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 * Symbols:: Symbolic objects.
672 * Numbers:: Numerical objects.
673 * Constants:: Pre-defined constants.
674 * Fundamental containers:: The power, add and mul classes.
675 * Lists:: Lists of expressions.
676 * Mathematical functions:: Mathematical functions.
677 * Relations:: Equality, Inequality and all that.
678 * Matrices:: Matrices.
679 * Indexed objects:: Handling indexed quantities.
680 * Non-commutative objects:: Algebras with non-commutative products.
684 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
685 @c node-name, next, previous, up
687 @cindex expression (class @code{ex})
690 The most common class of objects a user deals with is the expression
691 @code{ex}, representing a mathematical object like a variable, number,
692 function, sum, product, etc@dots{} Expressions may be put together to form
693 new expressions, passed as arguments to functions, and so on. Here is a
694 little collection of valid expressions:
697 ex MyEx1 = 5; // simple number
698 ex MyEx2 = x + 2*y; // polynomial in x and y
699 ex MyEx3 = (x + 1)/(x - 1); // rational expression
700 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
701 ex MyEx5 = MyEx4 + 1; // similar to above
704 Expressions are handles to other more fundamental objects, that often
705 contain other expressions thus creating a tree of expressions
706 (@xref{Internal Structures}, for particular examples). Most methods on
707 @code{ex} therefore run top-down through such an expression tree. For
708 example, the method @code{has()} scans recursively for occurrences of
709 something inside an expression. Thus, if you have declared @code{MyEx4}
710 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
711 the argument of @code{sin} and hence return @code{true}.
713 The next sections will outline the general picture of GiNaC's class
714 hierarchy and describe the classes of objects that are handled by
718 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
719 @c node-name, next, previous, up
720 @section The Class Hierarchy
722 GiNaC's class hierarchy consists of several classes representing
723 mathematical objects, all of which (except for @code{ex} and some
724 helpers) are internally derived from one abstract base class called
725 @code{basic}. You do not have to deal with objects of class
726 @code{basic}, instead you'll be dealing with symbols, numbers,
727 containers of expressions and so on.
731 To get an idea about what kinds of symbolic composits may be built we
732 have a look at the most important classes in the class hierarchy and
733 some of the relations among the classes:
735 @image{classhierarchy}
737 The abstract classes shown here (the ones without drop-shadow) are of no
738 interest for the user. They are used internally in order to avoid code
739 duplication if two or more classes derived from them share certain
740 features. An example is @code{expairseq}, a container for a sequence of
741 pairs each consisting of one expression and a number (@code{numeric}).
742 What @emph{is} visible to the user are the derived classes @code{add}
743 and @code{mul}, representing sums and products. @xref{Internal
744 Structures}, where these two classes are described in more detail. The
745 following table shortly summarizes what kinds of mathematical objects
746 are stored in the different classes:
749 @multitable @columnfractions .22 .78
750 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
751 @item @code{constant} @tab Constants like
758 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
759 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
760 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
761 @item @code{ncmul} @tab Products of non-commutative objects
762 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
767 @code{sqrt(}@math{2}@code{)}
770 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
771 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
772 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
773 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
774 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
775 @item @code{indexed} @tab Indexed object like @math{A_ij}
776 @item @code{tensor} @tab Special tensor like the delta and metric tensors
777 @item @code{idx} @tab Index of an indexed object
778 @item @code{varidx} @tab Index with variance
779 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
780 @item @code{wildcard} @tab Wildcard for pattern matching
784 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
785 @c node-name, next, previous, up
787 @cindex @code{symbol} (class)
788 @cindex hierarchy of classes
791 Symbols are for symbolic manipulation what atoms are for chemistry. You
792 can declare objects of class @code{symbol} as any other object simply by
793 saying @code{symbol x,y;}. There is, however, a catch in here having to
794 do with the fact that C++ is a compiled language. The information about
795 the symbol's name is thrown away by the compiler but at a later stage
796 you may want to print expressions holding your symbols. In order to
797 avoid confusion GiNaC's symbols are able to know their own name. This
798 is accomplished by declaring its name for output at construction time in
799 the fashion @code{symbol x("x");}. If you declare a symbol using the
800 default constructor (i.e. without string argument) the system will deal
801 out a unique name. That name may not be suitable for printing but for
802 internal routines when no output is desired it is often enough. We'll
803 come across examples of such symbols later in this tutorial.
805 This implies that the strings passed to symbols at construction time may
806 not be used for comparing two of them. It is perfectly legitimate to
807 write @code{symbol x("x"),y("x");} but it is likely to lead into
808 trouble. Here, @code{x} and @code{y} are different symbols and
809 statements like @code{x-y} will not be simplified to zero although the
810 output @code{x-x} looks funny. Such output may also occur when there
811 are two different symbols in two scopes, for instance when you call a
812 function that declares a symbol with a name already existent in a symbol
813 in the calling function. Again, comparing them (using @code{operator==}
814 for instance) will always reveal their difference. Watch out, please.
816 @cindex @code{subs()}
817 Although symbols can be assigned expressions for internal reasons, you
818 should not do it (and we are not going to tell you how it is done). If
819 you want to replace a symbol with something else in an expression, you
820 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
823 @node Numbers, Constants, Symbols, Basic Concepts
824 @c node-name, next, previous, up
826 @cindex @code{numeric} (class)
832 For storing numerical things, GiNaC uses Bruno Haible's library
833 @acronym{CLN}. The classes therein serve as foundation classes for
834 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
835 alternatively for Common Lisp Numbers. In order to find out more about
836 @acronym{CLN}'s internals the reader is refered to the documentation of
837 that library. @inforef{Introduction, , cln}, for more
838 information. Suffice to say that it is by itself build on top of another
839 library, the GNU Multiple Precision library @acronym{GMP}, which is an
840 extremely fast library for arbitrary long integers and rationals as well
841 as arbitrary precision floating point numbers. It is very commonly used
842 by several popular cryptographic applications. @acronym{CLN} extends
843 @acronym{GMP} by several useful things: First, it introduces the complex
844 number field over either reals (i.e. floating point numbers with
845 arbitrary precision) or rationals. Second, it automatically converts
846 rationals to integers if the denominator is unity and complex numbers to
847 real numbers if the imaginary part vanishes and also correctly treats
848 algebraic functions. Third it provides good implementations of
849 state-of-the-art algorithms for all trigonometric and hyperbolic
850 functions as well as for calculation of some useful constants.
852 The user can construct an object of class @code{numeric} in several
853 ways. The following example shows the four most important constructors.
854 It uses construction from C-integer, construction of fractions from two
855 integers, construction from C-float and construction from a string:
858 #include <ginac/ginac.h>
859 using namespace GiNaC;
863 numeric two = 2; // exact integer 2
864 numeric r(2,3); // exact fraction 2/3
865 numeric e(2.71828); // floating point number
866 numeric p = "3.14159265358979323846"; // constructor from string
867 // Trott's constant in scientific notation:
868 numeric trott("1.0841015122311136151E-2");
870 std::cout << two*p << std::endl; // floating point 6.283...
874 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
875 This would, however, call C's built-in operator @code{/} for integers
876 first and result in a numeric holding a plain integer 1. @strong{Never
877 use the operator @code{/} on integers} unless you know exactly what you
878 are doing! Use the constructor from two integers instead, as shown in
879 the example above. Writing @code{numeric(1)/2} may look funny but works
882 @cindex @code{Digits}
884 We have seen now the distinction between exact numbers and floating
885 point numbers. Clearly, the user should never have to worry about
886 dynamically created exact numbers, since their `exactness' always
887 determines how they ought to be handled, i.e. how `long' they are. The
888 situation is different for floating point numbers. Their accuracy is
889 controlled by one @emph{global} variable, called @code{Digits}. (For
890 those readers who know about Maple: it behaves very much like Maple's
891 @code{Digits}). All objects of class numeric that are constructed from
892 then on will be stored with a precision matching that number of decimal
896 #include <ginac/ginac.h>
898 using namespace GiNaC;
902 numeric three(3.0), one(1.0);
903 numeric x = one/three;
905 cout << "in " << Digits << " digits:" << endl;
907 cout << Pi.evalf() << endl;
919 The above example prints the following output to screen:
926 0.333333333333333333333333333333333333333333333333333333333333333333
927 3.14159265358979323846264338327950288419716939937510582097494459231
930 It should be clear that objects of class @code{numeric} should be used
931 for constructing numbers or for doing arithmetic with them. The objects
932 one deals with most of the time are the polymorphic expressions @code{ex}.
934 @subsection Tests on numbers
936 Once you have declared some numbers, assigned them to expressions and
937 done some arithmetic with them it is frequently desired to retrieve some
938 kind of information from them like asking whether that number is
939 integer, rational, real or complex. For those cases GiNaC provides
940 several useful methods. (Internally, they fall back to invocations of
941 certain CLN functions.)
943 As an example, let's construct some rational number, multiply it with
944 some multiple of its denominator and test what comes out:
947 #include <ginac/ginac.h>
949 using namespace GiNaC;
951 // some very important constants:
952 const numeric twentyone(21);
953 const numeric ten(10);
954 const numeric five(5);
958 numeric answer = twentyone;
961 cout << answer.is_integer() << endl; // false, it's 21/5
963 cout << answer.is_integer() << endl; // true, it's 42 now!
967 Note that the variable @code{answer} is constructed here as an integer
968 by @code{numeric}'s copy constructor but in an intermediate step it
969 holds a rational number represented as integer numerator and integer
970 denominator. When multiplied by 10, the denominator becomes unity and
971 the result is automatically converted to a pure integer again.
972 Internally, the underlying @acronym{CLN} is responsible for this
973 behaviour and we refer the reader to @acronym{CLN}'s documentation.
974 Suffice to say that the same behaviour applies to complex numbers as
975 well as return values of certain functions. Complex numbers are
976 automatically converted to real numbers if the imaginary part becomes
977 zero. The full set of tests that can be applied is listed in the
981 @multitable @columnfractions .30 .70
982 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
983 @item @code{.is_zero()}
984 @tab @dots{}equal to zero
985 @item @code{.is_positive()}
986 @tab @dots{}not complex and greater than 0
987 @item @code{.is_integer()}
988 @tab @dots{}a (non-complex) integer
989 @item @code{.is_pos_integer()}
990 @tab @dots{}an integer and greater than 0
991 @item @code{.is_nonneg_integer()}
992 @tab @dots{}an integer and greater equal 0
993 @item @code{.is_even()}
994 @tab @dots{}an even integer
995 @item @code{.is_odd()}
996 @tab @dots{}an odd integer
997 @item @code{.is_prime()}
998 @tab @dots{}a prime integer (probabilistic primality test)
999 @item @code{.is_rational()}
1000 @tab @dots{}an exact rational number (integers are rational, too)
1001 @item @code{.is_real()}
1002 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1003 @item @code{.is_cinteger()}
1004 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1005 @item @code{.is_crational()}
1006 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1011 @node Constants, Fundamental containers, Numbers, Basic Concepts
1012 @c node-name, next, previous, up
1014 @cindex @code{constant} (class)
1017 @cindex @code{Catalan}
1018 @cindex @code{Euler}
1019 @cindex @code{evalf()}
1020 Constants behave pretty much like symbols except that they return some
1021 specific number when the method @code{.evalf()} is called.
1023 The predefined known constants are:
1026 @multitable @columnfractions .14 .30 .56
1027 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1029 @tab Archimedes' constant
1030 @tab 3.14159265358979323846264338327950288
1031 @item @code{Catalan}
1032 @tab Catalan's constant
1033 @tab 0.91596559417721901505460351493238411
1035 @tab Euler's (or Euler-Mascheroni) constant
1036 @tab 0.57721566490153286060651209008240243
1041 @node Fundamental containers, Lists, Constants, Basic Concepts
1042 @c node-name, next, previous, up
1043 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1047 @cindex @code{power}
1049 Simple polynomial expressions are written down in GiNaC pretty much like
1050 in other CAS or like expressions involving numerical variables in C.
1051 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1052 been overloaded to achieve this goal. When you run the following
1053 code snippet, the constructor for an object of type @code{mul} is
1054 automatically called to hold the product of @code{a} and @code{b} and
1055 then the constructor for an object of type @code{add} is called to hold
1056 the sum of that @code{mul} object and the number one:
1060 symbol a("a"), b("b");
1065 @cindex @code{pow()}
1066 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1067 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1068 construction is necessary since we cannot safely overload the constructor
1069 @code{^} in C++ to construct a @code{power} object. If we did, it would
1070 have several counterintuitive and undesired effects:
1074 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1076 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1077 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1078 interpret this as @code{x^(a^b)}.
1080 Also, expressions involving integer exponents are very frequently used,
1081 which makes it even more dangerous to overload @code{^} since it is then
1082 hard to distinguish between the semantics as exponentiation and the one
1083 for exclusive or. (It would be embarassing to return @code{1} where one
1084 has requested @code{2^3}.)
1087 @cindex @command{ginsh}
1088 All effects are contrary to mathematical notation and differ from the
1089 way most other CAS handle exponentiation, therefore overloading @code{^}
1090 is ruled out for GiNaC's C++ part. The situation is different in
1091 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1092 that the other frequently used exponentiation operator @code{**} does
1093 not exist at all in C++).
1095 To be somewhat more precise, objects of the three classes described
1096 here, are all containers for other expressions. An object of class
1097 @code{power} is best viewed as a container with two slots, one for the
1098 basis, one for the exponent. All valid GiNaC expressions can be
1099 inserted. However, basic transformations like simplifying
1100 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1101 when this is mathematically possible. If we replace the outer exponent
1102 three in the example by some symbols @code{a}, the simplification is not
1103 safe and will not be performed, since @code{a} might be @code{1/2} and
1106 Objects of type @code{add} and @code{mul} are containers with an
1107 arbitrary number of slots for expressions to be inserted. Again, simple
1108 and safe simplifications are carried out like transforming
1109 @code{3*x+4-x} to @code{2*x+4}.
1111 The general rule is that when you construct such objects, GiNaC
1112 automatically creates them in canonical form, which might differ from
1113 the form you typed in your program. This allows for rapid comparison of
1114 expressions, since after all @code{a-a} is simply zero. Note, that the
1115 canonical form is not necessarily lexicographical ordering or in any way
1116 easily guessable. It is only guaranteed that constructing the same
1117 expression twice, either implicitly or explicitly, results in the same
1121 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1122 @c node-name, next, previous, up
1123 @section Lists of expressions
1124 @cindex @code{lst} (class)
1126 @cindex @code{nops()}
1128 @cindex @code{append()}
1129 @cindex @code{prepend()}
1130 @cindex @code{remove_first()}
1131 @cindex @code{remove_last()}
1133 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1134 expressions. These are sometimes used to supply a variable number of
1135 arguments of the same type to GiNaC methods such as @code{subs()} and
1136 @code{to_rational()}, so you should have a basic understanding about them.
1138 Lists of up to 16 expressions can be directly constructed from single
1143 symbol x("x"), y("y");
1144 lst l(x, 2, y, x+y);
1145 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1149 Use the @code{nops()} method to determine the size (number of expressions) of
1150 a list and the @code{op()} method to access individual elements:
1154 cout << l.nops() << endl; // prints '4'
1155 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1159 You can append or prepend an expression to a list with the @code{append()}
1160 and @code{prepend()} methods:
1164 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1165 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1169 Finally you can remove the first or last element of a list with
1170 @code{remove_first()} and @code{remove_last()}:
1174 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1175 l.remove_last(); // l is now @{x, 2, y, x+y@}
1180 @node Mathematical functions, Relations, Lists, Basic Concepts
1181 @c node-name, next, previous, up
1182 @section Mathematical functions
1183 @cindex @code{function} (class)
1184 @cindex trigonometric function
1185 @cindex hyperbolic function
1187 There are quite a number of useful functions hard-wired into GiNaC. For
1188 instance, all trigonometric and hyperbolic functions are implemented
1189 (@xref{Built-in Functions}, for a complete list).
