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) 1999-2002 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{} 1999-2002 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{} 1999-2002 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 * The Class Hierarchy:: Overview of GiNaC's classes.
676 * Error handling:: How the library reports errors.
677 * Symbols:: Symbolic objects.
678 * Numbers:: Numerical objects.
679 * Constants:: Pre-defined constants.
680 * Fundamental containers:: The power, add and mul classes.
681 * Lists:: Lists of expressions.
682 * Mathematical functions:: Mathematical functions.
683 * Relations:: Equality, Inequality and all that.
684 * Matrices:: Matrices.
685 * Indexed objects:: Handling indexed quantities.
686 * Non-commutative objects:: Algebras with non-commutative products.
690 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
691 @c node-name, next, previous, up
693 @cindex expression (class @code{ex})
696 The most common class of objects a user deals with is the expression
697 @code{ex}, representing a mathematical object like a variable, number,
698 function, sum, product, etc@dots{} Expressions may be put together to form
699 new expressions, passed as arguments to functions, and so on. Here is a
700 little collection of valid expressions:
703 ex MyEx1 = 5; // simple number
704 ex MyEx2 = x + 2*y; // polynomial in x and y
705 ex MyEx3 = (x + 1)/(x - 1); // rational expression
706 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
707 ex MyEx5 = MyEx4 + 1; // similar to above
710 Expressions are handles to other more fundamental objects, that often
711 contain other expressions thus creating a tree of expressions
712 (@xref{Internal Structures}, for particular examples). Most methods on
713 @code{ex} therefore run top-down through such an expression tree. For
714 example, the method @code{has()} scans recursively for occurrences of
715 something inside an expression. Thus, if you have declared @code{MyEx4}
716 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
717 the argument of @code{sin} and hence return @code{true}.
719 The next sections will outline the general picture of GiNaC's class
720 hierarchy and describe the classes of objects that are handled by
724 @node The Class Hierarchy, Error handling, Expressions, Basic Concepts
725 @c node-name, next, previous, up
726 @section The Class Hierarchy
728 GiNaC's class hierarchy consists of several classes representing
729 mathematical objects, all of which (except for @code{ex} and some
730 helpers) are internally derived from one abstract base class called
731 @code{basic}. You do not have to deal with objects of class
732 @code{basic}, instead you'll be dealing with symbols, numbers,
733 containers of expressions and so on.
737 To get an idea about what kinds of symbolic composites may be built we
738 have a look at the most important classes in the class hierarchy and
739 some of the relations among the classes:
741 @image{classhierarchy}
743 The abstract classes shown here (the ones without drop-shadow) are of no
744 interest for the user. They are used internally in order to avoid code
745 duplication if two or more classes derived from them share certain
746 features. An example is @code{expairseq}, a container for a sequence of
747 pairs each consisting of one expression and a number (@code{numeric}).
748 What @emph{is} visible to the user are the derived classes @code{add}
749 and @code{mul}, representing sums and products. @xref{Internal
750 Structures}, where these two classes are described in more detail. The
751 following table shortly summarizes what kinds of mathematical objects
752 are stored in the different classes:
755 @multitable @columnfractions .22 .78
756 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
757 @item @code{constant} @tab Constants like
764 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
765 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
766 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
767 @item @code{ncmul} @tab Products of non-commutative objects
768 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
773 @code{sqrt(}@math{2}@code{)}
776 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
777 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
778 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
779 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
780 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
781 @item @code{indexed} @tab Indexed object like @math{A_ij}
782 @item @code{tensor} @tab Special tensor like the delta and metric tensors
783 @item @code{idx} @tab Index of an indexed object
784 @item @code{varidx} @tab Index with variance
785 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
786 @item @code{wildcard} @tab Wildcard for pattern matching
791 @node Error handling, Symbols, The Class Hierarchy, Basic Concepts
792 @c node-name, next, previous, up
793 @section Error handling
795 @cindex @code{pole_error} (class)
797 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
798 generated by GiNaC are subclassed from the standard @code{exception} class
799 defined in the @file{<stdexcept>} header. In addition to the predefined
800 @code{logic_error}, @code{domain_error}, @code{out_of_range},
801 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
802 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
803 exception that gets thrown when trying to evaluate a mathematical function
806 The @code{pole_error} class has a member function
809 int pole_error::degree(void) const;
812 that returns the order of the singularity (or 0 when the pole is
813 logarithmic or the order is undefined).
815 When using GiNaC it is useful to arrange for exceptions to be catched in
816 the main program even if you don't want to do any special error handling.
817 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
818 default exception handler of your C++ compiler's run-time system which
819 usually only aborts the program without giving any information what went
822 Here is an example for a @code{main()} function that catches and prints
823 exceptions generated by GiNaC:
828 #include <ginac/ginac.h>
830 using namespace GiNaC;
838 @} catch (exception &p) @{
839 cerr << p.what() << endl;
847 @node Symbols, Numbers, Error handling, Basic Concepts
848 @c node-name, next, previous, up
850 @cindex @code{symbol} (class)
851 @cindex hierarchy of classes
854 Symbols are for symbolic manipulation what atoms are for chemistry. You
855 can declare objects of class @code{symbol} as any other object simply by
856 saying @code{symbol x,y;}. There is, however, a catch in here having to
857 do with the fact that C++ is a compiled language. The information about
858 the symbol's name is thrown away by the compiler but at a later stage
859 you may want to print expressions holding your symbols. In order to
860 avoid confusion GiNaC's symbols are able to know their own name. This
861 is accomplished by declaring its name for output at construction time in
862 the fashion @code{symbol x("x");}. If you declare a symbol using the
863 default constructor (i.e. without string argument) the system will deal
864 out a unique name. That name may not be suitable for printing but for
865 internal routines when no output is desired it is often enough. We'll
866 come across examples of such symbols later in this tutorial.
868 This implies that the strings passed to symbols at construction time may
869 not be used for comparing two of them. It is perfectly legitimate to
870 write @code{symbol x("x"),y("x");} but it is likely to lead into
871 trouble. Here, @code{x} and @code{y} are different symbols and
872 statements like @code{x-y} will not be simplified to zero although the
873 output @code{x-x} looks funny. Such output may also occur when there
874 are two different symbols in two scopes, for instance when you call a
875 function that declares a symbol with a name already existent in a symbol
876 in the calling function. Again, comparing them (using @code{operator==}
877 for instance) will always reveal their difference. Watch out, please.
879 @cindex @code{subs()}
880 Although symbols can be assigned expressions for internal reasons, you
881 should not do it (and we are not going to tell you how it is done). If
882 you want to replace a symbol with something else in an expression, you
883 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
886 @node Numbers, Constants, Symbols, Basic Concepts
887 @c node-name, next, previous, up
889 @cindex @code{numeric} (class)
895 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
896 The classes therein serve as foundation classes for GiNaC. CLN stands
897 for Class Library for Numbers or alternatively for Common Lisp Numbers.
898 In order to find out more about CLN's internals, the reader is referred to
899 the documentation of that library. @inforef{Introduction, , cln}, for
900 more information. Suffice to say that it is by itself build on top of
901 another library, the GNU Multiple Precision library GMP, which is an
902 extremely fast library for arbitrary long integers and rationals as well
903 as arbitrary precision floating point numbers. It is very commonly used
904 by several popular cryptographic applications. CLN extends GMP by
905 several useful things: First, it introduces the complex number field
906 over either reals (i.e. floating point numbers with arbitrary precision)
907 or rationals. Second, it automatically converts rationals to integers
908 if the denominator is unity and complex numbers to real numbers if the
909 imaginary part vanishes and also correctly treats algebraic functions.
910 Third it provides good implementations of state-of-the-art algorithms
911 for all trigonometric and hyperbolic functions as well as for
912 calculation of some useful constants.
914 The user can construct an object of class @code{numeric} in several
915 ways. The following example shows the four most important constructors.
916 It uses construction from C-integer, construction of fractions from two
917 integers, construction from C-float and construction from a string:
921 #include <ginac/ginac.h>
922 using namespace GiNaC;
926 numeric two = 2; // exact integer 2
927 numeric r(2,3); // exact fraction 2/3
928 numeric e(2.71828); // floating point number
929 numeric p = "3.14159265358979323846"; // constructor from string
930 // Trott's constant in scientific notation:
931 numeric trott("1.0841015122311136151E-2");
933 std::cout << two*p << std::endl; // floating point 6.283...
938 @cindex complex numbers
939 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
944 numeric z1 = 2-3*I; // exact complex number 2-3i
945 numeric z2 = 5.9+1.6*I; // complex floating point number
949 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
950 This would, however, call C's built-in operator @code{/} for integers
951 first and result in a numeric holding a plain integer 1. @strong{Never
952 use the operator @code{/} on integers} unless you know exactly what you
953 are doing! Use the constructor from two integers instead, as shown in
954 the example above. Writing @code{numeric(1)/2} may look funny but works
957 @cindex @code{Digits}
959 We have seen now the distinction between exact numbers and floating
960 point numbers. Clearly, the user should never have to worry about
961 dynamically created exact numbers, since their `exactness' always
962 determines how they ought to be handled, i.e. how `long' they are. The
963 situation is different for floating point numbers. Their accuracy is
964 controlled by one @emph{global} variable, called @code{Digits}. (For
965 those readers who know about Maple: it behaves very much like Maple's
966 @code{Digits}). All objects of class numeric that are constructed from
967 then on will be stored with a precision matching that number of decimal
972 #include <ginac/ginac.h>
974 using namespace GiNaC;
978 numeric three(3.0), one(1.0);
979 numeric x = one/three;
981 cout << "in " << Digits << " digits:" << endl;
983 cout << Pi.evalf() << endl;
995 The above example prints the following output to screen:
999 0.33333333333333333334
1000 3.1415926535897932385
1002 0.33333333333333333333333333333333333333333333333333333333333333333334
1003 3.1415926535897932384626433832795028841971693993751058209749445923078
1007 Note that the last number is not necessarily rounded as you would
1008 naively expect it to be rounded in the decimal system. But note also,
1009 that in both cases you got a couple of extra digits. This is because
1010 numbers are internally stored by CLN as chunks of binary digits in order
1011 to match your machine's word size and to not waste precision. Thus, on
1012 architectures with different word size, the above output might even
1013 differ with regard to actually computed digits.
1015 It should be clear that objects of class @code{numeric} should be used
1016 for constructing numbers or for doing arithmetic with them. The objects
1017 one deals with most of the time are the polymorphic expressions @code{ex}.
1019 @subsection Tests on numbers
1021 Once you have declared some numbers, assigned them to expressions and
1022 done some arithmetic with them it is frequently desired to retrieve some
1023 kind of information from them like asking whether that number is
1024 integer, rational, real or complex. For those cases GiNaC provides
1025 several useful methods. (Internally, they fall back to invocations of
1026 certain CLN functions.)
1028 As an example, let's construct some rational number, multiply it with
1029 some multiple of its denominator and test what comes out:
1033 #include <ginac/ginac.h>
1034 using namespace std;
1035 using namespace GiNaC;
1037 // some very important constants:
1038 const numeric twentyone(21);
1039 const numeric ten(10);
1040 const numeric five(5);
1044 numeric answer = twentyone;
1047 cout << answer.is_integer() << endl; // false, it's 21/5
1049 cout << answer.is_integer() << endl; // true, it's 42 now!
1053 Note that the variable @code{answer} is constructed here as an integer
1054 by @code{numeric}'s copy constructor but in an intermediate step it
1055 holds a rational number represented as integer numerator and integer
1056 denominator. When multiplied by 10, the denominator becomes unity and
1057 the result is automatically converted to a pure integer again.
1058 Internally, the underlying CLN is responsible for this behavior and we
1059 refer the reader to CLN's documentation. Suffice to say that
1060 the same behavior applies to complex numbers as well as return values of
1061 certain functions. Complex numbers are automatically converted to real
1062 numbers if the imaginary part becomes zero. The full set of tests that
1063 can be applied is listed in the following table.
1066 @multitable @columnfractions .30 .70
1067 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1068 @item @code{.is_zero()}
1069 @tab @dots{}equal to zero
1070 @item @code{.is_positive()}
1071 @tab @dots{}not complex and greater than 0
1072 @item @code{.is_integer()}
1073 @tab @dots{}a (non-complex) integer
1074 @item @code{.is_pos_integer()}
1075 @tab @dots{}an integer and greater than 0
1076 @item @code{.is_nonneg_integer()}
1077 @tab @dots{}an integer and greater equal 0
1078 @item @code{.is_even()}
1079 @tab @dots{}an even integer
1080 @item @code{.is_odd()}
1081 @tab @dots{}an odd integer
1082 @item @code{.is_prime()}
1083 @tab @dots{}a prime integer (probabilistic primality test)
1084 @item @code{.is_rational()}
1085 @tab @dots{}an exact rational number (integers are rational, too)
1086 @item @code{.is_real()}
1087 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1088 @item @code{.is_cinteger()}
1089 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1090 @item @code{.is_crational()}
1091 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1096 @node Constants, Fundamental containers, Numbers, Basic Concepts
1097 @c node-name, next, previous, up
1099 @cindex @code{constant} (class)
1102 @cindex @code{Catalan}
1103 @cindex @code{Euler}
1104 @cindex @code{evalf()}
1105 Constants behave pretty much like symbols except that they return some
1106 specific number when the method @code{.evalf()} is called.
1108 The predefined known constants are:
1111 @multitable @columnfractions .14 .30 .56
1112 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1114 @tab Archimedes' constant
1115 @tab 3.14159265358979323846264338327950288
1116 @item @code{Catalan}
1117 @tab Catalan's constant
1118 @tab 0.91596559417721901505460351493238411
1120 @tab Euler's (or Euler-Mascheroni) constant
1121 @tab 0.57721566490153286060651209008240243
1126 @node Fundamental containers, Lists, Constants, Basic Concepts
1127 @c node-name, next, previous, up
1128 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1132 @cindex @code{power}
1134 Simple polynomial expressions are written down in GiNaC pretty much like
1135 in other CAS or like expressions involving numerical variables in C.
1136 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1137 been overloaded to achieve this goal. When you run the following
1138 code snippet, the constructor for an object of type @code{mul} is
1139 automatically called to hold the product of @code{a} and @code{b} and
1140 then the constructor for an object of type @code{add} is called to hold
1141 the sum of that @code{mul} object and the number one:
1145 symbol a("a"), b("b");
1150 @cindex @code{pow()}
1151 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1152 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1153 construction is necessary since we cannot safely overload the constructor
1154 @code{^} in C++ to construct a @code{power} object. If we did, it would
1155 have several counterintuitive and undesired effects:
1159 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1161 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1162 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1163 interpret this as @code{x^(a^b)}.
1165 Also, expressions involving integer exponents are very frequently used,
1166 which makes it even more dangerous to overload @code{^} since it is then
1167 hard to distinguish between the semantics as exponentiation and the one
1168 for exclusive or. (It would be embarrassing to return @code{1} where one
1169 has requested @code{2^3}.)
1172 @cindex @command{ginsh}
1173 All effects are contrary to mathematical notation and differ from the
1174 way most other CAS handle exponentiation, therefore overloading @code{^}
1175 is ruled out for GiNaC's C++ part. The situation is different in
1176 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1177 that the other frequently used exponentiation operator @code{**} does
1178 not exist at all in C++).
1180 To be somewhat more precise, objects of the three classes described
1181 here, are all containers for other expressions. An object of class
1182 @code{power} is best viewed as a container with two slots, one for the
1183 basis, one for the exponent. All valid GiNaC expressions can be
1184 inserted. However, basic transformations like simplifying
1185 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1186 when this is mathematically possible. If we replace the outer exponent
1187 three in the example by some symbols @code{a}, the simplification is not
1188 safe and will not be performed, since @code{a} might be @code{1/2} and
1191 Objects of type @code{add} and @code{mul} are containers with an
1192 arbitrary number of slots for expressions to be inserted. Again, simple
1193 and safe simplifications are carried out like transforming
1194 @code{3*x+4-x} to @code{2*x+4}.
1196 The general rule is that when you construct such objects, GiNaC
1197 automatically creates them in canonical form, which might differ from
1198 the form you typed in your program. This allows for rapid comparison of
1199 expressions, since after all @code{a-a} is simply zero. Note, that the
1200 canonical form is not necessarily lexicographical ordering or in any way
1201 easily guessable. It is only guaranteed that constructing the same
1202 expression twice, either implicitly or explicitly, results in the same
1206 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1207 @c node-name, next, previous, up
1208 @section Lists of expressions
1209 @cindex @code{lst} (class)
1211 @cindex @code{nops()}
1213 @cindex @code{append()}
1214 @cindex @code{prepend()}
1215 @cindex @code{remove_first()}
1216 @cindex @code{remove_last()}
1218 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1219 expressions. These are sometimes used to supply a variable number of
1220 arguments of the same type to GiNaC methods such as @code{subs()} and
1221 @code{to_rational()}, so you should have a basic understanding about them.
