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 @math{O(n log n)}
747 algebraically correct, possibly except for a set of measure zero (e.g.
748 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
751 There are two types of automatic transformations in GiNaC that may not
752 behave in an entirely obvious way at first glance:
756 The terms of sums and products (and some other things like the arguments of
757 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
758 into a canonical form that is deterministic, but not lexicographical or in
759 any other way easily guessable (it almost always depends on the number and
760 order of the symbols you define). However, constructing the same expression
761 twice, either implicitly or explicitly, will always result in the same
764 Expressions of the form 'number times sum' are automatically expanded (this
765 has to do with GiNaC's internal representation of sums and products). For
768 ex MyEx5 = 2*(x + y); // 2*x+2*y
769 ex MyEx6 = z*(x + y); // z*(x+y)
773 The general rule is that when you construct expressions, GiNaC automatically
774 creates them in canonical form, which might differ from the form you typed in
775 your program. This may create some awkward looking output (@samp{-y+x} instead
776 of @samp{x-y}) but allows for more efficient operation and usually yields
777 some immediate simplifications.
779 @cindex @code{eval()}
780 Internally, the anonymous evaluator in GiNaC is implemented by the methods
783 ex ex::eval(int level = 0) const;
784 ex basic::eval(int level = 0) const;
787 but unless you are extending GiNaC with your own classes or functions, there
788 should never be any reason to call them explicitly. All GiNaC methods that
789 transform expressions, like @code{subs()} or @code{normal()}, automatically
790 re-evaluate their results.
793 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
794 @c node-name, next, previous, up
795 @section Error handling
797 @cindex @code{pole_error} (class)
799 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
800 generated by GiNaC are subclassed from the standard @code{exception} class
801 defined in the @file{<stdexcept>} header. In addition to the predefined
802 @code{logic_error}, @code{domain_error}, @code{out_of_range},
803 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
804 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
805 exception that gets thrown when trying to evaluate a mathematical function
808 The @code{pole_error} class has a member function
811 int pole_error::degree() const;
814 that returns the order of the singularity (or 0 when the pole is
815 logarithmic or the order is undefined).
817 When using GiNaC it is useful to arrange for exceptions to be catched in
818 the main program even if you don't want to do any special error handling.
819 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
820 default exception handler of your C++ compiler's run-time system which
821 usually only aborts the program without giving any information what went
824 Here is an example for a @code{main()} function that catches and prints
825 exceptions generated by GiNaC:
830 #include <ginac/ginac.h>
832 using namespace GiNaC;
840 @} catch (exception &p) @{
841 cerr << p.what() << endl;
849 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
850 @c node-name, next, previous, up
851 @section The Class Hierarchy
853 GiNaC's class hierarchy consists of several classes representing
854 mathematical objects, all of which (except for @code{ex} and some
855 helpers) are internally derived from one abstract base class called
856 @code{basic}. You do not have to deal with objects of class
857 @code{basic}, instead you'll be dealing with symbols, numbers,
858 containers of expressions and so on.
862 To get an idea about what kinds of symbolic composites may be built we
863 have a look at the most important classes in the class hierarchy and
864 some of the relations among the classes:
866 @image{classhierarchy}
868 The abstract classes shown here (the ones without drop-shadow) are of no
869 interest for the user. They are used internally in order to avoid code
870 duplication if two or more classes derived from them share certain
871 features. An example is @code{expairseq}, a container for a sequence of
872 pairs each consisting of one expression and a number (@code{numeric}).
873 What @emph{is} visible to the user are the derived classes @code{add}
874 and @code{mul}, representing sums and products. @xref{Internal
875 Structures}, where these two classes are described in more detail. The
876 following table shortly summarizes what kinds of mathematical objects
877 are stored in the different classes:
880 @multitable @columnfractions .22 .78
881 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
882 @item @code{constant} @tab Constants like
889 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
890 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
891 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
892 @item @code{ncmul} @tab Products of non-commutative objects
893 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
898 @code{sqrt(}@math{2}@code{)}
901 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
902 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
903 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
904 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
905 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
906 @item @code{indexed} @tab Indexed object like @math{A_ij}
907 @item @code{tensor} @tab Special tensor like the delta and metric tensors
908 @item @code{idx} @tab Index of an indexed object
909 @item @code{varidx} @tab Index with variance
910 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
911 @item @code{wildcard} @tab Wildcard for pattern matching
912 @item @code{structure} @tab Template for user-defined classes
917 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
918 @c node-name, next, previous, up
920 @cindex @code{symbol} (class)
921 @cindex hierarchy of classes
924 Symbols are for symbolic manipulation what atoms are for chemistry. You
925 can declare objects of class @code{symbol} as any other object simply by
926 saying @code{symbol x,y;}. There is, however, a catch in here having to
927 do with the fact that C++ is a compiled language. The information about
928 the symbol's name is thrown away by the compiler but at a later stage
929 you may want to print expressions holding your symbols. In order to
930 avoid confusion GiNaC's symbols are able to know their own name. This
931 is accomplished by declaring its name for output at construction time in
932 the fashion @code{symbol x("x");}. If you declare a symbol using the
933 default constructor (i.e. without string argument) the system will deal
934 out a unique name. That name may not be suitable for printing but for
935 internal routines when no output is desired it is often enough. We'll
936 come across examples of such symbols later in this tutorial.
938 This implies that the strings passed to symbols at construction time may
939 not be used for comparing two of them. It is perfectly legitimate to
940 write @code{symbol x("x"),y("x");} but it is likely to lead into
941 trouble. Here, @code{x} and @code{y} are different symbols and
942 statements like @code{x-y} will not be simplified to zero although the
943 output @code{x-x} looks funny. Such output may also occur when there
944 are two different symbols in two scopes, for instance when you call a
945 function that declares a symbol with a name already existent in a symbol
946 in the calling function. Again, comparing them (using @code{operator==}
947 for instance) will always reveal their difference. Watch out, please.
949 @cindex @code{subs()}
950 Although symbols can be assigned expressions for internal reasons, you
951 should not do it (and we are not going to tell you how it is done). If
952 you want to replace a symbol with something else in an expression, you
953 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
956 @node Numbers, Constants, Symbols, Basic Concepts
957 @c node-name, next, previous, up
959 @cindex @code{numeric} (class)
965 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
966 The classes therein serve as foundation classes for GiNaC. CLN stands
967 for Class Library for Numbers or alternatively for Common Lisp Numbers.
968 In order to find out more about CLN's internals, the reader is referred to
969 the documentation of that library. @inforef{Introduction, , cln}, for
970 more information. Suffice to say that it is by itself build on top of
971 another library, the GNU Multiple Precision library GMP, which is an
972 extremely fast library for arbitrary long integers and rationals as well
973 as arbitrary precision floating point numbers. It is very commonly used
974 by several popular cryptographic applications. CLN extends GMP by
975 several useful things: First, it introduces the complex number field
976 over either reals (i.e. floating point numbers with arbitrary precision)
977 or rationals. Second, it automatically converts rationals to integers
978 if the denominator is unity and complex numbers to real numbers if the
979 imaginary part vanishes and also correctly treats algebraic functions.
980 Third it provides good implementations of state-of-the-art algorithms
981 for all trigonometric and hyperbolic functions as well as for
982 calculation of some useful constants.
984 The user can construct an object of class @code{numeric} in several
985 ways. The following example shows the four most important constructors.
986 It uses construction from C-integer, construction of fractions from two
987 integers, construction from C-float and construction from a string:
991 #include <ginac/ginac.h>
992 using namespace GiNaC;
996 numeric two = 2; // exact integer 2
997 numeric r(2,3); // exact fraction 2/3
998 numeric e(2.71828); // floating point number
999 numeric p = "3.14159265358979323846"; // constructor from string
1000 // Trott's constant in scientific notation:
1001 numeric trott("1.0841015122311136151E-2");
1003 std::cout << two*p << std::endl; // floating point 6.283...
1008 @cindex complex numbers
1009 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1014 numeric z1 = 2-3*I; // exact complex number 2-3i
1015 numeric z2 = 5.9+1.6*I; // complex floating point number
1019 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1020 This would, however, call C's built-in operator @code{/} for integers
1021 first and result in a numeric holding a plain integer 1. @strong{Never
1022 use the operator @code{/} on integers} unless you know exactly what you
1023 are doing! Use the constructor from two integers instead, as shown in
1024 the example above. Writing @code{numeric(1)/2} may look funny but works
1027 @cindex @code{Digits}
1029 We have seen now the distinction between exact numbers and floating
1030 point numbers. Clearly, the user should never have to worry about
1031 dynamically created exact numbers, since their `exactness' always
1032 determines how they ought to be handled, i.e. how `long' they are. The
1033 situation is different for floating point numbers. Their accuracy is
1034 controlled by one @emph{global} variable, called @code{Digits}. (For
1035 those readers who know about Maple: it behaves very much like Maple's
1036 @code{Digits}). All objects of class numeric that are constructed from
1037 then on will be stored with a precision matching that number of decimal
1042 #include <ginac/ginac.h>
1043 using namespace std;
1044 using namespace GiNaC;
1048 numeric three(3.0), one(1.0);
1049 numeric x = one/three;
1051 cout << "in " << Digits << " digits:" << endl;
1053 cout << Pi.evalf() << endl;
1065 The above example prints the following output to screen:
1069 0.33333333333333333334
1070 3.1415926535897932385
1072 0.33333333333333333333333333333333333333333333333333333333333333333334
1073 3.1415926535897932384626433832795028841971693993751058209749445923078
1077 Note that the last number is not necessarily rounded as you would
1078 naively expect it to be rounded in the decimal system. But note also,
1079 that in both cases you got a couple of extra digits. This is because
1080 numbers are internally stored by CLN as chunks of binary digits in order
1081 to match your machine's word size and to not waste precision. Thus, on
1082 architectures with different word size, the above output might even
1083 differ with regard to actually computed digits.
1085 It should be clear that objects of class @code{numeric} should be used
1086 for constructing numbers or for doing arithmetic with them. The objects
1087 one deals with most of the time are the polymorphic expressions @code{ex}.
