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) 1999-2006 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{} 1999-2006 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{} 1999-2006 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. Perl is needed by the built
482 process as well, since some of the source files are automatically
483 generated by Perl scripts. Last but not least, the CLN library
484 is used extensively and needs to be installed on your system.
485 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
486 (it is covered by GPL) and install it prior to trying to install
487 GiNaC. The configure script checks if it can find it and if it cannot
488 it will refuse to continue.
491 @node Configuration, Building GiNaC, Prerequisites, Installation
492 @c node-name, next, previous, up
493 @section Configuration
494 @cindex configuration
497 To configure GiNaC means to prepare the source distribution for
498 building. It is done via a shell script called @command{configure} that
499 is shipped with the sources and was originally generated by GNU
500 Autoconf. Since a configure script generated by GNU Autoconf never
501 prompts, all customization must be done either via command line
502 parameters or environment variables. It accepts a list of parameters,
503 the complete set of which can be listed by calling it with the
504 @option{--help} option. The most important ones will be shortly
505 described in what follows:
510 @option{--disable-shared}: When given, this option switches off the
511 build of a shared library, i.e. a @file{.so} file. This may be convenient
512 when developing because it considerably speeds up compilation.
515 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
516 and headers are installed. It defaults to @file{/usr/local} which means
517 that the library is installed in the directory @file{/usr/local/lib},
518 the header files in @file{/usr/local/include/ginac} and the documentation
519 (like this one) into @file{/usr/local/share/doc/GiNaC}.
522 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
523 the library installed in some other directory than
524 @file{@var{PREFIX}/lib/}.
527 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
528 to have the header files installed in some other directory than
529 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
530 @option{--includedir=/usr/include} you will end up with the header files
531 sitting in the directory @file{/usr/include/ginac/}. Note that the
532 subdirectory @file{ginac} is enforced by this process in order to
533 keep the header files separated from others. This avoids some
534 clashes and allows for an easier deinstallation of GiNaC. This ought
535 to be considered A Good Thing (tm).
538 @option{--datadir=@var{DATADIR}}: This option may be given in case you
539 want to have the documentation installed in some other directory than
540 @file{@var{PREFIX}/share/doc/GiNaC/}.
544 In addition, you may specify some environment variables. @env{CXX}
545 holds the path and the name of the C++ compiler in case you want to
546 override the default in your path. (The @command{configure} script
547 searches your path for @command{c++}, @command{g++}, @command{gcc},
548 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
549 be very useful to define some compiler flags with the @env{CXXFLAGS}
550 environment variable, like optimization, debugging information and
551 warning levels. If omitted, it defaults to @option{-g
552 -O2}.@footnote{The @command{configure} script is itself generated from
553 the file @file{configure.ac}. It is only distributed in packaged
554 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
555 must generate @command{configure} along with the various
556 @file{Makefile.in} by using the @command{autogen.sh} script. This will
557 require a fair amount of support from your local toolchain, though.}
559 The whole process is illustrated in the following two
560 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
561 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
564 Here is a simple configuration for a site-wide GiNaC library assuming
565 everything is in default paths:
568 $ export CXXFLAGS="-Wall -O2"
572 And here is a configuration for a private static GiNaC library with
573 several components sitting in custom places (site-wide GCC and private
574 CLN). The compiler is persuaded to be picky and full assertions and
575 debugging information are switched on:
578 $ export CXX=/usr/local/gnu/bin/c++
579 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
580 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
581 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
582 $ ./configure --disable-shared --prefix=$(HOME)
586 @node Building GiNaC, Installing GiNaC, Configuration, Installation
587 @c node-name, next, previous, up
588 @section Building GiNaC
589 @cindex building GiNaC
591 After proper configuration you should just build the whole
596 at the command prompt and go for a cup of coffee. The exact time it
597 takes to compile GiNaC depends not only on the speed of your machines
598 but also on other parameters, for instance what value for @env{CXXFLAGS}
599 you entered. Optimization may be very time-consuming.
601 Just to make sure GiNaC works properly you may run a collection of
602 regression tests by typing
608 This will compile some sample programs, run them and check the output
609 for correctness. The regression tests fall in three categories. First,
610 the so called @emph{exams} are performed, simple tests where some
611 predefined input is evaluated (like a pupils' exam). Second, the
612 @emph{checks} test the coherence of results among each other with
613 possible random input. Third, some @emph{timings} are performed, which
614 benchmark some predefined problems with different sizes and display the
615 CPU time used in seconds. Each individual test should return a message
616 @samp{passed}. This is mostly intended to be a QA-check if something
617 was broken during development, not a sanity check of your system. Some
618 of the tests in sections @emph{checks} and @emph{timings} may require
619 insane amounts of memory and CPU time. Feel free to kill them if your
620 machine catches fire. Another quite important intent is to allow people
621 to fiddle around with optimization.
623 By default, the only documentation that will be built is this tutorial
624 in @file{.info} format. To build the GiNaC tutorial and reference manual
625 in HTML, DVI, PostScript, or PDF formats, use one of
634 Generally, the top-level Makefile runs recursively to the
635 subdirectories. It is therefore safe to go into any subdirectory
636 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
637 @var{target} there in case something went wrong.
640 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
641 @c node-name, next, previous, up
642 @section Installing GiNaC
645 To install GiNaC on your system, simply type
651 As described in the section about configuration the files will be
652 installed in the following directories (the directories will be created
653 if they don't already exist):
658 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
659 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
660 So will @file{libginac.so} unless the configure script was
661 given the option @option{--disable-shared}. The proper symlinks
662 will be established as well.
665 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
666 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
669 All documentation (info) will be stuffed into
670 @file{@var{PREFIX}/share/doc/GiNaC/} (or
671 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
675 For the sake of completeness we will list some other useful make
676 targets: @command{make clean} deletes all files generated by
677 @command{make}, i.e. all the object files. In addition @command{make
678 distclean} removes all files generated by the configuration and
679 @command{make maintainer-clean} goes one step further and deletes files
680 that may require special tools to rebuild (like the @command{libtool}
681 for instance). Finally @command{make uninstall} removes the installed
682 library, header files and documentation@footnote{Uninstallation does not
683 work after you have called @command{make distclean} since the
684 @file{Makefile} is itself generated by the configuration from
685 @file{Makefile.in} and hence deleted by @command{make distclean}. There
686 are two obvious ways out of this dilemma. First, you can run the
687 configuration again with the same @var{PREFIX} thus creating a
688 @file{Makefile} with a working @samp{uninstall} target. Second, you can
689 do it by hand since you now know where all the files went during
693 @node Basic concepts, Expressions, Installing GiNaC, Top
694 @c node-name, next, previous, up
695 @chapter Basic concepts
697 This chapter will describe the different fundamental objects that can be
698 handled by GiNaC. But before doing so, it is worthwhile introducing you
699 to the more commonly used class of expressions, representing a flexible
700 meta-class for storing all mathematical objects.
703 * Expressions:: The fundamental GiNaC class.
704 * Automatic evaluation:: Evaluation and canonicalization.
705 * Error handling:: How the library reports errors.
706 * The class hierarchy:: Overview of GiNaC's classes.
707 * Symbols:: Symbolic objects.
708 * Numbers:: Numerical objects.
709 * Constants:: Pre-defined constants.
710 * Fundamental containers:: Sums, products and powers.
711 * Lists:: Lists of expressions.
712 * Mathematical functions:: Mathematical functions.
713 * Relations:: Equality, Inequality and all that.
714 * Integrals:: Symbolic integrals.
715 * Matrices:: Matrices.
716 * Indexed objects:: Handling indexed quantities.
717 * Non-commutative objects:: Algebras with non-commutative products.
718 * Hash maps:: A faster alternative to std::map<>.
722 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
723 @c node-name, next, previous, up
725 @cindex expression (class @code{ex})
728 The most common class of objects a user deals with is the expression
729 @code{ex}, representing a mathematical object like a variable, number,
730 function, sum, product, etc@dots{} Expressions may be put together to form
731 new expressions, passed as arguments to functions, and so on. Here is a
732 little collection of valid expressions:
735 ex MyEx1 = 5; // simple number
736 ex MyEx2 = x + 2*y; // polynomial in x and y
737 ex MyEx3 = (x + 1)/(x - 1); // rational expression
738 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
739 ex MyEx5 = MyEx4 + 1; // similar to above
742 Expressions are handles to other more fundamental objects, that often
743 contain other expressions thus creating a tree of expressions
744 (@xref{Internal structures}, for particular examples). Most methods on
745 @code{ex} therefore run top-down through such an expression tree. For
746 example, the method @code{has()} scans recursively for occurrences of
747 something inside an expression. Thus, if you have declared @code{MyEx4}
748 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
749 the argument of @code{sin} and hence return @code{true}.
751 The next sections will outline the general picture of GiNaC's class
752 hierarchy and describe the classes of objects that are handled by
755 @subsection Note: Expressions and STL containers
757 GiNaC expressions (@code{ex} objects) have value semantics (they can be
758 assigned, reassigned and copied like integral types) but the operator
759 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
760 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
762 This implies that in order to use expressions in sorted containers such as
763 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
764 comparison predicate. GiNaC provides such a predicate, called
765 @code{ex_is_less}. For example, a set of expressions should be defined
766 as @code{std::set<ex, ex_is_less>}.
768 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
769 don't pose a problem. A @code{std::vector<ex>} works as expected.
771 @xref{Information about expressions}, for more about comparing and ordering
775 @node Automatic evaluation, Error handling, Expressions, Basic concepts
776 @c node-name, next, previous, up
777 @section Automatic evaluation and canonicalization of expressions
780 GiNaC performs some automatic transformations on expressions, to simplify
781 them and put them into a canonical form. Some examples:
784 ex MyEx1 = 2*x - 1 + x; // 3*x-1
785 ex MyEx2 = x - x; // 0
786 ex MyEx3 = cos(2*Pi); // 1
787 ex MyEx4 = x*y/x; // y
790 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
791 evaluation}. GiNaC only performs transformations that are
795 at most of complexity
803 algebraically correct, possibly except for a set of measure zero (e.g.
804 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
807 There are two types of automatic transformations in GiNaC that may not
808 behave in an entirely obvious way at first glance:
812 The terms of sums and products (and some other things like the arguments of
813 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
814 into a canonical form that is deterministic, but not lexicographical or in
815 any other way easy to guess (it almost always depends on the number and
816 order of the symbols you define). However, constructing the same expression
817 twice, either implicitly or explicitly, will always result in the same
820 Expressions of the form 'number times sum' are automatically expanded (this
821 has to do with GiNaC's internal representation of sums and products). For
824 ex MyEx5 = 2*(x + y); // 2*x+2*y
825 ex MyEx6 = z*(x + y); // z*(x+y)
829 The general rule is that when you construct expressions, GiNaC automatically
830 creates them in canonical form, which might differ from the form you typed in
831 your program. This may create some awkward looking output (@samp{-y+x} instead
832 of @samp{x-y}) but allows for more efficient operation and usually yields
833 some immediate simplifications.
