the expressions matched by them. @code{subs()} doesn't
3107 know anything about algebra; it performs purely syntactic substitutions.
3112 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3114 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3116 > subs((a+b+c)^2,a+b=x);
3118 > subs((a+b+c)^2,a+b+$1==x+$1);
3120 > subs(a+2*b,a+b=x);
3122 > subs(4*x^3-2*x^2+5*x-1,x==a);
3124 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3126 > subs(sin(1+sin(x)),sin($1)==cos($1));
3128 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3132 The last example would be written in C++ in this way:
3136 symbol a("a"), b("b"), x("x"), y("y");
3137 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3138 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3139 cout << e.expand() << endl;
3145 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3146 @c node-name, next, previous, up
3147 @section Applying a Function on Subexpressions
3148 @cindex Tree traversal
3149 @cindex @code{map()}
3151 Sometimes you may want to perform an operation on specific parts of an
3152 expression while leaving the general structure of it intact. An example
3153 of this would be a matrix trace operation: the trace of a sum is the sum
3154 of the traces of the individual terms. That is, the trace should @dfn{map}
3155 on the sum, by applying itself to each of the sum's operands. It is possible
3156 to do this manually which usually results in code like this:
3161 if (is_a<matrix>(e))
3162 return ex_to<matrix>(e).trace();
3163 else if (is_a<add>(e)) @{
3165 for (unsigned i=0; i<e.nops(); i++)
3166 sum += calc_trace(e.op(i));
3168 @} else if (is_a<mul>)(e)) @{
3176 This is, however, slightly inefficient (if the sum is very large it can take
3177 a long time to add the terms one-by-one), and its applicability is limited to
3178 a rather small class of expressions. If @code{calc_trace()} is called with
3179 a relation or a list as its argument, you will probably want the trace to
3180 be taken on both sides of the relation or of all elements of the list.
3182 GiNaC offers the @code{map()} method to aid in the implementation of such
3186 static ex ex::map(map_function & f) const;
3187 static ex ex::map(ex (*f)(const ex & e)) const;
3190 In the first (preferred) form, @code{map()} takes a function object that
3191 is subclassed from the @code{map_function} class. In the second form, it
3192 takes a pointer to a function that accepts and returns an expression.
3193 @code{map()} constructs a new expression of the same type, applying the
3194 specified function on all subexpressions (in the sense of @code{op()}),
3197 The use of a function object makes it possible to supply more arguments to
3198 the function that is being mapped, or to keep local state information.
3199 The @code{map_function} class declares a virtual function call operator
3200 that you can overload. Here is a sample implementation of @code{calc_trace()}
3201 that uses @code{map()} in a recursive fashion:
3204 struct calc_trace : public map_function @{
3205 ex operator()(const ex &e)
3207 if (is_a<matrix>(e))
3208 return ex_to<matrix>(e).trace();
3209 else if (is_a<mul>(e)) @{
3212 return e.map(*this);
3217 This function object could then be used like this:
3221 ex M = ... // expression with matrices
3222 calc_trace do_trace;
3223 ex tr = do_trace(M);
3227 Here is another example for you to meditate over. It removes quadratic
3228 terms in a variable from an expanded polynomial:
3231 struct map_rem_quad : public map_function @{
3233 map_rem_quad(const ex & var_) : var(var_) @{@}
3235 ex operator()(const ex & e)
3237 if (is_a<add>(e) || is_a<mul>(e))
3238 return e.map(*this);
3239 else if (is_a<power>(e) && e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3249 symbol x("x"), y("y");
3252 for (int i=0; i<8; i++)
3253 e += pow(x, i) * pow(y, 8-i) * (i+1);
3255 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
3257 map_rem_quad rem_quad(x);
3258 cout << rem_quad(e) << endl;
3259 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3263 @command{ginsh} offers a slightly different implementation of @code{map()}
3264 that allows applying algebraic functions to operands. The second argument
3265 to @code{map()} is an expression containing the wildcard @samp{$0} which
3266 acts as the placeholder for the operands:
3271 > map(a+2*b,sin($0));
3273 > map(@{a,b,c@},$0^2+$0);
3274 @{a^2+a,b^2+b,c^2+c@}
3277 Note that it is only possible to use algebraic functions in the second
3278 argument. You can not use functions like @samp{diff()}, @samp{op()},
3279 @samp{subs()} etc. because these are evaluated immediately:
3282 > map(@{a,b,c@},diff($0,a));
3284 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3285 to "map(@{a,b,c@},0)".
3289 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3290 @c node-name, next, previous, up
3291 @section Polynomial arithmetic
3293 @subsection Expanding and collecting
3294 @cindex @code{expand()}
3295 @cindex @code{collect()}
3297 A polynomial in one or more variables has many equivalent
3298 representations. Some useful ones serve a specific purpose. Consider
3299 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3300 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3301 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3302 representations are the recursive ones where one collects for exponents
3303 in one of the three variable. Since the factors are themselves
3304 polynomials in the remaining two variables the procedure can be
3305 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
3306 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3309 To bring an expression into expanded form, its method
3315 may be called. In our example above, this corresponds to @math{4*x*y +
3316 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3317 GiNaC is not easily guessable you should be prepared to see different
3318 orderings of terms in such sums!
3320 Another useful representation of multivariate polynomials is as a
3321 univariate polynomial in one of the variables with the coefficients
3322 being polynomials in the remaining variables. The method
3323 @code{collect()} accomplishes this task:
3326 ex ex::collect(const ex & s, bool distributed = false);
3329 The first argument to @code{collect()} can also be a list of objects in which
3330 case the result is either a recursively collected polynomial, or a polynomial
3331 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3332 by the @code{distributed} flag.
3334 Note that the original polynomial needs to be in expanded form (for the
3335 variables concerned) in order for @code{collect()} to be able to find the
3336 coefficients properly.
3338 The following @command{ginsh} transcript shows an application of @code{collect()}
3339 together with @code{find()}:
3342 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3343 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
3344 > collect(a,@{p,q@});
3345 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
3346 > collect(a,find(a,sin($1)));
3347 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3348 > collect(a,@{find(a,sin($1)),p,q@});
3349 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3350 > collect(a,@{find(a,sin($1)),d@});
3351 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3354 @subsection Degree and coefficients
3355 @cindex @code{degree()}
3356 @cindex @code{ldegree()}
3357 @cindex @code{coeff()}
3359 The degree and low degree of a polynomial can be obtained using the two
3363 int ex::degree(const ex & s);
3364 int ex::ldegree(const ex & s);
3367 which also work reliably on non-expanded input polynomials (they even work
3368 on rational functions, returning the asymptotic degree). To extract
3369 a coefficient with a certain power from an expanded polynomial you use
3372 ex ex::coeff(const ex & s, int n);
3375 You can also obtain the leading and trailing coefficients with the methods
3378 ex ex::lcoeff(const ex & s);
3379 ex ex::tcoeff(const ex & s);
3382 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3385 An application is illustrated in the next example, where a multivariate
3386 polynomial is analyzed:
3389 #include <ginac/ginac.h>
3390 using namespace std;
3391 using namespace GiNaC;
3395 symbol x("x"), y("y");
3396 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3397 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3398 ex Poly = PolyInp.expand();
3400 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3401 cout << "The x^" << i << "-coefficient is "
3402 << Poly.coeff(x,i) << endl;
3404 cout << "As polynomial in y: "
3405 << Poly.collect(y) << endl;
3409 When run, it returns an output in the following fashion:
3412 The x^0-coefficient is y^2+11*y
3413 The x^1-coefficient is 5*y^2-2*y
3414 The x^2-coefficient is -1
3415 The x^3-coefficient is 4*y
3416 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3419 As always, the exact output may vary between different versions of GiNaC
3420 or even from run to run since the internal canonical ordering is not
3421 within the user's sphere of influence.
