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) 1999-2001 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{} 1999-2001 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 linguistical 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{} 1999-2001 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:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package Tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
216 #include <ginac/ginac.h>
218 using namespace GiNaC;
220 ex HermitePoly(const symbol & x, int n)
222 ex HKer=exp(-pow(x, 2));
223 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
224 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
231 for (int i=0; i<6; ++i)
232 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
238 When run, this will type out
244 H_3(z) == -12*z+8*z^3
245 H_4(z) == -48*z^2+16*z^4+12
246 H_5(z) == 120*z-160*z^3+32*z^5
249 This method of generating the coefficients is of course far from optimal
250 for production purposes.
252 In order to show some more examples of what GiNaC can do we will now use
253 the @command{ginsh}, a simple GiNaC interactive shell that provides a
254 convenient window into GiNaC's capabilities.
257 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
258 @c node-name, next, previous, up
259 @section What it can do for you
261 @cindex @command{ginsh}
262 After invoking @command{ginsh} one can test and experiment with GiNaC's
263 features much like in other Computer Algebra Systems except that it does
264 not provide programming constructs like loops or conditionals. For a
265 concise description of the @command{ginsh} syntax we refer to its
266 accompanied man page. Suffice to say that assignments and comparisons in
267 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
270 It can manipulate arbitrary precision integers in a very fast way.
271 Rational numbers are automatically converted to fractions of coprime
276 369988485035126972924700782451696644186473100389722973815184405301748249
278 123329495011708990974900260817232214728824366796574324605061468433916083
285 Exact numbers are always retained as exact numbers and only evaluated as
286 floating point numbers if requested. For instance, with numeric
287 radicals is dealt pretty much as with symbols. Products of sums of them
291 > expand((1+a^(1/5)-a^(2/5))^3);
292 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
293 > expand((1+3^(1/5)-3^(2/5))^3);
295 > evalf((1+3^(1/5)-3^(2/5))^3);
296 0.33408977534118624228
299 The function @code{evalf} that was used above converts any number in
300 GiNaC's expressions into floating point numbers. This can be done to
301 arbitrary predefined accuracy:
305 0.14285714285714285714
309 0.1428571428571428571428571428571428571428571428571428571428571428571428
310 5714285714285714285714285714285714285
313 Exact numbers other than rationals that can be manipulated in GiNaC
314 include predefined constants like Archimedes' @code{Pi}. They can both
315 be used in symbolic manipulations (as an exact number) as well as in
316 numeric expressions (as an inexact number):
322 9.869604401089358619+x
326 11.869604401089358619
329 Built-in functions evaluate immediately to exact numbers if
330 this is possible. Conversions that can be safely performed are done
331 immediately; conversions that are not generally valid are not done:
342 (Note that converting the last input to @code{x} would allow one to
343 conclude that @code{42*Pi} is equal to @code{0}.)
345 Linear equation systems can be solved along with basic linear
346 algebra manipulations over symbolic expressions. In C++ GiNaC offers
347 a matrix class for this purpose but we can see what it can do using
348 @command{ginsh}'s bracket notation to type them in:
351 > lsolve(a+x*y==z,x);
353 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
355 > M = [ [1, 3], [-3, 2] ];
359 > charpoly(M,lambda);
361 > A = [ [1, 1], [2, -1] ];
364 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 Multivariate polynomials and rational functions may be expanded,
370 collected and normalized (i.e. converted to a ratio of two coprime
374 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
375 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
376 > b = x^2 + 4*x*y - y^2;
379 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
381 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
383 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
388 You can differentiate functions and expand them as Taylor or Laurent
389 series in a very natural syntax (the second argument of @code{series} is
390 a relation defining the evaluation point, the third specifies the
393 @cindex Zeta function
397 > series(sin(x),x==0,4);
399 > series(1/tan(x),x==0,4);
400 x^(-1)-1/3*x+Order(x^2)
401 > series(tgamma(x),x==0,3);
402 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
403 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
405 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
406 -(0.90747907608088628905)*x^2+Order(x^3)
407 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
408 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
409 -Euler-1/12+Order((x-1/2*Pi)^3)
412 Here we have made use of the @command{ginsh}-command @code{"} to pop the
413 previously evaluated element from @command{ginsh}'s internal stack.
415 If you ever wanted to convert units in C or C++ and found this is
416 cumbersome, here is the solution. Symbolic types can always be used as
417 tags for different types of objects. Converting from wrong units to the
418 metric system is now easy:
426 140613.91592783185568*kg*m^(-2)
430 @node Installation, Prerequisites, What it can do for you, Top
431 @c node-name, next, previous, up
432 @chapter Installation
435 GiNaC's installation follows the spirit of most GNU software. It is
436 easily installed on your system by three steps: configuration, build,
440 * Prerequisites:: Packages upon which GiNaC depends.
441 * Configuration:: How to configure GiNaC.
442 * Building GiNaC:: How to compile GiNaC.
443 * Installing GiNaC:: How to install GiNaC on your system.
447 @node Prerequisites, Configuration, Installation, Installation
448 @c node-name, next, previous, up
449 @section Prerequisites
451 In order to install GiNaC on your system, some prerequisites need to be
452 met. First of all, you need to have a C++-compiler adhering to the
453 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
454 development so if you have a different compiler you are on your own.
455 For the configuration to succeed you need a Posix compliant shell
456 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
457 by the built process as well, since some of the source files are
458 automatically generated by Perl scripts. Last but not least, Bruno
459 Haible's library @acronym{CLN} is extensively used and needs to be
460 installed on your system. Please get it either from
461 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
462 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
463 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
464 site} (it is covered by GPL) and install it prior to trying to install
465 GiNaC. The configure script checks if it can find it and if it cannot
466 it will refuse to continue.
469 @node Configuration, Building GiNaC, Prerequisites, Installation
470 @c node-name, next, previous, up
471 @section Configuration
472 @cindex configuration
475 To configure GiNaC means to prepare the source distribution for
476 building. It is done via a shell script called @command{configure} that
477 is shipped with the sources and was originally generated by GNU
478 Autoconf. Since a configure script generated by GNU Autoconf never
479 prompts, all customization must be done either via command line
480 parameters or environment variables. It accepts a list of parameters,
481 the complete set of which can be listed by calling it with the
482 @option{--help} option. The most important ones will be shortly
483 described in what follows:
488 @option{--disable-shared}: When given, this option switches off the
489 build of a shared library, i.e. a @file{.so} file. This may be convenient
490 when developing because it considerably speeds up compilation.
493 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
494 and headers are installed. It defaults to @file{/usr/local} which means
495 that the library is installed in the directory @file{/usr/local/lib},
496 the header files in @file{/usr/local/include/ginac} and the documentation
497 (like this one) into @file{/usr/local/share/doc/GiNaC}.
500 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
501 the library installed in some other directory than
502 @file{@var{PREFIX}/lib/}.
505 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
506 to have the header files installed in some other directory than
507 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
508 @option{--includedir=/usr/include} you will end up with the header files
509 sitting in the directory @file{/usr/include/ginac/}. Note that the
510 subdirectory @file{ginac} is enforced by this process in order to
511 keep the header files separated from others. This avoids some
512 clashes and allows for an easier deinstallation of GiNaC. This ought
513 to be considered A Good Thing (tm).
516 @option{--datadir=@var{DATADIR}}: This option may be given in case you
517 want to have the documentation installed in some other directory than
518 @file{@var{PREFIX}/share/doc/GiNaC/}.
522 In addition, you may specify some environment variables. @env{CXX}
523 holds the path and the name of the C++ compiler in case you want to
524 override the default in your path. (The @command{configure} script
525 searches your path for @command{c++}, @command{g++}, @command{gcc},
526 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
527 be very useful to define some compiler flags with the @env{CXXFLAGS}
528 environment variable, like optimization, debugging information and
529 warning levels. If omitted, it defaults to @option{-g
530 -O2}.@footnote{The @command{configure} script is itself generated from
531 the file @file{configure.in}. It is only distributed in packaged
532 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
533 must generate @command{configure} along with the various
534 @file{Makefile.in} by using the @command{autogen.sh} script.}
536 The whole process is illustrated in the following two
537 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
538 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
541 Here is a simple configuration for a site-wide GiNaC library assuming
542 everything is in default paths:
545 $ export CXXFLAGS="-Wall -O2"
549 And here is a configuration for a private static GiNaC library with
550 several components sitting in custom places (site-wide @acronym{GCC} and
551 private @acronym{CLN}). The compiler is pursuaded to be picky and full
552 assertions and debugging information are switched on:
555 $ export CXX=/usr/local/gnu/bin/c++
556 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
557 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
558 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
559 $ ./configure --disable-shared --prefix=$(HOME)
563 @node Building GiNaC, Installing GiNaC, Configuration, Installation
564 @c node-name, next, previous, up
565 @section Building GiNaC
566 @cindex building GiNaC
568 After proper configuration you should just build the whole
573 at the command prompt and go for a cup of coffee. The exact time it
574 takes to compile GiNaC depends not only on the speed of your machines
575 but also on other parameters, for instance what value for @env{CXXFLAGS}
576 you entered. Optimization may be very time-consuming.
578 Just to make sure GiNaC works properly you may run a collection of
579 regression tests by typing
585 This will compile some sample programs, run them and check the output
586 for correctness. The regression tests fall in three categories. First,
587 the so called @emph{exams} are performed, simple tests where some
588 predefined input is evaluated (like a pupils' exam). Second, the
589 @emph{checks} test the coherence of results among each other with
590 possible random input. Third, some @emph{timings} are performed, which
591 benchmark some predefined problems with different sizes and display the
592 CPU time used in seconds. Each individual test should return a message
593 @samp{passed}. This is mostly intended to be a QA-check if something
594 was broken during development, not a sanity check of your system. Some
595 of the tests in sections @emph{checks} and @emph{timings} may require
596 insane amounts of memory and CPU time. Feel free to kill them if your
597 machine catches fire. Another quite important intent is to allow people
598 to fiddle around with optimization.
600 Generally, the top-level Makefile runs recursively to the
601 subdirectories. It is therfore safe to go into any subdirectory
602 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
603 @var{target} there in case something went wrong.
606 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
607 @c node-name, next, previous, up
608 @section Installing GiNaC
611 To install GiNaC on your system, simply type
617 As described in the section about configuration the files will be
618 installed in the following directories (the directories will be created
619 if they don't already exist):
624 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
625 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
626 So will @file{libginac.so} unless the configure script was
627 given the option @option{--disable-shared}. The proper symlinks
628 will be established as well.
631 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
632 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
635 All documentation (HTML and Postscript) will be stuffed into
636 @file{@var{PREFIX}/share/doc/GiNaC/} (or
637 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
641 For the sake of completeness we will list some other useful make
642 targets: @command{make clean} deletes all files generated by
643 @command{make}, i.e. all the object files. In addition @command{make
644 distclean} removes all files generated by the configuration and
645 @command{make maintainer-clean} goes one step further and deletes files
646 that may require special tools to rebuild (like the @command{libtool}
647 for instance). Finally @command{make uninstall} removes the installed
648 library, header files and documentation@footnote{Uninstallation does not
649 work after you have called @command{make distclean} since the
650 @file{Makefile} is itself generated by the configuration from
651 @file{Makefile.in} and hence deleted by @command{make distclean}. There
652 are two obvious ways out of this dilemma. First, you can run the
653 configuration again with the same @var{PREFIX} thus creating a
654 @file{Makefile} with a working @samp{uninstall} target. Second, you can
655 do it by hand since you now know where all the files went during
659 @node Basic Concepts, Expressions, Installing GiNaC, Top
660 @c node-name, next, previous, up
661 @chapter Basic Concepts
663 This chapter will describe the different fundamental objects that can be
664 handled by GiNaC. But before doing so, it is worthwhile introducing you
665 to the more commonly used class of expressions, representing a flexible
666 meta-class for storing all mathematical objects.
669 * Expressions:: The fundamental GiNaC class.
670 * The Class Hierarchy:: Overview of GiNaC's classes.
671 * Error handling:: How the library reports errors.
672 * Symbols:: Symbolic objects.
673 * Numbers:: Numerical objects.
674 * Constants:: Pre-defined constants.
675 * Fundamental containers:: The power, add and mul classes.
676 * Lists:: Lists of expressions.
677 * Mathematical functions:: Mathematical functions.
678 * Relations:: Equality, Inequality and all that.
679 * Matrices:: Matrices.
680 * Indexed objects:: Handling indexed quantities.
681 * Non-commutative objects:: Algebras with non-commutative products.
685 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
686 @c node-name, next, previous, up
688 @cindex expression (class @code{ex})
691 The most common class of objects a user deals with is the expression
692 @code{ex}, representing a mathematical object like a variable, number,
693 function, sum, product, etc@dots{} Expressions may be put together to form
694 new expressions, passed as arguments to functions, and so on. Here is a
695 little collection of valid expressions:
698 ex MyEx1 = 5; // simple number
699 ex MyEx2 = x + 2*y; // polynomial in x and y
700 ex MyEx3 = (x + 1)/(x - 1); // rational expression
701 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
702 ex MyEx5 = MyEx4 + 1; // similar to above
705 Expressions are handles to other more fundamental objects, that often
706 contain other expressions thus creating a tree of expressions
707 (@xref{Internal Structures}, for particular examples). Most methods on
708 @code{ex} therefore run top-down through such an expression tree. For
709 example, the method @code{has()} scans recursively for occurrences of
710 something inside an expression. Thus, if you have declared @code{MyEx4}
711 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
712 the argument of @code{sin} and hence return @code{true}.
714 The next sections will outline the general picture of GiNaC's class
715 hierarchy and describe the classes of objects that are handled by
719 @node The Class Hierarchy, Error handling, Expressions, Basic Concepts
720 @c node-name, next, previous, up
721 @section The Class Hierarchy
723 GiNaC's class hierarchy consists of several classes representing
724 mathematical objects, all of which (except for @code{ex} and some
725 helpers) are internally derived from one abstract base class called
726 @code{basic}. You do not have to deal with objects of class
727 @code{basic}, instead you'll be dealing with symbols, numbers,
728 containers of expressions and so on.
732 To get an idea about what kinds of symbolic composits may be built we
733 have a look at the most important classes in the class hierarchy and
734 some of the relations among the classes:
736 @image{classhierarchy}
738 The abstract classes shown here (the ones without drop-shadow) are of no
739 interest for the user. They are used internally in order to avoid code
740 duplication if two or more classes derived from them share certain
741 features. An example is @code{expairseq}, a container for a sequence of
742 pairs each consisting of one expression and a number (@code{numeric}).
743 What @emph{is} visible to the user are the derived classes @code{add}
744 and @code{mul}, representing sums and products. @xref{Internal
745 Structures}, where these two classes are described in more detail. The
746 following table shortly summarizes what kinds of mathematical objects
747 are stored in the different classes:
750 @multitable @columnfractions .22 .78
751 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
752 @item @code{constant} @tab Constants like
759 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
760 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
761 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
762 @item @code{ncmul} @tab Products of non-commutative objects
763 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
768 @code{sqrt(}@math{2}@code{)}
771 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
772 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
773 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
774 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
775 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
776 @item @code{indexed} @tab Indexed object like @math{A_ij}
777 @item @code{tensor} @tab Special tensor like the delta and metric tensors
778 @item @code{idx} @tab Index of an indexed object
779 @item @code{varidx} @tab Index with variance
780 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
781 @item @code{wildcard} @tab Wildcard for pattern matching
786 @node Error handling, Symbols, The Class Hierarchy, Basic Concepts
787 @c node-name, next, previous, up
788 @section Error handling
790 @cindex @code{pole_error} (class)
792 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
793 generated by GiNaC are subclassed from the standard @code{exception} class
794 defined in the @file{<stdexcept>} header. In addition to the predefined
795 @code{logic_error}, @code{domain_error}, @code{out_of_range},
796 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
797 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
798 exception that gets thrown when trying to evaluate a mathematical function
801 The @code{pole_error} class has a member function
804 int pole_error::degree(void) const;
807 that returns the order of the singularity (or 0 when the pole is
808 logarithmic or the order is undefined).
810 When using GiNaC it is useful to arrange for exceptions to be catched in
811 the main program even if you don't want to do any special error handling.
812 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
813 default exception handler of your C++ compiler's run-time system which
814 ususally only aborts the program without giving any information what went
817 Here is an example for a @code{main()} function that catches and prints
818 exceptions generated by GiNaC:
823 #include <ginac/ginac.h>
825 using namespace GiNaC;
833 @} catch (exception &p) @{
834 cerr << p.what() << endl;
842 @node Symbols, Numbers, Error handling, Basic Concepts
843 @c node-name, next, previous, up
845 @cindex @code{symbol} (class)
846 @cindex hierarchy of classes
849 Symbols are for symbolic manipulation what atoms are for chemistry. You
850 can declare objects of class @code{symbol} as any other object simply by
851 saying @code{symbol x,y;}. There is, however, a catch in here having to
852 do with the fact that C++ is a compiled language. The information about
853 the symbol's name is thrown away by the compiler but at a later stage
854 you may want to print expressions holding your symbols. In order to
855 avoid confusion GiNaC's symbols are able to know their own name. This
856 is accomplished by declaring its name for output at construction time in
857 the fashion @code{symbol x("x");}. If you declare a symbol using the
858 default constructor (i.e. without string argument) the system will deal
859 out a unique name. That name may not be suitable for printing but for
860 internal routines when no output is desired it is often enough. We'll
861 come across examples of such symbols later in this tutorial.
863 This implies that the strings passed to symbols at construction time may
864 not be used for comparing two of them. It is perfectly legitimate to
865 write @code{symbol x("x"),y("x");} but it is likely to lead into
866 trouble. Here, @code{x} and @code{y} are different symbols and
867 statements like @code{x-y} will not be simplified to zero although the
868 output @code{x-x} looks funny. Such output may also occur when there
869 are two different symbols in two scopes, for instance when you call a
870 function that declares a symbol with a name already existent in a symbol
871 in the calling function. Again, comparing them (using @code{operator==}
872 for instance) will always reveal their difference. Watch out, please.
