1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2001 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistical structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package Tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
216 #include <ginac/ginac.h>
218 using namespace GiNaC;
220 ex HermitePoly(const symbol & x, int n)
222 ex HKer=exp(-pow(x, 2));
223 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
224 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
231 for (int i=0; i<6; ++i)
232 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
238 When run, this will type out
244 H_3(z) == -12*z+8*z^3
245 H_4(z) == -48*z^2+16*z^4+12
246 H_5(z) == 120*z-160*z^3+32*z^5
249 This method of generating the coefficients is of course far from optimal
250 for production purposes.
252 In order to show some more examples of what GiNaC can do we will now use
253 the @command{ginsh}, a simple GiNaC interactive shell that provides a
254 convenient window into GiNaC's capabilities.
257 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
258 @c node-name, next, previous, up
259 @section What it can do for you
261 @cindex @command{ginsh}
262 After invoking @command{ginsh} one can test and experiment with GiNaC's
263 features much like in other Computer Algebra Systems except that it does
264 not provide programming constructs like loops or conditionals. For a
265 concise description of the @command{ginsh} syntax we refer to its
266 accompanied man page. Suffice to say that assignments and comparisons in
267 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
270 It can manipulate arbitrary precision integers in a very fast way.
271 Rational numbers are automatically converted to fractions of coprime
276 369988485035126972924700782451696644186473100389722973815184405301748249
278 123329495011708990974900260817232214728824366796574324605061468433916083
285 Exact numbers are always retained as exact numbers and only evaluated as
286 floating point numbers if requested. For instance, with numeric
287 radicals is dealt pretty much as with symbols. Products of sums of them
291 > expand((1+a^(1/5)-a^(2/5))^3);
292 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
293 > expand((1+3^(1/5)-3^(2/5))^3);
295 > evalf((1+3^(1/5)-3^(2/5))^3);
296 0.33408977534118624228
299 The function @code{evalf} that was used above converts any number in
300 GiNaC's expressions into floating point numbers. This can be done to
301 arbitrary predefined accuracy:
305 0.14285714285714285714
309 0.1428571428571428571428571428571428571428571428571428571428571428571428
310 5714285714285714285714285714285714285
313 Exact numbers other than rationals that can be manipulated in GiNaC
314 include predefined constants like Archimedes' @code{Pi}. They can both
315 be used in symbolic manipulations (as an exact number) as well as in
316 numeric expressions (as an inexact number):
322 9.869604401089358619+x
326 11.869604401089358619
329 Built-in functions evaluate immediately to exact numbers if
330 this is possible. Conversions that can be safely performed are done
331 immediately; conversions that are not generally valid are not done:
342 (Note that converting the last input to @code{x} would allow one to
343 conclude that @code{42*Pi} is equal to @code{0}.)
345 Linear equation systems can be solved along with basic linear
346 algebra manipulations over symbolic expressions. In C++ GiNaC offers
347 a matrix class for this purpose but we can see what it can do using
348 @command{ginsh}'s bracket notation to type them in:
351 > lsolve(a+x*y==z,x);
353 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
355 > M = [ [1, 3], [-3, 2] ];
359 > charpoly(M,lambda);
361 > A = [ [1, 1], [2, -1] ];
364 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 Multivariate polynomials and rational functions may be expanded,
370 collected and normalized (i.e. converted to a ratio of two coprime
374 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
375 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
376 > b = x^2 + 4*x*y - y^2;
379 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
381 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
383 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
388 You can differentiate functions and expand them as Taylor or Laurent
389 series in a very natural syntax (the second argument of @code{series} is
390 a relation defining the evaluation point, the third specifies the
393 @cindex Zeta function
397 > series(sin(x),x==0,4);
399 > series(1/tan(x),x==0,4);
400 x^(-1)-1/3*x+Order(x^2)
401 > series(tgamma(x),x==0,3);
402 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
403 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
405 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
406 -(0.90747907608088628905)*x^2+Order(x^3)
407 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
408 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
409 -Euler-1/12+Order((x-1/2*Pi)^3)
412 Here we have made use of the @command{ginsh}-command @code{"} to pop the
413 previously evaluated element from @command{ginsh}'s internal stack.
415 If you ever wanted to convert units in C or C++ and found this is
416 cumbersome, here is the solution. Symbolic types can always be used as
417 tags for different types of objects. Converting from wrong units to the
418 metric system is now easy:
426 140613.91592783185568*kg*m^(-2)
430 @node Installation, Prerequisites, What it can do for you, Top
431 @c node-name, next, previous, up
432 @chapter Installation
435 GiNaC's installation follows the spirit of most GNU software. It is
436 easily installed on your system by three steps: configuration, build,
440 * Prerequisites:: Packages upon which GiNaC depends.
441 * Configuration:: How to configure GiNaC.
442 * Building GiNaC:: How to compile GiNaC.
443 * Installing GiNaC:: How to install GiNaC on your system.
447 @node Prerequisites, Configuration, Installation, Installation
448 @c node-name, next, previous, up
449 @section Prerequisites
451 In order to install GiNaC on your system, some prerequisites need to be
452 met. First of all, you need to have a C++-compiler adhering to the
453 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
454 development so if you have a different compiler you are on your own.
455 For the configuration to succeed you need a Posix compliant shell
456 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
457 by the built process as well, since some of the source files are
458 automatically generated by Perl scripts. Last but not least, Bruno
459 Haible's library @acronym{CLN} is extensively used and needs to be
460 installed on your system. Please get it either from
461 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
462 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
463 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
464 site} (it is covered by GPL) and install it prior to trying to install
465 GiNaC. The configure script checks if it can find it and if it cannot
466 it will refuse to continue.
469 @node Configuration, Building GiNaC, Prerequisites, Installation
470 @c node-name, next, previous, up
471 @section Configuration
472 @cindex configuration
475 To configure GiNaC means to prepare the source distribution for
476 building. It is done via a shell script called @command{configure} that
477 is shipped with the sources and was originally generated by GNU
478 Autoconf. Since a configure script generated by GNU Autoconf never
479 prompts, all customization must be done either via command line
480 parameters or environment variables. It accepts a list of parameters,
481 the complete set of which can be listed by calling it with the
482 @option{--help} option. The most important ones will be shortly
483 described in what follows:
488 @option{--disable-shared}: When given, this option switches off the
489 build of a shared library, i.e. a @file{.so} file. This may be convenient
490 when developing because it considerably speeds up compilation.
493 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
494 and headers are installed. It defaults to @file{/usr/local} which means
495 that the library is installed in the directory @file{/usr/local/lib},
496 the header files in @file{/usr/local/include/ginac} and the documentation
497 (like this one) into @file{/usr/local/share/doc/GiNaC}.
500 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
501 the library installed in some other directory than
502 @file{@var{PREFIX}/lib/}.
505 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
506 to have the header files installed in some other directory than
507 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
508 @option{--includedir=/usr/include} you will end up with the header files
509 sitting in the directory @file{/usr/include/ginac/}. Note that the
510 subdirectory @file{ginac} is enforced by this process in order to
511 keep the header files separated from others. This avoids some
512 clashes and allows for an easier deinstallation of GiNaC. This ought
513 to be considered A Good Thing (tm).
516 @option{--datadir=@var{DATADIR}}: This option may be given in case you
517 want to have the documentation installed in some other directory than
518 @file{@var{PREFIX}/share/doc/GiNaC/}.
522 In addition, you may specify some environment variables.
523 @env{CXX} holds the path and the name of the C++ compiler
524 in case you want to override the default in your path. (The
525 @command{configure} script searches your path for @command{c++},
526 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
527 and @command{cc++} in that order.) It may be very useful to
528 define some compiler flags with the @env{CXXFLAGS} environment
529 variable, like optimization, debugging information and warning
530 levels. If omitted, it defaults to @option{-g -O2}.
532 The whole process is illustrated in the following two
533 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
534 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
537 Here is a simple configuration for a site-wide GiNaC library assuming
538 everything is in default paths:
541 $ export CXXFLAGS="-Wall -O2"
545 And here is a configuration for a private static GiNaC library with
546 several components sitting in custom places (site-wide @acronym{GCC} and
547 private @acronym{CLN}). The compiler is pursuaded to be picky and full
548 assertions and debugging information are switched on:
551 $ export CXX=/usr/local/gnu/bin/c++
552 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
553 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
554 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
555 $ ./configure --disable-shared --prefix=$(HOME)
559 @node Building GiNaC, Installing GiNaC, Configuration, Installation
560 @c node-name, next, previous, up
561 @section Building GiNaC
562 @cindex building GiNaC
564 After proper configuration you should just build the whole
569 at the command prompt and go for a cup of coffee. The exact time it
570 takes to compile GiNaC depends not only on the speed of your machines
571 but also on other parameters, for instance what value for @env{CXXFLAGS}
572 you entered. Optimization may be very time-consuming.
574 Just to make sure GiNaC works properly you may run a collection of
575 regression tests by typing
581 This will compile some sample programs, run them and check the output
582 for correctness. The regression tests fall in three categories. First,
583 the so called @emph{exams} are performed, simple tests where some
584 predefined input is evaluated (like a pupils' exam). Second, the
585 @emph{checks} test the coherence of results among each other with
586 possible random input. Third, some @emph{timings} are performed, which
587 benchmark some predefined problems with different sizes and display the
588 CPU time used in seconds. Each individual test should return a message
589 @samp{passed}. This is mostly intended to be a QA-check if something
590 was broken during development, not a sanity check of your system. Some
591 of the tests in sections @emph{checks} and @emph{timings} may require
592 insane amounts of memory and CPU time. Feel free to kill them if your
593 machine catches fire. Another quite important intent is to allow people
594 to fiddle around with optimization.
596 Generally, the top-level Makefile runs recursively to the
597 subdirectories. It is therfore safe to go into any subdirectory
598 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
599 @var{target} there in case something went wrong.
602 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
603 @c node-name, next, previous, up
604 @section Installing GiNaC
607 To install GiNaC on your system, simply type
613 As described in the section about configuration the files will be
614 installed in the following directories (the directories will be created
615 if they don't already exist):
620 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
621 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
622 So will @file{libginac.so} unless the configure script was
623 given the option @option{--disable-shared}. The proper symlinks
624 will be established as well.
627 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
628 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
631 All documentation (HTML and Postscript) will be stuffed into
632 @file{@var{PREFIX}/share/doc/GiNaC/} (or
633 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
637 For the sake of completeness we will list some other useful make
638 targets: @command{make clean} deletes all files generated by
639 @command{make}, i.e. all the object files. In addition @command{make
640 distclean} removes all files generated by the configuration and
641 @command{make maintainer-clean} goes one step further and deletes files
642 that may require special tools to rebuild (like the @command{libtool}
643 for instance). Finally @command{make uninstall} removes the installed
644 library, header files and documentation@footnote{Uninstallation does not
645 work after you have called @command{make distclean} since the
646 @file{Makefile} is itself generated by the configuration from
647 @file{Makefile.in} and hence deleted by @command{make distclean}. There
648 are two obvious ways out of this dilemma. First, you can run the
649 configuration again with the same @var{PREFIX} thus creating a
650 @file{Makefile} with a working @samp{uninstall} target. Second, you can
651 do it by hand since you now know where all the files went during
655 @node Basic Concepts, Expressions, Installing GiNaC, Top
656 @c node-name, next, previous, up
657 @chapter Basic Concepts
659 This chapter will describe the different fundamental objects that can be
660 handled by GiNaC. But before doing so, it is worthwhile introducing you
661 to the more commonly used class of expressions, representing a flexible
662 meta-class for storing all mathematical objects.
665 * Expressions:: The fundamental GiNaC class.
666 * The Class Hierarchy:: Overview of GiNaC's classes.
667 * Symbols:: Symbolic objects.
668 * Numbers:: Numerical objects.
669 * Constants:: Pre-defined constants.
670 * Fundamental containers:: The power, add and mul classes.
671 * Lists:: Lists of expressions.
672 * Mathematical functions:: Mathematical functions.
673 * Relations:: Equality, Inequality and all that.
674 * Matrices:: Matrices.
675 * Indexed objects:: Handling indexed quantities.
676 * Non-commutative objects:: Algebras with non-commutative products.
680 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
681 @c node-name, next, previous, up
683 @cindex expression (class @code{ex})
686 The most common class of objects a user deals with is the expression
687 @code{ex}, representing a mathematical object like a variable, number,
688 function, sum, product, etc@dots{} Expressions may be put together to form
689 new expressions, passed as arguments to functions, and so on. Here is a
690 little collection of valid expressions:
693 ex MyEx1 = 5; // simple number
694 ex MyEx2 = x + 2*y; // polynomial in x and y
695 ex MyEx3 = (x + 1)/(x - 1); // rational expression
696 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
697 ex MyEx5 = MyEx4 + 1; // similar to above
700 Expressions are handles to other more fundamental objects, that often
701 contain other expressions thus creating a tree of expressions
702 (@xref{Internal Structures}, for particular examples). Most methods on
703 @code{ex} therefore run top-down through such an expression tree. For
704 example, the method @code{has()} scans recursively for occurrences of
705 something inside an expression. Thus, if you have declared @code{MyEx4}
706 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
707 the argument of @code{sin} and hence return @code{true}.
709 The next sections will outline the general picture of GiNaC's class
710 hierarchy and describe the classes of objects that are handled by
714 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
715 @c node-name, next, previous, up
716 @section The Class Hierarchy
718 GiNaC's class hierarchy consists of several classes representing
719 mathematical objects, all of which (except for @code{ex} and some
720 helpers) are internally derived from one abstract base class called
721 @code{basic}. You do not have to deal with objects of class
722 @code{basic}, instead you'll be dealing with symbols, numbers,
723 containers of expressions and so on.
727 To get an idea about what kinds of symbolic composits may be built we
728 have a look at the most important classes in the class hierarchy and
729 some of the relations among the classes:
731 @image{classhierarchy}
733 The abstract classes shown here (the ones without drop-shadow) are of no
734 interest for the user. They are used internally in order to avoid code
735 duplication if two or more classes derived from them share certain
736 features. An example is @code{expairseq}, a container for a sequence of
737 pairs each consisting of one expression and a number (@code{numeric}).
738 What @emph{is} visible to the user are the derived classes @code{add}
739 and @code{mul}, representing sums and products. @xref{Internal
740 Structures}, where these two classes are described in more detail. The
741 following table shortly summarizes what kinds of mathematical objects
742 are stored in the different classes:
745 @multitable @columnfractions .22 .78
746 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
747 @item @code{constant} @tab Constants like
754 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
755 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
756 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
757 @item @code{ncmul} @tab Products of non-commutative objects
758 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
763 @code{sqrt(}@math{2}@code{)}
766 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
767 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
768 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
769 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
770 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
771 @item @code{indexed} @tab Indexed object like @math{A_ij}
772 @item @code{tensor} @tab Special tensor like the delta and metric tensors
773 @item @code{idx} @tab Index of an indexed object
774 @item @code{varidx} @tab Index with variance
775 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
776 @item @code{wildcard} @tab Wildcard for pattern matching
780 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
781 @c node-name, next, previous, up
783 @cindex @code{symbol} (class)
784 @cindex hierarchy of classes
787 Symbols are for symbolic manipulation what atoms are for chemistry. You
788 can declare objects of class @code{symbol} as any other object simply by
789 saying @code{symbol x,y;}. There is, however, a catch in here having to
790 do with the fact that C++ is a compiled language. The information about
791 the symbol's name is thrown away by the compiler but at a later stage
792 you may want to print expressions holding your symbols. In order to
793 avoid confusion GiNaC's symbols are able to know their own name. This
794 is accomplished by declaring its name for output at construction time in
795 the fashion @code{symbol x("x");}. If you declare a symbol using the
796 default constructor (i.e. without string argument) the system will deal
797 out a unique name. That name may not be suitable for printing but for
798 internal routines when no output is desired it is often enough. We'll
799 come across examples of such symbols later in this tutorial.
801 This implies that the strings passed to symbols at construction time may
802 not be used for comparing two of them. It is perfectly legitimate to
803 write @code{symbol x("x"),y("x");} but it is likely to lead into
804 trouble. Here, @code{x} and @code{y} are different symbols and
805 statements like @code{x-y} will not be simplified to zero although the
806 output @code{x-x} looks funny. Such output may also occur when there
807 are two different symbols in two scopes, for instance when you call a
808 function that declares a symbol with a name already existent in a symbol
809 in the calling function. Again, comparing them (using @code{operator==}
810 for instance) will always reveal their difference. Watch out, please.
812 @cindex @code{subs()}
813 Although symbols can be assigned expressions for internal reasons, you
814 should not do it (and we are not going to tell you how it is done). If
815 you want to replace a symbol with something else in an expression, you
816 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
819 @node Numbers, Constants, Symbols, Basic Concepts
820 @c node-name, next, previous, up
822 @cindex @code{numeric} (class)
828 For storing numerical things, GiNaC uses Bruno Haible's library
829 @acronym{CLN}. The classes therein serve as foundation classes for
830 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
831 alternatively for Common Lisp Numbers. In order to find out more about
832 @acronym{CLN}'s internals the reader is refered to the documentation of
833 that library. @inforef{Introduction, , cln}, for more
834 information. Suffice to say that it is by itself build on top of another
835 library, the GNU Multiple Precision library @acronym{GMP}, which is an
836 extremely fast library for arbitrary long integers and rationals as well
837 as arbitrary precision floating point numbers. It is very commonly used
838 by several popular cryptographic applications. @acronym{CLN} extends
839 @acronym{GMP} by several useful things: First, it introduces the complex
840 number field over either reals (i.e. floating point numbers with
841 arbitrary precision) or rationals. Second, it automatically converts
842 rationals to integers if the denominator is unity and complex numbers to
843 real numbers if the imaginary part vanishes and also correctly treats
844 algebraic functions. Third it provides good implementations of
845 state-of-the-art algorithms for all trigonometric and hyperbolic
846 functions as well as for calculation of some useful constants.
