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) 1999-2005 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2005 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2005 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 By default, the only documentation that will be built is this tutorial
606 in @file{.info} format. To build the GiNaC tutorial and reference manual
607 in HTML, DVI, PostScript, or PDF formats, use one of
616 Generally, the top-level Makefile runs recursively to the
617 subdirectories. It is therefore safe to go into any subdirectory
618 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
619 @var{target} there in case something went wrong.
622 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
623 @c node-name, next, previous, up
624 @section Installing GiNaC
627 To install GiNaC on your system, simply type
633 As described in the section about configuration the files will be
634 installed in the following directories (the directories will be created
635 if they don't already exist):
640 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
641 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
642 So will @file{libginac.so} unless the configure script was
643 given the option @option{--disable-shared}. The proper symlinks
644 will be established as well.
647 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
648 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
651 All documentation (info) will be stuffed into
652 @file{@var{PREFIX}/share/doc/GiNaC/} (or
653 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
657 For the sake of completeness we will list some other useful make
658 targets: @command{make clean} deletes all files generated by
659 @command{make}, i.e. all the object files. In addition @command{make
660 distclean} removes all files generated by the configuration and
661 @command{make maintainer-clean} goes one step further and deletes files
662 that may require special tools to rebuild (like the @command{libtool}
663 for instance). Finally @command{make uninstall} removes the installed
664 library, header files and documentation@footnote{Uninstallation does not
665 work after you have called @command{make distclean} since the
666 @file{Makefile} is itself generated by the configuration from
667 @file{Makefile.in} and hence deleted by @command{make distclean}. There
668 are two obvious ways out of this dilemma. First, you can run the
669 configuration again with the same @var{PREFIX} thus creating a
670 @file{Makefile} with a working @samp{uninstall} target. Second, you can
671 do it by hand since you now know where all the files went during
675 @node Basic Concepts, Expressions, Installing GiNaC, Top
676 @c node-name, next, previous, up
677 @chapter Basic Concepts
679 This chapter will describe the different fundamental objects that can be
680 handled by GiNaC. But before doing so, it is worthwhile introducing you
681 to the more commonly used class of expressions, representing a flexible
682 meta-class for storing all mathematical objects.
685 * Expressions:: The fundamental GiNaC class.
686 * Automatic evaluation:: Evaluation and canonicalization.
687 * Error handling:: How the library reports errors.
688 * The Class Hierarchy:: Overview of GiNaC's classes.
689 * Symbols:: Symbolic objects.
690 * Numbers:: Numerical objects.
691 * Constants:: Pre-defined constants.
692 * Fundamental containers:: Sums, products and powers.
693 * Lists:: Lists of expressions.
694 * Mathematical functions:: Mathematical functions.
695 * Relations:: Equality, Inequality and all that.
696 * Integrals:: Symbolic integrals.
697 * Matrices:: Matrices.
698 * Indexed objects:: Handling indexed quantities.
699 * Non-commutative objects:: Algebras with non-commutative products.
700 * Hash Maps:: A faster alternative to std::map<>.
704 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
705 @c node-name, next, previous, up
707 @cindex expression (class @code{ex})
710 The most common class of objects a user deals with is the expression
711 @code{ex}, representing a mathematical object like a variable, number,
712 function, sum, product, etc@dots{} Expressions may be put together to form
713 new expressions, passed as arguments to functions, and so on. Here is a
714 little collection of valid expressions:
717 ex MyEx1 = 5; // simple number
718 ex MyEx2 = x + 2*y; // polynomial in x and y
719 ex MyEx3 = (x + 1)/(x - 1); // rational expression
720 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
721 ex MyEx5 = MyEx4 + 1; // similar to above
724 Expressions are handles to other more fundamental objects, that often
725 contain other expressions thus creating a tree of expressions
726 (@xref{Internal Structures}, for particular examples). Most methods on
727 @code{ex} therefore run top-down through such an expression tree. For
728 example, the method @code{has()} scans recursively for occurrences of
729 something inside an expression. Thus, if you have declared @code{MyEx4}
730 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
731 the argument of @code{sin} and hence return @code{true}.
733 The next sections will outline the general picture of GiNaC's class
734 hierarchy and describe the classes of objects that are handled by
737 @subsection Note: Expressions and STL containers
739 GiNaC expressions (@code{ex} objects) have value semantics (they can be
740 assigned, reassigned and copied like integral types) but the operator
741 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
742 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
744 This implies that in order to use expressions in sorted containers such as
745 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
746 comparison predicate. GiNaC provides such a predicate, called
747 @code{ex_is_less}. For example, a set of expressions should be defined
748 as @code{std::set<ex, ex_is_less>}.
750 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
751 don't pose a problem. A @code{std::vector<ex>} works as expected.
753 @xref{Information About Expressions}, for more about comparing and ordering
757 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
758 @c node-name, next, previous, up
759 @section Automatic evaluation and canonicalization of expressions
762 GiNaC performs some automatic transformations on expressions, to simplify
763 them and put them into a canonical form. Some examples:
766 ex MyEx1 = 2*x - 1 + x; // 3*x-1
767 ex MyEx2 = x - x; // 0
768 ex MyEx3 = cos(2*Pi); // 1
769 ex MyEx4 = x*y/x; // y
772 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
773 evaluation}. GiNaC only performs transformations that are
777 at most of complexity
785 algebraically correct, possibly except for a set of measure zero (e.g.
786 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
789 There are two types of automatic transformations in GiNaC that may not
790 behave in an entirely obvious way at first glance:
794 The terms of sums and products (and some other things like the arguments of
795 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
796 into a canonical form that is deterministic, but not lexicographical or in
797 any other way easy to guess (it almost always depends on the number and
798 order of the symbols you define). However, constructing the same expression
799 twice, either implicitly or explicitly, will always result in the same
802 Expressions of the form 'number times sum' are automatically expanded (this
803 has to do with GiNaC's internal representation of sums and products). For
806 ex MyEx5 = 2*(x + y); // 2*x+2*y
807 ex MyEx6 = z*(x + y); // z*(x+y)
811 The general rule is that when you construct expressions, GiNaC automatically
812 creates them in canonical form, which might differ from the form you typed in
813 your program. This may create some awkward looking output (@samp{-y+x} instead
814 of @samp{x-y}) but allows for more efficient operation and usually yields
815 some immediate simplifications.
817 @cindex @code{eval()}
818 Internally, the anonymous evaluator in GiNaC is implemented by the methods
821 ex ex::eval(int level = 0) const;
822 ex basic::eval(int level = 0) const;
825 but unless you are extending GiNaC with your own classes or functions, there
826 should never be any reason to call them explicitly. All GiNaC methods that
827 transform expressions, like @code{subs()} or @code{normal()}, automatically
828 re-evaluate their results.
831 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
832 @c node-name, next, previous, up
833 @section Error handling
835 @cindex @code{pole_error} (class)
837 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
838 generated by GiNaC are subclassed from the standard @code{exception} class
839 defined in the @file{<stdexcept>} header. In addition to the predefined
840 @code{logic_error}, @code{domain_error}, @code{out_of_range},
841 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
842 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
843 exception that gets thrown when trying to evaluate a mathematical function
846 The @code{pole_error} class has a member function
849 int pole_error::degree() const;
852 that returns the order of the singularity (or 0 when the pole is
853 logarithmic or the order is undefined).
855 When using GiNaC it is useful to arrange for exceptions to be caught in
856 the main program even if you don't want to do any special error handling.
857 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
858 default exception handler of your C++ compiler's run-time system which
859 usually only aborts the program without giving any information what went
862 Here is an example for a @code{main()} function that catches and prints
863 exceptions generated by GiNaC:
868 #include <ginac/ginac.h>
870 using namespace GiNaC;
878 @} catch (exception &p) @{
879 cerr << p.what() << endl;
887 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
888 @c node-name, next, previous, up
889 @section The Class Hierarchy
891 GiNaC's class hierarchy consists of several classes representing
892 mathematical objects, all of which (except for @code{ex} and some
893 helpers) are internally derived from one abstract base class called
894 @code{basic}. You do not have to deal with objects of class
895 @code{basic}, instead you'll be dealing with symbols, numbers,
896 containers of expressions and so on.
900 To get an idea about what kinds of symbolic composites may be built we
901 have a look at the most important classes in the class hierarchy and
902 some of the relations among the classes:
904 @image{classhierarchy}
906 The abstract classes shown here (the ones without drop-shadow) are of no
907 interest for the user. They are used internally in order to avoid code
908 duplication if two or more classes derived from them share certain
909 features. An example is @code{expairseq}, a container for a sequence of
910 pairs each consisting of one expression and a number (@code{numeric}).
911 What @emph{is} visible to the user are the derived classes @code{add}
912 and @code{mul}, representing sums and products. @xref{Internal
913 Structures}, where these two classes are described in more detail. The
914 following table shortly summarizes what kinds of mathematical objects
915 are stored in the different classes:
918 @multitable @columnfractions .22 .78
919 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
920 @item @code{constant} @tab Constants like
927 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
928 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
929 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
930 @item @code{ncmul} @tab Products of non-commutative objects
931 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
936 @code{sqrt(}@math{2}@code{)}
939 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
940 @item @code{function} @tab A symbolic function like
947 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
948 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
949 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
950 @item @code{indexed} @tab Indexed object like @math{A_ij}
951 @item @code{tensor} @tab Special tensor like the delta and metric tensors
952 @item @code{idx} @tab Index of an indexed object
953 @item @code{varidx} @tab Index with variance
954 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
955 @item @code{wildcard} @tab Wildcard for pattern matching
956 @item @code{structure} @tab Template for user-defined classes
961 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
962 @c node-name, next, previous, up
964 @cindex @code{symbol} (class)
965 @cindex hierarchy of classes
968 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
969 manipulation what atoms are for chemistry.
971 A typical symbol definition looks like this:
976 This definition actually contains three very different things:
978 @item a C++ variable named @code{x}
979 @item a @code{symbol} object stored in this C++ variable; this object
980 represents the symbol in a GiNaC expression
981 @item the string @code{"x"} which is the name of the symbol, used (almost)
982 exclusively for printing expressions holding the symbol
985 Symbols have an explicit name, supplied as a string during construction,
986 because in C++, variable names can't be used as values, and the C++ compiler
987 throws them away during compilation.