835 @cindex @code{eval()}
836 Internally, the anonymous evaluator in GiNaC is implemented by the methods
839 ex ex::eval(int level = 0) const;
840 ex basic::eval(int level = 0) const;
843 but unless you are extending GiNaC with your own classes or functions, there
844 should never be any reason to call them explicitly. All GiNaC methods that
845 transform expressions, like @code{subs()} or @code{normal()}, automatically
846 re-evaluate their results.
849 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
850 @c node-name, next, previous, up
851 @section Error handling
853 @cindex @code{pole_error} (class)
855 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
856 generated by GiNaC are subclassed from the standard @code{exception} class
857 defined in the @file{<stdexcept>} header. In addition to the predefined
858 @code{logic_error}, @code{domain_error}, @code{out_of_range},
859 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
860 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
861 exception that gets thrown when trying to evaluate a mathematical function
864 The @code{pole_error} class has a member function
867 int pole_error::degree() const;
870 that returns the order of the singularity (or 0 when the pole is
871 logarithmic or the order is undefined).
873 When using GiNaC it is useful to arrange for exceptions to be caught in
874 the main program even if you don't want to do any special error handling.
875 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
876 default exception handler of your C++ compiler's run-time system which
877 usually only aborts the program without giving any information what went
880 Here is an example for a @code{main()} function that catches and prints
881 exceptions generated by GiNaC:
886 #include <ginac/ginac.h>
888 using namespace GiNaC;
896 @} catch (exception &p) @{
897 cerr << p.what() << endl;
905 @node The class hierarchy, Symbols, Error handling, Basic concepts
906 @c node-name, next, previous, up
907 @section The class hierarchy
909 GiNaC's class hierarchy consists of several classes representing
910 mathematical objects, all of which (except for @code{ex} and some
911 helpers) are internally derived from one abstract base class called
912 @code{basic}. You do not have to deal with objects of class
913 @code{basic}, instead you'll be dealing with symbols, numbers,
914 containers of expressions and so on.
918 To get an idea about what kinds of symbolic composites may be built we
919 have a look at the most important classes in the class hierarchy and
920 some of the relations among the classes:
922 @image{classhierarchy}
924 The abstract classes shown here (the ones without drop-shadow) are of no
925 interest for the user. They are used internally in order to avoid code
926 duplication if two or more classes derived from them share certain
927 features. An example is @code{expairseq}, a container for a sequence of
928 pairs each consisting of one expression and a number (@code{numeric}).
929 What @emph{is} visible to the user are the derived classes @code{add}
930 and @code{mul}, representing sums and products. @xref{Internal
931 structures}, where these two classes are described in more detail. The
932 following table shortly summarizes what kinds of mathematical objects
933 are stored in the different classes:
936 @multitable @columnfractions .22 .78
937 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
938 @item @code{constant} @tab Constants like
945 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
946 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
947 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
948 @item @code{ncmul} @tab Products of non-commutative objects
949 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
954 @code{sqrt(}@math{2}@code{)}
957 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
958 @item @code{function} @tab A symbolic function like
965 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
966 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
967 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
968 @item @code{indexed} @tab Indexed object like @math{A_ij}
969 @item @code{tensor} @tab Special tensor like the delta and metric tensors
970 @item @code{idx} @tab Index of an indexed object
971 @item @code{varidx} @tab Index with variance
972 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
973 @item @code{wildcard} @tab Wildcard for pattern matching
974 @item @code{structure} @tab Template for user-defined classes
979 @node Symbols, Numbers, The class hierarchy, Basic concepts
980 @c node-name, next, previous, up
982 @cindex @code{symbol} (class)
983 @cindex hierarchy of classes
986 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
987 manipulation what atoms are for chemistry.
989 A typical symbol definition looks like this:
994 This definition actually contains three very different things:
996 @item a C++ variable named @code{x}
997 @item a @code{symbol} object stored in this C++ variable; this object
998 represents the symbol in a GiNaC expression
999 @item the string @code{"x"} which is the name of the symbol, used (almost)
1000 exclusively for printing expressions holding the symbol
1003 Symbols have an explicit name, supplied as a string during construction,
1004 because in C++, variable names can't be used as values, and the C++ compiler
1005 throws them away during compilation.
1007 It is possible to omit the symbol name in the definition:
1012 In this case, GiNaC will assign the symbol an internal, unique name of the
1013 form @code{symbolNNN}. This won't affect the usability of the symbol but
1014 the output of your calculations will become more readable if you give your
1015 symbols sensible names (for intermediate expressions that are only used
1016 internally such anonymous symbols can be quite useful, however).
1018 Now, here is one important property of GiNaC that differentiates it from
1019 other computer algebra programs you may have used: GiNaC does @emph{not} use
1020 the names of symbols to tell them apart, but a (hidden) serial number that
1021 is unique for each newly created @code{symbol} object. In you want to use
1022 one and the same symbol in different places in your program, you must only
1023 create one @code{symbol} object and pass that around. If you create another
1024 symbol, even if it has the same name, GiNaC will treat it as a different
1041 // prints "x^6" which looks right, but...
1043 cout << e.degree(x) << endl;
1044 // ...this doesn't work. The symbol "x" here is different from the one
1045 // in f() and in the expression returned by f(). Consequently, it
1050 One possibility to ensure that @code{f()} and @code{main()} use the same
1051 symbol is to pass the symbol as an argument to @code{f()}:
1053 ex f(int n, const ex & x)
1062 // Now, f() uses the same symbol.
1065 cout << e.degree(x) << endl;
1066 // prints "6", as expected
1070 Another possibility would be to define a global symbol @code{x} that is used
1071 by both @code{f()} and @code{main()}. If you are using global symbols and
1072 multiple compilation units you must take special care, however. Suppose
1073 that you have a header file @file{globals.h} in your program that defines
1074 a @code{symbol x("x");}. In this case, every unit that includes
1075 @file{globals.h} would also get its own definition of @code{x} (because
1076 header files are just inlined into the source code by the C++ preprocessor),
1077 and hence you would again end up with multiple equally-named, but different,
1078 symbols. Instead, the @file{globals.h} header should only contain a
1079 @emph{declaration} like @code{extern symbol x;}, with the definition of
1080 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1082 A different approach to ensuring that symbols used in different parts of
1083 your program are identical is to create them with a @emph{factory} function
1086 const symbol & get_symbol(const string & s)
1088 static map<string, symbol> directory;
1089 map<string, symbol>::iterator i = directory.find(s);
1090 if (i != directory.end())
1093 return directory.insert(make_pair(s, symbol(s))).first->second;
1097 This function returns one newly constructed symbol for each name that is
1098 passed in, and it returns the same symbol when called multiple times with
1099 the same name. Using this symbol factory, we can rewrite our example like
1104 return pow(get_symbol("x"), n);
1111 // Both calls of get_symbol("x") yield the same symbol.
1112 cout << e.degree(get_symbol("x")) << endl;
1117 Instead of creating symbols from strings we could also have
1118 @code{get_symbol()} take, for example, an integer number as its argument.
1119 In this case, we would probably want to give the generated symbols names
1120 that include this number, which can be accomplished with the help of an
1121 @code{ostringstream}.
1123 In general, if you're getting weird results from GiNaC such as an expression
1124 @samp{x-x} that is not simplified to zero, you should check your symbol
1127 As we said, the names of symbols primarily serve for purposes of expression
1128 output. But there are actually two instances where GiNaC uses the names for
1129 identifying symbols: When constructing an expression from a string, and when
1130 recreating an expression from an archive (@pxref{Input/output}).
1132 In addition to its name, a symbol may contain a special string that is used
1135 symbol x("x", "\\Box");
1138 This creates a symbol that is printed as "@code{x}" in normal output, but
1139 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1140 information about the different output formats of expressions in GiNaC).
1141 GiNaC automatically creates proper LaTeX code for symbols having names of
1142 greek letters (@samp{alpha}, @samp{mu}, etc.).
1144 @cindex @code{subs()}
1145 Symbols in GiNaC can't be assigned values. If you need to store results of
1146 calculations and give them a name, use C++ variables of type @code{ex}.
1147 If you want to replace a symbol in an expression with something else, you
1148 can invoke the expression's @code{.subs()} method
1149 (@pxref{Substituting expressions}).
1151 @cindex @code{realsymbol()}
1152 By default, symbols are expected to stand in for complex values, i.e. they live
1153 in the complex domain. As a consequence, operations like complex conjugation,
1154 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1155 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1156 because of the unknown imaginary part of @code{x}.
1157 On the other hand, if you are sure that your symbols will hold only real
1158 values, you would like to have such functions evaluated. Therefore GiNaC
1159 allows you to specify
1160 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1161 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1163 @cindex @code{possymbol()}
1164 Furthermore, it is also possible to declare a symbol as positive. This will,
1165 for instance, enable the automatic simplification of @code{abs(x)} into
1166 @code{x}. This is done by declaying the symbol as @code{possymbol x("x");}.
1169 @node Numbers, Constants, Symbols, Basic concepts
1170 @c node-name, next, previous, up
1172 @cindex @code{numeric} (class)
1178 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1179 The classes therein serve as foundation classes for GiNaC. CLN stands
1180 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1181 In order to find out more about CLN's internals, the reader is referred to
1182 the documentation of that library. @inforef{Introduction, , cln}, for
1183 more information. Suffice to say that it is by itself build on top of
1184 another library, the GNU Multiple Precision library GMP, which is an
1185 extremely fast library for arbitrary long integers and rationals as well
1186 as arbitrary precision floating point numbers. It is very commonly used
1187 by several popular cryptographic applications. CLN extends GMP by
1188 several useful things: First, it introduces the complex number field
1189 over either reals (i.e. floating point numbers with arbitrary precision)
1190 or rationals. Second, it automatically converts rationals to integers
1191 if the denominator is unity and complex numbers to real numbers if the
1192 imaginary part vanishes and also correctly treats algebraic functions.
1193 Third it provides good implementations of state-of-the-art algorithms
1194 for all trigonometric and hyperbolic functions as well as for
1195 calculation of some useful constants.
1197 The user can construct an object of class @code{numeric} in several
1198 ways. The following example shows the four most important constructors.
1199 It uses construction from C-integer, construction of fractions from two
1200 integers, construction from C-float and construction from a string:
1204 #include <ginac/ginac.h>
1205 using namespace GiNaC;
1209 numeric two = 2; // exact integer 2
1210 numeric r(2,3); // exact fraction 2/3
1211 numeric e(2.71828); // floating point number
1212 numeric p = "3.14159265358979323846"; // constructor from string
1213 // Trott's constant in scientific notation:
1214 numeric trott("1.0841015122311136151E-2");
1216 std::cout << two*p << std::endl; // floating point 6.283...