848 The user can construct an object of class @code{numeric} in several
849 ways. The following example shows the four most important constructors.
850 It uses construction from C-integer, construction of fractions from two
851 integers, construction from C-float and construction from a string:
854 #include <ginac/ginac.h>
855 using namespace GiNaC;
859 numeric two(2); // exact integer 2
860 numeric r(2,3); // exact fraction 2/3
861 numeric e(2.71828); // floating point number
862 numeric p("3.1415926535897932385"); // floating point number
863 // Trott's constant in scientific notation:
864 numeric trott("1.0841015122311136151E-2");
866 std::cout << two*p << std::endl; // floating point 6.283...
870 Note that all those constructors are @emph{explicit} which means you are
871 not allowed to write @code{numeric two=2;}. This is because the basic
872 objects to be handled by GiNaC are the expressions @code{ex} and we want
873 to keep things simple and wish objects like @code{pow(x,2)} to be
874 handled the same way as @code{pow(x,a)}, which means that we need to
875 allow a general @code{ex} as base and exponent. Therefore there is an
876 implicit constructor from C-integers directly to expressions handling
877 numerics at work in most of our examples. This design really becomes
878 convenient when one declares own functions having more than one
879 parameter but it forbids using implicit constructors because that would
880 lead to compile-time ambiguities.
882 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
883 This would, however, call C's built-in operator @code{/} for integers
884 first and result in a numeric holding a plain integer 1. @strong{Never
885 use the operator @code{/} on integers} unless you know exactly what you
886 are doing! Use the constructor from two integers instead, as shown in
887 the example above. Writing @code{numeric(1)/2} may look funny but works
890 @cindex @code{Digits}
892 We have seen now the distinction between exact numbers and floating
893 point numbers. Clearly, the user should never have to worry about
894 dynamically created exact numbers, since their `exactness' always
895 determines how they ought to be handled, i.e. how `long' they are. The
896 situation is different for floating point numbers. Their accuracy is
897 controlled by one @emph{global} variable, called @code{Digits}. (For
898 those readers who know about Maple: it behaves very much like Maple's
899 @code{Digits}). All objects of class numeric that are constructed from
900 then on will be stored with a precision matching that number of decimal
904 #include <ginac/ginac.h>
906 using namespace GiNaC;
910 numeric three(3.0), one(1.0);
911 numeric x = one/three;
913 cout << "in " << Digits << " digits:" << endl;
915 cout << Pi.evalf() << endl;
927 The above example prints the following output to screen:
934 0.333333333333333333333333333333333333333333333333333333333333333333
935 3.14159265358979323846264338327950288419716939937510582097494459231
938 It should be clear that objects of class @code{numeric} should be used
939 for constructing numbers or for doing arithmetic with them. The objects
940 one deals with most of the time are the polymorphic expressions @code{ex}.
942 @subsection Tests on numbers
944 Once you have declared some numbers, assigned them to expressions and
945 done some arithmetic with them it is frequently desired to retrieve some
946 kind of information from them like asking whether that number is
947 integer, rational, real or complex. For those cases GiNaC provides
948 several useful methods. (Internally, they fall back to invocations of
949 certain CLN functions.)
951 As an example, let's construct some rational number, multiply it with
952 some multiple of its denominator and test what comes out:
955 #include <ginac/ginac.h>
957 using namespace GiNaC;
959 // some very important constants:
960 const numeric twentyone(21);
961 const numeric ten(10);
962 const numeric five(5);
966 numeric answer = twentyone;
969 cout << answer.is_integer() << endl; // false, it's 21/5
971 cout << answer.is_integer() << endl; // true, it's 42 now!
975 Note that the variable @code{answer} is constructed here as an integer
976 by @code{numeric}'s copy constructor but in an intermediate step it
977 holds a rational number represented as integer numerator and integer
978 denominator. When multiplied by 10, the denominator becomes unity and
979 the result is automatically converted to a pure integer again.
980 Internally, the underlying @acronym{CLN} is responsible for this
981 behaviour and we refer the reader to @acronym{CLN}'s documentation.
982 Suffice to say that the same behaviour applies to complex numbers as
983 well as return values of certain functions. Complex numbers are
984 automatically converted to real numbers if the imaginary part becomes
985 zero. The full set of tests that can be applied is listed in the
989 @multitable @columnfractions .30 .70
990 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
991 @item @code{.is_zero()}
992 @tab @dots{}equal to zero
993 @item @code{.is_positive()}
994 @tab @dots{}not complex and greater than 0
995 @item @code{.is_integer()}
996 @tab @dots{}a (non-complex) integer
997 @item @code{.is_pos_integer()}
998 @tab @dots{}an integer and greater than 0
999 @item @code{.is_nonneg_integer()}
1000 @tab @dots{}an integer and greater equal 0
1001 @item @code{.is_even()}
1002 @tab @dots{}an even integer
1003 @item @code{.is_odd()}
1004 @tab @dots{}an odd integer
1005 @item @code{.is_prime()}
1006 @tab @dots{}a prime integer (probabilistic primality test)
1007 @item @code{.is_rational()}
1008 @tab @dots{}an exact rational number (integers are rational, too)
1009 @item @code{.is_real()}
1010 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1011 @item @code{.is_cinteger()}
1012 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1013 @item @code{.is_crational()}
1014 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1019 @node Constants, Fundamental containers, Numbers, Basic Concepts
1020 @c node-name, next, previous, up
1022 @cindex @code{constant} (class)
1025 @cindex @code{Catalan}
1026 @cindex @code{Euler}
1027 @cindex @code{evalf()}
1028 Constants behave pretty much like symbols except that they return some
1029 specific number when the method @code{.evalf()} is called.
1031 The predefined known constants are:
1034 @multitable @columnfractions .14 .30 .56
1035 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1037 @tab Archimedes' constant
1038 @tab 3.14159265358979323846264338327950288
1039 @item @code{Catalan}
1040 @tab Catalan's constant
1041 @tab 0.91596559417721901505460351493238411
1043 @tab Euler's (or Euler-Mascheroni) constant
1044 @tab 0.57721566490153286060651209008240243
1049 @node Fundamental containers, Lists, Constants, Basic Concepts
1050 @c node-name, next, previous, up
1051 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1055 @cindex @code{power}
1057 Simple polynomial expressions are written down in GiNaC pretty much like
1058 in other CAS or like expressions involving numerical variables in C.
1059 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1060 been overloaded to achieve this goal. When you run the following
1061 code snippet, the constructor for an object of type @code{mul} is
1062 automatically called to hold the product of @code{a} and @code{b} and
1063 then the constructor for an object of type @code{add} is called to hold
1064 the sum of that @code{mul} object and the number one:
1068 symbol a("a"), b("b");
1073 @cindex @code{pow()}
1074 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1075 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1076 construction is necessary since we cannot safely overload the constructor
1077 @code{^} in C++ to construct a @code{power} object. If we did, it would
1078 have several counterintuitive and undesired effects:
1082 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1084 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1085 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1086 interpret this as @code{x^(a^b)}.
1088 Also, expressions involving integer exponents are very frequently used,
1089 which makes it even more dangerous to overload @code{^} since it is then
1090 hard to distinguish between the semantics as exponentiation and the one
1091 for exclusive or. (It would be embarassing to return @code{1} where one
1092 has requested @code{2^3}.)
1095 @cindex @command{ginsh}
1096 All effects are contrary to mathematical notation and differ from the
1097 way most other CAS handle exponentiation, therefore overloading @code{^}
1098 is ruled out for GiNaC's C++ part. The situation is different in
1099 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1100 that the other frequently used exponentiation operator @code{**} does
1101 not exist at all in C++).
1103 To be somewhat more precise, objects of the three classes described
1104 here, are all containers for other expressions. An object of class
1105 @code{power} is best viewed as a container with two slots, one for the
1106 basis, one for the exponent. All valid GiNaC expressions can be
1107 inserted. However, basic transformations like simplifying
1108 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1109 when this is mathematically possible. If we replace the outer exponent
1110 three in the example by some symbols @code{a}, the simplification is not
1111 safe and will not be performed, since @code{a} might be @code{1/2} and
1114 Objects of type @code{add} and @code{mul} are containers with an
1115 arbitrary number of slots for expressions to be inserted. Again, simple
1116 and safe simplifications are carried out like transforming
1117 @code{3*x+4-x} to @code{2*x+4}.
1119 The general rule is that when you construct such objects, GiNaC
1120 automatically creates them in canonical form, which might differ from
1121 the form you typed in your program. This allows for rapid comparison of
1122 expressions, since after all @code{a-a} is simply zero. Note, that the
1123 canonical form is not necessarily lexicographical ordering or in any way
1124 easily guessable. It is only guaranteed that constructing the same
1125 expression twice, either implicitly or explicitly, results in the same
1129 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1130 @c node-name, next, previous, up
1131 @section Lists of expressions
1132 @cindex @code{lst} (class)
1134 @cindex @code{nops()}
1136 @cindex @code{append()}
1137 @cindex @code{prepend()}
1138 @cindex @code{remove_first()}
1139 @cindex @code{remove_last()}
1141 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1142 expressions. These are sometimes used to supply a variable number of
1143 arguments of the same type to GiNaC methods such as @code{subs()} and
1144 @code{to_rational()}, so you should have a basic understanding about them.
1146 Lists of up to 16 expressions can be directly constructed from single
1151 symbol x("x"), y("y");
1152 lst l(x, 2, y, x+y);
1153 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1157 Use the @code{nops()} method to determine the size (number of expressions) of
1158 a list and the @code{op()} method to access individual elements:
1162 cout << l.nops() << endl; // prints '4'
1163 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1167 You can append or prepend an expression to a list with the @code{append()}
1168 and @code{prepend()} methods:
1172 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1173 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1177 Finally you can remove the first or last element of a list with
1178 @code{remove_first()} and @code{remove_last()}:
1182 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1183 l.remove_last(); // l is now @{x, 2, y, x+y@}
1188 @node Mathematical functions, Relations, Lists, Basic Concepts
1189 @c node-name, next, previous, up
1190 @section Mathematical functions
1191 @cindex @code{function} (class)
1192 @cindex trigonometric function
1193 @cindex hyperbolic function
1195 There are quite a number of useful functions hard-wired into GiNaC. For
1196 instance, all trigonometric and hyperbolic functions are implemented
1197 (@xref{Built-in Functions}, for a complete list).
1199 These functions are all objects of class @code{function}. They accept
1200 one or more expressions as arguments and return one expression. If the
1201 arguments are not numerical, the evaluation of the function may be
1202 halted, as it does in the next example, showing how a function returns
1203 itself twice and finally an expression that may be really useful:
1205 @cindex Gamma function
1206 @cindex @code{subs()}
1209 symbol x("x"), y("y");
1211 cout << tgamma(foo) << endl;
1212 // -> tgamma(x+(1/2)*y)
1213 ex bar = foo.subs(y==1);
1214 cout << tgamma(bar) << endl;
1216 ex foobar = bar.subs(x==7);
1217 cout << tgamma(foobar) << endl;
1218 // -> (135135/128)*Pi^(1/2)
1222 Besides evaluation most of these functions allow differentiation, series
1223 expansion and so on. Read the next chapter in order to learn more about
1227 @node Relations, Matrices, Mathematical functions, Basic Concepts
1228 @c node-name, next, previous, up
1230 @cindex @code{relational} (class)
1232 Sometimes, a relation holding between two expressions must be stored
1233 somehow. The class @code{relational} is a convenient container for such
1234 purposes. A relation is by definition a container for two @code{ex} and
1235 a relation between them that signals equality, inequality and so on.
1236 They are created by simply using the C++ operators @code{==}, @code{!=},
1237 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1239 @xref{Mathematical functions}, for examples where various applications
1240 of the @code{.subs()} method show how objects of class relational are
1241 used as arguments. There they provide an intuitive syntax for
1242 substitutions. They are also used as arguments to the @code{ex::series}
1243 method, where the left hand side of the relation specifies the variable
1244 to expand in and the right hand side the expansion point. They can also
1245 be used for creating systems of equations that are to be solved for
1246 unknown variables. But the most common usage of objects of this class
1247 is rather inconspicuous in statements of the form @code{if
1248 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1249 conversion from @code{relational} to @code{bool} takes place. Note,
1250 however, that @code{==} here does not perform any simplifications, hence
1251 @code{expand()} must be called explicitly.
1254 @node Matrices, Indexed objects, Relations, Basic Concepts
1255 @c node-name, next, previous, up
1257 @cindex @code{matrix} (class)
1259 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1260 matrix with @math{m} rows and @math{n} columns are accessed with two
1261 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1262 second one in the range 0@dots{}@math{n-1}.
1264 There are a couple of ways to construct matrices, with or without preset
1268 matrix::matrix(unsigned r, unsigned c);
1269 matrix::matrix(unsigned r, unsigned c, const lst & l);
1270 ex lst_to_matrix(const lst & l);
1271 ex diag_matrix(const lst & l);
1274 The first two functions are @code{matrix} constructors which create a matrix
1275 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1276 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1277 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1278 from a list of lists, each list representing a matrix row. Finally,
1279 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1280 elements. Note that the last two functions return expressions, not matrix
1283 Matrix elements can be accessed and set using the parenthesis (function call)
1287 const ex & matrix::operator()(unsigned r, unsigned c) const;
1288 ex & matrix::operator()(unsigned r, unsigned c);
1291 It is also possible to access the matrix elements in a linear fashion with
1292 the @code{op()} method. But C++-style subscripting with square brackets
1293 @samp{[]} is not available.
1295 Here are a couple of examples that all construct the same 2x2 diagonal
1300 symbol a("a"), b("b");
1308 e = matrix(2, 2, lst(a, 0, 0, b));
1310 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1312 e = diag_matrix(lst(a, b));
1319 @cindex @code{transpose()}
1320 @cindex @code{inverse()}
1321 There are three ways to do arithmetic with matrices. The first (and most
1322 efficient one) is to use the methods provided by the @code{matrix} class:
1325 matrix matrix::add(const matrix & other) const;
1326 matrix matrix::sub(const matrix & other) const;
1327 matrix matrix::mul(const matrix & other) const;
1328 matrix matrix::mul_scalar(const ex & other) const;
1329 matrix matrix::pow(const ex & expn) const;
1330 matrix matrix::transpose(void) const;
1331 matrix matrix::inverse(void) const;
1334 All of these methods return the result as a new matrix object. Here is an
1335 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1340 matrix A(2, 2, lst(1, 2, 3, 4));
1341 matrix B(2, 2, lst(-1, 0, 2, 1));
1342 matrix C(2, 2, lst(8, 4, 2, 1));
1344 matrix result = A.mul(B).sub(C.mul_scalar(2));
1345 cout << result << endl;
1346 // -> [[-13,-6],[1,2]]
1351 @cindex @code{evalm()}
1352 The second (and probably the most natural) way is to construct an expression
1353 containing matrices with the usual arithmetic operators and @code{pow()}.
1354 For efficiency reasons, expressions with sums, products and powers of
1355 matrices are not automatically evaluated in GiNaC. You have to call the
1359 ex ex::evalm() const;
1362 to obtain the result:
1369 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1370 cout << e.evalm() << endl;
1371 // -> [[-13,-6],[1,2]]
1376 The non-commutativity of the product @code{A*B} in this example is
1377 automatically recognized by GiNaC. There is no need to use a special
1378 operator here. @xref{Non-commutative objects}, for more information about
1379 dealing with non-commutative expressions.
1381 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1382 to perform the arithmetic:
1387 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1388 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1390 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1391 cout << e.simplify_indexed() << endl;
1392 // -> [[-13,-6],[1,2]].i.j
1396 Using indices is most useful when working with rectangular matrices and
1397 one-dimensional vectors because you don't have to worry about having to
1398 transpose matrices before multiplying them. @xref{Indexed objects}, for
1399 more information about using matrices with indices, and about indices in
1402 The @code{matrix} class provides a couple of additional methods for
1403 computing determinants, traces, and characteristic polynomials:
1406 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1407 ex matrix::trace(void) const;
1408 ex matrix::charpoly(const symbol & lambda) const;
1411 The @samp{algo} argument of @code{determinant()} allows to select between
1412 different algorithms for calculating the determinant. The possible values
1413 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1414 heuristic to automatically select an algorithm that is likely to give the
1415 result most quickly.
1418 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1419 @c node-name, next, previous, up
1420 @section Indexed objects
1422 GiNaC allows you to handle expressions containing general indexed objects in
1423 arbitrary spaces. It is also able to canonicalize and simplify such
1424 expressions and perform symbolic dummy index summations. There are a number
1425 of predefined indexed objects provided, like delta and metric tensors.
1427 There are few restrictions placed on indexed objects and their indices and
1428 it is easy to construct nonsense expressions, but our intention is to
1429 provide a general framework that allows you to implement algorithms with
1430 indexed quantities, getting in the way as little as possible.
1432 @cindex @code{idx} (class)
1433 @cindex @code{indexed} (class)
1434 @subsection Indexed quantities and their indices
1436 Indexed expressions in GiNaC are constructed of two special types of objects,
1437 @dfn{index objects} and @dfn{indexed objects}.
1441 @cindex contravariant
1444 @item Index objects are of class @code{idx} or a subclass. Every index has
1445 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1446 the index lives in) which can both be arbitrary expressions but are usually
1447 a number or a simple symbol. In addition, indices of class @code{varidx} have
1448 a @dfn{variance} (they can be co- or contravariant), and indices of class
1449 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1451 @item Indexed objects are of class @code{indexed} or a subclass. They
1452 contain a @dfn{base expression} (which is the expression being indexed), and
1453 one or more indices.