989 It is possible to omit the symbol name in the definition:
994 In this case, GiNaC will assign the symbol an internal, unique name of the
995 form @code{symbolNNN}. This won't affect the usability of the symbol but
996 the output of your calculations will become more readable if you give your
997 symbols sensible names (for intermediate expressions that are only used
998 internally such anonymous symbols can be quite useful, however).
1000 Now, here is one important property of GiNaC that differentiates it from
1001 other computer algebra programs you may have used: GiNaC does @emph{not} use
1002 the names of symbols to tell them apart, but a (hidden) serial number that
1003 is unique for each newly created @code{symbol} object. In you want to use
1004 one and the same symbol in different places in your program, you must only
1005 create one @code{symbol} object and pass that around. If you create another
1006 symbol, even if it has the same name, GiNaC will treat it as a different
1023 // prints "x^6" which looks right, but...
1025 cout << e.degree(x) << endl;
1026 // ...this doesn't work. The symbol "x" here is different from the one
1027 // in f() and in the expression returned by f(). Consequently, it
1032 One possibility to ensure that @code{f()} and @code{main()} use the same
1033 symbol is to pass the symbol as an argument to @code{f()}:
1035 ex f(int n, const ex & x)
1044 // Now, f() uses the same symbol.
1047 cout << e.degree(x) << endl;
1048 // prints "6", as expected
1052 Another possibility would be to define a global symbol @code{x} that is used
1053 by both @code{f()} and @code{main()}. If you are using global symbols and
1054 multiple compilation units you must take special care, however. Suppose
1055 that you have a header file @file{globals.h} in your program that defines
1056 a @code{symbol x("x");}. In this case, every unit that includes
1057 @file{globals.h} would also get its own definition of @code{x} (because
1058 header files are just inlined into the source code by the C++ preprocessor),
1059 and hence you would again end up with multiple equally-named, but different,
1060 symbols. Instead, the @file{globals.h} header should only contain a
1061 @emph{declaration} like @code{extern symbol x;}, with the definition of
1062 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1064 A different approach to ensuring that symbols used in different parts of
1065 your program are identical is to create them with a @emph{factory} function
1068 const symbol & get_symbol(const string & s)
1070 static map<string, symbol> directory;
1071 map<string, symbol>::iterator i = directory.find(s);
1072 if (i != directory.end())
1075 return directory.insert(make_pair(s, symbol(s))).first->second;
1079 This function returns one newly constructed symbol for each name that is
1080 passed in, and it returns the same symbol when called multiple times with
1081 the same name. Using this symbol factory, we can rewrite our example like
1086 return pow(get_symbol("x"), n);
1093 // Both calls of get_symbol("x") yield the same symbol.
1094 cout << e.degree(get_symbol("x")) << endl;
1099 Instead of creating symbols from strings we could also have
1100 @code{get_symbol()} take, for example, an integer number as its argument.
1101 In this case, we would probably want to give the generated symbols names
1102 that include this number, which can be accomplished with the help of an
1103 @code{ostringstream}.
1105 In general, if you're getting weird results from GiNaC such as an expression
1106 @samp{x-x} that is not simplified to zero, you should check your symbol
1109 As we said, the names of symbols primarily serve for purposes of expression
1110 output. But there are actually two instances where GiNaC uses the names for
1111 identifying symbols: When constructing an expression from a string, and when
1112 recreating an expression from an archive (@pxref{Input/Output}).
1114 In addition to its name, a symbol may contain a special string that is used
1117 symbol x("x", "\\Box");
1120 This creates a symbol that is printed as "@code{x}" in normal output, but
1121 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1122 information about the different output formats of expressions in GiNaC).
1123 GiNaC automatically creates proper LaTeX code for symbols having names of
1124 greek letters (@samp{alpha}, @samp{mu}, etc.).
1126 @cindex @code{subs()}
1127 Symbols in GiNaC can't be assigned values. If you need to store results of
1128 calculations and give them a name, use C++ variables of type @code{ex}.
1129 If you want to replace a symbol in an expression with something else, you
1130 can invoke the expression's @code{.subs()} method
1131 (@pxref{Substituting Expressions}).
1133 @cindex @code{realsymbol()}
1134 By default, symbols are expected to stand in for complex values, i.e. they live
1135 in the complex domain. As a consequence, operations like complex conjugation,
1136 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1137 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1138 because of the unknown imaginary part of @code{x}.
1139 On the other hand, if you are sure that your symbols will hold only real values, you
1140 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1141 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1142 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1145 @node Numbers, Constants, Symbols, Basic Concepts
1146 @c node-name, next, previous, up
1148 @cindex @code{numeric} (class)
1154 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1155 The classes therein serve as foundation classes for GiNaC. CLN stands
1156 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1157 In order to find out more about CLN's internals, the reader is referred to
1158 the documentation of that library. @inforef{Introduction, , cln}, for
1159 more information. Suffice to say that it is by itself build on top of
1160 another library, the GNU Multiple Precision library GMP, which is an
1161 extremely fast library for arbitrary long integers and rationals as well
1162 as arbitrary precision floating point numbers. It is very commonly used
1163 by several popular cryptographic applications. CLN extends GMP by
1164 several useful things: First, it introduces the complex number field
1165 over either reals (i.e. floating point numbers with arbitrary precision)
1166 or rationals. Second, it automatically converts rationals to integers
1167 if the denominator is unity and complex numbers to real numbers if the
1168 imaginary part vanishes and also correctly treats algebraic functions.
1169 Third it provides good implementations of state-of-the-art algorithms
1170 for all trigonometric and hyperbolic functions as well as for
1171 calculation of some useful constants.
1173 The user can construct an object of class @code{numeric} in several
1174 ways. The following example shows the four most important constructors.
1175 It uses construction from C-integer, construction of fractions from two
1176 integers, construction from C-float and construction from a string:
1180 #include <ginac/ginac.h>
1181 using namespace GiNaC;
1185 numeric two = 2; // exact integer 2
1186 numeric r(2,3); // exact fraction 2/3
1187 numeric e(2.71828); // floating point number
1188 numeric p = "3.14159265358979323846"; // constructor from string
1189 // Trott's constant in scientific notation:
1190 numeric trott("1.0841015122311136151E-2");
1192 std::cout << two*p << std::endl; // floating point 6.283...
1197 @cindex complex numbers
1198 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1203 numeric z1 = 2-3*I; // exact complex number 2-3i
1204 numeric z2 = 5.9+1.6*I; // complex floating point number
1208 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1209 This would, however, call C's built-in operator @code{/} for integers
1210 first and result in a numeric holding a plain integer 1. @strong{Never
1211 use the operator @code{/} on integers} unless you know exactly what you
1212 are doing! Use the constructor from two integers instead, as shown in
1213 the example above. Writing @code{numeric(1)/2} may look funny but works
1216 @cindex @code{Digits}
1218 We have seen now the distinction between exact numbers and floating
1219 point numbers. Clearly, the user should never have to worry about
1220 dynamically created exact numbers, since their `exactness' always
1221 determines how they ought to be handled, i.e. how `long' they are. The
1222 situation is different for floating point numbers. Their accuracy is
1223 controlled by one @emph{global} variable, called @code{Digits}. (For
1224 those readers who know about Maple: it behaves very much like Maple's
1225 @code{Digits}). All objects of class numeric that are constructed from
1226 then on will be stored with a precision matching that number of decimal
1231 #include <ginac/ginac.h>
1232 using namespace std;
1233 using namespace GiNaC;
1237 numeric three(3.0), one(1.0);
1238 numeric x = one/three;
1240 cout << "in " << Digits << " digits:" << endl;
1242 cout << Pi.evalf() << endl;
1254 The above example prints the following output to screen:
1258 0.33333333333333333334
1259 3.1415926535897932385
1261 0.33333333333333333333333333333333333333333333333333333333333333333334
1262 3.1415926535897932384626433832795028841971693993751058209749445923078
1266 Note that the last number is not necessarily rounded as you would
1267 naively expect it to be rounded in the decimal system. But note also,
1268 that in both cases you got a couple of extra digits. This is because
1269 numbers are internally stored by CLN as chunks of binary digits in order
1270 to match your machine's word size and to not waste precision. Thus, on
1271 architectures with different word size, the above output might even
1272 differ with regard to actually computed digits.
1274 It should be clear that objects of class @code{numeric} should be used
1275 for constructing numbers or for doing arithmetic with them. The objects
1276 one deals with most of the time are the polymorphic expressions @code{ex}.
1278 @subsection Tests on numbers
1280 Once you have declared some numbers, assigned them to expressions and
1281 done some arithmetic with them it is frequently desired to retrieve some
1282 kind of information from them like asking whether that number is
1283 integer, rational, real or complex. For those cases GiNaC provides
1284 several useful methods. (Internally, they fall back to invocations of
1285 certain CLN functions.)
1287 As an example, let's construct some rational number, multiply it with
1288 some multiple of its denominator and test what comes out:
1292 #include <ginac/ginac.h>
1293 using namespace std;
1294 using namespace GiNaC;
1296 // some very important constants:
1297 const numeric twentyone(21);
1298 const numeric ten(10);
1299 const numeric five(5);
1303 numeric answer = twentyone;
1306 cout << answer.is_integer() << endl; // false, it's 21/5
1308 cout << answer.is_integer() << endl; // true, it's 42 now!
1312 Note that the variable @code{answer} is constructed here as an integer
1313 by @code{numeric}'s copy constructor but in an intermediate step it
1314 holds a rational number represented as integer numerator and integer
1315 denominator. When multiplied by 10, the denominator becomes unity and
1316 the result is automatically converted to a pure integer again.
1317 Internally, the underlying CLN is responsible for this behavior and we
1318 refer the reader to CLN's documentation. Suffice to say that
1319 the same behavior applies to complex numbers as well as return values of
1320 certain functions. Complex numbers are automatically converted to real
1321 numbers if the imaginary part becomes zero. The full set of tests that
1322 can be applied is listed in the following table.