1221 @cindex complex numbers
1222 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1227 numeric z1 = 2-3*I; // exact complex number 2-3i
1228 numeric z2 = 5.9+1.6*I; // complex floating point number
1232 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1233 This would, however, call C's built-in operator @code{/} for integers
1234 first and result in a numeric holding a plain integer 1. @strong{Never
1235 use the operator @code{/} on integers} unless you know exactly what you
1236 are doing! Use the constructor from two integers instead, as shown in
1237 the example above. Writing @code{numeric(1)/2} may look funny but works
1240 @cindex @code{Digits}
1242 We have seen now the distinction between exact numbers and floating
1243 point numbers. Clearly, the user should never have to worry about
1244 dynamically created exact numbers, since their `exactness' always
1245 determines how they ought to be handled, i.e. how `long' they are. The
1246 situation is different for floating point numbers. Their accuracy is
1247 controlled by one @emph{global} variable, called @code{Digits}. (For
1248 those readers who know about Maple: it behaves very much like Maple's
1249 @code{Digits}). All objects of class numeric that are constructed from
1250 then on will be stored with a precision matching that number of decimal
1255 #include <ginac/ginac.h>
1256 using namespace std;
1257 using namespace GiNaC;
1261 numeric three(3.0), one(1.0);
1262 numeric x = one/three;
1264 cout << "in " << Digits << " digits:" << endl;
1266 cout << Pi.evalf() << endl;
1278 The above example prints the following output to screen:
1282 0.33333333333333333334
1283 3.1415926535897932385
1285 0.33333333333333333333333333333333333333333333333333333333333333333334
1286 3.1415926535897932384626433832795028841971693993751058209749445923078
1290 Note that the last number is not necessarily rounded as you would
1291 naively expect it to be rounded in the decimal system. But note also,
1292 that in both cases you got a couple of extra digits. This is because
1293 numbers are internally stored by CLN as chunks of binary digits in order
1294 to match your machine's word size and to not waste precision. Thus, on
1295 architectures with different word size, the above output might even
1296 differ with regard to actually computed digits.
1298 It should be clear that objects of class @code{numeric} should be used
1299 for constructing numbers or for doing arithmetic with them. The objects
1300 one deals with most of the time are the polymorphic expressions @code{ex}.
1302 @subsection Tests on numbers
1304 Once you have declared some numbers, assigned them to expressions and
1305 done some arithmetic with them it is frequently desired to retrieve some
1306 kind of information from them like asking whether that number is
1307 integer, rational, real or complex. For those cases GiNaC provides
1308 several useful methods. (Internally, they fall back to invocations of
1309 certain CLN functions.)
1311 As an example, let's construct some rational number, multiply it with
1312 some multiple of its denominator and test what comes out:
1316 #include <ginac/ginac.h>
1317 using namespace std;
1318 using namespace GiNaC;
1320 // some very important constants:
1321 const numeric twentyone(21);
1322 const numeric ten(10);
1323 const numeric five(5);
1327 numeric answer = twentyone;
1330 cout << answer.is_integer() << endl; // false, it's 21/5
1332 cout << answer.is_integer() << endl; // true, it's 42 now!
1336 Note that the variable @code{answer} is constructed here as an integer
1337 by @code{numeric}'s copy constructor but in an intermediate step it
1338 holds a rational number represented as integer numerator and integer
1339 denominator. When multiplied by 10, the denominator becomes unity and
1340 the result is automatically converted to a pure integer again.
1341 Internally, the underlying CLN is responsible for this behavior and we
1342 refer the reader to CLN's documentation. Suffice to say that
1343 the same behavior applies to complex numbers as well as return values of
1344 certain functions. Complex numbers are automatically converted to real
1345 numbers if the imaginary part becomes zero. The full set of tests that
1346 can be applied is listed in the following table.
1349 @multitable @columnfractions .30 .70
1350 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1351 @item @code{.is_zero()}
1352 @tab @dots{}equal to zero
1353 @item @code{.is_positive()}
1354 @tab @dots{}not complex and greater than 0
1355 @item @code{.is_integer()}
1356 @tab @dots{}a (non-complex) integer
1357 @item @code{.is_pos_integer()}
1358 @tab @dots{}an integer and greater than 0
1359 @item @code{.is_nonneg_integer()}
1360 @tab @dots{}an integer and greater equal 0
1361 @item @code{.is_even()}
1362 @tab @dots{}an even integer
1363 @item @code{.is_odd()}
1364 @tab @dots{}an odd integer
1365 @item @code{.is_prime()}
1366 @tab @dots{}a prime integer (probabilistic primality test)
1367 @item @code{.is_rational()}
1368 @tab @dots{}an exact rational number (integers are rational, too)
1369 @item @code{.is_real()}
1370 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1371 @item @code{.is_cinteger()}
1372 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1373 @item @code{.is_crational()}
1374 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1378 @subsection Numeric functions
1380 The following functions can be applied to @code{numeric} objects and will be
1381 evaluated immediately:
1384 @multitable @columnfractions .30 .70
1385 @item @strong{Name} @tab @strong{Function}
1386 @item @code{inverse(z)}
1387 @tab returns @math{1/z}
1388 @cindex @code{inverse()} (numeric)
1389 @item @code{pow(a, b)}
1390 @tab exponentiation @math{a^b}
1393 @item @code{real(z)}
1395 @cindex @code{real()}
1396 @item @code{imag(z)}
1398 @cindex @code{imag()}
1399 @item @code{csgn(z)}
1400 @tab complex sign (returns an @code{int})
1401 @item @code{step(x)}
1402 @tab step function (returns an @code{numeric})
1403 @item @code{numer(z)}
1404 @tab numerator of rational or complex rational number
1405 @item @code{denom(z)}
1406 @tab denominator of rational or complex rational number
1407 @item @code{sqrt(z)}
1409 @item @code{isqrt(n)}
1410 @tab integer square root
1411 @cindex @code{isqrt()}
1418 @item @code{asin(z)}
1420 @item @code{acos(z)}
1422 @item @code{atan(z)}
1423 @tab inverse tangent
1424 @item @code{atan(y, x)}
1425 @tab inverse tangent with two arguments
1426 @item @code{sinh(z)}
1427 @tab hyperbolic sine
1428 @item @code{cosh(z)}
1429 @tab hyperbolic cosine
1430 @item @code{tanh(z)}
1431 @tab hyperbolic tangent
1432 @item @code{asinh(z)}
1433 @tab inverse hyperbolic sine
1434 @item @code{acosh(z)}
1435 @tab inverse hyperbolic cosine
1436 @item @code{atanh(z)}
1437 @tab inverse hyperbolic tangent
1439 @tab exponential function
1441 @tab natural logarithm
1444 @item @code{zeta(z)}
1445 @tab Riemann's zeta function
1446 @item @code{tgamma(z)}
1448 @item @code{lgamma(z)}
1449 @tab logarithm of gamma function
1451 @tab psi (digamma) function
1452 @item @code{psi(n, z)}
1453 @tab derivatives of psi function (polygamma functions)
1454 @item @code{factorial(n)}
1455 @tab factorial function @math{n!}
1456 @item @code{doublefactorial(n)}
1457 @tab double factorial function @math{n!!}
1458 @cindex @code{doublefactorial()}
1459 @item @code{binomial(n, k)}
1460 @tab binomial coefficients
1461 @item @code{bernoulli(n)}
1462 @tab Bernoulli numbers
1463 @cindex @code{bernoulli()}
1464 @item @code{fibonacci(n)}
1465 @tab Fibonacci numbers
1466 @cindex @code{fibonacci()}
1467 @item @code{mod(a, b)}
1468 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1469 @cindex @code{mod()}
1470 @item @code{smod(a, b)}
1471 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1472 @cindex @code{smod()}
1473 @item @code{irem(a, b)}
1474 @tab integer remainder (has the sign of @math{a}, or is zero)
1475 @cindex @code{irem()}
1476 @item @code{irem(a, b, q)}
1477 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1478 @item @code{iquo(a, b)}
1479 @tab integer quotient
1480 @cindex @code{iquo()}
1481 @item @code{iquo(a, b, r)}
1482 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1483 @item @code{gcd(a, b)}
1484 @tab greatest common divisor
1485 @item @code{lcm(a, b)}
1486 @tab least common multiple
1490 Most of these functions are also available as symbolic functions that can be
1491 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1492 as polynomial algorithms.
1494 @subsection Converting numbers
1496 Sometimes it is desirable to convert a @code{numeric} object back to a
1497 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1498 class provides a couple of methods for this purpose:
1500 @cindex @code{to_int()}
1501 @cindex @code{to_long()}
1502 @cindex @code{to_double()}
1503 @cindex @code{to_cl_N()}
1505 int numeric::to_int() const;
1506 long numeric::to_long() const;
1507 double numeric::to_double() const;
1508 cln::cl_N numeric::to_cl_N() const;
1511 @code{to_int()} and @code{to_long()} only work when the number they are
1512 applied on is an exact integer. Otherwise the program will halt with a
1513 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1514 rational number will return a floating-point approximation. Both
1515 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1516 part of complex numbers.
1519 @node Constants, Fundamental containers, Numbers, Basic concepts
1520 @c node-name, next, previous, up
1522 @cindex @code{constant} (class)
1525 @cindex @code{Catalan}
1526 @cindex @code{Euler}
1527 @cindex @code{evalf()}
1528 Constants behave pretty much like symbols except that they return some
1529 specific number when the method @code{.evalf()} is called.
1531 The predefined known constants are:
1534 @multitable @columnfractions .14 .30 .56
1535 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1537 @tab Archimedes' constant
1538 @tab 3.14159265358979323846264338327950288
1539 @item @code{Catalan}
1540 @tab Catalan's constant
1541 @tab 0.91596559417721901505460351493238411
1543 @tab Euler's (or Euler-Mascheroni) constant
1544 @tab 0.57721566490153286060651209008240243
1549 @node Fundamental containers, Lists, Constants, Basic concepts
1550 @c node-name, next, previous, up
1551 @section Sums, products and powers
1555 @cindex @code{power}
1557 Simple rational expressions are written down in GiNaC pretty much like
1558 in other CAS or like expressions involving numerical variables in C.
1559 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1560 been overloaded to achieve this goal. When you run the following
1561 code snippet, the constructor for an object of type @code{mul} is
1562 automatically called to hold the product of @code{a} and @code{b} and
1563 then the constructor for an object of type @code{add} is called to hold
1564 the sum of that @code{mul} object and the number one:
1568 symbol a("a"), b("b");
1573 @cindex @code{pow()}
1574 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1575 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1576 construction is necessary since we cannot safely overload the constructor
1577 @code{^} in C++ to construct a @code{power} object. If we did, it would
1578 have several counterintuitive and undesired effects:
1582 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1584 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1585 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1586 interpret this as @code{x^(a^b)}.
