1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author @uref{http://www.ginac.de}
53 @vskip 0pt plus 1filll
54 Copyright @copyright{} 1999-2007 Johannes Gutenberg University Mainz, Germany
56 Permission is granted to make and distribute verbatim copies of
57 this manual provided the copyright notice and this permission notice
58 are preserved on all copies.
60 Permission is granted to copy and distribute modified versions of this
61 manual under the conditions for verbatim copying, provided that the entire
62 resulting derived work is distributed under the terms of a permission
63 notice identical to this one.
72 @node Top, Introduction, (dir), (dir)
73 @c node-name, next, previous, up
76 This is a tutorial that documents GiNaC @value{VERSION}, an open
77 framework for symbolic computation within the C++ programming language.
80 * Introduction:: GiNaC's purpose.
81 * A tour of GiNaC:: A quick tour of the library.
82 * Installation:: How to install the package.
83 * Basic concepts:: Description of fundamental classes.
84 * Methods and functions:: Algorithms for symbolic manipulations.
85 * Extending GiNaC:: How to extend the library.
86 * A comparison with other CAS:: Compares GiNaC to traditional CAS.
87 * Internal structures:: Description of some internal structures.
88 * Package tools:: Configuring packages to work with GiNaC.
94 @node Introduction, A tour of GiNaC, Top, Top
95 @c node-name, next, previous, up
97 @cindex history of GiNaC
99 The motivation behind GiNaC derives from the observation that most
100 present day computer algebra systems (CAS) are linguistically and
101 semantically impoverished. Although they are quite powerful tools for
102 learning math and solving particular problems they lack modern
103 linguistic structures that allow for the creation of large-scale
104 projects. GiNaC is an attempt to overcome this situation by extending a
105 well established and standardized computer language (C++) by some
106 fundamental symbolic capabilities, thus allowing for integrated systems
107 that embed symbolic manipulations together with more established areas
108 of computer science (like computation-intense numeric applications,
109 graphical interfaces, etc.) under one roof.
111 The particular problem that led to the writing of the GiNaC framework is
112 still a very active field of research, namely the calculation of higher
113 order corrections to elementary particle interactions. There,
114 theoretical physicists are interested in matching present day theories
115 against experiments taking place at particle accelerators. The
116 computations involved are so complex they call for a combined symbolical
117 and numerical approach. This turned out to be quite difficult to
118 accomplish with the present day CAS we have worked with so far and so we
119 tried to fill the gap by writing GiNaC. But of course its applications
120 are in no way restricted to theoretical physics.
122 This tutorial is intended for the novice user who is new to GiNaC but
123 already has some background in C++ programming. However, since a
124 hand-made documentation like this one is difficult to keep in sync with
125 the development, the actual documentation is inside the sources in the
126 form of comments. That documentation may be parsed by one of the many
127 Javadoc-like documentation systems. If you fail at generating it you
128 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
129 page}. It is an invaluable resource not only for the advanced user who
130 wishes to extend the system (or chase bugs) but for everybody who wants
131 to comprehend the inner workings of GiNaC. This little tutorial on the
132 other hand only covers the basic things that are unlikely to change in
136 The GiNaC framework for symbolic computation within the C++ programming
137 language is Copyright @copyright{} 1999-2007 Johannes Gutenberg
138 University Mainz, Germany.
140 This program is free software; you can redistribute it and/or
141 modify it under the terms of the GNU General Public License as
142 published by the Free Software Foundation; either version 2 of the
143 License, or (at your option) any later version.
145 This program is distributed in the hope that it will be useful, but
146 WITHOUT ANY WARRANTY; without even the implied warranty of
147 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
148 General Public License for more details.
150 You should have received a copy of the GNU General Public License
151 along with this program; see the file COPYING. If not, write to the
152 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
156 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
157 @c node-name, next, previous, up
158 @chapter A Tour of GiNaC
160 This quick tour of GiNaC wants to arise your interest in the
161 subsequent chapters by showing off a bit. Please excuse us if it
162 leaves many open questions.
165 * How to use it from within C++:: Two simple examples.
166 * What it can do for you:: A Tour of GiNaC's features.
170 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
171 @c node-name, next, previous, up
172 @section How to use it from within C++
174 The GiNaC open framework for symbolic computation within the C++ programming
175 language does not try to define a language of its own as conventional
176 CAS do. Instead, it extends the capabilities of C++ by symbolic
177 manipulations. Here is how to generate and print a simple (and rather
178 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
217 #include <ginac/ginac.h>
219 using namespace GiNaC;
221 ex HermitePoly(const symbol & x, int n)
223 ex HKer=exp(-pow(x, 2));
224 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
225 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
232 for (int i=0; i<6; ++i)
233 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
239 When run, this will type out
245 H_3(z) == -12*z+8*z^3
246 H_4(z) == -48*z^2+16*z^4+12
247 H_5(z) == 120*z-160*z^3+32*z^5
250 This method of generating the coefficients is of course far from optimal
251 for production purposes.
253 In order to show some more examples of what GiNaC can do we will now use
254 the @command{ginsh}, a simple GiNaC interactive shell that provides a
255 convenient window into GiNaC's capabilities.
258 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
259 @c node-name, next, previous, up
260 @section What it can do for you
262 @cindex @command{ginsh}
263 After invoking @command{ginsh} one can test and experiment with GiNaC's
264 features much like in other Computer Algebra Systems except that it does
265 not provide programming constructs like loops or conditionals. For a
266 concise description of the @command{ginsh} syntax we refer to its
267 accompanied man page. Suffice to say that assignments and comparisons in
268 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
271 It can manipulate arbitrary precision integers in a very fast way.
272 Rational numbers are automatically converted to fractions of coprime
277 369988485035126972924700782451696644186473100389722973815184405301748249
279 123329495011708990974900260817232214728824366796574324605061468433916083
286 Exact numbers are always retained as exact numbers and only evaluated as
287 floating point numbers if requested. For instance, with numeric
288 radicals is dealt pretty much as with symbols. Products of sums of them
292 > expand((1+a^(1/5)-a^(2/5))^3);
293 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
294 > expand((1+3^(1/5)-3^(2/5))^3);
296 > evalf((1+3^(1/5)-3^(2/5))^3);
297 0.33408977534118624228
300 The function @code{evalf} that was used above converts any number in
301 GiNaC's expressions into floating point numbers. This can be done to
302 arbitrary predefined accuracy:
306 0.14285714285714285714
310 0.1428571428571428571428571428571428571428571428571428571428571428571428
311 5714285714285714285714285714285714285
314 Exact numbers other than rationals that can be manipulated in GiNaC
315 include predefined constants like Archimedes' @code{Pi}. They can both
316 be used in symbolic manipulations (as an exact number) as well as in
317 numeric expressions (as an inexact number):
323 9.869604401089358619+x
327 11.869604401089358619
330 Built-in functions evaluate immediately to exact numbers if
331 this is possible. Conversions that can be safely performed are done
332 immediately; conversions that are not generally valid are not done:
343 (Note that converting the last input to @code{x} would allow one to
344 conclude that @code{42*Pi} is equal to @code{0}.)
346 Linear equation systems can be solved along with basic linear
347 algebra manipulations over symbolic expressions. In C++ GiNaC offers
348 a matrix class for this purpose but we can see what it can do using
349 @command{ginsh}'s bracket notation to type them in:
352 > lsolve(a+x*y==z,x);
354 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
356 > M = [ [1, 3], [-3, 2] ];
360 > charpoly(M,lambda);
362 > A = [ [1, 1], [2, -1] ];
365 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
368 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
369 > evalm(B^(2^12345));
370 [[1,0,0],[0,1,0],[0,0,1]]
373 Multivariate polynomials and rational functions may be expanded,
374 collected and normalized (i.e. converted to a ratio of two coprime
378 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
379 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
380 > b = x^2 + 4*x*y - y^2;
383 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
385 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
387 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
392 You can differentiate functions and expand them as Taylor or Laurent
393 series in a very natural syntax (the second argument of @code{series} is
394 a relation defining the evaluation point, the third specifies the
397 @cindex Zeta function
401 > series(sin(x),x==0,4);
403 > series(1/tan(x),x==0,4);
404 x^(-1)-1/3*x+Order(x^2)
405 > series(tgamma(x),x==0,3);
406 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
407 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
409 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
410 -(0.90747907608088628905)*x^2+Order(x^3)
411 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
412 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
413 -Euler-1/12+Order((x-1/2*Pi)^3)
416 Here we have made use of the @command{ginsh}-command @code{%} to pop the
417 previously evaluated element from @command{ginsh}'s internal stack.
419 Often, functions don't have roots in closed form. Nevertheless, it's
420 quite easy to compute a solution numerically, to arbitrary precision:
425 > fsolve(cos(x)==x,x,0,2);
426 0.7390851332151606416553120876738734040134117589007574649658
428 > X=fsolve(f,x,-10,10);
429 2.2191071489137460325957851882042901681753665565320678854155
431 -6.372367644529809108115521591070847222364418220770475144296E-58
434 Notice how the final result above differs slightly from zero by about
435 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
436 root cannot be represented more accurately than @code{X}. Such
437 inaccuracies are to be expected when computing with finite floating
440 If you ever wanted to convert units in C or C++ and found this is
441 cumbersome, here is the solution. Symbolic types can always be used as
442 tags for different types of objects. Converting from wrong units to the
443 metric system is now easy:
451 140613.91592783185568*kg*m^(-2)
455 @node Installation, Prerequisites, What it can do for you, Top
456 @c node-name, next, previous, up
457 @chapter Installation
460 GiNaC's installation follows the spirit of most GNU software. It is
461 easily installed on your system by three steps: configuration, build,
465 * Prerequisites:: Packages upon which GiNaC depends.
466 * Configuration:: How to configure GiNaC.
467 * Building GiNaC:: How to compile GiNaC.
468 * Installing GiNaC:: How to install GiNaC on your system.
472 @node Prerequisites, Configuration, Installation, Installation
473 @c node-name, next, previous, up
474 @section Prerequisites
476 In order to install GiNaC on your system, some prerequisites need to be
477 met. First of all, you need to have a C++-compiler adhering to the
478 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
479 so if you have a different compiler you are on your own. For the
480 configuration to succeed you need a Posix compliant shell installed in
481 @file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
482 required for the configuration, it can be downloaded from
483 @uref{http://pkg-config.freedesktop.org}.
484 Last but not least, the CLN library
485 is used extensively and needs to be installed on your system.
486 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
487 (it is covered by GPL) and install it prior to trying to install
488 GiNaC. The configure script checks if it can find it and if it cannot
489 it will refuse to continue.
492 @node Configuration, Building GiNaC, Prerequisites, Installation
493 @c node-name, next, previous, up
494 @section Configuration
495 @cindex configuration
498 To configure GiNaC means to prepare the source distribution for
499 building. It is done via a shell script called @command{configure} that
500 is shipped with the sources and was originally generated by GNU
501 Autoconf. Since a configure script generated by GNU Autoconf never
502 prompts, all customization must be done either via command line
503 parameters or environment variables. It accepts a list of parameters,
504 the complete set of which can be listed by calling it with the
505 @option{--help} option. The most important ones will be shortly
506 described in what follows:
511 @option{--disable-shared}: When given, this option switches off the
512 build of a shared library, i.e. a @file{.so} file. This may be convenient
513 when developing because it considerably speeds up compilation.
516 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
517 and headers are installed. It defaults to @file{/usr/local} which means
518 that the library is installed in the directory @file{/usr/local/lib},
519 the header files in @file{/usr/local/include/ginac} and the documentation
520 (like this one) into @file{/usr/local/share/doc/GiNaC}.
523 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
524 the library installed in some other directory than
525 @file{@var{PREFIX}/lib/}.
528 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
529 to have the header files installed in some other directory than
530 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
531 @option{--includedir=/usr/include} you will end up with the header files
532 sitting in the directory @file{/usr/include/ginac/}. Note that the
533 subdirectory @file{ginac} is enforced by this process in order to
534 keep the header files separated from others. This avoids some
535 clashes and allows for an easier deinstallation of GiNaC. This ought
536 to be considered A Good Thing (tm).