1191 These functions (better called @emph{pseudofunctions}) are all objects
1192 of class @code{function}. They accept one or more expressions as
1193 arguments and return one expression. If the arguments are not
1194 numerical, the evaluation of the function may be halted, as it does in
1195 the next example, showing how a function returns itself twice and
1196 finally an expression that may be really useful:
1198 @cindex Gamma function
1199 @cindex @code{subs()}
1202 symbol x("x"), y("y");
1204 cout << tgamma(foo) << endl;
1205 // -> tgamma(x+(1/2)*y)
1206 ex bar = foo.subs(y==1);
1207 cout << tgamma(bar) << endl;
1209 ex foobar = bar.subs(x==7);
1210 cout << tgamma(foobar) << endl;
1211 // -> (135135/128)*Pi^(1/2)
1215 Besides evaluation most of these functions allow differentiation, series
1216 expansion and so on. Read the next chapter in order to learn more about
1219 It must be noted that these pseudofunctions are created by inline
1220 functions, where the argument list is templated. This means that
1221 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1222 @code{sin(ex(1))} and will therefore not result in a floating point
1223 numeber. Unless of course the function prototype is explicitly
1224 overridden -- which is the case for arguments of type @code{numeric}
1225 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1226 point number of class @code{numeric} you should call
1227 @code{sin(numeric(1))}. This is almost the same as calling
1228 @code{sin(1).evalf()} except that the latter will return a numeric
1229 wrapped inside an @code{ex}.
1232 @node Relations, Matrices, Mathematical functions, Basic Concepts
1233 @c node-name, next, previous, up
1235 @cindex @code{relational} (class)
1237 Sometimes, a relation holding between two expressions must be stored
1238 somehow. The class @code{relational} is a convenient container for such
1239 purposes. A relation is by definition a container for two @code{ex} and
1240 a relation between them that signals equality, inequality and so on.
1241 They are created by simply using the C++ operators @code{==}, @code{!=},
1242 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1244 @xref{Mathematical functions}, for examples where various applications
1245 of the @code{.subs()} method show how objects of class relational are
1246 used as arguments. There they provide an intuitive syntax for
1247 substitutions. They are also used as arguments to the @code{ex::series}
1248 method, where the left hand side of the relation specifies the variable
1249 to expand in and the right hand side the expansion point. They can also
1250 be used for creating systems of equations that are to be solved for
1251 unknown variables. But the most common usage of objects of this class
1252 is rather inconspicuous in statements of the form @code{if
1253 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1254 conversion from @code{relational} to @code{bool} takes place. Note,
1255 however, that @code{==} here does not perform any simplifications, hence
1256 @code{expand()} must be called explicitly.
1259 @node Matrices, Indexed objects, Relations, Basic Concepts
1260 @c node-name, next, previous, up
1262 @cindex @code{matrix} (class)
1264 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1265 matrix with @math{m} rows and @math{n} columns are accessed with two
1266 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1267 second one in the range 0@dots{}@math{n-1}.
1269 There are a couple of ways to construct matrices, with or without preset
1273 matrix::matrix(unsigned r, unsigned c);
1274 matrix::matrix(unsigned r, unsigned c, const lst & l);
1275 ex lst_to_matrix(const lst & l);
1276 ex diag_matrix(const lst & l);
1279 The first two functions are @code{matrix} constructors which create a matrix
1280 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1281 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1282 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1283 from a list of lists, each list representing a matrix row. Finally,
1284 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1285 elements. Note that the last two functions return expressions, not matrix
1288 Matrix elements can be accessed and set using the parenthesis (function call)
1292 const ex & matrix::operator()(unsigned r, unsigned c) const;
1293 ex & matrix::operator()(unsigned r, unsigned c);
1296 It is also possible to access the matrix elements in a linear fashion with
1297 the @code{op()} method. But C++-style subscripting with square brackets
1298 @samp{[]} is not available.
1300 Here are a couple of examples that all construct the same 2x2 diagonal
1305 symbol a("a"), b("b");
1313 e = matrix(2, 2, lst(a, 0, 0, b));
1315 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1317 e = diag_matrix(lst(a, b));
1324 @cindex @code{transpose()}
1325 @cindex @code{inverse()}
1326 There are three ways to do arithmetic with matrices. The first (and most
1327 efficient one) is to use the methods provided by the @code{matrix} class:
1330 matrix matrix::add(const matrix & other) const;
1331 matrix matrix::sub(const matrix & other) const;
1332 matrix matrix::mul(const matrix & other) const;
1333 matrix matrix::mul_scalar(const ex & other) const;
1334 matrix matrix::pow(const ex & expn) const;
1335 matrix matrix::transpose(void) const;
1336 matrix matrix::inverse(void) const;
1339 All of these methods return the result as a new matrix object. Here is an
1340 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1345 matrix A(2, 2, lst(1, 2, 3, 4));
1346 matrix B(2, 2, lst(-1, 0, 2, 1));
1347 matrix C(2, 2, lst(8, 4, 2, 1));
1349 matrix result = A.mul(B).sub(C.mul_scalar(2));
1350 cout << result << endl;
1351 // -> [[-13,-6],[1,2]]
1356 @cindex @code{evalm()}
1357 The second (and probably the most natural) way is to construct an expression
1358 containing matrices with the usual arithmetic operators and @code{pow()}.
1359 For efficiency reasons, expressions with sums, products and powers of
1360 matrices are not automatically evaluated in GiNaC. You have to call the
1364 ex ex::evalm() const;
1367 to obtain the result:
1374 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1375 cout << e.evalm() << endl;
1376 // -> [[-13,-6],[1,2]]
1381 The non-commutativity of the product @code{A*B} in this example is
1382 automatically recognized by GiNaC. There is no need to use a special
1383 operator here. @xref{Non-commutative objects}, for more information about
1384 dealing with non-commutative expressions.
1386 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1387 to perform the arithmetic:
1392 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1393 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1395 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1396 cout << e.simplify_indexed() << endl;
1397 // -> [[-13,-6],[1,2]].i.j
1401 Using indices is most useful when working with rectangular matrices and
1402 one-dimensional vectors because you don't have to worry about having to
1403 transpose matrices before multiplying them. @xref{Indexed objects}, for
1404 more information about using matrices with indices, and about indices in
1407 The @code{matrix} class provides a couple of additional methods for
1408 computing determinants, traces, and characteristic polynomials:
1411 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1412 ex matrix::trace(void) const;
1413 ex matrix::charpoly(const symbol & lambda) const;
1416 The @samp{algo} argument of @code{determinant()} allows to select between
1417 different algorithms for calculating the determinant. The possible values
1418 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1419 heuristic to automatically select an algorithm that is likely to give the
1420 result most quickly.
1423 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1424 @c node-name, next, previous, up
1425 @section Indexed objects
1427 GiNaC allows you to handle expressions containing general indexed objects in
1428 arbitrary spaces. It is also able to canonicalize and simplify such
1429 expressions and perform symbolic dummy index summations. There are a number
1430 of predefined indexed objects provided, like delta and metric tensors.
1432 There are few restrictions placed on indexed objects and their indices and
1433 it is easy to construct nonsense expressions, but our intention is to
1434 provide a general framework that allows you to implement algorithms with
1435 indexed quantities, getting in the way as little as possible.
1437 @cindex @code{idx} (class)
1438 @cindex @code{indexed} (class)
1439 @subsection Indexed quantities and their indices
1441 Indexed expressions in GiNaC are constructed of two special types of objects,
1442 @dfn{index objects} and @dfn{indexed objects}.
1446 @cindex contravariant
1449 @item Index objects are of class @code{idx} or a subclass. Every index has
1450 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1451 the index lives in) which can both be arbitrary expressions but are usually
1452 a number or a simple symbol. In addition, indices of class @code{varidx} have
1453 a @dfn{variance} (they can be co- or contravariant), and indices of class
1454 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1456 @item Indexed objects are of class @code{indexed} or a subclass. They
1457 contain a @dfn{base expression} (which is the expression being indexed), and
1458 one or more indices.
1462 @strong{Note:} when printing expressions, covariant indices and indices
1463 without variance are denoted @samp{.i} while contravariant indices are
1464 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1465 value. In the following, we are going to use that notation in the text so
1466 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1467 not visible in the output.
1469 A simple example shall illustrate the concepts:
1472 #include <ginac/ginac.h>
1473 using namespace std;
1474 using namespace GiNaC;
1478 symbol i_sym("i"), j_sym("j");
1479 idx i(i_sym, 3), j(j_sym, 3);
1482 cout << indexed(A, i, j) << endl;
1487 The @code{idx} constructor takes two arguments, the index value and the
1488 index dimension. First we define two index objects, @code{i} and @code{j},
1489 both with the numeric dimension 3. The value of the index @code{i} is the
1490 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1491 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1492 construct an expression containing one indexed object, @samp{A.i.j}. It has
1493 the symbol @code{A} as its base expression and the two indices @code{i} and
1496 Note the difference between the indices @code{i} and @code{j} which are of
1497 class @code{idx}, and the index values which are the sybols @code{i_sym}
1498 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1499 or numbers but must be index objects. For example, the following is not
1500 correct and will raise an exception:
1503 symbol i("i"), j("j");
1504 e = indexed(A, i, j); // ERROR: indices must be of type idx
1507 You can have multiple indexed objects in an expression, index values can
1508 be numeric, and index dimensions symbolic:
1512 symbol B("B"), dim("dim");
1513 cout << 4 * indexed(A, i)
1514 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1519 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1520 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1521 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1522 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1523 @code{simplify_indexed()} for that, see below).
1525 In fact, base expressions, index values and index dimensions can be
1526 arbitrary expressions:
1530 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1535 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1536 get an error message from this but you will probably not be able to do
1537 anything useful with it.
1539 @cindex @code{get_value()}
1540 @cindex @code{get_dimension()}
1544 ex idx::get_value(void);
1545 ex idx::get_dimension(void);
1548 return the value and dimension of an @code{idx} object. If you have an index
1549 in an expression, such as returned by calling @code{.op()} on an indexed
1550 object, you can get a reference to the @code{idx} object with the function
1551 @code{ex_to<idx>()} on the expression.
1553 There are also the methods
1556 bool idx::is_numeric(void);
1557 bool idx::is_symbolic(void);
1558 bool idx::is_dim_numeric(void);
1559 bool idx::is_dim_symbolic(void);
1562 for checking whether the value and dimension are numeric or symbolic
1563 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1564 About Expressions}) returns information about the index value.
1566 @cindex @code{varidx} (class)
1567 If you need co- and contravariant indices, use the @code{varidx} class:
1571 symbol mu_sym("mu"), nu_sym("nu");
1572 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1573 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1575 cout << indexed(A, mu, nu) << endl;
1577 cout << indexed(A, mu_co, nu) << endl;
1579 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1584 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1585 co- or contravariant. The default is a contravariant (upper) index, but
1586 this can be overridden by supplying a third argument to the @code{varidx}
1587 constructor. The two methods
1590 bool varidx::is_covariant(void);
1591 bool varidx::is_contravariant(void);
1594 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1595 to get the object reference from an expression). There's also the very useful
1599 ex varidx::toggle_variance(void);
1602 which makes a new index with the same value and dimension but the opposite
1603 variance. By using it you only have to define the index once.
1605 @cindex @code{spinidx} (class)
1606 The @code{spinidx} class provides dotted and undotted variant indices, as
1607 used in the Weyl-van-der-Waerden spinor formalism:
1611 symbol K("K"), C_sym("C"), D_sym("D");
1612 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1613 // contravariant, undotted
1614 spinidx C_co(C_sym, 2, true); // covariant index
1615 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1616 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1618 cout << indexed(K, C, D) << endl;
1620 cout << indexed(K, C_co, D_dot) << endl;
1622 cout << indexed(K, D_co_dot, D) << endl;
1627 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1628 dotted or undotted. The default is undotted but this can be overridden by
1629 supplying a fourth argument to the @code{spinidx} constructor. The two
1633 bool spinidx::is_dotted(void);
1634 bool spinidx::is_undotted(void);
1637 allow you to check whether or not a @code{spinidx} object is dotted (use
1638 @code{ex_to<spinidx>()} to get the object reference from an expression).
1639 Finally, the two methods
1642 ex spinidx::toggle_dot(void);
1643 ex spinidx::toggle_variance_dot(void);
1646 create a new index with the same value and dimension but opposite dottedness
1647 and the same or opposite variance.
1649 @subsection Substituting indices
1651 @cindex @code{subs()}
1652 Sometimes you will want to substitute one symbolic index with another
1653 symbolic or numeric index, for example when calculating one specific element
1654 of a tensor expression. This is done with the @code{.subs()} method, as it
1655 is done for symbols (see @ref{Substituting Expressions}).
1657 You have two possibilities here. You can either substitute the whole index
1658 by another index or expression:
1662 ex e = indexed(A, mu_co);
1663 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1664 // -> A.mu becomes A~nu
1665 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1666 // -> A.mu becomes A~0
1667 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1668 // -> A.mu becomes A.0
1672 The third example shows that trying to replace an index with something that
1673 is not an index will substitute the index value instead.
1675 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1680 ex e = indexed(A, mu_co);
1681 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1682 // -> A.mu becomes A.nu
1683 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1684 // -> A.mu becomes A.0
1688 As you see, with the second method only the value of the index will get
1689 substituted. Its other properties, including its dimension, remain unchanged.
1690 If you want to change the dimension of an index you have to substitute the
1691 whole index by another one with the new dimension.
1693 Finally, substituting the base expression of an indexed object works as
1698 ex e = indexed(A, mu_co);
1699 cout << e << " becomes " << e.subs(A == A+B) << endl;
1700 // -> A.mu becomes (B+A).mu
1704 @subsection Symmetries
1705 @cindex @code{symmetry} (class)
1706 @cindex @code{sy_none()}
1707 @cindex @code{sy_symm()}
1708 @cindex @code{sy_anti()}
1709 @cindex @code{sy_cycl()}
1711 Indexed objects can have certain symmetry properties with respect to their
1712 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1713 that is constructed with the helper functions
1716 symmetry sy_none(...);
1717 symmetry sy_symm(...);
1718 symmetry sy_anti(...);
1719 symmetry sy_cycl(...);
1722 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1723 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1724 represents a cyclic symmetry. Each of these functions accepts up to four
1725 arguments which can be either symmetry objects themselves or unsigned integer
1726 numbers that represent an index position (counting from 0). A symmetry
1727 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1728 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1731 Here are some examples of symmetry definitions:
1736 e = indexed(A, i, j);
1737 e = indexed(A, sy_none(), i, j); // equivalent
1738 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1740 // Symmetric in all three indices:
1741 e = indexed(A, sy_symm(), i, j, k);
1742 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1743 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1744 // different canonical order
1746 // Symmetric in the first two indices only:
1747 e = indexed(A, sy_symm(0, 1), i, j, k);
1748 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1750 // Antisymmetric in the first and last index only (index ranges need not
1752 e = indexed(A, sy_anti(0, 2), i, j, k);
1753 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1755 // An example of a mixed symmetry: antisymmetric in the first two and
1756 // last two indices, symmetric when swapping the first and last index
1757 // pairs (like the Riemann curvature tensor):
1758 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1760 // Cyclic symmetry in all three indices:
1761 e = indexed(A, sy_cycl(), i, j, k);
1762 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1764 // The following examples are invalid constructions that will throw
1765 // an exception at run time.
1767 // An index may not appear multiple times:
1768 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1769 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1771 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1772 // same number of indices:
1773 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1775 // And of course, you cannot specify indices which are not there:
1776 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1780 If you need to specify more than four indices, you have to use the
1781 @code{.add()} method of the @code{symmetry} class. For example, to specify
1782 full symmetry in the first six indices you would write
1783 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1785 If an indexed object has a symmetry, GiNaC will automatically bring the
1786 indices into a canonical order which allows for some immediate simplifications:
1790 cout << indexed(A, sy_symm(), i, j)
1791 + indexed(A, sy_symm(), j, i) << endl;
1793 cout << indexed(B, sy_anti(), i, j)
1794 + indexed(B, sy_anti(), j, i) << endl;
1796 cout << indexed(B, sy_anti(), i, j, k)
1797 + indexed(B, sy_anti(), j, i, k) << endl;
1802 @cindex @code{get_free_indices()}
1804 @subsection Dummy indices
1806 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1807 that a summation over the index range is implied. Symbolic indices which are
1808 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1809 dummy nor free indices.
1811 To be recognized as a dummy index pair, the two indices must be of the same
1812 class and dimension and their value must be the same single symbol (an index
1813 like @samp{2*n+1} is never a dummy index). If the indices are of class
1814 @code{varidx} they must also be of opposite variance; if they are of class
1815 @code{spinidx} they must be both dotted or both undotted.