1588 Also, expressions involving integer exponents are very frequently used,
1589 which makes it even more dangerous to overload @code{^} since it is then
1590 hard to distinguish between the semantics as exponentiation and the one
1591 for exclusive or. (It would be embarrassing to return @code{1} where one
1592 has requested @code{2^3}.)
1595 @cindex @command{ginsh}
1596 All effects are contrary to mathematical notation and differ from the
1597 way most other CAS handle exponentiation, therefore overloading @code{^}
1598 is ruled out for GiNaC's C++ part. The situation is different in
1599 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1600 that the other frequently used exponentiation operator @code{**} does
1601 not exist at all in C++).
1603 To be somewhat more precise, objects of the three classes described
1604 here, are all containers for other expressions. An object of class
1605 @code{power} is best viewed as a container with two slots, one for the
1606 basis, one for the exponent. All valid GiNaC expressions can be
1607 inserted. However, basic transformations like simplifying
1608 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1609 when this is mathematically possible. If we replace the outer exponent
1610 three in the example by some symbols @code{a}, the simplification is not
1611 safe and will not be performed, since @code{a} might be @code{1/2} and
1614 Objects of type @code{add} and @code{mul} are containers with an
1615 arbitrary number of slots for expressions to be inserted. Again, simple
1616 and safe simplifications are carried out like transforming
1617 @code{3*x+4-x} to @code{2*x+4}.
1620 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1621 @c node-name, next, previous, up
1622 @section Lists of expressions
1623 @cindex @code{lst} (class)
1625 @cindex @code{nops()}
1627 @cindex @code{append()}
1628 @cindex @code{prepend()}
1629 @cindex @code{remove_first()}
1630 @cindex @code{remove_last()}
1631 @cindex @code{remove_all()}
1633 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1634 expressions. They are not as ubiquitous as in many other computer algebra
1635 packages, but are sometimes used to supply a variable number of arguments of
1636 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1637 constructors, so you should have a basic understanding of them.
1639 Lists can be constructed by assigning a comma-separated sequence of
1644 symbol x("x"), y("y");
1647 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1652 There are also constructors that allow direct creation of lists of up to
1653 16 expressions, which is often more convenient but slightly less efficient:
1657 // This produces the same list 'l' as above:
1658 // lst l(x, 2, y, x+y);
1659 // lst l = lst(x, 2, y, x+y);
1663 Use the @code{nops()} method to determine the size (number of expressions) of
1664 a list and the @code{op()} method or the @code{[]} operator to access
1665 individual elements:
1669 cout << l.nops() << endl; // prints '4'
1670 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1674 As with the standard @code{list<T>} container, accessing random elements of a
1675 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1676 sequential access to the elements of a list is possible with the
1677 iterator types provided by the @code{lst} class:
1680 typedef ... lst::const_iterator;
1681 typedef ... lst::const_reverse_iterator;
1682 lst::const_iterator lst::begin() const;
1683 lst::const_iterator lst::end() const;
1684 lst::const_reverse_iterator lst::rbegin() const;
1685 lst::const_reverse_iterator lst::rend() const;
1688 For example, to print the elements of a list individually you can use:
1693 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1698 which is one order faster than
1703 for (size_t i = 0; i < l.nops(); ++i)
1704 cout << l.op(i) << endl;
1708 These iterators also allow you to use some of the algorithms provided by
1709 the C++ standard library:
1713 // print the elements of the list (requires #include <iterator>)
1714 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1716 // sum up the elements of the list (requires #include <numeric>)
1717 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1718 cout << sum << endl; // prints '2+2*x+2*y'
1722 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1723 (the only other one is @code{matrix}). You can modify single elements:
1727 l[1] = 42; // l is now @{x, 42, y, x+y@}
1728 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1732 You can append or prepend an expression to a list with the @code{append()}
1733 and @code{prepend()} methods:
1737 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1738 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1742 You can remove the first or last element of a list with @code{remove_first()}
1743 and @code{remove_last()}:
1747 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1748 l.remove_last(); // l is now @{x, 7, y, x+y@}
1752 You can remove all the elements of a list with @code{remove_all()}:
1756 l.remove_all(); // l is now empty
1760 You can bring the elements of a list into a canonical order with @code{sort()}:
1769 // l1 and l2 are now equal
1773 Finally, you can remove all but the first element of consecutive groups of
1774 elements with @code{unique()}:
1779 l3 = x, 2, 2, 2, y, x+y, y+x;
1780 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1785 @node Mathematical functions, Relations, Lists, Basic concepts
1786 @c node-name, next, previous, up
1787 @section Mathematical functions
1788 @cindex @code{function} (class)
1789 @cindex trigonometric function
1790 @cindex hyperbolic function
1792 There are quite a number of useful functions hard-wired into GiNaC. For
1793 instance, all trigonometric and hyperbolic functions are implemented
1794 (@xref{Built-in functions}, for a complete list).
1796 These functions (better called @emph{pseudofunctions}) are all objects
1797 of class @code{function}. They accept one or more expressions as
1798 arguments and return one expression. If the arguments are not
1799 numerical, the evaluation of the function may be halted, as it does in
1800 the next example, showing how a function returns itself twice and
1801 finally an expression that may be really useful:
1803 @cindex Gamma function
1804 @cindex @code{subs()}
1807 symbol x("x"), y("y");
1809 cout << tgamma(foo) << endl;
1810 // -> tgamma(x+(1/2)*y)
1811 ex bar = foo.subs(y==1);
1812 cout << tgamma(bar) << endl;
1814 ex foobar = bar.subs(x==7);
1815 cout << tgamma(foobar) << endl;
1816 // -> (135135/128)*Pi^(1/2)
1820 Besides evaluation most of these functions allow differentiation, series
1821 expansion and so on. Read the next chapter in order to learn more about
1824 It must be noted that these pseudofunctions are created by inline
1825 functions, where the argument list is templated. This means that
1826 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1827 @code{sin(ex(1))} and will therefore not result in a floating point
1828 number. Unless of course the function prototype is explicitly
1829 overridden -- which is the case for arguments of type @code{numeric}
1830 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1831 point number of class @code{numeric} you should call
1832 @code{sin(numeric(1))}. This is almost the same as calling
1833 @code{sin(1).evalf()} except that the latter will return a numeric
1834 wrapped inside an @code{ex}.
1837 @node Relations, Integrals, Mathematical functions, Basic concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{relational} (class)
1842 Sometimes, a relation holding between two expressions must be stored
1843 somehow. The class @code{relational} is a convenient container for such
1844 purposes. A relation is by definition a container for two @code{ex} and
1845 a relation between them that signals equality, inequality and so on.
1846 They are created by simply using the C++ operators @code{==}, @code{!=},
1847 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1849 @xref{Mathematical functions}, for examples where various applications
1850 of the @code{.subs()} method show how objects of class relational are
1851 used as arguments. There they provide an intuitive syntax for
1852 substitutions. They are also used as arguments to the @code{ex::series}
1853 method, where the left hand side of the relation specifies the variable
1854 to expand in and the right hand side the expansion point. They can also
1855 be used for creating systems of equations that are to be solved for
1856 unknown variables. But the most common usage of objects of this class
1857 is rather inconspicuous in statements of the form @code{if
1858 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1859 conversion from @code{relational} to @code{bool} takes place. Note,
1860 however, that @code{==} here does not perform any simplifications, hence
1861 @code{expand()} must be called explicitly.
1863 @node Integrals, Matrices, Relations, Basic concepts
1864 @c node-name, next, previous, up
1866 @cindex @code{integral} (class)
1868 An object of class @dfn{integral} can be used to hold a symbolic integral.
1869 If you want to symbolically represent the integral of @code{x*x} from 0 to
1870 1, you would write this as
1872 integral(x, 0, 1, x*x)
1874 The first argument is the integration variable. It should be noted that
1875 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1876 fact, it can only integrate polynomials. An expression containing integrals
1877 can be evaluated symbolically by calling the
1881 method on it. Numerical evaluation is available by calling the
1885 method on an expression containing the integral. This will only evaluate
1886 integrals into a number if @code{subs}ing the integration variable by a
1887 number in the fourth argument of an integral and then @code{evalf}ing the
1888 result always results in a number. Of course, also the boundaries of the
1889 integration domain must @code{evalf} into numbers. It should be noted that
1890 trying to @code{evalf} a function with discontinuities in the integration
1891 domain is not recommended. The accuracy of the numeric evaluation of
1892 integrals is determined by the static member variable
1894 ex integral::relative_integration_error
1896 of the class @code{integral}. The default value of this is 10^-8.
1897 The integration works by halving the interval of integration, until numeric
1898 stability of the answer indicates that the requested accuracy has been
1899 reached. The maximum depth of the halving can be set via the static member
1902 int integral::max_integration_level
1904 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1905 return the integral unevaluated. The function that performs the numerical
1906 evaluation, is also available as
1908 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1911 This function will throw an exception if the maximum depth is exceeded. The
1912 last parameter of the function is optional and defaults to the
1913 @code{relative_integration_error}. To make sure that we do not do too
1914 much work if an expression contains the same integral multiple times,
1915 a lookup table is used.
1917 If you know that an expression holds an integral, you can get the
1918 integration variable, the left boundary, right boundary and integrand by
1919 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1920 @code{.op(3)}. Differentiating integrals with respect to variables works
1921 as expected. Note that it makes no sense to differentiate an integral
1922 with respect to the integration variable.
1924 @node Matrices, Indexed objects, Integrals, Basic concepts
1925 @c node-name, next, previous, up
1927 @cindex @code{matrix} (class)
1929 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1930 matrix with @math{m} rows and @math{n} columns are accessed with two
1931 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1932 second one in the range 0@dots{}@math{n-1}.
1934 There are a couple of ways to construct matrices, with or without preset
1935 elements. The constructor
1938 matrix::matrix(unsigned r, unsigned c);
1941 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1944 The fastest way to create a matrix with preinitialized elements is to assign
1945 a list of comma-separated expressions to an empty matrix (see below for an
1946 example). But you can also specify the elements as a (flat) list with
1949 matrix::matrix(unsigned r, unsigned c, const lst & l);
1954 @cindex @code{lst_to_matrix()}
1956 ex lst_to_matrix(const lst & l);
1959 constructs a matrix from a list of lists, each list representing a matrix row.
