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 composits 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 refered 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 differnt 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 @cindex @code{match()}
3037 The most basic application of patterns is to check whether an expression
3038 matches a given pattern. This is done by the function
3041 bool ex::match(const ex & pattern);
3042 bool ex::match(const ex & pattern, lst & repls);
3045 This function returns @code{true} when the expression matches the pattern
3046 and @code{false} if it doesn't. If used in the second form, the actual
3047 subexpressions matched by the wildcards get returned in the @code{repls}
3048 object as a list of relations of the form @samp{wildcard == expression}.
3049 If @code{match()} returns false, the state of @code{repls} is undefined.
3050 For reproducible results, the list should be empty when passed to
3051 @code{match()}, but it is also possible to find similarities in multiple
3052 expressions by passing in the result of a previous match.
3054 The matching algorithm works as follows:
3057 @item A single wildcard matches any expression. If one wildcard appears
3058 multiple times in a pattern, it must match the same expression in all
3059 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3060 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3061 @item If the expression is not of the same class as the pattern, the match
3062 fails (i.e. a sum only matches a sum, a function only matches a function,
3064 @item If the pattern is a function, it only matches the same function
3065 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3066 @item Except for sums and products, the match fails if the number of
3067 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3069 @item If there are no subexpressions, the expressions and the pattern must
3070 be equal (in the sense of @code{is_equal()}).
3071 @item Except for sums and products, each subexpression (@code{op()}) must
3072 match the corresponding subexpression of the pattern.
3075 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3076 account for their commutativity and associativity:
3079 @item If the pattern contains a term or factor that is a single wildcard,
3080 this one is used as the @dfn{global wildcard}. If there is more than one
3081 such wildcard, one of them is chosen as the global wildcard in a random
3083 @item Every term/factor of the pattern, except the global wildcard, is
3084 matched against every term of the expression in sequence. If no match is
3085 found, the whole match fails. Terms that did match are not considered in
3087 @item If there are no unmatched terms left, the match succeeds. Otherwise
3088 the match fails unless there is a global wildcard in the pattern, in
3089 which case this wildcard matches the remaining terms.
3092 In general, having more than one single wildcard as a term of a sum or a
3093 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3096 Here are some examples in @command{ginsh} to demonstrate how it works (the
3097 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3098 match fails, and the list of wildcard replacements otherwise):
3101 > match((x+y)^a,(x+y)^a);
3103 > match((x+y)^a,(x+y)^b);
3105 > match((x+y)^a,$1^$2);
3107 > match((x+y)^a,$1^$1);
3109 > match((x+y)^(x+y),$1^$1);
3111 > match((x+y)^(x+y),$1^$2);
3113 > match((a+b)*(a+c),($1+b)*($1+c));
3115 > match((a+b)*(a+c),(a+$1)*(a+$2));
3117 (Unpredictable. The result might also be [$1==c,$2==b].)
3118 > match((a+b)*(a+c),($1+$2)*($1+$3));
3119 (The result is undefined. Due to the sequential nature of the algorithm
3120 and the re-ordering of terms in GiNaC, the match for the first factor
3121 may be @{$1==a,$2==b@} in which case the match for the second factor
3122 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3124 > match(a*(x+y)+a*z+b,a*$1+$2);
3125 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3126 @{$1=x+y,$2=a*z+b@}.)
3127 > match(a+b+c+d+e+f,c);
3129 > match(a+b+c+d+e+f,c+$0);
3131 > match(a+b+c+d+e+f,c+e+$0);
3133 > match(a+b,a+b+$0);
3135 > match(a*b^2,a^$1*b^$2);
3137 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3138 even though a==a^1.)
3139 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3141 > match(atan2(y,x^2),atan2(y,$0));
3145 @cindex @code{has()}
3146 A more general way to look for patterns in expressions is provided by the
3150 bool ex::has(const ex & pattern);
3153 This function checks whether a pattern is matched by an expression itself or
3154 by any of its subexpressions.
3156 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3157 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3160 > has(x*sin(x+y+2*a),y);
3162 > has(x*sin(x+y+2*a),x+y);
3164 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3165 has the subexpressions "x", "y" and "2*a".)
3166 > has(x*sin(x+y+2*a),x+y+$1);
3168 (But this is possible.)
3169 > has(x*sin(2*(x+y)+2*a),x+y);
3171 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3172 which "x+y" is not a subexpression.)
3175 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3177 > has(4*x^2-x+3,$1*x);
3179 > has(4*x^2+x+3,$1*x);
3181 (Another possible pitfall. The first expression matches because the term
3182 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3183 contains a linear term you should use the coeff() function instead.)
3186 @cindex @code{find()}
3190 bool ex::find(const ex & pattern, lst & found);
3193 works a bit like @code{has()} but it doesn't stop upon finding the first
3194 match. Instead, it appends all found matches to the specified list. If there
3195 are multiple occurrences of the same expression, it is entered only once to
3196 the list. @code{find()} returns false if no matches were found (in
3197 @command{ginsh}, it returns an empty list):
3200 > find(1+x+x^2+x^3,x);
3202 > find(1+x+x^2+x^3,y);
3204 > find(1+x+x^2+x^3,x^$1);
3206 (Note the absence of "x".)
3207 > expand((sin(x)+sin(y))*(a+b));
3208 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3213 @cindex @code{subs()}
3214 Probably the most useful application of patterns is to use them for
3215 substituting expressions with the @code{subs()} method. Wildcards can be
3216 used in the search patterns as well as in the replacement expressions, where
3217 they get replaced by the expressions matched by them. @code{subs()} doesn't
3218 know anything about algebra; it performs purely syntactic substitutions.
3223 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3225 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3227 > subs((a+b+c)^2,a+b==x);
3229 > subs((a+b+c)^2,a+b+$1==x+$1);
3231 > subs(a+2*b,a+b==x);
3233 > subs(4*x^3-2*x^2+5*x-1,x==a);
3235 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3237 > subs(sin(1+sin(x)),sin($1)==cos($1));
3239 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3243 The last example would be written in C++ in this way:
3247 symbol a("a"), b("b"), x("x"), y("y");
3248 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3249 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3250 cout << e.expand() << endl;
3256 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3257 @c node-name, next, previous, up
3258 @section Applying a Function on Subexpressions
3259 @cindex Tree traversal
3260 @cindex @code{map()}
3262 Sometimes you may want to perform an operation on specific parts of an
3263 expression while leaving the general structure of it intact. An example
3264 of this would be a matrix trace operation: the trace of a sum is the sum
3265 of the traces of the individual terms. That is, the trace should @dfn{map}
3266 on the sum, by applying itself to each of the sum's operands. It is possible
3267 to do this manually which usually results in code like this:
3272 if (is_a<matrix>(e))
3273 return ex_to<matrix>(e).trace();
3274 else if (is_a<add>(e)) @{
3276 for (unsigned i=0; i<e.nops(); i++)
3277 sum += calc_trace(e.op(i));
3279 @} else if (is_a<mul>)(e)) @{
3287 This is, however, slightly inefficient (if the sum is very large it can take
3288 a long time to add the terms one-by-one), and its applicability is limited to
3289 a rather small class of expressions. If @code{calc_trace()} is called with
3290 a relation or a list as its argument, you will probably want the trace to
3291 be taken on both sides of the relation or of all elements of the list.
3293 GiNaC offers the @code{map()} method to aid in the implementation of such
3297 ex ex::map(map_function & f) const;
3298 ex ex::map(ex (*f)(const ex & e)) const;
3301 In the first (preferred) form, @code{map()} takes a function object that
3302 is subclassed from the @code{map_function} class. In the second form, it
3303 takes a pointer to a function that accepts and returns an expression.
3304 @code{map()} constructs a new expression of the same type, applying the
3305 specified function on all subexpressions (in the sense of @code{op()}),
3308 The use of a function object makes it possible to supply more arguments to
3309 the function that is being mapped, or to keep local state information.
3310 The @code{map_function} class declares a virtual function call operator
3311 that you can overload. Here is a sample implementation of @code{calc_trace()}
3312 that uses @code{map()} in a recursive fashion:
3315 struct calc_trace : public map_function @{
3316 ex operator()(const ex &e)
3318 if (is_a<matrix>(e))
3319 return ex_to<matrix>(e).trace();
3320 else if (is_a<mul>(e)) @{
3323 return e.map(*this);
3328 This function object could then be used like this:
3332 ex M = ... // expression with matrices
3333 calc_trace do_trace;
3334 ex tr = do_trace(M);
3338 Here is another example for you to meditate over. It removes quadratic
3339 terms in a variable from an expanded polynomial:
3342 struct map_rem_quad : public map_function @{
3344 map_rem_quad(const ex & var_) : var(var_) @{@}
3346 ex operator()(const ex & e)
3348 if (is_a<add>(e) || is_a<mul>(e))
3349 return e.map(*this);
3350 else if (is_a<power>(e) &&
3351 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3361 symbol x("x"), y("y");
3364 for (int i=0; i<8; i++)
3365 e += pow(x, i) * pow(y, 8-i) * (i+1);
3367 // -> 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
3369 map_rem_quad rem_quad(x);
3370 cout << rem_quad(e) << endl;
3371 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3375 @command{ginsh} offers a slightly different implementation of @code{map()}
3376 that allows applying algebraic functions to operands. The second argument
3377 to @code{map()} is an expression containing the wildcard @samp{$0} which
3378 acts as the placeholder for the operands:
3383 > map(a+2*b,sin($0));
3385 > map(@{a,b,c@},$0^2+$0);
3386 @{a^2+a,b^2+b,c^2+c@}
3389 Note that it is only possible to use algebraic functions in the second
3390 argument. You can not use functions like @samp{diff()}, @samp{op()},
3391 @samp{subs()} etc. because these are evaluated immediately:
3394 > map(@{a,b,c@},diff($0,a));
3396 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3397 to "map(@{a,b,c@},0)".