1817 The method @code{.get_free_indices()} returns a vector containing the free
1818 indices of an expression. It also checks that the free indices of the terms
1819 of a sum are consistent:
1823 symbol A("A"), B("B"), C("C");
1825 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1826 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1828 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1829 cout << exprseq(e.get_free_indices()) << endl;
1831 // 'j' and 'l' are dummy indices
1833 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1834 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1836 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1837 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1838 cout << exprseq(e.get_free_indices()) << endl;
1840 // 'nu' is a dummy index, but 'sigma' is not
1842 e = indexed(A, mu, mu);
1843 cout << exprseq(e.get_free_indices()) << endl;
1845 // 'mu' is not a dummy index because it appears twice with the same
1848 e = indexed(A, mu, nu) + 42;
1849 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1850 // this will throw an exception:
1851 // "add::get_free_indices: inconsistent indices in sum"
1855 @cindex @code{simplify_indexed()}
1856 @subsection Simplifying indexed expressions
1858 In addition to the few automatic simplifications that GiNaC performs on
1859 indexed expressions (such as re-ordering the indices of symmetric tensors
1860 and calculating traces and convolutions of matrices and predefined tensors)
1864 ex ex::simplify_indexed(void);
1865 ex ex::simplify_indexed(const scalar_products & sp);
1868 that performs some more expensive operations:
1871 @item it checks the consistency of free indices in sums in the same way
1872 @code{get_free_indices()} does
1873 @item it tries to give dumy indices that appear in different terms of a sum
1874 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1875 @item it (symbolically) calculates all possible dummy index summations/contractions
1876 with the predefined tensors (this will be explained in more detail in the
1878 @item it detects contractions that vanish for symmetry reasons, for example
1879 the contraction of a symmetric and a totally antisymmetric tensor
1880 @item as a special case of dummy index summation, it can replace scalar products
1881 of two tensors with a user-defined value
1884 The last point is done with the help of the @code{scalar_products} class
1885 which is used to store scalar products with known values (this is not an
1886 arithmetic class, you just pass it to @code{simplify_indexed()}):
1890 symbol A("A"), B("B"), C("C"), i_sym("i");
1894 sp.add(A, B, 0); // A and B are orthogonal
1895 sp.add(A, C, 0); // A and C are orthogonal
1896 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1898 e = indexed(A + B, i) * indexed(A + C, i);
1900 // -> (B+A).i*(A+C).i
1902 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1908 The @code{scalar_products} object @code{sp} acts as a storage for the
1909 scalar products added to it with the @code{.add()} method. This method
1910 takes three arguments: the two expressions of which the scalar product is
1911 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1912 @code{simplify_indexed()} will replace all scalar products of indexed
1913 objects that have the symbols @code{A} and @code{B} as base expressions
1914 with the single value 0. The number, type and dimension of the indices
1915 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1917 @cindex @code{expand()}
1918 The example above also illustrates a feature of the @code{expand()} method:
1919 if passed the @code{expand_indexed} option it will distribute indices
1920 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1922 @cindex @code{tensor} (class)
1923 @subsection Predefined tensors
1925 Some frequently used special tensors such as the delta, epsilon and metric
1926 tensors are predefined in GiNaC. They have special properties when
1927 contracted with other tensor expressions and some of them have constant
1928 matrix representations (they will evaluate to a number when numeric
1929 indices are specified).
1931 @cindex @code{delta_tensor()}
1932 @subsubsection Delta tensor
1934 The delta tensor takes two indices, is symmetric and has the matrix
1935 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
1936 @code{delta_tensor()}:
1940 symbol A("A"), B("B");
1942 idx i(symbol("i"), 3), j(symbol("j"), 3),
1943 k(symbol("k"), 3), l(symbol("l"), 3);
1945 ex e = indexed(A, i, j) * indexed(B, k, l)
1946 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1947 cout << e.simplify_indexed() << endl;
1950 cout << delta_tensor(i, i) << endl;
1955 @cindex @code{metric_tensor()}
1956 @subsubsection General metric tensor
1958 The function @code{metric_tensor()} creates a general symmetric metric
1959 tensor with two indices that can be used to raise/lower tensor indices. The
1960 metric tensor is denoted as @samp{g} in the output and if its indices are of
1961 mixed variance it is automatically replaced by a delta tensor:
1967 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1969 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1970 cout << e.simplify_indexed() << endl;
1973 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1974 cout << e.simplify_indexed() << endl;
1977 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1978 * metric_tensor(nu, rho);
1979 cout << e.simplify_indexed() << endl;
1982 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1983 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1984 + indexed(A, mu.toggle_variance(), rho));
1985 cout << e.simplify_indexed() << endl;
1990 @cindex @code{lorentz_g()}
1991 @subsubsection Minkowski metric tensor
1993 The Minkowski metric tensor is a special metric tensor with a constant
1994 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1995 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1996 It is created with the function @code{lorentz_g()} (although it is output as
2001 varidx mu(symbol("mu"), 4);
2003 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2004 * lorentz_g(mu, varidx(0, 4)); // negative signature
2005 cout << e.simplify_indexed() << endl;
2008 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2009 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2010 cout << e.simplify_indexed() << endl;
2015 @cindex @code{spinor_metric()}
2016 @subsubsection Spinor metric tensor
2018 The function @code{spinor_metric()} creates an antisymmetric tensor with
2019 two indices that is used to raise/lower indices of 2-component spinors.
2020 It is output as @samp{eps}:
2026 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2027 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2029 e = spinor_metric(A, B) * indexed(psi, B_co);
2030 cout << e.simplify_indexed() << endl;
2033 e = spinor_metric(A, B) * indexed(psi, A_co);
2034 cout << e.simplify_indexed() << endl;
2037 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2038 cout << e.simplify_indexed() << endl;
2041 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2042 cout << e.simplify_indexed() << endl;
2045 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2046 cout << e.simplify_indexed() << endl;
2049 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2050 cout << e.simplify_indexed() << endl;
2055 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2057 @cindex @code{epsilon_tensor()}
2058 @cindex @code{lorentz_eps()}
2059 @subsubsection Epsilon tensor
2061 The epsilon tensor is totally antisymmetric, its number of indices is equal
2062 to the dimension of the index space (the indices must all be of the same
2063 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2064 defined to be 1. Its behaviour with indices that have a variance also
2065 depends on the signature of the metric. Epsilon tensors are output as
2068 There are three functions defined to create epsilon tensors in 2, 3 and 4
2072 ex epsilon_tensor(const ex & i1, const ex & i2);
2073 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2074 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2077 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2078 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2079 Minkowski space (the last @code{bool} argument specifies whether the metric
2080 has negative or positive signature, as in the case of the Minkowski metric
2085 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2086 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2087 e = lorentz_eps(mu, nu, rho, sig) *
2088 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2089 cout << simplify_indexed(e) << endl;
2090 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2092 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2093 symbol A("A"), B("B");
2094 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2095 cout << simplify_indexed(e) << endl;
2096 // -> -B.k*A.j*eps.i.k.j
2097 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2098 cout << simplify_indexed(e) << endl;
2103 @subsection Linear algebra
2105 The @code{matrix} class can be used with indices to do some simple linear
2106 algebra (linear combinations and products of vectors and matrices, traces
2107 and scalar products):
2111 idx i(symbol("i"), 2), j(symbol("j"), 2);
2112 symbol x("x"), y("y");
2114 // A is a 2x2 matrix, X is a 2x1 vector
2115 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2117 cout << indexed(A, i, i) << endl;
2120 ex e = indexed(A, i, j) * indexed(X, j);
2121 cout << e.simplify_indexed() << endl;
2122 // -> [[2*y+x],[4*y+3*x]].i
2124 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2125 cout << e.simplify_indexed() << endl;
2126 // -> [[3*y+3*x,6*y+2*x]].j
2130 You can of course obtain the same results with the @code{matrix::add()},
2131 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2132 but with indices you don't have to worry about transposing matrices.
2134 Matrix indices always start at 0 and their dimension must match the number
2135 of rows/columns of the matrix. Matrices with one row or one column are
2136 vectors and can have one or two indices (it doesn't matter whether it's a
2137 row or a column vector). Other matrices must have two indices.
2139 You should be careful when using indices with variance on matrices. GiNaC
2140 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2141 @samp{F.mu.nu} are different matrices. In this case you should use only
2142 one form for @samp{F} and explicitly multiply it with a matrix representation
2143 of the metric tensor.
2146 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2147 @c node-name, next, previous, up
2148 @section Non-commutative objects
2150 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2151 non-commutative objects are built-in which are mostly of use in high energy
2155 @item Clifford (Dirac) algebra (class @code{clifford})
2156 @item su(3) Lie algebra (class @code{color})
2157 @item Matrices (unindexed) (class @code{matrix})
2160 The @code{clifford} and @code{color} classes are subclasses of
2161 @code{indexed} because the elements of these algebras ususally carry
2162 indices. The @code{matrix} class is described in more detail in
2165 Unlike most computer algebra systems, GiNaC does not primarily provide an
2166 operator (often denoted @samp{&*}) for representing inert products of
2167 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2168 classes of objects involved, and non-commutative products are formed with
2169 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2170 figuring out by itself which objects commute and will group the factors
2171 by their class. Consider this example:
2175 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2176 idx a(symbol("a"), 8), b(symbol("b"), 8);
2177 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2179 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2183 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2184 groups the non-commutative factors (the gammas and the su(3) generators)
2185 together while preserving the order of factors within each class (because
2186 Clifford objects commute with color objects). The resulting expression is a
2187 @emph{commutative} product with two factors that are themselves non-commutative
2188 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2189 parentheses are placed around the non-commutative products in the output.
2191 @cindex @code{ncmul} (class)
2192 Non-commutative products are internally represented by objects of the class
2193 @code{ncmul}, as opposed to commutative products which are handled by the
2194 @code{mul} class. You will normally not have to worry about this distinction,
2197 The advantage of this approach is that you never have to worry about using
2198 (or forgetting to use) a special operator when constructing non-commutative
2199 expressions. Also, non-commutative products in GiNaC are more intelligent
2200 than in other computer algebra systems; they can, for example, automatically
2201 canonicalize themselves according to rules specified in the implementation
2202 of the non-commutative classes. The drawback is that to work with other than
2203 the built-in algebras you have to implement new classes yourself. Symbols
2204 always commute and it's not possible to construct non-commutative products
2205 using symbols to represent the algebra elements or generators. User-defined
2206 functions can, however, be specified as being non-commutative.
2208 @cindex @code{return_type()}
2209 @cindex @code{return_type_tinfo()}
2210 Information about the commutativity of an object or expression can be
2211 obtained with the two member functions
2214 unsigned ex::return_type(void) const;
2215 unsigned ex::return_type_tinfo(void) const;
2218 The @code{return_type()} function returns one of three values (defined in
2219 the header file @file{flags.h}), corresponding to three categories of
2220 expressions in GiNaC:
2223 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2224 classes are of this kind.
2225 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2226 certain class of non-commutative objects which can be determined with the
2227 @code{return_type_tinfo()} method. Expressions of this category commute
2228 with everything except @code{noncommutative} expressions of the same
2230 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2231 of non-commutative objects of different classes. Expressions of this
2232 category don't commute with any other @code{noncommutative} or
2233 @code{noncommutative_composite} expressions.
2236 The value returned by the @code{return_type_tinfo()} method is valid only
2237 when the return type of the expression is @code{noncommutative}. It is a
2238 value that is unique to the class of the object and usually one of the
2239 constants in @file{tinfos.h}, or derived therefrom.
2241 Here are a couple of examples:
2244 @multitable @columnfractions 0.33 0.33 0.34
2245 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2246 @item @code{42} @tab @code{commutative} @tab -
2247 @item @code{2*x-y} @tab @code{commutative} @tab -
2248 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2249 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2250 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2251 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2255 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2256 @code{TINFO_clifford} for objects with a representation label of zero.
2257 Other representation labels yield a different @code{return_type_tinfo()},
2258 but it's the same for any two objects with the same label. This is also true
2261 A last note: With the exception of matrices, positive integer powers of
2262 non-commutative objects are automatically expanded in GiNaC. For example,
2263 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2264 non-commutative expressions).
2267 @cindex @code{clifford} (class)
2268 @subsection Clifford algebra
2270 @cindex @code{dirac_gamma()}
2271 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2272 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2273 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2274 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2277 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2280 which takes two arguments: the index and a @dfn{representation label} in the
2281 range 0 to 255 which is used to distinguish elements of different Clifford
2282 algebras (this is also called a @dfn{spin line index}). Gammas with different
2283 labels commute with each other. The dimension of the index can be 4 or (in
2284 the framework of dimensional regularization) any symbolic value. Spinor
2285 indices on Dirac gammas are not supported in GiNaC.
2287 @cindex @code{dirac_ONE()}
2288 The unity element of a Clifford algebra is constructed by
2291 ex dirac_ONE(unsigned char rl = 0);
2294 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2295 multiples of the unity element, even though it's customary to omit it.
2296 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2297 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2298 GiNaC may produce incorrect results.
2300 @cindex @code{dirac_gamma5()}
2301 There's a special element @samp{gamma5} that commutes with all other
2302 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2306 ex dirac_gamma5(unsigned char rl = 0);
2309 @cindex @code{dirac_gamma6()}
2310 @cindex @code{dirac_gamma7()}
2311 The two additional functions
2314 ex dirac_gamma6(unsigned char rl = 0);
2315 ex dirac_gamma7(unsigned char rl = 0);
2318 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2321 @cindex @code{dirac_slash()}
2322 Finally, the function
2325 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2328 creates a term that represents a contraction of @samp{e} with the Dirac
2329 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2330 with a unique index whose dimension is given by the @code{dim} argument).
2331 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2333 In products of dirac gammas, superfluous unity elements are automatically
2334 removed, squares are replaced by their values and @samp{gamma5} is
2335 anticommuted to the front. The @code{simplify_indexed()} function performs
2336 contractions in gamma strings, for example
2341 symbol a("a"), b("b"), D("D");
2342 varidx mu(symbol("mu"), D);
2343 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2344 * dirac_gamma(mu.toggle_variance());
2346 // -> gamma~mu*a\*gamma.mu
2347 e = e.simplify_indexed();
2350 cout << e.subs(D == 4) << endl;
2356 @cindex @code{dirac_trace()}
2357 To calculate the trace of an expression containing strings of Dirac gammas
2358 you use the function
2361 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2364 This function takes the trace of all gammas with the specified representation
2365 label; gammas with other labels are left standing. The last argument to
2366 @code{dirac_trace()} is the value to be returned for the trace of the unity
2367 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2368 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2369 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2370 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2371 This @samp{gamma5} scheme is described in greater detail in
2372 @cite{The Role of gamma5 in Dimensional Regularization}.
2374 The value of the trace itself is also usually different in 4 and in
2375 @math{D != 4} dimensions:
2380 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2381 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2382 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2383 cout << dirac_trace(e).simplify_indexed() << endl;
2390 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2391 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2392 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2393 cout << dirac_trace(e).simplify_indexed() << endl;
2394 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2398 Here is an example for using @code{dirac_trace()} to compute a value that
2399 appears in the calculation of the one-loop vacuum polarization amplitude in
2404 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2405 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2408 sp.add(l, l, pow(l, 2));
2409 sp.add(l, q, ldotq);
2411 ex e = dirac_gamma(mu) *
2412 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2413 dirac_gamma(mu.toggle_variance()) *
2414 (dirac_slash(l, D) + m * dirac_ONE());
2415 e = dirac_trace(e).simplify_indexed(sp);
2416 e = e.collect(lst(l, ldotq, m));
2418 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2422 The @code{canonicalize_clifford()} function reorders all gamma products that
2423 appear in an expression to a canonical (but not necessarily simple) form.
2424 You can use this to compare two expressions or for further simplifications:
2428 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2429 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2431 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2433 e = canonicalize_clifford(e);
2440 @cindex @code{color} (class)
2441 @subsection Color algebra
2443 @cindex @code{color_T()}
2444 For computations in quantum chromodynamics, GiNaC implements the base elements
2445 and structure constants of the su(3) Lie algebra (color algebra). The base
2446 elements @math{T_a} are constructed by the function
2449 ex color_T(const ex & a, unsigned char rl = 0);
2452 which takes two arguments: the index and a @dfn{representation label} in the
2453 range 0 to 255 which is used to distinguish elements of different color
2454 algebras. Objects with different labels commute with each other. The
2455 dimension of the index must be exactly 8 and it should be of class @code{idx},
2458 @cindex @code{color_ONE()}
2459 The unity element of a color algebra is constructed by
2462 ex color_ONE(unsigned char rl = 0);
2465 @strong{Note:} You must always use @code{color_ONE()} when referring to
2466 multiples of the unity element, even though it's customary to omit it.