1457 @strong{Note:} when printing expressions, covariant indices and indices
1458 without variance are denoted @samp{.i} while contravariant indices are
1459 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1460 value. In the following, we are going to use that notation in the text so
1461 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1462 not visible in the output.
1464 A simple example shall illustrate the concepts:
1467 #include <ginac/ginac.h>
1468 using namespace std;
1469 using namespace GiNaC;
1473 symbol i_sym("i"), j_sym("j");
1474 idx i(i_sym, 3), j(j_sym, 3);
1477 cout << indexed(A, i, j) << endl;
1482 The @code{idx} constructor takes two arguments, the index value and the
1483 index dimension. First we define two index objects, @code{i} and @code{j},
1484 both with the numeric dimension 3. The value of the index @code{i} is the
1485 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1486 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1487 construct an expression containing one indexed object, @samp{A.i.j}. It has
1488 the symbol @code{A} as its base expression and the two indices @code{i} and
1491 Note the difference between the indices @code{i} and @code{j} which are of
1492 class @code{idx}, and the index values which are the sybols @code{i_sym}
1493 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1494 or numbers but must be index objects. For example, the following is not
1495 correct and will raise an exception:
1498 symbol i("i"), j("j");
1499 e = indexed(A, i, j); // ERROR: indices must be of type idx
1502 You can have multiple indexed objects in an expression, index values can
1503 be numeric, and index dimensions symbolic:
1507 symbol B("B"), dim("dim");
1508 cout << 4 * indexed(A, i)
1509 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1514 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1515 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1516 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1517 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1518 @code{simplify_indexed()} for that, see below).
1520 In fact, base expressions, index values and index dimensions can be
1521 arbitrary expressions:
1525 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1530 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1531 get an error message from this but you will probably not be able to do
1532 anything useful with it.
1534 @cindex @code{get_value()}
1535 @cindex @code{get_dimension()}
1539 ex idx::get_value(void);
1540 ex idx::get_dimension(void);
1543 return the value and dimension of an @code{idx} object. If you have an index
1544 in an expression, such as returned by calling @code{.op()} on an indexed
1545 object, you can get a reference to the @code{idx} object with the function
1546 @code{ex_to<idx>()} on the expression.
1548 There are also the methods
1551 bool idx::is_numeric(void);
1552 bool idx::is_symbolic(void);
1553 bool idx::is_dim_numeric(void);
1554 bool idx::is_dim_symbolic(void);
1557 for checking whether the value and dimension are numeric or symbolic
1558 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1559 About Expressions}) returns information about the index value.
1561 @cindex @code{varidx} (class)
1562 If you need co- and contravariant indices, use the @code{varidx} class:
1566 symbol mu_sym("mu"), nu_sym("nu");
1567 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1568 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1570 cout << indexed(A, mu, nu) << endl;
1572 cout << indexed(A, mu_co, nu) << endl;
1574 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1579 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1580 co- or contravariant. The default is a contravariant (upper) index, but
1581 this can be overridden by supplying a third argument to the @code{varidx}
1582 constructor. The two methods
1585 bool varidx::is_covariant(void);
1586 bool varidx::is_contravariant(void);
1589 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1590 to get the object reference from an expression). There's also the very useful
1594 ex varidx::toggle_variance(void);
1597 which makes a new index with the same value and dimension but the opposite
1598 variance. By using it you only have to define the index once.
1600 @cindex @code{spinidx} (class)
1601 The @code{spinidx} class provides dotted and undotted variant indices, as
1602 used in the Weyl-van-der-Waerden spinor formalism:
1606 symbol K("K"), C_sym("C"), D_sym("D");
1607 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1608 // contravariant, undotted
1609 spinidx C_co(C_sym, 2, true); // covariant index
1610 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1611 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1613 cout << indexed(K, C, D) << endl;
1615 cout << indexed(K, C_co, D_dot) << endl;
1617 cout << indexed(K, D_co_dot, D) << endl;
1622 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1623 dotted or undotted. The default is undotted but this can be overridden by
1624 supplying a fourth argument to the @code{spinidx} constructor. The two
1628 bool spinidx::is_dotted(void);
1629 bool spinidx::is_undotted(void);
1632 allow you to check whether or not a @code{spinidx} object is dotted (use
1633 @code{ex_to<spinidx>()} to get the object reference from an expression).
1634 Finally, the two methods
1637 ex spinidx::toggle_dot(void);
1638 ex spinidx::toggle_variance_dot(void);
1641 create a new index with the same value and dimension but opposite dottedness
1642 and the same or opposite variance.
1644 @subsection Substituting indices
1646 @cindex @code{subs()}
1647 Sometimes you will want to substitute one symbolic index with another
1648 symbolic or numeric index, for example when calculating one specific element
1649 of a tensor expression. This is done with the @code{.subs()} method, as it
1650 is done for symbols (see @ref{Substituting Expressions}).
1652 You have two possibilities here. You can either substitute the whole index
1653 by another index or expression:
1657 ex e = indexed(A, mu_co);
1658 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1659 // -> A.mu becomes A~nu
1660 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1661 // -> A.mu becomes A~0
1662 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1663 // -> A.mu becomes A.0
1667 The third example shows that trying to replace an index with something that
1668 is not an index will substitute the index value instead.
1670 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1675 ex e = indexed(A, mu_co);
1676 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1677 // -> A.mu becomes A.nu
1678 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1679 // -> A.mu becomes A.0
1683 As you see, with the second method only the value of the index will get
1684 substituted. Its other properties, including its dimension, remain unchanged.
1685 If you want to change the dimension of an index you have to substitute the
1686 whole index by another one with the new dimension.
1688 Finally, substituting the base expression of an indexed object works as
1693 ex e = indexed(A, mu_co);
1694 cout << e << " becomes " << e.subs(A == A+B) << endl;
1695 // -> A.mu becomes (B+A).mu
1699 @subsection Symmetries
1700 @cindex @code{symmetry} (class)
1701 @cindex @code{sy_none()}
1702 @cindex @code{sy_symm()}
1703 @cindex @code{sy_anti()}
1704 @cindex @code{sy_cycl()}
1706 Indexed objects can have certain symmetry properties with respect to their
1707 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1708 that is constructed with the helper functions
1711 symmetry sy_none(...);
1712 symmetry sy_symm(...);
1713 symmetry sy_anti(...);
1714 symmetry sy_cycl(...);
1717 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1718 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1719 represents a cyclic symmetry. Each of these functions accepts up to four
1720 arguments which can be either symmetry objects themselves or unsigned integer
1721 numbers that represent an index position (counting from 0). A symmetry
1722 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1723 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1726 Here are some examples of symmetry definitions:
1731 e = indexed(A, i, j);
1732 e = indexed(A, sy_none(), i, j); // equivalent
1733 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1735 // Symmetric in all three indices:
1736 e = indexed(A, sy_symm(), i, j, k);
1737 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1738 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1739 // different canonical order
1741 // Symmetric in the first two indices only:
1742 e = indexed(A, sy_symm(0, 1), i, j, k);
1743 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1745 // Antisymmetric in the first and last index only (index ranges need not
1747 e = indexed(A, sy_anti(0, 2), i, j, k);
1748 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1750 // An example of a mixed symmetry: antisymmetric in the first two and
1751 // last two indices, symmetric when swapping the first and last index
1752 // pairs (like the Riemann curvature tensor):
1753 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1755 // Cyclic symmetry in all three indices:
1756 e = indexed(A, sy_cycl(), i, j, k);
1757 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1759 // The following examples are invalid constructions that will throw
1760 // an exception at run time.
1762 // An index may not appear multiple times:
1763 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1764 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1766 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1767 // same number of indices:
1768 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1770 // And of course, you cannot specify indices which are not there:
1771 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1775 If you need to specify more than four indices, you have to use the
1776 @code{.add()} method of the @code{symmetry} class. For example, to specify
1777 full symmetry in the first six indices you would write
1778 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1780 If an indexed object has a symmetry, GiNaC will automatically bring the
1781 indices into a canonical order which allows for some immediate simplifications:
1785 cout << indexed(A, sy_symm(), i, j)
1786 + indexed(A, sy_symm(), j, i) << endl;
1788 cout << indexed(B, sy_anti(), i, j)
1789 + indexed(B, sy_anti(), j, i) << endl;
1791 cout << indexed(B, sy_anti(), i, j, k)
1792 + indexed(B, sy_anti(), j, i, k) << endl;
1797 @cindex @code{get_free_indices()}
1799 @subsection Dummy indices
1801 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1802 that a summation over the index range is implied. Symbolic indices which are
1803 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1804 dummy nor free indices.
1806 To be recognized as a dummy index pair, the two indices must be of the same
1807 class and dimension and their value must be the same single symbol (an index
1808 like @samp{2*n+1} is never a dummy index). If the indices are of class
1809 @code{varidx} they must also be of opposite variance; if they are of class
1810 @code{spinidx} they must be both dotted or both undotted.
1812 The method @code{.get_free_indices()} returns a vector containing the free
1813 indices of an expression. It also checks that the free indices of the terms
1814 of a sum are consistent:
1818 symbol A("A"), B("B"), C("C");
1820 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1821 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1823 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1824 cout << exprseq(e.get_free_indices()) << endl;
1826 // 'j' and 'l' are dummy indices
1828 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1829 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1831 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1832 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1833 cout << exprseq(e.get_free_indices()) << endl;
1835 // 'nu' is a dummy index, but 'sigma' is not
1837 e = indexed(A, mu, mu);
1838 cout << exprseq(e.get_free_indices()) << endl;
1840 // 'mu' is not a dummy index because it appears twice with the same
1843 e = indexed(A, mu, nu) + 42;
1844 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1845 // this will throw an exception:
1846 // "add::get_free_indices: inconsistent indices in sum"
1850 @cindex @code{simplify_indexed()}
1851 @subsection Simplifying indexed expressions
1853 In addition to the few automatic simplifications that GiNaC performs on
1854 indexed expressions (such as re-ordering the indices of symmetric tensors
1855 and calculating traces and convolutions of matrices and predefined tensors)
1859 ex ex::simplify_indexed(void);
1860 ex ex::simplify_indexed(const scalar_products & sp);
1863 that performs some more expensive operations:
1866 @item it checks the consistency of free indices in sums in the same way
1867 @code{get_free_indices()} does
1868 @item it tries to give dumy indices that appear in different terms of a sum
1869 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1870 @item it (symbolically) calculates all possible dummy index summations/contractions
1871 with the predefined tensors (this will be explained in more detail in the
1873 @item as a special case of dummy index summation, it can replace scalar products
1874 of two tensors with a user-defined value
1877 The last point is done with the help of the @code{scalar_products} class
1878 which is used to store scalar products with known values (this is not an
1879 arithmetic class, you just pass it to @code{simplify_indexed()}):
1883 symbol A("A"), B("B"), C("C"), i_sym("i");
1887 sp.add(A, B, 0); // A and B are orthogonal
1888 sp.add(A, C, 0); // A and C are orthogonal
1889 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1891 e = indexed(A + B, i) * indexed(A + C, i);
1893 // -> (B+A).i*(A+C).i
1895 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1901 The @code{scalar_products} object @code{sp} acts as a storage for the
1902 scalar products added to it with the @code{.add()} method. This method
1903 takes three arguments: the two expressions of which the scalar product is
1904 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1905 @code{simplify_indexed()} will replace all scalar products of indexed
1906 objects that have the symbols @code{A} and @code{B} as base expressions
1907 with the single value 0. The number, type and dimension of the indices
1908 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1910 @cindex @code{expand()}
1911 The example above also illustrates a feature of the @code{expand()} method:
1912 if passed the @code{expand_indexed} option it will distribute indices
1913 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1915 @cindex @code{tensor} (class)
1916 @subsection Predefined tensors
1918 Some frequently used special tensors such as the delta, epsilon and metric
1919 tensors are predefined in GiNaC. They have special properties when
1920 contracted with other tensor expressions and some of them have constant
1921 matrix representations (they will evaluate to a number when numeric
1922 indices are specified).
1924 @cindex @code{delta_tensor()}
1925 @subsubsection Delta tensor
1927 The delta tensor takes two indices, is symmetric and has the matrix
1928 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
1929 @code{delta_tensor()}:
1933 symbol A("A"), B("B");
1935 idx i(symbol("i"), 3), j(symbol("j"), 3),
1936 k(symbol("k"), 3), l(symbol("l"), 3);
1938 ex e = indexed(A, i, j) * indexed(B, k, l)
1939 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1940 cout << e.simplify_indexed() << endl;
1943 cout << delta_tensor(i, i) << endl;
1948 @cindex @code{metric_tensor()}
1949 @subsubsection General metric tensor
1951 The function @code{metric_tensor()} creates a general symmetric metric
1952 tensor with two indices that can be used to raise/lower tensor indices. The
1953 metric tensor is denoted as @samp{g} in the output and if its indices are of
1954 mixed variance it is automatically replaced by a delta tensor:
1960 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1962 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1963 cout << e.simplify_indexed() << endl;
1966 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1967 cout << e.simplify_indexed() << endl;
1970 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1971 * metric_tensor(nu, rho);
1972 cout << e.simplify_indexed() << endl;
1975 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1976 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1977 + indexed(A, mu.toggle_variance(), rho));
1978 cout << e.simplify_indexed() << endl;
1983 @cindex @code{lorentz_g()}
1984 @subsubsection Minkowski metric tensor
1986 The Minkowski metric tensor is a special metric tensor with a constant
1987 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1988 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1989 It is created with the function @code{lorentz_g()} (although it is output as
1994 varidx mu(symbol("mu"), 4);
1996 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1997 * lorentz_g(mu, varidx(0, 4)); // negative signature
1998 cout << e.simplify_indexed() << endl;
2001 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2002 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2003 cout << e.simplify_indexed() << endl;
2008 @cindex @code{spinor_metric()}
2009 @subsubsection Spinor metric tensor
2011 The function @code{spinor_metric()} creates an antisymmetric tensor with
2012 two indices that is used to raise/lower indices of 2-component spinors.
2013 It is output as @samp{eps}:
2019 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2020 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2022 e = spinor_metric(A, B) * indexed(psi, B_co);
2023 cout << e.simplify_indexed() << endl;
2026 e = spinor_metric(A, B) * indexed(psi, A_co);
2027 cout << e.simplify_indexed() << endl;
2030 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2031 cout << e.simplify_indexed() << endl;
2034 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2035 cout << e.simplify_indexed() << endl;
2038 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2039 cout << e.simplify_indexed() << endl;
2042 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2043 cout << e.simplify_indexed() << endl;
2048 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2050 @cindex @code{epsilon_tensor()}
2051 @cindex @code{lorentz_eps()}
2052 @subsubsection Epsilon tensor
2054 The epsilon tensor is totally antisymmetric, its number of indices is equal
2055 to the dimension of the index space (the indices must all be of the same
2056 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2057 defined to be 1. Its behaviour with indices that have a variance also
2058 depends on the signature of the metric. Epsilon tensors are output as
2061 There are three functions defined to create epsilon tensors in 2, 3 and 4
2065 ex epsilon_tensor(const ex & i1, const ex & i2);
2066 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2067 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2070 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2071 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2072 Minkowski space (the last @code{bool} argument specifies whether the metric
2073 has negative or positive signature, as in the case of the Minkowski metric
2076 @subsection Linear algebra
2078 The @code{matrix} class can be used with indices to do some simple linear
2079 algebra (linear combinations and products of vectors and matrices, traces
2080 and scalar products):
2084 idx i(symbol("i"), 2), j(symbol("j"), 2);
2085 symbol x("x"), y("y");
2087 // A is a 2x2 matrix, X is a 2x1 vector
2088 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2090 cout << indexed(A, i, i) << endl;
2093 ex e = indexed(A, i, j) * indexed(X, j);
2094 cout << e.simplify_indexed() << endl;
2095 // -> [[2*y+x],[4*y+3*x]].i
2097 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2098 cout << e.simplify_indexed() << endl;
2099 // -> [[3*y+3*x,6*y+2*x]].j
2103 You can of course obtain the same results with the @code{matrix::add()},
2104 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2105 but with indices you don't have to worry about transposing matrices.
2107 Matrix indices always start at 0 and their dimension must match the number
2108 of rows/columns of the matrix. Matrices with one row or one column are
2109 vectors and can have one or two indices (it doesn't matter whether it's a
2110 row or a column vector). Other matrices must have two indices.
2112 You should be careful when using indices with variance on matrices. GiNaC
2113 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2114 @samp{F.mu.nu} are different matrices. In this case you should use only
2115 one form for @samp{F} and explicitly multiply it with a matrix representation
2116 of the metric tensor.
2119 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2120 @c node-name, next, previous, up
2121 @section Non-commutative objects
2123 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2124 non-commutative objects are built-in which are mostly of use in high energy
2128 @item Clifford (Dirac) algebra (class @code{clifford})
2129 @item su(3) Lie algebra (class @code{color})
2130 @item Matrices (unindexed) (class @code{matrix})
2133 The @code{clifford} and @code{color} classes are subclasses of
2134 @code{indexed} because the elements of these algebras ususally carry
2135 indices. The @code{matrix} class is described in more detail in
2138 Unlike most computer algebra systems, GiNaC does not primarily provide an
2139 operator (often denoted @samp{&*}) for representing inert products of
2140 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2141 classes of objects involved, and non-commutative products are formed with
2142 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2143 figuring out by itself which objects commute and will group the factors
2144 by their class. Consider this example:
2148 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2149 idx a(symbol("a"), 8), b(symbol("b"), 8);
2150 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2152 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2156 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2157 groups the non-commutative factors (the gammas and the su(3) generators)
2158 together while preserving the order of factors within each class (because
2159 Clifford objects commute with color objects). The resulting expression is a
2160 @emph{commutative} product with two factors that are themselves non-commutative
2161 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2162 parentheses are placed around the non-commutative products in the output.