1325 @multitable @columnfractions .30 .70
1326 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1327 @item @code{.is_zero()}
1328 @tab @dots{}equal to zero
1329 @item @code{.is_positive()}
1330 @tab @dots{}not complex and greater than 0
1331 @item @code{.is_integer()}
1332 @tab @dots{}a (non-complex) integer
1333 @item @code{.is_pos_integer()}
1334 @tab @dots{}an integer and greater than 0
1335 @item @code{.is_nonneg_integer()}
1336 @tab @dots{}an integer and greater equal 0
1337 @item @code{.is_even()}
1338 @tab @dots{}an even integer
1339 @item @code{.is_odd()}
1340 @tab @dots{}an odd integer
1341 @item @code{.is_prime()}
1342 @tab @dots{}a prime integer (probabilistic primality test)
1343 @item @code{.is_rational()}
1344 @tab @dots{}an exact rational number (integers are rational, too)
1345 @item @code{.is_real()}
1346 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1347 @item @code{.is_cinteger()}
1348 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1349 @item @code{.is_crational()}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1354 @subsection Numeric functions
1356 The following functions can be applied to @code{numeric} objects and will be
1357 evaluated immediately:
1360 @multitable @columnfractions .30 .70
1361 @item @strong{Name} @tab @strong{Function}
1362 @item @code{inverse(z)}
1363 @tab returns @math{1/z}
1364 @cindex @code{inverse()} (numeric)
1365 @item @code{pow(a, b)}
1366 @tab exponentiation @math{a^b}
1369 @item @code{real(z)}
1371 @cindex @code{real()}
1372 @item @code{imag(z)}
1374 @cindex @code{imag()}
1375 @item @code{csgn(z)}
1376 @tab complex sign (returns an @code{int})
1377 @item @code{numer(z)}
1378 @tab numerator of rational or complex rational number
1379 @item @code{denom(z)}
1380 @tab denominator of rational or complex rational number
1381 @item @code{sqrt(z)}
1383 @item @code{isqrt(n)}
1384 @tab integer square root
1385 @cindex @code{isqrt()}
1392 @item @code{asin(z)}
1394 @item @code{acos(z)}
1396 @item @code{atan(z)}
1397 @tab inverse tangent
1398 @item @code{atan(y, x)}
1399 @tab inverse tangent with two arguments
1400 @item @code{sinh(z)}
1401 @tab hyperbolic sine
1402 @item @code{cosh(z)}
1403 @tab hyperbolic cosine
1404 @item @code{tanh(z)}
1405 @tab hyperbolic tangent
1406 @item @code{asinh(z)}
1407 @tab inverse hyperbolic sine
1408 @item @code{acosh(z)}
1409 @tab inverse hyperbolic cosine
1410 @item @code{atanh(z)}
1411 @tab inverse hyperbolic tangent
1413 @tab exponential function
1415 @tab natural logarithm
1418 @item @code{zeta(z)}
1419 @tab Riemann's zeta function
1420 @item @code{tgamma(z)}
1422 @item @code{lgamma(z)}
1423 @tab logarithm of gamma function
1425 @tab psi (digamma) function
1426 @item @code{psi(n, z)}
1427 @tab derivatives of psi function (polygamma functions)
1428 @item @code{factorial(n)}
1429 @tab factorial function @math{n!}
1430 @item @code{doublefactorial(n)}
1431 @tab double factorial function @math{n!!}
1432 @cindex @code{doublefactorial()}
1433 @item @code{binomial(n, k)}
1434 @tab binomial coefficients
1435 @item @code{bernoulli(n)}
1436 @tab Bernoulli numbers
1437 @cindex @code{bernoulli()}
1438 @item @code{fibonacci(n)}
1439 @tab Fibonacci numbers
1440 @cindex @code{fibonacci()}
1441 @item @code{mod(a, b)}
1442 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1443 @cindex @code{mod()}
1444 @item @code{smod(a, b)}
1445 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1446 @cindex @code{smod()}
1447 @item @code{irem(a, b)}
1448 @tab integer remainder (has the sign of @math{a}, or is zero)
1449 @cindex @code{irem()}
1450 @item @code{irem(a, b, q)}
1451 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1452 @item @code{iquo(a, b)}
1453 @tab integer quotient
1454 @cindex @code{iquo()}
1455 @item @code{iquo(a, b, r)}
1456 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1457 @item @code{gcd(a, b)}
1458 @tab greatest common divisor
1459 @item @code{lcm(a, b)}
1460 @tab least common multiple
1464 Most of these functions are also available as symbolic functions that can be
1465 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1466 as polynomial algorithms.
1468 @subsection Converting numbers
1470 Sometimes it is desirable to convert a @code{numeric} object back to a
1471 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1472 class provides a couple of methods for this purpose:
1474 @cindex @code{to_int()}
1475 @cindex @code{to_long()}
1476 @cindex @code{to_double()}
1477 @cindex @code{to_cl_N()}
1479 int numeric::to_int() const;
1480 long numeric::to_long() const;
1481 double numeric::to_double() const;
1482 cln::cl_N numeric::to_cl_N() const;
1485 @code{to_int()} and @code{to_long()} only work when the number they are
1486 applied on is an exact integer. Otherwise the program will halt with a
1487 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1488 rational number will return a floating-point approximation. Both
1489 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1490 part of complex numbers.
1493 @node Constants, Fundamental containers, Numbers, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{constant} (class)
1499 @cindex @code{Catalan}
1500 @cindex @code{Euler}
1501 @cindex @code{evalf()}
1502 Constants behave pretty much like symbols except that they return some
1503 specific number when the method @code{.evalf()} is called.
1505 The predefined known constants are:
1508 @multitable @columnfractions .14 .30 .56
1509 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1511 @tab Archimedes' constant
1512 @tab 3.14159265358979323846264338327950288
1513 @item @code{Catalan}
1514 @tab Catalan's constant
1515 @tab 0.91596559417721901505460351493238411
1517 @tab Euler's (or Euler-Mascheroni) constant
1518 @tab 0.57721566490153286060651209008240243
1523 @node Fundamental containers, Lists, Constants, Basic Concepts
1524 @c node-name, next, previous, up
1525 @section Sums, products and powers
1529 @cindex @code{power}
1531 Simple rational expressions are written down in GiNaC pretty much like
1532 in other CAS or like expressions involving numerical variables in C.
1533 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1534 been overloaded to achieve this goal. When you run the following
1535 code snippet, the constructor for an object of type @code{mul} is
1536 automatically called to hold the product of @code{a} and @code{b} and
1537 then the constructor for an object of type @code{add} is called to hold
1538 the sum of that @code{mul} object and the number one:
1542 symbol a("a"), b("b");
1547 @cindex @code{pow()}
1548 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1549 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1550 construction is necessary since we cannot safely overload the constructor
1551 @code{^} in C++ to construct a @code{power} object. If we did, it would
1552 have several counterintuitive and undesired effects:
1556 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1558 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1559 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1560 interpret this as @code{x^(a^b)}.
1562 Also, expressions involving integer exponents are very frequently used,
1563 which makes it even more dangerous to overload @code{^} since it is then
1564 hard to distinguish between the semantics as exponentiation and the one
1565 for exclusive or. (It would be embarrassing to return @code{1} where one
1566 has requested @code{2^3}.)
1569 @cindex @command{ginsh}
1570 All effects are contrary to mathematical notation and differ from the
1571 way most other CAS handle exponentiation, therefore overloading @code{^}
1572 is ruled out for GiNaC's C++ part. The situation is different in
1573 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1574 that the other frequently used exponentiation operator @code{**} does
1575 not exist at all in C++).
1577 To be somewhat more precise, objects of the three classes described
1578 here, are all containers for other expressions. An object of class
1579 @code{power} is best viewed as a container with two slots, one for the
1580 basis, one for the exponent. All valid GiNaC expressions can be
1581 inserted. However, basic transformations like simplifying
1582 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1583 when this is mathematically possible. If we replace the outer exponent
1584 three in the example by some symbols @code{a}, the simplification is not
1585 safe and will not be performed, since @code{a} might be @code{1/2} and
1588 Objects of type @code{add} and @code{mul} are containers with an
1589 arbitrary number of slots for expressions to be inserted. Again, simple
1590 and safe simplifications are carried out like transforming
1591 @code{3*x+4-x} to @code{2*x+4}.
1594 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1595 @c node-name, next, previous, up
1596 @section Lists of expressions
1597 @cindex @code{lst} (class)
1599 @cindex @code{nops()}
1601 @cindex @code{append()}
1602 @cindex @code{prepend()}
1603 @cindex @code{remove_first()}
1604 @cindex @code{remove_last()}
1605 @cindex @code{remove_all()}
1607 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1608 expressions. They are not as ubiquitous as in many other computer algebra
1609 packages, but are sometimes used to supply a variable number of arguments of
1610 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1611 constructors, so you should have a basic understanding of them.
1613 Lists can be constructed by assigning a comma-separated sequence of
1618 symbol x("x"), y("y");
1621 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1626 There are also constructors that allow direct creation of lists of up to
1627 16 expressions, which is often more convenient but slightly less efficient:
1631 // This produces the same list 'l' as above:
1632 // lst l(x, 2, y, x+y);
1633 // lst l = lst(x, 2, y, x+y);
1637 Use the @code{nops()} method to determine the size (number of expressions) of
1638 a list and the @code{op()} method or the @code{[]} operator to access
1639 individual elements:
1643 cout << l.nops() << endl; // prints '4'
1644 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1648 As with the standard @code{list<T>} container, accessing random elements of a
1649 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1650 sequential access to the elements of a list is possible with the
1651 iterator types provided by the @code{lst} class:
1654 typedef ... lst::const_iterator;
1655 typedef ... lst::const_reverse_iterator;
1656 lst::const_iterator lst::begin() const;
1657 lst::const_iterator lst::end() const;
1658 lst::const_reverse_iterator lst::rbegin() const;
1659 lst::const_reverse_iterator lst::rend() const;
1662 For example, to print the elements of a list individually you can use:
1667 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1672 which is one order faster than
1677 for (size_t i = 0; i < l.nops(); ++i)
1678 cout << l.op(i) << endl;
1682 These iterators also allow you to use some of the algorithms provided by
1683 the C++ standard library:
1687 // print the elements of the list (requires #include <iterator>)
1688 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1690 // sum up the elements of the list (requires #include <numeric>)
1691 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1692 cout << sum << endl; // prints '2+2*x+2*y'
1696 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1697 (the only other one is @code{matrix}). You can modify single elements:
1701 l[1] = 42; // l is now @{x, 42, y, x+y@}
1702 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1706 You can append or prepend an expression to a list with the @code{append()}
1707 and @code{prepend()} methods:
1711 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1712 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1716 You can remove the first or last element of a list with @code{remove_first()}
1717 and @code{remove_last()}:
1721 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1722 l.remove_last(); // l is now @{x, 7, y, x+y@}
1726 You can remove all the elements of a list with @code{remove_all()}:
1730 l.remove_all(); // l is now empty
1734 You can bring the elements of a list into a canonical order with @code{sort()}:
1743 // l1 and l2 are now equal
1747 Finally, you can remove all but the first element of consecutive groups of
1748 elements with @code{unique()}:
1753 l3 = x, 2, 2, 2, y, x+y, y+x;
1754 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1759 @node Mathematical functions, Relations, Lists, Basic Concepts
1760 @c node-name, next, previous, up
1761 @section Mathematical functions
1762 @cindex @code{function} (class)
1763 @cindex trigonometric function
1764 @cindex hyperbolic function
1766 There are quite a number of useful functions hard-wired into GiNaC. For
1767 instance, all trigonometric and hyperbolic functions are implemented
1768 (@xref{Built-in Functions}, for a complete list).