2467 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2468 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2469 GiNaC may produce incorrect results.
2471 @cindex @code{color_d()}
2472 @cindex @code{color_f()}
2476 ex color_d(const ex & a, const ex & b, const ex & c);
2477 ex color_f(const ex & a, const ex & b, const ex & c);
2480 create the symmetric and antisymmetric structure constants @math{d_abc} and
2481 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2482 and @math{[T_a, T_b] = i f_abc T_c}.
2484 @cindex @code{color_h()}
2485 There's an additional function
2488 ex color_h(const ex & a, const ex & b, const ex & c);
2491 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2493 The function @code{simplify_indexed()} performs some simplifications on
2494 expressions containing color objects:
2499 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2500 k(symbol("k"), 8), l(symbol("l"), 8);
2502 e = color_d(a, b, l) * color_f(a, b, k);
2503 cout << e.simplify_indexed() << endl;
2506 e = color_d(a, b, l) * color_d(a, b, k);
2507 cout << e.simplify_indexed() << endl;
2510 e = color_f(l, a, b) * color_f(a, b, k);
2511 cout << e.simplify_indexed() << endl;
2514 e = color_h(a, b, c) * color_h(a, b, c);
2515 cout << e.simplify_indexed() << endl;
2518 e = color_h(a, b, c) * color_T(b) * color_T(c);
2519 cout << e.simplify_indexed() << endl;
2522 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2523 cout << e.simplify_indexed() << endl;
2526 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2527 cout << e.simplify_indexed() << endl;
2528 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2532 @cindex @code{color_trace()}
2533 To calculate the trace of an expression containing color objects you use the
2537 ex color_trace(const ex & e, unsigned char rl = 0);
2540 This function takes the trace of all color @samp{T} objects with the
2541 specified representation label; @samp{T}s with other labels are left
2542 standing. For example:
2546 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2548 // -> -I*f.a.c.b+d.a.c.b
2553 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2554 @c node-name, next, previous, up
2555 @chapter Methods and Functions
2558 In this chapter the most important algorithms provided by GiNaC will be
2559 described. Some of them are implemented as functions on expressions,
2560 others are implemented as methods provided by expression objects. If
2561 they are methods, there exists a wrapper function around it, so you can
2562 alternatively call it in a functional way as shown in the simple
2567 cout << "As method: " << sin(1).evalf() << endl;
2568 cout << "As function: " << evalf(sin(1)) << endl;
2572 @cindex @code{subs()}
2573 The general rule is that wherever methods accept one or more parameters
2574 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2575 wrapper accepts is the same but preceded by the object to act on
2576 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2577 most natural one in an OO model but it may lead to confusion for MapleV
2578 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2579 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2580 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2581 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2582 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2583 here. Also, users of MuPAD will in most cases feel more comfortable
2584 with GiNaC's convention. All function wrappers are implemented
2585 as simple inline functions which just call the corresponding method and
2586 are only provided for users uncomfortable with OO who are dead set to
2587 avoid method invocations. Generally, nested function wrappers are much
2588 harder to read than a sequence of methods and should therefore be
2589 avoided if possible. On the other hand, not everything in GiNaC is a
2590 method on class @code{ex} and sometimes calling a function cannot be
2594 * Information About Expressions::
2595 * Substituting Expressions::
2596 * Pattern Matching and Advanced Substitutions::
2597 * Applying a Function on Subexpressions::
2598 * Polynomial Arithmetic:: Working with polynomials.
2599 * Rational Expressions:: Working with rational functions.
2600 * Symbolic Differentiation::
2601 * Series Expansion:: Taylor and Laurent expansion.
2603 * Built-in Functions:: List of predefined mathematical functions.
2604 * Input/Output:: Input and output of expressions.
2608 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2609 @c node-name, next, previous, up
2610 @section Getting information about expressions
2612 @subsection Checking expression types
2613 @cindex @code{is_a<@dots{}>()}
2614 @cindex @code{is_exactly_a<@dots{}>()}
2615 @cindex @code{ex_to<@dots{}>()}
2616 @cindex Converting @code{ex} to other classes
2617 @cindex @code{info()}
2618 @cindex @code{return_type()}
2619 @cindex @code{return_type_tinfo()}
2621 Sometimes it's useful to check whether a given expression is a plain number,
2622 a sum, a polynomial with integer coefficients, or of some other specific type.
2623 GiNaC provides a couple of functions for this:
2626 bool is_a<T>(const ex & e);
2627 bool is_exactly_a<T>(const ex & e);
2628 bool ex::info(unsigned flag);
2629 unsigned ex::return_type(void) const;
2630 unsigned ex::return_type_tinfo(void) const;
2633 When the test made by @code{is_a<T>()} returns true, it is safe to call
2634 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2635 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2636 example, assuming @code{e} is an @code{ex}:
2641 if (is_a<numeric>(e))
2642 numeric n = ex_to<numeric>(e);
2647 @code{is_a<T>(e)} allows you to check whether the top-level object of
2648 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2649 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2650 e.g., for checking whether an expression is a number, a sum, or a product:
2657 is_a<numeric>(e1); // true
2658 is_a<numeric>(e2); // false
2659 is_a<add>(e1); // false
2660 is_a<add>(e2); // true
2661 is_a<mul>(e1); // false
2662 is_a<mul>(e2); // false
2666 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2667 top-level object of an expression @samp{e} is an instance of the GiNaC
2668 class @samp{T}, not including parent classes.
2670 The @code{info()} method is used for checking certain attributes of
2671 expressions. The possible values for the @code{flag} argument are defined
2672 in @file{ginac/flags.h}, the most important being explained in the following
2676 @multitable @columnfractions .30 .70
2677 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2678 @item @code{numeric}
2679 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2681 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2682 @item @code{rational}
2683 @tab @dots{}an exact rational number (integers are rational, too)
2684 @item @code{integer}
2685 @tab @dots{}a (non-complex) integer
2686 @item @code{crational}
2687 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2688 @item @code{cinteger}
2689 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2690 @item @code{positive}
2691 @tab @dots{}not complex and greater than 0
2692 @item @code{negative}
2693 @tab @dots{}not complex and less than 0
2694 @item @code{nonnegative}
2695 @tab @dots{}not complex and greater than or equal to 0
2697 @tab @dots{}an integer greater than 0
2699 @tab @dots{}an integer less than 0
2700 @item @code{nonnegint}
2701 @tab @dots{}an integer greater than or equal to 0
2703 @tab @dots{}an even integer
2705 @tab @dots{}an odd integer
2707 @tab @dots{}a prime integer (probabilistic primality test)
2708 @item @code{relation}
2709 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2710 @item @code{relation_equal}
2711 @tab @dots{}a @code{==} relation
2712 @item @code{relation_not_equal}
2713 @tab @dots{}a @code{!=} relation
2714 @item @code{relation_less}
2715 @tab @dots{}a @code{<} relation
2716 @item @code{relation_less_or_equal}
2717 @tab @dots{}a @code{<=} relation
2718 @item @code{relation_greater}
2719 @tab @dots{}a @code{>} relation
2720 @item @code{relation_greater_or_equal}
2721 @tab @dots{}a @code{>=} relation
2723 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2725 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2726 @item @code{polynomial}
2727 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2728 @item @code{integer_polynomial}
2729 @tab @dots{}a polynomial with (non-complex) integer coefficients
2730 @item @code{cinteger_polynomial}
2731 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2732 @item @code{rational_polynomial}
2733 @tab @dots{}a polynomial with (non-complex) rational coefficients
2734 @item @code{crational_polynomial}
2735 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2736 @item @code{rational_function}
2737 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2738 @item @code{algebraic}
2739 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2743 To determine whether an expression is commutative or non-commutative and if
2744 so, with which other expressions it would commute, you use the methods
2745 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2746 for an explanation of these.
2749 @subsection Accessing subexpressions
2750 @cindex @code{nops()}
2753 @cindex @code{relational} (class)
2755 GiNaC provides the two methods
2758 unsigned ex::nops();
2759 ex ex::op(unsigned i);
2762 for accessing the subexpressions in the container-like GiNaC classes like
2763 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2764 determines the number of subexpressions (@samp{operands}) contained, while
2765 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2766 In the case of a @code{power} object, @code{op(0)} will return the basis
2767 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2768 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2770 The left-hand and right-hand side expressions of objects of class
2771 @code{relational} (and only of these) can also be accessed with the methods
2779 @subsection Comparing expressions
2780 @cindex @code{is_equal()}
2781 @cindex @code{is_zero()}
2783 Expressions can be compared with the usual C++ relational operators like
2784 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2785 the result is usually not determinable and the result will be @code{false},
2786 except in the case of the @code{!=} operator. You should also be aware that
2787 GiNaC will only do the most trivial test for equality (subtracting both
2788 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2791 Actually, if you construct an expression like @code{a == b}, this will be
2792 represented by an object of the @code{relational} class (@pxref{Relations})
2793 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2795 There are also two methods
2798 bool ex::is_equal(const ex & other);
2802 for checking whether one expression is equal to another, or equal to zero,
2805 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2806 GiNaC header files. This method is however only to be used internally by
2807 GiNaC to establish a canonical sort order for terms, and using it to compare
2808 expressions will give very surprising results.
2811 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2812 @c node-name, next, previous, up
2813 @section Substituting expressions
2814 @cindex @code{subs()}
2816 Algebraic objects inside expressions can be replaced with arbitrary
2817 expressions via the @code{.subs()} method:
2820 ex ex::subs(const ex & e);
2821 ex ex::subs(const lst & syms, const lst & repls);
2824 In the first form, @code{subs()} accepts a relational of the form
2825 @samp{object == expression} or a @code{lst} of such relationals:
2829 symbol x("x"), y("y");
2831 ex e1 = 2*x^2-4*x+3;
2832 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2836 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2841 If you specify multiple substitutions, they are performed in parallel, so e.g.
2842 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2844 The second form of @code{subs()} takes two lists, one for the objects to be
2845 replaced and one for the expressions to be substituted (both lists must
2846 contain the same number of elements). Using this form, you would write
2847 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2849 @code{subs()} performs syntactic substitution of any complete algebraic
2850 object; it does not try to match sub-expressions as is demonstrated by the
2855 symbol x("x"), y("y"), z("z");
2857 ex e1 = pow(x+y, 2);
2858 cout << e1.subs(x+y == 4) << endl;
2861 ex e2 = sin(x)*sin(y)*cos(x);
2862 cout << e2.subs(sin(x) == cos(x)) << endl;
2863 // -> cos(x)^2*sin(y)
2866 cout << e3.subs(x+y == 4) << endl;
2868 // (and not 4+z as one might expect)
2872 A more powerful form of substitution using wildcards is described in the
2876 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2877 @c node-name, next, previous, up
2878 @section Pattern matching and advanced substitutions
2879 @cindex @code{wildcard} (class)
2880 @cindex Pattern matching
2882 GiNaC allows the use of patterns for checking whether an expression is of a
2883 certain form or contains subexpressions of a certain form, and for
2884 substituting expressions in a more general way.
2886 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2887 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2888 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2889 an unsigned integer number to allow having multiple different wildcards in a
2890 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2891 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2895 ex wild(unsigned label = 0);
2898 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2901 Some examples for patterns:
2903 @multitable @columnfractions .5 .5
2904 @item @strong{Constructed as} @tab @strong{Output as}
2905 @item @code{wild()} @tab @samp{$0}
2906 @item @code{pow(x,wild())} @tab @samp{x^$0}
2907 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2908 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2914 @item Wildcards behave like symbols and are subject to the same algebraic
2915 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2916 @item As shown in the last example, to use wildcards for indices you have to
2917 use them as the value of an @code{idx} object. This is because indices must
2918 always be of class @code{idx} (or a subclass).
2919 @item Wildcards only represent expressions or subexpressions. It is not
2920 possible to use them as placeholders for other properties like index
2921 dimension or variance, representation labels, symmetry of indexed objects
2923 @item Because wildcards are commutative, it is not possible to use wildcards
2924 as part of noncommutative products.
2925 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2926 are also valid patterns.
2929 @cindex @code{match()}
2930 The most basic application of patterns is to check whether an expression
2931 matches a given pattern. This is done by the function
2934 bool ex::match(const ex & pattern);
2935 bool ex::match(const ex & pattern, lst & repls);
2938 This function returns @code{true} when the expression matches the pattern
2939 and @code{false} if it doesn't. If used in the second form, the actual
2940 subexpressions matched by the wildcards get returned in the @code{repls}
2941 object as a list of relations of the form @samp{wildcard == expression}.
2942 If @code{match()} returns false, the state of @code{repls} is undefined.
2943 For reproducible results, the list should be empty when passed to
2944 @code{match()}, but it is also possible to find similarities in multiple
2945 expressions by passing in the result of a previous match.
2947 The matching algorithm works as follows:
2950 @item A single wildcard matches any expression. If one wildcard appears
2951 multiple times in a pattern, it must match the same expression in all
2952 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2953 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2954 @item If the expression is not of the same class as the pattern, the match
2955 fails (i.e. a sum only matches a sum, a function only matches a function,
2957 @item If the pattern is a function, it only matches the same function
2958 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2959 @item Except for sums and products, the match fails if the number of
2960 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2962 @item If there are no subexpressions, the expressions and the pattern must
2963 be equal (in the sense of @code{is_equal()}).
2964 @item Except for sums and products, each subexpression (@code{op()}) must
2965 match the corresponding subexpression of the pattern.
2968 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2969 account for their commutativity and associativity:
2972 @item If the pattern contains a term or factor that is a single wildcard,
2973 this one is used as the @dfn{global wildcard}. If there is more than one
2974 such wildcard, one of them is chosen as the global wildcard in a random
2976 @item Every term/factor of the pattern, except the global wildcard, is
2977 matched against every term of the expression in sequence. If no match is
2978 found, the whole match fails. Terms that did match are not considered in
2980 @item If there are no unmatched terms left, the match succeeds. Otherwise
2981 the match fails unless there is a global wildcard in the pattern, in
2982 which case this wildcard matches the remaining terms.
2985 In general, having more than one single wildcard as a term of a sum or a
2986 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2989 Here are some examples in @command{ginsh} to demonstrate how it works (the
2990 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2991 match fails, and the list of wildcard replacements otherwise):
2994 > match((x+y)^a,(x+y)^a);
2996 > match((x+y)^a,(x+y)^b);
2998 > match((x+y)^a,$1^$2);
3000 > match((x+y)^a,$1^$1);
3002 > match((x+y)^(x+y),$1^$1);
3004 > match((x+y)^(x+y),$1^$2);
3006 > match((a+b)*(a+c),($1+b)*($1+c));
3008 > match((a+b)*(a+c),(a+$1)*(a+$2));
3010 (Unpredictable. The result might also be [$1==c,$2==b].)
3011 > match((a+b)*(a+c),($1+$2)*($1+$3));
3012 (The result is undefined. Due to the sequential nature of the algorithm
3013 and the re-ordering of terms in GiNaC, the match for the first factor
3014 may be @{$1==a,$2==b@} in which case the match for the second factor
3015 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3017 > match(a*(x+y)+a*z+b,a*$1+$2);
3018 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3019 @{$1=x+y,$2=a*z+b@}.)
3020 > match(a+b+c+d+e+f,c);
3022 > match(a+b+c+d+e+f,c+$0);
3024 > match(a+b+c+d+e+f,c+e+$0);
3026 > match(a+b,a+b+$0);
3028 > match(a*b^2,a^$1*b^$2);
3030 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3031 even though a==a^1.)
3032 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3034 > match(atan2(y,x^2),atan2(y,$0));
3038 @cindex @code{has()}
3039 A more general way to look for patterns in expressions is provided by the
3043 bool ex::has(const ex & pattern);
3046 This function checks whether a pattern is matched by an expression itself or
3047 by any of its subexpressions.
3049 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3050 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3053 > has(x*sin(x+y+2*a),y);
3055 > has(x*sin(x+y+2*a),x+y);
3057 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3058 has the subexpressions "x", "y" and "2*a".)
3059 > has(x*sin(x+y+2*a),x+y+$1);
3061 (But this is possible.)