3423 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3424 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3425 with non-polynomial expressions as they not only work with symbols but with
3426 constants, functions and indexed objects as well:
3430 symbol a("a"), b("b"), c("c");
3431 idx i(symbol("i"), 3);
3433 ex e = pow(sin(x) - cos(x), 4);
3434 cout << e.degree(cos(x)) << endl;
3436 cout << e.expand().coeff(sin(x), 3) << endl;
3439 e = indexed(a+b, i) * indexed(b+c, i);
3440 e = e.expand(expand_options::expand_indexed);
3441 cout << e.collect(indexed(b, i)) << endl;
3442 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3447 @subsection Polynomial division
3448 @cindex polynomial division
3451 @cindex pseudo-remainder
3452 @cindex @code{quo()}
3453 @cindex @code{rem()}
3454 @cindex @code{prem()}
3455 @cindex @code{divide()}
3460 ex quo(const ex & a, const ex & b, const symbol & x);
3461 ex rem(const ex & a, const ex & b, const symbol & x);
3464 compute the quotient and remainder of univariate polynomials in the variable
3465 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3467 The additional function
3470 ex prem(const ex & a, const ex & b, const symbol & x);
3473 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3474 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3476 Exact division of multivariate polynomials is performed by the function
3479 bool divide(const ex & a, const ex & b, ex & q);
3482 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3483 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3484 in which case the value of @code{q} is undefined.
3487 @subsection Unit, content and primitive part
3488 @cindex @code{unit()}
3489 @cindex @code{content()}
3490 @cindex @code{primpart()}
3495 ex ex::unit(const symbol & x);
3496 ex ex::content(const symbol & x);
3497 ex ex::primpart(const symbol & x);
3500 return the unit part, content part, and primitive polynomial of a multivariate
3501 polynomial with respect to the variable @samp{x} (the unit part being the sign
3502 of the leading coefficient, the content part being the GCD of the coefficients,
3503 and the primitive polynomial being the input polynomial divided by the unit and
3504 content parts). The product of unit, content, and primitive part is the
3505 original polynomial.
3508 @subsection GCD and LCM
3511 @cindex @code{gcd()}
3512 @cindex @code{lcm()}
3514 The functions for polynomial greatest common divisor and least common
3515 multiple have the synopsis
3518 ex gcd(const ex & a, const ex & b);
3519 ex lcm(const ex & a, const ex & b);
3522 The functions @code{gcd()} and @code{lcm()} accept two expressions
3523 @code{a} and @code{b} as arguments and return a new expression, their
3524 greatest common divisor or least common multiple, respectively. If the
3525 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3526 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3529 #include <ginac/ginac.h>
3530 using namespace GiNaC;
3534 symbol x("x"), y("y"), z("z");
3535 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3536 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3538 ex P_gcd = gcd(P_a, P_b);
3540 ex P_lcm = lcm(P_a, P_b);
3541 // 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
3546 @subsection Square-free decomposition
3547 @cindex square-free decomposition
3548 @cindex factorization
3549 @cindex @code{sqrfree()}
3551 GiNaC still lacks proper factorization support. Some form of
3552 factorization is, however, easily implemented by noting that factors
3553 appearing in a polynomial with power two or more also appear in the
3554 derivative and hence can easily be found by computing the GCD of the
3555 original polynomial and its derivatives. Any system has an interface
3556 for this so called square-free factorization. So we provide one, too:
3558 ex sqrfree(const ex & a, const lst & l = lst());
3560 Here is an example that by the way illustrates how the result may depend
3561 on the order of differentiation:
3564 symbol x("x"), y("y");
3565 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3567 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3568 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3570 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3571 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3573 cout << sqrfree(BiVarPol) << endl;
3574 // -> depending on luck, any of the above
3579 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3580 @c node-name, next, previous, up
3581 @section Rational expressions
3583 @subsection The @code{normal} method
3584 @cindex @code{normal()}
3585 @cindex simplification
3586 @cindex temporary replacement
3588 Some basic form of simplification of expressions is called for frequently.
3589 GiNaC provides the method @code{.normal()}, which converts a rational function
3590 into an equivalent rational function of the form @samp{numerator/denominator}
3591 where numerator and denominator are coprime. If the input expression is already
3592 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3593 otherwise it performs fraction addition and multiplication.
3595 @code{.normal()} can also be used on expressions which are not rational functions
3596 as it will replace all non-rational objects (like functions or non-integer
3597 powers) by temporary symbols to bring the expression to the domain of rational
3598 functions before performing the normalization, and re-substituting these
3599 symbols afterwards. This algorithm is also available as a separate method
3600 @code{.to_rational()}, described below.
3602 This means that both expressions @code{t1} and @code{t2} are indeed
3603 simplified in this little program:
3606 #include <ginac/ginac.h>
3607 using namespace GiNaC;
3612 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3613 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3614 std::cout << "t1 is " << t1.normal() << std::endl;
3615 std::cout << "t2 is " << t2.normal() << std::endl;
3619 Of course this works for multivariate polynomials too, so the ratio of
3620 the sample-polynomials from the section about GCD and LCM above would be
3621 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3624 @subsection Numerator and denominator
3627 @cindex @code{numer()}
3628 @cindex @code{denom()}
3629 @cindex @code{numer_denom()}
3631 The numerator and denominator of an expression can be obtained with
3636 ex ex::numer_denom();
3639 These functions will first normalize the expression as described above and
3640 then return the numerator, denominator, or both as a list, respectively.
3641 If you need both numerator and denominator, calling @code{numer_denom()} is
3642 faster than using @code{numer()} and @code{denom()} separately.
3645 @subsection Converting to a rational expression
3646 @cindex @code{to_rational()}
3648 Some of the methods described so far only work on polynomials or rational
3649 functions. GiNaC provides a way to extend the domain of these functions to
3650 general expressions by using the temporary replacement algorithm described
3651 above. You do this by calling
3654 ex ex::to_rational(lst &l);
3657 on the expression to be converted. The supplied @code{lst} will be filled
3658 with the generated temporary symbols and their replacement expressions in
3659 a format that can be used directly for the @code{subs()} method. It can also
3660 already contain a list of replacements from an earlier application of
3661 @code{.to_rational()}, so it's possible to use it on multiple expressions
3662 and get consistent results.
3669 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3670 ex b = sin(x) + cos(x);
3673 divide(a.to_rational(l), b.to_rational(l), q);
3674 cout << q.subs(l) << endl;
3678 will print @samp{sin(x)-cos(x)}.
3681 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3682 @c node-name, next, previous, up
3683 @section Symbolic differentiation
3684 @cindex differentiation
3685 @cindex @code{diff()}
3687 @cindex product rule
3689 GiNaC's objects know how to differentiate themselves. Thus, a
3690 polynomial (class @code{add}) knows that its derivative is the sum of
3691 the derivatives of all the monomials:
3694 #include <ginac/ginac.h>
3695 using namespace GiNaC;
3699 symbol x("x"), y("y"), z("z");
3700 ex P = pow(x, 5) + pow(x, 2) + y;
3702 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3703 cout << P.diff(y) << endl; // 1
3704 cout << P.diff(z) << endl; // 0
3708 If a second integer parameter @var{n} is given, the @code{diff} method
3709 returns the @var{n}th derivative.
3711 If @emph{every} object and every function is told what its derivative
3712 is, all derivatives of composed objects can be calculated using the
3713 chain rule and the product rule. Consider, for instance the expression
3714 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3715 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3716 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3717 out that the composition is the generating function for Euler Numbers,
3718 i.e. the so called @var{n}th Euler number is the coefficient of
3719 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3720 identity to code a function that generates Euler numbers in just three
3723 @cindex Euler numbers
3725 #include <ginac/ginac.h>
3726 using namespace GiNaC;
3728 ex EulerNumber(unsigned n)
3731 const ex generator = pow(cosh(x),-1);
3732 return generator.diff(x,n).subs(x==0);
3737 for (unsigned i=0; i<11; i+=2)
3738 std::cout << EulerNumber(i) << std::endl;
3743 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3744 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3745 @code{i} by two since all odd Euler numbers vanish anyways.
3748 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3749 @c node-name, next, previous, up
3750 @section Series expansion
3751 @cindex @code{series()}
3752 @cindex Taylor expansion
3753 @cindex Laurent expansion
3754 @cindex @code{pseries} (class)
3756 Expressions know how to expand themselves as a Taylor series or (more
3757 generally) a Laurent series. As in most conventional Computer Algebra
3758 Systems, no distinction is made between those two. There is a class of
3759 its own for storing such series (@code{class pseries}) and a built-in
3760 function (called @code{Order}) for storing the order term of the series.