1223 Lists of up to 16 expressions can be directly constructed from single
1228 symbol x("x"), y("y");
1229 lst l(x, 2, y, x+y);
1230 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1234 Use the @code{nops()} method to determine the size (number of expressions) of
1235 a list and the @code{op()} method to access individual elements:
1239 cout << l.nops() << endl; // prints '4'
1240 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1244 You can append or prepend an expression to a list with the @code{append()}
1245 and @code{prepend()} methods:
1249 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1250 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1254 Finally you can remove the first or last element of a list with
1255 @code{remove_first()} and @code{remove_last()}:
1259 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1260 l.remove_last(); // l is now @{x, 2, y, x+y@}
1265 @node Mathematical functions, Relations, Lists, Basic Concepts
1266 @c node-name, next, previous, up
1267 @section Mathematical functions
1268 @cindex @code{function} (class)
1269 @cindex trigonometric function
1270 @cindex hyperbolic function
1272 There are quite a number of useful functions hard-wired into GiNaC. For
1273 instance, all trigonometric and hyperbolic functions are implemented
1274 (@xref{Built-in Functions}, for a complete list).
1276 These functions (better called @emph{pseudofunctions}) are all objects
1277 of class @code{function}. They accept one or more expressions as
1278 arguments and return one expression. If the arguments are not
1279 numerical, the evaluation of the function may be halted, as it does in
1280 the next example, showing how a function returns itself twice and
1281 finally an expression that may be really useful:
1283 @cindex Gamma function
1284 @cindex @code{subs()}
1287 symbol x("x"), y("y");
1289 cout << tgamma(foo) << endl;
1290 // -> tgamma(x+(1/2)*y)
1291 ex bar = foo.subs(y==1);
1292 cout << tgamma(bar) << endl;
1294 ex foobar = bar.subs(x==7);
1295 cout << tgamma(foobar) << endl;
1296 // -> (135135/128)*Pi^(1/2)
1300 Besides evaluation most of these functions allow differentiation, series
1301 expansion and so on. Read the next chapter in order to learn more about
1304 It must be noted that these pseudofunctions are created by inline
1305 functions, where the argument list is templated. This means that
1306 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1307 @code{sin(ex(1))} and will therefore not result in a floating point
1308 number. Unless of course the function prototype is explicitly
1309 overridden -- which is the case for arguments of type @code{numeric}
1310 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1311 point number of class @code{numeric} you should call
1312 @code{sin(numeric(1))}. This is almost the same as calling
1313 @code{sin(1).evalf()} except that the latter will return a numeric
1314 wrapped inside an @code{ex}.
1317 @node Relations, Matrices, Mathematical functions, Basic Concepts
1318 @c node-name, next, previous, up
1320 @cindex @code{relational} (class)
1322 Sometimes, a relation holding between two expressions must be stored
1323 somehow. The class @code{relational} is a convenient container for such
1324 purposes. A relation is by definition a container for two @code{ex} and
1325 a relation between them that signals equality, inequality and so on.
1326 They are created by simply using the C++ operators @code{==}, @code{!=},
1327 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1329 @xref{Mathematical functions}, for examples where various applications
1330 of the @code{.subs()} method show how objects of class relational are
1331 used as arguments. There they provide an intuitive syntax for
1332 substitutions. They are also used as arguments to the @code{ex::series}
1333 method, where the left hand side of the relation specifies the variable
1334 to expand in and the right hand side the expansion point. They can also
1335 be used for creating systems of equations that are to be solved for
1336 unknown variables. But the most common usage of objects of this class
1337 is rather inconspicuous in statements of the form @code{if
1338 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1339 conversion from @code{relational} to @code{bool} takes place. Note,
1340 however, that @code{==} here does not perform any simplifications, hence
1341 @code{expand()} must be called explicitly.
1344 @node Matrices, Indexed objects, Relations, Basic Concepts
1345 @c node-name, next, previous, up
1347 @cindex @code{matrix} (class)
1349 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1350 matrix with @math{m} rows and @math{n} columns are accessed with two
1351 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1352 second one in the range 0@dots{}@math{n-1}.
1354 There are a couple of ways to construct matrices, with or without preset
1357 @cindex @code{lst_to_matrix()}
1358 @cindex @code{diag_matrix()}
1359 @cindex @code{unit_matrix()}
1360 @cindex @code{symbolic_matrix()}
1362 matrix::matrix(unsigned r, unsigned c);
1363 matrix::matrix(unsigned r, unsigned c, const lst & l);
1364 ex lst_to_matrix(const lst & l);
1365 ex diag_matrix(const lst & l);
1366 ex unit_matrix(unsigned x);
1367 ex unit_matrix(unsigned r, unsigned c);
1368 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1369 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1372 The first two functions are @code{matrix} constructors which create a matrix
1373 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1374 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1375 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1376 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1377 constructs a diagonal matrix given the list of diagonal elements.
1378 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1379 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1380 with newly generated symbols made of the specified base name and the
1381 position of each element in the matrix.
1383 Matrix elements can be accessed and set using the parenthesis (function call)
1387 const ex & matrix::operator()(unsigned r, unsigned c) const;
1388 ex & matrix::operator()(unsigned r, unsigned c);
1391 It is also possible to access the matrix elements in a linear fashion with
1392 the @code{op()} method. But C++-style subscripting with square brackets
1393 @samp{[]} is not available.
1395 Here are a couple of examples of constructing matrices:
1399 symbol a("a"), b("b");
1407 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1410 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1413 cout << diag_matrix(lst(a, b)) << endl;
1416 cout << unit_matrix(3) << endl;
1417 // -> [[1,0,0],[0,1,0],[0,0,1]]
1419 cout << symbolic_matrix(2, 3, "x") << endl;
1420 // -> [[x00,x01,x02],[x10,x11,x12]]
1424 @cindex @code{transpose()}
1425 @cindex @code{inverse()}
1426 There are three ways to do arithmetic with matrices. The first (and most
1427 efficient one) is to use the methods provided by the @code{matrix} class:
1430 matrix matrix::add(const matrix & other) const;
1431 matrix matrix::sub(const matrix & other) const;
1432 matrix matrix::mul(const matrix & other) const;
1433 matrix matrix::mul_scalar(const ex & other) const;
1434 matrix matrix::pow(const ex & expn) const;
1435 matrix matrix::transpose(void) const;
1436 matrix matrix::inverse(void) const;
1439 All of these methods return the result as a new matrix object. Here is an
1440 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1445 matrix A(2, 2, lst(1, 2, 3, 4));
1446 matrix B(2, 2, lst(-1, 0, 2, 1));
1447 matrix C(2, 2, lst(8, 4, 2, 1));
1449 matrix result = A.mul(B).sub(C.mul_scalar(2));
1450 cout << result << endl;
1451 // -> [[-13,-6],[1,2]]
1456 @cindex @code{evalm()}
1457 The second (and probably the most natural) way is to construct an expression
1458 containing matrices with the usual arithmetic operators and @code{pow()}.
1459 For efficiency reasons, expressions with sums, products and powers of
1460 matrices are not automatically evaluated in GiNaC. You have to call the
1464 ex ex::evalm() const;
1467 to obtain the result:
1474 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1475 cout << e.evalm() << endl;
1476 // -> [[-13,-6],[1,2]]
1481 The non-commutativity of the product @code{A*B} in this example is
1482 automatically recognized by GiNaC. There is no need to use a special
1483 operator here. @xref{Non-commutative objects}, for more information about
1484 dealing with non-commutative expressions.
1486 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1487 to perform the arithmetic:
1492 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1493 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1495 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1496 cout << e.simplify_indexed() << endl;
1497 // -> [[-13,-6],[1,2]].i.j
1501 Using indices is most useful when working with rectangular matrices and
1502 one-dimensional vectors because you don't have to worry about having to
1503 transpose matrices before multiplying them. @xref{Indexed objects}, for
1504 more information about using matrices with indices, and about indices in
1507 The @code{matrix} class provides a couple of additional methods for
1508 computing determinants, traces, and characteristic polynomials:
1510 @cindex @code{determinant()}
1511 @cindex @code{trace()}
1512 @cindex @code{charpoly()}
1514 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1515 ex matrix::trace(void) const;
1516 ex matrix::charpoly(const symbol & lambda) const;
1519 The @samp{algo} argument of @code{determinant()} allows to select between
1520 different algorithms for calculating the determinant. The possible values
1521 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1522 heuristic to automatically select an algorithm that is likely to give the
1523 result most quickly.
1526 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1527 @c node-name, next, previous, up
1528 @section Indexed objects
1530 GiNaC allows you to handle expressions containing general indexed objects in
1531 arbitrary spaces. It is also able to canonicalize and simplify such
1532 expressions and perform symbolic dummy index summations. There are a number
1533 of predefined indexed objects provided, like delta and metric tensors.
1535 There are few restrictions placed on indexed objects and their indices and
1536 it is easy to construct nonsense expressions, but our intention is to
1537 provide a general framework that allows you to implement algorithms with
1538 indexed quantities, getting in the way as little as possible.
1540 @cindex @code{idx} (class)
1541 @cindex @code{indexed} (class)
1542 @subsection Indexed quantities and their indices
1544 Indexed expressions in GiNaC are constructed of two special types of objects,
1545 @dfn{index objects} and @dfn{indexed objects}.
1549 @cindex contravariant
1552 @item Index objects are of class @code{idx} or a subclass. Every index has
1553 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1554 the index lives in) which can both be arbitrary expressions but are usually
1555 a number or a simple symbol. In addition, indices of class @code{varidx} have
1556 a @dfn{variance} (they can be co- or contravariant), and indices of class
1557 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1559 @item Indexed objects are of class @code{indexed} or a subclass. They
1560 contain a @dfn{base expression} (which is the expression being indexed), and
1561 one or more indices.
1565 @strong{Note:} when printing expressions, covariant indices and indices
1566 without variance are denoted @samp{.i} while contravariant indices are
1567 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1568 value. In the following, we are going to use that notation in the text so
1569 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1570 not visible in the output.
1572 A simple example shall illustrate the concepts:
1576 #include <ginac/ginac.h>
1577 using namespace std;
1578 using namespace GiNaC;
1582 symbol i_sym("i"), j_sym("j");
1583 idx i(i_sym, 3), j(j_sym, 3);
1586 cout << indexed(A, i, j) << endl;
1591 The @code{idx} constructor takes two arguments, the index value and the
1592 index dimension. First we define two index objects, @code{i} and @code{j},
1593 both with the numeric dimension 3. The value of the index @code{i} is the
1594 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1595 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1596 construct an expression containing one indexed object, @samp{A.i.j}. It has
1597 the symbol @code{A} as its base expression and the two indices @code{i} and
1600 Note the difference between the indices @code{i} and @code{j} which are of
1601 class @code{idx}, and the index values which are the symbols @code{i_sym}
1602 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1603 or numbers but must be index objects. For example, the following is not
1604 correct and will raise an exception:
1607 symbol i("i"), j("j");
1608 e = indexed(A, i, j); // ERROR: indices must be of type idx
1611 You can have multiple indexed objects in an expression, index values can
1612 be numeric, and index dimensions symbolic:
1616 symbol B("B"), dim("dim");
1617 cout << 4 * indexed(A, i)
1618 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1623 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1624 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1625 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1626 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1627 @code{simplify_indexed()} for that, see below).
1629 In fact, base expressions, index values and index dimensions can be
1630 arbitrary expressions:
1634 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1639 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1640 get an error message from this but you will probably not be able to do
1641 anything useful with it.
1643 @cindex @code{get_value()}
1644 @cindex @code{get_dimension()}
1648 ex idx::get_value(void);
1649 ex idx::get_dimension(void);
1652 return the value and dimension of an @code{idx} object. If you have an index
1653 in an expression, such as returned by calling @code{.op()} on an indexed
1654 object, you can get a reference to the @code{idx} object with the function
1655 @code{ex_to<idx>()} on the expression.
1657 There are also the methods
1660 bool idx::is_numeric(void);
1661 bool idx::is_symbolic(void);
1662 bool idx::is_dim_numeric(void);
1663 bool idx::is_dim_symbolic(void);
1666 for checking whether the value and dimension are numeric or symbolic
1667 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1668 About Expressions}) returns information about the index value.
1670 @cindex @code{varidx} (class)
1671 If you need co- and contravariant indices, use the @code{varidx} class:
1675 symbol mu_sym("mu"), nu_sym("nu");
1676 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1677 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1679 cout << indexed(A, mu, nu) << endl;
1681 cout << indexed(A, mu_co, nu) << endl;
1683 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1688 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1689 co- or contravariant. The default is a contravariant (upper) index, but
1690 this can be overridden by supplying a third argument to the @code{varidx}
1691 constructor. The two methods
1694 bool varidx::is_covariant(void);
1695 bool varidx::is_contravariant(void);
1698 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1699 to get the object reference from an expression). There's also the very useful
1703 ex varidx::toggle_variance(void);
1706 which makes a new index with the same value and dimension but the opposite
1707 variance. By using it you only have to define the index once.
1709 @cindex @code{spinidx} (class)
1710 The @code{spinidx} class provides dotted and undotted variant indices, as
1711 used in the Weyl-van-der-Waerden spinor formalism:
1715 symbol K("K"), C_sym("C"), D_sym("D");
1716 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1717 // contravariant, undotted
1718 spinidx C_co(C_sym, 2, true); // covariant index
1719 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1720 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1722 cout << indexed(K, C, D) << endl;
1724 cout << indexed(K, C_co, D_dot) << endl;
1726 cout << indexed(K, D_co_dot, D) << endl;
1731 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1732 dotted or undotted. The default is undotted but this can be overridden by
1733 supplying a fourth argument to the @code{spinidx} constructor. The two
1737 bool spinidx::is_dotted(void);
1738 bool spinidx::is_undotted(void);
1741 allow you to check whether or not a @code{spinidx} object is dotted (use
1742 @code{ex_to<spinidx>()} to get the object reference from an expression).
1743 Finally, the two methods
1746 ex spinidx::toggle_dot(void);
1747 ex spinidx::toggle_variance_dot(void);
1750 create a new index with the same value and dimension but opposite dottedness
1751 and the same or opposite variance.
1753 @subsection Substituting indices
1755 @cindex @code{subs()}
1756 Sometimes you will want to substitute one symbolic index with another
1757 symbolic or numeric index, for example when calculating one specific element
1758 of a tensor expression. This is done with the @code{.subs()} method, as it
1759 is done for symbols (see @ref{Substituting Expressions}).
1761 You have two possibilities here. You can either substitute the whole index
1762 by another index or expression:
1766 ex e = indexed(A, mu_co);
1767 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1768 // -> A.mu becomes A~nu
1769 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1770 // -> A.mu becomes A~0
1771 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1772 // -> A.mu becomes A.0
1776 The third example shows that trying to replace an index with something that
1777 is not an index will substitute the index value instead.
1779 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1784 ex e = indexed(A, mu_co);
1785 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1786 // -> A.mu becomes A.nu
1787 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1788 // -> A.mu becomes A.0
1792 As you see, with the second method only the value of the index will get
1793 substituted. Its other properties, including its dimension, remain unchanged.
1794 If you want to change the dimension of an index you have to substitute the
1795 whole index by another one with the new dimension.
1797 Finally, substituting the base expression of an indexed object works as
1802 ex e = indexed(A, mu_co);
1803 cout << e << " becomes " << e.subs(A == A+B) << endl;
1804 // -> A.mu becomes (B+A).mu
1808 @subsection Symmetries
1809 @cindex @code{symmetry} (class)
1810 @cindex @code{sy_none()}
1811 @cindex @code{sy_symm()}
1812 @cindex @code{sy_anti()}
1813 @cindex @code{sy_cycl()}
1815 Indexed objects can have certain symmetry properties with respect to their
1816 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1817 that is constructed with the helper functions
1820 symmetry sy_none(...);
1821 symmetry sy_symm(...);
1822 symmetry sy_anti(...);
1823 symmetry sy_cycl(...);
1826 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1827 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1828 represents a cyclic symmetry. Each of these functions accepts up to four
1829 arguments which can be either symmetry objects themselves or unsigned integer
1830 numbers that represent an index position (counting from 0). A symmetry
1831 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1832 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1835 Here are some examples of symmetry definitions:
1840 e = indexed(A, i, j);
1841 e = indexed(A, sy_none(), i, j); // equivalent
1842 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1844 // Symmetric in all three indices:
1845 e = indexed(A, sy_symm(), i, j, k);
1846 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1847 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1848 // different canonical order
1850 // Symmetric in the first two indices only:
1851 e = indexed(A, sy_symm(0, 1), i, j, k);
1852 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1854 // Antisymmetric in the first and last index only (index ranges need not
1856 e = indexed(A, sy_anti(0, 2), i, j, k);
1857 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1859 // An example of a mixed symmetry: antisymmetric in the first two and
1860 // last two indices, symmetric when swapping the first and last index
1861 // pairs (like the Riemann curvature tensor):
1862 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1864 // Cyclic symmetry in all three indices:
1865 e = indexed(A, sy_cycl(), i, j, k);
1866 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1868 // The following examples are invalid constructions that will throw
1869 // an exception at run time.