539 @option{--datadir=@var{DATADIR}}: This option may be given in case you
540 want to have the documentation installed in some other directory than
541 @file{@var{PREFIX}/share/doc/GiNaC/}.
545 In addition, you may specify some environment variables. @env{CXX}
546 holds the path and the name of the C++ compiler in case you want to
547 override the default in your path. (The @command{configure} script
548 searches your path for @command{c++}, @command{g++}, @command{gcc},
549 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
550 be very useful to define some compiler flags with the @env{CXXFLAGS}
551 environment variable, like optimization, debugging information and
552 warning levels. If omitted, it defaults to @option{-g
553 -O2}.@footnote{The @command{configure} script is itself generated from
554 the file @file{configure.ac}. It is only distributed in packaged
555 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
556 must generate @command{configure} along with the various
557 @file{Makefile.in} by using the @command{autoreconf} utility. This will
558 require a fair amount of support from your local toolchain, though.}
560 The whole process is illustrated in the following two
561 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
562 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
565 Here is a simple configuration for a site-wide GiNaC library assuming
566 everything is in default paths:
569 $ export CXXFLAGS="-Wall -O2"
573 And here is a configuration for a private static GiNaC library with
574 several components sitting in custom places (site-wide GCC and private
575 CLN). The compiler is persuaded to be picky and full assertions and
576 debugging information are switched on:
579 $ export CXX=/usr/local/gnu/bin/c++
580 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
581 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
582 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
583 $ ./configure --disable-shared --prefix=$(HOME)
587 @node Building GiNaC, Installing GiNaC, Configuration, Installation
588 @c node-name, next, previous, up
589 @section Building GiNaC
590 @cindex building GiNaC
592 After proper configuration you should just build the whole
597 at the command prompt and go for a cup of coffee. The exact time it
598 takes to compile GiNaC depends not only on the speed of your machines
599 but also on other parameters, for instance what value for @env{CXXFLAGS}
600 you entered. Optimization may be very time-consuming.
602 Just to make sure GiNaC works properly you may run a collection of
603 regression tests by typing
609 This will compile some sample programs, run them and check the output
610 for correctness. The regression tests fall in three categories. First,
611 the so called @emph{exams} are performed, simple tests where some
612 predefined input is evaluated (like a pupils' exam). Second, the
613 @emph{checks} test the coherence of results among each other with
614 possible random input. Third, some @emph{timings} are performed, which
615 benchmark some predefined problems with different sizes and display the
616 CPU time used in seconds. Each individual test should return a message
617 @samp{passed}. This is mostly intended to be a QA-check if something
618 was broken during development, not a sanity check of your system. Some
619 of the tests in sections @emph{checks} and @emph{timings} may require
620 insane amounts of memory and CPU time. Feel free to kill them if your
621 machine catches fire. Another quite important intent is to allow people
622 to fiddle around with optimization.
624 By default, the only documentation that will be built is this tutorial
625 in @file{.info} format. To build the GiNaC tutorial and reference manual
626 in HTML, DVI, PostScript, or PDF formats, use one of
635 Generally, the top-level Makefile runs recursively to the
636 subdirectories. It is therefore safe to go into any subdirectory
637 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
638 @var{target} there in case something went wrong.
641 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
642 @c node-name, next, previous, up
643 @section Installing GiNaC
646 To install GiNaC on your system, simply type
652 As described in the section about configuration the files will be
653 installed in the following directories (the directories will be created
654 if they don't already exist):
659 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
660 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
661 So will @file{libginac.so} unless the configure script was
662 given the option @option{--disable-shared}. The proper symlinks
663 will be established as well.
666 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
667 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
670 All documentation (info) will be stuffed into
671 @file{@var{PREFIX}/share/doc/GiNaC/} (or
672 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
676 For the sake of completeness we will list some other useful make
677 targets: @command{make clean} deletes all files generated by
678 @command{make}, i.e. all the object files. In addition @command{make
679 distclean} removes all files generated by the configuration and
680 @command{make maintainer-clean} goes one step further and deletes files
681 that may require special tools to rebuild (like the @command{libtool}
682 for instance). Finally @command{make uninstall} removes the installed
683 library, header files and documentation@footnote{Uninstallation does not
684 work after you have called @command{make distclean} since the
685 @file{Makefile} is itself generated by the configuration from
686 @file{Makefile.in} and hence deleted by @command{make distclean}. There
687 are two obvious ways out of this dilemma. First, you can run the
688 configuration again with the same @var{PREFIX} thus creating a
689 @file{Makefile} with a working @samp{uninstall} target. Second, you can
690 do it by hand since you now know where all the files went during
694 @node Basic concepts, Expressions, Installing GiNaC, Top
695 @c node-name, next, previous, up
696 @chapter Basic concepts
698 This chapter will describe the different fundamental objects that can be
699 handled by GiNaC. But before doing so, it is worthwhile introducing you
700 to the more commonly used class of expressions, representing a flexible
701 meta-class for storing all mathematical objects.
704 * Expressions:: The fundamental GiNaC class.
705 * Automatic evaluation:: Evaluation and canonicalization.
706 * Error handling:: How the library reports errors.
707 * The class hierarchy:: Overview of GiNaC's classes.
708 * Symbols:: Symbolic objects.
709 * Numbers:: Numerical objects.
710 * Constants:: Pre-defined constants.
711 * Fundamental containers:: Sums, products and powers.
712 * Lists:: Lists of expressions.
713 * Mathematical functions:: Mathematical functions.
714 * Relations:: Equality, Inequality and all that.
715 * Integrals:: Symbolic integrals.
716 * Matrices:: Matrices.
717 * Indexed objects:: Handling indexed quantities.
718 * Non-commutative objects:: Algebras with non-commutative products.
719 * Hash maps:: A faster alternative to std::map<>.
723 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
724 @c node-name, next, previous, up
726 @cindex expression (class @code{ex})
729 The most common class of objects a user deals with is the expression
730 @code{ex}, representing a mathematical object like a variable, number,
731 function, sum, product, etc@dots{} Expressions may be put together to form
732 new expressions, passed as arguments to functions, and so on. Here is a
733 little collection of valid expressions:
736 ex MyEx1 = 5; // simple number
737 ex MyEx2 = x + 2*y; // polynomial in x and y
738 ex MyEx3 = (x + 1)/(x - 1); // rational expression
739 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
740 ex MyEx5 = MyEx4 + 1; // similar to above
743 Expressions are handles to other more fundamental objects, that often
744 contain other expressions thus creating a tree of expressions
745 (@xref{Internal structures}, for particular examples). Most methods on
746 @code{ex} therefore run top-down through such an expression tree. For
747 example, the method @code{has()} scans recursively for occurrences of
748 something inside an expression. Thus, if you have declared @code{MyEx4}
749 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
750 the argument of @code{sin} and hence return @code{true}.
752 The next sections will outline the general picture of GiNaC's class
753 hierarchy and describe the classes of objects that are handled by
756 @subsection Note: Expressions and STL containers
758 GiNaC expressions (@code{ex} objects) have value semantics (they can be
759 assigned, reassigned and copied like integral types) but the operator
760 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
761 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
763 This implies that in order to use expressions in sorted containers such as
764 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
765 comparison predicate. GiNaC provides such a predicate, called
766 @code{ex_is_less}. For example, a set of expressions should be defined
767 as @code{std::set<ex, ex_is_less>}.
769 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
770 don't pose a problem. A @code{std::vector<ex>} works as expected.
772 @xref{Information about expressions}, for more about comparing and ordering
776 @node Automatic evaluation, Error handling, Expressions, Basic concepts
777 @c node-name, next, previous, up
778 @section Automatic evaluation and canonicalization of expressions
781 GiNaC performs some automatic transformations on expressions, to simplify
782 them and put them into a canonical form. Some examples:
785 ex MyEx1 = 2*x - 1 + x; // 3*x-1
786 ex MyEx2 = x - x; // 0
787 ex MyEx3 = cos(2*Pi); // 1
788 ex MyEx4 = x*y/x; // y
791 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
792 evaluation}. GiNaC only performs transformations that are
796 at most of complexity
804 algebraically correct, possibly except for a set of measure zero (e.g.
805 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
808 There are two types of automatic transformations in GiNaC that may not
809 behave in an entirely obvious way at first glance:
813 The terms of sums and products (and some other things like the arguments of
814 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
815 into a canonical form that is deterministic, but not lexicographical or in
816 any other way easy to guess (it almost always depends on the number and
817 order of the symbols you define). However, constructing the same expression
818 twice, either implicitly or explicitly, will always result in the same
821 Expressions of the form 'number times sum' are automatically expanded (this
822 has to do with GiNaC's internal representation of sums and products). For
825 ex MyEx5 = 2*(x + y); // 2*x+2*y
826 ex MyEx6 = z*(x + y); // z*(x+y)
830 The general rule is that when you construct expressions, GiNaC automatically
831 creates them in canonical form, which might differ from the form you typed in
832 your program. This may create some awkward looking output (@samp{-y+x} instead
833 of @samp{x-y}) but allows for more efficient operation and usually yields
834 some immediate simplifications.
836 @cindex @code{eval()}
837 Internally, the anonymous evaluator in GiNaC is implemented by the methods
840 ex ex::eval(int level = 0) const;
841 ex basic::eval(int level = 0) const;
844 but unless you are extending GiNaC with your own classes or functions, there
845 should never be any reason to call them explicitly. All GiNaC methods that
846 transform expressions, like @code{subs()} or @code{normal()}, automatically
847 re-evaluate their results.
850 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
851 @c node-name, next, previous, up
852 @section Error handling
854 @cindex @code{pole_error} (class)
856 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
857 generated by GiNaC are subclassed from the standard @code{exception} class
858 defined in the @file{<stdexcept>} header. In addition to the predefined
859 @code{logic_error}, @code{domain_error}, @code{out_of_range},
860 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
861 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
862 exception that gets thrown when trying to evaluate a mathematical function
865 The @code{pole_error} class has a member function
868 int pole_error::degree() const;
871 that returns the order of the singularity (or 0 when the pole is
872 logarithmic or the order is undefined).
874 When using GiNaC it is useful to arrange for exceptions to be caught in
875 the main program even if you don't want to do any special error handling.
876 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
877 default exception handler of your C++ compiler's run-time system which
878 usually only aborts the program without giving any information what went
881 Here is an example for a @code{main()} function that catches and prints
882 exceptions generated by GiNaC:
887 #include <ginac/ginac.h>
889 using namespace GiNaC;
897 @} catch (exception &p) @{
898 cerr << p.what() << endl;
906 @node The class hierarchy, Symbols, Error handling, Basic concepts
907 @c node-name, next, previous, up
908 @section The class hierarchy
910 GiNaC's class hierarchy consists of several classes representing
911 mathematical objects, all of which (except for @code{ex} and some
912 helpers) are internally derived from one abstract base class called
913 @code{basic}. You do not have to deal with objects of class
914 @code{basic}, instead you'll be dealing with symbols, numbers,
915 containers of expressions and so on.
919 To get an idea about what kinds of symbolic composites may be built we
920 have a look at the most important classes in the class hierarchy and
921 some of the relations among the classes:
923 @image{classhierarchy}
925 The abstract classes shown here (the ones without drop-shadow) are of no
926 interest for the user. They are used internally in order to avoid code
927 duplication if two or more classes derived from them share certain
928 features. An example is @code{expairseq}, a container for a sequence of
929 pairs each consisting of one expression and a number (@code{numeric}).