874 @cindex @code{subs()}
875 Although symbols can be assigned expressions for internal reasons, you
876 should not do it (and we are not going to tell you how it is done). If
877 you want to replace a symbol with something else in an expression, you
878 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
881 @node Numbers, Constants, Symbols, Basic Concepts
882 @c node-name, next, previous, up
884 @cindex @code{numeric} (class)
890 For storing numerical things, GiNaC uses Bruno Haible's library
891 @acronym{CLN}. The classes therein serve as foundation classes for
892 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
893 alternatively for Common Lisp Numbers. In order to find out more about
894 @acronym{CLN}'s internals the reader is refered to the documentation of
895 that library. @inforef{Introduction, , cln}, for more
896 information. Suffice to say that it is by itself build on top of another
897 library, the GNU Multiple Precision library @acronym{GMP}, which is an
898 extremely fast library for arbitrary long integers and rationals as well
899 as arbitrary precision floating point numbers. It is very commonly used
900 by several popular cryptographic applications. @acronym{CLN} extends
901 @acronym{GMP} by several useful things: First, it introduces the complex
902 number field over either reals (i.e. floating point numbers with
903 arbitrary precision) or rationals. Second, it automatically converts
904 rationals to integers if the denominator is unity and complex numbers to
905 real numbers if the imaginary part vanishes and also correctly treats
906 algebraic functions. Third it provides good implementations of
907 state-of-the-art algorithms for all trigonometric and hyperbolic
908 functions as well as for calculation of some useful constants.
910 The user can construct an object of class @code{numeric} in several
911 ways. The following example shows the four most important constructors.
912 It uses construction from C-integer, construction of fractions from two
913 integers, construction from C-float and construction from a string:
916 #include <ginac/ginac.h>
917 using namespace GiNaC;
921 numeric two = 2; // exact integer 2
922 numeric r(2,3); // exact fraction 2/3
923 numeric e(2.71828); // floating point number
924 numeric p = "3.14159265358979323846"; // constructor from string
925 // Trott's constant in scientific notation:
926 numeric trott("1.0841015122311136151E-2");
928 std::cout << two*p << std::endl; // floating point 6.283...
932 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
933 This would, however, call C's built-in operator @code{/} for integers
934 first and result in a numeric holding a plain integer 1. @strong{Never
935 use the operator @code{/} on integers} unless you know exactly what you
936 are doing! Use the constructor from two integers instead, as shown in
937 the example above. Writing @code{numeric(1)/2} may look funny but works
940 @cindex @code{Digits}
942 We have seen now the distinction between exact numbers and floating
943 point numbers. Clearly, the user should never have to worry about
944 dynamically created exact numbers, since their `exactness' always
945 determines how they ought to be handled, i.e. how `long' they are. The
946 situation is different for floating point numbers. Their accuracy is
947 controlled by one @emph{global} variable, called @code{Digits}. (For
948 those readers who know about Maple: it behaves very much like Maple's
949 @code{Digits}). All objects of class numeric that are constructed from
950 then on will be stored with a precision matching that number of decimal
954 #include <ginac/ginac.h>
956 using namespace GiNaC;
960 numeric three(3.0), one(1.0);
961 numeric x = one/three;
963 cout << "in " << Digits << " digits:" << endl;
965 cout << Pi.evalf() << endl;
977 The above example prints the following output to screen:
984 0.333333333333333333333333333333333333333333333333333333333333333333
985 3.14159265358979323846264338327950288419716939937510582097494459231
988 It should be clear that objects of class @code{numeric} should be used
989 for constructing numbers or for doing arithmetic with them. The objects
990 one deals with most of the time are the polymorphic expressions @code{ex}.
992 @subsection Tests on numbers
994 Once you have declared some numbers, assigned them to expressions and
995 done some arithmetic with them it is frequently desired to retrieve some
996 kind of information from them like asking whether that number is
997 integer, rational, real or complex. For those cases GiNaC provides
998 several useful methods. (Internally, they fall back to invocations of
999 certain CLN functions.)
1001 As an example, let's construct some rational number, multiply it with
1002 some multiple of its denominator and test what comes out:
1005 #include <ginac/ginac.h>
1006 using namespace std;
1007 using namespace GiNaC;
1009 // some very important constants:
1010 const numeric twentyone(21);
1011 const numeric ten(10);
1012 const numeric five(5);
1016 numeric answer = twentyone;
1019 cout << answer.is_integer() << endl; // false, it's 21/5
1021 cout << answer.is_integer() << endl; // true, it's 42 now!
1025 Note that the variable @code{answer} is constructed here as an integer
1026 by @code{numeric}'s copy constructor but in an intermediate step it
1027 holds a rational number represented as integer numerator and integer
1028 denominator. When multiplied by 10, the denominator becomes unity and
1029 the result is automatically converted to a pure integer again.
1030 Internally, the underlying @acronym{CLN} is responsible for this
1031 behaviour and we refer the reader to @acronym{CLN}'s documentation.
1032 Suffice to say that the same behaviour applies to complex numbers as
1033 well as return values of certain functions. Complex numbers are
1034 automatically converted to real numbers if the imaginary part becomes
1035 zero. The full set of tests that can be applied is listed in the
1039 @multitable @columnfractions .30 .70
1040 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1041 @item @code{.is_zero()}
1042 @tab @dots{}equal to zero
1043 @item @code{.is_positive()}
1044 @tab @dots{}not complex and greater than 0
1045 @item @code{.is_integer()}
1046 @tab @dots{}a (non-complex) integer
1047 @item @code{.is_pos_integer()}
1048 @tab @dots{}an integer and greater than 0
1049 @item @code{.is_nonneg_integer()}
1050 @tab @dots{}an integer and greater equal 0
1051 @item @code{.is_even()}
1052 @tab @dots{}an even integer
1053 @item @code{.is_odd()}
1054 @tab @dots{}an odd integer
1055 @item @code{.is_prime()}
1056 @tab @dots{}a prime integer (probabilistic primality test)
1057 @item @code{.is_rational()}
1058 @tab @dots{}an exact rational number (integers are rational, too)
1059 @item @code{.is_real()}
1060 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1061 @item @code{.is_cinteger()}
1062 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1063 @item @code{.is_crational()}
1064 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1069 @node Constants, Fundamental containers, Numbers, Basic Concepts
1070 @c node-name, next, previous, up
1072 @cindex @code{constant} (class)
1075 @cindex @code{Catalan}
1076 @cindex @code{Euler}
1077 @cindex @code{evalf()}
1078 Constants behave pretty much like symbols except that they return some
1079 specific number when the method @code{.evalf()} is called.
1081 The predefined known constants are:
1084 @multitable @columnfractions .14 .30 .56
1085 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1087 @tab Archimedes' constant
1088 @tab 3.14159265358979323846264338327950288
1089 @item @code{Catalan}
1090 @tab Catalan's constant
1091 @tab 0.91596559417721901505460351493238411
1093 @tab Euler's (or Euler-Mascheroni) constant
1094 @tab 0.57721566490153286060651209008240243
1099 @node Fundamental containers, Lists, Constants, Basic Concepts
1100 @c node-name, next, previous, up
1101 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1105 @cindex @code{power}
1107 Simple polynomial expressions are written down in GiNaC pretty much like
1108 in other CAS or like expressions involving numerical variables in C.
1109 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1110 been overloaded to achieve this goal. When you run the following
1111 code snippet, the constructor for an object of type @code{mul} is
1112 automatically called to hold the product of @code{a} and @code{b} and
1113 then the constructor for an object of type @code{add} is called to hold
1114 the sum of that @code{mul} object and the number one:
1118 symbol a("a"), b("b");
1123 @cindex @code{pow()}
1124 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1125 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1126 construction is necessary since we cannot safely overload the constructor
1127 @code{^} in C++ to construct a @code{power} object. If we did, it would
1128 have several counterintuitive and undesired effects:
1132 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1134 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1135 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1136 interpret this as @code{x^(a^b)}.
1138 Also, expressions involving integer exponents are very frequently used,
1139 which makes it even more dangerous to overload @code{^} since it is then
1140 hard to distinguish between the semantics as exponentiation and the one
1141 for exclusive or. (It would be embarassing to return @code{1} where one
1142 has requested @code{2^3}.)
1145 @cindex @command{ginsh}
1146 All effects are contrary to mathematical notation and differ from the
1147 way most other CAS handle exponentiation, therefore overloading @code{^}
1148 is ruled out for GiNaC's C++ part. The situation is different in
1149 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1150 that the other frequently used exponentiation operator @code{**} does
1151 not exist at all in C++).
1153 To be somewhat more precise, objects of the three classes described
1154 here, are all containers for other expressions. An object of class
1155 @code{power} is best viewed as a container with two slots, one for the
1156 basis, one for the exponent. All valid GiNaC expressions can be
1157 inserted. However, basic transformations like simplifying
1158 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1159 when this is mathematically possible. If we replace the outer exponent
1160 three in the example by some symbols @code{a}, the simplification is not
1161 safe and will not be performed, since @code{a} might be @code{1/2} and
1164 Objects of type @code{add} and @code{mul} are containers with an
1165 arbitrary number of slots for expressions to be inserted. Again, simple
1166 and safe simplifications are carried out like transforming
1167 @code{3*x+4-x} to @code{2*x+4}.
1169 The general rule is that when you construct such objects, GiNaC
1170 automatically creates them in canonical form, which might differ from
1171 the form you typed in your program. This allows for rapid comparison of
1172 expressions, since after all @code{a-a} is simply zero. Note, that the
1173 canonical form is not necessarily lexicographical ordering or in any way
1174 easily guessable. It is only guaranteed that constructing the same
1175 expression twice, either implicitly or explicitly, results in the same
1179 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1180 @c node-name, next, previous, up
1181 @section Lists of expressions
1182 @cindex @code{lst} (class)
1184 @cindex @code{nops()}
1186 @cindex @code{append()}
1187 @cindex @code{prepend()}
1188 @cindex @code{remove_first()}
1189 @cindex @code{remove_last()}
1191 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1192 expressions. These are sometimes used to supply a variable number of
1193 arguments of the same type to GiNaC methods such as @code{subs()} and
1194 @code{to_rational()}, so you should have a basic understanding about them.
1196 Lists of up to 16 expressions can be directly constructed from single
1201 symbol x("x"), y("y");
1202 lst l(x, 2, y, x+y);
1203 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1207 Use the @code{nops()} method to determine the size (number of expressions) of
1208 a list and the @code{op()} method to access individual elements:
1212 cout << l.nops() << endl; // prints '4'
1213 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1217 You can append or prepend an expression to a list with the @code{append()}
1218 and @code{prepend()} methods:
1222 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1223 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1227 Finally you can remove the first or last element of a list with
1228 @code{remove_first()} and @code{remove_last()}:
1232 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1233 l.remove_last(); // l is now @{x, 2, y, x+y@}
1238 @node Mathematical functions, Relations, Lists, Basic Concepts
1239 @c node-name, next, previous, up
1240 @section Mathematical functions
1241 @cindex @code{function} (class)
1242 @cindex trigonometric function
1243 @cindex hyperbolic function
1245 There are quite a number of useful functions hard-wired into GiNaC. For
1246 instance, all trigonometric and hyperbolic functions are implemented
1247 (@xref{Built-in Functions}, for a complete list).
1249 These functions (better called @emph{pseudofunctions}) are all objects
1250 of class @code{function}. They accept one or more expressions as
1251 arguments and return one expression. If the arguments are not
1252 numerical, the evaluation of the function may be halted, as it does in
1253 the next example, showing how a function returns itself twice and
1254 finally an expression that may be really useful:
1256 @cindex Gamma function
1257 @cindex @code{subs()}
1260 symbol x("x"), y("y");
1262 cout << tgamma(foo) << endl;
1263 // -> tgamma(x+(1/2)*y)
1264 ex bar = foo.subs(y==1);
1265 cout << tgamma(bar) << endl;
1267 ex foobar = bar.subs(x==7);
1268 cout << tgamma(foobar) << endl;
1269 // -> (135135/128)*Pi^(1/2)
1273 Besides evaluation most of these functions allow differentiation, series
1274 expansion and so on. Read the next chapter in order to learn more about
1277 It must be noted that these pseudofunctions are created by inline
1278 functions, where the argument list is templated. This means that
1279 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1280 @code{sin(ex(1))} and will therefore not result in a floating point
1281 numeber. Unless of course the function prototype is explicitly
1282 overridden -- which is the case for arguments of type @code{numeric}
1283 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1284 point number of class @code{numeric} you should call
1285 @code{sin(numeric(1))}. This is almost the same as calling
1286 @code{sin(1).evalf()} except that the latter will return a numeric
1287 wrapped inside an @code{ex}.
1290 @node Relations, Matrices, Mathematical functions, Basic Concepts
1291 @c node-name, next, previous, up
1293 @cindex @code{relational} (class)
1295 Sometimes, a relation holding between two expressions must be stored
1296 somehow. The class @code{relational} is a convenient container for such
1297 purposes. A relation is by definition a container for two @code{ex} and
1298 a relation between them that signals equality, inequality and so on.
1299 They are created by simply using the C++ operators @code{==}, @code{!=},
1300 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1302 @xref{Mathematical functions}, for examples where various applications
1303 of the @code{.subs()} method show how objects of class relational are
1304 used as arguments. There they provide an intuitive syntax for
1305 substitutions. They are also used as arguments to the @code{ex::series}
1306 method, where the left hand side of the relation specifies the variable
1307 to expand in and the right hand side the expansion point. They can also
1308 be used for creating systems of equations that are to be solved for
1309 unknown variables. But the most common usage of objects of this class
1310 is rather inconspicuous in statements of the form @code{if
1311 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1312 conversion from @code{relational} to @code{bool} takes place. Note,
1313 however, that @code{==} here does not perform any simplifications, hence
1314 @code{expand()} must be called explicitly.
1317 @node Matrices, Indexed objects, Relations, Basic Concepts
1318 @c node-name, next, previous, up
1320 @cindex @code{matrix} (class)
1322 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1323 matrix with @math{m} rows and @math{n} columns are accessed with two
1324 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1325 second one in the range 0@dots{}@math{n-1}.
1327 There are a couple of ways to construct matrices, with or without preset
1331 matrix::matrix(unsigned r, unsigned c);
1332 matrix::matrix(unsigned r, unsigned c, const lst & l);
1333 ex lst_to_matrix(const lst & l);
1334 ex diag_matrix(const lst & l);
1337 The first two functions are @code{matrix} constructors which create a matrix
1338 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1339 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1340 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1341 from a list of lists, each list representing a matrix row. Finally,
1342 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1343 elements. Note that the last two functions return expressions, not matrix
1346 Matrix elements can be accessed and set using the parenthesis (function call)
1350 const ex & matrix::operator()(unsigned r, unsigned c) const;
1351 ex & matrix::operator()(unsigned r, unsigned c);
1354 It is also possible to access the matrix elements in a linear fashion with
1355 the @code{op()} method. But C++-style subscripting with square brackets
1356 @samp{[]} is not available.
1358 Here are a couple of examples that all construct the same 2x2 diagonal
1363 symbol a("a"), b("b");
1371 e = matrix(2, 2, lst(a, 0, 0, b));
1373 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1375 e = diag_matrix(lst(a, b));
1382 @cindex @code{transpose()}
1383 @cindex @code{inverse()}
1384 There are three ways to do arithmetic with matrices. The first (and most
1385 efficient one) is to use the methods provided by the @code{matrix} class:
1388 matrix matrix::add(const matrix & other) const;
1389 matrix matrix::sub(const matrix & other) const;
1390 matrix matrix::mul(const matrix & other) const;
1391 matrix matrix::mul_scalar(const ex & other) const;
1392 matrix matrix::pow(const ex & expn) const;
1393 matrix matrix::transpose(void) const;
1394 matrix matrix::inverse(void) const;
1397 All of these methods return the result as a new matrix object. Here is an
1398 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1403 matrix A(2, 2, lst(1, 2, 3, 4));
1404 matrix B(2, 2, lst(-1, 0, 2, 1));
1405 matrix C(2, 2, lst(8, 4, 2, 1));
1407 matrix result = A.mul(B).sub(C.mul_scalar(2));
1408 cout << result << endl;
1409 // -> [[-13,-6],[1,2]]
1414 @cindex @code{evalm()}
1415 The second (and probably the most natural) way is to construct an expression
1416 containing matrices with the usual arithmetic operators and @code{pow()}.
1417 For efficiency reasons, expressions with sums, products and powers of
1418 matrices are not automatically evaluated in GiNaC. You have to call the
1422 ex ex::evalm() const;
1425 to obtain the result:
1432 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1433 cout << e.evalm() << endl;
1434 // -> [[-13,-6],[1,2]]
1439 The non-commutativity of the product @code{A*B} in this example is
1440 automatically recognized by GiNaC. There is no need to use a special
1441 operator here. @xref{Non-commutative objects}, for more information about
1442 dealing with non-commutative expressions.
1444 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1445 to perform the arithmetic:
1450 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1451 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1453 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1454 cout << e.simplify_indexed() << endl;
1455 // -> [[-13,-6],[1,2]].i.j
1459 Using indices is most useful when working with rectangular matrices and
1460 one-dimensional vectors because you don't have to worry about having to
1461 transpose matrices before multiplying them. @xref{Indexed objects}, for
1462 more information about using matrices with indices, and about indices in
1465 The @code{matrix} class provides a couple of additional methods for
1466 computing determinants, traces, and characteristic polynomials:
1469 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1470 ex matrix::trace(void) const;
1471 ex matrix::charpoly(const symbol & lambda) const;
1474 The @samp{algo} argument of @code{determinant()} allows to select between
1475 different algorithms for calculating the determinant. The possible values
1476 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1477 heuristic to automatically select an algorithm that is likely to give the
1478 result most quickly.
1481 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1482 @c node-name, next, previous, up
1483 @section Indexed objects
1485 GiNaC allows you to handle expressions containing general indexed objects in
1486 arbitrary spaces. It is also able to canonicalize and simplify such
1487 expressions and perform symbolic dummy index summations. There are a number
1488 of predefined indexed objects provided, like delta and metric tensors.
