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) 1999-2003 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{} 1999-2003 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 linguistic 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{} 1999-2003 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:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 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
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
691 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
692 @c node-name, next, previous, up
694 @cindex expression (class @code{ex})
697 The most common class of objects a user deals with is the expression
698 @code{ex}, representing a mathematical object like a variable, number,
699 function, sum, product, etc@dots{} Expressions may be put together to form
700 new expressions, passed as arguments to functions, and so on. Here is a
701 little collection of valid expressions:
704 ex MyEx1 = 5; // simple number
705 ex MyEx2 = x + 2*y; // polynomial in x and y
706 ex MyEx3 = (x + 1)/(x - 1); // rational expression
707 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
708 ex MyEx5 = MyEx4 + 1; // similar to above
711 Expressions are handles to other more fundamental objects, that often
712 contain other expressions thus creating a tree of expressions
713 (@xref{Internal Structures}, for particular examples). Most methods on
714 @code{ex} therefore run top-down through such an expression tree. For
715 example, the method @code{has()} scans recursively for occurrences of
716 something inside an expression. Thus, if you have declared @code{MyEx4}
717 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
718 the argument of @code{sin} and hence return @code{true}.
720 The next sections will outline the general picture of GiNaC's class
721 hierarchy and describe the classes of objects that are handled by
725 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
726 @c node-name, next, previous, up
727 @section Automatic evaluation and canonicalization of expressions
730 GiNaC performs some automatic transformations on expressions, to simplify
731 them and put them into a canonical form. Some examples:
734 ex MyEx1 = 2*x - 1 + x; // 3*x-1
735 ex MyEx2 = x - x; // 0
736 ex MyEx3 = cos(2*Pi); // 1
737 ex MyEx4 = x*y/x; // y
740 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
741 evaluation}. GiNaC only performs transformations that are
745 at most of complexity
753 algebraically correct, possibly except for a set of measure zero (e.g.
754 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
757 There are two types of automatic transformations in GiNaC that may not
758 behave in an entirely obvious way at first glance:
762 The terms of sums and products (and some other things like the arguments of
763 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
764 into a canonical form that is deterministic, but not lexicographical or in
765 any other way easily guessable (it almost always depends on the number and
766 order of the symbols you define). However, constructing the same expression
767 twice, either implicitly or explicitly, will always result in the same
770 Expressions of the form 'number times sum' are automatically expanded (this
771 has to do with GiNaC's internal representation of sums and products). For
774 ex MyEx5 = 2*(x + y); // 2*x+2*y
775 ex MyEx6 = z*(x + y); // z*(x+y)
779 The general rule is that when you construct expressions, GiNaC automatically
780 creates them in canonical form, which might differ from the form you typed in
781 your program. This may create some awkward looking output (@samp{-y+x} instead
782 of @samp{x-y}) but allows for more efficient operation and usually yields
783 some immediate simplifications.
785 @cindex @code{eval()}
786 Internally, the anonymous evaluator in GiNaC is implemented by the methods
789 ex ex::eval(int level = 0) const;
790 ex basic::eval(int level = 0) const;
793 but unless you are extending GiNaC with your own classes or functions, there
794 should never be any reason to call them explicitly. All GiNaC methods that
795 transform expressions, like @code{subs()} or @code{normal()}, automatically
796 re-evaluate their results.
799 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
800 @c node-name, next, previous, up
801 @section Error handling
803 @cindex @code{pole_error} (class)
805 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
806 generated by GiNaC are subclassed from the standard @code{exception} class
807 defined in the @file{<stdexcept>} header. In addition to the predefined
808 @code{logic_error}, @code{domain_error}, @code{out_of_range},
809 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
810 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
811 exception that gets thrown when trying to evaluate a mathematical function
814 The @code{pole_error} class has a member function
817 int pole_error::degree() const;
820 that returns the order of the singularity (or 0 when the pole is
821 logarithmic or the order is undefined).
823 When using GiNaC it is useful to arrange for exceptions to be catched in
824 the main program even if you don't want to do any special error handling.
825 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
826 default exception handler of your C++ compiler's run-time system which
827 usually only aborts the program without giving any information what went
830 Here is an example for a @code{main()} function that catches and prints
831 exceptions generated by GiNaC:
836 #include <ginac/ginac.h>
838 using namespace GiNaC;
846 @} catch (exception &p) @{
847 cerr << p.what() << endl;
855 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
856 @c node-name, next, previous, up
857 @section The Class Hierarchy
859 GiNaC's class hierarchy consists of several classes representing
860 mathematical objects, all of which (except for @code{ex} and some
861 helpers) are internally derived from one abstract base class called
862 @code{basic}. You do not have to deal with objects of class
863 @code{basic}, instead you'll be dealing with symbols, numbers,
864 containers of expressions and so on.
868 To get an idea about what kinds of symbolic composites may be built we
869 have a look at the most important classes in the class hierarchy and
870 some of the relations among the classes:
872 @image{classhierarchy}
874 The abstract classes shown here (the ones without drop-shadow) are of no
875 interest for the user. They are used internally in order to avoid code
876 duplication if two or more classes derived from them share certain
877 features. An example is @code{expairseq}, a container for a sequence of
878 pairs each consisting of one expression and a number (@code{numeric}).
879 What @emph{is} visible to the user are the derived classes @code{add}
880 and @code{mul}, representing sums and products. @xref{Internal
881 Structures}, where these two classes are described in more detail. The
882 following table shortly summarizes what kinds of mathematical objects
883 are stored in the different classes:
886 @multitable @columnfractions .22 .78
887 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
888 @item @code{constant} @tab Constants like
895 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
896 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
897 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
898 @item @code{ncmul} @tab Products of non-commutative objects
899 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
904 @code{sqrt(}@math{2}@code{)}
907 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
908 @item @code{function} @tab A symbolic function like
915 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
916 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
917 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
918 @item @code{indexed} @tab Indexed object like @math{A_ij}
919 @item @code{tensor} @tab Special tensor like the delta and metric tensors
920 @item @code{idx} @tab Index of an indexed object
921 @item @code{varidx} @tab Index with variance
922 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
923 @item @code{wildcard} @tab Wildcard for pattern matching
924 @item @code{structure} @tab Template for user-defined classes
929 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
930 @c node-name, next, previous, up
932 @cindex @code{symbol} (class)
933 @cindex hierarchy of classes
936 Symbols are for symbolic manipulation what atoms are for chemistry. You
937 can declare objects of class @code{symbol} as any other object simply by
938 saying @code{symbol x,y;}. There is, however, a catch in here having to
939 do with the fact that C++ is a compiled language. The information about
940 the symbol's name is thrown away by the compiler but at a later stage
941 you may want to print expressions holding your symbols. In order to
942 avoid confusion GiNaC's symbols are able to know their own name. This
943 is accomplished by declaring its name for output at construction time in
944 the fashion @code{symbol x("x");}. If you declare a symbol using the
945 default constructor (i.e. without string argument) the system will deal
946 out a unique name. That name may not be suitable for printing but for
947 internal routines when no output is desired it is often enough. We'll
948 come across examples of such symbols later in this tutorial.
950 This implies that the strings passed to symbols at construction time may
951 not be used for comparing two of them. It is perfectly legitimate to
952 write @code{symbol x("x"),y("x");} but it is likely to lead into
953 trouble. Here, @code{x} and @code{y} are different symbols and
954 statements like @code{x-y} will not be simplified to zero although the
955 output @code{x-x} looks funny. Such output may also occur when there
956 are two different symbols in two scopes, for instance when you call a
957 function that declares a symbol with a name already existent in a symbol
958 in the calling function. Again, comparing them (using @code{operator==}
959 for instance) will always reveal their difference. Watch out, please.
961 @cindex @code{subs()}
962 Although symbols can be assigned expressions for internal reasons, you
963 should not do it (and we are not going to tell you how it is done). If
964 you want to replace a symbol with something else in an expression, you
965 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
968 @node Numbers, Constants, Symbols, Basic Concepts
969 @c node-name, next, previous, up
971 @cindex @code{numeric} (class)
977 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
978 The classes therein serve as foundation classes for GiNaC. CLN stands
979 for Class Library for Numbers or alternatively for Common Lisp Numbers.
980 In order to find out more about CLN's internals, the reader is referred to
981 the documentation of that library. @inforef{Introduction, , cln}, for
982 more information. Suffice to say that it is by itself build on top of
983 another library, the GNU Multiple Precision library GMP, which is an
984 extremely fast library for arbitrary long integers and rationals as well
985 as arbitrary precision floating point numbers. It is very commonly used
986 by several popular cryptographic applications. CLN extends GMP by
987 several useful things: First, it introduces the complex number field
988 over either reals (i.e. floating point numbers with arbitrary precision)
989 or rationals. Second, it automatically converts rationals to integers
990 if the denominator is unity and complex numbers to real numbers if the
991 imaginary part vanishes and also correctly treats algebraic functions.
992 Third it provides good implementations of state-of-the-art algorithms
993 for all trigonometric and hyperbolic functions as well as for
994 calculation of some useful constants.
996 The user can construct an object of class @code{numeric} in several
997 ways. The following example shows the four most important constructors.
998 It uses construction from C-integer, construction of fractions from two
999 integers, construction from C-float and construction from a string:
1003 #include <ginac/ginac.h>
1004 using namespace GiNaC;
1008 numeric two = 2; // exact integer 2
1009 numeric r(2,3); // exact fraction 2/3
1010 numeric e(2.71828); // floating point number
1011 numeric p = "3.14159265358979323846"; // constructor from string
1012 // Trott's constant in scientific notation:
1013 numeric trott("1.0841015122311136151E-2");
1015 std::cout << two*p << std::endl; // floating point 6.283...