3062 > has(x*sin(2*(x+y)+2*a),x+y);
3064 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3065 which "x+y" is not a subexpression.)
3068 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3070 > has(4*x^2-x+3,$1*x);
3072 > has(4*x^2+x+3,$1*x);
3074 (Another possible pitfall. The first expression matches because the term
3075 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3076 contains a linear term you should use the coeff() function instead.)
3079 @cindex @code{find()}
3083 bool ex::find(const ex & pattern, lst & found);
3086 works a bit like @code{has()} but it doesn't stop upon finding the first
3087 match. Instead, it appends all found matches to the specified list. If there
3088 are multiple occurrences of the same expression, it is entered only once to
3089 the list. @code{find()} returns false if no matches were found (in
3090 @command{ginsh}, it returns an empty list):
3093 > find(1+x+x^2+x^3,x);
3095 > find(1+x+x^2+x^3,y);
3097 > find(1+x+x^2+x^3,x^$1);
3099 (Note the absence of "x".)
3100 > expand((sin(x)+sin(y))*(a+b));
3101 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3106 @cindex @code{subs()}
3107 Probably the most useful application of patterns is to use them for
3108 substituting expressions with the @code{subs()} method. Wildcards can be
3109 used in the search patterns as well as in the replacement expressions, where
3110 they get replaced by the expressions matched by them. @code{subs()} doesn't
3111 know anything about algebra; it performs purely syntactic substitutions.
3116 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3118 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3120 > subs((a+b+c)^2,a+b=x);
3122 > subs((a+b+c)^2,a+b+$1==x+$1);
3124 > subs(a+2*b,a+b=x);
3126 > subs(4*x^3-2*x^2+5*x-1,x==a);
3128 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3130 > subs(sin(1+sin(x)),sin($1)==cos($1));
3132 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3136 The last example would be written in C++ in this way:
3140 symbol a("a"), b("b"), x("x"), y("y");
3141 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3142 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3143 cout << e.expand() << endl;
3149 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3150 @c node-name, next, previous, up
3151 @section Applying a Function on Subexpressions
3152 @cindex Tree traversal
3153 @cindex @code{map()}
3155 Sometimes you may want to perform an operation on specific parts of an
3156 expression while leaving the general structure of it intact. An example
3157 of this would be a matrix trace operation: the trace of a sum is the sum
3158 of the traces of the individual terms. That is, the trace should @dfn{map}
3159 on the sum, by applying itself to each of the sum's operands. It is possible
3160 to do this manually which usually results in code like this:
3165 if (is_a<matrix>(e))
3166 return ex_to<matrix>(e).trace();
3167 else if (is_a<add>(e)) @{
3169 for (unsigned i=0; i<e.nops(); i++)
3170 sum += calc_trace(e.op(i));
3172 @} else if (is_a<mul>)(e)) @{
3180 This is, however, slightly inefficient (if the sum is very large it can take
3181 a long time to add the terms one-by-one), and its applicability is limited to
3182 a rather small class of expressions. If @code{calc_trace()} is called with
3183 a relation or a list as its argument, you will probably want the trace to
3184 be taken on both sides of the relation or of all elements of the list.
3186 GiNaC offers the @code{map()} method to aid in the implementation of such
3190 static ex ex::map(map_function & f) const;
3191 static ex ex::map(ex (*f)(const ex & e)) const;
3194 In the first (preferred) form, @code{map()} takes a function object that
3195 is subclassed from the @code{map_function} class. In the second form, it
3196 takes a pointer to a function that accepts and returns an expression.
3197 @code{map()} constructs a new expression of the same type, applying the
3198 specified function on all subexpressions (in the sense of @code{op()}),
3201 The use of a function object makes it possible to supply more arguments to
3202 the function that is being mapped, or to keep local state information.
3203 The @code{map_function} class declares a virtual function call operator
3204 that you can overload. Here is a sample implementation of @code{calc_trace()}
3205 that uses @code{map()} in a recursive fashion:
3208 struct calc_trace : public map_function @{
3209 ex operator()(const ex &e)
3211 if (is_a<matrix>(e))
3212 return ex_to<matrix>(e).trace();
3213 else if (is_a<mul>(e)) @{
3216 return e.map(*this);
3221 This function object could then be used like this:
3225 ex M = ... // expression with matrices
3226 calc_trace do_trace;
3227 ex tr = do_trace(M);
3231 Here is another example for you to meditate over. It removes quadratic
3232 terms in a variable from an expanded polynomial:
3235 struct map_rem_quad : public map_function @{
3237 map_rem_quad(const ex & var_) : var(var_) @{@}
3239 ex operator()(const ex & e)
3241 if (is_a<add>(e) || is_a<mul>(e))
3242 return e.map(*this);
3243 else if (is_a<power>(e) && e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3253 symbol x("x"), y("y");
3256 for (int i=0; i<8; i++)
3257 e += pow(x, i) * pow(y, 8-i) * (i+1);
3259 // -> 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
3261 map_rem_quad rem_quad(x);
3262 cout << rem_quad(e) << endl;
3263 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3267 @command{ginsh} offers a slightly different implementation of @code{map()}
3268 that allows applying algebraic functions to operands. The second argument
3269 to @code{map()} is an expression containing the wildcard @samp{$0} which
3270 acts as the placeholder for the operands:
3275 > map(a+2*b,sin($0));
3277 > map(@{a,b,c@},$0^2+$0);
3278 @{a^2+a,b^2+b,c^2+c@}
3281 Note that it is only possible to use algebraic functions in the second
3282 argument. You can not use functions like @samp{diff()}, @samp{op()},
3283 @samp{subs()} etc. because these are evaluated immediately:
3286 > map(@{a,b,c@},diff($0,a));
3288 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3289 to "map(@{a,b,c@},0)".
3293 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3294 @c node-name, next, previous, up
3295 @section Polynomial arithmetic
3297 @subsection Expanding and collecting
3298 @cindex @code{expand()}
3299 @cindex @code{collect()}
3301 A polynomial in one or more variables has many equivalent
3302 representations. Some useful ones serve a specific purpose. Consider
3303 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3304 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3305 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3306 representations are the recursive ones where one collects for exponents
3307 in one of the three variable. Since the factors are themselves
3308 polynomials in the remaining two variables the procedure can be
3309 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
3310 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3313 To bring an expression into expanded form, its method
3319 may be called. In our example above, this corresponds to @math{4*x*y +
3320 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3321 GiNaC is not easily guessable you should be prepared to see different
3322 orderings of terms in such sums!
3324 Another useful representation of multivariate polynomials is as a
3325 univariate polynomial in one of the variables with the coefficients
3326 being polynomials in the remaining variables. The method
3327 @code{collect()} accomplishes this task:
3330 ex ex::collect(const ex & s, bool distributed = false);
3333 The first argument to @code{collect()} can also be a list of objects in which
3334 case the result is either a recursively collected polynomial, or a polynomial
3335 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3336 by the @code{distributed} flag.
3338 Note that the original polynomial needs to be in expanded form (for the
3339 variables concerned) in order for @code{collect()} to be able to find the
3340 coefficients properly.
3342 The following @command{ginsh} transcript shows an application of @code{collect()}
3343 together with @code{find()}:
3346 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3347 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
3348 > collect(a,@{p,q@});
3349 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)
3350 > collect(a,find(a,sin($1)));
3351 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3352 > collect(a,@{find(a,sin($1)),p,q@});
3353 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3354 > collect(a,@{find(a,sin($1)),d@});
3355 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3358 @subsection Degree and coefficients
3359 @cindex @code{degree()}
3360 @cindex @code{ldegree()}
3361 @cindex @code{coeff()}
3363 The degree and low degree of a polynomial can be obtained using the two
3367 int ex::degree(const ex & s);
3368 int ex::ldegree(const ex & s);
3371 which also work reliably on non-expanded input polynomials (they even work
3372 on rational functions, returning the asymptotic degree). To extract
3373 a coefficient with a certain power from an expanded polynomial you use
3376 ex ex::coeff(const ex & s, int n);
3379 You can also obtain the leading and trailing coefficients with the methods
3382 ex ex::lcoeff(const ex & s);
3383 ex ex::tcoeff(const ex & s);
3386 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3389 An application is illustrated in the next example, where a multivariate
3390 polynomial is analyzed:
3393 #include <ginac/ginac.h>
3394 using namespace std;
3395 using namespace GiNaC;
3399 symbol x("x"), y("y");
3400 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3401 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3402 ex Poly = PolyInp.expand();
3404 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3405 cout << "The x^" << i << "-coefficient is "
3406 << Poly.coeff(x,i) << endl;
3408 cout << "As polynomial in y: "
3409 << Poly.collect(y) << endl;
3413 When run, it returns an output in the following fashion:
3416 The x^0-coefficient is y^2+11*y
3417 The x^1-coefficient is 5*y^2-2*y
3418 The x^2-coefficient is -1
3419 The x^3-coefficient is 4*y
3420 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3423 As always, the exact output may vary between different versions of GiNaC
3424 or even from run to run since the internal canonical ordering is not
3425 within the user's sphere of influence.
3427 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3428 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3429 with non-polynomial expressions as they not only work with symbols but with
3430 constants, functions and indexed objects as well:
3434 symbol a("a"), b("b"), c("c");
3435 idx i(symbol("i"), 3);
3437 ex e = pow(sin(x) - cos(x), 4);
3438 cout << e.degree(cos(x)) << endl;
3440 cout << e.expand().coeff(sin(x), 3) << endl;
3443 e = indexed(a+b, i) * indexed(b+c, i);
3444 e = e.expand(expand_options::expand_indexed);
3445 cout << e.collect(indexed(b, i)) << endl;
3446 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3451 @subsection Polynomial division
3452 @cindex polynomial division
3455 @cindex pseudo-remainder
3456 @cindex @code{quo()}
3457 @cindex @code{rem()}
3458 @cindex @code{prem()}
3459 @cindex @code{divide()}
3464 ex quo(const ex & a, const ex & b, const symbol & x);
3465 ex rem(const ex & a, const ex & b, const symbol & x);
3468 compute the quotient and remainder of univariate polynomials in the variable
3469 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3471 The additional function
3474 ex prem(const ex & a, const ex & b, const symbol & x);
3477 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3478 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3480 Exact division of multivariate polynomials is performed by the function
3483 bool divide(const ex & a, const ex & b, ex & q);
3486 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3487 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3488 in which case the value of @code{q} is undefined.
3491 @subsection Unit, content and primitive part
3492 @cindex @code{unit()}
3493 @cindex @code{content()}
3494 @cindex @code{primpart()}
3499 ex ex::unit(const symbol & x);
3500 ex ex::content(const symbol & x);
3501 ex ex::primpart(const symbol & x);
3504 return the unit part, content part, and primitive polynomial of a multivariate
3505 polynomial with respect to the variable @samp{x} (the unit part being the sign
3506 of the leading coefficient, the content part being the GCD of the coefficients,
3507 and the primitive polynomial being the input polynomial divided by the unit and
3508 content parts). The product of unit, content, and primitive part is the
3509 original polynomial.
3512 @subsection GCD and LCM
3515 @cindex @code{gcd()}
3516 @cindex @code{lcm()}
3518 The functions for polynomial greatest common divisor and least common
3519 multiple have the synopsis
3522 ex gcd(const ex & a, const ex & b);
3523 ex lcm(const ex & a, const ex & b);
3526 The functions @code{gcd()} and @code{lcm()} accept two expressions
3527 @code{a} and @code{b} as arguments and return a new expression, their
3528 greatest common divisor or least common multiple, respectively. If the
3529 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3530 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3533 #include <ginac/ginac.h>
3534 using namespace GiNaC;
3538 symbol x("x"), y("y"), z("z");
3539 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3540 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3542 ex P_gcd = gcd(P_a, P_b);
3544 ex P_lcm = lcm(P_a, P_b);
3545 // 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
3550 @subsection Square-free decomposition
3551 @cindex square-free decomposition
3552 @cindex factorization
3553 @cindex @code{sqrfree()}
3555 GiNaC still lacks proper factorization support. Some form of
3556 factorization is, however, easily implemented by noting that factors
3557 appearing in a polynomial with power two or more also appear in the
3558 derivative and hence can easily be found by computing the GCD of the
3559 original polynomial and its derivatives. Any system has an interface
3560 for this so called square-free factorization. So we provide one, too:
3562 ex sqrfree(const ex & a, const lst & l = lst());
3564 Here is an example that by the way illustrates how the result may depend
3565 on the order of differentiation:
3568 symbol x("x"), y("y");
3569 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3571 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3572 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3574 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3575 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3577 cout << sqrfree(BiVarPol) << endl;
3578 // -> depending on luck, any of the above
3583 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3584 @c node-name, next, previous, up
3585 @section Rational expressions
3587 @subsection The @code{normal} method
3588 @cindex @code{normal()}
3589 @cindex simplification
3590 @cindex temporary replacement
3592 Some basic form of simplification of expressions is called for frequently.
3593 GiNaC provides the method @code{.normal()}, which converts a rational function
3594 into an equivalent rational function of the form @samp{numerator/denominator}
3595 where numerator and denominator are coprime. If the input expression is already
3596 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3597 otherwise it performs fraction addition and multiplication.
3599 @code{.normal()} can also be used on expressions which are not rational functions
3600 as it will replace all non-rational objects (like functions or non-integer
3601 powers) by temporary symbols to bring the expression to the domain of rational
3602 functions before performing the normalization, and re-substituting these
3603 symbols afterwards. This algorithm is also available as a separate method
3604 @code{.to_rational()}, described below.
3606 This means that both expressions @code{t1} and @code{t2} are indeed
3607 simplified in this little program:
3610 #include <ginac/ginac.h>
3611 using namespace GiNaC;
3616 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3617 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3618 std::cout << "t1 is " << t1.normal() << std::endl;
3619 std::cout << "t2 is " << t2.normal() << std::endl;
3623 Of course this works for multivariate polynomials too, so the ratio of
3624 the sample-polynomials from the section about GCD and LCM above would be
3625 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3628 @subsection Numerator and denominator
3631 @cindex @code{numer()}
3632 @cindex @code{denom()}
3633 @cindex @code{numer_denom()}
3635 The numerator and denominator of an expression can be obtained with
3640 ex ex::numer_denom();
3643 These functions will first normalize the expression as described above and
3644 then return the numerator, denominator, or both as a list, respectively.
3645 If you need both numerator and denominator, calling @code{numer_denom()} is
3646 faster than using @code{numer()} and @code{denom()} separately.
3649 @subsection Converting to a rational expression
3650 @cindex @code{to_rational()}
3652 Some of the methods described so far only work on polynomials or rational
3653 functions. GiNaC provides a way to extend the domain of these functions to
3654 general expressions by using the temporary replacement algorithm described
3655 above. You do this by calling
3658 ex ex::to_rational(lst &l);
3661 on the expression to be converted. The supplied @code{lst} will be filled
3662 with the generated temporary symbols and their replacement expressions in
3663 a format that can be used directly for the @code{subs()} method. It can also
3664 already contain a list of replacements from an earlier application of
3665 @code{.to_rational()}, so it's possible to use it on multiple expressions
3666 and get consistent results.
3673 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3674 ex b = sin(x) + cos(x);
3677 divide(a.to_rational(l), b.to_rational(l), q);
3678 cout << q.subs(l) << endl;
3682 will print @samp{sin(x)-cos(x)}.
3685 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3686 @c node-name, next, previous, up
3687 @section Symbolic differentiation
3688 @cindex differentiation
3689 @cindex @code{diff()}
3691 @cindex product rule
3693 GiNaC's objects know how to differentiate themselves. Thus, a
3694 polynomial (class @code{add}) knows that its derivative is the sum of
3695 the derivatives of all the monomials:
3698 #include <ginac/ginac.h>
3699 using namespace GiNaC;
3703 symbol x("x"), y("y"), z("z");
3704 ex P = pow(x, 5) + pow(x, 2) + y;
3706 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3707 cout << P.diff(y) << endl; // 1
3708 cout << P.diff(z) << endl; // 0
3712 If a second integer parameter @var{n} is given, the @code{diff} method
3713 returns the @var{n}th derivative.