930 What @emph{is} visible to the user are the derived classes @code{add}
931 and @code{mul}, representing sums and products. @xref{Internal
932 structures}, where these two classes are described in more detail. The
933 following table shortly summarizes what kinds of mathematical objects
934 are stored in the different classes:
937 @multitable @columnfractions .22 .78
938 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
939 @item @code{constant} @tab Constants like
946 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
947 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
948 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
949 @item @code{ncmul} @tab Products of non-commutative objects
950 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
955 @code{sqrt(}@math{2}@code{)}
958 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
959 @item @code{function} @tab A symbolic function like
966 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
967 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
968 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
969 @item @code{indexed} @tab Indexed object like @math{A_ij}
970 @item @code{tensor} @tab Special tensor like the delta and metric tensors
971 @item @code{idx} @tab Index of an indexed object
972 @item @code{varidx} @tab Index with variance
973 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
974 @item @code{wildcard} @tab Wildcard for pattern matching
975 @item @code{structure} @tab Template for user-defined classes
980 @node Symbols, Numbers, The class hierarchy, Basic concepts
981 @c node-name, next, previous, up
983 @cindex @code{symbol} (class)
984 @cindex hierarchy of classes
987 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
988 manipulation what atoms are for chemistry.
990 A typical symbol definition looks like this:
995 This definition actually contains three very different things:
997 @item a C++ variable named @code{x}
998 @item a @code{symbol} object stored in this C++ variable; this object
999 represents the symbol in a GiNaC expression
1000 @item the string @code{"x"} which is the name of the symbol, used (almost)
1001 exclusively for printing expressions holding the symbol
1004 Symbols have an explicit name, supplied as a string during construction,
1005 because in C++, variable names can't be used as values, and the C++ compiler
1006 throws them away during compilation.
1008 It is possible to omit the symbol name in the definition:
1013 In this case, GiNaC will assign the symbol an internal, unique name of the
1014 form @code{symbolNNN}. This won't affect the usability of the symbol but
1015 the output of your calculations will become more readable if you give your
1016 symbols sensible names (for intermediate expressions that are only used
1017 internally such anonymous symbols can be quite useful, however).
1019 Now, here is one important property of GiNaC that differentiates it from
1020 other computer algebra programs you may have used: GiNaC does @emph{not} use
1021 the names of symbols to tell them apart, but a (hidden) serial number that
1022 is unique for each newly created @code{symbol} object. If you want to use
1023 one and the same symbol in different places in your program, you must only
1024 create one @code{symbol} object and pass that around. If you create another
1025 symbol, even if it has the same name, GiNaC will treat it as a different
1042 // prints "x^6" which looks right, but...
1044 cout << e.degree(x) << endl;
1045 // ...this doesn't work. The symbol "x" here is different from the one
1046 // in f() and in the expression returned by f(). Consequently, it
1051 One possibility to ensure that @code{f()} and @code{main()} use the same
1052 symbol is to pass the symbol as an argument to @code{f()}:
1054 ex f(int n, const ex & x)
1063 // Now, f() uses the same symbol.
1066 cout << e.degree(x) << endl;
1067 // prints "6", as expected
1071 Another possibility would be to define a global symbol @code{x} that is used
1072 by both @code{f()} and @code{main()}. If you are using global symbols and
1073 multiple compilation units you must take special care, however. Suppose
1074 that you have a header file @file{globals.h} in your program that defines
1075 a @code{symbol x("x");}. In this case, every unit that includes
1076 @file{globals.h} would also get its own definition of @code{x} (because
1077 header files are just inlined into the source code by the C++ preprocessor),
1078 and hence you would again end up with multiple equally-named, but different,
1079 symbols. Instead, the @file{globals.h} header should only contain a
1080 @emph{declaration} like @code{extern symbol x;}, with the definition of
1081 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1083 A different approach to ensuring that symbols used in different parts of
1084 your program are identical is to create them with a @emph{factory} function
1087 const symbol & get_symbol(const string & s)
1089 static map<string, symbol> directory;
1090 map<string, symbol>::iterator i = directory.find(s);
1091 if (i != directory.end())
1094 return directory.insert(make_pair(s, symbol(s))).first->second;
1098 This function returns one newly constructed symbol for each name that is
1099 passed in, and it returns the same symbol when called multiple times with
1100 the same name. Using this symbol factory, we can rewrite our example like
1105 return pow(get_symbol("x"), n);
1112 // Both calls of get_symbol("x") yield the same symbol.
1113 cout << e.degree(get_symbol("x")) << endl;
1118 Instead of creating symbols from strings we could also have
1119 @code{get_symbol()} take, for example, an integer number as its argument.
1120 In this case, we would probably want to give the generated symbols names
1121 that include this number, which can be accomplished with the help of an
1122 @code{ostringstream}.
1124 In general, if you're getting weird results from GiNaC such as an expression
1125 @samp{x-x} that is not simplified to zero, you should check your symbol
1128 As we said, the names of symbols primarily serve for purposes of expression
1129 output. But there are actually two instances where GiNaC uses the names for
1130 identifying symbols: When constructing an expression from a string, and when
1131 recreating an expression from an archive (@pxref{Input/output}).
1133 In addition to its name, a symbol may contain a special string that is used
1136 symbol x("x", "\\Box");
1139 This creates a symbol that is printed as "@code{x}" in normal output, but
1140 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1141 information about the different output formats of expressions in GiNaC).
1142 GiNaC automatically creates proper LaTeX code for symbols having names of
1143 greek letters (@samp{alpha}, @samp{mu}, etc.).
1145 @cindex @code{subs()}
1146 Symbols in GiNaC can't be assigned values. If you need to store results of
1147 calculations and give them a name, use C++ variables of type @code{ex}.
1148 If you want to replace a symbol in an expression with something else, you
1149 can invoke the expression's @code{.subs()} method
1150 (@pxref{Substituting expressions}).
1152 @cindex @code{realsymbol()}
1153 By default, symbols are expected to stand in for complex values, i.e. they live
1154 in the complex domain. As a consequence, operations like complex conjugation,
1155 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1156 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1157 because of the unknown imaginary part of @code{x}.
1158 On the other hand, if you are sure that your symbols will hold only real
1159 values, you would like to have such functions evaluated. Therefore GiNaC
1160 allows you to specify
1161 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1162 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @cindex @code{possymbol()}
1165 Furthermore, it is also possible to declare a symbol as positive. This will,
1166 for instance, enable the automatic simplification of @code{abs(x)} into
1167 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1170 @node Numbers, Constants, Symbols, Basic concepts
1171 @c node-name, next, previous, up
1173 @cindex @code{numeric} (class)
1179 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1180 The classes therein serve as foundation classes for GiNaC. CLN stands
1181 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1182 In order to find out more about CLN's internals, the reader is referred to
1183 the documentation of that library. @inforef{Introduction, , cln}, for
1184 more information. Suffice to say that it is by itself build on top of
1185 another library, the GNU Multiple Precision library GMP, which is an
1186 extremely fast library for arbitrary long integers and rationals as well
1187 as arbitrary precision floating point numbers. It is very commonly used
1188 by several popular cryptographic applications. CLN extends GMP by
1189 several useful things: First, it introduces the complex number field
1190 over either reals (i.e. floating point numbers with arbitrary precision)
1191 or rationals. Second, it automatically converts rationals to integers
1192 if the denominator is unity and complex numbers to real numbers if the
1193 imaginary part vanishes and also correctly treats algebraic functions.
1194 Third it provides good implementations of state-of-the-art algorithms
1195 for all trigonometric and hyperbolic functions as well as for
1196 calculation of some useful constants.
1198 The user can construct an object of class @code{numeric} in several
1199 ways. The following example shows the four most important constructors.
1200 It uses construction from C-integer, construction of fractions from two
1201 integers, construction from C-float and construction from a string:
1205 #include <ginac/ginac.h>
1206 using namespace GiNaC;
1210 numeric two = 2; // exact integer 2
1211 numeric r(2,3); // exact fraction 2/3
1212 numeric e(2.71828); // floating point number
1213 numeric p = "3.14159265358979323846"; // constructor from string
1214 // Trott's constant in scientific notation:
1215 numeric trott("1.0841015122311136151E-2");
1217 std::cout << two*p << std::endl; // floating point 6.283...
1222 @cindex complex numbers
1223 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1228 numeric z1 = 2-3*I; // exact complex number 2-3i
1229 numeric z2 = 5.9+1.6*I; // complex floating point number
1233 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1234 This would, however, call C's built-in operator @code{/} for integers
1235 first and result in a numeric holding a plain integer 1. @strong{Never
1236 use the operator @code{/} on integers} unless you know exactly what you
1237 are doing! Use the constructor from two integers instead, as shown in
1238 the example above. Writing @code{numeric(1)/2} may look funny but works
1241 @cindex @code{Digits}
1243 We have seen now the distinction between exact numbers and floating
1244 point numbers. Clearly, the user should never have to worry about
1245 dynamically created exact numbers, since their `exactness' always
1246 determines how they ought to be handled, i.e. how `long' they are. The
1247 situation is different for floating point numbers. Their accuracy is
1248 controlled by one @emph{global} variable, called @code{Digits}. (For
1249 those readers who know about Maple: it behaves very much like Maple's
1250 @code{Digits}). All objects of class numeric that are constructed from
1251 then on will be stored with a precision matching that number of decimal
1256 #include <ginac/ginac.h>
1257 using namespace std;
1258 using namespace GiNaC;
1262 numeric three(3.0), one(1.0);
1263 numeric x = one/three;
1265 cout << "in " << Digits << " digits:" << endl;
1267 cout << Pi.evalf() << endl;
1279 The above example prints the following output to screen:
1283 0.33333333333333333334
1284 3.1415926535897932385
1286 0.33333333333333333333333333333333333333333333333333333333333333333334
1287 3.1415926535897932384626433832795028841971693993751058209749445923078
1291 Note that the last number is not necessarily rounded as you would
1292 naively expect it to be rounded in the decimal system. But note also,
1293 that in both cases you got a couple of extra digits. This is because
1294 numbers are internally stored by CLN as chunks of binary digits in order
1295 to match your machine's word size and to not waste precision. Thus, on
1296 architectures with different word size, the above output might even
1297 differ with regard to actually computed digits.
1299 It should be clear that objects of class @code{numeric} should be used
1300 for constructing numbers or for doing arithmetic with them. The objects
1301 one deals with most of the time are the polymorphic expressions @code{ex}.
1303 @subsection Tests on numbers
1305 Once you have declared some numbers, assigned them to expressions and
1306 done some arithmetic with them it is frequently desired to retrieve some
1307 kind of information from them like asking whether that number is
1308 integer, rational, real or complex. For those cases GiNaC provides
1309 several useful methods. (Internally, they fall back to invocations of
1310 certain CLN functions.)
1312 As an example, let's construct some rational number, multiply it with
1313 some multiple of its denominator and test what comes out:
1317 #include <ginac/ginac.h>
1318 using namespace std;
1319 using namespace GiNaC;
1321 // some very important constants:
1322 const numeric twentyone(21);
1323 const numeric ten(10);
1324 const numeric five(5);
1328 numeric answer = twentyone;
1331 cout << answer.is_integer() << endl; // false, it's 21/5
1333 cout << answer.is_integer() << endl; // true, it's 42 now!
1337 Note that the variable @code{answer} is constructed here as an integer
1338 by @code{numeric}'s copy constructor, but in an intermediate step it
1339 holds a rational number represented as integer numerator and integer
1340 denominator. When multiplied by 10, the denominator becomes unity and
1341 the result is automatically converted to a pure integer again.
1342 Internally, the underlying CLN is responsible for this behavior and we
1343 refer the reader to CLN's documentation. Suffice to say that
1344 the same behavior applies to complex numbers as well as return values of
1345 certain functions. Complex numbers are automatically converted to real
1346 numbers if the imaginary part becomes zero. The full set of tests that
1347 can be applied is listed in the following table.
