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) 1999-2003 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
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2003 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{} 1999-2003 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 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
691 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
692 @c node-name, next, previous, up
694 @cindex expression (class @code{ex})
697 The most common class of objects a user deals with is the expression
698 @code{ex}, representing a mathematical object like a variable, number,
699 function, sum, product, etc@dots{} Expressions may be put together to form
700 new expressions, passed as arguments to functions, and so on. Here is a
701 little collection of valid expressions:
704 ex MyEx1 = 5; // simple number
705 ex MyEx2 = x + 2*y; // polynomial in x and y
706 ex MyEx3 = (x + 1)/(x - 1); // rational expression
707 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
708 ex MyEx5 = MyEx4 + 1; // similar to above
711 Expressions are handles to other more fundamental objects, that often
712 contain other expressions thus creating a tree of expressions
713 (@xref{Internal Structures}, for particular examples). Most methods on
714 @code{ex} therefore run top-down through such an expression tree. For
715 example, the method @code{has()} scans recursively for occurrences of
716 something inside an expression. Thus, if you have declared @code{MyEx4}
717 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
718 the argument of @code{sin} and hence return @code{true}.
720 The next sections will outline the general picture of GiNaC's class
721 hierarchy and describe the classes of objects that are handled by
724 @subsection Note: Expressions and STL containers
726 GiNaC expressions (@code{ex} objects) have value semantics (they can be
727 assigned, reassigned and copied like integral types) but the operator
728 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
729 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
731 This implies that in order to use expressions in sorted containers such as
732 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
733 comparison predicate. GiNaC provides such a predicate, called
734 @code{ex_is_less}. For example, a set of expressions should be defined
735 as @code{std::set<ex, ex_is_less>}.
737 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
738 don't pose a problem. A @code{std::vector<ex>} works as expected.
740 @xref{Information About Expressions}, for more about comparing and ordering
744 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
745 @c node-name, next, previous, up
746 @section Automatic evaluation and canonicalization of expressions
749 GiNaC performs some automatic transformations on expressions, to simplify
750 them and put them into a canonical form. Some examples:
753 ex MyEx1 = 2*x - 1 + x; // 3*x-1
754 ex MyEx2 = x - x; // 0
755 ex MyEx3 = cos(2*Pi); // 1
756 ex MyEx4 = x*y/x; // y
759 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
760 evaluation}. GiNaC only performs transformations that are
764 at most of complexity
772 algebraically correct, possibly except for a set of measure zero (e.g.
773 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
776 There are two types of automatic transformations in GiNaC that may not
777 behave in an entirely obvious way at first glance:
781 The terms of sums and products (and some other things like the arguments of
782 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
783 into a canonical form that is deterministic, but not lexicographical or in
784 any other way easily guessable (it almost always depends on the number and
785 order of the symbols you define). However, constructing the same expression
786 twice, either implicitly or explicitly, will always result in the same
789 Expressions of the form 'number times sum' are automatically expanded (this
790 has to do with GiNaC's internal representation of sums and products). For
793 ex MyEx5 = 2*(x + y); // 2*x+2*y
794 ex MyEx6 = z*(x + y); // z*(x+y)
798 The general rule is that when you construct expressions, GiNaC automatically
799 creates them in canonical form, which might differ from the form you typed in
800 your program. This may create some awkward looking output (@samp{-y+x} instead
801 of @samp{x-y}) but allows for more efficient operation and usually yields
802 some immediate simplifications.
804 @cindex @code{eval()}
805 Internally, the anonymous evaluator in GiNaC is implemented by the methods
808 ex ex::eval(int level = 0) const;
809 ex basic::eval(int level = 0) const;
812 but unless you are extending GiNaC with your own classes or functions, there
813 should never be any reason to call them explicitly. All GiNaC methods that
814 transform expressions, like @code{subs()} or @code{normal()}, automatically
815 re-evaluate their results.
818 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
819 @c node-name, next, previous, up
820 @section Error handling
822 @cindex @code{pole_error} (class)
824 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
825 generated by GiNaC are subclassed from the standard @code{exception} class
826 defined in the @file{<stdexcept>} header. In addition to the predefined
827 @code{logic_error}, @code{domain_error}, @code{out_of_range},
828 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
829 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
830 exception that gets thrown when trying to evaluate a mathematical function
833 The @code{pole_error} class has a member function
836 int pole_error::degree() const;
839 that returns the order of the singularity (or 0 when the pole is
840 logarithmic or the order is undefined).
842 When using GiNaC it is useful to arrange for exceptions to be catched in
843 the main program even if you don't want to do any special error handling.
844 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
845 default exception handler of your C++ compiler's run-time system which
846 usually only aborts the program without giving any information what went
849 Here is an example for a @code{main()} function that catches and prints
850 exceptions generated by GiNaC:
855 #include <ginac/ginac.h>
857 using namespace GiNaC;
865 @} catch (exception &p) @{
866 cerr << p.what() << endl;
874 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
875 @c node-name, next, previous, up
876 @section The Class Hierarchy
878 GiNaC's class hierarchy consists of several classes representing
879 mathematical objects, all of which (except for @code{ex} and some
880 helpers) are internally derived from one abstract base class called
881 @code{basic}. You do not have to deal with objects of class
882 @code{basic}, instead you'll be dealing with symbols, numbers,
883 containers of expressions and so on.
887 To get an idea about what kinds of symbolic composites may be built we
888 have a look at the most important classes in the class hierarchy and
889 some of the relations among the classes:
891 @image{classhierarchy}
893 The abstract classes shown here (the ones without drop-shadow) are of no
894 interest for the user. They are used internally in order to avoid code
895 duplication if two or more classes derived from them share certain
896 features. An example is @code{expairseq}, a container for a sequence of
897 pairs each consisting of one expression and a number (@code{numeric}).
898 What @emph{is} visible to the user are the derived classes @code{add}
899 and @code{mul}, representing sums and products. @xref{Internal
900 Structures}, where these two classes are described in more detail. The
901 following table shortly summarizes what kinds of mathematical objects
902 are stored in the different classes:
905 @multitable @columnfractions .22 .78
906 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
907 @item @code{constant} @tab Constants like
914 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
915 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
916 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
917 @item @code{ncmul} @tab Products of non-commutative objects
918 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
923 @code{sqrt(}@math{2}@code{)}
926 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
927 @item @code{function} @tab A symbolic function like
934 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
935 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
936 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
937 @item @code{indexed} @tab Indexed object like @math{A_ij}
938 @item @code{tensor} @tab Special tensor like the delta and metric tensors
939 @item @code{idx} @tab Index of an indexed object
940 @item @code{varidx} @tab Index with variance
941 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
942 @item @code{wildcard} @tab Wildcard for pattern matching
943 @item @code{structure} @tab Template for user-defined classes
948 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
949 @c node-name, next, previous, up
951 @cindex @code{symbol} (class)
952 @cindex hierarchy of classes
955 Symbols are for symbolic manipulation what atoms are for chemistry. You
956 can declare objects of class @code{symbol} as any other object simply by
957 saying @code{symbol x,y;}. There is, however, a catch in here having to
958 do with the fact that C++ is a compiled language. The information about
959 the symbol's name is thrown away by the compiler but at a later stage
960 you may want to print expressions holding your symbols. In order to
961 avoid confusion GiNaC's symbols are able to know their own name. This
962 is accomplished by declaring its name for output at construction time in
963 the fashion @code{symbol x("x");}. If you declare a symbol using the
964 default constructor (i.e. without string argument) the system will deal
965 out a unique name. That name may not be suitable for printing but for
966 internal routines when no output is desired it is often enough. We'll
967 come across examples of such symbols later in this tutorial.
969 This implies that the strings passed to symbols at construction time may
970 not be used for comparing two of them. It is perfectly legitimate to
971 write @code{symbol x("x"),y("x");} but it is likely to lead into
972 trouble. Here, @code{x} and @code{y} are different symbols and
973 statements like @code{x-y} will not be simplified to zero although the
974 output @code{x-x} looks funny. Such output may also occur when there
975 are two different symbols in two scopes, for instance when you call a
976 function that declares a symbol with a name already existent in a symbol
977 in the calling function. Again, comparing them (using @code{operator==}
978 for instance) will always reveal their difference. Watch out, please.
980 @cindex @code{subs()}
981 Although symbols can be assigned expressions for internal reasons, you
982 should not do it (and we are not going to tell you how it is done). If
983 you want to replace a symbol with something else in an expression, you
984 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
987 @node Numbers, Constants, Symbols, Basic Concepts
988 @c node-name, next, previous, up
990 @cindex @code{numeric} (class)
996 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
997 The classes therein serve as foundation classes for GiNaC. CLN stands
998 for Class Library for Numbers or alternatively for Common Lisp Numbers.
999 In order to find out more about CLN's internals, the reader is referred to
1000 the documentation of that library. @inforef{Introduction, , cln}, for
1001 more information. Suffice to say that it is by itself build on top of
1002 another library, the GNU Multiple Precision library GMP, which is an
1003 extremely fast library for arbitrary long integers and rationals as well
1004 as arbitrary precision floating point numbers. It is very commonly used
1005 by several popular cryptographic applications. CLN extends GMP by
1006 several useful things: First, it introduces the complex number field
1007 over either reals (i.e. floating point numbers with arbitrary precision)
1008 or rationals. Second, it automatically converts rationals to integers
1009 if the denominator is unity and complex numbers to real numbers if the
1010 imaginary part vanishes and also correctly treats algebraic functions.
1011 Third it provides good implementations of state-of-the-art algorithms
1012 for all trigonometric and hyperbolic functions as well as for
1013 calculation of some useful constants.
1015 The user can construct an object of class @code{numeric} in several
1016 ways. The following example shows the four most important constructors.
1017 It uses construction from C-integer, construction of fractions from two
1018 integers, construction from C-float and construction from a string:
1022 #include <ginac/ginac.h>
1023 using namespace GiNaC;
1027 numeric two = 2; // exact integer 2
1028 numeric r(2,3); // exact fraction 2/3
1029 numeric e(2.71828); // floating point number
1030 numeric p = "3.14159265358979323846"; // constructor from string
1031 // Trott's constant in scientific notation:
1032 numeric trott("1.0841015122311136151E-2");
1034 std::cout << two*p << std::endl; // floating point 6.283...
1039 @cindex complex numbers
1040 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1045 numeric z1 = 2-3*I; // exact complex number 2-3i
1046 numeric z2 = 5.9+1.6*I; // complex floating point number
1050 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1051 This would, however, call C's built-in operator @code{/} for integers
1052 first and result in a numeric holding a plain integer 1. @strong{Never
1053 use the operator @code{/} on integers} unless you know exactly what you
1054 are doing! Use the constructor from two integers instead, as shown in
1055 the example above. Writing @code{numeric(1)/2} may look funny but works
1058 @cindex @code{Digits}
1060 We have seen now the distinction between exact numbers and floating
1061 point numbers. Clearly, the user should never have to worry about
1062 dynamically created exact numbers, since their `exactness' always
1063 determines how they ought to be handled, i.e. how `long' they are. The
1064 situation is different for floating point numbers. Their accuracy is
1065 controlled by one @emph{global} variable, called @code{Digits}. (For
1066 those readers who know about Maple: it behaves very much like Maple's
1067 @code{Digits}). All objects of class numeric that are constructed from
1068 then on will be stored with a precision matching that number of decimal
1073 #include <ginac/ginac.h>
1074 using namespace std;
1075 using namespace GiNaC;
1079 numeric three(3.0), one(1.0);
1080 numeric x = one/three;
1082 cout << "in " << Digits << " digits:" << endl;
1084 cout << Pi.evalf() << endl;
1096 The above example prints the following output to screen:
1100 0.33333333333333333334
1101 3.1415926535897932385
1103 0.33333333333333333333333333333333333333333333333333333333333333333334
1104 3.1415926535897932384626433832795028841971693993751058209749445923078
1108 Note that the last number is not necessarily rounded as you would
1109 naively expect it to be rounded in the decimal system. But note also,
1110 that in both cases you got a couple of extra digits. This is because
1111 numbers are internally stored by CLN as chunks of binary digits in order
1112 to match your machine's word size and to not waste precision. Thus, on
1113 architectures with different word size, the above output might even
1114 differ with regard to actually computed digits.
1116 It should be clear that objects of class @code{numeric} should be used
1117 for constructing numbers or for doing arithmetic with them. The objects
1118 one deals with most of the time are the polymorphic expressions @code{ex}.
1120 @subsection Tests on numbers
1122 Once you have declared some numbers, assigned them to expressions and
1123 done some arithmetic with them it is frequently desired to retrieve some
1124 kind of information from them like asking whether that number is
1125 integer, rational, real or complex. For those cases GiNaC provides
1126 several useful methods. (Internally, they fall back to invocations of
1127 certain CLN functions.)
1129 As an example, let's construct some rational number, multiply it with
1130 some multiple of its denominator and test what comes out:
1134 #include <ginac/ginac.h>
1135 using namespace std;
1136 using namespace GiNaC;
1138 // some very important constants:
1139 const numeric twentyone(21);
1140 const numeric ten(10);
1141 const numeric five(5);
1145 numeric answer = twentyone;
1148 cout << answer.is_integer() << endl; // false, it's 21/5
1150 cout << answer.is_integer() << endl; // true, it's 42 now!
1154 Note that the variable @code{answer} is constructed here as an integer
1155 by @code{numeric}'s copy constructor but in an intermediate step it
1156 holds a rational number represented as integer numerator and integer
1157 denominator. When multiplied by 10, the denominator becomes unity and
1158 the result is automatically converted to a pure integer again.
1159 Internally, the underlying CLN is responsible for this behavior and we
1160 refer the reader to CLN's documentation. Suffice to say that
1161 the same behavior applies to complex numbers as well as return values of
1162 certain functions. Complex numbers are automatically converted to real
1163 numbers if the imaginary part becomes zero. The full set of tests that
1164 can be applied is listed in the following table.
1167 @multitable @columnfractions .30 .70
1168 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1169 @item @code{.is_zero()}
1170 @tab @dots{}equal to zero
1171 @item @code{.is_positive()}
1172 @tab @dots{}not complex and greater than 0
1173 @item @code{.is_integer()}
1174 @tab @dots{}a (non-complex) integer
1175 @item @code{.is_pos_integer()}
1176 @tab @dots{}an integer and greater than 0
1177 @item @code{.is_nonneg_integer()}
1178 @tab @dots{}an integer and greater equal 0
1179 @item @code{.is_even()}
1180 @tab @dots{}an even integer
1181 @item @code{.is_odd()}
1182 @tab @dots{}an odd integer
1183 @item @code{.is_prime()}
1184 @tab @dots{}a prime integer (probabilistic primality test)
1185 @item @code{.is_rational()}
1186 @tab @dots{}an exact rational number (integers are rational, too)
1187 @item @code{.is_real()}
1188 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1189 @item @code{.is_cinteger()}
1190 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1191 @item @code{.is_crational()}
1192 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1196 @subsection Converting numbers
1198 Sometimes it is desirable to convert a @code{numeric} object back to a
1199 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1200 class provides a couple of methods for this purpose:
1202 @cindex @code{to_int()}
1203 @cindex @code{to_long()}
1204 @cindex @code{to_double()}
1205 @cindex @code{to_cl_N()}
1207 int numeric::to_int() const;
1208 long numeric::to_long() const;
1209 double numeric::to_double() const;
1210 cln::cl_N numeric::to_cl_N() const;
1213 @code{to_int()} and @code{to_long()} only work when the number they are
1214 applied on is an exact integer. Otherwise the program will halt with a
1215 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1216 rational number will return a floating-point approximation. Both
1217 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1218 part of complex numbers.
1221 @node Constants, Fundamental containers, Numbers, Basic Concepts
1222 @c node-name, next, previous, up
1224 @cindex @code{constant} (class)
1227 @cindex @code{Catalan}
1228 @cindex @code{Euler}
1229 @cindex @code{evalf()}
1230 Constants behave pretty much like symbols except that they return some
1231 specific number when the method @code{.evalf()} is called.
1233 The predefined known constants are:
1236 @multitable @columnfractions .14 .30 .56
1237 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1239 @tab Archimedes' constant
1240 @tab 3.14159265358979323846264338327950288
1241 @item @code{Catalan}
1242 @tab Catalan's constant
1243 @tab 0.91596559417721901505460351493238411
1245 @tab Euler's (or Euler-Mascheroni) constant
1246 @tab 0.57721566490153286060651209008240243
1251 @node Fundamental containers, Lists, Constants, Basic Concepts
1252 @c node-name, next, previous, up
1253 @section Sums, products and powers
1257 @cindex @code{power}
1259 Simple rational expressions are written down in GiNaC pretty much like
1260 in other CAS or like expressions involving numerical variables in C.
1261 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1262 been overloaded to achieve this goal. When you run the following
1263 code snippet, the constructor for an object of type @code{mul} is
1264 automatically called to hold the product of @code{a} and @code{b} and
1265 then the constructor for an object of type @code{add} is called to hold
1266 the sum of that @code{mul} object and the number one:
1270 symbol a("a"), b("b");
1275 @cindex @code{pow()}
1276 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1277 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1278 construction is necessary since we cannot safely overload the constructor
1279 @code{^} in C++ to construct a @code{power} object. If we did, it would
1280 have several counterintuitive and undesired effects:
1284 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1286 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1287 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1288 interpret this as @code{x^(a^b)}.
1290 Also, expressions involving integer exponents are very frequently used,
1291 which makes it even more dangerous to overload @code{^} since it is then
1292 hard to distinguish between the semantics as exponentiation and the one
1293 for exclusive or. (It would be embarrassing to return @code{1} where one
1294 has requested @code{2^3}.)
1297 @cindex @command{ginsh}
1298 All effects are contrary to mathematical notation and differ from the
1299 way most other CAS handle exponentiation, therefore overloading @code{^}
1300 is ruled out for GiNaC's C++ part. The situation is different in
1301 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1302 that the other frequently used exponentiation operator @code{**} does
1303 not exist at all in C++).
1305 To be somewhat more precise, objects of the three classes described
1306 here, are all containers for other expressions. An object of class
1307 @code{power} is best viewed as a container with two slots, one for the
1308 basis, one for the exponent. All valid GiNaC expressions can be
1309 inserted. However, basic transformations like simplifying
1310 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1311 when this is mathematically possible. If we replace the outer exponent
1312 three in the example by some symbols @code{a}, the simplification is not
1313 safe and will not be performed, since @code{a} might be @code{1/2} and
1316 Objects of type @code{add} and @code{mul} are containers with an
1317 arbitrary number of slots for expressions to be inserted. Again, simple
1318 and safe simplifications are carried out like transforming
1319 @code{3*x+4-x} to @code{2*x+4}.
1322 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1323 @c node-name, next, previous, up
1324 @section Lists of expressions
1325 @cindex @code{lst} (class)
1327 @cindex @code{nops()}
1329 @cindex @code{append()}
1330 @cindex @code{prepend()}
1331 @cindex @code{remove_first()}
1332 @cindex @code{remove_last()}
1333 @cindex @code{remove_all()}
1335 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1336 expressions. They are not as ubiquitous as in many other computer algebra
1337 packages, but are sometimes used to supply a variable number of arguments of
1338 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1339 constructors, so you should have a basic understanding of them.
1341 Lists can be constructed by assigning a comma-separated sequence of
1346 symbol x("x"), y("y");
1349 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1354 There are also constructors that allow direct creation of lists of up to
1355 16 expressions, which is often more convenient but slightly less efficient:
1359 // This produces the same list 'l' as above:
1360 // lst l(x, 2, y, x+y);
1361 // lst l = lst(x, 2, y, x+y);
1365 Use the @code{nops()} method to determine the size (number of expressions) of
1366 a list and the @code{op()} method or the @code{[]} operator to access
1367 individual elements:
1371 cout << l.nops() << endl; // prints '4'
1372 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1376 As with the standard @code{list<T>} container, accessing random elements of a
1377 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1378 sequential access to the elements of a list is possible with the
1379 iterator types provided by the @code{lst} class:
1382 typedef ... lst::const_iterator;
1383 typedef ... lst::const_reverse_iterator;
1384 lst::const_iterator lst::begin() const;
1385 lst::const_iterator lst::end() const;
1386 lst::const_reverse_iterator lst::rbegin() const;
1387 lst::const_reverse_iterator lst::rend() const;
1390 For example, to print the elements of a list individually you can use:
1395 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1400 which is one order faster than
1405 for (size_t i = 0; i < l.nops(); ++i)
1406 cout << l.op(i) << endl;
1410 These iterators also allow you to use some of the algorithms provided by
1411 the C++ standard library:
1415 // print the elements of the list (requires #include <iterator>)
1416 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1418 // sum up the elements of the list (requires #include <numeric>)
1419 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1420 cout << sum << endl; // prints '2+2*x+2*y'
1424 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1425 (the only other one is @code{matrix}). You can modify single elements:
1429 l[1] = 42; // l is now @{x, 42, y, x+y@}
1430 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1434 You can append or prepend an expression to a list with the @code{append()}
1435 and @code{prepend()} methods:
1439 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1440 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1444 You can remove the first or last element of a list with @code{remove_first()}
1445 and @code{remove_last()}:
1449 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1450 l.remove_last(); // l is now @{x, 7, y, x+y@}
1454 You can remove all the elements of a list with @code{remove_all()}:
1458 l.remove_all(); // l is now empty
1462 You can bring the elements of a list into a canonical order with @code{sort()}:
1471 // l1 and l2 are now equal
1475 Finally, you can remove all but the first element of consecutive groups of
1476 elements with @code{unique()}:
1481 l3 = x, 2, 2, 2, y, x+y, y+x;
1482 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1487 @node Mathematical functions, Relations, Lists, Basic Concepts
1488 @c node-name, next, previous, up
1489 @section Mathematical functions
1490 @cindex @code{function} (class)
1491 @cindex trigonometric function
1492 @cindex hyperbolic function
1494 There are quite a number of useful functions hard-wired into GiNaC. For
1495 instance, all trigonometric and hyperbolic functions are implemented
1496 (@xref{Built-in Functions}, for a complete list).
