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.
18 @dircategory Mathematics
20 * ginac: (ginac). C++ library for symbolic computation.
24 This is a tutorial that documents GiNaC @value{VERSION}, an open
25 framework for symbolic computation within the C++ programming language.
27 Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
29 Permission is granted to make and distribute verbatim copies of
30 this manual provided the copyright notice and this permission notice
31 are preserved on all copies.
34 Permission is granted to process this file through TeX and print the
35 results, provided the printed document carries copying permission
36 notice identical to this one except for the removal of this paragraph
39 Permission is granted to copy and distribute modified versions of this
40 manual under the conditions for verbatim copying, provided that the entire
41 resulting derived work is distributed under the terms of a permission
42 notice identical to this one.
46 @c finalout prevents ugly black rectangles on overfull hbox lines
48 @title GiNaC @value{VERSION}
49 @subtitle An open framework for symbolic computation within the C++ programming language
50 @subtitle @value{UPDATED}
51 @author @uref{http://www.ginac.de}
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2008 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{} 1999-2008 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., 51 Franklin Street, Fifth Floor, 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 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
483 required for the configuration, it can be downloaded from
484 @uref{http://pkg-config.freedesktop.org}.
485 Last but not least, the CLN library
486 is used extensively and needs to be installed on your system.
487 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
488 (it is covered by GPL) and install it prior to trying to install
489 GiNaC. The configure script checks if it can find it and if it cannot
490 it will refuse to continue.
493 @node Configuration, Building GiNaC, Prerequisites, Installation
494 @c node-name, next, previous, up
495 @section Configuration
496 @cindex configuration
499 To configure GiNaC means to prepare the source distribution for
500 building. It is done via a shell script called @command{configure} that
501 is shipped with the sources and was originally generated by GNU
502 Autoconf. Since a configure script generated by GNU Autoconf never
503 prompts, all customization must be done either via command line
504 parameters or environment variables. It accepts a list of parameters,
505 the complete set of which can be listed by calling it with the
506 @option{--help} option. The most important ones will be shortly
507 described in what follows:
512 @option{--disable-shared}: When given, this option switches off the
513 build of a shared library, i.e. a @file{.so} file. This may be convenient
514 when developing because it considerably speeds up compilation.
517 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
518 and headers are installed. It defaults to @file{/usr/local} which means
519 that the library is installed in the directory @file{/usr/local/lib},
520 the header files in @file{/usr/local/include/ginac} and the documentation
521 (like this one) into @file{/usr/local/share/doc/GiNaC}.
524 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
525 the library installed in some other directory than
526 @file{@var{PREFIX}/lib/}.
529 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
530 to have the header files installed in some other directory than
531 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
532 @option{--includedir=/usr/include} you will end up with the header files
533 sitting in the directory @file{/usr/include/ginac/}. Note that the
534 subdirectory @file{ginac} is enforced by this process in order to
535 keep the header files separated from others. This avoids some
536 clashes and allows for an easier deinstallation of GiNaC. This ought
537 to be considered A Good Thing (tm).
540 @option{--datadir=@var{DATADIR}}: This option may be given in case you
541 want to have the documentation installed in some other directory than
542 @file{@var{PREFIX}/share/doc/GiNaC/}.
546 In addition, you may specify some environment variables. @env{CXX}
547 holds the path and the name of the C++ compiler in case you want to
548 override the default in your path. (The @command{configure} script
549 searches your path for @command{c++}, @command{g++}, @command{gcc},
550 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
551 be very useful to define some compiler flags with the @env{CXXFLAGS}
552 environment variable, like optimization, debugging information and
553 warning levels. If omitted, it defaults to @option{-g
554 -O2}.@footnote{The @command{configure} script is itself generated from
555 the file @file{configure.ac}. It is only distributed in packaged
556 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
557 must generate @command{configure} along with the various
558 @file{Makefile.in} by using the @command{autoreconf} utility. This will
559 require a fair amount of support from your local toolchain, though.}
561 The whole process is illustrated in the following two
562 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
563 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
566 Here is a simple configuration for a site-wide GiNaC library assuming
567 everything is in default paths:
570 $ export CXXFLAGS="-Wall -O2"
574 And here is a configuration for a private static GiNaC library with
575 several components sitting in custom places (site-wide GCC and private
576 CLN). The compiler is persuaded to be picky and full assertions and
577 debugging information are switched on:
580 $ export CXX=/usr/local/gnu/bin/c++
581 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
582 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
583 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
584 $ ./configure --disable-shared --prefix=$(HOME)
588 @node Building GiNaC, Installing GiNaC, Configuration, Installation
589 @c node-name, next, previous, up
590 @section Building GiNaC
591 @cindex building GiNaC
593 After proper configuration you should just build the whole
598 at the command prompt and go for a cup of coffee. The exact time it
599 takes to compile GiNaC depends not only on the speed of your machines
600 but also on other parameters, for instance what value for @env{CXXFLAGS}
601 you entered. Optimization may be very time-consuming.
603 Just to make sure GiNaC works properly you may run a collection of
604 regression tests by typing
610 This will compile some sample programs, run them and check the output
611 for correctness. The regression tests fall in three categories. First,
612 the so called @emph{exams} are performed, simple tests where some
613 predefined input is evaluated (like a pupils' exam). Second, the
614 @emph{checks} test the coherence of results among each other with
615 possible random input. Third, some @emph{timings} are performed, which
616 benchmark some predefined problems with different sizes and display the
617 CPU time used in seconds. Each individual test should return a message
618 @samp{passed}. This is mostly intended to be a QA-check if something
619 was broken during development, not a sanity check of your system. Some
620 of the tests in sections @emph{checks} and @emph{timings} may require
621 insane amounts of memory and CPU time. Feel free to kill them if your
622 machine catches fire. Another quite important intent is to allow people
623 to fiddle around with optimization.
625 By default, the only documentation that will be built is this tutorial
626 in @file{.info} format. To build the GiNaC tutorial and reference manual
627 in HTML, DVI, PostScript, or PDF formats, use one of
636 Generally, the top-level Makefile runs recursively to the
637 subdirectories. It is therefore safe to go into any subdirectory
638 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
639 @var{target} there in case something went wrong.
642 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
643 @c node-name, next, previous, up
644 @section Installing GiNaC
647 To install GiNaC on your system, simply type
653 As described in the section about configuration the files will be
654 installed in the following directories (the directories will be created
655 if they don't already exist):
660 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
661 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
662 So will @file{libginac.so} unless the configure script was
663 given the option @option{--disable-shared}. The proper symlinks
664 will be established as well.
667 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
668 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
671 All documentation (info) will be stuffed into
672 @file{@var{PREFIX}/share/doc/GiNaC/} (or
673 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
677 For the sake of completeness we will list some other useful make
678 targets: @command{make clean} deletes all files generated by
679 @command{make}, i.e. all the object files. In addition @command{make
680 distclean} removes all files generated by the configuration and
681 @command{make maintainer-clean} goes one step further and deletes files
682 that may require special tools to rebuild (like the @command{libtool}
683 for instance). Finally @command{make uninstall} removes the installed
684 library, header files and documentation@footnote{Uninstallation does not
685 work after you have called @command{make distclean} since the
686 @file{Makefile} is itself generated by the configuration from
687 @file{Makefile.in} and hence deleted by @command{make distclean}. There
688 are two obvious ways out of this dilemma. First, you can run the
689 configuration again with the same @var{PREFIX} thus creating a
690 @file{Makefile} with a working @samp{uninstall} target. Second, you can
691 do it by hand since you now know where all the files went during
695 @node Basic concepts, Expressions, Installing GiNaC, Top
696 @c node-name, next, previous, up
697 @chapter Basic concepts
699 This chapter will describe the different fundamental objects that can be
700 handled by GiNaC. But before doing so, it is worthwhile introducing you
701 to the more commonly used class of expressions, representing a flexible
702 meta-class for storing all mathematical objects.
705 * Expressions:: The fundamental GiNaC class.
706 * Automatic evaluation:: Evaluation and canonicalization.
707 * Error handling:: How the library reports errors.
708 * The class hierarchy:: Overview of GiNaC's classes.
709 * Symbols:: Symbolic objects.
710 * Numbers:: Numerical objects.
711 * Constants:: Pre-defined constants.
712 * Fundamental containers:: Sums, products and powers.
713 * Lists:: Lists of expressions.
714 * Mathematical functions:: Mathematical functions.
715 * Relations:: Equality, Inequality and all that.
716 * Integrals:: Symbolic integrals.
717 * Matrices:: Matrices.
718 * Indexed objects:: Handling indexed quantities.
719 * Non-commutative objects:: Algebras with non-commutative products.
720 * Hash maps:: A faster alternative to std::map<>.
724 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
725 @c node-name, next, previous, up
727 @cindex expression (class @code{ex})
730 The most common class of objects a user deals with is the expression
731 @code{ex}, representing a mathematical object like a variable, number,
732 function, sum, product, etc@dots{} Expressions may be put together to form
733 new expressions, passed as arguments to functions, and so on. Here is a
734 little collection of valid expressions:
737 ex MyEx1 = 5; // simple number
738 ex MyEx2 = x + 2*y; // polynomial in x and y
739 ex MyEx3 = (x + 1)/(x - 1); // rational expression
740 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
741 ex MyEx5 = MyEx4 + 1; // similar to above
744 Expressions are handles to other more fundamental objects, that often
745 contain other expressions thus creating a tree of expressions
746 (@xref{Internal structures}, for particular examples). Most methods on
747 @code{ex} therefore run top-down through such an expression tree. For
748 example, the method @code{has()} scans recursively for occurrences of
749 something inside an expression. Thus, if you have declared @code{MyEx4}
750 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
751 the argument of @code{sin} and hence return @code{true}.
753 The next sections will outline the general picture of GiNaC's class
754 hierarchy and describe the classes of objects that are handled by
757 @subsection Note: Expressions and STL containers
759 GiNaC expressions (@code{ex} objects) have value semantics (they can be
760 assigned, reassigned and copied like integral types) but the operator
761 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
762 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
764 This implies that in order to use expressions in sorted containers such as
765 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
766 comparison predicate. GiNaC provides such a predicate, called
767 @code{ex_is_less}. For example, a set of expressions should be defined
768 as @code{std::set<ex, ex_is_less>}.
770 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
771 don't pose a problem. A @code{std::vector<ex>} works as expected.
773 @xref{Information about expressions}, for more about comparing and ordering
777 @node Automatic evaluation, Error handling, Expressions, Basic concepts
778 @c node-name, next, previous, up
779 @section Automatic evaluation and canonicalization of expressions
782 GiNaC performs some automatic transformations on expressions, to simplify
783 them and put them into a canonical form. Some examples:
786 ex MyEx1 = 2*x - 1 + x; // 3*x-1
787 ex MyEx2 = x - x; // 0
788 ex MyEx3 = cos(2*Pi); // 1
789 ex MyEx4 = x*y/x; // y
792 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
793 evaluation}. GiNaC only performs transformations that are
797 at most of complexity
805 algebraically correct, possibly except for a set of measure zero (e.g.