3761 As a consequence, if you want to work with series, i.e. multiply two
3762 series, you need to call the method @code{ex::series} again to convert
3763 it to a series object with the usual structure (expansion plus order
3764 term). A sample application from special relativity could read:
3767 #include <ginac/ginac.h>
3768 using namespace std;
3769 using namespace GiNaC;
3773 symbol v("v"), c("c");
3775 ex gamma = 1/sqrt(1 - pow(v/c,2));
3776 ex mass_nonrel = gamma.series(v==0, 10);
3778 cout << "the relativistic mass increase with v is " << endl
3779 << mass_nonrel << endl;
3781 cout << "the inverse square of this series is " << endl
3782 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3786 Only calling the series method makes the last output simplify to
3787 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3788 series raised to the power @math{-2}.
3790 @cindex M@'echain's formula
3791 As another instructive application, let us calculate the numerical
3792 value of Archimedes' constant
3796 (for which there already exists the built-in constant @code{Pi})
3797 using M@'echain's amazing formula
3799 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3802 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3804 We may expand the arcus tangent around @code{0} and insert the fractions
3805 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3806 carries an order term with it and the question arises what the system is
3807 supposed to do when the fractions are plugged into that order term. The
3808 solution is to use the function @code{series_to_poly()} to simply strip
3812 #include <ginac/ginac.h>
3813 using namespace GiNaC;
3815 ex mechain_pi(int degr)
3818 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3819 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3820 -4*pi_expansion.subs(x==numeric(1,239));
3826 using std::cout; // just for fun, another way of...
3827 using std::endl; // ...dealing with this namespace std.
3829 for (int i=2; i<12; i+=2) @{
3830 pi_frac = mechain_pi(i);
3831 cout << i << ":\t" << pi_frac << endl
3832 << "\t" << pi_frac.evalf() << endl;
3838 Note how we just called @code{.series(x,degr)} instead of
3839 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3840 method @code{series()}: if the first argument is a symbol the expression
3841 is expanded in that symbol around point @code{0}. When you run this
3842 program, it will type out:
3846 3.1832635983263598326
3847 4: 5359397032/1706489875
3848 3.1405970293260603143
3849 6: 38279241713339684/12184551018734375
3850 3.141621029325034425
3851 8: 76528487109180192540976/24359780855939418203125
3852 3.141591772182177295
3853 10: 327853873402258685803048818236/104359128170408663038552734375
3854 3.1415926824043995174
3858 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3859 @c node-name, next, previous, up
3860 @section Symmetrization
3861 @cindex @code{symmetrize()}
3862 @cindex @code{antisymmetrize()}
3863 @cindex @code{symmetrize_cyclic()}
3868 ex ex::symmetrize(const lst & l);
3869 ex ex::antisymmetrize(const lst & l);
3870 ex ex::symmetrize_cyclic(const lst & l);
3873 symmetrize an expression by returning the sum over all symmetric,
3874 antisymmetric or cyclic permutations of the specified list of objects,
3875 weighted by the number of permutations.
3877 The three additional methods
3880 ex ex::symmetrize();
3881 ex ex::antisymmetrize();
3882 ex ex::symmetrize_cyclic();
3885 symmetrize or antisymmetrize an expression over its free indices.
3887 Symmetrization is most useful with indexed expressions but can be used with
3888 almost any kind of object (anything that is @code{subs()}able):
3892 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3893 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3895 cout << indexed(A, i, j).symmetrize() << endl;
3896 // -> 1/2*A.j.i+1/2*A.i.j
3897 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3898 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3899 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3900 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
3905 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3906 @c node-name, next, previous, up
3907 @section Predefined mathematical functions
3909 GiNaC contains the following predefined mathematical functions:
3912 @multitable @columnfractions .30 .70
3913 @item @strong{Name} @tab @strong{Function}
3916 @item @code{csgn(x)}
3918 @item @code{sqrt(x)}
3919 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
3926 @item @code{asin(x)}
3928 @item @code{acos(x)}
3930 @item @code{atan(x)}
3931 @tab inverse tangent
3932 @item @code{atan2(y, x)}
3933 @tab inverse tangent with two arguments
3934 @item @code{sinh(x)}
3935 @tab hyperbolic sine
3936 @item @code{cosh(x)}
3937 @tab hyperbolic cosine
3938 @item @code{tanh(x)}
3939 @tab hyperbolic tangent
3940 @item @code{asinh(x)}
3941 @tab inverse hyperbolic sine
3942 @item @code{acosh(x)}
3943 @tab inverse hyperbolic cosine
3944 @item @code{atanh(x)}
3945 @tab inverse hyperbolic tangent
3947 @tab exponential function
3949 @tab natural logarithm
3952 @item @code{zeta(x)}
3953 @tab Riemann's zeta function
3954 @item @code{zeta(n, x)}
3955 @tab derivatives of Riemann's zeta function
3956 @item @code{tgamma(x)}
3958 @item @code{lgamma(x)}
3959 @tab logarithm of Gamma function
3960 @item @code{beta(x, y)}
3961 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3963 @tab psi (digamma) function
3964 @item @code{psi(n, x)}
3965 @tab derivatives of psi function (polygamma functions)
3966 @item @code{factorial(n)}
3967 @tab factorial function
3968 @item @code{binomial(n, m)}
3969 @tab binomial coefficients
3970 @item @code{Order(x)}
3971 @tab order term function in truncated power series
3976 For functions that have a branch cut in the complex plane GiNaC follows
3977 the conventions for C++ as defined in the ANSI standard as far as
3978 possible. In particular: the natural logarithm (@code{log}) and the
3979 square root (@code{sqrt}) both have their branch cuts running along the
3980 negative real axis where the points on the axis itself belong to the
3981 upper part (i.e. continuous with quadrant II). The inverse
3982 trigonometric and hyperbolic functions are not defined for complex
3983 arguments by the C++ standard, however. In GiNaC we follow the
3984 conventions used by CLN, which in turn follow the carefully designed
3985 definitions in the Common Lisp standard. It should be noted that this
3986 convention is identical to the one used by the C99 standard and by most
3987 serious CAS. It is to be expected that future revisions of the C++
3988 standard incorporate these functions in the complex domain in a manner
3989 compatible with C99.
3992 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3993 @c node-name, next, previous, up
3994 @section Input and output of expressions
3997 @subsection Expression output
3999 @cindex output of expressions
4001 The easiest way to print an expression is to write it to a stream:
4006 ex e = 4.5+pow(x,2)*3/2;
4007 cout << e << endl; // prints '(4.5)+3/2*x^2'
4011 The output format is identical to the @command{ginsh} input syntax and
4012 to that used by most computer algebra systems, but not directly pastable
4013 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4014 is printed as @samp{x^2}).
4016 It is possible to print expressions in a number of different formats with
4020 void ex::print(const print_context & c, unsigned level = 0);
4023 @cindex @code{print_context} (class)
4024 The type of @code{print_context} object passed in determines the format
4025 of the output. The possible types are defined in @file{ginac/print.h}.
4026 All constructors of @code{print_context} and derived classes take an
4027 @code{ostream &} as their first argument.
4029 To print an expression in a way that can be directly used in a C or C++
4030 program, you pass a @code{print_csrc} object like this:
4034 cout << "float f = ";
4035 e.print(print_csrc_float(cout));
4038 cout << "double d = ";
4039 e.print(print_csrc_double(cout));
4042 cout << "cl_N n = ";
4043 e.print(print_csrc_cl_N(cout));
4048 The three possible types mostly affect the way in which floating point
4049 numbers are written.
4051 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4054 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4055 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4056 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4059 The @code{print_context} type @code{print_tree} provides a dump of the
4060 internal structure of an expression for debugging purposes:
4064 e.print(print_tree(cout));
4071 add, hash=0x0, flags=0x3, nops=2
4072 power, hash=0x9, flags=0x3, nops=2
4073 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4074 2 (numeric), hash=0x80000042, flags=0xf
4075 3/2 (numeric), hash=0x80000061, flags=0xf
4078 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4082 This kind of output is also available in @command{ginsh} as the @code{print()}
4085 Another useful output format is for LaTeX parsing in mathematical mode.
4086 It is rather similar to the default @code{print_context} but provides
4087 some braces needed by LaTeX for delimiting boxes and also converts some
4088 common objects to conventional LaTeX names. It is possible to give symbols
4089 a special name for LaTeX output by supplying it as a second argument to
4090 the @code{symbol} constructor.