1350 @multitable @columnfractions .30 .70
1351 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1352 @item @code{.is_zero()}
1353 @tab @dots{}equal to zero
1354 @item @code{.is_positive()}
1355 @tab @dots{}not complex and greater than 0
1356 @item @code{.is_negative()}
1357 @tab @dots{}not complex and smaller than 0
1358 @item @code{.is_integer()}
1359 @tab @dots{}a (non-complex) integer
1360 @item @code{.is_pos_integer()}
1361 @tab @dots{}an integer and greater than 0
1362 @item @code{.is_nonneg_integer()}
1363 @tab @dots{}an integer and greater equal 0
1364 @item @code{.is_even()}
1365 @tab @dots{}an even integer
1366 @item @code{.is_odd()}
1367 @tab @dots{}an odd integer
1368 @item @code{.is_prime()}
1369 @tab @dots{}a prime integer (probabilistic primality test)
1370 @item @code{.is_rational()}
1371 @tab @dots{}an exact rational number (integers are rational, too)
1372 @item @code{.is_real()}
1373 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1374 @item @code{.is_cinteger()}
1375 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1376 @item @code{.is_crational()}
1377 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1383 @subsection Numeric functions
1385 The following functions can be applied to @code{numeric} objects and will be
1386 evaluated immediately:
1389 @multitable @columnfractions .30 .70
1390 @item @strong{Name} @tab @strong{Function}
1391 @item @code{inverse(z)}
1392 @tab returns @math{1/z}
1393 @cindex @code{inverse()} (numeric)
1394 @item @code{pow(a, b)}
1395 @tab exponentiation @math{a^b}
1398 @item @code{real(z)}
1400 @cindex @code{real()}
1401 @item @code{imag(z)}
1403 @cindex @code{imag()}
1404 @item @code{csgn(z)}
1405 @tab complex sign (returns an @code{int})
1406 @item @code{step(x)}
1407 @tab step function (returns an @code{numeric})
1408 @item @code{numer(z)}
1409 @tab numerator of rational or complex rational number
1410 @item @code{denom(z)}
1411 @tab denominator of rational or complex rational number
1412 @item @code{sqrt(z)}
1414 @item @code{isqrt(n)}
1415 @tab integer square root
1416 @cindex @code{isqrt()}
1423 @item @code{asin(z)}
1425 @item @code{acos(z)}
1427 @item @code{atan(z)}
1428 @tab inverse tangent
1429 @item @code{atan(y, x)}
1430 @tab inverse tangent with two arguments
1431 @item @code{sinh(z)}
1432 @tab hyperbolic sine
1433 @item @code{cosh(z)}
1434 @tab hyperbolic cosine
1435 @item @code{tanh(z)}
1436 @tab hyperbolic tangent
1437 @item @code{asinh(z)}
1438 @tab inverse hyperbolic sine
1439 @item @code{acosh(z)}
1440 @tab inverse hyperbolic cosine
1441 @item @code{atanh(z)}
1442 @tab inverse hyperbolic tangent
1444 @tab exponential function
1446 @tab natural logarithm
1449 @item @code{zeta(z)}
1450 @tab Riemann's zeta function
1451 @item @code{tgamma(z)}
1453 @item @code{lgamma(z)}
1454 @tab logarithm of gamma function
1456 @tab psi (digamma) function
1457 @item @code{psi(n, z)}
1458 @tab derivatives of psi function (polygamma functions)
1459 @item @code{factorial(n)}
1460 @tab factorial function @math{n!}
1461 @item @code{doublefactorial(n)}
1462 @tab double factorial function @math{n!!}
1463 @cindex @code{doublefactorial()}
1464 @item @code{binomial(n, k)}
1465 @tab binomial coefficients
1466 @item @code{bernoulli(n)}
1467 @tab Bernoulli numbers
1468 @cindex @code{bernoulli()}
1469 @item @code{fibonacci(n)}
1470 @tab Fibonacci numbers
1471 @cindex @code{fibonacci()}
1472 @item @code{mod(a, b)}
1473 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1474 @cindex @code{mod()}
1475 @item @code{smod(a, b)}
1476 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1477 @cindex @code{smod()}
1478 @item @code{irem(a, b)}
1479 @tab integer remainder (has the sign of @math{a}, or is zero)
1480 @cindex @code{irem()}
1481 @item @code{irem(a, b, q)}
1482 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1483 @item @code{iquo(a, b)}
1484 @tab integer quotient
1485 @cindex @code{iquo()}
1486 @item @code{iquo(a, b, r)}
1487 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1488 @item @code{gcd(a, b)}
1489 @tab greatest common divisor
1490 @item @code{lcm(a, b)}
1491 @tab least common multiple
1495 Most of these functions are also available as symbolic functions that can be
1496 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1497 as polynomial algorithms.
1499 @subsection Converting numbers
1501 Sometimes it is desirable to convert a @code{numeric} object back to a
1502 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1503 class provides a couple of methods for this purpose:
1505 @cindex @code{to_int()}
1506 @cindex @code{to_long()}
1507 @cindex @code{to_double()}
1508 @cindex @code{to_cl_N()}
1510 int numeric::to_int() const;
1511 long numeric::to_long() const;
1512 double numeric::to_double() const;
1513 cln::cl_N numeric::to_cl_N() const;
1516 @code{to_int()} and @code{to_long()} only work when the number they are
1517 applied on is an exact integer. Otherwise the program will halt with a
1518 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1519 rational number will return a floating-point approximation. Both
1520 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1521 part of complex numbers.
1524 @node Constants, Fundamental containers, Numbers, Basic concepts
1525 @c node-name, next, previous, up
1527 @cindex @code{constant} (class)
1530 @cindex @code{Catalan}
1531 @cindex @code{Euler}
1532 @cindex @code{evalf()}
1533 Constants behave pretty much like symbols except that they return some
1534 specific number when the method @code{.evalf()} is called.
1536 The predefined known constants are:
1539 @multitable @columnfractions .14 .32 .54
1540 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1542 @tab Archimedes' constant
1543 @tab 3.14159265358979323846264338327950288
1544 @item @code{Catalan}
1545 @tab Catalan's constant
1546 @tab 0.91596559417721901505460351493238411
1548 @tab Euler's (or Euler-Mascheroni) constant
1549 @tab 0.57721566490153286060651209008240243
1554 @node Fundamental containers, Lists, Constants, Basic concepts
1555 @c node-name, next, previous, up
1556 @section Sums, products and powers
1560 @cindex @code{power}
1562 Simple rational expressions are written down in GiNaC pretty much like
1563 in other CAS or like expressions involving numerical variables in C.
1564 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1565 been overloaded to achieve this goal. When you run the following
1566 code snippet, the constructor for an object of type @code{mul} is
1567 automatically called to hold the product of @code{a} and @code{b} and
1568 then the constructor for an object of type @code{add} is called to hold
1569 the sum of that @code{mul} object and the number one:
1573 symbol a("a"), b("b");
1578 @cindex @code{pow()}
1579 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1580 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1581 construction is necessary since we cannot safely overload the constructor
1582 @code{^} in C++ to construct a @code{power} object. If we did, it would
1583 have several counterintuitive and undesired effects:
1587 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1589 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1590 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1591 interpret this as @code{x^(a^b)}.
1593 Also, expressions involving integer exponents are very frequently used,
1594 which makes it even more dangerous to overload @code{^} since it is then
1595 hard to distinguish between the semantics as exponentiation and the one
1596 for exclusive or. (It would be embarrassing to return @code{1} where one
1597 has requested @code{2^3}.)
1600 @cindex @command{ginsh}
1601 All effects are contrary to mathematical notation and differ from the
1602 way most other CAS handle exponentiation, therefore overloading @code{^}
1603 is ruled out for GiNaC's C++ part. The situation is different in
1604 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1605 that the other frequently used exponentiation operator @code{**} does
1606 not exist at all in C++).
1608 To be somewhat more precise, objects of the three classes described
1609 here, are all containers for other expressions. An object of class
1610 @code{power} is best viewed as a container with two slots, one for the
1611 basis, one for the exponent. All valid GiNaC expressions can be
1612 inserted. However, basic transformations like simplifying
1613 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1614 when this is mathematically possible. If we replace the outer exponent
1615 three in the example by some symbols @code{a}, the simplification is not
1616 safe and will not be performed, since @code{a} might be @code{1/2} and
1619 Objects of type @code{add} and @code{mul} are containers with an
1620 arbitrary number of slots for expressions to be inserted. Again, simple
1621 and safe simplifications are carried out like transforming
1622 @code{3*x+4-x} to @code{2*x+4}.
1625 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1626 @c node-name, next, previous, up
1627 @section Lists of expressions
1628 @cindex @code{lst} (class)
1630 @cindex @code{nops()}
1632 @cindex @code{append()}
1633 @cindex @code{prepend()}
1634 @cindex @code{remove_first()}
1635 @cindex @code{remove_last()}
1636 @cindex @code{remove_all()}
1638 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1639 expressions. They are not as ubiquitous as in many other computer algebra
1640 packages, but are sometimes used to supply a variable number of arguments of
1641 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1642 constructors, so you should have a basic understanding of them.
1644 Lists can be constructed by assigning a comma-separated sequence of
1649 symbol x("x"), y("y");
1652 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1657 There are also constructors that allow direct creation of lists of up to
1658 16 expressions, which is often more convenient but slightly less efficient:
1662 // This produces the same list 'l' as above:
1663 // lst l(x, 2, y, x+y);
1664 // lst l = lst(x, 2, y, x+y);
1668 Use the @code{nops()} method to determine the size (number of expressions) of
1669 a list and the @code{op()} method or the @code{[]} operator to access
1670 individual elements:
1674 cout << l.nops() << endl; // prints '4'
1675 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1679 As with the standard @code{list<T>} container, accessing random elements of a
1680 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1681 sequential access to the elements of a list is possible with the
1682 iterator types provided by the @code{lst} class:
1685 typedef ... lst::const_iterator;
1686 typedef ... lst::const_reverse_iterator;
1687 lst::const_iterator lst::begin() const;
1688 lst::const_iterator lst::end() const;
1689 lst::const_reverse_iterator lst::rbegin() const;
1690 lst::const_reverse_iterator lst::rend() const;
1693 For example, to print the elements of a list individually you can use:
1698 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1703 which is one order faster than
1708 for (size_t i = 0; i < l.nops(); ++i)
1709 cout << l.op(i) << endl;
1713 These iterators also allow you to use some of the algorithms provided by
1714 the C++ standard library:
1718 // print the elements of the list (requires #include <iterator>)
1719 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1721 // sum up the elements of the list (requires #include <numeric>)
1722 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1723 cout << sum << endl; // prints '2+2*x+2*y'
1727 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1728 (the only other one is @code{matrix}). You can modify single elements:
1732 l[1] = 42; // l is now @{x, 42, y, x+y@}
1733 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1737 You can append or prepend an expression to a list with the @code{append()}
1738 and @code{prepend()} methods:
1742 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1743 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1747 You can remove the first or last element of a list with @code{remove_first()}
1748 and @code{remove_last()}:
1752 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1753 l.remove_last(); // l is now @{x, 7, y, x+y@}
1757 You can remove all the elements of a list with @code{remove_all()}:
1761 l.remove_all(); // l is now empty
1765 You can bring the elements of a list into a canonical order with @code{sort()}:
1774 // l1 and l2 are now equal
1778 Finally, you can remove all but the first element of consecutive groups of
1779 elements with @code{unique()}:
1784 l3 = x, 2, 2, 2, y, x+y, y+x;
1785 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1790 @node Mathematical functions, Relations, Lists, Basic concepts
1791 @c node-name, next, previous, up
1792 @section Mathematical functions
1793 @cindex @code{function} (class)
1794 @cindex trigonometric function
1795 @cindex hyperbolic function
1797 There are quite a number of useful functions hard-wired into GiNaC. For
1798 instance, all trigonometric and hyperbolic functions are implemented
1799 (@xref{Built-in functions}, for a complete list).