3401 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3402 @c node-name, next, previous, up
3403 @section Polynomial arithmetic
3405 @subsection Expanding and collecting
3406 @cindex @code{expand()}
3407 @cindex @code{collect()}
3408 @cindex @code{collect_common_factors()}
3410 A polynomial in one or more variables has many equivalent
3411 representations. Some useful ones serve a specific purpose. Consider
3412 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3413 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3414 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3415 representations are the recursive ones where one collects for exponents
3416 in one of the three variable. Since the factors are themselves
3417 polynomials in the remaining two variables the procedure can be
3418 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3419 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3422 To bring an expression into expanded form, its method
3428 may be called. In our example above, this corresponds to @math{4*x*y +
3429 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3430 GiNaC is not easily guessable you should be prepared to see different
3431 orderings of terms in such sums!
3433 Another useful representation of multivariate polynomials is as a
3434 univariate polynomial in one of the variables with the coefficients
3435 being polynomials in the remaining variables. The method
3436 @code{collect()} accomplishes this task:
3439 ex ex::collect(const ex & s, bool distributed = false);
3442 The first argument to @code{collect()} can also be a list of objects in which
3443 case the result is either a recursively collected polynomial, or a polynomial
3444 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3445 by the @code{distributed} flag.
3447 Note that the original polynomial needs to be in expanded form (for the
3448 variables concerned) in order for @code{collect()} to be able to find the
3449 coefficients properly.
3451 The following @command{ginsh} transcript shows an application of @code{collect()}
3452 together with @code{find()}:
3455 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3456 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
3457 > collect(a,@{p,q@});
3458 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)
3459 > collect(a,find(a,sin($1)));
3460 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3461 > collect(a,@{find(a,sin($1)),p,q@});
3462 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3463 > collect(a,@{find(a,sin($1)),d@});
3464 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3467 Polynomials can often be brought into a more compact form by collecting
3468 common factors from the terms of sums. This is accomplished by the function
3471 ex collect_common_factors(const ex & e);
3474 This function doesn't perform a full factorization but only looks for
3475 factors which are already explicitly present:
3478 > collect_common_factors(a*x+a*y);
3480 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
3482 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
3483 (c+a)*a*(x*y+y^2+x)*b
3486 @subsection Degree and coefficients
3487 @cindex @code{degree()}
3488 @cindex @code{ldegree()}
3489 @cindex @code{coeff()}
3491 The degree and low degree of a polynomial can be obtained using the two
3495 int ex::degree(const ex & s);
3496 int ex::ldegree(const ex & s);
3499 These functions only work reliably if the input polynomial is collected in
3500 terms of the object @samp{s}. Otherwise, they are only guaranteed to return
3501 the upper/lower bounds of the exponents. If you need accurate results, you
3502 have to call @code{expand()} and/or @code{collect()} on the input polynomial.
3510 > degree(expand(a),x);
3514 @code{degree()} also works on rational functions, returning the asymptotic
3518 > degree((x+1)/(x^3+1),x);
3522 If the input is not a polynomial or rational function in the variable @samp{s},
3523 the behavior of @code{degree()} and @code{ldegree()} is undefined.
3525 To extract a coefficient with a certain power from an expanded
3529 ex ex::coeff(const ex & s, int n);
3532 You can also obtain the leading and trailing coefficients with the methods
3535 ex ex::lcoeff(const ex & s);
3536 ex ex::tcoeff(const ex & s);
3539 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3542 An application is illustrated in the next example, where a multivariate
3543 polynomial is analyzed:
3547 symbol x("x"), y("y");
3548 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3549 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3550 ex Poly = PolyInp.expand();
3552 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3553 cout << "The x^" << i << "-coefficient is "
3554 << Poly.coeff(x,i) << endl;
3556 cout << "As polynomial in y: "
3557 << Poly.collect(y) << endl;
3561 When run, it returns an output in the following fashion:
3564 The x^0-coefficient is y^2+11*y
3565 The x^1-coefficient is 5*y^2-2*y
3566 The x^2-coefficient is -1
3567 The x^3-coefficient is 4*y
3568 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3571 As always, the exact output may vary between different versions of GiNaC
3572 or even from run to run since the internal canonical ordering is not
3573 within the user's sphere of influence.
3575 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3576 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3577 with non-polynomial expressions as they not only work with symbols but with
3578 constants, functions and indexed objects as well:
3582 symbol a("a"), b("b"), c("c");
3583 idx i(symbol("i"), 3);
3585 ex e = pow(sin(x) - cos(x), 4);
3586 cout << e.degree(cos(x)) << endl;
3588 cout << e.expand().coeff(sin(x), 3) << endl;
3591 e = indexed(a+b, i) * indexed(b+c, i);
3592 e = e.expand(expand_options::expand_indexed);
3593 cout << e.collect(indexed(b, i)) << endl;
3594 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3599 @subsection Polynomial division
3600 @cindex polynomial division
3603 @cindex pseudo-remainder
3604 @cindex @code{quo()}
3605 @cindex @code{rem()}
3606 @cindex @code{prem()}
3607 @cindex @code{divide()}
3612 ex quo(const ex & a, const ex & b, const symbol & x);
3613 ex rem(const ex & a, const ex & b, const symbol & x);
3616 compute the quotient and remainder of univariate polynomials in the variable
3617 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3619 The additional function
3622 ex prem(const ex & a, const ex & b, const symbol & x);
3625 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3626 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3628 Exact division of multivariate polynomials is performed by the function
3631 bool divide(const ex & a, const ex & b, ex & q);
3634 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3635 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3636 in which case the value of @code{q} is undefined.
3639 @subsection Unit, content and primitive part
3640 @cindex @code{unit()}
3641 @cindex @code{content()}
3642 @cindex @code{primpart()}
3647 ex ex::unit(const symbol & x);
3648 ex ex::content(const symbol & x);
3649 ex ex::primpart(const symbol & x);
3652 return the unit part, content part, and primitive polynomial of a multivariate
3653 polynomial with respect to the variable @samp{x} (the unit part being the sign
3654 of the leading coefficient, the content part being the GCD of the coefficients,
3655 and the primitive polynomial being the input polynomial divided by the unit and
3656 content parts). The product of unit, content, and primitive part is the
3657 original polynomial.
3660 @subsection GCD and LCM
3663 @cindex @code{gcd()}
3664 @cindex @code{lcm()}
3666 The functions for polynomial greatest common divisor and least common
3667 multiple have the synopsis
3670 ex gcd(const ex & a, const ex & b);
3671 ex lcm(const ex & a, const ex & b);
3674 The functions @code{gcd()} and @code{lcm()} accept two expressions
3675 @code{a} and @code{b} as arguments and return a new expression, their
3676 greatest common divisor or least common multiple, respectively. If the
3677 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3678 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3681 #include <ginac/ginac.h>
3682 using namespace GiNaC;
3686 symbol x("x"), y("y"), z("z");
3687 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3688 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3690 ex P_gcd = gcd(P_a, P_b);
3692 ex P_lcm = lcm(P_a, P_b);
3693 // 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
3698 @subsection Square-free decomposition
3699 @cindex square-free decomposition
3700 @cindex factorization
3701 @cindex @code{sqrfree()}
3703 GiNaC still lacks proper factorization support. Some form of
3704 factorization is, however, easily implemented by noting that factors
3705 appearing in a polynomial with power two or more also appear in the
3706 derivative and hence can easily be found by computing the GCD of the
3707 original polynomial and its derivatives. Any decent system has an
3708 interface for this so called square-free factorization. So we provide
3711 ex sqrfree(const ex & a, const lst & l = lst());
3713 Here is an example that by the way illustrates how the exact form of the
3714 result may slightly depend on the order of differentiation, calling for
3715 some care with subsequent processing of the result:
3718 symbol x("x"), y("y");
3719 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
3721 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3722 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
3724 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3725 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
3727 cout << sqrfree(BiVarPol) << endl;
3728 // -> depending on luck, any of the above
3731 Note also, how factors with the same exponents are not fully factorized
3735 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3736 @c node-name, next, previous, up
3737 @section Rational expressions
3739 @subsection The @code{normal} method
3740 @cindex @code{normal()}
3741 @cindex simplification
3742 @cindex temporary replacement
3744 Some basic form of simplification of expressions is called for frequently.
3745 GiNaC provides the method @code{.normal()}, which converts a rational function
3746 into an equivalent rational function of the form @samp{numerator/denominator}
3747 where numerator and denominator are coprime. If the input expression is already
3748 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3749 otherwise it performs fraction addition and multiplication.