3715 If @emph{every} object and every function is told what its derivative
3716 is, all derivatives of composed objects can be calculated using the
3717 chain rule and the product rule. Consider, for instance the expression
3718 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3719 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3720 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3721 out that the composition is the generating function for Euler Numbers,
3722 i.e. the so called @var{n}th Euler number is the coefficient of
3723 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3724 identity to code a function that generates Euler numbers in just three
3727 @cindex Euler numbers
3729 #include <ginac/ginac.h>
3730 using namespace GiNaC;
3732 ex EulerNumber(unsigned n)
3735 const ex generator = pow(cosh(x),-1);
3736 return generator.diff(x,n).subs(x==0);
3741 for (unsigned i=0; i<11; i+=2)
3742 std::cout << EulerNumber(i) << std::endl;
3747 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3748 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3749 @code{i} by two since all odd Euler numbers vanish anyways.
3752 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3753 @c node-name, next, previous, up
3754 @section Series expansion
3755 @cindex @code{series()}
3756 @cindex Taylor expansion
3757 @cindex Laurent expansion
3758 @cindex @code{pseries} (class)
3760 Expressions know how to expand themselves as a Taylor series or (more
3761 generally) a Laurent series. As in most conventional Computer Algebra
3762 Systems, no distinction is made between those two. There is a class of
3763 its own for storing such series (@code{class pseries}) and a built-in
3764 function (called @code{Order}) for storing the order term of the series.
3765 As a consequence, if you want to work with series, i.e. multiply two
3766 series, you need to call the method @code{ex::series} again to convert
3767 it to a series object with the usual structure (expansion plus order
3768 term). A sample application from special relativity could read:
3771 #include <ginac/ginac.h>
3772 using namespace std;
3773 using namespace GiNaC;
3777 symbol v("v"), c("c");
3779 ex gamma = 1/sqrt(1 - pow(v/c,2));
3780 ex mass_nonrel = gamma.series(v==0, 10);
3782 cout << "the relativistic mass increase with v is " << endl
3783 << mass_nonrel << endl;
3785 cout << "the inverse square of this series is " << endl
3786 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3790 Only calling the series method makes the last output simplify to
3791 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3792 series raised to the power @math{-2}.
3794 @cindex M@'echain's formula
3795 As another instructive application, let us calculate the numerical
3796 value of Archimedes' constant
3800 (for which there already exists the built-in constant @code{Pi})
3801 using M@'echain's amazing formula
3803 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3806 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3808 We may expand the arcus tangent around @code{0} and insert the fractions
3809 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3810 carries an order term with it and the question arises what the system is
3811 supposed to do when the fractions are plugged into that order term. The
3812 solution is to use the function @code{series_to_poly()} to simply strip
3816 #include <ginac/ginac.h>
3817 using namespace GiNaC;
3819 ex mechain_pi(int degr)
3822 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3823 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3824 -4*pi_expansion.subs(x==numeric(1,239));
3830 using std::cout; // just for fun, another way of...
3831 using std::endl; // ...dealing with this namespace std.
3833 for (int i=2; i<12; i+=2) @{
3834 pi_frac = mechain_pi(i);
3835 cout << i << ":\t" << pi_frac << endl
3836 << "\t" << pi_frac.evalf() << endl;
3842 Note how we just called @code{.series(x,degr)} instead of
3843 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3844 method @code{series()}: if the first argument is a symbol the expression
3845 is expanded in that symbol around point @code{0}. When you run this
3846 program, it will type out:
3850 3.1832635983263598326
3851 4: 5359397032/1706489875
3852 3.1405970293260603143
3853 6: 38279241713339684/12184551018734375
3854 3.141621029325034425
3855 8: 76528487109180192540976/24359780855939418203125
3856 3.141591772182177295
3857 10: 327853873402258685803048818236/104359128170408663038552734375
3858 3.1415926824043995174
3862 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3863 @c node-name, next, previous, up
3864 @section Symmetrization
3865 @cindex @code{symmetrize()}
3866 @cindex @code{antisymmetrize()}
3867 @cindex @code{symmetrize_cyclic()}
3872 ex ex::symmetrize(const lst & l);
3873 ex ex::antisymmetrize(const lst & l);
3874 ex ex::symmetrize_cyclic(const lst & l);
3877 symmetrize an expression by returning the sum over all symmetric,
3878 antisymmetric or cyclic permutations of the specified list of objects,
3879 weighted by the number of permutations.
3881 The three additional methods
3884 ex ex::symmetrize();
3885 ex ex::antisymmetrize();
3886 ex ex::symmetrize_cyclic();
3889 symmetrize or antisymmetrize an expression over its free indices.
3891 Symmetrization is most useful with indexed expressions but can be used with
3892 almost any kind of object (anything that is @code{subs()}able):
3896 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3897 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3899 cout << indexed(A, i, j).symmetrize() << endl;
3900 // -> 1/2*A.j.i+1/2*A.i.j
3901 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3902 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3903 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3904 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
3909 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3910 @c node-name, next, previous, up
3911 @section Predefined mathematical functions
3913 GiNaC contains the following predefined mathematical functions:
3916 @multitable @columnfractions .30 .70
3917 @item @strong{Name} @tab @strong{Function}
3920 @item @code{csgn(x)}
3922 @item @code{sqrt(x)}
3923 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
3930 @item @code{asin(x)}
3932 @item @code{acos(x)}
3934 @item @code{atan(x)}
3935 @tab inverse tangent
3936 @item @code{atan2(y, x)}
3937 @tab inverse tangent with two arguments
3938 @item @code{sinh(x)}
3939 @tab hyperbolic sine
3940 @item @code{cosh(x)}
3941 @tab hyperbolic cosine
3942 @item @code{tanh(x)}
3943 @tab hyperbolic tangent
3944 @item @code{asinh(x)}
3945 @tab inverse hyperbolic sine
3946 @item @code{acosh(x)}
3947 @tab inverse hyperbolic cosine
3948 @item @code{atanh(x)}
3949 @tab inverse hyperbolic tangent
3951 @tab exponential function
3953 @tab natural logarithm
3956 @item @code{zeta(x)}
3957 @tab Riemann's zeta function
3958 @item @code{zeta(n, x)}
3959 @tab derivatives of Riemann's zeta function
3960 @item @code{tgamma(x)}
3962 @item @code{lgamma(x)}
3963 @tab logarithm of Gamma function
3964 @item @code{beta(x, y)}
3965 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3967 @tab psi (digamma) function
3968 @item @code{psi(n, x)}
3969 @tab derivatives of psi function (polygamma functions)
3970 @item @code{factorial(n)}
3971 @tab factorial function
3972 @item @code{binomial(n, m)}
3973 @tab binomial coefficients
3974 @item @code{Order(x)}
3975 @tab order term function in truncated power series
3980 For functions that have a branch cut in the complex plane GiNaC follows
3981 the conventions for C++ as defined in the ANSI standard as far as
3982 possible. In particular: the natural logarithm (@code{log}) and the
3983 square root (@code{sqrt}) both have their branch cuts running along the
3984 negative real axis where the points on the axis itself belong to the
3985 upper part (i.e. continuous with quadrant II). The inverse
3986 trigonometric and hyperbolic functions are not defined for complex
3987 arguments by the C++ standard, however. In GiNaC we follow the
3988 conventions used by CLN, which in turn follow the carefully designed
3989 definitions in the Common Lisp standard. It should be noted that this
3990 convention is identical to the one used by the C99 standard and by most
3991 serious CAS. It is to be expected that future revisions of the C++
3992 standard incorporate these functions in the complex domain in a manner
3993 compatible with C99.
3996 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3997 @c node-name, next, previous, up
3998 @section Input and output of expressions
4001 @subsection Expression output
4003 @cindex output of expressions
4005 The easiest way to print an expression is to write it to a stream:
4010 ex e = 4.5+pow(x,2)*3/2;
4011 cout << e << endl; // prints '(4.5)+3/2*x^2'
4015 The output format is identical to the @command{ginsh} input syntax and
4016 to that used by most computer algebra systems, but not directly pastable
4017 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4018 is printed as @samp{x^2}).
4020 It is possible to print expressions in a number of different formats with
4024 void ex::print(const print_context & c, unsigned level = 0);
4027 @cindex @code{print_context} (class)
4028 The type of @code{print_context} object passed in determines the format
4029 of the output. The possible types are defined in @file{ginac/print.h}.
4030 All constructors of @code{print_context} and derived classes take an
4031 @code{ostream &} as their first argument.
4033 To print an expression in a way that can be directly used in a C or C++
4034 program, you pass a @code{print_csrc} object like this:
4038 cout << "float f = ";
4039 e.print(print_csrc_float(cout));
4042 cout << "double d = ";
4043 e.print(print_csrc_double(cout));
4046 cout << "cl_N n = ";
4047 e.print(print_csrc_cl_N(cout));
4052 The three possible types mostly affect the way in which floating point
4053 numbers are written.
4055 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4058 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4059 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4060 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4063 The @code{print_context} type @code{print_tree} provides a dump of the
4064 internal structure of an expression for debugging purposes:
4068 e.print(print_tree(cout));
4075 add, hash=0x0, flags=0x3, nops=2
4076 power, hash=0x9, flags=0x3, nops=2
4077 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4078 2 (numeric), hash=0x80000042, flags=0xf
4079 3/2 (numeric), hash=0x80000061, flags=0xf
4082 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4086 This kind of output is also available in @command{ginsh} as the @code{print()}
4089 Another useful output format is for LaTeX parsing in mathematical mode.
4090 It is rather similar to the default @code{print_context} but provides
4091 some braces needed by LaTeX for delimiting boxes and also converts some
4092 common objects to conventional LaTeX names. It is possible to give symbols
4093 a special name for LaTeX output by supplying it as a second argument to
4094 the @code{symbol} constructor.
4096 For example, the code snippet
4101 ex foo = lgamma(x).series(x==0,3);
4102 foo.print(print_latex(std::cout));
4108 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4111 @cindex Tree traversal
4112 If you need any fancy special output format, e.g. for interfacing GiNaC
4113 with other algebra systems or for producing code for different
4114 programming languages, you can always traverse the expression tree yourself:
4117 static void my_print(const ex & e)
4119 if (is_a<function>(e))
4120 cout << ex_to<function>(e).get_name();
4122 cout << e.bp->class_name();
4124 unsigned n = e.nops();
4126 for (unsigned i=0; i<n; i++) @{
4138 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4146 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4147 symbol(y))),numeric(-2)))
4150 If you need an output format that makes it possible to accurately
4151 reconstruct an expression by feeding the output to a suitable parser or
4152 object factory, you should consider storing the expression in an
4153 @code{archive} object and reading the object properties from there.
4154 See the section on archiving for more information.
4157 @subsection Expression input
4158 @cindex input of expressions
4160 GiNaC provides no way to directly read an expression from a stream because
4161 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4162 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4163 @code{y} you defined in your program and there is no way to specify the
4164 desired symbols to the @code{>>} stream input operator.
4166 Instead, GiNaC lets you construct an expression from a string, specifying the
4167 list of symbols to be used:
4171 symbol x("x"), y("y");
4172 ex e("2*x+sin(y)", lst(x, y));
4176 The input syntax is the same as that used by @command{ginsh} and the stream
4177 output operator @code{<<}. The symbols in the string are matched by name to
4178 the symbols in the list and if GiNaC encounters a symbol not specified in
4179 the list it will throw an exception.
4181 With this constructor, it's also easy to implement interactive GiNaC programs:
4186 #include <stdexcept>
4187 #include <ginac/ginac.h>
4188 using namespace std;
4189 using namespace GiNaC;
4196 cout << "Enter an expression containing 'x': ";
4201 cout << "The derivative of " << e << " with respect to x is ";
4202 cout << e.diff(x) << ".\n";
4203 @} catch (exception &p) @{
4204 cerr << p.what() << endl;
4210 @subsection Archiving
4211 @cindex @code{archive} (class)
4214 GiNaC allows creating @dfn{archives} of expressions which can be stored
4215 to or retrieved from files. To create an archive, you declare an object
4216 of class @code{archive} and archive expressions in it, giving each
4217 expression a unique name:
4221 using namespace std;
4222 #include <ginac/ginac.h>
4223 using namespace GiNaC;
4227 symbol x("x"), y("y"), z("z");
4229 ex foo = sin(x + 2*y) + 3*z + 41;
4233 a.archive_ex(foo, "foo");
4234 a.archive_ex(bar, "the second one");
4238 The archive can then be written to a file:
4242 ofstream out("foobar.gar");
4248 The file @file{foobar.gar} contains all information that is needed to
4249 reconstruct the expressions @code{foo} and @code{bar}.
4251 @cindex @command{viewgar}
4252 The tool @command{viewgar} that comes with GiNaC can be used to view
4253 the contents of GiNaC archive files:
4256 $ viewgar foobar.gar
4257 foo = 41+sin(x+2*y)+3*z
4258 the second one = 42+sin(x+2*y)+3*z
4261 The point of writing archive files is of course that they can later be
4267 ifstream in("foobar.gar");
4272 And the stored expressions can be retrieved by their name:
4278 ex ex1 = a2.unarchive_ex(syms, "foo");
4279 ex ex2 = a2.unarchive_ex(syms, "the second one");
4281 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4282 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4283 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4287 Note that you have to supply a list of the symbols which are to be inserted
4288 in the expressions. Symbols in archives are stored by their name only and
4289 if you don't specify which symbols you have, unarchiving the expression will
4290 create new symbols with that name. E.g. if you hadn't included @code{x} in
4291 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4292 have had no effect because the @code{x} in @code{ex1} would have been a
4293 different symbol than the @code{x} which was defined at the beginning of
4294 the program, altough both would appear as @samp{x} when printed.
4296 You can also use the information stored in an @code{archive} object to
4297 output expressions in a format suitable for exact reconstruction. The
4298 @code{archive} and @code{archive_node} classes have a couple of member
4299 functions that let you access the stored properties:
4302 static void my_print2(const archive_node & n)
4305 n.find_string("class", class_name);
4306 cout << class_name << "(";
4308 archive_node::propinfovector p;
4309 n.get_properties(p);
4311 unsigned num = p.size();
4312 for (unsigned i=0; i<num; i++) @{
4313 const string &name = p[i].name;
4314 if (name == "class")
4316 cout << name << "=";
4318 unsigned count = p[i].count;
4322 for (unsigned j=0; j<count; j++) @{
4323 switch (p[i].type) @{
4324 case archive_node::PTYPE_BOOL: @{
4326 n.find_bool(name, x);
4327 cout << (x ? "true" : "false");
4330 case archive_node::PTYPE_UNSIGNED: @{
4332 n.find_unsigned(name, x);
4336 case archive_node::PTYPE_STRING: @{
4338 n.find_string(name, x);
4339 cout << '\"' << x << '\"';
4342 case archive_node::PTYPE_NODE: @{
4343 const archive_node &x = n.find_ex_node(name, j);
4365 ex e = pow(2, x) - y;
4367 my_print2(ar.get_top_node(0)); cout << endl;
4375 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4376 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4377 overall_coeff=numeric(number="0"))
4380 Be warned, however, that the set of properties and their meaning for each
4381 class may change between GiNaC versions.
4384 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4385 @c node-name, next, previous, up
4386 @chapter Extending GiNaC
4388 By reading so far you should have gotten a fairly good understanding of
4389 GiNaC's design-patterns. From here on you should start reading the
4390 sources. All we can do now is issue some recommendations how to tackle
4391 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4392 develop some useful extension please don't hesitate to contact the GiNaC
4393 authors---they will happily incorporate them into future versions.
4396 * What does not belong into GiNaC:: What to avoid.
4397 * Symbolic functions:: Implementing symbolic functions.
4398 * Adding classes:: Defining new algebraic classes.
4402 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4403 @c node-name, next, previous, up
4404 @section What doesn't belong into GiNaC
4406 @cindex @command{ginsh}
4407 First of all, GiNaC's name must be read literally. It is designed to be
4408 a library for use within C++. The tiny @command{ginsh} accompanying
4409 GiNaC makes this even more clear: it doesn't even attempt to provide a
4410 language. There are no loops or conditional expressions in
4411 @command{ginsh}, it is merely a window into the library for the
4412 programmer to test stuff (or to show off). Still, the design of a
4413 complete CAS with a language of its own, graphical capabilites and all
4414 this on top of GiNaC is possible and is without doubt a nice project for
4417 There are many built-in functions in GiNaC that do not know how to
4418 evaluate themselves numerically to a precision declared at runtime
4419 (using @code{Digits}). Some may be evaluated at certain points, but not
4420 generally. This ought to be fixed. However, doing numerical
4421 computations with GiNaC's quite abstract classes is doomed to be
4422 inefficient. For this purpose, the underlying foundation classes
4423 provided by @acronym{CLN} are much better suited.