1801 These functions (better called @emph{pseudofunctions}) are all objects
1802 of class @code{function}. They accept one or more expressions as
1803 arguments and return one expression. If the arguments are not
1804 numerical, the evaluation of the function may be halted, as it does in
1805 the next example, showing how a function returns itself twice and
1806 finally an expression that may be really useful:
1808 @cindex Gamma function
1809 @cindex @code{subs()}
1812 symbol x("x"), y("y");
1814 cout << tgamma(foo) << endl;
1815 // -> tgamma(x+(1/2)*y)
1816 ex bar = foo.subs(y==1);
1817 cout << tgamma(bar) << endl;
1819 ex foobar = bar.subs(x==7);
1820 cout << tgamma(foobar) << endl;
1821 // -> (135135/128)*Pi^(1/2)
1825 Besides evaluation most of these functions allow differentiation, series
1826 expansion and so on. Read the next chapter in order to learn more about
1829 It must be noted that these pseudofunctions are created by inline
1830 functions, where the argument list is templated. This means that
1831 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1832 @code{sin(ex(1))} and will therefore not result in a floating point
1833 number. Unless of course the function prototype is explicitly
1834 overridden -- which is the case for arguments of type @code{numeric}
1835 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1836 point number of class @code{numeric} you should call
1837 @code{sin(numeric(1))}. This is almost the same as calling
1838 @code{sin(1).evalf()} except that the latter will return a numeric
1839 wrapped inside an @code{ex}.
1842 @node Relations, Integrals, Mathematical functions, Basic concepts
1843 @c node-name, next, previous, up
1845 @cindex @code{relational} (class)
1847 Sometimes, a relation holding between two expressions must be stored
1848 somehow. The class @code{relational} is a convenient container for such
1849 purposes. A relation is by definition a container for two @code{ex} and
1850 a relation between them that signals equality, inequality and so on.
1851 They are created by simply using the C++ operators @code{==}, @code{!=},
1852 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1854 @xref{Mathematical functions}, for examples where various applications
1855 of the @code{.subs()} method show how objects of class relational are
1856 used as arguments. There they provide an intuitive syntax for
1857 substitutions. They are also used as arguments to the @code{ex::series}
1858 method, where the left hand side of the relation specifies the variable
1859 to expand in and the right hand side the expansion point. They can also
1860 be used for creating systems of equations that are to be solved for
1861 unknown variables. But the most common usage of objects of this class
1862 is rather inconspicuous in statements of the form @code{if
1863 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1864 conversion from @code{relational} to @code{bool} takes place. Note,
1865 however, that @code{==} here does not perform any simplifications, hence
1866 @code{expand()} must be called explicitly.
1868 @node Integrals, Matrices, Relations, Basic concepts
1869 @c node-name, next, previous, up
1871 @cindex @code{integral} (class)
1873 An object of class @dfn{integral} can be used to hold a symbolic integral.
1874 If you want to symbolically represent the integral of @code{x*x} from 0 to
1875 1, you would write this as
1877 integral(x, 0, 1, x*x)
1879 The first argument is the integration variable. It should be noted that
1880 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1881 fact, it can only integrate polynomials. An expression containing integrals
1882 can be evaluated symbolically by calling the
1886 method on it. Numerical evaluation is available by calling the
1890 method on an expression containing the integral. This will only evaluate
1891 integrals into a number if @code{subs}ing the integration variable by a
1892 number in the fourth argument of an integral and then @code{evalf}ing the
1893 result always results in a number. Of course, also the boundaries of the
1894 integration domain must @code{evalf} into numbers. It should be noted that
1895 trying to @code{evalf} a function with discontinuities in the integration
1896 domain is not recommended. The accuracy of the numeric evaluation of
1897 integrals is determined by the static member variable
1899 ex integral::relative_integration_error
1901 of the class @code{integral}. The default value of this is 10^-8.
1902 The integration works by halving the interval of integration, until numeric
1903 stability of the answer indicates that the requested accuracy has been
1904 reached. The maximum depth of the halving can be set via the static member
1907 int integral::max_integration_level
1909 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1910 return the integral unevaluated. The function that performs the numerical
1911 evaluation, is also available as
1913 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1916 This function will throw an exception if the maximum depth is exceeded. The
1917 last parameter of the function is optional and defaults to the
1918 @code{relative_integration_error}. To make sure that we do not do too
1919 much work if an expression contains the same integral multiple times,
1920 a lookup table is used.
1922 If you know that an expression holds an integral, you can get the
1923 integration variable, the left boundary, right boundary and integrand by
1924 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1925 @code{.op(3)}. Differentiating integrals with respect to variables works
1926 as expected. Note that it makes no sense to differentiate an integral
1927 with respect to the integration variable.
1929 @node Matrices, Indexed objects, Integrals, Basic concepts
1930 @c node-name, next, previous, up
1932 @cindex @code{matrix} (class)
1934 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1935 matrix with @math{m} rows and @math{n} columns are accessed with two
1936 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1937 second one in the range 0@dots{}@math{n-1}.
1939 There are a couple of ways to construct matrices, with or without preset
1940 elements. The constructor
1943 matrix::matrix(unsigned r, unsigned c);
1946 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1949 The fastest way to create a matrix with preinitialized elements is to assign
1950 a list of comma-separated expressions to an empty matrix (see below for an
1951 example). But you can also specify the elements as a (flat) list with
1954 matrix::matrix(unsigned r, unsigned c, const lst & l);
1959 @cindex @code{lst_to_matrix()}
1961 ex lst_to_matrix(const lst & l);
1964 constructs a matrix from a list of lists, each list representing a matrix row.
1966 There is also a set of functions for creating some special types of
1969 @cindex @code{diag_matrix()}
1970 @cindex @code{unit_matrix()}
1971 @cindex @code{symbolic_matrix()}
1973 ex diag_matrix(const lst & l);
1974 ex unit_matrix(unsigned x);
1975 ex unit_matrix(unsigned r, unsigned c);
1976 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1977 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1978 const string & tex_base_name);
1981 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1982 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1983 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1984 matrix filled with newly generated symbols made of the specified base name
1985 and the position of each element in the matrix.
1987 Matrices often arise by omitting elements of another matrix. For
1988 instance, the submatrix @code{S} of a matrix @code{M} takes a
1989 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1990 by removing one row and one column from a matrix @code{M}. (The
1991 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1992 can be used for computing the inverse using Cramer's rule.)
1994 @cindex @code{sub_matrix()}
1995 @cindex @code{reduced_matrix()}
1997 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1998 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2001 The function @code{sub_matrix()} takes a row offset @code{r} and a
2002 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2003 columns. The function @code{reduced_matrix()} has two integer arguments
2004 that specify which row and column to remove:
2012 cout << reduced_matrix(m, 1, 1) << endl;
2013 // -> [[11,13],[31,33]]
2014 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2015 // -> [[22,23],[32,33]]
2019 Matrix elements can be accessed and set using the parenthesis (function call)
2023 const ex & matrix::operator()(unsigned r, unsigned c) const;
2024 ex & matrix::operator()(unsigned r, unsigned c);
2027 It is also possible to access the matrix elements in a linear fashion with
2028 the @code{op()} method. But C++-style subscripting with square brackets
2029 @samp{[]} is not available.
2031 Here are a couple of examples for constructing matrices:
2035 symbol a("a"), b("b");
2049 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2052 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2055 cout << diag_matrix(lst(a, b)) << endl;
2058 cout << unit_matrix(3) << endl;
2059 // -> [[1,0,0],[0,1,0],[0,0,1]]
2061 cout << symbolic_matrix(2, 3, "x") << endl;
2062 // -> [[x00,x01,x02],[x10,x11,x12]]
2066 @cindex @code{is_zero_matrix()}
2067 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2068 all entries of the matrix are zeros. There is also method
2069 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2070 expression is zero or a zero matrix.
2072 @cindex @code{transpose()}
2073 There are three ways to do arithmetic with matrices. The first (and most
2074 direct one) is to use the methods provided by the @code{matrix} class:
2077 matrix matrix::add(const matrix & other) const;
2078 matrix matrix::sub(const matrix & other) const;
2079 matrix matrix::mul(const matrix & other) const;
2080 matrix matrix::mul_scalar(const ex & other) const;
2081 matrix matrix::pow(const ex & expn) const;
2082 matrix matrix::transpose() const;
2085 All of these methods return the result as a new matrix object. Here is an
2086 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2091 matrix A(2, 2), B(2, 2), C(2, 2);
2099 matrix result = A.mul(B).sub(C.mul_scalar(2));
2100 cout << result << endl;
2101 // -> [[-13,-6],[1,2]]
2106 @cindex @code{evalm()}
2107 The second (and probably the most natural) way is to construct an expression
2108 containing matrices with the usual arithmetic operators and @code{pow()}.
2109 For efficiency reasons, expressions with sums, products and powers of
2110 matrices are not automatically evaluated in GiNaC. You have to call the
2114 ex ex::evalm() const;
2117 to obtain the result:
2124 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2125 cout << e.evalm() << endl;
2126 // -> [[-13,-6],[1,2]]
2131 The non-commutativity of the product @code{A*B} in this example is
2132 automatically recognized by GiNaC. There is no need to use a special
2133 operator here. @xref{Non-commutative objects}, for more information about
2134 dealing with non-commutative expressions.
2136 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2137 to perform the arithmetic:
2142 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2143 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2145 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2146 cout << e.simplify_indexed() << endl;
2147 // -> [[-13,-6],[1,2]].i.j
2151 Using indices is most useful when working with rectangular matrices and
2152 one-dimensional vectors because you don't have to worry about having to
2153 transpose matrices before multiplying them. @xref{Indexed objects}, for
2154 more information about using matrices with indices, and about indices in
2157 The @code{matrix} class provides a couple of additional methods for
2158 computing determinants, traces, characteristic polynomials and ranks:
2160 @cindex @code{determinant()}
2161 @cindex @code{trace()}
2162 @cindex @code{charpoly()}
2163 @cindex @code{rank()}
2165 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2166 ex matrix::trace() const;
2167 ex matrix::charpoly(const ex & lambda) const;
2168 unsigned matrix::rank() const;
2171 The @samp{algo} argument of @code{determinant()} allows to select
2172 between different algorithms for calculating the determinant. The
2173 asymptotic speed (as parametrized by the matrix size) can greatly differ
2174 between those algorithms, depending on the nature of the matrix'
2175 entries. The possible values are defined in the @file{flags.h} header
2176 file. By default, GiNaC uses a heuristic to automatically select an
2177 algorithm that is likely (but not guaranteed) to give the result most
2180 @cindex @code{inverse()} (matrix)
2181 @cindex @code{solve()}
2182 Matrices may also be inverted using the @code{ex matrix::inverse()}
2183 method and linear systems may be solved with:
2186 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2187 unsigned algo=solve_algo::automatic) const;
2190 Assuming the matrix object this method is applied on is an @code{m}
2191 times @code{n} matrix, then @code{vars} must be a @code{n} times
2192 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2193 times @code{p} matrix. The returned matrix then has dimension @code{n}
2194 times @code{p} and in the case of an underdetermined system will still
2195 contain some of the indeterminates from @code{vars}. If the system is
2196 overdetermined, an exception is thrown.
2199 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2200 @c node-name, next, previous, up
2201 @section Indexed objects
2203 GiNaC allows you to handle expressions containing general indexed objects in
2204 arbitrary spaces. It is also able to canonicalize and simplify such
2205 expressions and perform symbolic dummy index summations. There are a number
2206 of predefined indexed objects provided, like delta and metric tensors.
2208 There are few restrictions placed on indexed objects and their indices and
2209 it is easy to construct nonsense expressions, but our intention is to
2210 provide a general framework that allows you to implement algorithms with
2211 indexed quantities, getting in the way as little as possible.
2213 @cindex @code{idx} (class)
2214 @cindex @code{indexed} (class)
2215 @subsection Indexed quantities and their indices
2217 Indexed expressions in GiNaC are constructed of two special types of objects,
2218 @dfn{index objects} and @dfn{indexed objects}.
2222 @cindex contravariant
2225 @item Index objects are of class @code{idx} or a subclass. Every index has
2226 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2227 the index lives in) which can both be arbitrary expressions but are usually
2228 a number or a simple symbol. In addition, indices of class @code{varidx} have
2229 a @dfn{variance} (they can be co- or contravariant), and indices of class
2230 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2232 @item Indexed objects are of class @code{indexed} or a subclass. They
2233 contain a @dfn{base expression} (which is the expression being indexed), and
2234 one or more indices.
