1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2003 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2003 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
691 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
692 @c node-name, next, previous, up
694 @cindex expression (class @code{ex})
697 The most common class of objects a user deals with is the expression
698 @code{ex}, representing a mathematical object like a variable, number,
699 function, sum, product, etc@dots{} Expressions may be put together to form
700 new expressions, passed as arguments to functions, and so on. Here is a
701 little collection of valid expressions:
704 ex MyEx1 = 5; // simple number
705 ex MyEx2 = x + 2*y; // polynomial in x and y
706 ex MyEx3 = (x + 1)/(x - 1); // rational expression
707 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
708 ex MyEx5 = MyEx4 + 1; // similar to above
711 Expressions are handles to other more fundamental objects, that often
712 contain other expressions thus creating a tree of expressions
713 (@xref{Internal Structures}, for particular examples). Most methods on
714 @code{ex} therefore run top-down through such an expression tree. For
715 example, the method @code{has()} scans recursively for occurrences of
716 something inside an expression. Thus, if you have declared @code{MyEx4}
717 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
718 the argument of @code{sin} and hence return @code{true}.
720 The next sections will outline the general picture of GiNaC's class
721 hierarchy and describe the classes of objects that are handled by
725 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
726 @c node-name, next, previous, up
727 @section Automatic evaluation and canonicalization of expressions
730 GiNaC performs some automatic transformations on expressions, to simplify
731 them and put them into a canonical form. Some examples:
734 ex MyEx1 = 2*x - 1 + x; // 3*x-1
735 ex MyEx2 = x - x; // 0
736 ex MyEx3 = cos(2*Pi); // 1
737 ex MyEx4 = x*y/x; // y
740 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
741 evaluation}. GiNaC only performs transformations that are
745 at most of complexity @math{O(n log n)}
747 algebraically correct, possibly except for a set of measure zero (e.g.
748 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
751 There are two types of automatic transformations in GiNaC that may not
752 behave in an entirely obvious way at first glance:
756 The terms of sums and products (and some other things like the arguments of
757 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
758 into a canonical form that is deterministic, but not lexicographical or in
759 any other way easily guessable (it almost always depends on the number and
760 order of the symbols you define). However, constructing the same expression
761 twice, either implicitly or explicitly, will always result in the same
764 Expressions of the form 'number times sum' are automatically expanded (this
765 has to do with GiNaC's internal representation of sums and products). For
768 ex MyEx5 = 2*(x + y); // 2*x+2*y
769 ex MyEx6 = z*(x + y); // z*(x+y)
773 The general rule is that when you construct expressions, GiNaC automatically
774 creates them in canonical form, which might differ from the form you typed in
775 your program. This may create some awkward looking output (@samp{-y+x} instead
776 of @samp{y-x}) but allows for more efficient operation and usually yields
777 some immediate simplifications.
779 @cindex @code{eval()}
780 Internally, the anonymous evaluator in GiNaC is implemented by the methods
783 ex ex::eval(int level = 0) const;
784 ex basic::eval(int level = 0) const;
787 but unless you are extending GiNaC with your own classes or functions, there
788 should never be any reason to call them explicitly. All GiNaC methods that
789 transform expressions, like @code{subs()} or @code{normal()}, automatically
790 re-evaluate their results.
793 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
794 @c node-name, next, previous, up
795 @section Error handling
797 @cindex @code{pole_error} (class)
799 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
800 generated by GiNaC are subclassed from the standard @code{exception} class
801 defined in the @file{<stdexcept>} header. In addition to the predefined
802 @code{logic_error}, @code{domain_error}, @code{out_of_range},
803 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
804 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
805 exception that gets thrown when trying to evaluate a mathematical function
808 The @code{pole_error} class has a member function
811 int pole_error::degree() const;
814 that returns the order of the singularity (or 0 when the pole is
815 logarithmic or the order is undefined).
817 When using GiNaC it is useful to arrange for exceptions to be catched in
818 the main program even if you don't want to do any special error handling.
819 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
820 default exception handler of your C++ compiler's run-time system which
821 usually only aborts the program without giving any information what went
824 Here is an example for a @code{main()} function that catches and prints
825 exceptions generated by GiNaC:
830 #include <ginac/ginac.h>
832 using namespace GiNaC;
840 @} catch (exception &p) @{
841 cerr << p.what() << endl;
849 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
850 @c node-name, next, previous, up
851 @section The Class Hierarchy
853 GiNaC's class hierarchy consists of several classes representing
854 mathematical objects, all of which (except for @code{ex} and some
855 helpers) are internally derived from one abstract base class called
856 @code{basic}. You do not have to deal with objects of class
857 @code{basic}, instead you'll be dealing with symbols, numbers,
858 containers of expressions and so on.