4092 For example, the code snippet
4097 ex foo = lgamma(x).series(x==0,3);
4098 foo.print(print_latex(std::cout));
4104 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4107 @cindex Tree traversal
4108 If you need any fancy special output format, e.g. for interfacing GiNaC
4109 with other algebra systems or for producing code for different
4110 programming languages, you can always traverse the expression tree yourself:
4113 static void my_print(const ex & e)
4115 if (is_a<function>(e))
4116 cout << ex_to<function>(e).get_name();
4118 cout << e.bp->class_name();
4120 unsigned n = e.nops();
4122 for (unsigned i=0; i<n; i++) @{
4134 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4142 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4143 symbol(y))),numeric(-2)))
4146 If you need an output format that makes it possible to accurately
4147 reconstruct an expression by feeding the output to a suitable parser or
4148 object factory, you should consider storing the expression in an
4149 @code{archive} object and reading the object properties from there.
4150 See the section on archiving for more information.
4153 @subsection Expression input
4154 @cindex input of expressions
4156 GiNaC provides no way to directly read an expression from a stream because
4157 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4158 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4159 @code{y} you defined in your program and there is no way to specify the
4160 desired symbols to the @code{>>} stream input operator.
4162 Instead, GiNaC lets you construct an expression from a string, specifying the
4163 list of symbols to be used:
4167 symbol x("x"), y("y");
4168 ex e("2*x+sin(y)", lst(x, y));
4172 The input syntax is the same as that used by @command{ginsh} and the stream
4173 output operator @code{<<}. The symbols in the string are matched by name to
4174 the symbols in the list and if GiNaC encounters a symbol not specified in
4175 the list it will throw an exception.
4177 With this constructor, it's also easy to implement interactive GiNaC programs:
4182 #include <stdexcept>
4183 #include <ginac/ginac.h>
4184 using namespace std;
4185 using namespace GiNaC;
4192 cout << "Enter an expression containing 'x': ";
4197 cout << "The derivative of " << e << " with respect to x is ";
4198 cout << e.diff(x) << ".\n";
4199 @} catch (exception &p) @{
4200 cerr << p.what() << endl;
4206 @subsection Archiving
4207 @cindex @code{archive} (class)
4210 GiNaC allows creating @dfn{archives} of expressions which can be stored
4211 to or retrieved from files. To create an archive, you declare an object
4212 of class @code{archive} and archive expressions in it, giving each
4213 expression a unique name:
4217 using namespace std;
4218 #include <ginac/ginac.h>
4219 using namespace GiNaC;
4223 symbol x("x"), y("y"), z("z");
4225 ex foo = sin(x + 2*y) + 3*z + 41;
4229 a.archive_ex(foo, "foo");
4230 a.archive_ex(bar, "the second one");
4234 The archive can then be written to a file:
4238 ofstream out("foobar.gar");
4244 The file @file{foobar.gar} contains all information that is needed to
4245 reconstruct the expressions @code{foo} and @code{bar}.
4247 @cindex @command{viewgar}
4248 The tool @command{viewgar} that comes with GiNaC can be used to view
4249 the contents of GiNaC archive files:
4252 $ viewgar foobar.gar
4253 foo = 41+sin(x+2*y)+3*z
4254 the second one = 42+sin(x+2*y)+3*z
4257 The point of writing archive files is of course that they can later be
4263 ifstream in("foobar.gar");
4268 And the stored expressions can be retrieved by their name:
4274 ex ex1 = a2.unarchive_ex(syms, "foo");
4275 ex ex2 = a2.unarchive_ex(syms, "the second one");
4277 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4278 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4279 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4283 Note that you have to supply a list of the symbols which are to be inserted
4284 in the expressions. Symbols in archives are stored by their name only and
4285 if you don't specify which symbols you have, unarchiving the expression will
4286 create new symbols with that name. E.g. if you hadn't included @code{x} in
4287 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4288 have had no effect because the @code{x} in @code{ex1} would have been a
4289 different symbol than the @code{x} which was defined at the beginning of
4290 the program, altough both would appear as @samp{x} when printed.
4292 You can also use the information stored in an @code{archive} object to
4293 output expressions in a format suitable for exact reconstruction. The
4294 @code{archive} and @code{archive_node} classes have a couple of member
4295 functions that let you access the stored properties:
4298 static void my_print2(const archive_node & n)
4301 n.find_string("class", class_name);
4302 cout << class_name << "(";
4304 archive_node::propinfovector p;
4305 n.get_properties(p);
4307 unsigned num = p.size();
4308 for (unsigned i=0; i<num; i++) @{
4309 const string &name = p[i].name;
4310 if (name == "class")
4312 cout << name << "=";
4314 unsigned count = p[i].count;
4318 for (unsigned j=0; j<count; j++) @{
4319 switch (p[i].type) @{
4320 case archive_node::PTYPE_BOOL: @{
4322 n.find_bool(name, x);
4323 cout << (x ? "true" : "false");
4326 case archive_node::PTYPE_UNSIGNED: @{
4328 n.find_unsigned(name, x);
4332 case archive_node::PTYPE_STRING: @{
4334 n.find_string(name, x);
4335 cout << '\"' << x << '\"';
4338 case archive_node::PTYPE_NODE: @{
4339 const archive_node &x = n.find_ex_node(name, j);
4361 ex e = pow(2, x) - y;
4363 my_print2(ar.get_top_node(0)); cout << endl;
4371 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4372 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4373 overall_coeff=numeric(number="0"))
4376 Be warned, however, that the set of properties and their meaning for each
4377 class may change between GiNaC versions.
4380 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4381 @c node-name, next, previous, up
4382 @chapter Extending GiNaC
4384 By reading so far you should have gotten a fairly good understanding of
4385 GiNaC's design-patterns. From here on you should start reading the
4386 sources. All we can do now is issue some recommendations how to tackle
4387 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4388 develop some useful extension please don't hesitate to contact the GiNaC
4389 authors---they will happily incorporate them into future versions.
4392 * What does not belong into GiNaC:: What to avoid.
4393 * Symbolic functions:: Implementing symbolic functions.
4394 * Adding classes:: Defining new algebraic classes.
4398 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4399 @c node-name, next, previous, up
4400 @section What doesn't belong into GiNaC
4402 @cindex @command{ginsh}
4403 First of all, GiNaC's name must be read literally. It is designed to be
4404 a library for use within C++. The tiny @command{ginsh} accompanying
4405 GiNaC makes this even more clear: it doesn't even attempt to provide a
4406 language. There are no loops or conditional expressions in
4407 @command{ginsh}, it is merely a window into the library for the
4408 programmer to test stuff (or to show off). Still, the design of a
4409 complete CAS with a language of its own, graphical capabilites and all
4410 this on top of GiNaC is possible and is without doubt a nice project for
4413 There are many built-in functions in GiNaC that do not know how to
4414 evaluate themselves numerically to a precision declared at runtime
4415 (using @code{Digits}). Some may be evaluated at certain points, but not
4416 generally. This ought to be fixed. However, doing numerical
4417 computations with GiNaC's quite abstract classes is doomed to be
4418 inefficient. For this purpose, the underlying foundation classes
4419 provided by @acronym{CLN} are much better suited.
4422 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4423 @c node-name, next, previous, up
4424 @section Symbolic functions
4426 The easiest and most instructive way to start with is probably to
4427 implement your own function. GiNaC's functions are objects of class
4428 @code{function}. The preprocessor is then used to convert the function
4429 names to objects with a corresponding serial number that is used
4430 internally to identify them. You usually need not worry about this
4431 number. New functions may be inserted into the system via a kind of
4432 `registry'. It is your responsibility to care for some functions that
4433 are called when the user invokes certain methods. These are usual
4434 C++-functions accepting a number of @code{ex} as arguments and returning
4435 one @code{ex}. As an example, if we have a look at a simplified
4436 implementation of the cosine trigonometric function, we first need a
4437 function that is called when one wishes to @code{eval} it. It could
4438 look something like this:
4441 static ex cos_eval_method(const ex & x)
4443 // if (!x%(2*Pi)) return 1
4444 // if (!x%Pi) return -1
4445 // if (!x%Pi/2) return 0
4446 // care for other cases...