1871 // An index may not appear multiple times:
1872 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1873 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1875 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1876 // same number of indices:
1877 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1879 // And of course, you cannot specify indices which are not there:
1880 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1884 If you need to specify more than four indices, you have to use the
1885 @code{.add()} method of the @code{symmetry} class. For example, to specify
1886 full symmetry in the first six indices you would write
1887 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1889 If an indexed object has a symmetry, GiNaC will automatically bring the
1890 indices into a canonical order which allows for some immediate simplifications:
1894 cout << indexed(A, sy_symm(), i, j)
1895 + indexed(A, sy_symm(), j, i) << endl;
1897 cout << indexed(B, sy_anti(), i, j)
1898 + indexed(B, sy_anti(), j, i) << endl;
1900 cout << indexed(B, sy_anti(), i, j, k)
1901 - indexed(B, sy_anti(), j, k, i) << endl;
1906 @cindex @code{get_free_indices()}
1908 @subsection Dummy indices
1910 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1911 that a summation over the index range is implied. Symbolic indices which are
1912 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1913 dummy nor free indices.
1915 To be recognized as a dummy index pair, the two indices must be of the same
1916 class and their value must be the same single symbol (an index like
1917 @samp{2*n+1} is never a dummy index). If the indices are of class
1918 @code{varidx} they must also be of opposite variance; if they are of class
1919 @code{spinidx} they must be both dotted or both undotted.
1921 The method @code{.get_free_indices()} returns a vector containing the free
1922 indices of an expression. It also checks that the free indices of the terms
1923 of a sum are consistent:
1927 symbol A("A"), B("B"), C("C");
1929 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1930 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1932 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1933 cout << exprseq(e.get_free_indices()) << endl;
1935 // 'j' and 'l' are dummy indices
1937 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1938 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1940 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1941 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1942 cout << exprseq(e.get_free_indices()) << endl;
1944 // 'nu' is a dummy index, but 'sigma' is not
1946 e = indexed(A, mu, mu);
1947 cout << exprseq(e.get_free_indices()) << endl;
1949 // 'mu' is not a dummy index because it appears twice with the same
1952 e = indexed(A, mu, nu) + 42;
1953 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1954 // this will throw an exception:
1955 // "add::get_free_indices: inconsistent indices in sum"
1959 @cindex @code{simplify_indexed()}
1960 @subsection Simplifying indexed expressions
1962 In addition to the few automatic simplifications that GiNaC performs on
1963 indexed expressions (such as re-ordering the indices of symmetric tensors
1964 and calculating traces and convolutions of matrices and predefined tensors)
1968 ex ex::simplify_indexed(void);
1969 ex ex::simplify_indexed(const scalar_products & sp);
1972 that performs some more expensive operations:
1975 @item it checks the consistency of free indices in sums in the same way
1976 @code{get_free_indices()} does
1977 @item it tries to give dummy indices that appear in different terms of a sum
1978 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1979 @item it (symbolically) calculates all possible dummy index summations/contractions
1980 with the predefined tensors (this will be explained in more detail in the
1982 @item it detects contractions that vanish for symmetry reasons, for example
1983 the contraction of a symmetric and a totally antisymmetric tensor
1984 @item as a special case of dummy index summation, it can replace scalar products
1985 of two tensors with a user-defined value
1988 The last point is done with the help of the @code{scalar_products} class
1989 which is used to store scalar products with known values (this is not an
1990 arithmetic class, you just pass it to @code{simplify_indexed()}):
1994 symbol A("A"), B("B"), C("C"), i_sym("i");
1998 sp.add(A, B, 0); // A and B are orthogonal
1999 sp.add(A, C, 0); // A and C are orthogonal
2000 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2002 e = indexed(A + B, i) * indexed(A + C, i);
2004 // -> (B+A).i*(A+C).i
2006 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2012 The @code{scalar_products} object @code{sp} acts as a storage for the
2013 scalar products added to it with the @code{.add()} method. This method
2014 takes three arguments: the two expressions of which the scalar product is
2015 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2016 @code{simplify_indexed()} will replace all scalar products of indexed
2017 objects that have the symbols @code{A} and @code{B} as base expressions
2018 with the single value 0. The number, type and dimension of the indices
2019 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2021 @cindex @code{expand()}
2022 The example above also illustrates a feature of the @code{expand()} method:
2023 if passed the @code{expand_indexed} option it will distribute indices
2024 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2026 @cindex @code{tensor} (class)
2027 @subsection Predefined tensors
2029 Some frequently used special tensors such as the delta, epsilon and metric
2030 tensors are predefined in GiNaC. They have special properties when
2031 contracted with other tensor expressions and some of them have constant
2032 matrix representations (they will evaluate to a number when numeric
2033 indices are specified).
2035 @cindex @code{delta_tensor()}
2036 @subsubsection Delta tensor
2038 The delta tensor takes two indices, is symmetric and has the matrix
2039 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2040 @code{delta_tensor()}:
2044 symbol A("A"), B("B");
2046 idx i(symbol("i"), 3), j(symbol("j"), 3),
2047 k(symbol("k"), 3), l(symbol("l"), 3);
2049 ex e = indexed(A, i, j) * indexed(B, k, l)
2050 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2051 cout << e.simplify_indexed() << endl;
2054 cout << delta_tensor(i, i) << endl;
2059 @cindex @code{metric_tensor()}
2060 @subsubsection General metric tensor
2062 The function @code{metric_tensor()} creates a general symmetric metric
2063 tensor with two indices that can be used to raise/lower tensor indices. The
2064 metric tensor is denoted as @samp{g} in the output and if its indices are of
2065 mixed variance it is automatically replaced by a delta tensor:
2071 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2073 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2074 cout << e.simplify_indexed() << endl;
2077 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2078 cout << e.simplify_indexed() << endl;
2081 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2082 * metric_tensor(nu, rho);
2083 cout << e.simplify_indexed() << endl;
2086 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2087 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2088 + indexed(A, mu.toggle_variance(), rho));
2089 cout << e.simplify_indexed() << endl;
2094 @cindex @code{lorentz_g()}
2095 @subsubsection Minkowski metric tensor
2097 The Minkowski metric tensor is a special metric tensor with a constant
2098 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2099 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2100 It is created with the function @code{lorentz_g()} (although it is output as
2105 varidx mu(symbol("mu"), 4);
2107 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2108 * lorentz_g(mu, varidx(0, 4)); // negative signature
2109 cout << e.simplify_indexed() << endl;
2112 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2113 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2114 cout << e.simplify_indexed() << endl;
2119 @cindex @code{spinor_metric()}
2120 @subsubsection Spinor metric tensor
2122 The function @code{spinor_metric()} creates an antisymmetric tensor with
2123 two indices that is used to raise/lower indices of 2-component spinors.
2124 It is output as @samp{eps}:
2130 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2131 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2133 e = spinor_metric(A, B) * indexed(psi, B_co);
2134 cout << e.simplify_indexed() << endl;
2137 e = spinor_metric(A, B) * indexed(psi, A_co);
2138 cout << e.simplify_indexed() << endl;
2141 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2142 cout << e.simplify_indexed() << endl;
2145 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2146 cout << e.simplify_indexed() << endl;
2149 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2150 cout << e.simplify_indexed() << endl;
2153 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2154 cout << e.simplify_indexed() << endl;
2159 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2161 @cindex @code{epsilon_tensor()}
2162 @cindex @code{lorentz_eps()}
2163 @subsubsection Epsilon tensor
2165 The epsilon tensor is totally antisymmetric, its number of indices is equal
2166 to the dimension of the index space (the indices must all be of the same
2167 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2168 defined to be 1. Its behavior with indices that have a variance also
2169 depends on the signature of the metric. Epsilon tensors are output as
2172 There are three functions defined to create epsilon tensors in 2, 3 and 4
2176 ex epsilon_tensor(const ex & i1, const ex & i2);
2177 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2178 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2181 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2182 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2183 Minkowski space (the last @code{bool} argument specifies whether the metric
2184 has negative or positive signature, as in the case of the Minkowski metric
2189 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2190 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2191 e = lorentz_eps(mu, nu, rho, sig) *
2192 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2193 cout << simplify_indexed(e) << endl;
2194 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2196 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2197 symbol A("A"), B("B");
2198 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2199 cout << simplify_indexed(e) << endl;
2200 // -> -B.k*A.j*eps.i.k.j
2201 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2202 cout << simplify_indexed(e) << endl;
2207 @subsection Linear algebra
2209 The @code{matrix} class can be used with indices to do some simple linear
2210 algebra (linear combinations and products of vectors and matrices, traces
2211 and scalar products):
2215 idx i(symbol("i"), 2), j(symbol("j"), 2);
2216 symbol x("x"), y("y");
2218 // A is a 2x2 matrix, X is a 2x1 vector
2219 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2221 cout << indexed(A, i, i) << endl;
2224 ex e = indexed(A, i, j) * indexed(X, j);
2225 cout << e.simplify_indexed() << endl;
2226 // -> [[2*y+x],[4*y+3*x]].i
2228 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2229 cout << e.simplify_indexed() << endl;
2230 // -> [[3*y+3*x,6*y+2*x]].j
2234 You can of course obtain the same results with the @code{matrix::add()},
2235 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2236 but with indices you don't have to worry about transposing matrices.
2238 Matrix indices always start at 0 and their dimension must match the number
2239 of rows/columns of the matrix. Matrices with one row or one column are
2240 vectors and can have one or two indices (it doesn't matter whether it's a
2241 row or a column vector). Other matrices must have two indices.
2243 You should be careful when using indices with variance on matrices. GiNaC
2244 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2245 @samp{F.mu.nu} are different matrices. In this case you should use only
2246 one form for @samp{F} and explicitly multiply it with a matrix representation
2247 of the metric tensor.
2250 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2251 @c node-name, next, previous, up
2252 @section Non-commutative objects
2254 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2255 non-commutative objects are built-in which are mostly of use in high energy
2259 @item Clifford (Dirac) algebra (class @code{clifford})
2260 @item su(3) Lie algebra (class @code{color})
2261 @item Matrices (unindexed) (class @code{matrix})
2264 The @code{clifford} and @code{color} classes are subclasses of
2265 @code{indexed} because the elements of these algebras usually carry
2266 indices. The @code{matrix} class is described in more detail in
2269 Unlike most computer algebra systems, GiNaC does not primarily provide an
2270 operator (often denoted @samp{&*}) for representing inert products of
2271 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2272 classes of objects involved, and non-commutative products are formed with
2273 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2274 figuring out by itself which objects commute and will group the factors
2275 by their class. Consider this example:
2279 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2280 idx a(symbol("a"), 8), b(symbol("b"), 8);
2281 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2283 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2287 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2288 groups the non-commutative factors (the gammas and the su(3) generators)
2289 together while preserving the order of factors within each class (because
2290 Clifford objects commute with color objects). The resulting expression is a
2291 @emph{commutative} product with two factors that are themselves non-commutative
2292 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2293 parentheses are placed around the non-commutative products in the output.
2295 @cindex @code{ncmul} (class)
2296 Non-commutative products are internally represented by objects of the class
2297 @code{ncmul}, as opposed to commutative products which are handled by the
2298 @code{mul} class. You will normally not have to worry about this distinction,
2301 The advantage of this approach is that you never have to worry about using
2302 (or forgetting to use) a special operator when constructing non-commutative
2303 expressions. Also, non-commutative products in GiNaC are more intelligent
2304 than in other computer algebra systems; they can, for example, automatically
2305 canonicalize themselves according to rules specified in the implementation
2306 of the non-commutative classes. The drawback is that to work with other than
2307 the built-in algebras you have to implement new classes yourself. Symbols
2308 always commute and it's not possible to construct non-commutative products
2309 using symbols to represent the algebra elements or generators. User-defined
2310 functions can, however, be specified as being non-commutative.
2312 @cindex @code{return_type()}
2313 @cindex @code{return_type_tinfo()}
2314 Information about the commutativity of an object or expression can be
2315 obtained with the two member functions
2318 unsigned ex::return_type(void) const;
2319 unsigned ex::return_type_tinfo(void) const;
2322 The @code{return_type()} function returns one of three values (defined in
2323 the header file @file{flags.h}), corresponding to three categories of
2324 expressions in GiNaC:
2327 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2328 classes are of this kind.
2329 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2330 certain class of non-commutative objects which can be determined with the
2331 @code{return_type_tinfo()} method. Expressions of this category commute
2332 with everything except @code{noncommutative} expressions of the same
2334 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2335 of non-commutative objects of different classes. Expressions of this
2336 category don't commute with any other @code{noncommutative} or
2337 @code{noncommutative_composite} expressions.
2340 The value returned by the @code{return_type_tinfo()} method is valid only
2341 when the return type of the expression is @code{noncommutative}. It is a
2342 value that is unique to the class of the object and usually one of the
2343 constants in @file{tinfos.h}, or derived therefrom.
2345 Here are a couple of examples:
2348 @multitable @columnfractions 0.33 0.33 0.34
2349 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2350 @item @code{42} @tab @code{commutative} @tab -
2351 @item @code{2*x-y} @tab @code{commutative} @tab -
2352 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2353 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2354 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2355 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2359 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2360 @code{TINFO_clifford} for objects with a representation label of zero.
2361 Other representation labels yield a different @code{return_type_tinfo()},
2362 but it's the same for any two objects with the same label. This is also true
2365 A last note: With the exception of matrices, positive integer powers of
2366 non-commutative objects are automatically expanded in GiNaC. For example,
2367 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2368 non-commutative expressions).
2371 @cindex @code{clifford} (class)
2372 @subsection Clifford algebra
2374 @cindex @code{dirac_gamma()}
2375 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2376 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2377 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2378 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2381 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2384 which takes two arguments: the index and a @dfn{representation label} in the
2385 range 0 to 255 which is used to distinguish elements of different Clifford
2386 algebras (this is also called a @dfn{spin line index}). Gammas with different
2387 labels commute with each other. The dimension of the index can be 4 or (in
2388 the framework of dimensional regularization) any symbolic value. Spinor
2389 indices on Dirac gammas are not supported in GiNaC.
2391 @cindex @code{dirac_ONE()}
2392 The unity element of a Clifford algebra is constructed by
2395 ex dirac_ONE(unsigned char rl = 0);
2398 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2399 multiples of the unity element, even though it's customary to omit it.
2400 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2401 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2402 GiNaC will complain and/or produce incorrect results.
2404 @cindex @code{dirac_gamma5()}
2405 There is a special element @samp{gamma5} that commutes with all other
2406 gammas, has a unit square, and in 4 dimensions equals
2407 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2410 ex dirac_gamma5(unsigned char rl = 0);
2413 @cindex @code{dirac_gammaL()}
2414 @cindex @code{dirac_gammaR()}
2415 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2416 objects, constructed by
2419 ex dirac_gammaL(unsigned char rl = 0);
2420 ex dirac_gammaR(unsigned char rl = 0);
2423 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2424 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2426 @cindex @code{dirac_slash()}
2427 Finally, the function
2430 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2433 creates a term that represents a contraction of @samp{e} with the Dirac
2434 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2435 with a unique index whose dimension is given by the @code{dim} argument).
2436 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2438 In products of dirac gammas, superfluous unity elements are automatically
2439 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2440 and @samp{gammaR} are moved to the front.
2442 The @code{simplify_indexed()} function performs contractions in gamma strings,
2448 symbol a("a"), b("b"), D("D");
2449 varidx mu(symbol("mu"), D);
2450 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2451 * dirac_gamma(mu.toggle_variance());
2453 // -> gamma~mu*a\*gamma.mu
2454 e = e.simplify_indexed();
2457 cout << e.subs(D == 4) << endl;
2463 @cindex @code{dirac_trace()}
2464 To calculate the trace of an expression containing strings of Dirac gammas
2465 you use the function
2468 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2471 This function takes the trace of all gammas with the specified representation
2472 label; gammas with other labels are left standing. The last argument to
2473 @code{dirac_trace()} is the value to be returned for the trace of the unity
2474 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2475 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2476 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2477 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2478 This @samp{gamma5} scheme is described in greater detail in
2479 @cite{The Role of gamma5 in Dimensional Regularization}.
2481 The value of the trace itself is also usually different in 4 and in
2482 @math{D != 4} dimensions:
2487 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2488 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2489 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2490 cout << dirac_trace(e).simplify_indexed() << endl;
2497 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2498 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2499 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2500 cout << dirac_trace(e).simplify_indexed() << endl;
2501 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2505 Here is an example for using @code{dirac_trace()} to compute a value that
2506 appears in the calculation of the one-loop vacuum polarization amplitude in
2511 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2512 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2515 sp.add(l, l, pow(l, 2));
2516 sp.add(l, q, ldotq);
2518 ex e = dirac_gamma(mu) *
2519 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2520 dirac_gamma(mu.toggle_variance()) *
2521 (dirac_slash(l, D) + m * dirac_ONE());
2522 e = dirac_trace(e).simplify_indexed(sp);
2523 e = e.collect(lst(l, ldotq, m));
2525 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2529 The @code{canonicalize_clifford()} function reorders all gamma products that
2530 appear in an expression to a canonical (but not necessarily simple) form.