862 To get an idea about what kinds of symbolic composites may be built we
863 have a look at the most important classes in the class hierarchy and
864 some of the relations among the classes:
866 @image{classhierarchy}
868 The abstract classes shown here (the ones without drop-shadow) are of no
869 interest for the user. They are used internally in order to avoid code
870 duplication if two or more classes derived from them share certain
871 features. An example is @code{expairseq}, a container for a sequence of
872 pairs each consisting of one expression and a number (@code{numeric}).
873 What @emph{is} visible to the user are the derived classes @code{add}
874 and @code{mul}, representing sums and products. @xref{Internal
875 Structures}, where these two classes are described in more detail. The
876 following table shortly summarizes what kinds of mathematical objects
877 are stored in the different classes:
880 @multitable @columnfractions .22 .78
881 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
882 @item @code{constant} @tab Constants like
889 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
890 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
891 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
892 @item @code{ncmul} @tab Products of non-commutative objects
893 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
898 @code{sqrt(}@math{2}@code{)}
901 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
902 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
903 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
904 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
905 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
906 @item @code{indexed} @tab Indexed object like @math{A_ij}
907 @item @code{tensor} @tab Special tensor like the delta and metric tensors
908 @item @code{idx} @tab Index of an indexed object
909 @item @code{varidx} @tab Index with variance
910 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
911 @item @code{wildcard} @tab Wildcard for pattern matching
916 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
917 @c node-name, next, previous, up
919 @cindex @code{symbol} (class)
920 @cindex hierarchy of classes
923 Symbols are for symbolic manipulation what atoms are for chemistry. You
924 can declare objects of class @code{symbol} as any other object simply by
925 saying @code{symbol x,y;}. There is, however, a catch in here having to
926 do with the fact that C++ is a compiled language. The information about
927 the symbol's name is thrown away by the compiler but at a later stage
928 you may want to print expressions holding your symbols. In order to
929 avoid confusion GiNaC's symbols are able to know their own name. This
930 is accomplished by declaring its name for output at construction time in
931 the fashion @code{symbol x("x");}. If you declare a symbol using the
932 default constructor (i.e. without string argument) the system will deal
933 out a unique name. That name may not be suitable for printing but for
934 internal routines when no output is desired it is often enough. We'll
935 come across examples of such symbols later in this tutorial.
937 This implies that the strings passed to symbols at construction time may
938 not be used for comparing two of them. It is perfectly legitimate to
939 write @code{symbol x("x"),y("x");} but it is likely to lead into
940 trouble. Here, @code{x} and @code{y} are different symbols and
941 statements like @code{x-y} will not be simplified to zero although the
942 output @code{x-x} looks funny. Such output may also occur when there
943 are two different symbols in two scopes, for instance when you call a
944 function that declares a symbol with a name already existent in a symbol
945 in the calling function. Again, comparing them (using @code{operator==}
946 for instance) will always reveal their difference. Watch out, please.
948 @cindex @code{subs()}
949 Although symbols can be assigned expressions for internal reasons, you
950 should not do it (and we are not going to tell you how it is done). If
951 you want to replace a symbol with something else in an expression, you
952 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
955 @node Numbers, Constants, Symbols, Basic Concepts
956 @c node-name, next, previous, up
958 @cindex @code{numeric} (class)
964 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
965 The classes therein serve as foundation classes for GiNaC. CLN stands
966 for Class Library for Numbers or alternatively for Common Lisp Numbers.
967 In order to find out more about CLN's internals, the reader is referred to
968 the documentation of that library. @inforef{Introduction, , cln}, for
969 more information. Suffice to say that it is by itself build on top of
970 another library, the GNU Multiple Precision library GMP, which is an
971 extremely fast library for arbitrary long integers and rationals as well
972 as arbitrary precision floating point numbers. It is very commonly used
973 by several popular cryptographic applications. CLN extends GMP by
974 several useful things: First, it introduces the complex number field
975 over either reals (i.e. floating point numbers with arbitrary precision)
976 or rationals. Second, it automatically converts rationals to integers
977 if the denominator is unity and complex numbers to real numbers if the
978 imaginary part vanishes and also correctly treats algebraic functions.
979 Third it provides good implementations of state-of-the-art algorithms
980 for all trigonometric and hyperbolic functions as well as for
981 calculation of some useful constants.
983 The user can construct an object of class @code{numeric} in several
984 ways. The following example shows the four most important constructors.