1089 @subsection Tests on numbers
1091 Once you have declared some numbers, assigned them to expressions and
1092 done some arithmetic with them it is frequently desired to retrieve some
1093 kind of information from them like asking whether that number is
1094 integer, rational, real or complex. For those cases GiNaC provides
1095 several useful methods. (Internally, they fall back to invocations of
1096 certain CLN functions.)
1098 As an example, let's construct some rational number, multiply it with
1099 some multiple of its denominator and test what comes out:
1103 #include <ginac/ginac.h>
1104 using namespace std;
1105 using namespace GiNaC;
1107 // some very important constants:
1108 const numeric twentyone(21);
1109 const numeric ten(10);
1110 const numeric five(5);
1114 numeric answer = twentyone;
1117 cout << answer.is_integer() << endl; // false, it's 21/5
1119 cout << answer.is_integer() << endl; // true, it's 42 now!
1123 Note that the variable @code{answer} is constructed here as an integer
1124 by @code{numeric}'s copy constructor but in an intermediate step it
1125 holds a rational number represented as integer numerator and integer
1126 denominator. When multiplied by 10, the denominator becomes unity and
1127 the result is automatically converted to a pure integer again.
1128 Internally, the underlying CLN is responsible for this behavior and we
1129 refer the reader to CLN's documentation. Suffice to say that
1130 the same behavior applies to complex numbers as well as return values of
1131 certain functions. Complex numbers are automatically converted to real
1132 numbers if the imaginary part becomes zero. The full set of tests that
1133 can be applied is listed in the following table.
1136 @multitable @columnfractions .30 .70
1137 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1138 @item @code{.is_zero()}
1139 @tab @dots{}equal to zero
1140 @item @code{.is_positive()}
1141 @tab @dots{}not complex and greater than 0
1142 @item @code{.is_integer()}
1143 @tab @dots{}a (non-complex) integer
1144 @item @code{.is_pos_integer()}
1145 @tab @dots{}an integer and greater than 0
1146 @item @code{.is_nonneg_integer()}
1147 @tab @dots{}an integer and greater equal 0
1148 @item @code{.is_even()}
1149 @tab @dots{}an even integer
1150 @item @code{.is_odd()}
1151 @tab @dots{}an odd integer
1152 @item @code{.is_prime()}
1153 @tab @dots{}a prime integer (probabilistic primality test)
1154 @item @code{.is_rational()}
1155 @tab @dots{}an exact rational number (integers are rational, too)
1156 @item @code{.is_real()}
1157 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1158 @item @code{.is_cinteger()}
1159 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1160 @item @code{.is_crational()}
1161 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1166 @node Constants, Fundamental containers, Numbers, Basic Concepts
1167 @c node-name, next, previous, up
1169 @cindex @code{constant} (class)
1172 @cindex @code{Catalan}
1173 @cindex @code{Euler}
1174 @cindex @code{evalf()}
1175 Constants behave pretty much like symbols except that they return some
1176 specific number when the method @code{.evalf()} is called.
1178 The predefined known constants are:
1181 @multitable @columnfractions .14 .30 .56
1182 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1184 @tab Archimedes' constant
1185 @tab 3.14159265358979323846264338327950288
1186 @item @code{Catalan}
1187 @tab Catalan's constant
1188 @tab 0.91596559417721901505460351493238411
1190 @tab Euler's (or Euler-Mascheroni) constant
1191 @tab 0.57721566490153286060651209008240243
1196 @node Fundamental containers, Lists, Constants, Basic Concepts
1197 @c node-name, next, previous, up
1198 @section Sums, products and powers
1202 @cindex @code{power}
1204 Simple rational expressions are written down in GiNaC pretty much like
1205 in other CAS or like expressions involving numerical variables in C.
1206 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1207 been overloaded to achieve this goal. When you run the following
1208 code snippet, the constructor for an object of type @code{mul} is
1209 automatically called to hold the product of @code{a} and @code{b} and
1210 then the constructor for an object of type @code{add} is called to hold
1211 the sum of that @code{mul} object and the number one:
1215 symbol a("a"), b("b");
1220 @cindex @code{pow()}
1221 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1222 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1223 construction is necessary since we cannot safely overload the constructor
1224 @code{^} in C++ to construct a @code{power} object. If we did, it would
1225 have several counterintuitive and undesired effects:
1229 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1231 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1232 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1233 interpret this as @code{x^(a^b)}.
1235 Also, expressions involving integer exponents are very frequently used,
1236 which makes it even more dangerous to overload @code{^} since it is then
1237 hard to distinguish between the semantics as exponentiation and the one
1238 for exclusive or. (It would be embarrassing to return @code{1} where one
1239 has requested @code{2^3}.)
1242 @cindex @command{ginsh}
1243 All effects are contrary to mathematical notation and differ from the
1244 way most other CAS handle exponentiation, therefore overloading @code{^}
1245 is ruled out for GiNaC's C++ part. The situation is different in
1246 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1247 that the other frequently used exponentiation operator @code{**} does
1248 not exist at all in C++).
1250 To be somewhat more precise, objects of the three classes described
1251 here, are all containers for other expressions. An object of class
1252 @code{power} is best viewed as a container with two slots, one for the
1253 basis, one for the exponent. All valid GiNaC expressions can be
1254 inserted. However, basic transformations like simplifying
1255 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1256 when this is mathematically possible. If we replace the outer exponent
1257 three in the example by some symbols @code{a}, the simplification is not
1258 safe and will not be performed, since @code{a} might be @code{1/2} and
1261 Objects of type @code{add} and @code{mul} are containers with an
1262 arbitrary number of slots for expressions to be inserted. Again, simple
1263 and safe simplifications are carried out like transforming
1264 @code{3*x+4-x} to @code{2*x+4}.
1267 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1268 @c node-name, next, previous, up
1269 @section Lists of expressions
1270 @cindex @code{lst} (class)
1272 @cindex @code{nops()}
1274 @cindex @code{append()}
1275 @cindex @code{prepend()}
1276 @cindex @code{remove_first()}
1277 @cindex @code{remove_last()}
1278 @cindex @code{remove_all()}
1280 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1281 expressions. They are not as ubiquitous as in many other computer algebra
1282 packages, but are sometimes used to supply a variable number of arguments of
1283 the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
1284 so you should have a basic understanding of them.
1286 Lists of up to 16 expressions can be directly constructed from single
1291 symbol x("x"), y("y");
1292 lst l(x, 2, y, x+y);
1293 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1297 Use the @code{nops()} method to determine the size (number of expressions) of
1298 a list and the @code{op()} method or the @code{[]} operator to access
1299 individual elements:
1303 cout << l.nops() << endl; // prints '4'
1304 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1308 As with the standard @code{list<T>} container, accessing random elements of a
1309 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1310 sequential access to the elements of a list is possible with the
1311 iterator types provided by the @code{lst} class:
1314 typedef ... lst::const_iterator;
1315 typedef ... lst::const_reverse_iterator;
1316 lst::const_iterator lst::begin() const;
1317 lst::const_iterator lst::end() const;
1318 lst::const_reverse_iterator lst::rbegin() const;
1319 lst::const_reverse_iterator lst::rend() const;
1322 For example, to print the elements of a list individually you can use:
1327 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1332 which is one order faster than
1337 for (size_t i = 0; i < l.nops(); ++i)
1338 cout << l.op(i) << endl;
1342 These iterators also allow you to use some of the algorithms provided by
1343 the C++ standard library:
1347 // print the elements of the list (requires #include <iterator>)
1348 copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1350 // sum up the elements of the list (requires #include <numeric>)
1351 ex sum = accumulate(l.begin(), l.end(), ex(0));
1352 cout << sum << endl; // prints '2+2*x+2*y'
1356 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1357 (the only other one is @code{matrix}). You can modify single elements:
1361 l[1] = 42; // l is now @{x, 42, y, x+y@}
1362 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1366 You can append or prepend an expression to a list with the @code{append()}
1367 and @code{prepend()} methods:
1371 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1372 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1376 You can remove the first or last element of a list with @code{remove_first()}
1377 and @code{remove_last()}:
1381 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1382 l.remove_last(); // l is now @{x, 7, y, x+y@}
1386 You can remove all the elements of a list with @code{remove_all()}:
1390 l.remove_all(); // l is now empty
1394 You can bring the elements of a list into a canonical order with @code{sort()}:
1398 lst l1(x, 2, y, x+y);
1399 lst l2(2, x+y, x, y);
1402 // l1 and l2 are now equal
1406 Finally, you can remove all but the first element of consecutive groups of
1407 elements with @code{unique()}:
1411 lst l3(x, 2, 2, 2, y, x+y, y+x);
1412 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1417 @node Mathematical functions, Relations, Lists, Basic Concepts
1418 @c node-name, next, previous, up
1419 @section Mathematical functions
1420 @cindex @code{function} (class)
1421 @cindex trigonometric function
1422 @cindex hyperbolic function
1424 There are quite a number of useful functions hard-wired into GiNaC. For
1425 instance, all trigonometric and hyperbolic functions are implemented
1426 (@xref{Built-in Functions}, for a complete list).
1428 These functions (better called @emph{pseudofunctions}) are all objects
1429 of class @code{function}. They accept one or more expressions as
1430 arguments and return one expression. If the arguments are not
1431 numerical, the evaluation of the function may be halted, as it does in
1432 the next example, showing how a function returns itself twice and
1433 finally an expression that may be really useful:
1435 @cindex Gamma function
1436 @cindex @code{subs()}
1439 symbol x("x"), y("y");
1441 cout << tgamma(foo) << endl;
1442 // -> tgamma(x+(1/2)*y)
1443 ex bar = foo.subs(y==1);
1444 cout << tgamma(bar) << endl;
1446 ex foobar = bar.subs(x==7);
1447 cout << tgamma(foobar) << endl;
1448 // -> (135135/128)*Pi^(1/2)
1452 Besides evaluation most of these functions allow differentiation, series
1453 expansion and so on. Read the next chapter in order to learn more about
1456 It must be noted that these pseudofunctions are created by inline
1457 functions, where the argument list is templated. This means that
1458 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1459 @code{sin(ex(1))} and will therefore not result in a floating point
1460 number. Unless of course the function prototype is explicitly
1461 overridden -- which is the case for arguments of type @code{numeric}
1462 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1463 point number of class @code{numeric} you should call
1464 @code{sin(numeric(1))}. This is almost the same as calling
1465 @code{sin(1).evalf()} except that the latter will return a numeric
1466 wrapped inside an @code{ex}.