1020 @cindex complex numbers
1021 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1026 numeric z1 = 2-3*I; // exact complex number 2-3i
1027 numeric z2 = 5.9+1.6*I; // complex floating point number
1031 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1032 This would, however, call C's built-in operator @code{/} for integers
1033 first and result in a numeric holding a plain integer 1. @strong{Never
1034 use the operator @code{/} on integers} unless you know exactly what you
1035 are doing! Use the constructor from two integers instead, as shown in
1036 the example above. Writing @code{numeric(1)/2} may look funny but works
1039 @cindex @code{Digits}
1041 We have seen now the distinction between exact numbers and floating
1042 point numbers. Clearly, the user should never have to worry about
1043 dynamically created exact numbers, since their `exactness' always
1044 determines how they ought to be handled, i.e. how `long' they are. The
1045 situation is different for floating point numbers. Their accuracy is
1046 controlled by one @emph{global} variable, called @code{Digits}. (For
1047 those readers who know about Maple: it behaves very much like Maple's
1048 @code{Digits}). All objects of class numeric that are constructed from
1049 then on will be stored with a precision matching that number of decimal
1054 #include <ginac/ginac.h>
1055 using namespace std;
1056 using namespace GiNaC;
1060 numeric three(3.0), one(1.0);
1061 numeric x = one/three;
1063 cout << "in " << Digits << " digits:" << endl;
1065 cout << Pi.evalf() << endl;
1077 The above example prints the following output to screen:
1081 0.33333333333333333334
1082 3.1415926535897932385
1084 0.33333333333333333333333333333333333333333333333333333333333333333334
1085 3.1415926535897932384626433832795028841971693993751058209749445923078
1089 Note that the last number is not necessarily rounded as you would
1090 naively expect it to be rounded in the decimal system. But note also,
1091 that in both cases you got a couple of extra digits. This is because
1092 numbers are internally stored by CLN as chunks of binary digits in order
1093 to match your machine's word size and to not waste precision. Thus, on
1094 architectures with different word size, the above output might even
1095 differ with regard to actually computed digits.
1097 It should be clear that objects of class @code{numeric} should be used
1098 for constructing numbers or for doing arithmetic with them. The objects
1099 one deals with most of the time are the polymorphic expressions @code{ex}.
1101 @subsection Tests on numbers
1103 Once you have declared some numbers, assigned them to expressions and
1104 done some arithmetic with them it is frequently desired to retrieve some
1105 kind of information from them like asking whether that number is
1106 integer, rational, real or complex. For those cases GiNaC provides
1107 several useful methods. (Internally, they fall back to invocations of
1108 certain CLN functions.)
1110 As an example, let's construct some rational number, multiply it with
1111 some multiple of its denominator and test what comes out:
1115 #include <ginac/ginac.h>
1116 using namespace std;
1117 using namespace GiNaC;
1119 // some very important constants:
1120 const numeric twentyone(21);
1121 const numeric ten(10);
1122 const numeric five(5);
1126 numeric answer = twentyone;
1129 cout << answer.is_integer() << endl; // false, it's 21/5
1131 cout << answer.is_integer() << endl; // true, it's 42 now!
1135 Note that the variable @code{answer} is constructed here as an integer
1136 by @code{numeric}'s copy constructor but in an intermediate step it
1137 holds a rational number represented as integer numerator and integer
1138 denominator. When multiplied by 10, the denominator becomes unity and
1139 the result is automatically converted to a pure integer again.
1140 Internally, the underlying CLN is responsible for this behavior and we
1141 refer the reader to CLN's documentation. Suffice to say that
1142 the same behavior applies to complex numbers as well as return values of
1143 certain functions. Complex numbers are automatically converted to real
1144 numbers if the imaginary part becomes zero. The full set of tests that
1145 can be applied is listed in the following table.
1148 @multitable @columnfractions .30 .70
1149 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1150 @item @code{.is_zero()}
1151 @tab @dots{}equal to zero
1152 @item @code{.is_positive()}
1153 @tab @dots{}not complex and greater than 0
1154 @item @code{.is_integer()}
1155 @tab @dots{}a (non-complex) integer
1156 @item @code{.is_pos_integer()}
1157 @tab @dots{}an integer and greater than 0
1158 @item @code{.is_nonneg_integer()}
1159 @tab @dots{}an integer and greater equal 0
1160 @item @code{.is_even()}
1161 @tab @dots{}an even integer
1162 @item @code{.is_odd()}
1163 @tab @dots{}an odd integer
1164 @item @code{.is_prime()}
1165 @tab @dots{}a prime integer (probabilistic primality test)
1166 @item @code{.is_rational()}
1167 @tab @dots{}an exact rational number (integers are rational, too)
1168 @item @code{.is_real()}
1169 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1170 @item @code{.is_cinteger()}
1171 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1172 @item @code{.is_crational()}
1173 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1177 @subsection Converting numbers
1179 Sometimes it is desirable to convert a @code{numeric} object back to a
1180 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1181 class provides a couple of methods for this purpose:
1183 @cindex @code{to_int()}
1184 @cindex @code{to_long()}
1185 @cindex @code{to_double()}
1186 @cindex @code{to_cl_N()}
1188 int numeric::to_int() const;
1189 long numeric::to_long() const;
1190 double numeric::to_double() const;
1191 cln::cl_N numeric::to_cl_N() const;
1194 @code{to_int()} and @code{to_long()} only work when the number they are
1195 applied on is an exact integer. Otherwise the program will halt with a
1196 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1197 rational number will return a floating-point approximation. Both
1198 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1199 part of complex numbers.
1202 @node Constants, Fundamental containers, Numbers, Basic Concepts
1203 @c node-name, next, previous, up
1205 @cindex @code{constant} (class)
1208 @cindex @code{Catalan}
1209 @cindex @code{Euler}
1210 @cindex @code{evalf()}
1211 Constants behave pretty much like symbols except that they return some
1212 specific number when the method @code{.evalf()} is called.
1214 The predefined known constants are:
1217 @multitable @columnfractions .14 .30 .56
1218 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1220 @tab Archimedes' constant
1221 @tab 3.14159265358979323846264338327950288
1222 @item @code{Catalan}
1223 @tab Catalan's constant
1224 @tab 0.91596559417721901505460351493238411
1226 @tab Euler's (or Euler-Mascheroni) constant
1227 @tab 0.57721566490153286060651209008240243
1232 @node Fundamental containers, Lists, Constants, Basic Concepts
1233 @c node-name, next, previous, up
1234 @section Sums, products and powers
1238 @cindex @code{power}
1240 Simple rational expressions are written down in GiNaC pretty much like
1241 in other CAS or like expressions involving numerical variables in C.
1242 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1243 been overloaded to achieve this goal. When you run the following
1244 code snippet, the constructor for an object of type @code{mul} is
1245 automatically called to hold the product of @code{a} and @code{b} and
1246 then the constructor for an object of type @code{add} is called to hold
1247 the sum of that @code{mul} object and the number one:
1251 symbol a("a"), b("b");
1256 @cindex @code{pow()}
1257 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1258 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1259 construction is necessary since we cannot safely overload the constructor
1260 @code{^} in C++ to construct a @code{power} object. If we did, it would
1261 have several counterintuitive and undesired effects:
1265 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1267 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1268 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1269 interpret this as @code{x^(a^b)}.
1271 Also, expressions involving integer exponents are very frequently used,
1272 which makes it even more dangerous to overload @code{^} since it is then
1273 hard to distinguish between the semantics as exponentiation and the one
1274 for exclusive or. (It would be embarrassing to return @code{1} where one
1275 has requested @code{2^3}.)
1278 @cindex @command{ginsh}
1279 All effects are contrary to mathematical notation and differ from the
1280 way most other CAS handle exponentiation, therefore overloading @code{^}
1281 is ruled out for GiNaC's C++ part. The situation is different in
1282 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1283 that the other frequently used exponentiation operator @code{**} does
1284 not exist at all in C++).
1286 To be somewhat more precise, objects of the three classes described
1287 here, are all containers for other expressions. An object of class
1288 @code{power} is best viewed as a container with two slots, one for the
1289 basis, one for the exponent. All valid GiNaC expressions can be
1290 inserted. However, basic transformations like simplifying
1291 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1292 when this is mathematically possible. If we replace the outer exponent
1293 three in the example by some symbols @code{a}, the simplification is not
1294 safe and will not be performed, since @code{a} might be @code{1/2} and
1297 Objects of type @code{add} and @code{mul} are containers with an
1298 arbitrary number of slots for expressions to be inserted. Again, simple
1299 and safe simplifications are carried out like transforming
1300 @code{3*x+4-x} to @code{2*x+4}.
1303 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1304 @c node-name, next, previous, up
1305 @section Lists of expressions
1306 @cindex @code{lst} (class)
1308 @cindex @code{nops()}
1310 @cindex @code{append()}
1311 @cindex @code{prepend()}
1312 @cindex @code{remove_first()}
1313 @cindex @code{remove_last()}
1314 @cindex @code{remove_all()}
1316 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1317 expressions. They are not as ubiquitous as in many other computer algebra
1318 packages, but are sometimes used to supply a variable number of arguments of
1319 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1320 constructors, so you should have a basic understanding of them.
1322 Lists of up to 16 expressions can be directly constructed from single
1327 symbol x("x"), y("y");
1328 lst l(x, 2, y, x+y);
1329 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1333 Use the @code{nops()} method to determine the size (number of expressions) of
1334 a list and the @code{op()} method or the @code{[]} operator to access
1335 individual elements:
1339 cout << l.nops() << endl; // prints '4'
1340 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1344 As with the standard @code{list<T>} container, accessing random elements of a
1345 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1346 sequential access to the elements of a list is possible with the
1347 iterator types provided by the @code{lst} class:
1350 typedef ... lst::const_iterator;
1351 typedef ... lst::const_reverse_iterator;
1352 lst::const_iterator lst::begin() const;
1353 lst::const_iterator lst::end() const;
1354 lst::const_reverse_iterator lst::rbegin() const;
1355 lst::const_reverse_iterator lst::rend() const;
1358 For example, to print the elements of a list individually you can use:
1363 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1368 which is one order faster than
1373 for (size_t i = 0; i < l.nops(); ++i)
1374 cout << l.op(i) << endl;
1378 These iterators also allow you to use some of the algorithms provided by
1379 the C++ standard library:
1383 // print the elements of the list (requires #include <iterator>)
1384 copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1386 // sum up the elements of the list (requires #include <numeric>)
1387 ex sum = accumulate(l.begin(), l.end(), ex(0));
1388 cout << sum << endl; // prints '2+2*x+2*y'
1392 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1393 (the only other one is @code{matrix}). You can modify single elements:
1397 l[1] = 42; // l is now @{x, 42, y, x+y@}
1398 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1402 You can append or prepend an expression to a list with the @code{append()}
1403 and @code{prepend()} methods:
1407 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1408 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1412 You can remove the first or last element of a list with @code{remove_first()}
1413 and @code{remove_last()}:
1417 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1418 l.remove_last(); // l is now @{x, 7, y, x+y@}
1422 You can remove all the elements of a list with @code{remove_all()}:
1426 l.remove_all(); // l is now empty
1430 You can bring the elements of a list into a canonical order with @code{sort()}:
1434 lst l1(x, 2, y, x+y);
1435 lst l2(2, x+y, x, y);
1438 // l1 and l2 are now equal
1442 Finally, you can remove all but the first element of consecutive groups of
1443 elements with @code{unique()}:
1447 lst l3(x, 2, 2, 2, y, x+y, y+x);
1448 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1453 @node Mathematical functions, Relations, Lists, Basic Concepts
1454 @c node-name, next, previous, up
1455 @section Mathematical functions
1456 @cindex @code{function} (class)
1457 @cindex trigonometric function
1458 @cindex hyperbolic function
1460 There are quite a number of useful functions hard-wired into GiNaC. For
1461 instance, all trigonometric and hyperbolic functions are implemented
1462 (@xref{Built-in Functions}, for a complete list).
