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) 1999-2004 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{} 1999-2004 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{} 1999-2004 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 * Matrices:: Matrices.
697 * Indexed objects:: Handling indexed quantities.
698 * Non-commutative objects:: Algebras with non-commutative products.
699 * Hash Maps:: A faster alternative to std::map<>.
703 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
704 @c node-name, next, previous, up
706 @cindex expression (class @code{ex})
709 The most common class of objects a user deals with is the expression
710 @code{ex}, representing a mathematical object like a variable, number,
711 function, sum, product, etc@dots{} Expressions may be put together to form
712 new expressions, passed as arguments to functions, and so on. Here is a
713 little collection of valid expressions:
716 ex MyEx1 = 5; // simple number
717 ex MyEx2 = x + 2*y; // polynomial in x and y
718 ex MyEx3 = (x + 1)/(x - 1); // rational expression
719 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
720 ex MyEx5 = MyEx4 + 1; // similar to above
723 Expressions are handles to other more fundamental objects, that often
724 contain other expressions thus creating a tree of expressions
725 (@xref{Internal Structures}, for particular examples). Most methods on
726 @code{ex} therefore run top-down through such an expression tree. For
727 example, the method @code{has()} scans recursively for occurrences of
728 something inside an expression. Thus, if you have declared @code{MyEx4}
729 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
730 the argument of @code{sin} and hence return @code{true}.
732 The next sections will outline the general picture of GiNaC's class
733 hierarchy and describe the classes of objects that are handled by
736 @subsection Note: Expressions and STL containers
738 GiNaC expressions (@code{ex} objects) have value semantics (they can be
739 assigned, reassigned and copied like integral types) but the operator
740 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
741 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
743 This implies that in order to use expressions in sorted containers such as
744 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
745 comparison predicate. GiNaC provides such a predicate, called
746 @code{ex_is_less}. For example, a set of expressions should be defined
747 as @code{std::set<ex, ex_is_less>}.
749 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
750 don't pose a problem. A @code{std::vector<ex>} works as expected.
752 @xref{Information About Expressions}, for more about comparing and ordering
756 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
757 @c node-name, next, previous, up
758 @section Automatic evaluation and canonicalization of expressions
761 GiNaC performs some automatic transformations on expressions, to simplify
762 them and put them into a canonical form. Some examples:
765 ex MyEx1 = 2*x - 1 + x; // 3*x-1
766 ex MyEx2 = x - x; // 0
767 ex MyEx3 = cos(2*Pi); // 1
768 ex MyEx4 = x*y/x; // y
771 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
772 evaluation}. GiNaC only performs transformations that are
776 at most of complexity
784 algebraically correct, possibly except for a set of measure zero (e.g.
785 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
788 There are two types of automatic transformations in GiNaC that may not
789 behave in an entirely obvious way at first glance:
793 The terms of sums and products (and some other things like the arguments of
794 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
795 into a canonical form that is deterministic, but not lexicographical or in
796 any other way easy to guess (it almost always depends on the number and
797 order of the symbols you define). However, constructing the same expression
798 twice, either implicitly or explicitly, will always result in the same
801 Expressions of the form 'number times sum' are automatically expanded (this
802 has to do with GiNaC's internal representation of sums and products). For
805 ex MyEx5 = 2*(x + y); // 2*x+2*y
806 ex MyEx6 = z*(x + y); // z*(x+y)
810 The general rule is that when you construct expressions, GiNaC automatically
811 creates them in canonical form, which might differ from the form you typed in
812 your program. This may create some awkward looking output (@samp{-y+x} instead
813 of @samp{x-y}) but allows for more efficient operation and usually yields
814 some immediate simplifications.
816 @cindex @code{eval()}
817 Internally, the anonymous evaluator in GiNaC is implemented by the methods
820 ex ex::eval(int level = 0) const;
821 ex basic::eval(int level = 0) const;
824 but unless you are extending GiNaC with your own classes or functions, there
825 should never be any reason to call them explicitly. All GiNaC methods that
826 transform expressions, like @code{subs()} or @code{normal()}, automatically
827 re-evaluate their results.
830 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
831 @c node-name, next, previous, up
832 @section Error handling
834 @cindex @code{pole_error} (class)
836 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
837 generated by GiNaC are subclassed from the standard @code{exception} class
838 defined in the @file{<stdexcept>} header. In addition to the predefined
839 @code{logic_error}, @code{domain_error}, @code{out_of_range},
840 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
841 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
842 exception that gets thrown when trying to evaluate a mathematical function
845 The @code{pole_error} class has a member function
848 int pole_error::degree() const;
851 that returns the order of the singularity (or 0 when the pole is
852 logarithmic or the order is undefined).
854 When using GiNaC it is useful to arrange for exceptions to be caught in
855 the main program even if you don't want to do any special error handling.
856 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
857 default exception handler of your C++ compiler's run-time system which
858 usually only aborts the program without giving any information what went
861 Here is an example for a @code{main()} function that catches and prints
862 exceptions generated by GiNaC:
867 #include <ginac/ginac.h>
869 using namespace GiNaC;
877 @} catch (exception &p) @{
878 cerr << p.what() << endl;
886 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
887 @c node-name, next, previous, up
888 @section The Class Hierarchy
890 GiNaC's class hierarchy consists of several classes representing
891 mathematical objects, all of which (except for @code{ex} and some
892 helpers) are internally derived from one abstract base class called
893 @code{basic}. You do not have to deal with objects of class
894 @code{basic}, instead you'll be dealing with symbols, numbers,
895 containers of expressions and so on.
899 To get an idea about what kinds of symbolic composites may be built we
900 have a look at the most important classes in the class hierarchy and
901 some of the relations among the classes:
903 @image{classhierarchy}
905 The abstract classes shown here (the ones without drop-shadow) are of no
906 interest for the user. They are used internally in order to avoid code
907 duplication if two or more classes derived from them share certain
908 features. An example is @code{expairseq}, a container for a sequence of
909 pairs each consisting of one expression and a number (@code{numeric}).
910 What @emph{is} visible to the user are the derived classes @code{add}
911 and @code{mul}, representing sums and products. @xref{Internal
912 Structures}, where these two classes are described in more detail. The
913 following table shortly summarizes what kinds of mathematical objects
914 are stored in the different classes:
917 @multitable @columnfractions .22 .78
918 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
919 @item @code{constant} @tab Constants like
926 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
927 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
928 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
929 @item @code{ncmul} @tab Products of non-commutative objects
930 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
935 @code{sqrt(}@math{2}@code{)}
938 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
939 @item @code{function} @tab A symbolic function like
946 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
947 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
948 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
949 @item @code{indexed} @tab Indexed object like @math{A_ij}
950 @item @code{tensor} @tab Special tensor like the delta and metric tensors
951 @item @code{idx} @tab Index of an indexed object
952 @item @code{varidx} @tab Index with variance
953 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
954 @item @code{wildcard} @tab Wildcard for pattern matching
955 @item @code{structure} @tab Template for user-defined classes
960 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
961 @c node-name, next, previous, up
963 @cindex @code{symbol} (class)
964 @cindex hierarchy of classes
967 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
968 manipulation what atoms are for chemistry.
970 A typical symbol definition looks like this:
975 This definition actually contains three very different things:
977 @item a C++ variable named @code{x}
978 @item a @code{symbol} object stored in this C++ variable; this object
979 represents the symbol in a GiNaC expression
980 @item the string @code{"x"} which is the name of the symbol, used (almost)
981 exclusively for printing expressions holding the symbol
984 Symbols have an explicit name, supplied as a string during construction,
985 because in C++, variable names can't be used as values, and the C++ compiler
986 throws them away during compilation.
988 It is possible to omit the symbol name in the definition:
993 In this case, GiNaC will assign the symbol an internal, unique name of the
994 form @code{symbolNNN}. This won't affect the usability of the symbol but
995 the output of your calculations will become more readable if you give your
996 symbols sensible names (for intermediate expressions that are only used
997 internally such anonymous symbols can be quite useful, however).
999 Now, here is one important property of GiNaC that differentiates it from
1000 other computer algebra programs you may have used: GiNaC does @emph{not} use
1001 the names of symbols to tell them apart, but a (hidden) serial number that
1002 is unique for each newly created @code{symbol} object. In you want to use
1003 one and the same symbol in different places in your program, you must only
1004 create one @code{symbol} object and pass that around. If you create another
1005 symbol, even if it has the same name, GiNaC will treat it as a different
1022 // prints "x^6" which looks right, but...
1024 cout << e.degree(x) << endl;
1025 // ...this doesn't work. The symbol "x" here is different from the one
1026 // in f() and in the expression returned by f(). Consequently, it
1031 One possibility to ensure that @code{f()} and @code{main()} use the same
1032 symbol is to pass the symbol as an argument to @code{f()}:
1034 ex f(int n, const ex & x)
1043 // Now, f() uses the same symbol.
1046 cout << e.degree(x) << endl;
1047 // prints "6", as expected
1051 Another possibility would be to define a global symbol @code{x} that is used
1052 by both @code{f()} and @code{main()}. If you are using global symbols and
1053 multiple compilation units you must take special care, however. Suppose
1054 that you have a header file @file{globals.h} in your program that defines
1055 a @code{symbol x("x");}. In this case, every unit that includes
1056 @file{globals.h} would also get its own definition of @code{x} (because
1057 header files are just inlined into the source code by the C++ preprocessor),
1058 and hence you would again end up with multiple equally-named, but different,
1059 symbols. Instead, the @file{globals.h} header should only contain a
1060 @emph{declaration} like @code{extern symbol x;}, with the definition of
1061 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1063 A different approach to ensuring that symbols used in different parts of
1064 your program are identical is to create them with a @emph{factory} function
1067 const symbol & get_symbol(const string & s)
1069 static map<string, symbol> directory;
1070 map<string, symbol>::iterator i = directory.find(s);
1071 if (i != directory.end())
1074 return directory.insert(make_pair(s, symbol(s))).first->second;
1078 This function returns one newly constructed symbol for each name that is
1079 passed in, and it returns the same symbol when called multiple times with
1080 the same name. Using this symbol factory, we can rewrite our example like
1085 return pow(get_symbol("x"), n);
1092 // Both calls of get_symbol("x") yield the same symbol.
1093 cout << e.degree(get_symbol("x")) << endl;
1098 Instead of creating symbols from strings we could also have
1099 @code{get_symbol()} take, for example, an integer number as its argument.
1100 In this case, we would probably want to give the generated symbols names
1101 that include this number, which can be accomplished with the help of an
1102 @code{ostringstream}.
1104 In general, if you're getting weird results from GiNaC such as an expression
1105 @samp{x-x} that is not simplified to zero, you should check your symbol
1108 As we said, the names of symbols primarily serve for purposes of expression
1109 output. But there are actually two instances where GiNaC uses the names for
1110 identifying symbols: When constructing an expression from a string, and when
1111 recreating an expression from an archive (@pxref{Input/Output}).
1113 In addition to its name, a symbol may contain a special string that is used
1116 symbol x("x", "\\Box");
1119 This creates a symbol that is printed as "@code{x}" in normal output, but
1120 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1121 information about the different output formats of expressions in GiNaC).
1122 GiNaC automatically creates proper LaTeX code for symbols having names of
1123 greek letters (@samp{alpha}, @samp{mu}, etc.).
1125 @cindex @code{subs()}
1126 Symbols in GiNaC can't be assigned values. If you need to store results of
1127 calculations and give them a name, use C++ variables of type @code{ex}.
1128 If you want to replace a symbol in an expression with something else, you
1129 can invoke the expression's @code{.subs()} method
1130 (@pxref{Substituting Expressions}).
1132 @cindex @code{realsymbol()}
1133 By default, symbols are expected to stand in for complex values, i.e. they live
1134 in the complex domain. As a consequence, operations like complex conjugation,
1135 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1136 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1137 because of the unknown imaginary part of @code{x}.
1138 On the other hand, if you are sure that your symbols will hold only real values, you
1139 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1140 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1141 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1144 @node Numbers, Constants, Symbols, Basic Concepts
1145 @c node-name, next, previous, up
1147 @cindex @code{numeric} (class)
1153 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1154 The classes therein serve as foundation classes for GiNaC. CLN stands
1155 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1156 In order to find out more about CLN's internals, the reader is referred to
1157 the documentation of that library. @inforef{Introduction, , cln}, for
1158 more information. Suffice to say that it is by itself build on top of
1159 another library, the GNU Multiple Precision library GMP, which is an
1160 extremely fast library for arbitrary long integers and rationals as well
1161 as arbitrary precision floating point numbers. It is very commonly used
1162 by several popular cryptographic applications. CLN extends GMP by
1163 several useful things: First, it introduces the complex number field
1164 over either reals (i.e. floating point numbers with arbitrary precision)
1165 or rationals. Second, it automatically converts rationals to integers
1166 if the denominator is unity and complex numbers to real numbers if the
1167 imaginary part vanishes and also correctly treats algebraic functions.
1168 Third it provides good implementations of state-of-the-art algorithms
1169 for all trigonometric and hyperbolic functions as well as for
1170 calculation of some useful constants.
1172 The user can construct an object of class @code{numeric} in several
1173 ways. The following example shows the four most important constructors.
1174 It uses construction from C-integer, construction of fractions from two
1175 integers, construction from C-float and construction from a string:
1179 #include <ginac/ginac.h>
1180 using namespace GiNaC;
1184 numeric two = 2; // exact integer 2
1185 numeric r(2,3); // exact fraction 2/3
1186 numeric e(2.71828); // floating point number
1187 numeric p = "3.14159265358979323846"; // constructor from string
1188 // Trott's constant in scientific notation:
1189 numeric trott("1.0841015122311136151E-2");
1191 std::cout << two*p << std::endl; // floating point 6.283...
1196 @cindex complex numbers
1197 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1202 numeric z1 = 2-3*I; // exact complex number 2-3i
1203 numeric z2 = 5.9+1.6*I; // complex floating point number
1207 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1208 This would, however, call C's built-in operator @code{/} for integers
1209 first and result in a numeric holding a plain integer 1. @strong{Never
1210 use the operator @code{/} on integers} unless you know exactly what you
1211 are doing! Use the constructor from two integers instead, as shown in
1212 the example above. Writing @code{numeric(1)/2} may look funny but works
1215 @cindex @code{Digits}
1217 We have seen now the distinction between exact numbers and floating
1218 point numbers. Clearly, the user should never have to worry about
1219 dynamically created exact numbers, since their `exactness' always
1220 determines how they ought to be handled, i.e. how `long' they are. The
1221 situation is different for floating point numbers. Their accuracy is
1222 controlled by one @emph{global} variable, called @code{Digits}. (For
1223 those readers who know about Maple: it behaves very much like Maple's
1224 @code{Digits}). All objects of class numeric that are constructed from
1225 then on will be stored with a precision matching that number of decimal
1230 #include <ginac/ginac.h>
1231 using namespace std;
1232 using namespace GiNaC;
1236 numeric three(3.0), one(1.0);
1237 numeric x = one/three;
1239 cout << "in " << Digits << " digits:" << endl;
1241 cout << Pi.evalf() << endl;
1253 The above example prints the following output to screen:
1257 0.33333333333333333334
1258 3.1415926535897932385
1260 0.33333333333333333333333333333333333333333333333333333333333333333334
1261 3.1415926535897932384626433832795028841971693993751058209749445923078
1265 Note that the last number is not necessarily rounded as you would
1266 naively expect it to be rounded in the decimal system. But note also,
1267 that in both cases you got a couple of extra digits. This is because
1268 numbers are internally stored by CLN as chunks of binary digits in order
1269 to match your machine's word size and to not waste precision. Thus, on
1270 architectures with different word size, the above output might even
1271 differ with regard to actually computed digits.
1273 It should be clear that objects of class @code{numeric} should be used
1274 for constructing numbers or for doing arithmetic with them. The objects
1275 one deals with most of the time are the polymorphic expressions @code{ex}.
1277 @subsection Tests on numbers
1279 Once you have declared some numbers, assigned them to expressions and
1280 done some arithmetic with them it is frequently desired to retrieve some
1281 kind of information from them like asking whether that number is
1282 integer, rational, real or complex. For those cases GiNaC provides
1283 several useful methods. (Internally, they fall back to invocations of
1284 certain CLN functions.)
1286 As an example, let's construct some rational number, multiply it with
1287 some multiple of its denominator and test what comes out:
1291 #include <ginac/ginac.h>
1292 using namespace std;
1293 using namespace GiNaC;
1295 // some very important constants:
1296 const numeric twentyone(21);
1297 const numeric ten(10);
1298 const numeric five(5);
1302 numeric answer = twentyone;
1305 cout << answer.is_integer() << endl; // false, it's 21/5
1307 cout << answer.is_integer() << endl; // true, it's 42 now!
1311 Note that the variable @code{answer} is constructed here as an integer
1312 by @code{numeric}'s copy constructor but in an intermediate step it
1313 holds a rational number represented as integer numerator and integer
1314 denominator. When multiplied by 10, the denominator becomes unity and
1315 the result is automatically converted to a pure integer again.
1316 Internally, the underlying CLN is responsible for this behavior and we
1317 refer the reader to CLN's documentation. Suffice to say that
1318 the same behavior applies to complex numbers as well as return values of
1319 certain functions. Complex numbers are automatically converted to real
1320 numbers if the imaginary part becomes zero. The full set of tests that
1321 can be applied is listed in the following table.
1324 @multitable @columnfractions .30 .70
1325 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1326 @item @code{.is_zero()}
1327 @tab @dots{}equal to zero
1328 @item @code{.is_positive()}
1329 @tab @dots{}not complex and greater than 0
1330 @item @code{.is_integer()}
1331 @tab @dots{}a (non-complex) integer
1332 @item @code{.is_pos_integer()}
1333 @tab @dots{}an integer and greater than 0
1334 @item @code{.is_nonneg_integer()}
1335 @tab @dots{}an integer and greater equal 0
1336 @item @code{.is_even()}
1337 @tab @dots{}an even integer
1338 @item @code{.is_odd()}
1339 @tab @dots{}an odd integer
1340 @item @code{.is_prime()}
1341 @tab @dots{}a prime integer (probabilistic primality test)
1342 @item @code{.is_rational()}
1343 @tab @dots{}an exact rational number (integers are rational, too)
1344 @item @code{.is_real()}
1345 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1346 @item @code{.is_cinteger()}
1347 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1348 @item @code{.is_crational()}
1349 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1353 @subsection Numeric functions
1355 The following functions can be applied to @code{numeric} objects and will be
1356 evaluated immediately:
1359 @multitable @columnfractions .30 .70
1360 @item @strong{Name} @tab @strong{Function}
1361 @item @code{inverse(z)}
1362 @tab returns @math{1/z}
1363 @cindex @code{inverse()} (numeric)
1364 @item @code{pow(a, b)}
1365 @tab exponentiation @math{a^b}
1368 @item @code{real(z)}
1370 @cindex @code{real()}
1371 @item @code{imag(z)}
1373 @cindex @code{imag()}
1374 @item @code{csgn(z)}
1375 @tab complex sign (returns an @code{int})
1376 @item @code{numer(z)}
1377 @tab numerator of rational or complex rational number
1378 @item @code{denom(z)}
1379 @tab denominator of rational or complex rational number
1380 @item @code{sqrt(z)}
1382 @item @code{isqrt(n)}
1383 @tab integer square root
1384 @cindex @code{isqrt()}
1391 @item @code{asin(z)}
1393 @item @code{acos(z)}
1395 @item @code{atan(z)}
1396 @tab inverse tangent
1397 @item @code{atan(y, x)}
1398 @tab inverse tangent with two arguments
1399 @item @code{sinh(z)}
1400 @tab hyperbolic sine
1401 @item @code{cosh(z)}
1402 @tab hyperbolic cosine
1403 @item @code{tanh(z)}
1404 @tab hyperbolic tangent
1405 @item @code{asinh(z)}
1406 @tab inverse hyperbolic sine
1407 @item @code{acosh(z)}
1408 @tab inverse hyperbolic cosine
1409 @item @code{atanh(z)}
1410 @tab inverse hyperbolic tangent
1412 @tab exponential function
1414 @tab natural logarithm
1417 @item @code{zeta(z)}
1418 @tab Riemann's zeta function
1419 @item @code{tgamma(z)}
1421 @item @code{lgamma(z)}
1422 @tab logarithm of gamma function
1424 @tab psi (digamma) function
1425 @item @code{psi(n, z)}
1426 @tab derivatives of psi function (polygamma functions)
1427 @item @code{factorial(n)}
1428 @tab factorial function @math{n!}
1429 @item @code{doublefactorial(n)}
1430 @tab double factorial function @math{n!!}
1431 @cindex @code{doublefactorial()}
1432 @item @code{binomial(n, k)}
1433 @tab binomial coefficients
1434 @item @code{bernoulli(n)}
1435 @tab Bernoulli numbers
1436 @cindex @code{bernoulli()}
1437 @item @code{fibonacci(n)}
1438 @tab Fibonacci numbers
1439 @cindex @code{fibonacci()}
1440 @item @code{mod(a, b)}
1441 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1442 @cindex @code{mod()}
1443 @item @code{smod(a, b)}
1444 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1445 @cindex @code{smod()}
1446 @item @code{irem(a, b)}
1447 @tab integer remainder (has the sign of @math{a}, or is zero)
1448 @cindex @code{irem()}
1449 @item @code{irem(a, b, q)}
1450 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1451 @item @code{iquo(a, b)}
1452 @tab integer quotient
1453 @cindex @code{iquo()}
1454 @item @code{iquo(a, b, r)}
1455 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1456 @item @code{gcd(a, b)}
1457 @tab greatest common divisor
1458 @item @code{lcm(a, b)}
1459 @tab least common multiple
1463 Most of these functions are also available as symbolic functions that can be
1464 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1465 as polynomial algorithms.
1467 @subsection Converting numbers
1469 Sometimes it is desirable to convert a @code{numeric} object back to a
1470 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1471 class provides a couple of methods for this purpose:
1473 @cindex @code{to_int()}
1474 @cindex @code{to_long()}
1475 @cindex @code{to_double()}
1476 @cindex @code{to_cl_N()}
1478 int numeric::to_int() const;
1479 long numeric::to_long() const;
1480 double numeric::to_double() const;
1481 cln::cl_N numeric::to_cl_N() const;
1484 @code{to_int()} and @code{to_long()} only work when the number they are
1485 applied on is an exact integer. Otherwise the program will halt with a
1486 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1487 rational number will return a floating-point approximation. Both
1488 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1489 part of complex numbers.
1492 @node Constants, Fundamental containers, Numbers, Basic Concepts
1493 @c node-name, next, previous, up
1495 @cindex @code{constant} (class)
1498 @cindex @code{Catalan}
1499 @cindex @code{Euler}
1500 @cindex @code{evalf()}
1501 Constants behave pretty much like symbols except that they return some
1502 specific number when the method @code{.evalf()} is called.
1504 The predefined known constants are:
1507 @multitable @columnfractions .14 .30 .56
1508 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1510 @tab Archimedes' constant
1511 @tab 3.14159265358979323846264338327950288
1512 @item @code{Catalan}
1513 @tab Catalan's constant
1514 @tab 0.91596559417721901505460351493238411
1516 @tab Euler's (or Euler-Mascheroni) constant
1517 @tab 0.57721566490153286060651209008240243
1522 @node Fundamental containers, Lists, Constants, Basic Concepts
1523 @c node-name, next, previous, up
1524 @section Sums, products and powers
1528 @cindex @code{power}
1530 Simple rational expressions are written down in GiNaC pretty much like
1531 in other CAS or like expressions involving numerical variables in C.
1532 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1533 been overloaded to achieve this goal. When you run the following
1534 code snippet, the constructor for an object of type @code{mul} is
1535 automatically called to hold the product of @code{a} and @code{b} and
1536 then the constructor for an object of type @code{add} is called to hold
1537 the sum of that @code{mul} object and the number one:
1541 symbol a("a"), b("b");
1546 @cindex @code{pow()}
1547 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1548 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1549 construction is necessary since we cannot safely overload the constructor
1550 @code{^} in C++ to construct a @code{power} object. If we did, it would
1551 have several counterintuitive and undesired effects:
1555 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1557 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1558 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1559 interpret this as @code{x^(a^b)}.