806 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
809 There are two types of automatic transformations in GiNaC that may not
810 behave in an entirely obvious way at first glance:
814 The terms of sums and products (and some other things like the arguments of
815 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
816 into a canonical form that is deterministic, but not lexicographical or in
817 any other way easy to guess (it almost always depends on the number and
818 order of the symbols you define). However, constructing the same expression
819 twice, either implicitly or explicitly, will always result in the same
822 Expressions of the form 'number times sum' are automatically expanded (this
823 has to do with GiNaC's internal representation of sums and products). For
826 ex MyEx5 = 2*(x + y); // 2*x+2*y
827 ex MyEx6 = z*(x + y); // z*(x+y)
831 The general rule is that when you construct expressions, GiNaC automatically
832 creates them in canonical form, which might differ from the form you typed in
833 your program. This may create some awkward looking output (@samp{-y+x} instead
834 of @samp{x-y}) but allows for more efficient operation and usually yields
835 some immediate simplifications.
837 @cindex @code{eval()}
838 Internally, the anonymous evaluator in GiNaC is implemented by the methods
841 ex ex::eval(int level = 0) const;
842 ex basic::eval(int level = 0) const;
845 but unless you are extending GiNaC with your own classes or functions, there
846 should never be any reason to call them explicitly. All GiNaC methods that
847 transform expressions, like @code{subs()} or @code{normal()}, automatically
848 re-evaluate their results.
851 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
852 @c node-name, next, previous, up
853 @section Error handling
855 @cindex @code{pole_error} (class)
857 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
858 generated by GiNaC are subclassed from the standard @code{exception} class
859 defined in the @file{<stdexcept>} header. In addition to the predefined
860 @code{logic_error}, @code{domain_error}, @code{out_of_range},
861 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
862 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
863 exception that gets thrown when trying to evaluate a mathematical function
866 The @code{pole_error} class has a member function
869 int pole_error::degree() const;
872 that returns the order of the singularity (or 0 when the pole is
873 logarithmic or the order is undefined).
875 When using GiNaC it is useful to arrange for exceptions to be caught in
876 the main program even if you don't want to do any special error handling.
877 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
878 default exception handler of your C++ compiler's run-time system which
879 usually only aborts the program without giving any information what went
882 Here is an example for a @code{main()} function that catches and prints
883 exceptions generated by GiNaC:
888 #include <ginac/ginac.h>
890 using namespace GiNaC;
898 @} catch (exception &p) @{
899 cerr << p.what() << endl;
907 @node The class hierarchy, Symbols, Error handling, Basic concepts
908 @c node-name, next, previous, up
909 @section The class hierarchy
911 GiNaC's class hierarchy consists of several classes representing
912 mathematical objects, all of which (except for @code{ex} and some
913 helpers) are internally derived from one abstract base class called
914 @code{basic}. You do not have to deal with objects of class
915 @code{basic}, instead you'll be dealing with symbols, numbers,
916 containers of expressions and so on.
920 To get an idea about what kinds of symbolic composites may be built we
921 have a look at the most important classes in the class hierarchy and
922 some of the relations among the classes:
925 @image{classhierarchy}
931 The abstract classes shown here (the ones without drop-shadow) are of no
932 interest for the user. They are used internally in order to avoid code
933 duplication if two or more classes derived from them share certain
934 features. An example is @code{expairseq}, a container for a sequence of
935 pairs each consisting of one expression and a number (@code{numeric}).
936 What @emph{is} visible to the user are the derived classes @code{add}
937 and @code{mul}, representing sums and products. @xref{Internal
938 structures}, where these two classes are described in more detail. The
939 following table shortly summarizes what kinds of mathematical objects
940 are stored in the different classes:
943 @multitable @columnfractions .22 .78
944 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
945 @item @code{constant} @tab Constants like
952 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
953 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
954 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
955 @item @code{ncmul} @tab Products of non-commutative objects
956 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
961 @code{sqrt(}@math{2}@code{)}
964 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
965 @item @code{function} @tab A symbolic function like
972 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
973 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
974 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
975 @item @code{indexed} @tab Indexed object like @math{A_ij}
976 @item @code{tensor} @tab Special tensor like the delta and metric tensors
977 @item @code{idx} @tab Index of an indexed object
978 @item @code{varidx} @tab Index with variance
979 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
980 @item @code{wildcard} @tab Wildcard for pattern matching
981 @item @code{structure} @tab Template for user-defined classes
986 @node Symbols, Numbers, The class hierarchy, Basic concepts
987 @c node-name, next, previous, up
989 @cindex @code{symbol} (class)
990 @cindex hierarchy of classes
993 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
994 manipulation what atoms are for chemistry.
996 A typical symbol definition looks like this:
1001 This definition actually contains three very different things:
1003 @item a C++ variable named @code{x}
1004 @item a @code{symbol} object stored in this C++ variable; this object
1005 represents the symbol in a GiNaC expression
1006 @item the string @code{"x"} which is the name of the symbol, used (almost)
1007 exclusively for printing expressions holding the symbol
1010 Symbols have an explicit name, supplied as a string during construction,
1011 because in C++, variable names can't be used as values, and the C++ compiler
1012 throws them away during compilation.
1014 It is possible to omit the symbol name in the definition:
1019 In this case, GiNaC will assign the symbol an internal, unique name of the
1020 form @code{symbolNNN}. This won't affect the usability of the symbol but
1021 the output of your calculations will become more readable if you give your
1022 symbols sensible names (for intermediate expressions that are only used
1023 internally such anonymous symbols can be quite useful, however).
1025 Now, here is one important property of GiNaC that differentiates it from
1026 other computer algebra programs you may have used: GiNaC does @emph{not} use
1027 the names of symbols to tell them apart, but a (hidden) serial number that
1028 is unique for each newly created @code{symbol} object. If you want to use
1029 one and the same symbol in different places in your program, you must only
1030 create one @code{symbol} object and pass that around. If you create another
1031 symbol, even if it has the same name, GiNaC will treat it as a different
1048 // prints "x^6" which looks right, but...
1050 cout << e.degree(x) << endl;
1051 // ...this doesn't work. The symbol "x" here is different from the one
1052 // in f() and in the expression returned by f(). Consequently, it
1057 One possibility to ensure that @code{f()} and @code{main()} use the same
1058 symbol is to pass the symbol as an argument to @code{f()}:
1060 ex f(int n, const ex & x)
1069 // Now, f() uses the same symbol.
1072 cout << e.degree(x) << endl;
1073 // prints "6", as expected
1077 Another possibility would be to define a global symbol @code{x} that is used
1078 by both @code{f()} and @code{main()}. If you are using global symbols and
1079 multiple compilation units you must take special care, however. Suppose
1080 that you have a header file @file{globals.h} in your program that defines
1081 a @code{symbol x("x");}. In this case, every unit that includes
1082 @file{globals.h} would also get its own definition of @code{x} (because
1083 header files are just inlined into the source code by the C++ preprocessor),
1084 and hence you would again end up with multiple equally-named, but different,
1085 symbols. Instead, the @file{globals.h} header should only contain a
1086 @emph{declaration} like @code{extern symbol x;}, with the definition of
1087 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1089 A different approach to ensuring that symbols used in different parts of
1090 your program are identical is to create them with a @emph{factory} function
1093 const symbol & get_symbol(const string & s)
1095 static map<string, symbol> directory;
1096 map<string, symbol>::iterator i = directory.find(s);
1097 if (i != directory.end())
1100 return directory.insert(make_pair(s, symbol(s))).first->second;
1104 This function returns one newly constructed symbol for each name that is
1105 passed in, and it returns the same symbol when called multiple times with
1106 the same name. Using this symbol factory, we can rewrite our example like
1111 return pow(get_symbol("x"), n);
1118 // Both calls of get_symbol("x") yield the same symbol.
1119 cout << e.degree(get_symbol("x")) << endl;
1124 Instead of creating symbols from strings we could also have
1125 @code{get_symbol()} take, for example, an integer number as its argument.
1126 In this case, we would probably want to give the generated symbols names
1127 that include this number, which can be accomplished with the help of an
1128 @code{ostringstream}.
1130 In general, if you're getting weird results from GiNaC such as an expression
1131 @samp{x-x} that is not simplified to zero, you should check your symbol
1134 As we said, the names of symbols primarily serve for purposes of expression
1135 output. But there are actually two instances where GiNaC uses the names for
1136 identifying symbols: When constructing an expression from a string, and when
1137 recreating an expression from an archive (@pxref{Input/output}).
1139 In addition to its name, a symbol may contain a special string that is used
1142 symbol x("x", "\\Box");
1145 This creates a symbol that is printed as "@code{x}" in normal output, but
1146 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1147 information about the different output formats of expressions in GiNaC).
1148 GiNaC automatically creates proper LaTeX code for symbols having names of
1149 greek letters (@samp{alpha}, @samp{mu}, etc.).
1151 @cindex @code{subs()}
1152 Symbols in GiNaC can't be assigned values. If you need to store results of
1153 calculations and give them a name, use C++ variables of type @code{ex}.
1154 If you want to replace a symbol in an expression with something else, you
1155 can invoke the expression's @code{.subs()} method
1156 (@pxref{Substituting expressions}).
1158 @cindex @code{realsymbol()}
1159 By default, symbols are expected to stand in for complex values, i.e. they live
1160 in the complex domain. As a consequence, operations like complex conjugation,
1161 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1162 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1163 because of the unknown imaginary part of @code{x}.
1164 On the other hand, if you are sure that your symbols will hold only real
1165 values, you would like to have such functions evaluated. Therefore GiNaC
1166 allows you to specify
1167 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1168 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1170 @cindex @code{possymbol()}
1171 Furthermore, it is also possible to declare a symbol as positive. This will,
1172 for instance, enable the automatic simplification of @code{abs(x)} into
1173 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1176 @node Numbers, Constants, Symbols, Basic concepts
1177 @c node-name, next, previous, up
1179 @cindex @code{numeric} (class)
1185 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1186 The classes therein serve as foundation classes for GiNaC. CLN stands
1187 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1188 In order to find out more about CLN's internals, the reader is referred to
1189 the documentation of that library. @inforef{Introduction, , cln}, for
1190 more information. Suffice to say that it is by itself build on top of
1191 another library, the GNU Multiple Precision library GMP, which is an
1192 extremely fast library for arbitrary long integers and rationals as well
1193 as arbitrary precision floating point numbers. It is very commonly used
1194 by several popular cryptographic applications. CLN extends GMP by
1195 several useful things: First, it introduces the complex number field
1196 over either reals (i.e. floating point numbers with arbitrary precision)
1197 or rationals. Second, it automatically converts rationals to integers
1198 if the denominator is unity and complex numbers to real numbers if the
1199 imaginary part vanishes and also correctly treats algebraic functions.