1464 These functions (better called @emph{pseudofunctions}) are all objects
1465 of class @code{function}. They accept one or more expressions as
1466 arguments and return one expression. If the arguments are not
1467 numerical, the evaluation of the function may be halted, as it does in
1468 the next example, showing how a function returns itself twice and
1469 finally an expression that may be really useful:
1471 @cindex Gamma function
1472 @cindex @code{subs()}
1475 symbol x("x"), y("y");
1477 cout << tgamma(foo) << endl;
1478 // -> tgamma(x+(1/2)*y)
1479 ex bar = foo.subs(y==1);
1480 cout << tgamma(bar) << endl;
1482 ex foobar = bar.subs(x==7);
1483 cout << tgamma(foobar) << endl;
1484 // -> (135135/128)*Pi^(1/2)
1488 Besides evaluation most of these functions allow differentiation, series
1489 expansion and so on. Read the next chapter in order to learn more about
1492 It must be noted that these pseudofunctions are created by inline
1493 functions, where the argument list is templated. This means that
1494 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1495 @code{sin(ex(1))} and will therefore not result in a floating point
1496 number. Unless of course the function prototype is explicitly
1497 overridden -- which is the case for arguments of type @code{numeric}
1498 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1499 point number of class @code{numeric} you should call
1500 @code{sin(numeric(1))}. This is almost the same as calling
1501 @code{sin(1).evalf()} except that the latter will return a numeric
1502 wrapped inside an @code{ex}.
1505 @node Relations, Matrices, Mathematical functions, Basic Concepts
1506 @c node-name, next, previous, up
1508 @cindex @code{relational} (class)
1510 Sometimes, a relation holding between two expressions must be stored
1511 somehow. The class @code{relational} is a convenient container for such
1512 purposes. A relation is by definition a container for two @code{ex} and
1513 a relation between them that signals equality, inequality and so on.
1514 They are created by simply using the C++ operators @code{==}, @code{!=},
1515 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1517 @xref{Mathematical functions}, for examples where various applications
1518 of the @code{.subs()} method show how objects of class relational are
1519 used as arguments. There they provide an intuitive syntax for
1520 substitutions. They are also used as arguments to the @code{ex::series}
1521 method, where the left hand side of the relation specifies the variable
1522 to expand in and the right hand side the expansion point. They can also
1523 be used for creating systems of equations that are to be solved for
1524 unknown variables. But the most common usage of objects of this class
1525 is rather inconspicuous in statements of the form @code{if
1526 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1527 conversion from @code{relational} to @code{bool} takes place. Note,
1528 however, that @code{==} here does not perform any simplifications, hence
1529 @code{expand()} must be called explicitly.
1532 @node Matrices, Indexed objects, Relations, Basic Concepts
1533 @c node-name, next, previous, up
1535 @cindex @code{matrix} (class)
1537 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1538 matrix with @math{m} rows and @math{n} columns are accessed with two
1539 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1540 second one in the range 0@dots{}@math{n-1}.
1542 There are a couple of ways to construct matrices, with or without preset
1545 @cindex @code{lst_to_matrix()}
1546 @cindex @code{diag_matrix()}
1547 @cindex @code{unit_matrix()}
1548 @cindex @code{symbolic_matrix()}
1550 matrix::matrix(unsigned r, unsigned c);
1551 matrix::matrix(unsigned r, unsigned c, const lst & l);
1552 ex lst_to_matrix(const lst & l);
1553 ex diag_matrix(const lst & l);
1554 ex unit_matrix(unsigned x);
1555 ex unit_matrix(unsigned r, unsigned c);
1556 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1557 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1560 The first two functions are @code{matrix} constructors which create a matrix
1561 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1562 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1563 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1564 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1565 constructs a diagonal matrix given the list of diagonal elements.
1566 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1567 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1568 with newly generated symbols made of the specified base name and the
1569 position of each element in the matrix.
1571 Matrix elements can be accessed and set using the parenthesis (function call)
1575 const ex & matrix::operator()(unsigned r, unsigned c) const;
1576 ex & matrix::operator()(unsigned r, unsigned c);
1579 It is also possible to access the matrix elements in a linear fashion with
1580 the @code{op()} method. But C++-style subscripting with square brackets
1581 @samp{[]} is not available.
1583 Here are a couple of examples of constructing matrices:
1587 symbol a("a"), b("b");
1595 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1598 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1601 cout << diag_matrix(lst(a, b)) << endl;
1604 cout << unit_matrix(3) << endl;
1605 // -> [[1,0,0],[0,1,0],[0,0,1]]
1607 cout << symbolic_matrix(2, 3, "x") << endl;
1608 // -> [[x00,x01,x02],[x10,x11,x12]]
1612 @cindex @code{transpose()}
1613 There are three ways to do arithmetic with matrices. The first (and most
1614 direct one) is to use the methods provided by the @code{matrix} class:
1617 matrix matrix::add(const matrix & other) const;
1618 matrix matrix::sub(const matrix & other) const;
1619 matrix matrix::mul(const matrix & other) const;
1620 matrix matrix::mul_scalar(const ex & other) const;
1621 matrix matrix::pow(const ex & expn) const;
1622 matrix matrix::transpose() const;
1625 All of these methods return the result as a new matrix object. Here is an
1626 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1631 matrix A(2, 2, lst(1, 2, 3, 4));
1632 matrix B(2, 2, lst(-1, 0, 2, 1));
1633 matrix C(2, 2, lst(8, 4, 2, 1));
1635 matrix result = A.mul(B).sub(C.mul_scalar(2));
1636 cout << result << endl;
1637 // -> [[-13,-6],[1,2]]
1642 @cindex @code{evalm()}
1643 The second (and probably the most natural) way is to construct an expression
1644 containing matrices with the usual arithmetic operators and @code{pow()}.
1645 For efficiency reasons, expressions with sums, products and powers of
1646 matrices are not automatically evaluated in GiNaC. You have to call the
1650 ex ex::evalm() const;
1653 to obtain the result:
1660 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1661 cout << e.evalm() << endl;
1662 // -> [[-13,-6],[1,2]]
1667 The non-commutativity of the product @code{A*B} in this example is
1668 automatically recognized by GiNaC. There is no need to use a special
1669 operator here. @xref{Non-commutative objects}, for more information about
1670 dealing with non-commutative expressions.
1672 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1673 to perform the arithmetic:
1678 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1679 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1681 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1682 cout << e.simplify_indexed() << endl;
1683 // -> [[-13,-6],[1,2]].i.j
1687 Using indices is most useful when working with rectangular matrices and
1688 one-dimensional vectors because you don't have to worry about having to
1689 transpose matrices before multiplying them. @xref{Indexed objects}, for
1690 more information about using matrices with indices, and about indices in
1693 The @code{matrix} class provides a couple of additional methods for
1694 computing determinants, traces, and characteristic polynomials:
1696 @cindex @code{determinant()}
1697 @cindex @code{trace()}
1698 @cindex @code{charpoly()}
1700 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1701 ex matrix::trace() const;
1702 ex matrix::charpoly(const ex & lambda) const;
1705 The @samp{algo} argument of @code{determinant()} allows to select
1706 between different algorithms for calculating the determinant. The
1707 asymptotic speed (as parametrized by the matrix size) can greatly differ
1708 between those algorithms, depending on the nature of the matrix'
1709 entries. The possible values are defined in the @file{flags.h} header
1710 file. By default, GiNaC uses a heuristic to automatically select an
1711 algorithm that is likely (but not guaranteed) to give the result most
1714 @cindex @code{inverse()}
1715 @cindex @code{solve()}
1716 Matrices may also be inverted using the @code{ex matrix::inverse()}
1717 method and linear systems may be solved with:
1720 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1723 Assuming the matrix object this method is applied on is an @code{m}
1724 times @code{n} matrix, then @code{vars} must be a @code{n} times
1725 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1726 times @code{p} matrix. The returned matrix then has dimension @code{n}
1727 times @code{p} and in the case of an underdetermined system will still
1728 contain some of the indeterminates from @code{vars}. If the system is
1729 overdetermined, an exception is thrown.
1732 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1733 @c node-name, next, previous, up
1734 @section Indexed objects
1736 GiNaC allows you to handle expressions containing general indexed objects in
1737 arbitrary spaces. It is also able to canonicalize and simplify such
1738 expressions and perform symbolic dummy index summations. There are a number
1739 of predefined indexed objects provided, like delta and metric tensors.
1741 There are few restrictions placed on indexed objects and their indices and
1742 it is easy to construct nonsense expressions, but our intention is to
1743 provide a general framework that allows you to implement algorithms with
1744 indexed quantities, getting in the way as little as possible.
1746 @cindex @code{idx} (class)
1747 @cindex @code{indexed} (class)
1748 @subsection Indexed quantities and their indices
1750 Indexed expressions in GiNaC are constructed of two special types of objects,
1751 @dfn{index objects} and @dfn{indexed objects}.