4426 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4427 @c node-name, next, previous, up
4428 @section Symbolic functions
4430 The easiest and most instructive way to start with is probably to
4431 implement your own function. GiNaC's functions are objects of class
4432 @code{function}. The preprocessor is then used to convert the function
4433 names to objects with a corresponding serial number that is used
4434 internally to identify them. You usually need not worry about this
4435 number. New functions may be inserted into the system via a kind of
4436 `registry'. It is your responsibility to care for some functions that
4437 are called when the user invokes certain methods. These are usual
4438 C++-functions accepting a number of @code{ex} as arguments and returning
4439 one @code{ex}. As an example, if we have a look at a simplified
4440 implementation of the cosine trigonometric function, we first need a
4441 function that is called when one wishes to @code{eval} it. It could
4442 look something like this:
4445 static ex cos_eval_method(const ex & x)
4447 // if (!x%(2*Pi)) return 1
4448 // if (!x%Pi) return -1
4449 // if (!x%Pi/2) return 0
4450 // care for other cases...
4451 return cos(x).hold();
4455 @cindex @code{hold()}
4457 The last line returns @code{cos(x)} if we don't know what else to do and
4458 stops a potential recursive evaluation by saying @code{.hold()}, which
4459 sets a flag to the expression signaling that it has been evaluated. We
4460 should also implement a method for numerical evaluation and since we are
4461 lazy we sweep the problem under the rug by calling someone else's
4462 function that does so, in this case the one in class @code{numeric}:
4465 static ex cos_evalf(const ex & x)
4467 if (is_a<numeric>(x))
4468 return cos(ex_to<numeric>(x));
4470 return cos(x).hold();
4474 Differentiation will surely turn up and so we need to tell @code{cos}
4475 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4476 instance are then handled automatically by @code{basic::diff} and
4480 static ex cos_deriv(const ex & x, unsigned diff_param)
4486 @cindex product rule
4487 The second parameter is obligatory but uninteresting at this point. It
4488 specifies which parameter to differentiate in a partial derivative in
4489 case the function has more than one parameter and its main application
4490 is for correct handling of the chain rule. For Taylor expansion, it is
4491 enough to know how to differentiate. But if the function you want to
4492 implement does have a pole somewhere in the complex plane, you need to
4493 write another method for Laurent expansion around that point.
4495 Now that all the ingredients for @code{cos} have been set up, we need
4496 to tell the system about it. This is done by a macro and we are not
4497 going to descibe how it expands, please consult your preprocessor if you
4501 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4502 evalf_func(cos_evalf).
4503 derivative_func(cos_deriv));
4506 The first argument is the function's name used for calling it and for
4507 output. The second binds the corresponding methods as options to this
4508 object. Options are separated by a dot and can be given in an arbitrary
4509 order. GiNaC functions understand several more options which are always
4510 specified as @code{.option(params)}, for example a method for series
4511 expansion @code{.series_func(cos_series)}. Again, if no series
4512 expansion method is given, GiNaC defaults to simple Taylor expansion,
4513 which is correct if there are no poles involved as is the case for the
4514 @code{cos} function. The way GiNaC handles poles in case there are any
4515 is best understood by studying one of the examples, like the Gamma
4516 (@code{tgamma}) function for instance. (In essence the function first
4517 checks if there is a pole at the evaluation point and falls back to
4518 Taylor expansion if there isn't. Then, the pole is regularized by some
4519 suitable transformation.) Also, the new function needs to be declared
4520 somewhere. This may also be done by a convenient preprocessor macro:
4523 DECLARE_FUNCTION_1P(cos)
4526 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4527 implementation of @code{cos} is very incomplete and lacks several safety
4528 mechanisms. Please, have a look at the real implementation in GiNaC.
4529 (By the way: in case you are worrying about all the macros above we can
4530 assure you that functions are GiNaC's most macro-intense classes. We
4531 have done our best to avoid macros where we can.)
4534 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4535 @c node-name, next, previous, up
4536 @section Adding classes
4538 If you are doing some very specialized things with GiNaC you may find that
4539 you have to implement your own algebraic classes to fit your needs. This
4540 section will explain how to do this by giving the example of a simple
4541 'string' class. After reading this section you will know how to properly
4542 declare a GiNaC class and what the minimum required member functions are
4543 that you have to implement. We only cover the implementation of a 'leaf'
4544 class here (i.e. one that doesn't contain subexpressions). Creating a
4545 container class like, for example, a class representing tensor products is
4546 more involved but this section should give you enough information so you can
4547 consult the source to GiNaC's predefined classes if you want to implement
4548 something more complicated.
4550 @subsection GiNaC's run-time type information system
4552 @cindex hierarchy of classes
4554 All algebraic classes (that is, all classes that can appear in expressions)
4555 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4556 @code{basic *} (which is essentially what an @code{ex} is) represents a
4557 generic pointer to an algebraic class. Occasionally it is necessary to find
4558 out what the class of an object pointed to by a @code{basic *} really is.
4559 Also, for the unarchiving of expressions it must be possible to find the
4560 @code{unarchive()} function of a class given the class name (as a string). A
4561 system that provides this kind of information is called a run-time type
4562 information (RTTI) system. The C++ language provides such a thing (see the
4563 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4564 implements its own, simpler RTTI.
4566 The RTTI in GiNaC is based on two mechanisms:
4571 The @code{basic} class declares a member variable @code{tinfo_key} which
4572 holds an unsigned integer that identifies the object's class. These numbers
4573 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4574 classes. They all start with @code{TINFO_}.
4577 By means of some clever tricks with static members, GiNaC maintains a list
4578 of information for all classes derived from @code{basic}. The information
4579 available includes the class names, the @code{tinfo_key}s, and pointers
4580 to the unarchiving functions. This class registry is defined in the
4581 @file{registrar.h} header file.
4585 The disadvantage of this proprietary RTTI implementation is that there's
4586 a little more to do when implementing new classes (C++'s RTTI works more
4587 or less automatic) but don't worry, most of the work is simplified by
4590 @subsection A minimalistic example
4592 Now we will start implementing a new class @code{mystring} that allows
4593 placing character strings in algebraic expressions (this is not very useful,
4594 but it's just an example). This class will be a direct subclass of
4595 @code{basic}. You can use this sample implementation as a starting point
4596 for your own classes.
4598 The code snippets given here assume that you have included some header files
4604 #include <stdexcept>
4605 using namespace std;
4607 #include <ginac/ginac.h>
4608 using namespace GiNaC;
4611 The first thing we have to do is to define a @code{tinfo_key} for our new
4612 class. This can be any arbitrary unsigned number that is not already taken
4613 by one of the existing classes but it's better to come up with something
4614 that is unlikely to clash with keys that might be added in the future. The
4615 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4616 which is not a requirement but we are going to stick with this scheme:
4619 const unsigned TINFO_mystring = 0x42420001U;
4622 Now we can write down the class declaration. The class stores a C++
4623 @code{string} and the user shall be able to construct a @code{mystring}
4624 object from a C or C++ string:
4627 class mystring : public basic
4629 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4632 mystring(const string &s);
4633 mystring(const char *s);
4639 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4642 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4643 macros are defined in @file{registrar.h}. They take the name of the class
4644 and its direct superclass as arguments and insert all required declarations
4645 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4646 the first line after the opening brace of the class definition. The
4647 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4648 source (at global scope, of course, not inside a function).
4650 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4651 declarations of the default and copy constructor, the destructor, the
4652 assignment operator and a couple of other functions that are required. It
4653 also defines a type @code{inherited} which refers to the superclass so you
4654 don't have to modify your code every time you shuffle around the class
4655 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4656 constructor, the destructor and the assignment operator.
4658 Now there are nine member functions we have to implement to get a working
4664 @code{mystring()}, the default constructor.
4667 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4668 assignment operator to free dynamically allocated members. The @code{call_parent}
4669 specifies whether the @code{destroy()} function of the superclass is to be
4673 @code{void copy(const mystring &other)}, which is used in the copy constructor
4674 and assignment operator to copy the member variables over from another
4675 object of the same class.
4678 @code{void archive(archive_node &n)}, the archiving function. This stores all
4679 information needed to reconstruct an object of this class inside an
4680 @code{archive_node}.
4683 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4684 constructor. This constructs an instance of the class from the information
4685 found in an @code{archive_node}.
4688 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4689 unarchiving function. It constructs a new instance by calling the unarchiving
4693 @code{int compare_same_type(const basic &other)}, which is used internally
4694 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4695 -1, depending on the relative order of this object and the @code{other}
4696 object. If it returns 0, the objects are considered equal.
4697 @strong{Note:} This has nothing to do with the (numeric) ordering
4698 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4699 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4700 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4701 must provide a @code{compare_same_type()} function, even those representing
4702 objects for which no reasonable algebraic ordering relationship can be
4706 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4707 which are the two constructors we declared.
4711 Let's proceed step-by-step. The default constructor looks like this:
4714 mystring::mystring() : inherited(TINFO_mystring)
4716 // dynamically allocate resources here if required
4720 The golden rule is that in all constructors you have to set the
4721 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4722 it will be set by the constructor of the superclass and all hell will break
4723 loose in the RTTI. For your convenience, the @code{basic} class provides
4724 a constructor that takes a @code{tinfo_key} value, which we are using here
4725 (remember that in our case @code{inherited = basic}). If the superclass
4726 didn't have such a constructor, we would have to set the @code{tinfo_key}
4727 to the right value manually.
4729 In the default constructor you should set all other member variables to
4730 reasonable default values (we don't need that here since our @code{str}
4731 member gets set to an empty string automatically). The constructor(s) are of
4732 course also the right place to allocate any dynamic resources you require.
4734 Next, the @code{destroy()} function:
4737 void mystring::destroy(bool call_parent)
4739 // free dynamically allocated resources here if required
4741 inherited::destroy(call_parent);
4745 This function is where we free all dynamically allocated resources. We don't
4746 have any so we're not doing anything here, but if we had, for example, used
4747 a C-style @code{char *} to store our string, this would be the place to
4748 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4749 to call the @code{destroy()} function of the superclass after we're done
4750 (to mimic C++'s automatic invocation of superclass destructors where
4751 @code{destroy()} is called from outside a destructor).
4753 The @code{copy()} function just copies over the member variables from
4757 void mystring::copy(const mystring &other)
4759 inherited::copy(other);
4764 We can simply overwrite the member variables here. There's no need to worry
4765 about dynamically allocated storage. The assignment operator (which is
4766 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4767 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4768 explicitly call the @code{copy()} function of the superclass here so
4769 all the member variables will get copied.
4771 Next are the three functions for archiving. You have to implement them even
4772 if you don't plan to use archives, but the minimum required implementation
4773 is really simple. First, the archiving function:
4776 void mystring::archive(archive_node &n) const
4778 inherited::archive(n);
4779 n.add_string("string", str);
4783 The only thing that is really required is calling the @code{archive()}
4784 function of the superclass. Optionally, you can store all information you
4785 deem necessary for representing the object into the passed
4786 @code{archive_node}. We are just storing our string here. For more
4787 information on how the archiving works, consult the @file{archive.h} header
4790 The unarchiving constructor is basically the inverse of the archiving
4794 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4796 n.find_string("string", str);
4800 If you don't need archiving, just leave this function empty (but you must
4801 invoke the unarchiving constructor of the superclass). Note that we don't
4802 have to set the @code{tinfo_key} here because it is done automatically
4803 by the unarchiving constructor of the @code{basic} class.
4805 Finally, the unarchiving function:
4808 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4810 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4814 You don't have to understand how exactly this works. Just copy these four
4815 lines into your code literally (replacing the class name, of course). It
4816 calls the unarchiving constructor of the class and unless you are doing
4817 something very special (like matching @code{archive_node}s to global
4818 objects) you don't need a different implementation. For those who are
4819 interested: setting the @code{dynallocated} flag puts the object under
4820 the control of GiNaC's garbage collection. It will get deleted automatically
4821 once it is no longer referenced.
4823 Our @code{compare_same_type()} function uses a provided function to compare
4827 int mystring::compare_same_type(const basic &other) const
4829 const mystring &o = static_cast<const mystring &>(other);
4830 int cmpval = str.compare(o.str);
4833 else if (cmpval < 0)
4840 Although this function takes a @code{basic &}, it will always be a reference
4841 to an object of exactly the same class (objects of different classes are not
4842 comparable), so the cast is safe. If this function returns 0, the two objects
4843 are considered equal (in the sense that @math{A-B=0}), so you should compare
4844 all relevant member variables.
4846 Now the only thing missing is our two new constructors:
4849 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4851 // dynamically allocate resources here if required
4854 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4856 // dynamically allocate resources here if required
4860 No surprises here. We set the @code{str} member from the argument and
4861 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4863 That's it! We now have a minimal working GiNaC class that can store
4864 strings in algebraic expressions. Let's confirm that the RTTI works:
4867 ex e = mystring("Hello, world!");
4868 cout << is_a<mystring>(e) << endl;
4871 cout << e.bp->class_name() << endl;
4875 Obviously it does. Let's see what the expression @code{e} looks like:
4879 // -> [mystring object]
4882 Hm, not exactly what we expect, but of course the @code{mystring} class
4883 doesn't yet know how to print itself. This is done in the @code{print()}
4884 member function. Let's say that we wanted to print the string surrounded
4888 class mystring : public basic
4892 void print(const print_context &c, unsigned level = 0) const;
4896 void mystring::print(const print_context &c, unsigned level) const
4898 // print_context::s is a reference to an ostream
4899 c.s << '\"' << str << '\"';
4903 The @code{level} argument is only required for container classes to
4904 correctly parenthesize the output. Let's try again to print the expression:
4908 // -> "Hello, world!"
4911 Much better. The @code{mystring} class can be used in arbitrary expressions:
4914 e += mystring("GiNaC rulez");
4916 // -> "GiNaC rulez"+"Hello, world!"
4919 (GiNaC's automatic term reordering is in effect here), or even
4922 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4924 // -> "One string"^(2*sin(-"Another string"+Pi))
4927 Whether this makes sense is debatable but remember that this is only an
4928 example. At least it allows you to implement your own symbolic algorithms
4931 Note that GiNaC's algebraic rules remain unchanged:
4934 e = mystring("Wow") * mystring("Wow");
4938 e = pow(mystring("First")-mystring("Second"), 2);
4939 cout << e.expand() << endl;
4940 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4943 There's no way to, for example, make GiNaC's @code{add} class perform string
4944 concatenation. You would have to implement this yourself.
4946 @subsection Automatic evaluation
4948 @cindex @code{hold()}
4950 When dealing with objects that are just a little more complicated than the
4951 simple string objects we have implemented, chances are that you will want to
4952 have some automatic simplifications or canonicalizations performed on them.
4953 This is done in the evaluation member function @code{eval()}. Let's say that
4954 we wanted all strings automatically converted to lowercase with
4955 non-alphabetic characters stripped, and empty strings removed:
4958 class mystring : public basic
4962 ex eval(int level = 0) const;
4966 ex mystring::eval(int level) const
4969 for (int i=0; i<str.length(); i++) @{
4971 if (c >= 'A' && c <= 'Z')
4972 new_str += tolower(c);
4973 else if (c >= 'a' && c <= 'z')
4977 if (new_str.length() == 0)
4980 return mystring(new_str).hold();
4984 The @code{level} argument is used to limit the recursion depth of the
4985 evaluation. We don't have any subexpressions in the @code{mystring} class
4986 so we are not concerned with this. If we had, we would call the @code{eval()}
4987 functions of the subexpressions with @code{level - 1} as the argument if
4988 @code{level != 1}. The @code{hold()} member function sets a flag in the
4989 object that prevents further evaluation. Otherwise we might end up in an
4990 endless loop. When you want to return the object unmodified, use
4991 @code{return this->hold();}.
4993 Let's confirm that it works:
4996 ex e = mystring("Hello, world!") + mystring("!?#");
5000 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5005 @subsection Other member functions
5007 We have implemented only a small set of member functions to make the class
5008 work in the GiNaC framework. For a real algebraic class, there are probably
5009 some more functions that you will want to re-implement, such as
5010 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5011 or the header file of the class you want to make a subclass of to see
5012 what's there. One member function that you will most likely want to
5013 implement for terminal classes like the described string class is
5014 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5015 which will allow GiNaC to compare and canonicalize expressions much more
5018 You can, of course, also add your own new member functions. Remember,
5019 that the RTTI may be used to get information about what kinds of objects
5020 you are dealing with (the position in the class hierarchy) and that you
5021 can always extract the bare object from an @code{ex} by stripping the
5022 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5023 should become a need.