1200 Third it provides good implementations of state-of-the-art algorithms
1201 for all trigonometric and hyperbolic functions as well as for
1202 calculation of some useful constants.
1204 The user can construct an object of class @code{numeric} in several
1205 ways. The following example shows the four most important constructors.
1206 It uses construction from C-integer, construction of fractions from two
1207 integers, construction from C-float and construction from a string:
1211 #include <ginac/ginac.h>
1212 using namespace GiNaC;
1216 numeric two = 2; // exact integer 2
1217 numeric r(2,3); // exact fraction 2/3
1218 numeric e(2.71828); // floating point number
1219 numeric p = "3.14159265358979323846"; // constructor from string
1220 // Trott's constant in scientific notation:
1221 numeric trott("1.0841015122311136151E-2");
1223 std::cout << two*p << std::endl; // floating point 6.283...
1228 @cindex complex numbers
1229 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1234 numeric z1 = 2-3*I; // exact complex number 2-3i
1235 numeric z2 = 5.9+1.6*I; // complex floating point number
1239 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1240 This would, however, call C's built-in operator @code{/} for integers
1241 first and result in a numeric holding a plain integer 1. @strong{Never
1242 use the operator @code{/} on integers} unless you know exactly what you
1243 are doing! Use the constructor from two integers instead, as shown in
1244 the example above. Writing @code{numeric(1)/2} may look funny but works
1247 @cindex @code{Digits}
1249 We have seen now the distinction between exact numbers and floating
1250 point numbers. Clearly, the user should never have to worry about
1251 dynamically created exact numbers, since their `exactness' always
1252 determines how they ought to be handled, i.e. how `long' they are. The
1253 situation is different for floating point numbers. Their accuracy is
1254 controlled by one @emph{global} variable, called @code{Digits}. (For
1255 those readers who know about Maple: it behaves very much like Maple's
1256 @code{Digits}). All objects of class numeric that are constructed from
1257 then on will be stored with a precision matching that number of decimal
1262 #include <ginac/ginac.h>
1263 using namespace std;
1264 using namespace GiNaC;
1268 numeric three(3.0), one(1.0);
1269 numeric x = one/three;
1271 cout << "in " << Digits << " digits:" << endl;
1273 cout << Pi.evalf() << endl;
1285 The above example prints the following output to screen:
1289 0.33333333333333333334
1290 3.1415926535897932385
1292 0.33333333333333333333333333333333333333333333333333333333333333333334
1293 3.1415926535897932384626433832795028841971693993751058209749445923078
1297 Note that the last number is not necessarily rounded as you would
1298 naively expect it to be rounded in the decimal system. But note also,
1299 that in both cases you got a couple of extra digits. This is because
1300 numbers are internally stored by CLN as chunks of binary digits in order
1301 to match your machine's word size and to not waste precision. Thus, on
1302 architectures with different word size, the above output might even
1303 differ with regard to actually computed digits.
1305 It should be clear that objects of class @code{numeric} should be used
1306 for constructing numbers or for doing arithmetic with them. The objects
1307 one deals with most of the time are the polymorphic expressions @code{ex}.
1309 @subsection Tests on numbers
1311 Once you have declared some numbers, assigned them to expressions and
1312 done some arithmetic with them it is frequently desired to retrieve some
1313 kind of information from them like asking whether that number is
1314 integer, rational, real or complex. For those cases GiNaC provides
1315 several useful methods. (Internally, they fall back to invocations of
1316 certain CLN functions.)
1318 As an example, let's construct some rational number, multiply it with
1319 some multiple of its denominator and test what comes out:
1323 #include <ginac/ginac.h>
1324 using namespace std;
1325 using namespace GiNaC;
1327 // some very important constants:
1328 const numeric twentyone(21);
1329 const numeric ten(10);
1330 const numeric five(5);
1334 numeric answer = twentyone;
1337 cout << answer.is_integer() << endl; // false, it's 21/5
1339 cout << answer.is_integer() << endl; // true, it's 42 now!
1343 Note that the variable @code{answer} is constructed here as an integer
1344 by @code{numeric}'s copy constructor, but in an intermediate step it
1345 holds a rational number represented as integer numerator and integer
1346 denominator. When multiplied by 10, the denominator becomes unity and
1347 the result is automatically converted to a pure integer again.
1348 Internally, the underlying CLN is responsible for this behavior and we
1349 refer the reader to CLN's documentation. Suffice to say that
1350 the same behavior applies to complex numbers as well as return values of
1351 certain functions. Complex numbers are automatically converted to real
1352 numbers if the imaginary part becomes zero. The full set of tests that
1353 can be applied is listed in the following table.
1356 @multitable @columnfractions .30 .70
1357 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1358 @item @code{.is_zero()}
1359 @tab @dots{}equal to zero
1360 @item @code{.is_positive()}
1361 @tab @dots{}not complex and greater than 0
1362 @item @code{.is_negative()}
1363 @tab @dots{}not complex and smaller than 0
1364 @item @code{.is_integer()}
1365 @tab @dots{}a (non-complex) integer
1366 @item @code{.is_pos_integer()}
1367 @tab @dots{}an integer and greater than 0
1368 @item @code{.is_nonneg_integer()}
1369 @tab @dots{}an integer and greater equal 0
1370 @item @code{.is_even()}
1371 @tab @dots{}an even integer
1372 @item @code{.is_odd()}
1373 @tab @dots{}an odd integer
1374 @item @code{.is_prime()}
1375 @tab @dots{}a prime integer (probabilistic primality test)
1376 @item @code{.is_rational()}
1377 @tab @dots{}an exact rational number (integers are rational, too)
1378 @item @code{.is_real()}
1379 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1380 @item @code{.is_cinteger()}
1381 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1382 @item @code{.is_crational()}
1383 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1389 @subsection Numeric functions
1391 The following functions can be applied to @code{numeric} objects and will be
1392 evaluated immediately:
1395 @multitable @columnfractions .30 .70
1396 @item @strong{Name} @tab @strong{Function}
1397 @item @code{inverse(z)}
1398 @tab returns @math{1/z}
1399 @cindex @code{inverse()} (numeric)
1400 @item @code{pow(a, b)}
1401 @tab exponentiation @math{a^b}
1404 @item @code{real(z)}
1406 @cindex @code{real()}
1407 @item @code{imag(z)}
1409 @cindex @code{imag()}
1410 @item @code{csgn(z)}
1411 @tab complex sign (returns an @code{int})
1412 @item @code{step(x)}
1413 @tab step function (returns an @code{numeric})
1414 @item @code{numer(z)}
1415 @tab numerator of rational or complex rational number
1416 @item @code{denom(z)}
1417 @tab denominator of rational or complex rational number
1418 @item @code{sqrt(z)}
1420 @item @code{isqrt(n)}
1421 @tab integer square root
1422 @cindex @code{isqrt()}
1429 @item @code{asin(z)}
1431 @item @code{acos(z)}
1433 @item @code{atan(z)}
1434 @tab inverse tangent
1435 @item @code{atan(y, x)}
1436 @tab inverse tangent with two arguments
1437 @item @code{sinh(z)}
1438 @tab hyperbolic sine
1439 @item @code{cosh(z)}
1440 @tab hyperbolic cosine
1441 @item @code{tanh(z)}
1442 @tab hyperbolic tangent
1443 @item @code{asinh(z)}
1444 @tab inverse hyperbolic sine
1445 @item @code{acosh(z)}
1446 @tab inverse hyperbolic cosine
1447 @item @code{atanh(z)}
1448 @tab inverse hyperbolic tangent
1450 @tab exponential function
1452 @tab natural logarithm
1455 @item @code{zeta(z)}
1456 @tab Riemann's zeta function
1457 @item @code{tgamma(z)}
1459 @item @code{lgamma(z)}
1460 @tab logarithm of gamma function
1462 @tab psi (digamma) function
1463 @item @code{psi(n, z)}
1464 @tab derivatives of psi function (polygamma functions)
1465 @item @code{factorial(n)}
1466 @tab factorial function @math{n!}
1467 @item @code{doublefactorial(n)}
1468 @tab double factorial function @math{n!!}
1469 @cindex @code{doublefactorial()}
1470 @item @code{binomial(n, k)}
1471 @tab binomial coefficients
1472 @item @code{bernoulli(n)}
1473 @tab Bernoulli numbers
1474 @cindex @code{bernoulli()}
1475 @item @code{fibonacci(n)}
1476 @tab Fibonacci numbers
1477 @cindex @code{fibonacci()}
1478 @item @code{mod(a, b)}
1479 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1480 @cindex @code{mod()}
1481 @item @code{smod(a, b)}
1482 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1483 @cindex @code{smod()}
1484 @item @code{irem(a, b)}
1485 @tab integer remainder (has the sign of @math{a}, or is zero)
1486 @cindex @code{irem()}
1487 @item @code{irem(a, b, q)}
1488 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1489 @item @code{iquo(a, b)}
1490 @tab integer quotient
1491 @cindex @code{iquo()}
1492 @item @code{iquo(a, b, r)}
1493 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1494 @item @code{gcd(a, b)}
1495 @tab greatest common divisor
1496 @item @code{lcm(a, b)}
1497 @tab least common multiple
1501 Most of these functions are also available as symbolic functions that can be
1502 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1503 as polynomial algorithms.
1505 @subsection Converting numbers
1507 Sometimes it is desirable to convert a @code{numeric} object back to a
1508 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1509 class provides a couple of methods for this purpose:
1511 @cindex @code{to_int()}
1512 @cindex @code{to_long()}
1513 @cindex @code{to_double()}
1514 @cindex @code{to_cl_N()}
1516 int numeric::to_int() const;
1517 long numeric::to_long() const;
1518 double numeric::to_double() const;
1519 cln::cl_N numeric::to_cl_N() const;
1522 @code{to_int()} and @code{to_long()} only work when the number they are
1523 applied on is an exact integer. Otherwise the program will halt with a
1524 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1525 rational number will return a floating-point approximation. Both
1526 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1527 part of complex numbers.
1530 @node Constants, Fundamental containers, Numbers, Basic concepts
1531 @c node-name, next, previous, up
1533 @cindex @code{constant} (class)
1536 @cindex @code{Catalan}
1537 @cindex @code{Euler}
1538 @cindex @code{evalf()}
1539 Constants behave pretty much like symbols except that they return some
1540 specific number when the method @code{.evalf()} is called.