4447 return cos(x).hold();
4451 @cindex @code{hold()}
4453 The last line returns @code{cos(x)} if we don't know what else to do and
4454 stops a potential recursive evaluation by saying @code{.hold()}, which
4455 sets a flag to the expression signaling that it has been evaluated. We
4456 should also implement a method for numerical evaluation and since we are
4457 lazy we sweep the problem under the rug by calling someone else's
4458 function that does so, in this case the one in class @code{numeric}:
4461 static ex cos_evalf(const ex & x)
4463 if (is_a<numeric>(x))
4464 return cos(ex_to<numeric>(x));
4466 return cos(x).hold();
4470 Differentiation will surely turn up and so we need to tell @code{cos}
4471 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4472 instance are then handled automatically by @code{basic::diff} and
4476 static ex cos_deriv(const ex & x, unsigned diff_param)
4482 @cindex product rule
4483 The second parameter is obligatory but uninteresting at this point. It
4484 specifies which parameter to differentiate in a partial derivative in
4485 case the function has more than one parameter and its main application
4486 is for correct handling of the chain rule. For Taylor expansion, it is
4487 enough to know how to differentiate. But if the function you want to
4488 implement does have a pole somewhere in the complex plane, you need to
4489 write another method for Laurent expansion around that point.
4491 Now that all the ingredients for @code{cos} have been set up, we need
4492 to tell the system about it. This is done by a macro and we are not
4493 going to descibe how it expands, please consult your preprocessor if you
4497 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4498 evalf_func(cos_evalf).
4499 derivative_func(cos_deriv));
4502 The first argument is the function's name used for calling it and for
4503 output. The second binds the corresponding methods as options to this
4504 object. Options are separated by a dot and can be given in an arbitrary
4505 order. GiNaC functions understand several more options which are always
4506 specified as @code{.option(params)}, for example a method for series
4507 expansion @code{.series_func(cos_series)}. Again, if no series
4508 expansion method is given, GiNaC defaults to simple Taylor expansion,
4509 which is correct if there are no poles involved as is the case for the
4510 @code{cos} function. The way GiNaC handles poles in case there are any
4511 is best understood by studying one of the examples, like the Gamma
4512 (@code{tgamma}) function for instance. (In essence the function first
4513 checks if there is a pole at the evaluation point and falls back to
4514 Taylor expansion if there isn't. Then, the pole is regularized by some
4515 suitable transformation.) Also, the new function needs to be declared
4516 somewhere. This may also be done by a convenient preprocessor macro:
4519 DECLARE_FUNCTION_1P(cos)
4522 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4523 implementation of @code{cos} is very incomplete and lacks several safety
4524 mechanisms. Please, have a look at the real implementation in GiNaC.
4525 (By the way: in case you are worrying about all the macros above we can
4526 assure you that functions are GiNaC's most macro-intense classes. We
4527 have done our best to avoid macros where we can.)
4530 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4531 @c node-name, next, previous, up
4532 @section Adding classes
4534 If you are doing some very specialized things with GiNaC you may find that
4535 you have to implement your own algebraic classes to fit your needs. This
4536 section will explain how to do this by giving the example of a simple
4537 'string' class. After reading this section you will know how to properly
4538 declare a GiNaC class and what the minimum required member functions are
4539 that you have to implement. We only cover the implementation of a 'leaf'
4540 class here (i.e. one that doesn't contain subexpressions). Creating a
4541 container class like, for example, a class representing tensor products is
4542 more involved but this section should give you enough information so you can
4543 consult the source to GiNaC's predefined classes if you want to implement
4544 something more complicated.
4546 @subsection GiNaC's run-time type information system
4548 @cindex hierarchy of classes
4550 All algebraic classes (that is, all classes that can appear in expressions)
4551 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4552 @code{basic *} (which is essentially what an @code{ex} is) represents a
4553 generic pointer to an algebraic class. Occasionally it is necessary to find
4554 out what the class of an object pointed to by a @code{basic *} really is.
4555 Also, for the unarchiving of expressions it must be possible to find the
4556 @code{unarchive()} function of a class given the class name (as a string). A
4557 system that provides this kind of information is called a run-time type
4558 information (RTTI) system. The C++ language provides such a thing (see the
4559 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4560 implements its own, simpler RTTI.
4562 The RTTI in GiNaC is based on two mechanisms:
4567 The @code{basic} class declares a member variable @code{tinfo_key} which
4568 holds an unsigned integer that identifies the object's class. These numbers
4569 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4570 classes. They all start with @code{TINFO_}.
4573 By means of some clever tricks with static members, GiNaC maintains a list
4574 of information for all classes derived from @code{basic}. The information
4575 available includes the class names, the @code{tinfo_key}s, and pointers
4576 to the unarchiving functions. This class registry is defined in the
4577 @file{registrar.h} header file.
4581 The disadvantage of this proprietary RTTI implementation is that there's
4582 a little more to do when implementing new classes (C++'s RTTI works more
4583 or less automatic) but don't worry, most of the work is simplified by
4586 @subsection A minimalistic example
4588 Now we will start implementing a new class @code{mystring} that allows
4589 placing character strings in algebraic expressions (this is not very useful,
4590 but it's just an example). This class will be a direct subclass of
4591 @code{basic}. You can use this sample implementation as a starting point
4592 for your own classes.
4594 The code snippets given here assume that you have included some header files
4600 #include <stdexcept>
4601 using namespace std;
4603 #include <ginac/ginac.h>
4604 using namespace GiNaC;
4607 The first thing we have to do is to define a @code{tinfo_key} for our new
4608 class. This can be any arbitrary unsigned number that is not already taken
4609 by one of the existing classes but it's better to come up with something
4610 that is unlikely to clash with keys that might be added in the future. The
4611 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4612 which is not a requirement but we are going to stick with this scheme:
4615 const unsigned TINFO_mystring = 0x42420001U;
4618 Now we can write down the class declaration. The class stores a C++
4619 @code{string} and the user shall be able to construct a @code{mystring}
4620 object from a C or C++ string:
4623 class mystring : public basic
4625 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4628 mystring(const string &s);
4629 mystring(const char *s);
4635 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4638 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4639 macros are defined in @file{registrar.h}. They take the name of the class
4640 and its direct superclass as arguments and insert all required declarations
4641 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4642 the first line after the opening brace of the class definition. The
4643 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4644 source (at global scope, of course, not inside a function).
4646 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4647 declarations of the default and copy constructor, the destructor, the
4648 assignment operator and a couple of other functions that are required. It
4649 also defines a type @code{inherited} which refers to the superclass so you
4650 don't have to modify your code every time you shuffle around the class
4651 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4652 constructor, the destructor and the assignment operator.
4654 Now there are nine member functions we have to implement to get a working
4660 @code{mystring()}, the default constructor.
4663 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4664 assignment operator to free dynamically allocated members. The @code{call_parent}
4665 specifies whether the @code{destroy()} function of the superclass is to be
4669 @code{void copy(const mystring &other)}, which is used in the copy constructor
4670 and assignment operator to copy the member variables over from another
4671 object of the same class.
4674 @code{void archive(archive_node &n)}, the archiving function. This stores all
4675 information needed to reconstruct an object of this class inside an
4676 @code{archive_node}.
4679 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4680 constructor. This constructs an instance of the class from the information
4681 found in an @code{archive_node}.
4684 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4685 unarchiving function. It constructs a new instance by calling the unarchiving
4689 @code{int compare_same_type(const basic &other)}, which is used internally
4690 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4691 -1, depending on the relative order of this object and the @code{other}
4692 object. If it returns 0, the objects are considered equal.