1469 @node Relations, Matrices, Mathematical functions, Basic Concepts
1470 @c node-name, next, previous, up
1472 @cindex @code{relational} (class)
1474 Sometimes, a relation holding between two expressions must be stored
1475 somehow. The class @code{relational} is a convenient container for such
1476 purposes. A relation is by definition a container for two @code{ex} and
1477 a relation between them that signals equality, inequality and so on.
1478 They are created by simply using the C++ operators @code{==}, @code{!=},
1479 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1481 @xref{Mathematical functions}, for examples where various applications
1482 of the @code{.subs()} method show how objects of class relational are
1483 used as arguments. There they provide an intuitive syntax for
1484 substitutions. They are also used as arguments to the @code{ex::series}
1485 method, where the left hand side of the relation specifies the variable
1486 to expand in and the right hand side the expansion point. They can also
1487 be used for creating systems of equations that are to be solved for
1488 unknown variables. But the most common usage of objects of this class
1489 is rather inconspicuous in statements of the form @code{if
1490 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1491 conversion from @code{relational} to @code{bool} takes place. Note,
1492 however, that @code{==} here does not perform any simplifications, hence
1493 @code{expand()} must be called explicitly.
1496 @node Matrices, Indexed objects, Relations, Basic Concepts
1497 @c node-name, next, previous, up
1499 @cindex @code{matrix} (class)
1501 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1502 matrix with @math{m} rows and @math{n} columns are accessed with two
1503 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1504 second one in the range 0@dots{}@math{n-1}.
1506 There are a couple of ways to construct matrices, with or without preset
1509 @cindex @code{lst_to_matrix()}
1510 @cindex @code{diag_matrix()}
1511 @cindex @code{unit_matrix()}
1512 @cindex @code{symbolic_matrix()}
1514 matrix::matrix(unsigned r, unsigned c);
1515 matrix::matrix(unsigned r, unsigned c, const lst & l);
1516 ex lst_to_matrix(const lst & l);
1517 ex diag_matrix(const lst & l);
1518 ex unit_matrix(unsigned x);
1519 ex unit_matrix(unsigned r, unsigned c);
1520 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1521 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1524 The first two functions are @code{matrix} constructors which create a matrix
1525 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1526 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1527 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1528 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1529 constructs a diagonal matrix given the list of diagonal elements.
1530 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1531 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1532 with newly generated symbols made of the specified base name and the
1533 position of each element in the matrix.
1535 Matrix elements can be accessed and set using the parenthesis (function call)
1539 const ex & matrix::operator()(unsigned r, unsigned c) const;
1540 ex & matrix::operator()(unsigned r, unsigned c);
1543 It is also possible to access the matrix elements in a linear fashion with
1544 the @code{op()} method. But C++-style subscripting with square brackets
1545 @samp{[]} is not available.
1547 Here are a couple of examples of constructing matrices:
1551 symbol a("a"), b("b");
1559 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1562 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1565 cout << diag_matrix(lst(a, b)) << endl;
1568 cout << unit_matrix(3) << endl;
1569 // -> [[1,0,0],[0,1,0],[0,0,1]]
1571 cout << symbolic_matrix(2, 3, "x") << endl;
1572 // -> [[x00,x01,x02],[x10,x11,x12]]
1576 @cindex @code{transpose()}
1577 There are three ways to do arithmetic with matrices. The first (and most
1578 direct one) is to use the methods provided by the @code{matrix} class:
1581 matrix matrix::add(const matrix & other) const;
1582 matrix matrix::sub(const matrix & other) const;
1583 matrix matrix::mul(const matrix & other) const;
1584 matrix matrix::mul_scalar(const ex & other) const;
1585 matrix matrix::pow(const ex & expn) const;
1586 matrix matrix::transpose() const;
1589 All of these methods return the result as a new matrix object. Here is an
1590 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1595 matrix A(2, 2, lst(1, 2, 3, 4));
1596 matrix B(2, 2, lst(-1, 0, 2, 1));
1597 matrix C(2, 2, lst(8, 4, 2, 1));
1599 matrix result = A.mul(B).sub(C.mul_scalar(2));
1600 cout << result << endl;
1601 // -> [[-13,-6],[1,2]]
1606 @cindex @code{evalm()}
1607 The second (and probably the most natural) way is to construct an expression
1608 containing matrices with the usual arithmetic operators and @code{pow()}.
1609 For efficiency reasons, expressions with sums, products and powers of
1610 matrices are not automatically evaluated in GiNaC. You have to call the
1614 ex ex::evalm() const;
1617 to obtain the result:
1624 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1625 cout << e.evalm() << endl;
1626 // -> [[-13,-6],[1,2]]
1631 The non-commutativity of the product @code{A*B} in this example is
1632 automatically recognized by GiNaC. There is no need to use a special
1633 operator here. @xref{Non-commutative objects}, for more information about
1634 dealing with non-commutative expressions.
1636 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1637 to perform the arithmetic:
1642 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1643 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1645 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1646 cout << e.simplify_indexed() << endl;
1647 // -> [[-13,-6],[1,2]].i.j
1651 Using indices is most useful when working with rectangular matrices and
1652 one-dimensional vectors because you don't have to worry about having to
1653 transpose matrices before multiplying them. @xref{Indexed objects}, for
1654 more information about using matrices with indices, and about indices in
1657 The @code{matrix} class provides a couple of additional methods for
1658 computing determinants, traces, and characteristic polynomials:
1660 @cindex @code{determinant()}
1661 @cindex @code{trace()}
1662 @cindex @code{charpoly()}
1664 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1665 ex matrix::trace() const;
1666 ex matrix::charpoly(const ex & lambda) const;
1669 The @samp{algo} argument of @code{determinant()} allows to select
1670 between different algorithms for calculating the determinant. The
1671 asymptotic speed (as parametrized by the matrix size) can greatly differ
1672 between those algorithms, depending on the nature of the matrix'
1673 entries. The possible values are defined in the @file{flags.h} header
1674 file. By default, GiNaC uses a heuristic to automatically select an
1675 algorithm that is likely (but not guaranteed) to give the result most
1678 @cindex @code{inverse()}
1679 @cindex @code{solve()}
1680 Matrices may also be inverted using the @code{ex matrix::inverse()}
1681 method and linear systems may be solved with:
1684 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1687 Assuming the matrix object this method is applied on is an @code{m}
1688 times @code{n} matrix, then @code{vars} must be a @code{n} times
1689 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1690 times @code{p} matrix. The returned matrix then has dimension @code{n}
1691 times @code{p} and in the case of an underdetermined system will still
1692 contain some of the indeterminates from @code{vars}. If the system is
1693 overdetermined, an exception is thrown.
1696 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1697 @c node-name, next, previous, up
1698 @section Indexed objects
1700 GiNaC allows you to handle expressions containing general indexed objects in
1701 arbitrary spaces. It is also able to canonicalize and simplify such
1702 expressions and perform symbolic dummy index summations. There are a number
1703 of predefined indexed objects provided, like delta and metric tensors.
1705 There are few restrictions placed on indexed objects and their indices and
1706 it is easy to construct nonsense expressions, but our intention is to
1707 provide a general framework that allows you to implement algorithms with
1708 indexed quantities, getting in the way as little as possible.
1710 @cindex @code{idx} (class)
1711 @cindex @code{indexed} (class)
1712 @subsection Indexed quantities and their indices
1714 Indexed expressions in GiNaC are constructed of two special types of objects,
1715 @dfn{index objects} and @dfn{indexed objects}.
1719 @cindex contravariant
1722 @item Index objects are of class @code{idx} or a subclass. Every index has
1723 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1724 the index lives in) which can both be arbitrary expressions but are usually
1725 a number or a simple symbol. In addition, indices of class @code{varidx} have
1726 a @dfn{variance} (they can be co- or contravariant), and indices of class
1727 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1729 @item Indexed objects are of class @code{indexed} or a subclass. They
1730 contain a @dfn{base expression} (which is the expression being indexed), and
1731 one or more indices.
1735 @strong{Note:} when printing expressions, covariant indices and indices
1736 without variance are denoted @samp{.i} while contravariant indices are
1737 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1738 value. In the following, we are going to use that notation in the text so
1739 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1740 not visible in the output.
1742 A simple example shall illustrate the concepts:
1746 #include <ginac/ginac.h>
1747 using namespace std;
1748 using namespace GiNaC;
1752 symbol i_sym("i"), j_sym("j");
1753 idx i(i_sym, 3), j(j_sym, 3);
1756 cout << indexed(A, i, j) << endl;
1758 cout << index_dimensions << indexed(A, i, j) << endl;
1760 cout << dflt; // reset cout to default output format (dimensions hidden)
1764 The @code{idx} constructor takes two arguments, the index value and the
1765 index dimension. First we define two index objects, @code{i} and @code{j},
1766 both with the numeric dimension 3. The value of the index @code{i} is the
1767 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1768 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1769 construct an expression containing one indexed object, @samp{A.i.j}. It has
1770 the symbol @code{A} as its base expression and the two indices @code{i} and
1773 The dimensions of indices are normally not visible in the output, but one
1774 can request them to be printed with the @code{index_dimensions} manipulator,
1777 Note the difference between the indices @code{i} and @code{j} which are of
1778 class @code{idx}, and the index values which are the symbols @code{i_sym}
1779 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1780 or numbers but must be index objects. For example, the following is not
1781 correct and will raise an exception:
1784 symbol i("i"), j("j");
1785 e = indexed(A, i, j); // ERROR: indices must be of type idx
1788 You can have multiple indexed objects in an expression, index values can
1789 be numeric, and index dimensions symbolic:
1793 symbol B("B"), dim("dim");
1794 cout << 4 * indexed(A, i)
1795 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1800 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1801 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1802 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1803 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1804 @code{simplify_indexed()} for that, see below).