1561 Also, expressions involving integer exponents are very frequently used,
1562 which makes it even more dangerous to overload @code{^} since it is then
1563 hard to distinguish between the semantics as exponentiation and the one
1564 for exclusive or. (It would be embarrassing to return @code{1} where one
1565 has requested @code{2^3}.)
1568 @cindex @command{ginsh}
1569 All effects are contrary to mathematical notation and differ from the
1570 way most other CAS handle exponentiation, therefore overloading @code{^}
1571 is ruled out for GiNaC's C++ part. The situation is different in
1572 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1573 that the other frequently used exponentiation operator @code{**} does
1574 not exist at all in C++).
1576 To be somewhat more precise, objects of the three classes described
1577 here, are all containers for other expressions. An object of class
1578 @code{power} is best viewed as a container with two slots, one for the
1579 basis, one for the exponent. All valid GiNaC expressions can be
1580 inserted. However, basic transformations like simplifying
1581 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1582 when this is mathematically possible. If we replace the outer exponent
1583 three in the example by some symbols @code{a}, the simplification is not
1584 safe and will not be performed, since @code{a} might be @code{1/2} and
1587 Objects of type @code{add} and @code{mul} are containers with an
1588 arbitrary number of slots for expressions to be inserted. Again, simple
1589 and safe simplifications are carried out like transforming
1590 @code{3*x+4-x} to @code{2*x+4}.
1593 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1594 @c node-name, next, previous, up
1595 @section Lists of expressions
1596 @cindex @code{lst} (class)
1598 @cindex @code{nops()}
1600 @cindex @code{append()}
1601 @cindex @code{prepend()}
1602 @cindex @code{remove_first()}
1603 @cindex @code{remove_last()}
1604 @cindex @code{remove_all()}
1606 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1607 expressions. They are not as ubiquitous as in many other computer algebra
1608 packages, but are sometimes used to supply a variable number of arguments of
1609 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1610 constructors, so you should have a basic understanding of them.
1612 Lists can be constructed by assigning a comma-separated sequence of
1617 symbol x("x"), y("y");
1620 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1625 There are also constructors that allow direct creation of lists of up to
1626 16 expressions, which is often more convenient but slightly less efficient:
1630 // This produces the same list 'l' as above:
1631 // lst l(x, 2, y, x+y);
1632 // lst l = lst(x, 2, y, x+y);
1636 Use the @code{nops()} method to determine the size (number of expressions) of
1637 a list and the @code{op()} method or the @code{[]} operator to access
1638 individual elements:
1642 cout << l.nops() << endl; // prints '4'
1643 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1647 As with the standard @code{list<T>} container, accessing random elements of a
1648 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1649 sequential access to the elements of a list is possible with the
1650 iterator types provided by the @code{lst} class:
1653 typedef ... lst::const_iterator;
1654 typedef ... lst::const_reverse_iterator;
1655 lst::const_iterator lst::begin() const;
1656 lst::const_iterator lst::end() const;
1657 lst::const_reverse_iterator lst::rbegin() const;
1658 lst::const_reverse_iterator lst::rend() const;
1661 For example, to print the elements of a list individually you can use:
1666 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1671 which is one order faster than
1676 for (size_t i = 0; i < l.nops(); ++i)
1677 cout << l.op(i) << endl;
1681 These iterators also allow you to use some of the algorithms provided by
1682 the C++ standard library:
1686 // print the elements of the list (requires #include <iterator>)
1687 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1689 // sum up the elements of the list (requires #include <numeric>)
1690 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1691 cout << sum << endl; // prints '2+2*x+2*y'
1695 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1696 (the only other one is @code{matrix}). You can modify single elements:
1700 l[1] = 42; // l is now @{x, 42, y, x+y@}
1701 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1705 You can append or prepend an expression to a list with the @code{append()}
1706 and @code{prepend()} methods:
1710 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1711 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1715 You can remove the first or last element of a list with @code{remove_first()}
1716 and @code{remove_last()}:
1720 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1721 l.remove_last(); // l is now @{x, 7, y, x+y@}
1725 You can remove all the elements of a list with @code{remove_all()}:
1729 l.remove_all(); // l is now empty
1733 You can bring the elements of a list into a canonical order with @code{sort()}:
1742 // l1 and l2 are now equal
1746 Finally, you can remove all but the first element of consecutive groups of
1747 elements with @code{unique()}:
1752 l3 = x, 2, 2, 2, y, x+y, y+x;
1753 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1758 @node Mathematical functions, Relations, Lists, Basic Concepts
1759 @c node-name, next, previous, up
1760 @section Mathematical functions
1761 @cindex @code{function} (class)
1762 @cindex trigonometric function
1763 @cindex hyperbolic function
1765 There are quite a number of useful functions hard-wired into GiNaC. For
1766 instance, all trigonometric and hyperbolic functions are implemented
1767 (@xref{Built-in Functions}, for a complete list).
1769 These functions (better called @emph{pseudofunctions}) are all objects
1770 of class @code{function}. They accept one or more expressions as
1771 arguments and return one expression. If the arguments are not
1772 numerical, the evaluation of the function may be halted, as it does in
1773 the next example, showing how a function returns itself twice and
1774 finally an expression that may be really useful:
1776 @cindex Gamma function
1777 @cindex @code{subs()}
1780 symbol x("x"), y("y");
1782 cout << tgamma(foo) << endl;
1783 // -> tgamma(x+(1/2)*y)
1784 ex bar = foo.subs(y==1);
1785 cout << tgamma(bar) << endl;
1787 ex foobar = bar.subs(x==7);
1788 cout << tgamma(foobar) << endl;
1789 // -> (135135/128)*Pi^(1/2)
1793 Besides evaluation most of these functions allow differentiation, series
1794 expansion and so on. Read the next chapter in order to learn more about
1797 It must be noted that these pseudofunctions are created by inline
1798 functions, where the argument list is templated. This means that
1799 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1800 @code{sin(ex(1))} and will therefore not result in a floating point
1801 number. Unless of course the function prototype is explicitly
1802 overridden -- which is the case for arguments of type @code{numeric}
1803 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1804 point number of class @code{numeric} you should call
1805 @code{sin(numeric(1))}. This is almost the same as calling
1806 @code{sin(1).evalf()} except that the latter will return a numeric
1807 wrapped inside an @code{ex}.
1810 @node Relations, Matrices, Mathematical functions, Basic Concepts
1811 @c node-name, next, previous, up
1813 @cindex @code{relational} (class)
1815 Sometimes, a relation holding between two expressions must be stored
1816 somehow. The class @code{relational} is a convenient container for such
1817 purposes. A relation is by definition a container for two @code{ex} and
1818 a relation between them that signals equality, inequality and so on.
1819 They are created by simply using the C++ operators @code{==}, @code{!=},
1820 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1822 @xref{Mathematical functions}, for examples where various applications
1823 of the @code{.subs()} method show how objects of class relational are
1824 used as arguments. There they provide an intuitive syntax for
1825 substitutions. They are also used as arguments to the @code{ex::series}
1826 method, where the left hand side of the relation specifies the variable
1827 to expand in and the right hand side the expansion point. They can also
1828 be used for creating systems of equations that are to be solved for
1829 unknown variables. But the most common usage of objects of this class
1830 is rather inconspicuous in statements of the form @code{if
1831 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1832 conversion from @code{relational} to @code{bool} takes place. Note,
1833 however, that @code{==} here does not perform any simplifications, hence
1834 @code{expand()} must be called explicitly.
1837 @node Matrices, Indexed objects, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{matrix} (class)
1842 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1843 matrix with @math{m} rows and @math{n} columns are accessed with two
1844 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1845 second one in the range 0@dots{}@math{n-1}.
1847 There are a couple of ways to construct matrices, with or without preset
1848 elements. The constructor
1851 matrix::matrix(unsigned r, unsigned c);
1854 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1857 The fastest way to create a matrix with preinitialized elements is to assign
1858 a list of comma-separated expressions to an empty matrix (see below for an
1859 example). But you can also specify the elements as a (flat) list with
1862 matrix::matrix(unsigned r, unsigned c, const lst & l);
1867 @cindex @code{lst_to_matrix()}
1869 ex lst_to_matrix(const lst & l);
1872 constructs a matrix from a list of lists, each list representing a matrix row.
1874 There is also a set of functions for creating some special types of
1877 @cindex @code{diag_matrix()}
1878 @cindex @code{unit_matrix()}
1879 @cindex @code{symbolic_matrix()}
1881 ex diag_matrix(const lst & l);
1882 ex unit_matrix(unsigned x);
1883 ex unit_matrix(unsigned r, unsigned c);
1884 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1885 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1888 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1889 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1890 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1891 matrix filled with newly generated symbols made of the specified base name
1892 and the position of each element in the matrix.
1894 Matrix elements can be accessed and set using the parenthesis (function call)
1898 const ex & matrix::operator()(unsigned r, unsigned c) const;
1899 ex & matrix::operator()(unsigned r, unsigned c);
1902 It is also possible to access the matrix elements in a linear fashion with
1903 the @code{op()} method. But C++-style subscripting with square brackets
1904 @samp{[]} is not available.
1906 Here are a couple of examples for constructing matrices:
1910 symbol a("a"), b("b");
1924 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1927 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1930 cout << diag_matrix(lst(a, b)) << endl;
1933 cout << unit_matrix(3) << endl;
1934 // -> [[1,0,0],[0,1,0],[0,0,1]]
1936 cout << symbolic_matrix(2, 3, "x") << endl;
1937 // -> [[x00,x01,x02],[x10,x11,x12]]
1941 @cindex @code{transpose()}
1942 There are three ways to do arithmetic with matrices. The first (and most
1943 direct one) is to use the methods provided by the @code{matrix} class:
1946 matrix matrix::add(const matrix & other) const;
1947 matrix matrix::sub(const matrix & other) const;
1948 matrix matrix::mul(const matrix & other) const;
1949 matrix matrix::mul_scalar(const ex & other) const;
1950 matrix matrix::pow(const ex & expn) const;
1951 matrix matrix::transpose() const;
1954 All of these methods return the result as a new matrix object. Here is an
1955 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1960 matrix A(2, 2), B(2, 2), C(2, 2);
1968 matrix result = A.mul(B).sub(C.mul_scalar(2));
1969 cout << result << endl;
1970 // -> [[-13,-6],[1,2]]
1975 @cindex @code{evalm()}
1976 The second (and probably the most natural) way is to construct an expression
1977 containing matrices with the usual arithmetic operators and @code{pow()}.
1978 For efficiency reasons, expressions with sums, products and powers of
1979 matrices are not automatically evaluated in GiNaC. You have to call the
1983 ex ex::evalm() const;
1986 to obtain the result:
1993 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1994 cout << e.evalm() << endl;
1995 // -> [[-13,-6],[1,2]]
2000 The non-commutativity of the product @code{A*B} in this example is
2001 automatically recognized by GiNaC. There is no need to use a special
2002 operator here. @xref{Non-commutative objects}, for more information about
2003 dealing with non-commutative expressions.
2005 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2006 to perform the arithmetic:
2011 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2012 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2014 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2015 cout << e.simplify_indexed() << endl;
2016 // -> [[-13,-6],[1,2]].i.j
2020 Using indices is most useful when working with rectangular matrices and
2021 one-dimensional vectors because you don't have to worry about having to
2022 transpose matrices before multiplying them. @xref{Indexed objects}, for
2023 more information about using matrices with indices, and about indices in
2026 The @code{matrix} class provides a couple of additional methods for
2027 computing determinants, traces, characteristic polynomials and ranks:
2029 @cindex @code{determinant()}
2030 @cindex @code{trace()}
2031 @cindex @code{charpoly()}
2032 @cindex @code{rank()}
2034 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2035 ex matrix::trace() const;
2036 ex matrix::charpoly(const ex & lambda) const;
2037 unsigned matrix::rank() const;
2040 The @samp{algo} argument of @code{determinant()} allows to select
2041 between different algorithms for calculating the determinant. The
2042 asymptotic speed (as parametrized by the matrix size) can greatly differ
2043 between those algorithms, depending on the nature of the matrix'
2044 entries. The possible values are defined in the @file{flags.h} header
2045 file. By default, GiNaC uses a heuristic to automatically select an
2046 algorithm that is likely (but not guaranteed) to give the result most
2049 @cindex @code{inverse()} (matrix)
2050 @cindex @code{solve()}
2051 Matrices may also be inverted using the @code{ex matrix::inverse()}
2052 method and linear systems may be solved with:
2055 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2058 Assuming the matrix object this method is applied on is an @code{m}
2059 times @code{n} matrix, then @code{vars} must be a @code{n} times
2060 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2061 times @code{p} matrix. The returned matrix then has dimension @code{n}
2062 times @code{p} and in the case of an underdetermined system will still
2063 contain some of the indeterminates from @code{vars}. If the system is
2064 overdetermined, an exception is thrown.
2067 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2068 @c node-name, next, previous, up
2069 @section Indexed objects
2071 GiNaC allows you to handle expressions containing general indexed objects in
2072 arbitrary spaces. It is also able to canonicalize and simplify such
2073 expressions and perform symbolic dummy index summations. There are a number
2074 of predefined indexed objects provided, like delta and metric tensors.
2076 There are few restrictions placed on indexed objects and their indices and
2077 it is easy to construct nonsense expressions, but our intention is to
2078 provide a general framework that allows you to implement algorithms with
2079 indexed quantities, getting in the way as little as possible.
2081 @cindex @code{idx} (class)
2082 @cindex @code{indexed} (class)
2083 @subsection Indexed quantities and their indices
2085 Indexed expressions in GiNaC are constructed of two special types of objects,
2086 @dfn{index objects} and @dfn{indexed objects}.
2090 @cindex contravariant
2093 @item Index objects are of class @code{idx} or a subclass. Every index has
2094 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2095 the index lives in) which can both be arbitrary expressions but are usually
2096 a number or a simple symbol. In addition, indices of class @code{varidx} have
2097 a @dfn{variance} (they can be co- or contravariant), and indices of class
2098 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2100 @item Indexed objects are of class @code{indexed} or a subclass. They
2101 contain a @dfn{base expression} (which is the expression being indexed), and
2102 one or more indices.
2106 @strong{Note:} when printing expressions, covariant indices and indices
2107 without variance are denoted @samp{.i} while contravariant indices are
2108 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2109 value. In the following, we are going to use that notation in the text so
2110 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2111 not visible in the output.
2113 A simple example shall illustrate the concepts:
2117 #include <ginac/ginac.h>
2118 using namespace std;
2119 using namespace GiNaC;
2123 symbol i_sym("i"), j_sym("j");
2124 idx i(i_sym, 3), j(j_sym, 3);
2127 cout << indexed(A, i, j) << endl;
2129 cout << index_dimensions << indexed(A, i, j) << endl;
2131 cout << dflt; // reset cout to default output format (dimensions hidden)
2135 The @code{idx} constructor takes two arguments, the index value and the
2136 index dimension. First we define two index objects, @code{i} and @code{j},
2137 both with the numeric dimension 3. The value of the index @code{i} is the
2138 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2139 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2140 construct an expression containing one indexed object, @samp{A.i.j}. It has
2141 the symbol @code{A} as its base expression and the two indices @code{i} and
2144 The dimensions of indices are normally not visible in the output, but one
2145 can request them to be printed with the @code{index_dimensions} manipulator,
2148 Note the difference between the indices @code{i} and @code{j} which are of
2149 class @code{idx}, and the index values which are the symbols @code{i_sym}
2150 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2151 or numbers but must be index objects. For example, the following is not
2152 correct and will raise an exception:
2155 symbol i("i"), j("j");
2156 e = indexed(A, i, j); // ERROR: indices must be of type idx
2159 You can have multiple indexed objects in an expression, index values can
2160 be numeric, and index dimensions symbolic:
2164 symbol B("B"), dim("dim");
2165 cout << 4 * indexed(A, i)
2166 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2171 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2172 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2173 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2174 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2175 @code{simplify_indexed()} for that, see below).
2177 In fact, base expressions, index values and index dimensions can be
2178 arbitrary expressions:
2182 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2187 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2188 get an error message from this but you will probably not be able to do
2189 anything useful with it.
2191 @cindex @code{get_value()}
2192 @cindex @code{get_dimension()}
2196 ex idx::get_value();
2197 ex idx::get_dimension();
2200 return the value and dimension of an @code{idx} object. If you have an index
2201 in an expression, such as returned by calling @code{.op()} on an indexed
2202 object, you can get a reference to the @code{idx} object with the function
2203 @code{ex_to<idx>()} on the expression.
2205 There are also the methods
2208 bool idx::is_numeric();
2209 bool idx::is_symbolic();
2210 bool idx::is_dim_numeric();
2211 bool idx::is_dim_symbolic();
2214 for checking whether the value and dimension are numeric or symbolic
2215 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2216 About Expressions}) returns information about the index value.
2218 @cindex @code{varidx} (class)
2219 If you need co- and contravariant indices, use the @code{varidx} class:
2223 symbol mu_sym("mu"), nu_sym("nu");
2224 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2225 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2227 cout << indexed(A, mu, nu) << endl;
2229 cout << indexed(A, mu_co, nu) << endl;
2231 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2236 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2237 co- or contravariant. The default is a contravariant (upper) index, but
2238 this can be overridden by supplying a third argument to the @code{varidx}
2239 constructor. The two methods
2242 bool varidx::is_covariant();
2243 bool varidx::is_contravariant();
2246 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2247 to get the object reference from an expression). There's also the very useful
2251 ex varidx::toggle_variance();
2254 which makes a new index with the same value and dimension but the opposite
2255 variance. By using it you only have to define the index once.
2257 @cindex @code{spinidx} (class)
2258 The @code{spinidx} class provides dotted and undotted variant indices, as
2259 used in the Weyl-van-der-Waerden spinor formalism:
2263 symbol K("K"), C_sym("C"), D_sym("D");
2264 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2265 // contravariant, undotted
2266 spinidx C_co(C_sym, 2, true); // covariant index
2267 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2268 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2270 cout << indexed(K, C, D) << endl;
2272 cout << indexed(K, C_co, D_dot) << endl;
2274 cout << indexed(K, D_co_dot, D) << endl;
2279 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2280 dotted or undotted. The default is undotted but this can be overridden by
2281 supplying a fourth argument to the @code{spinidx} constructor. The two
2285 bool spinidx::is_dotted();
2286 bool spinidx::is_undotted();
2289 allow you to check whether or not a @code{spinidx} object is dotted (use
2290 @code{ex_to<spinidx>()} to get the object reference from an expression).
2291 Finally, the two methods
2294 ex spinidx::toggle_dot();
2295 ex spinidx::toggle_variance_dot();
2298 create a new index with the same value and dimension but opposite dottedness
2299 and the same or opposite variance.
2301 @subsection Substituting indices
2303 @cindex @code{subs()}
2304 Sometimes you will want to substitute one symbolic index with another
2305 symbolic or numeric index, for example when calculating one specific element
2306 of a tensor expression. This is done with the @code{.subs()} method, as it
2307 is done for symbols (see @ref{Substituting Expressions}).
2309 You have two possibilities here. You can either substitute the whole index
2310 by another index or expression:
2314 ex e = indexed(A, mu_co);
2315 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2316 // -> A.mu becomes A~nu
2317 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2318 // -> A.mu becomes A~0
2319 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2320 // -> A.mu becomes A.0
2324 The third example shows that trying to replace an index with something that
2325 is not an index will substitute the index value instead.
2327 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2332 ex e = indexed(A, mu_co);
2333 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2334 // -> A.mu becomes A.nu
2335 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2336 // -> A.mu becomes A.0
2340 As you see, with the second method only the value of the index will get
2341 substituted. Its other properties, including its dimension, remain unchanged.
2342 If you want to change the dimension of an index you have to substitute the
2343 whole index by another one with the new dimension.
2345 Finally, substituting the base expression of an indexed object works as
2350 ex e = indexed(A, mu_co);
2351 cout << e << " becomes " << e.subs(A == A+B) << endl;
2352 // -> A.mu becomes (B+A).mu
2356 @subsection Symmetries
2357 @cindex @code{symmetry} (class)
2358 @cindex @code{sy_none()}
2359 @cindex @code{sy_symm()}
2360 @cindex @code{sy_anti()}
2361 @cindex @code{sy_cycl()}
2363 Indexed objects can have certain symmetry properties with respect to their
2364 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2365 that is constructed with the helper functions
2368 symmetry sy_none(...);
2369 symmetry sy_symm(...);
2370 symmetry sy_anti(...);
2371 symmetry sy_cycl(...);
2374 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2375 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2376 represents a cyclic symmetry. Each of these functions accepts up to four
2377 arguments which can be either symmetry objects themselves or unsigned integer
2378 numbers that represent an index position (counting from 0). A symmetry
2379 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2380 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2383 Here are some examples of symmetry definitions:
2388 e = indexed(A, i, j);
2389 e = indexed(A, sy_none(), i, j); // equivalent
2390 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2392 // Symmetric in all three indices:
2393 e = indexed(A, sy_symm(), i, j, k);
2394 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2395 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2396 // different canonical order
2398 // Symmetric in the first two indices only:
2399 e = indexed(A, sy_symm(0, 1), i, j, k);
2400 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2402 // Antisymmetric in the first and last index only (index ranges need not
2404 e = indexed(A, sy_anti(0, 2), i, j, k);
2405 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2407 // An example of a mixed symmetry: antisymmetric in the first two and
2408 // last two indices, symmetric when swapping the first and last index
2409 // pairs (like the Riemann curvature tensor):
2410 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2412 // Cyclic symmetry in all three indices:
2413 e = indexed(A, sy_cycl(), i, j, k);
2414 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2416 // The following examples are invalid constructions that will throw
2417 // an exception at run time.
2419 // An index may not appear multiple times:
2420 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2421 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2423 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2424 // same number of indices:
2425 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2427 // And of course, you cannot specify indices which are not there:
2428 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2432 If you need to specify more than four indices, you have to use the
2433 @code{.add()} method of the @code{symmetry} class. For example, to specify
2434 full symmetry in the first six indices you would write
2435 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2437 If an indexed object has a symmetry, GiNaC will automatically bring the
2438 indices into a canonical order which allows for some immediate simplifications:
2442 cout << indexed(A, sy_symm(), i, j)
2443 + indexed(A, sy_symm(), j, i) << endl;
2445 cout << indexed(B, sy_anti(), i, j)
2446 + indexed(B, sy_anti(), j, i) << endl;
2448 cout << indexed(B, sy_anti(), i, j, k)
2449 - indexed(B, sy_anti(), j, k, i) << endl;
2454 @cindex @code{get_free_indices()}
2456 @subsection Dummy indices
2458 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2459 that a summation over the index range is implied. Symbolic indices which are
2460 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2461 dummy nor free indices.
2463 To be recognized as a dummy index pair, the two indices must be of the same
2464 class and their value must be the same single symbol (an index like
2465 @samp{2*n+1} is never a dummy index). If the indices are of class
2466 @code{varidx} they must also be of opposite variance; if they are of class
2467 @code{spinidx} they must be both dotted or both undotted.