2531 You can use this to compare two expressions or for further simplifications:
2535 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2536 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2538 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2540 e = canonicalize_clifford(e);
2547 @cindex @code{color} (class)
2548 @subsection Color algebra
2550 @cindex @code{color_T()}
2551 For computations in quantum chromodynamics, GiNaC implements the base elements
2552 and structure constants of the su(3) Lie algebra (color algebra). The base
2553 elements @math{T_a} are constructed by the function
2556 ex color_T(const ex & a, unsigned char rl = 0);
2559 which takes two arguments: the index and a @dfn{representation label} in the
2560 range 0 to 255 which is used to distinguish elements of different color
2561 algebras. Objects with different labels commute with each other. The
2562 dimension of the index must be exactly 8 and it should be of class @code{idx},
2565 @cindex @code{color_ONE()}
2566 The unity element of a color algebra is constructed by
2569 ex color_ONE(unsigned char rl = 0);
2572 @strong{Note:} You must always use @code{color_ONE()} when referring to
2573 multiples of the unity element, even though it's customary to omit it.
2574 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2575 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2576 GiNaC may produce incorrect results.
2578 @cindex @code{color_d()}
2579 @cindex @code{color_f()}
2583 ex color_d(const ex & a, const ex & b, const ex & c);
2584 ex color_f(const ex & a, const ex & b, const ex & c);
2587 create the symmetric and antisymmetric structure constants @math{d_abc} and
2588 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2589 and @math{[T_a, T_b] = i f_abc T_c}.
2591 @cindex @code{color_h()}
2592 There's an additional function
2595 ex color_h(const ex & a, const ex & b, const ex & c);
2598 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2600 The function @code{simplify_indexed()} performs some simplifications on
2601 expressions containing color objects:
2606 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2607 k(symbol("k"), 8), l(symbol("l"), 8);
2609 e = color_d(a, b, l) * color_f(a, b, k);
2610 cout << e.simplify_indexed() << endl;
2613 e = color_d(a, b, l) * color_d(a, b, k);
2614 cout << e.simplify_indexed() << endl;
2617 e = color_f(l, a, b) * color_f(a, b, k);
2618 cout << e.simplify_indexed() << endl;
2621 e = color_h(a, b, c) * color_h(a, b, c);
2622 cout << e.simplify_indexed() << endl;
2625 e = color_h(a, b, c) * color_T(b) * color_T(c);
2626 cout << e.simplify_indexed() << endl;
2629 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2630 cout << e.simplify_indexed() << endl;
2633 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2634 cout << e.simplify_indexed() << endl;
2635 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2639 @cindex @code{color_trace()}
2640 To calculate the trace of an expression containing color objects you use the
2644 ex color_trace(const ex & e, unsigned char rl = 0);
2647 This function takes the trace of all color @samp{T} objects with the
2648 specified representation label; @samp{T}s with other labels are left
2649 standing. For example:
2653 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2655 // -> -I*f.a.c.b+d.a.c.b
2660 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2661 @c node-name, next, previous, up
2662 @chapter Methods and Functions
2665 In this chapter the most important algorithms provided by GiNaC will be
2666 described. Some of them are implemented as functions on expressions,
2667 others are implemented as methods provided by expression objects. If
2668 they are methods, there exists a wrapper function around it, so you can
2669 alternatively call it in a functional way as shown in the simple
2674 cout << "As method: " << sin(1).evalf() << endl;
2675 cout << "As function: " << evalf(sin(1)) << endl;
2679 @cindex @code{subs()}
2680 The general rule is that wherever methods accept one or more parameters
2681 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2682 wrapper accepts is the same but preceded by the object to act on
2683 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2684 most natural one in an OO model but it may lead to confusion for MapleV
2685 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2686 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2687 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2688 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2689 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2690 here. Also, users of MuPAD will in most cases feel more comfortable
2691 with GiNaC's convention. All function wrappers are implemented
2692 as simple inline functions which just call the corresponding method and
2693 are only provided for users uncomfortable with OO who are dead set to
2694 avoid method invocations. Generally, nested function wrappers are much
2695 harder to read than a sequence of methods and should therefore be
2696 avoided if possible. On the other hand, not everything in GiNaC is a
2697 method on class @code{ex} and sometimes calling a function cannot be
2701 * Information About Expressions::
2702 * Substituting Expressions::
2703 * Pattern Matching and Advanced Substitutions::
2704 * Applying a Function on Subexpressions::
2705 * Polynomial Arithmetic:: Working with polynomials.
2706 * Rational Expressions:: Working with rational functions.
2707 * Symbolic Differentiation::
2708 * Series Expansion:: Taylor and Laurent expansion.
2710 * Built-in Functions:: List of predefined mathematical functions.
2711 * Input/Output:: Input and output of expressions.
2715 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2716 @c node-name, next, previous, up
2717 @section Getting information about expressions
2719 @subsection Checking expression types
2720 @cindex @code{is_a<@dots{}>()}
2721 @cindex @code{is_exactly_a<@dots{}>()}
2722 @cindex @code{ex_to<@dots{}>()}
2723 @cindex Converting @code{ex} to other classes
2724 @cindex @code{info()}
2725 @cindex @code{return_type()}
2726 @cindex @code{return_type_tinfo()}
2728 Sometimes it's useful to check whether a given expression is a plain number,
2729 a sum, a polynomial with integer coefficients, or of some other specific type.
2730 GiNaC provides a couple of functions for this:
2733 bool is_a<T>(const ex & e);
2734 bool is_exactly_a<T>(const ex & e);
2735 bool ex::info(unsigned flag);
2736 unsigned ex::return_type(void) const;
2737 unsigned ex::return_type_tinfo(void) const;
2740 When the test made by @code{is_a<T>()} returns true, it is safe to call
2741 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2742 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2743 example, assuming @code{e} is an @code{ex}:
2748 if (is_a<numeric>(e))
2749 numeric n = ex_to<numeric>(e);
2754 @code{is_a<T>(e)} allows you to check whether the top-level object of
2755 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2756 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2757 e.g., for checking whether an expression is a number, a sum, or a product:
2764 is_a<numeric>(e1); // true
2765 is_a<numeric>(e2); // false
2766 is_a<add>(e1); // false
2767 is_a<add>(e2); // true
2768 is_a<mul>(e1); // false
2769 is_a<mul>(e2); // false
2773 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2774 top-level object of an expression @samp{e} is an instance of the GiNaC
2775 class @samp{T}, not including parent classes.
2777 The @code{info()} method is used for checking certain attributes of
2778 expressions. The possible values for the @code{flag} argument are defined
2779 in @file{ginac/flags.h}, the most important being explained in the following
2783 @multitable @columnfractions .30 .70
2784 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2785 @item @code{numeric}
2786 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2788 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2789 @item @code{rational}
2790 @tab @dots{}an exact rational number (integers are rational, too)
2791 @item @code{integer}
2792 @tab @dots{}a (non-complex) integer
2793 @item @code{crational}
2794 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2795 @item @code{cinteger}
2796 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2797 @item @code{positive}
2798 @tab @dots{}not complex and greater than 0
2799 @item @code{negative}
2800 @tab @dots{}not complex and less than 0
2801 @item @code{nonnegative}
2802 @tab @dots{}not complex and greater than or equal to 0
2804 @tab @dots{}an integer greater than 0
2806 @tab @dots{}an integer less than 0
2807 @item @code{nonnegint}
2808 @tab @dots{}an integer greater than or equal to 0
2810 @tab @dots{}an even integer
2812 @tab @dots{}an odd integer
2814 @tab @dots{}a prime integer (probabilistic primality test)
2815 @item @code{relation}
2816 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2817 @item @code{relation_equal}
2818 @tab @dots{}a @code{==} relation
2819 @item @code{relation_not_equal}
2820 @tab @dots{}a @code{!=} relation
2821 @item @code{relation_less}
2822 @tab @dots{}a @code{<} relation
2823 @item @code{relation_less_or_equal}
2824 @tab @dots{}a @code{<=} relation
2825 @item @code{relation_greater}
2826 @tab @dots{}a @code{>} relation
2827 @item @code{relation_greater_or_equal}
2828 @tab @dots{}a @code{>=} relation
2830 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2832 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2833 @item @code{polynomial}
2834 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2835 @item @code{integer_polynomial}
2836 @tab @dots{}a polynomial with (non-complex) integer coefficients
2837 @item @code{cinteger_polynomial}
2838 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2839 @item @code{rational_polynomial}
2840 @tab @dots{}a polynomial with (non-complex) rational coefficients
2841 @item @code{crational_polynomial}
2842 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2843 @item @code{rational_function}
2844 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2845 @item @code{algebraic}
2846 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2850 To determine whether an expression is commutative or non-commutative and if
2851 so, with which other expressions it would commute, you use the methods
2852 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2853 for an explanation of these.
2856 @subsection Accessing subexpressions
2857 @cindex @code{nops()}
2860 @cindex @code{relational} (class)
2862 GiNaC provides the two methods
2865 unsigned ex::nops();
2866 ex ex::op(unsigned i);
2869 for accessing the subexpressions in the container-like GiNaC classes like
2870 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2871 determines the number of subexpressions (@samp{operands}) contained, while
2872 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2873 In the case of a @code{power} object, @code{op(0)} will return the basis
2874 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2875 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2877 The left-hand and right-hand side expressions of objects of class
2878 @code{relational} (and only of these) can also be accessed with the methods
2886 @subsection Comparing expressions
2887 @cindex @code{is_equal()}
2888 @cindex @code{is_zero()}
2890 Expressions can be compared with the usual C++ relational operators like
2891 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2892 the result is usually not determinable and the result will be @code{false},
2893 except in the case of the @code{!=} operator. You should also be aware that
2894 GiNaC will only do the most trivial test for equality (subtracting both
2895 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2898 Actually, if you construct an expression like @code{a == b}, this will be
2899 represented by an object of the @code{relational} class (@pxref{Relations})
2900 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
2902 There are also two methods
2905 bool ex::is_equal(const ex & other);
2909 for checking whether one expression is equal to another, or equal to zero,
2912 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2913 GiNaC header files. This method is however only to be used internally by
2914 GiNaC to establish a canonical sort order for terms, and using it to compare
2915 expressions will give very surprising results.
2918 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2919 @c node-name, next, previous, up
2920 @section Substituting expressions
2921 @cindex @code{subs()}
2923 Algebraic objects inside expressions can be replaced with arbitrary
2924 expressions via the @code{.subs()} method:
2927 ex ex::subs(const ex & e);
2928 ex ex::subs(const lst & syms, const lst & repls);
2931 In the first form, @code{subs()} accepts a relational of the form
2932 @samp{object == expression} or a @code{lst} of such relationals:
2936 symbol x("x"), y("y");
2938 ex e1 = 2*x^2-4*x+3;
2939 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2943 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2948 If you specify multiple substitutions, they are performed in parallel, so e.g.
2949 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2951 The second form of @code{subs()} takes two lists, one for the objects to be
2952 replaced and one for the expressions to be substituted (both lists must
2953 contain the same number of elements). Using this form, you would write
2954 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2956 @code{subs()} performs syntactic substitution of any complete algebraic
2957 object; it does not try to match sub-expressions as is demonstrated by the
2962 symbol x("x"), y("y"), z("z");
2964 ex e1 = pow(x+y, 2);
2965 cout << e1.subs(x+y == 4) << endl;
2968 ex e2 = sin(x)*sin(y)*cos(x);
2969 cout << e2.subs(sin(x) == cos(x)) << endl;
2970 // -> cos(x)^2*sin(y)
2973 cout << e3.subs(x+y == 4) << endl;
2975 // (and not 4+z as one might expect)
2979 A more powerful form of substitution using wildcards is described in the
2983 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2984 @c node-name, next, previous, up
2985 @section Pattern matching and advanced substitutions
2986 @cindex @code{wildcard} (class)
2987 @cindex Pattern matching
2989 GiNaC allows the use of patterns for checking whether an expression is of a
2990 certain form or contains subexpressions of a certain form, and for
2991 substituting expressions in a more general way.
2993 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2994 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2995 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2996 an unsigned integer number to allow having multiple different wildcards in a
2997 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2998 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3002 ex wild(unsigned label = 0);
3005 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3008 Some examples for patterns:
3010 @multitable @columnfractions .5 .5
3011 @item @strong{Constructed as} @tab @strong{Output as}
3012 @item @code{wild()} @tab @samp{$0}
3013 @item @code{pow(x,wild())} @tab @samp{x^$0}
3014 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3015 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3021 @item Wildcards behave like symbols and are subject to the same algebraic
3022 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3023 @item As shown in the last example, to use wildcards for indices you have to
3024 use them as the value of an @code{idx} object. This is because indices must
3025 always be of class @code{idx} (or a subclass).
3026 @item Wildcards only represent expressions or subexpressions. It is not
3027 possible to use them as placeholders for other properties like index
3028 dimension or variance, representation labels, symmetry of indexed objects
3030 @item Because wildcards are commutative, it is not possible to use wildcards
3031 as part of noncommutative products.
3032 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3033 are also valid patterns.
3036 @subsection Matching expressions
3037 @cindex @code{match()}
3038 The most basic application of patterns is to check whether an expression
3039 matches a given pattern. This is done by the function
3042 bool ex::match(const ex & pattern);
3043 bool ex::match(const ex & pattern, lst & repls);
3046 This function returns @code{true} when the expression matches the pattern
3047 and @code{false} if it doesn't. If used in the second form, the actual
3048 subexpressions matched by the wildcards get returned in the @code{repls}
3049 object as a list of relations of the form @samp{wildcard == expression}.
3050 If @code{match()} returns false, the state of @code{repls} is undefined.
3051 For reproducible results, the list should be empty when passed to
3052 @code{match()}, but it is also possible to find similarities in multiple
3053 expressions by passing in the result of a previous match.
3055 The matching algorithm works as follows:
3058 @item A single wildcard matches any expression. If one wildcard appears
3059 multiple times in a pattern, it must match the same expression in all
3060 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3061 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3062 @item If the expression is not of the same class as the pattern, the match
3063 fails (i.e. a sum only matches a sum, a function only matches a function,
3065 @item If the pattern is a function, it only matches the same function
3066 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3067 @item Except for sums and products, the match fails if the number of
3068 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3070 @item If there are no subexpressions, the expressions and the pattern must
3071 be equal (in the sense of @code{is_equal()}).
3072 @item Except for sums and products, each subexpression (@code{op()}) must
3073 match the corresponding subexpression of the pattern.
3076 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3077 account for their commutativity and associativity:
3080 @item If the pattern contains a term or factor that is a single wildcard,
3081 this one is used as the @dfn{global wildcard}. If there is more than one
3082 such wildcard, one of them is chosen as the global wildcard in a random
3084 @item Every term/factor of the pattern, except the global wildcard, is
3085 matched against every term of the expression in sequence. If no match is
3086 found, the whole match fails. Terms that did match are not considered in
3088 @item If there are no unmatched terms left, the match succeeds. Otherwise
3089 the match fails unless there is a global wildcard in the pattern, in
3090 which case this wildcard matches the remaining terms.
3093 In general, having more than one single wildcard as a term of a sum or a
3094 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3097 Here are some examples in @command{ginsh} to demonstrate how it works (the
3098 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3099 match fails, and the list of wildcard replacements otherwise):
3102 > match((x+y)^a,(x+y)^a);
3104 > match((x+y)^a,(x+y)^b);
3106 > match((x+y)^a,$1^$2);
3108 > match((x+y)^a,$1^$1);
3110 > match((x+y)^(x+y),$1^$1);
3112 > match((x+y)^(x+y),$1^$2);
3114 > match((a+b)*(a+c),($1+b)*($1+c));
3116 > match((a+b)*(a+c),(a+$1)*(a+$2));
3118 (Unpredictable. The result might also be [$1==c,$2==b].)
3119 > match((a+b)*(a+c),($1+$2)*($1+$3));
3120 (The result is undefined. Due to the sequential nature of the algorithm
3121 and the re-ordering of terms in GiNaC, the match for the first factor
3122 may be @{$1==a,$2==b@} in which case the match for the second factor
3123 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3125 > match(a*(x+y)+a*z+b,a*$1+$2);
3126 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3127 @{$1=x+y,$2=a*z+b@}.)
3128 > match(a+b+c+d+e+f,c);
3130 > match(a+b+c+d+e+f,c+$0);
3132 > match(a+b+c+d+e+f,c+e+$0);
3134 > match(a+b,a+b+$0);
3136 > match(a*b^2,a^$1*b^$2);
3138 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3139 even though a==a^1.)
3140 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3142 > match(atan2(y,x^2),atan2(y,$0));
3146 @subsection Matching parts of expressions
3147 @cindex @code{has()}
3148 A more general way to look for patterns in expressions is provided by the
3152 bool ex::has(const ex & pattern);
3155 This function checks whether a pattern is matched by an expression itself or
3156 by any of its subexpressions.