2238 @strong{Please notice:} when printing expressions, covariant indices and indices
2239 without variance are denoted @samp{.i} while contravariant indices are
2240 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2241 value. In the following, we are going to use that notation in the text so
2242 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2243 not visible in the output.
2245 A simple example shall illustrate the concepts:
2249 #include <ginac/ginac.h>
2250 using namespace std;
2251 using namespace GiNaC;
2255 symbol i_sym("i"), j_sym("j");
2256 idx i(i_sym, 3), j(j_sym, 3);
2259 cout << indexed(A, i, j) << endl;
2261 cout << index_dimensions << indexed(A, i, j) << endl;
2263 cout << dflt; // reset cout to default output format (dimensions hidden)
2267 The @code{idx} constructor takes two arguments, the index value and the
2268 index dimension. First we define two index objects, @code{i} and @code{j},
2269 both with the numeric dimension 3. The value of the index @code{i} is the
2270 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2271 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2272 construct an expression containing one indexed object, @samp{A.i.j}. It has
2273 the symbol @code{A} as its base expression and the two indices @code{i} and
2276 The dimensions of indices are normally not visible in the output, but one
2277 can request them to be printed with the @code{index_dimensions} manipulator,
2280 Note the difference between the indices @code{i} and @code{j} which are of
2281 class @code{idx}, and the index values which are the symbols @code{i_sym}
2282 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2283 or numbers but must be index objects. For example, the following is not
2284 correct and will raise an exception:
2287 symbol i("i"), j("j");
2288 e = indexed(A, i, j); // ERROR: indices must be of type idx
2291 You can have multiple indexed objects in an expression, index values can
2292 be numeric, and index dimensions symbolic:
2296 symbol B("B"), dim("dim");
2297 cout << 4 * indexed(A, i)
2298 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2303 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2304 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2305 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2306 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2307 @code{simplify_indexed()} for that, see below).
2309 In fact, base expressions, index values and index dimensions can be
2310 arbitrary expressions:
2314 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2319 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2320 get an error message from this but you will probably not be able to do
2321 anything useful with it.
2323 @cindex @code{get_value()}
2324 @cindex @code{get_dimension()}
2328 ex idx::get_value();
2329 ex idx::get_dimension();
2332 return the value and dimension of an @code{idx} object. If you have an index
2333 in an expression, such as returned by calling @code{.op()} on an indexed
2334 object, you can get a reference to the @code{idx} object with the function
2335 @code{ex_to<idx>()} on the expression.
2337 There are also the methods
2340 bool idx::is_numeric();
2341 bool idx::is_symbolic();
2342 bool idx::is_dim_numeric();
2343 bool idx::is_dim_symbolic();
2346 for checking whether the value and dimension are numeric or symbolic
2347 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2348 about expressions}) returns information about the index value.
2350 @cindex @code{varidx} (class)
2351 If you need co- and contravariant indices, use the @code{varidx} class:
2355 symbol mu_sym("mu"), nu_sym("nu");
2356 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2357 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2359 cout << indexed(A, mu, nu) << endl;
2361 cout << indexed(A, mu_co, nu) << endl;
2363 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2368 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2369 co- or contravariant. The default is a contravariant (upper) index, but
2370 this can be overridden by supplying a third argument to the @code{varidx}
2371 constructor. The two methods
2374 bool varidx::is_covariant();
2375 bool varidx::is_contravariant();
2378 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2379 to get the object reference from an expression). There's also the very useful
2383 ex varidx::toggle_variance();
2386 which makes a new index with the same value and dimension but the opposite
2387 variance. By using it you only have to define the index once.
2389 @cindex @code{spinidx} (class)
2390 The @code{spinidx} class provides dotted and undotted variant indices, as
2391 used in the Weyl-van-der-Waerden spinor formalism:
2395 symbol K("K"), C_sym("C"), D_sym("D");
2396 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2397 // contravariant, undotted
2398 spinidx C_co(C_sym, 2, true); // covariant index
2399 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2400 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2402 cout << indexed(K, C, D) << endl;
2404 cout << indexed(K, C_co, D_dot) << endl;
2406 cout << indexed(K, D_co_dot, D) << endl;
2411 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2412 dotted or undotted. The default is undotted but this can be overridden by
2413 supplying a fourth argument to the @code{spinidx} constructor. The two
2417 bool spinidx::is_dotted();
2418 bool spinidx::is_undotted();
2421 allow you to check whether or not a @code{spinidx} object is dotted (use
2422 @code{ex_to<spinidx>()} to get the object reference from an expression).
2423 Finally, the two methods
2426 ex spinidx::toggle_dot();
2427 ex spinidx::toggle_variance_dot();
2430 create a new index with the same value and dimension but opposite dottedness
2431 and the same or opposite variance.
2433 @subsection Substituting indices
2435 @cindex @code{subs()}
2436 Sometimes you will want to substitute one symbolic index with another
2437 symbolic or numeric index, for example when calculating one specific element
2438 of a tensor expression. This is done with the @code{.subs()} method, as it
2439 is done for symbols (see @ref{Substituting expressions}).
2441 You have two possibilities here. You can either substitute the whole index
2442 by another index or expression:
2446 ex e = indexed(A, mu_co);
2447 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2448 // -> A.mu becomes A~nu
2449 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2450 // -> A.mu becomes A~0
2451 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2452 // -> A.mu becomes A.0
2456 The third example shows that trying to replace an index with something that
2457 is not an index will substitute the index value instead.
2459 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2464 ex e = indexed(A, mu_co);
2465 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2466 // -> A.mu becomes A.nu
2467 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2468 // -> A.mu becomes A.0
2472 As you see, with the second method only the value of the index will get
2473 substituted. Its other properties, including its dimension, remain unchanged.
2474 If you want to change the dimension of an index you have to substitute the
2475 whole index by another one with the new dimension.
2477 Finally, substituting the base expression of an indexed object works as
2482 ex e = indexed(A, mu_co);
2483 cout << e << " becomes " << e.subs(A == A+B) << endl;
2484 // -> A.mu becomes (B+A).mu
2488 @subsection Symmetries
2489 @cindex @code{symmetry} (class)
2490 @cindex @code{sy_none()}
2491 @cindex @code{sy_symm()}
2492 @cindex @code{sy_anti()}
2493 @cindex @code{sy_cycl()}
2495 Indexed objects can have certain symmetry properties with respect to their
2496 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2497 that is constructed with the helper functions
2500 symmetry sy_none(...);
2501 symmetry sy_symm(...);
2502 symmetry sy_anti(...);
2503 symmetry sy_cycl(...);
2506 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2507 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2508 represents a cyclic symmetry. Each of these functions accepts up to four
2509 arguments which can be either symmetry objects themselves or unsigned integer
2510 numbers that represent an index position (counting from 0). A symmetry
2511 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2512 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2515 Here are some examples of symmetry definitions:
2520 e = indexed(A, i, j);
2521 e = indexed(A, sy_none(), i, j); // equivalent
2522 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2524 // Symmetric in all three indices:
2525 e = indexed(A, sy_symm(), i, j, k);
2526 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2527 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2528 // different canonical order
2530 // Symmetric in the first two indices only:
2531 e = indexed(A, sy_symm(0, 1), i, j, k);
2532 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2534 // Antisymmetric in the first and last index only (index ranges need not
2536 e = indexed(A, sy_anti(0, 2), i, j, k);
2537 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2539 // An example of a mixed symmetry: antisymmetric in the first two and
2540 // last two indices, symmetric when swapping the first and last index
2541 // pairs (like the Riemann curvature tensor):
2542 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2544 // Cyclic symmetry in all three indices:
2545 e = indexed(A, sy_cycl(), i, j, k);
2546 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2548 // The following examples are invalid constructions that will throw
2549 // an exception at run time.
2551 // An index may not appear multiple times:
2552 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2553 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2555 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2556 // same number of indices:
2557 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2559 // And of course, you cannot specify indices which are not there:
2560 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2564 If you need to specify more than four indices, you have to use the
2565 @code{.add()} method of the @code{symmetry} class. For example, to specify
2566 full symmetry in the first six indices you would write
2567 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2569 If an indexed object has a symmetry, GiNaC will automatically bring the
2570 indices into a canonical order which allows for some immediate simplifications:
2574 cout << indexed(A, sy_symm(), i, j)
2575 + indexed(A, sy_symm(), j, i) << endl;
2577 cout << indexed(B, sy_anti(), i, j)
2578 + indexed(B, sy_anti(), j, i) << endl;
2580 cout << indexed(B, sy_anti(), i, j, k)
2581 - indexed(B, sy_anti(), j, k, i) << endl;
2586 @cindex @code{get_free_indices()}
2588 @subsection Dummy indices
2590 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2591 that a summation over the index range is implied. Symbolic indices which are
2592 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2593 dummy nor free indices.
2595 To be recognized as a dummy index pair, the two indices must be of the same
2596 class and their value must be the same single symbol (an index like
2597 @samp{2*n+1} is never a dummy index). If the indices are of class
2598 @code{varidx} they must also be of opposite variance; if they are of class
2599 @code{spinidx} they must be both dotted or both undotted.
2601 The method @code{.get_free_indices()} returns a vector containing the free
2602 indices of an expression. It also checks that the free indices of the terms
2603 of a sum are consistent:
2607 symbol A("A"), B("B"), C("C");
2609 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2610 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2612 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2613 cout << exprseq(e.get_free_indices()) << endl;
2615 // 'j' and 'l' are dummy indices
2617 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2618 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2620 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2621 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2622 cout << exprseq(e.get_free_indices()) << endl;
2624 // 'nu' is a dummy index, but 'sigma' is not
2626 e = indexed(A, mu, mu);
2627 cout << exprseq(e.get_free_indices()) << endl;
2629 // 'mu' is not a dummy index because it appears twice with the same
2632 e = indexed(A, mu, nu) + 42;
2633 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2634 // this will throw an exception:
2635 // "add::get_free_indices: inconsistent indices in sum"
2639 @cindex @code{expand_dummy_sum()}
2640 A dummy index summation like
2647 can be expanded for indices with numeric
2648 dimensions (e.g. 3) into the explicit sum like
2650 $a_1b^1+a_2b^2+a_3b^3 $.
2653 a.1 b~1 + a.2 b~2 + a.3 b~3.
2655 This is performed by the function
2658 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2661 which takes an expression @code{e} and returns the expanded sum for all
2662 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2663 is set to @code{true} then all substitutions are made by @code{idx} class
2664 indices, i.e. without variance. In this case the above sum
2673 $a_1b_1+a_2b_2+a_3b_3 $.
2676 a.1 b.1 + a.2 b.2 + a.3 b.3.
2680 @cindex @code{simplify_indexed()}
2681 @subsection Simplifying indexed expressions
2683 In addition to the few automatic simplifications that GiNaC performs on
2684 indexed expressions (such as re-ordering the indices of symmetric tensors
2685 and calculating traces and convolutions of matrices and predefined tensors)
2689 ex ex::simplify_indexed();
2690 ex ex::simplify_indexed(const scalar_products & sp);
2693 that performs some more expensive operations:
2696 @item it checks the consistency of free indices in sums in the same way
2697 @code{get_free_indices()} does
2698 @item it tries to give dummy indices that appear in different terms of a sum
2699 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2700 @item it (symbolically) calculates all possible dummy index summations/contractions
2701 with the predefined tensors (this will be explained in more detail in the
2703 @item it detects contractions that vanish for symmetry reasons, for example
2704 the contraction of a symmetric and a totally antisymmetric tensor
2705 @item as a special case of dummy index summation, it can replace scalar products
2706 of two tensors with a user-defined value
2709 The last point is done with the help of the @code{scalar_products} class
2710 which is used to store scalar products with known values (this is not an
2711 arithmetic class, you just pass it to @code{simplify_indexed()}):
2715 symbol A("A"), B("B"), C("C"), i_sym("i");
2719 sp.add(A, B, 0); // A and B are orthogonal
2720 sp.add(A, C, 0); // A and C are orthogonal
2721 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2723 e = indexed(A + B, i) * indexed(A + C, i);
2725 // -> (B+A).i*(A+C).i
2727 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2733 The @code{scalar_products} object @code{sp} acts as a storage for the
2734 scalar products added to it with the @code{.add()} method. This method
2735 takes three arguments: the two expressions of which the scalar product is
2736 taken, and the expression to replace it with.