1498 These functions (better called @emph{pseudofunctions}) are all objects
1499 of class @code{function}. They accept one or more expressions as
1500 arguments and return one expression. If the arguments are not
1501 numerical, the evaluation of the function may be halted, as it does in
1502 the next example, showing how a function returns itself twice and
1503 finally an expression that may be really useful:
1505 @cindex Gamma function
1506 @cindex @code{subs()}
1509 symbol x("x"), y("y");
1511 cout << tgamma(foo) << endl;
1512 // -> tgamma(x+(1/2)*y)
1513 ex bar = foo.subs(y==1);
1514 cout << tgamma(bar) << endl;
1516 ex foobar = bar.subs(x==7);
1517 cout << tgamma(foobar) << endl;
1518 // -> (135135/128)*Pi^(1/2)
1522 Besides evaluation most of these functions allow differentiation, series
1523 expansion and so on. Read the next chapter in order to learn more about
1526 It must be noted that these pseudofunctions are created by inline
1527 functions, where the argument list is templated. This means that
1528 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1529 @code{sin(ex(1))} and will therefore not result in a floating point
1530 number. Unless of course the function prototype is explicitly
1531 overridden -- which is the case for arguments of type @code{numeric}
1532 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1533 point number of class @code{numeric} you should call
1534 @code{sin(numeric(1))}. This is almost the same as calling
1535 @code{sin(1).evalf()} except that the latter will return a numeric
1536 wrapped inside an @code{ex}.
1539 @node Relations, Matrices, Mathematical functions, Basic Concepts
1540 @c node-name, next, previous, up
1542 @cindex @code{relational} (class)
1544 Sometimes, a relation holding between two expressions must be stored
1545 somehow. The class @code{relational} is a convenient container for such
1546 purposes. A relation is by definition a container for two @code{ex} and
1547 a relation between them that signals equality, inequality and so on.
1548 They are created by simply using the C++ operators @code{==}, @code{!=},
1549 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1551 @xref{Mathematical functions}, for examples where various applications
1552 of the @code{.subs()} method show how objects of class relational are
1553 used as arguments. There they provide an intuitive syntax for
1554 substitutions. They are also used as arguments to the @code{ex::series}
1555 method, where the left hand side of the relation specifies the variable
1556 to expand in and the right hand side the expansion point. They can also
1557 be used for creating systems of equations that are to be solved for
1558 unknown variables. But the most common usage of objects of this class
1559 is rather inconspicuous in statements of the form @code{if
1560 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1561 conversion from @code{relational} to @code{bool} takes place. Note,
1562 however, that @code{==} here does not perform any simplifications, hence
1563 @code{expand()} must be called explicitly.
1566 @node Matrices, Indexed objects, Relations, Basic Concepts
1567 @c node-name, next, previous, up
1569 @cindex @code{matrix} (class)
1571 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1572 matrix with @math{m} rows and @math{n} columns are accessed with two
1573 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1574 second one in the range 0@dots{}@math{n-1}.
1576 There are a couple of ways to construct matrices, with or without preset
1577 elements. The constructor
1580 matrix::matrix(unsigned r, unsigned c);
1583 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1586 The fastest way to create a matrix with preinitialized elements is to assign
1587 a list of comma-separated expressions to an empty matrix (see below for an
1588 example). But you can also specify the elements as a (flat) list with
1591 matrix::matrix(unsigned r, unsigned c, const lst & l);
1596 @cindex @code{lst_to_matrix()}
1598 ex lst_to_matrix(const lst & l);
1601 constructs a matrix from a list of lists, each list representing a matrix row.
1603 There is also a set of functions for creating some special types of
1606 @cindex @code{diag_matrix()}
1607 @cindex @code{unit_matrix()}
1608 @cindex @code{symbolic_matrix()}
1610 ex diag_matrix(const lst & l);
1611 ex unit_matrix(unsigned x);
1612 ex unit_matrix(unsigned r, unsigned c);
1613 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1614 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1617 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1618 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1619 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1620 matrix filled with newly generated symbols made of the specified base name
1621 and the position of each element in the matrix.
1623 Matrix elements can be accessed and set using the parenthesis (function call)
1627 const ex & matrix::operator()(unsigned r, unsigned c) const;
1628 ex & matrix::operator()(unsigned r, unsigned c);
1631 It is also possible to access the matrix elements in a linear fashion with
1632 the @code{op()} method. But C++-style subscripting with square brackets
1633 @samp{[]} is not available.
1635 Here are a couple of examples for constructing matrices:
1639 symbol a("a"), b("b");
1653 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1656 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1659 cout << diag_matrix(lst(a, b)) << endl;
1662 cout << unit_matrix(3) << endl;
1663 // -> [[1,0,0],[0,1,0],[0,0,1]]
1665 cout << symbolic_matrix(2, 3, "x") << endl;
1666 // -> [[x00,x01,x02],[x10,x11,x12]]
1670 @cindex @code{transpose()}
1671 There are three ways to do arithmetic with matrices. The first (and most
1672 direct one) is to use the methods provided by the @code{matrix} class:
1675 matrix matrix::add(const matrix & other) const;
1676 matrix matrix::sub(const matrix & other) const;
1677 matrix matrix::mul(const matrix & other) const;
1678 matrix matrix::mul_scalar(const ex & other) const;
1679 matrix matrix::pow(const ex & expn) const;
1680 matrix matrix::transpose() const;
1683 All of these methods return the result as a new matrix object. Here is an
1684 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1689 matrix A(2, 2), B(2, 2), C(2, 2);
1697 matrix result = A.mul(B).sub(C.mul_scalar(2));
1698 cout << result << endl;
1699 // -> [[-13,-6],[1,2]]
1704 @cindex @code{evalm()}
1705 The second (and probably the most natural) way is to construct an expression
1706 containing matrices with the usual arithmetic operators and @code{pow()}.
1707 For efficiency reasons, expressions with sums, products and powers of
1708 matrices are not automatically evaluated in GiNaC. You have to call the
1712 ex ex::evalm() const;
1715 to obtain the result:
1722 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1723 cout << e.evalm() << endl;
1724 // -> [[-13,-6],[1,2]]
1729 The non-commutativity of the product @code{A*B} in this example is
1730 automatically recognized by GiNaC. There is no need to use a special
1731 operator here. @xref{Non-commutative objects}, for more information about
1732 dealing with non-commutative expressions.
1734 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1735 to perform the arithmetic:
1740 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1741 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1743 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1744 cout << e.simplify_indexed() << endl;
1745 // -> [[-13,-6],[1,2]].i.j
1749 Using indices is most useful when working with rectangular matrices and
1750 one-dimensional vectors because you don't have to worry about having to
1751 transpose matrices before multiplying them. @xref{Indexed objects}, for
1752 more information about using matrices with indices, and about indices in
1755 The @code{matrix} class provides a couple of additional methods for
1756 computing determinants, traces, and characteristic polynomials:
1758 @cindex @code{determinant()}
1759 @cindex @code{trace()}
1760 @cindex @code{charpoly()}
1762 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1763 ex matrix::trace() const;
1764 ex matrix::charpoly(const ex & lambda) const;
1767 The @samp{algo} argument of @code{determinant()} allows to select
1768 between different algorithms for calculating the determinant. The
1769 asymptotic speed (as parametrized by the matrix size) can greatly differ
1770 between those algorithms, depending on the nature of the matrix'
1771 entries. The possible values are defined in the @file{flags.h} header
1772 file. By default, GiNaC uses a heuristic to automatically select an
1773 algorithm that is likely (but not guaranteed) to give the result most
1776 @cindex @code{inverse()}
1777 @cindex @code{solve()}
1778 Matrices may also be inverted using the @code{ex matrix::inverse()}
1779 method and linear systems may be solved with:
1782 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1785 Assuming the matrix object this method is applied on is an @code{m}
1786 times @code{n} matrix, then @code{vars} must be a @code{n} times
1787 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1788 times @code{p} matrix. The returned matrix then has dimension @code{n}
1789 times @code{p} and in the case of an underdetermined system will still
1790 contain some of the indeterminates from @code{vars}. If the system is
1791 overdetermined, an exception is thrown.
1794 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1795 @c node-name, next, previous, up
1796 @section Indexed objects
1798 GiNaC allows you to handle expressions containing general indexed objects in
1799 arbitrary spaces. It is also able to canonicalize and simplify such
1800 expressions and perform symbolic dummy index summations. There are a number
1801 of predefined indexed objects provided, like delta and metric tensors.
1803 There are few restrictions placed on indexed objects and their indices and
1804 it is easy to construct nonsense expressions, but our intention is to
1805 provide a general framework that allows you to implement algorithms with
1806 indexed quantities, getting in the way as little as possible.
1808 @cindex @code{idx} (class)
1809 @cindex @code{indexed} (class)
1810 @subsection Indexed quantities and their indices
1812 Indexed expressions in GiNaC are constructed of two special types of objects,
1813 @dfn{index objects} and @dfn{indexed objects}.
1817 @cindex contravariant
1820 @item Index objects are of class @code{idx} or a subclass. Every index has
1821 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1822 the index lives in) which can both be arbitrary expressions but are usually
1823 a number or a simple symbol. In addition, indices of class @code{varidx} have
1824 a @dfn{variance} (they can be co- or contravariant), and indices of class
1825 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1827 @item Indexed objects are of class @code{indexed} or a subclass. They
1828 contain a @dfn{base expression} (which is the expression being indexed), and
1829 one or more indices.
1833 @strong{Note:} when printing expressions, covariant indices and indices
1834 without variance are denoted @samp{.i} while contravariant indices are
1835 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1836 value. In the following, we are going to use that notation in the text so
1837 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1838 not visible in the output.
1840 A simple example shall illustrate the concepts:
1844 #include <ginac/ginac.h>
1845 using namespace std;
1846 using namespace GiNaC;
1850 symbol i_sym("i"), j_sym("j");
1851 idx i(i_sym, 3), j(j_sym, 3);
1854 cout << indexed(A, i, j) << endl;
1856 cout << index_dimensions << indexed(A, i, j) << endl;
1858 cout << dflt; // reset cout to default output format (dimensions hidden)
1862 The @code{idx} constructor takes two arguments, the index value and the
1863 index dimension. First we define two index objects, @code{i} and @code{j},
1864 both with the numeric dimension 3. The value of the index @code{i} is the
1865 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1866 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1867 construct an expression containing one indexed object, @samp{A.i.j}. It has
1868 the symbol @code{A} as its base expression and the two indices @code{i} and
1871 The dimensions of indices are normally not visible in the output, but one
1872 can request them to be printed with the @code{index_dimensions} manipulator,
1875 Note the difference between the indices @code{i} and @code{j} which are of
1876 class @code{idx}, and the index values which are the symbols @code{i_sym}
1877 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1878 or numbers but must be index objects. For example, the following is not
1879 correct and will raise an exception:
1882 symbol i("i"), j("j");
1883 e = indexed(A, i, j); // ERROR: indices must be of type idx
1886 You can have multiple indexed objects in an expression, index values can
1887 be numeric, and index dimensions symbolic:
1891 symbol B("B"), dim("dim");
1892 cout << 4 * indexed(A, i)
1893 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1898 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1899 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1900 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1901 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1902 @code{simplify_indexed()} for that, see below).
1904 In fact, base expressions, index values and index dimensions can be
1905 arbitrary expressions:
1909 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1914 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1915 get an error message from this but you will probably not be able to do
1916 anything useful with it.
1918 @cindex @code{get_value()}
1919 @cindex @code{get_dimension()}
1923 ex idx::get_value();
1924 ex idx::get_dimension();
1927 return the value and dimension of an @code{idx} object. If you have an index
1928 in an expression, such as returned by calling @code{.op()} on an indexed
1929 object, you can get a reference to the @code{idx} object with the function
1930 @code{ex_to<idx>()} on the expression.
1932 There are also the methods
1935 bool idx::is_numeric();
1936 bool idx::is_symbolic();
1937 bool idx::is_dim_numeric();
1938 bool idx::is_dim_symbolic();
1941 for checking whether the value and dimension are numeric or symbolic
1942 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1943 About Expressions}) returns information about the index value.
1945 @cindex @code{varidx} (class)
1946 If you need co- and contravariant indices, use the @code{varidx} class:
1950 symbol mu_sym("mu"), nu_sym("nu");
1951 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1952 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1954 cout << indexed(A, mu, nu) << endl;
1956 cout << indexed(A, mu_co, nu) << endl;
1958 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1963 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1964 co- or contravariant. The default is a contravariant (upper) index, but
1965 this can be overridden by supplying a third argument to the @code{varidx}
1966 constructor. The two methods
1969 bool varidx::is_covariant();
1970 bool varidx::is_contravariant();
1973 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1974 to get the object reference from an expression). There's also the very useful
1978 ex varidx::toggle_variance();
1981 which makes a new index with the same value and dimension but the opposite
1982 variance. By using it you only have to define the index once.
1984 @cindex @code{spinidx} (class)
1985 The @code{spinidx} class provides dotted and undotted variant indices, as
1986 used in the Weyl-van-der-Waerden spinor formalism:
1990 symbol K("K"), C_sym("C"), D_sym("D");
1991 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1992 // contravariant, undotted
1993 spinidx C_co(C_sym, 2, true); // covariant index
1994 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1995 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1997 cout << indexed(K, C, D) << endl;
1999 cout << indexed(K, C_co, D_dot) << endl;
2001 cout << indexed(K, D_co_dot, D) << endl;
2006 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2007 dotted or undotted. The default is undotted but this can be overridden by
2008 supplying a fourth argument to the @code{spinidx} constructor. The two
2012 bool spinidx::is_dotted();
2013 bool spinidx::is_undotted();
2016 allow you to check whether or not a @code{spinidx} object is dotted (use
2017 @code{ex_to<spinidx>()} to get the object reference from an expression).
2018 Finally, the two methods
2021 ex spinidx::toggle_dot();
2022 ex spinidx::toggle_variance_dot();
2025 create a new index with the same value and dimension but opposite dottedness
2026 and the same or opposite variance.
2028 @subsection Substituting indices
2030 @cindex @code{subs()}
2031 Sometimes you will want to substitute one symbolic index with another
2032 symbolic or numeric index, for example when calculating one specific element
2033 of a tensor expression. This is done with the @code{.subs()} method, as it
2034 is done for symbols (see @ref{Substituting Expressions}).
2036 You have two possibilities here. You can either substitute the whole index
2037 by another index or expression:
2041 ex e = indexed(A, mu_co);
2042 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2043 // -> A.mu becomes A~nu
2044 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2045 // -> A.mu becomes A~0
2046 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2047 // -> A.mu becomes A.0
2051 The third example shows that trying to replace an index with something that
2052 is not an index will substitute the index value instead.
2054 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2059 ex e = indexed(A, mu_co);
2060 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2061 // -> A.mu becomes A.nu
2062 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2063 // -> A.mu becomes A.0
2067 As you see, with the second method only the value of the index will get
2068 substituted. Its other properties, including its dimension, remain unchanged.
2069 If you want to change the dimension of an index you have to substitute the
2070 whole index by another one with the new dimension.
2072 Finally, substituting the base expression of an indexed object works as
2077 ex e = indexed(A, mu_co);
2078 cout << e << " becomes " << e.subs(A == A+B) << endl;
2079 // -> A.mu becomes (B+A).mu
2083 @subsection Symmetries
2084 @cindex @code{symmetry} (class)
2085 @cindex @code{sy_none()}
2086 @cindex @code{sy_symm()}
2087 @cindex @code{sy_anti()}
2088 @cindex @code{sy_cycl()}
2090 Indexed objects can have certain symmetry properties with respect to their
2091 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2092 that is constructed with the helper functions
2095 symmetry sy_none(...);
2096 symmetry sy_symm(...);
2097 symmetry sy_anti(...);
2098 symmetry sy_cycl(...);
2101 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2102 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2103 represents a cyclic symmetry. Each of these functions accepts up to four
2104 arguments which can be either symmetry objects themselves or unsigned integer
2105 numbers that represent an index position (counting from 0). A symmetry
2106 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2107 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2110 Here are some examples of symmetry definitions:
2115 e = indexed(A, i, j);
2116 e = indexed(A, sy_none(), i, j); // equivalent
2117 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2119 // Symmetric in all three indices:
2120 e = indexed(A, sy_symm(), i, j, k);
2121 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2122 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2123 // different canonical order
2125 // Symmetric in the first two indices only:
2126 e = indexed(A, sy_symm(0, 1), i, j, k);
2127 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2129 // Antisymmetric in the first and last index only (index ranges need not
2131 e = indexed(A, sy_anti(0, 2), i, j, k);
2132 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2134 // An example of a mixed symmetry: antisymmetric in the first two and
2135 // last two indices, symmetric when swapping the first and last index
2136 // pairs (like the Riemann curvature tensor):
2137 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2139 // Cyclic symmetry in all three indices:
2140 e = indexed(A, sy_cycl(), i, j, k);
2141 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2143 // The following examples are invalid constructions that will throw
2144 // an exception at run time.
2146 // An index may not appear multiple times:
2147 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2148 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2150 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2151 // same number of indices:
2152 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2154 // And of course, you cannot specify indices which are not there:
2155 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2159 If you need to specify more than four indices, you have to use the
2160 @code{.add()} method of the @code{symmetry} class. For example, to specify
2161 full symmetry in the first six indices you would write
2162 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2164 If an indexed object has a symmetry, GiNaC will automatically bring the
2165 indices into a canonical order which allows for some immediate simplifications:
2169 cout << indexed(A, sy_symm(), i, j)
2170 + indexed(A, sy_symm(), j, i) << endl;
2172 cout << indexed(B, sy_anti(), i, j)
2173 + indexed(B, sy_anti(), j, i) << endl;
2175 cout << indexed(B, sy_anti(), i, j, k)
2176 - indexed(B, sy_anti(), j, k, i) << endl;
2181 @cindex @code{get_free_indices()}
2183 @subsection Dummy indices
2185 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2186 that a summation over the index range is implied. Symbolic indices which are
2187 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2188 dummy nor free indices.
2190 To be recognized as a dummy index pair, the two indices must be of the same
2191 class and their value must be the same single symbol (an index like
2192 @samp{2*n+1} is never a dummy index). If the indices are of class
2193 @code{varidx} they must also be of opposite variance; if they are of class
2194 @code{spinidx} they must be both dotted or both undotted.
2196 The method @code{.get_free_indices()} returns a vector containing the free
2197 indices of an expression. It also checks that the free indices of the terms
2198 of a sum are consistent:
2202 symbol A("A"), B("B"), C("C");
2204 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2205 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2207 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2208 cout << exprseq(e.get_free_indices()) << endl;
2210 // 'j' and 'l' are dummy indices
2212 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2213 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2215 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2216 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2217 cout << exprseq(e.get_free_indices()) << endl;
2219 // 'nu' is a dummy index, but 'sigma' is not
2221 e = indexed(A, mu, mu);
2222 cout << exprseq(e.get_free_indices()) << endl;
2224 // 'mu' is not a dummy index because it appears twice with the same
2227 e = indexed(A, mu, nu) + 42;
2228 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2229 // this will throw an exception:
2230 // "add::get_free_indices: inconsistent indices in sum"
2234 @cindex @code{simplify_indexed()}
2235 @subsection Simplifying indexed expressions
2237 In addition to the few automatic simplifications that GiNaC performs on
2238 indexed expressions (such as re-ordering the indices of symmetric tensors
2239 and calculating traces and convolutions of matrices and predefined tensors)
2243 ex ex::simplify_indexed();
2244 ex ex::simplify_indexed(const scalar_products & sp);
2247 that performs some more expensive operations:
2250 @item it checks the consistency of free indices in sums in the same way
2251 @code{get_free_indices()} does
2252 @item it tries to give dummy indices that appear in different terms of a sum
2253 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2254 @item it (symbolically) calculates all possible dummy index summations/contractions
2255 with the predefined tensors (this will be explained in more detail in the
2257 @item it detects contractions that vanish for symmetry reasons, for example
2258 the contraction of a symmetric and a totally antisymmetric tensor
2259 @item as a special case of dummy index summation, it can replace scalar products
2260 of two tensors with a user-defined value
2263 The last point is done with the help of the @code{scalar_products} class
2264 which is used to store scalar products with known values (this is not an
2265 arithmetic class, you just pass it to @code{simplify_indexed()}):
2269 symbol A("A"), B("B"), C("C"), i_sym("i");
2273 sp.add(A, B, 0); // A and B are orthogonal
2274 sp.add(A, C, 0); // A and C are orthogonal
2275 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2277 e = indexed(A + B, i) * indexed(A + C, i);
2279 // -> (B+A).i*(A+C).i
2281 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2287 The @code{scalar_products} object @code{sp} acts as a storage for the
2288 scalar products added to it with the @code{.add()} method. This method
2289 takes three arguments: the two expressions of which the scalar product is
2290 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2291 @code{simplify_indexed()} will replace all scalar products of indexed
2292 objects that have the symbols @code{A} and @code{B} as base expressions
2293 with the single value 0. The number, type and dimension of the indices
2294 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2296 @cindex @code{expand()}
2297 The example above also illustrates a feature of the @code{expand()} method:
2298 if passed the @code{expand_indexed} option it will distribute indices
2299 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2301 @cindex @code{tensor} (class)
2302 @subsection Predefined tensors
2304 Some frequently used special tensors such as the delta, epsilon and metric
2305 tensors are predefined in GiNaC. They have special properties when
2306 contracted with other tensor expressions and some of them have constant
2307 matrix representations (they will evaluate to a number when numeric
2308 indices are specified).