2469 The method @code{.get_free_indices()} returns a vector containing the free
2470 indices of an expression. It also checks that the free indices of the terms
2471 of a sum are consistent:
2475 symbol A("A"), B("B"), C("C");
2477 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2478 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2480 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2481 cout << exprseq(e.get_free_indices()) << endl;
2483 // 'j' and 'l' are dummy indices
2485 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2486 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2488 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2489 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2490 cout << exprseq(e.get_free_indices()) << endl;
2492 // 'nu' is a dummy index, but 'sigma' is not
2494 e = indexed(A, mu, mu);
2495 cout << exprseq(e.get_free_indices()) << endl;
2497 // 'mu' is not a dummy index because it appears twice with the same
2500 e = indexed(A, mu, nu) + 42;
2501 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2502 // this will throw an exception:
2503 // "add::get_free_indices: inconsistent indices in sum"
2507 @cindex @code{simplify_indexed()}
2508 @subsection Simplifying indexed expressions
2510 In addition to the few automatic simplifications that GiNaC performs on
2511 indexed expressions (such as re-ordering the indices of symmetric tensors
2512 and calculating traces and convolutions of matrices and predefined tensors)
2516 ex ex::simplify_indexed();
2517 ex ex::simplify_indexed(const scalar_products & sp);
2520 that performs some more expensive operations:
2523 @item it checks the consistency of free indices in sums in the same way
2524 @code{get_free_indices()} does
2525 @item it tries to give dummy indices that appear in different terms of a sum
2526 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2527 @item it (symbolically) calculates all possible dummy index summations/contractions
2528 with the predefined tensors (this will be explained in more detail in the
2530 @item it detects contractions that vanish for symmetry reasons, for example
2531 the contraction of a symmetric and a totally antisymmetric tensor
2532 @item as a special case of dummy index summation, it can replace scalar products
2533 of two tensors with a user-defined value
2536 The last point is done with the help of the @code{scalar_products} class
2537 which is used to store scalar products with known values (this is not an
2538 arithmetic class, you just pass it to @code{simplify_indexed()}):
2542 symbol A("A"), B("B"), C("C"), i_sym("i");
2546 sp.add(A, B, 0); // A and B are orthogonal
2547 sp.add(A, C, 0); // A and C are orthogonal
2548 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2550 e = indexed(A + B, i) * indexed(A + C, i);
2552 // -> (B+A).i*(A+C).i
2554 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2560 The @code{scalar_products} object @code{sp} acts as a storage for the
2561 scalar products added to it with the @code{.add()} method. This method
2562 takes three arguments: the two expressions of which the scalar product is
2563 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2564 @code{simplify_indexed()} will replace all scalar products of indexed
2565 objects that have the symbols @code{A} and @code{B} as base expressions
2566 with the single value 0. The number, type and dimension of the indices
2567 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2569 @cindex @code{expand()}
2570 The example above also illustrates a feature of the @code{expand()} method:
2571 if passed the @code{expand_indexed} option it will distribute indices
2572 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2574 @cindex @code{tensor} (class)
2575 @subsection Predefined tensors
2577 Some frequently used special tensors such as the delta, epsilon and metric
2578 tensors are predefined in GiNaC. They have special properties when
2579 contracted with other tensor expressions and some of them have constant
2580 matrix representations (they will evaluate to a number when numeric
2581 indices are specified).
2583 @cindex @code{delta_tensor()}
2584 @subsubsection Delta tensor
2586 The delta tensor takes two indices, is symmetric and has the matrix
2587 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2588 @code{delta_tensor()}:
2592 symbol A("A"), B("B");
2594 idx i(symbol("i"), 3), j(symbol("j"), 3),
2595 k(symbol("k"), 3), l(symbol("l"), 3);
2597 ex e = indexed(A, i, j) * indexed(B, k, l)
2598 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2599 cout << e.simplify_indexed() << endl;
2602 cout << delta_tensor(i, i) << endl;
2607 @cindex @code{metric_tensor()}
2608 @subsubsection General metric tensor
2610 The function @code{metric_tensor()} creates a general symmetric metric
2611 tensor with two indices that can be used to raise/lower tensor indices. The
2612 metric tensor is denoted as @samp{g} in the output and if its indices are of
2613 mixed variance it is automatically replaced by a delta tensor:
2619 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2621 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2622 cout << e.simplify_indexed() << endl;
2625 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2626 cout << e.simplify_indexed() << endl;
2629 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2630 * metric_tensor(nu, rho);
2631 cout << e.simplify_indexed() << endl;
2634 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2635 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2636 + indexed(A, mu.toggle_variance(), rho));
2637 cout << e.simplify_indexed() << endl;
2642 @cindex @code{lorentz_g()}
2643 @subsubsection Minkowski metric tensor
2645 The Minkowski metric tensor is a special metric tensor with a constant
2646 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2647 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2648 It is created with the function @code{lorentz_g()} (although it is output as
2653 varidx mu(symbol("mu"), 4);
2655 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2656 * lorentz_g(mu, varidx(0, 4)); // negative signature
2657 cout << e.simplify_indexed() << endl;
2660 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2661 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2662 cout << e.simplify_indexed() << endl;
2667 @cindex @code{spinor_metric()}
2668 @subsubsection Spinor metric tensor
2670 The function @code{spinor_metric()} creates an antisymmetric tensor with
2671 two indices that is used to raise/lower indices of 2-component spinors.
2672 It is output as @samp{eps}:
2678 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2679 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2681 e = spinor_metric(A, B) * indexed(psi, B_co);
2682 cout << e.simplify_indexed() << endl;
2685 e = spinor_metric(A, B) * indexed(psi, A_co);
2686 cout << e.simplify_indexed() << endl;
2689 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2690 cout << e.simplify_indexed() << endl;
2693 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2694 cout << e.simplify_indexed() << endl;
2697 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2698 cout << e.simplify_indexed() << endl;
2701 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2702 cout << e.simplify_indexed() << endl;
2707 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2709 @cindex @code{epsilon_tensor()}
2710 @cindex @code{lorentz_eps()}
2711 @subsubsection Epsilon tensor
2713 The epsilon tensor is totally antisymmetric, its number of indices is equal
2714 to the dimension of the index space (the indices must all be of the same
2715 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2716 defined to be 1. Its behavior with indices that have a variance also
2717 depends on the signature of the metric. Epsilon tensors are output as
2720 There are three functions defined to create epsilon tensors in 2, 3 and 4
2724 ex epsilon_tensor(const ex & i1, const ex & i2);
2725 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2726 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2729 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2730 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2731 Minkowski space (the last @code{bool} argument specifies whether the metric
2732 has negative or positive signature, as in the case of the Minkowski metric
2737 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2738 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2739 e = lorentz_eps(mu, nu, rho, sig) *
2740 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2741 cout << simplify_indexed(e) << endl;
2742 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2744 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2745 symbol A("A"), B("B");
2746 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2747 cout << simplify_indexed(e) << endl;
2748 // -> -B.k*A.j*eps.i.k.j
2749 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2750 cout << simplify_indexed(e) << endl;
2755 @subsection Linear algebra
2757 The @code{matrix} class can be used with indices to do some simple linear
2758 algebra (linear combinations and products of vectors and matrices, traces
2759 and scalar products):
2763 idx i(symbol("i"), 2), j(symbol("j"), 2);
2764 symbol x("x"), y("y");
2766 // A is a 2x2 matrix, X is a 2x1 vector
2767 matrix A(2, 2), X(2, 1);
2772 cout << indexed(A, i, i) << endl;
2775 ex e = indexed(A, i, j) * indexed(X, j);
2776 cout << e.simplify_indexed() << endl;
2777 // -> [[2*y+x],[4*y+3*x]].i
2779 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2780 cout << e.simplify_indexed() << endl;
2781 // -> [[3*y+3*x,6*y+2*x]].j
2785 You can of course obtain the same results with the @code{matrix::add()},
2786 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2787 but with indices you don't have to worry about transposing matrices.
2789 Matrix indices always start at 0 and their dimension must match the number
2790 of rows/columns of the matrix. Matrices with one row or one column are
2791 vectors and can have one or two indices (it doesn't matter whether it's a
2792 row or a column vector). Other matrices must have two indices.
2794 You should be careful when using indices with variance on matrices. GiNaC
2795 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2796 @samp{F.mu.nu} are different matrices. In this case you should use only
2797 one form for @samp{F} and explicitly multiply it with a matrix representation
2798 of the metric tensor.
2801 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2802 @c node-name, next, previous, up
2803 @section Non-commutative objects
2805 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2806 non-commutative objects are built-in which are mostly of use in high energy
2810 @item Clifford (Dirac) algebra (class @code{clifford})
2811 @item su(3) Lie algebra (class @code{color})
2812 @item Matrices (unindexed) (class @code{matrix})
2815 The @code{clifford} and @code{color} classes are subclasses of
2816 @code{indexed} because the elements of these algebras usually carry
2817 indices. The @code{matrix} class is described in more detail in
2820 Unlike most computer algebra systems, GiNaC does not primarily provide an
2821 operator (often denoted @samp{&*}) for representing inert products of
2822 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2823 classes of objects involved, and non-commutative products are formed with
2824 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2825 figuring out by itself which objects commutate and will group the factors
2826 by their class. Consider this example:
2830 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2831 idx a(symbol("a"), 8), b(symbol("b"), 8);
2832 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2834 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2838 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2839 groups the non-commutative factors (the gammas and the su(3) generators)
2840 together while preserving the order of factors within each class (because
2841 Clifford objects commutate with color objects). The resulting expression is a
2842 @emph{commutative} product with two factors that are themselves non-commutative
2843 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2844 parentheses are placed around the non-commutative products in the output.
2846 @cindex @code{ncmul} (class)
2847 Non-commutative products are internally represented by objects of the class
2848 @code{ncmul}, as opposed to commutative products which are handled by the
2849 @code{mul} class. You will normally not have to worry about this distinction,
2852 The advantage of this approach is that you never have to worry about using
2853 (or forgetting to use) a special operator when constructing non-commutative
2854 expressions. Also, non-commutative products in GiNaC are more intelligent
2855 than in other computer algebra systems; they can, for example, automatically
2856 canonicalize themselves according to rules specified in the implementation
2857 of the non-commutative classes. The drawback is that to work with other than
2858 the built-in algebras you have to implement new classes yourself. Symbols
2859 always commutate and it's not possible to construct non-commutative products
2860 using symbols to represent the algebra elements or generators. User-defined
2861 functions can, however, be specified as being non-commutative.
2863 @cindex @code{return_type()}
2864 @cindex @code{return_type_tinfo()}
2865 Information about the commutativity of an object or expression can be
2866 obtained with the two member functions
2869 unsigned ex::return_type() const;
2870 unsigned ex::return_type_tinfo() const;
2873 The @code{return_type()} function returns one of three values (defined in
2874 the header file @file{flags.h}), corresponding to three categories of
2875 expressions in GiNaC:
2878 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2879 classes are of this kind.
2880 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2881 certain class of non-commutative objects which can be determined with the
2882 @code{return_type_tinfo()} method. Expressions of this category commutate
2883 with everything except @code{noncommutative} expressions of the same
2885 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2886 of non-commutative objects of different classes. Expressions of this
2887 category don't commutate with any other @code{noncommutative} or
2888 @code{noncommutative_composite} expressions.
2891 The value returned by the @code{return_type_tinfo()} method is valid only
2892 when the return type of the expression is @code{noncommutative}. It is a
2893 value that is unique to the class of the object and usually one of the
2894 constants in @file{tinfos.h}, or derived therefrom.
2896 Here are a couple of examples:
2899 @multitable @columnfractions 0.33 0.33 0.34
2900 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2901 @item @code{42} @tab @code{commutative} @tab -
2902 @item @code{2*x-y} @tab @code{commutative} @tab -
2903 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2904 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2905 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2906 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2910 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2911 @code{TINFO_clifford} for objects with a representation label of zero.
2912 Other representation labels yield a different @code{return_type_tinfo()},
2913 but it's the same for any two objects with the same label. This is also true
2916 A last note: With the exception of matrices, positive integer powers of
2917 non-commutative objects are automatically expanded in GiNaC. For example,
2918 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2919 non-commutative expressions).
2922 @cindex @code{clifford} (class)
2923 @subsection Clifford algebra
2925 @cindex @code{dirac_gamma()}
2926 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2927 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2928 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2929 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2932 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2935 which takes two arguments: the index and a @dfn{representation label} in the
2936 range 0 to 255 which is used to distinguish elements of different Clifford
2937 algebras (this is also called a @dfn{spin line index}). Gammas with different
2938 labels commutate with each other. The dimension of the index can be 4 or (in
2939 the framework of dimensional regularization) any symbolic value. Spinor
2940 indices on Dirac gammas are not supported in GiNaC.
2942 @cindex @code{dirac_ONE()}
2943 The unity element of a Clifford algebra is constructed by
2946 ex dirac_ONE(unsigned char rl = 0);
2949 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2950 multiples of the unity element, even though it's customary to omit it.
2951 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2952 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2953 GiNaC will complain and/or produce incorrect results.
2955 @cindex @code{dirac_gamma5()}
2956 There is a special element @samp{gamma5} that commutates with all other
2957 gammas, has a unit square, and in 4 dimensions equals
2958 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2961 ex dirac_gamma5(unsigned char rl = 0);
2964 @cindex @code{dirac_gammaL()}
2965 @cindex @code{dirac_gammaR()}
2966 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2967 objects, constructed by
2970 ex dirac_gammaL(unsigned char rl = 0);
2971 ex dirac_gammaR(unsigned char rl = 0);
2974 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2975 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2977 @cindex @code{dirac_slash()}
2978 Finally, the function
2981 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2984 creates a term that represents a contraction of @samp{e} with the Dirac
2985 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2986 with a unique index whose dimension is given by the @code{dim} argument).
2987 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2989 In products of dirac gammas, superfluous unity elements are automatically
2990 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2991 and @samp{gammaR} are moved to the front.
2993 The @code{simplify_indexed()} function performs contractions in gamma strings,
2999 symbol a("a"), b("b"), D("D");
3000 varidx mu(symbol("mu"), D);
3001 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3002 * dirac_gamma(mu.toggle_variance());
3004 // -> gamma~mu*a\*gamma.mu
3005 e = e.simplify_indexed();
3008 cout << e.subs(D == 4) << endl;
3014 @cindex @code{dirac_trace()}
3015 To calculate the trace of an expression containing strings of Dirac gammas
3016 you use one of the functions
3019 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
3020 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3021 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3024 These functions take the trace over all gammas in the specified set @code{rls}
3025 or list @code{rll} of representation labels, or the single label @code{rl};
3026 gammas with other labels are left standing. The last argument to
3027 @code{dirac_trace()} is the value to be returned for the trace of the unity
3028 element, which defaults to 4.
3030 The @code{dirac_trace()} function is a linear functional that is equal to the
3031 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3032 functional is not cyclic in @math{D != 4} dimensions when acting on
3033 expressions containing @samp{gamma5}, so it's not a proper trace. This
3034 @samp{gamma5} scheme is described in greater detail in
3035 @cite{The Role of gamma5 in Dimensional Regularization}.
3037 The value of the trace itself is also usually different in 4 and in
3038 @math{D != 4} dimensions:
3043 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3044 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3045 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3046 cout << dirac_trace(e).simplify_indexed() << endl;
3053 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3054 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3055 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3056 cout << dirac_trace(e).simplify_indexed() << endl;
3057 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3061 Here is an example for using @code{dirac_trace()} to compute a value that
3062 appears in the calculation of the one-loop vacuum polarization amplitude in
3067 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3068 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3071 sp.add(l, l, pow(l, 2));
3072 sp.add(l, q, ldotq);
3074 ex e = dirac_gamma(mu) *
3075 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3076 dirac_gamma(mu.toggle_variance()) *
3077 (dirac_slash(l, D) + m * dirac_ONE());
3078 e = dirac_trace(e).simplify_indexed(sp);
3079 e = e.collect(lst(l, ldotq, m));
3081 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3085 The @code{canonicalize_clifford()} function reorders all gamma products that
3086 appear in an expression to a canonical (but not necessarily simple) form.
3087 You can use this to compare two expressions or for further simplifications:
3091 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3092 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3094 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3096 e = canonicalize_clifford(e);
3098 // -> 2*ONE*eta~mu~nu
3103 @cindex @code{color} (class)
3104 @subsection Color algebra
3106 @cindex @code{color_T()}
3107 For computations in quantum chromodynamics, GiNaC implements the base elements
3108 and structure constants of the su(3) Lie algebra (color algebra). The base
3109 elements @math{T_a} are constructed by the function
3112 ex color_T(const ex & a, unsigned char rl = 0);
3115 which takes two arguments: the index and a @dfn{representation label} in the
3116 range 0 to 255 which is used to distinguish elements of different color
3117 algebras. Objects with different labels commutate with each other. The
3118 dimension of the index must be exactly 8 and it should be of class @code{idx},
3121 @cindex @code{color_ONE()}
3122 The unity element of a color algebra is constructed by
3125 ex color_ONE(unsigned char rl = 0);
3128 @strong{Note:} You must always use @code{color_ONE()} when referring to
3129 multiples of the unity element, even though it's customary to omit it.
3130 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3131 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3132 GiNaC may produce incorrect results.
3134 @cindex @code{color_d()}
3135 @cindex @code{color_f()}
3139 ex color_d(const ex & a, const ex & b, const ex & c);
3140 ex color_f(const ex & a, const ex & b, const ex & c);
3143 create the symmetric and antisymmetric structure constants @math{d_abc} and
3144 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3145 and @math{[T_a, T_b] = i f_abc T_c}.
3147 @cindex @code{color_h()}
3148 There's an additional function
3151 ex color_h(const ex & a, const ex & b, const ex & c);
3154 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3156 The function @code{simplify_indexed()} performs some simplifications on
3157 expressions containing color objects:
3162 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3163 k(symbol("k"), 8), l(symbol("l"), 8);
3165 e = color_d(a, b, l) * color_f(a, b, k);
3166 cout << e.simplify_indexed() << endl;
3169 e = color_d(a, b, l) * color_d(a, b, k);
3170 cout << e.simplify_indexed() << endl;
3173 e = color_f(l, a, b) * color_f(a, b, k);
3174 cout << e.simplify_indexed() << endl;
3177 e = color_h(a, b, c) * color_h(a, b, c);
3178 cout << e.simplify_indexed() << endl;
3181 e = color_h(a, b, c) * color_T(b) * color_T(c);
3182 cout << e.simplify_indexed() << endl;
3185 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3186 cout << e.simplify_indexed() << endl;
3189 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3190 cout << e.simplify_indexed() << endl;
3191 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3195 @cindex @code{color_trace()}
3196 To calculate the trace of an expression containing color objects you use one
3200 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3201 ex color_trace(const ex & e, const lst & rll);
3202 ex color_trace(const ex & e, unsigned char rl = 0);
3205 These functions take the trace over all color @samp{T} objects in the
3206 specified set @code{rls} or list @code{rll} of representation labels, or the
3207 single label @code{rl}; @samp{T}s with other labels are left standing. For
3212 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3214 // -> -I*f.a.c.b+d.a.c.b
3219 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3220 @c node-name, next, previous, up
3223 @cindex @code{exhashmap} (class)
3225 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3226 that can be used as a drop-in replacement for the STL
3227 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3228 typically constant-time, element look-up than @code{map<>}.
3230 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3231 following differences:
3235 no @code{lower_bound()} and @code{upper_bound()} methods
3237 no reverse iterators, no @code{rbegin()}/@code{rend()}
3239 no @code{operator<(exhashmap, exhashmap)}
3241 the comparison function object @code{key_compare} is hardcoded to
3244 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3245 initial hash table size (the actual table size after construction may be
3246 larger than the specified value)
3248 the method @code{size_t bucket_count()} returns the current size of the hash
3251 @code{insert()} and @code{erase()} operations invalidate all iterators
3255 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3256 @c node-name, next, previous, up
3257 @chapter Methods and Functions
3260 In this chapter the most important algorithms provided by GiNaC will be
3261 described. Some of them are implemented as functions on expressions,
3262 others are implemented as methods provided by expression objects. If
3263 they are methods, there exists a wrapper function around it, so you can
3264 alternatively call it in a functional way as shown in the simple
3269 cout << "As method: " << sin(1).evalf() << endl;
3270 cout << "As function: " << evalf(sin(1)) << endl;
3274 @cindex @code{subs()}
3275 The general rule is that wherever methods accept one or more parameters
3276 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3277 wrapper accepts is the same but preceded by the object to act on
3278 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3279 most natural one in an OO model but it may lead to confusion for MapleV
3280 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3281 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3282 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3283 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3284 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3285 here. Also, users of MuPAD will in most cases feel more comfortable
3286 with GiNaC's convention. All function wrappers are implemented
3287 as simple inline functions which just call the corresponding method and
3288 are only provided for users uncomfortable with OO who are dead set to
3289 avoid method invocations. Generally, nested function wrappers are much
3290 harder to read than a sequence of methods and should therefore be
3291 avoided if possible. On the other hand, not everything in GiNaC is a
3292 method on class @code{ex} and sometimes calling a function cannot be
3296 * Information About Expressions::
3297 * Numerical Evaluation::
3298 * Substituting Expressions::
3299 * Pattern Matching and Advanced Substitutions::
3300 * Applying a Function on Subexpressions::
3301 * Visitors and Tree Traversal::
3302 * Polynomial Arithmetic:: Working with polynomials.
3303 * Rational Expressions:: Working with rational functions.
3304 * Symbolic Differentiation::
3305 * Series Expansion:: Taylor and Laurent expansion.
3307 * Built-in Functions:: List of predefined mathematical functions.
3308 * Multiple polylogarithms::
3309 * Complex Conjugation::
3310 * Built-in Functions:: List of predefined mathematical functions.
3311 * Solving Linear Systems of Equations::
3312 * Input/Output:: Input and output of expressions.
3316 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3317 @c node-name, next, previous, up
3318 @section Getting information about expressions
3320 @subsection Checking expression types
3321 @cindex @code{is_a<@dots{}>()}
3322 @cindex @code{is_exactly_a<@dots{}>()}
3323 @cindex @code{ex_to<@dots{}>()}
3324 @cindex Converting @code{ex} to other classes
3325 @cindex @code{info()}
3326 @cindex @code{return_type()}
3327 @cindex @code{return_type_tinfo()}
3329 Sometimes it's useful to check whether a given expression is a plain number,
3330 a sum, a polynomial with integer coefficients, or of some other specific type.
3331 GiNaC provides a couple of functions for this:
3334 bool is_a<T>(const ex & e);
3335 bool is_exactly_a<T>(const ex & e);
3336 bool ex::info(unsigned flag);
3337 unsigned ex::return_type() const;
3338 unsigned ex::return_type_tinfo() const;
3341 When the test made by @code{is_a<T>()} returns true, it is safe to call
3342 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3343 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3344 example, assuming @code{e} is an @code{ex}:
3349 if (is_a<numeric>(e))
3350 numeric n = ex_to<numeric>(e);
3355 @code{is_a<T>(e)} allows you to check whether the top-level object of
3356 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3357 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3358 e.g., for checking whether an expression is a number, a sum, or a product:
3365 is_a<numeric>(e1); // true
3366 is_a<numeric>(e2); // false
3367 is_a<add>(e1); // false
3368 is_a<add>(e2); // true
3369 is_a<mul>(e1); // false
3370 is_a<mul>(e2); // false
3374 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3375 top-level object of an expression @samp{e} is an instance of the GiNaC
3376 class @samp{T}, not including parent classes.