2738 @cindex @code{expand()}
2739 The example above also illustrates a feature of the @code{expand()} method:
2740 if passed the @code{expand_indexed} option it will distribute indices
2741 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2743 @cindex @code{tensor} (class)
2744 @subsection Predefined tensors
2746 Some frequently used special tensors such as the delta, epsilon and metric
2747 tensors are predefined in GiNaC. They have special properties when
2748 contracted with other tensor expressions and some of them have constant
2749 matrix representations (they will evaluate to a number when numeric
2750 indices are specified).
2752 @cindex @code{delta_tensor()}
2753 @subsubsection Delta tensor
2755 The delta tensor takes two indices, is symmetric and has the matrix
2756 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2757 @code{delta_tensor()}:
2761 symbol A("A"), B("B");
2763 idx i(symbol("i"), 3), j(symbol("j"), 3),
2764 k(symbol("k"), 3), l(symbol("l"), 3);
2766 ex e = indexed(A, i, j) * indexed(B, k, l)
2767 * delta_tensor(i, k) * delta_tensor(j, l);
2768 cout << e.simplify_indexed() << endl;
2771 cout << delta_tensor(i, i) << endl;
2776 @cindex @code{metric_tensor()}
2777 @subsubsection General metric tensor
2779 The function @code{metric_tensor()} creates a general symmetric metric
2780 tensor with two indices that can be used to raise/lower tensor indices. The
2781 metric tensor is denoted as @samp{g} in the output and if its indices are of
2782 mixed variance it is automatically replaced by a delta tensor:
2788 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2790 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2791 cout << e.simplify_indexed() << endl;
2794 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2795 cout << e.simplify_indexed() << endl;
2798 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2799 * metric_tensor(nu, rho);
2800 cout << e.simplify_indexed() << endl;
2803 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2804 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2805 + indexed(A, mu.toggle_variance(), rho));
2806 cout << e.simplify_indexed() << endl;
2811 @cindex @code{lorentz_g()}
2812 @subsubsection Minkowski metric tensor
2814 The Minkowski metric tensor is a special metric tensor with a constant
2815 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2816 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2817 It is created with the function @code{lorentz_g()} (although it is output as
2822 varidx mu(symbol("mu"), 4);
2824 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2825 * lorentz_g(mu, varidx(0, 4)); // negative signature
2826 cout << e.simplify_indexed() << endl;
2829 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2830 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2831 cout << e.simplify_indexed() << endl;
2836 @cindex @code{spinor_metric()}
2837 @subsubsection Spinor metric tensor
2839 The function @code{spinor_metric()} creates an antisymmetric tensor with
2840 two indices that is used to raise/lower indices of 2-component spinors.
2841 It is output as @samp{eps}:
2847 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2848 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2850 e = spinor_metric(A, B) * indexed(psi, B_co);
2851 cout << e.simplify_indexed() << endl;
2854 e = spinor_metric(A, B) * indexed(psi, A_co);
2855 cout << e.simplify_indexed() << endl;
2858 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2859 cout << e.simplify_indexed() << endl;
2862 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2863 cout << e.simplify_indexed() << endl;
2866 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2867 cout << e.simplify_indexed() << endl;
2870 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2871 cout << e.simplify_indexed() << endl;
2876 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2878 @cindex @code{epsilon_tensor()}
2879 @cindex @code{lorentz_eps()}
2880 @subsubsection Epsilon tensor
2882 The epsilon tensor is totally antisymmetric, its number of indices is equal
2883 to the dimension of the index space (the indices must all be of the same
2884 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2885 defined to be 1. Its behavior with indices that have a variance also
2886 depends on the signature of the metric. Epsilon tensors are output as
2889 There are three functions defined to create epsilon tensors in 2, 3 and 4
2893 ex epsilon_tensor(const ex & i1, const ex & i2);
2894 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2895 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2896 bool pos_sig = false);
2899 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2900 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2901 Minkowski space (the last @code{bool} argument specifies whether the metric
2902 has negative or positive signature, as in the case of the Minkowski metric
2907 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2908 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2909 e = lorentz_eps(mu, nu, rho, sig) *
2910 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2911 cout << simplify_indexed(e) << endl;
2912 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2914 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2915 symbol A("A"), B("B");
2916 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2917 cout << simplify_indexed(e) << endl;
2918 // -> -B.k*A.j*eps.i.k.j
2919 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2920 cout << simplify_indexed(e) << endl;
2925 @subsection Linear algebra
2927 The @code{matrix} class can be used with indices to do some simple linear
2928 algebra (linear combinations and products of vectors and matrices, traces
2929 and scalar products):
2933 idx i(symbol("i"), 2), j(symbol("j"), 2);
2934 symbol x("x"), y("y");
2936 // A is a 2x2 matrix, X is a 2x1 vector
2937 matrix A(2, 2), X(2, 1);
2942 cout << indexed(A, i, i) << endl;
2945 ex e = indexed(A, i, j) * indexed(X, j);
2946 cout << e.simplify_indexed() << endl;
2947 // -> [[2*y+x],[4*y+3*x]].i
2949 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2950 cout << e.simplify_indexed() << endl;
2951 // -> [[3*y+3*x,6*y+2*x]].j
2955 You can of course obtain the same results with the @code{matrix::add()},
2956 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2957 but with indices you don't have to worry about transposing matrices.
2959 Matrix indices always start at 0 and their dimension must match the number
2960 of rows/columns of the matrix. Matrices with one row or one column are
2961 vectors and can have one or two indices (it doesn't matter whether it's a
2962 row or a column vector). Other matrices must have two indices.
2964 You should be careful when using indices with variance on matrices. GiNaC
2965 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2966 @samp{F.mu.nu} are different matrices. In this case you should use only
2967 one form for @samp{F} and explicitly multiply it with a matrix representation
2968 of the metric tensor.
2971 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2972 @c node-name, next, previous, up
2973 @section Non-commutative objects
2975 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2976 non-commutative objects are built-in which are mostly of use in high energy
2980 @item Clifford (Dirac) algebra (class @code{clifford})
2981 @item su(3) Lie algebra (class @code{color})
2982 @item Matrices (unindexed) (class @code{matrix})
2985 The @code{clifford} and @code{color} classes are subclasses of
2986 @code{indexed} because the elements of these algebras usually carry
2987 indices. The @code{matrix} class is described in more detail in
2990 Unlike most computer algebra systems, GiNaC does not primarily provide an
2991 operator (often denoted @samp{&*}) for representing inert products of
2992 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2993 classes of objects involved, and non-commutative products are formed with
2994 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2995 figuring out by itself which objects commutate and will group the factors
2996 by their class. Consider this example:
3000 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3001 idx a(symbol("a"), 8), b(symbol("b"), 8);
3002 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3004 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3008 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3009 groups the non-commutative factors (the gammas and the su(3) generators)
3010 together while preserving the order of factors within each class (because
3011 Clifford objects commutate with color objects). The resulting expression is a
3012 @emph{commutative} product with two factors that are themselves non-commutative
3013 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3014 parentheses are placed around the non-commutative products in the output.
3016 @cindex @code{ncmul} (class)
3017 Non-commutative products are internally represented by objects of the class
3018 @code{ncmul}, as opposed to commutative products which are handled by the
3019 @code{mul} class. You will normally not have to worry about this distinction,
3022 The advantage of this approach is that you never have to worry about using
3023 (or forgetting to use) a special operator when constructing non-commutative
3024 expressions. Also, non-commutative products in GiNaC are more intelligent
3025 than in other computer algebra systems; they can, for example, automatically
3026 canonicalize themselves according to rules specified in the implementation
3027 of the non-commutative classes. The drawback is that to work with other than
3028 the built-in algebras you have to implement new classes yourself. Both
3029 symbols and user-defined functions can be specified as being non-commutative.
3031 @cindex @code{return_type()}
3032 @cindex @code{return_type_tinfo()}
3033 Information about the commutativity of an object or expression can be
3034 obtained with the two member functions
3037 unsigned ex::return_type() const;
3038 unsigned ex::return_type_tinfo() const;
3041 The @code{return_type()} function returns one of three values (defined in
3042 the header file @file{flags.h}), corresponding to three categories of
3043 expressions in GiNaC:
3046 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3047 classes are of this kind.
3048 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3049 certain class of non-commutative objects which can be determined with the
3050 @code{return_type_tinfo()} method. Expressions of this category commutate
3051 with everything except @code{noncommutative} expressions of the same
3053 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3054 of non-commutative objects of different classes. Expressions of this
3055 category don't commutate with any other @code{noncommutative} or
3056 @code{noncommutative_composite} expressions.
3059 The value returned by the @code{return_type_tinfo()} method is valid only
3060 when the return type of the expression is @code{noncommutative}. It is a
3061 value that is unique to the class of the object, but may vary every time a
3062 GiNaC program is being run (it is dynamically assigned on start-up).
3064 Here are a couple of examples:
3067 @multitable @columnfractions 0.33 0.33 0.34
3068 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3069 @item @code{42} @tab @code{commutative} @tab -
3070 @item @code{2*x-y} @tab @code{commutative} @tab -
3071 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3072 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3073 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3074 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3078 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3079 @code{TINFO_clifford} for objects with a representation label of zero.
3080 Other representation labels yield a different @code{return_type_tinfo()},
3081 but it's the same for any two objects with the same label. This is also true
3084 A last note: With the exception of matrices, positive integer powers of
3085 non-commutative objects are automatically expanded in GiNaC. For example,
3086 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3087 non-commutative expressions).
3090 @cindex @code{clifford} (class)
3091 @subsection Clifford algebra
3094 Clifford algebras are supported in two flavours: Dirac gamma
3095 matrices (more physical) and generic Clifford algebras (more
3098 @cindex @code{dirac_gamma()}
3099 @subsubsection Dirac gamma matrices
3100 Dirac gamma matrices (note that GiNaC doesn't treat them
3101 as matrices) are designated as @samp{gamma~mu} and satisfy
3102 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3103 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3104 constructed by the function
3107 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3110 which takes two arguments: the index and a @dfn{representation label} in the
3111 range 0 to 255 which is used to distinguish elements of different Clifford
3112 algebras (this is also called a @dfn{spin line index}). Gammas with different
3113 labels commutate with each other. The dimension of the index can be 4 or (in
3114 the framework of dimensional regularization) any symbolic value. Spinor
3115 indices on Dirac gammas are not supported in GiNaC.
3117 @cindex @code{dirac_ONE()}
3118 The unity element of a Clifford algebra is constructed by
3121 ex dirac_ONE(unsigned char rl = 0);
3124 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3125 multiples of the unity element, even though it's customary to omit it.
3126 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3127 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3128 GiNaC will complain and/or produce incorrect results.
3130 @cindex @code{dirac_gamma5()}
3131 There is a special element @samp{gamma5} that commutates with all other
3132 gammas, has a unit square, and in 4 dimensions equals
3133 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3136 ex dirac_gamma5(unsigned char rl = 0);
3139 @cindex @code{dirac_gammaL()}
3140 @cindex @code{dirac_gammaR()}
3141 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3142 objects, constructed by
3145 ex dirac_gammaL(unsigned char rl = 0);
3146 ex dirac_gammaR(unsigned char rl = 0);
3149 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3150 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3152 @cindex @code{dirac_slash()}
3153 Finally, the function
3156 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3159 creates a term that represents a contraction of @samp{e} with the Dirac
3160 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3161 with a unique index whose dimension is given by the @code{dim} argument).