4693 @strong{Note:} This has nothing to do with the (numeric) ordering
4694 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4695 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4696 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4697 must provide a @code{compare_same_type()} function, even those representing
4698 objects for which no reasonable algebraic ordering relationship can be
4702 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4703 which are the two constructors we declared.
4707 Let's proceed step-by-step. The default constructor looks like this:
4710 mystring::mystring() : inherited(TINFO_mystring)
4712 // dynamically allocate resources here if required
4716 The golden rule is that in all constructors you have to set the
4717 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4718 it will be set by the constructor of the superclass and all hell will break
4719 loose in the RTTI. For your convenience, the @code{basic} class provides
4720 a constructor that takes a @code{tinfo_key} value, which we are using here
4721 (remember that in our case @code{inherited = basic}). If the superclass
4722 didn't have such a constructor, we would have to set the @code{tinfo_key}
4723 to the right value manually.
4725 In the default constructor you should set all other member variables to
4726 reasonable default values (we don't need that here since our @code{str}
4727 member gets set to an empty string automatically). The constructor(s) are of
4728 course also the right place to allocate any dynamic resources you require.
4730 Next, the @code{destroy()} function:
4733 void mystring::destroy(bool call_parent)
4735 // free dynamically allocated resources here if required
4737 inherited::destroy(call_parent);
4741 This function is where we free all dynamically allocated resources. We don't
4742 have any so we're not doing anything here, but if we had, for example, used
4743 a C-style @code{char *} to store our string, this would be the place to
4744 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4745 to call the @code{destroy()} function of the superclass after we're done
4746 (to mimic C++'s automatic invocation of superclass destructors where
4747 @code{destroy()} is called from outside a destructor).
4749 The @code{copy()} function just copies over the member variables from
4753 void mystring::copy(const mystring &other)
4755 inherited::copy(other);
4760 We can simply overwrite the member variables here. There's no need to worry
4761 about dynamically allocated storage. The assignment operator (which is
4762 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4763 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4764 explicitly call the @code{copy()} function of the superclass here so
4765 all the member variables will get copied.
4767 Next are the three functions for archiving. You have to implement them even
4768 if you don't plan to use archives, but the minimum required implementation
4769 is really simple. First, the archiving function:
4772 void mystring::archive(archive_node &n) const
4774 inherited::archive(n);
4775 n.add_string("string", str);
4779 The only thing that is really required is calling the @code{archive()}
4780 function of the superclass. Optionally, you can store all information you
4781 deem necessary for representing the object into the passed
4782 @code{archive_node}. We are just storing our string here. For more
4783 information on how the archiving works, consult the @file{archive.h} header
4786 The unarchiving constructor is basically the inverse of the archiving
4790 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4792 n.find_string("string", str);
4796 If you don't need archiving, just leave this function empty (but you must
4797 invoke the unarchiving constructor of the superclass). Note that we don't
4798 have to set the @code{tinfo_key} here because it is done automatically
4799 by the unarchiving constructor of the @code{basic} class.
4801 Finally, the unarchiving function:
4804 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4806 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4810 You don't have to understand how exactly this works. Just copy these four
4811 lines into your code literally (replacing the class name, of course). It
4812 calls the unarchiving constructor of the class and unless you are doing
4813 something very special (like matching @code{archive_node}s to global
4814 objects) you don't need a different implementation. For those who are
4815 interested: setting the @code{dynallocated} flag puts the object under
4816 the control of GiNaC's garbage collection. It will get deleted automatically
4817 once it is no longer referenced.
4819 Our @code{compare_same_type()} function uses a provided function to compare
4823 int mystring::compare_same_type(const basic &other) const
4825 const mystring &o = static_cast<const mystring &>(other);
4826 int cmpval = str.compare(o.str);
4829 else if (cmpval < 0)
4836 Although this function takes a @code{basic &}, it will always be a reference
4837 to an object of exactly the same class (objects of different classes are not
4838 comparable), so the cast is safe. If this function returns 0, the two objects
4839 are considered equal (in the sense that @math{A-B=0}), so you should compare
4840 all relevant member variables.
4842 Now the only thing missing is our two new constructors:
4845 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4847 // dynamically allocate resources here if required
4850 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4852 // dynamically allocate resources here if required
4856 No surprises here. We set the @code{str} member from the argument and
4857 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4859 That's it! We now have a minimal working GiNaC class that can store
4860 strings in algebraic expressions. Let's confirm that the RTTI works:
4863 ex e = mystring("Hello, world!");
4864 cout << is_a<mystring>(e) << endl;
4867 cout << e.bp->class_name() << endl;
4871 Obviously it does. Let's see what the expression @code{e} looks like:
4875 // -> [mystring object]
4878 Hm, not exactly what we expect, but of course the @code{mystring} class
4879 doesn't yet know how to print itself. This is done in the @code{print()}
4880 member function. Let's say that we wanted to print the string surrounded
4884 class mystring : public basic
4888 void print(const print_context &c, unsigned level = 0) const;
4892 void mystring::print(const print_context &c, unsigned level) const
4894 // print_context::s is a reference to an ostream
4895 c.s << '\"' << str << '\"';
4899 The @code{level} argument is only required for container classes to
4900 correctly parenthesize the output. Let's try again to print the expression:
4904 // -> "Hello, world!"
4907 Much better. The @code{mystring} class can be used in arbitrary expressions:
4910 e += mystring("GiNaC rulez");
4912 // -> "GiNaC rulez"+"Hello, world!"
4915 (GiNaC's automatic term reordering is in effect here), or even
4918 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4920 // -> "One string"^(2*sin(-"Another string"+Pi))
4923 Whether this makes sense is debatable but remember that this is only an
4924 example. At least it allows you to implement your own symbolic algorithms
4927 Note that GiNaC's algebraic rules remain unchanged:
4930 e = mystring("Wow") * mystring("Wow");
4934 e = pow(mystring("First")-mystring("Second"), 2);
4935 cout << e.expand() << endl;
4936 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4939 There's no way to, for example, make GiNaC's @code{add} class perform string
4940 concatenation. You would have to implement this yourself.
4942 @subsection Automatic evaluation
4944 @cindex @code{hold()}
4946 When dealing with objects that are just a little more complicated than the
4947 simple string objects we have implemented, chances are that you will want to
4948 have some automatic simplifications or canonicalizations performed on them.
4949 This is done in the evaluation member function @code{eval()}. Let's say that
4950 we wanted all strings automatically converted to lowercase with
4951 non-alphabetic characters stripped, and empty strings removed:
4954 class mystring : public basic
4958 ex eval(int level = 0) const;
4962 ex mystring::eval(int level) const
4965 for (int i=0; i<str.length(); i++) @{
4967 if (c >= 'A' && c <= 'Z')
4968 new_str += tolower(c);
4969 else if (c >= 'a' && c <= 'z')
4973 if (new_str.length() == 0)
4976 return mystring(new_str).hold();
4980 The @code{level} argument is used to limit the recursion depth of the
4981 evaluation. We don't have any subexpressions in the @code{mystring} class
4982 so we are not concerned with this. If we had, we would call the @code{eval()}
4983 functions of the subexpressions with @code{level - 1} as the argument if
4984 @code{level != 1}. The @code{hold()} member function sets a flag in the
4985 object that prevents further evaluation. Otherwise we might end up in an
4986 endless loop. When you want to return the object unmodified, use
4987 @code{return this->hold();}.
4989 Let's confirm that it works:
4992 ex e = mystring("Hello, world!") + mystring("!?#");
4996 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5001 @subsection Other member functions
5003 We have implemented only a small set of member functions to make the class
5004 work in the GiNaC framework. For a real algebraic class, there are probably
5005 some more functions that you will want to re-implement, such as
5006 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5007 or the header file of the class you want to make a subclass of to see
5008 what's there. One member function that you will most likely want to
5009 implement for terminal classes like the described string class is
5010 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5011 which will allow GiNaC to compare and canonicalize expressions much more
5014 You can, of course, also add your own new member functions. Remember,
5015 that the RTTI may be used to get information about what kinds of objects
5016 you are dealing with (the position in the class hierarchy) and that you
5017 can always extract the bare object from an @code{ex} by stripping the
5018 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5019 should become a need.