985 It uses construction from C-integer, construction of fractions from two
986 integers, construction from C-float and construction from a string:
990 #include <ginac/ginac.h>
991 using namespace GiNaC;
995 numeric two = 2; // exact integer 2
996 numeric r(2,3); // exact fraction 2/3
997 numeric e(2.71828); // floating point number
998 numeric p = "3.14159265358979323846"; // constructor from string
999 // Trott's constant in scientific notation:
1000 numeric trott("1.0841015122311136151E-2");
1002 std::cout << two*p << std::endl; // floating point 6.283...
1007 @cindex complex numbers
1008 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1013 numeric z1 = 2-3*I; // exact complex number 2-3i
1014 numeric z2 = 5.9+1.6*I; // complex floating point number
1018 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1019 This would, however, call C's built-in operator @code{/} for integers
1020 first and result in a numeric holding a plain integer 1. @strong{Never
1021 use the operator @code{/} on integers} unless you know exactly what you
1022 are doing! Use the constructor from two integers instead, as shown in
1023 the example above. Writing @code{numeric(1)/2} may look funny but works
1026 @cindex @code{Digits}
1028 We have seen now the distinction between exact numbers and floating
1029 point numbers. Clearly, the user should never have to worry about
1030 dynamically created exact numbers, since their `exactness' always
1031 determines how they ought to be handled, i.e. how `long' they are. The
1032 situation is different for floating point numbers. Their accuracy is
1033 controlled by one @emph{global} variable, called @code{Digits}. (For
1034 those readers who know about Maple: it behaves very much like Maple's
1035 @code{Digits}). All objects of class numeric that are constructed from
1036 then on will be stored with a precision matching that number of decimal
1041 #include <ginac/ginac.h>
1042 using namespace std;
1043 using namespace GiNaC;
1047 numeric three(3.0), one(1.0);
1048 numeric x = one/three;
1050 cout << "in " << Digits << " digits:" << endl;
1052 cout << Pi.evalf() << endl;
1064 The above example prints the following output to screen:
1068 0.33333333333333333334
1069 3.1415926535897932385
1071 0.33333333333333333333333333333333333333333333333333333333333333333334
1072 3.1415926535897932384626433832795028841971693993751058209749445923078
1076 Note that the last number is not necessarily rounded as you would
1077 naively expect it to be rounded in the decimal system. But note also,
1078 that in both cases you got a couple of extra digits. This is because
1079 numbers are internally stored by CLN as chunks of binary digits in order
1080 to match your machine's word size and to not waste precision. Thus, on
1081 architectures with different word size, the above output might even
1082 differ with regard to actually computed digits.
1084 It should be clear that objects of class @code{numeric} should be used
1085 for constructing numbers or for doing arithmetic with them. The objects
1086 one deals with most of the time are the polymorphic expressions @code{ex}.
1088 @subsection Tests on numbers
1090 Once you have declared some numbers, assigned them to expressions and
1091 done some arithmetic with them it is frequently desired to retrieve some
1092 kind of information from them like asking whether that number is
1093 integer, rational, real or complex. For those cases GiNaC provides
1094 several useful methods. (Internally, they fall back to invocations of
1095 certain CLN functions.)
1097 As an example, let's construct some rational number, multiply it with
1098 some multiple of its denominator and test what comes out:
1102 #include <ginac/ginac.h>
1103 using namespace std;
1104 using namespace GiNaC;
1106 // some very important constants:
1107 const numeric twentyone(21);
1108 const numeric ten(10);
1109 const numeric five(5);
1113 numeric answer = twentyone;
1116 cout << answer.is_integer() << endl; // false, it's 21/5
1118 cout << answer.is_integer() << endl; // true, it's 42 now!
1122 Note that the variable @code{answer} is constructed here as an integer
1123 by @code{numeric}'s copy constructor but in an intermediate step it
1124 holds a rational number represented as integer numerator and integer
1125 denominator. When multiplied by 10, the denominator becomes unity and
1126 the result is automatically converted to a pure integer again.
1127 Internally, the underlying CLN is responsible for this behavior and we
1128 refer the reader to CLN's documentation. Suffice to say that
1129 the same behavior applies to complex numbers as well as return values of
1130 certain functions. Complex numbers are automatically converted to real
1131 numbers if the imaginary part becomes zero. The full set of tests that
1132 can be applied is listed in the following table.