1490 There are few restrictions placed on indexed objects and their indices and
1491 it is easy to construct nonsense expressions, but our intention is to
1492 provide a general framework that allows you to implement algorithms with
1493 indexed quantities, getting in the way as little as possible.
1495 @cindex @code{idx} (class)
1496 @cindex @code{indexed} (class)
1497 @subsection Indexed quantities and their indices
1499 Indexed expressions in GiNaC are constructed of two special types of objects,
1500 @dfn{index objects} and @dfn{indexed objects}.
1504 @cindex contravariant
1507 @item Index objects are of class @code{idx} or a subclass. Every index has
1508 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1509 the index lives in) which can both be arbitrary expressions but are usually
1510 a number or a simple symbol. In addition, indices of class @code{varidx} have
1511 a @dfn{variance} (they can be co- or contravariant), and indices of class
1512 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1514 @item Indexed objects are of class @code{indexed} or a subclass. They
1515 contain a @dfn{base expression} (which is the expression being indexed), and
1516 one or more indices.
1520 @strong{Note:} when printing expressions, covariant indices and indices
1521 without variance are denoted @samp{.i} while contravariant indices are
1522 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1523 value. In the following, we are going to use that notation in the text so
1524 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1525 not visible in the output.
1527 A simple example shall illustrate the concepts:
1530 #include <ginac/ginac.h>
1531 using namespace std;
1532 using namespace GiNaC;
1536 symbol i_sym("i"), j_sym("j");
1537 idx i(i_sym, 3), j(j_sym, 3);
1540 cout << indexed(A, i, j) << endl;
1545 The @code{idx} constructor takes two arguments, the index value and the
1546 index dimension. First we define two index objects, @code{i} and @code{j},
1547 both with the numeric dimension 3. The value of the index @code{i} is the
1548 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1549 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1550 construct an expression containing one indexed object, @samp{A.i.j}. It has
1551 the symbol @code{A} as its base expression and the two indices @code{i} and
1554 Note the difference between the indices @code{i} and @code{j} which are of
1555 class @code{idx}, and the index values which are the sybols @code{i_sym}
1556 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1557 or numbers but must be index objects. For example, the following is not
1558 correct and will raise an exception:
1561 symbol i("i"), j("j");
1562 e = indexed(A, i, j); // ERROR: indices must be of type idx
1565 You can have multiple indexed objects in an expression, index values can
1566 be numeric, and index dimensions symbolic:
1570 symbol B("B"), dim("dim");
1571 cout << 4 * indexed(A, i)
1572 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1577 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1578 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1579 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1580 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1581 @code{simplify_indexed()} for that, see below).
1583 In fact, base expressions, index values and index dimensions can be
1584 arbitrary expressions:
1588 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1593 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1594 get an error message from this but you will probably not be able to do
1595 anything useful with it.
1597 @cindex @code{get_value()}
1598 @cindex @code{get_dimension()}
1602 ex idx::get_value(void);
1603 ex idx::get_dimension(void);
1606 return the value and dimension of an @code{idx} object. If you have an index
1607 in an expression, such as returned by calling @code{.op()} on an indexed
1608 object, you can get a reference to the @code{idx} object with the function
1609 @code{ex_to<idx>()} on the expression.
1611 There are also the methods
1614 bool idx::is_numeric(void);
1615 bool idx::is_symbolic(void);
1616 bool idx::is_dim_numeric(void);
1617 bool idx::is_dim_symbolic(void);
1620 for checking whether the value and dimension are numeric or symbolic
1621 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1622 About Expressions}) returns information about the index value.
1624 @cindex @code{varidx} (class)
1625 If you need co- and contravariant indices, use the @code{varidx} class:
1629 symbol mu_sym("mu"), nu_sym("nu");
1630 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1631 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1633 cout << indexed(A, mu, nu) << endl;
1635 cout << indexed(A, mu_co, nu) << endl;
1637 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1642 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1643 co- or contravariant. The default is a contravariant (upper) index, but
1644 this can be overridden by supplying a third argument to the @code{varidx}
1645 constructor. The two methods
1648 bool varidx::is_covariant(void);
1649 bool varidx::is_contravariant(void);
1652 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1653 to get the object reference from an expression). There's also the very useful
1657 ex varidx::toggle_variance(void);
1660 which makes a new index with the same value and dimension but the opposite
1661 variance. By using it you only have to define the index once.
1663 @cindex @code{spinidx} (class)
1664 The @code{spinidx} class provides dotted and undotted variant indices, as
1665 used in the Weyl-van-der-Waerden spinor formalism:
1669 symbol K("K"), C_sym("C"), D_sym("D");
1670 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1671 // contravariant, undotted
1672 spinidx C_co(C_sym, 2, true); // covariant index
1673 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1674 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1676 cout << indexed(K, C, D) << endl;
1678 cout << indexed(K, C_co, D_dot) << endl;
1680 cout << indexed(K, D_co_dot, D) << endl;
1685 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1686 dotted or undotted. The default is undotted but this can be overridden by
1687 supplying a fourth argument to the @code{spinidx} constructor. The two
1691 bool spinidx::is_dotted(void);
1692 bool spinidx::is_undotted(void);
1695 allow you to check whether or not a @code{spinidx} object is dotted (use
1696 @code{ex_to<spinidx>()} to get the object reference from an expression).
1697 Finally, the two methods
1700 ex spinidx::toggle_dot(void);
1701 ex spinidx::toggle_variance_dot(void);
1704 create a new index with the same value and dimension but opposite dottedness
1705 and the same or opposite variance.
1707 @subsection Substituting indices
1709 @cindex @code{subs()}
1710 Sometimes you will want to substitute one symbolic index with another
1711 symbolic or numeric index, for example when calculating one specific element
1712 of a tensor expression. This is done with the @code{.subs()} method, as it
1713 is done for symbols (see @ref{Substituting Expressions}).
1715 You have two possibilities here. You can either substitute the whole index
1716 by another index or expression:
1720 ex e = indexed(A, mu_co);
1721 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1722 // -> A.mu becomes A~nu
1723 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1724 // -> A.mu becomes A~0
1725 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1726 // -> A.mu becomes A.0
1730 The third example shows that trying to replace an index with something that
1731 is not an index will substitute the index value instead.
1733 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1738 ex e = indexed(A, mu_co);
1739 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1740 // -> A.mu becomes A.nu
1741 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1742 // -> A.mu becomes A.0
1746 As you see, with the second method only the value of the index will get
1747 substituted. Its other properties, including its dimension, remain unchanged.
1748 If you want to change the dimension of an index you have to substitute the
1749 whole index by another one with the new dimension.
1751 Finally, substituting the base expression of an indexed object works as
1756 ex e = indexed(A, mu_co);
1757 cout << e << " becomes " << e.subs(A == A+B) << endl;
1758 // -> A.mu becomes (B+A).mu
1762 @subsection Symmetries
1763 @cindex @code{symmetry} (class)
1764 @cindex @code{sy_none()}
1765 @cindex @code{sy_symm()}
1766 @cindex @code{sy_anti()}
1767 @cindex @code{sy_cycl()}
1769 Indexed objects can have certain symmetry properties with respect to their
1770 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1771 that is constructed with the helper functions
1774 symmetry sy_none(...);
1775 symmetry sy_symm(...);
1776 symmetry sy_anti(...);
1777 symmetry sy_cycl(...);
1780 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1781 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1782 represents a cyclic symmetry. Each of these functions accepts up to four
1783 arguments which can be either symmetry objects themselves or unsigned integer
1784 numbers that represent an index position (counting from 0). A symmetry
1785 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1786 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1789 Here are some examples of symmetry definitions:
1794 e = indexed(A, i, j);
1795 e = indexed(A, sy_none(), i, j); // equivalent
1796 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1798 // Symmetric in all three indices:
1799 e = indexed(A, sy_symm(), i, j, k);
1800 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1801 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1802 // different canonical order
1804 // Symmetric in the first two indices only:
1805 e = indexed(A, sy_symm(0, 1), i, j, k);
1806 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1808 // Antisymmetric in the first and last index only (index ranges need not
1810 e = indexed(A, sy_anti(0, 2), i, j, k);
1811 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1813 // An example of a mixed symmetry: antisymmetric in the first two and
1814 // last two indices, symmetric when swapping the first and last index
1815 // pairs (like the Riemann curvature tensor):
1816 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1818 // Cyclic symmetry in all three indices:
1819 e = indexed(A, sy_cycl(), i, j, k);
1820 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1822 // The following examples are invalid constructions that will throw
1823 // an exception at run time.
1825 // An index may not appear multiple times:
1826 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1827 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1829 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1830 // same number of indices:
1831 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1833 // And of course, you cannot specify indices which are not there:
1834 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1838 If you need to specify more than four indices, you have to use the
1839 @code{.add()} method of the @code{symmetry} class. For example, to specify
1840 full symmetry in the first six indices you would write
1841 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1843 If an indexed object has a symmetry, GiNaC will automatically bring the
1844 indices into a canonical order which allows for some immediate simplifications:
1848 cout << indexed(A, sy_symm(), i, j)
1849 + indexed(A, sy_symm(), j, i) << endl;
1851 cout << indexed(B, sy_anti(), i, j)
1852 + indexed(B, sy_anti(), j, i) << endl;
1854 cout << indexed(B, sy_anti(), i, j, k)
1855 + indexed(B, sy_anti(), j, i, k) << endl;
1860 @cindex @code{get_free_indices()}
1862 @subsection Dummy indices
1864 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1865 that a summation over the index range is implied. Symbolic indices which are
1866 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1867 dummy nor free indices.
1869 To be recognized as a dummy index pair, the two indices must be of the same
1870 class and dimension and their value must be the same single symbol (an index
1871 like @samp{2*n+1} is never a dummy index). If the indices are of class
1872 @code{varidx} they must also be of opposite variance; if they are of class
1873 @code{spinidx} they must be both dotted or both undotted.
1875 The method @code{.get_free_indices()} returns a vector containing the free
1876 indices of an expression. It also checks that the free indices of the terms
1877 of a sum are consistent:
1881 symbol A("A"), B("B"), C("C");
1883 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1884 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1886 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1887 cout << exprseq(e.get_free_indices()) << endl;
1889 // 'j' and 'l' are dummy indices
1891 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1892 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1894 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1895 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1896 cout << exprseq(e.get_free_indices()) << endl;
1898 // 'nu' is a dummy index, but 'sigma' is not
1900 e = indexed(A, mu, mu);
1901 cout << exprseq(e.get_free_indices()) << endl;
1903 // 'mu' is not a dummy index because it appears twice with the same
1906 e = indexed(A, mu, nu) + 42;
1907 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1908 // this will throw an exception:
1909 // "add::get_free_indices: inconsistent indices in sum"
1913 @cindex @code{simplify_indexed()}
1914 @subsection Simplifying indexed expressions
1916 In addition to the few automatic simplifications that GiNaC performs on
1917 indexed expressions (such as re-ordering the indices of symmetric tensors
1918 and calculating traces and convolutions of matrices and predefined tensors)
1922 ex ex::simplify_indexed(void);
1923 ex ex::simplify_indexed(const scalar_products & sp);
1926 that performs some more expensive operations:
1929 @item it checks the consistency of free indices in sums in the same way
1930 @code{get_free_indices()} does
1931 @item it tries to give dumy indices that appear in different terms of a sum
1932 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1933 @item it (symbolically) calculates all possible dummy index summations/contractions
1934 with the predefined tensors (this will be explained in more detail in the
1936 @item it detects contractions that vanish for symmetry reasons, for example
1937 the contraction of a symmetric and a totally antisymmetric tensor
1938 @item as a special case of dummy index summation, it can replace scalar products
1939 of two tensors with a user-defined value
1942 The last point is done with the help of the @code{scalar_products} class
1943 which is used to store scalar products with known values (this is not an
1944 arithmetic class, you just pass it to @code{simplify_indexed()}):
1948 symbol A("A"), B("B"), C("C"), i_sym("i");
1952 sp.add(A, B, 0); // A and B are orthogonal
1953 sp.add(A, C, 0); // A and C are orthogonal
1954 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1956 e = indexed(A + B, i) * indexed(A + C, i);
1958 // -> (B+A).i*(A+C).i
1960 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1966 The @code{scalar_products} object @code{sp} acts as a storage for the
1967 scalar products added to it with the @code{.add()} method. This method
1968 takes three arguments: the two expressions of which the scalar product is
1969 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1970 @code{simplify_indexed()} will replace all scalar products of indexed
1971 objects that have the symbols @code{A} and @code{B} as base expressions
1972 with the single value 0. The number, type and dimension of the indices
1973 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1975 @cindex @code{expand()}
1976 The example above also illustrates a feature of the @code{expand()} method:
1977 if passed the @code{expand_indexed} option it will distribute indices
1978 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1980 @cindex @code{tensor} (class)
1981 @subsection Predefined tensors
1983 Some frequently used special tensors such as the delta, epsilon and metric
1984 tensors are predefined in GiNaC. They have special properties when
1985 contracted with other tensor expressions and some of them have constant
1986 matrix representations (they will evaluate to a number when numeric
1987 indices are specified).
1989 @cindex @code{delta_tensor()}
1990 @subsubsection Delta tensor
1992 The delta tensor takes two indices, is symmetric and has the matrix
1993 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
1994 @code{delta_tensor()}:
1998 symbol A("A"), B("B");
2000 idx i(symbol("i"), 3), j(symbol("j"), 3),
2001 k(symbol("k"), 3), l(symbol("l"), 3);
2003 ex e = indexed(A, i, j) * indexed(B, k, l)
2004 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2005 cout << e.simplify_indexed() << endl;
2008 cout << delta_tensor(i, i) << endl;
2013 @cindex @code{metric_tensor()}
2014 @subsubsection General metric tensor
2016 The function @code{metric_tensor()} creates a general symmetric metric
2017 tensor with two indices that can be used to raise/lower tensor indices. The
2018 metric tensor is denoted as @samp{g} in the output and if its indices are of
2019 mixed variance it is automatically replaced by a delta tensor:
2025 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2027 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2028 cout << e.simplify_indexed() << endl;
2031 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2032 cout << e.simplify_indexed() << endl;
2035 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2036 * metric_tensor(nu, rho);
2037 cout << e.simplify_indexed() << endl;
2040 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2041 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2042 + indexed(A, mu.toggle_variance(), rho));
2043 cout << e.simplify_indexed() << endl;
2048 @cindex @code{lorentz_g()}
2049 @subsubsection Minkowski metric tensor
2051 The Minkowski metric tensor is a special metric tensor with a constant
2052 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2053 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2054 It is created with the function @code{lorentz_g()} (although it is output as
2059 varidx mu(symbol("mu"), 4);
2061 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2062 * lorentz_g(mu, varidx(0, 4)); // negative signature
2063 cout << e.simplify_indexed() << endl;
2066 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2067 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2068 cout << e.simplify_indexed() << endl;
2073 @cindex @code{spinor_metric()}
2074 @subsubsection Spinor metric tensor
2076 The function @code{spinor_metric()} creates an antisymmetric tensor with
2077 two indices that is used to raise/lower indices of 2-component spinors.
2078 It is output as @samp{eps}:
2084 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2085 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2087 e = spinor_metric(A, B) * indexed(psi, B_co);
2088 cout << e.simplify_indexed() << endl;
2091 e = spinor_metric(A, B) * indexed(psi, A_co);
2092 cout << e.simplify_indexed() << endl;
2095 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2096 cout << e.simplify_indexed() << endl;
2099 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2100 cout << e.simplify_indexed() << endl;
2103 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2104 cout << e.simplify_indexed() << endl;
2107 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2108 cout << e.simplify_indexed() << endl;
2113 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2115 @cindex @code{epsilon_tensor()}
2116 @cindex @code{lorentz_eps()}
2117 @subsubsection Epsilon tensor
2119 The epsilon tensor is totally antisymmetric, its number of indices is equal
2120 to the dimension of the index space (the indices must all be of the same
2121 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2122 defined to be 1. Its behaviour with indices that have a variance also
2123 depends on the signature of the metric. Epsilon tensors are output as
2126 There are three functions defined to create epsilon tensors in 2, 3 and 4
2130 ex epsilon_tensor(const ex & i1, const ex & i2);
2131 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2132 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2135 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2136 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2137 Minkowski space (the last @code{bool} argument specifies whether the metric
2138 has negative or positive signature, as in the case of the Minkowski metric
2143 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2144 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2145 e = lorentz_eps(mu, nu, rho, sig) *
2146 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2147 cout << simplify_indexed(e) << endl;
2148 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2150 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2151 symbol A("A"), B("B");
2152 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2153 cout << simplify_indexed(e) << endl;
2154 // -> -B.k*A.j*eps.i.k.j
2155 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2156 cout << simplify_indexed(e) << endl;
2161 @subsection Linear algebra
2163 The @code{matrix} class can be used with indices to do some simple linear
2164 algebra (linear combinations and products of vectors and matrices, traces
2165 and scalar products):
2169 idx i(symbol("i"), 2), j(symbol("j"), 2);
2170 symbol x("x"), y("y");
2172 // A is a 2x2 matrix, X is a 2x1 vector
2173 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2175 cout << indexed(A, i, i) << endl;
2178 ex e = indexed(A, i, j) * indexed(X, j);
2179 cout << e.simplify_indexed() << endl;
2180 // -> [[2*y+x],[4*y+3*x]].i
2182 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2183 cout << e.simplify_indexed() << endl;
2184 // -> [[3*y+3*x,6*y+2*x]].j
2188 You can of course obtain the same results with the @code{matrix::add()},
2189 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2190 but with indices you don't have to worry about transposing matrices.
2192 Matrix indices always start at 0 and their dimension must match the number
2193 of rows/columns of the matrix. Matrices with one row or one column are
2194 vectors and can have one or two indices (it doesn't matter whether it's a
2195 row or a column vector). Other matrices must have two indices.