1542 The predefined known constants are:
1545 @multitable @columnfractions .14 .32 .54
1546 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1548 @tab Archimedes' constant
1549 @tab 3.14159265358979323846264338327950288
1550 @item @code{Catalan}
1551 @tab Catalan's constant
1552 @tab 0.91596559417721901505460351493238411
1554 @tab Euler's (or Euler-Mascheroni) constant
1555 @tab 0.57721566490153286060651209008240243
1560 @node Fundamental containers, Lists, Constants, Basic concepts
1561 @c node-name, next, previous, up
1562 @section Sums, products and powers
1566 @cindex @code{power}
1568 Simple rational expressions are written down in GiNaC pretty much like
1569 in other CAS or like expressions involving numerical variables in C.
1570 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1571 been overloaded to achieve this goal. When you run the following
1572 code snippet, the constructor for an object of type @code{mul} is
1573 automatically called to hold the product of @code{a} and @code{b} and
1574 then the constructor for an object of type @code{add} is called to hold
1575 the sum of that @code{mul} object and the number one:
1579 symbol a("a"), b("b");
1584 @cindex @code{pow()}
1585 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1586 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1587 construction is necessary since we cannot safely overload the constructor
1588 @code{^} in C++ to construct a @code{power} object. If we did, it would
1589 have several counterintuitive and undesired effects:
1593 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1595 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1596 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1597 interpret this as @code{x^(a^b)}.
1599 Also, expressions involving integer exponents are very frequently used,
1600 which makes it even more dangerous to overload @code{^} since it is then
1601 hard to distinguish between the semantics as exponentiation and the one
1602 for exclusive or. (It would be embarrassing to return @code{1} where one
1603 has requested @code{2^3}.)
1606 @cindex @command{ginsh}
1607 All effects are contrary to mathematical notation and differ from the
1608 way most other CAS handle exponentiation, therefore overloading @code{^}
1609 is ruled out for GiNaC's C++ part. The situation is different in
1610 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1611 that the other frequently used exponentiation operator @code{**} does
1612 not exist at all in C++).
1614 To be somewhat more precise, objects of the three classes described
1615 here, are all containers for other expressions. An object of class
1616 @code{power} is best viewed as a container with two slots, one for the
1617 basis, one for the exponent. All valid GiNaC expressions can be
1618 inserted. However, basic transformations like simplifying
1619 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1620 when this is mathematically possible. If we replace the outer exponent
1621 three in the example by some symbols @code{a}, the simplification is not
1622 safe and will not be performed, since @code{a} might be @code{1/2} and
1625 Objects of type @code{add} and @code{mul} are containers with an
1626 arbitrary number of slots for expressions to be inserted. Again, simple
1627 and safe simplifications are carried out like transforming
1628 @code{3*x+4-x} to @code{2*x+4}.
1631 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1632 @c node-name, next, previous, up
1633 @section Lists of expressions
1634 @cindex @code{lst} (class)
1636 @cindex @code{nops()}
1638 @cindex @code{append()}
1639 @cindex @code{prepend()}
1640 @cindex @code{remove_first()}
1641 @cindex @code{remove_last()}
1642 @cindex @code{remove_all()}
1644 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1645 expressions. They are not as ubiquitous as in many other computer algebra
1646 packages, but are sometimes used to supply a variable number of arguments of
1647 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1648 constructors, so you should have a basic understanding of them.
1650 Lists can be constructed by assigning a comma-separated sequence of
1655 symbol x("x"), y("y");
1658 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1663 There are also constructors that allow direct creation of lists of up to
1664 16 expressions, which is often more convenient but slightly less efficient:
1668 // This produces the same list 'l' as above:
1669 // lst l(x, 2, y, x+y);
1670 // lst l = lst(x, 2, y, x+y);
1674 Use the @code{nops()} method to determine the size (number of expressions) of
1675 a list and the @code{op()} method or the @code{[]} operator to access
1676 individual elements:
1680 cout << l.nops() << endl; // prints '4'
1681 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1685 As with the standard @code{list<T>} container, accessing random elements of a
1686 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1687 sequential access to the elements of a list is possible with the
1688 iterator types provided by the @code{lst} class:
1691 typedef ... lst::const_iterator;
1692 typedef ... lst::const_reverse_iterator;
1693 lst::const_iterator lst::begin() const;
1694 lst::const_iterator lst::end() const;
1695 lst::const_reverse_iterator lst::rbegin() const;
1696 lst::const_reverse_iterator lst::rend() const;
1699 For example, to print the elements of a list individually you can use:
1704 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1709 which is one order faster than
1714 for (size_t i = 0; i < l.nops(); ++i)
1715 cout << l.op(i) << endl;
1719 These iterators also allow you to use some of the algorithms provided by
1720 the C++ standard library:
1724 // print the elements of the list (requires #include <iterator>)
1725 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1727 // sum up the elements of the list (requires #include <numeric>)
1728 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1729 cout << sum << endl; // prints '2+2*x+2*y'
1733 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1734 (the only other one is @code{matrix}). You can modify single elements:
1738 l[1] = 42; // l is now @{x, 42, y, x+y@}
1739 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1743 You can append or prepend an expression to a list with the @code{append()}
1744 and @code{prepend()} methods:
1748 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1749 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1753 You can remove the first or last element of a list with @code{remove_first()}
1754 and @code{remove_last()}:
1758 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1759 l.remove_last(); // l is now @{x, 7, y, x+y@}
1763 You can remove all the elements of a list with @code{remove_all()}:
1767 l.remove_all(); // l is now empty
1771 You can bring the elements of a list into a canonical order with @code{sort()}:
1780 // l1 and l2 are now equal
1784 Finally, you can remove all but the first element of consecutive groups of
1785 elements with @code{unique()}:
1790 l3 = x, 2, 2, 2, y, x+y, y+x;
1791 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1796 @node Mathematical functions, Relations, Lists, Basic concepts
1797 @c node-name, next, previous, up
1798 @section Mathematical functions
1799 @cindex @code{function} (class)
1800 @cindex trigonometric function
1801 @cindex hyperbolic function
1803 There are quite a number of useful functions hard-wired into GiNaC. For
1804 instance, all trigonometric and hyperbolic functions are implemented
1805 (@xref{Built-in functions}, for a complete list).
1807 These functions (better called @emph{pseudofunctions}) are all objects
1808 of class @code{function}. They accept one or more expressions as
1809 arguments and return one expression. If the arguments are not
1810 numerical, the evaluation of the function may be halted, as it does in
1811 the next example, showing how a function returns itself twice and
1812 finally an expression that may be really useful:
1814 @cindex Gamma function
1815 @cindex @code{subs()}
1818 symbol x("x"), y("y");
1820 cout << tgamma(foo) << endl;
1821 // -> tgamma(x+(1/2)*y)
1822 ex bar = foo.subs(y==1);
1823 cout << tgamma(bar) << endl;
1825 ex foobar = bar.subs(x==7);
1826 cout << tgamma(foobar) << endl;
1827 // -> (135135/128)*Pi^(1/2)
1831 Besides evaluation most of these functions allow differentiation, series
1832 expansion and so on. Read the next chapter in order to learn more about
1835 It must be noted that these pseudofunctions are created by inline
1836 functions, where the argument list is templated. This means that
1837 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1838 @code{sin(ex(1))} and will therefore not result in a floating point
1839 number. Unless of course the function prototype is explicitly
1840 overridden -- which is the case for arguments of type @code{numeric}
1841 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1842 point number of class @code{numeric} you should call
1843 @code{sin(numeric(1))}. This is almost the same as calling
1844 @code{sin(1).evalf()} except that the latter will return a numeric
1845 wrapped inside an @code{ex}.
1848 @node Relations, Integrals, Mathematical functions, Basic concepts
1849 @c node-name, next, previous, up
1851 @cindex @code{relational} (class)
1853 Sometimes, a relation holding between two expressions must be stored
1854 somehow. The class @code{relational} is a convenient container for such
1855 purposes. A relation is by definition a container for two @code{ex} and
1856 a relation between them that signals equality, inequality and so on.
1857 They are created by simply using the C++ operators @code{==}, @code{!=},
1858 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1860 @xref{Mathematical functions}, for examples where various applications
1861 of the @code{.subs()} method show how objects of class relational are
1862 used as arguments. There they provide an intuitive syntax for
1863 substitutions. They are also used as arguments to the @code{ex::series}
1864 method, where the left hand side of the relation specifies the variable
1865 to expand in and the right hand side the expansion point. They can also
1866 be used for creating systems of equations that are to be solved for
1867 unknown variables. But the most common usage of objects of this class
1868 is rather inconspicuous in statements of the form @code{if
1869 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1870 conversion from @code{relational} to @code{bool} takes place. Note,
1871 however, that @code{==} here does not perform any simplifications, hence
1872 @code{expand()} must be called explicitly.
1874 @node Integrals, Matrices, Relations, Basic concepts
1875 @c node-name, next, previous, up
1877 @cindex @code{integral} (class)
1879 An object of class @dfn{integral} can be used to hold a symbolic integral.
1880 If you want to symbolically represent the integral of @code{x*x} from 0 to
1881 1, you would write this as
1883 integral(x, 0, 1, x*x)
1885 The first argument is the integration variable. It should be noted that
1886 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1887 fact, it can only integrate polynomials. An expression containing integrals
1888 can be evaluated symbolically by calling the
1892 method on it. Numerical evaluation is available by calling the
1896 method on an expression containing the integral. This will only evaluate
1897 integrals into a number if @code{subs}ing the integration variable by a
1898 number in the fourth argument of an integral and then @code{evalf}ing the
1899 result always results in a number. Of course, also the boundaries of the
1900 integration domain must @code{evalf} into numbers. It should be noted that
1901 trying to @code{evalf} a function with discontinuities in the integration
1902 domain is not recommended. The accuracy of the numeric evaluation of
1903 integrals is determined by the static member variable
1905 ex integral::relative_integration_error
1907 of the class @code{integral}. The default value of this is 10^-8.
1908 The integration works by halving the interval of integration, until numeric
1909 stability of the answer indicates that the requested accuracy has been
1910 reached. The maximum depth of the halving can be set via the static member
1913 int integral::max_integration_level
1915 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1916 return the integral unevaluated. The function that performs the numerical
1917 evaluation, is also available as
1919 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1922 This function will throw an exception if the maximum depth is exceeded. The
1923 last parameter of the function is optional and defaults to the
1924 @code{relative_integration_error}. To make sure that we do not do too
1925 much work if an expression contains the same integral multiple times,
1926 a lookup table is used.
1928 If you know that an expression holds an integral, you can get the
1929 integration variable, the left boundary, right boundary and integrand by
1930 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1931 @code{.op(3)}. Differentiating integrals with respect to variables works
1932 as expected. Note that it makes no sense to differentiate an integral
1933 with respect to the integration variable.
1935 @node Matrices, Indexed objects, Integrals, Basic concepts
1936 @c node-name, next, previous, up
1938 @cindex @code{matrix} (class)
1940 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1941 matrix with @math{m} rows and @math{n} columns are accessed with two
1942 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1943 second one in the range 0@dots{}@math{n-1}.