3751 @code{.normal()} can also be used on expressions which are not rational functions
3752 as it will replace all non-rational objects (like functions or non-integer
3753 powers) by temporary symbols to bring the expression to the domain of rational
3754 functions before performing the normalization, and re-substituting these
3755 symbols afterwards. This algorithm is also available as a separate method
3756 @code{.to_rational()}, described below.
3758 This means that both expressions @code{t1} and @code{t2} are indeed
3759 simplified in this little code snippet:
3764 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3765 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3766 std::cout << "t1 is " << t1.normal() << std::endl;
3767 std::cout << "t2 is " << t2.normal() << std::endl;
3771 Of course this works for multivariate polynomials too, so the ratio of
3772 the sample-polynomials from the section about GCD and LCM above would be
3773 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3776 @subsection Numerator and denominator
3779 @cindex @code{numer()}
3780 @cindex @code{denom()}
3781 @cindex @code{numer_denom()}
3783 The numerator and denominator of an expression can be obtained with
3788 ex ex::numer_denom();
3791 These functions will first normalize the expression as described above and
3792 then return the numerator, denominator, or both as a list, respectively.
3793 If you need both numerator and denominator, calling @code{numer_denom()} is
3794 faster than using @code{numer()} and @code{denom()} separately.
3797 @subsection Converting to a rational expression
3798 @cindex @code{to_rational()}
3800 Some of the methods described so far only work on polynomials or rational
3801 functions. GiNaC provides a way to extend the domain of these functions to
3802 general expressions by using the temporary replacement algorithm described
3803 above. You do this by calling
3806 ex ex::to_rational(lst &l);
3809 on the expression to be converted. The supplied @code{lst} will be filled
3810 with the generated temporary symbols and their replacement expressions in
3811 a format that can be used directly for the @code{subs()} method. It can also
3812 already contain a list of replacements from an earlier application of
3813 @code{.to_rational()}, so it's possible to use it on multiple expressions
3814 and get consistent results.
3821 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3822 ex b = sin(x) + cos(x);
3825 divide(a.to_rational(l), b.to_rational(l), q);
3826 cout << q.subs(l) << endl;
3830 will print @samp{sin(x)-cos(x)}.
3833 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3834 @c node-name, next, previous, up
3835 @section Symbolic differentiation
3836 @cindex differentiation
3837 @cindex @code{diff()}
3839 @cindex product rule
3841 GiNaC's objects know how to differentiate themselves. Thus, a
3842 polynomial (class @code{add}) knows that its derivative is the sum of
3843 the derivatives of all the monomials:
3847 symbol x("x"), y("y"), z("z");
3848 ex P = pow(x, 5) + pow(x, 2) + y;
3850 cout << P.diff(x,2) << endl;
3852 cout << P.diff(y) << endl; // 1
3854 cout << P.diff(z) << endl; // 0
3859 If a second integer parameter @var{n} is given, the @code{diff} method
3860 returns the @var{n}th derivative.
3862 If @emph{every} object and every function is told what its derivative
3863 is, all derivatives of composed objects can be calculated using the
3864 chain rule and the product rule. Consider, for instance the expression
3865 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3866 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3867 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3868 out that the composition is the generating function for Euler Numbers,
3869 i.e. the so called @var{n}th Euler number is the coefficient of
3870 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3871 identity to code a function that generates Euler numbers in just three
3874 @cindex Euler numbers
3876 #include <ginac/ginac.h>
3877 using namespace GiNaC;
3879 ex EulerNumber(unsigned n)
3882 const ex generator = pow(cosh(x),-1);
3883 return generator.diff(x,n).subs(x==0);
3888 for (unsigned i=0; i<11; i+=2)
3889 std::cout << EulerNumber(i) << std::endl;
3894 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3895 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3896 @code{i} by two since all odd Euler numbers vanish anyways.
3899 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3900 @c node-name, next, previous, up
3901 @section Series expansion
3902 @cindex @code{series()}
3903 @cindex Taylor expansion
3904 @cindex Laurent expansion
3905 @cindex @code{pseries} (class)
3906 @cindex @code{Order()}
3908 Expressions know how to expand themselves as a Taylor series or (more
3909 generally) a Laurent series. As in most conventional Computer Algebra
3910 Systems, no distinction is made between those two. There is a class of
3911 its own for storing such series (@code{class pseries}) and a built-in
3912 function (called @code{Order}) for storing the order term of the series.
3913 As a consequence, if you want to work with series, i.e. multiply two
3914 series, you need to call the method @code{ex::series} again to convert
3915 it to a series object with the usual structure (expansion plus order
3916 term). A sample application from special relativity could read:
3919 #include <ginac/ginac.h>
3920 using namespace std;
3921 using namespace GiNaC;
3925 symbol v("v"), c("c");
3927 ex gamma = 1/sqrt(1 - pow(v/c,2));
3928 ex mass_nonrel = gamma.series(v==0, 10);
3930 cout << "the relativistic mass increase with v is " << endl
3931 << mass_nonrel << endl;
3933 cout << "the inverse square of this series is " << endl
3934 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3938 Only calling the series method makes the last output simplify to
3939 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3940 series raised to the power @math{-2}.
3942 @cindex Machin's formula
3943 As another instructive application, let us calculate the numerical
3944 value of Archimedes' constant
3948 (for which there already exists the built-in constant @code{Pi})
3949 using Machin's amazing formula
3951 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3954 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3956 We may expand the arcus tangent around @code{0} and insert the fractions
3957 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3958 carries an order term with it and the question arises what the system is
3959 supposed to do when the fractions are plugged into that order term. The
3960 solution is to use the function @code{series_to_poly()} to simply strip
3964 #include <ginac/ginac.h>
3965 using namespace GiNaC;
3967 ex machin_pi(int degr)
3970 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3971 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3972 -4*pi_expansion.subs(x==numeric(1,239));
3978 using std::cout; // just for fun, another way of...
3979 using std::endl; // ...dealing with this namespace std.
3981 for (int i=2; i<12; i+=2) @{
3982 pi_frac = machin_pi(i);
3983 cout << i << ":\t" << pi_frac << endl
3984 << "\t" << pi_frac.evalf() << endl;
3990 Note how we just called @code{.series(x,degr)} instead of
3991 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3992 method @code{series()}: if the first argument is a symbol the expression
3993 is expanded in that symbol around point @code{0}. When you run this
3994 program, it will type out:
3998 3.1832635983263598326
3999 4: 5359397032/1706489875
4000 3.1405970293260603143
4001 6: 38279241713339684/12184551018734375
4002 3.141621029325034425
4003 8: 76528487109180192540976/24359780855939418203125
4004 3.141591772182177295
4005 10: 327853873402258685803048818236/104359128170408663038552734375
4006 3.1415926824043995174
4010 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4011 @c node-name, next, previous, up
4012 @section Symmetrization
4013 @cindex @code{symmetrize()}
4014 @cindex @code{antisymmetrize()}
4015 @cindex @code{symmetrize_cyclic()}
4020 ex ex::symmetrize(const lst & l);
4021 ex ex::antisymmetrize(const lst & l);
4022 ex ex::symmetrize_cyclic(const lst & l);
4025 symmetrize an expression by returning the sum over all symmetric,
4026 antisymmetric or cyclic permutations of the specified list of objects,
4027 weighted by the number of permutations.
4029 The three additional methods
4032 ex ex::symmetrize();
4033 ex ex::antisymmetrize();
4034 ex ex::symmetrize_cyclic();
4037 symmetrize or antisymmetrize an expression over its free indices.