5021 That's it. May the source be with you!
5024 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5025 @c node-name, next, previous, up
5026 @chapter A Comparison With Other CAS
5029 This chapter will give you some information on how GiNaC compares to
5030 other, traditional Computer Algebra Systems, like @emph{Maple},
5031 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5032 disadvantages over these systems.
5035 * Advantages:: Stengths of the GiNaC approach.
5036 * Disadvantages:: Weaknesses of the GiNaC approach.
5037 * Why C++?:: Attractiveness of C++.
5040 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5041 @c node-name, next, previous, up
5044 GiNaC has several advantages over traditional Computer
5045 Algebra Systems, like
5050 familiar language: all common CAS implement their own proprietary
5051 grammar which you have to learn first (and maybe learn again when your
5052 vendor decides to `enhance' it). With GiNaC you can write your program
5053 in common C++, which is standardized.
5057 structured data types: you can build up structured data types using
5058 @code{struct}s or @code{class}es together with STL features instead of
5059 using unnamed lists of lists of lists.
5062 strongly typed: in CAS, you usually have only one kind of variables
5063 which can hold contents of an arbitrary type. This 4GL like feature is
5064 nice for novice programmers, but dangerous.
5067 development tools: powerful development tools exist for C++, like fancy
5068 editors (e.g. with automatic indentation and syntax highlighting),
5069 debuggers, visualization tools, documentation generators@dots{}
5072 modularization: C++ programs can easily be split into modules by
5073 separating interface and implementation.
5076 price: GiNaC is distributed under the GNU Public License which means
5077 that it is free and available with source code. And there are excellent
5078 C++-compilers for free, too.
5081 extendable: you can add your own classes to GiNaC, thus extending it on
5082 a very low level. Compare this to a traditional CAS that you can
5083 usually only extend on a high level by writing in the language defined
5084 by the parser. In particular, it turns out to be almost impossible to
5085 fix bugs in a traditional system.
5088 multiple interfaces: Though real GiNaC programs have to be written in
5089 some editor, then be compiled, linked and executed, there are more ways
5090 to work with the GiNaC engine. Many people want to play with
5091 expressions interactively, as in traditional CASs. Currently, two such
5092 windows into GiNaC have been implemented and many more are possible: the
5093 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5094 types to a command line and second, as a more consistent approach, an
5095 interactive interface to the @acronym{Cint} C++ interpreter has been put
5096 together (called @acronym{GiNaC-cint}) that allows an interactive
5097 scripting interface consistent with the C++ language.
5100 seemless integration: it is somewhere between difficult and impossible
5101 to call CAS functions from within a program written in C++ or any other
5102 programming language and vice versa. With GiNaC, your symbolic routines
5103 are part of your program. You can easily call third party libraries,
5104 e.g. for numerical evaluation or graphical interaction. All other
5105 approaches are much more cumbersome: they range from simply ignoring the
5106 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5107 system (i.e. @emph{Yacas}).
5110 efficiency: often large parts of a program do not need symbolic
5111 calculations at all. Why use large integers for loop variables or
5112 arbitrary precision arithmetics where @code{int} and @code{double} are
5113 sufficient? For pure symbolic applications, GiNaC is comparable in
5114 speed with other CAS.
5119 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5120 @c node-name, next, previous, up
5121 @section Disadvantages
5123 Of course it also has some disadvantages:
5128 advanced features: GiNaC cannot compete with a program like
5129 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5130 which grows since 1981 by the work of dozens of programmers, with
5131 respect to mathematical features. Integration, factorization,
5132 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5133 not planned for the near future).
5136 portability: While the GiNaC library itself is designed to avoid any
5137 platform dependent features (it should compile on any ANSI compliant C++
5138 compiler), the currently used version of the CLN library (fast large
5139 integer and arbitrary precision arithmetics) can be compiled only on
5140 systems with a recently new C++ compiler from the GNU Compiler
5141 Collection (@acronym{GCC}).@footnote{This is because CLN uses
5142 PROVIDE/REQUIRE like macros to let the compiler gather all static
5143 initializations, which works for GNU C++ only.} GiNaC uses recent
5144 language features like explicit constructors, mutable members, RTTI,
5145 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
5146 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
5147 ANSI compliant, support all needed features.
5152 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5153 @c node-name, next, previous, up
5156 Why did we choose to implement GiNaC in C++ instead of Java or any other
5157 language? C++ is not perfect: type checking is not strict (casting is
5158 possible), separation between interface and implementation is not
5159 complete, object oriented design is not enforced. The main reason is
5160 the often scolded feature of operator overloading in C++. While it may
5161 be true that operating on classes with a @code{+} operator is rarely
5162 meaningful, it is perfectly suited for algebraic expressions. Writing
5163 @math{3x+5y} as @code{3*x+5*y} instead of
5164 @code{x.times(3).plus(y.times(5))} looks much more natural.
5165 Furthermore, the main developers are more familiar with C++ than with
5166 any other programming language.
5169 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5170 @c node-name, next, previous, up
5171 @appendix Internal Structures
5174 * Expressions are reference counted::
5175 * Internal representation of products and sums::
5178 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5179 @c node-name, next, previous, up
5180 @appendixsection Expressions are reference counted
5182 @cindex reference counting
5183 @cindex copy-on-write
5184 @cindex garbage collection
5185 An expression is extremely light-weight since internally it works like a
5186 handle to the actual representation and really holds nothing more than a
5187 pointer to some other object. What this means in practice is that
5188 whenever you create two @code{ex} and set the second equal to the first
5189 no copying process is involved. Instead, the copying takes place as soon
5190 as you try to change the second. Consider the simple sequence of code:
5193 #include <ginac/ginac.h>
5194 using namespace std;
5195 using namespace GiNaC;
5199 symbol x("x"), y("y"), z("z");
5202 e1 = sin(x + 2*y) + 3*z + 41;
5203 e2 = e1; // e2 points to same object as e1
5204 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5205 e2 += 1; // e2 is copied into a new object
5206 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5210 The line @code{e2 = e1;} creates a second expression pointing to the
5211 object held already by @code{e1}. The time involved for this operation
5212 is therefore constant, no matter how large @code{e1} was. Actual
5213 copying, however, must take place in the line @code{e2 += 1;} because
5214 @code{e1} and @code{e2} are not handles for the same object any more.
5215 This concept is called @dfn{copy-on-write semantics}. It increases
5216 performance considerably whenever one object occurs multiple times and
5217 represents a simple garbage collection scheme because when an @code{ex}
5218 runs out of scope its destructor checks whether other expressions handle
5219 the object it points to too and deletes the object from memory if that
5220 turns out not to be the case. A slightly less trivial example of
5221 differentiation using the chain-rule should make clear how powerful this
5225 #include <ginac/ginac.h>
5226 using namespace std;
5227 using namespace GiNaC;
5231 symbol x("x"), y("y");
5235 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5236 cout << e1 << endl // prints x+3*y
5237 << e2 << endl // prints (x+3*y)^3
5238 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5242 Here, @code{e1} will actually be referenced three times while @code{e2}
5243 will be referenced two times. When the power of an expression is built,
5244 that expression needs not be copied. Likewise, since the derivative of
5245 a power of an expression can be easily expressed in terms of that
5246 expression, no copying of @code{e1} is involved when @code{e3} is
5247 constructed. So, when @code{e3} is constructed it will print as
5248 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5249 holds a reference to @code{e2} and the factor in front is just
5252 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5253 semantics. When you insert an expression into a second expression, the
5254 result behaves exactly as if the contents of the first expression were
5255 inserted. But it may be useful to remember that this is not what
5256 happens. Knowing this will enable you to write much more efficient
5257 code. If you still have an uncertain feeling with copy-on-write
5258 semantics, we recommend you have a look at the
5259 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5260 Marshall Cline. Chapter 16 covers this issue and presents an
5261 implementation which is pretty close to the one in GiNaC.