1945 There are a couple of ways to construct matrices, with or without preset
1946 elements. The constructor
1949 matrix::matrix(unsigned r, unsigned c);
1952 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1955 The fastest way to create a matrix with preinitialized elements is to assign
1956 a list of comma-separated expressions to an empty matrix (see below for an
1957 example). But you can also specify the elements as a (flat) list with
1960 matrix::matrix(unsigned r, unsigned c, const lst & l);
1965 @cindex @code{lst_to_matrix()}
1967 ex lst_to_matrix(const lst & l);
1970 constructs a matrix from a list of lists, each list representing a matrix row.
1972 There is also a set of functions for creating some special types of
1975 @cindex @code{diag_matrix()}
1976 @cindex @code{unit_matrix()}
1977 @cindex @code{symbolic_matrix()}
1979 ex diag_matrix(const lst & l);
1980 ex unit_matrix(unsigned x);
1981 ex unit_matrix(unsigned r, unsigned c);
1982 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1983 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1984 const string & tex_base_name);
1987 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1988 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1989 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1990 matrix filled with newly generated symbols made of the specified base name
1991 and the position of each element in the matrix.
1993 Matrices often arise by omitting elements of another matrix. For
1994 instance, the submatrix @code{S} of a matrix @code{M} takes a
1995 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1996 by removing one row and one column from a matrix @code{M}. (The
1997 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1998 can be used for computing the inverse using Cramer's rule.)
2000 @cindex @code{sub_matrix()}
2001 @cindex @code{reduced_matrix()}
2003 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2004 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2007 The function @code{sub_matrix()} takes a row offset @code{r} and a
2008 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2009 columns. The function @code{reduced_matrix()} has two integer arguments
2010 that specify which row and column to remove:
2018 cout << reduced_matrix(m, 1, 1) << endl;
2019 // -> [[11,13],[31,33]]
2020 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2021 // -> [[22,23],[32,33]]
2025 Matrix elements can be accessed and set using the parenthesis (function call)
2029 const ex & matrix::operator()(unsigned r, unsigned c) const;
2030 ex & matrix::operator()(unsigned r, unsigned c);
2033 It is also possible to access the matrix elements in a linear fashion with
2034 the @code{op()} method. But C++-style subscripting with square brackets
2035 @samp{[]} is not available.
2037 Here are a couple of examples for constructing matrices:
2041 symbol a("a"), b("b");
2055 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2058 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2061 cout << diag_matrix(lst(a, b)) << endl;
2064 cout << unit_matrix(3) << endl;
2065 // -> [[1,0,0],[0,1,0],[0,0,1]]
2067 cout << symbolic_matrix(2, 3, "x") << endl;
2068 // -> [[x00,x01,x02],[x10,x11,x12]]
2072 @cindex @code{is_zero_matrix()}
2073 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2074 all entries of the matrix are zeros. There is also method
2075 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2076 expression is zero or a zero matrix.
2078 @cindex @code{transpose()}
2079 There are three ways to do arithmetic with matrices. The first (and most
2080 direct one) is to use the methods provided by the @code{matrix} class:
2083 matrix matrix::add(const matrix & other) const;
2084 matrix matrix::sub(const matrix & other) const;
2085 matrix matrix::mul(const matrix & other) const;
2086 matrix matrix::mul_scalar(const ex & other) const;
2087 matrix matrix::pow(const ex & expn) const;
2088 matrix matrix::transpose() const;
2091 All of these methods return the result as a new matrix object. Here is an
2092 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2097 matrix A(2, 2), B(2, 2), C(2, 2);
2105 matrix result = A.mul(B).sub(C.mul_scalar(2));
2106 cout << result << endl;
2107 // -> [[-13,-6],[1,2]]
2112 @cindex @code{evalm()}
2113 The second (and probably the most natural) way is to construct an expression
2114 containing matrices with the usual arithmetic operators and @code{pow()}.
2115 For efficiency reasons, expressions with sums, products and powers of
2116 matrices are not automatically evaluated in GiNaC. You have to call the
2120 ex ex::evalm() const;
2123 to obtain the result:
2130 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2131 cout << e.evalm() << endl;
2132 // -> [[-13,-6],[1,2]]
2137 The non-commutativity of the product @code{A*B} in this example is
2138 automatically recognized by GiNaC. There is no need to use a special
2139 operator here. @xref{Non-commutative objects}, for more information about
2140 dealing with non-commutative expressions.
2142 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2143 to perform the arithmetic:
2148 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2149 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2151 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2152 cout << e.simplify_indexed() << endl;
2153 // -> [[-13,-6],[1,2]].i.j
2157 Using indices is most useful when working with rectangular matrices and
2158 one-dimensional vectors because you don't have to worry about having to
2159 transpose matrices before multiplying them. @xref{Indexed objects}, for
2160 more information about using matrices with indices, and about indices in
2163 The @code{matrix} class provides a couple of additional methods for
2164 computing determinants, traces, characteristic polynomials and ranks:
2166 @cindex @code{determinant()}
2167 @cindex @code{trace()}
2168 @cindex @code{charpoly()}
2169 @cindex @code{rank()}
2171 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2172 ex matrix::trace() const;
2173 ex matrix::charpoly(const ex & lambda) const;
2174 unsigned matrix::rank() const;
2177 The @samp{algo} argument of @code{determinant()} allows to select
2178 between different algorithms for calculating the determinant. The
2179 asymptotic speed (as parametrized by the matrix size) can greatly differ
2180 between those algorithms, depending on the nature of the matrix'
2181 entries. The possible values are defined in the @file{flags.h} header
2182 file. By default, GiNaC uses a heuristic to automatically select an
2183 algorithm that is likely (but not guaranteed) to give the result most
2186 @cindex @code{inverse()} (matrix)
2187 @cindex @code{solve()}
2188 Matrices may also be inverted using the @code{ex matrix::inverse()}
2189 method and linear systems may be solved with:
2192 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2193 unsigned algo=solve_algo::automatic) const;
2196 Assuming the matrix object this method is applied on is an @code{m}
2197 times @code{n} matrix, then @code{vars} must be a @code{n} times
2198 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2199 times @code{p} matrix. The returned matrix then has dimension @code{n}
2200 times @code{p} and in the case of an underdetermined system will still
2201 contain some of the indeterminates from @code{vars}. If the system is
2202 overdetermined, an exception is thrown.
2205 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2206 @c node-name, next, previous, up
2207 @section Indexed objects
2209 GiNaC allows you to handle expressions containing general indexed objects in
2210 arbitrary spaces. It is also able to canonicalize and simplify such
2211 expressions and perform symbolic dummy index summations. There are a number
2212 of predefined indexed objects provided, like delta and metric tensors.
2214 There are few restrictions placed on indexed objects and their indices and
2215 it is easy to construct nonsense expressions, but our intention is to
2216 provide a general framework that allows you to implement algorithms with
2217 indexed quantities, getting in the way as little as possible.
2219 @cindex @code{idx} (class)
2220 @cindex @code{indexed} (class)
2221 @subsection Indexed quantities and their indices
2223 Indexed expressions in GiNaC are constructed of two special types of objects,
2224 @dfn{index objects} and @dfn{indexed objects}.
2228 @cindex contravariant
2231 @item Index objects are of class @code{idx} or a subclass. Every index has
2232 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2233 the index lives in) which can both be arbitrary expressions but are usually
2234 a number or a simple symbol. In addition, indices of class @code{varidx} have
2235 a @dfn{variance} (they can be co- or contravariant), and indices of class
2236 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2238 @item Indexed objects are of class @code{indexed} or a subclass. They
2239 contain a @dfn{base expression} (which is the expression being indexed), and
2240 one or more indices.
2244 @strong{Please notice:} when printing expressions, covariant indices and indices
2245 without variance are denoted @samp{.i} while contravariant indices are
2246 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2247 value. In the following, we are going to use that notation in the text so
2248 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2249 not visible in the output.
2251 A simple example shall illustrate the concepts:
2255 #include <ginac/ginac.h>
2256 using namespace std;
2257 using namespace GiNaC;
2261 symbol i_sym("i"), j_sym("j");
2262 idx i(i_sym, 3), j(j_sym, 3);
2265 cout << indexed(A, i, j) << endl;
2267 cout << index_dimensions << indexed(A, i, j) << endl;
2269 cout << dflt; // reset cout to default output format (dimensions hidden)
2273 The @code{idx} constructor takes two arguments, the index value and the
2274 index dimension. First we define two index objects, @code{i} and @code{j},
2275 both with the numeric dimension 3. The value of the index @code{i} is the
2276 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2277 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2278 construct an expression containing one indexed object, @samp{A.i.j}. It has
2279 the symbol @code{A} as its base expression and the two indices @code{i} and
2282 The dimensions of indices are normally not visible in the output, but one
2283 can request them to be printed with the @code{index_dimensions} manipulator,
2286 Note the difference between the indices @code{i} and @code{j} which are of
2287 class @code{idx}, and the index values which are the symbols @code{i_sym}
2288 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2289 or numbers but must be index objects. For example, the following is not
2290 correct and will raise an exception:
2293 symbol i("i"), j("j");
2294 e = indexed(A, i, j); // ERROR: indices must be of type idx
2297 You can have multiple indexed objects in an expression, index values can
2298 be numeric, and index dimensions symbolic:
2302 symbol B("B"), dim("dim");
2303 cout << 4 * indexed(A, i)
2304 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2309 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2310 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2311 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2312 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2313 @code{simplify_indexed()} for that, see below).
2315 In fact, base expressions, index values and index dimensions can be
2316 arbitrary expressions:
2320 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2325 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2326 get an error message from this but you will probably not be able to do
2327 anything useful with it.
2329 @cindex @code{get_value()}
2330 @cindex @code{get_dimension()}
2334 ex idx::get_value();
2335 ex idx::get_dimension();
2338 return the value and dimension of an @code{idx} object. If you have an index
2339 in an expression, such as returned by calling @code{.op()} on an indexed
2340 object, you can get a reference to the @code{idx} object with the function
2341 @code{ex_to<idx>()} on the expression.
2343 There are also the methods
2346 bool idx::is_numeric();
2347 bool idx::is_symbolic();
2348 bool idx::is_dim_numeric();
2349 bool idx::is_dim_symbolic();
2352 for checking whether the value and dimension are numeric or symbolic
2353 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2354 about expressions}) returns information about the index value.
2356 @cindex @code{varidx} (class)
2357 If you need co- and contravariant indices, use the @code{varidx} class:
2361 symbol mu_sym("mu"), nu_sym("nu");
2362 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2363 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2365 cout << indexed(A, mu, nu) << endl;
2367 cout << indexed(A, mu_co, nu) << endl;
2369 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2374 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2375 co- or contravariant. The default is a contravariant (upper) index, but
2376 this can be overridden by supplying a third argument to the @code{varidx}
2377 constructor. The two methods
2380 bool varidx::is_covariant();
2381 bool varidx::is_contravariant();
2384 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2385 to get the object reference from an expression). There's also the very useful
2389 ex varidx::toggle_variance();
2392 which makes a new index with the same value and dimension but the opposite
2393 variance. By using it you only have to define the index once.