2197 You should be careful when using indices with variance on matrices. GiNaC
2198 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2199 @samp{F.mu.nu} are different matrices. In this case you should use only
2200 one form for @samp{F} and explicitly multiply it with a matrix representation
2201 of the metric tensor.
2204 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2205 @c node-name, next, previous, up
2206 @section Non-commutative objects
2208 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2209 non-commutative objects are built-in which are mostly of use in high energy
2213 @item Clifford (Dirac) algebra (class @code{clifford})
2214 @item su(3) Lie algebra (class @code{color})
2215 @item Matrices (unindexed) (class @code{matrix})
2218 The @code{clifford} and @code{color} classes are subclasses of
2219 @code{indexed} because the elements of these algebras ususally carry
2220 indices. The @code{matrix} class is described in more detail in
2223 Unlike most computer algebra systems, GiNaC does not primarily provide an
2224 operator (often denoted @samp{&*}) for representing inert products of
2225 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2226 classes of objects involved, and non-commutative products are formed with
2227 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2228 figuring out by itself which objects commute and will group the factors
2229 by their class. Consider this example:
2233 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2234 idx a(symbol("a"), 8), b(symbol("b"), 8);
2235 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2237 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2241 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2242 groups the non-commutative factors (the gammas and the su(3) generators)
2243 together while preserving the order of factors within each class (because
2244 Clifford objects commute with color objects). The resulting expression is a
2245 @emph{commutative} product with two factors that are themselves non-commutative
2246 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2247 parentheses are placed around the non-commutative products in the output.
2249 @cindex @code{ncmul} (class)
2250 Non-commutative products are internally represented by objects of the class
2251 @code{ncmul}, as opposed to commutative products which are handled by the
2252 @code{mul} class. You will normally not have to worry about this distinction,
2255 The advantage of this approach is that you never have to worry about using
2256 (or forgetting to use) a special operator when constructing non-commutative
2257 expressions. Also, non-commutative products in GiNaC are more intelligent
2258 than in other computer algebra systems; they can, for example, automatically
2259 canonicalize themselves according to rules specified in the implementation
2260 of the non-commutative classes. The drawback is that to work with other than
2261 the built-in algebras you have to implement new classes yourself. Symbols
2262 always commute and it's not possible to construct non-commutative products
2263 using symbols to represent the algebra elements or generators. User-defined
2264 functions can, however, be specified as being non-commutative.
2266 @cindex @code{return_type()}
2267 @cindex @code{return_type_tinfo()}
2268 Information about the commutativity of an object or expression can be
2269 obtained with the two member functions
2272 unsigned ex::return_type(void) const;
2273 unsigned ex::return_type_tinfo(void) const;
2276 The @code{return_type()} function returns one of three values (defined in
2277 the header file @file{flags.h}), corresponding to three categories of
2278 expressions in GiNaC:
2281 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2282 classes are of this kind.
2283 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2284 certain class of non-commutative objects which can be determined with the
2285 @code{return_type_tinfo()} method. Expressions of this category commute
2286 with everything except @code{noncommutative} expressions of the same
2288 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2289 of non-commutative objects of different classes. Expressions of this
2290 category don't commute with any other @code{noncommutative} or
2291 @code{noncommutative_composite} expressions.
2294 The value returned by the @code{return_type_tinfo()} method is valid only
2295 when the return type of the expression is @code{noncommutative}. It is a
2296 value that is unique to the class of the object and usually one of the
2297 constants in @file{tinfos.h}, or derived therefrom.
2299 Here are a couple of examples:
2302 @multitable @columnfractions 0.33 0.33 0.34
2303 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2304 @item @code{42} @tab @code{commutative} @tab -
2305 @item @code{2*x-y} @tab @code{commutative} @tab -
2306 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2307 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2308 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2309 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2313 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2314 @code{TINFO_clifford} for objects with a representation label of zero.
2315 Other representation labels yield a different @code{return_type_tinfo()},
2316 but it's the same for any two objects with the same label. This is also true
2319 A last note: With the exception of matrices, positive integer powers of
2320 non-commutative objects are automatically expanded in GiNaC. For example,
2321 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2322 non-commutative expressions).
2325 @cindex @code{clifford} (class)
2326 @subsection Clifford algebra
2328 @cindex @code{dirac_gamma()}
2329 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2330 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2331 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2332 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2335 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2338 which takes two arguments: the index and a @dfn{representation label} in the
2339 range 0 to 255 which is used to distinguish elements of different Clifford
2340 algebras (this is also called a @dfn{spin line index}). Gammas with different
2341 labels commute with each other. The dimension of the index can be 4 or (in
2342 the framework of dimensional regularization) any symbolic value. Spinor
2343 indices on Dirac gammas are not supported in GiNaC.
2345 @cindex @code{dirac_ONE()}
2346 The unity element of a Clifford algebra is constructed by
2349 ex dirac_ONE(unsigned char rl = 0);
2352 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2353 multiples of the unity element, even though it's customary to omit it.
2354 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2355 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2356 GiNaC may produce incorrect results.
2358 @cindex @code{dirac_gamma5()}
2359 There's a special element @samp{gamma5} that commutes with all other
2360 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2364 ex dirac_gamma5(unsigned char rl = 0);
2367 @cindex @code{dirac_gamma6()}
2368 @cindex @code{dirac_gamma7()}
2369 The two additional functions
2372 ex dirac_gamma6(unsigned char rl = 0);
2373 ex dirac_gamma7(unsigned char rl = 0);
2376 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2379 @cindex @code{dirac_slash()}
2380 Finally, the function
2383 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2386 creates a term that represents a contraction of @samp{e} with the Dirac
2387 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2388 with a unique index whose dimension is given by the @code{dim} argument).
2389 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2391 In products of dirac gammas, superfluous unity elements are automatically
2392 removed, squares are replaced by their values and @samp{gamma5} is
2393 anticommuted to the front. The @code{simplify_indexed()} function performs
2394 contractions in gamma strings, for example
2399 symbol a("a"), b("b"), D("D");
2400 varidx mu(symbol("mu"), D);
2401 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2402 * dirac_gamma(mu.toggle_variance());
2404 // -> gamma~mu*a\*gamma.mu
2405 e = e.simplify_indexed();
2408 cout << e.subs(D == 4) << endl;
2414 @cindex @code{dirac_trace()}
2415 To calculate the trace of an expression containing strings of Dirac gammas
2416 you use the function
2419 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2422 This function takes the trace of all gammas with the specified representation
2423 label; gammas with other labels are left standing. The last argument to
2424 @code{dirac_trace()} is the value to be returned for the trace of the unity
2425 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2426 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2427 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2428 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2429 This @samp{gamma5} scheme is described in greater detail in
2430 @cite{The Role of gamma5 in Dimensional Regularization}.
2432 The value of the trace itself is also usually different in 4 and in
2433 @math{D != 4} dimensions:
2438 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2439 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2440 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2441 cout << dirac_trace(e).simplify_indexed() << endl;
2448 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2449 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2450 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2451 cout << dirac_trace(e).simplify_indexed() << endl;
2452 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2456 Here is an example for using @code{dirac_trace()} to compute a value that
2457 appears in the calculation of the one-loop vacuum polarization amplitude in
2462 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2463 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2466 sp.add(l, l, pow(l, 2));
2467 sp.add(l, q, ldotq);
2469 ex e = dirac_gamma(mu) *
2470 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2471 dirac_gamma(mu.toggle_variance()) *
2472 (dirac_slash(l, D) + m * dirac_ONE());
2473 e = dirac_trace(e).simplify_indexed(sp);
2474 e = e.collect(lst(l, ldotq, m));
2476 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2480 The @code{canonicalize_clifford()} function reorders all gamma products that
2481 appear in an expression to a canonical (but not necessarily simple) form.
2482 You can use this to compare two expressions or for further simplifications:
2486 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2487 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2489 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2491 e = canonicalize_clifford(e);
2498 @cindex @code{color} (class)
2499 @subsection Color algebra
2501 @cindex @code{color_T()}
2502 For computations in quantum chromodynamics, GiNaC implements the base elements
2503 and structure constants of the su(3) Lie algebra (color algebra). The base
2504 elements @math{T_a} are constructed by the function
2507 ex color_T(const ex & a, unsigned char rl = 0);
2510 which takes two arguments: the index and a @dfn{representation label} in the
2511 range 0 to 255 which is used to distinguish elements of different color
2512 algebras. Objects with different labels commute with each other. The
2513 dimension of the index must be exactly 8 and it should be of class @code{idx},
2516 @cindex @code{color_ONE()}
2517 The unity element of a color algebra is constructed by
2520 ex color_ONE(unsigned char rl = 0);
2523 @strong{Note:} You must always use @code{color_ONE()} when referring to
2524 multiples of the unity element, even though it's customary to omit it.
2525 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2526 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2527 GiNaC may produce incorrect results.
2529 @cindex @code{color_d()}
2530 @cindex @code{color_f()}
2534 ex color_d(const ex & a, const ex & b, const ex & c);
2535 ex color_f(const ex & a, const ex & b, const ex & c);
2538 create the symmetric and antisymmetric structure constants @math{d_abc} and
2539 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2540 and @math{[T_a, T_b] = i f_abc T_c}.
2542 @cindex @code{color_h()}
2543 There's an additional function
2546 ex color_h(const ex & a, const ex & b, const ex & c);
2549 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2551 The function @code{simplify_indexed()} performs some simplifications on
2552 expressions containing color objects:
2557 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2558 k(symbol("k"), 8), l(symbol("l"), 8);
2560 e = color_d(a, b, l) * color_f(a, b, k);
2561 cout << e.simplify_indexed() << endl;
2564 e = color_d(a, b, l) * color_d(a, b, k);
2565 cout << e.simplify_indexed() << endl;
2568 e = color_f(l, a, b) * color_f(a, b, k);
2569 cout << e.simplify_indexed() << endl;
2572 e = color_h(a, b, c) * color_h(a, b, c);
2573 cout << e.simplify_indexed() << endl;
2576 e = color_h(a, b, c) * color_T(b) * color_T(c);
2577 cout << e.simplify_indexed() << endl;
2580 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2581 cout << e.simplify_indexed() << endl;
2584 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2585 cout << e.simplify_indexed() << endl;
2586 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2590 @cindex @code{color_trace()}
2591 To calculate the trace of an expression containing color objects you use the
2595 ex color_trace(const ex & e, unsigned char rl = 0);
2598 This function takes the trace of all color @samp{T} objects with the
2599 specified representation label; @samp{T}s with other labels are left
2600 standing. For example:
2604 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2606 // -> -I*f.a.c.b+d.a.c.b
2611 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2612 @c node-name, next, previous, up
2613 @chapter Methods and Functions
2616 In this chapter the most important algorithms provided by GiNaC will be
2617 described. Some of them are implemented as functions on expressions,
2618 others are implemented as methods provided by expression objects. If
2619 they are methods, there exists a wrapper function around it, so you can
2620 alternatively call it in a functional way as shown in the simple
2625 cout << "As method: " << sin(1).evalf() << endl;
2626 cout << "As function: " << evalf(sin(1)) << endl;
2630 @cindex @code{subs()}
2631 The general rule is that wherever methods accept one or more parameters
2632 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2633 wrapper accepts is the same but preceded by the object to act on
2634 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2635 most natural one in an OO model but it may lead to confusion for MapleV
2636 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2637 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2638 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2639 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2640 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2641 here. Also, users of MuPAD will in most cases feel more comfortable
2642 with GiNaC's convention. All function wrappers are implemented
2643 as simple inline functions which just call the corresponding method and
2644 are only provided for users uncomfortable with OO who are dead set to
2645 avoid method invocations. Generally, nested function wrappers are much
2646 harder to read than a sequence of methods and should therefore be
2647 avoided if possible. On the other hand, not everything in GiNaC is a
2648 method on class @code{ex} and sometimes calling a function cannot be
2652 * Information About Expressions::
2653 * Substituting Expressions::
2654 * Pattern Matching and Advanced Substitutions::
2655 * Applying a Function on Subexpressions::
2656 * Polynomial Arithmetic:: Working with polynomials.
2657 * Rational Expressions:: Working with rational functions.
2658 * Symbolic Differentiation::
2659 * Series Expansion:: Taylor and Laurent expansion.
2661 * Built-in Functions:: List of predefined mathematical functions.
2662 * Input/Output:: Input and output of expressions.
2666 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2667 @c node-name, next, previous, up
2668 @section Getting information about expressions
2670 @subsection Checking expression types
2671 @cindex @code{is_a<@dots{}>()}
2672 @cindex @code{is_exactly_a<@dots{}>()}
2673 @cindex @code{ex_to<@dots{}>()}
2674 @cindex Converting @code{ex} to other classes
2675 @cindex @code{info()}
2676 @cindex @code{return_type()}
2677 @cindex @code{return_type_tinfo()}
2679 Sometimes it's useful to check whether a given expression is a plain number,
2680 a sum, a polynomial with integer coefficients, or of some other specific type.
2681 GiNaC provides a couple of functions for this:
2684 bool is_a<T>(const ex & e);
2685 bool is_exactly_a<T>(const ex & e);
2686 bool ex::info(unsigned flag);
2687 unsigned ex::return_type(void) const;
2688 unsigned ex::return_type_tinfo(void) const;
2691 When the test made by @code{is_a<T>()} returns true, it is safe to call
2692 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2693 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2694 example, assuming @code{e} is an @code{ex}:
2699 if (is_a<numeric>(e))
2700 numeric n = ex_to<numeric>(e);
2705 @code{is_a<T>(e)} allows you to check whether the top-level object of
2706 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2707 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2708 e.g., for checking whether an expression is a number, a sum, or a product:
2715 is_a<numeric>(e1); // true
2716 is_a<numeric>(e2); // false
2717 is_a<add>(e1); // false
2718 is_a<add>(e2); // true
2719 is_a<mul>(e1); // false
2720 is_a<mul>(e2); // false
2724 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2725 top-level object of an expression @samp{e} is an instance of the GiNaC
2726 class @samp{T}, not including parent classes.
2728 The @code{info()} method is used for checking certain attributes of
2729 expressions. The possible values for the @code{flag} argument are defined
2730 in @file{ginac/flags.h}, the most important being explained in the following
2734 @multitable @columnfractions .30 .70
2735 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2736 @item @code{numeric}
2737 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2739 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2740 @item @code{rational}
2741 @tab @dots{}an exact rational number (integers are rational, too)
2742 @item @code{integer}
2743 @tab @dots{}a (non-complex) integer
2744 @item @code{crational}
2745 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2746 @item @code{cinteger}
2747 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2748 @item @code{positive}
2749 @tab @dots{}not complex and greater than 0
2750 @item @code{negative}
2751 @tab @dots{}not complex and less than 0
2752 @item @code{nonnegative}
2753 @tab @dots{}not complex and greater than or equal to 0
2755 @tab @dots{}an integer greater than 0
2757 @tab @dots{}an integer less than 0
2758 @item @code{nonnegint}
2759 @tab @dots{}an integer greater than or equal to 0
2761 @tab @dots{}an even integer
2763 @tab @dots{}an odd integer
2765 @tab @dots{}a prime integer (probabilistic primality test)
2766 @item @code{relation}
2767 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2768 @item @code{relation_equal}
2769 @tab @dots{}a @code{==} relation
2770 @item @code{relation_not_equal}
2771 @tab @dots{}a @code{!=} relation
2772 @item @code{relation_less}
2773 @tab @dots{}a @code{<} relation
2774 @item @code{relation_less_or_equal}
2775 @tab @dots{}a @code{<=} relation
2776 @item @code{relation_greater}
2777 @tab @dots{}a @code{>} relation
2778 @item @code{relation_greater_or_equal}
2779 @tab @dots{}a @code{>=} relation
2781 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2783 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2784 @item @code{polynomial}
2785 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2786 @item @code{integer_polynomial}
2787 @tab @dots{}a polynomial with (non-complex) integer coefficients
2788 @item @code{cinteger_polynomial}
2789 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2790 @item @code{rational_polynomial}
2791 @tab @dots{}a polynomial with (non-complex) rational coefficients
2792 @item @code{crational_polynomial}
2793 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2794 @item @code{rational_function}
2795 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2796 @item @code{algebraic}
2797 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2801 To determine whether an expression is commutative or non-commutative and if
2802 so, with which other expressions it would commute, you use the methods
2803 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2804 for an explanation of these.
2807 @subsection Accessing subexpressions
2808 @cindex @code{nops()}
2811 @cindex @code{relational} (class)
2813 GiNaC provides the two methods
2816 unsigned ex::nops();
2817 ex ex::op(unsigned i);
2820 for accessing the subexpressions in the container-like GiNaC classes like
2821 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2822 determines the number of subexpressions (@samp{operands}) contained, while
2823 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2824 In the case of a @code{power} object, @code{op(0)} will return the basis
2825 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2826 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2828 The left-hand and right-hand side expressions of objects of class
2829 @code{relational} (and only of these) can also be accessed with the methods
2837 @subsection Comparing expressions
2838 @cindex @code{is_equal()}
2839 @cindex @code{is_zero()}
2841 Expressions can be compared with the usual C++ relational operators like
2842 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2843 the result is usually not determinable and the result will be @code{false},
2844 except in the case of the @code{!=} operator. You should also be aware that
2845 GiNaC will only do the most trivial test for equality (subtracting both
2846 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2849 Actually, if you construct an expression like @code{a == b}, this will be
2850 represented by an object of the @code{relational} class (@pxref{Relations})
2851 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2853 There are also two methods
2856 bool ex::is_equal(const ex & other);
2860 for checking whether one expression is equal to another, or equal to zero,
2863 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2864 GiNaC header files. This method is however only to be used internally by
2865 GiNaC to establish a canonical sort order for terms, and using it to compare
2866 expressions will give very surprising results.
2869 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2870 @c node-name, next, previous, up
2871 @section Substituting expressions
2872 @cindex @code{subs()}
2874 Algebraic objects inside expressions can be replaced with arbitrary
2875 expressions via the @code{.subs()} method:
2878 ex ex::subs(const ex & e);
2879 ex ex::subs(const lst & syms, const lst & repls);
2882 In the first form, @code{subs()} accepts a relational of the form
2883 @samp{object == expression} or a @code{lst} of such relationals:
2887 symbol x("x"), y("y");
2889 ex e1 = 2*x^2-4*x+3;
2890 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2894 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2899 If you specify multiple substitutions, they are performed in parallel, so e.g.