1806 In fact, base expressions, index values and index dimensions can be
1807 arbitrary expressions:
1811 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1816 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1817 get an error message from this but you will probably not be able to do
1818 anything useful with it.
1820 @cindex @code{get_value()}
1821 @cindex @code{get_dimension()}
1825 ex idx::get_value();
1826 ex idx::get_dimension();
1829 return the value and dimension of an @code{idx} object. If you have an index
1830 in an expression, such as returned by calling @code{.op()} on an indexed
1831 object, you can get a reference to the @code{idx} object with the function
1832 @code{ex_to<idx>()} on the expression.
1834 There are also the methods
1837 bool idx::is_numeric();
1838 bool idx::is_symbolic();
1839 bool idx::is_dim_numeric();
1840 bool idx::is_dim_symbolic();
1843 for checking whether the value and dimension are numeric or symbolic
1844 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1845 About Expressions}) returns information about the index value.
1847 @cindex @code{varidx} (class)
1848 If you need co- and contravariant indices, use the @code{varidx} class:
1852 symbol mu_sym("mu"), nu_sym("nu");
1853 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1854 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1856 cout << indexed(A, mu, nu) << endl;
1858 cout << indexed(A, mu_co, nu) << endl;
1860 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1865 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1866 co- or contravariant. The default is a contravariant (upper) index, but
1867 this can be overridden by supplying a third argument to the @code{varidx}
1868 constructor. The two methods
1871 bool varidx::is_covariant();
1872 bool varidx::is_contravariant();
1875 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1876 to get the object reference from an expression). There's also the very useful
1880 ex varidx::toggle_variance();
1883 which makes a new index with the same value and dimension but the opposite
1884 variance. By using it you only have to define the index once.
1886 @cindex @code{spinidx} (class)
1887 The @code{spinidx} class provides dotted and undotted variant indices, as
1888 used in the Weyl-van-der-Waerden spinor formalism:
1892 symbol K("K"), C_sym("C"), D_sym("D");
1893 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1894 // contravariant, undotted
1895 spinidx C_co(C_sym, 2, true); // covariant index
1896 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1897 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1899 cout << indexed(K, C, D) << endl;
1901 cout << indexed(K, C_co, D_dot) << endl;
1903 cout << indexed(K, D_co_dot, D) << endl;
1908 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1909 dotted or undotted. The default is undotted but this can be overridden by
1910 supplying a fourth argument to the @code{spinidx} constructor. The two
1914 bool spinidx::is_dotted();
1915 bool spinidx::is_undotted();
1918 allow you to check whether or not a @code{spinidx} object is dotted (use
1919 @code{ex_to<spinidx>()} to get the object reference from an expression).
1920 Finally, the two methods
1923 ex spinidx::toggle_dot();
1924 ex spinidx::toggle_variance_dot();
1927 create a new index with the same value and dimension but opposite dottedness
1928 and the same or opposite variance.
1930 @subsection Substituting indices
1932 @cindex @code{subs()}
1933 Sometimes you will want to substitute one symbolic index with another
1934 symbolic or numeric index, for example when calculating one specific element
1935 of a tensor expression. This is done with the @code{.subs()} method, as it
1936 is done for symbols (see @ref{Substituting Expressions}).
1938 You have two possibilities here. You can either substitute the whole index
1939 by another index or expression:
1943 ex e = indexed(A, mu_co);
1944 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1945 // -> A.mu becomes A~nu
1946 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1947 // -> A.mu becomes A~0
1948 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1949 // -> A.mu becomes A.0
1953 The third example shows that trying to replace an index with something that
1954 is not an index will substitute the index value instead.
1956 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1961 ex e = indexed(A, mu_co);
1962 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1963 // -> A.mu becomes A.nu
1964 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1965 // -> A.mu becomes A.0
1969 As you see, with the second method only the value of the index will get
1970 substituted. Its other properties, including its dimension, remain unchanged.
1971 If you want to change the dimension of an index you have to substitute the
1972 whole index by another one with the new dimension.
1974 Finally, substituting the base expression of an indexed object works as
1979 ex e = indexed(A, mu_co);
1980 cout << e << " becomes " << e.subs(A == A+B) << endl;
1981 // -> A.mu becomes (B+A).mu
1985 @subsection Symmetries
1986 @cindex @code{symmetry} (class)
1987 @cindex @code{sy_none()}
1988 @cindex @code{sy_symm()}
1989 @cindex @code{sy_anti()}
1990 @cindex @code{sy_cycl()}
1992 Indexed objects can have certain symmetry properties with respect to their
1993 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1994 that is constructed with the helper functions
1997 symmetry sy_none(...);
1998 symmetry sy_symm(...);
1999 symmetry sy_anti(...);
2000 symmetry sy_cycl(...);
2003 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2004 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2005 represents a cyclic symmetry. Each of these functions accepts up to four
2006 arguments which can be either symmetry objects themselves or unsigned integer
2007 numbers that represent an index position (counting from 0). A symmetry
2008 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2009 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2012 Here are some examples of symmetry definitions:
2017 e = indexed(A, i, j);
2018 e = indexed(A, sy_none(), i, j); // equivalent
2019 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2021 // Symmetric in all three indices:
2022 e = indexed(A, sy_symm(), i, j, k);
2023 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2024 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2025 // different canonical order
2027 // Symmetric in the first two indices only:
2028 e = indexed(A, sy_symm(0, 1), i, j, k);
2029 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2031 // Antisymmetric in the first and last index only (index ranges need not
2033 e = indexed(A, sy_anti(0, 2), i, j, k);
2034 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2036 // An example of a mixed symmetry: antisymmetric in the first two and
2037 // last two indices, symmetric when swapping the first and last index
2038 // pairs (like the Riemann curvature tensor):
2039 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2041 // Cyclic symmetry in all three indices:
2042 e = indexed(A, sy_cycl(), i, j, k);
2043 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2045 // The following examples are invalid constructions that will throw
2046 // an exception at run time.
2048 // An index may not appear multiple times:
2049 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2050 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2052 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2053 // same number of indices:
2054 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2056 // And of course, you cannot specify indices which are not there:
2057 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2061 If you need to specify more than four indices, you have to use the
2062 @code{.add()} method of the @code{symmetry} class. For example, to specify
2063 full symmetry in the first six indices you would write
2064 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2066 If an indexed object has a symmetry, GiNaC will automatically bring the
2067 indices into a canonical order which allows for some immediate simplifications:
2071 cout << indexed(A, sy_symm(), i, j)
2072 + indexed(A, sy_symm(), j, i) << endl;
2074 cout << indexed(B, sy_anti(), i, j)
2075 + indexed(B, sy_anti(), j, i) << endl;
2077 cout << indexed(B, sy_anti(), i, j, k)
2078 - indexed(B, sy_anti(), j, k, i) << endl;
2083 @cindex @code{get_free_indices()}
2085 @subsection Dummy indices
2087 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2088 that a summation over the index range is implied. Symbolic indices which are
2089 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2090 dummy nor free indices.
2092 To be recognized as a dummy index pair, the two indices must be of the same
2093 class and their value must be the same single symbol (an index like
2094 @samp{2*n+1} is never a dummy index). If the indices are of class
2095 @code{varidx} they must also be of opposite variance; if they are of class
2096 @code{spinidx} they must be both dotted or both undotted.
2098 The method @code{.get_free_indices()} returns a vector containing the free
2099 indices of an expression. It also checks that the free indices of the terms
2100 of a sum are consistent:
2104 symbol A("A"), B("B"), C("C");
2106 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2107 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2109 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2110 cout << exprseq(e.get_free_indices()) << endl;
2112 // 'j' and 'l' are dummy indices
2114 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2115 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2117 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2118 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2119 cout << exprseq(e.get_free_indices()) << endl;
2121 // 'nu' is a dummy index, but 'sigma' is not
2123 e = indexed(A, mu, mu);
2124 cout << exprseq(e.get_free_indices()) << endl;
2126 // 'mu' is not a dummy index because it appears twice with the same
2129 e = indexed(A, mu, nu) + 42;
2130 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2131 // this will throw an exception:
2132 // "add::get_free_indices: inconsistent indices in sum"
2136 @cindex @code{simplify_indexed()}
2137 @subsection Simplifying indexed expressions
2139 In addition to the few automatic simplifications that GiNaC performs on
2140 indexed expressions (such as re-ordering the indices of symmetric tensors
2141 and calculating traces and convolutions of matrices and predefined tensors)
2145 ex ex::simplify_indexed();
2146 ex ex::simplify_indexed(const scalar_products & sp);
2149 that performs some more expensive operations:
2152 @item it checks the consistency of free indices in sums in the same way
2153 @code{get_free_indices()} does
2154 @item it tries to give dummy indices that appear in different terms of a sum
2155 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2156 @item it (symbolically) calculates all possible dummy index summations/contractions
2157 with the predefined tensors (this will be explained in more detail in the
2159 @item it detects contractions that vanish for symmetry reasons, for example
2160 the contraction of a symmetric and a totally antisymmetric tensor
2161 @item as a special case of dummy index summation, it can replace scalar products
2162 of two tensors with a user-defined value
2165 The last point is done with the help of the @code{scalar_products} class
2166 which is used to store scalar products with known values (this is not an
2167 arithmetic class, you just pass it to @code{simplify_indexed()}):
2171 symbol A("A"), B("B"), C("C"), i_sym("i");
2175 sp.add(A, B, 0); // A and B are orthogonal
2176 sp.add(A, C, 0); // A and C are orthogonal
2177 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2179 e = indexed(A + B, i) * indexed(A + C, i);
2181 // -> (B+A).i*(A+C).i
2183 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2189 The @code{scalar_products} object @code{sp} acts as a storage for the
2190 scalar products added to it with the @code{.add()} method. This method
2191 takes three arguments: the two expressions of which the scalar product is
2192 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2193 @code{simplify_indexed()} will replace all scalar products of indexed
2194 objects that have the symbols @code{A} and @code{B} as base expressions
2195 with the single value 0. The number, type and dimension of the indices
2196 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2198 @cindex @code{expand()}
2199 The example above also illustrates a feature of the @code{expand()} method:
2200 if passed the @code{expand_indexed} option it will distribute indices
2201 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2203 @cindex @code{tensor} (class)
2204 @subsection Predefined tensors
2206 Some frequently used special tensors such as the delta, epsilon and metric
2207 tensors are predefined in GiNaC. They have special properties when
2208 contracted with other tensor expressions and some of them have constant
2209 matrix representations (they will evaluate to a number when numeric
2210 indices are specified).