3158 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3159 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3162 > has(x*sin(x+y+2*a),y);
3164 > has(x*sin(x+y+2*a),x+y);
3166 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3167 has the subexpressions "x", "y" and "2*a".)
3168 > has(x*sin(x+y+2*a),x+y+$1);
3170 (But this is possible.)
3171 > has(x*sin(2*(x+y)+2*a),x+y);
3173 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3174 which "x+y" is not a subexpression.)
3177 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3179 > has(4*x^2-x+3,$1*x);
3181 > has(4*x^2+x+3,$1*x);
3183 (Another possible pitfall. The first expression matches because the term
3184 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3185 contains a linear term you should use the coeff() function instead.)
3188 @cindex @code{find()}
3192 bool ex::find(const ex & pattern, lst & found);
3195 works a bit like @code{has()} but it doesn't stop upon finding the first
3196 match. Instead, it inserts all found matches into the specified list. If
3197 there are multiple occurrences of the same expression, it is entered only
3198 once to the list. @code{find()} returns false if no matches were found (in
3199 @command{ginsh}, it returns an empty list):
3202 > find(1+x+x^2+x^3,x);
3204 > find(1+x+x^2+x^3,y);
3206 > find(1+x+x^2+x^3,x^$1);
3208 (Note the absence of "x".)
3209 > expand((sin(x)+sin(y))*(a+b));
3210 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3215 @subsection Substituting expressions
3216 @cindex @code{subs()}
3217 Probably the most useful application of patterns is to use them for
3218 substituting expressions with the @code{subs()} method. Wildcards can be
3219 used in the search patterns as well as in the replacement expressions, where
3220 they get replaced by the expressions matched by them. @code{subs()} doesn't
3221 know anything about algebra; it performs purely syntactic substitutions.
3226 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3228 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3230 > subs((a+b+c)^2,a+b==x);
3232 > subs((a+b+c)^2,a+b+$1==x+$1);
3234 > subs(a+2*b,a+b==x);
3236 > subs(4*x^3-2*x^2+5*x-1,x==a);
3238 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3240 > subs(sin(1+sin(x)),sin($1)==cos($1));
3242 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3246 The last example would be written in C++ in this way:
3250 symbol a("a"), b("b"), x("x"), y("y");
3251 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3252 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3253 cout << e.expand() << endl;
3259 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3260 @c node-name, next, previous, up
3261 @section Applying a Function on Subexpressions
3262 @cindex Tree traversal
3263 @cindex @code{map()}
3265 Sometimes you may want to perform an operation on specific parts of an
3266 expression while leaving the general structure of it intact. An example
3267 of this would be a matrix trace operation: the trace of a sum is the sum
3268 of the traces of the individual terms. That is, the trace should @dfn{map}
3269 on the sum, by applying itself to each of the sum's operands. It is possible
3270 to do this manually which usually results in code like this:
3275 if (is_a<matrix>(e))
3276 return ex_to<matrix>(e).trace();
3277 else if (is_a<add>(e)) @{
3279 for (unsigned i=0; i<e.nops(); i++)
3280 sum += calc_trace(e.op(i));
3282 @} else if (is_a<mul>)(e)) @{
3290 This is, however, slightly inefficient (if the sum is very large it can take
3291 a long time to add the terms one-by-one), and its applicability is limited to
3292 a rather small class of expressions. If @code{calc_trace()} is called with
3293 a relation or a list as its argument, you will probably want the trace to
3294 be taken on both sides of the relation or of all elements of the list.
3296 GiNaC offers the @code{map()} method to aid in the implementation of such
3300 ex ex::map(map_function & f) const;
3301 ex ex::map(ex (*f)(const ex & e)) const;
3304 In the first (preferred) form, @code{map()} takes a function object that
3305 is subclassed from the @code{map_function} class. In the second form, it
3306 takes a pointer to a function that accepts and returns an expression.
3307 @code{map()} constructs a new expression of the same type, applying the
3308 specified function on all subexpressions (in the sense of @code{op()}),
3311 The use of a function object makes it possible to supply more arguments to
3312 the function that is being mapped, or to keep local state information.
3313 The @code{map_function} class declares a virtual function call operator
3314 that you can overload. Here is a sample implementation of @code{calc_trace()}
3315 that uses @code{map()} in a recursive fashion:
3318 struct calc_trace : public map_function @{
3319 ex operator()(const ex &e)
3321 if (is_a<matrix>(e))
3322 return ex_to<matrix>(e).trace();
3323 else if (is_a<mul>(e)) @{
3326 return e.map(*this);
3331 This function object could then be used like this:
3335 ex M = ... // expression with matrices
3336 calc_trace do_trace;
3337 ex tr = do_trace(M);
3341 Here is another example for you to meditate over. It removes quadratic
3342 terms in a variable from an expanded polynomial:
3345 struct map_rem_quad : public map_function @{
3347 map_rem_quad(const ex & var_) : var(var_) @{@}
3349 ex operator()(const ex & e)
3351 if (is_a<add>(e) || is_a<mul>(e))
3352 return e.map(*this);
3353 else if (is_a<power>(e) &&
3354 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3364 symbol x("x"), y("y");
3367 for (int i=0; i<8; i++)
3368 e += pow(x, i) * pow(y, 8-i) * (i+1);
3370 // -> 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
3372 map_rem_quad rem_quad(x);
3373 cout << rem_quad(e) << endl;
3374 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3378 @command{ginsh} offers a slightly different implementation of @code{map()}
3379 that allows applying algebraic functions to operands. The second argument
3380 to @code{map()} is an expression containing the wildcard @samp{$0} which
3381 acts as the placeholder for the operands:
3386 > map(a+2*b,sin($0));
3388 > map(@{a,b,c@},$0^2+$0);
3389 @{a^2+a,b^2+b,c^2+c@}
3392 Note that it is only possible to use algebraic functions in the second
3393 argument. You can not use functions like @samp{diff()}, @samp{op()},
3394 @samp{subs()} etc. because these are evaluated immediately:
3397 > map(@{a,b,c@},diff($0,a));
3399 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3400 to "map(@{a,b,c@},0)".
3404 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3405 @c node-name, next, previous, up
3406 @section Polynomial arithmetic
3408 @subsection Expanding and collecting
3409 @cindex @code{expand()}
3410 @cindex @code{collect()}
3411 @cindex @code{collect_common_factors()}
3413 A polynomial in one or more variables has many equivalent
3414 representations. Some useful ones serve a specific purpose. Consider
3415 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3416 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3417 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3418 representations are the recursive ones where one collects for exponents
3419 in one of the three variable. Since the factors are themselves
3420 polynomials in the remaining two variables the procedure can be
3421 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3422 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3425 To bring an expression into expanded form, its method
3431 may be called. In our example above, this corresponds to @math{4*x*y +
3432 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3433 GiNaC is not easily guessable you should be prepared to see different
3434 orderings of terms in such sums!
3436 Another useful representation of multivariate polynomials is as a
3437 univariate polynomial in one of the variables with the coefficients
3438 being polynomials in the remaining variables. The method
3439 @code{collect()} accomplishes this task:
3442 ex ex::collect(const ex & s, bool distributed = false);
3445 The first argument to @code{collect()} can also be a list of objects in which
3446 case the result is either a recursively collected polynomial, or a polynomial
3447 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3448 by the @code{distributed} flag.
3450 Note that the original polynomial needs to be in expanded form (for the
3451 variables concerned) in order for @code{collect()} to be able to find the
3452 coefficients properly.
3454 The following @command{ginsh} transcript shows an application of @code{collect()}
3455 together with @code{find()}:
3458 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3459 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
3460 > collect(a,@{p,q@});
3461 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)
3462 > collect(a,find(a,sin($1)));
3463 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3464 > collect(a,@{find(a,sin($1)),p,q@});
3465 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3466 > collect(a,@{find(a,sin($1)),d@});
3467 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3470 Polynomials can often be brought into a more compact form by collecting
3471 common factors from the terms of sums. This is accomplished by the function
3474 ex collect_common_factors(const ex & e);
3477 This function doesn't perform a full factorization but only looks for
3478 factors which are already explicitly present:
3481 > collect_common_factors(a*x+a*y);
3483 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
3485 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
3486 (c+a)*a*(x*y+y^2+x)*b
3489 @subsection Degree and coefficients
3490 @cindex @code{degree()}
3491 @cindex @code{ldegree()}
3492 @cindex @code{coeff()}
3494 The degree and low degree of a polynomial can be obtained using the two
3498 int ex::degree(const ex & s);
3499 int ex::ldegree(const ex & s);
3502 These functions only work reliably if the input polynomial is collected in
3503 terms of the object @samp{s}. Otherwise, they are only guaranteed to return
3504 the upper/lower bounds of the exponents. If you need accurate results, you
3505 have to call @code{expand()} and/or @code{collect()} on the input polynomial.
3513 > degree(expand(a),x);
3517 @code{degree()} also works on rational functions, returning the asymptotic
3521 > degree((x+1)/(x^3+1),x);
3525 If the input is not a polynomial or rational function in the variable @samp{s},
3526 the behavior of @code{degree()} and @code{ldegree()} is undefined.
3528 To extract a coefficient with a certain power from an expanded
3532 ex ex::coeff(const ex & s, int n);
3535 You can also obtain the leading and trailing coefficients with the methods
3538 ex ex::lcoeff(const ex & s);
3539 ex ex::tcoeff(const ex & s);
3542 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3545 An application is illustrated in the next example, where a multivariate
3546 polynomial is analyzed:
3550 symbol x("x"), y("y");
3551 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3552 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3553 ex Poly = PolyInp.expand();
3555 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3556 cout << "The x^" << i << "-coefficient is "
3557 << Poly.coeff(x,i) << endl;
3559 cout << "As polynomial in y: "
3560 << Poly.collect(y) << endl;
3564 When run, it returns an output in the following fashion:
3567 The x^0-coefficient is y^2+11*y
3568 The x^1-coefficient is 5*y^2-2*y
3569 The x^2-coefficient is -1
3570 The x^3-coefficient is 4*y
3571 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3574 As always, the exact output may vary between different versions of GiNaC
3575 or even from run to run since the internal canonical ordering is not
3576 within the user's sphere of influence.
3578 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3579 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3580 with non-polynomial expressions as they not only work with symbols but with
3581 constants, functions and indexed objects as well:
3585 symbol a("a"), b("b"), c("c");
3586 idx i(symbol("i"), 3);
3588 ex e = pow(sin(x) - cos(x), 4);
3589 cout << e.degree(cos(x)) << endl;
3591 cout << e.expand().coeff(sin(x), 3) << endl;
3594 e = indexed(a+b, i) * indexed(b+c, i);
3595 e = e.expand(expand_options::expand_indexed);
3596 cout << e.collect(indexed(b, i)) << endl;
3597 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3602 @subsection Polynomial division
3603 @cindex polynomial division
3606 @cindex pseudo-remainder
3607 @cindex @code{quo()}
3608 @cindex @code{rem()}
3609 @cindex @code{prem()}
3610 @cindex @code{divide()}
3615 ex quo(const ex & a, const ex & b, const symbol & x);
3616 ex rem(const ex & a, const ex & b, const symbol & x);
3619 compute the quotient and remainder of univariate polynomials in the variable
3620 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3622 The additional function
3625 ex prem(const ex & a, const ex & b, const symbol & x);
3628 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3629 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3631 Exact division of multivariate polynomials is performed by the function
3634 bool divide(const ex & a, const ex & b, ex & q);
3637 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3638 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3639 in which case the value of @code{q} is undefined.
3642 @subsection Unit, content and primitive part
3643 @cindex @code{unit()}
3644 @cindex @code{content()}
3645 @cindex @code{primpart()}
3650 ex ex::unit(const symbol & x);
3651 ex ex::content(const symbol & x);
3652 ex ex::primpart(const symbol & x);
3655 return the unit part, content part, and primitive polynomial of a multivariate
3656 polynomial with respect to the variable @samp{x} (the unit part being the sign
3657 of the leading coefficient, the content part being the GCD of the coefficients,
3658 and the primitive polynomial being the input polynomial divided by the unit and
3659 content parts). The product of unit, content, and primitive part is the
3660 original polynomial.
3663 @subsection GCD and LCM
3666 @cindex @code{gcd()}
3667 @cindex @code{lcm()}
3669 The functions for polynomial greatest common divisor and least common
3670 multiple have the synopsis
3673 ex gcd(const ex & a, const ex & b);
3674 ex lcm(const ex & a, const ex & b);
3677 The functions @code{gcd()} and @code{lcm()} accept two expressions
3678 @code{a} and @code{b} as arguments and return a new expression, their
3679 greatest common divisor or least common multiple, respectively. If the
3680 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3681 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3684 #include <ginac/ginac.h>
3685 using namespace GiNaC;
3689 symbol x("x"), y("y"), z("z");
3690 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3691 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3693 ex P_gcd = gcd(P_a, P_b);
3695 ex P_lcm = lcm(P_a, P_b);
3696 // 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
3701 @subsection Square-free decomposition
3702 @cindex square-free decomposition
3703 @cindex factorization
3704 @cindex @code{sqrfree()}
3706 GiNaC still lacks proper factorization support. Some form of
3707 factorization is, however, easily implemented by noting that factors
3708 appearing in a polynomial with power two or more also appear in the
3709 derivative and hence can easily be found by computing the GCD of the
3710 original polynomial and its derivatives. Any decent system has an
3711 interface for this so called square-free factorization. So we provide
3714 ex sqrfree(const ex & a, const lst & l = lst());
3716 Here is an example that by the way illustrates how the exact form of the
3717 result may slightly depend on the order of differentiation, calling for
3718 some care with subsequent processing of the result:
3721 symbol x("x"), y("y");
3722 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
3724 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3725 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
3727 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3728 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
3730 cout << sqrfree(BiVarPol) << endl;
3731 // -> depending on luck, any of the above
3734 Note also, how factors with the same exponents are not fully factorized
3738 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3739 @c node-name, next, previous, up
3740 @section Rational expressions
3742 @subsection The @code{normal} method
3743 @cindex @code{normal()}
3744 @cindex simplification
3745 @cindex temporary replacement
3747 Some basic form of simplification of expressions is called for frequently.
3748 GiNaC provides the method @code{.normal()}, which converts a rational function
3749 into an equivalent rational function of the form @samp{numerator/denominator}
3750 where numerator and denominator are coprime. If the input expression is already
3751 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3752 otherwise it performs fraction addition and multiplication.
3754 @code{.normal()} can also be used on expressions which are not rational functions
3755 as it will replace all non-rational objects (like functions or non-integer
3756 powers) by temporary symbols to bring the expression to the domain of rational
3757 functions before performing the normalization, and re-substituting these
3758 symbols afterwards. This algorithm is also available as a separate method
3759 @code{.to_rational()}, described below.
3761 This means that both expressions @code{t1} and @code{t2} are indeed
3762 simplified in this little code snippet:
3767 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3768 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3769 std::cout << "t1 is " << t1.normal() << std::endl;
3770 std::cout << "t2 is " << t2.normal() << std::endl;
3774 Of course this works for multivariate polynomials too, so the ratio of
3775 the sample-polynomials from the section about GCD and LCM above would be
3776 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3779 @subsection Numerator and denominator
3782 @cindex @code{numer()}
3783 @cindex @code{denom()}
3784 @cindex @code{numer_denom()}
3786 The numerator and denominator of an expression can be obtained with
3791 ex ex::numer_denom();
3794 These functions will first normalize the expression as described above and
3795 then return the numerator, denominator, or both as a list, respectively.
3796 If you need both numerator and denominator, calling @code{numer_denom()} is
3797 faster than using @code{numer()} and @code{denom()} separately.
3800 @subsection Converting to a rational expression
3801 @cindex @code{to_rational()}
3803 Some of the methods described so far only work on polynomials or rational
3804 functions. GiNaC provides a way to extend the domain of these functions to
3805 general expressions by using the temporary replacement algorithm described
3806 above. You do this by calling
3809 ex ex::to_rational(lst &l);
3812 on the expression to be converted. The supplied @code{lst} will be filled
3813 with the generated temporary symbols and their replacement expressions in
3814 a format that can be used directly for the @code{subs()} method. It can also
3815 already contain a list of replacements from an earlier application of
3816 @code{.to_rational()}, so it's possible to use it on multiple expressions
3817 and get consistent results.
3824 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3825 ex b = sin(x) + cos(x);
3828 divide(a.to_rational(l), b.to_rational(l), q);
3829 cout << q.subs(l) << endl;
3833 will print @samp{sin(x)-cos(x)}.
3836 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3837 @c node-name, next, previous, up
3838 @section Symbolic differentiation
3839 @cindex differentiation
3840 @cindex @code{diff()}
3842 @cindex product rule
3844 GiNaC's objects know how to differentiate themselves. Thus, a
3845 polynomial (class @code{add}) knows that its derivative is the sum of
3846 the derivatives of all the monomials:
3850 symbol x("x"), y("y"), z("z");
3851 ex P = pow(x, 5) + pow(x, 2) + y;
3853 cout << P.diff(x,2) << endl;
3855 cout << P.diff(y) << endl; // 1
3857 cout << P.diff(z) << endl; // 0
3862 If a second integer parameter @var{n} is given, the @code{diff} method
3863 returns the @var{n}th derivative.