2310 @cindex @code{delta_tensor()}
2311 @subsubsection Delta tensor
2313 The delta tensor takes two indices, is symmetric and has the matrix
2314 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2315 @code{delta_tensor()}:
2319 symbol A("A"), B("B");
2321 idx i(symbol("i"), 3), j(symbol("j"), 3),
2322 k(symbol("k"), 3), l(symbol("l"), 3);
2324 ex e = indexed(A, i, j) * indexed(B, k, l)
2325 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2326 cout << e.simplify_indexed() << endl;
2329 cout << delta_tensor(i, i) << endl;
2334 @cindex @code{metric_tensor()}
2335 @subsubsection General metric tensor
2337 The function @code{metric_tensor()} creates a general symmetric metric
2338 tensor with two indices that can be used to raise/lower tensor indices. The
2339 metric tensor is denoted as @samp{g} in the output and if its indices are of
2340 mixed variance it is automatically replaced by a delta tensor:
2346 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2348 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2349 cout << e.simplify_indexed() << endl;
2352 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2353 cout << e.simplify_indexed() << endl;
2356 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2357 * metric_tensor(nu, rho);
2358 cout << e.simplify_indexed() << endl;
2361 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2362 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2363 + indexed(A, mu.toggle_variance(), rho));
2364 cout << e.simplify_indexed() << endl;
2369 @cindex @code{lorentz_g()}
2370 @subsubsection Minkowski metric tensor
2372 The Minkowski metric tensor is a special metric tensor with a constant
2373 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2374 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2375 It is created with the function @code{lorentz_g()} (although it is output as
2380 varidx mu(symbol("mu"), 4);
2382 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2383 * lorentz_g(mu, varidx(0, 4)); // negative signature
2384 cout << e.simplify_indexed() << endl;
2387 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2388 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2389 cout << e.simplify_indexed() << endl;
2394 @cindex @code{spinor_metric()}
2395 @subsubsection Spinor metric tensor
2397 The function @code{spinor_metric()} creates an antisymmetric tensor with
2398 two indices that is used to raise/lower indices of 2-component spinors.
2399 It is output as @samp{eps}:
2405 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2406 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2408 e = spinor_metric(A, B) * indexed(psi, B_co);
2409 cout << e.simplify_indexed() << endl;
2412 e = spinor_metric(A, B) * indexed(psi, A_co);
2413 cout << e.simplify_indexed() << endl;
2416 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2417 cout << e.simplify_indexed() << endl;
2420 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2421 cout << e.simplify_indexed() << endl;
2424 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2425 cout << e.simplify_indexed() << endl;
2428 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2429 cout << e.simplify_indexed() << endl;
2434 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2436 @cindex @code{epsilon_tensor()}
2437 @cindex @code{lorentz_eps()}
2438 @subsubsection Epsilon tensor
2440 The epsilon tensor is totally antisymmetric, its number of indices is equal
2441 to the dimension of the index space (the indices must all be of the same
2442 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2443 defined to be 1. Its behavior with indices that have a variance also
2444 depends on the signature of the metric. Epsilon tensors are output as
2447 There are three functions defined to create epsilon tensors in 2, 3 and 4
2451 ex epsilon_tensor(const ex & i1, const ex & i2);
2452 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2453 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2456 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2457 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2458 Minkowski space (the last @code{bool} argument specifies whether the metric
2459 has negative or positive signature, as in the case of the Minkowski metric
2464 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2465 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2466 e = lorentz_eps(mu, nu, rho, sig) *
2467 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2468 cout << simplify_indexed(e) << endl;
2469 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2471 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2472 symbol A("A"), B("B");
2473 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2474 cout << simplify_indexed(e) << endl;
2475 // -> -B.k*A.j*eps.i.k.j
2476 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2477 cout << simplify_indexed(e) << endl;
2482 @subsection Linear algebra
2484 The @code{matrix} class can be used with indices to do some simple linear
2485 algebra (linear combinations and products of vectors and matrices, traces
2486 and scalar products):
2490 idx i(symbol("i"), 2), j(symbol("j"), 2);
2491 symbol x("x"), y("y");
2493 // A is a 2x2 matrix, X is a 2x1 vector
2494 matrix A(2, 2), X(2, 1);
2499 cout << indexed(A, i, i) << endl;
2502 ex e = indexed(A, i, j) * indexed(X, j);
2503 cout << e.simplify_indexed() << endl;
2504 // -> [[2*y+x],[4*y+3*x]].i
2506 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2507 cout << e.simplify_indexed() << endl;
2508 // -> [[3*y+3*x,6*y+2*x]].j
2512 You can of course obtain the same results with the @code{matrix::add()},
2513 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2514 but with indices you don't have to worry about transposing matrices.
2516 Matrix indices always start at 0 and their dimension must match the number
2517 of rows/columns of the matrix. Matrices with one row or one column are
2518 vectors and can have one or two indices (it doesn't matter whether it's a
2519 row or a column vector). Other matrices must have two indices.
2521 You should be careful when using indices with variance on matrices. GiNaC
2522 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2523 @samp{F.mu.nu} are different matrices. In this case you should use only
2524 one form for @samp{F} and explicitly multiply it with a matrix representation
2525 of the metric tensor.
2528 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2529 @c node-name, next, previous, up
2530 @section Non-commutative objects
2532 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2533 non-commutative objects are built-in which are mostly of use in high energy
2537 @item Clifford (Dirac) algebra (class @code{clifford})
2538 @item su(3) Lie algebra (class @code{color})
2539 @item Matrices (unindexed) (class @code{matrix})
2542 The @code{clifford} and @code{color} classes are subclasses of
2543 @code{indexed} because the elements of these algebras usually carry
2544 indices. The @code{matrix} class is described in more detail in
2547 Unlike most computer algebra systems, GiNaC does not primarily provide an
2548 operator (often denoted @samp{&*}) for representing inert products of
2549 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2550 classes of objects involved, and non-commutative products are formed with
2551 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2552 figuring out by itself which objects commute and will group the factors
2553 by their class. Consider this example:
2557 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2558 idx a(symbol("a"), 8), b(symbol("b"), 8);
2559 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2561 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2565 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2566 groups the non-commutative factors (the gammas and the su(3) generators)
2567 together while preserving the order of factors within each class (because
2568 Clifford objects commute with color objects). The resulting expression is a
2569 @emph{commutative} product with two factors that are themselves non-commutative
2570 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2571 parentheses are placed around the non-commutative products in the output.
2573 @cindex @code{ncmul} (class)
2574 Non-commutative products are internally represented by objects of the class
2575 @code{ncmul}, as opposed to commutative products which are handled by the
2576 @code{mul} class. You will normally not have to worry about this distinction,
2579 The advantage of this approach is that you never have to worry about using
2580 (or forgetting to use) a special operator when constructing non-commutative
2581 expressions. Also, non-commutative products in GiNaC are more intelligent
2582 than in other computer algebra systems; they can, for example, automatically
2583 canonicalize themselves according to rules specified in the implementation
2584 of the non-commutative classes. The drawback is that to work with other than
2585 the built-in algebras you have to implement new classes yourself. Symbols
2586 always commute and it's not possible to construct non-commutative products
2587 using symbols to represent the algebra elements or generators. User-defined
2588 functions can, however, be specified as being non-commutative.
2590 @cindex @code{return_type()}
2591 @cindex @code{return_type_tinfo()}
2592 Information about the commutativity of an object or expression can be
2593 obtained with the two member functions
2596 unsigned ex::return_type() const;
2597 unsigned ex::return_type_tinfo() const;
2600 The @code{return_type()} function returns one of three values (defined in
2601 the header file @file{flags.h}), corresponding to three categories of
2602 expressions in GiNaC:
2605 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2606 classes are of this kind.
2607 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2608 certain class of non-commutative objects which can be determined with the
2609 @code{return_type_tinfo()} method. Expressions of this category commute
2610 with everything except @code{noncommutative} expressions of the same
2612 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2613 of non-commutative objects of different classes. Expressions of this
2614 category don't commute with any other @code{noncommutative} or
2615 @code{noncommutative_composite} expressions.
2618 The value returned by the @code{return_type_tinfo()} method is valid only
2619 when the return type of the expression is @code{noncommutative}. It is a
2620 value that is unique to the class of the object and usually one of the
2621 constants in @file{tinfos.h}, or derived therefrom.
2623 Here are a couple of examples:
2626 @multitable @columnfractions 0.33 0.33 0.34
2627 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2628 @item @code{42} @tab @code{commutative} @tab -
2629 @item @code{2*x-y} @tab @code{commutative} @tab -
2630 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2631 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2632 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2633 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2637 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2638 @code{TINFO_clifford} for objects with a representation label of zero.
2639 Other representation labels yield a different @code{return_type_tinfo()},
2640 but it's the same for any two objects with the same label. This is also true
2643 A last note: With the exception of matrices, positive integer powers of
2644 non-commutative objects are automatically expanded in GiNaC. For example,
2645 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2646 non-commutative expressions).
2649 @cindex @code{clifford} (class)
2650 @subsection Clifford algebra
2652 @cindex @code{dirac_gamma()}
2653 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2654 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2655 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2656 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2659 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2662 which takes two arguments: the index and a @dfn{representation label} in the
2663 range 0 to 255 which is used to distinguish elements of different Clifford
2664 algebras (this is also called a @dfn{spin line index}). Gammas with different
2665 labels commute with each other. The dimension of the index can be 4 or (in
2666 the framework of dimensional regularization) any symbolic value. Spinor
2667 indices on Dirac gammas are not supported in GiNaC.
2669 @cindex @code{dirac_ONE()}
2670 The unity element of a Clifford algebra is constructed by
2673 ex dirac_ONE(unsigned char rl = 0);
2676 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2677 multiples of the unity element, even though it's customary to omit it.
2678 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2679 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2680 GiNaC will complain and/or produce incorrect results.
2682 @cindex @code{dirac_gamma5()}
2683 There is a special element @samp{gamma5} that commutes with all other
2684 gammas, has a unit square, and in 4 dimensions equals
2685 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2688 ex dirac_gamma5(unsigned char rl = 0);
2691 @cindex @code{dirac_gammaL()}
2692 @cindex @code{dirac_gammaR()}
2693 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2694 objects, constructed by
2697 ex dirac_gammaL(unsigned char rl = 0);
2698 ex dirac_gammaR(unsigned char rl = 0);
2701 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2702 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2704 @cindex @code{dirac_slash()}
2705 Finally, the function
2708 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2711 creates a term that represents a contraction of @samp{e} with the Dirac
2712 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2713 with a unique index whose dimension is given by the @code{dim} argument).
2714 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2716 In products of dirac gammas, superfluous unity elements are automatically
2717 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2718 and @samp{gammaR} are moved to the front.
2720 The @code{simplify_indexed()} function performs contractions in gamma strings,
2726 symbol a("a"), b("b"), D("D");
2727 varidx mu(symbol("mu"), D);
2728 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2729 * dirac_gamma(mu.toggle_variance());
2731 // -> gamma~mu*a\*gamma.mu
2732 e = e.simplify_indexed();
2735 cout << e.subs(D == 4) << endl;
2741 @cindex @code{dirac_trace()}
2742 To calculate the trace of an expression containing strings of Dirac gammas
2743 you use the function
2746 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2749 This function takes the trace of all gammas with the specified representation
2750 label; gammas with other labels are left standing. The last argument to
2751 @code{dirac_trace()} is the value to be returned for the trace of the unity
2752 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2753 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2754 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2755 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2756 This @samp{gamma5} scheme is described in greater detail in
2757 @cite{The Role of gamma5 in Dimensional Regularization}.
2759 The value of the trace itself is also usually different in 4 and in
2760 @math{D != 4} dimensions:
2765 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2766 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2767 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2768 cout << dirac_trace(e).simplify_indexed() << endl;
2775 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2776 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2777 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2778 cout << dirac_trace(e).simplify_indexed() << endl;
2779 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2783 Here is an example for using @code{dirac_trace()} to compute a value that
2784 appears in the calculation of the one-loop vacuum polarization amplitude in
2789 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2790 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2793 sp.add(l, l, pow(l, 2));
2794 sp.add(l, q, ldotq);
2796 ex e = dirac_gamma(mu) *
2797 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2798 dirac_gamma(mu.toggle_variance()) *
2799 (dirac_slash(l, D) + m * dirac_ONE());
2800 e = dirac_trace(e).simplify_indexed(sp);
2801 e = e.collect(lst(l, ldotq, m));
2803 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2807 The @code{canonicalize_clifford()} function reorders all gamma products that
2808 appear in an expression to a canonical (but not necessarily simple) form.
2809 You can use this to compare two expressions or for further simplifications:
2813 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2814 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2816 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2818 e = canonicalize_clifford(e);
2825 @cindex @code{color} (class)
2826 @subsection Color algebra
2828 @cindex @code{color_T()}
2829 For computations in quantum chromodynamics, GiNaC implements the base elements
2830 and structure constants of the su(3) Lie algebra (color algebra). The base
2831 elements @math{T_a} are constructed by the function
2834 ex color_T(const ex & a, unsigned char rl = 0);
2837 which takes two arguments: the index and a @dfn{representation label} in the
2838 range 0 to 255 which is used to distinguish elements of different color
2839 algebras. Objects with different labels commute with each other. The
2840 dimension of the index must be exactly 8 and it should be of class @code{idx},
2843 @cindex @code{color_ONE()}
2844 The unity element of a color algebra is constructed by
2847 ex color_ONE(unsigned char rl = 0);
2850 @strong{Note:} You must always use @code{color_ONE()} when referring to
2851 multiples of the unity element, even though it's customary to omit it.
2852 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2853 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2854 GiNaC may produce incorrect results.
2856 @cindex @code{color_d()}
2857 @cindex @code{color_f()}
2861 ex color_d(const ex & a, const ex & b, const ex & c);
2862 ex color_f(const ex & a, const ex & b, const ex & c);
2865 create the symmetric and antisymmetric structure constants @math{d_abc} and
2866 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2867 and @math{[T_a, T_b] = i f_abc T_c}.
2869 @cindex @code{color_h()}
2870 There's an additional function
2873 ex color_h(const ex & a, const ex & b, const ex & c);
2876 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2878 The function @code{simplify_indexed()} performs some simplifications on
2879 expressions containing color objects:
2884 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2885 k(symbol("k"), 8), l(symbol("l"), 8);
2887 e = color_d(a, b, l) * color_f(a, b, k);
2888 cout << e.simplify_indexed() << endl;
2891 e = color_d(a, b, l) * color_d(a, b, k);
2892 cout << e.simplify_indexed() << endl;
2895 e = color_f(l, a, b) * color_f(a, b, k);
2896 cout << e.simplify_indexed() << endl;
2899 e = color_h(a, b, c) * color_h(a, b, c);
2900 cout << e.simplify_indexed() << endl;
2903 e = color_h(a, b, c) * color_T(b) * color_T(c);
2904 cout << e.simplify_indexed() << endl;
2907 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2908 cout << e.simplify_indexed() << endl;
2911 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2912 cout << e.simplify_indexed() << endl;
2913 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2917 @cindex @code{color_trace()}
2918 To calculate the trace of an expression containing color objects you use the
2922 ex color_trace(const ex & e, unsigned char rl = 0);
2925 This function takes the trace of all color @samp{T} objects with the
2926 specified representation label; @samp{T}s with other labels are left
2927 standing. For example:
2931 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2933 // -> -I*f.a.c.b+d.a.c.b
2938 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2939 @c node-name, next, previous, up
2940 @chapter Methods and Functions
2943 In this chapter the most important algorithms provided by GiNaC will be
2944 described. Some of them are implemented as functions on expressions,
2945 others are implemented as methods provided by expression objects. If
2946 they are methods, there exists a wrapper function around it, so you can
2947 alternatively call it in a functional way as shown in the simple
2952 cout << "As method: " << sin(1).evalf() << endl;
2953 cout << "As function: " << evalf(sin(1)) << endl;
2957 @cindex @code{subs()}
2958 The general rule is that wherever methods accept one or more parameters
2959 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2960 wrapper accepts is the same but preceded by the object to act on
2961 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2962 most natural one in an OO model but it may lead to confusion for MapleV
2963 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2964 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2965 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2966 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2967 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2968 here. Also, users of MuPAD will in most cases feel more comfortable
2969 with GiNaC's convention. All function wrappers are implemented
2970 as simple inline functions which just call the corresponding method and
2971 are only provided for users uncomfortable with OO who are dead set to
2972 avoid method invocations. Generally, nested function wrappers are much
2973 harder to read than a sequence of methods and should therefore be
2974 avoided if possible. On the other hand, not everything in GiNaC is a
2975 method on class @code{ex} and sometimes calling a function cannot be
2979 * Information About Expressions::
2980 * Numerical Evaluation::
2981 * Substituting Expressions::
2982 * Pattern Matching and Advanced Substitutions::
2983 * Applying a Function on Subexpressions::
2984 * Visitors and Tree Traversal::
2985 * Polynomial Arithmetic:: Working with polynomials.
2986 * Rational Expressions:: Working with rational functions.
2987 * Symbolic Differentiation::
2988 * Series Expansion:: Taylor and Laurent expansion.
2990 * Built-in Functions:: List of predefined mathematical functions.
2991 * Solving Linear Systems of Equations::
2992 * Input/Output:: Input and output of expressions.
2996 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
2997 @c node-name, next, previous, up
2998 @section Getting information about expressions
3000 @subsection Checking expression types
3001 @cindex @code{is_a<@dots{}>()}
3002 @cindex @code{is_exactly_a<@dots{}>()}
3003 @cindex @code{ex_to<@dots{}>()}
3004 @cindex Converting @code{ex} to other classes
3005 @cindex @code{info()}
3006 @cindex @code{return_type()}
3007 @cindex @code{return_type_tinfo()}
3009 Sometimes it's useful to check whether a given expression is a plain number,
3010 a sum, a polynomial with integer coefficients, or of some other specific type.
3011 GiNaC provides a couple of functions for this:
3014 bool is_a<T>(const ex & e);
3015 bool is_exactly_a<T>(const ex & e);
3016 bool ex::info(unsigned flag);
3017 unsigned ex::return_type() const;
3018 unsigned ex::return_type_tinfo() const;
3021 When the test made by @code{is_a<T>()} returns true, it is safe to call
3022 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3023 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3024 example, assuming @code{e} is an @code{ex}:
3029 if (is_a<numeric>(e))
3030 numeric n = ex_to<numeric>(e);
3035 @code{is_a<T>(e)} allows you to check whether the top-level object of
3036 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3037 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3038 e.g., for checking whether an expression is a number, a sum, or a product:
3045 is_a<numeric>(e1); // true
3046 is_a<numeric>(e2); // false
3047 is_a<add>(e1); // false
3048 is_a<add>(e2); // true
3049 is_a<mul>(e1); // false
3050 is_a<mul>(e2); // false
3054 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3055 top-level object of an expression @samp{e} is an instance of the GiNaC
3056 class @samp{T}, not including parent classes.
3058 The @code{info()} method is used for checking certain attributes of
3059 expressions. The possible values for the @code{flag} argument are defined
3060 in @file{ginac/flags.h}, the most important being explained in the following
3064 @multitable @columnfractions .30 .70
3065 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3066 @item @code{numeric}
3067 @tab @dots{}a number (same as @code{is_<numeric>(...)})
3069 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3070 @item @code{rational}
3071 @tab @dots{}an exact rational number (integers are rational, too)
3072 @item @code{integer}
3073 @tab @dots{}a (non-complex) integer
3074 @item @code{crational}
3075 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3076 @item @code{cinteger}
3077 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3078 @item @code{positive}
3079 @tab @dots{}not complex and greater than 0
3080 @item @code{negative}
3081 @tab @dots{}not complex and less than 0
3082 @item @code{nonnegative}
3083 @tab @dots{}not complex and greater than or equal to 0
3085 @tab @dots{}an integer greater than 0
3087 @tab @dots{}an integer less than 0
3088 @item @code{nonnegint}
3089 @tab @dots{}an integer greater than or equal to 0
3091 @tab @dots{}an even integer
3093 @tab @dots{}an odd integer
3095 @tab @dots{}a prime integer (probabilistic primality test)
3096 @item @code{relation}
3097 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3098 @item @code{relation_equal}
3099 @tab @dots{}a @code{==} relation
3100 @item @code{relation_not_equal}
3101 @tab @dots{}a @code{!=} relation
3102 @item @code{relation_less}
3103 @tab @dots{}a @code{<} relation
3104 @item @code{relation_less_or_equal}
3105 @tab @dots{}a @code{<=} relation
3106 @item @code{relation_greater}
3107 @tab @dots{}a @code{>} relation
3108 @item @code{relation_greater_or_equal}
3109 @tab @dots{}a @code{>=} relation
3111 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3113 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3114 @item @code{polynomial}
3115 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3116 @item @code{integer_polynomial}
3117 @tab @dots{}a polynomial with (non-complex) integer coefficients
3118 @item @code{cinteger_polynomial}
3119 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3120 @item @code{rational_polynomial}
3121 @tab @dots{}a polynomial with (non-complex) rational coefficients
3122 @item @code{crational_polynomial}
3123 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3124 @item @code{rational_function}
3125 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3126 @item @code{algebraic}
3127 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3131 To determine whether an expression is commutative or non-commutative and if
3132 so, with which other expressions it would commute, you use the methods
3133 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3134 for an explanation of these.