4039 Symmetrization is most useful with indexed expressions but can be used with
4040 almost any kind of object (anything that is @code{subs()}able):
4044 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4045 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4047 cout << indexed(A, i, j).symmetrize() << endl;
4048 // -> 1/2*A.j.i+1/2*A.i.j
4049 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4050 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4051 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4052 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4057 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
4058 @c node-name, next, previous, up
4059 @section Predefined mathematical functions
4061 GiNaC contains the following predefined mathematical functions:
4064 @multitable @columnfractions .30 .70
4065 @item @strong{Name} @tab @strong{Function}
4068 @cindex @code{abs()}
4069 @item @code{csgn(x)}
4071 @cindex @code{csgn()}
4072 @item @code{sqrt(x)}
4073 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4074 @cindex @code{sqrt()}
4077 @cindex @code{sin()}
4080 @cindex @code{cos()}
4083 @cindex @code{tan()}
4084 @item @code{asin(x)}
4086 @cindex @code{asin()}
4087 @item @code{acos(x)}
4089 @cindex @code{acos()}
4090 @item @code{atan(x)}
4091 @tab inverse tangent
4092 @cindex @code{atan()}
4093 @item @code{atan2(y, x)}
4094 @tab inverse tangent with two arguments
4095 @item @code{sinh(x)}
4096 @tab hyperbolic sine
4097 @cindex @code{sinh()}
4098 @item @code{cosh(x)}
4099 @tab hyperbolic cosine
4100 @cindex @code{cosh()}
4101 @item @code{tanh(x)}
4102 @tab hyperbolic tangent
4103 @cindex @code{tanh()}
4104 @item @code{asinh(x)}
4105 @tab inverse hyperbolic sine
4106 @cindex @code{asinh()}
4107 @item @code{acosh(x)}
4108 @tab inverse hyperbolic cosine
4109 @cindex @code{acosh()}
4110 @item @code{atanh(x)}
4111 @tab inverse hyperbolic tangent
4112 @cindex @code{atanh()}
4114 @tab exponential function
4115 @cindex @code{exp()}
4117 @tab natural logarithm
4118 @cindex @code{log()}
4121 @cindex @code{Li2()}
4122 @item @code{zeta(x)}
4123 @tab Riemann's zeta function
4124 @cindex @code{zeta()}
4125 @item @code{zeta(n, x)}
4126 @tab derivatives of Riemann's zeta function
4127 @item @code{tgamma(x)}
4129 @cindex @code{tgamma()}
4130 @cindex Gamma function
4131 @item @code{lgamma(x)}
4132 @tab logarithm of Gamma function
4133 @cindex @code{lgamma()}
4134 @item @code{beta(x, y)}
4135 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4136 @cindex @code{beta()}
4138 @tab psi (digamma) function
4139 @cindex @code{psi()}
4140 @item @code{psi(n, x)}
4141 @tab derivatives of psi function (polygamma functions)
4142 @item @code{factorial(n)}
4143 @tab factorial function
4144 @cindex @code{factorial()}
4145 @item @code{binomial(n, m)}
4146 @tab binomial coefficients
4147 @cindex @code{binomial()}
4148 @item @code{Order(x)}
4149 @tab order term function in truncated power series
4150 @cindex @code{Order()}
4155 For functions that have a branch cut in the complex plane GiNaC follows
4156 the conventions for C++ as defined in the ANSI standard as far as
4157 possible. In particular: the natural logarithm (@code{log}) and the
4158 square root (@code{sqrt}) both have their branch cuts running along the
4159 negative real axis where the points on the axis itself belong to the
4160 upper part (i.e. continuous with quadrant II). The inverse
4161 trigonometric and hyperbolic functions are not defined for complex
4162 arguments by the C++ standard, however. In GiNaC we follow the
4163 conventions used by CLN, which in turn follow the carefully designed
4164 definitions in the Common Lisp standard. It should be noted that this
4165 convention is identical to the one used by the C99 standard and by most
4166 serious CAS. It is to be expected that future revisions of the C++
4167 standard incorporate these functions in the complex domain in a manner
4168 compatible with C99.
4171 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
4172 @c node-name, next, previous, up
4173 @section Input and output of expressions
4176 @subsection Expression output
4178 @cindex output of expressions
4180 The easiest way to print an expression is to write it to a stream:
4185 ex e = 4.5+pow(x,2)*3/2;
4186 cout << e << endl; // prints '(4.5)+3/2*x^2'
4190 The output format is identical to the @command{ginsh} input syntax and
4191 to that used by most computer algebra systems, but not directly pastable
4192 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4193 is printed as @samp{x^2}).
4195 It is possible to print expressions in a number of different formats with
4199 void ex::print(const print_context & c, unsigned level = 0);
4202 @cindex @code{print_context} (class)
4203 The type of @code{print_context} object passed in determines the format
4204 of the output. The possible types are defined in @file{ginac/print.h}.
4205 All constructors of @code{print_context} and derived classes take an
4206 @code{ostream &} as their first argument.
4208 To print an expression in a way that can be directly used in a C or C++
4209 program, you pass a @code{print_csrc} object like this:
4213 cout << "float f = ";
4214 e.print(print_csrc_float(cout));
4217 cout << "double d = ";
4218 e.print(print_csrc_double(cout));
4221 cout << "cl_N n = ";
4222 e.print(print_csrc_cl_N(cout));
4227 The three possible types mostly affect the way in which floating point
4228 numbers are written.
4230 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4233 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4234 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4235 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4238 The @code{print_context} type @code{print_tree} provides a dump of the
4239 internal structure of an expression for debugging purposes:
4243 e.print(print_tree(cout));
4250 add, hash=0x0, flags=0x3, nops=2
4251 power, hash=0x9, flags=0x3, nops=2
4252 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4253 2 (numeric), hash=0x80000042, flags=0xf
4254 3/2 (numeric), hash=0x80000061, flags=0xf
4257 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4261 This kind of output is also available in @command{ginsh} as the @code{print()}
4264 Another useful output format is for LaTeX parsing in mathematical mode.
4265 It is rather similar to the default @code{print_context} but provides
4266 some braces needed by LaTeX for delimiting boxes and also converts some
4267 common objects to conventional LaTeX names. It is possible to give symbols
4268 a special name for LaTeX output by supplying it as a second argument to
4269 the @code{symbol} constructor.
4271 For example, the code snippet
4276 ex foo = lgamma(x).series(x==0,3);
4277 foo.print(print_latex(std::cout));
4283 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4286 @cindex Tree traversal
4287 If you need any fancy special output format, e.g. for interfacing GiNaC
4288 with other algebra systems or for producing code for different
4289 programming languages, you can always traverse the expression tree yourself:
4292 static void my_print(const ex & e)
4294 if (is_a<function>(e))
4295 cout << ex_to<function>(e).get_name();
4297 cout << e.bp->class_name();
4299 unsigned n = e.nops();
4301 for (unsigned i=0; i<n; i++) @{
4313 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4321 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4322 symbol(y))),numeric(-2)))
4325 If you need an output format that makes it possible to accurately
4326 reconstruct an expression by feeding the output to a suitable parser or
4327 object factory, you should consider storing the expression in an
4328 @code{archive} object and reading the object properties from there.
4329 See the section on archiving for more information.
4332 @subsection Expression input
4333 @cindex input of expressions
4335 GiNaC provides no way to directly read an expression from a stream because
4336 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4337 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4338 @code{y} you defined in your program and there is no way to specify the
4339 desired symbols to the @code{>>} stream input operator.
4341 Instead, GiNaC lets you construct an expression from a string, specifying the
4342 list of symbols and indices to be used:
4346 symbol x("x"), y("y"), p("p");
4347 idx i(symbol("i"), 3);
4348 ex e("2*x+sin(y)+p.i", lst(x, y, p, i));
4352 The input syntax is the same as that used by @command{ginsh} and the stream
4353 output operator @code{<<}. The symbols and indices in the string are matched
4354 by name to the symbols and indices in the list and if GiNaC encounters a
4355 symbol or index not specified in the list it will throw an exception. Only
4356 indices whose values are single symbols can be used (i.e. numeric indices
4357 or compound indices as in "A.(2*n+1)" are not allowed).
4359 With this constructor, it's also easy to implement interactive GiNaC programs:
4364 #include <stdexcept>
4365 #include <ginac/ginac.h>
4366 using namespace std;
4367 using namespace GiNaC;
4374 cout << "Enter an expression containing 'x': ";
4379 cout << "The derivative of " << e << " with respect to x is ";
4380 cout << e.diff(x) << ".\n";
4381 @} catch (exception &p) @{
4382 cerr << p.what() << endl;
4388 @subsection Archiving
4389 @cindex @code{archive} (class)
4392 GiNaC allows creating @dfn{archives} of expressions which can be stored
4393 to or retrieved from files. To create an archive, you declare an object
4394 of class @code{archive} and archive expressions in it, giving each
4395 expression a unique name:
4399 using namespace std;
4400 #include <ginac/ginac.h>
4401 using namespace GiNaC;
4405 symbol x("x"), y("y"), z("z");
4407 ex foo = sin(x + 2*y) + 3*z + 41;
4411 a.archive_ex(foo, "foo");
4412 a.archive_ex(bar, "the second one");
4416 The archive can then be written to a file:
4420 ofstream out("foobar.gar");
4426 The file @file{foobar.gar} contains all information that is needed to
4427 reconstruct the expressions @code{foo} and @code{bar}.
4429 @cindex @command{viewgar}
4430 The tool @command{viewgar} that comes with GiNaC can be used to view
4431 the contents of GiNaC archive files:
4434 $ viewgar foobar.gar
4435 foo = 41+sin(x+2*y)+3*z
4436 the second one = 42+sin(x+2*y)+3*z
4439 The point of writing archive files is of course that they can later be
4445 ifstream in("foobar.gar");
4450 And the stored expressions can be retrieved by their name:
4456 ex ex1 = a2.unarchive_ex(syms, "foo");
4457 ex ex2 = a2.unarchive_ex(syms, "the second one");
4459 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4460 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4461 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4465 Note that you have to supply a list of the symbols which are to be inserted
4466 in the expressions. Symbols in archives are stored by their name only and
4467 if you don't specify which symbols you have, unarchiving the expression will
4468 create new symbols with that name. E.g. if you hadn't included @code{x} in
4469 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4470 have had no effect because the @code{x} in @code{ex1} would have been a
4471 different symbol than the @code{x} which was defined at the beginning of
4472 the program, although both would appear as @samp{x} when printed.