2164 @cindex @code{ncmul} (class)
2165 Non-commutative products are internally represented by objects of the class
2166 @code{ncmul}, as opposed to commutative products which are handled by the
2167 @code{mul} class. You will normally not have to worry about this distinction,
2170 The advantage of this approach is that you never have to worry about using
2171 (or forgetting to use) a special operator when constructing non-commutative
2172 expressions. Also, non-commutative products in GiNaC are more intelligent
2173 than in other computer algebra systems; they can, for example, automatically
2174 canonicalize themselves according to rules specified in the implementation
2175 of the non-commutative classes. The drawback is that to work with other than
2176 the built-in algebras you have to implement new classes yourself. Symbols
2177 always commute and it's not possible to construct non-commutative products
2178 using symbols to represent the algebra elements or generators. User-defined
2179 functions can, however, be specified as being non-commutative.
2181 @cindex @code{return_type()}
2182 @cindex @code{return_type_tinfo()}
2183 Information about the commutativity of an object or expression can be
2184 obtained with the two member functions
2187 unsigned ex::return_type(void) const;
2188 unsigned ex::return_type_tinfo(void) const;
2191 The @code{return_type()} function returns one of three values (defined in
2192 the header file @file{flags.h}), corresponding to three categories of
2193 expressions in GiNaC:
2196 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2197 classes are of this kind.
2198 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2199 certain class of non-commutative objects which can be determined with the
2200 @code{return_type_tinfo()} method. Expressions of this category commute
2201 with everything except @code{noncommutative} expressions of the same
2203 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2204 of non-commutative objects of different classes. Expressions of this
2205 category don't commute with any other @code{noncommutative} or
2206 @code{noncommutative_composite} expressions.
2209 The value returned by the @code{return_type_tinfo()} method is valid only
2210 when the return type of the expression is @code{noncommutative}. It is a
2211 value that is unique to the class of the object and usually one of the
2212 constants in @file{tinfos.h}, or derived therefrom.
2214 Here are a couple of examples:
2217 @multitable @columnfractions 0.33 0.33 0.34
2218 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2219 @item @code{42} @tab @code{commutative} @tab -
2220 @item @code{2*x-y} @tab @code{commutative} @tab -
2221 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2222 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2223 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2224 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2228 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2229 @code{TINFO_clifford} for objects with a representation label of zero.
2230 Other representation labels yield a different @code{return_type_tinfo()},
2231 but it's the same for any two objects with the same label. This is also true
2234 A last note: With the exception of matrices, positive integer powers of
2235 non-commutative objects are automatically expanded in GiNaC. For example,
2236 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2237 non-commutative expressions).
2240 @cindex @code{clifford} (class)
2241 @subsection Clifford algebra
2243 @cindex @code{dirac_gamma()}
2244 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2245 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2246 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2247 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2250 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2253 which takes two arguments: the index and a @dfn{representation label} in the
2254 range 0 to 255 which is used to distinguish elements of different Clifford
2255 algebras (this is also called a @dfn{spin line index}). Gammas with different
2256 labels commute with each other. The dimension of the index can be 4 or (in
2257 the framework of dimensional regularization) any symbolic value. Spinor
2258 indices on Dirac gammas are not supported in GiNaC.
2260 @cindex @code{dirac_ONE()}
2261 The unity element of a Clifford algebra is constructed by
2264 ex dirac_ONE(unsigned char rl = 0);
2267 @cindex @code{dirac_gamma5()}
2268 and there's a special element @samp{gamma5} that commutes with all other
2269 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2273 ex dirac_gamma5(unsigned char rl = 0);
2276 @cindex @code{dirac_gamma6()}
2277 @cindex @code{dirac_gamma7()}
2278 The two additional functions
2281 ex dirac_gamma6(unsigned char rl = 0);
2282 ex dirac_gamma7(unsigned char rl = 0);
2285 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2288 @cindex @code{dirac_slash()}
2289 Finally, the function
2292 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2295 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2296 whose dimension is given by the @code{dim} argument.
2298 In products of dirac gammas, superfluous unity elements are automatically
2299 removed, squares are replaced by their values and @samp{gamma5} is
2300 anticommuted to the front. The @code{simplify_indexed()} function performs
2301 contractions in gamma strings, for example
2306 symbol a("a"), b("b"), D("D");
2307 varidx mu(symbol("mu"), D);
2308 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2309 * dirac_gamma(mu.toggle_variance());
2311 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2312 e = e.simplify_indexed();
2314 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2315 cout << e.subs(D == 4) << endl;
2316 // -> -2*gamma~symbol10*a.symbol10
2317 // [ == -2 * dirac_slash(a, D) ]
2322 @cindex @code{dirac_trace()}
2323 To calculate the trace of an expression containing strings of Dirac gammas
2324 you use the function
2327 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2330 This function takes the trace of all gammas with the specified representation
2331 label; gammas with other labels are left standing. The last argument to
2332 @code{dirac_trace()} is the value to be returned for the trace of the unity
2333 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2334 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2335 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2336 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2337 This @samp{gamma5} scheme is described in greater detail in
2338 @cite{The Role of gamma5 in Dimensional Regularization}.
2340 The value of the trace itself is also usually different in 4 and in
2341 @math{D != 4} dimensions:
2346 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2347 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2348 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2349 cout << dirac_trace(e).simplify_indexed() << endl;
2356 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2357 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2358 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2359 cout << dirac_trace(e).simplify_indexed() << endl;
2360 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2364 Here is an example for using @code{dirac_trace()} to compute a value that
2365 appears in the calculation of the one-loop vacuum polarization amplitude in
2370 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2371 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2374 sp.add(l, l, pow(l, 2));
2375 sp.add(l, q, ldotq);
2377 ex e = dirac_gamma(mu) *
2378 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2379 dirac_gamma(mu.toggle_variance()) *
2380 (dirac_slash(l, D) + m * dirac_ONE());
2381 e = dirac_trace(e).simplify_indexed(sp);
2382 e = e.collect(lst(l, ldotq, m));
2384 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2388 The @code{canonicalize_clifford()} function reorders all gamma products that
2389 appear in an expression to a canonical (but not necessarily simple) form.
2390 You can use this to compare two expressions or for further simplifications:
2394 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2395 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2397 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2399 e = canonicalize_clifford(e);
2406 @cindex @code{color} (class)
2407 @subsection Color algebra
2409 @cindex @code{color_T()}
2410 For computations in quantum chromodynamics, GiNaC implements the base elements
2411 and structure constants of the su(3) Lie algebra (color algebra). The base
2412 elements @math{T_a} are constructed by the function
2415 ex color_T(const ex & a, unsigned char rl = 0);
2418 which takes two arguments: the index and a @dfn{representation label} in the
2419 range 0 to 255 which is used to distinguish elements of different color
2420 algebras. Objects with different labels commute with each other. The
2421 dimension of the index must be exactly 8 and it should be of class @code{idx},
2424 @cindex @code{color_ONE()}
2425 The unity element of a color algebra is constructed by
2428 ex color_ONE(unsigned char rl = 0);
2431 @cindex @code{color_d()}
2432 @cindex @code{color_f()}
2436 ex color_d(const ex & a, const ex & b, const ex & c);
2437 ex color_f(const ex & a, const ex & b, const ex & c);
2440 create the symmetric and antisymmetric structure constants @math{d_abc} and
2441 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2442 and @math{[T_a, T_b] = i f_abc T_c}.
2444 @cindex @code{color_h()}
2445 There's an additional function
2448 ex color_h(const ex & a, const ex & b, const ex & c);
2451 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2453 The function @code{simplify_indexed()} performs some simplifications on
2454 expressions containing color objects:
2459 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2460 k(symbol("k"), 8), l(symbol("l"), 8);
2462 e = color_d(a, b, l) * color_f(a, b, k);
2463 cout << e.simplify_indexed() << endl;
2466 e = color_d(a, b, l) * color_d(a, b, k);
2467 cout << e.simplify_indexed() << endl;
2470 e = color_f(l, a, b) * color_f(a, b, k);
2471 cout << e.simplify_indexed() << endl;
2474 e = color_h(a, b, c) * color_h(a, b, c);
2475 cout << e.simplify_indexed() << endl;
2478 e = color_h(a, b, c) * color_T(b) * color_T(c);
2479 cout << e.simplify_indexed() << endl;
2482 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2483 cout << e.simplify_indexed() << endl;
2486 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2487 cout << e.simplify_indexed() << endl;
2488 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2492 @cindex @code{color_trace()}
2493 To calculate the trace of an expression containing color objects you use the
2497 ex color_trace(const ex & e, unsigned char rl = 0);
2500 This function takes the trace of all color @samp{T} objects with the
2501 specified representation label; @samp{T}s with other labels are left
2502 standing. For example:
2506 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2508 // -> -I*f.a.c.b+d.a.c.b
2513 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2514 @c node-name, next, previous, up
2515 @chapter Methods and Functions
2518 In this chapter the most important algorithms provided by GiNaC will be
2519 described. Some of them are implemented as functions on expressions,
2520 others are implemented as methods provided by expression objects. If
2521 they are methods, there exists a wrapper function around it, so you can
2522 alternatively call it in a functional way as shown in the simple
2527 cout << "As method: " << sin(1).evalf() << endl;
2528 cout << "As function: " << evalf(sin(1)) << endl;
2532 @cindex @code{subs()}
2533 The general rule is that wherever methods accept one or more parameters
2534 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2535 wrapper accepts is the same but preceded by the object to act on
2536 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2537 most natural one in an OO model but it may lead to confusion for MapleV
2538 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2539 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2540 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2541 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2542 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2543 here. Also, users of MuPAD will in most cases feel more comfortable
2544 with GiNaC's convention. All function wrappers are implemented
2545 as simple inline functions which just call the corresponding method and
2546 are only provided for users uncomfortable with OO who are dead set to
2547 avoid method invocations. Generally, nested function wrappers are much
2548 harder to read than a sequence of methods and should therefore be
2549 avoided if possible. On the other hand, not everything in GiNaC is a
2550 method on class @code{ex} and sometimes calling a function cannot be
2554 * Information About Expressions::
2555 * Substituting Expressions::
2556 * Pattern Matching and Advanced Substitutions::
2557 * Applying a Function on Subexpressions::
2558 * Polynomial Arithmetic:: Working with polynomials.
2559 * Rational Expressions:: Working with rational functions.
2560 * Symbolic Differentiation::
2561 * Series Expansion:: Taylor and Laurent expansion.
2563 * Built-in Functions:: List of predefined mathematical functions.
2564 * Input/Output:: Input and output of expressions.
2568 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2569 @c node-name, next, previous, up
2570 @section Getting information about expressions
2572 @subsection Checking expression types
2573 @cindex @code{is_a<@dots{}>()}
2574 @cindex @code{is_exactly_a<@dots{}>()}
2575 @cindex @code{ex_to<@dots{}>()}
2576 @cindex Converting @code{ex} to other classes
2577 @cindex @code{info()}
2578 @cindex @code{return_type()}
2579 @cindex @code{return_type_tinfo()}
2581 Sometimes it's useful to check whether a given expression is a plain number,
2582 a sum, a polynomial with integer coefficients, or of some other specific type.
2583 GiNaC provides a couple of functions for this:
2586 bool is_a<T>(const ex & e);
2587 bool is_exactly_a<T>(const ex & e);
2588 bool ex::info(unsigned flag);
2589 unsigned ex::return_type(void) const;
2590 unsigned ex::return_type_tinfo(void) const;
2593 When the test made by @code{is_a<T>()} returns true, it is safe to call
2594 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2595 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2596 example, assuming @code{e} is an @code{ex}:
2601 if (is_a<numeric>(e))
2602 numeric n = ex_to<numeric>(e);
2607 @code{is_a<T>(e)} allows you to check whether the top-level object of
2608 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2609 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2610 e.g., for checking whether an expression is a number, a sum, or a product:
2617 is_a<numeric>(e1); // true
2618 is_a<numeric>(e2); // false
2619 is_a<add>(e1); // false
2620 is_a<add>(e2); // true
2621 is_a<mul>(e1); // false
2622 is_a<mul>(e2); // false
2626 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2627 top-level object of an expression @samp{e} is an instance of the GiNaC
2628 class @samp{T}, not including parent classes.
2630 The @code{info()} method is used for checking certain attributes of
2631 expressions. The possible values for the @code{flag} argument are defined
2632 in @file{ginac/flags.h}, the most important being explained in the following
2636 @multitable @columnfractions .30 .70
2637 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2638 @item @code{numeric}
2639 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2641 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2642 @item @code{rational}
2643 @tab @dots{}an exact rational number (integers are rational, too)
2644 @item @code{integer}
2645 @tab @dots{}a (non-complex) integer
2646 @item @code{crational}
2647 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2648 @item @code{cinteger}
2649 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2650 @item @code{positive}
2651 @tab @dots{}not complex and greater than 0
2652 @item @code{negative}
2653 @tab @dots{}not complex and less than 0
2654 @item @code{nonnegative}
2655 @tab @dots{}not complex and greater than or equal to 0
2657 @tab @dots{}an integer greater than 0
2659 @tab @dots{}an integer less than 0
2660 @item @code{nonnegint}
2661 @tab @dots{}an integer greater than or equal to 0
2663 @tab @dots{}an even integer
2665 @tab @dots{}an odd integer
2667 @tab @dots{}a prime integer (probabilistic primality test)
2668 @item @code{relation}
2669 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2670 @item @code{relation_equal}
2671 @tab @dots{}a @code{==} relation
2672 @item @code{relation_not_equal}
2673 @tab @dots{}a @code{!=} relation
2674 @item @code{relation_less}
2675 @tab @dots{}a @code{<} relation
2676 @item @code{relation_less_or_equal}
2677 @tab @dots{}a @code{<=} relation
2678 @item @code{relation_greater}
2679 @tab @dots{}a @code{>} relation
2680 @item @code{relation_greater_or_equal}
2681 @tab @dots{}a @code{>=} relation
2683 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2685 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2686 @item @code{polynomial}
2687 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2688 @item @code{integer_polynomial}
2689 @tab @dots{}a polynomial with (non-complex) integer coefficients
2690 @item @code{cinteger_polynomial}
2691 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2692 @item @code{rational_polynomial}
2693 @tab @dots{}a polynomial with (non-complex) rational coefficients
2694 @item @code{crational_polynomial}
2695 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2696 @item @code{rational_function}
2697 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2698 @item @code{algebraic}
2699 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2703 To determine whether an expression is commutative or non-commutative and if
2704 so, with which other expressions it would commute, you use the methods
2705 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2706 for an explanation of these.
2709 @subsection Accessing subexpressions
2710 @cindex @code{nops()}
2713 @cindex @code{relational} (class)
2715 GiNaC provides the two methods
2718 unsigned ex::nops();
2719 ex ex::op(unsigned i);
2722 for accessing the subexpressions in the container-like GiNaC classes like
2723 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2724 determines the number of subexpressions (@samp{operands}) contained, while
2725 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2726 In the case of a @code{power} object, @code{op(0)} will return the basis
2727 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2728 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2730 The left-hand and right-hand side expressions of objects of class
2731 @code{relational} (and only of these) can also be accessed with the methods
2739 @subsection Comparing expressions
2740 @cindex @code{is_equal()}
2741 @cindex @code{is_zero()}
2743 Expressions can be compared with the usual C++ relational operators like
2744 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2745 the result is usually not determinable and the result will be @code{false},
2746 except in the case of the @code{!=} operator. You should also be aware that
2747 GiNaC will only do the most trivial test for equality (subtracting both
2748 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2751 Actually, if you construct an expression like @code{a == b}, this will be
2752 represented by an object of the @code{relational} class (@pxref{Relations})
2753 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2755 There are also two methods
2758 bool ex::is_equal(const ex & other);
2762 for checking whether one expression is equal to another, or equal to zero,
2765 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2766 GiNaC header files. This method is however only to be used internally by
2767 GiNaC to establish a canonical sort order for terms, and using it to compare
2768 expressions will give very surprising results.
2771 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2772 @c node-name, next, previous, up
2773 @section Substituting expressions
2774 @cindex @code{subs()}
2776 Algebraic objects inside expressions can be replaced with arbitrary
2777 expressions via the @code{.subs()} method:
2780 ex ex::subs(const ex & e);
2781 ex ex::subs(const lst & syms, const lst & repls);
2784 In the first form, @code{subs()} accepts a relational of the form
2785 @samp{object == expression} or a @code{lst} of such relationals:
2789 symbol x("x"), y("y");
2791 ex e1 = 2*x^2-4*x+3;
2792 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2796 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2801 If you specify multiple substitutions, they are performed in parallel, so e.g.
2802 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2804 The second form of @code{subs()} takes two lists, one for the objects to be
2805 replaced and one for the expressions to be substituted (both lists must
2806 contain the same number of elements). Using this form, you would write
2807 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2809 @code{subs()} performs syntactic substitution of any complete algebraic
2810 object; it does not try to match sub-expressions as is demonstrated by the
2815 symbol x("x"), y("y"), z("z");
2817 ex e1 = pow(x+y, 2);
2818 cout << e1.subs(x+y == 4) << endl;
2821 ex e2 = sin(x)*sin(y)*cos(x);
2822 cout << e2.subs(sin(x) == cos(x)) << endl;
2823 // -> cos(x)^2*sin(y)
2826 cout << e3.subs(x+y == 4) << endl;
2828 // (and not 4+z as one might expect)
2832 A more powerful form of substitution using wildcards is described in the
2836 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2837 @c node-name, next, previous, up
2838 @section Pattern matching and advanced substitutions
2839 @cindex @code{wildcard} (class)
2840 @cindex Pattern matching
2842 GiNaC allows the use of patterns for checking whether an expression is of a
2843 certain form or contains subexpressions of a certain form, and for
2844 substituting expressions in a more general way.