3137 @subsection Accessing subexpressions
3138 @cindex @code{nops()}
3141 @cindex @code{relational} (class)
3143 GiNaC provides the two methods
3147 ex ex::op(size_t i);
3150 for accessing the subexpressions in the container-like GiNaC classes like
3151 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3152 determines the number of subexpressions (@samp{operands}) contained, while
3153 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3154 In the case of a @code{power} object, @code{op(0)} will return the basis
3155 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3156 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3158 The left-hand and right-hand side expressions of objects of class
3159 @code{relational} (and only of these) can also be accessed with the methods
3167 @subsection Comparing expressions
3168 @cindex @code{is_equal()}
3169 @cindex @code{is_zero()}
3171 Expressions can be compared with the usual C++ relational operators like
3172 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3173 the result is usually not determinable and the result will be @code{false},
3174 except in the case of the @code{!=} operator. You should also be aware that
3175 GiNaC will only do the most trivial test for equality (subtracting both
3176 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3179 Actually, if you construct an expression like @code{a == b}, this will be
3180 represented by an object of the @code{relational} class (@pxref{Relations})
3181 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3183 There are also two methods
3186 bool ex::is_equal(const ex & other);
3190 for checking whether one expression is equal to another, or equal to zero,
3194 @subsection Ordering expressions
3195 @cindex @code{ex_is_less} (class)
3196 @cindex @code{ex_is_equal} (class)
3197 @cindex @code{compare()}
3199 Sometimes it is necessary to establish a mathematically well-defined ordering
3200 on a set of arbitrary expressions, for example to use expressions as keys
3201 in a @code{std::map<>} container, or to bring a vector of expressions into
3202 a canonical order (which is done internally by GiNaC for sums and products).
3204 The operators @code{<}, @code{>} etc. described in the last section cannot
3205 be used for this, as they don't implement an ordering relation in the
3206 mathematical sense. In particular, they are not guaranteed to be
3207 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3208 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3211 By default, STL classes and algorithms use the @code{<} and @code{==}
3212 operators to compare objects, which are unsuitable for expressions, but GiNaC
3213 provides two functors that can be supplied as proper binary comparison
3214 predicates to the STL:
3217 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3219 bool operator()(const ex &lh, const ex &rh) const;
3222 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3224 bool operator()(const ex &lh, const ex &rh) const;
3228 For example, to define a @code{map} that maps expressions to strings you
3232 std::map<ex, std::string, ex_is_less> myMap;
3235 Omitting the @code{ex_is_less} template parameter will introduce spurious
3236 bugs because the map operates improperly.
3238 Other examples for the use of the functors:
3246 std::sort(v.begin(), v.end(), ex_is_less());
3248 // count the number of expressions equal to '1'
3249 unsigned num_ones = std::count_if(v.begin(), v.end(),
3250 std::bind2nd(ex_is_equal(), 1));
3253 The implementation of @code{ex_is_less} uses the member function
3256 int ex::compare(const ex & other) const;
3259 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3260 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3264 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3265 @c node-name, next, previous, up
3266 @section Numercial Evaluation
3267 @cindex @code{evalf()}
3269 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3270 To evaluate them using floating-point arithmetic you need to call
3273 ex ex::evalf(int level = 0) const;
3276 @cindex @code{Digits}
3277 The accuracy of the evaluation is controlled by the global object @code{Digits}
3278 which can be assigned an integer value. The default value of @code{Digits}
3279 is 17. @xref{Numbers}, for more information and examples.
3281 To evaluate an expression to a @code{double} floating-point number you can
3282 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3286 // Approximate sin(x/Pi)
3288 ex e = series(sin(x/Pi), x == 0, 6);
3290 // Evaluate numerically at x=0.1
3291 ex f = evalf(e.subs(x == 0.1));
3293 // ex_to<numeric> is an unsafe cast, so check the type first
3294 if (is_a<numeric>(f)) @{
3295 double d = ex_to<numeric>(f).to_double();
3304 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3305 @c node-name, next, previous, up
3306 @section Substituting expressions
3307 @cindex @code{subs()}
3309 Algebraic objects inside expressions can be replaced with arbitrary
3310 expressions via the @code{.subs()} method:
3313 ex ex::subs(const ex & e, unsigned options = 0);
3314 ex ex::subs(const exmap & m, unsigned options = 0);
3315 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3318 In the first form, @code{subs()} accepts a relational of the form
3319 @samp{object == expression} or a @code{lst} of such relationals:
3323 symbol x("x"), y("y");
3325 ex e1 = 2*x^2-4*x+3;
3326 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3330 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3335 If you specify multiple substitutions, they are performed in parallel, so e.g.
3336 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3338 The second form of @code{subs()} takes an @code{exmap} object which is a
3339 pair associative container that maps expressions to expressions (currently
3340 implemented as a @code{std::map}). This is the most efficient one of the
3341 three @code{subs()} forms and should be used when the number of objects to
3342 be substituted is large or unknown.
3344 Using this form, the second example from above would look like this:
3348 symbol x("x"), y("y");
3354 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3358 The third form of @code{subs()} takes two lists, one for the objects to be
3359 replaced and one for the expressions to be substituted (both lists must
3360 contain the same number of elements). Using this form, you would write
3364 symbol x("x"), y("y");
3367 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3371 The optional last argument to @code{subs()} is a combination of
3372 @code{subs_options} flags. There are two options available:
3373 @code{subs_options::no_pattern} disables pattern matching, which makes
3374 large @code{subs()} operations significantly faster if you are not using
3375 patterns. The second option, @code{subs_options::algebraic} enables
3376 algebraic substitutions in products and powers.
3377 @ref{Pattern Matching and Advanced Substitutions}, for more information
3378 about patterns and algebraic substitutions.
3380 @code{subs()} performs syntactic substitution of any complete algebraic
3381 object; it does not try to match sub-expressions as is demonstrated by the
3386 symbol x("x"), y("y"), z("z");
3388 ex e1 = pow(x+y, 2);
3389 cout << e1.subs(x+y == 4) << endl;
3392 ex e2 = sin(x)*sin(y)*cos(x);
3393 cout << e2.subs(sin(x) == cos(x)) << endl;
3394 // -> cos(x)^2*sin(y)
3397 cout << e3.subs(x+y == 4) << endl;
3399 // (and not 4+z as one might expect)
3403 A more powerful form of substitution using wildcards is described in the
3407 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3408 @c node-name, next, previous, up
3409 @section Pattern matching and advanced substitutions
3410 @cindex @code{wildcard} (class)
3411 @cindex Pattern matching
3413 GiNaC allows the use of patterns for checking whether an expression is of a
3414 certain form or contains subexpressions of a certain form, and for
3415 substituting expressions in a more general way.
3417 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3418 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3419 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3420 an unsigned integer number to allow having multiple different wildcards in a
3421 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3422 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3426 ex wild(unsigned label = 0);
3429 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3432 Some examples for patterns:
3434 @multitable @columnfractions .5 .5
3435 @item @strong{Constructed as} @tab @strong{Output as}
3436 @item @code{wild()} @tab @samp{$0}
3437 @item @code{pow(x,wild())} @tab @samp{x^$0}
3438 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3439 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3445 @item Wildcards behave like symbols and are subject to the same algebraic
3446 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3447 @item As shown in the last example, to use wildcards for indices you have to
3448 use them as the value of an @code{idx} object. This is because indices must
3449 always be of class @code{idx} (or a subclass).
3450 @item Wildcards only represent expressions or subexpressions. It is not
3451 possible to use them as placeholders for other properties like index
3452 dimension or variance, representation labels, symmetry of indexed objects
3454 @item Because wildcards are commutative, it is not possible to use wildcards
3455 as part of noncommutative products.
3456 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3457 are also valid patterns.
3460 @subsection Matching expressions
3461 @cindex @code{match()}
3462 The most basic application of patterns is to check whether an expression
3463 matches a given pattern. This is done by the function
3466 bool ex::match(const ex & pattern);
3467 bool ex::match(const ex & pattern, lst & repls);
3470 This function returns @code{true} when the expression matches the pattern
3471 and @code{false} if it doesn't. If used in the second form, the actual
3472 subexpressions matched by the wildcards get returned in the @code{repls}
3473 object as a list of relations of the form @samp{wildcard == expression}.
3474 If @code{match()} returns false, the state of @code{repls} is undefined.
3475 For reproducible results, the list should be empty when passed to
3476 @code{match()}, but it is also possible to find similarities in multiple
3477 expressions by passing in the result of a previous match.
3479 The matching algorithm works as follows:
3482 @item A single wildcard matches any expression. If one wildcard appears
3483 multiple times in a pattern, it must match the same expression in all
3484 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3485 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3486 @item If the expression is not of the same class as the pattern, the match
3487 fails (i.e. a sum only matches a sum, a function only matches a function,
3489 @item If the pattern is a function, it only matches the same function
3490 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3491 @item Except for sums and products, the match fails if the number of
3492 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3494 @item If there are no subexpressions, the expressions and the pattern must
3495 be equal (in the sense of @code{is_equal()}).
3496 @item Except for sums and products, each subexpression (@code{op()}) must
3497 match the corresponding subexpression of the pattern.
3500 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3501 account for their commutativity and associativity:
3504 @item If the pattern contains a term or factor that is a single wildcard,
3505 this one is used as the @dfn{global wildcard}. If there is more than one
3506 such wildcard, one of them is chosen as the global wildcard in a random
3508 @item Every term/factor of the pattern, except the global wildcard, is
3509 matched against every term of the expression in sequence. If no match is
3510 found, the whole match fails. Terms that did match are not considered in
3512 @item If there are no unmatched terms left, the match succeeds. Otherwise
3513 the match fails unless there is a global wildcard in the pattern, in
3514 which case this wildcard matches the remaining terms.
3517 In general, having more than one single wildcard as a term of a sum or a
3518 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3521 Here are some examples in @command{ginsh} to demonstrate how it works (the
3522 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3523 match fails, and the list of wildcard replacements otherwise):
3526 > match((x+y)^a,(x+y)^a);
3528 > match((x+y)^a,(x+y)^b);
3530 > match((x+y)^a,$1^$2);
3532 > match((x+y)^a,$1^$1);
3534 > match((x+y)^(x+y),$1^$1);
3536 > match((x+y)^(x+y),$1^$2);
3538 > match((a+b)*(a+c),($1+b)*($1+c));
3540 > match((a+b)*(a+c),(a+$1)*(a+$2));
3542 (Unpredictable. The result might also be [$1==c,$2==b].)
3543 > match((a+b)*(a+c),($1+$2)*($1+$3));
3544 (The result is undefined. Due to the sequential nature of the algorithm
3545 and the re-ordering of terms in GiNaC, the match for the first factor
3546 may be @{$1==a,$2==b@} in which case the match for the second factor
3547 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3549 > match(a*(x+y)+a*z+b,a*$1+$2);
3550 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3551 @{$1=x+y,$2=a*z+b@}.)
3552 > match(a+b+c+d+e+f,c);
3554 > match(a+b+c+d+e+f,c+$0);
3556 > match(a+b+c+d+e+f,c+e+$0);
3558 > match(a+b,a+b+$0);
3560 > match(a*b^2,a^$1*b^$2);
3562 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3563 even though a==a^1.)
3564 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3566 > match(atan2(y,x^2),atan2(y,$0));
3570 @subsection Matching parts of expressions
3571 @cindex @code{has()}
3572 A more general way to look for patterns in expressions is provided by the
3576 bool ex::has(const ex & pattern);
3579 This function checks whether a pattern is matched by an expression itself or
3580 by any of its subexpressions.
3582 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3583 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3586 > has(x*sin(x+y+2*a),y);
3588 > has(x*sin(x+y+2*a),x+y);
3590 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3591 has the subexpressions "x", "y" and "2*a".)
3592 > has(x*sin(x+y+2*a),x+y+$1);
3594 (But this is possible.)
3595 > has(x*sin(2*(x+y)+2*a),x+y);
3597 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3598 which "x+y" is not a subexpression.)
3601 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3603 > has(4*x^2-x+3,$1*x);
3605 > has(4*x^2+x+3,$1*x);
3607 (Another possible pitfall. The first expression matches because the term
3608 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3609 contains a linear term you should use the coeff() function instead.)
3612 @cindex @code{find()}
3616 bool ex::find(const ex & pattern, lst & found);
3619 works a bit like @code{has()} but it doesn't stop upon finding the first
3620 match. Instead, it appends all found matches to the specified list. If there
3621 are multiple occurrences of the same expression, it is entered only once to
3622 the list. @code{find()} returns false if no matches were found (in
3623 @command{ginsh}, it returns an empty list):
3626 > find(1+x+x^2+x^3,x);
3628 > find(1+x+x^2+x^3,y);
3630 > find(1+x+x^2+x^3,x^$1);
3632 (Note the absence of "x".)
3633 > expand((sin(x)+sin(y))*(a+b));
3634 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3639 @subsection Substituting expressions
3640 @cindex @code{subs()}
3641 Probably the most useful application of patterns is to use them for
3642 substituting expressions with the @code{subs()} method. Wildcards can be
3643 used in the search patterns as well as in the replacement expressions, where
3644 they get replaced by the expressions matched by them. @code{subs()} doesn't
3645 know anything about algebra; it performs purely syntactic substitutions.
3650 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3652 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3654 > subs((a+b+c)^2,a+b==x);
3656 > subs((a+b+c)^2,a+b+$1==x+$1);
3658 > subs(a+2*b,a+b==x);
3660 > subs(4*x^3-2*x^2+5*x-1,x==a);
3662 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3664 > subs(sin(1+sin(x)),sin($1)==cos($1));
3666 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3670 The last example would be written in C++ in this way:
3674 symbol a("a"), b("b"), x("x"), y("y");
3675 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3676 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3677 cout << e.expand() << endl;
3682 @subsection Algebraic substitutions
3683 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3684 enables smarter, algebraic substitutions in products and powers. If you want
3685 to substitute some factors of a product, you only need to list these factors
3686 in your pattern. Furthermore, if an (integer) power of some expression occurs
3687 in your pattern and in the expression that you want the substitution to occur
3688 in, it can be substituted as many times as possible, without getting negative
3691 An example clarifies it all (hopefully):
3694 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3695 subs_options::algebraic) << endl;
3696 // --> (y+x)^6+b^6+a^6
3698 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3700 // Powers and products are smart, but addition is just the same.
3702 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3705 // As I said: addition is just the same.
3707 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3708 // --> x^3*b*a^2+2*b
3710 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3712 // --> 2*b+x^3*b^(-1)*a^(-2)
3714 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3715 // --> -1-2*a^2+4*a^3+5*a
3717 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3718 subs_options::algebraic) << endl;
3719 // --> -1+5*x+4*x^3-2*x^2
3720 // You should not really need this kind of patterns very often now.
3721 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3723 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3724 subs_options::algebraic) << endl;
3725 // --> cos(1+cos(x))
3727 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3728 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3729 subs_options::algebraic)) << endl;
3734 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3735 @c node-name, next, previous, up
3736 @section Applying a Function on Subexpressions
3737 @cindex tree traversal
3738 @cindex @code{map()}
3740 Sometimes you may want to perform an operation on specific parts of an
3741 expression while leaving the general structure of it intact. An example
3742 of this would be a matrix trace operation: the trace of a sum is the sum
3743 of the traces of the individual terms. That is, the trace should @dfn{map}
3744 on the sum, by applying itself to each of the sum's operands. It is possible
3745 to do this manually which usually results in code like this:
3750 if (is_a<matrix>(e))
3751 return ex_to<matrix>(e).trace();
3752 else if (is_a<add>(e)) @{
3754 for (size_t i=0; i<e.nops(); i++)
3755 sum += calc_trace(e.op(i));
3757 @} else if (is_a<mul>)(e)) @{
3765 This is, however, slightly inefficient (if the sum is very large it can take
3766 a long time to add the terms one-by-one), and its applicability is limited to
3767 a rather small class of expressions. If @code{calc_trace()} is called with
3768 a relation or a list as its argument, you will probably want the trace to
3769 be taken on both sides of the relation or of all elements of the list.
3771 GiNaC offers the @code{map()} method to aid in the implementation of such
3775 ex ex::map(map_function & f) const;
3776 ex ex::map(ex (*f)(const ex & e)) const;
3779 In the first (preferred) form, @code{map()} takes a function object that
3780 is subclassed from the @code{map_function} class. In the second form, it
3781 takes a pointer to a function that accepts and returns an expression.
3782 @code{map()} constructs a new expression of the same type, applying the
3783 specified function on all subexpressions (in the sense of @code{op()}),
3786 The use of a function object makes it possible to supply more arguments to
3787 the function that is being mapped, or to keep local state information.
3788 The @code{map_function} class declares a virtual function call operator
3789 that you can overload. Here is a sample implementation of @code{calc_trace()}
3790 that uses @code{map()} in a recursive fashion:
3793 struct calc_trace : public map_function @{
3794 ex operator()(const ex &e)
3796 if (is_a<matrix>(e))
3797 return ex_to<matrix>(e).trace();
3798 else if (is_a<mul>(e)) @{
3801 return e.map(*this);
3806 This function object could then be used like this:
3810 ex M = ... // expression with matrices
3811 calc_trace do_trace;
3812 ex tr = do_trace(M);
3816 Here is another example for you to meditate over. It removes quadratic
3817 terms in a variable from an expanded polynomial:
3820 struct map_rem_quad : public map_function @{
3822 map_rem_quad(const ex & var_) : var(var_) @{@}
3824 ex operator()(const ex & e)
3826 if (is_a<add>(e) || is_a<mul>(e))
3827 return e.map(*this);
3828 else if (is_a<power>(e) &&
3829 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3839 symbol x("x"), y("y");
3842 for (int i=0; i<8; i++)
3843 e += pow(x, i) * pow(y, 8-i) * (i+1);
3845 // -> 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
3847 map_rem_quad rem_quad(x);
3848 cout << rem_quad(e) << endl;
3849 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3853 @command{ginsh} offers a slightly different implementation of @code{map()}
3854 that allows applying algebraic functions to operands. The second argument
3855 to @code{map()} is an expression containing the wildcard @samp{$0} which
3856 acts as the placeholder for the operands:
3861 > map(a+2*b,sin($0));
3863 > map(@{a,b,c@},$0^2+$0);
3864 @{a^2+a,b^2+b,c^2+c@}
3867 Note that it is only possible to use algebraic functions in the second
3868 argument. You can not use functions like @samp{diff()}, @samp{op()},
3869 @samp{subs()} etc. because these are evaluated immediately:
3872 > map(@{a,b,c@},diff($0,a));
3874 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3875 to "map(@{a,b,c@},0)".
3879 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3880 @c node-name, next, previous, up
3881 @section Visitors and Tree Traversal
3882 @cindex tree traversal
3883 @cindex @code{visitor} (class)
3884 @cindex @code{accept()}
3885 @cindex @code{visit()}
3886 @cindex @code{traverse()}
3887 @cindex @code{traverse_preorder()}
3888 @cindex @code{traverse_postorder()}
3890 Suppose that you need a function that returns a list of all indices appearing
3891 in an arbitrary expression. The indices can have any dimension, and for
3892 indices with variance you always want the covariant version returned.
3894 You can't use @code{get_free_indices()} because you also want to include
3895 dummy indices in the list, and you can't use @code{find()} as it needs
3896 specific index dimensions (and it would require two passes: one for indices
3897 with variance, one for plain ones).
3899 The obvious solution to this problem is a tree traversal with a type switch,
3900 such as the following:
3903 void gather_indices_helper(const ex & e, lst & l)
3905 if (is_a<varidx>(e)) @{
3906 const varidx & vi = ex_to<varidx>(e);
3907 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3908 @} else if (is_a<idx>(e)) @{
3911 size_t n = e.nops();
3912 for (size_t i = 0; i < n; ++i)
3913 gather_indices_helper(e.op(i), l);
3917 lst gather_indices(const ex & e)
3920 gather_indices_helper(e, l);
3927 This works fine but fans of object-oriented programming will feel
3928 uncomfortable with the type switch. One reason is that there is a possibility
3929 for subtle bugs regarding derived classes. If we had, for example, written
3932 if (is_a<idx>(e)) @{
3934 @} else if (is_a<varidx>(e)) @{
3938 in @code{gather_indices_helper}, the code wouldn't have worked because the
3939 first line "absorbs" all classes derived from @code{idx}, including
3940 @code{varidx}, so the special case for @code{varidx} would never have been
3943 Also, for a large number of classes, a type switch like the above can get
3944 unwieldy and inefficient (it's a linear search, after all).
3945 @code{gather_indices_helper} only checks for two classes, but if you had to
3946 write a function that required a different implementation for nearly
3947 every GiNaC class, the result would be very hard to maintain and extend.
3949 The cleanest approach to the problem would be to add a new virtual function
3950 to GiNaC's class hierarchy. In our example, there would be specializations
3951 for @code{idx} and @code{varidx} while the default implementation in
3952 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
3953 impossible to add virtual member functions to existing classes without
3954 changing their source and recompiling everything. GiNaC comes with source,
3955 so you could actually do this, but for a small algorithm like the one
3956 presented this would be impractical.
3958 One solution to this dilemma is the @dfn{Visitor} design pattern,
3959 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
3960 variation, described in detail in
3961 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
3962 virtual functions to the class hierarchy to implement operations, GiNaC
3963 provides a single "bouncing" method @code{accept()} that takes an instance
3964 of a special @code{visitor} class and redirects execution to the one
3965 @code{visit()} virtual function of the visitor that matches the type of
3966 object that @code{accept()} was being invoked on.
3968 Visitors in GiNaC must derive from the global @code{visitor} class as well
3969 as from the class @code{T::visitor} of each class @code{T} they want to
3970 visit, and implement the member functions @code{void visit(const T &)} for
3976 void ex::accept(visitor & v) const;
3979 will then dispatch to the correct @code{visit()} member function of the
3980 specified visitor @code{v} for the type of GiNaC object at the root of the
3981 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
3983 Here is an example of a visitor:
3987 : public visitor, // this is required
3988 public add::visitor, // visit add objects
3989 public numeric::visitor, // visit numeric objects
3990 public basic::visitor // visit basic objects
3992 void visit(const add & x)
3993 @{ cout << "called with an add object" << endl; @}
3995 void visit(const numeric & x)
3996 @{ cout << "called with a numeric object" << endl; @}
3998 void visit(const basic & x)
3999 @{ cout << "called with a basic object" << endl; @}
4003 which can be used as follows:
4014 // prints "called with a numeric object"
4016 // prints "called with an add object"
4018 // prints "called with a basic object"
4022 The @code{visit(const basic &)} method gets called for all objects that are
4023 not @code{numeric} or @code{add} and acts as an (optional) default.