1961 There is also a set of functions for creating some special types of
1964 @cindex @code{diag_matrix()}
1965 @cindex @code{unit_matrix()}
1966 @cindex @code{symbolic_matrix()}
1968 ex diag_matrix(const lst & l);
1969 ex unit_matrix(unsigned x);
1970 ex unit_matrix(unsigned r, unsigned c);
1971 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1972 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1973 const string & tex_base_name);
1976 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1977 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1978 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1979 matrix filled with newly generated symbols made of the specified base name
1980 and the position of each element in the matrix.
1982 Matrices often arise by omitting elements of another matrix. For
1983 instance, the submatrix @code{S} of a matrix @code{M} takes a
1984 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1985 by removing one row and one column from a matrix @code{M}. (The
1986 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1987 can be used for computing the inverse using Cramer's rule.)
1989 @cindex @code{sub_matrix()}
1990 @cindex @code{reduced_matrix()}
1992 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1993 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1996 The function @code{sub_matrix()} takes a row offset @code{r} and a
1997 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1998 columns. The function @code{reduced_matrix()} has two integer arguments
1999 that specify which row and column to remove:
2007 cout << reduced_matrix(m, 1, 1) << endl;
2008 // -> [[11,13],[31,33]]
2009 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2010 // -> [[22,23],[32,33]]
2014 Matrix elements can be accessed and set using the parenthesis (function call)
2018 const ex & matrix::operator()(unsigned r, unsigned c) const;
2019 ex & matrix::operator()(unsigned r, unsigned c);
2022 It is also possible to access the matrix elements in a linear fashion with
2023 the @code{op()} method. But C++-style subscripting with square brackets
2024 @samp{[]} is not available.
2026 Here are a couple of examples for constructing matrices:
2030 symbol a("a"), b("b");
2044 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2047 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2050 cout << diag_matrix(lst(a, b)) << endl;
2053 cout << unit_matrix(3) << endl;
2054 // -> [[1,0,0],[0,1,0],[0,0,1]]
2056 cout << symbolic_matrix(2, 3, "x") << endl;
2057 // -> [[x00,x01,x02],[x10,x11,x12]]
2061 @cindex @code{is_zero_matrix()}
2062 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2063 all entries of the matrix are zeros. There is also method
2064 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2065 expression is zero or a zero matrix.
2067 @cindex @code{transpose()}
2068 There are three ways to do arithmetic with matrices. The first (and most
2069 direct one) is to use the methods provided by the @code{matrix} class:
2072 matrix matrix::add(const matrix & other) const;
2073 matrix matrix::sub(const matrix & other) const;
2074 matrix matrix::mul(const matrix & other) const;
2075 matrix matrix::mul_scalar(const ex & other) const;
2076 matrix matrix::pow(const ex & expn) const;
2077 matrix matrix::transpose() const;
2080 All of these methods return the result as a new matrix object. Here is an
2081 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2086 matrix A(2, 2), B(2, 2), C(2, 2);
2094 matrix result = A.mul(B).sub(C.mul_scalar(2));
2095 cout << result << endl;
2096 // -> [[-13,-6],[1,2]]
2101 @cindex @code{evalm()}
2102 The second (and probably the most natural) way is to construct an expression
2103 containing matrices with the usual arithmetic operators and @code{pow()}.
2104 For efficiency reasons, expressions with sums, products and powers of
2105 matrices are not automatically evaluated in GiNaC. You have to call the
2109 ex ex::evalm() const;
2112 to obtain the result:
2119 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2120 cout << e.evalm() << endl;
2121 // -> [[-13,-6],[1,2]]
2126 The non-commutativity of the product @code{A*B} in this example is
2127 automatically recognized by GiNaC. There is no need to use a special
2128 operator here. @xref{Non-commutative objects}, for more information about
2129 dealing with non-commutative expressions.
2131 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2132 to perform the arithmetic:
2137 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2138 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2140 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2141 cout << e.simplify_indexed() << endl;
2142 // -> [[-13,-6],[1,2]].i.j
2146 Using indices is most useful when working with rectangular matrices and
2147 one-dimensional vectors because you don't have to worry about having to
2148 transpose matrices before multiplying them. @xref{Indexed objects}, for
2149 more information about using matrices with indices, and about indices in
2152 The @code{matrix} class provides a couple of additional methods for
2153 computing determinants, traces, characteristic polynomials and ranks:
2155 @cindex @code{determinant()}
2156 @cindex @code{trace()}
2157 @cindex @code{charpoly()}
2158 @cindex @code{rank()}
2160 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2161 ex matrix::trace() const;
2162 ex matrix::charpoly(const ex & lambda) const;
2163 unsigned matrix::rank() const;
2166 The @samp{algo} argument of @code{determinant()} allows to select
2167 between different algorithms for calculating the determinant. The
2168 asymptotic speed (as parametrized by the matrix size) can greatly differ
2169 between those algorithms, depending on the nature of the matrix'
2170 entries. The possible values are defined in the @file{flags.h} header
2171 file. By default, GiNaC uses a heuristic to automatically select an
2172 algorithm that is likely (but not guaranteed) to give the result most
2175 @cindex @code{inverse()} (matrix)
2176 @cindex @code{solve()}
2177 Matrices may also be inverted using the @code{ex matrix::inverse()}
2178 method and linear systems may be solved with:
2181 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2182 unsigned algo=solve_algo::automatic) const;
2185 Assuming the matrix object this method is applied on is an @code{m}
2186 times @code{n} matrix, then @code{vars} must be a @code{n} times
2187 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2188 times @code{p} matrix. The returned matrix then has dimension @code{n}
2189 times @code{p} and in the case of an underdetermined system will still
2190 contain some of the indeterminates from @code{vars}. If the system is
2191 overdetermined, an exception is thrown.
2194 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2195 @c node-name, next, previous, up
2196 @section Indexed objects
2198 GiNaC allows you to handle expressions containing general indexed objects in
2199 arbitrary spaces. It is also able to canonicalize and simplify such
2200 expressions and perform symbolic dummy index summations. There are a number
2201 of predefined indexed objects provided, like delta and metric tensors.
2203 There are few restrictions placed on indexed objects and their indices and
2204 it is easy to construct nonsense expressions, but our intention is to
2205 provide a general framework that allows you to implement algorithms with
2206 indexed quantities, getting in the way as little as possible.
2208 @cindex @code{idx} (class)
2209 @cindex @code{indexed} (class)
2210 @subsection Indexed quantities and their indices
2212 Indexed expressions in GiNaC are constructed of two special types of objects,
2213 @dfn{index objects} and @dfn{indexed objects}.
2217 @cindex contravariant
2220 @item Index objects are of class @code{idx} or a subclass. Every index has
2221 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2222 the index lives in) which can both be arbitrary expressions but are usually
2223 a number or a simple symbol. In addition, indices of class @code{varidx} have
2224 a @dfn{variance} (they can be co- or contravariant), and indices of class
2225 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2227 @item Indexed objects are of class @code{indexed} or a subclass. They
2228 contain a @dfn{base expression} (which is the expression being indexed), and
2229 one or more indices.
2233 @strong{Please notice:} when printing expressions, covariant indices and indices
2234 without variance are denoted @samp{.i} while contravariant indices are
2235 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2236 value. In the following, we are going to use that notation in the text so
2237 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2238 not visible in the output.
2240 A simple example shall illustrate the concepts:
2244 #include <ginac/ginac.h>
2245 using namespace std;
2246 using namespace GiNaC;
2250 symbol i_sym("i"), j_sym("j");
2251 idx i(i_sym, 3), j(j_sym, 3);
2254 cout << indexed(A, i, j) << endl;
2256 cout << index_dimensions << indexed(A, i, j) << endl;
2258 cout << dflt; // reset cout to default output format (dimensions hidden)
2262 The @code{idx} constructor takes two arguments, the index value and the
2263 index dimension. First we define two index objects, @code{i} and @code{j},
2264 both with the numeric dimension 3. The value of the index @code{i} is the
2265 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2266 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2267 construct an expression containing one indexed object, @samp{A.i.j}. It has
2268 the symbol @code{A} as its base expression and the two indices @code{i} and
2271 The dimensions of indices are normally not visible in the output, but one
2272 can request them to be printed with the @code{index_dimensions} manipulator,
2275 Note the difference between the indices @code{i} and @code{j} which are of
2276 class @code{idx}, and the index values which are the symbols @code{i_sym}
2277 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2278 or numbers but must be index objects. For example, the following is not
2279 correct and will raise an exception:
2282 symbol i("i"), j("j");
2283 e = indexed(A, i, j); // ERROR: indices must be of type idx
2286 You can have multiple indexed objects in an expression, index values can
2287 be numeric, and index dimensions symbolic:
2291 symbol B("B"), dim("dim");
2292 cout << 4 * indexed(A, i)
2293 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2298 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2299 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2300 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2301 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2302 @code{simplify_indexed()} for that, see below).
2304 In fact, base expressions, index values and index dimensions can be
2305 arbitrary expressions:
2309 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2314 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2315 get an error message from this but you will probably not be able to do
2316 anything useful with it.
2318 @cindex @code{get_value()}
2319 @cindex @code{get_dimension()}
2323 ex idx::get_value();
2324 ex idx::get_dimension();
2327 return the value and dimension of an @code{idx} object. If you have an index
2328 in an expression, such as returned by calling @code{.op()} on an indexed
2329 object, you can get a reference to the @code{idx} object with the function
2330 @code{ex_to<idx>()} on the expression.
2332 There are also the methods
2335 bool idx::is_numeric();
2336 bool idx::is_symbolic();
2337 bool idx::is_dim_numeric();
2338 bool idx::is_dim_symbolic();
2341 for checking whether the value and dimension are numeric or symbolic
2342 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2343 about expressions}) returns information about the index value.
2345 @cindex @code{varidx} (class)
2346 If you need co- and contravariant indices, use the @code{varidx} class:
2350 symbol mu_sym("mu"), nu_sym("nu");
2351 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2352 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2354 cout << indexed(A, mu, nu) << endl;
2356 cout << indexed(A, mu_co, nu) << endl;
2358 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2363 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2364 co- or contravariant. The default is a contravariant (upper) index, but
2365 this can be overridden by supplying a third argument to the @code{varidx}
2366 constructor. The two methods
2369 bool varidx::is_covariant();
2370 bool varidx::is_contravariant();
2373 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2374 to get the object reference from an expression). There's also the very useful
2378 ex varidx::toggle_variance();
2381 which makes a new index with the same value and dimension but the opposite
2382 variance. By using it you only have to define the index once.