2900 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2902 The second form of @code{subs()} takes two lists, one for the objects to be
2903 replaced and one for the expressions to be substituted (both lists must
2904 contain the same number of elements). Using this form, you would write
2905 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2907 @code{subs()} performs syntactic substitution of any complete algebraic
2908 object; it does not try to match sub-expressions as is demonstrated by the
2913 symbol x("x"), y("y"), z("z");
2915 ex e1 = pow(x+y, 2);
2916 cout << e1.subs(x+y == 4) << endl;
2919 ex e2 = sin(x)*sin(y)*cos(x);
2920 cout << e2.subs(sin(x) == cos(x)) << endl;
2921 // -> cos(x)^2*sin(y)
2924 cout << e3.subs(x+y == 4) << endl;
2926 // (and not 4+z as one might expect)
2930 A more powerful form of substitution using wildcards is described in the
2934 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2935 @c node-name, next, previous, up
2936 @section Pattern matching and advanced substitutions
2937 @cindex @code{wildcard} (class)
2938 @cindex Pattern matching
2940 GiNaC allows the use of patterns for checking whether an expression is of a
2941 certain form or contains subexpressions of a certain form, and for
2942 substituting expressions in a more general way.
2944 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2945 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2946 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2947 an unsigned integer number to allow having multiple different wildcards in a
2948 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2949 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2953 ex wild(unsigned label = 0);
2956 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2959 Some examples for patterns:
2961 @multitable @columnfractions .5 .5
2962 @item @strong{Constructed as} @tab @strong{Output as}
2963 @item @code{wild()} @tab @samp{$0}
2964 @item @code{pow(x,wild())} @tab @samp{x^$0}
2965 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2966 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2972 @item Wildcards behave like symbols and are subject to the same algebraic
2973 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2974 @item As shown in the last example, to use wildcards for indices you have to
2975 use them as the value of an @code{idx} object. This is because indices must
2976 always be of class @code{idx} (or a subclass).
2977 @item Wildcards only represent expressions or subexpressions. It is not
2978 possible to use them as placeholders for other properties like index
2979 dimension or variance, representation labels, symmetry of indexed objects
2981 @item Because wildcards are commutative, it is not possible to use wildcards
2982 as part of noncommutative products.
2983 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2984 are also valid patterns.
2987 @cindex @code{match()}
2988 The most basic application of patterns is to check whether an expression
2989 matches a given pattern. This is done by the function
2992 bool ex::match(const ex & pattern);
2993 bool ex::match(const ex & pattern, lst & repls);
2996 This function returns @code{true} when the expression matches the pattern
2997 and @code{false} if it doesn't. If used in the second form, the actual
2998 subexpressions matched by the wildcards get returned in the @code{repls}
2999 object as a list of relations of the form @samp{wildcard == expression}.
3000 If @code{match()} returns false, the state of @code{repls} is undefined.
3001 For reproducible results, the list should be empty when passed to
3002 @code{match()}, but it is also possible to find similarities in multiple
3003 expressions by passing in the result of a previous match.
3005 The matching algorithm works as follows:
3008 @item A single wildcard matches any expression. If one wildcard appears
3009 multiple times in a pattern, it must match the same expression in all
3010 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3011 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3012 @item If the expression is not of the same class as the pattern, the match
3013 fails (i.e. a sum only matches a sum, a function only matches a function,
3015 @item If the pattern is a function, it only matches the same function
3016 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3017 @item Except for sums and products, the match fails if the number of
3018 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3020 @item If there are no subexpressions, the expressions and the pattern must
3021 be equal (in the sense of @code{is_equal()}).
3022 @item Except for sums and products, each subexpression (@code{op()}) must
3023 match the corresponding subexpression of the pattern.
3026 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3027 account for their commutativity and associativity:
3030 @item If the pattern contains a term or factor that is a single wildcard,
3031 this one is used as the @dfn{global wildcard}. If there is more than one
3032 such wildcard, one of them is chosen as the global wildcard in a random
3034 @item Every term/factor of the pattern, except the global wildcard, is
3035 matched against every term of the expression in sequence. If no match is
3036 found, the whole match fails. Terms that did match are not considered in
3038 @item If there are no unmatched terms left, the match succeeds. Otherwise
3039 the match fails unless there is a global wildcard in the pattern, in
3040 which case this wildcard matches the remaining terms.
3043 In general, having more than one single wildcard as a term of a sum or a
3044 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3047 Here are some examples in @command{ginsh} to demonstrate how it works (the
3048 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3049 match fails, and the list of wildcard replacements otherwise):
3052 > match((x+y)^a,(x+y)^a);
3054 > match((x+y)^a,(x+y)^b);
3056 > match((x+y)^a,$1^$2);
3058 > match((x+y)^a,$1^$1);
3060 > match((x+y)^(x+y),$1^$1);
3062 > match((x+y)^(x+y),$1^$2);
3064 > match((a+b)*(a+c),($1+b)*($1+c));
3066 > match((a+b)*(a+c),(a+$1)*(a+$2));
3068 (Unpredictable. The result might also be [$1==c,$2==b].)
3069 > match((a+b)*(a+c),($1+$2)*($1+$3));
3070 (The result is undefined. Due to the sequential nature of the algorithm
3071 and the re-ordering of terms in GiNaC, the match for the first factor
3072 may be @{$1==a,$2==b@} in which case the match for the second factor
3073 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3075 > match(a*(x+y)+a*z+b,a*$1+$2);
3076 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3077 @{$1=x+y,$2=a*z+b@}.)
3078 > match(a+b+c+d+e+f,c);
3080 > match(a+b+c+d+e+f,c+$0);
3082 > match(a+b+c+d+e+f,c+e+$0);
3084 > match(a+b,a+b+$0);
3086 > match(a*b^2,a^$1*b^$2);
3088 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3089 even though a==a^1.)
3090 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3092 > match(atan2(y,x^2),atan2(y,$0));
3096 @cindex @code{has()}
3097 A more general way to look for patterns in expressions is provided by the
3101 bool ex::has(const ex & pattern);
3104 This function checks whether a pattern is matched by an expression itself or
3105 by any of its subexpressions.
3107 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3108 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3111 > has(x*sin(x+y+2*a),y);
3113 > has(x*sin(x+y+2*a),x+y);
3115 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3116 has the subexpressions "x", "y" and "2*a".)
3117 > has(x*sin(x+y+2*a),x+y+$1);
3119 (But this is possible.)
3120 > has(x*sin(2*(x+y)+2*a),x+y);
3122 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3123 which "x+y" is not a subexpression.)
3126 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3128 > has(4*x^2-x+3,$1*x);
3130 > has(4*x^2+x+3,$1*x);
3132 (Another possible pitfall. The first expression matches because the term
3133 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3134 contains a linear term you should use the coeff() function instead.)
3137 @cindex @code{find()}
3141 bool ex::find(const ex & pattern, lst & found);
3144 works a bit like @code{has()} but it doesn't stop upon finding the first
3145 match. Instead, it appends all found matches to the specified list. If there
3146 are multiple occurrences of the same expression, it is entered only once to
3147 the list. @code{find()} returns false if no matches were found (in
3148 @command{ginsh}, it returns an empty list):
3151 > find(1+x+x^2+x^3,x);
3153 > find(1+x+x^2+x^3,y);
3155 > find(1+x+x^2+x^3,x^$1);
3157 (Note the absence of "x".)
3158 > expand((sin(x)+sin(y))*(a+b));
3159 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3164 @cindex @code{subs()}
3165 Probably the most useful application of patterns is to use them for
3166 substituting expressions with the @code{subs()} method. Wildcards can be
3167 used in the search patterns as well as in the replacement expressions, where
3168 they get replaced by the expressions matched by them. @code{subs()} doesn't
3169 know anything about algebra; it performs purely syntactic substitutions.
3174 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3176 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3178 > subs((a+b+c)^2,a+b=x);
3180 > subs((a+b+c)^2,a+b+$1==x+$1);
3182 > subs(a+2*b,a+b=x);
3184 > subs(4*x^3-2*x^2+5*x-1,x==a);
3186 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3188 > subs(sin(1+sin(x)),sin($1)==cos($1));
3190 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3194 The last example would be written in C++ in this way:
3198 symbol a("a"), b("b"), x("x"), y("y");
3199 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3200 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3201 cout << e.expand() << endl;
3207 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3208 @c node-name, next, previous, up
3209 @section Applying a Function on Subexpressions
3210 @cindex Tree traversal
3211 @cindex @code{map()}
3213 Sometimes you may want to perform an operation on specific parts of an
3214 expression while leaving the general structure of it intact. An example
3215 of this would be a matrix trace operation: the trace of a sum is the sum
3216 of the traces of the individual terms. That is, the trace should @dfn{map}
3217 on the sum, by applying itself to each of the sum's operands. It is possible
3218 to do this manually which usually results in code like this:
3223 if (is_a<matrix>(e))
3224 return ex_to<matrix>(e).trace();
3225 else if (is_a<add>(e)) @{
3227 for (unsigned i=0; i<e.nops(); i++)
3228 sum += calc_trace(e.op(i));
3230 @} else if (is_a<mul>)(e)) @{
3238 This is, however, slightly inefficient (if the sum is very large it can take
3239 a long time to add the terms one-by-one), and its applicability is limited to
3240 a rather small class of expressions. If @code{calc_trace()} is called with
3241 a relation or a list as its argument, you will probably want the trace to
3242 be taken on both sides of the relation or of all elements of the list.
3244 GiNaC offers the @code{map()} method to aid in the implementation of such
3248 ex ex::map(map_function & f) const;
3249 ex ex::map(ex (*f)(const ex & e)) const;
3252 In the first (preferred) form, @code{map()} takes a function object that
3253 is subclassed from the @code{map_function} class. In the second form, it
3254 takes a pointer to a function that accepts and returns an expression.
3255 @code{map()} constructs a new expression of the same type, applying the
3256 specified function on all subexpressions (in the sense of @code{op()}),
3259 The use of a function object makes it possible to supply more arguments to
3260 the function that is being mapped, or to keep local state information.
3261 The @code{map_function} class declares a virtual function call operator
3262 that you can overload. Here is a sample implementation of @code{calc_trace()}
3263 that uses @code{map()} in a recursive fashion:
3266 struct calc_trace : public map_function @{
3267 ex operator()(const ex &e)
3269 if (is_a<matrix>(e))
3270 return ex_to<matrix>(e).trace();
3271 else if (is_a<mul>(e)) @{
3274 return e.map(*this);
3279 This function object could then be used like this:
3283 ex M = ... // expression with matrices
3284 calc_trace do_trace;
3285 ex tr = do_trace(M);
3289 Here is another example for you to meditate over. It removes quadratic
3290 terms in a variable from an expanded polynomial:
3293 struct map_rem_quad : public map_function @{
3295 map_rem_quad(const ex & var_) : var(var_) @{@}
3297 ex operator()(const ex & e)
3299 if (is_a<add>(e) || is_a<mul>(e))
3300 return e.map(*this);
3301 else if (is_a<power>(e) && e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3311 symbol x("x"), y("y");
3314 for (int i=0; i<8; i++)
3315 e += pow(x, i) * pow(y, 8-i) * (i+1);
3317 // -> 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
3319 map_rem_quad rem_quad(x);
3320 cout << rem_quad(e) << endl;
3321 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3325 @command{ginsh} offers a slightly different implementation of @code{map()}
3326 that allows applying algebraic functions to operands. The second argument
3327 to @code{map()} is an expression containing the wildcard @samp{$0} which
3328 acts as the placeholder for the operands:
3333 > map(a+2*b,sin($0));
3335 > map(@{a,b,c@},$0^2+$0);
3336 @{a^2+a,b^2+b,c^2+c@}
3339 Note that it is only possible to use algebraic functions in the second
3340 argument. You can not use functions like @samp{diff()}, @samp{op()},
3341 @samp{subs()} etc. because these are evaluated immediately:
3344 > map(@{a,b,c@},diff($0,a));
3346 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3347 to "map(@{a,b,c@},0)".
3351 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3352 @c node-name, next, previous, up
3353 @section Polynomial arithmetic
3355 @subsection Expanding and collecting
3356 @cindex @code{expand()}
3357 @cindex @code{collect()}
3359 A polynomial in one or more variables has many equivalent
3360 representations. Some useful ones serve a specific purpose. Consider
3361 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3362 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3363 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3364 representations are the recursive ones where one collects for exponents
3365 in one of the three variable. Since the factors are themselves
3366 polynomials in the remaining two variables the procedure can be
3367 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
3368 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3371 To bring an expression into expanded form, its method
3377 may be called. In our example above, this corresponds to @math{4*x*y +
3378 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3379 GiNaC is not easily guessable you should be prepared to see different
3380 orderings of terms in such sums!
3382 Another useful representation of multivariate polynomials is as a
3383 univariate polynomial in one of the variables with the coefficients
3384 being polynomials in the remaining variables. The method
3385 @code{collect()} accomplishes this task:
3388 ex ex::collect(const ex & s, bool distributed = false);
3391 The first argument to @code{collect()} can also be a list of objects in which
3392 case the result is either a recursively collected polynomial, or a polynomial
3393 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3394 by the @code{distributed} flag.
3396 Note that the original polynomial needs to be in expanded form (for the
3397 variables concerned) in order for @code{collect()} to be able to find the
3398 coefficients properly.
3400 The following @command{ginsh} transcript shows an application of @code{collect()}
3401 together with @code{find()}:
3404 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3405 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
3406 > collect(a,@{p,q@});
3407 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)
3408 > collect(a,find(a,sin($1)));
3409 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3410 > collect(a,@{find(a,sin($1)),p,q@});
3411 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3412 > collect(a,@{find(a,sin($1)),d@});
3413 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3416 @subsection Degree and coefficients
3417 @cindex @code{degree()}
3418 @cindex @code{ldegree()}
3419 @cindex @code{coeff()}
3421 The degree and low degree of a polynomial can be obtained using the two
3425 int ex::degree(const ex & s);
3426 int ex::ldegree(const ex & s);
3429 which also work reliably on non-expanded input polynomials (they even work
3430 on rational functions, returning the asymptotic degree). To extract
3431 a coefficient with a certain power from an expanded polynomial you use
3434 ex ex::coeff(const ex & s, int n);
3437 You can also obtain the leading and trailing coefficients with the methods
3440 ex ex::lcoeff(const ex & s);
3441 ex ex::tcoeff(const ex & s);
3444 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3447 An application is illustrated in the next example, where a multivariate
3448 polynomial is analyzed:
3451 #include <ginac/ginac.h>
3452 using namespace std;
3453 using namespace GiNaC;
3457 symbol x("x"), y("y");
3458 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3459 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3460 ex Poly = PolyInp.expand();
3462 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3463 cout << "The x^" << i << "-coefficient is "
3464 << Poly.coeff(x,i) << endl;
3466 cout << "As polynomial in y: "
3467 << Poly.collect(y) << endl;
3471 When run, it returns an output in the following fashion:
3474 The x^0-coefficient is y^2+11*y
3475 The x^1-coefficient is 5*y^2-2*y
3476 The x^2-coefficient is -1
3477 The x^3-coefficient is 4*y
3478 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3481 As always, the exact output may vary between different versions of GiNaC
3482 or even from run to run since the internal canonical ordering is not
3483 within the user's sphere of influence.
3485 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3486 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3487 with non-polynomial expressions as they not only work with symbols but with
3488 constants, functions and indexed objects as well:
3492 symbol a("a"), b("b"), c("c");
3493 idx i(symbol("i"), 3);
3495 ex e = pow(sin(x) - cos(x), 4);
3496 cout << e.degree(cos(x)) << endl;
3498 cout << e.expand().coeff(sin(x), 3) << endl;
3501 e = indexed(a+b, i) * indexed(b+c, i);
3502 e = e.expand(expand_options::expand_indexed);
3503 cout << e.collect(indexed(b, i)) << endl;
3504 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3509 @subsection Polynomial division
3510 @cindex polynomial division
3513 @cindex pseudo-remainder
3514 @cindex @code{quo()}
3515 @cindex @code{rem()}
3516 @cindex @code{prem()}
3517 @cindex @code{divide()}
3522 ex quo(const ex & a, const ex & b, const symbol & x);
3523 ex rem(const ex & a, const ex & b, const symbol & x);
3526 compute the quotient and remainder of univariate polynomials in the variable
3527 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3529 The additional function
3532 ex prem(const ex & a, const ex & b, const symbol & x);
3535 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3536 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3538 Exact division of multivariate polynomials is performed by the function
3541 bool divide(const ex & a, const ex & b, ex & q);
3544 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3545 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3546 in which case the value of @code{q} is undefined.
3549 @subsection Unit, content and primitive part
3550 @cindex @code{unit()}
3551 @cindex @code{content()}
3552 @cindex @code{primpart()}
3557 ex ex::unit(const symbol & x);
3558 ex ex::content(const symbol & x);
3559 ex ex::primpart(const symbol & x);
3562 return the unit part, content part, and primitive polynomial of a multivariate
3563 polynomial with respect to the variable @samp{x} (the unit part being the sign
3564 of the leading coefficient, the content part being the GCD of the coefficients,
3565 and the primitive polynomial being the input polynomial divided by the unit and
3566 content parts). The product of unit, content, and primitive part is the
3567 original polynomial.