4025 From a conceptual point of view, the @code{visit()} methods of the visitor
4026 behave like a newly added virtual function of the visited hierarchy.
4027 In addition, visitors can store state in member variables, and they can
4028 be extended by deriving a new visitor from an existing one, thus building
4029 hierarchies of visitors.
4031 We can now rewrite our index example from above with a visitor:
4034 class gather_indices_visitor
4035 : public visitor, public idx::visitor, public varidx::visitor
4039 void visit(const idx & i)
4044 void visit(const varidx & vi)
4046 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4050 const lst & get_result() // utility function
4059 What's missing is the tree traversal. We could implement it in
4060 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4063 void ex::traverse_preorder(visitor & v) const;
4064 void ex::traverse_postorder(visitor & v) const;
4065 void ex::traverse(visitor & v) const;
4068 @code{traverse_preorder()} visits a node @emph{before} visiting its
4069 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4070 visiting its subexpressions. @code{traverse()} is a synonym for
4071 @code{traverse_preorder()}.
4073 Here is a new implementation of @code{gather_indices()} that uses the visitor
4074 and @code{traverse()}:
4077 lst gather_indices(const ex & e)
4079 gather_indices_visitor v;
4081 return v.get_result();
4086 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4087 @c node-name, next, previous, up
4088 @section Polynomial arithmetic
4090 @subsection Expanding and collecting
4091 @cindex @code{expand()}
4092 @cindex @code{collect()}
4093 @cindex @code{collect_common_factors()}
4095 A polynomial in one or more variables has many equivalent
4096 representations. Some useful ones serve a specific purpose. Consider
4097 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4098 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4099 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4100 representations are the recursive ones where one collects for exponents
4101 in one of the three variable. Since the factors are themselves
4102 polynomials in the remaining two variables the procedure can be
4103 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4104 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4107 To bring an expression into expanded form, its method
4110 ex ex::expand(unsigned options = 0);
4113 may be called. In our example above, this corresponds to @math{4*x*y +
4114 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4115 GiNaC is not easily guessable you should be prepared to see different
4116 orderings of terms in such sums!
4118 Another useful representation of multivariate polynomials is as a
4119 univariate polynomial in one of the variables with the coefficients
4120 being polynomials in the remaining variables. The method
4121 @code{collect()} accomplishes this task:
4124 ex ex::collect(const ex & s, bool distributed = false);
4127 The first argument to @code{collect()} can also be a list of objects in which
4128 case the result is either a recursively collected polynomial, or a polynomial
4129 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4130 by the @code{distributed} flag.
4132 Note that the original polynomial needs to be in expanded form (for the
4133 variables concerned) in order for @code{collect()} to be able to find the
4134 coefficients properly.
4136 The following @command{ginsh} transcript shows an application of @code{collect()}
4137 together with @code{find()}:
4140 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4141 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
4142 > collect(a,@{p,q@});
4143 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)
4144 > collect(a,find(a,sin($1)));
4145 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4146 > collect(a,@{find(a,sin($1)),p,q@});
4147 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4148 > collect(a,@{find(a,sin($1)),d@});
4149 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4152 Polynomials can often be brought into a more compact form by collecting
4153 common factors from the terms of sums. This is accomplished by the function
4156 ex collect_common_factors(const ex & e);
4159 This function doesn't perform a full factorization but only looks for
4160 factors which are already explicitly present:
4163 > collect_common_factors(a*x+a*y);
4165 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4167 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4168 (c+a)*a*(x*y+y^2+x)*b
4171 @subsection Degree and coefficients
4172 @cindex @code{degree()}
4173 @cindex @code{ldegree()}
4174 @cindex @code{coeff()}
4176 The degree and low degree of a polynomial can be obtained using the two
4180 int ex::degree(const ex & s);
4181 int ex::ldegree(const ex & s);
4184 which also work reliably on non-expanded input polynomials (they even work
4185 on rational functions, returning the asymptotic degree). To extract
4186 a coefficient with a certain power from an expanded polynomial you use
4189 ex ex::coeff(const ex & s, int n);
4192 You can also obtain the leading and trailing coefficients with the methods
4195 ex ex::lcoeff(const ex & s);
4196 ex ex::tcoeff(const ex & s);
4199 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4202 An application is illustrated in the next example, where a multivariate
4203 polynomial is analyzed:
4207 symbol x("x"), y("y");
4208 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4209 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4210 ex Poly = PolyInp.expand();
4212 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4213 cout << "The x^" << i << "-coefficient is "
4214 << Poly.coeff(x,i) << endl;
4216 cout << "As polynomial in y: "
4217 << Poly.collect(y) << endl;
4221 When run, it returns an output in the following fashion:
4224 The x^0-coefficient is y^2+11*y
4225 The x^1-coefficient is 5*y^2-2*y
4226 The x^2-coefficient is -1
4227 The x^3-coefficient is 4*y
4228 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4231 As always, the exact output may vary between different versions of GiNaC
4232 or even from run to run since the internal canonical ordering is not
4233 within the user's sphere of influence.
4235 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4236 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4237 with non-polynomial expressions as they not only work with symbols but with
4238 constants, functions and indexed objects as well:
4242 symbol a("a"), b("b"), c("c");
4243 idx i(symbol("i"), 3);
4245 ex e = pow(sin(x) - cos(x), 4);
4246 cout << e.degree(cos(x)) << endl;
4248 cout << e.expand().coeff(sin(x), 3) << endl;
4251 e = indexed(a+b, i) * indexed(b+c, i);
4252 e = e.expand(expand_options::expand_indexed);
4253 cout << e.collect(indexed(b, i)) << endl;
4254 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4259 @subsection Polynomial division
4260 @cindex polynomial division
4263 @cindex pseudo-remainder
4264 @cindex @code{quo()}
4265 @cindex @code{rem()}
4266 @cindex @code{prem()}
4267 @cindex @code{divide()}
4272 ex quo(const ex & a, const ex & b, const ex & x);
4273 ex rem(const ex & a, const ex & b, const ex & x);
4276 compute the quotient and remainder of univariate polynomials in the variable
4277 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4279 The additional function
4282 ex prem(const ex & a, const ex & b, const ex & x);
4285 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4286 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4288 Exact division of multivariate polynomials is performed by the function
4291 bool divide(const ex & a, const ex & b, ex & q);
4294 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4295 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4296 in which case the value of @code{q} is undefined.
4299 @subsection Unit, content and primitive part
4300 @cindex @code{unit()}
4301 @cindex @code{content()}
4302 @cindex @code{primpart()}
4307 ex ex::unit(const ex & x);
4308 ex ex::content(const ex & x);
4309 ex ex::primpart(const ex & x);
4312 return the unit part, content part, and primitive polynomial of a multivariate
4313 polynomial with respect to the variable @samp{x} (the unit part being the sign
4314 of the leading coefficient, the content part being the GCD of the coefficients,
4315 and the primitive polynomial being the input polynomial divided by the unit and
4316 content parts). The product of unit, content, and primitive part is the
4317 original polynomial.
4320 @subsection GCD and LCM
4323 @cindex @code{gcd()}
4324 @cindex @code{lcm()}
4326 The functions for polynomial greatest common divisor and least common
4327 multiple have the synopsis
4330 ex gcd(const ex & a, const ex & b);
4331 ex lcm(const ex & a, const ex & b);
4334 The functions @code{gcd()} and @code{lcm()} accept two expressions
4335 @code{a} and @code{b} as arguments and return a new expression, their
4336 greatest common divisor or least common multiple, respectively. If the
4337 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4338 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4341 #include <ginac/ginac.h>
4342 using namespace GiNaC;
4346 symbol x("x"), y("y"), z("z");
4347 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4348 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4350 ex P_gcd = gcd(P_a, P_b);
4352 ex P_lcm = lcm(P_a, P_b);
4353 // 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
4358 @subsection Square-free decomposition
4359 @cindex square-free decomposition
4360 @cindex factorization
4361 @cindex @code{sqrfree()}
4363 GiNaC still lacks proper factorization support. Some form of
4364 factorization is, however, easily implemented by noting that factors
4365 appearing in a polynomial with power two or more also appear in the
4366 derivative and hence can easily be found by computing the GCD of the
4367 original polynomial and its derivatives. Any decent system has an
4368 interface for this so called square-free factorization. So we provide
4371 ex sqrfree(const ex & a, const lst & l = lst());
4373 Here is an example that by the way illustrates how the exact form of the
4374 result may slightly depend on the order of differentiation, calling for
4375 some care with subsequent processing of the result:
4378 symbol x("x"), y("y");
4379 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4381 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4382 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4384 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4385 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4387 cout << sqrfree(BiVarPol) << endl;
4388 // -> depending on luck, any of the above
4391 Note also, how factors with the same exponents are not fully factorized
4395 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4396 @c node-name, next, previous, up
4397 @section Rational expressions
4399 @subsection The @code{normal} method
4400 @cindex @code{normal()}
4401 @cindex simplification
4402 @cindex temporary replacement
4404 Some basic form of simplification of expressions is called for frequently.
4405 GiNaC provides the method @code{.normal()}, which converts a rational function
4406 into an equivalent rational function of the form @samp{numerator/denominator}
4407 where numerator and denominator are coprime. If the input expression is already
4408 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4409 otherwise it performs fraction addition and multiplication.
4411 @code{.normal()} can also be used on expressions which are not rational functions
4412 as it will replace all non-rational objects (like functions or non-integer
4413 powers) by temporary symbols to bring the expression to the domain of rational
4414 functions before performing the normalization, and re-substituting these
4415 symbols afterwards. This algorithm is also available as a separate method
4416 @code{.to_rational()}, described below.
4418 This means that both expressions @code{t1} and @code{t2} are indeed
4419 simplified in this little code snippet:
4424 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4425 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4426 std::cout << "t1 is " << t1.normal() << std::endl;
4427 std::cout << "t2 is " << t2.normal() << std::endl;
4431 Of course this works for multivariate polynomials too, so the ratio of
4432 the sample-polynomials from the section about GCD and LCM above would be
4433 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4436 @subsection Numerator and denominator
4439 @cindex @code{numer()}
4440 @cindex @code{denom()}
4441 @cindex @code{numer_denom()}
4443 The numerator and denominator of an expression can be obtained with
4448 ex ex::numer_denom();
4451 These functions will first normalize the expression as described above and
4452 then return the numerator, denominator, or both as a list, respectively.
4453 If you need both numerator and denominator, calling @code{numer_denom()} is
4454 faster than using @code{numer()} and @code{denom()} separately.
4457 @subsection Converting to a polynomial or rational expression
4458 @cindex @code{to_polynomial()}
4459 @cindex @code{to_rational()}
4461 Some of the methods described so far only work on polynomials or rational
4462 functions. GiNaC provides a way to extend the domain of these functions to
4463 general expressions by using the temporary replacement algorithm described
4464 above. You do this by calling
4467 ex ex::to_polynomial(exmap & m);
4468 ex ex::to_polynomial(lst & l);
4472 ex ex::to_rational(exmap & m);
4473 ex ex::to_rational(lst & l);
4476 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4477 will be filled with the generated temporary symbols and their replacement
4478 expressions in a format that can be used directly for the @code{subs()}
4479 method. It can also already contain a list of replacements from an earlier
4480 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4481 possible to use it on multiple expressions and get consistent results.
4483 The difference betwerrn @code{.to_polynomial()} and @code{.to_rational()}
4484 is probably best illustrated with an example:
4488 symbol x("x"), y("y");
4489 ex a = 2*x/sin(x) - y/(3*sin(x));
4493 ex p = a.to_polynomial(lp);
4494 cout << " = " << p << "\n with " << lp << endl;
4495 // = symbol3*symbol2*y+2*symbol2*x
4496 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4499 ex r = a.to_rational(lr);
4500 cout << " = " << r << "\n with " << lr << endl;
4501 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4502 // with @{symbol4==sin(x)@}
4506 The following more useful example will print @samp{sin(x)-cos(x)}:
4511 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4512 ex b = sin(x) + cos(x);
4515 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4516 cout << q.subs(m) << endl;
4521 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4522 @c node-name, next, previous, up
4523 @section Symbolic differentiation
4524 @cindex differentiation
4525 @cindex @code{diff()}
4527 @cindex product rule
4529 GiNaC's objects know how to differentiate themselves. Thus, a
4530 polynomial (class @code{add}) knows that its derivative is the sum of
4531 the derivatives of all the monomials:
4535 symbol x("x"), y("y"), z("z");
4536 ex P = pow(x, 5) + pow(x, 2) + y;
4538 cout << P.diff(x,2) << endl;
4540 cout << P.diff(y) << endl; // 1
4542 cout << P.diff(z) << endl; // 0
4547 If a second integer parameter @var{n} is given, the @code{diff} method
4548 returns the @var{n}th derivative.
4550 If @emph{every} object and every function is told what its derivative
4551 is, all derivatives of composed objects can be calculated using the
4552 chain rule and the product rule. Consider, for instance the expression
4553 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4554 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4555 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4556 out that the composition is the generating function for Euler Numbers,
4557 i.e. the so called @var{n}th Euler number is the coefficient of
4558 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4559 identity to code a function that generates Euler numbers in just three
4562 @cindex Euler numbers
4564 #include <ginac/ginac.h>
4565 using namespace GiNaC;
4567 ex EulerNumber(unsigned n)
4570 const ex generator = pow(cosh(x),-1);
4571 return generator.diff(x,n).subs(x==0);
4576 for (unsigned i=0; i<11; i+=2)
4577 std::cout << EulerNumber(i) << std::endl;
4582 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4583 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4584 @code{i} by two since all odd Euler numbers vanish anyways.
4587 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4588 @c node-name, next, previous, up
4589 @section Series expansion
4590 @cindex @code{series()}
4591 @cindex Taylor expansion
4592 @cindex Laurent expansion
4593 @cindex @code{pseries} (class)
4594 @cindex @code{Order()}
4596 Expressions know how to expand themselves as a Taylor series or (more
4597 generally) a Laurent series. As in most conventional Computer Algebra
4598 Systems, no distinction is made between those two. There is a class of
4599 its own for storing such series (@code{class pseries}) and a built-in
4600 function (called @code{Order}) for storing the order term of the series.
4601 As a consequence, if you want to work with series, i.e. multiply two
4602 series, you need to call the method @code{ex::series} again to convert
4603 it to a series object with the usual structure (expansion plus order
4604 term). A sample application from special relativity could read:
4607 #include <ginac/ginac.h>
4608 using namespace std;
4609 using namespace GiNaC;
4613 symbol v("v"), c("c");
4615 ex gamma = 1/sqrt(1 - pow(v/c,2));
4616 ex mass_nonrel = gamma.series(v==0, 10);
4618 cout << "the relativistic mass increase with v is " << endl
4619 << mass_nonrel << endl;
4621 cout << "the inverse square of this series is " << endl
4622 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4626 Only calling the series method makes the last output simplify to
4627 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4628 series raised to the power @math{-2}.
4630 @cindex Machin's formula
4631 As another instructive application, let us calculate the numerical
4632 value of Archimedes' constant
4636 (for which there already exists the built-in constant @code{Pi})
4637 using John Machin's amazing formula
4639 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4642 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4644 This equation (and similar ones) were used for over 200 years for
4645 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4646 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4647 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4648 order term with it and the question arises what the system is supposed
4649 to do when the fractions are plugged into that order term. The solution
4650 is to use the function @code{series_to_poly()} to simply strip the order
4654 #include <ginac/ginac.h>
4655 using namespace GiNaC;
4657 ex machin_pi(int degr)
4660 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4661 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4662 -4*pi_expansion.subs(x==numeric(1,239));
4668 using std::cout; // just for fun, another way of...
4669 using std::endl; // ...dealing with this namespace std.
4671 for (int i=2; i<12; i+=2) @{
4672 pi_frac = machin_pi(i);
4673 cout << i << ":\t" << pi_frac << endl
4674 << "\t" << pi_frac.evalf() << endl;
4680 Note how we just called @code{.series(x,degr)} instead of
4681 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4682 method @code{series()}: if the first argument is a symbol the expression
4683 is expanded in that symbol around point @code{0}. When you run this
4684 program, it will type out:
4688 3.1832635983263598326
4689 4: 5359397032/1706489875
4690 3.1405970293260603143
4691 6: 38279241713339684/12184551018734375
4692 3.141621029325034425
4693 8: 76528487109180192540976/24359780855939418203125
4694 3.141591772182177295
4695 10: 327853873402258685803048818236/104359128170408663038552734375
4696 3.1415926824043995174
4700 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4701 @c node-name, next, previous, up
4702 @section Symmetrization
4703 @cindex @code{symmetrize()}
4704 @cindex @code{antisymmetrize()}
4705 @cindex @code{symmetrize_cyclic()}
4710 ex ex::symmetrize(const lst & l);
4711 ex ex::antisymmetrize(const lst & l);
4712 ex ex::symmetrize_cyclic(const lst & l);
4715 symmetrize an expression by returning the sum over all symmetric,
4716 antisymmetric or cyclic permutations of the specified list of objects,
4717 weighted by the number of permutations.
4719 The three additional methods
4722 ex ex::symmetrize();
4723 ex ex::antisymmetrize();
4724 ex ex::symmetrize_cyclic();
4727 symmetrize or antisymmetrize an expression over its free indices.
4729 Symmetrization is most useful with indexed expressions but can be used with
4730 almost any kind of object (anything that is @code{subs()}able):
4734 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4735 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4737 cout << indexed(A, i, j).symmetrize() << endl;
4738 // -> 1/2*A.j.i+1/2*A.i.j
4739 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4740 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4741 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4742 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4747 @node Built-in Functions, Solving Linear Systems of Equations, Symmetrization, Methods and Functions
4748 @c node-name, next, previous, up
4749 @section Predefined mathematical functions
4751 GiNaC contains the following predefined mathematical functions:
4754 @multitable @columnfractions .30 .70
4755 @item @strong{Name} @tab @strong{Function}
4758 @cindex @code{abs()}
4759 @item @code{csgn(x)}
4761 @cindex @code{csgn()}
4762 @item @code{sqrt(x)}
4763 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4764 @cindex @code{sqrt()}
4767 @cindex @code{sin()}
4770 @cindex @code{cos()}
4773 @cindex @code{tan()}
4774 @item @code{asin(x)}
4776 @cindex @code{asin()}
4777 @item @code{acos(x)}
4779 @cindex @code{acos()}
4780 @item @code{atan(x)}
4781 @tab inverse tangent
4782 @cindex @code{atan()}
4783 @item @code{atan2(y, x)}
4784 @tab inverse tangent with two arguments
4785 @item @code{sinh(x)}
4786 @tab hyperbolic sine
4787 @cindex @code{sinh()}
4788 @item @code{cosh(x)}
4789 @tab hyperbolic cosine
4790 @cindex @code{cosh()}
4791 @item @code{tanh(x)}
4792 @tab hyperbolic tangent
4793 @cindex @code{tanh()}
4794 @item @code{asinh(x)}
4795 @tab inverse hyperbolic sine
4796 @cindex @code{asinh()}
4797 @item @code{acosh(x)}
4798 @tab inverse hyperbolic cosine
4799 @cindex @code{acosh()}
4800 @item @code{atanh(x)}
4801 @tab inverse hyperbolic tangent
4802 @cindex @code{atanh()}
4804 @tab exponential function
4805 @cindex @code{exp()}
4807 @tab natural logarithm
4808 @cindex @code{log()}
4811 @cindex @code{Li2()}
4812 @item @code{zeta(x)}
4813 @tab Riemann's zeta function
4814 @cindex @code{zeta()}
4815 @item @code{zeta(n, x)}
4816 @tab derivatives of Riemann's zeta function
4817 @item @code{tgamma(x)}
4819 @cindex @code{tgamma()}
4820 @cindex Gamma function
4821 @item @code{lgamma(x)}
4822 @tab logarithm of Gamma function
4823 @cindex @code{lgamma()}
4824 @item @code{beta(x, y)}
4825 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4826 @cindex @code{beta()}
4828 @tab psi (digamma) function
4829 @cindex @code{psi()}
4830 @item @code{psi(n, x)}
4831 @tab derivatives of psi function (polygamma functions)
4832 @item @code{factorial(n)}
4833 @tab factorial function
4834 @cindex @code{factorial()}
4835 @item @code{binomial(n, m)}
4836 @tab binomial coefficients
4837 @cindex @code{binomial()}
4838 @item @code{Order(x)}
4839 @tab order term function in truncated power series
4840 @cindex @code{Order()}
4841 @item @code{Li(n, x)}
4844 @item @code{S(n, p, x)}
4845 @tab Nielsen's generalized polylogarithm
4847 @item @code{H(m_lst, x)}
4848 @tab harmonic polylogarithm
4850 @item @code{Li(m_lst, x_lst)}
4851 @tab multiple polylogarithm
4853 @item @code{mZeta(m_lst)}
4854 @tab multiple zeta value
4855 @cindex @code{mZeta()}
4860 For functions that have a branch cut in the complex plane GiNaC follows
4861 the conventions for C++ as defined in the ANSI standard as far as
4862 possible. In particular: the natural logarithm (@code{log}) and the
4863 square root (@code{sqrt}) both have their branch cuts running along the
4864 negative real axis where the points on the axis itself belong to the
4865 upper part (i.e. continuous with quadrant II). The inverse
4866 trigonometric and hyperbolic functions are not defined for complex
4867 arguments by the C++ standard, however. In GiNaC we follow the
4868 conventions used by CLN, which in turn follow the carefully designed
4869 definitions in the Common Lisp standard. It should be noted that this
4870 convention is identical to the one used by the C99 standard and by most
4871 serious CAS. It is to be expected that future revisions of the C++
4872 standard incorporate these functions in the complex domain in a manner
4873 compatible with C99.