3378 The @code{info()} method is used for checking certain attributes of
3379 expressions. The possible values for the @code{flag} argument are defined
3380 in @file{ginac/flags.h}, the most important being explained in the following
3384 @multitable @columnfractions .30 .70
3385 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3386 @item @code{numeric}
3387 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3389 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3390 @item @code{rational}
3391 @tab @dots{}an exact rational number (integers are rational, too)
3392 @item @code{integer}
3393 @tab @dots{}a (non-complex) integer
3394 @item @code{crational}
3395 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3396 @item @code{cinteger}
3397 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3398 @item @code{positive}
3399 @tab @dots{}not complex and greater than 0
3400 @item @code{negative}
3401 @tab @dots{}not complex and less than 0
3402 @item @code{nonnegative}
3403 @tab @dots{}not complex and greater than or equal to 0
3405 @tab @dots{}an integer greater than 0
3407 @tab @dots{}an integer less than 0
3408 @item @code{nonnegint}
3409 @tab @dots{}an integer greater than or equal to 0
3411 @tab @dots{}an even integer
3413 @tab @dots{}an odd integer
3415 @tab @dots{}a prime integer (probabilistic primality test)
3416 @item @code{relation}
3417 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3418 @item @code{relation_equal}
3419 @tab @dots{}a @code{==} relation
3420 @item @code{relation_not_equal}
3421 @tab @dots{}a @code{!=} relation
3422 @item @code{relation_less}
3423 @tab @dots{}a @code{<} relation
3424 @item @code{relation_less_or_equal}
3425 @tab @dots{}a @code{<=} relation
3426 @item @code{relation_greater}
3427 @tab @dots{}a @code{>} relation
3428 @item @code{relation_greater_or_equal}
3429 @tab @dots{}a @code{>=} relation
3431 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3433 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3434 @item @code{polynomial}
3435 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3436 @item @code{integer_polynomial}
3437 @tab @dots{}a polynomial with (non-complex) integer coefficients
3438 @item @code{cinteger_polynomial}
3439 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3440 @item @code{rational_polynomial}
3441 @tab @dots{}a polynomial with (non-complex) rational coefficients
3442 @item @code{crational_polynomial}
3443 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3444 @item @code{rational_function}
3445 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3446 @item @code{algebraic}
3447 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3451 To determine whether an expression is commutative or non-commutative and if
3452 so, with which other expressions it would commutate, you use the methods
3453 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3454 for an explanation of these.
3457 @subsection Accessing subexpressions
3460 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3461 @code{function}, act as containers for subexpressions. For example, the
3462 subexpressions of a sum (an @code{add} object) are the individual terms,
3463 and the subexpressions of a @code{function} are the function's arguments.
3465 @cindex @code{nops()}
3467 GiNaC provides several ways of accessing subexpressions. The first way is to
3472 ex ex::op(size_t i);
3475 @code{nops()} determines the number of subexpressions (operands) contained
3476 in the expression, while @code{op(i)} returns the @code{i}-th
3477 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3478 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3479 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3480 @math{i>0} are the indices.
3483 @cindex @code{const_iterator}
3484 The second way to access subexpressions is via the STL-style random-access
3485 iterator class @code{const_iterator} and the methods
3488 const_iterator ex::begin();
3489 const_iterator ex::end();
3492 @code{begin()} returns an iterator referring to the first subexpression;
3493 @code{end()} returns an iterator which is one-past the last subexpression.
3494 If the expression has no subexpressions, then @code{begin() == end()}. These
3495 iterators can also be used in conjunction with non-modifying STL algorithms.
3497 Here is an example that (non-recursively) prints the subexpressions of a
3498 given expression in three different ways:
3505 for (size_t i = 0; i != e.nops(); ++i)
3506 cout << e.op(i) << endl;
3509 for (const_iterator i = e.begin(); i != e.end(); ++i)
3512 // with iterators and STL copy()
3513 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3517 @cindex @code{const_preorder_iterator}
3518 @cindex @code{const_postorder_iterator}
3519 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3520 expression's immediate children. GiNaC provides two additional iterator
3521 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3522 that iterate over all objects in an expression tree, in preorder or postorder,
3523 respectively. They are STL-style forward iterators, and are created with the
3527 const_preorder_iterator ex::preorder_begin();
3528 const_preorder_iterator ex::preorder_end();
3529 const_postorder_iterator ex::postorder_begin();
3530 const_postorder_iterator ex::postorder_end();
3533 The following example illustrates the differences between
3534 @code{const_iterator}, @code{const_preorder_iterator}, and
3535 @code{const_postorder_iterator}:
3539 symbol A("A"), B("B"), C("C");
3540 ex e = lst(lst(A, B), C);
3542 std::copy(e.begin(), e.end(),
3543 std::ostream_iterator<ex>(cout, "\n"));
3547 std::copy(e.preorder_begin(), e.preorder_end(),
3548 std::ostream_iterator<ex>(cout, "\n"));
3555 std::copy(e.postorder_begin(), e.postorder_end(),
3556 std::ostream_iterator<ex>(cout, "\n"));
3565 @cindex @code{relational} (class)
3566 Finally, the left-hand side and right-hand side expressions of objects of
3567 class @code{relational} (and only of these) can also be accessed with the
3576 @subsection Comparing expressions
3577 @cindex @code{is_equal()}
3578 @cindex @code{is_zero()}
3580 Expressions can be compared with the usual C++ relational operators like
3581 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3582 the result is usually not determinable and the result will be @code{false},
3583 except in the case of the @code{!=} operator. You should also be aware that
3584 GiNaC will only do the most trivial test for equality (subtracting both
3585 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3588 Actually, if you construct an expression like @code{a == b}, this will be
3589 represented by an object of the @code{relational} class (@pxref{Relations})
3590 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3592 There are also two methods
3595 bool ex::is_equal(const ex & other);
3599 for checking whether one expression is equal to another, or equal to zero,
3603 @subsection Ordering expressions
3604 @cindex @code{ex_is_less} (class)
3605 @cindex @code{ex_is_equal} (class)
3606 @cindex @code{compare()}
3608 Sometimes it is necessary to establish a mathematically well-defined ordering
3609 on a set of arbitrary expressions, for example to use expressions as keys
3610 in a @code{std::map<>} container, or to bring a vector of expressions into
3611 a canonical order (which is done internally by GiNaC for sums and products).
3613 The operators @code{<}, @code{>} etc. described in the last section cannot
3614 be used for this, as they don't implement an ordering relation in the
3615 mathematical sense. In particular, they are not guaranteed to be
3616 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3617 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3620 By default, STL classes and algorithms use the @code{<} and @code{==}
3621 operators to compare objects, which are unsuitable for expressions, but GiNaC
3622 provides two functors that can be supplied as proper binary comparison
3623 predicates to the STL:
3626 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3628 bool operator()(const ex &lh, const ex &rh) const;
3631 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3633 bool operator()(const ex &lh, const ex &rh) const;
3637 For example, to define a @code{map} that maps expressions to strings you
3641 std::map<ex, std::string, ex_is_less> myMap;
3644 Omitting the @code{ex_is_less} template parameter will introduce spurious
3645 bugs because the map operates improperly.
3647 Other examples for the use of the functors:
3655 std::sort(v.begin(), v.end(), ex_is_less());
3657 // count the number of expressions equal to '1'
3658 unsigned num_ones = std::count_if(v.begin(), v.end(),
3659 std::bind2nd(ex_is_equal(), 1));
3662 The implementation of @code{ex_is_less} uses the member function
3665 int ex::compare(const ex & other) const;
3668 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3669 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3673 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3674 @c node-name, next, previous, up
3675 @section Numerical Evaluation
3676 @cindex @code{evalf()}
3678 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3679 To evaluate them using floating-point arithmetic you need to call
3682 ex ex::evalf(int level = 0) const;
3685 @cindex @code{Digits}
3686 The accuracy of the evaluation is controlled by the global object @code{Digits}
3687 which can be assigned an integer value. The default value of @code{Digits}
3688 is 17. @xref{Numbers}, for more information and examples.
3690 To evaluate an expression to a @code{double} floating-point number you can
3691 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3695 // Approximate sin(x/Pi)
3697 ex e = series(sin(x/Pi), x == 0, 6);
3699 // Evaluate numerically at x=0.1
3700 ex f = evalf(e.subs(x == 0.1));
3702 // ex_to<numeric> is an unsafe cast, so check the type first
3703 if (is_a<numeric>(f)) @{
3704 double d = ex_to<numeric>(f).to_double();
3713 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3714 @c node-name, next, previous, up
3715 @section Substituting expressions
3716 @cindex @code{subs()}
3718 Algebraic objects inside expressions can be replaced with arbitrary
3719 expressions via the @code{.subs()} method:
3722 ex ex::subs(const ex & e, unsigned options = 0);
3723 ex ex::subs(const exmap & m, unsigned options = 0);
3724 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3727 In the first form, @code{subs()} accepts a relational of the form
3728 @samp{object == expression} or a @code{lst} of such relationals:
3732 symbol x("x"), y("y");
3734 ex e1 = 2*x^2-4*x+3;
3735 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3739 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3744 If you specify multiple substitutions, they are performed in parallel, so e.g.
3745 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3747 The second form of @code{subs()} takes an @code{exmap} object which is a
3748 pair associative container that maps expressions to expressions (currently
3749 implemented as a @code{std::map}). This is the most efficient one of the
3750 three @code{subs()} forms and should be used when the number of objects to
3751 be substituted is large or unknown.
3753 Using this form, the second example from above would look like this:
3757 symbol x("x"), y("y");
3763 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3767 The third form of @code{subs()} takes two lists, one for the objects to be
3768 replaced and one for the expressions to be substituted (both lists must
3769 contain the same number of elements). Using this form, you would write
3773 symbol x("x"), y("y");
3776 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3780 The optional last argument to @code{subs()} is a combination of
3781 @code{subs_options} flags. There are two options available:
3782 @code{subs_options::no_pattern} disables pattern matching, which makes
3783 large @code{subs()} operations significantly faster if you are not using
3784 patterns. The second option, @code{subs_options::algebraic} enables
3785 algebraic substitutions in products and powers.
3786 @ref{Pattern Matching and Advanced Substitutions}, for more information
3787 about patterns and algebraic substitutions.
3789 @code{subs()} performs syntactic substitution of any complete algebraic
3790 object; it does not try to match sub-expressions as is demonstrated by the
3795 symbol x("x"), y("y"), z("z");
3797 ex e1 = pow(x+y, 2);
3798 cout << e1.subs(x+y == 4) << endl;
3801 ex e2 = sin(x)*sin(y)*cos(x);
3802 cout << e2.subs(sin(x) == cos(x)) << endl;
3803 // -> cos(x)^2*sin(y)
3806 cout << e3.subs(x+y == 4) << endl;
3808 // (and not 4+z as one might expect)
3812 A more powerful form of substitution using wildcards is described in the
3816 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3817 @c node-name, next, previous, up
3818 @section Pattern matching and advanced substitutions
3819 @cindex @code{wildcard} (class)
3820 @cindex Pattern matching
3822 GiNaC allows the use of patterns for checking whether an expression is of a
3823 certain form or contains subexpressions of a certain form, and for
3824 substituting expressions in a more general way.
3826 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3827 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3828 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3829 an unsigned integer number to allow having multiple different wildcards in a
3830 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3831 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3835 ex wild(unsigned label = 0);
3838 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3841 Some examples for patterns:
3843 @multitable @columnfractions .5 .5
3844 @item @strong{Constructed as} @tab @strong{Output as}
3845 @item @code{wild()} @tab @samp{$0}
3846 @item @code{pow(x,wild())} @tab @samp{x^$0}
3847 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3848 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3854 @item Wildcards behave like symbols and are subject to the same algebraic
3855 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3856 @item As shown in the last example, to use wildcards for indices you have to
3857 use them as the value of an @code{idx} object. This is because indices must
3858 always be of class @code{idx} (or a subclass).
3859 @item Wildcards only represent expressions or subexpressions. It is not
3860 possible to use them as placeholders for other properties like index
3861 dimension or variance, representation labels, symmetry of indexed objects
3863 @item Because wildcards are commutative, it is not possible to use wildcards
3864 as part of noncommutative products.
3865 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3866 are also valid patterns.
3869 @subsection Matching expressions
3870 @cindex @code{match()}
3871 The most basic application of patterns is to check whether an expression
3872 matches a given pattern. This is done by the function
3875 bool ex::match(const ex & pattern);
3876 bool ex::match(const ex & pattern, lst & repls);
3879 This function returns @code{true} when the expression matches the pattern
3880 and @code{false} if it doesn't. If used in the second form, the actual
3881 subexpressions matched by the wildcards get returned in the @code{repls}
3882 object as a list of relations of the form @samp{wildcard == expression}.
3883 If @code{match()} returns false, the state of @code{repls} is undefined.
3884 For reproducible results, the list should be empty when passed to
3885 @code{match()}, but it is also possible to find similarities in multiple
3886 expressions by passing in the result of a previous match.
3888 The matching algorithm works as follows:
3891 @item A single wildcard matches any expression. If one wildcard appears
3892 multiple times in a pattern, it must match the same expression in all
3893 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3894 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3895 @item If the expression is not of the same class as the pattern, the match
3896 fails (i.e. a sum only matches a sum, a function only matches a function,
3898 @item If the pattern is a function, it only matches the same function
3899 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3900 @item Except for sums and products, the match fails if the number of
3901 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3903 @item If there are no subexpressions, the expressions and the pattern must
3904 be equal (in the sense of @code{is_equal()}).
3905 @item Except for sums and products, each subexpression (@code{op()}) must
3906 match the corresponding subexpression of the pattern.
3909 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3910 account for their commutativity and associativity:
3913 @item If the pattern contains a term or factor that is a single wildcard,
3914 this one is used as the @dfn{global wildcard}. If there is more than one
3915 such wildcard, one of them is chosen as the global wildcard in a random
3917 @item Every term/factor of the pattern, except the global wildcard, is
3918 matched against every term of the expression in sequence. If no match is
3919 found, the whole match fails. Terms that did match are not considered in
3921 @item If there are no unmatched terms left, the match succeeds. Otherwise
3922 the match fails unless there is a global wildcard in the pattern, in
3923 which case this wildcard matches the remaining terms.
3926 In general, having more than one single wildcard as a term of a sum or a
3927 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3930 Here are some examples in @command{ginsh} to demonstrate how it works (the
3931 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3932 match fails, and the list of wildcard replacements otherwise):
3935 > match((x+y)^a,(x+y)^a);
3937 > match((x+y)^a,(x+y)^b);
3939 > match((x+y)^a,$1^$2);
3941 > match((x+y)^a,$1^$1);
3943 > match((x+y)^(x+y),$1^$1);
3945 > match((x+y)^(x+y),$1^$2);
3947 > match((a+b)*(a+c),($1+b)*($1+c));
3949 > match((a+b)*(a+c),(a+$1)*(a+$2));
3951 (Unpredictable. The result might also be [$1==c,$2==b].)
3952 > match((a+b)*(a+c),($1+$2)*($1+$3));
3953 (The result is undefined. Due to the sequential nature of the algorithm
3954 and the re-ordering of terms in GiNaC, the match for the first factor
3955 may be @{$1==a,$2==b@} in which case the match for the second factor
3956 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3958 > match(a*(x+y)+a*z+b,a*$1+$2);
3959 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3960 @{$1=x+y,$2=a*z+b@}.)
3961 > match(a+b+c+d+e+f,c);
3963 > match(a+b+c+d+e+f,c+$0);
3965 > match(a+b+c+d+e+f,c+e+$0);
3967 > match(a+b,a+b+$0);
3969 > match(a*b^2,a^$1*b^$2);
3971 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3972 even though a==a^1.)
3973 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3975 > match(atan2(y,x^2),atan2(y,$0));
3979 @subsection Matching parts of expressions
3980 @cindex @code{has()}
3981 A more general way to look for patterns in expressions is provided by the
3985 bool ex::has(const ex & pattern);
3988 This function checks whether a pattern is matched by an expression itself or
3989 by any of its subexpressions.
3991 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3992 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3995 > has(x*sin(x+y+2*a),y);
3997 > has(x*sin(x+y+2*a),x+y);
3999 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4000 has the subexpressions "x", "y" and "2*a".)
4001 > has(x*sin(x+y+2*a),x+y+$1);
4003 (But this is possible.)
4004 > has(x*sin(2*(x+y)+2*a),x+y);
4006 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4007 which "x+y" is not a subexpression.)
4010 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4012 > has(4*x^2-x+3,$1*x);
4014 > has(4*x^2+x+3,$1*x);
4016 (Another possible pitfall. The first expression matches because the term
4017 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4018 contains a linear term you should use the coeff() function instead.)
4021 @cindex @code{find()}
4025 bool ex::find(const ex & pattern, lst & found);
4028 works a bit like @code{has()} but it doesn't stop upon finding the first
4029 match. Instead, it appends all found matches to the specified list. If there
4030 are multiple occurrences of the same expression, it is entered only once to
4031 the list. @code{find()} returns false if no matches were found (in
4032 @command{ginsh}, it returns an empty list):
4035 > find(1+x+x^2+x^3,x);
4037 > find(1+x+x^2+x^3,y);
4039 > find(1+x+x^2+x^3,x^$1);
4041 (Note the absence of "x".)
4042 > expand((sin(x)+sin(y))*(a+b));
4043 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4048 @subsection Substituting expressions
4049 @cindex @code{subs()}
4050 Probably the most useful application of patterns is to use them for
4051 substituting expressions with the @code{subs()} method. Wildcards can be
4052 used in the search patterns as well as in the replacement expressions, where
4053 they get replaced by the expressions matched by them. @code{subs()} doesn't
4054 know anything about algebra; it performs purely syntactic substitutions.
4059 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4061 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4063 > subs((a+b+c)^2,a+b==x);
4065 > subs((a+b+c)^2,a+b+$1==x+$1);
4067 > subs(a+2*b,a+b==x);
4069 > subs(4*x^3-2*x^2+5*x-1,x==a);
4071 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4073 > subs(sin(1+sin(x)),sin($1)==cos($1));
4075 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4079 The last example would be written in C++ in this way:
4083 symbol a("a"), b("b"), x("x"), y("y");
4084 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4085 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4086 cout << e.expand() << endl;
4091 @subsection Algebraic substitutions
4092 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4093 enables smarter, algebraic substitutions in products and powers. If you want
4094 to substitute some factors of a product, you only need to list these factors
4095 in your pattern. Furthermore, if an (integer) power of some expression occurs
4096 in your pattern and in the expression that you want the substitution to occur
4097 in, it can be substituted as many times as possible, without getting negative
4100 An example clarifies it all (hopefully):
4103 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4104 subs_options::algebraic) << endl;
4105 // --> (y+x)^6+b^6+a^6
4107 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4109 // Powers and products are smart, but addition is just the same.
4111 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4114 // As I said: addition is just the same.
4116 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4117 // --> x^3*b*a^2+2*b
4119 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4121 // --> 2*b+x^3*b^(-1)*a^(-2)
4123 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4124 // --> -1-2*a^2+4*a^3+5*a
4126 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4127 subs_options::algebraic) << endl;
4128 // --> -1+5*x+4*x^3-2*x^2
4129 // You should not really need this kind of patterns very often now.
4130 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4132 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4133 subs_options::algebraic) << endl;
4134 // --> cos(1+cos(x))
4136 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4137 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4138 subs_options::algebraic)) << endl;
4143 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4144 @c node-name, next, previous, up
4145 @section Applying a Function on Subexpressions
4146 @cindex tree traversal
4147 @cindex @code{map()}
4149 Sometimes you may want to perform an operation on specific parts of an
4150 expression while leaving the general structure of it intact. An example
4151 of this would be a matrix trace operation: the trace of a sum is the sum
4152 of the traces of the individual terms. That is, the trace should @dfn{map}
4153 on the sum, by applying itself to each of the sum's operands. It is possible
4154 to do this manually which usually results in code like this:
4159 if (is_a<matrix>(e))
4160 return ex_to<matrix>(e).trace();
4161 else if (is_a<add>(e)) @{
4163 for (size_t i=0; i<e.nops(); i++)
4164 sum += calc_trace(e.op(i));
4166 @} else if (is_a<mul>)(e)) @{
4174 This is, however, slightly inefficient (if the sum is very large it can take
4175 a long time to add the terms one-by-one), and its applicability is limited to
4176 a rather small class of expressions. If @code{calc_trace()} is called with
4177 a relation or a list as its argument, you will probably want the trace to
4178 be taken on both sides of the relation or of all elements of the list.
4180 GiNaC offers the @code{map()} method to aid in the implementation of such
4184 ex ex::map(map_function & f) const;
4185 ex ex::map(ex (*f)(const ex & e)) const;
4188 In the first (preferred) form, @code{map()} takes a function object that
4189 is subclassed from the @code{map_function} class. In the second form, it
4190 takes a pointer to a function that accepts and returns an expression.
4191 @code{map()} constructs a new expression of the same type, applying the
4192 specified function on all subexpressions (in the sense of @code{op()}),
4195 The use of a function object makes it possible to supply more arguments to
4196 the function that is being mapped, or to keep local state information.
4197 The @code{map_function} class declares a virtual function call operator
4198 that you can overload. Here is a sample implementation of @code{calc_trace()}
4199 that uses @code{map()} in a recursive fashion:
4202 struct calc_trace : public map_function @{
4203 ex operator()(const ex &e)
4205 if (is_a<matrix>(e))
4206 return ex_to<matrix>(e).trace();
4207 else if (is_a<mul>(e)) @{
4210 return e.map(*this);
4215 This function object could then be used like this:
4219 ex M = ... // expression with matrices
4220 calc_trace do_trace;
4221 ex tr = do_trace(M);
4225 Here is another example for you to meditate over. It removes quadratic
4226 terms in a variable from an expanded polynomial:
4229 struct map_rem_quad : public map_function @{
4231 map_rem_quad(const ex & var_) : var(var_) @{@}
4233 ex operator()(const ex & e)
4235 if (is_a<add>(e) || is_a<mul>(e))
4236 return e.map(*this);
4237 else if (is_a<power>(e) &&
4238 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4248 symbol x("x"), y("y");
4251 for (int i=0; i<8; i++)
4252 e += pow(x, i) * pow(y, 8-i) * (i+1);
4254 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
4256 map_rem_quad rem_quad(x);
4257 cout << rem_quad(e) << endl;
4258 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4262 @command{ginsh} offers a slightly different implementation of @code{map()}
4263 that allows applying algebraic functions to operands. The second argument
4264 to @code{map()} is an expression containing the wildcard @samp{$0} which
4265 acts as the placeholder for the operands:
4270 > map(a+2*b,sin($0));
4272 > map(@{a,b,c@},$0^2+$0);
4273 @{a^2+a,b^2+b,c^2+c@}
4276 Note that it is only possible to use algebraic functions in the second
4277 argument. You can not use functions like @samp{diff()}, @samp{op()},
4278 @samp{subs()} etc. because these are evaluated immediately:
4281 > map(@{a,b,c@},diff($0,a));
4283 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4284 to "map(@{a,b,c@},0)".
4288 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4289 @c node-name, next, previous, up
4290 @section Visitors and Tree Traversal
4291 @cindex tree traversal
4292 @cindex @code{visitor} (class)
4293 @cindex @code{accept()}
4294 @cindex @code{visit()}
4295 @cindex @code{traverse()}
4296 @cindex @code{traverse_preorder()}
4297 @cindex @code{traverse_postorder()}
4299 Suppose that you need a function that returns a list of all indices appearing
4300 in an arbitrary expression. The indices can have any dimension, and for
4301 indices with variance you always want the covariant version returned.
4303 You can't use @code{get_free_indices()} because you also want to include
4304 dummy indices in the list, and you can't use @code{find()} as it needs
4305 specific index dimensions (and it would require two passes: one for indices
4306 with variance, one for plain ones).
4308 The obvious solution to this problem is a tree traversal with a type switch,
4309 such as the following:
4312 void gather_indices_helper(const ex & e, lst & l)
4314 if (is_a<varidx>(e)) @{
4315 const varidx & vi = ex_to<varidx>(e);
4316 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4317 @} else if (is_a<idx>(e)) @{
4320 size_t n = e.nops();
4321 for (size_t i = 0; i < n; ++i)
4322 gather_indices_helper(e.op(i), l);
4326 lst gather_indices(const ex & e)
4329 gather_indices_helper(e, l);
4336 This works fine but fans of object-oriented programming will feel
4337 uncomfortable with the type switch. One reason is that there is a possibility
4338 for subtle bugs regarding derived classes. If we had, for example, written
4341 if (is_a<idx>(e)) @{
4343 @} else if (is_a<varidx>(e)) @{
4347 in @code{gather_indices_helper}, the code wouldn't have worked because the
4348 first line "absorbs" all classes derived from @code{idx}, including
4349 @code{varidx}, so the special case for @code{varidx} would never have been
4352 Also, for a large number of classes, a type switch like the above can get
4353 unwieldy and inefficient (it's a linear search, after all).