3570 @subsection GCD and LCM
3573 @cindex @code{gcd()}
3574 @cindex @code{lcm()}
3576 The functions for polynomial greatest common divisor and least common
3577 multiple have the synopsis
3580 ex gcd(const ex & a, const ex & b);
3581 ex lcm(const ex & a, const ex & b);
3584 The functions @code{gcd()} and @code{lcm()} accept two expressions
3585 @code{a} and @code{b} as arguments and return a new expression, their
3586 greatest common divisor or least common multiple, respectively. If the
3587 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3588 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3591 #include <ginac/ginac.h>
3592 using namespace GiNaC;
3596 symbol x("x"), y("y"), z("z");
3597 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3598 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3600 ex P_gcd = gcd(P_a, P_b);
3602 ex P_lcm = lcm(P_a, P_b);
3603 // 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
3608 @subsection Square-free decomposition
3609 @cindex square-free decomposition
3610 @cindex factorization
3611 @cindex @code{sqrfree()}
3613 GiNaC still lacks proper factorization support. Some form of
3614 factorization is, however, easily implemented by noting that factors
3615 appearing in a polynomial with power two or more also appear in the
3616 derivative and hence can easily be found by computing the GCD of the
3617 original polynomial and its derivatives. Any system has an interface
3618 for this so called square-free factorization. So we provide one, too:
3620 ex sqrfree(const ex & a, const lst & l = lst());
3622 Here is an example that by the way illustrates how the result may depend
3623 on the order of differentiation:
3626 symbol x("x"), y("y");
3627 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3629 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3630 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3632 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3633 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3635 cout << sqrfree(BiVarPol) << endl;
3636 // -> depending on luck, any of the above
3641 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3642 @c node-name, next, previous, up
3643 @section Rational expressions
3645 @subsection The @code{normal} method
3646 @cindex @code{normal()}
3647 @cindex simplification
3648 @cindex temporary replacement
3650 Some basic form of simplification of expressions is called for frequently.
3651 GiNaC provides the method @code{.normal()}, which converts a rational function
3652 into an equivalent rational function of the form @samp{numerator/denominator}
3653 where numerator and denominator are coprime. If the input expression is already
3654 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3655 otherwise it performs fraction addition and multiplication.
3657 @code{.normal()} can also be used on expressions which are not rational functions
3658 as it will replace all non-rational objects (like functions or non-integer
3659 powers) by temporary symbols to bring the expression to the domain of rational
3660 functions before performing the normalization, and re-substituting these
3661 symbols afterwards. This algorithm is also available as a separate method
3662 @code{.to_rational()}, described below.
3664 This means that both expressions @code{t1} and @code{t2} are indeed
3665 simplified in this little program:
3668 #include <ginac/ginac.h>
3669 using namespace GiNaC;
3674 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3675 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3676 std::cout << "t1 is " << t1.normal() << std::endl;
3677 std::cout << "t2 is " << t2.normal() << std::endl;
3681 Of course this works for multivariate polynomials too, so the ratio of
3682 the sample-polynomials from the section about GCD and LCM above would be
3683 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3686 @subsection Numerator and denominator
3689 @cindex @code{numer()}
3690 @cindex @code{denom()}
3691 @cindex @code{numer_denom()}
3693 The numerator and denominator of an expression can be obtained with
3698 ex ex::numer_denom();
3701 These functions will first normalize the expression as described above and
3702 then return the numerator, denominator, or both as a list, respectively.
3703 If you need both numerator and denominator, calling @code{numer_denom()} is
3704 faster than using @code{numer()} and @code{denom()} separately.
3707 @subsection Converting to a rational expression
3708 @cindex @code{to_rational()}
3710 Some of the methods described so far only work on polynomials or rational
3711 functions. GiNaC provides a way to extend the domain of these functions to
3712 general expressions by using the temporary replacement algorithm described
3713 above. You do this by calling
3716 ex ex::to_rational(lst &l);
3719 on the expression to be converted. The supplied @code{lst} will be filled
3720 with the generated temporary symbols and their replacement expressions in
3721 a format that can be used directly for the @code{subs()} method. It can also
3722 already contain a list of replacements from an earlier application of
3723 @code{.to_rational()}, so it's possible to use it on multiple expressions
3724 and get consistent results.
3731 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3732 ex b = sin(x) + cos(x);
3735 divide(a.to_rational(l), b.to_rational(l), q);
3736 cout << q.subs(l) << endl;
3740 will print @samp{sin(x)-cos(x)}.
3743 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3744 @c node-name, next, previous, up
3745 @section Symbolic differentiation
3746 @cindex differentiation
3747 @cindex @code{diff()}
3749 @cindex product rule
3751 GiNaC's objects know how to differentiate themselves. Thus, a
3752 polynomial (class @code{add}) knows that its derivative is the sum of
3753 the derivatives of all the monomials:
3756 #include <ginac/ginac.h>
3757 using namespace GiNaC;
3761 symbol x("x"), y("y"), z("z");
3762 ex P = pow(x, 5) + pow(x, 2) + y;
3764 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3765 cout << P.diff(y) << endl; // 1
3766 cout << P.diff(z) << endl; // 0
3770 If a second integer parameter @var{n} is given, the @code{diff} method
3771 returns the @var{n}th derivative.
3773 If @emph{every} object and every function is told what its derivative
3774 is, all derivatives of composed objects can be calculated using the
3775 chain rule and the product rule. Consider, for instance the expression
3776 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3777 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3778 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3779 out that the composition is the generating function for Euler Numbers,
3780 i.e. the so called @var{n}th Euler number is the coefficient of
3781 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3782 identity to code a function that generates Euler numbers in just three
3785 @cindex Euler numbers
3787 #include <ginac/ginac.h>
3788 using namespace GiNaC;
3790 ex EulerNumber(unsigned n)
3793 const ex generator = pow(cosh(x),-1);
3794 return generator.diff(x,n).subs(x==0);
3799 for (unsigned i=0; i<11; i+=2)
3800 std::cout << EulerNumber(i) << std::endl;
3805 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3806 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3807 @code{i} by two since all odd Euler numbers vanish anyways.
3810 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3811 @c node-name, next, previous, up
3812 @section Series expansion
3813 @cindex @code{series()}
3814 @cindex Taylor expansion
3815 @cindex Laurent expansion
3816 @cindex @code{pseries} (class)
3818 Expressions know how to expand themselves as a Taylor series or (more
3819 generally) a Laurent series. As in most conventional Computer Algebra
3820 Systems, no distinction is made between those two. There is a class of
3821 its own for storing such series (@code{class pseries}) and a built-in
3822 function (called @code{Order}) for storing the order term of the series.
3823 As a consequence, if you want to work with series, i.e. multiply two
3824 series, you need to call the method @code{ex::series} again to convert
3825 it to a series object with the usual structure (expansion plus order
3826 term). A sample application from special relativity could read:
3829 #include <ginac/ginac.h>
3830 using namespace std;
3831 using namespace GiNaC;
3835 symbol v("v"), c("c");
3837 ex gamma = 1/sqrt(1 - pow(v/c,2));
3838 ex mass_nonrel = gamma.series(v==0, 10);
3840 cout << "the relativistic mass increase with v is " << endl
3841 << mass_nonrel << endl;
3843 cout << "the inverse square of this series is " << endl
3844 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3848 Only calling the series method makes the last output simplify to
3849 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3850 series raised to the power @math{-2}.
3852 @cindex M@'echain's formula
3853 As another instructive application, let us calculate the numerical
3854 value of Archimedes' constant
3858 (for which there already exists the built-in constant @code{Pi})
3859 using M@'echain's amazing formula
3861 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3864 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3866 We may expand the arcus tangent around @code{0} and insert the fractions
3867 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3868 carries an order term with it and the question arises what the system is
3869 supposed to do when the fractions are plugged into that order term. The
3870 solution is to use the function @code{series_to_poly()} to simply strip
3874 #include <ginac/ginac.h>
3875 using namespace GiNaC;
3877 ex mechain_pi(int degr)
3880 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3881 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3882 -4*pi_expansion.subs(x==numeric(1,239));
3888 using std::cout; // just for fun, another way of...
3889 using std::endl; // ...dealing with this namespace std.
3891 for (int i=2; i<12; i+=2) @{
3892 pi_frac = mechain_pi(i);
3893 cout << i << ":\t" << pi_frac << endl
3894 << "\t" << pi_frac.evalf() << endl;
3900 Note how we just called @code{.series(x,degr)} instead of
3901 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3902 method @code{series()}: if the first argument is a symbol the expression
3903 is expanded in that symbol around point @code{0}. When you run this
3904 program, it will type out:
3908 3.1832635983263598326
3909 4: 5359397032/1706489875
3910 3.1405970293260603143
3911 6: 38279241713339684/12184551018734375
3912 3.141621029325034425
3913 8: 76528487109180192540976/24359780855939418203125
3914 3.141591772182177295
3915 10: 327853873402258685803048818236/104359128170408663038552734375
3916 3.1415926824043995174
3920 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3921 @c node-name, next, previous, up
3922 @section Symmetrization
3923 @cindex @code{symmetrize()}
3924 @cindex @code{antisymmetrize()}
3925 @cindex @code{symmetrize_cyclic()}
3930 ex ex::symmetrize(const lst & l);
3931 ex ex::antisymmetrize(const lst & l);
3932 ex ex::symmetrize_cyclic(const lst & l);
3935 symmetrize an expression by returning the sum over all symmetric,
3936 antisymmetric or cyclic permutations of the specified list of objects,
3937 weighted by the number of permutations.
3939 The three additional methods
3942 ex ex::symmetrize();
3943 ex ex::antisymmetrize();
3944 ex ex::symmetrize_cyclic();
3947 symmetrize or antisymmetrize an expression over its free indices.
3949 Symmetrization is most useful with indexed expressions but can be used with
3950 almost any kind of object (anything that is @code{subs()}able):
3954 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3955 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3957 cout << indexed(A, i, j).symmetrize() << endl;
3958 // -> 1/2*A.j.i+1/2*A.i.j
3959 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3960 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3961 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3962 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
3967 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3968 @c node-name, next, previous, up
3969 @section Predefined mathematical functions
3971 GiNaC contains the following predefined mathematical functions:
3974 @multitable @columnfractions .30 .70
3975 @item @strong{Name} @tab @strong{Function}
3978 @item @code{csgn(x)}
3980 @item @code{sqrt(x)}
3981 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
3988 @item @code{asin(x)}
3990 @item @code{acos(x)}
3992 @item @code{atan(x)}
3993 @tab inverse tangent
3994 @item @code{atan2(y, x)}
3995 @tab inverse tangent with two arguments
3996 @item @code{sinh(x)}
3997 @tab hyperbolic sine
3998 @item @code{cosh(x)}
3999 @tab hyperbolic cosine
4000 @item @code{tanh(x)}
4001 @tab hyperbolic tangent
4002 @item @code{asinh(x)}
4003 @tab inverse hyperbolic sine
4004 @item @code{acosh(x)}
4005 @tab inverse hyperbolic cosine
4006 @item @code{atanh(x)}
4007 @tab inverse hyperbolic tangent
4009 @tab exponential function
4011 @tab natural logarithm
4014 @item @code{zeta(x)}
4015 @tab Riemann's zeta function
4016 @item @code{zeta(n, x)}
4017 @tab derivatives of Riemann's zeta function
4018 @item @code{tgamma(x)}
4020 @item @code{lgamma(x)}
4021 @tab logarithm of Gamma function
4022 @item @code{beta(x, y)}
4023 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4025 @tab psi (digamma) function
4026 @item @code{psi(n, x)}
4027 @tab derivatives of psi function (polygamma functions)
4028 @item @code{factorial(n)}
4029 @tab factorial function
4030 @item @code{binomial(n, m)}
4031 @tab binomial coefficients
4032 @item @code{Order(x)}
4033 @tab order term function in truncated power series
4038 For functions that have a branch cut in the complex plane GiNaC follows
4039 the conventions for C++ as defined in the ANSI standard as far as
4040 possible. In particular: the natural logarithm (@code{log}) and the
4041 square root (@code{sqrt}) both have their branch cuts running along the
4042 negative real axis where the points on the axis itself belong to the
4043 upper part (i.e. continuous with quadrant II). The inverse
4044 trigonometric and hyperbolic functions are not defined for complex
4045 arguments by the C++ standard, however. In GiNaC we follow the
4046 conventions used by CLN, which in turn follow the carefully designed
4047 definitions in the Common Lisp standard. It should be noted that this
4048 convention is identical to the one used by the C99 standard and by most
4049 serious CAS. It is to be expected that future revisions of the C++
4050 standard incorporate these functions in the complex domain in a manner
4051 compatible with C99.
4054 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
4055 @c node-name, next, previous, up
4056 @section Input and output of expressions
4059 @subsection Expression output
4061 @cindex output of expressions
4063 The easiest way to print an expression is to write it to a stream:
4068 ex e = 4.5+pow(x,2)*3/2;
4069 cout << e << endl; // prints '(4.5)+3/2*x^2'
4073 The output format is identical to the @command{ginsh} input syntax and
4074 to that used by most computer algebra systems, but not directly pastable
4075 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4076 is printed as @samp{x^2}).
4078 It is possible to print expressions in a number of different formats with
4082 void ex::print(const print_context & c, unsigned level = 0);
4085 @cindex @code{print_context} (class)
4086 The type of @code{print_context} object passed in determines the format
4087 of the output. The possible types are defined in @file{ginac/print.h}.
4088 All constructors of @code{print_context} and derived classes take an
4089 @code{ostream &} as their first argument.
4091 To print an expression in a way that can be directly used in a C or C++
4092 program, you pass a @code{print_csrc} object like this:
4096 cout << "float f = ";
4097 e.print(print_csrc_float(cout));
4100 cout << "double d = ";
4101 e.print(print_csrc_double(cout));
4104 cout << "cl_N n = ";
4105 e.print(print_csrc_cl_N(cout));
4110 The three possible types mostly affect the way in which floating point
4111 numbers are written.
4113 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4116 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4117 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4118 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4121 The @code{print_context} type @code{print_tree} provides a dump of the
4122 internal structure of an expression for debugging purposes:
4126 e.print(print_tree(cout));
4133 add, hash=0x0, flags=0x3, nops=2
4134 power, hash=0x9, flags=0x3, nops=2
4135 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4136 2 (numeric), hash=0x80000042, flags=0xf
4137 3/2 (numeric), hash=0x80000061, flags=0xf
4140 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4144 This kind of output is also available in @command{ginsh} as the @code{print()}
4147 Another useful output format is for LaTeX parsing in mathematical mode.
4148 It is rather similar to the default @code{print_context} but provides
4149 some braces needed by LaTeX for delimiting boxes and also converts some
4150 common objects to conventional LaTeX names. It is possible to give symbols
4151 a special name for LaTeX output by supplying it as a second argument to
4152 the @code{symbol} constructor.
4154 For example, the code snippet
4159 ex foo = lgamma(x).series(x==0,3);
4160 foo.print(print_latex(std::cout));
4166 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4169 @cindex Tree traversal
4170 If you need any fancy special output format, e.g. for interfacing GiNaC
4171 with other algebra systems or for producing code for different
4172 programming languages, you can always traverse the expression tree yourself:
4175 static void my_print(const ex & e)
4177 if (is_a<function>(e))
4178 cout << ex_to<function>(e).get_name();
4180 cout << e.bp->class_name();
4182 unsigned n = e.nops();
4184 for (unsigned i=0; i<n; i++) @{
4196 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4204 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4205 symbol(y))),numeric(-2)))
4208 If you need an output format that makes it possible to accurately
4209 reconstruct an expression by feeding the output to a suitable parser or
4210 object factory, you should consider storing the expression in an
4211 @code{archive} object and reading the object properties from there.
4212 See the section on archiving for more information.
4215 @subsection Expression input
4216 @cindex input of expressions
4218 GiNaC provides no way to directly read an expression from a stream because
4219 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4220 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4221 @code{y} you defined in your program and there is no way to specify the
4222 desired symbols to the @code{>>} stream input operator.
4224 Instead, GiNaC lets you construct an expression from a string, specifying the
4225 list of symbols to be used:
4229 symbol x("x"), y("y");
4230 ex e("2*x+sin(y)", lst(x, y));
4234 The input syntax is the same as that used by @command{ginsh} and the stream
4235 output operator @code{<<}. The symbols in the string are matched by name to
4236 the symbols in the list and if GiNaC encounters a symbol not specified in
4237 the list it will throw an exception.
4239 With this constructor, it's also easy to implement interactive GiNaC programs:
4244 #include <stdexcept>
4245 #include <ginac/ginac.h>
4246 using namespace std;
4247 using namespace GiNaC;
4254 cout << "Enter an expression containing 'x': ";
4259 cout << "The derivative of " << e << " with respect to x is ";
4260 cout << e.diff(x) << ".\n";
4261 @} catch (exception &p) @{
4262 cerr << p.what() << endl;
4268 @subsection Archiving
4269 @cindex @code{archive} (class)
4272 GiNaC allows creating @dfn{archives} of expressions which can be stored
4273 to or retrieved from files. To create an archive, you declare an object
4274 of class @code{archive} and archive expressions in it, giving each
4275 expression a unique name:
4279 using namespace std;
4280 #include <ginac/ginac.h>
4281 using namespace GiNaC;
4285 symbol x("x"), y("y"), z("z");
4287 ex foo = sin(x + 2*y) + 3*z + 41;
4291 a.archive_ex(foo, "foo");
4292 a.archive_ex(bar, "the second one");
4296 The archive can then be written to a file:
4300 ofstream out("foobar.gar");
4306 The file @file{foobar.gar} contains all information that is needed to
4307 reconstruct the expressions @code{foo} and @code{bar}.
4309 @cindex @command{viewgar}
4310 The tool @command{viewgar} that comes with GiNaC can be used to view
4311 the contents of GiNaC archive files:
4314 $ viewgar foobar.gar
4315 foo = 41+sin(x+2*y)+3*z
4316 the second one = 42+sin(x+2*y)+3*z
4319 The point of writing archive files is of course that they can later be
4325 ifstream in("foobar.gar");
4330 And the stored expressions can be retrieved by their name:
4336 ex ex1 = a2.unarchive_ex(syms, "foo");
4337 ex ex2 = a2.unarchive_ex(syms, "the second one");
4339 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4340 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4341 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4345 Note that you have to supply a list of the symbols which are to be inserted
4346 in the expressions. Symbols in archives are stored by their name only and
4347 if you don't specify which symbols you have, unarchiving the expression will
4348 create new symbols with that name. E.g. if you hadn't included @code{x} in
4349 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4350 have had no effect because the @code{x} in @code{ex1} would have been a
4351 different symbol than the @code{x} which was defined at the beginning of
4352 the program, altough both would appear as @samp{x} when printed.