1755 @cindex contravariant
1758 @item Index objects are of class @code{idx} or a subclass. Every index has
1759 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1760 the index lives in) which can both be arbitrary expressions but are usually
1761 a number or a simple symbol. In addition, indices of class @code{varidx} have
1762 a @dfn{variance} (they can be co- or contravariant), and indices of class
1763 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1765 @item Indexed objects are of class @code{indexed} or a subclass. They
1766 contain a @dfn{base expression} (which is the expression being indexed), and
1767 one or more indices.
1771 @strong{Note:} when printing expressions, covariant indices and indices
1772 without variance are denoted @samp{.i} while contravariant indices are
1773 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1774 value. In the following, we are going to use that notation in the text so
1775 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1776 not visible in the output.
1778 A simple example shall illustrate the concepts:
1782 #include <ginac/ginac.h>
1783 using namespace std;
1784 using namespace GiNaC;
1788 symbol i_sym("i"), j_sym("j");
1789 idx i(i_sym, 3), j(j_sym, 3);
1792 cout << indexed(A, i, j) << endl;
1794 cout << index_dimensions << indexed(A, i, j) << endl;
1796 cout << dflt; // reset cout to default output format (dimensions hidden)
1800 The @code{idx} constructor takes two arguments, the index value and the
1801 index dimension. First we define two index objects, @code{i} and @code{j},
1802 both with the numeric dimension 3. The value of the index @code{i} is the
1803 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1804 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1805 construct an expression containing one indexed object, @samp{A.i.j}. It has
1806 the symbol @code{A} as its base expression and the two indices @code{i} and
1809 The dimensions of indices are normally not visible in the output, but one
1810 can request them to be printed with the @code{index_dimensions} manipulator,
1813 Note the difference between the indices @code{i} and @code{j} which are of
1814 class @code{idx}, and the index values which are the symbols @code{i_sym}
1815 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1816 or numbers but must be index objects. For example, the following is not
1817 correct and will raise an exception:
1820 symbol i("i"), j("j");
1821 e = indexed(A, i, j); // ERROR: indices must be of type idx
1824 You can have multiple indexed objects in an expression, index values can
1825 be numeric, and index dimensions symbolic:
1829 symbol B("B"), dim("dim");
1830 cout << 4 * indexed(A, i)
1831 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1836 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1837 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1838 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1839 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1840 @code{simplify_indexed()} for that, see below).
1842 In fact, base expressions, index values and index dimensions can be
1843 arbitrary expressions:
1847 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1852 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1853 get an error message from this but you will probably not be able to do
1854 anything useful with it.
1856 @cindex @code{get_value()}
1857 @cindex @code{get_dimension()}
1861 ex idx::get_value();
1862 ex idx::get_dimension();
1865 return the value and dimension of an @code{idx} object. If you have an index
1866 in an expression, such as returned by calling @code{.op()} on an indexed
1867 object, you can get a reference to the @code{idx} object with the function
1868 @code{ex_to<idx>()} on the expression.
1870 There are also the methods
1873 bool idx::is_numeric();
1874 bool idx::is_symbolic();
1875 bool idx::is_dim_numeric();
1876 bool idx::is_dim_symbolic();
1879 for checking whether the value and dimension are numeric or symbolic
1880 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1881 About Expressions}) returns information about the index value.
1883 @cindex @code{varidx} (class)
1884 If you need co- and contravariant indices, use the @code{varidx} class:
1888 symbol mu_sym("mu"), nu_sym("nu");
1889 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1890 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1892 cout << indexed(A, mu, nu) << endl;
1894 cout << indexed(A, mu_co, nu) << endl;
1896 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1901 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1902 co- or contravariant. The default is a contravariant (upper) index, but
1903 this can be overridden by supplying a third argument to the @code{varidx}
1904 constructor. The two methods
1907 bool varidx::is_covariant();
1908 bool varidx::is_contravariant();
1911 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1912 to get the object reference from an expression). There's also the very useful
1916 ex varidx::toggle_variance();
1919 which makes a new index with the same value and dimension but the opposite
1920 variance. By using it you only have to define the index once.
1922 @cindex @code{spinidx} (class)
1923 The @code{spinidx} class provides dotted and undotted variant indices, as
1924 used in the Weyl-van-der-Waerden spinor formalism:
1928 symbol K("K"), C_sym("C"), D_sym("D");
1929 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1930 // contravariant, undotted
1931 spinidx C_co(C_sym, 2, true); // covariant index
1932 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1933 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1935 cout << indexed(K, C, D) << endl;
1937 cout << indexed(K, C_co, D_dot) << endl;
1939 cout << indexed(K, D_co_dot, D) << endl;
1944 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1945 dotted or undotted. The default is undotted but this can be overridden by
1946 supplying a fourth argument to the @code{spinidx} constructor. The two
1950 bool spinidx::is_dotted();
1951 bool spinidx::is_undotted();
1954 allow you to check whether or not a @code{spinidx} object is dotted (use
1955 @code{ex_to<spinidx>()} to get the object reference from an expression).
1956 Finally, the two methods
1959 ex spinidx::toggle_dot();
1960 ex spinidx::toggle_variance_dot();
1963 create a new index with the same value and dimension but opposite dottedness
1964 and the same or opposite variance.
1966 @subsection Substituting indices
1968 @cindex @code{subs()}
1969 Sometimes you will want to substitute one symbolic index with another
1970 symbolic or numeric index, for example when calculating one specific element
1971 of a tensor expression. This is done with the @code{.subs()} method, as it
1972 is done for symbols (see @ref{Substituting Expressions}).
1974 You have two possibilities here. You can either substitute the whole index
1975 by another index or expression:
1979 ex e = indexed(A, mu_co);
1980 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1981 // -> A.mu becomes A~nu
1982 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1983 // -> A.mu becomes A~0
1984 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1985 // -> A.mu becomes A.0
1989 The third example shows that trying to replace an index with something that
1990 is not an index will substitute the index value instead.
1992 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1997 ex e = indexed(A, mu_co);
1998 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1999 // -> A.mu becomes A.nu
2000 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2001 // -> A.mu becomes A.0
2005 As you see, with the second method only the value of the index will get
2006 substituted. Its other properties, including its dimension, remain unchanged.
2007 If you want to change the dimension of an index you have to substitute the
2008 whole index by another one with the new dimension.
2010 Finally, substituting the base expression of an indexed object works as
2015 ex e = indexed(A, mu_co);
2016 cout << e << " becomes " << e.subs(A == A+B) << endl;
2017 // -> A.mu becomes (B+A).mu
2021 @subsection Symmetries
2022 @cindex @code{symmetry} (class)
2023 @cindex @code{sy_none()}
2024 @cindex @code{sy_symm()}
2025 @cindex @code{sy_anti()}
2026 @cindex @code{sy_cycl()}
2028 Indexed objects can have certain symmetry properties with respect to their
2029 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2030 that is constructed with the helper functions
2033 symmetry sy_none(...);
2034 symmetry sy_symm(...);
2035 symmetry sy_anti(...);
2036 symmetry sy_cycl(...);
2039 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2040 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2041 represents a cyclic symmetry. Each of these functions accepts up to four
2042 arguments which can be either symmetry objects themselves or unsigned integer
2043 numbers that represent an index position (counting from 0). A symmetry
2044 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2045 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2048 Here are some examples of symmetry definitions:
2053 e = indexed(A, i, j);
2054 e = indexed(A, sy_none(), i, j); // equivalent
2055 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2057 // Symmetric in all three indices:
2058 e = indexed(A, sy_symm(), i, j, k);
2059 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2060 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2061 // different canonical order
2063 // Symmetric in the first two indices only:
2064 e = indexed(A, sy_symm(0, 1), i, j, k);
2065 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2067 // Antisymmetric in the first and last index only (index ranges need not
2069 e = indexed(A, sy_anti(0, 2), i, j, k);
2070 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2072 // An example of a mixed symmetry: antisymmetric in the first two and
2073 // last two indices, symmetric when swapping the first and last index
2074 // pairs (like the Riemann curvature tensor):
2075 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2077 // Cyclic symmetry in all three indices:
2078 e = indexed(A, sy_cycl(), i, j, k);
2079 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2081 // The following examples are invalid constructions that will throw
2082 // an exception at run time.
2084 // An index may not appear multiple times:
2085 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2086 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2088 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2089 // same number of indices:
2090 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2092 // And of course, you cannot specify indices which are not there:
2093 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2097 If you need to specify more than four indices, you have to use the
2098 @code{.add()} method of the @code{symmetry} class. For example, to specify
2099 full symmetry in the first six indices you would write
2100 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2102 If an indexed object has a symmetry, GiNaC will automatically bring the
2103 indices into a canonical order which allows for some immediate simplifications:
2107 cout << indexed(A, sy_symm(), i, j)
2108 + indexed(A, sy_symm(), j, i) << endl;
2110 cout << indexed(B, sy_anti(), i, j)
2111 + indexed(B, sy_anti(), j, i) << endl;
2113 cout << indexed(B, sy_anti(), i, j, k)
2114 - indexed(B, sy_anti(), j, k, i) << endl;
2119 @cindex @code{get_free_indices()}
2121 @subsection Dummy indices
2123 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2124 that a summation over the index range is implied. Symbolic indices which are
2125 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2126 dummy nor free indices.
2128 To be recognized as a dummy index pair, the two indices must be of the same
2129 class and their value must be the same single symbol (an index like
2130 @samp{2*n+1} is never a dummy index). If the indices are of class
2131 @code{varidx} they must also be of opposite variance; if they are of class
2132 @code{spinidx} they must be both dotted or both undotted.