2384 @cindex @code{spinidx} (class)
2385 The @code{spinidx} class provides dotted and undotted variant indices, as
2386 used in the Weyl-van-der-Waerden spinor formalism:
2390 symbol K("K"), C_sym("C"), D_sym("D");
2391 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2392 // contravariant, undotted
2393 spinidx C_co(C_sym, 2, true); // covariant index
2394 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2395 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2397 cout << indexed(K, C, D) << endl;
2399 cout << indexed(K, C_co, D_dot) << endl;
2401 cout << indexed(K, D_co_dot, D) << endl;
2406 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2407 dotted or undotted. The default is undotted but this can be overridden by
2408 supplying a fourth argument to the @code{spinidx} constructor. The two
2412 bool spinidx::is_dotted();
2413 bool spinidx::is_undotted();
2416 allow you to check whether or not a @code{spinidx} object is dotted (use
2417 @code{ex_to<spinidx>()} to get the object reference from an expression).
2418 Finally, the two methods
2421 ex spinidx::toggle_dot();
2422 ex spinidx::toggle_variance_dot();
2425 create a new index with the same value and dimension but opposite dottedness
2426 and the same or opposite variance.
2428 @subsection Substituting indices
2430 @cindex @code{subs()}
2431 Sometimes you will want to substitute one symbolic index with another
2432 symbolic or numeric index, for example when calculating one specific element
2433 of a tensor expression. This is done with the @code{.subs()} method, as it
2434 is done for symbols (see @ref{Substituting expressions}).
2436 You have two possibilities here. You can either substitute the whole index
2437 by another index or expression:
2441 ex e = indexed(A, mu_co);
2442 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2443 // -> A.mu becomes A~nu
2444 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2445 // -> A.mu becomes A~0
2446 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2447 // -> A.mu becomes A.0
2451 The third example shows that trying to replace an index with something that
2452 is not an index will substitute the index value instead.
2454 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2459 ex e = indexed(A, mu_co);
2460 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2461 // -> A.mu becomes A.nu
2462 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2463 // -> A.mu becomes A.0
2467 As you see, with the second method only the value of the index will get
2468 substituted. Its other properties, including its dimension, remain unchanged.
2469 If you want to change the dimension of an index you have to substitute the
2470 whole index by another one with the new dimension.
2472 Finally, substituting the base expression of an indexed object works as
2477 ex e = indexed(A, mu_co);
2478 cout << e << " becomes " << e.subs(A == A+B) << endl;
2479 // -> A.mu becomes (B+A).mu
2483 @subsection Symmetries
2484 @cindex @code{symmetry} (class)
2485 @cindex @code{sy_none()}
2486 @cindex @code{sy_symm()}
2487 @cindex @code{sy_anti()}
2488 @cindex @code{sy_cycl()}
2490 Indexed objects can have certain symmetry properties with respect to their
2491 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2492 that is constructed with the helper functions
2495 symmetry sy_none(...);
2496 symmetry sy_symm(...);
2497 symmetry sy_anti(...);
2498 symmetry sy_cycl(...);
2501 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2502 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2503 represents a cyclic symmetry. Each of these functions accepts up to four
2504 arguments which can be either symmetry objects themselves or unsigned integer
2505 numbers that represent an index position (counting from 0). A symmetry
2506 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2507 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2510 Here are some examples of symmetry definitions:
2515 e = indexed(A, i, j);
2516 e = indexed(A, sy_none(), i, j); // equivalent
2517 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2519 // Symmetric in all three indices:
2520 e = indexed(A, sy_symm(), i, j, k);
2521 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2522 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2523 // different canonical order
2525 // Symmetric in the first two indices only:
2526 e = indexed(A, sy_symm(0, 1), i, j, k);
2527 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2529 // Antisymmetric in the first and last index only (index ranges need not
2531 e = indexed(A, sy_anti(0, 2), i, j, k);
2532 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2534 // An example of a mixed symmetry: antisymmetric in the first two and
2535 // last two indices, symmetric when swapping the first and last index
2536 // pairs (like the Riemann curvature tensor):
2537 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2539 // Cyclic symmetry in all three indices:
2540 e = indexed(A, sy_cycl(), i, j, k);
2541 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2543 // The following examples are invalid constructions that will throw
2544 // an exception at run time.
2546 // An index may not appear multiple times:
2547 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2548 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2550 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2551 // same number of indices:
2552 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2554 // And of course, you cannot specify indices which are not there:
2555 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2559 If you need to specify more than four indices, you have to use the
2560 @code{.add()} method of the @code{symmetry} class. For example, to specify
2561 full symmetry in the first six indices you would write
2562 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2564 If an indexed object has a symmetry, GiNaC will automatically bring the
2565 indices into a canonical order which allows for some immediate simplifications:
2569 cout << indexed(A, sy_symm(), i, j)
2570 + indexed(A, sy_symm(), j, i) << endl;
2572 cout << indexed(B, sy_anti(), i, j)
2573 + indexed(B, sy_anti(), j, i) << endl;
2575 cout << indexed(B, sy_anti(), i, j, k)
2576 - indexed(B, sy_anti(), j, k, i) << endl;
2581 @cindex @code{get_free_indices()}
2583 @subsection Dummy indices
2585 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2586 that a summation over the index range is implied. Symbolic indices which are
2587 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2588 dummy nor free indices.
2590 To be recognized as a dummy index pair, the two indices must be of the same
2591 class and their value must be the same single symbol (an index like
2592 @samp{2*n+1} is never a dummy index). If the indices are of class
2593 @code{varidx} they must also be of opposite variance; if they are of class
2594 @code{spinidx} they must be both dotted or both undotted.
2596 The method @code{.get_free_indices()} returns a vector containing the free
2597 indices of an expression. It also checks that the free indices of the terms
2598 of a sum are consistent:
2602 symbol A("A"), B("B"), C("C");
2604 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2605 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2607 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2608 cout << exprseq(e.get_free_indices()) << endl;
2610 // 'j' and 'l' are dummy indices
2612 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2613 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2615 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2616 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2617 cout << exprseq(e.get_free_indices()) << endl;
2619 // 'nu' is a dummy index, but 'sigma' is not
2621 e = indexed(A, mu, mu);
2622 cout << exprseq(e.get_free_indices()) << endl;
2624 // 'mu' is not a dummy index because it appears twice with the same
2627 e = indexed(A, mu, nu) + 42;
2628 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2629 // this will throw an exception:
2630 // "add::get_free_indices: inconsistent indices in sum"
2634 @cindex @code{expand_dummy_sum()}
2635 A dummy index summation like
2642 can be expanded for indices with numeric
2643 dimensions (e.g. 3) into the explicit sum like
2645 $a_1b^1+a_2b^2+a_3b^3 $.
2648 a.1 b~1 + a.2 b~2 + a.3 b~3.
2650 This is performed by the function
2653 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2656 which takes an expression @code{e} and returns the expanded sum for all
2657 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2658 is set to @code{true} then all substitutions are made by @code{idx} class
2659 indices, i.e. without variance. In this case the above sum
2668 $a_1b_1+a_2b_2+a_3b_3 $.
2671 a.1 b.1 + a.2 b.2 + a.3 b.3.
2675 @cindex @code{simplify_indexed()}
2676 @subsection Simplifying indexed expressions
2678 In addition to the few automatic simplifications that GiNaC performs on
2679 indexed expressions (such as re-ordering the indices of symmetric tensors
2680 and calculating traces and convolutions of matrices and predefined tensors)
2684 ex ex::simplify_indexed();
2685 ex ex::simplify_indexed(const scalar_products & sp);
2688 that performs some more expensive operations:
2691 @item it checks the consistency of free indices in sums in the same way
2692 @code{get_free_indices()} does
2693 @item it tries to give dummy indices that appear in different terms of a sum
2694 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2695 @item it (symbolically) calculates all possible dummy index summations/contractions
2696 with the predefined tensors (this will be explained in more detail in the
2698 @item it detects contractions that vanish for symmetry reasons, for example
2699 the contraction of a symmetric and a totally antisymmetric tensor
2700 @item as a special case of dummy index summation, it can replace scalar products
2701 of two tensors with a user-defined value
2704 The last point is done with the help of the @code{scalar_products} class
2705 which is used to store scalar products with known values (this is not an
2706 arithmetic class, you just pass it to @code{simplify_indexed()}):
2710 symbol A("A"), B("B"), C("C"), i_sym("i");
2714 sp.add(A, B, 0); // A and B are orthogonal
2715 sp.add(A, C, 0); // A and C are orthogonal
2716 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2718 e = indexed(A + B, i) * indexed(A + C, i);
2720 // -> (B+A).i*(A+C).i
2722 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2728 The @code{scalar_products} object @code{sp} acts as a storage for the
2729 scalar products added to it with the @code{.add()} method. This method
2730 takes three arguments: the two expressions of which the scalar product is
2731 taken, and the expression to replace it with.
2733 @cindex @code{expand()}
2734 The example above also illustrates a feature of the @code{expand()} method:
2735 if passed the @code{expand_indexed} option it will distribute indices
2736 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2738 @cindex @code{tensor} (class)
2739 @subsection Predefined tensors
2741 Some frequently used special tensors such as the delta, epsilon and metric
2742 tensors are predefined in GiNaC. They have special properties when
2743 contracted with other tensor expressions and some of them have constant
2744 matrix representations (they will evaluate to a number when numeric
2745 indices are specified).
2747 @cindex @code{delta_tensor()}
2748 @subsubsection Delta tensor
2750 The delta tensor takes two indices, is symmetric and has the matrix
2751 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2752 @code{delta_tensor()}:
2756 symbol A("A"), B("B");
2758 idx i(symbol("i"), 3), j(symbol("j"), 3),
2759 k(symbol("k"), 3), l(symbol("l"), 3);
2761 ex e = indexed(A, i, j) * indexed(B, k, l)
2762 * delta_tensor(i, k) * delta_tensor(j, l);
2763 cout << e.simplify_indexed() << endl;
2766 cout << delta_tensor(i, i) << endl;
2771 @cindex @code{metric_tensor()}
2772 @subsubsection General metric tensor
2774 The function @code{metric_tensor()} creates a general symmetric metric
2775 tensor with two indices that can be used to raise/lower tensor indices. The
2776 metric tensor is denoted as @samp{g} in the output and if its indices are of
2777 mixed variance it is automatically replaced by a delta tensor:
2783 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2785 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2786 cout << e.simplify_indexed() << endl;
2789 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2790 cout << e.simplify_indexed() << endl;
2793 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2794 * metric_tensor(nu, rho);
2795 cout << e.simplify_indexed() << endl;
2798 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2799 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2800 + indexed(A, mu.toggle_variance(), rho));
2801 cout << e.simplify_indexed() << endl;
2806 @cindex @code{lorentz_g()}
2807 @subsubsection Minkowski metric tensor
2809 The Minkowski metric tensor is a special metric tensor with a constant
2810 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2811 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2812 It is created with the function @code{lorentz_g()} (although it is output as
2817 varidx mu(symbol("mu"), 4);
2819 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2820 * lorentz_g(mu, varidx(0, 4)); // negative signature
2821 cout << e.simplify_indexed() << endl;
2824 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2825 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2826 cout << e.simplify_indexed() << endl;
2831 @cindex @code{spinor_metric()}
2832 @subsubsection Spinor metric tensor
2834 The function @code{spinor_metric()} creates an antisymmetric tensor with
2835 two indices that is used to raise/lower indices of 2-component spinors.