4354 @code{gather_indices_helper} only checks for two classes, but if you had to
4355 write a function that required a different implementation for nearly
4356 every GiNaC class, the result would be very hard to maintain and extend.
4358 The cleanest approach to the problem would be to add a new virtual function
4359 to GiNaC's class hierarchy. In our example, there would be specializations
4360 for @code{idx} and @code{varidx} while the default implementation in
4361 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4362 impossible to add virtual member functions to existing classes without
4363 changing their source and recompiling everything. GiNaC comes with source,
4364 so you could actually do this, but for a small algorithm like the one
4365 presented this would be impractical.
4367 One solution to this dilemma is the @dfn{Visitor} design pattern,
4368 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4369 variation, described in detail in
4370 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4371 virtual functions to the class hierarchy to implement operations, GiNaC
4372 provides a single "bouncing" method @code{accept()} that takes an instance
4373 of a special @code{visitor} class and redirects execution to the one
4374 @code{visit()} virtual function of the visitor that matches the type of
4375 object that @code{accept()} was being invoked on.
4377 Visitors in GiNaC must derive from the global @code{visitor} class as well
4378 as from the class @code{T::visitor} of each class @code{T} they want to
4379 visit, and implement the member functions @code{void visit(const T &)} for
4385 void ex::accept(visitor & v) const;
4388 will then dispatch to the correct @code{visit()} member function of the
4389 specified visitor @code{v} for the type of GiNaC object at the root of the
4390 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4392 Here is an example of a visitor:
4396 : public visitor, // this is required
4397 public add::visitor, // visit add objects
4398 public numeric::visitor, // visit numeric objects
4399 public basic::visitor // visit basic objects
4401 void visit(const add & x)
4402 @{ cout << "called with an add object" << endl; @}
4404 void visit(const numeric & x)
4405 @{ cout << "called with a numeric object" << endl; @}
4407 void visit(const basic & x)
4408 @{ cout << "called with a basic object" << endl; @}
4412 which can be used as follows:
4423 // prints "called with a numeric object"
4425 // prints "called with an add object"
4427 // prints "called with a basic object"
4431 The @code{visit(const basic &)} method gets called for all objects that are
4432 not @code{numeric} or @code{add} and acts as an (optional) default.
4434 From a conceptual point of view, the @code{visit()} methods of the visitor
4435 behave like a newly added virtual function of the visited hierarchy.
4436 In addition, visitors can store state in member variables, and they can
4437 be extended by deriving a new visitor from an existing one, thus building
4438 hierarchies of visitors.
4440 We can now rewrite our index example from above with a visitor:
4443 class gather_indices_visitor
4444 : public visitor, public idx::visitor, public varidx::visitor
4448 void visit(const idx & i)
4453 void visit(const varidx & vi)
4455 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4459 const lst & get_result() // utility function
4468 What's missing is the tree traversal. We could implement it in
4469 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4472 void ex::traverse_preorder(visitor & v) const;
4473 void ex::traverse_postorder(visitor & v) const;
4474 void ex::traverse(visitor & v) const;
4477 @code{traverse_preorder()} visits a node @emph{before} visiting its
4478 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4479 visiting its subexpressions. @code{traverse()} is a synonym for
4480 @code{traverse_preorder()}.
4482 Here is a new implementation of @code{gather_indices()} that uses the visitor
4483 and @code{traverse()}:
4486 lst gather_indices(const ex & e)
4488 gather_indices_visitor v;
4490 return v.get_result();
4494 Alternatively, you could use pre- or postorder iterators for the tree
4498 lst gather_indices(const ex & e)
4500 gather_indices_visitor v;
4501 for (const_preorder_iterator i = e.preorder_begin();
4502 i != e.preorder_end(); ++i) @{
4505 return v.get_result();
4510 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4511 @c node-name, next, previous, up
4512 @section Polynomial arithmetic
4514 @subsection Expanding and collecting
4515 @cindex @code{expand()}
4516 @cindex @code{collect()}
4517 @cindex @code{collect_common_factors()}
4519 A polynomial in one or more variables has many equivalent
4520 representations. Some useful ones serve a specific purpose. Consider
4521 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4522 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4523 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4524 representations are the recursive ones where one collects for exponents
4525 in one of the three variable. Since the factors are themselves
4526 polynomials in the remaining two variables the procedure can be
4527 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4528 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4531 To bring an expression into expanded form, its method
4534 ex ex::expand(unsigned options = 0);
4537 may be called. In our example above, this corresponds to @math{4*x*y +
4538 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4539 GiNaC is not easy to guess you should be prepared to see different
4540 orderings of terms in such sums!
4542 Another useful representation of multivariate polynomials is as a
4543 univariate polynomial in one of the variables with the coefficients
4544 being polynomials in the remaining variables. The method
4545 @code{collect()} accomplishes this task:
4548 ex ex::collect(const ex & s, bool distributed = false);
4551 The first argument to @code{collect()} can also be a list of objects in which
4552 case the result is either a recursively collected polynomial, or a polynomial
4553 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4554 by the @code{distributed} flag.
4556 Note that the original polynomial needs to be in expanded form (for the
4557 variables concerned) in order for @code{collect()} to be able to find the
4558 coefficients properly.
4560 The following @command{ginsh} transcript shows an application of @code{collect()}
4561 together with @code{find()}:
4564 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4565 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4566 > collect(a,@{p,q@});
4567 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4568 > collect(a,find(a,sin($1)));
4569 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4570 > collect(a,@{find(a,sin($1)),p,q@});
4571 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4572 > collect(a,@{find(a,sin($1)),d@});
4573 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4576 Polynomials can often be brought into a more compact form by collecting
4577 common factors from the terms of sums. This is accomplished by the function
4580 ex collect_common_factors(const ex & e);
4583 This function doesn't perform a full factorization but only looks for
4584 factors which are already explicitly present:
4587 > collect_common_factors(a*x+a*y);
4589 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4591 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4592 (c+a)*a*(x*y+y^2+x)*b
4595 @subsection Degree and coefficients
4596 @cindex @code{degree()}
4597 @cindex @code{ldegree()}
4598 @cindex @code{coeff()}
4600 The degree and low degree of a polynomial can be obtained using the two
4604 int ex::degree(const ex & s);
4605 int ex::ldegree(const ex & s);
4608 which also work reliably on non-expanded input polynomials (they even work
4609 on rational functions, returning the asymptotic degree). By definition, the
4610 degree of zero is zero. To extract a coefficient with a certain power from
4611 an expanded polynomial you use
4614 ex ex::coeff(const ex & s, int n);
4617 You can also obtain the leading and trailing coefficients with the methods
4620 ex ex::lcoeff(const ex & s);
4621 ex ex::tcoeff(const ex & s);
4624 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4627 An application is illustrated in the next example, where a multivariate
4628 polynomial is analyzed:
4632 symbol x("x"), y("y");
4633 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4634 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4635 ex Poly = PolyInp.expand();
4637 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4638 cout << "The x^" << i << "-coefficient is "
4639 << Poly.coeff(x,i) << endl;
4641 cout << "As polynomial in y: "
4642 << Poly.collect(y) << endl;
4646 When run, it returns an output in the following fashion:
4649 The x^0-coefficient is y^2+11*y
4650 The x^1-coefficient is 5*y^2-2*y
4651 The x^2-coefficient is -1
4652 The x^3-coefficient is 4*y
4653 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4656 As always, the exact output may vary between different versions of GiNaC
4657 or even from run to run since the internal canonical ordering is not
4658 within the user's sphere of influence.
4660 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4661 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4662 with non-polynomial expressions as they not only work with symbols but with
4663 constants, functions and indexed objects as well:
4667 symbol a("a"), b("b"), c("c"), x("x");
4668 idx i(symbol("i"), 3);
4670 ex e = pow(sin(x) - cos(x), 4);
4671 cout << e.degree(cos(x)) << endl;
4673 cout << e.expand().coeff(sin(x), 3) << endl;
4676 e = indexed(a+b, i) * indexed(b+c, i);
4677 e = e.expand(expand_options::expand_indexed);
4678 cout << e.collect(indexed(b, i)) << endl;
4679 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4684 @subsection Polynomial division
4685 @cindex polynomial division
4688 @cindex pseudo-remainder
4689 @cindex @code{quo()}
4690 @cindex @code{rem()}
4691 @cindex @code{prem()}
4692 @cindex @code{divide()}
4697 ex quo(const ex & a, const ex & b, const ex & x);
4698 ex rem(const ex & a, const ex & b, const ex & x);
4701 compute the quotient and remainder of univariate polynomials in the variable
4702 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4704 The additional function
4707 ex prem(const ex & a, const ex & b, const ex & x);
4710 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4711 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4713 Exact division of multivariate polynomials is performed by the function
4716 bool divide(const ex & a, const ex & b, ex & q);
4719 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4720 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4721 in which case the value of @code{q} is undefined.
4724 @subsection Unit, content and primitive part
4725 @cindex @code{unit()}
4726 @cindex @code{content()}
4727 @cindex @code{primpart()}
4732 ex ex::unit(const ex & x);
4733 ex ex::content(const ex & x);
4734 ex ex::primpart(const ex & x);
4737 return the unit part, content part, and primitive polynomial of a multivariate
4738 polynomial with respect to the variable @samp{x} (the unit part being the sign
4739 of the leading coefficient, the content part being the GCD of the coefficients,
4740 and the primitive polynomial being the input polynomial divided by the unit and
4741 content parts). The product of unit, content, and primitive part is the
4742 original polynomial.
4745 @subsection GCD, LCM and resultant
4748 @cindex @code{gcd()}
4749 @cindex @code{lcm()}
4751 The functions for polynomial greatest common divisor and least common
4752 multiple have the synopsis
4755 ex gcd(const ex & a, const ex & b);
4756 ex lcm(const ex & a, const ex & b);
4759 The functions @code{gcd()} and @code{lcm()} accept two expressions
4760 @code{a} and @code{b} as arguments and return a new expression, their
4761 greatest common divisor or least common multiple, respectively. If the
4762 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4763 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4766 #include <ginac/ginac.h>
4767 using namespace GiNaC;
4771 symbol x("x"), y("y"), z("z");
4772 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4773 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4775 ex P_gcd = gcd(P_a, P_b);
4777 ex P_lcm = lcm(P_a, P_b);
4778 // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
4783 @cindex @code{resultant()}
4785 The resultant of two expressions only makes sense with polynomials.
4786 It is always computed with respect to a specific symbol within the
4787 expressions. The function has the interface
4790 ex resultant(const ex & a, const ex & b, const ex & s);
4793 Resultants are symmetric in @code{a} and @code{b}. The following example
4794 computes the resultant of two expressions with respect to @code{x} and
4795 @code{y}, respectively:
4798 #include <ginac/ginac.h>
4799 using namespace GiNaC;
4803 symbol x("x"), y("y");
4805 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
4808 r = resultant(e1, e2, x);
4810 r = resultant(e1, e2, y);
4815 @subsection Square-free decomposition
4816 @cindex square-free decomposition
4817 @cindex factorization
4818 @cindex @code{sqrfree()}
4820 GiNaC still lacks proper factorization support. Some form of
4821 factorization is, however, easily implemented by noting that factors
4822 appearing in a polynomial with power two or more also appear in the
4823 derivative and hence can easily be found by computing the GCD of the
4824 original polynomial and its derivatives. Any decent system has an
4825 interface for this so called square-free factorization. So we provide
4828 ex sqrfree(const ex & a, const lst & l = lst());
4830 Here is an example that by the way illustrates how the exact form of the
4831 result may slightly depend on the order of differentiation, calling for
4832 some care with subsequent processing of the result:
4835 symbol x("x"), y("y");
4836 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4838 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4839 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4841 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4842 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4844 cout << sqrfree(BiVarPol) << endl;
4845 // -> depending on luck, any of the above
4848 Note also, how factors with the same exponents are not fully factorized
4852 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4853 @c node-name, next, previous, up
4854 @section Rational expressions
4856 @subsection The @code{normal} method
4857 @cindex @code{normal()}
4858 @cindex simplification
4859 @cindex temporary replacement
4861 Some basic form of simplification of expressions is called for frequently.
4862 GiNaC provides the method @code{.normal()}, which converts a rational function
4863 into an equivalent rational function of the form @samp{numerator/denominator}
4864 where numerator and denominator are coprime. If the input expression is already
4865 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4866 otherwise it performs fraction addition and multiplication.
4868 @code{.normal()} can also be used on expressions which are not rational functions
4869 as it will replace all non-rational objects (like functions or non-integer
4870 powers) by temporary symbols to bring the expression to the domain of rational
4871 functions before performing the normalization, and re-substituting these
4872 symbols afterwards. This algorithm is also available as a separate method
4873 @code{.to_rational()}, described below.
4875 This means that both expressions @code{t1} and @code{t2} are indeed
4876 simplified in this little code snippet:
4881 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4882 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4883 std::cout << "t1 is " << t1.normal() << std::endl;
4884 std::cout << "t2 is " << t2.normal() << std::endl;
4888 Of course this works for multivariate polynomials too, so the ratio of
4889 the sample-polynomials from the section about GCD and LCM above would be
4890 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4893 @subsection Numerator and denominator
4896 @cindex @code{numer()}
4897 @cindex @code{denom()}
4898 @cindex @code{numer_denom()}
4900 The numerator and denominator of an expression can be obtained with
4905 ex ex::numer_denom();
4908 These functions will first normalize the expression as described above and
4909 then return the numerator, denominator, or both as a list, respectively.
4910 If you need both numerator and denominator, calling @code{numer_denom()} is
4911 faster than using @code{numer()} and @code{denom()} separately.
4914 @subsection Converting to a polynomial or rational expression
4915 @cindex @code{to_polynomial()}
4916 @cindex @code{to_rational()}
4918 Some of the methods described so far only work on polynomials or rational
4919 functions. GiNaC provides a way to extend the domain of these functions to
4920 general expressions by using the temporary replacement algorithm described
4921 above. You do this by calling
4924 ex ex::to_polynomial(exmap & m);
4925 ex ex::to_polynomial(lst & l);
4929 ex ex::to_rational(exmap & m);
4930 ex ex::to_rational(lst & l);
4933 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4934 will be filled with the generated temporary symbols and their replacement
4935 expressions in a format that can be used directly for the @code{subs()}
4936 method. It can also already contain a list of replacements from an earlier
4937 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4938 possible to use it on multiple expressions and get consistent results.
4940 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4941 is probably best illustrated with an example:
4945 symbol x("x"), y("y");
4946 ex a = 2*x/sin(x) - y/(3*sin(x));
4950 ex p = a.to_polynomial(lp);
4951 cout << " = " << p << "\n with " << lp << endl;
4952 // = symbol3*symbol2*y+2*symbol2*x
4953 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4956 ex r = a.to_rational(lr);
4957 cout << " = " << r << "\n with " << lr << endl;
4958 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4959 // with @{symbol4==sin(x)@}
4963 The following more useful example will print @samp{sin(x)-cos(x)}:
4968 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4969 ex b = sin(x) + cos(x);
4972 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4973 cout << q.subs(m) << endl;
4978 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4979 @c node-name, next, previous, up
4980 @section Symbolic differentiation
4981 @cindex differentiation
4982 @cindex @code{diff()}
4984 @cindex product rule
4986 GiNaC's objects know how to differentiate themselves. Thus, a
4987 polynomial (class @code{add}) knows that its derivative is the sum of
4988 the derivatives of all the monomials:
4992 symbol x("x"), y("y"), z("z");
4993 ex P = pow(x, 5) + pow(x, 2) + y;
4995 cout << P.diff(x,2) << endl;
4997 cout << P.diff(y) << endl; // 1
4999 cout << P.diff(z) << endl; // 0
5004 If a second integer parameter @var{n} is given, the @code{diff} method
5005 returns the @var{n}th derivative.
5007 If @emph{every} object and every function is told what its derivative
5008 is, all derivatives of composed objects can be calculated using the
5009 chain rule and the product rule. Consider, for instance the expression
5010 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5011 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5012 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5013 out that the composition is the generating function for Euler Numbers,
5014 i.e. the so called @var{n}th Euler number is the coefficient of
5015 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5016 identity to code a function that generates Euler numbers in just three
5019 @cindex Euler numbers
5021 #include <ginac/ginac.h>
5022 using namespace GiNaC;
5024 ex EulerNumber(unsigned n)
5027 const ex generator = pow(cosh(x),-1);
5028 return generator.diff(x,n).subs(x==0);
5033 for (unsigned i=0; i<11; i+=2)
5034 std::cout << EulerNumber(i) << std::endl;
5039 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5040 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5041 @code{i} by two since all odd Euler numbers vanish anyways.
5044 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5045 @c node-name, next, previous, up
5046 @section Series expansion
5047 @cindex @code{series()}
5048 @cindex Taylor expansion
5049 @cindex Laurent expansion
5050 @cindex @code{pseries} (class)
5051 @cindex @code{Order()}
5053 Expressions know how to expand themselves as a Taylor series or (more
5054 generally) a Laurent series. As in most conventional Computer Algebra
5055 Systems, no distinction is made between those two. There is a class of
5056 its own for storing such series (@code{class pseries}) and a built-in
5057 function (called @code{Order}) for storing the order term of the series.
5058 As a consequence, if you want to work with series, i.e. multiply two
5059 series, you need to call the method @code{ex::series} again to convert
5060 it to a series object with the usual structure (expansion plus order
5061 term). A sample application from special relativity could read:
5064 #include <ginac/ginac.h>
5065 using namespace std;
5066 using namespace GiNaC;
5070 symbol v("v"), c("c");
5072 ex gamma = 1/sqrt(1 - pow(v/c,2));
5073 ex mass_nonrel = gamma.series(v==0, 10);
5075 cout << "the relativistic mass increase with v is " << endl
5076 << mass_nonrel << endl;
5078 cout << "the inverse square of this series is " << endl
5079 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5083 Only calling the series method makes the last output simplify to
5084 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5085 series raised to the power @math{-2}.
5087 @cindex Machin's formula
5088 As another instructive application, let us calculate the numerical
5089 value of Archimedes' constant
5093 (for which there already exists the built-in constant @code{Pi})
5094 using John Machin's amazing formula
5096 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5099 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5101 This equation (and similar ones) were used for over 200 years for
5102 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5103 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5104 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5105 order term with it and the question arises what the system is supposed
5106 to do when the fractions are plugged into that order term. The solution
5107 is to use the function @code{series_to_poly()} to simply strip the order
5111 #include <ginac/ginac.h>
5112 using namespace GiNaC;
5114 ex machin_pi(int degr)
5117 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5118 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5119 -4*pi_expansion.subs(x==numeric(1,239));
5125 using std::cout; // just for fun, another way of...
5126 using std::endl; // ...dealing with this namespace std.
5128 for (int i=2; i<12; i+=2) @{
5129 pi_frac = machin_pi(i);
5130 cout << i << ":\t" << pi_frac << endl
5131 << "\t" << pi_frac.evalf() << endl;
5137 Note how we just called @code{.series(x,degr)} instead of
5138 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5139 method @code{series()}: if the first argument is a symbol the expression
5140 is expanded in that symbol around point @code{0}. When you run this
5141 program, it will type out:
5145 3.1832635983263598326
5146 4: 5359397032/1706489875
5147 3.1405970293260603143
5148 6: 38279241713339684/12184551018734375
5149 3.141621029325034425
5150 8: 76528487109180192540976/24359780855939418203125
5151 3.141591772182177295
5152 10: 327853873402258685803048818236/104359128170408663038552734375
5153 3.1415926824043995174
5157 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5158 @c node-name, next, previous, up
5159 @section Symmetrization
5160 @cindex @code{symmetrize()}
5161 @cindex @code{antisymmetrize()}
5162 @cindex @code{symmetrize_cyclic()}
5167 ex ex::symmetrize(const lst & l);
5168 ex ex::antisymmetrize(const lst & l);
5169 ex ex::symmetrize_cyclic(const lst & l);
5172 symmetrize an expression by returning the sum over all symmetric,
5173 antisymmetric or cyclic permutations of the specified list of objects,
5174 weighted by the number of permutations.
5176 The three additional methods
5179 ex ex::symmetrize();
5180 ex ex::antisymmetrize();
5181 ex ex::symmetrize_cyclic();
5184 symmetrize or antisymmetrize an expression over its free indices.
5186 Symmetrization is most useful with indexed expressions but can be used with
5187 almost any kind of object (anything that is @code{subs()}able):
5191 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5192 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5194 cout << indexed(A, i, j).symmetrize() << endl;
5195 // -> 1/2*A.j.i+1/2*A.i.j
5196 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5197 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5198 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5199 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5203 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5204 @c node-name, next, previous, up
5205 @section Predefined mathematical functions
5207 @subsection Overview
5209 GiNaC contains the following predefined mathematical functions:
5212 @multitable @columnfractions .30 .70
5213 @item @strong{Name} @tab @strong{Function}
5216 @cindex @code{abs()}
5217 @item @code{csgn(x)}
5219 @cindex @code{conjugate()}
5220 @item @code{conjugate(x)}
5221 @tab complex conjugation
5222 @cindex @code{csgn()}
5223 @item @code{sqrt(x)}
5224 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5225 @cindex @code{sqrt()}
5228 @cindex @code{sin()}
5231 @cindex @code{cos()}
5234 @cindex @code{tan()}
5235 @item @code{asin(x)}
5237 @cindex @code{asin()}
5238 @item @code{acos(x)}
5240 @cindex @code{acos()}
5241 @item @code{atan(x)}
5242 @tab inverse tangent
5243 @cindex @code{atan()}
5244 @item @code{atan2(y, x)}
5245 @tab inverse tangent with two arguments
5246 @item @code{sinh(x)}
5247 @tab hyperbolic sine
5248 @cindex @code{sinh()}
5249 @item @code{cosh(x)}
5250 @tab hyperbolic cosine
5251 @cindex @code{cosh()}
5252 @item @code{tanh(x)}
5253 @tab hyperbolic tangent
5254 @cindex @code{tanh()}
5255 @item @code{asinh(x)}
5256 @tab inverse hyperbolic sine
5257 @cindex @code{asinh()}
5258 @item @code{acosh(x)}
5259 @tab inverse hyperbolic cosine
5260 @cindex @code{acosh()}
5261 @item @code{atanh(x)}
5262 @tab inverse hyperbolic tangent
5263 @cindex @code{atanh()}
5265 @tab exponential function
5266 @cindex @code{exp()}
5268 @tab natural logarithm
5269 @cindex @code{log()}
5272 @cindex @code{Li2()}
5273 @item @code{Li(m, x)}
5274 @tab classical polylogarithm as well as multiple polylogarithm
5276 @item @code{S(n, p, x)}
5277 @tab Nielsen's generalized polylogarithm
5279 @item @code{H(m, x)}
5280 @tab harmonic polylogarithm
5282 @item @code{zeta(m)}
5283 @tab Riemann's zeta function as well as multiple zeta value
5284 @cindex @code{zeta()}
5285 @item @code{zeta(m, s)}
5286 @tab alternating Euler sum
5287 @cindex @code{zeta()}
5288 @item @code{zetaderiv(n, x)}
5289 @tab derivatives of Riemann's zeta function
5290 @item @code{tgamma(x)}
5292 @cindex @code{tgamma()}
5293 @cindex gamma function
5294 @item @code{lgamma(x)}
5295 @tab logarithm of gamma function
5296 @cindex @code{lgamma()}
5297 @item @code{beta(x, y)}
5298 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5299 @cindex @code{beta()}
5301 @tab psi (digamma) function
5302 @cindex @code{psi()}
5303 @item @code{psi(n, x)}
5304 @tab derivatives of psi function (polygamma functions)
5305 @item @code{factorial(n)}
5306 @tab factorial function @math{n!}
5307 @cindex @code{factorial()}
5308 @item @code{binomial(n, k)}
5309 @tab binomial coefficients
5310 @cindex @code{binomial()}
5311 @item @code{Order(x)}
5312 @tab order term function in truncated power series
5313 @cindex @code{Order()}
5318 For functions that have a branch cut in the complex plane GiNaC follows
5319 the conventions for C++ as defined in the ANSI standard as far as
5320 possible. In particular: the natural logarithm (@code{log}) and the
5321 square root (@code{sqrt}) both have their branch cuts running along the
5322 negative real axis where the points on the axis itself belong to the
5323 upper part (i.e. continuous with quadrant II). The inverse
5324 trigonometric and hyperbolic functions are not defined for complex
5325 arguments by the C++ standard, however. In GiNaC we follow the
5326 conventions used by CLN, which in turn follow the carefully designed
5327 definitions in the Common Lisp standard. It should be noted that this
5328 convention is identical to the one used by the C99 standard and by most
5329 serious CAS. It is to be expected that future revisions of the C++
5330 standard incorporate these functions in the complex domain in a manner
5331 compatible with C99.