4474 You can also use the information stored in an @code{archive} object to
4475 output expressions in a format suitable for exact reconstruction. The
4476 @code{archive} and @code{archive_node} classes have a couple of member
4477 functions that let you access the stored properties:
4480 static void my_print2(const archive_node & n)
4483 n.find_string("class", class_name);
4484 cout << class_name << "(";
4486 archive_node::propinfovector p;
4487 n.get_properties(p);
4489 unsigned num = p.size();
4490 for (unsigned i=0; i<num; i++) @{
4491 const string &name = p[i].name;
4492 if (name == "class")
4494 cout << name << "=";
4496 unsigned count = p[i].count;
4500 for (unsigned j=0; j<count; j++) @{
4501 switch (p[i].type) @{
4502 case archive_node::PTYPE_BOOL: @{
4504 n.find_bool(name, x, j);
4505 cout << (x ? "true" : "false");
4508 case archive_node::PTYPE_UNSIGNED: @{
4510 n.find_unsigned(name, x, j);
4514 case archive_node::PTYPE_STRING: @{
4516 n.find_string(name, x, j);
4517 cout << '\"' << x << '\"';
4520 case archive_node::PTYPE_NODE: @{
4521 const archive_node &x = n.find_ex_node(name, j);
4543 ex e = pow(2, x) - y;
4545 my_print2(ar.get_top_node(0)); cout << endl;
4553 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4554 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4555 overall_coeff=numeric(number="0"))
4558 Be warned, however, that the set of properties and their meaning for each
4559 class may change between GiNaC versions.
4562 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4563 @c node-name, next, previous, up
4564 @chapter Extending GiNaC
4566 By reading so far you should have gotten a fairly good understanding of
4567 GiNaC's design-patterns. From here on you should start reading the
4568 sources. All we can do now is issue some recommendations how to tackle
4569 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4570 develop some useful extension please don't hesitate to contact the GiNaC
4571 authors---they will happily incorporate them into future versions.
4574 * What does not belong into GiNaC:: What to avoid.
4575 * Symbolic functions:: Implementing symbolic functions.
4576 * Adding classes:: Defining new algebraic classes.
4580 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4581 @c node-name, next, previous, up
4582 @section What doesn't belong into GiNaC
4584 @cindex @command{ginsh}
4585 First of all, GiNaC's name must be read literally. It is designed to be
4586 a library for use within C++. The tiny @command{ginsh} accompanying
4587 GiNaC makes this even more clear: it doesn't even attempt to provide a
4588 language. There are no loops or conditional expressions in
4589 @command{ginsh}, it is merely a window into the library for the
4590 programmer to test stuff (or to show off). Still, the design of a
4591 complete CAS with a language of its own, graphical capabilities and all
4592 this on top of GiNaC is possible and is without doubt a nice project for
4595 There are many built-in functions in GiNaC that do not know how to
4596 evaluate themselves numerically to a precision declared at runtime
4597 (using @code{Digits}). Some may be evaluated at certain points, but not
4598 generally. This ought to be fixed. However, doing numerical
4599 computations with GiNaC's quite abstract classes is doomed to be
4600 inefficient. For this purpose, the underlying foundation classes
4601 provided by CLN are much better suited.
4604 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4605 @c node-name, next, previous, up
4606 @section Symbolic functions
4608 The easiest and most instructive way to start with is probably to
4609 implement your own function. GiNaC's functions are objects of class
4610 @code{function}. The preprocessor is then used to convert the function
4611 names to objects with a corresponding serial number that is used
4612 internally to identify them. You usually need not worry about this
4613 number. New functions may be inserted into the system via a kind of
4614 `registry'. It is your responsibility to care for some functions that
4615 are called when the user invokes certain methods. These are usual
4616 C++-functions accepting a number of @code{ex} as arguments and returning
4617 one @code{ex}. As an example, if we have a look at a simplified
4618 implementation of the cosine trigonometric function, we first need a
4619 function that is called when one wishes to @code{eval} it. It could
4620 look something like this:
4623 static ex cos_eval_method(const ex & x)
4625 // if (!x%(2*Pi)) return 1
4626 // if (!x%Pi) return -1
4627 // if (!x%Pi/2) return 0
4628 // care for other cases...
4629 return cos(x).hold();
4633 @cindex @code{hold()}
4635 The last line returns @code{cos(x)} if we don't know what else to do and
4636 stops a potential recursive evaluation by saying @code{.hold()}, which
4637 sets a flag to the expression signaling that it has been evaluated. We
4638 should also implement a method for numerical evaluation and since we are
4639 lazy we sweep the problem under the rug by calling someone else's
4640 function that does so, in this case the one in class @code{numeric}:
4643 static ex cos_evalf(const ex & x)
4645 if (is_a<numeric>(x))
4646 return cos(ex_to<numeric>(x));
4648 return cos(x).hold();
4652 Differentiation will surely turn up and so we need to tell @code{cos}
4653 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4654 instance are then handled automatically by @code{basic::diff} and
4658 static ex cos_deriv(const ex & x, unsigned diff_param)
4664 @cindex product rule
4665 The second parameter is obligatory but uninteresting at this point. It
4666 specifies which parameter to differentiate in a partial derivative in
4667 case the function has more than one parameter and its main application
4668 is for correct handling of the chain rule. For Taylor expansion, it is
4669 enough to know how to differentiate. But if the function you want to
4670 implement does have a pole somewhere in the complex plane, you need to
4671 write another method for Laurent expansion around that point.
4673 Now that all the ingredients for @code{cos} have been set up, we need
4674 to tell the system about it. This is done by a macro and we are not
4675 going to describe how it expands, please consult your preprocessor if you
4679 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4680 evalf_func(cos_evalf).
4681 derivative_func(cos_deriv));
4684 The first argument is the function's name used for calling it and for
4685 output. The second binds the corresponding methods as options to this
4686 object. Options are separated by a dot and can be given in an arbitrary
4687 order. GiNaC functions understand several more options which are always
4688 specified as @code{.option(params)}, for example a method for series
4689 expansion @code{.series_func(cos_series)}. Again, if no series
4690 expansion method is given, GiNaC defaults to simple Taylor expansion,
4691 which is correct if there are no poles involved as is the case for the
4692 @code{cos} function. The way GiNaC handles poles in case there are any
4693 is best understood by studying one of the examples, like the Gamma
4694 (@code{tgamma}) function for instance. (In essence the function first
4695 checks if there is a pole at the evaluation point and falls back to
4696 Taylor expansion if there isn't. Then, the pole is regularized by some
4697 suitable transformation.) Also, the new function needs to be declared
4698 somewhere. This may also be done by a convenient preprocessor macro:
4701 DECLARE_FUNCTION_1P(cos)
4704 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4705 implementation of @code{cos} is very incomplete and lacks several safety
4706 mechanisms. Please, have a look at the real implementation in GiNaC.
4707 (By the way: in case you are worrying about all the macros above we can
4708 assure you that functions are GiNaC's most macro-intense classes. We
4709 have done our best to avoid macros where we can.)
4712 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4713 @c node-name, next, previous, up
4714 @section Adding classes
4716 If you are doing some very specialized things with GiNaC you may find that
4717 you have to implement your own algebraic classes to fit your needs. This
4718 section will explain how to do this by giving the example of a simple
4719 'string' class. After reading this section you will know how to properly
4720 declare a GiNaC class and what the minimum required member functions are
4721 that you have to implement. We only cover the implementation of a 'leaf'
4722 class here (i.e. one that doesn't contain subexpressions). Creating a
4723 container class like, for example, a class representing tensor products is
4724 more involved but this section should give you enough information so you can
4725 consult the source to GiNaC's predefined classes if you want to implement
4726 something more complicated.
4728 @subsection GiNaC's run-time type information system
4730 @cindex hierarchy of classes
4732 All algebraic classes (that is, all classes that can appear in expressions)
4733 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4734 @code{basic *} (which is essentially what an @code{ex} is) represents a
4735 generic pointer to an algebraic class. Occasionally it is necessary to find
4736 out what the class of an object pointed to by a @code{basic *} really is.
4737 Also, for the unarchiving of expressions it must be possible to find the
4738 @code{unarchive()} function of a class given the class name (as a string). A
4739 system that provides this kind of information is called a run-time type
4740 information (RTTI) system. The C++ language provides such a thing (see the
4741 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4742 implements its own, simpler RTTI.
4744 The RTTI in GiNaC is based on two mechanisms:
4749 The @code{basic} class declares a member variable @code{tinfo_key} which
4750 holds an unsigned integer that identifies the object's class. These numbers
4751 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4752 classes. They all start with @code{TINFO_}.
4755 By means of some clever tricks with static members, GiNaC maintains a list
4756 of information for all classes derived from @code{basic}. The information
4757 available includes the class names, the @code{tinfo_key}s, and pointers
4758 to the unarchiving functions. This class registry is defined in the
4759 @file{registrar.h} header file.