5025 That's it. May the source be with you!
5028 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5029 @c node-name, next, previous, up
5030 @chapter A Comparison With Other CAS
5033 This chapter will give you some information on how GiNaC compares to
5034 other, traditional Computer Algebra Systems, like @emph{Maple},
5035 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5036 disadvantages over these systems.
5039 * Advantages:: Stengths of the GiNaC approach.
5040 * Disadvantages:: Weaknesses of the GiNaC approach.
5041 * Why C++?:: Attractiveness of C++.
5044 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5045 @c node-name, next, previous, up
5048 GiNaC has several advantages over traditional Computer
5049 Algebra Systems, like
5054 familiar language: all common CAS implement their own proprietary
5055 grammar which you have to learn first (and maybe learn again when your
5056 vendor decides to `enhance' it). With GiNaC you can write your program
5057 in common C++, which is standardized.
5061 structured data types: you can build up structured data types using
5062 @code{struct}s or @code{class}es together with STL features instead of
5063 using unnamed lists of lists of lists.
5066 strongly typed: in CAS, you usually have only one kind of variables
5067 which can hold contents of an arbitrary type. This 4GL like feature is
5068 nice for novice programmers, but dangerous.
5071 development tools: powerful development tools exist for C++, like fancy
5072 editors (e.g. with automatic indentation and syntax highlighting),
5073 debuggers, visualization tools, documentation generators@dots{}
5076 modularization: C++ programs can easily be split into modules by
5077 separating interface and implementation.
5080 price: GiNaC is distributed under the GNU Public License which means
5081 that it is free and available with source code. And there are excellent
5082 C++-compilers for free, too.
5085 extendable: you can add your own classes to GiNaC, thus extending it on
5086 a very low level. Compare this to a traditional CAS that you can
5087 usually only extend on a high level by writing in the language defined
5088 by the parser. In particular, it turns out to be almost impossible to
5089 fix bugs in a traditional system.
5092 multiple interfaces: Though real GiNaC programs have to be written in
5093 some editor, then be compiled, linked and executed, there are more ways
5094 to work with the GiNaC engine. Many people want to play with
5095 expressions interactively, as in traditional CASs. Currently, two such
5096 windows into GiNaC have been implemented and many more are possible: the
5097 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5098 types to a command line and second, as a more consistent approach, an
5099 interactive interface to the @acronym{Cint} C++ interpreter has been put
5100 together (called @acronym{GiNaC-cint}) that allows an interactive
5101 scripting interface consistent with the C++ language.
5104 seemless integration: it is somewhere between difficult and impossible
5105 to call CAS functions from within a program written in C++ or any other
5106 programming language and vice versa. With GiNaC, your symbolic routines
5107 are part of your program. You can easily call third party libraries,
5108 e.g. for numerical evaluation or graphical interaction. All other
5109 approaches are much more cumbersome: they range from simply ignoring the
5110 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5111 system (i.e. @emph{Yacas}).
5114 efficiency: often large parts of a program do not need symbolic
5115 calculations at all. Why use large integers for loop variables or
5116 arbitrary precision arithmetics where @code{int} and @code{double} are
5117 sufficient? For pure symbolic applications, GiNaC is comparable in
5118 speed with other CAS.
5123 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5124 @c node-name, next, previous, up
5125 @section Disadvantages
5127 Of course it also has some disadvantages:
5132 advanced features: GiNaC cannot compete with a program like
5133 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5134 which grows since 1981 by the work of dozens of programmers, with
5135 respect to mathematical features. Integration, factorization,
5136 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5137 not planned for the near future).
5140 portability: While the GiNaC library itself is designed to avoid any
5141 platform dependent features (it should compile on any ANSI compliant C++
5142 compiler), the currently used version of the CLN library (fast large
5143 integer and arbitrary precision arithmetics) can be compiled only on
5144 systems with a recently new C++ compiler from the GNU Compiler
5145 Collection (@acronym{GCC}).@footnote{This is because CLN uses
5146 PROVIDE/REQUIRE like macros to let the compiler gather all static
5147 initializations, which works for GNU C++ only.} GiNaC uses recent
5148 language features like explicit constructors, mutable members, RTTI,
5149 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
5150 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
5151 ANSI compliant, support all needed features.
5156 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5157 @c node-name, next, previous, up
5160 Why did we choose to implement GiNaC in C++ instead of Java or any other
5161 language? C++ is not perfect: type checking is not strict (casting is
5162 possible), separation between interface and implementation is not
5163 complete, object oriented design is not enforced. The main reason is
5164 the often scolded feature of operator overloading in C++. While it may
5165 be true that operating on classes with a @code{+} operator is rarely
5166 meaningful, it is perfectly suited for algebraic expressions. Writing
5167 @math{3x+5y} as @code{3*x+5*y} instead of
5168 @code{x.times(3).plus(y.times(5))} looks much more natural.
5169 Furthermore, the main developers are more familiar with C++ than with
5170 any other programming language.
5173 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5174 @c node-name, next, previous, up
5175 @appendix Internal Structures
5178 * Expressions are reference counted::
5179 * Internal representation of products and sums::
5182 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5183 @c node-name, next, previous, up
5184 @appendixsection Expressions are reference counted
5186 @cindex reference counting
5187 @cindex copy-on-write
5188 @cindex garbage collection
5189 An expression is extremely light-weight since internally it works like a
5190 handle to the actual representation and really holds nothing more than a
5191 pointer to some other object. What this means in practice is that
5192 whenever you create two @code{ex} and set the second equal to the first
5193 no copying process is involved. Instead, the copying takes place as soon
5194 as you try to change the second. Consider the simple sequence of code:
5197 #include <ginac/ginac.h>
5198 using namespace std;
5199 using namespace GiNaC;
5203 symbol x("x"), y("y"), z("z");
5206 e1 = sin(x + 2*y) + 3*z + 41;
5207 e2 = e1; // e2 points to same object as e1
5208 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5209 e2 += 1; // e2 is copied into a new object
5210 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5214 The line @code{e2 = e1;} creates a second expression pointing to the
5215 object held already by @code{e1}. The time involved for this operation
5216 is therefore constant, no matter how large @code{e1} was. Actual
5217 copying, however, must take place in the line @code{e2 += 1;} because
5218 @code{e1} and @code{e2} are not handles for the same object any more.
5219 This concept is called @dfn{copy-on-write semantics}. It increases
5220 performance considerably whenever one object occurs multiple times and
5221 represents a simple garbage collection scheme because when an @code{ex}
5222 runs out of scope its destructor checks whether other expressions handle
5223 the object it points to too and deletes the object from memory if that
5224 turns out not to be the case. A slightly less trivial example of
5225 differentiation using the chain-rule should make clear how powerful this
5229 #include <ginac/ginac.h>
5230 using namespace std;
5231 using namespace GiNaC;
5235 symbol x("x"), y("y");
5239 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5240 cout << e1 << endl // prints x+3*y
5241 << e2 << endl // prints (x+3*y)^3
5242 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5246 Here, @code{e1} will actually be referenced three times while @code{e2}
5247 will be referenced two times. When the power of an expression is built,
5248 that expression needs not be copied. Likewise, since the derivative of
5249 a power of an expression can be easily expressed in terms of that
5250 expression, no copying of @code{e1} is involved when @code{e3} is
5251 constructed. So, when @code{e3} is constructed it will print as
5252 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5253 holds a reference to @code{e2} and the factor in front is just
5256 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5257 semantics. When you insert an expression into a second expression, the
5258 result behaves exactly as if the contents of the first expression were
5259 inserted. But it may be useful to remember that this is not what
5260 happens. Knowing this will enable you to write much more efficient
5261 code. If you still have an uncertain feeling with copy-on-write
5262 semantics, we recommend you have a look at the
5263 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5264 Marshall Cline. Chapter 16 covers this issue and presents an
5265 implementation which is pretty close to the one in GiNaC.
5268 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5269 @c node-name, next, previous, up
5270 @appendixsection Internal representation of products and sums
5272 @cindex representation
5275 @cindex @code{power}
5276 Although it should be completely transparent for the user of
5277 GiNaC a short discussion of this topic helps to understand the sources
5278 and also explain performance to a large degree. Consider the
5279 unexpanded symbolic expression
5281 $2d^3 \left( 4a + 5b - 3 \right)$
5284 @math{2*d^3*(4*a+5*b-3)}
5286 which could naively be represented by a tree of linear containers for
5287 addition and multiplication, one container for exponentiation with base
5288 and exponent and some atomic leaves of symbols and numbers in this
5293 @cindex pair-wise representation
5294 However, doing so results in a rather deeply nested tree which will
5295 quickly become inefficient to manipulate. We can improve on this by
5296 representing the sum as a sequence of terms, each one being a pair of a
5297 purely numeric multiplicative coefficient and its rest. In the same
5298 spirit we can store the multiplication as a sequence of terms, each
5299 having a numeric exponent and a possibly complicated base, the tree
5300 becomes much more flat:
5304 The number @code{3} above the symbol @code{d} shows that @code{mul}
5305 objects are treated similarly where the coefficients are interpreted as
5306 @emph{exponents} now. Addition of sums of terms or multiplication of
5307 products with numerical exponents can be coded to be very efficient with
5308 such a pair-wise representation. Internally, this handling is performed
5309 by most CAS in this way. It typically speeds up manipulations by an
5310 order of magnitude. The overall multiplicative factor @code{2} and the
5311 additive term @code{-3} look somewhat out of place in this
5312 representation, however, since they are still carrying a trivial
5313 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5314 this is avoided by adding a field that carries an overall numeric
5315 coefficient. This results in the realistic picture of internal
5318 $2d^3 \left( 4a + 5b - 3 \right)$:
5321 @math{2*d^3*(4*a+5*b-3)}:
5327 This also allows for a better handling of numeric radicals, since
5328 @code{sqrt(2)} can now be carried along calculations. Now it should be
5329 clear, why both classes @code{add} and @code{mul} are derived from the
5330 same abstract class: the data representation is the same, only the
5331 semantics differs. In the class hierarchy, methods for polynomial
5332 expansion and the like are reimplemented for @code{add} and @code{mul},
5333 but the data structure is inherited from @code{expairseq}.
5336 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5337 @c node-name, next, previous, up
5338 @appendix Package Tools
5340 If you are creating a software package that uses the GiNaC library,
5341 setting the correct command line options for the compiler and linker
5342 can be difficult. GiNaC includes two tools to make this process easier.
5345 * ginac-config:: A shell script to detect compiler and linker flags.
5346 * AM_PATH_GINAC:: Macro for GNU automake.
5350 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5351 @c node-name, next, previous, up
5352 @section @command{ginac-config}
5353 @cindex ginac-config
5355 @command{ginac-config} is a shell script that you can use to determine
5356 the compiler and linker command line options required to compile and
5357 link a program with the GiNaC library.
5359 @command{ginac-config} takes the following flags:
5363 Prints out the version of GiNaC installed.
5365 Prints '-I' flags pointing to the installed header files.
5367 Prints out the linker flags necessary to link a program against GiNaC.
5368 @item --prefix[=@var{PREFIX}]
5369 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5370 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5371 Otherwise, prints out the configured value of @env{$prefix}.
5372 @item --exec-prefix[=@var{PREFIX}]
5373 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5374 Otherwise, prints out the configured value of @env{$exec_prefix}.
5377 Typically, @command{ginac-config} will be used within a configure
5378 script, as described below. It, however, can also be used directly from
5379 the command line using backquotes to compile a simple program. For
5383 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5386 This command line might expand to (for example):
5389 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5390 -lginac -lcln -lstdc++
5393 Not only is the form using @command{ginac-config} easier to type, it will
5394 work on any system, no matter how GiNaC was configured.
5397 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5398 @c node-name, next, previous, up
5399 @section @samp{AM_PATH_GINAC}
5400 @cindex AM_PATH_GINAC
5402 For packages configured using GNU automake, GiNaC also provides
5403 a macro to automate the process of checking for GiNaC.
5406 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5414 Determines the location of GiNaC using @command{ginac-config}, which is
5415 either found in the user's path, or from the environment variable
5416 @env{GINACLIB_CONFIG}.
5419 Tests the installed libraries to make sure that their version
5420 is later than @var{MINIMUM-VERSION}. (A default version will be used
5424 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5425 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5426 variable to the output of @command{ginac-config --libs}, and calls
5427 @samp{AC_SUBST()} for these variables so they can be used in generated
5428 makefiles, and then executes @var{ACTION-IF-FOUND}.
5431 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5432 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5436 This macro is in file @file{ginac.m4} which is installed in
5437 @file{$datadir/aclocal}. Note that if automake was installed with a
5438 different @samp{--prefix} than GiNaC, you will either have to manually
5439 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5440 aclocal the @samp{-I} option when running it.
5443 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5444 * Example package:: Example of a package using AM_PATH_GINAC.
5448 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5449 @c node-name, next, previous, up
5450 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5452 Simply make sure that @command{ginac-config} is in your path, and run
5453 the configure script.
5460 The directory where the GiNaC libraries are installed needs
5461 to be found by your system's dynamic linker.
5463 This is generally done by
5466 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5472 setting the environment variable @env{LD_LIBRARY_PATH},
5475 or, as a last resort,
5478 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5479 running configure, for instance:
5482 LDFLAGS=-R/home/cbauer/lib ./configure
5487 You can also specify a @command{ginac-config} not in your path by
5488 setting the @env{GINACLIB_CONFIG} environment variable to the
5489 name of the executable
5492 If you move the GiNaC package from its installed location,
5493 you will either need to modify @command{ginac-config} script
5494 manually to point to the new location or rebuild GiNaC.
5505 --with-ginac-prefix=@var{PREFIX}
5506 --with-ginac-exec-prefix=@var{PREFIX}
5509 are provided to override the prefix and exec-prefix that were stored
5510 in the @command{ginac-config} shell script by GiNaC's configure. You are
5511 generally better off configuring GiNaC with the right path to begin with.
5515 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5516 @c node-name, next, previous, up
5517 @subsection Example of a package using @samp{AM_PATH_GINAC}
5519 The following shows how to build a simple package using automake
5520 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5523 #include <ginac/ginac.h>
5527 GiNaC::symbol x("x");
5528 GiNaC::ex a = GiNaC::sin(x);
5529 std::cout << "Derivative of " << a
5530 << " is " << a.diff(x) << std::endl;
5535 You should first read the introductory portions of the automake
5536 Manual, if you are not already familiar with it.
5538 Two files are needed, @file{configure.in}, which is used to build the
5542 dnl Process this file with autoconf to produce a configure script.
5544 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5550 AM_PATH_GINAC(0.7.0, [
5551 LIBS="$LIBS $GINACLIB_LIBS"
5552 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5553 ], AC_MSG_ERROR([need to have GiNaC installed]))
5558 The only command in this which is not standard for automake
5559 is the @samp{AM_PATH_GINAC} macro.
5561 That command does the following: If a GiNaC version greater or equal
5562 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5563 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5564 the error message `need to have GiNaC installed'
5566 And the @file{Makefile.am}, which will be used to build the Makefile.
5569 ## Process this file with automake to produce Makefile.in
5570 bin_PROGRAMS = simple
5571 simple_SOURCES = simple.cpp
5574 This @file{Makefile.am}, says that we are building a single executable,
5575 from a single sourcefile @file{simple.cpp}. Since every program
5576 we are building uses GiNaC we simply added the GiNaC options
5577 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5578 want to specify them on a per-program basis: for instance by
5582 simple_LDADD = $(GINACLIB_LIBS)
5583 INCLUDES = $(GINACLIB_CPPFLAGS)
5586 to the @file{Makefile.am}.
5588 To try this example out, create a new directory and add the three
5591 Now execute the following commands:
5594 $ automake --add-missing
5599 You now have a package that can be built in the normal fashion
5608 @node Bibliography, Concept Index, Example package, Top
5609 @c node-name, next, previous, up
5610 @appendix Bibliography
5615 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5618 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5621 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5624 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5627 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5628 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5631 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5632 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5633 Academic Press, London
5636 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5637 D.E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5640 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5645 @node Concept Index, , Bibliography, Top
5646 @c node-name, next, previous, up
5647 @unnumbered Concept Index