3865 If @emph{every} object and every function is told what its derivative
3866 is, all derivatives of composed objects can be calculated using the
3867 chain rule and the product rule. Consider, for instance the expression
3868 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3869 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3870 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3871 out that the composition is the generating function for Euler Numbers,
3872 i.e. the so called @var{n}th Euler number is the coefficient of
3873 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3874 identity to code a function that generates Euler numbers in just three
3877 @cindex Euler numbers
3879 #include <ginac/ginac.h>
3880 using namespace GiNaC;
3882 ex EulerNumber(unsigned n)
3885 const ex generator = pow(cosh(x),-1);
3886 return generator.diff(x,n).subs(x==0);
3891 for (unsigned i=0; i<11; i+=2)
3892 std::cout << EulerNumber(i) << std::endl;
3897 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3898 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3899 @code{i} by two since all odd Euler numbers vanish anyways.
3902 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3903 @c node-name, next, previous, up
3904 @section Series expansion
3905 @cindex @code{series()}
3906 @cindex Taylor expansion
3907 @cindex Laurent expansion
3908 @cindex @code{pseries} (class)
3909 @cindex @code{Order()}
3911 Expressions know how to expand themselves as a Taylor series or (more
3912 generally) a Laurent series. As in most conventional Computer Algebra
3913 Systems, no distinction is made between those two. There is a class of
3914 its own for storing such series (@code{class pseries}) and a built-in
3915 function (called @code{Order}) for storing the order term of the series.
3916 As a consequence, if you want to work with series, i.e. multiply two
3917 series, you need to call the method @code{ex::series} again to convert
3918 it to a series object with the usual structure (expansion plus order
3919 term). A sample application from special relativity could read:
3922 #include <ginac/ginac.h>
3923 using namespace std;
3924 using namespace GiNaC;
3928 symbol v("v"), c("c");
3930 ex gamma = 1/sqrt(1 - pow(v/c,2));
3931 ex mass_nonrel = gamma.series(v==0, 10);
3933 cout << "the relativistic mass increase with v is " << endl
3934 << mass_nonrel << endl;
3936 cout << "the inverse square of this series is " << endl
3937 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3941 Only calling the series method makes the last output simplify to
3942 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3943 series raised to the power @math{-2}.
3945 @cindex Machin's formula
3946 As another instructive application, let us calculate the numerical
3947 value of Archimedes' constant
3951 (for which there already exists the built-in constant @code{Pi})
3952 using Machin's amazing formula
3954 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3957 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3959 We may expand the arcus tangent around @code{0} and insert the fractions
3960 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3961 carries an order term with it and the question arises what the system is
3962 supposed to do when the fractions are plugged into that order term. The
3963 solution is to use the function @code{series_to_poly()} to simply strip
3967 #include <ginac/ginac.h>
3968 using namespace GiNaC;
3970 ex machin_pi(int degr)
3973 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3974 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3975 -4*pi_expansion.subs(x==numeric(1,239));
3981 using std::cout; // just for fun, another way of...
3982 using std::endl; // ...dealing with this namespace std.
3984 for (int i=2; i<12; i+=2) @{
3985 pi_frac = machin_pi(i);
3986 cout << i << ":\t" << pi_frac << endl
3987 << "\t" << pi_frac.evalf() << endl;
3993 Note how we just called @code{.series(x,degr)} instead of
3994 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3995 method @code{series()}: if the first argument is a symbol the expression
3996 is expanded in that symbol around point @code{0}. When you run this
3997 program, it will type out:
4001 3.1832635983263598326
4002 4: 5359397032/1706489875
4003 3.1405970293260603143
4004 6: 38279241713339684/12184551018734375
4005 3.141621029325034425
4006 8: 76528487109180192540976/24359780855939418203125
4007 3.141591772182177295
4008 10: 327853873402258685803048818236/104359128170408663038552734375
4009 3.1415926824043995174
4013 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4014 @c node-name, next, previous, up
4015 @section Symmetrization
4016 @cindex @code{symmetrize()}
4017 @cindex @code{antisymmetrize()}
4018 @cindex @code{symmetrize_cyclic()}
4023 ex ex::symmetrize(const lst & l);
4024 ex ex::antisymmetrize(const lst & l);
4025 ex ex::symmetrize_cyclic(const lst & l);
4028 symmetrize an expression by returning the sum over all symmetric,
4029 antisymmetric or cyclic permutations of the specified list of objects,
4030 weighted by the number of permutations.
4032 The three additional methods
4035 ex ex::symmetrize();
4036 ex ex::antisymmetrize();
4037 ex ex::symmetrize_cyclic();
4040 symmetrize or antisymmetrize an expression over its free indices.
4042 Symmetrization is most useful with indexed expressions but can be used with
4043 almost any kind of object (anything that is @code{subs()}able):
4047 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4048 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4050 cout << indexed(A, i, j).symmetrize() << endl;
4051 // -> 1/2*A.j.i+1/2*A.i.j
4052 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4053 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4054 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4055 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4060 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
4061 @c node-name, next, previous, up
4062 @section Predefined mathematical functions
4064 GiNaC contains the following predefined mathematical functions:
4067 @multitable @columnfractions .30 .70
4068 @item @strong{Name} @tab @strong{Function}
4071 @cindex @code{abs()}
4072 @item @code{csgn(x)}
4074 @cindex @code{csgn()}
4075 @item @code{sqrt(x)}
4076 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4077 @cindex @code{sqrt()}
4080 @cindex @code{sin()}
4083 @cindex @code{cos()}
4086 @cindex @code{tan()}
4087 @item @code{asin(x)}
4089 @cindex @code{asin()}
4090 @item @code{acos(x)}
4092 @cindex @code{acos()}
4093 @item @code{atan(x)}
4094 @tab inverse tangent
4095 @cindex @code{atan()}
4096 @item @code{atan2(y, x)}
4097 @tab inverse tangent with two arguments
4098 @item @code{sinh(x)}
4099 @tab hyperbolic sine
4100 @cindex @code{sinh()}
4101 @item @code{cosh(x)}
4102 @tab hyperbolic cosine
4103 @cindex @code{cosh()}
4104 @item @code{tanh(x)}
4105 @tab hyperbolic tangent
4106 @cindex @code{tanh()}
4107 @item @code{asinh(x)}
4108 @tab inverse hyperbolic sine
4109 @cindex @code{asinh()}
4110 @item @code{acosh(x)}
4111 @tab inverse hyperbolic cosine
4112 @cindex @code{acosh()}
4113 @item @code{atanh(x)}
4114 @tab inverse hyperbolic tangent
4115 @cindex @code{atanh()}
4117 @tab exponential function
4118 @cindex @code{exp()}
4120 @tab natural logarithm
4121 @cindex @code{log()}
4124 @cindex @code{Li2()}
4125 @item @code{zeta(x)}
4126 @tab Riemann's zeta function
4127 @cindex @code{zeta()}
4128 @item @code{zeta(n, x)}
4129 @tab derivatives of Riemann's zeta function
4130 @item @code{tgamma(x)}
4132 @cindex @code{tgamma()}
4133 @cindex Gamma function
4134 @item @code{lgamma(x)}
4135 @tab logarithm of Gamma function
4136 @cindex @code{lgamma()}
4137 @item @code{beta(x, y)}
4138 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4139 @cindex @code{beta()}
4141 @tab psi (digamma) function
4142 @cindex @code{psi()}
4143 @item @code{psi(n, x)}
4144 @tab derivatives of psi function (polygamma functions)
4145 @item @code{factorial(n)}
4146 @tab factorial function
4147 @cindex @code{factorial()}
4148 @item @code{binomial(n, m)}
4149 @tab binomial coefficients
4150 @cindex @code{binomial()}
4151 @item @code{Order(x)}
4152 @tab order term function in truncated power series
4153 @cindex @code{Order()}
4158 For functions that have a branch cut in the complex plane GiNaC follows
4159 the conventions for C++ as defined in the ANSI standard as far as
4160 possible. In particular: the natural logarithm (@code{log}) and the
4161 square root (@code{sqrt}) both have their branch cuts running along the
4162 negative real axis where the points on the axis itself belong to the
4163 upper part (i.e. continuous with quadrant II). The inverse
4164 trigonometric and hyperbolic functions are not defined for complex
4165 arguments by the C++ standard, however. In GiNaC we follow the
4166 conventions used by CLN, which in turn follow the carefully designed
4167 definitions in the Common Lisp standard. It should be noted that this
4168 convention is identical to the one used by the C99 standard and by most
4169 serious CAS. It is to be expected that future revisions of the C++
4170 standard incorporate these functions in the complex domain in a manner
4171 compatible with C99.
4174 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
4175 @c node-name, next, previous, up
4176 @section Input and output of expressions
4179 @subsection Expression output
4181 @cindex output of expressions
4183 The easiest way to print an expression is to write it to a stream:
4188 ex e = 4.5+pow(x,2)*3/2;
4189 cout << e << endl; // prints '(4.5)+3/2*x^2'
4193 The output format is identical to the @command{ginsh} input syntax and
4194 to that used by most computer algebra systems, but not directly pastable
4195 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4196 is printed as @samp{x^2}).
4198 It is possible to print expressions in a number of different formats with
4202 void ex::print(const print_context & c, unsigned level = 0);
4205 @cindex @code{print_context} (class)
4206 The type of @code{print_context} object passed in determines the format
4207 of the output. The possible types are defined in @file{ginac/print.h}.
4208 All constructors of @code{print_context} and derived classes take an
4209 @code{ostream &} as their first argument.
4211 To print an expression in a way that can be directly used in a C or C++
4212 program, you pass a @code{print_csrc} object like this:
4216 cout << "float f = ";
4217 e.print(print_csrc_float(cout));
4220 cout << "double d = ";
4221 e.print(print_csrc_double(cout));
4224 cout << "cl_N n = ";
4225 e.print(print_csrc_cl_N(cout));
4230 The three possible types mostly affect the way in which floating point
4231 numbers are written.
4233 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4236 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4237 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4238 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4241 The @code{print_context} type @code{print_tree} provides a dump of the
4242 internal structure of an expression for debugging purposes:
4246 e.print(print_tree(cout));
4253 add, hash=0x0, flags=0x3, nops=2
4254 power, hash=0x9, flags=0x3, nops=2
4255 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4256 2 (numeric), hash=0x80000042, flags=0xf
4257 3/2 (numeric), hash=0x80000061, flags=0xf
4260 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4264 This kind of output is also available in @command{ginsh} as the @code{print()}
4267 Another useful output format is for LaTeX parsing in mathematical mode.
4268 It is rather similar to the default @code{print_context} but provides
4269 some braces needed by LaTeX for delimiting boxes and also converts some
4270 common objects to conventional LaTeX names. It is possible to give symbols
4271 a special name for LaTeX output by supplying it as a second argument to
4272 the @code{symbol} constructor.
4274 For example, the code snippet
4279 ex foo = lgamma(x).series(x==0,3);
4280 foo.print(print_latex(std::cout));
4286 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4289 @cindex Tree traversal
4290 If you need any fancy special output format, e.g. for interfacing GiNaC
4291 with other algebra systems or for producing code for different
4292 programming languages, you can always traverse the expression tree yourself:
4295 static void my_print(const ex & e)
4297 if (is_a<function>(e))
4298 cout << ex_to<function>(e).get_name();
4300 cout << e.bp->class_name();
4302 unsigned n = e.nops();
4304 for (unsigned i=0; i<n; i++) @{
4316 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4324 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4325 symbol(y))),numeric(-2)))
4328 If you need an output format that makes it possible to accurately
4329 reconstruct an expression by feeding the output to a suitable parser or
4330 object factory, you should consider storing the expression in an
4331 @code{archive} object and reading the object properties from there.
4332 See the section on archiving for more information.
4335 @subsection Expression input
4336 @cindex input of expressions
4338 GiNaC provides no way to directly read an expression from a stream because
4339 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4340 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4341 @code{y} you defined in your program and there is no way to specify the
4342 desired symbols to the @code{>>} stream input operator.
4344 Instead, GiNaC lets you construct an expression from a string, specifying the
4345 list of symbols and indices to be used:
4349 symbol x("x"), y("y"), p("p");
4350 idx i(symbol("i"), 3);
4351 ex e("2*x+sin(y)+p.i", lst(x, y, p, i));
4355 The input syntax is the same as that used by @command{ginsh} and the stream
4356 output operator @code{<<}. The symbols and indices in the string are matched
4357 by name to the symbols and indices in the list and if GiNaC encounters a
4358 symbol or index not specified in the list it will throw an exception. Only
4359 indices whose values are single symbols can be used (i.e. numeric indices
4360 or compound indices as in "A.(2*n+1)" are not allowed).
4362 With this constructor, it's also easy to implement interactive GiNaC programs:
4367 #include <stdexcept>
4368 #include <ginac/ginac.h>
4369 using namespace std;
4370 using namespace GiNaC;
4377 cout << "Enter an expression containing 'x': ";
4382 cout << "The derivative of " << e << " with respect to x is ";
4383 cout << e.diff(x) << ".\n";
4384 @} catch (exception &p) @{
4385 cerr << p.what() << endl;
4391 @subsection Archiving
4392 @cindex @code{archive} (class)
4395 GiNaC allows creating @dfn{archives} of expressions which can be stored
4396 to or retrieved from files. To create an archive, you declare an object
4397 of class @code{archive} and archive expressions in it, giving each
4398 expression a unique name:
4402 using namespace std;
4403 #include <ginac/ginac.h>
4404 using namespace GiNaC;
4408 symbol x("x"), y("y"), z("z");
4410 ex foo = sin(x + 2*y) + 3*z + 41;
4414 a.archive_ex(foo, "foo");
4415 a.archive_ex(bar, "the second one");
4419 The archive can then be written to a file:
4423 ofstream out("foobar.gar");
4429 The file @file{foobar.gar} contains all information that is needed to
4430 reconstruct the expressions @code{foo} and @code{bar}.
4432 @cindex @command{viewgar}
4433 The tool @command{viewgar} that comes with GiNaC can be used to view
4434 the contents of GiNaC archive files:
4437 $ viewgar foobar.gar
4438 foo = 41+sin(x+2*y)+3*z
4439 the second one = 42+sin(x+2*y)+3*z
4442 The point of writing archive files is of course that they can later be
4448 ifstream in("foobar.gar");
4453 And the stored expressions can be retrieved by their name:
4459 ex ex1 = a2.unarchive_ex(syms, "foo");
4460 ex ex2 = a2.unarchive_ex(syms, "the second one");
4462 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4463 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4464 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4468 Note that you have to supply a list of the symbols which are to be inserted
4469 in the expressions. Symbols in archives are stored by their name only and
4470 if you don't specify which symbols you have, unarchiving the expression will
4471 create new symbols with that name. E.g. if you hadn't included @code{x} in
4472 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4473 have had no effect because the @code{x} in @code{ex1} would have been a
4474 different symbol than the @code{x} which was defined at the beginning of
4475 the program, although both would appear as @samp{x} when printed.
4477 You can also use the information stored in an @code{archive} object to
4478 output expressions in a format suitable for exact reconstruction. The
4479 @code{archive} and @code{archive_node} classes have a couple of member
4480 functions that let you access the stored properties:
4483 static void my_print2(const archive_node & n)
4486 n.find_string("class", class_name);
4487 cout << class_name << "(";
4489 archive_node::propinfovector p;
4490 n.get_properties(p);
4492 unsigned num = p.size();
4493 for (unsigned i=0; i<num; i++) @{
4494 const string &name = p[i].name;
4495 if (name == "class")
4497 cout << name << "=";
4499 unsigned count = p[i].count;
4503 for (unsigned j=0; j<count; j++) @{
4504 switch (p[i].type) @{
4505 case archive_node::PTYPE_BOOL: @{
4507 n.find_bool(name, x, j);
4508 cout << (x ? "true" : "false");
4511 case archive_node::PTYPE_UNSIGNED: @{
4513 n.find_unsigned(name, x, j);
4517 case archive_node::PTYPE_STRING: @{
4519 n.find_string(name, x, j);
4520 cout << '\"' << x << '\"';
4523 case archive_node::PTYPE_NODE: @{
4524 const archive_node &x = n.find_ex_node(name, j);
4546 ex e = pow(2, x) - y;
4548 my_print2(ar.get_top_node(0)); cout << endl;
4556 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4557 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4558 overall_coeff=numeric(number="0"))
4561 Be warned, however, that the set of properties and their meaning for each
4562 class may change between GiNaC versions.
4565 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4566 @c node-name, next, previous, up
4567 @chapter Extending GiNaC
4569 By reading so far you should have gotten a fairly good understanding of
4570 GiNaC's design-patterns. From here on you should start reading the
4571 sources. All we can do now is issue some recommendations how to tackle
4572 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4573 develop some useful extension please don't hesitate to contact the GiNaC
4574 authors---they will happily incorporate them into future versions.
4577 * What does not belong into GiNaC:: What to avoid.