1770 These functions (better called @emph{pseudofunctions}) are all objects
1771 of class @code{function}. They accept one or more expressions as
1772 arguments and return one expression. If the arguments are not
1773 numerical, the evaluation of the function may be halted, as it does in
1774 the next example, showing how a function returns itself twice and
1775 finally an expression that may be really useful:
1777 @cindex Gamma function
1778 @cindex @code{subs()}
1781 symbol x("x"), y("y");
1783 cout << tgamma(foo) << endl;
1784 // -> tgamma(x+(1/2)*y)
1785 ex bar = foo.subs(y==1);
1786 cout << tgamma(bar) << endl;
1788 ex foobar = bar.subs(x==7);
1789 cout << tgamma(foobar) << endl;
1790 // -> (135135/128)*Pi^(1/2)
1794 Besides evaluation most of these functions allow differentiation, series
1795 expansion and so on. Read the next chapter in order to learn more about
1798 It must be noted that these pseudofunctions are created by inline
1799 functions, where the argument list is templated. This means that
1800 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1801 @code{sin(ex(1))} and will therefore not result in a floating point
1802 number. Unless of course the function prototype is explicitly
1803 overridden -- which is the case for arguments of type @code{numeric}
1804 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1805 point number of class @code{numeric} you should call
1806 @code{sin(numeric(1))}. This is almost the same as calling
1807 @code{sin(1).evalf()} except that the latter will return a numeric
1808 wrapped inside an @code{ex}.
1811 @node Relations, Integrals, Mathematical functions, Basic Concepts
1812 @c node-name, next, previous, up
1814 @cindex @code{relational} (class)
1816 Sometimes, a relation holding between two expressions must be stored
1817 somehow. The class @code{relational} is a convenient container for such
1818 purposes. A relation is by definition a container for two @code{ex} and
1819 a relation between them that signals equality, inequality and so on.
1820 They are created by simply using the C++ operators @code{==}, @code{!=},
1821 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1823 @xref{Mathematical functions}, for examples where various applications
1824 of the @code{.subs()} method show how objects of class relational are
1825 used as arguments. There they provide an intuitive syntax for
1826 substitutions. They are also used as arguments to the @code{ex::series}
1827 method, where the left hand side of the relation specifies the variable
1828 to expand in and the right hand side the expansion point. They can also
1829 be used for creating systems of equations that are to be solved for
1830 unknown variables. But the most common usage of objects of this class
1831 is rather inconspicuous in statements of the form @code{if
1832 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1833 conversion from @code{relational} to @code{bool} takes place. Note,
1834 however, that @code{==} here does not perform any simplifications, hence
1835 @code{expand()} must be called explicitly.
1837 @node Integrals, Matrices, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{integral} (class)
1842 An object of class @dfn{integral} can be used to hold a symbolic integral.
1843 If you want to symbolically represent the integral of @code{x*x} from 0 to
1844 1, you would write this as
1846 integral(x, 0, 1, x*x)
1848 The first argument is the integration variable. It should be noted that
1849 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1850 fact, it can only integrate polynomials. An expression containing integrals
1851 can be evaluated symbolically by calling the
1855 method on it. Numerical evaluation is available by calling the
1859 method on an expression containing the integral. This will only evaluate
1860 integrals into a number if @code{subs}ing the integration variable by a
1861 number in the fourth argument of an integral and then @code{evalf}ing the
1862 result always results in a number. Of course, also the boundaries of the
1863 integration domain must @code{evalf} into numbers. It should be noted that
1864 trying to @code{evalf} a function with discontinuities in the integration
1865 domain is not recommended. The accuracy of the numeric evaluation of
1866 integrals is determined by the static member variable
1868 ex integral::relative_integration_error
1870 of the class @code{integral}. The default value of this is 10^-8.
1871 The integration works by halving the interval of integration, until numeric
1872 stability of the answer indicates that the requested accuracy has been
1873 reached. The maximum depth of the halving can be set via the static member
1876 int integral::max_integration_level
1878 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1879 return the integral unevaluated. The function that performs the numerical
1880 evaluation, is also available as
1882 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1885 This function will throw an exception if the maximum depth is exceeded. The
1886 last parameter of the function is optional and defaults to the
1887 @code{relative_integration_error}. To make sure that we do not do too
1888 much work if an expression contains the same integral multiple times,
1889 a lookup table is used.
1891 If you know that an expression holds an integral, you can get the
1892 integration variable, the left boundary, right boundary and integrant by
1893 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1894 @code{.op(3)}. Differentiating integrals with respect to variables works
1895 as expected. Note that it makes no sense to differentiate an integral
1896 with respect to the integration variable.
1898 @node Matrices, Indexed objects, Integrals, Basic Concepts
1899 @c node-name, next, previous, up
1901 @cindex @code{matrix} (class)
1903 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1904 matrix with @math{m} rows and @math{n} columns are accessed with two
1905 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1906 second one in the range 0@dots{}@math{n-1}.
1908 There are a couple of ways to construct matrices, with or without preset
1909 elements. The constructor
1912 matrix::matrix(unsigned r, unsigned c);
1915 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1918 The fastest way to create a matrix with preinitialized elements is to assign
1919 a list of comma-separated expressions to an empty matrix (see below for an
1920 example). But you can also specify the elements as a (flat) list with
1923 matrix::matrix(unsigned r, unsigned c, const lst & l);
1928 @cindex @code{lst_to_matrix()}
1930 ex lst_to_matrix(const lst & l);
1933 constructs a matrix from a list of lists, each list representing a matrix row.
1935 There is also a set of functions for creating some special types of
1938 @cindex @code{diag_matrix()}
1939 @cindex @code{unit_matrix()}
1940 @cindex @code{symbolic_matrix()}
1942 ex diag_matrix(const lst & l);
1943 ex unit_matrix(unsigned x);
1944 ex unit_matrix(unsigned r, unsigned c);
1945 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1946 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1947 const string & tex_base_name);
1950 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1951 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1952 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1953 matrix filled with newly generated symbols made of the specified base name
1954 and the position of each element in the matrix.
1956 Matrix elements can be accessed and set using the parenthesis (function call)
1960 const ex & matrix::operator()(unsigned r, unsigned c) const;
1961 ex & matrix::operator()(unsigned r, unsigned c);
1964 It is also possible to access the matrix elements in a linear fashion with
1965 the @code{op()} method. But C++-style subscripting with square brackets
1966 @samp{[]} is not available.
1968 Here are a couple of examples for constructing matrices:
1972 symbol a("a"), b("b");
1986 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1989 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1992 cout << diag_matrix(lst(a, b)) << endl;
1995 cout << unit_matrix(3) << endl;
1996 // -> [[1,0,0],[0,1,0],[0,0,1]]
1998 cout << symbolic_matrix(2, 3, "x") << endl;
1999 // -> [[x00,x01,x02],[x10,x11,x12]]
2003 @cindex @code{transpose()}
2004 There are three ways to do arithmetic with matrices. The first (and most
2005 direct one) is to use the methods provided by the @code{matrix} class:
2008 matrix matrix::add(const matrix & other) const;
2009 matrix matrix::sub(const matrix & other) const;
2010 matrix matrix::mul(const matrix & other) const;
2011 matrix matrix::mul_scalar(const ex & other) const;
2012 matrix matrix::pow(const ex & expn) const;
2013 matrix matrix::transpose() const;
2016 All of these methods return the result as a new matrix object. Here is an
2017 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2022 matrix A(2, 2), B(2, 2), C(2, 2);
2030 matrix result = A.mul(B).sub(C.mul_scalar(2));
2031 cout << result << endl;
2032 // -> [[-13,-6],[1,2]]
2037 @cindex @code{evalm()}
2038 The second (and probably the most natural) way is to construct an expression
2039 containing matrices with the usual arithmetic operators and @code{pow()}.
2040 For efficiency reasons, expressions with sums, products and powers of
2041 matrices are not automatically evaluated in GiNaC. You have to call the
2045 ex ex::evalm() const;
2048 to obtain the result:
2055 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2056 cout << e.evalm() << endl;
2057 // -> [[-13,-6],[1,2]]
2062 The non-commutativity of the product @code{A*B} in this example is
2063 automatically recognized by GiNaC. There is no need to use a special
2064 operator here. @xref{Non-commutative objects}, for more information about
2065 dealing with non-commutative expressions.