2836 It is output as @samp{eps}:
2842 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2843 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2845 e = spinor_metric(A, B) * indexed(psi, B_co);
2846 cout << e.simplify_indexed() << endl;
2849 e = spinor_metric(A, B) * indexed(psi, A_co);
2850 cout << e.simplify_indexed() << endl;
2853 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2854 cout << e.simplify_indexed() << endl;
2857 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2858 cout << e.simplify_indexed() << endl;
2861 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2862 cout << e.simplify_indexed() << endl;
2865 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2866 cout << e.simplify_indexed() << endl;
2871 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2873 @cindex @code{epsilon_tensor()}
2874 @cindex @code{lorentz_eps()}
2875 @subsubsection Epsilon tensor
2877 The epsilon tensor is totally antisymmetric, its number of indices is equal
2878 to the dimension of the index space (the indices must all be of the same
2879 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2880 defined to be 1. Its behavior with indices that have a variance also
2881 depends on the signature of the metric. Epsilon tensors are output as
2884 There are three functions defined to create epsilon tensors in 2, 3 and 4
2888 ex epsilon_tensor(const ex & i1, const ex & i2);
2889 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2890 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2891 bool pos_sig = false);
2894 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2895 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2896 Minkowski space (the last @code{bool} argument specifies whether the metric
2897 has negative or positive signature, as in the case of the Minkowski metric
2902 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2903 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2904 e = lorentz_eps(mu, nu, rho, sig) *
2905 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2906 cout << simplify_indexed(e) << endl;
2907 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2909 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2910 symbol A("A"), B("B");
2911 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2912 cout << simplify_indexed(e) << endl;
2913 // -> -B.k*A.j*eps.i.k.j
2914 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2915 cout << simplify_indexed(e) << endl;
2920 @subsection Linear algebra
2922 The @code{matrix} class can be used with indices to do some simple linear
2923 algebra (linear combinations and products of vectors and matrices, traces
2924 and scalar products):
2928 idx i(symbol("i"), 2), j(symbol("j"), 2);
2929 symbol x("x"), y("y");
2931 // A is a 2x2 matrix, X is a 2x1 vector
2932 matrix A(2, 2), X(2, 1);
2937 cout << indexed(A, i, i) << endl;
2940 ex e = indexed(A, i, j) * indexed(X, j);
2941 cout << e.simplify_indexed() << endl;
2942 // -> [[2*y+x],[4*y+3*x]].i
2944 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2945 cout << e.simplify_indexed() << endl;
2946 // -> [[3*y+3*x,6*y+2*x]].j
2950 You can of course obtain the same results with the @code{matrix::add()},
2951 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2952 but with indices you don't have to worry about transposing matrices.
2954 Matrix indices always start at 0 and their dimension must match the number
2955 of rows/columns of the matrix. Matrices with one row or one column are
2956 vectors and can have one or two indices (it doesn't matter whether it's a
2957 row or a column vector). Other matrices must have two indices.
2959 You should be careful when using indices with variance on matrices. GiNaC
2960 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2961 @samp{F.mu.nu} are different matrices. In this case you should use only
2962 one form for @samp{F} and explicitly multiply it with a matrix representation
2963 of the metric tensor.
2966 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2967 @c node-name, next, previous, up
2968 @section Non-commutative objects
2970 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2971 non-commutative objects are built-in which are mostly of use in high energy
2975 @item Clifford (Dirac) algebra (class @code{clifford})
2976 @item su(3) Lie algebra (class @code{color})
2977 @item Matrices (unindexed) (class @code{matrix})
2980 The @code{clifford} and @code{color} classes are subclasses of
2981 @code{indexed} because the elements of these algebras usually carry
2982 indices. The @code{matrix} class is described in more detail in
2985 Unlike most computer algebra systems, GiNaC does not primarily provide an
2986 operator (often denoted @samp{&*}) for representing inert products of
2987 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2988 classes of objects involved, and non-commutative products are formed with
2989 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2990 figuring out by itself which objects commutate and will group the factors
2991 by their class. Consider this example:
2995 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2996 idx a(symbol("a"), 8), b(symbol("b"), 8);
2997 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2999 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3003 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3004 groups the non-commutative factors (the gammas and the su(3) generators)
3005 together while preserving the order of factors within each class (because
3006 Clifford objects commutate with color objects). The resulting expression is a
3007 @emph{commutative} product with two factors that are themselves non-commutative
3008 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3009 parentheses are placed around the non-commutative products in the output.
3011 @cindex @code{ncmul} (class)
3012 Non-commutative products are internally represented by objects of the class
3013 @code{ncmul}, as opposed to commutative products which are handled by the
3014 @code{mul} class. You will normally not have to worry about this distinction,
3017 The advantage of this approach is that you never have to worry about using
3018 (or forgetting to use) a special operator when constructing non-commutative
3019 expressions. Also, non-commutative products in GiNaC are more intelligent
3020 than in other computer algebra systems; they can, for example, automatically
3021 canonicalize themselves according to rules specified in the implementation
3022 of the non-commutative classes. The drawback is that to work with other than
3023 the built-in algebras you have to implement new classes yourself. Both
3024 symbols and user-defined functions can be specified as being non-commutative.
3026 @cindex @code{return_type()}
3027 @cindex @code{return_type_tinfo()}
3028 Information about the commutativity of an object or expression can be
3029 obtained with the two member functions
3032 unsigned ex::return_type() const;
3033 unsigned ex::return_type_tinfo() const;
3036 The @code{return_type()} function returns one of three values (defined in
3037 the header file @file{flags.h}), corresponding to three categories of
3038 expressions in GiNaC:
3041 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3042 classes are of this kind.
3043 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3044 certain class of non-commutative objects which can be determined with the
3045 @code{return_type_tinfo()} method. Expressions of this category commutate
3046 with everything except @code{noncommutative} expressions of the same
3048 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3049 of non-commutative objects of different classes. Expressions of this
3050 category don't commutate with any other @code{noncommutative} or
3051 @code{noncommutative_composite} expressions.
3054 The value returned by the @code{return_type_tinfo()} method is valid only
3055 when the return type of the expression is @code{noncommutative}. It is a
3056 value that is unique to the class of the object and usually one of the
3057 constants in @file{tinfos.h}, or derived therefrom.
3059 Here are a couple of examples:
3062 @multitable @columnfractions 0.33 0.33 0.34
3063 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3064 @item @code{42} @tab @code{commutative} @tab -
3065 @item @code{2*x-y} @tab @code{commutative} @tab -
3066 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3067 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3068 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3069 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3073 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3074 @code{TINFO_clifford} for objects with a representation label of zero.
3075 Other representation labels yield a different @code{return_type_tinfo()},
3076 but it's the same for any two objects with the same label. This is also true
3079 A last note: With the exception of matrices, positive integer powers of
3080 non-commutative objects are automatically expanded in GiNaC. For example,
3081 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3082 non-commutative expressions).
3085 @cindex @code{clifford} (class)
3086 @subsection Clifford algebra
3089 Clifford algebras are supported in two flavours: Dirac gamma
3090 matrices (more physical) and generic Clifford algebras (more
3093 @cindex @code{dirac_gamma()}
3094 @subsubsection Dirac gamma matrices
3095 Dirac gamma matrices (note that GiNaC doesn't treat them
3096 as matrices) are designated as @samp{gamma~mu} and satisfy
3097 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3098 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3099 constructed by the function
3102 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3105 which takes two arguments: the index and a @dfn{representation label} in the
3106 range 0 to 255 which is used to distinguish elements of different Clifford
3107 algebras (this is also called a @dfn{spin line index}). Gammas with different
3108 labels commutate with each other. The dimension of the index can be 4 or (in
3109 the framework of dimensional regularization) any symbolic value. Spinor
3110 indices on Dirac gammas are not supported in GiNaC.
3112 @cindex @code{dirac_ONE()}
3113 The unity element of a Clifford algebra is constructed by
3116 ex dirac_ONE(unsigned char rl = 0);
3119 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3120 multiples of the unity element, even though it's customary to omit it.
3121 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3122 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3123 GiNaC will complain and/or produce incorrect results.
3125 @cindex @code{dirac_gamma5()}
3126 There is a special element @samp{gamma5} that commutates with all other
3127 gammas, has a unit square, and in 4 dimensions equals
3128 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3131 ex dirac_gamma5(unsigned char rl = 0);
3134 @cindex @code{dirac_gammaL()}
3135 @cindex @code{dirac_gammaR()}
3136 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3137 objects, constructed by
3140 ex dirac_gammaL(unsigned char rl = 0);
3141 ex dirac_gammaR(unsigned char rl = 0);
3144 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3145 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3147 @cindex @code{dirac_slash()}
3148 Finally, the function
3151 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3154 creates a term that represents a contraction of @samp{e} with the Dirac
3155 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3156 with a unique index whose dimension is given by the @code{dim} argument).
3157 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3159 In products of dirac gammas, superfluous unity elements are automatically
3160 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3161 and @samp{gammaR} are moved to the front.
3163 The @code{simplify_indexed()} function performs contractions in gamma strings,
3169 symbol a("a"), b("b"), D("D");
3170 varidx mu(symbol("mu"), D);
3171 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3172 * dirac_gamma(mu.toggle_variance());
3174 // -> gamma~mu*a\*gamma.mu
3175 e = e.simplify_indexed();
3178 cout << e.subs(D == 4) << endl;
3184 @cindex @code{dirac_trace()}
3185 To calculate the trace of an expression containing strings of Dirac gammas
3186 you use one of the functions
3189 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3190 const ex & trONE = 4);
3191 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3192 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3195 These functions take the trace over all gammas in the specified set @code{rls}
3196 or list @code{rll} of representation labels, or the single label @code{rl};
3197 gammas with other labels are left standing. The last argument to
3198 @code{dirac_trace()} is the value to be returned for the trace of the unity
3199 element, which defaults to 4.
3201 The @code{dirac_trace()} function is a linear functional that is equal to the
3202 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3203 functional is not cyclic in
3206 dimensions when acting on
3207 expressions containing @samp{gamma5}, so it's not a proper trace. This
3208 @samp{gamma5} scheme is described in greater detail in
3209 @cite{The Role of gamma5 in Dimensional Regularization}.