4876 @node Solving Linear Systems of Equations, Input/Output, Built-in Functions, Methods and Functions
4877 @c node-name, next, previous, up
4878 @section Solving Linear Systems of Equations
4879 @cindex @code{lsolve()}
4881 The function @code{lsolve()} provides a convenient wrapper around some
4882 matrix operations that comes in handy when a system of linear equations
4886 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
4889 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
4890 @code{relational}) while @code{symbols} is a @code{lst} of
4891 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
4894 It returns the @code{lst} of solutions as an expression. As an example,
4895 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
4899 symbol a("a"), b("b"), x("x"), y("y");
4901 eqns = a*x+b*y==3, x-y==b;
4903 cout << lsolve(eqns, vars) << endl;
4904 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
4907 When the linear equations @code{eqns} are underdetermined, the solution
4908 will contain one or more tautological entries like @code{x==x},
4909 depending on the rank of the system. When they are overdetermined, the
4910 solution will be an empty @code{lst}. Note the third optional parameter
4911 to @code{lsolve()}: it accepts the same parameters as
4912 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
4916 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
4917 @c node-name, next, previous, up
4918 @section Input and output of expressions
4921 @subsection Expression output
4923 @cindex output of expressions
4925 Expressions can simply be written to any stream:
4930 ex e = 4.5*I+pow(x,2)*3/2;
4931 cout << e << endl; // prints '4.5*I+3/2*x^2'
4935 The default output format is identical to the @command{ginsh} input syntax and
4936 to that used by most computer algebra systems, but not directly pastable
4937 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4938 is printed as @samp{x^2}).
4940 It is possible to print expressions in a number of different formats with
4941 a set of stream manipulators;
4944 std::ostream & dflt(std::ostream & os);
4945 std::ostream & latex(std::ostream & os);
4946 std::ostream & tree(std::ostream & os);
4947 std::ostream & csrc(std::ostream & os);
4948 std::ostream & csrc_float(std::ostream & os);
4949 std::ostream & csrc_double(std::ostream & os);
4950 std::ostream & csrc_cl_N(std::ostream & os);
4951 std::ostream & index_dimensions(std::ostream & os);
4952 std::ostream & no_index_dimensions(std::ostream & os);
4955 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
4956 @command{ginsh} via the @code{print()}, @code{print_latex()} and
4957 @code{print_csrc()} functions, respectively.
4960 All manipulators affect the stream state permanently. To reset the output
4961 format to the default, use the @code{dflt} manipulator:
4965 cout << latex; // all output to cout will be in LaTeX format from now on
4966 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4967 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
4968 cout << dflt; // revert to default output format
4969 cout << e << endl; // prints '4.5*I+3/2*x^2'
4973 If you don't want to affect the format of the stream you're working with,
4974 you can output to a temporary @code{ostringstream} like this:
4979 s << latex << e; // format of cout remains unchanged
4980 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4985 @cindex @code{csrc_float}
4986 @cindex @code{csrc_double}
4987 @cindex @code{csrc_cl_N}
4988 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
4989 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
4990 format that can be directly used in a C or C++ program. The three possible
4991 formats select the data types used for numbers (@code{csrc_cl_N} uses the
4992 classes provided by the CLN library):
4996 cout << "f = " << csrc_float << e << ";\n";
4997 cout << "d = " << csrc_double << e << ";\n";
4998 cout << "n = " << csrc_cl_N << e << ";\n";
5002 The above example will produce (note the @code{x^2} being converted to
5006 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5007 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5008 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5012 The @code{tree} manipulator allows dumping the internal structure of an
5013 expression for debugging purposes:
5024 add, hash=0x0, flags=0x3, nops=2
5025 power, hash=0x0, flags=0x3, nops=2
5026 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5027 2 (numeric), hash=0x6526b0fa, flags=0xf
5028 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5031 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5035 @cindex @code{latex}
5036 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5037 It is rather similar to the default format but provides some braces needed
5038 by LaTeX for delimiting boxes and also converts some common objects to
5039 conventional LaTeX names. It is possible to give symbols a special name for
5040 LaTeX output by supplying it as a second argument to the @code{symbol}
5043 For example, the code snippet
5047 symbol x("x", "\\circ");
5048 ex e = lgamma(x).series(x==0,3);
5049 cout << latex << e << endl;
5056 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5059 @cindex @code{index_dimensions}
5060 @cindex @code{no_index_dimensions}
5061 Index dimensions are normally hidden in the output. To make them visible, use
5062 the @code{index_dimensions} manipulator. The dimensions will be written in
5063 square brackets behind each index value in the default and LaTeX output
5068 symbol x("x"), y("y");
5069 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5070 ex e = indexed(x, mu) * indexed(y, nu);
5073 // prints 'x~mu*y~nu'
5074 cout << index_dimensions << e << endl;
5075 // prints 'x~mu[4]*y~nu[4]'
5076 cout << no_index_dimensions << e << endl;
5077 // prints 'x~mu*y~nu'
5082 @cindex Tree traversal
5083 If you need any fancy special output format, e.g. for interfacing GiNaC
5084 with other algebra systems or for producing code for different
5085 programming languages, you can always traverse the expression tree yourself:
5088 static void my_print(const ex & e)
5090 if (is_a<function>(e))
5091 cout << ex_to<function>(e).get_name();
5093 cout << e.bp->class_name();
5095 size_t n = e.nops();
5097 for (size_t i=0; i<n; i++) @{
5109 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5117 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5118 symbol(y))),numeric(-2)))
5121 If you need an output format that makes it possible to accurately
5122 reconstruct an expression by feeding the output to a suitable parser or
5123 object factory, you should consider storing the expression in an
5124 @code{archive} object and reading the object properties from there.
5125 See the section on archiving for more information.
5128 @subsection Expression input
5129 @cindex input of expressions
5131 GiNaC provides no way to directly read an expression from a stream because
5132 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5133 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5134 @code{y} you defined in your program and there is no way to specify the
5135 desired symbols to the @code{>>} stream input operator.
5137 Instead, GiNaC lets you construct an expression from a string, specifying the
5138 list of symbols to be used:
5142 symbol x("x"), y("y");
5143 ex e("2*x+sin(y)", lst(x, y));
5147 The input syntax is the same as that used by @command{ginsh} and the stream
5148 output operator @code{<<}. The symbols in the string are matched by name to
5149 the symbols in the list and if GiNaC encounters a symbol not specified in
5150 the list it will throw an exception.
5152 With this constructor, it's also easy to implement interactive GiNaC programs:
5157 #include <stdexcept>
5158 #include <ginac/ginac.h>
5159 using namespace std;
5160 using namespace GiNaC;
5167 cout << "Enter an expression containing 'x': ";
5172 cout << "The derivative of " << e << " with respect to x is ";
5173 cout << e.diff(x) << ".\n";
5174 @} catch (exception &p) @{
5175 cerr << p.what() << endl;
5181 @subsection Archiving
5182 @cindex @code{archive} (class)
5185 GiNaC allows creating @dfn{archives} of expressions which can be stored
5186 to or retrieved from files. To create an archive, you declare an object
5187 of class @code{archive} and archive expressions in it, giving each
5188 expression a unique name:
5192 using namespace std;
5193 #include <ginac/ginac.h>
5194 using namespace GiNaC;
5198 symbol x("x"), y("y"), z("z");
5200 ex foo = sin(x + 2*y) + 3*z + 41;
5204 a.archive_ex(foo, "foo");
5205 a.archive_ex(bar, "the second one");
5209 The archive can then be written to a file:
5213 ofstream out("foobar.gar");
5219 The file @file{foobar.gar} contains all information that is needed to
5220 reconstruct the expressions @code{foo} and @code{bar}.
5222 @cindex @command{viewgar}
5223 The tool @command{viewgar} that comes with GiNaC can be used to view
5224 the contents of GiNaC archive files:
5227 $ viewgar foobar.gar
5228 foo = 41+sin(x+2*y)+3*z
5229 the second one = 42+sin(x+2*y)+3*z
5232 The point of writing archive files is of course that they can later be
5238 ifstream in("foobar.gar");
5243 And the stored expressions can be retrieved by their name:
5250 ex ex1 = a2.unarchive_ex(syms, "foo");
5251 ex ex2 = a2.unarchive_ex(syms, "the second one");
5253 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5254 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5255 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5259 Note that you have to supply a list of the symbols which are to be inserted
5260 in the expressions. Symbols in archives are stored by their name only and
5261 if you don't specify which symbols you have, unarchiving the expression will
5262 create new symbols with that name. E.g. if you hadn't included @code{x} in
5263 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5264 have had no effect because the @code{x} in @code{ex1} would have been a
5265 different symbol than the @code{x} which was defined at the beginning of
5266 the program, although both would appear as @samp{x} when printed.
5268 You can also use the information stored in an @code{archive} object to
5269 output expressions in a format suitable for exact reconstruction. The
5270 @code{archive} and @code{archive_node} classes have a couple of member
5271 functions that let you access the stored properties:
5274 static void my_print2(const archive_node & n)
5277 n.find_string("class", class_name);
5278 cout << class_name << "(";
5280 archive_node::propinfovector p;
5281 n.get_properties(p);
5283 size_t num = p.size();
5284 for (size_t i=0; i<num; i++) @{
5285 const string &name = p[i].name;
5286 if (name == "class")
5288 cout << name << "=";
5290 unsigned count = p[i].count;
5294 for (unsigned j=0; j<count; j++) @{
5295 switch (p[i].type) @{
5296 case archive_node::PTYPE_BOOL: @{
5298 n.find_bool(name, x, j);
5299 cout << (x ? "true" : "false");
5302 case archive_node::PTYPE_UNSIGNED: @{
5304 n.find_unsigned(name, x, j);
5308 case archive_node::PTYPE_STRING: @{
5310 n.find_string(name, x, j);
5311 cout << '\"' << x << '\"';
5314 case archive_node::PTYPE_NODE: @{
5315 const archive_node &x = n.find_ex_node(name, j);
5337 ex e = pow(2, x) - y;
5339 my_print2(ar.get_top_node(0)); cout << endl;
5347 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5348 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5349 overall_coeff=numeric(number="0"))
5352 Be warned, however, that the set of properties and their meaning for each
5353 class may change between GiNaC versions.
5356 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5357 @c node-name, next, previous, up
5358 @chapter Extending GiNaC
5360 By reading so far you should have gotten a fairly good understanding of
5361 GiNaC's design patterns. From here on you should start reading the
5362 sources. All we can do now is issue some recommendations how to tackle
5363 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5364 develop some useful extension please don't hesitate to contact the GiNaC
5365 authors---they will happily incorporate them into future versions.
5368 * What does not belong into GiNaC:: What to avoid.
5369 * Symbolic functions:: Implementing symbolic functions.
5370 * Printing:: Adding new output formats.
5371 * Structures:: Defining new algebraic classes (the easy way).
5372 * Adding classes:: Defining new algebraic classes (the hard way).
5376 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5377 @c node-name, next, previous, up
5378 @section What doesn't belong into GiNaC
5380 @cindex @command{ginsh}
5381 First of all, GiNaC's name must be read literally. It is designed to be
5382 a library for use within C++. The tiny @command{ginsh} accompanying
5383 GiNaC makes this even more clear: it doesn't even attempt to provide a
5384 language. There are no loops or conditional expressions in
5385 @command{ginsh}, it is merely a window into the library for the
5386 programmer to test stuff (or to show off). Still, the design of a
5387 complete CAS with a language of its own, graphical capabilities and all
5388 this on top of GiNaC is possible and is without doubt a nice project for
5391 There are many built-in functions in GiNaC that do not know how to
5392 evaluate themselves numerically to a precision declared at runtime
5393 (using @code{Digits}). Some may be evaluated at certain points, but not
5394 generally. This ought to be fixed. However, doing numerical
5395 computations with GiNaC's quite abstract classes is doomed to be
5396 inefficient. For this purpose, the underlying foundation classes
5397 provided by CLN are much better suited.
5400 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
5401 @c node-name, next, previous, up
5402 @section Symbolic functions
5404 The easiest and most instructive way to start extending GiNaC is probably to
5405 create your own symbolic functions. These are implemented with the help of
5406 two preprocessor macros:
5408 @cindex @code{DECLARE_FUNCTION}
5409 @cindex @code{REGISTER_FUNCTION}
5411 DECLARE_FUNCTION_<n>P(<name>)
5412 REGISTER_FUNCTION(<name>, <options>)
5415 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5416 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5417 parameters of type @code{ex} and returns a newly constructed GiNaC
5418 @code{function} object that represents your function.
5420 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5421 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5422 set of options that associate the symbolic function with C++ functions you
5423 provide to implement the various methods such as evaluation, derivative,
5424 series expansion etc. They also describe additional attributes the function
5425 might have, such as symmetry and commutation properties, and a name for
5426 LaTeX output. Multiple options are separated by the member access operator
5427 @samp{.} and can be given in an arbitrary order.
5429 (By the way: in case you are worrying about all the macros above we can
5430 assure you that functions are GiNaC's most macro-intense classes. We have
5431 done our best to avoid macros where we can.)
5433 @subsection A minimal example
5435 Here is an example for the implementation of a function with two arguments
5436 that is not further evaluated:
5439 DECLARE_FUNCTION_2P(myfcn)
5441 static ex myfcn_eval(const ex & x, const ex & y)
5443 return myfcn(x, y).hold();
5446 REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
5449 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5450 in algebraic expressions:
5456 ex e = 2*myfcn(42, 3*x+1) - x;
5457 // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
5458 // the actual expression
5460 // prints '2*myfcn(42,1+3*x)-x'
5465 @cindex @code{hold()}
5467 The @code{eval_func()} option specifies the C++ function that implements
5468 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5469 the same number of arguments as the associated symbolic function (two in this
5470 case) and returns the (possibly transformed or in some way simplified)
5471 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5472 of the automatic evaluation process). If no (further) evaluation is to take
5473 place, the @code{eval_func()} function must return the original function
5474 with @code{.hold()}, to avoid a potential infinite recursion. If your
5475 symbolic functions produce a segmentation fault or stack overflow when
5476 using them in expressions, you are probably missing a @code{.hold()}
5479 There is not much you can do with the @code{myfcn} function. It merely acts
5480 as a kind of container for its arguments (which is, however, sometimes
5481 perfectly sufficient). Let's have a look at the implementation of GiNaC's
5484 @subsection The cosine function
5486 The GiNaC header file @file{inifcns.h} contains the line
5489 DECLARE_FUNCTION_1P(cos)
5492 which declares to all programs using GiNaC that there is a function @samp{cos}
5493 that takes one @code{ex} as an argument. This is all they need to know to use
5494 this function in expressions.
5496 The implementation of the cosine function is in @file{inifcns_trans.cpp}. The
5497 @code{eval_func()} function looks something like this (actually, it doesn't
5498 look like this at all, but it should give you an idea what is going on):
5501 static ex cos_eval(const ex & x)
5503 if ("x is a multiple of 2*Pi")
5505 else if ("x is a multiple of Pi")
5507 else if ("x is a multiple of Pi/2")
5511 else if ("x has the form 'acos(y)'")
5513 else if ("x has the form 'asin(y)'")
5518 return cos(x).hold();
5522 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5523 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5524 symbolic transformation can be done, the unmodified function is returned
5525 with @code{.hold()}.
5527 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5528 The user has to call @code{evalf()} for that. This is implemented in a
5532 static ex cos_evalf(const ex & x)
5534 if (is_a<numeric>(x))
5535 return cos(ex_to<numeric>(x));
5537 return cos(x).hold();
5541 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5542 in this case the @code{cos()} function for @code{numeric} objects, which in
5543 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5544 isn't really needed here, but reminds us that the corresponding @code{eval()}
5545 function would require it in this place.
5547 Differentiation will surely turn up and so we need to tell @code{cos}
5548 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5549 instance, are then handled automatically by @code{basic::diff} and
5553 static ex cos_deriv(const ex & x, unsigned diff_param)
5559 @cindex product rule
5560 The second parameter is obligatory but uninteresting at this point. It
5561 specifies which parameter to differentiate in a partial derivative in
5562 case the function has more than one parameter, and its main application
5563 is for correct handling of the chain rule.
5565 An implementation of the series expansion is not needed for @code{cos()} as
5566 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5567 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5568 the other hand, does have poles and may need to do Laurent expansion:
5571 static ex tan_series(const ex & x, const relational & rel,
5572 int order, unsigned options)
5574 // Find the actual expansion point
5575 const ex x_pt = x.subs(rel);
5577 if ("x_pt is not an odd multiple of Pi/2")
5578 throw do_taylor(); // tell function::series() to do Taylor expansion
5580 // On a pole, expand sin()/cos()
5581 return (sin(x)/cos(x)).series(rel, order+2, options);
5585 The @code{series()} implementation of a function @emph{must} return a
5586 @code{pseries} object, otherwise your code will crash.
5588 Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
5589 macro is used to tell the system how the @code{cos()} function behaves:
5592 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5593 evalf_func(cos_evalf).
5594 derivative_func(cos_deriv).
5595 latex_name("\\cos"));
5598 This registers the @code{cos_eval()}, @code{cos_evalf()} and
5599 @code{cos_deriv()} C++ functions with the @code{cos()} function, and also
5600 gives it a proper LaTeX name.
5602 @subsection Function options
5604 GiNaC functions understand several more options which are always
5605 specified as @code{.option(params)}. None of them are required, but you
5606 need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
5607 the @code{eval()} method).
5610 eval_func(<C++ function>)
5611 evalf_func(<C++ function>)
5612 derivative_func(<C++ function>)
5613 series_func(<C++ function>)
5616 These specify the C++ functions that implement symbolic evaluation,
5617 numeric evaluation, partial derivatives, and series expansion, respectively.
5618 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5619 @code{diff()} and @code{series()}.
5621 The @code{eval_func()} function needs to use @code{.hold()} if no further
5622 automatic evaluation is desired or possible.
5624 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5625 expansion, which is correct if there are no poles involved. If the function
5626 has poles in the complex plane, the @code{series_func()} needs to check
5627 whether the expansion point is on a pole and fall back to Taylor expansion
5628 if it isn't. Otherwise, the pole usually needs to be regularized by some
5629 suitable transformation.
5632 latex_name(const string & n)
5635 specifies the LaTeX code that represents the name of the function in LaTeX
5636 output. The default is to put the function name in an @code{\mbox@{@}}.
5639 do_not_evalf_params()
5642 This tells @code{evalf()} to not recursively evaluate the parameters of the
5643 function before calling the @code{evalf_func()}.
5646 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5649 This allows you to explicitly specify the commutation properties of the
5650 function (@xref{Non-commutative objects}, for an explanation of
5651 (non)commutativity in GiNaC). For example, you can use
5652 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5653 GiNaC treat your function like a matrix. By default, functions inherit the
5654 commutation properties of their first argument.
5657 set_symmetry(const symmetry & s)
5660 specifies the symmetry properties of the function with respect to its
5661 arguments. @xref{Indexed objects}, for an explanation of symmetry
5662 specifications. GiNaC will automatically rearrange the arguments of
5663 symmetric functions into a canonical order.
5665 Sometimes you may want to have finer control over how functions are
5666 displayed in the output. For example, the @code{abs()} function prints
5667 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
5668 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
5672 print_func<C>(<C++ function>)
5675 option which is explained in the next section.
5678 @node Printing, Structures, Symbolic functions, Extending GiNaC
5679 @c node-name, next, previous, up
5680 @section GiNaC's expression output system
5682 GiNaC allows the output of expressions in a variety of different formats
5683 (@pxref{Input/Output}). This section will explain how expression output
5684 is implemented internally, and how to define your own output formats or
5685 change the output format of built-in algebraic objects. You will also want
5686 to read this section if you plan to write your own algebraic classes or
5689 @cindex @code{print_context} (class)
5690 @cindex @code{print_dflt} (class)
5691 @cindex @code{print_latex} (class)
5692 @cindex @code{print_tree} (class)
5693 @cindex @code{print_csrc} (class)
5694 All the different output formats are represented by a hierarchy of classes
5695 rooted in the @code{print_context} class, defined in the @file{print.h}
5700 the default output format
5702 output in LaTeX mathematical mode
5704 a dump of the internal expression structure (for debugging)
5706 the base class for C source output
5707 @item print_csrc_float
5708 C source output using the @code{float} type
5709 @item print_csrc_double
5710 C source output using the @code{double} type
5711 @item print_csrc_cl_N
5712 C source output using CLN types
5715 The @code{print_context} base class provides two public data members:
5727 @code{s} is a reference to the stream to output to, while @code{options}
5728 holds flags and modifiers. Currently, there is only one flag defined:
5729 @code{print_options::print_index_dimensions} instructs the @code{idx} class
5730 to print the index dimension which is normally hidden.
5732 When you write something like @code{std::cout << e}, where @code{e} is
5733 an object of class @code{ex}, GiNaC will construct an appropriate
5734 @code{print_context} object (of a class depending on the selected output
5735 format), fill in the @code{s} and @code{options} members, and call
5737 @cindex @code{print()}
5739 void ex::print(const print_context & c, unsigned level = 0) const;
5742 which in turn forwards the call to the @code{print()} method of the
5743 top-level algebraic object contained in the expression.
5745 Unlike other methods, GiNaC classes don't usually override their
5746 @code{print()} method to implement expression output. Instead, the default
5747 implementation @code{basic::print(c, level)} performs a run-time double
5748 dispatch to a function selected by the dynamic type of the object and the
5749 passed @code{print_context}. To this end, GiNaC maintains a separate method
5750 table for each class, similar to the virtual function table used for ordinary
5751 (single) virtual function dispatch.
5753 The method table contains one slot for each possible @code{print_context}
5754 type, indexed by the (internally assigned) serial number of the type. Slots
5755 may be empty, in which case GiNaC will retry the method lookup with the
5756 @code{print_context} object's parent class, possibly repeating the process
5757 until it reaches the @code{print_context} base class. If there's still no
5758 method defined, the method table of the algebraic object's parent class
5759 is consulted, and so on, until a matching method is found (eventually it
5760 will reach the combination @code{basic/print_context}, which prints the
5761 object's class name enclosed in square brackets).