4763 The disadvantage of this proprietary RTTI implementation is that there's
4764 a little more to do when implementing new classes (C++'s RTTI works more
4765 or less automatic) but don't worry, most of the work is simplified by
4768 @subsection A minimalistic example
4770 Now we will start implementing a new class @code{mystring} that allows
4771 placing character strings in algebraic expressions (this is not very useful,
4772 but it's just an example). This class will be a direct subclass of
4773 @code{basic}. You can use this sample implementation as a starting point
4774 for your own classes.
4776 The code snippets given here assume that you have included some header files
4782 #include <stdexcept>
4783 using namespace std;
4785 #include <ginac/ginac.h>
4786 using namespace GiNaC;
4789 The first thing we have to do is to define a @code{tinfo_key} for our new
4790 class. This can be any arbitrary unsigned number that is not already taken
4791 by one of the existing classes but it's better to come up with something
4792 that is unlikely to clash with keys that might be added in the future. The
4793 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4794 which is not a requirement but we are going to stick with this scheme:
4797 const unsigned TINFO_mystring = 0x42420001U;
4800 Now we can write down the class declaration. The class stores a C++
4801 @code{string} and the user shall be able to construct a @code{mystring}
4802 object from a C or C++ string:
4805 class mystring : public basic
4807 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4810 mystring(const string &s);
4811 mystring(const char *s);
4817 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4820 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4821 macros are defined in @file{registrar.h}. They take the name of the class
4822 and its direct superclass as arguments and insert all required declarations
4823 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4824 the first line after the opening brace of the class definition. The
4825 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4826 source (at global scope, of course, not inside a function).
4828 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4829 declarations of the default and copy constructor, the destructor, the
4830 assignment operator and a couple of other functions that are required. It
4831 also defines a type @code{inherited} which refers to the superclass so you
4832 don't have to modify your code every time you shuffle around the class
4833 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4834 constructor, the destructor and the assignment operator.
4836 Now there are nine member functions we have to implement to get a working
4842 @code{mystring()}, the default constructor.
4845 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4846 assignment operator to free dynamically allocated members. The @code{call_parent}
4847 specifies whether the @code{destroy()} function of the superclass is to be
4851 @code{void copy(const mystring &other)}, which is used in the copy constructor
4852 and assignment operator to copy the member variables over from another
4853 object of the same class.
4856 @code{void archive(archive_node &n)}, the archiving function. This stores all
4857 information needed to reconstruct an object of this class inside an
4858 @code{archive_node}.
4861 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4862 constructor. This constructs an instance of the class from the information
4863 found in an @code{archive_node}.
4866 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4867 unarchiving function. It constructs a new instance by calling the unarchiving
4871 @code{int compare_same_type(const basic &other)}, which is used internally
4872 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4873 -1, depending on the relative order of this object and the @code{other}
4874 object. If it returns 0, the objects are considered equal.
4875 @strong{Note:} This has nothing to do with the (numeric) ordering
4876 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4877 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4878 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4879 must provide a @code{compare_same_type()} function, even those representing
4880 objects for which no reasonable algebraic ordering relationship can be
4884 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4885 which are the two constructors we declared.
4889 Let's proceed step-by-step. The default constructor looks like this:
4892 mystring::mystring() : inherited(TINFO_mystring)
4894 // dynamically allocate resources here if required
4898 The golden rule is that in all constructors you have to set the
4899 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4900 it will be set by the constructor of the superclass and all hell will break
4901 loose in the RTTI. For your convenience, the @code{basic} class provides
4902 a constructor that takes a @code{tinfo_key} value, which we are using here
4903 (remember that in our case @code{inherited = basic}). If the superclass
4904 didn't have such a constructor, we would have to set the @code{tinfo_key}
4905 to the right value manually.
4907 In the default constructor you should set all other member variables to
4908 reasonable default values (we don't need that here since our @code{str}
4909 member gets set to an empty string automatically). The constructor(s) are of
4910 course also the right place to allocate any dynamic resources you require.
4912 Next, the @code{destroy()} function:
4915 void mystring::destroy(bool call_parent)
4917 // free dynamically allocated resources here if required
4919 inherited::destroy(call_parent);
4923 This function is where we free all dynamically allocated resources. We
4924 don't have any so we're not doing anything here, but if we had, for
4925 example, used a C-style @code{char *} to store our string, this would be
4926 the place to @code{delete[]} the string storage. If @code{call_parent}
4927 is true, we have to call the @code{destroy()} function of the superclass
4928 after we're done (to mimic C++'s automatic invocation of superclass
4929 destructors where @code{destroy()} is called from outside a destructor).
4931 The @code{copy()} function just copies over the member variables from
4935 void mystring::copy(const mystring &other)
4937 inherited::copy(other);
4942 We can simply overwrite the member variables here. There's no need to worry
4943 about dynamically allocated storage. The assignment operator (which is
4944 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4945 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4946 explicitly call the @code{copy()} function of the superclass here so
4947 all the member variables will get copied.
4949 Next are the three functions for archiving. You have to implement them even
4950 if you don't plan to use archives, but the minimum required implementation
4951 is really simple. First, the archiving function:
4954 void mystring::archive(archive_node &n) const
4956 inherited::archive(n);
4957 n.add_string("string", str);
4961 The only thing that is really required is calling the @code{archive()}
4962 function of the superclass. Optionally, you can store all information you
4963 deem necessary for representing the object into the passed
4964 @code{archive_node}. We are just storing our string here. For more
4965 information on how the archiving works, consult the @file{archive.h} header
4968 The unarchiving constructor is basically the inverse of the archiving
4972 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4974 n.find_string("string", str);
4978 If you don't need archiving, just leave this function empty (but you must
4979 invoke the unarchiving constructor of the superclass). Note that we don't
4980 have to set the @code{tinfo_key} here because it is done automatically
4981 by the unarchiving constructor of the @code{basic} class.
4983 Finally, the unarchiving function:
4986 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4988 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4992 You don't have to understand how exactly this works. Just copy these
4993 four lines into your code literally (replacing the class name, of
4994 course). It calls the unarchiving constructor of the class and unless
4995 you are doing something very special (like matching @code{archive_node}s
4996 to global objects) you don't need a different implementation. For those
4997 who are interested: setting the @code{dynallocated} flag puts the object
4998 under the control of GiNaC's garbage collection. It will get deleted
4999 automatically once it is no longer referenced.
5001 Our @code{compare_same_type()} function uses a provided function to compare
5005 int mystring::compare_same_type(const basic &other) const
5007 const mystring &o = static_cast<const mystring &>(other);
5008 int cmpval = str.compare(o.str);
5011 else if (cmpval < 0)
5018 Although this function takes a @code{basic &}, it will always be a reference
5019 to an object of exactly the same class (objects of different classes are not
5020 comparable), so the cast is safe. If this function returns 0, the two objects
5021 are considered equal (in the sense that @math{A-B=0}), so you should compare
5022 all relevant member variables.
5024 Now the only thing missing is our two new constructors:
5027 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
5029 // dynamically allocate resources here if required
5032 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
5034 // dynamically allocate resources here if required
5038 No surprises here. We set the @code{str} member from the argument and
5039 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
5041 That's it! We now have a minimal working GiNaC class that can store
5042 strings in algebraic expressions. Let's confirm that the RTTI works:
5045 ex e = mystring("Hello, world!");
5046 cout << is_a<mystring>(e) << endl;
5049 cout << e.bp->class_name() << endl;
5053 Obviously it does. Let's see what the expression @code{e} looks like:
5057 // -> [mystring object]
5060 Hm, not exactly what we expect, but of course the @code{mystring} class
5061 doesn't yet know how to print itself. This is done in the @code{print()}
5062 member function. Let's say that we wanted to print the string surrounded
5066 class mystring : public basic
5070 void print(const print_context &c, unsigned level = 0) const;
5074 void mystring::print(const print_context &c, unsigned level) const
5076 // print_context::s is a reference to an ostream
5077 c.s << '\"' << str << '\"';
5081 The @code{level} argument is only required for container classes to
5082 correctly parenthesize the output. Let's try again to print the expression:
5086 // -> "Hello, world!"
5089 Much better. The @code{mystring} class can be used in arbitrary expressions:
5092 e += mystring("GiNaC rulez");
5094 // -> "GiNaC rulez"+"Hello, world!"
5097 (GiNaC's automatic term reordering is in effect here), or even
5100 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
5102 // -> "One string"^(2*sin(-"Another string"+Pi))
5105 Whether this makes sense is debatable but remember that this is only an
5106 example. At least it allows you to implement your own symbolic algorithms
5109 Note that GiNaC's algebraic rules remain unchanged:
5112 e = mystring("Wow") * mystring("Wow");
5116 e = pow(mystring("First")-mystring("Second"), 2);
5117 cout << e.expand() << endl;
5118 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5121 There's no way to, for example, make GiNaC's @code{add} class perform string
5122 concatenation. You would have to implement this yourself.
5124 @subsection Automatic evaluation
5126 @cindex @code{hold()}
5127 @cindex @code{eval()}
5129 When dealing with objects that are just a little more complicated than the
5130 simple string objects we have implemented, chances are that you will want to
5131 have some automatic simplifications or canonicalizations performed on them.