2212 @cindex @code{delta_tensor()}
2213 @subsubsection Delta tensor
2215 The delta tensor takes two indices, is symmetric and has the matrix
2216 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2217 @code{delta_tensor()}:
2221 symbol A("A"), B("B");
2223 idx i(symbol("i"), 3), j(symbol("j"), 3),
2224 k(symbol("k"), 3), l(symbol("l"), 3);
2226 ex e = indexed(A, i, j) * indexed(B, k, l)
2227 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2228 cout << e.simplify_indexed() << endl;
2231 cout << delta_tensor(i, i) << endl;
2236 @cindex @code{metric_tensor()}
2237 @subsubsection General metric tensor
2239 The function @code{metric_tensor()} creates a general symmetric metric
2240 tensor with two indices that can be used to raise/lower tensor indices. The
2241 metric tensor is denoted as @samp{g} in the output and if its indices are of
2242 mixed variance it is automatically replaced by a delta tensor:
2248 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2250 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2251 cout << e.simplify_indexed() << endl;
2254 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2255 cout << e.simplify_indexed() << endl;
2258 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2259 * metric_tensor(nu, rho);
2260 cout << e.simplify_indexed() << endl;
2263 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2264 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2265 + indexed(A, mu.toggle_variance(), rho));
2266 cout << e.simplify_indexed() << endl;
2271 @cindex @code{lorentz_g()}
2272 @subsubsection Minkowski metric tensor
2274 The Minkowski metric tensor is a special metric tensor with a constant
2275 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2276 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2277 It is created with the function @code{lorentz_g()} (although it is output as
2282 varidx mu(symbol("mu"), 4);
2284 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2285 * lorentz_g(mu, varidx(0, 4)); // negative signature
2286 cout << e.simplify_indexed() << endl;
2289 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2290 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2291 cout << e.simplify_indexed() << endl;
2296 @cindex @code{spinor_metric()}
2297 @subsubsection Spinor metric tensor
2299 The function @code{spinor_metric()} creates an antisymmetric tensor with
2300 two indices that is used to raise/lower indices of 2-component spinors.
2301 It is output as @samp{eps}:
2307 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2308 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2310 e = spinor_metric(A, B) * indexed(psi, B_co);
2311 cout << e.simplify_indexed() << endl;
2314 e = spinor_metric(A, B) * indexed(psi, A_co);
2315 cout << e.simplify_indexed() << endl;
2318 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2319 cout << e.simplify_indexed() << endl;
2322 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2323 cout << e.simplify_indexed() << endl;
2326 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2327 cout << e.simplify_indexed() << endl;
2330 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2331 cout << e.simplify_indexed() << endl;
2336 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2338 @cindex @code{epsilon_tensor()}
2339 @cindex @code{lorentz_eps()}
2340 @subsubsection Epsilon tensor
2342 The epsilon tensor is totally antisymmetric, its number of indices is equal
2343 to the dimension of the index space (the indices must all be of the same
2344 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2345 defined to be 1. Its behavior with indices that have a variance also
2346 depends on the signature of the metric. Epsilon tensors are output as
2349 There are three functions defined to create epsilon tensors in 2, 3 and 4
2353 ex epsilon_tensor(const ex & i1, const ex & i2);
2354 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2355 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2358 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2359 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2360 Minkowski space (the last @code{bool} argument specifies whether the metric
2361 has negative or positive signature, as in the case of the Minkowski metric
2366 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2367 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2368 e = lorentz_eps(mu, nu, rho, sig) *
2369 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2370 cout << simplify_indexed(e) << endl;
2371 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2373 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2374 symbol A("A"), B("B");
2375 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2376 cout << simplify_indexed(e) << endl;
2377 // -> -B.k*A.j*eps.i.k.j
2378 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2379 cout << simplify_indexed(e) << endl;
2384 @subsection Linear algebra
2386 The @code{matrix} class can be used with indices to do some simple linear
2387 algebra (linear combinations and products of vectors and matrices, traces
2388 and scalar products):
2392 idx i(symbol("i"), 2), j(symbol("j"), 2);
2393 symbol x("x"), y("y");
2395 // A is a 2x2 matrix, X is a 2x1 vector
2396 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2398 cout << indexed(A, i, i) << endl;
2401 ex e = indexed(A, i, j) * indexed(X, j);
2402 cout << e.simplify_indexed() << endl;
2403 // -> [[2*y+x],[4*y+3*x]].i
2405 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2406 cout << e.simplify_indexed() << endl;
2407 // -> [[3*y+3*x,6*y+2*x]].j
2411 You can of course obtain the same results with the @code{matrix::add()},
2412 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2413 but with indices you don't have to worry about transposing matrices.
2415 Matrix indices always start at 0 and their dimension must match the number
2416 of rows/columns of the matrix. Matrices with one row or one column are
2417 vectors and can have one or two indices (it doesn't matter whether it's a
2418 row or a column vector). Other matrices must have two indices.
2420 You should be careful when using indices with variance on matrices. GiNaC
2421 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2422 @samp{F.mu.nu} are different matrices. In this case you should use only
2423 one form for @samp{F} and explicitly multiply it with a matrix representation
2424 of the metric tensor.
2427 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2428 @c node-name, next, previous, up
2429 @section Non-commutative objects
2431 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2432 non-commutative objects are built-in which are mostly of use in high energy
2436 @item Clifford (Dirac) algebra (class @code{clifford})
2437 @item su(3) Lie algebra (class @code{color})
2438 @item Matrices (unindexed) (class @code{matrix})
2441 The @code{clifford} and @code{color} classes are subclasses of
2442 @code{indexed} because the elements of these algebras usually carry
2443 indices. The @code{matrix} class is described in more detail in
2446 Unlike most computer algebra systems, GiNaC does not primarily provide an
2447 operator (often denoted @samp{&*}) for representing inert products of
2448 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2449 classes of objects involved, and non-commutative products are formed with
2450 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2451 figuring out by itself which objects commute and will group the factors
2452 by their class. Consider this example:
2456 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2457 idx a(symbol("a"), 8), b(symbol("b"), 8);
2458 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2460 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2464 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2465 groups the non-commutative factors (the gammas and the su(3) generators)
2466 together while preserving the order of factors within each class (because
2467 Clifford objects commute with color objects). The resulting expression is a
2468 @emph{commutative} product with two factors that are themselves non-commutative
2469 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2470 parentheses are placed around the non-commutative products in the output.
2472 @cindex @code{ncmul} (class)
2473 Non-commutative products are internally represented by objects of the class
2474 @code{ncmul}, as opposed to commutative products which are handled by the
2475 @code{mul} class. You will normally not have to worry about this distinction,
2478 The advantage of this approach is that you never have to worry about using
2479 (or forgetting to use) a special operator when constructing non-commutative
2480 expressions. Also, non-commutative products in GiNaC are more intelligent
2481 than in other computer algebra systems; they can, for example, automatically
2482 canonicalize themselves according to rules specified in the implementation
2483 of the non-commutative classes. The drawback is that to work with other than
2484 the built-in algebras you have to implement new classes yourself. Symbols
2485 always commute and it's not possible to construct non-commutative products
2486 using symbols to represent the algebra elements or generators. User-defined
2487 functions can, however, be specified as being non-commutative.
2489 @cindex @code{return_type()}
2490 @cindex @code{return_type_tinfo()}
2491 Information about the commutativity of an object or expression can be
2492 obtained with the two member functions
2495 unsigned ex::return_type() const;
2496 unsigned ex::return_type_tinfo() const;
2499 The @code{return_type()} function returns one of three values (defined in
2500 the header file @file{flags.h}), corresponding to three categories of
2501 expressions in GiNaC:
2504 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2505 classes are of this kind.
2506 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2507 certain class of non-commutative objects which can be determined with the
2508 @code{return_type_tinfo()} method. Expressions of this category commute
2509 with everything except @code{noncommutative} expressions of the same
2511 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2512 of non-commutative objects of different classes. Expressions of this
2513 category don't commute with any other @code{noncommutative} or
2514 @code{noncommutative_composite} expressions.
2517 The value returned by the @code{return_type_tinfo()} method is valid only
2518 when the return type of the expression is @code{noncommutative}. It is a
2519 value that is unique to the class of the object and usually one of the
2520 constants in @file{tinfos.h}, or derived therefrom.
2522 Here are a couple of examples:
2525 @multitable @columnfractions 0.33 0.33 0.34
2526 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2527 @item @code{42} @tab @code{commutative} @tab -
2528 @item @code{2*x-y} @tab @code{commutative} @tab -
2529 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2530 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2531 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2532 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2536 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2537 @code{TINFO_clifford} for objects with a representation label of zero.
2538 Other representation labels yield a different @code{return_type_tinfo()},
2539 but it's the same for any two objects with the same label. This is also true
2542 A last note: With the exception of matrices, positive integer powers of
2543 non-commutative objects are automatically expanded in GiNaC. For example,
2544 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2545 non-commutative expressions).