2134 The method @code{.get_free_indices()} returns a vector containing the free
2135 indices of an expression. It also checks that the free indices of the terms
2136 of a sum are consistent:
2140 symbol A("A"), B("B"), C("C");
2142 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2143 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2145 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2146 cout << exprseq(e.get_free_indices()) << endl;
2148 // 'j' and 'l' are dummy indices
2150 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2151 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2153 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2154 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2155 cout << exprseq(e.get_free_indices()) << endl;
2157 // 'nu' is a dummy index, but 'sigma' is not
2159 e = indexed(A, mu, mu);
2160 cout << exprseq(e.get_free_indices()) << endl;
2162 // 'mu' is not a dummy index because it appears twice with the same
2165 e = indexed(A, mu, nu) + 42;
2166 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2167 // this will throw an exception:
2168 // "add::get_free_indices: inconsistent indices in sum"
2172 @cindex @code{simplify_indexed()}
2173 @subsection Simplifying indexed expressions
2175 In addition to the few automatic simplifications that GiNaC performs on
2176 indexed expressions (such as re-ordering the indices of symmetric tensors
2177 and calculating traces and convolutions of matrices and predefined tensors)
2181 ex ex::simplify_indexed();
2182 ex ex::simplify_indexed(const scalar_products & sp);
2185 that performs some more expensive operations:
2188 @item it checks the consistency of free indices in sums in the same way
2189 @code{get_free_indices()} does
2190 @item it tries to give dummy indices that appear in different terms of a sum
2191 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2192 @item it (symbolically) calculates all possible dummy index summations/contractions
2193 with the predefined tensors (this will be explained in more detail in the
2195 @item it detects contractions that vanish for symmetry reasons, for example
2196 the contraction of a symmetric and a totally antisymmetric tensor
2197 @item as a special case of dummy index summation, it can replace scalar products
2198 of two tensors with a user-defined value
2201 The last point is done with the help of the @code{scalar_products} class
2202 which is used to store scalar products with known values (this is not an
2203 arithmetic class, you just pass it to @code{simplify_indexed()}):
2207 symbol A("A"), B("B"), C("C"), i_sym("i");
2211 sp.add(A, B, 0); // A and B are orthogonal
2212 sp.add(A, C, 0); // A and C are orthogonal
2213 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2215 e = indexed(A + B, i) * indexed(A + C, i);
2217 // -> (B+A).i*(A+C).i
2219 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2225 The @code{scalar_products} object @code{sp} acts as a storage for the
2226 scalar products added to it with the @code{.add()} method. This method
2227 takes three arguments: the two expressions of which the scalar product is
2228 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2229 @code{simplify_indexed()} will replace all scalar products of indexed
2230 objects that have the symbols @code{A} and @code{B} as base expressions
2231 with the single value 0. The number, type and dimension of the indices
2232 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2234 @cindex @code{expand()}
2235 The example above also illustrates a feature of the @code{expand()} method:
2236 if passed the @code{expand_indexed} option it will distribute indices
2237 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2239 @cindex @code{tensor} (class)
2240 @subsection Predefined tensors
2242 Some frequently used special tensors such as the delta, epsilon and metric
2243 tensors are predefined in GiNaC. They have special properties when
2244 contracted with other tensor expressions and some of them have constant
2245 matrix representations (they will evaluate to a number when numeric
2246 indices are specified).
2248 @cindex @code{delta_tensor()}
2249 @subsubsection Delta tensor
2251 The delta tensor takes two indices, is symmetric and has the matrix
2252 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2253 @code{delta_tensor()}:
2257 symbol A("A"), B("B");
2259 idx i(symbol("i"), 3), j(symbol("j"), 3),
2260 k(symbol("k"), 3), l(symbol("l"), 3);
2262 ex e = indexed(A, i, j) * indexed(B, k, l)
2263 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2264 cout << e.simplify_indexed() << endl;
2267 cout << delta_tensor(i, i) << endl;
2272 @cindex @code{metric_tensor()}
2273 @subsubsection General metric tensor
2275 The function @code{metric_tensor()} creates a general symmetric metric
2276 tensor with two indices that can be used to raise/lower tensor indices. The
2277 metric tensor is denoted as @samp{g} in the output and if its indices are of
2278 mixed variance it is automatically replaced by a delta tensor:
2284 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2286 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2287 cout << e.simplify_indexed() << endl;
2290 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2291 cout << e.simplify_indexed() << endl;
2294 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2295 * metric_tensor(nu, rho);
2296 cout << e.simplify_indexed() << endl;
2299 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2300 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2301 + indexed(A, mu.toggle_variance(), rho));
2302 cout << e.simplify_indexed() << endl;
2307 @cindex @code{lorentz_g()}
2308 @subsubsection Minkowski metric tensor
2310 The Minkowski metric tensor is a special metric tensor with a constant
2311 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2312 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2313 It is created with the function @code{lorentz_g()} (although it is output as
2318 varidx mu(symbol("mu"), 4);
2320 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2321 * lorentz_g(mu, varidx(0, 4)); // negative signature
2322 cout << e.simplify_indexed() << endl;
2325 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2326 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2327 cout << e.simplify_indexed() << endl;
2332 @cindex @code{spinor_metric()}
2333 @subsubsection Spinor metric tensor
2335 The function @code{spinor_metric()} creates an antisymmetric tensor with
2336 two indices that is used to raise/lower indices of 2-component spinors.
2337 It is output as @samp{eps}:
2343 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2344 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2346 e = spinor_metric(A, B) * indexed(psi, B_co);
2347 cout << e.simplify_indexed() << endl;
2350 e = spinor_metric(A, B) * indexed(psi, A_co);
2351 cout << e.simplify_indexed() << endl;
2354 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2355 cout << e.simplify_indexed() << endl;
2358 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2359 cout << e.simplify_indexed() << endl;
2362 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2363 cout << e.simplify_indexed() << endl;
2366 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2367 cout << e.simplify_indexed() << endl;
2372 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2374 @cindex @code{epsilon_tensor()}
2375 @cindex @code{lorentz_eps()}
2376 @subsubsection Epsilon tensor
2378 The epsilon tensor is totally antisymmetric, its number of indices is equal
2379 to the dimension of the index space (the indices must all be of the same
2380 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2381 defined to be 1. Its behavior with indices that have a variance also
2382 depends on the signature of the metric. Epsilon tensors are output as
2385 There are three functions defined to create epsilon tensors in 2, 3 and 4
2389 ex epsilon_tensor(const ex & i1, const ex & i2);
2390 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2391 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2394 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2395 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2396 Minkowski space (the last @code{bool} argument specifies whether the metric
2397 has negative or positive signature, as in the case of the Minkowski metric
2402 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2403 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2404 e = lorentz_eps(mu, nu, rho, sig) *
2405 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2406 cout << simplify_indexed(e) << endl;
2407 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2409 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2410 symbol A("A"), B("B");
2411 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2412 cout << simplify_indexed(e) << endl;
2413 // -> -B.k*A.j*eps.i.k.j
2414 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2415 cout << simplify_indexed(e) << endl;
2420 @subsection Linear algebra
2422 The @code{matrix} class can be used with indices to do some simple linear
2423 algebra (linear combinations and products of vectors and matrices, traces
2424 and scalar products):
2428 idx i(symbol("i"), 2), j(symbol("j"), 2);
2429 symbol x("x"), y("y");
2431 // A is a 2x2 matrix, X is a 2x1 vector
2432 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2434 cout << indexed(A, i, i) << endl;
2437 ex e = indexed(A, i, j) * indexed(X, j);
2438 cout << e.simplify_indexed() << endl;
2439 // -> [[2*y+x],[4*y+3*x]].i
2441 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2442 cout << e.simplify_indexed() << endl;
2443 // -> [[3*y+3*x,6*y+2*x]].j
2447 You can of course obtain the same results with the @code{matrix::add()},
2448 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2449 but with indices you don't have to worry about transposing matrices.
2451 Matrix indices always start at 0 and their dimension must match the number
2452 of rows/columns of the matrix. Matrices with one row or one column are
2453 vectors and can have one or two indices (it doesn't matter whether it's a
2454 row or a column vector). Other matrices must have two indices.
2456 You should be careful when using indices with variance on matrices. GiNaC
2457 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2458 @samp{F.mu.nu} are different matrices. In this case you should use only
2459 one form for @samp{F} and explicitly multiply it with a matrix representation
2460 of the metric tensor.
2463 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2464 @c node-name, next, previous, up
2465 @section Non-commutative objects
2467 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2468 non-commutative objects are built-in which are mostly of use in high energy
2472 @item Clifford (Dirac) algebra (class @code{clifford})
2473 @item su(3) Lie algebra (class @code{color})
2474 @item Matrices (unindexed) (class @code{matrix})
2477 The @code{clifford} and @code{color} classes are subclasses of
2478 @code{indexed} because the elements of these algebras usually carry
2479 indices. The @code{matrix} class is described in more detail in
2482 Unlike most computer algebra systems, GiNaC does not primarily provide an
2483 operator (often denoted @samp{&*}) for representing inert products of
2484 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2485 classes of objects involved, and non-commutative products are formed with
2486 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2487 figuring out by itself which objects commute and will group the factors
2488 by their class. Consider this example:
2492 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2493 idx a(symbol("a"), 8), b(symbol("b"), 8);
2494 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2496 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2500 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2501 groups the non-commutative factors (the gammas and the su(3) generators)
2502 together while preserving the order of factors within each class (because
2503 Clifford objects commute with color objects). The resulting expression is a
2504 @emph{commutative} product with two factors that are themselves non-commutative
2505 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2506 parentheses are placed around the non-commutative products in the output.
2508 @cindex @code{ncmul} (class)
2509 Non-commutative products are internally represented by objects of the class
2510 @code{ncmul}, as opposed to commutative products which are handled by the
2511 @code{mul} class. You will normally not have to worry about this distinction,
2514 The advantage of this approach is that you never have to worry about using
2515 (or forgetting to use) a special operator when constructing non-commutative
2516 expressions. Also, non-commutative products in GiNaC are more intelligent
2517 than in other computer algebra systems; they can, for example, automatically
2518 canonicalize themselves according to rules specified in the implementation
2519 of the non-commutative classes. The drawback is that to work with other than
2520 the built-in algebras you have to implement new classes yourself. Symbols
2521 always commute and it's not possible to construct non-commutative products
2522 using symbols to represent the algebra elements or generators. User-defined
2523 functions can, however, be specified as being non-commutative.
2525 @cindex @code{return_type()}
2526 @cindex @code{return_type_tinfo()}
2527 Information about the commutativity of an object or expression can be
2528 obtained with the two member functions
2531 unsigned ex::return_type() const;
2532 unsigned ex::return_type_tinfo() const;
2535 The @code{return_type()} function returns one of three values (defined in
2536 the header file @file{flags.h}), corresponding to three categories of
2537 expressions in GiNaC:
2540 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2541 classes are of this kind.