2067 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2068 to perform the arithmetic:
2073 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2074 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2076 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2077 cout << e.simplify_indexed() << endl;
2078 // -> [[-13,-6],[1,2]].i.j
2082 Using indices is most useful when working with rectangular matrices and
2083 one-dimensional vectors because you don't have to worry about having to
2084 transpose matrices before multiplying them. @xref{Indexed objects}, for
2085 more information about using matrices with indices, and about indices in
2088 The @code{matrix} class provides a couple of additional methods for
2089 computing determinants, traces, characteristic polynomials and ranks:
2091 @cindex @code{determinant()}
2092 @cindex @code{trace()}
2093 @cindex @code{charpoly()}
2094 @cindex @code{rank()}
2096 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2097 ex matrix::trace() const;
2098 ex matrix::charpoly(const ex & lambda) const;
2099 unsigned matrix::rank() const;
2102 The @samp{algo} argument of @code{determinant()} allows to select
2103 between different algorithms for calculating the determinant. The
2104 asymptotic speed (as parametrized by the matrix size) can greatly differ
2105 between those algorithms, depending on the nature of the matrix'
2106 entries. The possible values are defined in the @file{flags.h} header
2107 file. By default, GiNaC uses a heuristic to automatically select an
2108 algorithm that is likely (but not guaranteed) to give the result most
2111 @cindex @code{inverse()} (matrix)
2112 @cindex @code{solve()}
2113 Matrices may also be inverted using the @code{ex matrix::inverse()}
2114 method and linear systems may be solved with:
2117 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2118 unsigned algo=solve_algo::automatic) const;
2121 Assuming the matrix object this method is applied on is an @code{m}
2122 times @code{n} matrix, then @code{vars} must be a @code{n} times
2123 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2124 times @code{p} matrix. The returned matrix then has dimension @code{n}
2125 times @code{p} and in the case of an underdetermined system will still
2126 contain some of the indeterminates from @code{vars}. If the system is
2127 overdetermined, an exception is thrown.
2130 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2131 @c node-name, next, previous, up
2132 @section Indexed objects
2134 GiNaC allows you to handle expressions containing general indexed objects in
2135 arbitrary spaces. It is also able to canonicalize and simplify such
2136 expressions and perform symbolic dummy index summations. There are a number
2137 of predefined indexed objects provided, like delta and metric tensors.
2139 There are few restrictions placed on indexed objects and their indices and
2140 it is easy to construct nonsense expressions, but our intention is to
2141 provide a general framework that allows you to implement algorithms with
2142 indexed quantities, getting in the way as little as possible.
2144 @cindex @code{idx} (class)
2145 @cindex @code{indexed} (class)
2146 @subsection Indexed quantities and their indices
2148 Indexed expressions in GiNaC are constructed of two special types of objects,
2149 @dfn{index objects} and @dfn{indexed objects}.
2153 @cindex contravariant
2156 @item Index objects are of class @code{idx} or a subclass. Every index has
2157 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2158 the index lives in) which can both be arbitrary expressions but are usually
2159 a number or a simple symbol. In addition, indices of class @code{varidx} have
2160 a @dfn{variance} (they can be co- or contravariant), and indices of class
2161 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2163 @item Indexed objects are of class @code{indexed} or a subclass. They
2164 contain a @dfn{base expression} (which is the expression being indexed), and
2165 one or more indices.
2169 @strong{Please notice:} when printing expressions, covariant indices and indices
2170 without variance are denoted @samp{.i} while contravariant indices are
2171 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2172 value. In the following, we are going to use that notation in the text so
2173 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2174 not visible in the output.
2176 A simple example shall illustrate the concepts:
2180 #include <ginac/ginac.h>
2181 using namespace std;
2182 using namespace GiNaC;
2186 symbol i_sym("i"), j_sym("j");
2187 idx i(i_sym, 3), j(j_sym, 3);
2190 cout << indexed(A, i, j) << endl;
2192 cout << index_dimensions << indexed(A, i, j) << endl;
2194 cout << dflt; // reset cout to default output format (dimensions hidden)
2198 The @code{idx} constructor takes two arguments, the index value and the
2199 index dimension. First we define two index objects, @code{i} and @code{j},
2200 both with the numeric dimension 3. The value of the index @code{i} is the
2201 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2202 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2203 construct an expression containing one indexed object, @samp{A.i.j}. It has
2204 the symbol @code{A} as its base expression and the two indices @code{i} and
2207 The dimensions of indices are normally not visible in the output, but one
2208 can request them to be printed with the @code{index_dimensions} manipulator,
2211 Note the difference between the indices @code{i} and @code{j} which are of
2212 class @code{idx}, and the index values which are the symbols @code{i_sym}
2213 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2214 or numbers but must be index objects. For example, the following is not
2215 correct and will raise an exception:
2218 symbol i("i"), j("j");
2219 e = indexed(A, i, j); // ERROR: indices must be of type idx
2222 You can have multiple indexed objects in an expression, index values can
2223 be numeric, and index dimensions symbolic:
2227 symbol B("B"), dim("dim");
2228 cout << 4 * indexed(A, i)
2229 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2234 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2235 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2236 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2237 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2238 @code{simplify_indexed()} for that, see below).
2240 In fact, base expressions, index values and index dimensions can be
2241 arbitrary expressions:
2245 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2250 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2251 get an error message from this but you will probably not be able to do
2252 anything useful with it.
2254 @cindex @code{get_value()}
2255 @cindex @code{get_dimension()}
2259 ex idx::get_value();
2260 ex idx::get_dimension();
2263 return the value and dimension of an @code{idx} object. If you have an index
2264 in an expression, such as returned by calling @code{.op()} on an indexed
2265 object, you can get a reference to the @code{idx} object with the function
2266 @code{ex_to<idx>()} on the expression.
2268 There are also the methods
2271 bool idx::is_numeric();
2272 bool idx::is_symbolic();
2273 bool idx::is_dim_numeric();
2274 bool idx::is_dim_symbolic();
2277 for checking whether the value and dimension are numeric or symbolic
2278 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2279 About Expressions}) returns information about the index value.
2281 @cindex @code{varidx} (class)
2282 If you need co- and contravariant indices, use the @code{varidx} class:
2286 symbol mu_sym("mu"), nu_sym("nu");
2287 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2288 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2290 cout << indexed(A, mu, nu) << endl;
2292 cout << indexed(A, mu_co, nu) << endl;
2294 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2299 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2300 co- or contravariant. The default is a contravariant (upper) index, but
2301 this can be overridden by supplying a third argument to the @code{varidx}
2302 constructor. The two methods
2305 bool varidx::is_covariant();
2306 bool varidx::is_contravariant();
2309 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2310 to get the object reference from an expression). There's also the very useful
2314 ex varidx::toggle_variance();
2317 which makes a new index with the same value and dimension but the opposite
2318 variance. By using it you only have to define the index once.
2320 @cindex @code{spinidx} (class)
2321 The @code{spinidx} class provides dotted and undotted variant indices, as
2322 used in the Weyl-van-der-Waerden spinor formalism:
2326 symbol K("K"), C_sym("C"), D_sym("D");
2327 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2328 // contravariant, undotted
2329 spinidx C_co(C_sym, 2, true); // covariant index
2330 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2331 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2333 cout << indexed(K, C, D) << endl;
2335 cout << indexed(K, C_co, D_dot) << endl;
2337 cout << indexed(K, D_co_dot, D) << endl;
2342 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2343 dotted or undotted. The default is undotted but this can be overridden by
2344 supplying a fourth argument to the @code{spinidx} constructor. The two
2348 bool spinidx::is_dotted();
2349 bool spinidx::is_undotted();
2352 allow you to check whether or not a @code{spinidx} object is dotted (use
2353 @code{ex_to<spinidx>()} to get the object reference from an expression).
2354 Finally, the two methods
2357 ex spinidx::toggle_dot();
2358 ex spinidx::toggle_variance_dot();
2361 create a new index with the same value and dimension but opposite dottedness
2362 and the same or opposite variance.
2364 @subsection Substituting indices
2366 @cindex @code{subs()}
2367 Sometimes you will want to substitute one symbolic index with another
2368 symbolic or numeric index, for example when calculating one specific element
2369 of a tensor expression. This is done with the @code{.subs()} method, as it
2370 is done for symbols (see @ref{Substituting Expressions}).
2372 You have two possibilities here. You can either substitute the whole index
2373 by another index or expression:
2377 ex e = indexed(A, mu_co);
2378 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2379 // -> A.mu becomes A~nu
2380 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2381 // -> A.mu becomes A~0
2382 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2383 // -> A.mu becomes A.0
2387 The third example shows that trying to replace an index with something that
2388 is not an index will substitute the index value instead.
2390 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2395 ex e = indexed(A, mu_co);
2396 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2397 // -> A.mu becomes A.nu
2398 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2399 // -> A.mu becomes A.0
2403 As you see, with the second method only the value of the index will get
2404 substituted. Its other properties, including its dimension, remain unchanged.
2405 If you want to change the dimension of an index you have to substitute the
2406 whole index by another one with the new dimension.
2408 Finally, substituting the base expression of an indexed object works as
2413 ex e = indexed(A, mu_co);
2414 cout << e << " becomes " << e.subs(A == A+B) << endl;
2415 // -> A.mu becomes (B+A).mu
2419 @subsection Symmetries
2420 @cindex @code{symmetry} (class)
2421 @cindex @code{sy_none()}
2422 @cindex @code{sy_symm()}
2423 @cindex @code{sy_anti()}
2424 @cindex @code{sy_cycl()}
2426 Indexed objects can have certain symmetry properties with respect to their
2427 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2428 that is constructed with the helper functions
2431 symmetry sy_none(...);
2432 symmetry sy_symm(...);
2433 symmetry sy_anti(...);
2434 symmetry sy_cycl(...);
2437 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2438 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2439 represents a cyclic symmetry. Each of these functions accepts up to four
2440 arguments which can be either symmetry objects themselves or unsigned integer
2441 numbers that represent an index position (counting from 0). A symmetry
2442 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2443 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2446 Here are some examples of symmetry definitions:
2451 e = indexed(A, i, j);
2452 e = indexed(A, sy_none(), i, j); // equivalent
2453 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2455 // Symmetric in all three indices:
2456 e = indexed(A, sy_symm(), i, j, k);
2457 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2458 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2459 // different canonical order
2461 // Symmetric in the first two indices only:
2462 e = indexed(A, sy_symm(0, 1), i, j, k);
2463 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2465 // Antisymmetric in the first and last index only (index ranges need not
2467 e = indexed(A, sy_anti(0, 2), i, j, k);
2468 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2470 // An example of a mixed symmetry: antisymmetric in the first two and
2471 // last two indices, symmetric when swapping the first and last index
2472 // pairs (like the Riemann curvature tensor):
2473 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2475 // Cyclic symmetry in all three indices:
2476 e = indexed(A, sy_cycl(), i, j, k);
2477 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2479 // The following examples are invalid constructions that will throw
2480 // an exception at run time.