3162 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3164 In products of dirac gammas, superfluous unity elements are automatically
3165 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3166 and @samp{gammaR} are moved to the front.
3168 The @code{simplify_indexed()} function performs contractions in gamma strings,
3174 symbol a("a"), b("b"), D("D");
3175 varidx mu(symbol("mu"), D);
3176 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3177 * dirac_gamma(mu.toggle_variance());
3179 // -> gamma~mu*a\*gamma.mu
3180 e = e.simplify_indexed();
3183 cout << e.subs(D == 4) << endl;
3189 @cindex @code{dirac_trace()}
3190 To calculate the trace of an expression containing strings of Dirac gammas
3191 you use one of the functions
3194 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3195 const ex & trONE = 4);
3196 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3197 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3200 These functions take the trace over all gammas in the specified set @code{rls}
3201 or list @code{rll} of representation labels, or the single label @code{rl};
3202 gammas with other labels are left standing. The last argument to
3203 @code{dirac_trace()} is the value to be returned for the trace of the unity
3204 element, which defaults to 4.
3206 The @code{dirac_trace()} function is a linear functional that is equal to the
3207 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3208 functional is not cyclic in
3214 dimensions when acting on
3215 expressions containing @samp{gamma5}, so it's not a proper trace. This
3216 @samp{gamma5} scheme is described in greater detail in the article
3217 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3219 The value of the trace itself is also usually different in 4 and in
3230 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3231 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3232 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3233 cout << dirac_trace(e).simplify_indexed() << endl;
3240 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3241 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3242 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3243 cout << dirac_trace(e).simplify_indexed() << endl;
3244 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3248 Here is an example for using @code{dirac_trace()} to compute a value that
3249 appears in the calculation of the one-loop vacuum polarization amplitude in
3254 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3255 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3258 sp.add(l, l, pow(l, 2));
3259 sp.add(l, q, ldotq);
3261 ex e = dirac_gamma(mu) *
3262 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3263 dirac_gamma(mu.toggle_variance()) *
3264 (dirac_slash(l, D) + m * dirac_ONE());
3265 e = dirac_trace(e).simplify_indexed(sp);
3266 e = e.collect(lst(l, ldotq, m));
3268 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3272 The @code{canonicalize_clifford()} function reorders all gamma products that
3273 appear in an expression to a canonical (but not necessarily simple) form.
3274 You can use this to compare two expressions or for further simplifications:
3278 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3279 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3281 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3283 e = canonicalize_clifford(e);
3285 // -> 2*ONE*eta~mu~nu
3289 @cindex @code{clifford_unit()}
3290 @subsubsection A generic Clifford algebra
3292 A generic Clifford algebra, i.e. a
3298 dimensional algebra with
3305 satisfying the identities
3307 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3310 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3312 for some bilinear form (@code{metric})
3313 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3314 and contain symbolic entries. Such generators are created by the
3318 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3321 where @code{mu} should be a @code{idx} (or descendant) class object
3322 indexing the generators.
3323 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3324 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3325 object. In fact, any expression either with two free indices or without
3326 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3327 object with two newly created indices with @code{metr} as its
3328 @code{op(0)} will be used.
3329 Optional parameter @code{rl} allows to distinguish different
3330 Clifford algebras, which will commute with each other.
3332 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3333 something very close to @code{dirac_gamma(mu)}, although
3334 @code{dirac_gamma} have more efficient simplification mechanism.
3335 @cindex @code{clifford::get_metric()}
3336 The method @code{clifford::get_metric()} returns a metric defining this
3339 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3340 the Clifford algebra units with a call like that
3343 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3346 since this may yield some further automatic simplifications. Again, for a
3347 metric defined through a @code{matrix} such a symmetry is detected
3350 Individual generators of a Clifford algebra can be accessed in several
3356 idx i(symbol("i"), 4);
3358 ex M = diag_matrix(lst(1, -1, 0, s));
3359 ex e = clifford_unit(i, M);
3360 ex e0 = e.subs(i == 0);
3361 ex e1 = e.subs(i == 1);
3362 ex e2 = e.subs(i == 2);
3363 ex e3 = e.subs(i == 3);
3368 will produce four anti-commuting generators of a Clifford algebra with properties
3370 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3373 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3374 @code{pow(e3, 2) = s}.
3377 @cindex @code{lst_to_clifford()}
3378 A similar effect can be achieved from the function
3381 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3382 unsigned char rl = 0);
3383 ex lst_to_clifford(const ex & v, const ex & e);
3386 which converts a list or vector
3388 $v = (v^0, v^1, ..., v^n)$
3391 @samp{v = (v~0, v~1, ..., v~n)}
3396 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3399 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3402 directly supplied in the second form of the procedure. In the first form
3403 the Clifford unit @samp{e.k} is generated by the call of
3404 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3405 with the help of @code{lst_to_clifford()} as follows
3410 idx i(symbol("i"), 4);
3412 ex M = diag_matrix(lst(1, -1, 0, s));
3413 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3414 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3415 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3416 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3421 @cindex @code{clifford_to_lst()}
3422 There is the inverse function
3425 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3428 which takes an expression @code{e} and tries to find a list
3430 $v = (v^0, v^1, ..., v^n)$
3433 @samp{v = (v~0, v~1, ..., v~n)}
3437 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3440 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3442 with respect to the given Clifford units @code{c} and with none of the
3443 @samp{v~k} containing Clifford units @code{c} (of course, this
3444 may be impossible). This function can use an @code{algebraic} method
3445 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3447 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3450 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3452 is zero or is not @code{numeric} for some @samp{k}
3453 then the method will be automatically changed to symbolic. The same effect
3454 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3456 @cindex @code{clifford_prime()}
3457 @cindex @code{clifford_star()}
3458 @cindex @code{clifford_bar()}
3459 There are several functions for (anti-)automorphisms of Clifford algebras:
3462 ex clifford_prime(const ex & e)
3463 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3464 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3467 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3468 changes signs of all Clifford units in the expression. The reversion
3469 of a Clifford algebra @code{clifford_star()} coincides with the
3470 @code{conjugate()} method and effectively reverses the order of Clifford
3471 units in any product. Finally the main anti-automorphism
3472 of a Clifford algebra @code{clifford_bar()} is the composition of the
3473 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3474 in a product. These functions correspond to the notations
3489 used in Clifford algebra textbooks.
3491 @cindex @code{clifford_norm()}
3495 ex clifford_norm(const ex & e);
3498 @cindex @code{clifford_inverse()}
3499 calculates the norm of a Clifford number from the expression
3501 $||e||^2 = e\overline{e}$.
3504 @code{||e||^2 = e \bar@{e@}}
3506 The inverse of a Clifford expression is returned by the function
3509 ex clifford_inverse(const ex & e);
3512 which calculates it as
3514 $e^{-1} = \overline{e}/||e||^2$.
3517 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3526 then an exception is raised.
3528 @cindex @code{remove_dirac_ONE()}
3529 If a Clifford number happens to be a factor of
3530 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3531 expression by the function
3534 ex remove_dirac_ONE(const ex & e);
3537 @cindex @code{canonicalize_clifford()}
3538 The function @code{canonicalize_clifford()} works for a
3539 generic Clifford algebra in a similar way as for Dirac gammas.
3541 The next provided function is
3543 @cindex @code{clifford_moebius_map()}
3545 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3546 const ex & d, const ex & v, const ex & G,
3547 unsigned char rl = 0);
3548 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3549 unsigned char rl = 0);
3552 It takes a list or vector @code{v} and makes the Moebius (conformal or
3553 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3554 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3555 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3556 indexed object, tensormetric, matrix or a Clifford unit, in the later
3557 case the optional parameter @code{rl} is ignored even if supplied.
3558 Depending from the type of @code{v} the returned value of this function
3559 is either a vector or a list holding vector's components.
3561 @cindex @code{clifford_max_label()}
3562 Finally the function
3565 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3568 can detect a presence of Clifford objects in the expression @code{e}: if
3569 such objects are found it returns the maximal
3570 @code{representation_label} of them, otherwise @code{-1}. The optional
3571 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3572 be ignored during the search.
3574 LaTeX output for Clifford units looks like
3575 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3576 @code{representation_label} and @code{\nu} is the index of the
3577 corresponding unit. This provides a flexible typesetting with a suitable
3578 definition of the @code{\clifford} command. For example, the definition
3580 \newcommand@{\clifford@}[1][]@{@}
3582 typesets all Clifford units identically, while the alternative definition
3584 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3586 prints units with @code{representation_label=0} as
3593 with @code{representation_label=1} as
3600 and with @code{representation_label=2} as
3608 @cindex @code{color} (class)
3609 @subsection Color algebra
3611 @cindex @code{color_T()}
3612 For computations in quantum chromodynamics, GiNaC implements the base elements
3613 and structure constants of the su(3) Lie algebra (color algebra). The base
3614 elements @math{T_a} are constructed by the function
3617 ex color_T(const ex & a, unsigned char rl = 0);
3620 which takes two arguments: the index and a @dfn{representation label} in the
3621 range 0 to 255 which is used to distinguish elements of different color
3622 algebras. Objects with different labels commutate with each other. The
3623 dimension of the index must be exactly 8 and it should be of class @code{idx},
3626 @cindex @code{color_ONE()}
3627 The unity element of a color algebra is constructed by
3630 ex color_ONE(unsigned char rl = 0);
3633 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3634 multiples of the unity element, even though it's customary to omit it.
3635 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3636 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3637 GiNaC may produce incorrect results.
3639 @cindex @code{color_d()}
3640 @cindex @code{color_f()}
3644 ex color_d(const ex & a, const ex & b, const ex & c);
3645 ex color_f(const ex & a, const ex & b, const ex & c);
3648 create the symmetric and antisymmetric structure constants @math{d_abc} and
3649 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3650 and @math{[T_a, T_b] = i f_abc T_c}.
3652 These functions evaluate to their numerical values,
3653 if you supply numeric indices to them. The index values should be in
3654 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3655 goes along better with the notations used in physical literature.
3657 @cindex @code{color_h()}
3658 There's an additional function
3661 ex color_h(const ex & a, const ex & b, const ex & c);
3664 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3666 The function @code{simplify_indexed()} performs some simplifications on
3667 expressions containing color objects:
3672 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3673 k(symbol("k"), 8), l(symbol("l"), 8);
3675 e = color_d(a, b, l) * color_f(a, b, k);
3676 cout << e.simplify_indexed() << endl;
3679 e = color_d(a, b, l) * color_d(a, b, k);
3680 cout << e.simplify_indexed() << endl;
3683 e = color_f(l, a, b) * color_f(a, b, k);
3684 cout << e.simplify_indexed() << endl;
3687 e = color_h(a, b, c) * color_h(a, b, c);
3688 cout << e.simplify_indexed() << endl;
3691 e = color_h(a, b, c) * color_T(b) * color_T(c);
3692 cout << e.simplify_indexed() << endl;
3695 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3696 cout << e.simplify_indexed() << endl;
3699 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3700 cout << e.simplify_indexed() << endl;
3701 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3705 @cindex @code{color_trace()}
3706 To calculate the trace of an expression containing color objects you use one
3710 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3711 ex color_trace(const ex & e, const lst & rll);
3712 ex color_trace(const ex & e, unsigned char rl = 0);
3715 These functions take the trace over all color @samp{T} objects in the
3716 specified set @code{rls} or list @code{rll} of representation labels, or the
3717 single label @code{rl}; @samp{T}s with other labels are left standing. For
3722 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3724 // -> -I*f.a.c.b+d.a.c.b
3729 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3730 @c node-name, next, previous, up
3733 @cindex @code{exhashmap} (class)
3735 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3736 that can be used as a drop-in replacement for the STL
3737 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3738 typically constant-time, element look-up than @code{map<>}.
3740 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3741 following differences:
3745 no @code{lower_bound()} and @code{upper_bound()} methods
3747 no reverse iterators, no @code{rbegin()}/@code{rend()}
3749 no @code{operator<(exhashmap, exhashmap)}