5763 You can think of the print methods of all the different classes and output
5764 formats as being arranged in a two-dimensional matrix with one axis listing
5765 the algebraic classes and the other axis listing the @code{print_context}
5768 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
5769 to implement printing, but then they won't get any of the benefits of the
5770 double dispatch mechanism (such as the ability for derived classes to
5771 inherit only certain print methods from its parent, or the replacement of
5772 methods at run-time).
5774 @subsection Print methods for classes
5776 The method table for a class is set up either in the definition of the class,
5777 by passing the appropriate @code{print_func<C>()} option to
5778 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
5779 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
5780 can also be used to override existing methods dynamically.
5782 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
5783 be a member function of the class (or one of its parent classes), a static
5784 member function, or an ordinary (global) C++ function. The @code{C} template
5785 parameter specifies the appropriate @code{print_context} type for which the
5786 method should be invoked, while, in the case of @code{set_print_func<>()}, the
5787 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
5788 the class is the one being implemented by
5789 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
5791 For print methods that are member functions, their first argument must be of
5792 a type convertible to a @code{const C &}, and the second argument must be an
5795 For static members and global functions, the first argument must be of a type
5796 convertible to a @code{const T &}, the second argument must be of a type
5797 convertible to a @code{const C &}, and the third argument must be an
5798 @code{unsigned}. A global function will, of course, not have access to
5799 private and protected members of @code{T}.
5801 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
5802 and @code{basic::print()}) is used for proper parenthesizing of the output
5803 (and by @code{print_tree} for proper indentation). It can be used for similar
5804 purposes if you write your own output formats.
5806 The explanations given above may seem complicated, but in practice it's
5807 really simple, as shown in the following example. Suppose that we want to
5808 display exponents in LaTeX output not as superscripts but with little
5809 upwards-pointing arrows. This can be achieved in the following way:
5812 void my_print_power_as_latex(const power & p,
5813 const print_latex & c,
5816 // get the precedence of the 'power' class
5817 unsigned power_prec = p.precedence();
5819 // if the parent operator has the same or a higher precedence
5820 // we need parentheses around the power
5821 if (level >= power_prec)
5824 // print the basis and exponent, each enclosed in braces, and
5825 // separated by an uparrow
5827 p.op(0).print(c, power_prec);
5828 c.s << "@}\\uparrow@{";
5829 p.op(1).print(c, power_prec);
5832 // don't forget the closing parenthesis
5833 if (level >= power_prec)
5839 // a sample expression
5840 symbol x("x"), y("y");
5841 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
5843 // switch to LaTeX mode
5846 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
5849 // now we replace the method for the LaTeX output of powers with
5851 set_print_func<power, print_latex>(my_print_power_as_latex);
5853 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
5863 The first argument of @code{my_print_power_as_latex} could also have been
5864 a @code{const basic &}, the second one a @code{const print_context &}.
5867 The above code depends on @code{mul} objects converting their operands to
5868 @code{power} objects for the purpose of printing.
5871 The output of products including negative powers as fractions is also
5872 controlled by the @code{mul} class.
5875 The @code{power/print_latex} method provided by GiNaC prints square roots
5876 using @code{\sqrt}, but the above code doesn't.
5880 It's not possible to restore a method table entry to its previous or default
5881 value. Once you have called @code{set_print_func()}, you can only override
5882 it with another call to @code{set_print_func()}, but you can't easily go back
5883 to the default behavior again (you can, of course, dig around in the GiNaC
5884 sources, find the method that is installed at startup
5885 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
5886 one; that is, after you circumvent the C++ member access control@dots{}).
5888 @subsection Print methods for functions
5890 Symbolic functions employ a print method dispatch mechanism similar to the
5891 one used for classes. The methods are specified with @code{print_func<C>()}
5892 function options. If you don't specify any special print methods, the function
5893 will be printed with its name (or LaTeX name, if supplied), followed by a
5894 comma-separated list of arguments enclosed in parentheses.
5896 For example, this is what GiNaC's @samp{abs()} function is defined like:
5899 static ex abs_eval(const ex & arg) @{ ... @}
5900 static ex abs_evalf(const ex & arg) @{ ... @}
5902 static void abs_print_latex(const ex & arg, const print_context & c)
5904 c.s << "@{|"; arg.print(c); c.s << "|@}";
5907 static void abs_print_csrc_float(const ex & arg, const print_context & c)
5909 c.s << "fabs("; arg.print(c); c.s << ")";
5912 REGISTER_FUNCTION(abs, eval_func(abs_eval).
5913 evalf_func(abs_evalf).
5914 print_func<print_latex>(abs_print_latex).
5915 print_func<print_csrc_float>(abs_print_csrc_float).
5916 print_func<print_csrc_double>(abs_print_csrc_float));
5919 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
5920 in non-CLN C source output, but as @code{abs(x)} in all other formats.
5922 There is currently no equivalent of @code{set_print_func()} for functions.
5925 @node Structures, Adding classes, Printing, Extending GiNaC
5926 @c node-name, next, previous, up
5929 If you are doing some very specialized things with GiNaC, or if you just
5930 need some more organized way to store data in your expressions instead of
5931 anonymous lists, you may want to implement your own algebraic classes.
5932 ('algebraic class' means any class directly or indirectly derived from
5933 @code{basic} that can be used in GiNaC expressions).
5935 GiNaC offers two ways of accomplishing this: either by using the
5936 @code{structure<T>} template class, or by rolling your own class from
5937 scratch. This section will discuss the @code{structure<T>} template which
5938 is easier to use but more limited, while the implementation of custom
5939 GiNaC classes is the topic of the next section. However, you may want to
5940 read both sections because many common concepts and member functions are
5941 shared by both concepts, and it will also allow you to decide which approach
5942 is most suited to your needs.
5944 The @code{structure<T>} template, defined in the GiNaC header file
5945 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
5946 or @code{class}) into a GiNaC object that can be used in expressions.
5948 @subsection Example: scalar products
5950 Let's suppose that we need a way to handle some kind of abstract scalar
5951 product of the form @samp{<x|y>} in expressions. Objects of the scalar
5952 product class have to store their left and right operands, which can in turn
5953 be arbitrary expressions. Here is a possible way to represent such a
5954 product in a C++ @code{struct}:
5958 using namespace std;
5960 #include <ginac/ginac.h>
5961 using namespace GiNaC;
5967 sprod_s(ex l, ex r) : left(l), right(r) @{@}
5971 The default constructor is required. Now, to make a GiNaC class out of this
5972 data structure, we need only one line:
5975 typedef structure<sprod_s> sprod;
5978 That's it. This line constructs an algebraic class @code{sprod} which
5979 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
5980 expressions like any other GiNaC class:
5984 symbol a("a"), b("b");
5985 ex e = sprod(sprod_s(a, b));
5989 Note the difference between @code{sprod} which is the algebraic class, and
5990 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
5991 and @code{right} data members. As shown above, an @code{sprod} can be
5992 constructed from an @code{sprod_s} object.
5994 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
5995 you could define a little wrapper function like this:
5998 inline ex make_sprod(ex left, ex right)
6000 return sprod(sprod_s(left, right));
6004 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6005 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6006 @code{get_struct()}:
6010 cout << ex_to<sprod>(e)->left << endl;
6012 cout << ex_to<sprod>(e).get_struct().right << endl;
6017 You only have read access to the members of @code{sprod_s}.
6019 The type definition of @code{sprod} is enough to write your own algorithms
6020 that deal with scalar products, for example:
6025 if (is_a<sprod>(p)) @{
6026 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6027 return make_sprod(sp.right, sp.left);
6038 @subsection Structure output
6040 While the @code{sprod} type is useable it still leaves something to be
6041 desired, most notably proper output:
6046 // -> [structure object]
6050 By default, any structure types you define will be printed as
6051 @samp{[structure object]}. To override this you can either specialize the
6052 template's @code{print()} member function, or specify print methods with
6053 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6054 it's not possible to supply class options like @code{print_func<>()} to
6055 structures, so for a self-contained structure type you need to resort to
6056 overriding the @code{print()} function, which is also what we will do here.
6058 The member functions of GiNaC classes are described in more detail in the
6059 next section, but it shouldn't be hard to figure out what's going on here:
6062 void sprod::print(const print_context & c, unsigned level) const
6064 // tree debug output handled by superclass
6065 if (is_a<print_tree>(c))
6066 inherited::print(c, level);
6068 // get the contained sprod_s object
6069 const sprod_s & sp = get_struct();
6071 // print_context::s is a reference to an ostream
6072 c.s << "<" << sp.left << "|" << sp.right << ">";
6076 Now we can print expressions containing scalar products:
6082 cout << swap_sprod(e) << endl;
6087 @subsection Comparing structures
6089 The @code{sprod} class defined so far still has one important drawback: all
6090 scalar products are treated as being equal because GiNaC doesn't know how to
6091 compare objects of type @code{sprod_s}. This can lead to some confusing
6092 and undesired behavior:
6096 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6098 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6099 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6103 To remedy this, we first need to define the operators @code{==} and @code{<}
6104 for objects of type @code{sprod_s}:
6107 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6109 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6112 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6114 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6118 The ordering established by the @code{<} operator doesn't have to make any
6119 algebraic sense, but it needs to be well defined. Note that we can't use
6120 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6121 in the implementation of these operators because they would construct
6122 GiNaC @code{relational} objects which in the case of @code{<} do not
6123 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6124 decide which one is algebraically 'less').
6126 Next, we need to change our definition of the @code{sprod} type to let
6127 GiNaC know that an ordering relation exists for the embedded objects:
6130 typedef structure<sprod_s, compare_std_less> sprod;
6133 @code{sprod} objects then behave as expected:
6137 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6138 // -> <a|b>-<a^2|b^2>
6139 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6140 // -> <a|b>+<a^2|b^2>
6141 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6143 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6148 The @code{compare_std_less} policy parameter tells GiNaC to use the
6149 @code{std::less} and @code{std::equal_to} functors to compare objects of
6150 type @code{sprod_s}. By default, these functors forward their work to the
6151 standard @code{<} and @code{==} operators, which we have overloaded.
6152 Alternatively, we could have specialized @code{std::less} and
6153 @code{std::equal_to} for class @code{sprod_s}.
6155 GiNaC provides two other comparison policies for @code{structure<T>}
6156 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6157 which does a bit-wise comparison of the contained @code{T} objects.
6158 This should be used with extreme care because it only works reliably with
6159 built-in integral types, and it also compares any padding (filler bytes of
6160 undefined value) that the @code{T} class might have.
6162 @subsection Subexpressions
6164 Our scalar product class has two subexpressions: the left and right
6165 operands. It might be a good idea to make them accessible via the standard
6166 @code{nops()} and @code{op()} methods:
6169 size_t sprod::nops() const
6174 ex sprod::op(size_t i) const
6178 return get_struct().left;
6180 return get_struct().right;
6182 throw std::range_error("sprod::op(): no such operand");
6187 Implementing @code{nops()} and @code{op()} for container types such as
6188 @code{sprod} has two other nice side effects:
6192 @code{has()} works as expected
6194 GiNaC generates better hash keys for the objects (the default implementation
6195 of @code{calchash()} takes subexpressions into account)
6198 @cindex @code{let_op()}
6199 There is a non-const variant of @code{op()} called @code{let_op()} that
6200 allows replacing subexpressions:
6203 ex & sprod::let_op(size_t i)
6205 // every non-const member function must call this
6206 ensure_if_modifiable();
6210 return get_struct().left;
6212 return get_struct().right;
6214 throw std::range_error("sprod::let_op(): no such operand");
6219 Once we have provided @code{let_op()} we also get @code{subs()} and
6220 @code{map()} for free. In fact, every container class that returns a non-null
6221 @code{nops()} value must either implement @code{let_op()} or provide custom
6222 implementations of @code{subs()} and @code{map()}.
6224 In turn, the availability of @code{map()} enables the recursive behavior of a
6225 couple of other default method implementations, in particular @code{evalf()},
6226 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6227 we probably want to provide our own version of @code{expand()} for scalar
6228 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6229 This is left as an exercise for the reader.
6231 The @code{structure<T>} template defines many more member functions that
6232 you can override by specialization to customize the behavior of your
6233 structures. You are referred to the next section for a description of
6234 some of these (especially @code{eval()}). There is, however, one topic
6235 that shall be addressed here, as it demonstrates one peculiarity of the
6236 @code{structure<T>} template: archiving.
6238 @subsection Archiving structures
6240 If you don't know how the archiving of GiNaC objects is implemented, you
6241 should first read the next section and then come back here. You're back?
6244 To implement archiving for structures it is not enough to provide
6245 specializations for the @code{archive()} member function and the
6246 unarchiving constructor (the @code{unarchive()} function has a default
6247 implementation). You also need to provide a unique name (as a string literal)
6248 for each structure type you define. This is because in GiNaC archives,
6249 the class of an object is stored as a string, the class name.
6251 By default, this class name (as returned by the @code{class_name()} member
6252 function) is @samp{structure} for all structure classes. This works as long
6253 as you have only defined one structure type, but if you use two or more you
6254 need to provide a different name for each by specializing the
6255 @code{get_class_name()} member function. Here is a sample implementation
6256 for enabling archiving of the scalar product type defined above:
6259 const char *sprod::get_class_name() @{ return "sprod"; @}
6261 void sprod::archive(archive_node & n) const
6263 inherited::archive(n);
6264 n.add_ex("left", get_struct().left);
6265 n.add_ex("right", get_struct().right);
6268 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
6270 n.find_ex("left", get_struct().left, sym_lst);
6271 n.find_ex("right", get_struct().right, sym_lst);
6275 Note that the unarchiving constructor is @code{sprod::structure} and not
6276 @code{sprod::sprod}, and that we don't need to supply an
6277 @code{sprod::unarchive()} function.
6280 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
6281 @c node-name, next, previous, up
6282 @section Adding classes
6284 The @code{structure<T>} template provides an way to extend GiNaC with custom
6285 algebraic classes that is easy to use but has its limitations, the most
6286 severe of which being that you can't add any new member functions to
6287 structures. To be able to do this, you need to write a new class definition
6290 This section will explain how to implement new algebraic classes in GiNaC by
6291 giving the example of a simple 'string' class. After reading this section
6292 you will know how to properly declare a GiNaC class and what the minimum
6293 required member functions are that you have to implement. We only cover the
6294 implementation of a 'leaf' class here (i.e. one that doesn't contain
6295 subexpressions). Creating a container class like, for example, a class
6296 representing tensor products is more involved but this section should give
6297 you enough information so you can consult the source to GiNaC's predefined
6298 classes if you want to implement something more complicated.
6300 @subsection GiNaC's run-time type information system
6302 @cindex hierarchy of classes
6304 All algebraic classes (that is, all classes that can appear in expressions)
6305 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
6306 @code{basic *} (which is essentially what an @code{ex} is) represents a
6307 generic pointer to an algebraic class. Occasionally it is necessary to find
6308 out what the class of an object pointed to by a @code{basic *} really is.
6309 Also, for the unarchiving of expressions it must be possible to find the
6310 @code{unarchive()} function of a class given the class name (as a string). A
6311 system that provides this kind of information is called a run-time type
6312 information (RTTI) system. The C++ language provides such a thing (see the
6313 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
6314 implements its own, simpler RTTI.
6316 The RTTI in GiNaC is based on two mechanisms:
6321 The @code{basic} class declares a member variable @code{tinfo_key} which
6322 holds an unsigned integer that identifies the object's class. These numbers
6323 are defined in the @file{tinfos.h} header file for the built-in GiNaC
6324 classes. They all start with @code{TINFO_}.
6327 By means of some clever tricks with static members, GiNaC maintains a list
6328 of information for all classes derived from @code{basic}. The information
6329 available includes the class names, the @code{tinfo_key}s, and pointers
6330 to the unarchiving functions. This class registry is defined in the
6331 @file{registrar.h} header file.
6335 The disadvantage of this proprietary RTTI implementation is that there's
6336 a little more to do when implementing new classes (C++'s RTTI works more
6337 or less automatically) but don't worry, most of the work is simplified by
6340 @subsection A minimalistic example
6342 Now we will start implementing a new class @code{mystring} that allows
6343 placing character strings in algebraic expressions (this is not very useful,
6344 but it's just an example). This class will be a direct subclass of
6345 @code{basic}. You can use this sample implementation as a starting point
6346 for your own classes.
6348 The code snippets given here assume that you have included some header files
6354 #include <stdexcept>
6355 using namespace std;
6357 #include <ginac/ginac.h>
6358 using namespace GiNaC;
6361 The first thing we have to do is to define a @code{tinfo_key} for our new
6362 class. This can be any arbitrary unsigned number that is not already taken
6363 by one of the existing classes but it's better to come up with something
6364 that is unlikely to clash with keys that might be added in the future. The
6365 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
6366 which is not a requirement but we are going to stick with this scheme:
6369 const unsigned TINFO_mystring = 0x42420001U;
6372 Now we can write down the class declaration. The class stores a C++
6373 @code{string} and the user shall be able to construct a @code{mystring}
6374 object from a C or C++ string:
6377 class mystring : public basic
6379 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
6382 mystring(const string &s);
6383 mystring(const char *s);
6389 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6392 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
6393 macros are defined in @file{registrar.h}. They take the name of the class
6394 and its direct superclass as arguments and insert all required declarations
6395 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
6396 the first line after the opening brace of the class definition. The
6397 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
6398 source (at global scope, of course, not inside a function).
6400 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
6401 declarations of the default constructor and a couple of other functions that
6402 are required. It also defines a type @code{inherited} which refers to the
6403 superclass so you don't have to modify your code every time you shuffle around
6404 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
6405 class with the GiNaC RTTI (there is also a
6406 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
6407 options for the class, and which we will be using instead in a few minutes).
6409 Now there are seven member functions we have to implement to get a working
6415 @code{mystring()}, the default constructor.
6418 @code{void archive(archive_node &n)}, the archiving function. This stores all
6419 information needed to reconstruct an object of this class inside an
6420 @code{archive_node}.
6423 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
6424 constructor. This constructs an instance of the class from the information
6425 found in an @code{archive_node}.
6428 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
6429 unarchiving function. It constructs a new instance by calling the unarchiving
6433 @cindex @code{compare_same_type()}
6434 @code{int compare_same_type(const basic &other)}, which is used internally
6435 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
6436 -1, depending on the relative order of this object and the @code{other}
6437 object. If it returns 0, the objects are considered equal.
6438 @strong{Note:} This has nothing to do with the (numeric) ordering
6439 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
6440 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
6441 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
6442 must provide a @code{compare_same_type()} function, even those representing
6443 objects for which no reasonable algebraic ordering relationship can be
6447 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
6448 which are the two constructors we declared.
6452 Let's proceed step-by-step. The default constructor looks like this:
6455 mystring::mystring() : inherited(TINFO_mystring) @{@}
6458 The golden rule is that in all constructors you have to set the
6459 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
6460 it will be set by the constructor of the superclass and all hell will break
6461 loose in the RTTI. For your convenience, the @code{basic} class provides
6462 a constructor that takes a @code{tinfo_key} value, which we are using here
6463 (remember that in our case @code{inherited == basic}). If the superclass
6464 didn't have such a constructor, we would have to set the @code{tinfo_key}
6465 to the right value manually.
6467 In the default constructor you should set all other member variables to
6468 reasonable default values (we don't need that here since our @code{str}
6469 member gets set to an empty string automatically).
6471 Next are the three functions for archiving. You have to implement them even
6472 if you don't plan to use archives, but the minimum required implementation
6473 is really simple. First, the archiving function:
6476 void mystring::archive(archive_node &n) const
6478 inherited::archive(n);
6479 n.add_string("string", str);
6483 The only thing that is really required is calling the @code{archive()}
6484 function of the superclass. Optionally, you can store all information you
6485 deem necessary for representing the object into the passed
6486 @code{archive_node}. We are just storing our string here. For more
6487 information on how the archiving works, consult the @file{archive.h} header
6490 The unarchiving constructor is basically the inverse of the archiving
6494 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
6496 n.find_string("string", str);
6500 If you don't need archiving, just leave this function empty (but you must
6501 invoke the unarchiving constructor of the superclass). Note that we don't
6502 have to set the @code{tinfo_key} here because it is done automatically
6503 by the unarchiving constructor of the @code{basic} class.
6505 Finally, the unarchiving function:
6508 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
6510 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
6514 You don't have to understand how exactly this works. Just copy these
6515 four lines into your code literally (replacing the class name, of
6516 course). It calls the unarchiving constructor of the class and unless
6517 you are doing something very special (like matching @code{archive_node}s
6518 to global objects) you don't need a different implementation. For those
6519 who are interested: setting the @code{dynallocated} flag puts the object
6520 under the control of GiNaC's garbage collection. It will get deleted
6521 automatically once it is no longer referenced.
6523 Our @code{compare_same_type()} function uses a provided function to compare
6527 int mystring::compare_same_type(const basic &other) const
6529 const mystring &o = static_cast<const mystring &>(other);
6530 int cmpval = str.compare(o.str);
6533 else if (cmpval < 0)
6540 Although this function takes a @code{basic &}, it will always be a reference
6541 to an object of exactly the same class (objects of different classes are not
6542 comparable), so the cast is safe. If this function returns 0, the two objects
6543 are considered equal (in the sense that @math{A-B=0}), so you should compare
6544 all relevant member variables.
6546 Now the only thing missing is our two new constructors:
6549 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6550 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6553 No surprises here. We set the @code{str} member from the argument and
6554 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6556 That's it! We now have a minimal working GiNaC class that can store
6557 strings in algebraic expressions. Let's confirm that the RTTI works:
6560 ex e = mystring("Hello, world!");
6561 cout << is_a<mystring>(e) << endl;
6564 cout << e.bp->class_name() << endl;
6568 Obviously it does. Let's see what the expression @code{e} looks like:
6572 // -> [mystring object]
6575 Hm, not exactly what we expect, but of course the @code{mystring} class
6576 doesn't yet know how to print itself. This can be done either by implementing
6577 the @code{print()} member function, or, preferably, by specifying a
6578 @code{print_func<>()} class option. Let's say that we want to print the string
6579 surrounded by double quotes:
6582 class mystring : public basic
6586 void do_print(const print_context &c, unsigned level = 0) const;
6590 void mystring::do_print(const print_context &c, unsigned level) const
6592 // print_context::s is a reference to an ostream
6593 c.s << '\"' << str << '\"';
6597 The @code{level} argument is only required for container classes to
6598 correctly parenthesize the output.