5333 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5334 @c node-name, next, previous, up
5335 @subsection Multiple polylogarithms
5337 @cindex polylogarithm
5338 @cindex Nielsen's generalized polylogarithm
5339 @cindex harmonic polylogarithm
5340 @cindex multiple zeta value
5341 @cindex alternating Euler sum
5342 @cindex multiple polylogarithm
5344 The multiple polylogarithm is the most generic member of a family of functions,
5345 to which others like the harmonic polylogarithm, Nielsen's generalized
5346 polylogarithm and the multiple zeta value belong.
5347 Everyone of these functions can also be written as a multiple polylogarithm with specific
5348 parameters. This whole family of functions is therefore often referred to simply as
5349 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5351 To facilitate the discussion of these functions we distinguish between indices and
5352 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5353 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5355 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5356 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5357 for the argument @code{x} as well.
5358 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5359 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5360 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5361 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5362 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5364 The functions print in LaTeX format as
5366 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5372 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5375 $\zeta(m_1,m_2,\ldots,m_k)$.
5377 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5378 are printed with a line above, e.g.
5380 $\zeta(5,\overline{2})$.
5382 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5384 Definitions and analytical as well as numerical properties of multiple polylogarithms
5385 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5386 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5387 except for a few differences which will be explicitly stated in the following.
5389 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5390 that the indices and arguments are understood to be in the same order as in which they appear in
5391 the series representation. This means
5393 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5396 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5399 $\zeta(1,2)$ evaluates to infinity.
5401 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5404 The functions only evaluate if the indices are integers greater than zero, except for the indices
5405 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5406 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5407 @code{zeta(lst(3,4), lst(-1,1))} means
5409 $\zeta(\overline{3},4)$.
5411 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5412 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5413 e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
5414 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5415 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5416 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5417 evaluates also for negative integers and positive even integers. For example:
5420 > Li(@{3,1@},@{x,1@});
5423 -zeta(@{3,2@},@{-1,-1@})
5428 It is easy to tell for a given function into which other function it can be rewritten, may
5429 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5430 with negative indices or trailing zeros (the example above gives a hint). Signs can
5431 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5432 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5433 @code{Li} (@code{eval()} already cares for the possible downgrade):
5436 > convert_H_to_Li(@{0,-2,-1,3@},x);
5437 Li(@{3,1,3@},@{-x,1,-1@})
5438 > convert_H_to_Li(@{2,-1,0@},x);
5439 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5442 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5443 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5448 $x_1x_2\cdots x_i < 1$ holds.
5454 > evalf(zeta(@{3,1,3,1@}));
5455 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5458 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5459 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5461 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5466 In long expressions this helps a lot with debugging, because you can easily spot
5467 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5468 cancellations of divergencies happen.
5470 Useful publications:
5472 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5473 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5475 @cite{Harmonic Polylogarithms},
5476 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5478 @cite{Special Values of Multiple Polylogarithms},
5479 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5481 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5482 @c node-name, next, previous, up
5483 @section Complex Conjugation
5485 @cindex @code{conjugate()}
5493 returns the complex conjugate of the expression. For all built-in functions and objects the
5494 conjugation gives the expected results:
5498 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5502 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5503 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5504 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5505 // -> -gamma5*gamma~b*gamma~a
5509 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5510 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5511 arguments. This is the default strategy. If you want to define your own functions and want to
5512 change this behavior, you have to supply a specialized conjugation method for your function
5513 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5515 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5516 @c node-name, next, previous, up
5517 @section Solving Linear Systems of Equations
5518 @cindex @code{lsolve()}
5520 The function @code{lsolve()} provides a convenient wrapper around some
5521 matrix operations that comes in handy when a system of linear equations
5525 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5528 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5529 @code{relational}) while @code{symbols} is a @code{lst} of
5530 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5533 It returns the @code{lst} of solutions as an expression. As an example,
5534 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5538 symbol a("a"), b("b"), x("x"), y("y");
5540 eqns = a*x+b*y==3, x-y==b;
5542 cout << lsolve(eqns, vars) << endl;
5543 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5546 When the linear equations @code{eqns} are underdetermined, the solution
5547 will contain one or more tautological entries like @code{x==x},
5548 depending on the rank of the system. When they are overdetermined, the
5549 solution will be an empty @code{lst}. Note the third optional parameter
5550 to @code{lsolve()}: it accepts the same parameters as
5551 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5555 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5556 @c node-name, next, previous, up
5557 @section Input and output of expressions
5560 @subsection Expression output
5562 @cindex output of expressions
5564 Expressions can simply be written to any stream:
5569 ex e = 4.5*I+pow(x,2)*3/2;
5570 cout << e << endl; // prints '4.5*I+3/2*x^2'
5574 The default output format is identical to the @command{ginsh} input syntax and
5575 to that used by most computer algebra systems, but not directly pastable
5576 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5577 is printed as @samp{x^2}).
5579 It is possible to print expressions in a number of different formats with
5580 a set of stream manipulators;
5583 std::ostream & dflt(std::ostream & os);
5584 std::ostream & latex(std::ostream & os);
5585 std::ostream & tree(std::ostream & os);
5586 std::ostream & csrc(std::ostream & os);
5587 std::ostream & csrc_float(std::ostream & os);
5588 std::ostream & csrc_double(std::ostream & os);
5589 std::ostream & csrc_cl_N(std::ostream & os);
5590 std::ostream & index_dimensions(std::ostream & os);
5591 std::ostream & no_index_dimensions(std::ostream & os);
5594 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5595 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5596 @code{print_csrc()} functions, respectively.
5599 All manipulators affect the stream state permanently. To reset the output
5600 format to the default, use the @code{dflt} manipulator:
5604 cout << latex; // all output to cout will be in LaTeX format from now on
5605 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5606 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5607 cout << dflt; // revert to default output format
5608 cout << e << endl; // prints '4.5*I+3/2*x^2'
5612 If you don't want to affect the format of the stream you're working with,
5613 you can output to a temporary @code{ostringstream} like this:
5618 s << latex << e; // format of cout remains unchanged
5619 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5624 @cindex @code{csrc_float}
5625 @cindex @code{csrc_double}
5626 @cindex @code{csrc_cl_N}
5627 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5628 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5629 format that can be directly used in a C or C++ program. The three possible
5630 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5631 classes provided by the CLN library):
5635 cout << "f = " << csrc_float << e << ";\n";
5636 cout << "d = " << csrc_double << e << ";\n";
5637 cout << "n = " << csrc_cl_N << e << ";\n";
5641 The above example will produce (note the @code{x^2} being converted to
5645 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5646 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5647 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5651 The @code{tree} manipulator allows dumping the internal structure of an
5652 expression for debugging purposes:
5663 add, hash=0x0, flags=0x3, nops=2
5664 power, hash=0x0, flags=0x3, nops=2
5665 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5666 2 (numeric), hash=0x6526b0fa, flags=0xf
5667 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5670 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5674 @cindex @code{latex}
5675 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5676 It is rather similar to the default format but provides some braces needed
5677 by LaTeX for delimiting boxes and also converts some common objects to
5678 conventional LaTeX names. It is possible to give symbols a special name for
5679 LaTeX output by supplying it as a second argument to the @code{symbol}
5682 For example, the code snippet
5686 symbol x("x", "\\circ");
5687 ex e = lgamma(x).series(x==0,3);
5688 cout << latex << e << endl;
5695 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5698 @cindex @code{index_dimensions}
5699 @cindex @code{no_index_dimensions}
5700 Index dimensions are normally hidden in the output. To make them visible, use
5701 the @code{index_dimensions} manipulator. The dimensions will be written in
5702 square brackets behind each index value in the default and LaTeX output
5707 symbol x("x"), y("y");
5708 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5709 ex e = indexed(x, mu) * indexed(y, nu);
5712 // prints 'x~mu*y~nu'
5713 cout << index_dimensions << e << endl;
5714 // prints 'x~mu[4]*y~nu[4]'
5715 cout << no_index_dimensions << e << endl;
5716 // prints 'x~mu*y~nu'
5721 @cindex Tree traversal
5722 If you need any fancy special output format, e.g. for interfacing GiNaC
5723 with other algebra systems or for producing code for different
5724 programming languages, you can always traverse the expression tree yourself:
5727 static void my_print(const ex & e)
5729 if (is_a<function>(e))
5730 cout << ex_to<function>(e).get_name();
5732 cout << ex_to<basic>(e).class_name();
5734 size_t n = e.nops();
5736 for (size_t i=0; i<n; i++) @{
5748 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5756 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5757 symbol(y))),numeric(-2)))
5760 If you need an output format that makes it possible to accurately
5761 reconstruct an expression by feeding the output to a suitable parser or
5762 object factory, you should consider storing the expression in an
5763 @code{archive} object and reading the object properties from there.
5764 See the section on archiving for more information.
5767 @subsection Expression input
5768 @cindex input of expressions
5770 GiNaC provides no way to directly read an expression from a stream because
5771 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5772 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5773 @code{y} you defined in your program and there is no way to specify the
5774 desired symbols to the @code{>>} stream input operator.
5776 Instead, GiNaC lets you construct an expression from a string, specifying the
5777 list of symbols to be used:
5781 symbol x("x"), y("y");
5782 ex e("2*x+sin(y)", lst(x, y));
5786 The input syntax is the same as that used by @command{ginsh} and the stream
5787 output operator @code{<<}. The symbols in the string are matched by name to
5788 the symbols in the list and if GiNaC encounters a symbol not specified in
5789 the list it will throw an exception.
5791 With this constructor, it's also easy to implement interactive GiNaC programs:
5796 #include <stdexcept>
5797 #include <ginac/ginac.h>
5798 using namespace std;
5799 using namespace GiNaC;
5806 cout << "Enter an expression containing 'x': ";
5811 cout << "The derivative of " << e << " with respect to x is ";
5812 cout << e.diff(x) << ".\n";
5813 @} catch (exception &p) @{
5814 cerr << p.what() << endl;
5820 @subsection Archiving
5821 @cindex @code{archive} (class)
5824 GiNaC allows creating @dfn{archives} of expressions which can be stored
5825 to or retrieved from files. To create an archive, you declare an object
5826 of class @code{archive} and archive expressions in it, giving each
5827 expression a unique name:
5831 using namespace std;
5832 #include <ginac/ginac.h>
5833 using namespace GiNaC;
5837 symbol x("x"), y("y"), z("z");
5839 ex foo = sin(x + 2*y) + 3*z + 41;
5843 a.archive_ex(foo, "foo");
5844 a.archive_ex(bar, "the second one");
5848 The archive can then be written to a file:
5852 ofstream out("foobar.gar");
5858 The file @file{foobar.gar} contains all information that is needed to
5859 reconstruct the expressions @code{foo} and @code{bar}.
5861 @cindex @command{viewgar}
5862 The tool @command{viewgar} that comes with GiNaC can be used to view
5863 the contents of GiNaC archive files:
5866 $ viewgar foobar.gar
5867 foo = 41+sin(x+2*y)+3*z
5868 the second one = 42+sin(x+2*y)+3*z
5871 The point of writing archive files is of course that they can later be
5877 ifstream in("foobar.gar");
5882 And the stored expressions can be retrieved by their name:
5889 ex ex1 = a2.unarchive_ex(syms, "foo");
5890 ex ex2 = a2.unarchive_ex(syms, "the second one");
5892 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5893 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5894 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5898 Note that you have to supply a list of the symbols which are to be inserted
5899 in the expressions. Symbols in archives are stored by their name only and
5900 if you don't specify which symbols you have, unarchiving the expression will
5901 create new symbols with that name. E.g. if you hadn't included @code{x} in
5902 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5903 have had no effect because the @code{x} in @code{ex1} would have been a
5904 different symbol than the @code{x} which was defined at the beginning of
5905 the program, although both would appear as @samp{x} when printed.
5907 You can also use the information stored in an @code{archive} object to
5908 output expressions in a format suitable for exact reconstruction. The
5909 @code{archive} and @code{archive_node} classes have a couple of member
5910 functions that let you access the stored properties:
5913 static void my_print2(const archive_node & n)
5916 n.find_string("class", class_name);
5917 cout << class_name << "(";
5919 archive_node::propinfovector p;
5920 n.get_properties(p);
5922 size_t num = p.size();
5923 for (size_t i=0; i<num; i++) @{
5924 const string &name = p[i].name;
5925 if (name == "class")
5927 cout << name << "=";
5929 unsigned count = p[i].count;
5933 for (unsigned j=0; j<count; j++) @{
5934 switch (p[i].type) @{
5935 case archive_node::PTYPE_BOOL: @{
5937 n.find_bool(name, x, j);
5938 cout << (x ? "true" : "false");
5941 case archive_node::PTYPE_UNSIGNED: @{
5943 n.find_unsigned(name, x, j);
5947 case archive_node::PTYPE_STRING: @{
5949 n.find_string(name, x, j);
5950 cout << '\"' << x << '\"';
5953 case archive_node::PTYPE_NODE: @{
5954 const archive_node &x = n.find_ex_node(name, j);
5976 ex e = pow(2, x) - y;
5978 my_print2(ar.get_top_node(0)); cout << endl;
5986 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5987 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5988 overall_coeff=numeric(number="0"))
5991 Be warned, however, that the set of properties and their meaning for each
5992 class may change between GiNaC versions.
5995 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5996 @c node-name, next, previous, up
5997 @chapter Extending GiNaC
5999 By reading so far you should have gotten a fairly good understanding of
6000 GiNaC's design patterns. From here on you should start reading the
6001 sources. All we can do now is issue some recommendations how to tackle
6002 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6003 develop some useful extension please don't hesitate to contact the GiNaC
6004 authors---they will happily incorporate them into future versions.
6007 * What does not belong into GiNaC:: What to avoid.
6008 * Symbolic functions:: Implementing symbolic functions.
6009 * Printing:: Adding new output formats.
6010 * Structures:: Defining new algebraic classes (the easy way).
6011 * Adding classes:: Defining new algebraic classes (the hard way).
6015 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6016 @c node-name, next, previous, up
6017 @section What doesn't belong into GiNaC
6019 @cindex @command{ginsh}
6020 First of all, GiNaC's name must be read literally. It is designed to be
6021 a library for use within C++. The tiny @command{ginsh} accompanying
6022 GiNaC makes this even more clear: it doesn't even attempt to provide a
6023 language. There are no loops or conditional expressions in
6024 @command{ginsh}, it is merely a window into the library for the
6025 programmer to test stuff (or to show off). Still, the design of a
6026 complete CAS with a language of its own, graphical capabilities and all
6027 this on top of GiNaC is possible and is without doubt a nice project for
6030 There are many built-in functions in GiNaC that do not know how to
6031 evaluate themselves numerically to a precision declared at runtime
6032 (using @code{Digits}). Some may be evaluated at certain points, but not
6033 generally. This ought to be fixed. However, doing numerical
6034 computations with GiNaC's quite abstract classes is doomed to be
6035 inefficient. For this purpose, the underlying foundation classes
6036 provided by CLN are much better suited.
6039 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6040 @c node-name, next, previous, up
6041 @section Symbolic functions
6043 The easiest and most instructive way to start extending GiNaC is probably to
6044 create your own symbolic functions. These are implemented with the help of
6045 two preprocessor macros:
6047 @cindex @code{DECLARE_FUNCTION}
6048 @cindex @code{REGISTER_FUNCTION}
6050 DECLARE_FUNCTION_<n>P(<name>)
6051 REGISTER_FUNCTION(<name>, <options>)
6054 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6055 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6056 parameters of type @code{ex} and returns a newly constructed GiNaC
6057 @code{function} object that represents your function.
6059 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6060 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6061 set of options that associate the symbolic function with C++ functions you
6062 provide to implement the various methods such as evaluation, derivative,
6063 series expansion etc. They also describe additional attributes the function
6064 might have, such as symmetry and commutation properties, and a name for
6065 LaTeX output. Multiple options are separated by the member access operator
6066 @samp{.} and can be given in an arbitrary order.
6068 (By the way: in case you are worrying about all the macros above we can
6069 assure you that functions are GiNaC's most macro-intense classes. We have
6070 done our best to avoid macros where we can.)
6072 @subsection A minimal example
6074 Here is an example for the implementation of a function with two arguments
6075 that is not further evaluated:
6078 DECLARE_FUNCTION_2P(myfcn)
6080 REGISTER_FUNCTION(myfcn, dummy())
6083 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6084 in algebraic expressions:
6090 ex e = 2*myfcn(42, 1+3*x) - x;
6092 // prints '2*myfcn(42,1+3*x)-x'
6097 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6098 "no options". A function with no options specified merely acts as a kind of
6099 container for its arguments. It is a pure "dummy" function with no associated
6100 logic (which is, however, sometimes perfectly sufficient).
6102 Let's now have a look at the implementation of GiNaC's cosine function for an
6103 example of how to make an "intelligent" function.
6105 @subsection The cosine function
6107 The GiNaC header file @file{inifcns.h} contains the line
6110 DECLARE_FUNCTION_1P(cos)
6113 which declares to all programs using GiNaC that there is a function @samp{cos}
6114 that takes one @code{ex} as an argument. This is all they need to know to use
6115 this function in expressions.
6117 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6118 is its @code{REGISTER_FUNCTION} line:
6121 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6122 evalf_func(cos_evalf).
6123 derivative_func(cos_deriv).
6124 latex_name("\\cos"));
6127 There are four options defined for the cosine function. One of them
6128 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6129 other three indicate the C++ functions in which the "brains" of the cosine
6130 function are defined.
6132 @cindex @code{hold()}
6134 The @code{eval_func()} option specifies the C++ function that implements
6135 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6136 the same number of arguments as the associated symbolic function (one in this
6137 case) and returns the (possibly transformed or in some way simplified)
6138 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6139 of the automatic evaluation process). If no (further) evaluation is to take
6140 place, the @code{eval_func()} function must return the original function
6141 with @code{.hold()}, to avoid a potential infinite recursion. If your
6142 symbolic functions produce a segmentation fault or stack overflow when
6143 using them in expressions, you are probably missing a @code{.hold()}
6146 The @code{eval_func()} function for the cosine looks something like this
6147 (actually, it doesn't look like this at all, but it should give you an idea
6151 static ex cos_eval(const ex & x)
6153 if ("x is a multiple of 2*Pi")
6155 else if ("x is a multiple of Pi")
6157 else if ("x is a multiple of Pi/2")
6161 else if ("x has the form 'acos(y)'")
6163 else if ("x has the form 'asin(y)'")
6168 return cos(x).hold();
6172 This function is called every time the cosine is used in a symbolic expression:
6178 // this calls cos_eval(Pi), and inserts its return value into
6179 // the actual expression
6186 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6187 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6188 symbolic transformation can be done, the unmodified function is returned
6189 with @code{.hold()}.
6191 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6192 The user has to call @code{evalf()} for that. This is implemented in a
6196 static ex cos_evalf(const ex & x)
6198 if (is_a<numeric>(x))
6199 return cos(ex_to<numeric>(x));
6201 return cos(x).hold();
6205 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6206 in this case the @code{cos()} function for @code{numeric} objects, which in
6207 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6208 isn't really needed here, but reminds us that the corresponding @code{eval()}
6209 function would require it in this place.
6211 Differentiation will surely turn up and so we need to tell @code{cos}
6212 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6213 instance, are then handled automatically by @code{basic::diff} and
6217 static ex cos_deriv(const ex & x, unsigned diff_param)
6223 @cindex product rule
6224 The second parameter is obligatory but uninteresting at this point. It
6225 specifies which parameter to differentiate in a partial derivative in
6226 case the function has more than one parameter, and its main application
6227 is for correct handling of the chain rule.
6229 An implementation of the series expansion is not needed for @code{cos()} as
6230 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6231 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6232 the other hand, does have poles and may need to do Laurent expansion:
6235 static ex tan_series(const ex & x, const relational & rel,
6236 int order, unsigned options)
6238 // Find the actual expansion point
6239 const ex x_pt = x.subs(rel);
6241 if ("x_pt is not an odd multiple of Pi/2")
6242 throw do_taylor(); // tell function::series() to do Taylor expansion
6244 // On a pole, expand sin()/cos()
6245 return (sin(x)/cos(x)).series(rel, order+2, options);
6249 The @code{series()} implementation of a function @emph{must} return a
6250 @code{pseries} object, otherwise your code will crash.
6252 @subsection Function options
6254 GiNaC functions understand several more options which are always
6255 specified as @code{.option(params)}. None of them are required, but you
6256 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6257 is a do-nothing option called @code{dummy()} which you can use to define
6258 functions without any special options.
6261 eval_func(<C++ function>)
6262 evalf_func(<C++ function>)
6263 derivative_func(<C++ function>)
6264 series_func(<C++ function>)
6265 conjugate_func(<C++ function>)
6268 These specify the C++ functions that implement symbolic evaluation,
6269 numeric evaluation, partial derivatives, and series expansion, respectively.
6270 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6271 @code{diff()} and @code{series()}.
6273 The @code{eval_func()} function needs to use @code{.hold()} if no further
6274 automatic evaluation is desired or possible.
6276 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6277 expansion, which is correct if there are no poles involved. If the function
6278 has poles in the complex plane, the @code{series_func()} needs to check
6279 whether the expansion point is on a pole and fall back to Taylor expansion
6280 if it isn't. Otherwise, the pole usually needs to be regularized by some
6281 suitable transformation.
6284 latex_name(const string & n)
6287 specifies the LaTeX code that represents the name of the function in LaTeX
6288 output. The default is to put the function name in an @code{\mbox@{@}}.
6291 do_not_evalf_params()
6294 This tells @code{evalf()} to not recursively evaluate the parameters of the
6295 function before calling the @code{evalf_func()}.
6298 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6301 This allows you to explicitly specify the commutation properties of the
6302 function (@xref{Non-commutative objects}, for an explanation of
6303 (non)commutativity in GiNaC). For example, you can use
6304 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6305 GiNaC treat your function like a matrix. By default, functions inherit the
6306 commutation properties of their first argument.
6309 set_symmetry(const symmetry & s)
6312 specifies the symmetry properties of the function with respect to its
6313 arguments. @xref{Indexed objects}, for an explanation of symmetry
6314 specifications. GiNaC will automatically rearrange the arguments of
6315 symmetric functions into a canonical order.
6317 Sometimes you may want to have finer control over how functions are
6318 displayed in the output. For example, the @code{abs()} function prints
6319 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6320 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6324 print_func<C>(<C++ function>)
6327 option which is explained in the next section.
6329 @subsection Functions with a variable number of arguments
6331 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6332 functions with a fixed number of arguments. Sometimes, though, you may need
6333 to have a function that accepts a variable number of expressions. One way to
6334 accomplish this is to pass variable-length lists as arguments. The
6335 @code{Li()} function uses this method for multiple polylogarithms.