2846 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2847 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2848 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2849 an unsigned integer number to allow having multiple different wildcards in a
2850 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2851 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2855 ex wild(unsigned label = 0);
2858 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2861 Some examples for patterns:
2863 @multitable @columnfractions .5 .5
2864 @item @strong{Constructed as} @tab @strong{Output as}
2865 @item @code{wild()} @tab @samp{$0}
2866 @item @code{pow(x,wild())} @tab @samp{x^$0}
2867 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2868 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2874 @item Wildcards behave like symbols and are subject to the same algebraic
2875 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2876 @item As shown in the last example, to use wildcards for indices you have to
2877 use them as the value of an @code{idx} object. This is because indices must
2878 always be of class @code{idx} (or a subclass).
2879 @item Wildcards only represent expressions or subexpressions. It is not
2880 possible to use them as placeholders for other properties like index
2881 dimension or variance, representation labels, symmetry of indexed objects
2883 @item Because wildcards are commutative, it is not possible to use wildcards
2884 as part of noncommutative products.
2885 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2886 are also valid patterns.
2889 @cindex @code{match()}
2890 The most basic application of patterns is to check whether an expression
2891 matches a given pattern. This is done by the function
2894 bool ex::match(const ex & pattern);
2895 bool ex::match(const ex & pattern, lst & repls);
2898 This function returns @code{true} when the expression matches the pattern
2899 and @code{false} if it doesn't. If used in the second form, the actual
2900 subexpressions matched by the wildcards get returned in the @code{repls}
2901 object as a list of relations of the form @samp{wildcard == expression}.
2902 If @code{match()} returns false, the state of @code{repls} is undefined.
2903 For reproducible results, the list should be empty when passed to
2904 @code{match()}, but it is also possible to find similarities in multiple
2905 expressions by passing in the result of a previous match.
2907 The matching algorithm works as follows:
2910 @item A single wildcard matches any expression. If one wildcard appears
2911 multiple times in a pattern, it must match the same expression in all
2912 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2913 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2914 @item If the expression is not of the same class as the pattern, the match
2915 fails (i.e. a sum only matches a sum, a function only matches a function,
2917 @item If the pattern is a function, it only matches the same function
2918 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2919 @item Except for sums and products, the match fails if the number of
2920 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2922 @item If there are no subexpressions, the expressions and the pattern must
2923 be equal (in the sense of @code{is_equal()}).
2924 @item Except for sums and products, each subexpression (@code{op()}) must
2925 match the corresponding subexpression of the pattern.
2928 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2929 account for their commutativity and associativity:
2932 @item If the pattern contains a term or factor that is a single wildcard,
2933 this one is used as the @dfn{global wildcard}. If there is more than one
2934 such wildcard, one of them is chosen as the global wildcard in a random
2936 @item Every term/factor of the pattern, except the global wildcard, is
2937 matched against every term of the expression in sequence. If no match is
2938 found, the whole match fails. Terms that did match are not considered in
2940 @item If there are no unmatched terms left, the match succeeds. Otherwise
2941 the match fails unless there is a global wildcard in the pattern, in
2942 which case this wildcard matches the remaining terms.
2945 In general, having more than one single wildcard as a term of a sum or a
2946 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2949 Here are some examples in @command{ginsh} to demonstrate how it works (the
2950 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2951 match fails, and the list of wildcard replacements otherwise):
2954 > match((x+y)^a,(x+y)^a);
2956 > match((x+y)^a,(x+y)^b);
2958 > match((x+y)^a,$1^$2);
2960 > match((x+y)^a,$1^$1);
2962 > match((x+y)^(x+y),$1^$1);
2964 > match((x+y)^(x+y),$1^$2);
2966 > match((a+b)*(a+c),($1+b)*($1+c));
2968 > match((a+b)*(a+c),(a+$1)*(a+$2));
2970 (Unpredictable. The result might also be [$1==c,$2==b].)
2971 > match((a+b)*(a+c),($1+$2)*($1+$3));
2972 (The result is undefined. Due to the sequential nature of the algorithm
2973 and the re-ordering of terms in GiNaC, the match for the first factor
2974 may be @{$1==a,$2==b@} in which case the match for the second factor
2975 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
2977 > match(a*(x+y)+a*z+b,a*$1+$2);
2978 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
2979 @{$1=x+y,$2=a*z+b@}.)
2980 > match(a+b+c+d+e+f,c);
2982 > match(a+b+c+d+e+f,c+$0);
2984 > match(a+b+c+d+e+f,c+e+$0);
2986 > match(a+b,a+b+$0);
2988 > match(a*b^2,a^$1*b^$2);
2990 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
2991 even though a==a^1.)
2992 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
2994 > match(atan2(y,x^2),atan2(y,$0));
2998 @cindex @code{has()}
2999 A more general way to look for patterns in expressions is provided by the
3003 bool ex::has(const ex & pattern);
3006 This function checks whether a pattern is matched by an expression itself or
3007 by any of its subexpressions.
3009 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3010 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3013 > has(x*sin(x+y+2*a),y);
3015 > has(x*sin(x+y+2*a),x+y);
3017 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3018 has the subexpressions "x", "y" and "2*a".)
3019 > has(x*sin(x+y+2*a),x+y+$1);
3021 (But this is possible.)
3022 > has(x*sin(2*(x+y)+2*a),x+y);
3024 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3025 which "x+y" is not a subexpression.)
3028 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3030 > has(4*x^2-x+3,$1*x);
3032 > has(4*x^2+x+3,$1*x);
3034 (Another possible pitfall. The first expression matches because the term
3035 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3036 contains a linear term you should use the coeff() function instead.)
3039 @cindex @code{find()}
3043 bool ex::find(const ex & pattern, lst & found);
3046 works a bit like @code{has()} but it doesn't stop upon finding the first
3047 match. Instead, it appends all found matches to the specified list. If there
3048 are multiple occurrences of the same expression, it is entered only once to
3049 the list. @code{find()} returns false if no matches were found (in
3050 @command{ginsh}, it returns an empty list):
3053 > find(1+x+x^2+x^3,x);
3055 > find(1+x+x^2+x^3,y);
3057 > find(1+x+x^2+x^3,x^$1);
3059 (Note the absence of "x".)
3060 > expand((sin(x)+sin(y))*(a+b));
3061 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3066 @cindex @code{subs()}
3067 Probably the most useful application of patterns is to use them for
3068 substituting expressions with the @code{subs()} method. Wildcards can be
3069 used in the search patterns as well as in the replacement expressions, where
3070 they get replaced by the expressions matched by them. @code{subs()} doesn't
3071 know anything about algebra; it performs purely syntactic substitutions.
3076 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3078 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3080 > subs((a+b+c)^2,a+b=x);
3082 > subs((a+b+c)^2,a+b+$1==x+$1);
3084 > subs(a+2*b,a+b=x);
3086 > subs(4*x^3-2*x^2+5*x-1,x==a);
3088 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3090 > subs(sin(1+sin(x)),sin($1)==cos($1));
3092 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3096 The last example would be written in C++ in this way:
3100 symbol a("a"), b("b"), x("x"), y("y");
3101 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3102 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3103 cout << e.expand() << endl;
3109 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3110 @c node-name, next, previous, up
3111 @section Applying a Function on Subexpressions
3112 @cindex Tree traversal
3113 @cindex @code{map()}
3115 Sometimes you may want to perform an operation on specific parts of an
3116 expression while leaving the general structure of it intact. An example
3117 of this would be a matrix trace operation: the trace of a sum is the sum
3118 of the traces of the individual terms. That is, the trace should @dfn{map}
3119 on the sum, by applying itself to each of the sum's operands. It is possible
3120 to do this manually which usually results in code like this:
3125 if (is_a<matrix>(e))
3126 return ex_to<matrix>(e).trace();
3127 else if (is_a<add>(e)) @{
3129 for (unsigned i=0; i<e.nops(); i++)
3130 sum += calc_trace(e.op(i));
3132 @} else if (is_a<mul>)(e)) @{
3140 This is, however, slightly inefficient (if the sum is very large it can take
3141 a long time to add the terms one-by-one), and its applicability is limited to
3142 a rather small class of expressions. If @code{calc_trace()} is called with
3143 a relation or a list as its argument, you will probably want the trace to
3144 be taken on both sides of the relation or of all elements of the list.
3146 GiNaC offers the @code{map()} method to aid in the implementation of such
3150 static ex ex::map(map_function & f) const;
3151 static ex ex::map(ex (*f)(const ex & e)) const;
3154 In the first (preferred) form, @code{map()} takes a function object that
3155 is subclassed from the @code{map_function} class. In the second form, it
3156 takes a pointer to a function that accepts and returns an expression.
3157 @code{map()} constructs a new expression of the same type, applying the
3158 specified function on all subexpressions (in the sense of @code{op()}),
3161 The use of a function object makes it possible to supply more arguments to
3162 the function that is being mapped, or to keep local state information.
3163 The @code{map_function} class declares a virtual function call operator
3164 that you can overload. Here is a sample implementation of @code{calc_trace()}
3165 that uses @code{map()} in a recursive fashion:
3168 struct calc_trace : public map_function @{
3169 ex operator()(const ex &e)
3171 if (is_a<matrix>(e))
3172 return ex_to<matrix>(e).trace();
3173 else if (is_a<mul>(e)) @{
3176 return e.map(*this);
3181 This function object could then be used like this:
3185 ex M = ... // expression with matrices
3186 calc_trace do_trace;
3187 ex tr = do_trace(M);
3191 @command{ginsh} offers a slightly different implementation of @code{map()}
3192 that allows applying algebraic functions to operands. The second argument
3193 to @code{map()} is an expression containing the wildcard @samp{$0} which
3194 acts as the placeholder for the operands:
3199 > map(a+2*b,sin($0));
3201 > map(@{a,b,c@},$0^2+$0);
3202 @{a^2+a,b^2+b,c^2+c@}
3205 Note that it is only possible to use algebraic functions in the second
3206 argument. You can not use functions like @samp{diff()}, @samp{op()},
3207 @samp{subs()} etc. because these are evaluated immediately:
3210 > map(@{a,b,c@},diff($0,a));
3212 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3213 to "map(@{a,b,c@},0)".
3217 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3218 @c node-name, next, previous, up
3219 @section Polynomial arithmetic
3221 @subsection Expanding and collecting
3222 @cindex @code{expand()}
3223 @cindex @code{collect()}
3225 A polynomial in one or more variables has many equivalent
3226 representations. Some useful ones serve a specific purpose. Consider
3227 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3228 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3229 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3230 representations are the recursive ones where one collects for exponents
3231 in one of the three variable. Since the factors are themselves
3232 polynomials in the remaining two variables the procedure can be
3233 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
3234 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3237 To bring an expression into expanded form, its method
3243 may be called. In our example above, this corresponds to @math{4*x*y +
3244 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3245 GiNaC is not easily guessable you should be prepared to see different
3246 orderings of terms in such sums!
3248 Another useful representation of multivariate polynomials is as a
3249 univariate polynomial in one of the variables with the coefficients
3250 being polynomials in the remaining variables. The method
3251 @code{collect()} accomplishes this task:
3254 ex ex::collect(const ex & s, bool distributed = false);
3257 The first argument to @code{collect()} can also be a list of objects in which
3258 case the result is either a recursively collected polynomial, or a polynomial
3259 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3260 by the @code{distributed} flag.
3262 Note that the original polynomial needs to be in expanded form (for the
3263 variables concerned) in order for @code{collect()} to be able to find the
3264 coefficients properly.
3266 The following @command{ginsh} transcript shows an application of @code{collect()}
3267 together with @code{find()}:
3270 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3271 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
3272 > collect(a,@{p,q@});
3273 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)
3274 > collect(a,find(a,sin($1)));
3275 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3276 > collect(a,@{find(a,sin($1)),p,q@});
3277 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3278 > collect(a,@{find(a,sin($1)),d@});
3279 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3282 @subsection Degree and coefficients
3283 @cindex @code{degree()}
3284 @cindex @code{ldegree()}
3285 @cindex @code{coeff()}
3287 The degree and low degree of a polynomial can be obtained using the two
3291 int ex::degree(const ex & s);
3292 int ex::ldegree(const ex & s);
3295 which also work reliably on non-expanded input polynomials (they even work
3296 on rational functions, returning the asymptotic degree). To extract
3297 a coefficient with a certain power from an expanded polynomial you use
3300 ex ex::coeff(const ex & s, int n);
3303 You can also obtain the leading and trailing coefficients with the methods
3306 ex ex::lcoeff(const ex & s);
3307 ex ex::tcoeff(const ex & s);
3310 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3313 An application is illustrated in the next example, where a multivariate
3314 polynomial is analyzed:
3317 #include <ginac/ginac.h>
3318 using namespace std;
3319 using namespace GiNaC;
3323 symbol x("x"), y("y");
3324 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3325 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3326 ex Poly = PolyInp.expand();
3328 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3329 cout << "The x^" << i << "-coefficient is "
3330 << Poly.coeff(x,i) << endl;
3332 cout << "As polynomial in y: "
3333 << Poly.collect(y) << endl;
3337 When run, it returns an output in the following fashion:
3340 The x^0-coefficient is y^2+11*y
3341 The x^1-coefficient is 5*y^2-2*y
3342 The x^2-coefficient is -1
3343 The x^3-coefficient is 4*y
3344 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3347 As always, the exact output may vary between different versions of GiNaC
3348 or even from run to run since the internal canonical ordering is not
3349 within the user's sphere of influence.
3351 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3352 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3353 with non-polynomial expressions as they not only work with symbols but with
3354 constants, functions and indexed objects as well:
3358 symbol a("a"), b("b"), c("c");
3359 idx i(symbol("i"), 3);
3361 ex e = pow(sin(x) - cos(x), 4);
3362 cout << e.degree(cos(x)) << endl;
3364 cout << e.expand().coeff(sin(x), 3) << endl;
3367 e = indexed(a+b, i) * indexed(b+c, i);
3368 e = e.expand(expand_options::expand_indexed);
3369 cout << e.collect(indexed(b, i)) << endl;
3370 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3375 @subsection Polynomial division
3376 @cindex polynomial division
3379 @cindex pseudo-remainder
3380 @cindex @code{quo()}
3381 @cindex @code{rem()}
3382 @cindex @code{prem()}
3383 @cindex @code{divide()}
3388 ex quo(const ex & a, const ex & b, const symbol & x);
3389 ex rem(const ex & a, const ex & b, const symbol & x);
3392 compute the quotient and remainder of univariate polynomials in the variable
3393 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3395 The additional function
3398 ex prem(const ex & a, const ex & b, const symbol & x);
3401 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3402 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3404 Exact division of multivariate polynomials is performed by the function
3407 bool divide(const ex & a, const ex & b, ex & q);
3410 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3411 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3412 in which case the value of @code{q} is undefined.
3415 @subsection Unit, content and primitive part
3416 @cindex @code{unit()}
3417 @cindex @code{content()}
3418 @cindex @code{primpart()}
3423 ex ex::unit(const symbol & x);
3424 ex ex::content(const symbol & x);
3425 ex ex::primpart(const symbol & x);
3428 return the unit part, content part, and primitive polynomial of a multivariate
3429 polynomial with respect to the variable @samp{x} (the unit part being the sign
3430 of the leading coefficient, the content part being the GCD of the coefficients,
3431 and the primitive polynomial being the input polynomial divided by the unit and
3432 content parts). The product of unit, content, and primitive part is the
3433 original polynomial.
3436 @subsection GCD and LCM
3439 @cindex @code{gcd()}
3440 @cindex @code{lcm()}
3442 The functions for polynomial greatest common divisor and least common
3443 multiple have the synopsis
3446 ex gcd(const ex & a, const ex & b);
3447 ex lcm(const ex & a, const ex & b);
3450 The functions @code{gcd()} and @code{lcm()} accept two expressions
3451 @code{a} and @code{b} as arguments and return a new expression, their
3452 greatest common divisor or least common multiple, respectively. If the
3453 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3454 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3457 #include <ginac/ginac.h>
3458 using namespace GiNaC;
3462 symbol x("x"), y("y"), z("z");
3463 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3464 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3466 ex P_gcd = gcd(P_a, P_b);
3468 ex P_lcm = lcm(P_a, P_b);
3469 // 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
3474 @subsection Square-free decomposition
3475 @cindex square-free decomposition
3476 @cindex factorization
3477 @cindex @code{sqrfree()}
3479 GiNaC still lacks proper factorization support. Some form of
3480 factorization is, however, easily implemented by noting that factors
3481 appearing in a polynomial with power two or more also appear in the
3482 derivative and hence can easily be found by computing the GCD of the
3483 original polynomial and its derivatives. Any system has an interface
3484 for this so called square-free factorization. So we provide one, too:
3486 ex sqrfree(const ex & a, const lst & l = lst());
3488 Here is an example that by the way illustrates how the result may depend
3489 on the order of differentiation:
3492 symbol x("x"), y("y");
3493 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3495 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3496 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3498 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3499 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3501 cout << sqrfree(BiVarPol) << endl;
3502 // -> depending on luck, any of the above
3507 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3508 @c node-name, next, previous, up
3509 @section Rational expressions
3511 @subsection The @code{normal} method
3512 @cindex @code{normal()}
3513 @cindex simplification
3514 @cindex temporary replacement
3516 Some basic form of simplification of expressions is called for frequently.
3517 GiNaC provides the method @code{.normal()}, which converts a rational function
3518 into an equivalent rational function of the form @samp{numerator/denominator}
3519 where numerator and denominator are coprime. If the input expression is already
3520 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3521 otherwise it performs fraction addition and multiplication.