2548 @cindex @code{clifford} (class)
2549 @subsection Clifford algebra
2551 @cindex @code{dirac_gamma()}
2552 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2553 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2554 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2555 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2558 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2561 which takes two arguments: the index and a @dfn{representation label} in the
2562 range 0 to 255 which is used to distinguish elements of different Clifford
2563 algebras (this is also called a @dfn{spin line index}). Gammas with different
2564 labels commute with each other. The dimension of the index can be 4 or (in
2565 the framework of dimensional regularization) any symbolic value. Spinor
2566 indices on Dirac gammas are not supported in GiNaC.
2568 @cindex @code{dirac_ONE()}
2569 The unity element of a Clifford algebra is constructed by
2572 ex dirac_ONE(unsigned char rl = 0);
2575 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2576 multiples of the unity element, even though it's customary to omit it.
2577 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2578 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2579 GiNaC will complain and/or produce incorrect results.
2581 @cindex @code{dirac_gamma5()}
2582 There is a special element @samp{gamma5} that commutes with all other
2583 gammas, has a unit square, and in 4 dimensions equals
2584 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2587 ex dirac_gamma5(unsigned char rl = 0);
2590 @cindex @code{dirac_gammaL()}
2591 @cindex @code{dirac_gammaR()}
2592 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2593 objects, constructed by
2596 ex dirac_gammaL(unsigned char rl = 0);
2597 ex dirac_gammaR(unsigned char rl = 0);
2600 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2601 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2603 @cindex @code{dirac_slash()}
2604 Finally, the function
2607 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2610 creates a term that represents a contraction of @samp{e} with the Dirac
2611 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2612 with a unique index whose dimension is given by the @code{dim} argument).
2613 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2615 In products of dirac gammas, superfluous unity elements are automatically
2616 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2617 and @samp{gammaR} are moved to the front.
2619 The @code{simplify_indexed()} function performs contractions in gamma strings,
2625 symbol a("a"), b("b"), D("D");
2626 varidx mu(symbol("mu"), D);
2627 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2628 * dirac_gamma(mu.toggle_variance());
2630 // -> gamma~mu*a\*gamma.mu
2631 e = e.simplify_indexed();
2634 cout << e.subs(D == 4) << endl;
2640 @cindex @code{dirac_trace()}
2641 To calculate the trace of an expression containing strings of Dirac gammas
2642 you use the function
2645 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2648 This function takes the trace of all gammas with the specified representation
2649 label; gammas with other labels are left standing. The last argument to
2650 @code{dirac_trace()} is the value to be returned for the trace of the unity
2651 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2652 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2653 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2654 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2655 This @samp{gamma5} scheme is described in greater detail in
2656 @cite{The Role of gamma5 in Dimensional Regularization}.
2658 The value of the trace itself is also usually different in 4 and in
2659 @math{D != 4} dimensions:
2664 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2665 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2666 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2667 cout << dirac_trace(e).simplify_indexed() << endl;
2674 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2675 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2676 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2677 cout << dirac_trace(e).simplify_indexed() << endl;
2678 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2682 Here is an example for using @code{dirac_trace()} to compute a value that
2683 appears in the calculation of the one-loop vacuum polarization amplitude in
2688 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2689 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2692 sp.add(l, l, pow(l, 2));
2693 sp.add(l, q, ldotq);
2695 ex e = dirac_gamma(mu) *
2696 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2697 dirac_gamma(mu.toggle_variance()) *
2698 (dirac_slash(l, D) + m * dirac_ONE());
2699 e = dirac_trace(e).simplify_indexed(sp);
2700 e = e.collect(lst(l, ldotq, m));
2702 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2706 The @code{canonicalize_clifford()} function reorders all gamma products that
2707 appear in an expression to a canonical (but not necessarily simple) form.
2708 You can use this to compare two expressions or for further simplifications:
2712 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2713 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2715 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2717 e = canonicalize_clifford(e);
2724 @cindex @code{color} (class)
2725 @subsection Color algebra
2727 @cindex @code{color_T()}
2728 For computations in quantum chromodynamics, GiNaC implements the base elements
2729 and structure constants of the su(3) Lie algebra (color algebra). The base
2730 elements @math{T_a} are constructed by the function
2733 ex color_T(const ex & a, unsigned char rl = 0);
2736 which takes two arguments: the index and a @dfn{representation label} in the
2737 range 0 to 255 which is used to distinguish elements of different color
2738 algebras. Objects with different labels commute with each other. The
2739 dimension of the index must be exactly 8 and it should be of class @code{idx},
2742 @cindex @code{color_ONE()}
2743 The unity element of a color algebra is constructed by
2746 ex color_ONE(unsigned char rl = 0);
2749 @strong{Note:} You must always use @code{color_ONE()} when referring to
2750 multiples of the unity element, even though it's customary to omit it.
2751 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2752 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2753 GiNaC may produce incorrect results.
2755 @cindex @code{color_d()}
2756 @cindex @code{color_f()}
2760 ex color_d(const ex & a, const ex & b, const ex & c);
2761 ex color_f(const ex & a, const ex & b, const ex & c);
2764 create the symmetric and antisymmetric structure constants @math{d_abc} and
2765 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2766 and @math{[T_a, T_b] = i f_abc T_c}.
2768 @cindex @code{color_h()}
2769 There's an additional function
2772 ex color_h(const ex & a, const ex & b, const ex & c);
2775 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2777 The function @code{simplify_indexed()} performs some simplifications on
2778 expressions containing color objects:
2783 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2784 k(symbol("k"), 8), l(symbol("l"), 8);
2786 e = color_d(a, b, l) * color_f(a, b, k);
2787 cout << e.simplify_indexed() << endl;
2790 e = color_d(a, b, l) * color_d(a, b, k);
2791 cout << e.simplify_indexed() << endl;
2794 e = color_f(l, a, b) * color_f(a, b, k);
2795 cout << e.simplify_indexed() << endl;
2798 e = color_h(a, b, c) * color_h(a, b, c);
2799 cout << e.simplify_indexed() << endl;
2802 e = color_h(a, b, c) * color_T(b) * color_T(c);
2803 cout << e.simplify_indexed() << endl;
2806 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2807 cout << e.simplify_indexed() << endl;
2810 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2811 cout << e.simplify_indexed() << endl;
2812 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2816 @cindex @code{color_trace()}
2817 To calculate the trace of an expression containing color objects you use the
2821 ex color_trace(const ex & e, unsigned char rl = 0);
2824 This function takes the trace of all color @samp{T} objects with the
2825 specified representation label; @samp{T}s with other labels are left
2826 standing. For example:
2830 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2832 // -> -I*f.a.c.b+d.a.c.b
2837 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2838 @c node-name, next, previous, up
2839 @chapter Methods and Functions
2842 In this chapter the most important algorithms provided by GiNaC will be
2843 described. Some of them are implemented as functions on expressions,
2844 others are implemented as methods provided by expression objects. If
2845 they are methods, there exists a wrapper function around it, so you can
2846 alternatively call it in a functional way as shown in the simple
2851 cout << "As method: " << sin(1).evalf() << endl;
2852 cout << "As function: " << evalf(sin(1)) << endl;
2856 @cindex @code{subs()}
2857 The general rule is that wherever methods accept one or more parameters
2858 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2859 wrapper accepts is the same but preceded by the object to act on
2860 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2861 most natural one in an OO model but it may lead to confusion for MapleV
2862 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2863 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2864 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2865 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2866 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2867 here. Also, users of MuPAD will in most cases feel more comfortable
2868 with GiNaC's convention. All function wrappers are implemented
2869 as simple inline functions which just call the corresponding method and
2870 are only provided for users uncomfortable with OO who are dead set to
2871 avoid method invocations. Generally, nested function wrappers are much
2872 harder to read than a sequence of methods and should therefore be
2873 avoided if possible. On the other hand, not everything in GiNaC is a
2874 method on class @code{ex} and sometimes calling a function cannot be
2878 * Information About Expressions::
2879 * Substituting Expressions::
2880 * Pattern Matching and Advanced Substitutions::
2881 * Applying a Function on Subexpressions::
2882 * Visitors and Tree Traversal::
2883 * Polynomial Arithmetic:: Working with polynomials.
2884 * Rational Expressions:: Working with rational functions.
2885 * Symbolic Differentiation::
2886 * Series Expansion:: Taylor and Laurent expansion.
2888 * Built-in Functions:: List of predefined mathematical functions.
2889 * Solving Linear Systems of Equations::
2890 * Input/Output:: Input and output of expressions.
2894 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2895 @c node-name, next, previous, up
2896 @section Getting information about expressions
2898 @subsection Checking expression types
2899 @cindex @code{is_a<@dots{}>()}
2900 @cindex @code{is_exactly_a<@dots{}>()}
2901 @cindex @code{ex_to<@dots{}>()}
2902 @cindex Converting @code{ex} to other classes
2903 @cindex @code{info()}
2904 @cindex @code{return_type()}
2905 @cindex @code{return_type_tinfo()}
2907 Sometimes it's useful to check whether a given expression is a plain number,
2908 a sum, a polynomial with integer coefficients, or of some other specific type.
2909 GiNaC provides a couple of functions for this:
2912 bool is_a<T>(const ex & e);
2913 bool is_exactly_a<T>(const ex & e);
2914 bool ex::info(unsigned flag);
2915 unsigned ex::return_type() const;
2916 unsigned ex::return_type_tinfo() const;
2919 When the test made by @code{is_a<T>()} returns true, it is safe to call
2920 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2921 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2922 example, assuming @code{e} is an @code{ex}:
2927 if (is_a<numeric>(e))
2928 numeric n = ex_to<numeric>(e);
2933 @code{is_a<T>(e)} allows you to check whether the top-level object of
2934 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2935 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2936 e.g., for checking whether an expression is a number, a sum, or a product:
2943 is_a<numeric>(e1); // true
2944 is_a<numeric>(e2); // false
2945 is_a<add>(e1); // false
2946 is_a<add>(e2); // true
2947 is_a<mul>(e1); // false
2948 is_a<mul>(e2); // false
2952 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2953 top-level object of an expression @samp{e} is an instance of the GiNaC
2954 class @samp{T}, not including parent classes.