6337 It is also possible to define functions that accept a different number of
6338 parameters under the same function name, such as the @code{psi()} function
6339 which can be called either as @code{psi(z)} (the digamma function) or as
6340 @code{psi(n, z)} (polygamma functions). These are actually two different
6341 functions in GiNaC that, however, have the same name. Defining such
6342 functions is not possible with the macros but requires manually fiddling
6343 with GiNaC internals. If you are interested, please consult the GiNaC source
6344 code for the @code{psi()} function (@file{inifcns.h} and
6345 @file{inifcns_gamma.cpp}).
6348 @node Printing, Structures, Symbolic functions, Extending GiNaC
6349 @c node-name, next, previous, up
6350 @section GiNaC's expression output system
6352 GiNaC allows the output of expressions in a variety of different formats
6353 (@pxref{Input/Output}). This section will explain how expression output
6354 is implemented internally, and how to define your own output formats or
6355 change the output format of built-in algebraic objects. You will also want
6356 to read this section if you plan to write your own algebraic classes or
6359 @cindex @code{print_context} (class)
6360 @cindex @code{print_dflt} (class)
6361 @cindex @code{print_latex} (class)
6362 @cindex @code{print_tree} (class)
6363 @cindex @code{print_csrc} (class)
6364 All the different output formats are represented by a hierarchy of classes
6365 rooted in the @code{print_context} class, defined in the @file{print.h}
6370 the default output format
6372 output in LaTeX mathematical mode
6374 a dump of the internal expression structure (for debugging)
6376 the base class for C source output
6377 @item print_csrc_float
6378 C source output using the @code{float} type
6379 @item print_csrc_double
6380 C source output using the @code{double} type
6381 @item print_csrc_cl_N
6382 C source output using CLN types
6385 The @code{print_context} base class provides two public data members:
6397 @code{s} is a reference to the stream to output to, while @code{options}
6398 holds flags and modifiers. Currently, there is only one flag defined:
6399 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6400 to print the index dimension which is normally hidden.
6402 When you write something like @code{std::cout << e}, where @code{e} is
6403 an object of class @code{ex}, GiNaC will construct an appropriate
6404 @code{print_context} object (of a class depending on the selected output
6405 format), fill in the @code{s} and @code{options} members, and call
6407 @cindex @code{print()}
6409 void ex::print(const print_context & c, unsigned level = 0) const;
6412 which in turn forwards the call to the @code{print()} method of the
6413 top-level algebraic object contained in the expression.
6415 Unlike other methods, GiNaC classes don't usually override their
6416 @code{print()} method to implement expression output. Instead, the default
6417 implementation @code{basic::print(c, level)} performs a run-time double
6418 dispatch to a function selected by the dynamic type of the object and the
6419 passed @code{print_context}. To this end, GiNaC maintains a separate method
6420 table for each class, similar to the virtual function table used for ordinary
6421 (single) virtual function dispatch.
6423 The method table contains one slot for each possible @code{print_context}
6424 type, indexed by the (internally assigned) serial number of the type. Slots
6425 may be empty, in which case GiNaC will retry the method lookup with the
6426 @code{print_context} object's parent class, possibly repeating the process
6427 until it reaches the @code{print_context} base class. If there's still no
6428 method defined, the method table of the algebraic object's parent class
6429 is consulted, and so on, until a matching method is found (eventually it
6430 will reach the combination @code{basic/print_context}, which prints the
6431 object's class name enclosed in square brackets).
6433 You can think of the print methods of all the different classes and output
6434 formats as being arranged in a two-dimensional matrix with one axis listing
6435 the algebraic classes and the other axis listing the @code{print_context}
6438 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6439 to implement printing, but then they won't get any of the benefits of the
6440 double dispatch mechanism (such as the ability for derived classes to
6441 inherit only certain print methods from its parent, or the replacement of
6442 methods at run-time).
6444 @subsection Print methods for classes
6446 The method table for a class is set up either in the definition of the class,
6447 by passing the appropriate @code{print_func<C>()} option to
6448 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6449 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6450 can also be used to override existing methods dynamically.
6452 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6453 be a member function of the class (or one of its parent classes), a static
6454 member function, or an ordinary (global) C++ function. The @code{C} template
6455 parameter specifies the appropriate @code{print_context} type for which the
6456 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6457 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6458 the class is the one being implemented by
6459 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6461 For print methods that are member functions, their first argument must be of
6462 a type convertible to a @code{const C &}, and the second argument must be an
6465 For static members and global functions, the first argument must be of a type
6466 convertible to a @code{const T &}, the second argument must be of a type
6467 convertible to a @code{const C &}, and the third argument must be an
6468 @code{unsigned}. A global function will, of course, not have access to
6469 private and protected members of @code{T}.
6471 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6472 and @code{basic::print()}) is used for proper parenthesizing of the output
6473 (and by @code{print_tree} for proper indentation). It can be used for similar
6474 purposes if you write your own output formats.
6476 The explanations given above may seem complicated, but in practice it's
6477 really simple, as shown in the following example. Suppose that we want to
6478 display exponents in LaTeX output not as superscripts but with little
6479 upwards-pointing arrows. This can be achieved in the following way:
6482 void my_print_power_as_latex(const power & p,
6483 const print_latex & c,
6486 // get the precedence of the 'power' class
6487 unsigned power_prec = p.precedence();
6489 // if the parent operator has the same or a higher precedence
6490 // we need parentheses around the power
6491 if (level >= power_prec)
6494 // print the basis and exponent, each enclosed in braces, and
6495 // separated by an uparrow
6497 p.op(0).print(c, power_prec);
6498 c.s << "@}\\uparrow@{";
6499 p.op(1).print(c, power_prec);
6502 // don't forget the closing parenthesis
6503 if (level >= power_prec)
6509 // a sample expression
6510 symbol x("x"), y("y");
6511 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6513 // switch to LaTeX mode
6516 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6519 // now we replace the method for the LaTeX output of powers with
6521 set_print_func<power, print_latex>(my_print_power_as_latex);
6523 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6533 The first argument of @code{my_print_power_as_latex} could also have been
6534 a @code{const basic &}, the second one a @code{const print_context &}.
6537 The above code depends on @code{mul} objects converting their operands to
6538 @code{power} objects for the purpose of printing.
6541 The output of products including negative powers as fractions is also
6542 controlled by the @code{mul} class.
6545 The @code{power/print_latex} method provided by GiNaC prints square roots
6546 using @code{\sqrt}, but the above code doesn't.
6550 It's not possible to restore a method table entry to its previous or default
6551 value. Once you have called @code{set_print_func()}, you can only override
6552 it with another call to @code{set_print_func()}, but you can't easily go back
6553 to the default behavior again (you can, of course, dig around in the GiNaC
6554 sources, find the method that is installed at startup
6555 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6556 one; that is, after you circumvent the C++ member access control@dots{}).
6558 @subsection Print methods for functions
6560 Symbolic functions employ a print method dispatch mechanism similar to the
6561 one used for classes. The methods are specified with @code{print_func<C>()}
6562 function options. If you don't specify any special print methods, the function
6563 will be printed with its name (or LaTeX name, if supplied), followed by a
6564 comma-separated list of arguments enclosed in parentheses.
6566 For example, this is what GiNaC's @samp{abs()} function is defined like:
6569 static ex abs_eval(const ex & arg) @{ ... @}
6570 static ex abs_evalf(const ex & arg) @{ ... @}
6572 static void abs_print_latex(const ex & arg, const print_context & c)
6574 c.s << "@{|"; arg.print(c); c.s << "|@}";
6577 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6579 c.s << "fabs("; arg.print(c); c.s << ")";
6582 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6583 evalf_func(abs_evalf).
6584 print_func<print_latex>(abs_print_latex).
6585 print_func<print_csrc_float>(abs_print_csrc_float).
6586 print_func<print_csrc_double>(abs_print_csrc_float));
6589 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6590 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6592 There is currently no equivalent of @code{set_print_func()} for functions.
6594 @subsection Adding new output formats
6596 Creating a new output format involves subclassing @code{print_context},
6597 which is somewhat similar to adding a new algebraic class
6598 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6599 that needs to go into the class definition, and a corresponding macro
6600 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6601 Every @code{print_context} class needs to provide a default constructor
6602 and a constructor from an @code{std::ostream} and an @code{unsigned}
6605 Here is an example for a user-defined @code{print_context} class:
6608 class print_myformat : public print_dflt
6610 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6612 print_myformat(std::ostream & os, unsigned opt = 0)
6613 : print_dflt(os, opt) @{@}
6616 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6618 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6621 That's all there is to it. None of the actual expression output logic is
6622 implemented in this class. It merely serves as a selector for choosing
6623 a particular format. The algorithms for printing expressions in the new
6624 format are implemented as print methods, as described above.
6626 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6627 exactly like GiNaC's default output format:
6632 ex e = pow(x, 2) + 1;
6634 // this prints "1+x^2"
6637 // this also prints "1+x^2"
6638 e.print(print_myformat()); cout << endl;
6644 To fill @code{print_myformat} with life, we need to supply appropriate
6645 print methods with @code{set_print_func()}, like this:
6648 // This prints powers with '**' instead of '^'. See the LaTeX output
6649 // example above for explanations.
6650 void print_power_as_myformat(const power & p,
6651 const print_myformat & c,
6654 unsigned power_prec = p.precedence();
6655 if (level >= power_prec)
6657 p.op(0).print(c, power_prec);
6659 p.op(1).print(c, power_prec);
6660 if (level >= power_prec)
6666 // install a new print method for power objects
6667 set_print_func<power, print_myformat>(print_power_as_myformat);
6669 // now this prints "1+x**2"
6670 e.print(print_myformat()); cout << endl;
6672 // but the default format is still "1+x^2"
6678 @node Structures, Adding classes, Printing, Extending GiNaC
6679 @c node-name, next, previous, up
6682 If you are doing some very specialized things with GiNaC, or if you just
6683 need some more organized way to store data in your expressions instead of
6684 anonymous lists, you may want to implement your own algebraic classes.
6685 ('algebraic class' means any class directly or indirectly derived from
6686 @code{basic} that can be used in GiNaC expressions).
6688 GiNaC offers two ways of accomplishing this: either by using the
6689 @code{structure<T>} template class, or by rolling your own class from
6690 scratch. This section will discuss the @code{structure<T>} template which
6691 is easier to use but more limited, while the implementation of custom
6692 GiNaC classes is the topic of the next section. However, you may want to
6693 read both sections because many common concepts and member functions are
6694 shared by both concepts, and it will also allow you to decide which approach
6695 is most suited to your needs.
6697 The @code{structure<T>} template, defined in the GiNaC header file
6698 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6699 or @code{class}) into a GiNaC object that can be used in expressions.
6701 @subsection Example: scalar products
6703 Let's suppose that we need a way to handle some kind of abstract scalar
6704 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6705 product class have to store their left and right operands, which can in turn
6706 be arbitrary expressions. Here is a possible way to represent such a
6707 product in a C++ @code{struct}:
6711 using namespace std;
6713 #include <ginac/ginac.h>
6714 using namespace GiNaC;
6720 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6724 The default constructor is required. Now, to make a GiNaC class out of this
6725 data structure, we need only one line:
6728 typedef structure<sprod_s> sprod;
6731 That's it. This line constructs an algebraic class @code{sprod} which
6732 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6733 expressions like any other GiNaC class:
6737 symbol a("a"), b("b");
6738 ex e = sprod(sprod_s(a, b));
6742 Note the difference between @code{sprod} which is the algebraic class, and
6743 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6744 and @code{right} data members. As shown above, an @code{sprod} can be
6745 constructed from an @code{sprod_s} object.
6747 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6748 you could define a little wrapper function like this:
6751 inline ex make_sprod(ex left, ex right)
6753 return sprod(sprod_s(left, right));
6757 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6758 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6759 @code{get_struct()}:
6763 cout << ex_to<sprod>(e)->left << endl;
6765 cout << ex_to<sprod>(e).get_struct().right << endl;
6770 You only have read access to the members of @code{sprod_s}.
6772 The type definition of @code{sprod} is enough to write your own algorithms
6773 that deal with scalar products, for example:
6778 if (is_a<sprod>(p)) @{
6779 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6780 return make_sprod(sp.right, sp.left);
6791 @subsection Structure output
6793 While the @code{sprod} type is useable it still leaves something to be
6794 desired, most notably proper output:
6799 // -> [structure object]
6803 By default, any structure types you define will be printed as
6804 @samp{[structure object]}. To override this you can either specialize the
6805 template's @code{print()} member function, or specify print methods with
6806 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6807 it's not possible to supply class options like @code{print_func<>()} to
6808 structures, so for a self-contained structure type you need to resort to
6809 overriding the @code{print()} function, which is also what we will do here.
6811 The member functions of GiNaC classes are described in more detail in the
6812 next section, but it shouldn't be hard to figure out what's going on here:
6815 void sprod::print(const print_context & c, unsigned level) const
6817 // tree debug output handled by superclass
6818 if (is_a<print_tree>(c))
6819 inherited::print(c, level);
6821 // get the contained sprod_s object
6822 const sprod_s & sp = get_struct();
6824 // print_context::s is a reference to an ostream
6825 c.s << "<" << sp.left << "|" << sp.right << ">";
6829 Now we can print expressions containing scalar products:
6835 cout << swap_sprod(e) << endl;
6840 @subsection Comparing structures
6842 The @code{sprod} class defined so far still has one important drawback: all
6843 scalar products are treated as being equal because GiNaC doesn't know how to
6844 compare objects of type @code{sprod_s}. This can lead to some confusing
6845 and undesired behavior:
6849 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6851 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6852 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6856 To remedy this, we first need to define the operators @code{==} and @code{<}
6857 for objects of type @code{sprod_s}:
6860 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6862 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6865 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6867 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6871 The ordering established by the @code{<} operator doesn't have to make any
6872 algebraic sense, but it needs to be well defined. Note that we can't use
6873 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6874 in the implementation of these operators because they would construct
6875 GiNaC @code{relational} objects which in the case of @code{<} do not
6876 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6877 decide which one is algebraically 'less').
6879 Next, we need to change our definition of the @code{sprod} type to let
6880 GiNaC know that an ordering relation exists for the embedded objects:
6883 typedef structure<sprod_s, compare_std_less> sprod;
6886 @code{sprod} objects then behave as expected:
6890 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6891 // -> <a|b>-<a^2|b^2>
6892 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6893 // -> <a|b>+<a^2|b^2>
6894 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6896 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6901 The @code{compare_std_less} policy parameter tells GiNaC to use the
6902 @code{std::less} and @code{std::equal_to} functors to compare objects of
6903 type @code{sprod_s}. By default, these functors forward their work to the
6904 standard @code{<} and @code{==} operators, which we have overloaded.
6905 Alternatively, we could have specialized @code{std::less} and
6906 @code{std::equal_to} for class @code{sprod_s}.
6908 GiNaC provides two other comparison policies for @code{structure<T>}
6909 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6910 which does a bit-wise comparison of the contained @code{T} objects.
6911 This should be used with extreme care because it only works reliably with
6912 built-in integral types, and it also compares any padding (filler bytes of
6913 undefined value) that the @code{T} class might have.
6915 @subsection Subexpressions
6917 Our scalar product class has two subexpressions: the left and right
6918 operands. It might be a good idea to make them accessible via the standard
6919 @code{nops()} and @code{op()} methods:
6922 size_t sprod::nops() const
6927 ex sprod::op(size_t i) const
6931 return get_struct().left;
6933 return get_struct().right;
6935 throw std::range_error("sprod::op(): no such operand");
6940 Implementing @code{nops()} and @code{op()} for container types such as
6941 @code{sprod} has two other nice side effects:
6945 @code{has()} works as expected
6947 GiNaC generates better hash keys for the objects (the default implementation
6948 of @code{calchash()} takes subexpressions into account)
6951 @cindex @code{let_op()}
6952 There is a non-const variant of @code{op()} called @code{let_op()} that
6953 allows replacing subexpressions:
6956 ex & sprod::let_op(size_t i)
6958 // every non-const member function must call this
6959 ensure_if_modifiable();
6963 return get_struct().left;
6965 return get_struct().right;
6967 throw std::range_error("sprod::let_op(): no such operand");
6972 Once we have provided @code{let_op()} we also get @code{subs()} and
6973 @code{map()} for free. In fact, every container class that returns a non-null
6974 @code{nops()} value must either implement @code{let_op()} or provide custom
6975 implementations of @code{subs()} and @code{map()}.
6977 In turn, the availability of @code{map()} enables the recursive behavior of a
6978 couple of other default method implementations, in particular @code{evalf()},
6979 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6980 we probably want to provide our own version of @code{expand()} for scalar
6981 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6982 This is left as an exercise for the reader.
6984 The @code{structure<T>} template defines many more member functions that
6985 you can override by specialization to customize the behavior of your
6986 structures. You are referred to the next section for a description of
6987 some of these (especially @code{eval()}). There is, however, one topic
6988 that shall be addressed here, as it demonstrates one peculiarity of the
6989 @code{structure<T>} template: archiving.
6991 @subsection Archiving structures
6993 If you don't know how the archiving of GiNaC objects is implemented, you
6994 should first read the next section and then come back here. You're back?
6997 To implement archiving for structures it is not enough to provide
6998 specializations for the @code{archive()} member function and the
6999 unarchiving constructor (the @code{unarchive()} function has a default
7000 implementation). You also need to provide a unique name (as a string literal)
7001 for each structure type you define. This is because in GiNaC archives,
7002 the class of an object is stored as a string, the class name.
7004 By default, this class name (as returned by the @code{class_name()} member
7005 function) is @samp{structure} for all structure classes. This works as long
7006 as you have only defined one structure type, but if you use two or more you
7007 need to provide a different name for each by specializing the
7008 @code{get_class_name()} member function. Here is a sample implementation
7009 for enabling archiving of the scalar product type defined above:
7012 const char *sprod::get_class_name() @{ return "sprod"; @}
7014 void sprod::archive(archive_node & n) const
7016 inherited::archive(n);
7017 n.add_ex("left", get_struct().left);
7018 n.add_ex("right", get_struct().right);
7021 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7023 n.find_ex("left", get_struct().left, sym_lst);
7024 n.find_ex("right", get_struct().right, sym_lst);
7028 Note that the unarchiving constructor is @code{sprod::structure} and not
7029 @code{sprod::sprod}, and that we don't need to supply an
7030 @code{sprod::unarchive()} function.
7033 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7034 @c node-name, next, previous, up
7035 @section Adding classes
7037 The @code{structure<T>} template provides an way to extend GiNaC with custom
7038 algebraic classes that is easy to use but has its limitations, the most
7039 severe of which being that you can't add any new member functions to
7040 structures. To be able to do this, you need to write a new class definition
7043 This section will explain how to implement new algebraic classes in GiNaC by
7044 giving the example of a simple 'string' class. After reading this section
7045 you will know how to properly declare a GiNaC class and what the minimum
7046 required member functions are that you have to implement. We only cover the
7047 implementation of a 'leaf' class here (i.e. one that doesn't contain
7048 subexpressions). Creating a container class like, for example, a class
7049 representing tensor products is more involved but this section should give
7050 you enough information so you can consult the source to GiNaC's predefined
7051 classes if you want to implement something more complicated.
7053 @subsection GiNaC's run-time type information system
7055 @cindex hierarchy of classes
7057 All algebraic classes (that is, all classes that can appear in expressions)
7058 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7059 @code{basic *} (which is essentially what an @code{ex} is) represents a
7060 generic pointer to an algebraic class. Occasionally it is necessary to find
7061 out what the class of an object pointed to by a @code{basic *} really is.
7062 Also, for the unarchiving of expressions it must be possible to find the
7063 @code{unarchive()} function of a class given the class name (as a string). A
7064 system that provides this kind of information is called a run-time type
7065 information (RTTI) system. The C++ language provides such a thing (see the
7066 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7067 implements its own, simpler RTTI.
7069 The RTTI in GiNaC is based on two mechanisms:
7074 The @code{basic} class declares a member variable @code{tinfo_key} which
7075 holds an unsigned integer that identifies the object's class. These numbers
7076 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7077 classes. They all start with @code{TINFO_}.
7080 By means of some clever tricks with static members, GiNaC maintains a list
7081 of information for all classes derived from @code{basic}. The information
7082 available includes the class names, the @code{tinfo_key}s, and pointers
7083 to the unarchiving functions. This class registry is defined in the
7084 @file{registrar.h} header file.
7088 The disadvantage of this proprietary RTTI implementation is that there's
7089 a little more to do when implementing new classes (C++'s RTTI works more
7090 or less automatically) but don't worry, most of the work is simplified by
7093 @subsection A minimalistic example
7095 Now we will start implementing a new class @code{mystring} that allows
7096 placing character strings in algebraic expressions (this is not very useful,
7097 but it's just an example). This class will be a direct subclass of
7098 @code{basic}. You can use this sample implementation as a starting point
7099 for your own classes.
7101 The code snippets given here assume that you have included some header files
7107 #include <stdexcept>
7108 using namespace std;
7110 #include <ginac/ginac.h>
7111 using namespace GiNaC;
7114 The first thing we have to do is to define a @code{tinfo_key} for our new
7115 class. This can be any arbitrary unsigned number that is not already taken
7116 by one of the existing classes but it's better to come up with something
7117 that is unlikely to clash with keys that might be added in the future. The
7118 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7119 which is not a requirement but we are going to stick with this scheme:
7122 const unsigned TINFO_mystring = 0x42420001U;
7125 Now we can write down the class declaration. The class stores a C++
7126 @code{string} and the user shall be able to construct a @code{mystring}
7127 object from a C or C++ string:
7130 class mystring : public basic
7132 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7135 mystring(const string &s);
7136 mystring(const char *s);
7142 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7145 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7146 macros are defined in @file{registrar.h}. They take the name of the class
7147 and its direct superclass as arguments and insert all required declarations
7148 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7149 the first line after the opening brace of the class definition. The
7150 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7151 source (at global scope, of course, not inside a function).
7153 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7154 declarations of the default constructor and a couple of other functions that
7155 are required. It also defines a type @code{inherited} which refers to the
7156 superclass so you don't have to modify your code every time you shuffle around
7157 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7158 class with the GiNaC RTTI (there is also a
7159 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7160 options for the class, and which we will be using instead in a few minutes).
7162 Now there are seven member functions we have to implement to get a working
7168 @code{mystring()}, the default constructor.
7171 @code{void archive(archive_node &n)}, the archiving function. This stores all
7172 information needed to reconstruct an object of this class inside an
7173 @code{archive_node}.
7176 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7177 constructor. This constructs an instance of the class from the information
7178 found in an @code{archive_node}.
7181 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7182 unarchiving function. It constructs a new instance by calling the unarchiving
7186 @cindex @code{compare_same_type()}
7187 @code{int compare_same_type(const basic &other)}, which is used internally
7188 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7189 -1, depending on the relative order of this object and the @code{other}
7190 object. If it returns 0, the objects are considered equal.
7191 @strong{Note:} This has nothing to do with the (numeric) ordering
7192 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7193 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7194 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7195 must provide a @code{compare_same_type()} function, even those representing
7196 objects for which no reasonable algebraic ordering relationship can be
7200 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7201 which are the two constructors we declared.
7205 Let's proceed step-by-step. The default constructor looks like this:
7208 mystring::mystring() : inherited(TINFO_mystring) @{@}
7211 The golden rule is that in all constructors you have to set the
7212 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7213 it will be set by the constructor of the superclass and all hell will break
7214 loose in the RTTI. For your convenience, the @code{basic} class provides
7215 a constructor that takes a @code{tinfo_key} value, which we are using here
7216 (remember that in our case @code{inherited == basic}). If the superclass
7217 didn't have such a constructor, we would have to set the @code{tinfo_key}
7218 to the right value manually.
7220 In the default constructor you should set all other member variables to
7221 reasonable default values (we don't need that here since our @code{str}
7222 member gets set to an empty string automatically).