6600 Now we need to tell GiNaC that @code{mystring} objects should use the
6601 @code{do_print()} member function for printing themselves. For this, we
6605 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6611 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
6612 print_func<print_context>(&mystring::do_print))
6615 Let's try again to print the expression:
6619 // -> "Hello, world!"
6622 Much better. If we wanted to have @code{mystring} objects displayed in a
6623 different way depending on the output format (default, LaTeX, etc.), we
6624 would have supplied multiple @code{print_func<>()} options with different
6625 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
6626 separated by dots. This is similar to the way options are specified for
6627 symbolic functions. @xref{Printing}, for a more in-depth description of the
6628 way expression output is implemented in GiNaC.
6630 The @code{mystring} class can be used in arbitrary expressions:
6633 e += mystring("GiNaC rulez");
6635 // -> "GiNaC rulez"+"Hello, world!"
6638 (GiNaC's automatic term reordering is in effect here), or even
6641 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
6643 // -> "One string"^(2*sin(-"Another string"+Pi))
6646 Whether this makes sense is debatable but remember that this is only an
6647 example. At least it allows you to implement your own symbolic algorithms
6650 Note that GiNaC's algebraic rules remain unchanged:
6653 e = mystring("Wow") * mystring("Wow");
6657 e = pow(mystring("First")-mystring("Second"), 2);
6658 cout << e.expand() << endl;
6659 // -> -2*"First"*"Second"+"First"^2+"Second"^2
6662 There's no way to, for example, make GiNaC's @code{add} class perform string
6663 concatenation. You would have to implement this yourself.
6665 @subsection Automatic evaluation
6668 @cindex @code{eval()}
6669 @cindex @code{hold()}
6670 When dealing with objects that are just a little more complicated than the
6671 simple string objects we have implemented, chances are that you will want to
6672 have some automatic simplifications or canonicalizations performed on them.
6673 This is done in the evaluation member function @code{eval()}. Let's say that
6674 we wanted all strings automatically converted to lowercase with
6675 non-alphabetic characters stripped, and empty strings removed:
6678 class mystring : public basic
6682 ex eval(int level = 0) const;
6686 ex mystring::eval(int level) const
6689 for (int i=0; i<str.length(); i++) @{
6691 if (c >= 'A' && c <= 'Z')
6692 new_str += tolower(c);
6693 else if (c >= 'a' && c <= 'z')
6697 if (new_str.length() == 0)
6700 return mystring(new_str).hold();
6704 The @code{level} argument is used to limit the recursion depth of the
6705 evaluation. We don't have any subexpressions in the @code{mystring}
6706 class so we are not concerned with this. If we had, we would call the
6707 @code{eval()} functions of the subexpressions with @code{level - 1} as
6708 the argument if @code{level != 1}. The @code{hold()} member function
6709 sets a flag in the object that prevents further evaluation. Otherwise
6710 we might end up in an endless loop. When you want to return the object
6711 unmodified, use @code{return this->hold();}.
6713 Let's confirm that it works:
6716 ex e = mystring("Hello, world!") + mystring("!?#");
6720 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
6725 @subsection Optional member functions
6727 We have implemented only a small set of member functions to make the class
6728 work in the GiNaC framework. There are two functions that are not strictly
6729 required but will make operations with objects of the class more efficient:
6731 @cindex @code{calchash()}
6732 @cindex @code{is_equal_same_type()}
6734 unsigned calchash() const;
6735 bool is_equal_same_type(const basic &other) const;
6738 The @code{calchash()} method returns an @code{unsigned} hash value for the
6739 object which will allow GiNaC to compare and canonicalize expressions much
6740 more efficiently. You should consult the implementation of some of the built-in
6741 GiNaC classes for examples of hash functions. The default implementation of
6742 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
6743 class and all subexpressions that are accessible via @code{op()}.
6745 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
6746 tests for equality without establishing an ordering relation, which is often
6747 faster. The default implementation of @code{is_equal_same_type()} just calls
6748 @code{compare_same_type()} and tests its result for zero.
6750 @subsection Other member functions
6752 For a real algebraic class, there are probably some more functions that you
6753 might want to provide:
6756 bool info(unsigned inf) const;
6757 ex evalf(int level = 0) const;
6758 ex series(const relational & r, int order, unsigned options = 0) const;
6759 ex derivative(const symbol & s) const;
6762 If your class stores sub-expressions (see the scalar product example in the
6763 previous section) you will probably want to override
6765 @cindex @code{let_op()}
6768 ex op(size_t i) const;
6769 ex & let_op(size_t i);
6770 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
6771 ex map(map_function & f) const;
6774 @code{let_op()} is a variant of @code{op()} that allows write access. The
6775 default implementations of @code{subs()} and @code{map()} use it, so you have
6776 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
6778 You can, of course, also add your own new member functions. Remember
6779 that the RTTI may be used to get information about what kinds of objects
6780 you are dealing with (the position in the class hierarchy) and that you
6781 can always extract the bare object from an @code{ex} by stripping the
6782 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
6783 should become a need.
6785 That's it. May the source be with you!
6788 @node A Comparison With Other CAS, Advantages, Adding classes, Top
6789 @c node-name, next, previous, up
6790 @chapter A Comparison With Other CAS
6793 This chapter will give you some information on how GiNaC compares to
6794 other, traditional Computer Algebra Systems, like @emph{Maple},
6795 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
6796 disadvantages over these systems.
6799 * Advantages:: Strengths of the GiNaC approach.
6800 * Disadvantages:: Weaknesses of the GiNaC approach.
6801 * Why C++?:: Attractiveness of C++.
6804 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
6805 @c node-name, next, previous, up
6808 GiNaC has several advantages over traditional Computer
6809 Algebra Systems, like
6814 familiar language: all common CAS implement their own proprietary
6815 grammar which you have to learn first (and maybe learn again when your
6816 vendor decides to `enhance' it). With GiNaC you can write your program
6817 in common C++, which is standardized.
6821 structured data types: you can build up structured data types using
6822 @code{struct}s or @code{class}es together with STL features instead of
6823 using unnamed lists of lists of lists.
6826 strongly typed: in CAS, you usually have only one kind of variables
6827 which can hold contents of an arbitrary type. This 4GL like feature is
6828 nice for novice programmers, but dangerous.
6831 development tools: powerful development tools exist for C++, like fancy
6832 editors (e.g. with automatic indentation and syntax highlighting),
6833 debuggers, visualization tools, documentation generators@dots{}
6836 modularization: C++ programs can easily be split into modules by
6837 separating interface and implementation.
6840 price: GiNaC is distributed under the GNU Public License which means
6841 that it is free and available with source code. And there are excellent
6842 C++-compilers for free, too.
6845 extendable: you can add your own classes to GiNaC, thus extending it on
6846 a very low level. Compare this to a traditional CAS that you can
6847 usually only extend on a high level by writing in the language defined
6848 by the parser. In particular, it turns out to be almost impossible to
6849 fix bugs in a traditional system.
6852 multiple interfaces: Though real GiNaC programs have to be written in
6853 some editor, then be compiled, linked and executed, there are more ways
6854 to work with the GiNaC engine. Many people want to play with
6855 expressions interactively, as in traditional CASs. Currently, two such
6856 windows into GiNaC have been implemented and many more are possible: the
6857 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
6858 types to a command line and second, as a more consistent approach, an
6859 interactive interface to the Cint C++ interpreter has been put together
6860 (called GiNaC-cint) that allows an interactive scripting interface
6861 consistent with the C++ language. It is available from the usual GiNaC
6865 seamless integration: it is somewhere between difficult and impossible
6866 to call CAS functions from within a program written in C++ or any other
6867 programming language and vice versa. With GiNaC, your symbolic routines
6868 are part of your program. You can easily call third party libraries,
6869 e.g. for numerical evaluation or graphical interaction. All other
6870 approaches are much more cumbersome: they range from simply ignoring the
6871 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
6872 system (i.e. @emph{Yacas}).
6875 efficiency: often large parts of a program do not need symbolic
6876 calculations at all. Why use large integers for loop variables or
6877 arbitrary precision arithmetics where @code{int} and @code{double} are
6878 sufficient? For pure symbolic applications, GiNaC is comparable in
6879 speed with other CAS.
6884 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
6885 @c node-name, next, previous, up
6886 @section Disadvantages
6888 Of course it also has some disadvantages:
6893 advanced features: GiNaC cannot compete with a program like
6894 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
6895 which grows since 1981 by the work of dozens of programmers, with
6896 respect to mathematical features. Integration, factorization,
6897 non-trivial simplifications, limits etc. are missing in GiNaC (and are
6898 not planned for the near future).
6901 portability: While the GiNaC library itself is designed to avoid any
6902 platform dependent features (it should compile on any ANSI compliant C++
6903 compiler), the currently used version of the CLN library (fast large
6904 integer and arbitrary precision arithmetics) can only by compiled
6905 without hassle on systems with the C++ compiler from the GNU Compiler
6906 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
6907 macros to let the compiler gather all static initializations, which
6908 works for GNU C++ only. Feel free to contact the authors in case you
6909 really believe that you need to use a different compiler. We have
6910 occasionally used other compilers and may be able to give you advice.}
6911 GiNaC uses recent language features like explicit constructors, mutable
6912 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
6913 literally. Recent GCC versions starting at 2.95.3, although itself not
6914 yet ANSI compliant, support all needed features.
6919 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
6920 @c node-name, next, previous, up
6923 Why did we choose to implement GiNaC in C++ instead of Java or any other
6924 language? C++ is not perfect: type checking is not strict (casting is
6925 possible), separation between interface and implementation is not
6926 complete, object oriented design is not enforced. The main reason is
6927 the often scolded feature of operator overloading in C++. While it may
6928 be true that operating on classes with a @code{+} operator is rarely
6929 meaningful, it is perfectly suited for algebraic expressions. Writing
6930 @math{3x+5y} as @code{3*x+5*y} instead of
6931 @code{x.times(3).plus(y.times(5))} looks much more natural.
6932 Furthermore, the main developers are more familiar with C++ than with
6933 any other programming language.
6936 @node Internal Structures, Expressions are reference counted, Why C++? , Top
6937 @c node-name, next, previous, up
6938 @appendix Internal Structures
6941 * Expressions are reference counted::
6942 * Internal representation of products and sums::
6945 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
6946 @c node-name, next, previous, up
6947 @appendixsection Expressions are reference counted
6949 @cindex reference counting
6950 @cindex copy-on-write
6951 @cindex garbage collection
6952 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
6953 where the counter belongs to the algebraic objects derived from class
6954 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
6955 which @code{ex} contains an instance. If you understood that, you can safely
6956 skip the rest of this passage.
6958 Expressions are extremely light-weight since internally they work like
6959 handles to the actual representation. They really hold nothing more
6960 than a pointer to some other object. What this means in practice is
6961 that whenever you create two @code{ex} and set the second equal to the
6962 first no copying process is involved. Instead, the copying takes place
6963 as soon as you try to change the second. Consider the simple sequence
6968 #include <ginac/ginac.h>
6969 using namespace std;
6970 using namespace GiNaC;
6974 symbol x("x"), y("y"), z("z");
6977 e1 = sin(x + 2*y) + 3*z + 41;
6978 e2 = e1; // e2 points to same object as e1
6979 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
6980 e2 += 1; // e2 is copied into a new object
6981 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
6985 The line @code{e2 = e1;} creates a second expression pointing to the
6986 object held already by @code{e1}. The time involved for this operation
6987 is therefore constant, no matter how large @code{e1} was. Actual
6988 copying, however, must take place in the line @code{e2 += 1;} because
6989 @code{e1} and @code{e2} are not handles for the same object any more.
6990 This concept is called @dfn{copy-on-write semantics}. It increases
6991 performance considerably whenever one object occurs multiple times and
6992 represents a simple garbage collection scheme because when an @code{ex}
6993 runs out of scope its destructor checks whether other expressions handle
6994 the object it points to too and deletes the object from memory if that
6995 turns out not to be the case. A slightly less trivial example of
6996 differentiation using the chain-rule should make clear how powerful this
7001 symbol x("x"), y("y");
7005 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7006 cout << e1 << endl // prints x+3*y
7007 << e2 << endl // prints (x+3*y)^3
7008 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7012 Here, @code{e1} will actually be referenced three times while @code{e2}
7013 will be referenced two times. When the power of an expression is built,
7014 that expression needs not be copied. Likewise, since the derivative of
7015 a power of an expression can be easily expressed in terms of that
7016 expression, no copying of @code{e1} is involved when @code{e3} is
7017 constructed. So, when @code{e3} is constructed it will print as
7018 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7019 holds a reference to @code{e2} and the factor in front is just
7022 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7023 semantics. When you insert an expression into a second expression, the
7024 result behaves exactly as if the contents of the first expression were
7025 inserted. But it may be useful to remember that this is not what
7026 happens. Knowing this will enable you to write much more efficient
7027 code. If you still have an uncertain feeling with copy-on-write
7028 semantics, we recommend you have a look at the
7029 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7030 Marshall Cline. Chapter 16 covers this issue and presents an
7031 implementation which is pretty close to the one in GiNaC.
7034 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7035 @c node-name, next, previous, up
7036 @appendixsection Internal representation of products and sums
7038 @cindex representation
7041 @cindex @code{power}
7042 Although it should be completely transparent for the user of
7043 GiNaC a short discussion of this topic helps to understand the sources
7044 and also explain performance to a large degree. Consider the
7045 unexpanded symbolic expression
7047 $2d^3 \left( 4a + 5b - 3 \right)$
7050 @math{2*d^3*(4*a+5*b-3)}
7052 which could naively be represented by a tree of linear containers for
7053 addition and multiplication, one container for exponentiation with base
7054 and exponent and some atomic leaves of symbols and numbers in this
7059 @cindex pair-wise representation
7060 However, doing so results in a rather deeply nested tree which will
7061 quickly become inefficient to manipulate. We can improve on this by
7062 representing the sum as a sequence of terms, each one being a pair of a
7063 purely numeric multiplicative coefficient and its rest. In the same
7064 spirit we can store the multiplication as a sequence of terms, each
7065 having a numeric exponent and a possibly complicated base, the tree
7066 becomes much more flat:
7070 The number @code{3} above the symbol @code{d} shows that @code{mul}
7071 objects are treated similarly where the coefficients are interpreted as
7072 @emph{exponents} now. Addition of sums of terms or multiplication of
7073 products with numerical exponents can be coded to be very efficient with
7074 such a pair-wise representation. Internally, this handling is performed
7075 by most CAS in this way. It typically speeds up manipulations by an
7076 order of magnitude. The overall multiplicative factor @code{2} and the
7077 additive term @code{-3} look somewhat out of place in this
7078 representation, however, since they are still carrying a trivial
7079 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7080 this is avoided by adding a field that carries an overall numeric
7081 coefficient. This results in the realistic picture of internal
7084 $2d^3 \left( 4a + 5b - 3 \right)$:
7087 @math{2*d^3*(4*a+5*b-3)}:
7093 This also allows for a better handling of numeric radicals, since
7094 @code{sqrt(2)} can now be carried along calculations. Now it should be
7095 clear, why both classes @code{add} and @code{mul} are derived from the
7096 same abstract class: the data representation is the same, only the
7097 semantics differs. In the class hierarchy, methods for polynomial
7098 expansion and the like are reimplemented for @code{add} and @code{mul},
7099 but the data structure is inherited from @code{expairseq}.
7102 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7103 @c node-name, next, previous, up
7104 @appendix Package Tools
7106 If you are creating a software package that uses the GiNaC library,
7107 setting the correct command line options for the compiler and linker
7108 can be difficult. GiNaC includes two tools to make this process easier.
7111 * ginac-config:: A shell script to detect compiler and linker flags.
7112 * AM_PATH_GINAC:: Macro for GNU automake.
7116 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7117 @c node-name, next, previous, up
7118 @section @command{ginac-config}
7119 @cindex ginac-config
7121 @command{ginac-config} is a shell script that you can use to determine
7122 the compiler and linker command line options required to compile and
7123 link a program with the GiNaC library.
7125 @command{ginac-config} takes the following flags:
7129 Prints out the version of GiNaC installed.
7131 Prints '-I' flags pointing to the installed header files.
7133 Prints out the linker flags necessary to link a program against GiNaC.
7134 @item --prefix[=@var{PREFIX}]
7135 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7136 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7137 Otherwise, prints out the configured value of @env{$prefix}.
7138 @item --exec-prefix[=@var{PREFIX}]
7139 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7140 Otherwise, prints out the configured value of @env{$exec_prefix}.
7143 Typically, @command{ginac-config} will be used within a configure
7144 script, as described below. It, however, can also be used directly from
7145 the command line using backquotes to compile a simple program. For
7149 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7152 This command line might expand to (for example):
7155 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7156 -lginac -lcln -lstdc++
7159 Not only is the form using @command{ginac-config} easier to type, it will
7160 work on any system, no matter how GiNaC was configured.
7163 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7164 @c node-name, next, previous, up
7165 @section @samp{AM_PATH_GINAC}
7166 @cindex AM_PATH_GINAC
7168 For packages configured using GNU automake, GiNaC also provides
7169 a macro to automate the process of checking for GiNaC.
7172 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7180 Determines the location of GiNaC using @command{ginac-config}, which is
7181 either found in the user's path, or from the environment variable
7182 @env{GINACLIB_CONFIG}.
7185 Tests the installed libraries to make sure that their version
7186 is later than @var{MINIMUM-VERSION}. (A default version will be used
7190 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7191 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7192 variable to the output of @command{ginac-config --libs}, and calls
7193 @samp{AC_SUBST()} for these variables so they can be used in generated
7194 makefiles, and then executes @var{ACTION-IF-FOUND}.
7197 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7198 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7202 This macro is in file @file{ginac.m4} which is installed in
7203 @file{$datadir/aclocal}. Note that if automake was installed with a
7204 different @samp{--prefix} than GiNaC, you will either have to manually
7205 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7206 aclocal the @samp{-I} option when running it.
7209 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7210 * Example package:: Example of a package using AM_PATH_GINAC.
7214 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7215 @c node-name, next, previous, up
7216 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7218 Simply make sure that @command{ginac-config} is in your path, and run
7219 the configure script.
7226 The directory where the GiNaC libraries are installed needs
7227 to be found by your system's dynamic linker.
7229 This is generally done by
7232 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7238 setting the environment variable @env{LD_LIBRARY_PATH},
7241 or, as a last resort,
7244 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7245 running configure, for instance:
7248 LDFLAGS=-R/home/cbauer/lib ./configure
7253 You can also specify a @command{ginac-config} not in your path by
7254 setting the @env{GINACLIB_CONFIG} environment variable to the
7255 name of the executable
7258 If you move the GiNaC package from its installed location,
7259 you will either need to modify @command{ginac-config} script
7260 manually to point to the new location or rebuild GiNaC.
7271 --with-ginac-prefix=@var{PREFIX}
7272 --with-ginac-exec-prefix=@var{PREFIX}
7275 are provided to override the prefix and exec-prefix that were stored
7276 in the @command{ginac-config} shell script by GiNaC's configure. You are
7277 generally better off configuring GiNaC with the right path to begin with.
7281 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
7282 @c node-name, next, previous, up
7283 @subsection Example of a package using @samp{AM_PATH_GINAC}
7285 The following shows how to build a simple package using automake
7286 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
7289 #include <ginac/ginac.h>
7293 GiNaC::symbol x("x");
7294 GiNaC::ex a = GiNaC::sin(x);
7295 std::cout << "Derivative of " << a
7296 << " is " << a.diff(x) << std::endl;
7301 You should first read the introductory portions of the automake
7302 Manual, if you are not already familiar with it.
7304 Two files are needed, @file{configure.in}, which is used to build the
7308 dnl Process this file with autoconf to produce a configure script.
7310 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
7316 AM_PATH_GINAC(0.9.0, [
7317 LIBS="$LIBS $GINACLIB_LIBS"
7318 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
7319 ], AC_MSG_ERROR([need to have GiNaC installed]))
7324 The only command in this which is not standard for automake
7325 is the @samp{AM_PATH_GINAC} macro.
7327 That command does the following: If a GiNaC version greater or equal
7328 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
7329 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
7330 the error message `need to have GiNaC installed'
7332 And the @file{Makefile.am}, which will be used to build the Makefile.
7335 ## Process this file with automake to produce Makefile.in
7336 bin_PROGRAMS = simple
7337 simple_SOURCES = simple.cpp
7340 This @file{Makefile.am}, says that we are building a single executable,
7341 from a single source file @file{simple.cpp}. Since every program
7342 we are building uses GiNaC we simply added the GiNaC options
7343 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
7344 want to specify them on a per-program basis: for instance by
7348 simple_LDADD = $(GINACLIB_LIBS)
7349 INCLUDES = $(GINACLIB_CPPFLAGS)
7352 to the @file{Makefile.am}.
7354 To try this example out, create a new directory and add the three
7357 Now execute the following commands:
7360 $ automake --add-missing
7365 You now have a package that can be built in the normal fashion
7374 @node Bibliography, Concept Index, Example package, Top
7375 @c node-name, next, previous, up
7376 @appendix Bibliography
7381 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
7384 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
7387 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
7390 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
7393 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
7394 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
7397 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
7398 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
7399 Academic Press, London
7402 @cite{Computer Algebra Systems - A Practical Guide},
7403 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
7406 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
7407 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
7410 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
7411 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
7414 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
7419 @node Concept Index, , Bibliography, Top
7420 @c node-name, next, previous, up
7421 @unnumbered Concept Index