4578 * Symbolic functions:: Implementing symbolic functions.
4579 * Adding classes:: Defining new algebraic classes.
4583 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4584 @c node-name, next, previous, up
4585 @section What doesn't belong into GiNaC
4587 @cindex @command{ginsh}
4588 First of all, GiNaC's name must be read literally. It is designed to be
4589 a library for use within C++. The tiny @command{ginsh} accompanying
4590 GiNaC makes this even more clear: it doesn't even attempt to provide a
4591 language. There are no loops or conditional expressions in
4592 @command{ginsh}, it is merely a window into the library for the
4593 programmer to test stuff (or to show off). Still, the design of a
4594 complete CAS with a language of its own, graphical capabilities and all
4595 this on top of GiNaC is possible and is without doubt a nice project for
4598 There are many built-in functions in GiNaC that do not know how to
4599 evaluate themselves numerically to a precision declared at runtime
4600 (using @code{Digits}). Some may be evaluated at certain points, but not
4601 generally. This ought to be fixed. However, doing numerical
4602 computations with GiNaC's quite abstract classes is doomed to be
4603 inefficient. For this purpose, the underlying foundation classes
4604 provided by CLN are much better suited.
4607 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4608 @c node-name, next, previous, up
4609 @section Symbolic functions
4611 The easiest and most instructive way to start with is probably to
4612 implement your own function. GiNaC's functions are objects of class
4613 @code{function}. The preprocessor is then used to convert the function
4614 names to objects with a corresponding serial number that is used
4615 internally to identify them. You usually need not worry about this
4616 number. New functions may be inserted into the system via a kind of
4617 `registry'. It is your responsibility to care for some functions that
4618 are called when the user invokes certain methods. These are usual
4619 C++-functions accepting a number of @code{ex} as arguments and returning
4620 one @code{ex}. As an example, if we have a look at a simplified
4621 implementation of the cosine trigonometric function, we first need a
4622 function that is called when one wishes to @code{eval} it. It could
4623 look something like this:
4626 static ex cos_eval_method(const ex & x)
4628 // if (!x%(2*Pi)) return 1
4629 // if (!x%Pi) return -1
4630 // if (!x%Pi/2) return 0
4631 // care for other cases...
4632 return cos(x).hold();
4636 @cindex @code{hold()}
4638 The last line returns @code{cos(x)} if we don't know what else to do and
4639 stops a potential recursive evaluation by saying @code{.hold()}, which
4640 sets a flag to the expression signaling that it has been evaluated. We
4641 should also implement a method for numerical evaluation and since we are
4642 lazy we sweep the problem under the rug by calling someone else's
4643 function that does so, in this case the one in class @code{numeric}:
4646 static ex cos_evalf(const ex & x)
4648 if (is_a<numeric>(x))
4649 return cos(ex_to<numeric>(x));
4651 return cos(x).hold();
4655 Differentiation will surely turn up and so we need to tell @code{cos}
4656 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4657 instance are then handled automatically by @code{basic::diff} and
4661 static ex cos_deriv(const ex & x, unsigned diff_param)
4667 @cindex product rule
4668 The second parameter is obligatory but uninteresting at this point. It
4669 specifies which parameter to differentiate in a partial derivative in
4670 case the function has more than one parameter and its main application
4671 is for correct handling of the chain rule. For Taylor expansion, it is
4672 enough to know how to differentiate. But if the function you want to
4673 implement does have a pole somewhere in the complex plane, you need to
4674 write another method for Laurent expansion around that point.
4676 Now that all the ingredients for @code{cos} have been set up, we need
4677 to tell the system about it. This is done by a macro and we are not
4678 going to describe how it expands, please consult your preprocessor if you
4682 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4683 evalf_func(cos_evalf).
4684 derivative_func(cos_deriv));
4687 The first argument is the function's name used for calling it and for
4688 output. The second binds the corresponding methods as options to this
4689 object. Options are separated by a dot and can be given in an arbitrary
4690 order. GiNaC functions understand several more options which are always
4691 specified as @code{.option(params)}, for example a method for series
4692 expansion @code{.series_func(cos_series)}. Again, if no series
4693 expansion method is given, GiNaC defaults to simple Taylor expansion,
4694 which is correct if there are no poles involved as is the case for the
4695 @code{cos} function. The way GiNaC handles poles in case there are any
4696 is best understood by studying one of the examples, like the Gamma
4697 (@code{tgamma}) function for instance. (In essence the function first
4698 checks if there is a pole at the evaluation point and falls back to
4699 Taylor expansion if there isn't. Then, the pole is regularized by some
4700 suitable transformation.) Also, the new function needs to be declared
4701 somewhere. This may also be done by a convenient preprocessor macro:
4704 DECLARE_FUNCTION_1P(cos)
4707 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4708 implementation of @code{cos} is very incomplete and lacks several safety
4709 mechanisms. Please, have a look at the real implementation in GiNaC.
4710 (By the way: in case you are worrying about all the macros above we can
4711 assure you that functions are GiNaC's most macro-intense classes. We
4712 have done our best to avoid macros where we can.)
4715 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4716 @c node-name, next, previous, up
4717 @section Adding classes
4719 If you are doing some very specialized things with GiNaC you may find that
4720 you have to implement your own algebraic classes to fit your needs. This
4721 section will explain how to do this by giving the example of a simple
4722 'string' class. After reading this section you will know how to properly
4723 declare a GiNaC class and what the minimum required member functions are
4724 that you have to implement. We only cover the implementation of a 'leaf'
4725 class here (i.e. one that doesn't contain subexpressions). Creating a
4726 container class like, for example, a class representing tensor products is
4727 more involved but this section should give you enough information so you can
4728 consult the source to GiNaC's predefined classes if you want to implement
4729 something more complicated.
4731 @subsection GiNaC's run-time type information system
4733 @cindex hierarchy of classes
4735 All algebraic classes (that is, all classes that can appear in expressions)
4736 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4737 @code{basic *} (which is essentially what an @code{ex} is) represents a
4738 generic pointer to an algebraic class. Occasionally it is necessary to find
4739 out what the class of an object pointed to by a @code{basic *} really is.
4740 Also, for the unarchiving of expressions it must be possible to find the
4741 @code{unarchive()} function of a class given the class name (as a string). A
4742 system that provides this kind of information is called a run-time type
4743 information (RTTI) system. The C++ language provides such a thing (see the
4744 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4745 implements its own, simpler RTTI.
4747 The RTTI in GiNaC is based on two mechanisms:
4752 The @code{basic} class declares a member variable @code{tinfo_key} which
4753 holds an unsigned integer that identifies the object's class. These numbers
4754 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4755 classes. They all start with @code{TINFO_}.
4758 By means of some clever tricks with static members, GiNaC maintains a list
4759 of information for all classes derived from @code{basic}. The information
4760 available includes the class names, the @code{tinfo_key}s, and pointers
4761 to the unarchiving functions. This class registry is defined in the
4762 @file{registrar.h} header file.
4766 The disadvantage of this proprietary RTTI implementation is that there's
4767 a little more to do when implementing new classes (C++'s RTTI works more
4768 or less automatic) but don't worry, most of the work is simplified by
4771 @subsection A minimalistic example
4773 Now we will start implementing a new class @code{mystring} that allows
4774 placing character strings in algebraic expressions (this is not very useful,
4775 but it's just an example). This class will be a direct subclass of
4776 @code{basic}. You can use this sample implementation as a starting point
4777 for your own classes.
4779 The code snippets given here assume that you have included some header files
4785 #include <stdexcept>
4786 using namespace std;
4788 #include <ginac/ginac.h>
4789 using namespace GiNaC;
4792 The first thing we have to do is to define a @code{tinfo_key} for our new
4793 class. This can be any arbitrary unsigned number that is not already taken
4794 by one of the existing classes but it's better to come up with something
4795 that is unlikely to clash with keys that might be added in the future. The
4796 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4797 which is not a requirement but we are going to stick with this scheme:
4800 const unsigned TINFO_mystring = 0x42420001U;
4803 Now we can write down the class declaration. The class stores a C++
4804 @code{string} and the user shall be able to construct a @code{mystring}
4805 object from a C or C++ string:
4808 class mystring : public basic
4810 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4813 mystring(const string &s);
4814 mystring(const char *s);
4820 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4823 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4824 macros are defined in @file{registrar.h}. They take the name of the class
4825 and its direct superclass as arguments and insert all required declarations
4826 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4827 the first line after the opening brace of the class definition. The
4828 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4829 source (at global scope, of course, not inside a function).
4831 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4832 declarations of the default and copy constructor, the destructor, the
4833 assignment operator and a couple of other functions that are required. It
4834 also defines a type @code{inherited} which refers to the superclass so you
4835 don't have to modify your code every time you shuffle around the class
4836 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4837 constructor, the destructor and the assignment operator.
4839 Now there are nine member functions we have to implement to get a working
4845 @code{mystring()}, the default constructor.
4848 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4849 assignment operator to free dynamically allocated members. The @code{call_parent}
4850 specifies whether the @code{destroy()} function of the superclass is to be
4854 @code{void copy(const mystring &other)}, which is used in the copy constructor
4855 and assignment operator to copy the member variables over from another
4856 object of the same class.
4859 @code{void archive(archive_node &n)}, the archiving function. This stores all
4860 information needed to reconstruct an object of this class inside an
4861 @code{archive_node}.
4864 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4865 constructor. This constructs an instance of the class from the information
4866 found in an @code{archive_node}.
4869 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4870 unarchiving function. It constructs a new instance by calling the unarchiving
4874 @code{int compare_same_type(const basic &other)}, which is used internally
4875 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4876 -1, depending on the relative order of this object and the @code{other}
4877 object. If it returns 0, the objects are considered equal.
4878 @strong{Note:} This has nothing to do with the (numeric) ordering
4879 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4880 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4881 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4882 must provide a @code{compare_same_type()} function, even those representing
4883 objects for which no reasonable algebraic ordering relationship can be
4887 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4888 which are the two constructors we declared.
4892 Let's proceed step-by-step. The default constructor looks like this:
4895 mystring::mystring() : inherited(TINFO_mystring)
4897 // dynamically allocate resources here if required
4901 The golden rule is that in all constructors you have to set the
4902 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4903 it will be set by the constructor of the superclass and all hell will break
4904 loose in the RTTI. For your convenience, the @code{basic} class provides
4905 a constructor that takes a @code{tinfo_key} value, which we are using here
4906 (remember that in our case @code{inherited = basic}). If the superclass
4907 didn't have such a constructor, we would have to set the @code{tinfo_key}
4908 to the right value manually.
4910 In the default constructor you should set all other member variables to
4911 reasonable default values (we don't need that here since our @code{str}
4912 member gets set to an empty string automatically). The constructor(s) are of
4913 course also the right place to allocate any dynamic resources you require.
4915 Next, the @code{destroy()} function:
4918 void mystring::destroy(bool call_parent)
4920 // free dynamically allocated resources here if required
4922 inherited::destroy(call_parent);
4926 This function is where we free all dynamically allocated resources. We
4927 don't have any so we're not doing anything here, but if we had, for
4928 example, used a C-style @code{char *} to store our string, this would be
4929 the place to @code{delete[]} the string storage. If @code{call_parent}
4930 is true, we have to call the @code{destroy()} function of the superclass
4931 after we're done (to mimic C++'s automatic invocation of superclass
4932 destructors where @code{destroy()} is called from outside a destructor).
4934 The @code{copy()} function just copies over the member variables from
4938 void mystring::copy(const mystring &other)
4940 inherited::copy(other);
4945 We can simply overwrite the member variables here. There's no need to worry
4946 about dynamically allocated storage. The assignment operator (which is
4947 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4948 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4949 explicitly call the @code{copy()} function of the superclass here so
4950 all the member variables will get copied.
4952 Next are the three functions for archiving. You have to implement them even
4953 if you don't plan to use archives, but the minimum required implementation
4954 is really simple. First, the archiving function:
4957 void mystring::archive(archive_node &n) const
4959 inherited::archive(n);
4960 n.add_string("string", str);
4964 The only thing that is really required is calling the @code{archive()}
4965 function of the superclass. Optionally, you can store all information you
4966 deem necessary for representing the object into the passed
4967 @code{archive_node}. We are just storing our string here. For more
4968 information on how the archiving works, consult the @file{archive.h} header
4971 The unarchiving constructor is basically the inverse of the archiving
4975 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4977 n.find_string("string", str);
4981 If you don't need archiving, just leave this function empty (but you must
4982 invoke the unarchiving constructor of the superclass). Note that we don't
4983 have to set the @code{tinfo_key} here because it is done automatically
4984 by the unarchiving constructor of the @code{basic} class.
4986 Finally, the unarchiving function:
4989 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4991 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4995 You don't have to understand how exactly this works. Just copy these
4996 four lines into your code literally (replacing the class name, of
4997 course). It calls the unarchiving constructor of the class and unless
4998 you are doing something very special (like matching @code{archive_node}s
4999 to global objects) you don't need a different implementation. For those
5000 who are interested: setting the @code{dynallocated} flag puts the object
5001 under the control of GiNaC's garbage collection. It will get deleted
5002 automatically once it is no longer referenced.
5004 Our @code{compare_same_type()} function uses a provided function to compare
5008 int mystring::compare_same_type(const basic &other) const
5010 const mystring &o = static_cast<const mystring &>(other);
5011 int cmpval = str.compare(o.str);
5014 else if (cmpval < 0)
5021 Although this function takes a @code{basic &}, it will always be a reference
5022 to an object of exactly the same class (objects of different classes are not
5023 comparable), so the cast is safe. If this function returns 0, the two objects
5024 are considered equal (in the sense that @math{A-B=0}), so you should compare
5025 all relevant member variables.
5027 Now the only thing missing is our two new constructors:
5030 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
5032 // dynamically allocate resources here if required
5035 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
5037 // dynamically allocate resources here if required
5041 No surprises here. We set the @code{str} member from the argument and
5042 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
5044 That's it! We now have a minimal working GiNaC class that can store
5045 strings in algebraic expressions. Let's confirm that the RTTI works:
5048 ex e = mystring("Hello, world!");
5049 cout << is_a<mystring>(e) << endl;
5052 cout << e.bp->class_name() << endl;
5056 Obviously it does. Let's see what the expression @code{e} looks like:
5060 // -> [mystring object]
5063 Hm, not exactly what we expect, but of course the @code{mystring} class
5064 doesn't yet know how to print itself. This is done in the @code{print()}
5065 member function. Let's say that we wanted to print the string surrounded
5069 class mystring : public basic
5073 void print(const print_context &c, unsigned level = 0) const;
5077 void mystring::print(const print_context &c, unsigned level) const
5079 // print_context::s is a reference to an ostream
5080 c.s << '\"' << str << '\"';
5084 The @code{level} argument is only required for container classes to
5085 correctly parenthesize the output. Let's try again to print the expression:
5089 // -> "Hello, world!"
5092 Much better. The @code{mystring} class can be used in arbitrary expressions:
5095 e += mystring("GiNaC rulez");
5097 // -> "GiNaC rulez"+"Hello, world!"
5100 (GiNaC's automatic term reordering is in effect here), or even
5103 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
5105 // -> "One string"^(2*sin(-"Another string"+Pi))
5108 Whether this makes sense is debatable but remember that this is only an
5109 example. At least it allows you to implement your own symbolic algorithms
5112 Note that GiNaC's algebraic rules remain unchanged:
5115 e = mystring("Wow") * mystring("Wow");
5119 e = pow(mystring("First")-mystring("Second"), 2);
5120 cout << e.expand() << endl;
5121 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5124 There's no way to, for example, make GiNaC's @code{add} class perform string
5125 concatenation. You would have to implement this yourself.
5127 @subsection Automatic evaluation
5129 @cindex @code{hold()}
5130 @cindex @code{eval()}
5132 When dealing with objects that are just a little more complicated than the
5133 simple string objects we have implemented, chances are that you will want to
5134 have some automatic simplifications or canonicalizations performed on them.
5135 This is done in the evaluation member function @code{eval()}. Let's say that
5136 we wanted all strings automatically converted to lowercase with
5137 non-alphabetic characters stripped, and empty strings removed:
5140 class mystring : public basic
5144 ex eval(int level = 0) const;
5148 ex mystring::eval(int level) const
5151 for (int i=0; i<str.length(); i++) @{
5153 if (c >= 'A' && c <= 'Z')
5154 new_str += tolower(c);
5155 else if (c >= 'a' && c <= 'z')
5159 if (new_str.length() == 0)
5162 return mystring(new_str).hold();
5166 The @code{level} argument is used to limit the recursion depth of the
5167 evaluation. We don't have any subexpressions in the @code{mystring}
5168 class so we are not concerned with this. If we had, we would call the
5169 @code{eval()} functions of the subexpressions with @code{level - 1} as
5170 the argument if @code{level != 1}. The @code{hold()} member function
5171 sets a flag in the object that prevents further evaluation. Otherwise
5172 we might end up in an endless loop. When you want to return the object
5173 unmodified, use @code{return this->hold();}.