7224 Next are the three functions for archiving. You have to implement them even
7225 if you don't plan to use archives, but the minimum required implementation
7226 is really simple. First, the archiving function:
7229 void mystring::archive(archive_node &n) const
7231 inherited::archive(n);
7232 n.add_string("string", str);
7236 The only thing that is really required is calling the @code{archive()}
7237 function of the superclass. Optionally, you can store all information you
7238 deem necessary for representing the object into the passed
7239 @code{archive_node}. We are just storing our string here. For more
7240 information on how the archiving works, consult the @file{archive.h} header
7243 The unarchiving constructor is basically the inverse of the archiving
7247 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7249 n.find_string("string", str);
7253 If you don't need archiving, just leave this function empty (but you must
7254 invoke the unarchiving constructor of the superclass). Note that we don't
7255 have to set the @code{tinfo_key} here because it is done automatically
7256 by the unarchiving constructor of the @code{basic} class.
7258 Finally, the unarchiving function:
7261 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7263 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7267 You don't have to understand how exactly this works. Just copy these
7268 four lines into your code literally (replacing the class name, of
7269 course). It calls the unarchiving constructor of the class and unless
7270 you are doing something very special (like matching @code{archive_node}s
7271 to global objects) you don't need a different implementation. For those
7272 who are interested: setting the @code{dynallocated} flag puts the object
7273 under the control of GiNaC's garbage collection. It will get deleted
7274 automatically once it is no longer referenced.
7276 Our @code{compare_same_type()} function uses a provided function to compare
7280 int mystring::compare_same_type(const basic &other) const
7282 const mystring &o = static_cast<const mystring &>(other);
7283 int cmpval = str.compare(o.str);
7286 else if (cmpval < 0)
7293 Although this function takes a @code{basic &}, it will always be a reference
7294 to an object of exactly the same class (objects of different classes are not
7295 comparable), so the cast is safe. If this function returns 0, the two objects
7296 are considered equal (in the sense that @math{A-B=0}), so you should compare
7297 all relevant member variables.
7299 Now the only thing missing is our two new constructors:
7302 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7303 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7306 No surprises here. We set the @code{str} member from the argument and
7307 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7309 That's it! We now have a minimal working GiNaC class that can store
7310 strings in algebraic expressions. Let's confirm that the RTTI works:
7313 ex e = mystring("Hello, world!");
7314 cout << is_a<mystring>(e) << endl;
7317 cout << e.bp->class_name() << endl;
7321 Obviously it does. Let's see what the expression @code{e} looks like:
7325 // -> [mystring object]
7328 Hm, not exactly what we expect, but of course the @code{mystring} class
7329 doesn't yet know how to print itself. This can be done either by implementing
7330 the @code{print()} member function, or, preferably, by specifying a
7331 @code{print_func<>()} class option. Let's say that we want to print the string
7332 surrounded by double quotes:
7335 class mystring : public basic
7339 void do_print(const print_context &c, unsigned level = 0) const;
7343 void mystring::do_print(const print_context &c, unsigned level) const
7345 // print_context::s is a reference to an ostream
7346 c.s << '\"' << str << '\"';
7350 The @code{level} argument is only required for container classes to
7351 correctly parenthesize the output.
7353 Now we need to tell GiNaC that @code{mystring} objects should use the
7354 @code{do_print()} member function for printing themselves. For this, we
7358 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7364 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7365 print_func<print_context>(&mystring::do_print))
7368 Let's try again to print the expression:
7372 // -> "Hello, world!"
7375 Much better. If we wanted to have @code{mystring} objects displayed in a
7376 different way depending on the output format (default, LaTeX, etc.), we
7377 would have supplied multiple @code{print_func<>()} options with different
7378 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7379 separated by dots. This is similar to the way options are specified for
7380 symbolic functions. @xref{Printing}, for a more in-depth description of the
7381 way expression output is implemented in GiNaC.
7383 The @code{mystring} class can be used in arbitrary expressions:
7386 e += mystring("GiNaC rulez");
7388 // -> "GiNaC rulez"+"Hello, world!"
7391 (GiNaC's automatic term reordering is in effect here), or even
7394 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7396 // -> "One string"^(2*sin(-"Another string"+Pi))
7399 Whether this makes sense is debatable but remember that this is only an
7400 example. At least it allows you to implement your own symbolic algorithms
7403 Note that GiNaC's algebraic rules remain unchanged:
7406 e = mystring("Wow") * mystring("Wow");
7410 e = pow(mystring("First")-mystring("Second"), 2);
7411 cout << e.expand() << endl;
7412 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7415 There's no way to, for example, make GiNaC's @code{add} class perform string
7416 concatenation. You would have to implement this yourself.
7418 @subsection Automatic evaluation
7421 @cindex @code{eval()}
7422 @cindex @code{hold()}
7423 When dealing with objects that are just a little more complicated than the
7424 simple string objects we have implemented, chances are that you will want to
7425 have some automatic simplifications or canonicalizations performed on them.
7426 This is done in the evaluation member function @code{eval()}. Let's say that
7427 we wanted all strings automatically converted to lowercase with
7428 non-alphabetic characters stripped, and empty strings removed:
7431 class mystring : public basic
7435 ex eval(int level = 0) const;
7439 ex mystring::eval(int level) const
7442 for (int i=0; i<str.length(); i++) @{
7444 if (c >= 'A' && c <= 'Z')
7445 new_str += tolower(c);
7446 else if (c >= 'a' && c <= 'z')
7450 if (new_str.length() == 0)
7453 return mystring(new_str).hold();
7457 The @code{level} argument is used to limit the recursion depth of the
7458 evaluation. We don't have any subexpressions in the @code{mystring}
7459 class so we are not concerned with this. If we had, we would call the
7460 @code{eval()} functions of the subexpressions with @code{level - 1} as
7461 the argument if @code{level != 1}. The @code{hold()} member function
7462 sets a flag in the object that prevents further evaluation. Otherwise
7463 we might end up in an endless loop. When you want to return the object
7464 unmodified, use @code{return this->hold();}.
7466 Let's confirm that it works:
7469 ex e = mystring("Hello, world!") + mystring("!?#");
7473 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7478 @subsection Optional member functions
7480 We have implemented only a small set of member functions to make the class
7481 work in the GiNaC framework. There are two functions that are not strictly
7482 required but will make operations with objects of the class more efficient:
7484 @cindex @code{calchash()}
7485 @cindex @code{is_equal_same_type()}
7487 unsigned calchash() const;
7488 bool is_equal_same_type(const basic &other) const;
7491 The @code{calchash()} method returns an @code{unsigned} hash value for the
7492 object which will allow GiNaC to compare and canonicalize expressions much
7493 more efficiently. You should consult the implementation of some of the built-in
7494 GiNaC classes for examples of hash functions. The default implementation of
7495 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7496 class and all subexpressions that are accessible via @code{op()}.
7498 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7499 tests for equality without establishing an ordering relation, which is often
7500 faster. The default implementation of @code{is_equal_same_type()} just calls
7501 @code{compare_same_type()} and tests its result for zero.
7503 @subsection Other member functions
7505 For a real algebraic class, there are probably some more functions that you
7506 might want to provide:
7509 bool info(unsigned inf) const;
7510 ex evalf(int level = 0) const;
7511 ex series(const relational & r, int order, unsigned options = 0) const;
7512 ex derivative(const symbol & s) const;
7515 If your class stores sub-expressions (see the scalar product example in the
7516 previous section) you will probably want to override
7518 @cindex @code{let_op()}
7521 ex op(size_t i) const;
7522 ex & let_op(size_t i);
7523 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7524 ex map(map_function & f) const;
7527 @code{let_op()} is a variant of @code{op()} that allows write access. The
7528 default implementations of @code{subs()} and @code{map()} use it, so you have
7529 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7531 You can, of course, also add your own new member functions. Remember
7532 that the RTTI may be used to get information about what kinds of objects
7533 you are dealing with (the position in the class hierarchy) and that you
7534 can always extract the bare object from an @code{ex} by stripping the
7535 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7536 should become a need.
7538 That's it. May the source be with you!
7541 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7542 @c node-name, next, previous, up
7543 @chapter A Comparison With Other CAS
7546 This chapter will give you some information on how GiNaC compares to
7547 other, traditional Computer Algebra Systems, like @emph{Maple},
7548 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7549 disadvantages over these systems.
7552 * Advantages:: Strengths of the GiNaC approach.
7553 * Disadvantages:: Weaknesses of the GiNaC approach.
7554 * Why C++?:: Attractiveness of C++.
7557 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7558 @c node-name, next, previous, up
7561 GiNaC has several advantages over traditional Computer
7562 Algebra Systems, like
7567 familiar language: all common CAS implement their own proprietary
7568 grammar which you have to learn first (and maybe learn again when your
7569 vendor decides to `enhance' it). With GiNaC you can write your program
7570 in common C++, which is standardized.
7574 structured data types: you can build up structured data types using
7575 @code{struct}s or @code{class}es together with STL features instead of
7576 using unnamed lists of lists of lists.
7579 strongly typed: in CAS, you usually have only one kind of variables
7580 which can hold contents of an arbitrary type. This 4GL like feature is
7581 nice for novice programmers, but dangerous.
7584 development tools: powerful development tools exist for C++, like fancy
7585 editors (e.g. with automatic indentation and syntax highlighting),
7586 debuggers, visualization tools, documentation generators@dots{}
7589 modularization: C++ programs can easily be split into modules by
7590 separating interface and implementation.
7593 price: GiNaC is distributed under the GNU Public License which means
7594 that it is free and available with source code. And there are excellent
7595 C++-compilers for free, too.
7598 extendable: you can add your own classes to GiNaC, thus extending it on
7599 a very low level. Compare this to a traditional CAS that you can
7600 usually only extend on a high level by writing in the language defined
7601 by the parser. In particular, it turns out to be almost impossible to
7602 fix bugs in a traditional system.
7605 multiple interfaces: Though real GiNaC programs have to be written in
7606 some editor, then be compiled, linked and executed, there are more ways
7607 to work with the GiNaC engine. Many people want to play with
7608 expressions interactively, as in traditional CASs. Currently, two such
7609 windows into GiNaC have been implemented and many more are possible: the
7610 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7611 types to a command line and second, as a more consistent approach, an
7612 interactive interface to the Cint C++ interpreter has been put together
7613 (called GiNaC-cint) that allows an interactive scripting interface
7614 consistent with the C++ language. It is available from the usual GiNaC
7618 seamless integration: it is somewhere between difficult and impossible
7619 to call CAS functions from within a program written in C++ or any other
7620 programming language and vice versa. With GiNaC, your symbolic routines
7621 are part of your program. You can easily call third party libraries,
7622 e.g. for numerical evaluation or graphical interaction. All other
7623 approaches are much more cumbersome: they range from simply ignoring the
7624 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7625 system (i.e. @emph{Yacas}).
7628 efficiency: often large parts of a program do not need symbolic
7629 calculations at all. Why use large integers for loop variables or
7630 arbitrary precision arithmetics where @code{int} and @code{double} are
7631 sufficient? For pure symbolic applications, GiNaC is comparable in
7632 speed with other CAS.
7637 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7638 @c node-name, next, previous, up
7639 @section Disadvantages
7641 Of course it also has some disadvantages:
7646 advanced features: GiNaC cannot compete with a program like
7647 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7648 which grows since 1981 by the work of dozens of programmers, with
7649 respect to mathematical features. Integration, factorization,
7650 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7651 not planned for the near future).
7654 portability: While the GiNaC library itself is designed to avoid any
7655 platform dependent features (it should compile on any ANSI compliant C++
7656 compiler), the currently used version of the CLN library (fast large
7657 integer and arbitrary precision arithmetics) can only by compiled
7658 without hassle on systems with the C++ compiler from the GNU Compiler
7659 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7660 macros to let the compiler gather all static initializations, which
7661 works for GNU C++ only. Feel free to contact the authors in case you
7662 really believe that you need to use a different compiler. We have
7663 occasionally used other compilers and may be able to give you advice.}
7664 GiNaC uses recent language features like explicit constructors, mutable
7665 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7666 literally. Recent GCC versions starting at 2.95.3, although itself not
7667 yet ANSI compliant, support all needed features.
7672 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7673 @c node-name, next, previous, up
7676 Why did we choose to implement GiNaC in C++ instead of Java or any other
7677 language? C++ is not perfect: type checking is not strict (casting is
7678 possible), separation between interface and implementation is not
7679 complete, object oriented design is not enforced. The main reason is
7680 the often scolded feature of operator overloading in C++. While it may
7681 be true that operating on classes with a @code{+} operator is rarely
7682 meaningful, it is perfectly suited for algebraic expressions. Writing
7683 @math{3x+5y} as @code{3*x+5*y} instead of
7684 @code{x.times(3).plus(y.times(5))} looks much more natural.
7685 Furthermore, the main developers are more familiar with C++ than with
7686 any other programming language.
7689 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7690 @c node-name, next, previous, up
7691 @appendix Internal Structures
7694 * Expressions are reference counted::
7695 * Internal representation of products and sums::
7698 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7699 @c node-name, next, previous, up
7700 @appendixsection Expressions are reference counted
7702 @cindex reference counting
7703 @cindex copy-on-write
7704 @cindex garbage collection
7705 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7706 where the counter belongs to the algebraic objects derived from class
7707 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7708 which @code{ex} contains an instance. If you understood that, you can safely
7709 skip the rest of this passage.
7711 Expressions are extremely light-weight since internally they work like
7712 handles to the actual representation. They really hold nothing more
7713 than a pointer to some other object. What this means in practice is
7714 that whenever you create two @code{ex} and set the second equal to the
7715 first no copying process is involved. Instead, the copying takes place
7716 as soon as you try to change the second. Consider the simple sequence
7721 #include <ginac/ginac.h>
7722 using namespace std;
7723 using namespace GiNaC;
7727 symbol x("x"), y("y"), z("z");
7730 e1 = sin(x + 2*y) + 3*z + 41;
7731 e2 = e1; // e2 points to same object as e1
7732 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7733 e2 += 1; // e2 is copied into a new object
7734 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7738 The line @code{e2 = e1;} creates a second expression pointing to the
7739 object held already by @code{e1}. The time involved for this operation
7740 is therefore constant, no matter how large @code{e1} was. Actual
7741 copying, however, must take place in the line @code{e2 += 1;} because
7742 @code{e1} and @code{e2} are not handles for the same object any more.
7743 This concept is called @dfn{copy-on-write semantics}. It increases
7744 performance considerably whenever one object occurs multiple times and
7745 represents a simple garbage collection scheme because when an @code{ex}
7746 runs out of scope its destructor checks whether other expressions handle
7747 the object it points to too and deletes the object from memory if that
7748 turns out not to be the case. A slightly less trivial example of
7749 differentiation using the chain-rule should make clear how powerful this
7754 symbol x("x"), y("y");
7758 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7759 cout << e1 << endl // prints x+3*y
7760 << e2 << endl // prints (x+3*y)^3
7761 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7765 Here, @code{e1} will actually be referenced three times while @code{e2}
7766 will be referenced two times. When the power of an expression is built,
7767 that expression needs not be copied. Likewise, since the derivative of
7768 a power of an expression can be easily expressed in terms of that
7769 expression, no copying of @code{e1} is involved when @code{e3} is
7770 constructed. So, when @code{e3} is constructed it will print as
7771 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7772 holds a reference to @code{e2} and the factor in front is just
7775 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7776 semantics. When you insert an expression into a second expression, the
7777 result behaves exactly as if the contents of the first expression were
7778 inserted. But it may be useful to remember that this is not what
7779 happens. Knowing this will enable you to write much more efficient
7780 code. If you still have an uncertain feeling with copy-on-write
7781 semantics, we recommend you have a look at the
7782 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7783 Marshall Cline. Chapter 16 covers this issue and presents an
7784 implementation which is pretty close to the one in GiNaC.
7787 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7788 @c node-name, next, previous, up
7789 @appendixsection Internal representation of products and sums
7791 @cindex representation
7794 @cindex @code{power}
7795 Although it should be completely transparent for the user of
7796 GiNaC a short discussion of this topic helps to understand the sources
7797 and also explain performance to a large degree. Consider the
7798 unexpanded symbolic expression
7800 $2d^3 \left( 4a + 5b - 3 \right)$
7803 @math{2*d^3*(4*a+5*b-3)}
7805 which could naively be represented by a tree of linear containers for
7806 addition and multiplication, one container for exponentiation with base
7807 and exponent and some atomic leaves of symbols and numbers in this
7812 @cindex pair-wise representation
7813 However, doing so results in a rather deeply nested tree which will
7814 quickly become inefficient to manipulate. We can improve on this by
7815 representing the sum as a sequence of terms, each one being a pair of a
7816 purely numeric multiplicative coefficient and its rest. In the same
7817 spirit we can store the multiplication as a sequence of terms, each
7818 having a numeric exponent and a possibly complicated base, the tree
7819 becomes much more flat:
7823 The number @code{3} above the symbol @code{d} shows that @code{mul}
7824 objects are treated similarly where the coefficients are interpreted as
7825 @emph{exponents} now. Addition of sums of terms or multiplication of
7826 products with numerical exponents can be coded to be very efficient with
7827 such a pair-wise representation. Internally, this handling is performed
7828 by most CAS in this way. It typically speeds up manipulations by an
7829 order of magnitude. The overall multiplicative factor @code{2} and the
7830 additive term @code{-3} look somewhat out of place in this
7831 representation, however, since they are still carrying a trivial
7832 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7833 this is avoided by adding a field that carries an overall numeric
7834 coefficient. This results in the realistic picture of internal
7837 $2d^3 \left( 4a + 5b - 3 \right)$:
7840 @math{2*d^3*(4*a+5*b-3)}:
7846 This also allows for a better handling of numeric radicals, since
7847 @code{sqrt(2)} can now be carried along calculations. Now it should be
7848 clear, why both classes @code{add} and @code{mul} are derived from the
7849 same abstract class: the data representation is the same, only the
7850 semantics differs. In the class hierarchy, methods for polynomial
7851 expansion and the like are reimplemented for @code{add} and @code{mul},
7852 but the data structure is inherited from @code{expairseq}.
7855 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7856 @c node-name, next, previous, up
7857 @appendix Package Tools
7859 If you are creating a software package that uses the GiNaC library,
7860 setting the correct command line options for the compiler and linker
7861 can be difficult. GiNaC includes two tools to make this process easier.
7864 * ginac-config:: A shell script to detect compiler and linker flags.
7865 * AM_PATH_GINAC:: Macro for GNU automake.
7869 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7870 @c node-name, next, previous, up
7871 @section @command{ginac-config}
7872 @cindex ginac-config
7874 @command{ginac-config} is a shell script that you can use to determine
7875 the compiler and linker command line options required to compile and
7876 link a program with the GiNaC library.
7878 @command{ginac-config} takes the following flags:
7882 Prints out the version of GiNaC installed.
7884 Prints '-I' flags pointing to the installed header files.
7886 Prints out the linker flags necessary to link a program against GiNaC.
7887 @item --prefix[=@var{PREFIX}]
7888 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7889 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7890 Otherwise, prints out the configured value of @env{$prefix}.
7891 @item --exec-prefix[=@var{PREFIX}]
7892 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7893 Otherwise, prints out the configured value of @env{$exec_prefix}.
7896 Typically, @command{ginac-config} will be used within a configure
7897 script, as described below. It, however, can also be used directly from
7898 the command line using backquotes to compile a simple program. For
7902 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7905 This command line might expand to (for example):
7908 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7909 -lginac -lcln -lstdc++
7912 Not only is the form using @command{ginac-config} easier to type, it will
7913 work on any system, no matter how GiNaC was configured.
7916 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7917 @c node-name, next, previous, up
7918 @section @samp{AM_PATH_GINAC}
7919 @cindex AM_PATH_GINAC
7921 For packages configured using GNU automake, GiNaC also provides
7922 a macro to automate the process of checking for GiNaC.
7925 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7933 Determines the location of GiNaC using @command{ginac-config}, which is
7934 either found in the user's path, or from the environment variable
7935 @env{GINACLIB_CONFIG}.
7938 Tests the installed libraries to make sure that their version
7939 is later than @var{MINIMUM-VERSION}. (A default version will be used
7943 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7944 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7945 variable to the output of @command{ginac-config --libs}, and calls
7946 @samp{AC_SUBST()} for these variables so they can be used in generated
7947 makefiles, and then executes @var{ACTION-IF-FOUND}.
7950 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7951 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7955 This macro is in file @file{ginac.m4} which is installed in
7956 @file{$datadir/aclocal}. Note that if automake was installed with a
7957 different @samp{--prefix} than GiNaC, you will either have to manually
7958 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7959 aclocal the @samp{-I} option when running it.
7962 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7963 * Example package:: Example of a package using AM_PATH_GINAC.
7967 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7968 @c node-name, next, previous, up
7969 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7971 Simply make sure that @command{ginac-config} is in your path, and run
7972 the configure script.
7979 The directory where the GiNaC libraries are installed needs
7980 to be found by your system's dynamic linker.
7982 This is generally done by
7985 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7991 setting the environment variable @env{LD_LIBRARY_PATH},
7994 or, as a last resort,
7997 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7998 running configure, for instance:
8001 LDFLAGS=-R/home/cbauer/lib ./configure
8006 You can also specify a @command{ginac-config} not in your path by
8007 setting the @env{GINACLIB_CONFIG} environment variable to the
8008 name of the executable
8011 If you move the GiNaC package from its installed location,
8012 you will either need to modify @command{ginac-config} script
8013 manually to point to the new location or rebuild GiNaC.
8024 --with-ginac-prefix=@var{PREFIX}
8025 --with-ginac-exec-prefix=@var{PREFIX}
8028 are provided to override the prefix and exec-prefix that were stored
8029 in the @command{ginac-config} shell script by GiNaC's configure. You are
8030 generally better off configuring GiNaC with the right path to begin with.
8034 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8035 @c node-name, next, previous, up
8036 @subsection Example of a package using @samp{AM_PATH_GINAC}
8038 The following shows how to build a simple package using automake
8039 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8043 #include <ginac/ginac.h>
8047 GiNaC::symbol x("x");
8048 GiNaC::ex a = GiNaC::sin(x);
8049 std::cout << "Derivative of " << a
8050 << " is " << a.diff(x) << std::endl;
8055 You should first read the introductory portions of the automake
8056 Manual, if you are not already familiar with it.
8058 Two files are needed, @file{configure.in}, which is used to build the
8062 dnl Process this file with autoconf to produce a configure script.
8064 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8070 AM_PATH_GINAC(0.9.0, [
8071 LIBS="$LIBS $GINACLIB_LIBS"
8072 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8073 ], AC_MSG_ERROR([need to have GiNaC installed]))
8078 The only command in this which is not standard for automake
8079 is the @samp{AM_PATH_GINAC} macro.
8081 That command does the following: If a GiNaC version greater or equal
8082 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8083 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8084 the error message `need to have GiNaC installed'
8086 And the @file{Makefile.am}, which will be used to build the Makefile.
8089 ## Process this file with automake to produce Makefile.in
8090 bin_PROGRAMS = simple
8091 simple_SOURCES = simple.cpp
8094 This @file{Makefile.am}, says that we are building a single executable,
8095 from a single source file @file{simple.cpp}. Since every program
8096 we are building uses GiNaC we simply added the GiNaC options
8097 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8098 want to specify them on a per-program basis: for instance by
8102 simple_LDADD = $(GINACLIB_LIBS)
8103 INCLUDES = $(GINACLIB_CPPFLAGS)
8106 to the @file{Makefile.am}.
8108 To try this example out, create a new directory and add the three
8111 Now execute the following commands:
8114 $ automake --add-missing
8119 You now have a package that can be built in the normal fashion
8128 @node Bibliography, Concept Index, Example package, Top
8129 @c node-name, next, previous, up
8130 @appendix Bibliography
8135 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8138 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8141 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8144 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8147 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8148 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8151 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8152 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8153 Academic Press, London
8156 @cite{Computer Algebra Systems - A Practical Guide},
8157 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8160 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8161 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8164 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8165 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8168 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8173 @node Concept Index, , Bibliography, Top
8174 @c node-name, next, previous, up
8175 @unnumbered Concept Index