2395 @cindex @code{spinidx} (class)
2396 The @code{spinidx} class provides dotted and undotted variant indices, as
2397 used in the Weyl-van-der-Waerden spinor formalism:
2401 symbol K("K"), C_sym("C"), D_sym("D");
2402 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2403 // contravariant, undotted
2404 spinidx C_co(C_sym, 2, true); // covariant index
2405 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2406 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2408 cout << indexed(K, C, D) << endl;
2410 cout << indexed(K, C_co, D_dot) << endl;
2412 cout << indexed(K, D_co_dot, D) << endl;
2417 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2418 dotted or undotted. The default is undotted but this can be overridden by
2419 supplying a fourth argument to the @code{spinidx} constructor. The two
2423 bool spinidx::is_dotted();
2424 bool spinidx::is_undotted();
2427 allow you to check whether or not a @code{spinidx} object is dotted (use
2428 @code{ex_to<spinidx>()} to get the object reference from an expression).
2429 Finally, the two methods
2432 ex spinidx::toggle_dot();
2433 ex spinidx::toggle_variance_dot();
2436 create a new index with the same value and dimension but opposite dottedness
2437 and the same or opposite variance.
2439 @subsection Substituting indices
2441 @cindex @code{subs()}
2442 Sometimes you will want to substitute one symbolic index with another
2443 symbolic or numeric index, for example when calculating one specific element
2444 of a tensor expression. This is done with the @code{.subs()} method, as it
2445 is done for symbols (see @ref{Substituting expressions}).
2447 You have two possibilities here. You can either substitute the whole index
2448 by another index or expression:
2452 ex e = indexed(A, mu_co);
2453 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2454 // -> A.mu becomes A~nu
2455 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2456 // -> A.mu becomes A~0
2457 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2458 // -> A.mu becomes A.0
2462 The third example shows that trying to replace an index with something that
2463 is not an index will substitute the index value instead.
2465 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2470 ex e = indexed(A, mu_co);
2471 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2472 // -> A.mu becomes A.nu
2473 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2474 // -> A.mu becomes A.0
2478 As you see, with the second method only the value of the index will get
2479 substituted. Its other properties, including its dimension, remain unchanged.
2480 If you want to change the dimension of an index you have to substitute the
2481 whole index by another one with the new dimension.
2483 Finally, substituting the base expression of an indexed object works as
2488 ex e = indexed(A, mu_co);
2489 cout << e << " becomes " << e.subs(A == A+B) << endl;
2490 // -> A.mu becomes (B+A).mu
2494 @subsection Symmetries
2495 @cindex @code{symmetry} (class)
2496 @cindex @code{sy_none()}
2497 @cindex @code{sy_symm()}
2498 @cindex @code{sy_anti()}
2499 @cindex @code{sy_cycl()}
2501 Indexed objects can have certain symmetry properties with respect to their
2502 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2503 that is constructed with the helper functions
2506 symmetry sy_none(...);
2507 symmetry sy_symm(...);
2508 symmetry sy_anti(...);
2509 symmetry sy_cycl(...);
2512 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2513 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2514 represents a cyclic symmetry. Each of these functions accepts up to four
2515 arguments which can be either symmetry objects themselves or unsigned integer
2516 numbers that represent an index position (counting from 0). A symmetry
2517 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2518 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2521 Here are some examples of symmetry definitions:
2526 e = indexed(A, i, j);
2527 e = indexed(A, sy_none(), i, j); // equivalent
2528 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2530 // Symmetric in all three indices:
2531 e = indexed(A, sy_symm(), i, j, k);
2532 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2533 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2534 // different canonical order
2536 // Symmetric in the first two indices only:
2537 e = indexed(A, sy_symm(0, 1), i, j, k);
2538 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2540 // Antisymmetric in the first and last index only (index ranges need not
2542 e = indexed(A, sy_anti(0, 2), i, j, k);
2543 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2545 // An example of a mixed symmetry: antisymmetric in the first two and
2546 // last two indices, symmetric when swapping the first and last index
2547 // pairs (like the Riemann curvature tensor):
2548 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2550 // Cyclic symmetry in all three indices:
2551 e = indexed(A, sy_cycl(), i, j, k);
2552 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2554 // The following examples are invalid constructions that will throw
2555 // an exception at run time.
2557 // An index may not appear multiple times:
2558 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2559 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2561 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2562 // same number of indices:
2563 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2565 // And of course, you cannot specify indices which are not there:
2566 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2570 If you need to specify more than four indices, you have to use the
2571 @code{.add()} method of the @code{symmetry} class. For example, to specify
2572 full symmetry in the first six indices you would write
2573 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2575 If an indexed object has a symmetry, GiNaC will automatically bring the
2576 indices into a canonical order which allows for some immediate simplifications:
2580 cout << indexed(A, sy_symm(), i, j)
2581 + indexed(A, sy_symm(), j, i) << endl;
2583 cout << indexed(B, sy_anti(), i, j)
2584 + indexed(B, sy_anti(), j, i) << endl;
2586 cout << indexed(B, sy_anti(), i, j, k)
2587 - indexed(B, sy_anti(), j, k, i) << endl;
2592 @cindex @code{get_free_indices()}
2594 @subsection Dummy indices
2596 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2597 that a summation over the index range is implied. Symbolic indices which are
2598 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2599 dummy nor free indices.
2601 To be recognized as a dummy index pair, the two indices must be of the same
2602 class and their value must be the same single symbol (an index like
2603 @samp{2*n+1} is never a dummy index). If the indices are of class
2604 @code{varidx} they must also be of opposite variance; if they are of class
2605 @code{spinidx} they must be both dotted or both undotted.
2607 The method @code{.get_free_indices()} returns a vector containing the free
2608 indices of an expression. It also checks that the free indices of the terms
2609 of a sum are consistent:
2613 symbol A("A"), B("B"), C("C");
2615 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2616 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2618 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2619 cout << exprseq(e.get_free_indices()) << endl;
2621 // 'j' and 'l' are dummy indices
2623 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2624 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2626 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2627 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2628 cout << exprseq(e.get_free_indices()) << endl;
2630 // 'nu' is a dummy index, but 'sigma' is not
2632 e = indexed(A, mu, mu);
2633 cout << exprseq(e.get_free_indices()) << endl;
2635 // 'mu' is not a dummy index because it appears twice with the same
2638 e = indexed(A, mu, nu) + 42;
2639 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2640 // this will throw an exception:
2641 // "add::get_free_indices: inconsistent indices in sum"
2645 @cindex @code{expand_dummy_sum()}
2646 A dummy index summation like
2653 can be expanded for indices with numeric
2654 dimensions (e.g. 3) into the explicit sum like
2656 $a_1b^1+a_2b^2+a_3b^3 $.
2659 a.1 b~1 + a.2 b~2 + a.3 b~3.
2661 This is performed by the function
2664 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2667 which takes an expression @code{e} and returns the expanded sum for all
2668 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2669 is set to @code{true} then all substitutions are made by @code{idx} class
2670 indices, i.e. without variance. In this case the above sum
2679 $a_1b_1+a_2b_2+a_3b_3 $.
2682 a.1 b.1 + a.2 b.2 + a.3 b.3.
2686 @cindex @code{simplify_indexed()}
2687 @subsection Simplifying indexed expressions
2689 In addition to the few automatic simplifications that GiNaC performs on
2690 indexed expressions (such as re-ordering the indices of symmetric tensors
2691 and calculating traces and convolutions of matrices and predefined tensors)
2695 ex ex::simplify_indexed();
2696 ex ex::simplify_indexed(const scalar_products & sp);
2699 that performs some more expensive operations:
2702 @item it checks the consistency of free indices in sums in the same way
2703 @code{get_free_indices()} does
2704 @item it tries to give dummy indices that appear in different terms of a sum
2705 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2706 @item it (symbolically) calculates all possible dummy index summations/contractions
2707 with the predefined tensors (this will be explained in more detail in the
2709 @item it detects contractions that vanish for symmetry reasons, for example
2710 the contraction of a symmetric and a totally antisymmetric tensor
2711 @item as a special case of dummy index summation, it can replace scalar products
2712 of two tensors with a user-defined value
2715 The last point is done with the help of the @code{scalar_products} class
2716 which is used to store scalar products with known values (this is not an
2717 arithmetic class, you just pass it to @code{simplify_indexed()}):
2721 symbol A("A"), B("B"), C("C"), i_sym("i");
2725 sp.add(A, B, 0); // A and B are orthogonal
2726 sp.add(A, C, 0); // A and C are orthogonal
2727 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2729 e = indexed(A + B, i) * indexed(A + C, i);
2731 // -> (B+A).i*(A+C).i
2733 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2739 The @code{scalar_products} object @code{sp} acts as a storage for the
2740 scalar products added to it with the @code{.add()} method. This method
2741 takes three arguments: the two expressions of which the scalar product is
2742 taken, and the expression to replace it with.
2744 @cindex @code{expand()}
2745 The example above also illustrates a feature of the @code{expand()} method:
2746 if passed the @code{expand_indexed} option it will distribute indices
2747 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2749 @cindex @code{tensor} (class)
2750 @subsection Predefined tensors
2752 Some frequently used special tensors such as the delta, epsilon and metric
2753 tensors are predefined in GiNaC. They have special properties when
2754 contracted with other tensor expressions and some of them have constant
2755 matrix representations (they will evaluate to a number when numeric
2756 indices are specified).
2758 @cindex @code{delta_tensor()}
2759 @subsubsection Delta tensor
2761 The delta tensor takes two indices, is symmetric and has the matrix
2762 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2763 @code{delta_tensor()}:
2767 symbol A("A"), B("B");
2769 idx i(symbol("i"), 3), j(symbol("j"), 3),
2770 k(symbol("k"), 3), l(symbol("l"), 3);
2772 ex e = indexed(A, i, j) * indexed(B, k, l)
2773 * delta_tensor(i, k) * delta_tensor(j, l);
2774 cout << e.simplify_indexed() << endl;
2777 cout << delta_tensor(i, i) << endl;
2782 @cindex @code{metric_tensor()}
2783 @subsubsection General metric tensor
2785 The function @code{metric_tensor()} creates a general symmetric metric
2786 tensor with two indices that can be used to raise/lower tensor indices. The
2787 metric tensor is denoted as @samp{g} in the output and if its indices are of
2788 mixed variance it is automatically replaced by a delta tensor:
2794 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2796 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2797 cout << e.simplify_indexed() << endl;
2800 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2801 cout << e.simplify_indexed() << endl;
2804 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2805 * metric_tensor(nu, rho);
2806 cout << e.simplify_indexed() << endl;
2809 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2810 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2811 + indexed(A, mu.toggle_variance(), rho));
2812 cout << e.simplify_indexed() << endl;
2817 @cindex @code{lorentz_g()}
2818 @subsubsection Minkowski metric tensor
2820 The Minkowski metric tensor is a special metric tensor with a constant
2821 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2822 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2823 It is created with the function @code{lorentz_g()} (although it is output as
2828 varidx mu(symbol("mu"), 4);
2830 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2831 * lorentz_g(mu, varidx(0, 4)); // negative signature
2832 cout << e.simplify_indexed() << endl;
2835 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2836 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2837 cout << e.simplify_indexed() << endl;
2842 @cindex @code{spinor_metric()}
2843 @subsubsection Spinor metric tensor
2845 The function @code{spinor_metric()} creates an antisymmetric tensor with
2846 two indices that is used to raise/lower indices of 2-component spinors.