3523 @code{.normal()} can also be used on expressions which are not rational functions
3524 as it will replace all non-rational objects (like functions or non-integer
3525 powers) by temporary symbols to bring the expression to the domain of rational
3526 functions before performing the normalization, and re-substituting these
3527 symbols afterwards. This algorithm is also available as a separate method
3528 @code{.to_rational()}, described below.
3530 This means that both expressions @code{t1} and @code{t2} are indeed
3531 simplified in this little program:
3534 #include <ginac/ginac.h>
3535 using namespace GiNaC;
3540 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3541 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3542 std::cout << "t1 is " << t1.normal() << std::endl;
3543 std::cout << "t2 is " << t2.normal() << std::endl;
3547 Of course this works for multivariate polynomials too, so the ratio of
3548 the sample-polynomials from the section about GCD and LCM above would be
3549 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3552 @subsection Numerator and denominator
3555 @cindex @code{numer()}
3556 @cindex @code{denom()}
3557 @cindex @code{numer_denom()}
3559 The numerator and denominator of an expression can be obtained with
3564 ex ex::numer_denom();
3567 These functions will first normalize the expression as described above and
3568 then return the numerator, denominator, or both as a list, respectively.
3569 If you need both numerator and denominator, calling @code{numer_denom()} is
3570 faster than using @code{numer()} and @code{denom()} separately.
3573 @subsection Converting to a rational expression
3574 @cindex @code{to_rational()}
3576 Some of the methods described so far only work on polynomials or rational
3577 functions. GiNaC provides a way to extend the domain of these functions to
3578 general expressions by using the temporary replacement algorithm described
3579 above. You do this by calling
3582 ex ex::to_rational(lst &l);
3585 on the expression to be converted. The supplied @code{lst} will be filled
3586 with the generated temporary symbols and their replacement expressions in
3587 a format that can be used directly for the @code{subs()} method. It can also
3588 already contain a list of replacements from an earlier application of
3589 @code{.to_rational()}, so it's possible to use it on multiple expressions
3590 and get consistent results.
3597 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3598 ex b = sin(x) + cos(x);
3601 divide(a.to_rational(l), b.to_rational(l), q);
3602 cout << q.subs(l) << endl;
3606 will print @samp{sin(x)-cos(x)}.
3609 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3610 @c node-name, next, previous, up
3611 @section Symbolic differentiation
3612 @cindex differentiation
3613 @cindex @code{diff()}
3615 @cindex product rule
3617 GiNaC's objects know how to differentiate themselves. Thus, a
3618 polynomial (class @code{add}) knows that its derivative is the sum of
3619 the derivatives of all the monomials:
3622 #include <ginac/ginac.h>
3623 using namespace GiNaC;
3627 symbol x("x"), y("y"), z("z");
3628 ex P = pow(x, 5) + pow(x, 2) + y;
3630 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3631 cout << P.diff(y) << endl; // 1
3632 cout << P.diff(z) << endl; // 0
3636 If a second integer parameter @var{n} is given, the @code{diff} method
3637 returns the @var{n}th derivative.
3639 If @emph{every} object and every function is told what its derivative
3640 is, all derivatives of composed objects can be calculated using the
3641 chain rule and the product rule. Consider, for instance the expression
3642 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3643 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3644 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3645 out that the composition is the generating function for Euler Numbers,
3646 i.e. the so called @var{n}th Euler number is the coefficient of
3647 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3648 identity to code a function that generates Euler numbers in just three
3651 @cindex Euler numbers
3653 #include <ginac/ginac.h>
3654 using namespace GiNaC;
3656 ex EulerNumber(unsigned n)
3659 const ex generator = pow(cosh(x),-1);
3660 return generator.diff(x,n).subs(x==0);
3665 for (unsigned i=0; i<11; i+=2)
3666 std::cout << EulerNumber(i) << std::endl;
3671 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3672 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3673 @code{i} by two since all odd Euler numbers vanish anyways.
3676 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3677 @c node-name, next, previous, up
3678 @section Series expansion
3679 @cindex @code{series()}
3680 @cindex Taylor expansion
3681 @cindex Laurent expansion
3682 @cindex @code{pseries} (class)
3684 Expressions know how to expand themselves as a Taylor series or (more
3685 generally) a Laurent series. As in most conventional Computer Algebra
3686 Systems, no distinction is made between those two. There is a class of
3687 its own for storing such series (@code{class pseries}) and a built-in
3688 function (called @code{Order}) for storing the order term of the series.
3689 As a consequence, if you want to work with series, i.e. multiply two
3690 series, you need to call the method @code{ex::series} again to convert
3691 it to a series object with the usual structure (expansion plus order
3692 term). A sample application from special relativity could read:
3695 #include <ginac/ginac.h>
3696 using namespace std;
3697 using namespace GiNaC;
3701 symbol v("v"), c("c");
3703 ex gamma = 1/sqrt(1 - pow(v/c,2));
3704 ex mass_nonrel = gamma.series(v==0, 10);
3706 cout << "the relativistic mass increase with v is " << endl
3707 << mass_nonrel << endl;
3709 cout << "the inverse square of this series is " << endl
3710 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3714 Only calling the series method makes the last output simplify to
3715 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3716 series raised to the power @math{-2}.
3718 @cindex M@'echain's formula
3719 As another instructive application, let us calculate the numerical
3720 value of Archimedes' constant
3724 (for which there already exists the built-in constant @code{Pi})
3725 using M@'echain's amazing formula
3727 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3730 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3732 We may expand the arcus tangent around @code{0} and insert the fractions
3733 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3734 carries an order term with it and the question arises what the system is
3735 supposed to do when the fractions are plugged into that order term. The
3736 solution is to use the function @code{series_to_poly()} to simply strip
3740 #include <ginac/ginac.h>
3741 using namespace GiNaC;
3743 ex mechain_pi(int degr)
3746 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3747 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3748 -4*pi_expansion.subs(x==numeric(1,239));
3754 using std::cout; // just for fun, another way of...
3755 using std::endl; // ...dealing with this namespace std.
3757 for (int i=2; i<12; i+=2) @{
3758 pi_frac = mechain_pi(i);
3759 cout << i << ":\t" << pi_frac << endl
3760 << "\t" << pi_frac.evalf() << endl;
3766 Note how we just called @code{.series(x,degr)} instead of
3767 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3768 method @code{series()}: if the first argument is a symbol the expression
3769 is expanded in that symbol around point @code{0}. When you run this
3770 program, it will type out:
3774 3.1832635983263598326
3775 4: 5359397032/1706489875
3776 3.1405970293260603143
3777 6: 38279241713339684/12184551018734375
3778 3.141621029325034425
3779 8: 76528487109180192540976/24359780855939418203125
3780 3.141591772182177295
3781 10: 327853873402258685803048818236/104359128170408663038552734375
3782 3.1415926824043995174
3786 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3787 @c node-name, next, previous, up
3788 @section Symmetrization
3789 @cindex @code{symmetrize()}
3790 @cindex @code{antisymmetrize()}
3791 @cindex @code{symmetrize_cyclic()}
3796 ex ex::symmetrize(const lst & l);
3797 ex ex::antisymmetrize(const lst & l);
3798 ex ex::symmetrize_cyclic(const lst & l);
3801 symmetrize an expression by returning the sum over all symmetric,
3802 antisymmetric or cyclic permutations of the specified list of objects,
3803 weighted by the number of permutations.
3805 The three additional methods
3808 ex ex::symmetrize();
3809 ex ex::antisymmetrize();
3810 ex ex::symmetrize_cyclic();
3813 symmetrize or antisymmetrize an expression over its free indices.
3815 Symmetrization is most useful with indexed expressions but can be used with
3816 almost any kind of object (anything that is @code{subs()}able):
3820 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3821 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3823 cout << indexed(A, i, j).symmetrize() << endl;
3824 // -> 1/2*A.j.i+1/2*A.i.j
3825 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3826 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3827 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3828 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
3833 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3834 @c node-name, next, previous, up
3835 @section Predefined mathematical functions
3837 GiNaC contains the following predefined mathematical functions:
3840 @multitable @columnfractions .30 .70
3841 @item @strong{Name} @tab @strong{Function}
3844 @item @code{csgn(x)}
3846 @item @code{sqrt(x)}
3847 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3854 @item @code{asin(x)}
3856 @item @code{acos(x)}
3858 @item @code{atan(x)}
3859 @tab inverse tangent
3860 @item @code{atan2(y, x)}
3861 @tab inverse tangent with two arguments
3862 @item @code{sinh(x)}
3863 @tab hyperbolic sine
3864 @item @code{cosh(x)}
3865 @tab hyperbolic cosine
3866 @item @code{tanh(x)}
3867 @tab hyperbolic tangent
3868 @item @code{asinh(x)}
3869 @tab inverse hyperbolic sine
3870 @item @code{acosh(x)}
3871 @tab inverse hyperbolic cosine
3872 @item @code{atanh(x)}
3873 @tab inverse hyperbolic tangent
3875 @tab exponential function
3877 @tab natural logarithm
3880 @item @code{zeta(x)}
3881 @tab Riemann's zeta function
3882 @item @code{zeta(n, x)}
3883 @tab derivatives of Riemann's zeta function
3884 @item @code{tgamma(x)}
3886 @item @code{lgamma(x)}
3887 @tab logarithm of Gamma function
3888 @item @code{beta(x, y)}
3889 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3891 @tab psi (digamma) function
3892 @item @code{psi(n, x)}
3893 @tab derivatives of psi function (polygamma functions)
3894 @item @code{factorial(n)}
3895 @tab factorial function
3896 @item @code{binomial(n, m)}
3897 @tab binomial coefficients
3898 @item @code{Order(x)}
3899 @tab order term function in truncated power series
3900 @item @code{Derivative(x, l)}
3901 @tab inert partial differentiation operator (used internally)
3906 For functions that have a branch cut in the complex plane GiNaC follows
3907 the conventions for C++ as defined in the ANSI standard as far as
3908 possible. In particular: the natural logarithm (@code{log}) and the
3909 square root (@code{sqrt}) both have their branch cuts running along the
3910 negative real axis where the points on the axis itself belong to the
3911 upper part (i.e. continuous with quadrant II). The inverse
3912 trigonometric and hyperbolic functions are not defined for complex
3913 arguments by the C++ standard, however. In GiNaC we follow the
3914 conventions used by CLN, which in turn follow the carefully designed
3915 definitions in the Common Lisp standard. It should be noted that this
3916 convention is identical to the one used by the C99 standard and by most
3917 serious CAS. It is to be expected that future revisions of the C++
3918 standard incorporate these functions in the complex domain in a manner
3919 compatible with C99.
3922 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3923 @c node-name, next, previous, up
3924 @section Input and output of expressions
3927 @subsection Expression output
3929 @cindex output of expressions
3931 The easiest way to print an expression is to write it to a stream:
3936 ex e = 4.5+pow(x,2)*3/2;
3937 cout << e << endl; // prints '(4.5)+3/2*x^2'
3941 The output format is identical to the @command{ginsh} input syntax and
3942 to that used by most computer algebra systems, but not directly pastable
3943 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3944 is printed as @samp{x^2}).
3946 It is possible to print expressions in a number of different formats with
3950 void ex::print(const print_context & c, unsigned level = 0);
3953 @cindex @code{print_context} (class)
3954 The type of @code{print_context} object passed in determines the format
3955 of the output. The possible types are defined in @file{ginac/print.h}.
3956 All constructors of @code{print_context} and derived classes take an
3957 @code{ostream &} as their first argument.
3959 To print an expression in a way that can be directly used in a C or C++
3960 program, you pass a @code{print_csrc} object like this:
3964 cout << "float f = ";
3965 e.print(print_csrc_float(cout));
3968 cout << "double d = ";
3969 e.print(print_csrc_double(cout));
3972 cout << "cl_N n = ";
3973 e.print(print_csrc_cl_N(cout));
3978 The three possible types mostly affect the way in which floating point
3979 numbers are written.
3981 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3984 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3985 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3986 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3989 The @code{print_context} type @code{print_tree} provides a dump of the
3990 internal structure of an expression for debugging purposes:
3994 e.print(print_tree(cout));
4001 add, hash=0x0, flags=0x3, nops=2
4002 power, hash=0x9, flags=0x3, nops=2
4003 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4004 2 (numeric), hash=0x80000042, flags=0xf
4005 3/2 (numeric), hash=0x80000061, flags=0xf
4008 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4012 This kind of output is also available in @command{ginsh} as the @code{print()}
4015 Another useful output format is for LaTeX parsing in mathematical mode.
4016 It is rather similar to the default @code{print_context} but provides
4017 some braces needed by LaTeX for delimiting boxes and also converts some
4018 common objects to conventional LaTeX names. It is possible to give symbols
4019 a special name for LaTeX output by supplying it as a second argument to
4020 the @code{symbol} constructor.
4022 For example, the code snippet
4027 ex foo = lgamma(x).series(x==0,3);
4028 foo.print(print_latex(std::cout));
4034 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4037 @cindex Tree traversal
4038 If you need any fancy special output format, e.g. for interfacing GiNaC
4039 with other algebra systems or for producing code for different
4040 programming languages, you can always traverse the expression tree yourself:
4043 static void my_print(const ex & e)
4045 if (is_a<function>(e))
4046 cout << ex_to<function>(e).get_name();
4048 cout << e.bp->class_name();
4050 unsigned n = e.nops();
4052 for (unsigned i=0; i<n; i++) @{
4064 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4072 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4073 symbol(y))),numeric(-2)))
4076 If you need an output format that makes it possible to accurately
4077 reconstruct an expression by feeding the output to a suitable parser or
4078 object factory, you should consider storing the expression in an
4079 @code{archive} object and reading the object properties from there.
4080 See the section on archiving for more information.
4083 @subsection Expression input
4084 @cindex input of expressions
4086 GiNaC provides no way to directly read an expression from a stream because
4087 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4088 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4089 @code{y} you defined in your program and there is no way to specify the
4090 desired symbols to the @code{>>} stream input operator.
4092 Instead, GiNaC lets you construct an expression from a string, specifying the
4093 list of symbols to be used:
4097 symbol x("x"), y("y");
4098 ex e("2*x+sin(y)", lst(x, y));
4102 The input syntax is the same as that used by @command{ginsh} and the stream
4103 output operator @code{<<}. The symbols in the string are matched by name to
4104 the symbols in the list and if GiNaC encounters a symbol not specified in
4105 the list it will throw an exception.
4107 With this constructor, it's also easy to implement interactive GiNaC programs:
4112 #include <stdexcept>
4113 #include <ginac/ginac.h>
4114 using namespace std;
4115 using namespace GiNaC;
4122 cout << "Enter an expression containing 'x': ";
4127 cout << "The derivative of " << e << " with respect to x is ";
4128 cout << e.diff(x) << ".\n";
4129 @} catch (exception &p) @{
4130 cerr << p.what() << endl;
4136 @subsection Archiving
4137 @cindex @code{archive} (class)
4140 GiNaC allows creating @dfn{archives} of expressions which can be stored
4141 to or retrieved from files. To create an archive, you declare an object
4142 of class @code{archive} and archive expressions in it, giving each
4143 expression a unique name:
4147 using namespace std;
4148 #include <ginac/ginac.h>
4149 using namespace GiNaC;
4153 symbol x("x"), y("y"), z("z");
4155 ex foo = sin(x + 2*y) + 3*z + 41;
4159 a.archive_ex(foo, "foo");
4160 a.archive_ex(bar, "the second one");
4164 The archive can then be written to a file:
4168 ofstream out("foobar.gar");
4174 The file @file{foobar.gar} contains all information that is needed to
4175 reconstruct the expressions @code{foo} and @code{bar}.
4177 @cindex @command{viewgar}
4178 The tool @command{viewgar} that comes with GiNaC can be used to view
4179 the contents of GiNaC archive files:
4182 $ viewgar foobar.gar
4183 foo = 41+sin(x+2*y)+3*z
4184 the second one = 42+sin(x+2*y)+3*z
4187 The point of writing archive files is of course that they can later be
4193 ifstream in("foobar.gar");
4198 And the stored expressions can be retrieved by their name:
4204 ex ex1 = a2.unarchive_ex(syms, "foo");
4205 ex ex2 = a2.unarchive_ex(syms, "the second one");
4207 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4208 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4209 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4213 Note that you have to supply a list of the symbols which are to be inserted
4214 in the expressions. Symbols in archives are stored by their name only and
4215 if you don't specify which symbols you have, unarchiving the expression will
4216 create new symbols with that name. E.g. if you hadn't included @code{x} in
4217 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4218 have had no effect because the @code{x} in @code{ex1} would have been a
4219 different symbol than the @code{x} which was defined at the beginning of
4220 the program, altough both would appear as @samp{x} when printed.