2482 // An index may not appear multiple times:
2483 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2484 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2486 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2487 // same number of indices:
2488 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2490 // And of course, you cannot specify indices which are not there:
2491 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2495 If you need to specify more than four indices, you have to use the
2496 @code{.add()} method of the @code{symmetry} class. For example, to specify
2497 full symmetry in the first six indices you would write
2498 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2500 If an indexed object has a symmetry, GiNaC will automatically bring the
2501 indices into a canonical order which allows for some immediate simplifications:
2505 cout << indexed(A, sy_symm(), i, j)
2506 + indexed(A, sy_symm(), j, i) << endl;
2508 cout << indexed(B, sy_anti(), i, j)
2509 + indexed(B, sy_anti(), j, i) << endl;
2511 cout << indexed(B, sy_anti(), i, j, k)
2512 - indexed(B, sy_anti(), j, k, i) << endl;
2517 @cindex @code{get_free_indices()}
2519 @subsection Dummy indices
2521 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2522 that a summation over the index range is implied. Symbolic indices which are
2523 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2524 dummy nor free indices.
2526 To be recognized as a dummy index pair, the two indices must be of the same
2527 class and their value must be the same single symbol (an index like
2528 @samp{2*n+1} is never a dummy index). If the indices are of class
2529 @code{varidx} they must also be of opposite variance; if they are of class
2530 @code{spinidx} they must be both dotted or both undotted.
2532 The method @code{.get_free_indices()} returns a vector containing the free
2533 indices of an expression. It also checks that the free indices of the terms
2534 of a sum are consistent:
2538 symbol A("A"), B("B"), C("C");
2540 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2541 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2543 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2544 cout << exprseq(e.get_free_indices()) << endl;
2546 // 'j' and 'l' are dummy indices
2548 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2549 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2551 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2552 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2553 cout << exprseq(e.get_free_indices()) << endl;
2555 // 'nu' is a dummy index, but 'sigma' is not
2557 e = indexed(A, mu, mu);
2558 cout << exprseq(e.get_free_indices()) << endl;
2560 // 'mu' is not a dummy index because it appears twice with the same
2563 e = indexed(A, mu, nu) + 42;
2564 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2565 // this will throw an exception:
2566 // "add::get_free_indices: inconsistent indices in sum"
2570 @cindex @code{simplify_indexed()}
2571 @subsection Simplifying indexed expressions
2573 In addition to the few automatic simplifications that GiNaC performs on
2574 indexed expressions (such as re-ordering the indices of symmetric tensors
2575 and calculating traces and convolutions of matrices and predefined tensors)
2579 ex ex::simplify_indexed();
2580 ex ex::simplify_indexed(const scalar_products & sp);
2583 that performs some more expensive operations:
2586 @item it checks the consistency of free indices in sums in the same way
2587 @code{get_free_indices()} does
2588 @item it tries to give dummy indices that appear in different terms of a sum
2589 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2590 @item it (symbolically) calculates all possible dummy index summations/contractions
2591 with the predefined tensors (this will be explained in more detail in the
2593 @item it detects contractions that vanish for symmetry reasons, for example
2594 the contraction of a symmetric and a totally antisymmetric tensor
2595 @item as a special case of dummy index summation, it can replace scalar products
2596 of two tensors with a user-defined value
2599 The last point is done with the help of the @code{scalar_products} class
2600 which is used to store scalar products with known values (this is not an
2601 arithmetic class, you just pass it to @code{simplify_indexed()}):
2605 symbol A("A"), B("B"), C("C"), i_sym("i");
2609 sp.add(A, B, 0); // A and B are orthogonal
2610 sp.add(A, C, 0); // A and C are orthogonal
2611 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2613 e = indexed(A + B, i) * indexed(A + C, i);
2615 // -> (B+A).i*(A+C).i
2617 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2623 The @code{scalar_products} object @code{sp} acts as a storage for the
2624 scalar products added to it with the @code{.add()} method. This method
2625 takes three arguments: the two expressions of which the scalar product is
2626 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2627 @code{simplify_indexed()} will replace all scalar products of indexed
2628 objects that have the symbols @code{A} and @code{B} as base expressions
2629 with the single value 0. The number, type and dimension of the indices
2630 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2632 @cindex @code{expand()}
2633 The example above also illustrates a feature of the @code{expand()} method:
2634 if passed the @code{expand_indexed} option it will distribute indices
2635 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2637 @cindex @code{tensor} (class)
2638 @subsection Predefined tensors
2640 Some frequently used special tensors such as the delta, epsilon and metric
2641 tensors are predefined in GiNaC. They have special properties when
2642 contracted with other tensor expressions and some of them have constant
2643 matrix representations (they will evaluate to a number when numeric
2644 indices are specified).
2646 @cindex @code{delta_tensor()}
2647 @subsubsection Delta tensor
2649 The delta tensor takes two indices, is symmetric and has the matrix
2650 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2651 @code{delta_tensor()}:
2655 symbol A("A"), B("B");
2657 idx i(symbol("i"), 3), j(symbol("j"), 3),
2658 k(symbol("k"), 3), l(symbol("l"), 3);
2660 ex e = indexed(A, i, j) * indexed(B, k, l)
2661 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2662 cout << e.simplify_indexed() << endl;
2665 cout << delta_tensor(i, i) << endl;
2670 @cindex @code{metric_tensor()}
2671 @subsubsection General metric tensor
2673 The function @code{metric_tensor()} creates a general symmetric metric
2674 tensor with two indices that can be used to raise/lower tensor indices. The
2675 metric tensor is denoted as @samp{g} in the output and if its indices are of
2676 mixed variance it is automatically replaced by a delta tensor:
2682 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2684 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2685 cout << e.simplify_indexed() << endl;
2688 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2689 cout << e.simplify_indexed() << endl;
2692 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2693 * metric_tensor(nu, rho);
2694 cout << e.simplify_indexed() << endl;
2697 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2698 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2699 + indexed(A, mu.toggle_variance(), rho));
2700 cout << e.simplify_indexed() << endl;
2705 @cindex @code{lorentz_g()}
2706 @subsubsection Minkowski metric tensor
2708 The Minkowski metric tensor is a special metric tensor with a constant
2709 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2710 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2711 It is created with the function @code{lorentz_g()} (although it is output as
2716 varidx mu(symbol("mu"), 4);
2718 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2719 * lorentz_g(mu, varidx(0, 4)); // negative signature
2720 cout << e.simplify_indexed() << endl;
2723 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2724 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2725 cout << e.simplify_indexed() << endl;
2730 @cindex @code{spinor_metric()}
2731 @subsubsection Spinor metric tensor
2733 The function @code{spinor_metric()} creates an antisymmetric tensor with
2734 two indices that is used to raise/lower indices of 2-component spinors.
2735 It is output as @samp{eps}:
2741 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2742 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2744 e = spinor_metric(A, B) * indexed(psi, B_co);
2745 cout << e.simplify_indexed() << endl;
2748 e = spinor_metric(A, B) * indexed(psi, A_co);
2749 cout << e.simplify_indexed() << endl;
2752 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2753 cout << e.simplify_indexed() << endl;
2756 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2757 cout << e.simplify_indexed() << endl;
2760 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2761 cout << e.simplify_indexed() << endl;
2764 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2765 cout << e.simplify_indexed() << endl;
2770 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2772 @cindex @code{epsilon_tensor()}
2773 @cindex @code{lorentz_eps()}
2774 @subsubsection Epsilon tensor
2776 The epsilon tensor is totally antisymmetric, its number of indices is equal
2777 to the dimension of the index space (the indices must all be of the same
2778 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2779 defined to be 1. Its behavior with indices that have a variance also
2780 depends on the signature of the metric. Epsilon tensors are output as
2783 There are three functions defined to create epsilon tensors in 2, 3 and 4
2787 ex epsilon_tensor(const ex & i1, const ex & i2);
2788 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2789 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2790 bool pos_sig = false);
2793 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2794 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2795 Minkowski space (the last @code{bool} argument specifies whether the metric
2796 has negative or positive signature, as in the case of the Minkowski metric
2801 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2802 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2803 e = lorentz_eps(mu, nu, rho, sig) *
2804 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2805 cout << simplify_indexed(e) << endl;
2806 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2808 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2809 symbol A("A"), B("B");
2810 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2811 cout << simplify_indexed(e) << endl;
2812 // -> -B.k*A.j*eps.i.k.j
2813 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2814 cout << simplify_indexed(e) << endl;
2819 @subsection Linear algebra
2821 The @code{matrix} class can be used with indices to do some simple linear
2822 algebra (linear combinations and products of vectors and matrices, traces
2823 and scalar products):
2827 idx i(symbol("i"), 2), j(symbol("j"), 2);
2828 symbol x("x"), y("y");
2830 // A is a 2x2 matrix, X is a 2x1 vector
2831 matrix A(2, 2), X(2, 1);
2836 cout << indexed(A, i, i) << endl;
2839 ex e = indexed(A, i, j) * indexed(X, j);
2840 cout << e.simplify_indexed() << endl;
2841 // -> [[2*y+x],[4*y+3*x]].i
2843 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2844 cout << e.simplify_indexed() << endl;
2845 // -> [[3*y+3*x,6*y+2*x]].j
2849 You can of course obtain the same results with the @code{matrix::add()},
2850 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2851 but with indices you don't have to worry about transposing matrices.
2853 Matrix indices always start at 0 and their dimension must match the number
2854 of rows/columns of the matrix. Matrices with one row or one column are
2855 vectors and can have one or two indices (it doesn't matter whether it's a
2856 row or a column vector). Other matrices must have two indices.
2858 You should be careful when using indices with variance on matrices. GiNaC
2859 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2860 @samp{F.mu.nu} are different matrices. In this case you should use only
2861 one form for @samp{F} and explicitly multiply it with a matrix representation
2862 of the metric tensor.