4354 You can also use the information stored in an @code{archive} object to
4355 output expressions in a format suitable for exact reconstruction. The
4356 @code{archive} and @code{archive_node} classes have a couple of member
4357 functions that let you access the stored properties:
4360 static void my_print2(const archive_node & n)
4363 n.find_string("class", class_name);
4364 cout << class_name << "(";
4366 archive_node::propinfovector p;
4367 n.get_properties(p);
4369 unsigned num = p.size();
4370 for (unsigned i=0; i<num; i++) @{
4371 const string &name = p[i].name;
4372 if (name == "class")
4374 cout << name << "=";
4376 unsigned count = p[i].count;
4380 for (unsigned j=0; j<count; j++) @{
4381 switch (p[i].type) @{
4382 case archive_node::PTYPE_BOOL: @{
4384 n.find_bool(name, x, j);
4385 cout << (x ? "true" : "false");
4388 case archive_node::PTYPE_UNSIGNED: @{
4390 n.find_unsigned(name, x, j);
4394 case archive_node::PTYPE_STRING: @{
4396 n.find_string(name, x, j);
4397 cout << '\"' << x << '\"';
4400 case archive_node::PTYPE_NODE: @{
4401 const archive_node &x = n.find_ex_node(name, j);
4423 ex e = pow(2, x) - y;
4425 my_print2(ar.get_top_node(0)); cout << endl;
4433 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4434 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4435 overall_coeff=numeric(number="0"))
4438 Be warned, however, that the set of properties and their meaning for each
4439 class may change between GiNaC versions.
4442 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4443 @c node-name, next, previous, up
4444 @chapter Extending GiNaC
4446 By reading so far you should have gotten a fairly good understanding of
4447 GiNaC's design-patterns. From here on you should start reading the
4448 sources. All we can do now is issue some recommendations how to tackle
4449 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4450 develop some useful extension please don't hesitate to contact the GiNaC
4451 authors---they will happily incorporate them into future versions.
4454 * What does not belong into GiNaC:: What to avoid.
4455 * Symbolic functions:: Implementing symbolic functions.
4456 * Adding classes:: Defining new algebraic classes.
4460 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4461 @c node-name, next, previous, up
4462 @section What doesn't belong into GiNaC
4464 @cindex @command{ginsh}
4465 First of all, GiNaC's name must be read literally. It is designed to be
4466 a library for use within C++. The tiny @command{ginsh} accompanying
4467 GiNaC makes this even more clear: it doesn't even attempt to provide a
4468 language. There are no loops or conditional expressions in
4469 @command{ginsh}, it is merely a window into the library for the
4470 programmer to test stuff (or to show off). Still, the design of a
4471 complete CAS with a language of its own, graphical capabilites and all
4472 this on top of GiNaC is possible and is without doubt a nice project for
4475 There are many built-in functions in GiNaC that do not know how to
4476 evaluate themselves numerically to a precision declared at runtime
4477 (using @code{Digits}). Some may be evaluated at certain points, but not
4478 generally. This ought to be fixed. However, doing numerical
4479 computations with GiNaC's quite abstract classes is doomed to be
4480 inefficient. For this purpose, the underlying foundation classes
4481 provided by @acronym{CLN} are much better suited.
4484 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4485 @c node-name, next, previous, up
4486 @section Symbolic functions
4488 The easiest and most instructive way to start with is probably to
4489 implement your own function. GiNaC's functions are objects of class
4490 @code{function}. The preprocessor is then used to convert the function
4491 names to objects with a corresponding serial number that is used
4492 internally to identify them. You usually need not worry about this
4493 number. New functions may be inserted into the system via a kind of
4494 `registry'. It is your responsibility to care for some functions that
4495 are called when the user invokes certain methods. These are usual
4496 C++-functions accepting a number of @code{ex} as arguments and returning
4497 one @code{ex}. As an example, if we have a look at a simplified
4498 implementation of the cosine trigonometric function, we first need a
4499 function that is called when one wishes to @code{eval} it. It could
4500 look something like this:
4503 static ex cos_eval_method(const ex & x)
4505 // if (!x%(2*Pi)) return 1
4506 // if (!x%Pi) return -1
4507 // if (!x%Pi/2) return 0
4508 // care for other cases...
4509 return cos(x).hold();
4513 @cindex @code{hold()}
4515 The last line returns @code{cos(x)} if we don't know what else to do and
4516 stops a potential recursive evaluation by saying @code{.hold()}, which
4517 sets a flag to the expression signaling that it has been evaluated. We
4518 should also implement a method for numerical evaluation and since we are
4519 lazy we sweep the problem under the rug by calling someone else's
4520 function that does so, in this case the one in class @code{numeric}:
4523 static ex cos_evalf(const ex & x)
4525 if (is_a<numeric>(x))
4526 return cos(ex_to<numeric>(x));
4528 return cos(x).hold();
4532 Differentiation will surely turn up and so we need to tell @code{cos}
4533 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4534 instance are then handled automatically by @code{basic::diff} and
4538 static ex cos_deriv(const ex & x, unsigned diff_param)
4544 @cindex product rule
4545 The second parameter is obligatory but uninteresting at this point. It
4546 specifies which parameter to differentiate in a partial derivative in
4547 case the function has more than one parameter and its main application
4548 is for correct handling of the chain rule. For Taylor expansion, it is
4549 enough to know how to differentiate. But if the function you want to
4550 implement does have a pole somewhere in the complex plane, you need to
4551 write another method for Laurent expansion around that point.
4553 Now that all the ingredients for @code{cos} have been set up, we need
4554 to tell the system about it. This is done by a macro and we are not
4555 going to descibe how it expands, please consult your preprocessor if you
4559 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4560 evalf_func(cos_evalf).
4561 derivative_func(cos_deriv));
4564 The first argument is the function's name used for calling it and for
4565 output. The second binds the corresponding methods as options to this
4566 object. Options are separated by a dot and can be given in an arbitrary
4567 order. GiNaC functions understand several more options which are always
4568 specified as @code{.option(params)}, for example a method for series
4569 expansion @code{.series_func(cos_series)}. Again, if no series
4570 expansion method is given, GiNaC defaults to simple Taylor expansion,
4571 which is correct if there are no poles involved as is the case for the
4572 @code{cos} function. The way GiNaC handles poles in case there are any
4573 is best understood by studying one of the examples, like the Gamma
4574 (@code{tgamma}) function for instance. (In essence the function first
4575 checks if there is a pole at the evaluation point and falls back to
4576 Taylor expansion if there isn't. Then, the pole is regularized by some
4577 suitable transformation.) Also, the new function needs to be declared
4578 somewhere. This may also be done by a convenient preprocessor macro:
4581 DECLARE_FUNCTION_1P(cos)
4584 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4585 implementation of @code{cos} is very incomplete and lacks several safety
4586 mechanisms. Please, have a look at the real implementation in GiNaC.
4587 (By the way: in case you are worrying about all the macros above we can
4588 assure you that functions are GiNaC's most macro-intense classes. We
4589 have done our best to avoid macros where we can.)
4592 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4593 @c node-name, next, previous, up
4594 @section Adding classes
4596 If you are doing some very specialized things with GiNaC you may find that
4597 you have to implement your own algebraic classes to fit your needs. This
4598 section will explain how to do this by giving the example of a simple
4599 'string' class. After reading this section you will know how to properly
4600 declare a GiNaC class and what the minimum required member functions are
4601 that you have to implement. We only cover the implementation of a 'leaf'
4602 class here (i.e. one that doesn't contain subexpressions). Creating a
4603 container class like, for example, a class representing tensor products is
4604 more involved but this section should give you enough information so you can
4605 consult the source to GiNaC's predefined classes if you want to implement
4606 something more complicated.
4608 @subsection GiNaC's run-time type information system
4610 @cindex hierarchy of classes
4612 All algebraic classes (that is, all classes that can appear in expressions)
4613 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4614 @code{basic *} (which is essentially what an @code{ex} is) represents a
4615 generic pointer to an algebraic class. Occasionally it is necessary to find
4616 out what the class of an object pointed to by a @code{basic *} really is.
4617 Also, for the unarchiving of expressions it must be possible to find the
4618 @code{unarchive()} function of a class given the class name (as a string). A
4619 system that provides this kind of information is called a run-time type
4620 information (RTTI) system. The C++ language provides such a thing (see the
4621 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4622 implements its own, simpler RTTI.
4624 The RTTI in GiNaC is based on two mechanisms:
4629 The @code{basic} class declares a member variable @code{tinfo_key} which
4630 holds an unsigned integer that identifies the object's class. These numbers
4631 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4632 classes. They all start with @code{TINFO_}.
4635 By means of some clever tricks with static members, GiNaC maintains a list
4636 of information for all classes derived from @code{basic}. The information
4637 available includes the class names, the @code{tinfo_key}s, and pointers
4638 to the unarchiving functions. This class registry is defined in the
4639 @file{registrar.h} header file.
4643 The disadvantage of this proprietary RTTI implementation is that there's
4644 a little more to do when implementing new classes (C++'s RTTI works more
4645 or less automatic) but don't worry, most of the work is simplified by
4648 @subsection A minimalistic example
4650 Now we will start implementing a new class @code{mystring} that allows
4651 placing character strings in algebraic expressions (this is not very useful,
4652 but it's just an example). This class will be a direct subclass of
4653 @code{basic}. You can use this sample implementation as a starting point
4654 for your own classes.
4656 The code snippets given here assume that you have included some header files
4662 #include <stdexcept>
4663 using namespace std;
4665 #include <ginac/ginac.h>
4666 using namespace GiNaC;
4669 The first thing we have to do is to define a @code{tinfo_key} for our new
4670 class. This can be any arbitrary unsigned number that is not already taken
4671 by one of the existing classes but it's better to come up with something
4672 that is unlikely to clash with keys that might be added in the future. The
4673 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4674 which is not a requirement but we are going to stick with this scheme:
4677 const unsigned TINFO_mystring = 0x42420001U;
4680 Now we can write down the class declaration. The class stores a C++
4681 @code{string} and the user shall be able to construct a @code{mystring}
4682 object from a C or C++ string:
4685 class mystring : public basic
4687 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4690 mystring(const string &s);
4691 mystring(const char *s);
4697 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4700 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4701 macros are defined in @file{registrar.h}. They take the name of the class
4702 and its direct superclass as arguments and insert all required declarations
4703 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4704 the first line after the opening brace of the class definition. The
4705 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4706 source (at global scope, of course, not inside a function).
4708 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4709 declarations of the default and copy constructor, the destructor, the
4710 assignment operator and a couple of other functions that are required. It
4711 also defines a type @code{inherited} which refers to the superclass so you
4712 don't have to modify your code every time you shuffle around the class
4713 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4714 constructor, the destructor and the assignment operator.
4716 Now there are nine member functions we have to implement to get a working
4722 @code{mystring()}, the default constructor.
4725 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4726 assignment operator to free dynamically allocated members. The @code{call_parent}
4727 specifies whether the @code{destroy()} function of the superclass is to be
4731 @code{void copy(const mystring &other)}, which is used in the copy constructor
4732 and assignment operator to copy the member variables over from another
4733 object of the same class.
4736 @code{void archive(archive_node &n)}, the archiving function. This stores all
4737 information needed to reconstruct an object of this class inside an
4738 @code{archive_node}.
4741 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4742 constructor. This constructs an instance of the class from the information
4743 found in an @code{archive_node}.
4746 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4747 unarchiving function. It constructs a new instance by calling the unarchiving
4751 @code{int compare_same_type(const basic &other)}, which is used internally
4752 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4753 -1, depending on the relative order of this object and the @code{other}
4754 object. If it returns 0, the objects are considered equal.
4755 @strong{Note:} This has nothing to do with the (numeric) ordering
4756 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4757 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4758 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4759 must provide a @code{compare_same_type()} function, even those representing
4760 objects for which no reasonable algebraic ordering relationship can be
4764 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4765 which are the two constructors we declared.
4769 Let's proceed step-by-step. The default constructor looks like this:
4772 mystring::mystring() : inherited(TINFO_mystring)
4774 // dynamically allocate resources here if required
4778 The golden rule is that in all constructors you have to set the
4779 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4780 it will be set by the constructor of the superclass and all hell will break
4781 loose in the RTTI. For your convenience, the @code{basic} class provides
4782 a constructor that takes a @code{tinfo_key} value, which we are using here
4783 (remember that in our case @code{inherited = basic}). If the superclass
4784 didn't have such a constructor, we would have to set the @code{tinfo_key}
4785 to the right value manually.
4787 In the default constructor you should set all other member variables to
4788 reasonable default values (we don't need that here since our @code{str}
4789 member gets set to an empty string automatically). The constructor(s) are of
4790 course also the right place to allocate any dynamic resources you require.
4792 Next, the @code{destroy()} function:
4795 void mystring::destroy(bool call_parent)
4797 // free dynamically allocated resources here if required
4799 inherited::destroy(call_parent);
4803 This function is where we free all dynamically allocated resources. We don't
4804 have any so we're not doing anything here, but if we had, for example, used
4805 a C-style @code{char *} to store our string, this would be the place to
4806 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4807 to call the @code{destroy()} function of the superclass after we're done
4808 (to mimic C++'s automatic invocation of superclass destructors where
4809 @code{destroy()} is called from outside a destructor).
4811 The @code{copy()} function just copies over the member variables from
4815 void mystring::copy(const mystring &other)
4817 inherited::copy(other);
4822 We can simply overwrite the member variables here. There's no need to worry
4823 about dynamically allocated storage. The assignment operator (which is
4824 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4825 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4826 explicitly call the @code{copy()} function of the superclass here so
4827 all the member variables will get copied.
4829 Next are the three functions for archiving. You have to implement them even
4830 if you don't plan to use archives, but the minimum required implementation
4831 is really simple. First, the archiving function:
4834 void mystring::archive(archive_node &n) const
4836 inherited::archive(n);
4837 n.add_string("string", str);
4841 The only thing that is really required is calling the @code{archive()}
4842 function of the superclass. Optionally, you can store all information you
4843 deem necessary for representing the object into the passed
4844 @code{archive_node}. We are just storing our string here. For more
4845 information on how the archiving works, consult the @file{archive.h} header
4848 The unarchiving constructor is basically the inverse of the archiving
4852 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4854 n.find_string("string", str);
4858 If you don't need archiving, just leave this function empty (but you must
4859 invoke the unarchiving constructor of the superclass). Note that we don't
4860 have to set the @code{tinfo_key} here because it is done automatically
4861 by the unarchiving constructor of the @code{basic} class.
4863 Finally, the unarchiving function:
4866 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4868 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4872 You don't have to understand how exactly this works. Just copy these four
4873 lines into your code literally (replacing the class name, of course). It
4874 calls the unarchiving constructor of the class and unless you are doing
4875 something very special (like matching @code{archive_node}s to global
4876 objects) you don't need a different implementation. For those who are
4877 interested: setting the @code{dynallocated} flag puts the object under
4878 the control of GiNaC's garbage collection. It will get deleted automatically
4879 once it is no longer referenced.
4881 Our @code{compare_same_type()} function uses a provided function to compare
4885 int mystring::compare_same_type(const basic &other) const
4887 const mystring &o = static_cast<const mystring &>(other);
4888 int cmpval = str.compare(o.str);
4891 else if (cmpval < 0)
4898 Although this function takes a @code{basic &}, it will always be a reference
4899 to an object of exactly the same class (objects of different classes are not
4900 comparable), so the cast is safe. If this function returns 0, the two objects
4901 are considered equal (in the sense that @math{A-B=0}), so you should compare
4902 all relevant member variables.
4904 Now the only thing missing is our two new constructors:
4907 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4909 // dynamically allocate resources here if required
4912 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4914 // dynamically allocate resources here if required
4918 No surprises here. We set the @code{str} member from the argument and
4919 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4921 That's it! We now have a minimal working GiNaC class that can store
4922 strings in algebraic expressions. Let's confirm that the RTTI works:
4925 ex e = mystring("Hello, world!");
4926 cout << is_a<mystring>(e) << endl;
4929 cout << e.bp->class_name() << endl;
4933 Obviously it does. Let's see what the expression @code{e} looks like:
4937 // -> [mystring object]
4940 Hm, not exactly what we expect, but of course the @code{mystring} class
4941 doesn't yet know how to print itself. This is done in the @code{print()}
4942 member function. Let's say that we wanted to print the string surrounded
4946 class mystring : public basic
4950 void print(const print_context &c, unsigned level = 0) const;
4954 void mystring::print(const print_context &c, unsigned level) const
4956 // print_context::s is a reference to an ostream
4957 c.s << '\"' << str << '\"';
4961 The @code{level} argument is only required for container classes to
4962 correctly parenthesize the output. Let's try again to print the expression:
4966 // -> "Hello, world!"
4969 Much better. The @code{mystring} class can be used in arbitrary expressions:
4972 e += mystring("GiNaC rulez");
4974 // -> "GiNaC rulez"+"Hello, world!"
4977 (GiNaC's automatic term reordering is in effect here), or even
4980 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4982 // -> "One string"^(2*sin(-"Another string"+Pi))
4985 Whether this makes sense is debatable but remember that this is only an
4986 example. At least it allows you to implement your own symbolic algorithms
4989 Note that GiNaC's algebraic rules remain unchanged:
4992 e = mystring("Wow") * mystring("Wow");
4996 e = pow(mystring("First")-mystring("Second"), 2);
4997 cout << e.expand() << endl;
4998 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5001 There's no way to, for example, make GiNaC's @code{add} class perform string
5002 concatenation. You would have to implement this yourself.
5004 @subsection Automatic evaluation
5006 @cindex @code{hold()}
5008 When dealing with objects that are just a little more complicated than the
5009 simple string objects we have implemented, chances are that you will want to
5010 have some automatic simplifications or canonicalizations performed on them.