2956 The @code{info()} method is used for checking certain attributes of
2957 expressions. The possible values for the @code{flag} argument are defined
2958 in @file{ginac/flags.h}, the most important being explained in the following
2962 @multitable @columnfractions .30 .70
2963 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2964 @item @code{numeric}
2965 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2967 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2968 @item @code{rational}
2969 @tab @dots{}an exact rational number (integers are rational, too)
2970 @item @code{integer}
2971 @tab @dots{}a (non-complex) integer
2972 @item @code{crational}
2973 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2974 @item @code{cinteger}
2975 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2976 @item @code{positive}
2977 @tab @dots{}not complex and greater than 0
2978 @item @code{negative}
2979 @tab @dots{}not complex and less than 0
2980 @item @code{nonnegative}
2981 @tab @dots{}not complex and greater than or equal to 0
2983 @tab @dots{}an integer greater than 0
2985 @tab @dots{}an integer less than 0
2986 @item @code{nonnegint}
2987 @tab @dots{}an integer greater than or equal to 0
2989 @tab @dots{}an even integer
2991 @tab @dots{}an odd integer
2993 @tab @dots{}a prime integer (probabilistic primality test)
2994 @item @code{relation}
2995 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2996 @item @code{relation_equal}
2997 @tab @dots{}a @code{==} relation
2998 @item @code{relation_not_equal}
2999 @tab @dots{}a @code{!=} relation
3000 @item @code{relation_less}
3001 @tab @dots{}a @code{<} relation
3002 @item @code{relation_less_or_equal}
3003 @tab @dots{}a @code{<=} relation
3004 @item @code{relation_greater}
3005 @tab @dots{}a @code{>} relation
3006 @item @code{relation_greater_or_equal}
3007 @tab @dots{}a @code{>=} relation
3009 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3011 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3012 @item @code{polynomial}
3013 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3014 @item @code{integer_polynomial}
3015 @tab @dots{}a polynomial with (non-complex) integer coefficients
3016 @item @code{cinteger_polynomial}
3017 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3018 @item @code{rational_polynomial}
3019 @tab @dots{}a polynomial with (non-complex) rational coefficients
3020 @item @code{crational_polynomial}
3021 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3022 @item @code{rational_function}
3023 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3024 @item @code{algebraic}
3025 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3029 To determine whether an expression is commutative or non-commutative and if
3030 so, with which other expressions it would commute, you use the methods
3031 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3032 for an explanation of these.
3035 @subsection Accessing subexpressions
3036 @cindex @code{nops()}
3039 @cindex @code{relational} (class)
3041 GiNaC provides the two methods
3045 ex ex::op(size_t i);
3048 for accessing the subexpressions in the container-like GiNaC classes like
3049 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3050 determines the number of subexpressions (@samp{operands}) contained, while
3051 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3052 In the case of a @code{power} object, @code{op(0)} will return the basis
3053 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3054 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3056 The left-hand and right-hand side expressions of objects of class
3057 @code{relational} (and only of these) can also be accessed with the methods
3065 @subsection Comparing expressions
3066 @cindex @code{is_equal()}
3067 @cindex @code{is_zero()}
3069 Expressions can be compared with the usual C++ relational operators like
3070 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3071 the result is usually not determinable and the result will be @code{false},
3072 except in the case of the @code{!=} operator. You should also be aware that
3073 GiNaC will only do the most trivial test for equality (subtracting both
3074 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3077 Actually, if you construct an expression like @code{a == b}, this will be
3078 represented by an object of the @code{relational} class (@pxref{Relations})
3079 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3081 There are also two methods
3084 bool ex::is_equal(const ex & other);
3088 for checking whether one expression is equal to another, or equal to zero,
3092 @subsection Ordering expressions
3093 @cindex @code{ex_is_less} (class)
3094 @cindex @code{ex_is_equal} (class)
3095 @cindex @code{compare()}
3097 Sometimes it is necessary to establish a mathematically well-defined ordering
3098 on a set of arbitrary expressions, for example to use expressions as keys
3099 in a @code{std::map<>} container, or to bring a vector of expressions into
3100 a canonical order (which is done internally by GiNaC for sums and products).
3102 The operators @code{<}, @code{>} etc. described in the last section cannot
3103 be used for this, as they don't implement an ordering relation in the
3104 mathematical sense. In particular, they are not guaranteed to be
3105 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3106 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3109 By default, STL classes and algorithms use the @code{<} and @code{==}
3110 operators to compare objects, which are unsuitable for expressions, but GiNaC
3111 provides two functors that can be supplied as proper binary comparison
3112 predicates to the STL:
3115 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3117 bool operator()(const ex &lh, const ex &rh) const;
3120 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3122 bool operator()(const ex &lh, const ex &rh) const;
3126 For example, to define a @code{map} that maps expressions to strings you
3130 std::map<ex, std::string, ex_is_less> myMap;
3133 Omitting the @code{ex_is_less} template parameter will introduce spurious
3134 bugs because the map operates improperly.
3136 Other examples for the use of the functors:
3144 std::sort(v.begin(), v.end(), ex_is_less());
3146 // count the number of expressions equal to '1'
3147 unsigned num_ones = std::count_if(v.begin(), v.end(),
3148 std::bind2nd(ex_is_equal(), 1));
3151 The implementation of @code{ex_is_less} uses the member function
3154 int ex::compare(const ex & other) const;
3157 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3158 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3162 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
3163 @c node-name, next, previous, up
3164 @section Substituting expressions
3165 @cindex @code{subs()}
3167 Algebraic objects inside expressions can be replaced with arbitrary
3168 expressions via the @code{.subs()} method:
3171 ex ex::subs(const ex & e, unsigned options = 0);
3172 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3175 In the first form, @code{subs()} accepts a relational of the form
3176 @samp{object == expression} or a @code{lst} of such relationals:
3180 symbol x("x"), y("y");
3182 ex e1 = 2*x^2-4*x+3;
3183 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3187 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3192 If you specify multiple substitutions, they are performed in parallel, so e.g.
3193 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3195 The second form of @code{subs()} takes two lists, one for the objects to be
3196 replaced and one for the expressions to be substituted (both lists must
3197 contain the same number of elements). Using this form, you would write
3198 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
3200 The optional last argument to @code{subs()} is a combination of
3201 @code{subs_options} flags. There are two options available:
3202 @code{subs_options::no_pattern} disables pattern matching, which makes
3203 large @code{subs()} operations significantly faster if you are not using
3204 patterns. The second option, @code{subs_options::algebraic} enables
3205 algebraic substitutions in products and powers.
3206 @ref{Pattern Matching and Advanced Substitutions}, for more information
3207 about patterns and algebraic substitutions.
3209 @code{subs()} performs syntactic substitution of any complete algebraic
3210 object; it does not try to match sub-expressions as is demonstrated by the
3215 symbol x("x"), y("y"), z("z");
3217 ex e1 = pow(x+y, 2);
3218 cout << e1.subs(x+y == 4) << endl;
3221 ex e2 = sin(x)*sin(y)*cos(x);
3222 cout << e2.subs(sin(x) == cos(x)) << endl;
3223 // -> cos(x)^2*sin(y)
3226 cout << e3.subs(x+y == 4) << endl;
3228 // (and not 4+z as one might expect)
3232 A more powerful form of substitution using wildcards is described in the
3236 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3237 @c node-name, next, previous, up
3238 @section Pattern matching and advanced substitutions
3239 @cindex @code{wildcard} (class)
3240 @cindex Pattern matching
3242 GiNaC allows the use of patterns for checking whether an expression is of a
3243 certain form or contains subexpressions of a certain form, and for
3244 substituting expressions in a more general way.
3246 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3247 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3248 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3249 an unsigned integer number to allow having multiple different wildcards in a
3250 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3251 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3255 ex wild(unsigned label = 0);
3258 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3261 Some examples for patterns:
3263 @multitable @columnfractions .5 .5
3264 @item @strong{Constructed as} @tab @strong{Output as}
3265 @item @code{wild()} @tab @samp{$0}
3266 @item @code{pow(x,wild())} @tab @samp{x^$0}
3267 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3268 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3274 @item Wildcards behave like symbols and are subject to the same algebraic
3275 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3276 @item As shown in the last example, to use wildcards for indices you have to
3277 use them as the value of an @code{idx} object. This is because indices must
3278 always be of class @code{idx} (or a subclass).
3279 @item Wildcards only represent expressions or subexpressions. It is not
3280 possible to use them as placeholders for other properties like index
3281 dimension or variance, representation labels, symmetry of indexed objects
3283 @item Because wildcards are commutative, it is not possible to use wildcards
3284 as part of noncommutative products.
3285 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3286 are also valid patterns.
3289 @subsection Matching expressions
3290 @cindex @code{match()}
3291 The most basic application of patterns is to check whether an expression
3292 matches a given pattern. This is done by the function
3295 bool ex::match(const ex & pattern);
3296 bool ex::match(const ex & pattern, lst & repls);
3299 This function returns @code{true} when the expression matches the pattern
3300 and @code{false} if it doesn't. If used in the second form, the actual
3301 subexpressions matched by the wildcards get returned in the @code{repls}
3302 object as a list of relations of the form @samp{wildcard == expression}.
3303 If @code{match()} returns false, the state of @code{repls} is undefined.
3304 For reproducible results, the list should be empty when passed to
3305 @code{match()}, but it is also possible to find similarities in multiple
3306 expressions by passing in the result of a previous match.
3308 The matching algorithm works as follows:
3311 @item A single wildcard matches any expression. If one wildcard appears
3312 multiple times in a pattern, it must match the same expression in all
3313 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3314 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3315 @item If the expression is not of the same class as the pattern, the match
3316 fails (i.e. a sum only matches a sum, a function only matches a function,
3318 @item If the pattern is a function, it only matches the same function
3319 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3320 @item Except for sums and products, the match fails if the number of
3321 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3323 @item If there are no subexpressions, the expressions and the pattern must
3324 be equal (in the sense of @code{is_equal()}).
3325 @item Except for sums and products, each subexpression (@code{op()}) must
3326 match the corresponding subexpression of the pattern.