4222 You can also use the information stored in an @code{archive} object to
4223 output expressions in a format suitable for exact reconstruction. The
4224 @code{archive} and @code{archive_node} classes have a couple of member
4225 functions that let you access the stored properties:
4228 static void my_print2(const archive_node & n)
4231 n.find_string("class", class_name);
4232 cout << class_name << "(";
4234 archive_node::propinfovector p;
4235 n.get_properties(p);
4237 unsigned num = p.size();
4238 for (unsigned i=0; i<num; i++) @{
4239 const string &name = p[i].name;
4240 if (name == "class")
4242 cout << name << "=";
4244 unsigned count = p[i].count;
4248 for (unsigned j=0; j<count; j++) @{
4249 switch (p[i].type) @{
4250 case archive_node::PTYPE_BOOL: @{
4252 n.find_bool(name, x);
4253 cout << (x ? "true" : "false");
4256 case archive_node::PTYPE_UNSIGNED: @{
4258 n.find_unsigned(name, x);
4262 case archive_node::PTYPE_STRING: @{
4264 n.find_string(name, x);
4265 cout << '\"' << x << '\"';
4268 case archive_node::PTYPE_NODE: @{
4269 const archive_node &x = n.find_ex_node(name, j);
4291 ex e = pow(2, x) - y;
4293 my_print2(ar.get_top_node(0)); cout << endl;
4301 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4302 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4303 overall_coeff=numeric(number="0"))
4306 Be warned, however, that the set of properties and their meaning for each
4307 class may change between GiNaC versions.
4310 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4311 @c node-name, next, previous, up
4312 @chapter Extending GiNaC
4314 By reading so far you should have gotten a fairly good understanding of
4315 GiNaC's design-patterns. From here on you should start reading the
4316 sources. All we can do now is issue some recommendations how to tackle
4317 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4318 develop some useful extension please don't hesitate to contact the GiNaC
4319 authors---they will happily incorporate them into future versions.
4322 * What does not belong into GiNaC:: What to avoid.
4323 * Symbolic functions:: Implementing symbolic functions.
4324 * Adding classes:: Defining new algebraic classes.
4328 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4329 @c node-name, next, previous, up
4330 @section What doesn't belong into GiNaC
4332 @cindex @command{ginsh}
4333 First of all, GiNaC's name must be read literally. It is designed to be
4334 a library for use within C++. The tiny @command{ginsh} accompanying
4335 GiNaC makes this even more clear: it doesn't even attempt to provide a
4336 language. There are no loops or conditional expressions in
4337 @command{ginsh}, it is merely a window into the library for the
4338 programmer to test stuff (or to show off). Still, the design of a
4339 complete CAS with a language of its own, graphical capabilites and all
4340 this on top of GiNaC is possible and is without doubt a nice project for
4343 There are many built-in functions in GiNaC that do not know how to
4344 evaluate themselves numerically to a precision declared at runtime
4345 (using @code{Digits}). Some may be evaluated at certain points, but not
4346 generally. This ought to be fixed. However, doing numerical
4347 computations with GiNaC's quite abstract classes is doomed to be
4348 inefficient. For this purpose, the underlying foundation classes
4349 provided by @acronym{CLN} are much better suited.
4352 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4353 @c node-name, next, previous, up
4354 @section Symbolic functions
4356 The easiest and most instructive way to start with is probably to
4357 implement your own function. GiNaC's functions are objects of class
4358 @code{function}. The preprocessor is then used to convert the function
4359 names to objects with a corresponding serial number that is used
4360 internally to identify them. You usually need not worry about this
4361 number. New functions may be inserted into the system via a kind of
4362 `registry'. It is your responsibility to care for some functions that
4363 are called when the user invokes certain methods. These are usual
4364 C++-functions accepting a number of @code{ex} as arguments and returning
4365 one @code{ex}. As an example, if we have a look at a simplified
4366 implementation of the cosine trigonometric function, we first need a
4367 function that is called when one wishes to @code{eval} it. It could
4368 look something like this:
4371 static ex cos_eval_method(const ex & x)
4373 // if (!x%(2*Pi)) return 1
4374 // if (!x%Pi) return -1
4375 // if (!x%Pi/2) return 0
4376 // care for other cases...
4377 return cos(x).hold();
4381 @cindex @code{hold()}
4383 The last line returns @code{cos(x)} if we don't know what else to do and
4384 stops a potential recursive evaluation by saying @code{.hold()}, which
4385 sets a flag to the expression signaling that it has been evaluated. We
4386 should also implement a method for numerical evaluation and since we are
4387 lazy we sweep the problem under the rug by calling someone else's
4388 function that does so, in this case the one in class @code{numeric}:
4391 static ex cos_evalf(const ex & x)
4393 return cos(ex_to<numeric>(x));
4397 Differentiation will surely turn up and so we need to tell @code{cos}
4398 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4399 instance are then handled automatically by @code{basic::diff} and
4403 static ex cos_deriv(const ex & x, unsigned diff_param)
4409 @cindex product rule
4410 The second parameter is obligatory but uninteresting at this point. It
4411 specifies which parameter to differentiate in a partial derivative in
4412 case the function has more than one parameter and its main application
4413 is for correct handling of the chain rule. For Taylor expansion, it is
4414 enough to know how to differentiate. But if the function you want to
4415 implement does have a pole somewhere in the complex plane, you need to
4416 write another method for Laurent expansion around that point.
4418 Now that all the ingredients for @code{cos} have been set up, we need
4419 to tell the system about it. This is done by a macro and we are not
4420 going to descibe how it expands, please consult your preprocessor if you
4424 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4425 evalf_func(cos_evalf).
4426 derivative_func(cos_deriv));
4429 The first argument is the function's name used for calling it and for
4430 output. The second binds the corresponding methods as options to this
4431 object. Options are separated by a dot and can be given in an arbitrary
4432 order. GiNaC functions understand several more options which are always
4433 specified as @code{.option(params)}, for example a method for series
4434 expansion @code{.series_func(cos_series)}. Again, if no series
4435 expansion method is given, GiNaC defaults to simple Taylor expansion,
4436 which is correct if there are no poles involved as is the case for the
4437 @code{cos} function. The way GiNaC handles poles in case there are any
4438 is best understood by studying one of the examples, like the Gamma
4439 (@code{tgamma}) function for instance. (In essence the function first
4440 checks if there is a pole at the evaluation point and falls back to
4441 Taylor expansion if there isn't. Then, the pole is regularized by some
4442 suitable transformation.) Also, the new function needs to be declared
4443 somewhere. This may also be done by a convenient preprocessor macro:
4446 DECLARE_FUNCTION_1P(cos)
4449 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4450 implementation of @code{cos} is very incomplete and lacks several safety
4451 mechanisms. Please, have a look at the real implementation in GiNaC.
4452 (By the way: in case you are worrying about all the macros above we can
4453 assure you that functions are GiNaC's most macro-intense classes. We
4454 have done our best to avoid macros where we can.)
4457 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4458 @c node-name, next, previous, up
4459 @section Adding classes
4461 If you are doing some very specialized things with GiNaC you may find that
4462 you have to implement your own algebraic classes to fit your needs. This
4463 section will explain how to do this by giving the example of a simple
4464 'string' class. After reading this section you will know how to properly
4465 declare a GiNaC class and what the minimum required member functions are
4466 that you have to implement. We only cover the implementation of a 'leaf'
4467 class here (i.e. one that doesn't contain subexpressions). Creating a
4468 container class like, for example, a class representing tensor products is
4469 more involved but this section should give you enough information so you can
4470 consult the source to GiNaC's predefined classes if you want to implement
4471 something more complicated.
4473 @subsection GiNaC's run-time type information system
4475 @cindex hierarchy of classes
4477 All algebraic classes (that is, all classes that can appear in expressions)
4478 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4479 @code{basic *} (which is essentially what an @code{ex} is) represents a
4480 generic pointer to an algebraic class. Occasionally it is necessary to find
4481 out what the class of an object pointed to by a @code{basic *} really is.
4482 Also, for the unarchiving of expressions it must be possible to find the
4483 @code{unarchive()} function of a class given the class name (as a string). A
4484 system that provides this kind of information is called a run-time type
4485 information (RTTI) system. The C++ language provides such a thing (see the
4486 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4487 implements its own, simpler RTTI.
4489 The RTTI in GiNaC is based on two mechanisms:
4494 The @code{basic} class declares a member variable @code{tinfo_key} which
4495 holds an unsigned integer that identifies the object's class. These numbers
4496 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4497 classes. They all start with @code{TINFO_}.
4500 By means of some clever tricks with static members, GiNaC maintains a list
4501 of information for all classes derived from @code{basic}. The information
4502 available includes the class names, the @code{tinfo_key}s, and pointers
4503 to the unarchiving functions. This class registry is defined in the
4504 @file{registrar.h} header file.
4508 The disadvantage of this proprietary RTTI implementation is that there's
4509 a little more to do when implementing new classes (C++'s RTTI works more
4510 or less automatic) but don't worry, most of the work is simplified by
4513 @subsection A minimalistic example
4515 Now we will start implementing a new class @code{mystring} that allows
4516 placing character strings in algebraic expressions (this is not very useful,
4517 but it's just an example). This class will be a direct subclass of
4518 @code{basic}. You can use this sample implementation as a starting point
4519 for your own classes.
4521 The code snippets given here assume that you have included some header files
4527 #include <stdexcept>
4528 using namespace std;
4530 #include <ginac/ginac.h>
4531 using namespace GiNaC;
4534 The first thing we have to do is to define a @code{tinfo_key} for our new
4535 class. This can be any arbitrary unsigned number that is not already taken
4536 by one of the existing classes but it's better to come up with something
4537 that is unlikely to clash with keys that might be added in the future. The
4538 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4539 which is not a requirement but we are going to stick with this scheme:
4542 const unsigned TINFO_mystring = 0x42420001U;
4545 Now we can write down the class declaration. The class stores a C++
4546 @code{string} and the user shall be able to construct a @code{mystring}
4547 object from a C or C++ string:
4550 class mystring : public basic
4552 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4555 mystring(const string &s);
4556 mystring(const char *s);
4562 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4565 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4566 macros are defined in @file{registrar.h}. They take the name of the class
4567 and its direct superclass as arguments and insert all required declarations
4568 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4569 the first line after the opening brace of the class definition. The
4570 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4571 source (at global scope, of course, not inside a function).
4573 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4574 declarations of the default and copy constructor, the destructor, the
4575 assignment operator and a couple of other functions that are required. It
4576 also defines a type @code{inherited} which refers to the superclass so you
4577 don't have to modify your code every time you shuffle around the class
4578 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4579 constructor, the destructor and the assignment operator.
4581 Now there are nine member functions we have to implement to get a working
4587 @code{mystring()}, the default constructor.
4590 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4591 assignment operator to free dynamically allocated members. The @code{call_parent}
4592 specifies whether the @code{destroy()} function of the superclass is to be
4596 @code{void copy(const mystring &other)}, which is used in the copy constructor
4597 and assignment operator to copy the member variables over from another
4598 object of the same class.
4601 @code{void archive(archive_node &n)}, the archiving function. This stores all
4602 information needed to reconstruct an object of this class inside an
4603 @code{archive_node}.
4606 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4607 constructor. This constructs an instance of the class from the information
4608 found in an @code{archive_node}.
4611 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4612 unarchiving function. It constructs a new instance by calling the unarchiving
4616 @code{int compare_same_type(const basic &other)}, which is used internally
4617 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4618 -1, depending on the relative order of this object and the @code{other}
4619 object. If it returns 0, the objects are considered equal.
4620 @strong{Note:} This has nothing to do with the (numeric) ordering
4621 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4622 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4623 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4624 must provide a @code{compare_same_type()} function, even those representing
4625 objects for which no reasonable algebraic ordering relationship can be
4629 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4630 which are the two constructors we declared.
4634 Let's proceed step-by-step. The default constructor looks like this:
4637 mystring::mystring() : inherited(TINFO_mystring)
4639 // dynamically allocate resources here if required
4643 The golden rule is that in all constructors you have to set the
4644 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4645 it will be set by the constructor of the superclass and all hell will break
4646 loose in the RTTI. For your convenience, the @code{basic} class provides
4647 a constructor that takes a @code{tinfo_key} value, which we are using here
4648 (remember that in our case @code{inherited = basic}). If the superclass
4649 didn't have such a constructor, we would have to set the @code{tinfo_key}
4650 to the right value manually.
4652 In the default constructor you should set all other member variables to
4653 reasonable default values (we don't need that here since our @code{str}
4654 member gets set to an empty string automatically). The constructor(s) are of
4655 course also the right place to allocate any dynamic resources you require.
4657 Next, the @code{destroy()} function:
4660 void mystring::destroy(bool call_parent)
4662 // free dynamically allocated resources here if required
4664 inherited::destroy(call_parent);
4668 This function is where we free all dynamically allocated resources. We don't
4669 have any so we're not doing anything here, but if we had, for example, used
4670 a C-style @code{char *} to store our string, this would be the place to
4671 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4672 to call the @code{destroy()} function of the superclass after we're done
4673 (to mimic C++'s automatic invocation of superclass destructors where
4674 @code{destroy()} is called from outside a destructor).
4676 The @code{copy()} function just copies over the member variables from
4680 void mystring::copy(const mystring &other)
4682 inherited::copy(other);
4687 We can simply overwrite the member variables here. There's no need to worry
4688 about dynamically allocated storage. The assignment operator (which is
4689 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4690 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4691 explicitly call the @code{copy()} function of the superclass here so
4692 all the member variables will get copied.
4694 Next are the three functions for archiving. You have to implement them even
4695 if you don't plan to use archives, but the minimum required implementation
4696 is really simple. First, the archiving function:
4699 void mystring::archive(archive_node &n) const
4701 inherited::archive(n);
4702 n.add_string("string", str);
4706 The only thing that is really required is calling the @code{archive()}
4707 function of the superclass. Optionally, you can store all information you
4708 deem necessary for representing the object into the passed
4709 @code{archive_node}. We are just storing our string here. For more
4710 information on how the archiving works, consult the @file{archive.h} header
4713 The unarchiving constructor is basically the inverse of the archiving
4717 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4719 n.find_string("string", str);
4723 If you don't need archiving, just leave this function empty (but you must
4724 invoke the unarchiving constructor of the superclass). Note that we don't
4725 have to set the @code{tinfo_key} here because it is done automatically
4726 by the unarchiving constructor of the @code{basic} class.
4728 Finally, the unarchiving function:
4731 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4733 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4737 You don't have to understand how exactly this works. Just copy these four
4738 lines into your code literally (replacing the class name, of course). It
4739 calls the unarchiving constructor of the class and unless you are doing
4740 something very special (like matching @code{archive_node}s to global
4741 objects) you don't need a different implementation. For those who are
4742 interested: setting the @code{dynallocated} flag puts the object under
4743 the control of GiNaC's garbage collection. It will get deleted automatically
4744 once it is no longer referenced.
4746 Our @code{compare_same_type()} function uses a provided function to compare
4750 int mystring::compare_same_type(const basic &other) const
4752 const mystring &o = static_cast<const mystring &>(other);
4753 int cmpval = str.compare(o.str);
4756 else if (cmpval < 0)
4763 Although this function takes a @code{basic &}, it will always be a reference
4764 to an object of exactly the same class (objects of different classes are not
4765 comparable), so the cast is safe. If this function returns 0, the two objects
4766 are considered equal (in the sense that @math{A-B=0}), so you should compare
4767 all relevant member variables.
4769 Now the only thing missing is our two new constructors:
4772 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4774 // dynamically allocate resources here if required
4777 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4779 // dynamically allocate resources here if required
4783 No surprises here. We set the @code{str} member from the argument and
4784 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4786 That's it! We now have a minimal working GiNaC class that can store
4787 strings in algebraic expressions. Let's confirm that the RTTI works:
4790 ex e = mystring("Hello, world!");
4791 cout << is_a<mystring>(e) << endl;
4794 cout << e.bp->class_name() << endl;
4798 Obviously it does. Let's see what the expression @code{e} looks like:
4802 // -> [mystring object]
4805 Hm, not exactly what we expect, but of course the @code{mystring} class
4806 doesn't yet know how to print itself. This is done in the @code{print()}
4807 member function. Let's say that we wanted to print the string surrounded
4811 class mystring : public basic
4815 void print(const print_context &c, unsigned level = 0) const;
4819 void mystring::print(const print_context &c, unsigned level) const
4821 // print_context::s is a reference to an ostream
4822 c.s << '\"' << str << '\"';
4826 The @code{level} argument is only required for container classes to
4827 correctly parenthesize the output. Let's try again to print the expression:
4831 // -> "Hello, world!"
4834 Much better. The @code{mystring} class can be used in arbitrary expressions:
4837 e += mystring("GiNaC rulez");
4839 // -> "GiNaC rulez"+"Hello, world!"
4842 (GiNaC's automatic term reordering is in effect here), or even
4845 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4847 // -> "One string"^(2*sin(-"Another string"+Pi))
4850 Whether this makes sense is debatable but remember that this is only an
4851 example. At least it allows you to implement your own symbolic algorithms
4854 Note that GiNaC's algebraic rules remain unchanged:
4857 e = mystring("Wow") * mystring("Wow");
4861 e = pow(mystring("First")-mystring("Second"), 2);
4862 cout << e.expand() << endl;
4863 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4866 There's no way to, for example, make GiNaC's @code{add} class perform string
4867 concatenation. You would have to implement this yourself.
4869 @subsection Automatic evaluation
4871 @cindex @code{hold()}
4873 When dealing with objects that are just a little more complicated than the
4874 simple string objects we have implemented, chances are that you will want to
4875 have some automatic simplifications or canonicalizations performed on them.