2542 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2543 certain class of non-commutative objects which can be determined with the
2544 @code{return_type_tinfo()} method. Expressions of this category commute
2545 with everything except @code{noncommutative} expressions of the same
2547 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2548 of non-commutative objects of different classes. Expressions of this
2549 category don't commute with any other @code{noncommutative} or
2550 @code{noncommutative_composite} expressions.
2553 The value returned by the @code{return_type_tinfo()} method is valid only
2554 when the return type of the expression is @code{noncommutative}. It is a
2555 value that is unique to the class of the object and usually one of the
2556 constants in @file{tinfos.h}, or derived therefrom.
2558 Here are a couple of examples:
2561 @multitable @columnfractions 0.33 0.33 0.34
2562 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2563 @item @code{42} @tab @code{commutative} @tab -
2564 @item @code{2*x-y} @tab @code{commutative} @tab -
2565 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2566 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2567 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2568 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2572 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2573 @code{TINFO_clifford} for objects with a representation label of zero.
2574 Other representation labels yield a different @code{return_type_tinfo()},
2575 but it's the same for any two objects with the same label. This is also true
2578 A last note: With the exception of matrices, positive integer powers of
2579 non-commutative objects are automatically expanded in GiNaC. For example,
2580 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2581 non-commutative expressions).
2584 @cindex @code{clifford} (class)
2585 @subsection Clifford algebra
2587 @cindex @code{dirac_gamma()}
2588 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2589 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2590 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2591 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2594 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2597 which takes two arguments: the index and a @dfn{representation label} in the
2598 range 0 to 255 which is used to distinguish elements of different Clifford
2599 algebras (this is also called a @dfn{spin line index}). Gammas with different
2600 labels commute with each other. The dimension of the index can be 4 or (in
2601 the framework of dimensional regularization) any symbolic value. Spinor
2602 indices on Dirac gammas are not supported in GiNaC.
2604 @cindex @code{dirac_ONE()}
2605 The unity element of a Clifford algebra is constructed by
2608 ex dirac_ONE(unsigned char rl = 0);
2611 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2612 multiples of the unity element, even though it's customary to omit it.
2613 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2614 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2615 GiNaC will complain and/or produce incorrect results.
2617 @cindex @code{dirac_gamma5()}
2618 There is a special element @samp{gamma5} that commutes with all other
2619 gammas, has a unit square, and in 4 dimensions equals
2620 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2623 ex dirac_gamma5(unsigned char rl = 0);
2626 @cindex @code{dirac_gammaL()}
2627 @cindex @code{dirac_gammaR()}
2628 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2629 objects, constructed by
2632 ex dirac_gammaL(unsigned char rl = 0);
2633 ex dirac_gammaR(unsigned char rl = 0);
2636 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2637 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2639 @cindex @code{dirac_slash()}
2640 Finally, the function
2643 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2646 creates a term that represents a contraction of @samp{e} with the Dirac
2647 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2648 with a unique index whose dimension is given by the @code{dim} argument).
2649 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2651 In products of dirac gammas, superfluous unity elements are automatically
2652 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2653 and @samp{gammaR} are moved to the front.
2655 The @code{simplify_indexed()} function performs contractions in gamma strings,
2661 symbol a("a"), b("b"), D("D");
2662 varidx mu(symbol("mu"), D);
2663 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2664 * dirac_gamma(mu.toggle_variance());
2666 // -> gamma~mu*a\*gamma.mu
2667 e = e.simplify_indexed();
2670 cout << e.subs(D == 4) << endl;
2676 @cindex @code{dirac_trace()}
2677 To calculate the trace of an expression containing strings of Dirac gammas
2678 you use the function
2681 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2684 This function takes the trace of all gammas with the specified representation
2685 label; gammas with other labels are left standing. The last argument to
2686 @code{dirac_trace()} is the value to be returned for the trace of the unity
2687 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2688 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2689 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2690 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2691 This @samp{gamma5} scheme is described in greater detail in
2692 @cite{The Role of gamma5 in Dimensional Regularization}.
2694 The value of the trace itself is also usually different in 4 and in
2695 @math{D != 4} dimensions:
2700 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2701 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2702 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2703 cout << dirac_trace(e).simplify_indexed() << endl;
2710 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2711 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2712 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2713 cout << dirac_trace(e).simplify_indexed() << endl;
2714 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2718 Here is an example for using @code{dirac_trace()} to compute a value that
2719 appears in the calculation of the one-loop vacuum polarization amplitude in
2724 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2725 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2728 sp.add(l, l, pow(l, 2));
2729 sp.add(l, q, ldotq);
2731 ex e = dirac_gamma(mu) *
2732 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2733 dirac_gamma(mu.toggle_variance()) *
2734 (dirac_slash(l, D) + m * dirac_ONE());
2735 e = dirac_trace(e).simplify_indexed(sp);
2736 e = e.collect(lst(l, ldotq, m));
2738 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2742 The @code{canonicalize_clifford()} function reorders all gamma products that
2743 appear in an expression to a canonical (but not necessarily simple) form.
2744 You can use this to compare two expressions or for further simplifications:
2748 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2749 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2751 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2753 e = canonicalize_clifford(e);
2760 @cindex @code{color} (class)
2761 @subsection Color algebra
2763 @cindex @code{color_T()}
2764 For computations in quantum chromodynamics, GiNaC implements the base elements
2765 and structure constants of the su(3) Lie algebra (color algebra). The base
2766 elements @math{T_a} are constructed by the function
2769 ex color_T(const ex & a, unsigned char rl = 0);
2772 which takes two arguments: the index and a @dfn{representation label} in the
2773 range 0 to 255 which is used to distinguish elements of different color
2774 algebras. Objects with different labels commute with each other. The
2775 dimension of the index must be exactly 8 and it should be of class @code{idx},
2778 @cindex @code{color_ONE()}
2779 The unity element of a color algebra is constructed by
2782 ex color_ONE(unsigned char rl = 0);
2785 @strong{Note:} You must always use @code{color_ONE()} when referring to
2786 multiples of the unity element, even though it's customary to omit it.
2787 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2788 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2789 GiNaC may produce incorrect results.
2791 @cindex @code{color_d()}
2792 @cindex @code{color_f()}
2796 ex color_d(const ex & a, const ex & b, const ex & c);
2797 ex color_f(const ex & a, const ex & b, const ex & c);
2800 create the symmetric and antisymmetric structure constants @math{d_abc} and
2801 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2802 and @math{[T_a, T_b] = i f_abc T_c}.
2804 @cindex @code{color_h()}
2805 There's an additional function
2808 ex color_h(const ex & a, const ex & b, const ex & c);
2811 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2813 The function @code{simplify_indexed()} performs some simplifications on
2814 expressions containing color objects:
2819 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2820 k(symbol("k"), 8), l(symbol("l"), 8);
2822 e = color_d(a, b, l) * color_f(a, b, k);
2823 cout << e.simplify_indexed() << endl;
2826 e = color_d(a, b, l) * color_d(a, b, k);
2827 cout << e.simplify_indexed() << endl;
2830 e = color_f(l, a, b) * color_f(a, b, k);
2831 cout << e.simplify_indexed() << endl;
2834 e = color_h(a, b, c) * color_h(a, b, c);
2835 cout << e.simplify_indexed() << endl;
2838 e = color_h(a, b, c) * color_T(b) * color_T(c);
2839 cout << e.simplify_indexed() << endl;
2842 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2843 cout << e.simplify_indexed() << endl;
2846 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2847 cout << e.simplify_indexed() << endl;
2848 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2852 @cindex @code{color_trace()}
2853 To calculate the trace of an expression containing color objects you use the
2857 ex color_trace(const ex & e, unsigned char rl = 0);
2860 This function takes the trace of all color @samp{T} objects with the
2861 specified representation label; @samp{T}s with other labels are left
2862 standing. For example:
2866 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2868 // -> -I*f.a.c.b+d.a.c.b
2873 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2874 @c node-name, next, previous, up
2875 @chapter Methods and Functions
2878 In this chapter the most important algorithms provided by GiNaC will be
2879 described. Some of them are implemented as functions on expressions,
2880 others are implemented as methods provided by expression objects. If
2881 they are methods, there exists a wrapper function around it, so you can
2882 alternatively call it in a functional way as shown in the simple
2887 cout << "As method: " << sin(1).evalf() << endl;
2888 cout << "As function: " << evalf(sin(1)) << endl;
2892 @cindex @code{subs()}
2893 The general rule is that wherever methods accept one or more parameters
2894 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2895 wrapper accepts is the same but preceded by the object to act on
2896 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2897 most natural one in an OO model but it may lead to confusion for MapleV
2898 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2899 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2900 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2901 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2902 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2903 here. Also, users of MuPAD will in most cases feel more comfortable
2904 with GiNaC's convention. All function wrappers are implemented
2905 as simple inline functions which just call the corresponding method and
2906 are only provided for users uncomfortable with OO who are dead set to
2907 avoid method invocations. Generally, nested function wrappers are much
2908 harder to read than a sequence of methods and should therefore be
2909 avoided if possible. On the other hand, not everything in GiNaC is a
2910 method on class @code{ex} and sometimes calling a function cannot be
2914 * Information About Expressions::
2915 * Numerical Evaluation::
2916 * Substituting Expressions::
2917 * Pattern Matching and Advanced Substitutions::
2918 * Applying a Function on Subexpressions::
2919 * Visitors and Tree Traversal::
2920 * Polynomial Arithmetic:: Working with polynomials.
2921 * Rational Expressions:: Working with rational functions.
2922 * Symbolic Differentiation::
2923 * Series Expansion:: Taylor and Laurent expansion.
2925 * Built-in Functions:: List of predefined mathematical functions.
2926 * Solving Linear Systems of Equations::
2927 * Input/Output:: Input and output of expressions.
2931 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
2932 @c node-name, next, previous, up
2933 @section Getting information about expressions
2935 @subsection Checking expression types
2936 @cindex @code{is_a<@dots{}>()}
2937 @cindex @code{is_exactly_a<@dots{}>()}
2938 @cindex @code{ex_to<@dots{}>()}
2939 @cindex Converting @code{ex} to other classes
2940 @cindex @code{info()}
2941 @cindex @code{return_type()}
2942 @cindex @code{return_type_tinfo()}
2944 Sometimes it's useful to check whether a given expression is a plain number,
2945 a sum, a polynomial with integer coefficients, or of some other specific type.