5132 This is done in the evaluation member function @code{eval()}. Let's say that
5133 we wanted all strings automatically converted to lowercase with
5134 non-alphabetic characters stripped, and empty strings removed:
5137 class mystring : public basic
5141 ex eval(int level = 0) const;
5145 ex mystring::eval(int level) const
5148 for (int i=0; i<str.length(); i++) @{
5150 if (c >= 'A' && c <= 'Z')
5151 new_str += tolower(c);
5152 else if (c >= 'a' && c <= 'z')
5156 if (new_str.length() == 0)
5159 return mystring(new_str).hold();
5163 The @code{level} argument is used to limit the recursion depth of the
5164 evaluation. We don't have any subexpressions in the @code{mystring}
5165 class so we are not concerned with this. If we had, we would call the
5166 @code{eval()} functions of the subexpressions with @code{level - 1} as
5167 the argument if @code{level != 1}. The @code{hold()} member function
5168 sets a flag in the object that prevents further evaluation. Otherwise
5169 we might end up in an endless loop. When you want to return the object
5170 unmodified, use @code{return this->hold();}.
5172 Let's confirm that it works:
5175 ex e = mystring("Hello, world!") + mystring("!?#");
5179 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5184 @subsection Other member functions
5186 We have implemented only a small set of member functions to make the class
5187 work in the GiNaC framework. For a real algebraic class, there are probably
5188 some more functions that you will want to re-implement, such as
5189 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5190 or the header file of the class you want to make a subclass of to see
5191 what's there. One member function that you will most likely want to
5192 implement for terminal classes like the described string class is
5193 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5194 which will allow GiNaC to compare and canonicalize expressions much more
5197 You can, of course, also add your own new member functions. Remember,
5198 that the RTTI may be used to get information about what kinds of objects
5199 you are dealing with (the position in the class hierarchy) and that you
5200 can always extract the bare object from an @code{ex} by stripping the
5201 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5202 should become a need.
5204 That's it. May the source be with you!
5207 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5208 @c node-name, next, previous, up
5209 @chapter A Comparison With Other CAS
5212 This chapter will give you some information on how GiNaC compares to
5213 other, traditional Computer Algebra Systems, like @emph{Maple},
5214 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5215 disadvantages over these systems.
5218 * Advantages:: Strengths of the GiNaC approach.
5219 * Disadvantages:: Weaknesses of the GiNaC approach.
5220 * Why C++?:: Attractiveness of C++.
5223 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5224 @c node-name, next, previous, up
5227 GiNaC has several advantages over traditional Computer
5228 Algebra Systems, like
5233 familiar language: all common CAS implement their own proprietary
5234 grammar which you have to learn first (and maybe learn again when your
5235 vendor decides to `enhance' it). With GiNaC you can write your program
5236 in common C++, which is standardized.
5240 structured data types: you can build up structured data types using
5241 @code{struct}s or @code{class}es together with STL features instead of
5242 using unnamed lists of lists of lists.
5245 strongly typed: in CAS, you usually have only one kind of variables
5246 which can hold contents of an arbitrary type. This 4GL like feature is
5247 nice for novice programmers, but dangerous.
5250 development tools: powerful development tools exist for C++, like fancy
5251 editors (e.g. with automatic indentation and syntax highlighting),
5252 debuggers, visualization tools, documentation generators@dots{}
5255 modularization: C++ programs can easily be split into modules by
5256 separating interface and implementation.
5259 price: GiNaC is distributed under the GNU Public License which means
5260 that it is free and available with source code. And there are excellent
5261 C++-compilers for free, too.
5264 extendable: you can add your own classes to GiNaC, thus extending it on
5265 a very low level. Compare this to a traditional CAS that you can
5266 usually only extend on a high level by writing in the language defined
5267 by the parser. In particular, it turns out to be almost impossible to
5268 fix bugs in a traditional system.
5271 multiple interfaces: Though real GiNaC programs have to be written in
5272 some editor, then be compiled, linked and executed, there are more ways
5273 to work with the GiNaC engine. Many people want to play with
5274 expressions interactively, as in traditional CASs. Currently, two such
5275 windows into GiNaC have been implemented and many more are possible: the
5276 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5277 types to a command line and second, as a more consistent approach, an
5278 interactive interface to the Cint C++ interpreter has been put together
5279 (called GiNaC-cint) that allows an interactive scripting interface
5280 consistent with the C++ language. It is available from the usual GiNaC
5284 seamless integration: it is somewhere between difficult and impossible
5285 to call CAS functions from within a program written in C++ or any other
5286 programming language and vice versa. With GiNaC, your symbolic routines
5287 are part of your program. You can easily call third party libraries,
5288 e.g. for numerical evaluation or graphical interaction. All other
5289 approaches are much more cumbersome: they range from simply ignoring the
5290 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5291 system (i.e. @emph{Yacas}).
5294 efficiency: often large parts of a program do not need symbolic
5295 calculations at all. Why use large integers for loop variables or
5296 arbitrary precision arithmetics where @code{int} and @code{double} are
5297 sufficient? For pure symbolic applications, GiNaC is comparable in
5298 speed with other CAS.
5303 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5304 @c node-name, next, previous, up
5305 @section Disadvantages
5307 Of course it also has some disadvantages:
5312 advanced features: GiNaC cannot compete with a program like
5313 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5314 which grows since 1981 by the work of dozens of programmers, with
5315 respect to mathematical features. Integration, factorization,
5316 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5317 not planned for the near future).
5320 portability: While the GiNaC library itself is designed to avoid any
5321 platform dependent features (it should compile on any ANSI compliant C++
5322 compiler), the currently used version of the CLN library (fast large
5323 integer and arbitrary precision arithmetics) can only by compiled
5324 without hassle on systems with the C++ compiler from the GNU Compiler
5325 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
5326 macros to let the compiler gather all static initializations, which
5327 works for GNU C++ only. Feel free to contact the authors in case you
5328 really believe that you need to use a different compiler. We have
5329 occasionally used other compilers and may be able to give you advice.}
5330 GiNaC uses recent language features like explicit constructors, mutable
5331 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
5332 literally. Recent GCC versions starting at 2.95.3, although itself not
5333 yet ANSI compliant, support all needed features.
5338 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5339 @c node-name, next, previous, up
5342 Why did we choose to implement GiNaC in C++ instead of Java or any other
5343 language? C++ is not perfect: type checking is not strict (casting is
5344 possible), separation between interface and implementation is not
5345 complete, object oriented design is not enforced. The main reason is
5346 the often scolded feature of operator overloading in C++. While it may
5347 be true that operating on classes with a @code{+} operator is rarely
5348 meaningful, it is perfectly suited for algebraic expressions. Writing
5349 @math{3x+5y} as @code{3*x+5*y} instead of
5350 @code{x.times(3).plus(y.times(5))} looks much more natural.
5351 Furthermore, the main developers are more familiar with C++ than with
5352 any other programming language.
5355 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5356 @c node-name, next, previous, up
5357 @appendix Internal Structures
5360 * Expressions are reference counted::
5361 * Internal representation of products and sums::
5364 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5365 @c node-name, next, previous, up
5366 @appendixsection Expressions are reference counted
5368 @cindex reference counting
5369 @cindex copy-on-write
5370 @cindex garbage collection
5371 An expression is extremely light-weight since internally it works like a
5372 handle to the actual representation and really holds nothing more than a
5373 pointer to some other object. What this means in practice is that
5374 whenever you create two @code{ex} and set the second equal to the first
5375 no copying process is involved. Instead, the copying takes place as soon
5376 as you try to change the second. Consider the simple sequence of code:
5380 #include <ginac/ginac.h>
5381 using namespace std;
5382 using namespace GiNaC;
5386 symbol x("x"), y("y"), z("z");
5389 e1 = sin(x + 2*y) + 3*z + 41;
5390 e2 = e1; // e2 points to same object as e1
5391 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5392 e2 += 1; // e2 is copied into a new object
5393 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5397 The line @code{e2 = e1;} creates a second expression pointing to the
5398 object held already by @code{e1}. The time involved for this operation
5399 is therefore constant, no matter how large @code{e1} was. Actual
5400 copying, however, must take place in the line @code{e2 += 1;} because
5401 @code{e1} and @code{e2} are not handles for the same object any more.
5402 This concept is called @dfn{copy-on-write semantics}. It increases
5403 performance considerably whenever one object occurs multiple times and
5404 represents a simple garbage collection scheme because when an @code{ex}
5405 runs out of scope its destructor checks whether other expressions handle
5406 the object it points to too and deletes the object from memory if that
5407 turns out not to be the case. A slightly less trivial example of
5408 differentiation using the chain-rule should make clear how powerful this
5413 symbol x("x"), y("y");
5417 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5418 cout << e1 << endl // prints x+3*y
5419 << e2 << endl // prints (x+3*y)^3
5420 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5424 Here, @code{e1} will actually be referenced three times while @code{e2}
5425 will be referenced two times. When the power of an expression is built,
5426 that expression needs not be copied. Likewise, since the derivative of
5427 a power of an expression can be easily expressed in terms of that
5428 expression, no copying of @code{e1} is involved when @code{e3} is
5429 constructed. So, when @code{e3} is constructed it will print as
5430 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5431 holds a reference to @code{e2} and the factor in front is just
5434 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5435 semantics. When you insert an expression into a second expression, the
5436 result behaves exactly as if the contents of the first expression were
5437 inserted. But it may be useful to remember that this is not what
5438 happens. Knowing this will enable you to write much more efficient
5439 code. If you still have an uncertain feeling with copy-on-write
5440 semantics, we recommend you have a look at the
5441 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5442 Marshall Cline. Chapter 16 covers this issue and presents an
5443 implementation which is pretty close to the one in GiNaC.