3211 The value of the trace itself is also usually different in 4 and in
3219 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3220 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3221 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3222 cout << dirac_trace(e).simplify_indexed() << endl;
3229 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3230 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3231 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3232 cout << dirac_trace(e).simplify_indexed() << endl;
3233 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3237 Here is an example for using @code{dirac_trace()} to compute a value that
3238 appears in the calculation of the one-loop vacuum polarization amplitude in
3243 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3244 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3247 sp.add(l, l, pow(l, 2));
3248 sp.add(l, q, ldotq);
3250 ex e = dirac_gamma(mu) *
3251 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3252 dirac_gamma(mu.toggle_variance()) *
3253 (dirac_slash(l, D) + m * dirac_ONE());
3254 e = dirac_trace(e).simplify_indexed(sp);
3255 e = e.collect(lst(l, ldotq, m));
3257 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3261 The @code{canonicalize_clifford()} function reorders all gamma products that
3262 appear in an expression to a canonical (but not necessarily simple) form.
3263 You can use this to compare two expressions or for further simplifications:
3267 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3268 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3270 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3272 e = canonicalize_clifford(e);
3274 // -> 2*ONE*eta~mu~nu
3278 @cindex @code{clifford_unit()}
3279 @subsubsection A generic Clifford algebra
3281 A generic Clifford algebra, i.e. a
3285 dimensional algebra with
3289 satisfying the identities
3291 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3294 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3296 for some bilinear form (@code{metric})
3297 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3298 and contain symbolic entries. Such generators are created by the
3302 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3305 where @code{mu} should be a @code{idx} (or descendant) class object
3306 indexing the generators.
3307 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3308 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3309 object. In fact, any expression either with two free indices or without
3310 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3311 object with two newly created indices with @code{metr} as its
3312 @code{op(0)} will be used.
3313 Optional parameter @code{rl} allows to distinguish different
3314 Clifford algebras, which will commute with each other.
3316 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3317 something very close to @code{dirac_gamma(mu)}, although
3318 @code{dirac_gamma} have more efficient simplification mechanism.
3319 @cindex @code{clifford::get_metric()}
3320 The method @code{clifford::get_metric()} returns a metric defining this
3323 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3324 the Clifford algebra units with a call like that
3327 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3330 since this may yield some further automatic simplifications. Again, for a
3331 metric defined through a @code{matrix} such a symmetry is detected
3334 Individual generators of a Clifford algebra can be accessed in several
3340 idx i(symbol("i"), 4);
3342 ex M = diag_matrix(lst(1, -1, 0, s));
3343 ex e = clifford_unit(i, M);
3344 ex e0 = e.subs(i == 0);
3345 ex e1 = e.subs(i == 1);
3346 ex e2 = e.subs(i == 2);
3347 ex e3 = e.subs(i == 3);
3352 will produce four anti-commuting generators of a Clifford algebra with properties
3354 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3357 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3358 @code{pow(e3, 2) = s}.
3361 @cindex @code{lst_to_clifford()}
3362 A similar effect can be achieved from the function
3365 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3366 unsigned char rl = 0);
3367 ex lst_to_clifford(const ex & v, const ex & e);
3370 which converts a list or vector
3372 $v = (v^0, v^1, ..., v^n)$
3375 @samp{v = (v~0, v~1, ..., v~n)}
3380 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3383 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3386 directly supplied in the second form of the procedure. In the first form
3387 the Clifford unit @samp{e.k} is generated by the call of
3388 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3389 with the help of @code{lst_to_clifford()} as follows
3394 idx i(symbol("i"), 4);
3396 ex M = diag_matrix(lst(1, -1, 0, s));
3397 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3398 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3399 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3400 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3405 @cindex @code{clifford_to_lst()}
3406 There is the inverse function
3409 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3412 which takes an expression @code{e} and tries to find a list
3414 $v = (v^0, v^1, ..., v^n)$
3417 @samp{v = (v~0, v~1, ..., v~n)}
3421 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3424 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3426 with respect to the given Clifford units @code{c} and with none of the
3427 @samp{v~k} containing Clifford units @code{c} (of course, this
3428 may be impossible). This function can use an @code{algebraic} method
3429 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3431 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3434 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3436 is zero or is not @code{numeric} for some @samp{k}
3437 then the method will be automatically changed to symbolic. The same effect
3438 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3440 @cindex @code{clifford_prime()}
3441 @cindex @code{clifford_star()}
3442 @cindex @code{clifford_bar()}
3443 There are several functions for (anti-)automorphisms of Clifford algebras:
3446 ex clifford_prime(const ex & e)
3447 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3448 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3451 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3452 changes signs of all Clifford units in the expression. The reversion
3453 of a Clifford algebra @code{clifford_star()} coincides with the
3454 @code{conjugate()} method and effectively reverses the order of Clifford
3455 units in any product. Finally the main anti-automorphism
3456 of a Clifford algebra @code{clifford_bar()} is the composition of the
3457 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3458 in a product. These functions correspond to the notations
3473 used in Clifford algebra textbooks.
3475 @cindex @code{clifford_norm()}
3479 ex clifford_norm(const ex & e);
3482 @cindex @code{clifford_inverse()}
3483 calculates the norm of a Clifford number from the expression
3485 $||e||^2 = e\overline{e}$.
3488 @code{||e||^2 = e \bar@{e@}}
3490 The inverse of a Clifford expression is returned by the function
3493 ex clifford_inverse(const ex & e);
3496 which calculates it as
3498 $e^{-1} = \overline{e}/||e||^2$.
3501 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3510 then an exception is raised.
3512 @cindex @code{remove_dirac_ONE()}
3513 If a Clifford number happens to be a factor of
3514 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3515 expression by the function
3518 ex remove_dirac_ONE(const ex & e);
3521 @cindex @code{canonicalize_clifford()}
3522 The function @code{canonicalize_clifford()} works for a
3523 generic Clifford algebra in a similar way as for Dirac gammas.
3525 The next provided function is
3527 @cindex @code{clifford_moebius_map()}
3529 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3530 const ex & d, const ex & v, const ex & G,
3531 unsigned char rl = 0);
3532 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3533 unsigned char rl = 0);
3536 It takes a list or vector @code{v} and makes the Moebius (conformal or
3537 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3538 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3539 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3540 indexed object, tensormetric, matrix or a Clifford unit, in the later
3541 case the optional parameter @code{rl} is ignored even if supplied.
3542 Depending from the type of @code{v} the returned value of this function
3543 is either a vector or a list holding vector's components.
3545 @cindex @code{clifford_max_label()}
3546 Finally the function
3549 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3552 can detect a presence of Clifford objects in the expression @code{e}: if
3553 such objects are found it returns the maximal
3554 @code{representation_label} of them, otherwise @code{-1}. The optional
3555 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3556 be ignored during the search.
3558 LaTeX output for Clifford units looks like
3559 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3560 @code{representation_label} and @code{\nu} is the index of the
3561 corresponding unit. This provides a flexible typesetting with a suitable
3562 defintion of the @code{\clifford} command. For example, the definition
3564 \newcommand@{\clifford@}[1][]@{@}
3566 typesets all Clifford units identically, while the alternative definition
3568 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3570 prints units with @code{representation_label=0} as
3577 with @code{representation_label=1} as
3584 and with @code{representation_label=2} as
3592 @cindex @code{color} (class)
3593 @subsection Color algebra
3595 @cindex @code{color_T()}
3596 For computations in quantum chromodynamics, GiNaC implements the base elements
3597 and structure constants of the su(3) Lie algebra (color algebra). The base
3598 elements @math{T_a} are constructed by the function
3601 ex color_T(const ex & a, unsigned char rl = 0);
3604 which takes two arguments: the index and a @dfn{representation label} in the
3605 range 0 to 255 which is used to distinguish elements of different color
3606 algebras. Objects with different labels commutate with each other. The
3607 dimension of the index must be exactly 8 and it should be of class @code{idx},
3610 @cindex @code{color_ONE()}
3611 The unity element of a color algebra is constructed by
3614 ex color_ONE(unsigned char rl = 0);
3617 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3618 multiples of the unity element, even though it's customary to omit it.
3619 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3620 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3621 GiNaC may produce incorrect results.
3623 @cindex @code{color_d()}
3624 @cindex @code{color_f()}
3628 ex color_d(const ex & a, const ex & b, const ex & c);
3629 ex color_f(const ex & a, const ex & b, const ex & c);
3632 create the symmetric and antisymmetric structure constants @math{d_abc} and
3633 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3634 and @math{[T_a, T_b] = i f_abc T_c}.
3636 These functions evaluate to their numerical values,
3637 if you supply numeric indices to them. The index values should be in
3638 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3639 goes along better with the notations used in physical literature.
3641 @cindex @code{color_h()}
3642 There's an additional function
3645 ex color_h(const ex & a, const ex & b, const ex & c);
3648 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3650 The function @code{simplify_indexed()} performs some simplifications on
3651 expressions containing color objects:
3656 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3657 k(symbol("k"), 8), l(symbol("l"), 8);
3659 e = color_d(a, b, l) * color_f(a, b, k);
3660 cout << e.simplify_indexed() << endl;
3663 e = color_d(a, b, l) * color_d(a, b, k);
3664 cout << e.simplify_indexed() << endl;
3667 e = color_f(l, a, b) * color_f(a, b, k);
3668 cout << e.simplify_indexed() << endl;
3671 e = color_h(a, b, c) * color_h(a, b, c);
3672 cout << e.simplify_indexed() << endl;
3675 e = color_h(a, b, c) * color_T(b) * color_T(c);
3676 cout << e.simplify_indexed() << endl;
3679 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3680 cout << e.simplify_indexed() << endl;
3683 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3684 cout << e.simplify_indexed() << endl;
3685 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3689 @cindex @code{color_trace()}
3690 To calculate the trace of an expression containing color objects you use one
3694 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3695 ex color_trace(const ex & e, const lst & rll);
3696 ex color_trace(const ex & e, unsigned char rl = 0);
3699 These functions take the trace over all color @samp{T} objects in the
3700 specified set @code{rls} or list @code{rll} of representation labels, or the
3701 single label @code{rl}; @samp{T}s with other labels are left standing. For
3706 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3708 // -> -I*f.a.c.b+d.a.c.b
3713 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3714 @c node-name, next, previous, up
3717 @cindex @code{exhashmap} (class)
3719 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3720 that can be used as a drop-in replacement for the STL
3721 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3722 typically constant-time, element look-up than @code{map<>}.
3724 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3725 following differences:
3729 no @code{lower_bound()} and @code{upper_bound()} methods
3731 no reverse iterators, no @code{rbegin()}/@code{rend()}
3733 no @code{operator<(exhashmap, exhashmap)}
3735 the comparison function object @code{key_compare} is hardcoded to
3738 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3739 initial hash table size (the actual table size after construction may be
3740 larger than the specified value)
3742 the method @code{size_t bucket_count()} returns the current size of the hash
3745 @code{insert()} and @code{erase()} operations invalidate all iterators