2946 GiNaC provides a couple of functions for this:
2949 bool is_a<T>(const ex & e);
2950 bool is_exactly_a<T>(const ex & e);
2951 bool ex::info(unsigned flag);
2952 unsigned ex::return_type() const;
2953 unsigned ex::return_type_tinfo() const;
2956 When the test made by @code{is_a<T>()} returns true, it is safe to call
2957 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2958 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2959 example, assuming @code{e} is an @code{ex}:
2964 if (is_a<numeric>(e))
2965 numeric n = ex_to<numeric>(e);
2970 @code{is_a<T>(e)} allows you to check whether the top-level object of
2971 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2972 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2973 e.g., for checking whether an expression is a number, a sum, or a product:
2980 is_a<numeric>(e1); // true
2981 is_a<numeric>(e2); // false
2982 is_a<add>(e1); // false
2983 is_a<add>(e2); // true
2984 is_a<mul>(e1); // false
2985 is_a<mul>(e2); // false
2989 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2990 top-level object of an expression @samp{e} is an instance of the GiNaC
2991 class @samp{T}, not including parent classes.
2993 The @code{info()} method is used for checking certain attributes of
2994 expressions. The possible values for the @code{flag} argument are defined
2995 in @file{ginac/flags.h}, the most important being explained in the following
2999 @multitable @columnfractions .30 .70
3000 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3001 @item @code{numeric}
3002 @tab @dots{}a number (same as @code{is_<numeric>(...)})
3004 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3005 @item @code{rational}
3006 @tab @dots{}an exact rational number (integers are rational, too)
3007 @item @code{integer}
3008 @tab @dots{}a (non-complex) integer
3009 @item @code{crational}
3010 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3011 @item @code{cinteger}
3012 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3013 @item @code{positive}
3014 @tab @dots{}not complex and greater than 0
3015 @item @code{negative}
3016 @tab @dots{}not complex and less than 0
3017 @item @code{nonnegative}
3018 @tab @dots{}not complex and greater than or equal to 0
3020 @tab @dots{}an integer greater than 0
3022 @tab @dots{}an integer less than 0
3023 @item @code{nonnegint}
3024 @tab @dots{}an integer greater than or equal to 0
3026 @tab @dots{}an even integer
3028 @tab @dots{}an odd integer
3030 @tab @dots{}a prime integer (probabilistic primality test)
3031 @item @code{relation}
3032 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3033 @item @code{relation_equal}
3034 @tab @dots{}a @code{==} relation
3035 @item @code{relation_not_equal}
3036 @tab @dots{}a @code{!=} relation
3037 @item @code{relation_less}
3038 @tab @dots{}a @code{<} relation
3039 @item @code{relation_less_or_equal}
3040 @tab @dots{}a @code{<=} relation
3041 @item @code{relation_greater}
3042 @tab @dots{}a @code{>} relation
3043 @item @code{relation_greater_or_equal}
3044 @tab @dots{}a @code{>=} relation
3046 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3048 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3049 @item @code{polynomial}
3050 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3051 @item @code{integer_polynomial}
3052 @tab @dots{}a polynomial with (non-complex) integer coefficients
3053 @item @code{cinteger_polynomial}
3054 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3055 @item @code{rational_polynomial}
3056 @tab @dots{}a polynomial with (non-complex) rational coefficients
3057 @item @code{crational_polynomial}
3058 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3059 @item @code{rational_function}
3060 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3061 @item @code{algebraic}
3062 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3066 To determine whether an expression is commutative or non-commutative and if
3067 so, with which other expressions it would commute, you use the methods
3068 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3069 for an explanation of these.
3072 @subsection Accessing subexpressions
3073 @cindex @code{nops()}
3076 @cindex @code{relational} (class)
3078 GiNaC provides the two methods
3082 ex ex::op(size_t i);
3085 for accessing the subexpressions in the container-like GiNaC classes like
3086 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3087 determines the number of subexpressions (@samp{operands}) contained, while
3088 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3089 In the case of a @code{power} object, @code{op(0)} will return the basis
3090 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3091 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3093 The left-hand and right-hand side expressions of objects of class
3094 @code{relational} (and only of these) can also be accessed with the methods
3102 @subsection Comparing expressions
3103 @cindex @code{is_equal()}
3104 @cindex @code{is_zero()}
3106 Expressions can be compared with the usual C++ relational operators like
3107 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3108 the result is usually not determinable and the result will be @code{false},
3109 except in the case of the @code{!=} operator. You should also be aware that
3110 GiNaC will only do the most trivial test for equality (subtracting both
3111 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3114 Actually, if you construct an expression like @code{a == b}, this will be
3115 represented by an object of the @code{relational} class (@pxref{Relations})
3116 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3118 There are also two methods
3121 bool ex::is_equal(const ex & other);
3125 for checking whether one expression is equal to another, or equal to zero,
3129 @subsection Ordering expressions
3130 @cindex @code{ex_is_less} (class)
3131 @cindex @code{ex_is_equal} (class)
3132 @cindex @code{compare()}
3134 Sometimes it is necessary to establish a mathematically well-defined ordering
3135 on a set of arbitrary expressions, for example to use expressions as keys
3136 in a @code{std::map<>} container, or to bring a vector of expressions into
3137 a canonical order (which is done internally by GiNaC for sums and products).
3139 The operators @code{<}, @code{>} etc. described in the last section cannot
3140 be used for this, as they don't implement an ordering relation in the
3141 mathematical sense. In particular, they are not guaranteed to be
3142 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3143 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3146 By default, STL classes and algorithms use the @code{<} and @code{==}
3147 operators to compare objects, which are unsuitable for expressions, but GiNaC
3148 provides two functors that can be supplied as proper binary comparison
3149 predicates to the STL:
3152 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3154 bool operator()(const ex &lh, const ex &rh) const;
3157 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3159 bool operator()(const ex &lh, const ex &rh) const;
3163 For example, to define a @code{map} that maps expressions to strings you
3167 std::map<ex, std::string, ex_is_less> myMap;
3170 Omitting the @code{ex_is_less} template parameter will introduce spurious
3171 bugs because the map operates improperly.
3173 Other examples for the use of the functors:
3181 std::sort(v.begin(), v.end(), ex_is_less());
3183 // count the number of expressions equal to '1'
3184 unsigned num_ones = std::count_if(v.begin(), v.end(),
3185 std::bind2nd(ex_is_equal(), 1));
3188 The implementation of @code{ex_is_less} uses the member function
3191 int ex::compare(const ex & other) const;
3194 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3195 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3199 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3200 @c node-name, next, previous, up
3201 @section Numercial Evaluation
3202 @cindex @code{evalf()}
3204 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3205 To evaluate them using floating-point arithmetic you need to call
3208 ex ex::evalf(int level = 0) const;
3211 @cindex @code{Digits}
3212 The accuracy of the evaluation is controlled by the global object @code{Digits}
3213 which can be assigned an integer value. The default value of @code{Digits}
3214 is 17. @xref{Numbers}, for more information and examples.
3216 To evaluate an expression to a @code{double} floating-point number you can
3217 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3221 // Approximate sin(x/Pi)
3223 ex e = series(sin(x/Pi), x == 0, 6);
3225 // Evaluate numerically at x=0.1
3226 ex f = evalf(e.subs(x == 0.1));
3228 // ex_to<numeric> is an unsafe cast, so check the type first
3229 if (is_a<numeric>(f)) @{
3230 double d = ex_to<numeric>(f).to_double();
3239 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3240 @c node-name, next, previous, up
3241 @section Substituting expressions
3242 @cindex @code{subs()}
3244 Algebraic objects inside expressions can be replaced with arbitrary
3245 expressions via the @code{.subs()} method:
3248 ex ex::subs(const ex & e, unsigned options = 0);
3249 ex ex::subs(const exmap & m, unsigned options = 0);
3250 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3253 In the first form, @code{subs()} accepts a relational of the form
3254 @samp{object == expression} or a @code{lst} of such relationals:
3258 symbol x("x"), y("y");
3260 ex e1 = 2*x^2-4*x+3;
3261 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3265 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3270 If you specify multiple substitutions, they are performed in parallel, so e.g.
3271 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3273 The second form of @code{subs()} takes an @code{exmap} object which is a
3274 pair associative container that maps expressions to expressions (currently
3275 implemented as a @code{std::map}). This is the most efficient one of the
3276 three @code{subs()} forms and should be used when the number of objects to
3277 be substituted is large or unknown.
3279 Using this form, the second example from above would look like this:
3283 symbol x("x"), y("y");
3289 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3293 The third form of @code{subs()} takes two lists, one for the objects to be
3294 replaced and one for the expressions to be substituted (both lists must
3295 contain the same number of elements). Using this form, you would write
3299 symbol x("x"), y("y");
3302 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3306 The optional last argument to @code{subs()} is a combination of
3307 @code{subs_options} flags. There are two options available:
3308 @code{subs_options::no_pattern} disables pattern matching, which makes
3309 large @code{subs()} operations significantly faster if you are not using
3310 patterns. The second option, @code{subs_options::algebraic} enables
3311 algebraic substitutions in products and powers.
3312 @ref{Pattern Matching and Advanced Substitutions}, for more information
3313 about patterns and algebraic substitutions.
3315 @code{subs()} performs syntactic substitution of any complete algebraic
3316 object; it does not try to match sub-expressions as is demonstrated by the
3321 symbol x("x"), y("y"), z("z");
3323 ex e1 = pow(x+y, 2);
3324 cout << e1.subs(x+y == 4) << endl;
3327 ex e2 = sin(x)*sin(y)*cos(x);
3328 cout << e2.subs(sin(x) == cos(x)) << endl;
3329 // -> cos(x)^2*sin(y)
3332 cout << e3.subs(x+y == 4) << endl;
3334 // (and not 4+z as one might expect)
3338 A more powerful form of substitution using wildcards is described in the
3342 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3343 @c node-name, next, previous, up
3344 @section Pattern matching and advanced substitutions
3345 @cindex @code{wildcard} (class)
3346 @cindex Pattern matching
3348 GiNaC allows the use of patterns for checking whether an expression is of a
3349 certain form or contains subexpressions of a certain form, and for
3350 substituting expressions in a more general way.