2847 It is output as @samp{eps}:
2853 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2854 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2856 e = spinor_metric(A, B) * indexed(psi, B_co);
2857 cout << e.simplify_indexed() << endl;
2860 e = spinor_metric(A, B) * indexed(psi, A_co);
2861 cout << e.simplify_indexed() << endl;
2864 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2865 cout << e.simplify_indexed() << endl;
2868 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2869 cout << e.simplify_indexed() << endl;
2872 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2873 cout << e.simplify_indexed() << endl;
2876 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2877 cout << e.simplify_indexed() << endl;
2882 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2884 @cindex @code{epsilon_tensor()}
2885 @cindex @code{lorentz_eps()}
2886 @subsubsection Epsilon tensor
2888 The epsilon tensor is totally antisymmetric, its number of indices is equal
2889 to the dimension of the index space (the indices must all be of the same
2890 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2891 defined to be 1. Its behavior with indices that have a variance also
2892 depends on the signature of the metric. Epsilon tensors are output as
2895 There are three functions defined to create epsilon tensors in 2, 3 and 4
2899 ex epsilon_tensor(const ex & i1, const ex & i2);
2900 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2901 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2902 bool pos_sig = false);
2905 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2906 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2907 Minkowski space (the last @code{bool} argument specifies whether the metric
2908 has negative or positive signature, as in the case of the Minkowski metric
2913 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2914 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2915 e = lorentz_eps(mu, nu, rho, sig) *
2916 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2917 cout << simplify_indexed(e) << endl;
2918 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2920 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2921 symbol A("A"), B("B");
2922 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2923 cout << simplify_indexed(e) << endl;
2924 // -> -B.k*A.j*eps.i.k.j
2925 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2926 cout << simplify_indexed(e) << endl;
2931 @subsection Linear algebra
2933 The @code{matrix} class can be used with indices to do some simple linear
2934 algebra (linear combinations and products of vectors and matrices, traces
2935 and scalar products):
2939 idx i(symbol("i"), 2), j(symbol("j"), 2);
2940 symbol x("x"), y("y");
2942 // A is a 2x2 matrix, X is a 2x1 vector
2943 matrix A(2, 2), X(2, 1);
2948 cout << indexed(A, i, i) << endl;
2951 ex e = indexed(A, i, j) * indexed(X, j);
2952 cout << e.simplify_indexed() << endl;
2953 // -> [[2*y+x],[4*y+3*x]].i
2955 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2956 cout << e.simplify_indexed() << endl;
2957 // -> [[3*y+3*x,6*y+2*x]].j
2961 You can of course obtain the same results with the @code{matrix::add()},
2962 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2963 but with indices you don't have to worry about transposing matrices.
2965 Matrix indices always start at 0 and their dimension must match the number
2966 of rows/columns of the matrix. Matrices with one row or one column are
2967 vectors and can have one or two indices (it doesn't matter whether it's a
2968 row or a column vector). Other matrices must have two indices.
2970 You should be careful when using indices with variance on matrices. GiNaC
2971 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2972 @samp{F.mu.nu} are different matrices. In this case you should use only
2973 one form for @samp{F} and explicitly multiply it with a matrix representation
2974 of the metric tensor.
2977 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2978 @c node-name, next, previous, up
2979 @section Non-commutative objects
2981 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2982 non-commutative objects are built-in which are mostly of use in high energy
2986 @item Clifford (Dirac) algebra (class @code{clifford})
2987 @item su(3) Lie algebra (class @code{color})
2988 @item Matrices (unindexed) (class @code{matrix})
2991 The @code{clifford} and @code{color} classes are subclasses of
2992 @code{indexed} because the elements of these algebras usually carry
2993 indices. The @code{matrix} class is described in more detail in
2996 Unlike most computer algebra systems, GiNaC does not primarily provide an
2997 operator (often denoted @samp{&*}) for representing inert products of
2998 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2999 classes of objects involved, and non-commutative products are formed with
3000 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3001 figuring out by itself which objects commutate and will group the factors
3002 by their class. Consider this example:
3006 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3007 idx a(symbol("a"), 8), b(symbol("b"), 8);
3008 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3010 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3014 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3015 groups the non-commutative factors (the gammas and the su(3) generators)
3016 together while preserving the order of factors within each class (because
3017 Clifford objects commutate with color objects). The resulting expression is a
3018 @emph{commutative} product with two factors that are themselves non-commutative
3019 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3020 parentheses are placed around the non-commutative products in the output.
3022 @cindex @code{ncmul} (class)
3023 Non-commutative products are internally represented by objects of the class
3024 @code{ncmul}, as opposed to commutative products which are handled by the
3025 @code{mul} class. You will normally not have to worry about this distinction,
3028 The advantage of this approach is that you never have to worry about using
3029 (or forgetting to use) a special operator when constructing non-commutative
3030 expressions. Also, non-commutative products in GiNaC are more intelligent
3031 than in other computer algebra systems; they can, for example, automatically
3032 canonicalize themselves according to rules specified in the implementation
3033 of the non-commutative classes. The drawback is that to work with other than
3034 the built-in algebras you have to implement new classes yourself. Both
3035 symbols and user-defined functions can be specified as being non-commutative.
3037 @cindex @code{return_type()}
3038 @cindex @code{return_type_tinfo()}
3039 Information about the commutativity of an object or expression can be
3040 obtained with the two member functions
3043 unsigned ex::return_type() const;
3044 unsigned ex::return_type_tinfo() const;
3047 The @code{return_type()} function returns one of three values (defined in
3048 the header file @file{flags.h}), corresponding to three categories of
3049 expressions in GiNaC:
3052 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3053 classes are of this kind.
3054 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3055 certain class of non-commutative objects which can be determined with the
3056 @code{return_type_tinfo()} method. Expressions of this category commutate
3057 with everything except @code{noncommutative} expressions of the same
3059 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3060 of non-commutative objects of different classes. Expressions of this
3061 category don't commutate with any other @code{noncommutative} or
3062 @code{noncommutative_composite} expressions.
3065 The value returned by the @code{return_type_tinfo()} method is valid only
3066 when the return type of the expression is @code{noncommutative}. It is a
3067 value that is unique to the class of the object, but may vary every time a
3068 GiNaC program is being run (it is dynamically assigned on start-up).
3070 Here are a couple of examples:
3073 @multitable @columnfractions 0.33 0.33 0.34
3074 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3075 @item @code{42} @tab @code{commutative} @tab -
3076 @item @code{2*x-y} @tab @code{commutative} @tab -
3077 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3078 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3079 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3080 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3084 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3085 @code{TINFO_clifford} for objects with a representation label of zero.
3086 Other representation labels yield a different @code{return_type_tinfo()},
3087 but it's the same for any two objects with the same label. This is also true
3090 A last note: With the exception of matrices, positive integer powers of
3091 non-commutative objects are automatically expanded in GiNaC. For example,
3092 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3093 non-commutative expressions).
3096 @cindex @code{clifford} (class)
3097 @subsection Clifford algebra
3100 Clifford algebras are supported in two flavours: Dirac gamma
3101 matrices (more physical) and generic Clifford algebras (more
3104 @cindex @code{dirac_gamma()}
3105 @subsubsection Dirac gamma matrices
3106 Dirac gamma matrices (note that GiNaC doesn't treat them
3107 as matrices) are designated as @samp{gamma~mu} and satisfy
3108 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3109 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3110 constructed by the function
3113 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3116 which takes two arguments: the index and a @dfn{representation label} in the
3117 range 0 to 255 which is used to distinguish elements of different Clifford
3118 algebras (this is also called a @dfn{spin line index}). Gammas with different
3119 labels commutate with each other. The dimension of the index can be 4 or (in
3120 the framework of dimensional regularization) any symbolic value. Spinor
3121 indices on Dirac gammas are not supported in GiNaC.
3123 @cindex @code{dirac_ONE()}
3124 The unity element of a Clifford algebra is constructed by
3127 ex dirac_ONE(unsigned char rl = 0);
3130 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3131 multiples of the unity element, even though it's customary to omit it.
3132 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3133 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3134 GiNaC will complain and/or produce incorrect results.
3136 @cindex @code{dirac_gamma5()}
3137 There is a special element @samp{gamma5} that commutates with all other
3138 gammas, has a unit square, and in 4 dimensions equals
3139 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3142 ex dirac_gamma5(unsigned char rl = 0);
3145 @cindex @code{dirac_gammaL()}
3146 @cindex @code{dirac_gammaR()}
3147 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3148 objects, constructed by
3151 ex dirac_gammaL(unsigned char rl = 0);
3152 ex dirac_gammaR(unsigned char rl = 0);
3155 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3156 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3158 @cindex @code{dirac_slash()}
3159 Finally, the function
3162 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3165 creates a term that represents a contraction of @samp{e} with the Dirac
3166 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3167 with a unique index whose dimension is given by the @code{dim} argument).
3168 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3170 In products of dirac gammas, superfluous unity elements are automatically
3171 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3172 and @samp{gammaR} are moved to the front.
3174 The @code{simplify_indexed()} function performs contractions in gamma strings,
3180 symbol a("a"), b("b"), D("D");
3181 varidx mu(symbol("mu"), D);
3182 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3183 * dirac_gamma(mu.toggle_variance());
3185 // -> gamma~mu*a\*gamma.mu
3186 e = e.simplify_indexed();
3189 cout << e.subs(D == 4) << endl;
3195 @cindex @code{dirac_trace()}
3196 To calculate the trace of an expression containing strings of Dirac gammas
3197 you use one of the functions
3200 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3201 const ex & trONE = 4);
3202 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3203 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3206 These functions take the trace over all gammas in the specified set @code{rls}
3207 or list @code{rll} of representation labels, or the single label @code{rl};
3208 gammas with other labels are left standing. The last argument to
3209 @code{dirac_trace()} is the value to be returned for the trace of the unity
3210 element, which defaults to 4.