2865 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2866 @c node-name, next, previous, up
2867 @section Non-commutative objects
2869 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2870 non-commutative objects are built-in which are mostly of use in high energy
2874 @item Clifford (Dirac) algebra (class @code{clifford})
2875 @item su(3) Lie algebra (class @code{color})
2876 @item Matrices (unindexed) (class @code{matrix})
2879 The @code{clifford} and @code{color} classes are subclasses of
2880 @code{indexed} because the elements of these algebras usually carry
2881 indices. The @code{matrix} class is described in more detail in
2884 Unlike most computer algebra systems, GiNaC does not primarily provide an
2885 operator (often denoted @samp{&*}) for representing inert products of
2886 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2887 classes of objects involved, and non-commutative products are formed with
2888 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2889 figuring out by itself which objects commutate and will group the factors
2890 by their class. Consider this example:
2894 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2895 idx a(symbol("a"), 8), b(symbol("b"), 8);
2896 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2898 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2902 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2903 groups the non-commutative factors (the gammas and the su(3) generators)
2904 together while preserving the order of factors within each class (because
2905 Clifford objects commutate with color objects). The resulting expression is a
2906 @emph{commutative} product with two factors that are themselves non-commutative
2907 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2908 parentheses are placed around the non-commutative products in the output.
2910 @cindex @code{ncmul} (class)
2911 Non-commutative products are internally represented by objects of the class
2912 @code{ncmul}, as opposed to commutative products which are handled by the
2913 @code{mul} class. You will normally not have to worry about this distinction,
2916 The advantage of this approach is that you never have to worry about using
2917 (or forgetting to use) a special operator when constructing non-commutative
2918 expressions. Also, non-commutative products in GiNaC are more intelligent
2919 than in other computer algebra systems; they can, for example, automatically
2920 canonicalize themselves according to rules specified in the implementation
2921 of the non-commutative classes. The drawback is that to work with other than
2922 the built-in algebras you have to implement new classes yourself. Symbols
2923 always commutate and it's not possible to construct non-commutative products
2924 using symbols to represent the algebra elements or generators. User-defined
2925 functions can, however, be specified as being non-commutative.
2927 @cindex @code{return_type()}
2928 @cindex @code{return_type_tinfo()}
2929 Information about the commutativity of an object or expression can be
2930 obtained with the two member functions
2933 unsigned ex::return_type() const;
2934 unsigned ex::return_type_tinfo() const;
2937 The @code{return_type()} function returns one of three values (defined in
2938 the header file @file{flags.h}), corresponding to three categories of
2939 expressions in GiNaC:
2942 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2943 classes are of this kind.
2944 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2945 certain class of non-commutative objects which can be determined with the
2946 @code{return_type_tinfo()} method. Expressions of this category commutate
2947 with everything except @code{noncommutative} expressions of the same
2949 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2950 of non-commutative objects of different classes. Expressions of this
2951 category don't commutate with any other @code{noncommutative} or
2952 @code{noncommutative_composite} expressions.
2955 The value returned by the @code{return_type_tinfo()} method is valid only
2956 when the return type of the expression is @code{noncommutative}. It is a
2957 value that is unique to the class of the object and usually one of the
2958 constants in @file{tinfos.h}, or derived therefrom.
2960 Here are a couple of examples:
2963 @multitable @columnfractions 0.33 0.33 0.34
2964 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2965 @item @code{42} @tab @code{commutative} @tab -
2966 @item @code{2*x-y} @tab @code{commutative} @tab -
2967 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2968 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2969 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2970 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2974 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2975 @code{TINFO_clifford} for objects with a representation label of zero.
2976 Other representation labels yield a different @code{return_type_tinfo()},
2977 but it's the same for any two objects with the same label. This is also true
2980 A last note: With the exception of matrices, positive integer powers of
2981 non-commutative objects are automatically expanded in GiNaC. For example,
2982 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2983 non-commutative expressions).
2986 @cindex @code{clifford} (class)
2987 @subsection Clifford algebra
2990 Clifford algebras are supported in two flavours: Dirac gamma
2991 matrices (more physical) and generic Clifford algebras (more
2994 @cindex @code{dirac_gamma()}
2995 @subsubsection Dirac gamma matrices
2996 Dirac gamma matrices (note that GiNaC doesn't treat them
2997 as matrices) are designated as @samp{gamma~mu} and satisfy
2998 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
2999 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3000 constructed by the function
3003 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3006 which takes two arguments: the index and a @dfn{representation label} in the
3007 range 0 to 255 which is used to distinguish elements of different Clifford
3008 algebras (this is also called a @dfn{spin line index}). Gammas with different
3009 labels commutate with each other. The dimension of the index can be 4 or (in
3010 the framework of dimensional regularization) any symbolic value. Spinor
3011 indices on Dirac gammas are not supported in GiNaC.
3013 @cindex @code{dirac_ONE()}
3014 The unity element of a Clifford algebra is constructed by
3017 ex dirac_ONE(unsigned char rl = 0);
3020 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3021 multiples of the unity element, even though it's customary to omit it.
3022 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3023 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3024 GiNaC will complain and/or produce incorrect results.
3026 @cindex @code{dirac_gamma5()}
3027 There is a special element @samp{gamma5} that commutates with all other
3028 gammas, has a unit square, and in 4 dimensions equals
3029 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3032 ex dirac_gamma5(unsigned char rl = 0);
3035 @cindex @code{dirac_gammaL()}
3036 @cindex @code{dirac_gammaR()}
3037 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3038 objects, constructed by
3041 ex dirac_gammaL(unsigned char rl = 0);
3042 ex dirac_gammaR(unsigned char rl = 0);
3045 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3046 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3048 @cindex @code{dirac_slash()}
3049 Finally, the function
3052 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3055 creates a term that represents a contraction of @samp{e} with the Dirac
3056 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3057 with a unique index whose dimension is given by the @code{dim} argument).
3058 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3060 In products of dirac gammas, superfluous unity elements are automatically
3061 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3062 and @samp{gammaR} are moved to the front.
3064 The @code{simplify_indexed()} function performs contractions in gamma strings,
3070 symbol a("a"), b("b"), D("D");
3071 varidx mu(symbol("mu"), D);
3072 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3073 * dirac_gamma(mu.toggle_variance());
3075 // -> gamma~mu*a\*gamma.mu
3076 e = e.simplify_indexed();
3079 cout << e.subs(D == 4) << endl;
3085 @cindex @code{dirac_trace()}
3086 To calculate the trace of an expression containing strings of Dirac gammas
3087 you use one of the functions
3090 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3091 const ex & trONE = 4);
3092 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3093 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3096 These functions take the trace over all gammas in the specified set @code{rls}
3097 or list @code{rll} of representation labels, or the single label @code{rl};
3098 gammas with other labels are left standing. The last argument to
3099 @code{dirac_trace()} is the value to be returned for the trace of the unity
3100 element, which defaults to 4.
3102 The @code{dirac_trace()} function is a linear functional that is equal to the
3103 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3104 functional is not cyclic in
3107 dimensions when acting on
3108 expressions containing @samp{gamma5}, so it's not a proper trace. This
3109 @samp{gamma5} scheme is described in greater detail in
3110 @cite{The Role of gamma5 in Dimensional Regularization}.
3112 The value of the trace itself is also usually different in 4 and in
3120 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3121 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3122 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3123 cout << dirac_trace(e).simplify_indexed() << endl;
3130 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3131 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3132 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3133 cout << dirac_trace(e).simplify_indexed() << endl;
3134 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3138 Here is an example for using @code{dirac_trace()} to compute a value that
3139 appears in the calculation of the one-loop vacuum polarization amplitude in
3144 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3145 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3148 sp.add(l, l, pow(l, 2));
3149 sp.add(l, q, ldotq);
3151 ex e = dirac_gamma(mu) *
3152 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3153 dirac_gamma(mu.toggle_variance()) *
3154 (dirac_slash(l, D) + m * dirac_ONE());
3155 e = dirac_trace(e).simplify_indexed(sp);
3156 e = e.collect(lst(l, ldotq, m));
3158 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3162 The @code{canonicalize_clifford()} function reorders all gamma products that
3163 appear in an expression to a canonical (but not necessarily simple) form.
3164 You can use this to compare two expressions or for further simplifications:
3168 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3169 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3171 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3173 e = canonicalize_clifford(e);
3175 // -> 2*ONE*eta~mu~nu
3179 @cindex @code{clifford_unit()}
3180 @subsubsection A generic Clifford algebra
3182 A generic Clifford algebra, i.e. a
3186 dimensional algebra with
3190 satisfying the identities
3192 $e_i e_j + e_j e_i = M(i, j) $
3195 e~i e~j + e~j e~i = M(i, j)
3197 for some matrix (@code{metric})
3198 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3199 entries. Such generators are created by the function
3202 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3205 where @code{mu} should be a @code{varidx} class object indexing the
3206 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3207 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3208 object, optional parameter @code{rl} allows to distinguish different
3209 Clifford algebras (which will commute with each other). Note that the call
3210 @code{clifford_unit(mu, minkmetric())} creates something very close to
3211 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3212 metric defining this Clifford number.
3214 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3215 the Clifford algebra units with a call like that
3218 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3221 since this may yield some further automatic simplifications.
3223 Individual generators of a Clifford algebra can be accessed in several
3229 varidx nu(symbol("nu"), 4);
3231 ex M = diag_matrix(lst(1, -1, 0, s));
3232 ex e = clifford_unit(nu, M);
3233 ex e0 = e.subs(nu == 0);
3234 ex e1 = e.subs(nu == 1);
3235 ex e2 = e.subs(nu == 2);
3236 ex e3 = e.subs(nu == 3);
3241 will produce four anti-commuting generators of a Clifford algebra with properties
3243 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3246 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3249 @cindex @code{lst_to_clifford()}
3250 A similar effect can be achieved from the function
3253 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3254 unsigned char rl = 0);
3255 ex lst_to_clifford(const ex & v, const ex & e);
3258 which converts a list or vector
3260 $v = (v^0, v^1, ..., v^n)$
3263 @samp{v = (v~0, v~1, ..., v~n)}
3268 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3271 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3274 directly supplied in the second form of the procedure. In the first form
3275 the Clifford unit @samp{e.k} is generated by the call of
3276 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3277 with the help of @code{lst_to_clifford()} as follows
3282 varidx nu(symbol("nu"), 4);
3284 ex M = diag_matrix(lst(1, -1, 0, s));
3285 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3286 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3287 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3288 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3293 @cindex @code{clifford_to_lst()}
3294 There is the inverse function
3297 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3300 which takes an expression @code{e} and tries to find a list
3302 $v = (v^0, v^1, ..., v^n)$
3305 @samp{v = (v~0, v~1, ..., v~n)}
3309 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3312 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3314 with respect to the given Clifford units @code{c} and with none of the
3315 @samp{v~k} containing Clifford units @code{c} (of course, this
3316 may be impossible). This function can use an @code{algebraic} method
3317 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3319 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3322 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3324 is zero or is not a @code{numeric} for some @samp{k}
3325 then the method will be automatically changed to symbolic. The same effect
3326 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3328 @cindex @code{clifford_prime()}
3329 @cindex @code{clifford_star()}
3330 @cindex @code{clifford_bar()}
3331 There are several functions for (anti-)automorphisms of Clifford algebras:
3334 ex clifford_prime(const ex & e)
3335 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3336 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3339 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3340 changes signs of all Clifford units in the expression. The reversion
3341 of a Clifford algebra @code{clifford_star()} coincides with the
3342 @code{conjugate()} method and effectively reverses the order of Clifford
3343 units in any product. Finally the main anti-automorphism
3344 of a Clifford algebra @code{clifford_bar()} is the composition of the
3345 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3346 in a product. These functions correspond to the notations
3361 used in Clifford algebra textbooks.