3352 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3353 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3354 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3355 an unsigned integer number to allow having multiple different wildcards in a
3356 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3357 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3361 ex wild(unsigned label = 0);
3364 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3367 Some examples for patterns:
3369 @multitable @columnfractions .5 .5
3370 @item @strong{Constructed as} @tab @strong{Output as}
3371 @item @code{wild()} @tab @samp{$0}
3372 @item @code{pow(x,wild())} @tab @samp{x^$0}
3373 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3374 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3380 @item Wildcards behave like symbols and are subject to the same algebraic
3381 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3382 @item As shown in the last example, to use wildcards for indices you have to
3383 use them as the value of an @code{idx} object. This is because indices must
3384 always be of class @code{idx} (or a subclass).
3385 @item Wildcards only represent expressions or subexpressions. It is not
3386 possible to use them as placeholders for other properties like index
3387 dimension or variance, representation labels, symmetry of indexed objects
3389 @item Because wildcards are commutative, it is not possible to use wildcards
3390 as part of noncommutative products.
3391 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3392 are also valid patterns.
3395 @subsection Matching expressions
3396 @cindex @code{match()}
3397 The most basic application of patterns is to check whether an expression
3398 matches a given pattern. This is done by the function
3401 bool ex::match(const ex & pattern);
3402 bool ex::match(const ex & pattern, lst & repls);
3405 This function returns @code{true} when the expression matches the pattern
3406 and @code{false} if it doesn't. If used in the second form, the actual
3407 subexpressions matched by the wildcards get returned in the @code{repls}
3408 object as a list of relations of the form @samp{wildcard == expression}.
3409 If @code{match()} returns false, the state of @code{repls} is undefined.
3410 For reproducible results, the list should be empty when passed to
3411 @code{match()}, but it is also possible to find similarities in multiple
3412 expressions by passing in the result of a previous match.
3414 The matching algorithm works as follows:
3417 @item A single wildcard matches any expression. If one wildcard appears
3418 multiple times in a pattern, it must match the same expression in all
3419 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3420 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3421 @item If the expression is not of the same class as the pattern, the match
3422 fails (i.e. a sum only matches a sum, a function only matches a function,
3424 @item If the pattern is a function, it only matches the same function
3425 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3426 @item Except for sums and products, the match fails if the number of
3427 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3429 @item If there are no subexpressions, the expressions and the pattern must
3430 be equal (in the sense of @code{is_equal()}).
3431 @item Except for sums and products, each subexpression (@code{op()}) must
3432 match the corresponding subexpression of the pattern.
3435 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3436 account for their commutativity and associativity:
3439 @item If the pattern contains a term or factor that is a single wildcard,
3440 this one is used as the @dfn{global wildcard}. If there is more than one
3441 such wildcard, one of them is chosen as the global wildcard in a random
3443 @item Every term/factor of the pattern, except the global wildcard, is
3444 matched against every term of the expression in sequence. If no match is
3445 found, the whole match fails. Terms that did match are not considered in
3447 @item If there are no unmatched terms left, the match succeeds. Otherwise
3448 the match fails unless there is a global wildcard in the pattern, in
3449 which case this wildcard matches the remaining terms.
3452 In general, having more than one single wildcard as a term of a sum or a
3453 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3456 Here are some examples in @command{ginsh} to demonstrate how it works (the
3457 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3458 match fails, and the list of wildcard replacements otherwise):
3461 > match((x+y)^a,(x+y)^a);
3463 > match((x+y)^a,(x+y)^b);
3465 > match((x+y)^a,$1^$2);
3467 > match((x+y)^a,$1^$1);
3469 > match((x+y)^(x+y),$1^$1);
3471 > match((x+y)^(x+y),$1^$2);
3473 > match((a+b)*(a+c),($1+b)*($1+c));
3475 > match((a+b)*(a+c),(a+$1)*(a+$2));
3477 (Unpredictable. The result might also be [$1==c,$2==b].)
3478 > match((a+b)*(a+c),($1+$2)*($1+$3));
3479 (The result is undefined. Due to the sequential nature of the algorithm
3480 and the re-ordering of terms in GiNaC, the match for the first factor
3481 may be @{$1==a,$2==b@} in which case the match for the second factor
3482 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3484 > match(a*(x+y)+a*z+b,a*$1+$2);
3485 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3486 @{$1=x+y,$2=a*z+b@}.)
3487 > match(a+b+c+d+e+f,c);
3489 > match(a+b+c+d+e+f,c+$0);
3491 > match(a+b+c+d+e+f,c+e+$0);
3493 > match(a+b,a+b+$0);
3495 > match(a*b^2,a^$1*b^$2);
3497 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3498 even though a==a^1.)
3499 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3501 > match(atan2(y,x^2),atan2(y,$0));
3505 @subsection Matching parts of expressions
3506 @cindex @code{has()}
3507 A more general way to look for patterns in expressions is provided by the
3511 bool ex::has(const ex & pattern);
3514 This function checks whether a pattern is matched by an expression itself or
3515 by any of its subexpressions.
3517 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3518 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3521 > has(x*sin(x+y+2*a),y);
3523 > has(x*sin(x+y+2*a),x+y);
3525 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3526 has the subexpressions "x", "y" and "2*a".)
3527 > has(x*sin(x+y+2*a),x+y+$1);
3529 (But this is possible.)
3530 > has(x*sin(2*(x+y)+2*a),x+y);
3532 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3533 which "x+y" is not a subexpression.)
3536 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3538 > has(4*x^2-x+3,$1*x);
3540 > has(4*x^2+x+3,$1*x);
3542 (Another possible pitfall. The first expression matches because the term
3543 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3544 contains a linear term you should use the coeff() function instead.)
3547 @cindex @code{find()}
3551 bool ex::find(const ex & pattern, lst & found);
3554 works a bit like @code{has()} but it doesn't stop upon finding the first
3555 match. Instead, it appends all found matches to the specified list. If there
3556 are multiple occurrences of the same expression, it is entered only once to
3557 the list. @code{find()} returns false if no matches were found (in
3558 @command{ginsh}, it returns an empty list):
3561 > find(1+x+x^2+x^3,x);
3563 > find(1+x+x^2+x^3,y);
3565 > find(1+x+x^2+x^3,x^$1);
3567 (Note the absence of "x".)
3568 > expand((sin(x)+sin(y))*(a+b));
3569 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3574 @subsection Substituting expressions
3575 @cindex @code{subs()}
3576 Probably the most useful application of patterns is to use them for
3577 substituting expressions with the @code{subs()} method. Wildcards can be
3578 used in the search patterns as well as in the replacement expressions, where
3579 they get replaced by the expressions matched by them. @code{subs()} doesn't
3580 know anything about algebra; it performs purely syntactic substitutions.
3585 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3587 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3589 > subs((a+b+c)^2,a+b==x);
3591 > subs((a+b+c)^2,a+b+$1==x+$1);
3593 > subs(a+2*b,a+b==x);
3595 > subs(4*x^3-2*x^2+5*x-1,x==a);
3597 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3599 > subs(sin(1+sin(x)),sin($1)==cos($1));
3601 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3605 The last example would be written in C++ in this way:
3609 symbol a("a"), b("b"), x("x"), y("y");
3610 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3611 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3612 cout << e.expand() << endl;
3617 @subsection Algebraic substitutions
3618 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3619 enables smarter, algebraic substitutions in products and powers. If you want
3620 to substitute some factors of a product, you only need to list these factors
3621 in your pattern. Furthermore, if an (integer) power of some expression occurs
3622 in your pattern and in the expression that you want the substitution to occur
3623 in, it can be substituted as many times as possible, without getting negative
3626 An example clarifies it all (hopefully):
3629 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3630 subs_options::algebraic) << endl;
3631 // --> (y+x)^6+b^6+a^6
3633 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3635 // Powers and products are smart, but addition is just the same.
3637 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3640 // As I said: addition is just the same.
3642 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3643 // --> x^3*b*a^2+2*b
3645 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3647 // --> 2*b+x^3*b^(-1)*a^(-2)
3649 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3650 // --> -1-2*a^2+4*a^3+5*a
3652 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3653 subs_options::algebraic) << endl;
3654 // --> -1+5*x+4*x^3-2*x^2
3655 // You should not really need this kind of patterns very often now.
3656 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3658 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3659 subs_options::algebraic) << endl;
3660 // --> cos(1+cos(x))
3662 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3663 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3664 subs_options::algebraic)) << endl;
3669 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3670 @c node-name, next, previous, up
3671 @section Applying a Function on Subexpressions
3672 @cindex tree traversal
3673 @cindex @code{map()}
3675 Sometimes you may want to perform an operation on specific parts of an
3676 expression while leaving the general structure of it intact. An example
3677 of this would be a matrix trace operation: the trace of a sum is the sum
3678 of the traces of the individual terms. That is, the trace should @dfn{map}
3679 on the sum, by applying itself to each of the sum's operands. It is possible
3680 to do this manually which usually results in code like this:
3685 if (is_a<matrix>(e))
3686 return ex_to<matrix>(e).trace();
3687 else if (is_a<add>(e)) @{
3689 for (size_t i=0; i<e.nops(); i++)
3690 sum += calc_trace(e.op(i));
3692 @} else if (is_a<mul>)(e)) @{
3700 This is, however, slightly inefficient (if the sum is very large it can take
3701 a long time to add the terms one-by-one), and its applicability is limited to
3702 a rather small class of expressions. If @code{calc_trace()} is called with
3703 a relation or a list as its argument, you will probably want the trace to
3704 be taken on both sides of the relation or of all elements of the list.
3706 GiNaC offers the @code{map()} method to aid in the implementation of such
3710 ex ex::map(map_function & f) const;
3711 ex ex::map(ex (*f)(const ex & e)) const;
3714 In the first (preferred) form, @code{map()} takes a function object that
3715 is subclassed from the @code{map_function} class. In the second form, it
3716 takes a pointer to a function that accepts and returns an expression.
3717 @code{map()} constructs a new expression of the same type, applying the
3718 specified function on all subexpressions (in the sense of @code{op()}),