5175 Let's confirm that it works:
5178 ex e = mystring("Hello, world!") + mystring("!?#");
5182 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5187 @subsection Other member functions
5189 We have implemented only a small set of member functions to make the class
5190 work in the GiNaC framework. For a real algebraic class, there are probably
5191 some more functions that you will want to re-implement, such as
5192 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5193 or the header file of the class you want to make a subclass of to see
5194 what's there. One member function that you will most likely want to
5195 implement for terminal classes like the described string class is
5196 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5197 which will allow GiNaC to compare and canonicalize expressions much more
5200 You can, of course, also add your own new member functions. Remember,
5201 that the RTTI may be used to get information about what kinds of objects
5202 you are dealing with (the position in the class hierarchy) and that you
5203 can always extract the bare object from an @code{ex} by stripping the
5204 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5205 should become a need.
5207 That's it. May the source be with you!
5210 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5211 @c node-name, next, previous, up
5212 @chapter A Comparison With Other CAS
5215 This chapter will give you some information on how GiNaC compares to
5216 other, traditional Computer Algebra Systems, like @emph{Maple},
5217 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5218 disadvantages over these systems.
5221 * Advantages:: Strengths of the GiNaC approach.
5222 * Disadvantages:: Weaknesses of the GiNaC approach.
5223 * Why C++?:: Attractiveness of C++.
5226 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5227 @c node-name, next, previous, up
5230 GiNaC has several advantages over traditional Computer
5231 Algebra Systems, like
5236 familiar language: all common CAS implement their own proprietary
5237 grammar which you have to learn first (and maybe learn again when your
5238 vendor decides to `enhance' it). With GiNaC you can write your program
5239 in common C++, which is standardized.
5243 structured data types: you can build up structured data types using
5244 @code{struct}s or @code{class}es together with STL features instead of
5245 using unnamed lists of lists of lists.
5248 strongly typed: in CAS, you usually have only one kind of variables
5249 which can hold contents of an arbitrary type. This 4GL like feature is
5250 nice for novice programmers, but dangerous.
5253 development tools: powerful development tools exist for C++, like fancy
5254 editors (e.g. with automatic indentation and syntax highlighting),
5255 debuggers, visualization tools, documentation generators@dots{}
5258 modularization: C++ programs can easily be split into modules by
5259 separating interface and implementation.
5262 price: GiNaC is distributed under the GNU Public License which means
5263 that it is free and available with source code. And there are excellent
5264 C++-compilers for free, too.
5267 extendable: you can add your own classes to GiNaC, thus extending it on
5268 a very low level. Compare this to a traditional CAS that you can
5269 usually only extend on a high level by writing in the language defined
5270 by the parser. In particular, it turns out to be almost impossible to
5271 fix bugs in a traditional system.
5274 multiple interfaces: Though real GiNaC programs have to be written in
5275 some editor, then be compiled, linked and executed, there are more ways
5276 to work with the GiNaC engine. Many people want to play with
5277 expressions interactively, as in traditional CASs. Currently, two such
5278 windows into GiNaC have been implemented and many more are possible: the
5279 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5280 types to a command line and second, as a more consistent approach, an
5281 interactive interface to the Cint C++ interpreter has been put together
5282 (called GiNaC-cint) that allows an interactive scripting interface
5283 consistent with the C++ language. It is available from the usual GiNaC
5287 seamless integration: it is somewhere between difficult and impossible
5288 to call CAS functions from within a program written in C++ or any other
5289 programming language and vice versa. With GiNaC, your symbolic routines
5290 are part of your program. You can easily call third party libraries,
5291 e.g. for numerical evaluation or graphical interaction. All other
5292 approaches are much more cumbersome: they range from simply ignoring the
5293 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5294 system (i.e. @emph{Yacas}).
5297 efficiency: often large parts of a program do not need symbolic
5298 calculations at all. Why use large integers for loop variables or
5299 arbitrary precision arithmetics where @code{int} and @code{double} are
5300 sufficient? For pure symbolic applications, GiNaC is comparable in
5301 speed with other CAS.
5306 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5307 @c node-name, next, previous, up
5308 @section Disadvantages
5310 Of course it also has some disadvantages:
5315 advanced features: GiNaC cannot compete with a program like
5316 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5317 which grows since 1981 by the work of dozens of programmers, with
5318 respect to mathematical features. Integration, factorization,
5319 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5320 not planned for the near future).
5323 portability: While the GiNaC library itself is designed to avoid any
5324 platform dependent features (it should compile on any ANSI compliant C++
5325 compiler), the currently used version of the CLN library (fast large
5326 integer and arbitrary precision arithmetics) can only by compiled
5327 without hassle on systems with the C++ compiler from the GNU Compiler
5328 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
5329 macros to let the compiler gather all static initializations, which
5330 works for GNU C++ only. Feel free to contact the authors in case you
5331 really believe that you need to use a different compiler. We have
5332 occasionally used other compilers and may be able to give you advice.}
5333 GiNaC uses recent language features like explicit constructors, mutable
5334 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
5335 literally. Recent GCC versions starting at 2.95.3, although itself not
5336 yet ANSI compliant, support all needed features.
5341 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5342 @c node-name, next, previous, up
5345 Why did we choose to implement GiNaC in C++ instead of Java or any other
5346 language? C++ is not perfect: type checking is not strict (casting is
5347 possible), separation between interface and implementation is not
5348 complete, object oriented design is not enforced. The main reason is
5349 the often scolded feature of operator overloading in C++. While it may
5350 be true that operating on classes with a @code{+} operator is rarely
5351 meaningful, it is perfectly suited for algebraic expressions. Writing
5352 @math{3x+5y} as @code{3*x+5*y} instead of
5353 @code{x.times(3).plus(y.times(5))} looks much more natural.
5354 Furthermore, the main developers are more familiar with C++ than with
5355 any other programming language.
5358 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5359 @c node-name, next, previous, up
5360 @appendix Internal Structures
5363 * Expressions are reference counted::
5364 * Internal representation of products and sums::
5367 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5368 @c node-name, next, previous, up
5369 @appendixsection Expressions are reference counted
5371 @cindex reference counting
5372 @cindex copy-on-write
5373 @cindex garbage collection
5374 An expression is extremely light-weight since internally it works like a
5375 handle to the actual representation and really holds nothing more than a
5376 pointer to some other object. What this means in practice is that
5377 whenever you create two @code{ex} and set the second equal to the first
5378 no copying process is involved. Instead, the copying takes place as soon
5379 as you try to change the second. Consider the simple sequence of code:
5383 #include <ginac/ginac.h>
5384 using namespace std;
5385 using namespace GiNaC;
5389 symbol x("x"), y("y"), z("z");
5392 e1 = sin(x + 2*y) + 3*z + 41;
5393 e2 = e1; // e2 points to same object as e1
5394 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5395 e2 += 1; // e2 is copied into a new object
5396 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5400 The line @code{e2 = e1;} creates a second expression pointing to the
5401 object held already by @code{e1}. The time involved for this operation
5402 is therefore constant, no matter how large @code{e1} was. Actual
5403 copying, however, must take place in the line @code{e2 += 1;} because
5404 @code{e1} and @code{e2} are not handles for the same object any more.
5405 This concept is called @dfn{copy-on-write semantics}. It increases
5406 performance considerably whenever one object occurs multiple times and
5407 represents a simple garbage collection scheme because when an @code{ex}
5408 runs out of scope its destructor checks whether other expressions handle
5409 the object it points to too and deletes the object from memory if that
5410 turns out not to be the case. A slightly less trivial example of
5411 differentiation using the chain-rule should make clear how powerful this
5416 symbol x("x"), y("y");
5420 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5421 cout << e1 << endl // prints x+3*y
5422 << e2 << endl // prints (x+3*y)^3
5423 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5427 Here, @code{e1} will actually be referenced three times while @code{e2}
5428 will be referenced two times. When the power of an expression is built,
5429 that expression needs not be copied. Likewise, since the derivative of
5430 a power of an expression can be easily expressed in terms of that
5431 expression, no copying of @code{e1} is involved when @code{e3} is
5432 constructed. So, when @code{e3} is constructed it will print as
5433 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5434 holds a reference to @code{e2} and the factor in front is just
5437 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5438 semantics. When you insert an expression into a second expression, the
5439 result behaves exactly as if the contents of the first expression were
5440 inserted. But it may be useful to remember that this is not what
5441 happens. Knowing this will enable you to write much more efficient
5442 code. If you still have an uncertain feeling with copy-on-write
5443 semantics, we recommend you have a look at the
5444 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5445 Marshall Cline. Chapter 16 covers this issue and presents an
5446 implementation which is pretty close to the one in GiNaC.
5449 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5450 @c node-name, next, previous, up
5451 @appendixsection Internal representation of products and sums
5453 @cindex representation
5456 @cindex @code{power}
5457 Although it should be completely transparent for the user of
5458 GiNaC a short discussion of this topic helps to understand the sources
5459 and also explain performance to a large degree. Consider the
5460 unexpanded symbolic expression
5462 $2d^3 \left( 4a + 5b - 3 \right)$
5465 @math{2*d^3*(4*a+5*b-3)}
5467 which could naively be represented by a tree of linear containers for
5468 addition and multiplication, one container for exponentiation with base
5469 and exponent and some atomic leaves of symbols and numbers in this
5474 @cindex pair-wise representation
5475 However, doing so results in a rather deeply nested tree which will
5476 quickly become inefficient to manipulate. We can improve on this by
5477 representing the sum as a sequence of terms, each one being a pair of a
5478 purely numeric multiplicative coefficient and its rest. In the same
5479 spirit we can store the multiplication as a sequence of terms, each
5480 having a numeric exponent and a possibly complicated base, the tree
5481 becomes much more flat:
5485 The number @code{3} above the symbol @code{d} shows that @code{mul}
5486 objects are treated similarly where the coefficients are interpreted as
5487 @emph{exponents} now. Addition of sums of terms or multiplication of
5488 products with numerical exponents can be coded to be very efficient with
5489 such a pair-wise representation. Internally, this handling is performed
5490 by most CAS in this way. It typically speeds up manipulations by an
5491 order of magnitude. The overall multiplicative factor @code{2} and the
5492 additive term @code{-3} look somewhat out of place in this
5493 representation, however, since they are still carrying a trivial
5494 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5495 this is avoided by adding a field that carries an overall numeric
5496 coefficient. This results in the realistic picture of internal
5499 $2d^3 \left( 4a + 5b - 3 \right)$:
5502 @math{2*d^3*(4*a+5*b-3)}:
5508 This also allows for a better handling of numeric radicals, since
5509 @code{sqrt(2)} can now be carried along calculations. Now it should be
5510 clear, why both classes @code{add} and @code{mul} are derived from the
5511 same abstract class: the data representation is the same, only the
5512 semantics differs. In the class hierarchy, methods for polynomial
5513 expansion and the like are reimplemented for @code{add} and @code{mul},
5514 but the data structure is inherited from @code{expairseq}.
5517 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5518 @c node-name, next, previous, up
5519 @appendix Package Tools
5521 If you are creating a software package that uses the GiNaC library,
5522 setting the correct command line options for the compiler and linker
5523 can be difficult. GiNaC includes two tools to make this process easier.
5526 * ginac-config:: A shell script to detect compiler and linker flags.
5527 * AM_PATH_GINAC:: Macro for GNU automake.
5531 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5532 @c node-name, next, previous, up
5533 @section @command{ginac-config}
5534 @cindex ginac-config
5536 @command{ginac-config} is a shell script that you can use to determine
5537 the compiler and linker command line options required to compile and
5538 link a program with the GiNaC library.
5540 @command{ginac-config} takes the following flags:
5544 Prints out the version of GiNaC installed.
5546 Prints '-I' flags pointing to the installed header files.
5548 Prints out the linker flags necessary to link a program against GiNaC.
5549 @item --prefix[=@var{PREFIX}]
5550 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5551 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5552 Otherwise, prints out the configured value of @env{$prefix}.
5553 @item --exec-prefix[=@var{PREFIX}]
5554 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5555 Otherwise, prints out the configured value of @env{$exec_prefix}.
5558 Typically, @command{ginac-config} will be used within a configure
5559 script, as described below. It, however, can also be used directly from
5560 the command line using backquotes to compile a simple program. For
5564 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5567 This command line might expand to (for example):
5570 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5571 -lginac -lcln -lstdc++
5574 Not only is the form using @command{ginac-config} easier to type, it will
5575 work on any system, no matter how GiNaC was configured.
5578 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5579 @c node-name, next, previous, up
5580 @section @samp{AM_PATH_GINAC}
5581 @cindex AM_PATH_GINAC
5583 For packages configured using GNU automake, GiNaC also provides
5584 a macro to automate the process of checking for GiNaC.
5587 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5595 Determines the location of GiNaC using @command{ginac-config}, which is
5596 either found in the user's path, or from the environment variable
5597 @env{GINACLIB_CONFIG}.
5600 Tests the installed libraries to make sure that their version
5601 is later than @var{MINIMUM-VERSION}. (A default version will be used
5605 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5606 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5607 variable to the output of @command{ginac-config --libs}, and calls
5608 @samp{AC_SUBST()} for these variables so they can be used in generated
5609 makefiles, and then executes @var{ACTION-IF-FOUND}.
5612 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5613 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5617 This macro is in file @file{ginac.m4} which is installed in
5618 @file{$datadir/aclocal}. Note that if automake was installed with a
5619 different @samp{--prefix} than GiNaC, you will either have to manually
5620 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5621 aclocal the @samp{-I} option when running it.
5624 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5625 * Example package:: Example of a package using AM_PATH_GINAC.
5629 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5630 @c node-name, next, previous, up
5631 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5633 Simply make sure that @command{ginac-config} is in your path, and run
5634 the configure script.
5641 The directory where the GiNaC libraries are installed needs
5642 to be found by your system's dynamic linker.
5644 This is generally done by
5647 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5653 setting the environment variable @env{LD_LIBRARY_PATH},
5656 or, as a last resort,
5659 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5660 running configure, for instance:
5663 LDFLAGS=-R/home/cbauer/lib ./configure
5668 You can also specify a @command{ginac-config} not in your path by
5669 setting the @env{GINACLIB_CONFIG} environment variable to the
5670 name of the executable
5673 If you move the GiNaC package from its installed location,
5674 you will either need to modify @command{ginac-config} script
5675 manually to point to the new location or rebuild GiNaC.
5686 --with-ginac-prefix=@var{PREFIX}
5687 --with-ginac-exec-prefix=@var{PREFIX}
5690 are provided to override the prefix and exec-prefix that were stored
5691 in the @command{ginac-config} shell script by GiNaC's configure. You are
5692 generally better off configuring GiNaC with the right path to begin with.
5696 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5697 @c node-name, next, previous, up
5698 @subsection Example of a package using @samp{AM_PATH_GINAC}
5700 The following shows how to build a simple package using automake
5701 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5704 #include <ginac/ginac.h>
5708 GiNaC::symbol x("x");
5709 GiNaC::ex a = GiNaC::sin(x);
5710 std::cout << "Derivative of " << a
5711 << " is " << a.diff(x) << std::endl;
5716 You should first read the introductory portions of the automake
5717 Manual, if you are not already familiar with it.
5719 Two files are needed, @file{configure.in}, which is used to build the
5723 dnl Process this file with autoconf to produce a configure script.
5725 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5731 AM_PATH_GINAC(0.9.0, [
5732 LIBS="$LIBS $GINACLIB_LIBS"
5733 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5734 ], AC_MSG_ERROR([need to have GiNaC installed]))
5739 The only command in this which is not standard for automake
5740 is the @samp{AM_PATH_GINAC} macro.
5742 That command does the following: If a GiNaC version greater or equal
5743 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5744 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5745 the error message `need to have GiNaC installed'
5747 And the @file{Makefile.am}, which will be used to build the Makefile.
5750 ## Process this file with automake to produce Makefile.in
5751 bin_PROGRAMS = simple
5752 simple_SOURCES = simple.cpp
5755 This @file{Makefile.am}, says that we are building a single executable,
5756 from a single source file @file{simple.cpp}. Since every program
5757 we are building uses GiNaC we simply added the GiNaC options
5758 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5759 want to specify them on a per-program basis: for instance by
5763 simple_LDADD = $(GINACLIB_LIBS)
5764 INCLUDES = $(GINACLIB_CPPFLAGS)
5767 to the @file{Makefile.am}.
5769 To try this example out, create a new directory and add the three
5772 Now execute the following commands:
5775 $ automake --add-missing
5780 You now have a package that can be built in the normal fashion
5789 @node Bibliography, Concept Index, Example package, Top
5790 @c node-name, next, previous, up
5791 @appendix Bibliography
5796 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5799 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5802 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5805 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5808 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5809 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5812 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5813 James H. Davenport, Yvon Siret, and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
5814 Academic Press, London
5817 @cite{Computer Algebra Systems - A Practical Guide},
5818 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
5821 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5822 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5825 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
5830 @node Concept Index, , Bibliography, Top
5831 @c node-name, next, previous, up
5832 @unnumbered Concept Index