5264 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5265 @c node-name, next, previous, up
5266 @appendixsection Internal representation of products and sums
5268 @cindex representation
5271 @cindex @code{power}
5272 Although it should be completely transparent for the user of
5273 GiNaC a short discussion of this topic helps to understand the sources
5274 and also explain performance to a large degree. Consider the
5275 unexpanded symbolic expression
5277 $2d^3 \left( 4a + 5b - 3 \right)$
5280 @math{2*d^3*(4*a+5*b-3)}
5282 which could naively be represented by a tree of linear containers for
5283 addition and multiplication, one container for exponentiation with base
5284 and exponent and some atomic leaves of symbols and numbers in this
5289 @cindex pair-wise representation
5290 However, doing so results in a rather deeply nested tree which will
5291 quickly become inefficient to manipulate. We can improve on this by
5292 representing the sum as a sequence of terms, each one being a pair of a
5293 purely numeric multiplicative coefficient and its rest. In the same
5294 spirit we can store the multiplication as a sequence of terms, each
5295 having a numeric exponent and a possibly complicated base, the tree
5296 becomes much more flat:
5300 The number @code{3} above the symbol @code{d} shows that @code{mul}
5301 objects are treated similarly where the coefficients are interpreted as
5302 @emph{exponents} now. Addition of sums of terms or multiplication of
5303 products with numerical exponents can be coded to be very efficient with
5304 such a pair-wise representation. Internally, this handling is performed
5305 by most CAS in this way. It typically speeds up manipulations by an
5306 order of magnitude. The overall multiplicative factor @code{2} and the
5307 additive term @code{-3} look somewhat out of place in this
5308 representation, however, since they are still carrying a trivial
5309 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5310 this is avoided by adding a field that carries an overall numeric
5311 coefficient. This results in the realistic picture of internal
5314 $2d^3 \left( 4a + 5b - 3 \right)$:
5317 @math{2*d^3*(4*a+5*b-3)}:
5323 This also allows for a better handling of numeric radicals, since
5324 @code{sqrt(2)} can now be carried along calculations. Now it should be
5325 clear, why both classes @code{add} and @code{mul} are derived from the
5326 same abstract class: the data representation is the same, only the
5327 semantics differs. In the class hierarchy, methods for polynomial
5328 expansion and the like are reimplemented for @code{add} and @code{mul},
5329 but the data structure is inherited from @code{expairseq}.
5332 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5333 @c node-name, next, previous, up
5334 @appendix Package Tools
5336 If you are creating a software package that uses the GiNaC library,
5337 setting the correct command line options for the compiler and linker
5338 can be difficult. GiNaC includes two tools to make this process easier.
5341 * ginac-config:: A shell script to detect compiler and linker flags.
5342 * AM_PATH_GINAC:: Macro for GNU automake.
5346 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5347 @c node-name, next, previous, up
5348 @section @command{ginac-config}
5349 @cindex ginac-config
5351 @command{ginac-config} is a shell script that you can use to determine
5352 the compiler and linker command line options required to compile and
5353 link a program with the GiNaC library.
5355 @command{ginac-config} takes the following flags:
5359 Prints out the version of GiNaC installed.
5361 Prints '-I' flags pointing to the installed header files.
5363 Prints out the linker flags necessary to link a program against GiNaC.
5364 @item --prefix[=@var{PREFIX}]
5365 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5366 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5367 Otherwise, prints out the configured value of @env{$prefix}.
5368 @item --exec-prefix[=@var{PREFIX}]
5369 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5370 Otherwise, prints out the configured value of @env{$exec_prefix}.
5373 Typically, @command{ginac-config} will be used within a configure
5374 script, as described below. It, however, can also be used directly from
5375 the command line using backquotes to compile a simple program. For
5379 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5382 This command line might expand to (for example):
5385 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5386 -lginac -lcln -lstdc++
5389 Not only is the form using @command{ginac-config} easier to type, it will
5390 work on any system, no matter how GiNaC was configured.
5393 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5394 @c node-name, next, previous, up
5395 @section @samp{AM_PATH_GINAC}
5396 @cindex AM_PATH_GINAC
5398 For packages configured using GNU automake, GiNaC also provides
5399 a macro to automate the process of checking for GiNaC.
5402 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5410 Determines the location of GiNaC using @command{ginac-config}, which is
5411 either found in the user's path, or from the environment variable
5412 @env{GINACLIB_CONFIG}.
5415 Tests the installed libraries to make sure that their version
5416 is later than @var{MINIMUM-VERSION}. (A default version will be used
5420 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5421 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5422 variable to the output of @command{ginac-config --libs}, and calls
5423 @samp{AC_SUBST()} for these variables so they can be used in generated
5424 makefiles, and then executes @var{ACTION-IF-FOUND}.
5427 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5428 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5432 This macro is in file @file{ginac.m4} which is installed in
5433 @file{$datadir/aclocal}. Note that if automake was installed with a
5434 different @samp{--prefix} than GiNaC, you will either have to manually
5435 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5436 aclocal the @samp{-I} option when running it.
5439 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5440 * Example package:: Example of a package using AM_PATH_GINAC.
5444 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5445 @c node-name, next, previous, up
5446 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5448 Simply make sure that @command{ginac-config} is in your path, and run
5449 the configure script.
5456 The directory where the GiNaC libraries are installed needs
5457 to be found by your system's dynamic linker.
5459 This is generally done by
5462 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5468 setting the environment variable @env{LD_LIBRARY_PATH},
5471 or, as a last resort,
5474 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5475 running configure, for instance:
5478 LDFLAGS=-R/home/cbauer/lib ./configure
5483 You can also specify a @command{ginac-config} not in your path by
5484 setting the @env{GINACLIB_CONFIG} environment variable to the
5485 name of the executable
5488 If you move the GiNaC package from its installed location,
5489 you will either need to modify @command{ginac-config} script
5490 manually to point to the new location or rebuild GiNaC.
5501 --with-ginac-prefix=@var{PREFIX}
5502 --with-ginac-exec-prefix=@var{PREFIX}
5505 are provided to override the prefix and exec-prefix that were stored
5506 in the @command{ginac-config} shell script by GiNaC's configure. You are
5507 generally better off configuring GiNaC with the right path to begin with.
5511 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5512 @c node-name, next, previous, up
5513 @subsection Example of a package using @samp{AM_PATH_GINAC}
5515 The following shows how to build a simple package using automake
5516 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5519 #include <ginac/ginac.h>
5523 GiNaC::symbol x("x");
5524 GiNaC::ex a = GiNaC::sin(x);
5525 std::cout << "Derivative of " << a
5526 << " is " << a.diff(x) << std::endl;
5531 You should first read the introductory portions of the automake
5532 Manual, if you are not already familiar with it.
5534 Two files are needed, @file{configure.in}, which is used to build the
5538 dnl Process this file with autoconf to produce a configure script.
5540 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5546 AM_PATH_GINAC(0.7.0, [
5547 LIBS="$LIBS $GINACLIB_LIBS"
5548 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5549 ], AC_MSG_ERROR([need to have GiNaC installed]))
5554 The only command in this which is not standard for automake
5555 is the @samp{AM_PATH_GINAC} macro.
5557 That command does the following: If a GiNaC version greater or equal
5558 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5559 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5560 the error message `need to have GiNaC installed'
5562 And the @file{Makefile.am}, which will be used to build the Makefile.
5565 ## Process this file with automake to produce Makefile.in
5566 bin_PROGRAMS = simple
5567 simple_SOURCES = simple.cpp
5570 This @file{Makefile.am}, says that we are building a single executable,
5571 from a single sourcefile @file{simple.cpp}. Since every program
5572 we are building uses GiNaC we simply added the GiNaC options
5573 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5574 want to specify them on a per-program basis: for instance by
5578 simple_LDADD = $(GINACLIB_LIBS)
5579 INCLUDES = $(GINACLIB_CPPFLAGS)
5582 to the @file{Makefile.am}.
5584 To try this example out, create a new directory and add the three
5587 Now execute the following commands:
5590 $ automake --add-missing
5595 You now have a package that can be built in the normal fashion
5604 @node Bibliography, Concept Index, Example package, Top
5605 @c node-name, next, previous, up
5606 @appendix Bibliography
5611 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5614 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5617 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5620 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5623 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5624 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5627 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5628 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5629 Academic Press, London
5632 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5633 D.E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5636 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5641 @node Concept Index, , Bibliography, Top
5642 @c node-name, next, previous, up
5643 @unnumbered Concept Index