3363 @cindex @code{clifford_norm()}
3367 ex clifford_norm(const ex & e);
3370 @cindex @code{clifford_inverse()}
3371 calculates the norm of a Clifford number from the expression
3373 $||e||^2 = e\overline{e}$.
3376 @code{||e||^2 = e \bar@{e@}}
3378 The inverse of a Clifford expression is returned by the function
3381 ex clifford_inverse(const ex & e);
3384 which calculates it as
3386 $e^{-1} = \overline{e}/||e||^2$.
3389 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3398 then an exception is raised.
3400 @cindex @code{remove_dirac_ONE()}
3401 If a Clifford number happens to be a factor of
3402 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3403 expression by the function
3406 ex remove_dirac_ONE(const ex & e);
3409 @cindex @code{canonicalize_clifford()}
3410 The function @code{canonicalize_clifford()} works for a
3411 generic Clifford algebra in a similar way as for Dirac gammas.
3413 The last provided function is
3415 @cindex @code{clifford_moebius_map()}
3417 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3418 const ex & d, const ex & v, const ex & G,
3419 unsigned char rl = 0);
3420 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3421 unsigned char rl = 0);
3424 It takes a list or vector @code{v} and makes the Moebius (conformal or
3425 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3426 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3427 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3428 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3429 ignored even if supplied. The returned value of this function is a list
3430 of components of the resulting vector.
3432 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3433 where @code{1} is the @code{representation_label} and @code{\nu} is the
3434 index of the corresponding unit. This provides a flexible typesetting
3435 with a suitable defintion of the @code{\clifford} command. For example, the
3438 \newcommand@{\clifford@}[1][]@{@}
3440 typesets all Clifford units identically, while the alternative definition
3442 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3444 prints units with @code{representation_label=0} as
3451 with @code{representation_label=1} as
3458 and with @code{representation_label=2} as
3466 @cindex @code{color} (class)
3467 @subsection Color algebra
3469 @cindex @code{color_T()}
3470 For computations in quantum chromodynamics, GiNaC implements the base elements
3471 and structure constants of the su(3) Lie algebra (color algebra). The base
3472 elements @math{T_a} are constructed by the function
3475 ex color_T(const ex & a, unsigned char rl = 0);
3478 which takes two arguments: the index and a @dfn{representation label} in the
3479 range 0 to 255 which is used to distinguish elements of different color
3480 algebras. Objects with different labels commutate with each other. The
3481 dimension of the index must be exactly 8 and it should be of class @code{idx},
3484 @cindex @code{color_ONE()}
3485 The unity element of a color algebra is constructed by
3488 ex color_ONE(unsigned char rl = 0);
3491 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3492 multiples of the unity element, even though it's customary to omit it.
3493 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3494 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3495 GiNaC may produce incorrect results.
3497 @cindex @code{color_d()}
3498 @cindex @code{color_f()}
3502 ex color_d(const ex & a, const ex & b, const ex & c);
3503 ex color_f(const ex & a, const ex & b, const ex & c);
3506 create the symmetric and antisymmetric structure constants @math{d_abc} and
3507 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3508 and @math{[T_a, T_b] = i f_abc T_c}.
3510 @cindex @code{color_h()}
3511 There's an additional function
3514 ex color_h(const ex & a, const ex & b, const ex & c);
3517 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3519 The function @code{simplify_indexed()} performs some simplifications on
3520 expressions containing color objects:
3525 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3526 k(symbol("k"), 8), l(symbol("l"), 8);
3528 e = color_d(a, b, l) * color_f(a, b, k);
3529 cout << e.simplify_indexed() << endl;
3532 e = color_d(a, b, l) * color_d(a, b, k);
3533 cout << e.simplify_indexed() << endl;
3536 e = color_f(l, a, b) * color_f(a, b, k);
3537 cout << e.simplify_indexed() << endl;
3540 e = color_h(a, b, c) * color_h(a, b, c);
3541 cout << e.simplify_indexed() << endl;
3544 e = color_h(a, b, c) * color_T(b) * color_T(c);
3545 cout << e.simplify_indexed() << endl;
3548 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3549 cout << e.simplify_indexed() << endl;
3552 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3553 cout << e.simplify_indexed() << endl;
3554 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3558 @cindex @code{color_trace()}
3559 To calculate the trace of an expression containing color objects you use one
3563 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3564 ex color_trace(const ex & e, const lst & rll);
3565 ex color_trace(const ex & e, unsigned char rl = 0);
3568 These functions take the trace over all color @samp{T} objects in the
3569 specified set @code{rls} or list @code{rll} of representation labels, or the
3570 single label @code{rl}; @samp{T}s with other labels are left standing. For
3575 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3577 // -> -I*f.a.c.b+d.a.c.b
3582 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3583 @c node-name, next, previous, up
3586 @cindex @code{exhashmap} (class)
3588 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3589 that can be used as a drop-in replacement for the STL
3590 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3591 typically constant-time, element look-up than @code{map<>}.
3593 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3594 following differences:
3598 no @code{lower_bound()} and @code{upper_bound()} methods
3600 no reverse iterators, no @code{rbegin()}/@code{rend()}
3602 no @code{operator<(exhashmap, exhashmap)}
3604 the comparison function object @code{key_compare} is hardcoded to
3607 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3608 initial hash table size (the actual table size after construction may be
3609 larger than the specified value)
3611 the method @code{size_t bucket_count()} returns the current size of the hash
3614 @code{insert()} and @code{erase()} operations invalidate all iterators
3618 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3619 @c node-name, next, previous, up
3620 @chapter Methods and Functions
3623 In this chapter the most important algorithms provided by GiNaC will be
3624 described. Some of them are implemented as functions on expressions,
3625 others are implemented as methods provided by expression objects. If
3626 they are methods, there exists a wrapper function around it, so you can
3627 alternatively call it in a functional way as shown in the simple
3632 cout << "As method: " << sin(1).evalf() << endl;
3633 cout << "As function: " << evalf(sin(1)) << endl;
3637 @cindex @code{subs()}
3638 The general rule is that wherever methods accept one or more parameters
3639 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3640 wrapper accepts is the same but preceded by the object to act on
3641 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3642 most natural one in an OO model but it may lead to confusion for MapleV
3643 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3644 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3645 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3646 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3647 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3648 here. Also, users of MuPAD will in most cases feel more comfortable
3649 with GiNaC's convention. All function wrappers are implemented
3650 as simple inline functions which just call the corresponding method and
3651 are only provided for users uncomfortable with OO who are dead set to
3652 avoid method invocations. Generally, nested function wrappers are much
3653 harder to read than a sequence of methods and should therefore be
3654 avoided if possible. On the other hand, not everything in GiNaC is a
3655 method on class @code{ex} and sometimes calling a function cannot be
3659 * Information About Expressions::
3660 * Numerical Evaluation::
3661 * Substituting Expressions::
3662 * Pattern Matching and Advanced Substitutions::
3663 * Applying a Function on Subexpressions::
3664 * Visitors and Tree Traversal::
3665 * Polynomial Arithmetic:: Working with polynomials.
3666 * Rational Expressions:: Working with rational functions.
3667 * Symbolic Differentiation::
3668 * Series Expansion:: Taylor and Laurent expansion.
3670 * Built-in Functions:: List of predefined mathematical functions.
3671 * Multiple polylogarithms::
3672 * Complex Conjugation::
3673 * Built-in Functions:: List of predefined mathematical functions.
3674 * Solving Linear Systems of Equations::
3675 * Input/Output:: Input and output of expressions.
3679 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3680 @c node-name, next, previous, up
3681 @section Getting information about expressions
3683 @subsection Checking expression types
3684 @cindex @code{is_a<@dots{}>()}
3685 @cindex @code{is_exactly_a<@dots{}>()}
3686 @cindex @code{ex_to<@dots{}>()}
3687 @cindex Converting @code{ex} to other classes
3688 @cindex @code{info()}
3689 @cindex @code{return_type()}
3690 @cindex @code{return_type_tinfo()}
3692 Sometimes it's useful to check whether a given expression is a plain number,
3693 a sum, a polynomial with integer coefficients, or of some other specific type.
3694 GiNaC provides a couple of functions for this:
3697 bool is_a<T>(const ex & e);
3698 bool is_exactly_a<T>(const ex & e);
3699 bool ex::info(unsigned flag);
3700 unsigned ex::return_type() const;
3701 unsigned ex::return_type_tinfo() const;
3704 When the test made by @code{is_a<T>()} returns true, it is safe to call
3705 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3706 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3707 example, assuming @code{e} is an @code{ex}:
3712 if (is_a<numeric>(e))
3713 numeric n = ex_to<numeric>(e);
3718 @code{is_a<T>(e)} allows you to check whether the top-level object of
3719 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3720 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3721 e.g., for checking whether an expression is a number, a sum, or a product:
3728 is_a<numeric>(e1); // true
3729 is_a<numeric>(e2); // false
3730 is_a<add>(e1); // false
3731 is_a<add>(e2); // true
3732 is_a<mul>(e1); // false
3733 is_a<mul>(e2); // false
3737 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3738 top-level object of an expression @samp{e} is an instance of the GiNaC
3739 class @samp{T}, not including parent classes.