5446 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5447 @c node-name, next, previous, up
5448 @appendixsection Internal representation of products and sums
5450 @cindex representation
5453 @cindex @code{power}
5454 Although it should be completely transparent for the user of
5455 GiNaC a short discussion of this topic helps to understand the sources
5456 and also explain performance to a large degree. Consider the
5457 unexpanded symbolic expression
5459 $2d^3 \left( 4a + 5b - 3 \right)$
5462 @math{2*d^3*(4*a+5*b-3)}
5464 which could naively be represented by a tree of linear containers for
5465 addition and multiplication, one container for exponentiation with base
5466 and exponent and some atomic leaves of symbols and numbers in this
5471 @cindex pair-wise representation
5472 However, doing so results in a rather deeply nested tree which will
5473 quickly become inefficient to manipulate. We can improve on this by
5474 representing the sum as a sequence of terms, each one being a pair of a
5475 purely numeric multiplicative coefficient and its rest. In the same
5476 spirit we can store the multiplication as a sequence of terms, each
5477 having a numeric exponent and a possibly complicated base, the tree
5478 becomes much more flat:
5482 The number @code{3} above the symbol @code{d} shows that @code{mul}
5483 objects are treated similarly where the coefficients are interpreted as
5484 @emph{exponents} now. Addition of sums of terms or multiplication of
5485 products with numerical exponents can be coded to be very efficient with
5486 such a pair-wise representation. Internally, this handling is performed
5487 by most CAS in this way. It typically speeds up manipulations by an
5488 order of magnitude. The overall multiplicative factor @code{2} and the
5489 additive term @code{-3} look somewhat out of place in this
5490 representation, however, since they are still carrying a trivial
5491 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5492 this is avoided by adding a field that carries an overall numeric
5493 coefficient. This results in the realistic picture of internal
5496 $2d^3 \left( 4a + 5b - 3 \right)$:
5499 @math{2*d^3*(4*a+5*b-3)}:
5505 This also allows for a better handling of numeric radicals, since
5506 @code{sqrt(2)} can now be carried along calculations. Now it should be
5507 clear, why both classes @code{add} and @code{mul} are derived from the
5508 same abstract class: the data representation is the same, only the
5509 semantics differs. In the class hierarchy, methods for polynomial
5510 expansion and the like are reimplemented for @code{add} and @code{mul},
5511 but the data structure is inherited from @code{expairseq}.
5514 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5515 @c node-name, next, previous, up
5516 @appendix Package Tools
5518 If you are creating a software package that uses the GiNaC library,
5519 setting the correct command line options for the compiler and linker
5520 can be difficult. GiNaC includes two tools to make this process easier.
5523 * ginac-config:: A shell script to detect compiler and linker flags.
5524 * AM_PATH_GINAC:: Macro for GNU automake.
5528 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5529 @c node-name, next, previous, up
5530 @section @command{ginac-config}
5531 @cindex ginac-config
5533 @command{ginac-config} is a shell script that you can use to determine
5534 the compiler and linker command line options required to compile and
5535 link a program with the GiNaC library.
5537 @command{ginac-config} takes the following flags:
5541 Prints out the version of GiNaC installed.
5543 Prints '-I' flags pointing to the installed header files.
5545 Prints out the linker flags necessary to link a program against GiNaC.
5546 @item --prefix[=@var{PREFIX}]
5547 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5548 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5549 Otherwise, prints out the configured value of @env{$prefix}.
5550 @item --exec-prefix[=@var{PREFIX}]
5551 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5552 Otherwise, prints out the configured value of @env{$exec_prefix}.
5555 Typically, @command{ginac-config} will be used within a configure
5556 script, as described below. It, however, can also be used directly from
5557 the command line using backquotes to compile a simple program. For
5561 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5564 This command line might expand to (for example):
5567 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5568 -lginac -lcln -lstdc++
5571 Not only is the form using @command{ginac-config} easier to type, it will
5572 work on any system, no matter how GiNaC was configured.
5575 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5576 @c node-name, next, previous, up
5577 @section @samp{AM_PATH_GINAC}
5578 @cindex AM_PATH_GINAC
5580 For packages configured using GNU automake, GiNaC also provides
5581 a macro to automate the process of checking for GiNaC.
5584 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5592 Determines the location of GiNaC using @command{ginac-config}, which is
5593 either found in the user's path, or from the environment variable
5594 @env{GINACLIB_CONFIG}.
5597 Tests the installed libraries to make sure that their version
5598 is later than @var{MINIMUM-VERSION}. (A default version will be used
5602 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5603 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5604 variable to the output of @command{ginac-config --libs}, and calls
5605 @samp{AC_SUBST()} for these variables so they can be used in generated
5606 makefiles, and then executes @var{ACTION-IF-FOUND}.
5609 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5610 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5614 This macro is in file @file{ginac.m4} which is installed in
5615 @file{$datadir/aclocal}. Note that if automake was installed with a
5616 different @samp{--prefix} than GiNaC, you will either have to manually
5617 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5618 aclocal the @samp{-I} option when running it.
5621 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5622 * Example package:: Example of a package using AM_PATH_GINAC.
5626 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5627 @c node-name, next, previous, up
5628 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5630 Simply make sure that @command{ginac-config} is in your path, and run
5631 the configure script.
5638 The directory where the GiNaC libraries are installed needs
5639 to be found by your system's dynamic linker.
5641 This is generally done by
5644 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5650 setting the environment variable @env{LD_LIBRARY_PATH},
5653 or, as a last resort,
5656 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5657 running configure, for instance:
5660 LDFLAGS=-R/home/cbauer/lib ./configure
5665 You can also specify a @command{ginac-config} not in your path by
5666 setting the @env{GINACLIB_CONFIG} environment variable to the
5667 name of the executable
5670 If you move the GiNaC package from its installed location,
5671 you will either need to modify @command{ginac-config} script
5672 manually to point to the new location or rebuild GiNaC.
5683 --with-ginac-prefix=@var{PREFIX}
5684 --with-ginac-exec-prefix=@var{PREFIX}
5687 are provided to override the prefix and exec-prefix that were stored
5688 in the @command{ginac-config} shell script by GiNaC's configure. You are
5689 generally better off configuring GiNaC with the right path to begin with.
5693 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5694 @c node-name, next, previous, up
5695 @subsection Example of a package using @samp{AM_PATH_GINAC}
5697 The following shows how to build a simple package using automake
5698 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5701 #include <ginac/ginac.h>
5705 GiNaC::symbol x("x");
5706 GiNaC::ex a = GiNaC::sin(x);
5707 std::cout << "Derivative of " << a
5708 << " is " << a.diff(x) << std::endl;
5713 You should first read the introductory portions of the automake
5714 Manual, if you are not already familiar with it.
5716 Two files are needed, @file{configure.in}, which is used to build the
5720 dnl Process this file with autoconf to produce a configure script.
5722 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5728 AM_PATH_GINAC(0.9.0, [
5729 LIBS="$LIBS $GINACLIB_LIBS"
5730 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5731 ], AC_MSG_ERROR([need to have GiNaC installed]))
5736 The only command in this which is not standard for automake
5737 is the @samp{AM_PATH_GINAC} macro.
5739 That command does the following: If a GiNaC version greater or equal
5740 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5741 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5742 the error message `need to have GiNaC installed'
5744 And the @file{Makefile.am}, which will be used to build the Makefile.
5747 ## Process this file with automake to produce Makefile.in
5748 bin_PROGRAMS = simple
5749 simple_SOURCES = simple.cpp
5752 This @file{Makefile.am}, says that we are building a single executable,
5753 from a single sourcefile @file{simple.cpp}. Since every program
5754 we are building uses GiNaC we simply added the GiNaC options
5755 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5756 want to specify them on a per-program basis: for instance by
5760 simple_LDADD = $(GINACLIB_LIBS)
5761 INCLUDES = $(GINACLIB_CPPFLAGS)
5764 to the @file{Makefile.am}.
5766 To try this example out, create a new directory and add the three
5769 Now execute the following commands:
5772 $ automake --add-missing
5777 You now have a package that can be built in the normal fashion
5786 @node Bibliography, Concept Index, Example package, Top
5787 @c node-name, next, previous, up
5788 @appendix Bibliography
5793 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5796 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5799 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5802 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5805 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5806 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5809 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5810 James H. Davenport, Yvon Siret, and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
5811 Academic Press, London
5814 @cite{Computer Algebra Systems - A Practical Guide},
5815 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
5818 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5819 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5822 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
5827 @node Concept Index, , Bibliography, Top
5828 @c node-name, next, previous, up
5829 @unnumbered Concept Index