5011 This is done in the evaluation member function @code{eval()}. Let's say that
5012 we wanted all strings automatically converted to lowercase with
5013 non-alphabetic characters stripped, and empty strings removed:
5016 class mystring : public basic
5020 ex eval(int level = 0) const;
5024 ex mystring::eval(int level) const
5027 for (int i=0; i<str.length(); i++) @{
5029 if (c >= 'A' && c <= 'Z')
5030 new_str += tolower(c);
5031 else if (c >= 'a' && c <= 'z')
5035 if (new_str.length() == 0)
5038 return mystring(new_str).hold();
5042 The @code{level} argument is used to limit the recursion depth of the
5043 evaluation. We don't have any subexpressions in the @code{mystring} class
5044 so we are not concerned with this. If we had, we would call the @code{eval()}
5045 functions of the subexpressions with @code{level - 1} as the argument if
5046 @code{level != 1}. The @code{hold()} member function sets a flag in the
5047 object that prevents further evaluation. Otherwise we might end up in an
5048 endless loop. When you want to return the object unmodified, use
5049 @code{return this->hold();}.
5051 Let's confirm that it works:
5054 ex e = mystring("Hello, world!") + mystring("!?#");
5058 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5063 @subsection Other member functions
5065 We have implemented only a small set of member functions to make the class
5066 work in the GiNaC framework. For a real algebraic class, there are probably
5067 some more functions that you will want to re-implement, such as
5068 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5069 or the header file of the class you want to make a subclass of to see
5070 what's there. One member function that you will most likely want to
5071 implement for terminal classes like the described string class is
5072 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5073 which will allow GiNaC to compare and canonicalize expressions much more
5076 You can, of course, also add your own new member functions. Remember,
5077 that the RTTI may be used to get information about what kinds of objects
5078 you are dealing with (the position in the class hierarchy) and that you
5079 can always extract the bare object from an @code{ex} by stripping the
5080 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5081 should become a need.
5083 That's it. May the source be with you!
5086 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5087 @c node-name, next, previous, up
5088 @chapter A Comparison With Other CAS
5091 This chapter will give you some information on how GiNaC compares to
5092 other, traditional Computer Algebra Systems, like @emph{Maple},
5093 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5094 disadvantages over these systems.
5097 * Advantages:: Stengths of the GiNaC approach.
5098 * Disadvantages:: Weaknesses of the GiNaC approach.
5099 * Why C++?:: Attractiveness of C++.
5102 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5103 @c node-name, next, previous, up
5106 GiNaC has several advantages over traditional Computer
5107 Algebra Systems, like
5112 familiar language: all common CAS implement their own proprietary
5113 grammar which you have to learn first (and maybe learn again when your
5114 vendor decides to `enhance' it). With GiNaC you can write your program
5115 in common C++, which is standardized.
5119 structured data types: you can build up structured data types using
5120 @code{struct}s or @code{class}es together with STL features instead of
5121 using unnamed lists of lists of lists.
5124 strongly typed: in CAS, you usually have only one kind of variables
5125 which can hold contents of an arbitrary type. This 4GL like feature is
5126 nice for novice programmers, but dangerous.
5129 development tools: powerful development tools exist for C++, like fancy
5130 editors (e.g. with automatic indentation and syntax highlighting),
5131 debuggers, visualization tools, documentation generators@dots{}
5134 modularization: C++ programs can easily be split into modules by
5135 separating interface and implementation.
5138 price: GiNaC is distributed under the GNU Public License which means
5139 that it is free and available with source code. And there are excellent
5140 C++-compilers for free, too.
5143 extendable: you can add your own classes to GiNaC, thus extending it on
5144 a very low level. Compare this to a traditional CAS that you can
5145 usually only extend on a high level by writing in the language defined
5146 by the parser. In particular, it turns out to be almost impossible to
5147 fix bugs in a traditional system.
5150 multiple interfaces: Though real GiNaC programs have to be written in
5151 some editor, then be compiled, linked and executed, there are more ways
5152 to work with the GiNaC engine. Many people want to play with
5153 expressions interactively, as in traditional CASs. Currently, two such
5154 windows into GiNaC have been implemented and many more are possible: the
5155 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5156 types to a command line and second, as a more consistent approach, an
5157 interactive interface to the @acronym{Cint} C++ interpreter has been put
5158 together (called @acronym{GiNaC-cint}) that allows an interactive
5159 scripting interface consistent with the C++ language.
5162 seemless integration: it is somewhere between difficult and impossible
5163 to call CAS functions from within a program written in C++ or any other
5164 programming language and vice versa. With GiNaC, your symbolic routines
5165 are part of your program. You can easily call third party libraries,
5166 e.g. for numerical evaluation or graphical interaction. All other
5167 approaches are much more cumbersome: they range from simply ignoring the
5168 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5169 system (i.e. @emph{Yacas}).
5172 efficiency: often large parts of a program do not need symbolic
5173 calculations at all. Why use large integers for loop variables or
5174 arbitrary precision arithmetics where @code{int} and @code{double} are
5175 sufficient? For pure symbolic applications, GiNaC is comparable in
5176 speed with other CAS.
5181 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5182 @c node-name, next, previous, up
5183 @section Disadvantages
5185 Of course it also has some disadvantages:
5190 advanced features: GiNaC cannot compete with a program like
5191 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5192 which grows since 1981 by the work of dozens of programmers, with
5193 respect to mathematical features. Integration, factorization,
5194 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5195 not planned for the near future).
5198 portability: While the GiNaC library itself is designed to avoid any
5199 platform dependent features (it should compile on any ANSI compliant C++
5200 compiler), the currently used version of the CLN library (fast large
5201 integer and arbitrary precision arithmetics) can be compiled only on
5202 systems with a recently new C++ compiler from the GNU Compiler
5203 Collection (@acronym{GCC}).@footnote{This is because CLN uses
5204 PROVIDE/REQUIRE like macros to let the compiler gather all static
5205 initializations, which works for GNU C++ only.} GiNaC uses recent
5206 language features like explicit constructors, mutable members, RTTI,
5207 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
5208 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
5209 ANSI compliant, support all needed features.
5214 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5215 @c node-name, next, previous, up
5218 Why did we choose to implement GiNaC in C++ instead of Java or any other
5219 language? C++ is not perfect: type checking is not strict (casting is
5220 possible), separation between interface and implementation is not
5221 complete, object oriented design is not enforced. The main reason is
5222 the often scolded feature of operator overloading in C++. While it may
5223 be true that operating on classes with a @code{+} operator is rarely
5224 meaningful, it is perfectly suited for algebraic expressions. Writing
5225 @math{3x+5y} as @code{3*x+5*y} instead of
5226 @code{x.times(3).plus(y.times(5))} looks much more natural.
5227 Furthermore, the main developers are more familiar with C++ than with
5228 any other programming language.
5231 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5232 @c node-name, next, previous, up
5233 @appendix Internal Structures
5236 * Expressions are reference counted::
5237 * Internal representation of products and sums::
5240 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5241 @c node-name, next, previous, up
5242 @appendixsection Expressions are reference counted
5244 @cindex reference counting
5245 @cindex copy-on-write
5246 @cindex garbage collection
5247 An expression is extremely light-weight since internally it works like a
5248 handle to the actual representation and really holds nothing more than a
5249 pointer to some other object. What this means in practice is that
5250 whenever you create two @code{ex} and set the second equal to the first
5251 no copying process is involved. Instead, the copying takes place as soon
5252 as you try to change the second. Consider the simple sequence of code:
5255 #include <ginac/ginac.h>
5256 using namespace std;
5257 using namespace GiNaC;
5261 symbol x("x"), y("y"), z("z");
5264 e1 = sin(x + 2*y) + 3*z + 41;
5265 e2 = e1; // e2 points to same object as e1
5266 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5267 e2 += 1; // e2 is copied into a new object
5268 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5272 The line @code{e2 = e1;} creates a second expression pointing to the
5273 object held already by @code{e1}. The time involved for this operation
5274 is therefore constant, no matter how large @code{e1} was. Actual
5275 copying, however, must take place in the line @code{e2 += 1;} because
5276 @code{e1} and @code{e2} are not handles for the same object any more.
5277 This concept is called @dfn{copy-on-write semantics}. It increases
5278 performance considerably whenever one object occurs multiple times and
5279 represents a simple garbage collection scheme because when an @code{ex}
5280 runs out of scope its destructor checks whether other expressions handle
5281 the object it points to too and deletes the object from memory if that
5282 turns out not to be the case. A slightly less trivial example of
5283 differentiation using the chain-rule should make clear how powerful this
5287 #include <ginac/ginac.h>
5288 using namespace std;
5289 using namespace GiNaC;
5293 symbol x("x"), y("y");
5297 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5298 cout << e1 << endl // prints x+3*y
5299 << e2 << endl // prints (x+3*y)^3
5300 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5304 Here, @code{e1} will actually be referenced three times while @code{e2}
5305 will be referenced two times. When the power of an expression is built,
5306 that expression needs not be copied. Likewise, since the derivative of
5307 a power of an expression can be easily expressed in terms of that
5308 expression, no copying of @code{e1} is involved when @code{e3} is
5309 constructed. So, when @code{e3} is constructed it will print as
5310 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5311 holds a reference to @code{e2} and the factor in front is just
5314 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5315 semantics. When you insert an expression into a second expression, the
5316 result behaves exactly as if the contents of the first expression were
5317 inserted. But it may be useful to remember that this is not what
5318 happens. Knowing this will enable you to write much more efficient
5319 code. If you still have an uncertain feeling with copy-on-write
5320 semantics, we recommend you have a look at the
5321 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5322 Marshall Cline. Chapter 16 covers this issue and presents an
5323 implementation which is pretty close to the one in GiNaC.
5326 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5327 @c node-name, next, previous, up
5328 @appendixsection Internal representation of products and sums
5330 @cindex representation
5333 @cindex @code{power}
5334 Although it should be completely transparent for the user of
5335 GiNaC a short discussion of this topic helps to understand the sources
5336 and also explain performance to a large degree. Consider the
5337 unexpanded symbolic expression
5339 $2d^3 \left( 4a + 5b - 3 \right)$
5342 @math{2*d^3*(4*a+5*b-3)}
5344 which could naively be represented by a tree of linear containers for
5345 addition and multiplication, one container for exponentiation with base
5346 and exponent and some atomic leaves of symbols and numbers in this
5351 @cindex pair-wise representation
5352 However, doing so results in a rather deeply nested tree which will
5353 quickly become inefficient to manipulate. We can improve on this by
5354 representing the sum as a sequence of terms, each one being a pair of a
5355 purely numeric multiplicative coefficient and its rest. In the same
5356 spirit we can store the multiplication as a sequence of terms, each
5357 having a numeric exponent and a possibly complicated base, the tree
5358 becomes much more flat:
5362 The number @code{3} above the symbol @code{d} shows that @code{mul}
5363 objects are treated similarly where the coefficients are interpreted as
5364 @emph{exponents} now. Addition of sums of terms or multiplication of
5365 products with numerical exponents can be coded to be very efficient with
5366 such a pair-wise representation. Internally, this handling is performed
5367 by most CAS in this way. It typically speeds up manipulations by an
5368 order of magnitude. The overall multiplicative factor @code{2} and the
5369 additive term @code{-3} look somewhat out of place in this
5370 representation, however, since they are still carrying a trivial
5371 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5372 this is avoided by adding a field that carries an overall numeric
5373 coefficient. This results in the realistic picture of internal
5376 $2d^3 \left( 4a + 5b - 3 \right)$:
5379 @math{2*d^3*(4*a+5*b-3)}:
5385 This also allows for a better handling of numeric radicals, since
5386 @code{sqrt(2)} can now be carried along calculations. Now it should be
5387 clear, why both classes @code{add} and @code{mul} are derived from the
5388 same abstract class: the data representation is the same, only the
5389 semantics differs. In the class hierarchy, methods for polynomial
5390 expansion and the like are reimplemented for @code{add} and @code{mul},
5391 but the data structure is inherited from @code{expairseq}.
5394 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5395 @c node-name, next, previous, up
5396 @appendix Package Tools
5398 If you are creating a software package that uses the GiNaC library,
5399 setting the correct command line options for the compiler and linker
5400 can be difficult. GiNaC includes two tools to make this process easier.
5403 * ginac-config:: A shell script to detect compiler and linker flags.
5404 * AM_PATH_GINAC:: Macro for GNU automake.
5408 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5409 @c node-name, next, previous, up
5410 @section @command{ginac-config}
5411 @cindex ginac-config
5413 @command{ginac-config} is a shell script that you can use to determine
5414 the compiler and linker command line options required to compile and
5415 link a program with the GiNaC library.
5417 @command{ginac-config} takes the following flags:
5421 Prints out the version of GiNaC installed.
5423 Prints '-I' flags pointing to the installed header files.
5425 Prints out the linker flags necessary to link a program against GiNaC.
5426 @item --prefix[=@var{PREFIX}]
5427 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5428 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5429 Otherwise, prints out the configured value of @env{$prefix}.
5430 @item --exec-prefix[=@var{PREFIX}]
5431 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5432 Otherwise, prints out the configured value of @env{$exec_prefix}.
5435 Typically, @command{ginac-config} will be used within a configure
5436 script, as described below. It, however, can also be used directly from
5437 the command line using backquotes to compile a simple program. For
5441 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5444 This command line might expand to (for example):
5447 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5448 -lginac -lcln -lstdc++
5451 Not only is the form using @command{ginac-config} easier to type, it will
5452 work on any system, no matter how GiNaC was configured.
5455 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5456 @c node-name, next, previous, up
5457 @section @samp{AM_PATH_GINAC}
5458 @cindex AM_PATH_GINAC
5460 For packages configured using GNU automake, GiNaC also provides
5461 a macro to automate the process of checking for GiNaC.
5464 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5472 Determines the location of GiNaC using @command{ginac-config}, which is
5473 either found in the user's path, or from the environment variable
5474 @env{GINACLIB_CONFIG}.
5477 Tests the installed libraries to make sure that their version
5478 is later than @var{MINIMUM-VERSION}. (A default version will be used
5482 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5483 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5484 variable to the output of @command{ginac-config --libs}, and calls
5485 @samp{AC_SUBST()} for these variables so they can be used in generated
5486 makefiles, and then executes @var{ACTION-IF-FOUND}.
5489 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5490 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5494 This macro is in file @file{ginac.m4} which is installed in
5495 @file{$datadir/aclocal}. Note that if automake was installed with a
5496 different @samp{--prefix} than GiNaC, you will either have to manually
5497 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5498 aclocal the @samp{-I} option when running it.
5501 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5502 * Example package:: Example of a package using AM_PATH_GINAC.
5506 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5507 @c node-name, next, previous, up
5508 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5510 Simply make sure that @command{ginac-config} is in your path, and run
5511 the configure script.
5518 The directory where the GiNaC libraries are installed needs
5519 to be found by your system's dynamic linker.
5521 This is generally done by
5524 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5530 setting the environment variable @env{LD_LIBRARY_PATH},
5533 or, as a last resort,
5536 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5537 running configure, for instance:
5540 LDFLAGS=-R/home/cbauer/lib ./configure
5545 You can also specify a @command{ginac-config} not in your path by
5546 setting the @env{GINACLIB_CONFIG} environment variable to the
5547 name of the executable
5550 If you move the GiNaC package from its installed location,
5551 you will either need to modify @command{ginac-config} script
5552 manually to point to the new location or rebuild GiNaC.
5563 --with-ginac-prefix=@var{PREFIX}
5564 --with-ginac-exec-prefix=@var{PREFIX}
5567 are provided to override the prefix and exec-prefix that were stored
5568 in the @command{ginac-config} shell script by GiNaC's configure. You are
5569 generally better off configuring GiNaC with the right path to begin with.
5573 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5574 @c node-name, next, previous, up
5575 @subsection Example of a package using @samp{AM_PATH_GINAC}
5577 The following shows how to build a simple package using automake
5578 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5581 #include <ginac/ginac.h>
5585 GiNaC::symbol x("x");
5586 GiNaC::ex a = GiNaC::sin(x);
5587 std::cout << "Derivative of " << a
5588 << " is " << a.diff(x) << std::endl;
5593 You should first read the introductory portions of the automake
5594 Manual, if you are not already familiar with it.
5596 Two files are needed, @file{configure.in}, which is used to build the
5600 dnl Process this file with autoconf to produce a configure script.
5602 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5608 AM_PATH_GINAC(0.7.0, [
5609 LIBS="$LIBS $GINACLIB_LIBS"
5610 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5611 ], AC_MSG_ERROR([need to have GiNaC installed]))
5616 The only command in this which is not standard for automake
5617 is the @samp{AM_PATH_GINAC} macro.
5619 That command does the following: If a GiNaC version greater or equal
5620 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5621 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5622 the error message `need to have GiNaC installed'
5624 And the @file{Makefile.am}, which will be used to build the Makefile.
5627 ## Process this file with automake to produce Makefile.in
5628 bin_PROGRAMS = simple
5629 simple_SOURCES = simple.cpp
5632 This @file{Makefile.am}, says that we are building a single executable,
5633 from a single sourcefile @file{simple.cpp}. Since every program
5634 we are building uses GiNaC we simply added the GiNaC options
5635 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5636 want to specify them on a per-program basis: for instance by
5640 simple_LDADD = $(GINACLIB_LIBS)
5641 INCLUDES = $(GINACLIB_CPPFLAGS)
5644 to the @file{Makefile.am}.
5646 To try this example out, create a new directory and add the three
5649 Now execute the following commands:
5652 $ automake --add-missing
5657 You now have a package that can be built in the normal fashion
5666 @node Bibliography, Concept Index, Example package, Top
5667 @c node-name, next, previous, up
5668 @appendix Bibliography
5673 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5676 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5679 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5682 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5685 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5686 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5689 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5690 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5691 Academic Press, London
5694 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5695 D.E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5698 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5703 @node Concept Index, , Bibliography, Top
5704 @c node-name, next, previous, up
5705 @unnumbered Concept Index