3212 The @code{dirac_trace()} function is a linear functional that is equal to the
3213 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3214 functional is not cyclic in
3220 dimensions when acting on
3221 expressions containing @samp{gamma5}, so it's not a proper trace. This
3222 @samp{gamma5} scheme is described in greater detail in the article
3223 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3225 The value of the trace itself is also usually different in 4 and in
3236 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3237 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3238 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3239 cout << dirac_trace(e).simplify_indexed() << endl;
3246 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3247 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3248 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3249 cout << dirac_trace(e).simplify_indexed() << endl;
3250 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3254 Here is an example for using @code{dirac_trace()} to compute a value that
3255 appears in the calculation of the one-loop vacuum polarization amplitude in
3260 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3261 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3264 sp.add(l, l, pow(l, 2));
3265 sp.add(l, q, ldotq);
3267 ex e = dirac_gamma(mu) *
3268 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3269 dirac_gamma(mu.toggle_variance()) *
3270 (dirac_slash(l, D) + m * dirac_ONE());
3271 e = dirac_trace(e).simplify_indexed(sp);
3272 e = e.collect(lst(l, ldotq, m));
3274 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3278 The @code{canonicalize_clifford()} function reorders all gamma products that
3279 appear in an expression to a canonical (but not necessarily simple) form.
3280 You can use this to compare two expressions or for further simplifications:
3284 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3285 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3287 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3289 e = canonicalize_clifford(e);
3291 // -> 2*ONE*eta~mu~nu
3295 @cindex @code{clifford_unit()}
3296 @subsubsection A generic Clifford algebra
3298 A generic Clifford algebra, i.e. a
3304 dimensional algebra with
3311 satisfying the identities
3313 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3316 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3318 for some bilinear form (@code{metric})
3319 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3320 and contain symbolic entries. Such generators are created by the
3324 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3327 where @code{mu} should be a @code{idx} (or descendant) class object
3328 indexing the generators.
3329 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3330 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3331 object. In fact, any expression either with two free indices or without
3332 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3333 object with two newly created indices with @code{metr} as its
3334 @code{op(0)} will be used.
3335 Optional parameter @code{rl} allows to distinguish different
3336 Clifford algebras, which will commute with each other.
3338 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3339 something very close to @code{dirac_gamma(mu)}, although
3340 @code{dirac_gamma} have more efficient simplification mechanism.
3341 @cindex @code{clifford::get_metric()}
3342 The method @code{clifford::get_metric()} returns a metric defining this
3345 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3346 the Clifford algebra units with a call like that
3349 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3352 since this may yield some further automatic simplifications. Again, for a
3353 metric defined through a @code{matrix} such a symmetry is detected
3356 Individual generators of a Clifford algebra can be accessed in several
3362 idx i(symbol("i"), 4);
3364 ex M = diag_matrix(lst(1, -1, 0, s));
3365 ex e = clifford_unit(i, M);
3366 ex e0 = e.subs(i == 0);
3367 ex e1 = e.subs(i == 1);
3368 ex e2 = e.subs(i == 2);
3369 ex e3 = e.subs(i == 3);
3374 will produce four anti-commuting generators of a Clifford algebra with properties
3376 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3379 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3380 @code{pow(e3, 2) = s}.
3383 @cindex @code{lst_to_clifford()}
3384 A similar effect can be achieved from the function
3387 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3388 unsigned char rl = 0);
3389 ex lst_to_clifford(const ex & v, const ex & e);
3392 which converts a list or vector
3394 $v = (v^0, v^1, ..., v^n)$
3397 @samp{v = (v~0, v~1, ..., v~n)}
3402 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3405 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3408 directly supplied in the second form of the procedure. In the first form
3409 the Clifford unit @samp{e.k} is generated by the call of
3410 @code{clifford_unit(mu, metr, rl)}.
3411 @cindex pseudo-vector
3412 If the number of components supplied
3413 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3414 1 then function @code{lst_to_clifford()} uses the following
3415 pseudo-vector representation:
3417 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3420 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3423 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3428 idx i(symbol("i"), 4);
3430 ex M = diag_matrix(lst(1, -1, 0, s));
3431 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3432 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3433 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3434 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3439 @cindex @code{clifford_to_lst()}
3440 There is the inverse function
3443 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3446 which takes an expression @code{e} and tries to find a list
3448 $v = (v^0, v^1, ..., v^n)$
3451 @samp{v = (v~0, v~1, ..., v~n)}
3453 such that the expression is either vector
3455 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3458 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3462 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3465 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3467 with respect to the given Clifford units @code{c}. Here none of the
3468 @samp{v~k} should contain Clifford units @code{c} (of course, this
3469 may be impossible). This function can use an @code{algebraic} method
3470 (default) or a symbolic one. With the @code{algebraic} method the
3471 @samp{v~k} are calculated as
3473 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3476 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3478 is zero or is not @code{numeric} for some @samp{k}
3479 then the method will be automatically changed to symbolic. The same effect
3480 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3482 @cindex @code{clifford_prime()}
3483 @cindex @code{clifford_star()}
3484 @cindex @code{clifford_bar()}
3485 There are several functions for (anti-)automorphisms of Clifford algebras:
3488 ex clifford_prime(const ex & e)
3489 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3490 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3493 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3494 changes signs of all Clifford units in the expression. The reversion
3495 of a Clifford algebra @code{clifford_star()} coincides with the
3496 @code{conjugate()} method and effectively reverses the order of Clifford
3497 units in any product. Finally the main anti-automorphism
3498 of a Clifford algebra @code{clifford_bar()} is the composition of the
3499 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3500 in a product. These functions correspond to the notations
3515 used in Clifford algebra textbooks.
3517 @cindex @code{clifford_norm()}
3521 ex clifford_norm(const ex & e);
3524 @cindex @code{clifford_inverse()}
3525 calculates the norm of a Clifford number from the expression
3527 $||e||^2 = e\overline{e}$.
3530 @code{||e||^2 = e \bar@{e@}}
3532 The inverse of a Clifford expression is returned by the function
3535 ex clifford_inverse(const ex & e);
3538 which calculates it as
3540 $e^{-1} = \overline{e}/||e||^2$.
3543 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3552 then an exception is raised.
3554 @cindex @code{remove_dirac_ONE()}
3555 If a Clifford number happens to be a factor of
3556 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3557 expression by the function
3560 ex remove_dirac_ONE(const ex & e);
3563 @cindex @code{canonicalize_clifford()}
3564 The function @code{canonicalize_clifford()} works for a
3565 generic Clifford algebra in a similar way as for Dirac gammas.
3567 The next provided function is
3569 @cindex @code{clifford_moebius_map()}
3571 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3572 const ex & d, const ex & v, const ex & G,
3573 unsigned char rl = 0);
3574 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3575 unsigned char rl = 0);
3578 It takes a list or vector @code{v} and makes the Moebius (conformal or
3579 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3580 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3581 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3582 indexed object, tensormetric, matrix or a Clifford unit, in the later
3583 case the optional parameter @code{rl} is ignored even if supplied.
3584 Depending from the type of @code{v} the returned value of this function
3585 is either a vector or a list holding vector's components.
3587 @cindex @code{clifford_max_label()}
3588 Finally the function
3591 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3594 can detect a presence of Clifford objects in the expression @code{e}: if
3595 such objects are found it returns the maximal
3596 @code{representation_label} of them, otherwise @code{-1}. The optional
3597 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3598 be ignored during the search.
3600 LaTeX output for Clifford units looks like
3601 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3602 @code{representation_label} and @code{\nu} is the index of the
3603 corresponding unit. This provides a flexible typesetting with a suitable
3604 definition of the @code{\clifford} command. For example, the definition
3606 \newcommand@{\clifford@}[1][]@{@}
3608 typesets all Clifford units identically, while the alternative definition
3610 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3612 prints units with @code{representation_label=0} as
3619 with @code{representation_label=1} as
3626 and with @code{representation_label=2} as
3634 @cindex @code{color} (class)
3635 @subsection Color algebra
3637 @cindex @code{color_T()}
3638 For computations in quantum chromodynamics, GiNaC implements the base elements
3639 and structure constants of the su(3) Lie algebra (color algebra). The base
3640 elements @math{T_a} are constructed by the function
3643 ex color_T(const ex & a, unsigned char rl = 0);
3646 which takes two arguments: the index and a @dfn{representation label} in the
3647 range 0 to 255 which is used to distinguish elements of different color
3648 algebras. Objects with different labels commutate with each other. The
3649 dimension of the index must be exactly 8 and it should be of class @code{idx},
3652 @cindex @code{color_ONE()}
3653 The unity element of a color algebra is constructed by
3656 ex color_ONE(unsigned char rl = 0);
3659 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3660 multiples of the unity element, even though it's customary to omit it.
3661 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3662 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3663 GiNaC may produce incorrect results.
3665 @cindex @code{color_d()}
3666 @cindex @code{color_f()}
3670 ex color_d(const ex & a, const ex & b, const ex & c);
3671 ex color_f(const ex & a, const ex & b, const ex & c);
3674 create the symmetric and antisymmetric structure constants @math{d_abc} and
3675 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3676 and @math{[T_a, T_b] = i f_abc T_c}.
3678 These functions evaluate to their numerical values,
3679 if you supply numeric indices to them. The index values should be in
3680 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3681 goes along better with the notations used in physical literature.
3683 @cindex @code{color_h()}
3684 There's an additional function
3687 ex color_h(const ex & a, const ex & b, const ex & c);
3690 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3692 The function @code{simplify_indexed()} performs some simplifications on
3693 expressions containing color objects:
3698 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3699 k(symbol("k"), 8), l(symbol("l"), 8);
3701 e = color_d(a, b, l) * color_f(a, b, k);
3702 cout << e.simplify_indexed() << endl;
3705 e = color_d(a, b, l) * color_d(a, b, k);
3706 cout << e.simplify_indexed() << endl;
3709 e = color_f(l, a, b) * color_f(a, b, k);
3710 cout << e.simplify_indexed() << endl;
3713 e = color_h(a, b, c) * color_h(a, b, c);
3714 cout << e.simplify_indexed() << endl;
3717 e = color_h(a, b, c) * color_T(b) * color_T(c);
3718 cout << e.simplify_indexed() << endl;
3721 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3722 cout << e.simplify_indexed() << endl;
3725 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3726 cout << e.simplify_indexed() << endl;
3727 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3731 @cindex @code{color_trace()}
3732 To calculate the trace of an expression containing color objects you use one
3736 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3737 ex color_trace(const ex & e, const lst & rll);
3738 ex color_trace(const ex & e, unsigned char rl = 0);
3741 These functions take the trace over all color @samp{T} objects in the
3742 specified set @code{rls} or list @code{rll} of representation labels, or the
3743 single label @code{rl}; @samp{T}s with other labels are left standing. For
3748 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3750 // -> -I*f.a.c.b+d.a.c.b