3329 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3330 account for their commutativity and associativity:
3333 @item If the pattern contains a term or factor that is a single wildcard,
3334 this one is used as the @dfn{global wildcard}. If there is more than one
3335 such wildcard, one of them is chosen as the global wildcard in a random
3337 @item Every term/factor of the pattern, except the global wildcard, is
3338 matched against every term of the expression in sequence. If no match is
3339 found, the whole match fails. Terms that did match are not considered in
3341 @item If there are no unmatched terms left, the match succeeds. Otherwise
3342 the match fails unless there is a global wildcard in the pattern, in
3343 which case this wildcard matches the remaining terms.
3346 In general, having more than one single wildcard as a term of a sum or a
3347 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3350 Here are some examples in @command{ginsh} to demonstrate how it works (the
3351 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3352 match fails, and the list of wildcard replacements otherwise):
3355 > match((x+y)^a,(x+y)^a);
3357 > match((x+y)^a,(x+y)^b);
3359 > match((x+y)^a,$1^$2);
3361 > match((x+y)^a,$1^$1);
3363 > match((x+y)^(x+y),$1^$1);
3365 > match((x+y)^(x+y),$1^$2);
3367 > match((a+b)*(a+c),($1+b)*($1+c));
3369 > match((a+b)*(a+c),(a+$1)*(a+$2));
3371 (Unpredictable. The result might also be [$1==c,$2==b].)
3372 > match((a+b)*(a+c),($1+$2)*($1+$3));
3373 (The result is undefined. Due to the sequential nature of the algorithm
3374 and the re-ordering of terms in GiNaC, the match for the first factor
3375 may be @{$1==a,$2==b@} in which case the match for the second factor
3376 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3378 > match(a*(x+y)+a*z+b,a*$1+$2);
3379 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3380 @{$1=x+y,$2=a*z+b@}.)
3381 > match(a+b+c+d+e+f,c);
3383 > match(a+b+c+d+e+f,c+$0);
3385 > match(a+b+c+d+e+f,c+e+$0);
3387 > match(a+b,a+b+$0);
3389 > match(a*b^2,a^$1*b^$2);
3391 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3392 even though a==a^1.)
3393 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3395 > match(atan2(y,x^2),atan2(y,$0));
3399 @subsection Matching parts of expressions
3400 @cindex @code{has()}
3401 A more general way to look for patterns in expressions is provided by the
3405 bool ex::has(const ex & pattern);
3408 This function checks whether a pattern is matched by an expression itself or
3409 by any of its subexpressions.
3411 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3412 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3415 > has(x*sin(x+y+2*a),y);
3417 > has(x*sin(x+y+2*a),x+y);
3419 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3420 has the subexpressions "x", "y" and "2*a".)
3421 > has(x*sin(x+y+2*a),x+y+$1);
3423 (But this is possible.)
3424 > has(x*sin(2*(x+y)+2*a),x+y);
3426 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3427 which "x+y" is not a subexpression.)
3430 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3432 > has(4*x^2-x+3,$1*x);
3434 > has(4*x^2+x+3,$1*x);
3436 (Another possible pitfall. The first expression matches because the term
3437 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3438 contains a linear term you should use the coeff() function instead.)
3441 @cindex @code{find()}
3445 bool ex::find(const ex & pattern, lst & found);
3448 works a bit like @code{has()} but it doesn't stop upon finding the first
3449 match. Instead, it appends all found matches to the specified list. If there
3450 are multiple occurrences of the same expression, it is entered only once to
3451 the list. @code{find()} returns false if no matches were found (in
3452 @command{ginsh}, it returns an empty list):
3455 > find(1+x+x^2+x^3,x);
3457 > find(1+x+x^2+x^3,y);
3459 > find(1+x+x^2+x^3,x^$1);
3461 (Note the absence of "x".)
3462 > expand((sin(x)+sin(y))*(a+b));
3463 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3468 @subsection Substituting expressions
3469 @cindex @code{subs()}
3470 Probably the most useful application of patterns is to use them for
3471 substituting expressions with the @code{subs()} method. Wildcards can be
3472 used in the search patterns as well as in the replacement expressions, where
3473 they get replaced by the expressions matched by them. @code{subs()} doesn't
3474 know anything about algebra; it performs purely syntactic substitutions.
3479 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3481 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3483 > subs((a+b+c)^2,a+b==x);
3485 > subs((a+b+c)^2,a+b+$1==x+$1);
3487 > subs(a+2*b,a+b==x);
3489 > subs(4*x^3-2*x^2+5*x-1,x==a);
3491 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3493 > subs(sin(1+sin(x)),sin($1)==cos($1));
3495 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3499 The last example would be written in C++ in this way:
3503 symbol a("a"), b("b"), x("x"), y("y");
3504 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3505 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3506 cout << e.expand() << endl;
3511 @subsection Algebraic substitutions
3512 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3513 enables smarter, algebraic substitutions in products and powers. If you want
3514 to substitute some factors of a product, you only need to list these factors
3515 in your pattern. Furthermore, if an (integer) power of some expression occurs
3516 in your pattern and in the expression that you want the substitution to occur
3517 in, it can be substituted as many times as possible, without getting negative
3520 An example clarifies it all (hopefully):
3523 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3524 subs_options::algebraic) << endl;
3525 // --> (y+x)^6+b^6+a^6
3527 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3529 // Powers and products are smart, but addition is just the same.
3531 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3534 // As I said: addition is just the same.
3536 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3537 // --> x^3*b*a^2+2*b
3539 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3541 // --> 2*b+x^3*b^(-1)*a^(-2)
3543 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3544 // --> -1-2*a^2+4*a^3+5*a
3546 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3547 subs_options::algebraic) << endl;
3548 // --> -1+5*x+4*x^3-2*x^2
3549 // You should not really need this kind of patterns very often now.
3550 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3552 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3553 subs_options::algebraic) << endl;
3554 // --> cos(1+cos(x))
3556 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3557 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3558 subs_options::algebraic)) << endl;
3563 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3564 @c node-name, next, previous, up
3565 @section Applying a Function on Subexpressions
3566 @cindex tree traversal
3567 @cindex @code{map()}
3569 Sometimes you may want to perform an operation on specific parts of an
3570 expression while leaving the general structure of it intact. An example
3571 of this would be a matrix trace operation: the trace of a sum is the sum
3572 of the traces of the individual terms. That is, the trace should @dfn{map}
3573 on the sum, by applying itself to each of the sum's operands. It is possible
3574 to do this manually which usually results in code like this:
3579 if (is_a<matrix>(e))
3580 return ex_to<matrix>(e).trace();
3581 else if (is_a<add>(e)) @{
3583 for (size_t i=0; i<e.nops(); i++)
3584 sum += calc_trace(e.op(i));
3586 @} else if (is_a<mul>)(e)) @{
3594 This is, however, slightly inefficient (if the sum is very large it can take
3595 a long time to add the terms one-by-one), and its applicability is limited to
3596 a rather small class of expressions. If @code{calc_trace()} is called with
3597 a relation or a list as its argument, you will probably want the trace to
3598 be taken on both sides of the relation or of all elements of the list.
3600 GiNaC offers the @code{map()} method to aid in the implementation of such
3604 ex ex::map(map_function & f) const;
3605 ex ex::map(ex (*f)(const ex & e)) const;
3608 In the first (preferred) form, @code{map()} takes a function object that
3609 is subclassed from the @code{map_function} class. In the second form, it
3610 takes a pointer to a function that accepts and returns an expression.
3611 @code{map()} constructs a new expression of the same type, applying the
3612 specified function on all subexpressions (in the sense of @code{op()}),
3615 The use of a function object makes it possible to supply more arguments to
3616 the function that is being mapped, or to keep local state information.
3617 The @code{map_function} class declares a virtual function call operator
3618 that you can overload. Here is a sample implementation of @code{calc_trace()}
3619 that uses @code{map()} in a recursive fashion:
3622 struct calc_trace : public map_function @{
3623 ex operator()(const ex &e)
3625 if (is_a<matrix>(e))
3626 return ex_to<matrix>(e).trace();
3627 else if (is_a<mul>(e)) @{
3630 return e.map(*this);
3635 This function object could then be used like this:
3639 ex M = ... // expression with matrices
3640 calc_trace do_trace;
3641 ex tr = do_trace(M);
3645 Here is another example for you to meditate over. It removes quadratic
3646 terms in a variable from an expanded polynomial:
3649 struct map_rem_quad : public map_function @{
3651 map_rem_quad(const ex & var_) : var(var_) @{@}
3653 ex operator()(const ex & e)
3655 if (is_a<add>(e) || is_a<mul>(e))
3656 return e.map(*this);
3657 else if (is_a<power>(e) &&
3658 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3668 symbol x("x"), y("y");
3671 for (int i=0; i<8; i++)
3672 e += pow(x, i) * pow(y, 8-i) * (i+1);
3674 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
3676 map_rem_quad rem_quad(x);
3677 cout << rem_quad(e) << endl;
3678 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3682 @command{ginsh} offers a slightly different implementation of @code{map()}
3683 that allows applying algebraic functions to operands. The second argument
3684 to @code{map()} is an expression containing the wildcard @samp{$0} which
3685 acts as the placeholder for the operands:
3690 > map(a+2*b,sin($0));
3692 > map(@{a,b,c@},$0^2+$0);
3693 @{a^2+a,b^2+b,c^2+c@}
3696 Note that it is only possible to use algebraic functions in the second
3697 argument. You can not use functions like @samp{diff()}, @samp{op()},
3698 @samp{subs()} etc. because these are evaluated immediately:
3701 > map(@{a,b,c@},diff($0,a));
3703 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3704 to "map(@{a,b,c@},0)".
3708 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3709 @c node-name, next, previous, up
3710 @section Visitors and Tree Traversal
3711 @cindex tree traversal
3712 @cindex @code{visitor} (class)
3713 @cindex @code{accept()}
3714 @cindex @code{visit()}
3715 @cindex @code{traverse()}
3716 @cindex @code{traverse_preorder()}
3717 @cindex @code{traverse_postorder()}