4876 This is done in the evaluation member function @code{eval()}. Let's say that
4877 we wanted all strings automatically converted to lowercase with
4878 non-alphabetic characters stripped, and empty strings removed:
4881 class mystring : public basic
4885 ex eval(int level = 0) const;
4889 ex mystring::eval(int level) const
4892 for (int i=0; i<str.length(); i++) @{
4894 if (c >= 'A' && c <= 'Z')
4895 new_str += tolower(c);
4896 else if (c >= 'a' && c <= 'z')
4900 if (new_str.length() == 0)
4903 return mystring(new_str).hold();
4907 The @code{level} argument is used to limit the recursion depth of the
4908 evaluation. We don't have any subexpressions in the @code{mystring} class
4909 so we are not concerned with this. If we had, we would call the @code{eval()}
4910 functions of the subexpressions with @code{level - 1} as the argument if
4911 @code{level != 1}. The @code{hold()} member function sets a flag in the
4912 object that prevents further evaluation. Otherwise we might end up in an
4913 endless loop. When you want to return the object unmodified, use
4914 @code{return this->hold();}.
4916 Let's confirm that it works:
4919 ex e = mystring("Hello, world!") + mystring("!?#");
4923 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4928 @subsection Other member functions
4930 We have implemented only a small set of member functions to make the class
4931 work in the GiNaC framework. For a real algebraic class, there are probably
4932 some more functions that you will want to re-implement, such as
4933 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4934 or the header file of the class you want to make a subclass of to see
4935 what's there. One member function that you will most likely want to
4936 implement for terminal classes like the described string class is
4937 @code{calcchash()} that returns an @code{unsigned} hash value for the object
4938 which will allow GiNaC to compare and canonicalize expressions much more
4941 You can, of course, also add your own new member functions. Remember,
4942 that the RTTI may be used to get information about what kinds of objects
4943 you are dealing with (the position in the class hierarchy) and that you
4944 can always extract the bare object from an @code{ex} by stripping the
4945 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
4946 should become a need.
4948 That's it. May the source be with you!
4951 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4952 @c node-name, next, previous, up
4953 @chapter A Comparison With Other CAS
4956 This chapter will give you some information on how GiNaC compares to
4957 other, traditional Computer Algebra Systems, like @emph{Maple},
4958 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4959 disadvantages over these systems.
4962 * Advantages:: Stengths of the GiNaC approach.
4963 * Disadvantages:: Weaknesses of the GiNaC approach.
4964 * Why C++?:: Attractiveness of C++.
4967 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4968 @c node-name, next, previous, up
4971 GiNaC has several advantages over traditional Computer
4972 Algebra Systems, like
4977 familiar language: all common CAS implement their own proprietary
4978 grammar which you have to learn first (and maybe learn again when your
4979 vendor decides to `enhance' it). With GiNaC you can write your program
4980 in common C++, which is standardized.
4984 structured data types: you can build up structured data types using
4985 @code{struct}s or @code{class}es together with STL features instead of
4986 using unnamed lists of lists of lists.
4989 strongly typed: in CAS, you usually have only one kind of variables
4990 which can hold contents of an arbitrary type. This 4GL like feature is
4991 nice for novice programmers, but dangerous.
4994 development tools: powerful development tools exist for C++, like fancy
4995 editors (e.g. with automatic indentation and syntax highlighting),
4996 debuggers, visualization tools, documentation generators@dots{}
4999 modularization: C++ programs can easily be split into modules by
5000 separating interface and implementation.
5003 price: GiNaC is distributed under the GNU Public License which means
5004 that it is free and available with source code. And there are excellent
5005 C++-compilers for free, too.
5008 extendable: you can add your own classes to GiNaC, thus extending it on
5009 a very low level. Compare this to a traditional CAS that you can
5010 usually only extend on a high level by writing in the language defined
5011 by the parser. In particular, it turns out to be almost impossible to
5012 fix bugs in a traditional system.
5015 multiple interfaces: Though real GiNaC programs have to be written in
5016 some editor, then be compiled, linked and executed, there are more ways
5017 to work with the GiNaC engine. Many people want to play with
5018 expressions interactively, as in traditional CASs. Currently, two such
5019 windows into GiNaC have been implemented and many more are possible: the
5020 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5021 types to a command line and second, as a more consistent approach, an
5022 interactive interface to the @acronym{Cint} C++ interpreter has been put
5023 together (called @acronym{GiNaC-cint}) that allows an interactive
5024 scripting interface consistent with the C++ language.
5027 seemless integration: it is somewhere between difficult and impossible
5028 to call CAS functions from within a program written in C++ or any other
5029 programming language and vice versa. With GiNaC, your symbolic routines
5030 are part of your program. You can easily call third party libraries,
5031 e.g. for numerical evaluation or graphical interaction. All other
5032 approaches are much more cumbersome: they range from simply ignoring the
5033 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5034 system (i.e. @emph{Yacas}).
5037 efficiency: often large parts of a program do not need symbolic
5038 calculations at all. Why use large integers for loop variables or
5039 arbitrary precision arithmetics where @code{int} and @code{double} are
5040 sufficient? For pure symbolic applications, GiNaC is comparable in
5041 speed with other CAS.
5046 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5047 @c node-name, next, previous, up
5048 @section Disadvantages
5050 Of course it also has some disadvantages:
5055 advanced features: GiNaC cannot compete with a program like
5056 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5057 which grows since 1981 by the work of dozens of programmers, with
5058 respect to mathematical features. Integration, factorization,
5059 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5060 not planned for the near future).
5063 portability: While the GiNaC library itself is designed to avoid any
5064 platform dependent features (it should compile on any ANSI compliant C++
5065 compiler), the currently used version of the CLN library (fast large
5066 integer and arbitrary precision arithmetics) can be compiled only on
5067 systems with a recently new C++ compiler from the GNU Compiler
5068 Collection (@acronym{GCC}).@footnote{This is because CLN uses
5069 PROVIDE/REQUIRE like macros to let the compiler gather all static
5070 initializations, which works for GNU C++ only.} GiNaC uses recent
5071 language features like explicit constructors, mutable members, RTTI,
5072 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
5073 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
5074 ANSI compliant, support all needed features.
5079 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5080 @c node-name, next, previous, up
5083 Why did we choose to implement GiNaC in C++ instead of Java or any other
5084 language? C++ is not perfect: type checking is not strict (casting is
5085 possible), separation between interface and implementation is not
5086 complete, object oriented design is not enforced. The main reason is
5087 the often scolded feature of operator overloading in C++. While it may
5088 be true that operating on classes with a @code{+} operator is rarely
5089 meaningful, it is perfectly suited for algebraic expressions. Writing
5090 @math{3x+5y} as @code{3*x+5*y} instead of
5091 @code{x.times(3).plus(y.times(5))} looks much more natural.
5092 Furthermore, the main developers are more familiar with C++ than with
5093 any other programming language.
5096 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5097 @c node-name, next, previous, up
5098 @appendix Internal Structures
5101 * Expressions are reference counted::
5102 * Internal representation of products and sums::
5105 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5106 @c node-name, next, previous, up
5107 @appendixsection Expressions are reference counted
5109 @cindex reference counting
5110 @cindex copy-on-write
5111 @cindex garbage collection
5112 An expression is extremely light-weight since internally it works like a
5113 handle to the actual representation and really holds nothing more than a
5114 pointer to some other object. What this means in practice is that
5115 whenever you create two @code{ex} and set the second equal to the first
5116 no copying process is involved. Instead, the copying takes place as soon
5117 as you try to change the second. Consider the simple sequence of code:
5120 #include <ginac/ginac.h>
5121 using namespace std;
5122 using namespace GiNaC;
5126 symbol x("x"), y("y"), z("z");
5129 e1 = sin(x + 2*y) + 3*z + 41;
5130 e2 = e1; // e2 points to same object as e1
5131 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5132 e2 += 1; // e2 is copied into a new object
5133 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5137 The line @code{e2 = e1;} creates a second expression pointing to the
5138 object held already by @code{e1}. The time involved for this operation
5139 is therefore constant, no matter how large @code{e1} was. Actual
5140 copying, however, must take place in the line @code{e2 += 1;} because
5141 @code{e1} and @code{e2} are not handles for the same object any more.
5142 This concept is called @dfn{copy-on-write semantics}. It increases
5143 performance considerably whenever one object occurs multiple times and
5144 represents a simple garbage collection scheme because when an @code{ex}
5145 runs out of scope its destructor checks whether other expressions handle
5146 the object it points to too and deletes the object from memory if that
5147 turns out not to be the case. A slightly less trivial example of
5148 differentiation using the chain-rule should make clear how powerful this
5152 #include <ginac/ginac.h>
5153 using namespace std;
5154 using namespace GiNaC;
5158 symbol x("x"), y("y");
5162 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5163 cout << e1 << endl // prints x+3*y
5164 << e2 << endl // prints (x+3*y)^3
5165 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5169 Here, @code{e1} will actually be referenced three times while @code{e2}
5170 will be referenced two times. When the power of an expression is built,
5171 that expression needs not be copied. Likewise, since the derivative of
5172 a power of an expression can be easily expressed in terms of that
5173 expression, no copying of @code{e1} is involved when @code{e3} is
5174 constructed. So, when @code{e3} is constructed it will print as
5175 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5176 holds a reference to @code{e2} and the factor in front is just
5179 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5180 semantics. When you insert an expression into a second expression, the
5181 result behaves exactly as if the contents of the first expression were
5182 inserted. But it may be useful to remember that this is not what
5183 happens. Knowing this will enable you to write much more efficient
5184 code. If you still have an uncertain feeling with copy-on-write
5185 semantics, we recommend you have a look at the
5186 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5187 Marshall Cline. Chapter 16 covers this issue and presents an
5188 implementation which is pretty close to the one in GiNaC.
5191 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5192 @c node-name, next, previous, up
5193 @appendixsection Internal representation of products and sums
5195 @cindex representation
5198 @cindex @code{power}
5199 Although it should be completely transparent for the user of
5200 GiNaC a short discussion of this topic helps to understand the sources
5201 and also explain performance to a large degree. Consider the
5202 unexpanded symbolic expression
5204 $2d^3 \left( 4a + 5b - 3 \right)$
5207 @math{2*d^3*(4*a+5*b-3)}
5209 which could naively be represented by a tree of linear containers for
5210 addition and multiplication, one container for exponentiation with base
5211 and exponent and some atomic leaves of symbols and numbers in this
5216 @cindex pair-wise representation
5217 However, doing so results in a rather deeply nested tree which will
5218 quickly become inefficient to manipulate. We can improve on this by
5219 representing the sum as a sequence of terms, each one being a pair of a
5220 purely numeric multiplicative coefficient and its rest. In the same
5221 spirit we can store the multiplication as a sequence of terms, each
5222 having a numeric exponent and a possibly complicated base, the tree
5223 becomes much more flat:
5227 The number @code{3} above the symbol @code{d} shows that @code{mul}
5228 objects are treated similarly where the coefficients are interpreted as
5229 @emph{exponents} now. Addition of sums of terms or multiplication of
5230 products with numerical exponents can be coded to be very efficient with
5231 such a pair-wise representation. Internally, this handling is performed
5232 by most CAS in this way. It typically speeds up manipulations by an
5233 order of magnitude. The overall multiplicative factor @code{2} and the
5234 additive term @code{-3} look somewhat out of place in this
5235 representation, however, since they are still carrying a trivial
5236 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5237 this is avoided by adding a field that carries an overall numeric
5238 coefficient. This results in the realistic picture of internal
5241 $2d^3 \left( 4a + 5b - 3 \right)$:
5244 @math{2*d^3*(4*a+5*b-3)}:
5250 This also allows for a better handling of numeric radicals, since
5251 @code{sqrt(2)} can now be carried along calculations. Now it should be
5252 clear, why both classes @code{add} and @code{mul} are derived from the
5253 same abstract class: the data representation is the same, only the
5254 semantics differs. In the class hierarchy, methods for polynomial
5255 expansion and the like are reimplemented for @code{add} and @code{mul},
5256 but the data structure is inherited from @code{expairseq}.
5259 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5260 @c node-name, next, previous, up
5261 @appendix Package Tools
5263 If you are creating a software package that uses the GiNaC library,
5264 setting the correct command line options for the compiler and linker
5265 can be difficult. GiNaC includes two tools to make this process easier.
5268 * ginac-config:: A shell script to detect compiler and linker flags.
5269 * AM_PATH_GINAC:: Macro for GNU automake.
5273 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5274 @c node-name, next, previous, up
5275 @section @command{ginac-config}
5276 @cindex ginac-config
5278 @command{ginac-config} is a shell script that you can use to determine
5279 the compiler and linker command line options required to compile and
5280 link a program with the GiNaC library.
5282 @command{ginac-config} takes the following flags:
5286 Prints out the version of GiNaC installed.
5288 Prints '-I' flags pointing to the installed header files.
5290 Prints out the linker flags necessary to link a program against GiNaC.
5291 @item --prefix[=@var{PREFIX}]
5292 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5293 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5294 Otherwise, prints out the configured value of @env{$prefix}.
5295 @item --exec-prefix[=@var{PREFIX}]
5296 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5297 Otherwise, prints out the configured value of @env{$exec_prefix}.
5300 Typically, @command{ginac-config} will be used within a configure
5301 script, as described below. It, however, can also be used directly from
5302 the command line using backquotes to compile a simple program. For
5306 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5309 This command line might expand to (for example):
5312 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5313 -lginac -lcln -lstdc++
5316 Not only is the form using @command{ginac-config} easier to type, it will
5317 work on any system, no matter how GiNaC was configured.
5320 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5321 @c node-name, next, previous, up
5322 @section @samp{AM_PATH_GINAC}
5323 @cindex AM_PATH_GINAC
5325 For packages configured using GNU automake, GiNaC also provides
5326 a macro to automate the process of checking for GiNaC.
5329 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5337 Determines the location of GiNaC using @command{ginac-config}, which is
5338 either found in the user's path, or from the environment variable
5339 @env{GINACLIB_CONFIG}.
5342 Tests the installed libraries to make sure that their version
5343 is later than @var{MINIMUM-VERSION}. (A default version will be used
5347 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5348 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5349 variable to the output of @command{ginac-config --libs}, and calls
5350 @samp{AC_SUBST()} for these variables so they can be used in generated
5351 makefiles, and then executes @var{ACTION-IF-FOUND}.
5354 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5355 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5359 This macro is in file @file{ginac.m4} which is installed in
5360 @file{$datadir/aclocal}. Note that if automake was installed with a
5361 different @samp{--prefix} than GiNaC, you will either have to manually
5362 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5363 aclocal the @samp{-I} option when running it.
5366 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5367 * Example package:: Example of a package using AM_PATH_GINAC.
5371 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5372 @c node-name, next, previous, up
5373 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5375 Simply make sure that @command{ginac-config} is in your path, and run
5376 the configure script.
5383 The directory where the GiNaC libraries are installed needs
5384 to be found by your system's dynamic linker.
5386 This is generally done by
5389 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5395 setting the environment variable @env{LD_LIBRARY_PATH},
5398 or, as a last resort,
5401 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5402 running configure, for instance:
5405 LDFLAGS=-R/home/cbauer/lib ./configure
5410 You can also specify a @command{ginac-config} not in your path by
5411 setting the @env{GINACLIB_CONFIG} environment variable to the
5412 name of the executable
5415 If you move the GiNaC package from its installed location,
5416 you will either need to modify @command{ginac-config} script
5417 manually to point to the new location or rebuild GiNaC.
5428 --with-ginac-prefix=@var{PREFIX}
5429 --with-ginac-exec-prefix=@var{PREFIX}
5432 are provided to override the prefix and exec-prefix that were stored
5433 in the @command{ginac-config} shell script by GiNaC's configure. You are
5434 generally better off configuring GiNaC with the right path to begin with.
5438 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5439 @c node-name, next, previous, up
5440 @subsection Example of a package using @samp{AM_PATH_GINAC}
5442 The following shows how to build a simple package using automake
5443 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5446 #include <ginac/ginac.h>
5450 GiNaC::symbol x("x");
5451 GiNaC::ex a = GiNaC::sin(x);
5452 std::cout << "Derivative of " << a
5453 << " is " << a.diff(x) << std::endl;
5458 You should first read the introductory portions of the automake
5459 Manual, if you are not already familiar with it.
5461 Two files are needed, @file{configure.in}, which is used to build the
5465 dnl Process this file with autoconf to produce a configure script.
5467 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5473 AM_PATH_GINAC(0.7.0, [
5474 LIBS="$LIBS $GINACLIB_LIBS"
5475 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5476 ], AC_MSG_ERROR([need to have GiNaC installed]))
5481 The only command in this which is not standard for automake
5482 is the @samp{AM_PATH_GINAC} macro.
5484 That command does the following: If a GiNaC version greater or equal
5485 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5486 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5487 the error message `need to have GiNaC installed'
5489 And the @file{Makefile.am}, which will be used to build the Makefile.
5492 ## Process this file with automake to produce Makefile.in
5493 bin_PROGRAMS = simple
5494 simple_SOURCES = simple.cpp
5497 This @file{Makefile.am}, says that we are building a single executable,
5498 from a single sourcefile @file{simple.cpp}. Since every program
5499 we are building uses GiNaC we simply added the GiNaC options
5500 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5501 want to specify them on a per-program basis: for instance by
5505 simple_LDADD = $(GINACLIB_LIBS)
5506 INCLUDES = $(GINACLIB_CPPFLAGS)
5509 to the @file{Makefile.am}.
5511 To try this example out, create a new directory and add the three
5514 Now execute the following commands:
5517 $ automake --add-missing
5522 You now have a package that can be built in the normal fashion
5531 @node Bibliography, Concept Index, Example package, Top
5532 @c node-name, next, previous, up
5533 @appendix Bibliography
5538 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5541 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5544 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5547 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5550 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5551 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5554 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5555 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5556 Academic Press, London
5559 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5564 @node Concept Index, , Bibliography, Top
5565 @c node-name, next, previous, up
5566 @unnumbered Concept Index