1135 @multitable @columnfractions .30 .70
1136 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1137 @item @code{.is_zero()}
1138 @tab @dots{}equal to zero
1139 @item @code{.is_positive()}
1140 @tab @dots{}not complex and greater than 0
1141 @item @code{.is_integer()}
1142 @tab @dots{}a (non-complex) integer
1143 @item @code{.is_pos_integer()}
1144 @tab @dots{}an integer and greater than 0
1145 @item @code{.is_nonneg_integer()}
1146 @tab @dots{}an integer and greater equal 0
1147 @item @code{.is_even()}
1148 @tab @dots{}an even integer
1149 @item @code{.is_odd()}
1150 @tab @dots{}an odd integer
1151 @item @code{.is_prime()}
1152 @tab @dots{}a prime integer (probabilistic primality test)
1153 @item @code{.is_rational()}
1154 @tab @dots{}an exact rational number (integers are rational, too)
1155 @item @code{.is_real()}
1156 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1157 @item @code{.is_cinteger()}
1158 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1159 @item @code{.is_crational()}
1160 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1165 @node Constants, Fundamental containers, Numbers, Basic Concepts
1166 @c node-name, next, previous, up
1168 @cindex @code{constant} (class)
1171 @cindex @code{Catalan}
1172 @cindex @code{Euler}
1173 @cindex @code{evalf()}
1174 Constants behave pretty much like symbols except that they return some
1175 specific number when the method @code{.evalf()} is called.
1177 The predefined known constants are:
1180 @multitable @columnfractions .14 .30 .56
1181 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1183 @tab Archimedes' constant
1184 @tab 3.14159265358979323846264338327950288
1185 @item @code{Catalan}
1186 @tab Catalan's constant
1187 @tab 0.91596559417721901505460351493238411
1189 @tab Euler's (or Euler-Mascheroni) constant
1190 @tab 0.57721566490153286060651209008240243
1195 @node Fundamental containers, Lists, Constants, Basic Concepts
1196 @c node-name, next, previous, up
1197 @section Sums, products and powers
1201 @cindex @code{power}
1203 Simple rational expressions are written down in GiNaC pretty much like
1204 in other CAS or like expressions involving numerical variables in C.
1205 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1206 been overloaded to achieve this goal. When you run the following
1207 code snippet, the constructor for an object of type @code{mul} is
1208 automatically called to hold the product of @code{a} and @code{b} and
1209 then the constructor for an object of type @code{add} is called to hold
1210 the sum of that @code{mul} object and the number one:
1214 symbol a("a"), b("b");
1219 @cindex @code{pow()}
1220 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1221 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1222 construction is necessary since we cannot safely overload the constructor
1223 @code{^} in C++ to construct a @code{power} object. If we did, it would
1224 have several counterintuitive and undesired effects:
1228 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1230 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1231 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1232 interpret this as @code{x^(a^b)}.
1234 Also, expressions involving integer exponents are very frequently used,
1235 which makes it even more dangerous to overload @code{^} since it is then
1236 hard to distinguish between the semantics as exponentiation and the one
1237 for exclusive or. (It would be embarrassing to return @code{1} where one
1238 has requested @code{2^3}.)
1241 @cindex @command{ginsh}
1242 All effects are contrary to mathematical notation and differ from the
1243 way most other CAS handle exponentiation, therefore overloading @code{^}
1244 is ruled out for GiNaC's C++ part. The situation is different in
1245 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1246 that the other frequently used exponentiation operator @code{**} does
1247 not exist at all in C++).
1249 To be somewhat more precise, objects of the three classes described
1250 here, are all containers for other expressions. An object of class
1251 @code{power} is best viewed as a container with two slots, one for the
1252 basis, one for the exponent. All valid GiNaC expressions can be
1253 inserted. However, basic transformations like simplifying
1254 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1255 when this is mathematically possible. If we replace the outer exponent
1256 three in the example by some symbols @code{a}, the simplification is not
1257 safe and will not be performed, since @code{a} might be @code{1/2} and
1260 Objects of type @code{add} and @code{mul} are containers with an
1261 arbitrary number of slots for expressions to be inserted. Again, simple
1262 and safe simplifications are carried out like transforming
1263 @code{3*x+4-x} to @code{2*x+4}.
1266 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1267 @c node-name, next, previous, up
1268 @section Lists of expressions
1269 @cindex @code{lst} (class)
1271 @cindex @code{nops()}
1273 @cindex @code{append()}
1274 @cindex @code{prepend()}
1275 @cindex @code{remove_first()}
1276 @cindex @code{remove_last()}
1277 @cindex @code{remove_all()}
1279 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1280 expressions. They are not as ubiquitous as in many other computer algebra
1281 packages, but are sometimes used to supply a variable number of arguments of
1282 the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
1283 so you should have a basic understanding of them.
1285 Lists of up to 16 expressions can be directly constructed from single
1290 symbol x("x"), y("y");
1291 lst l(x, 2, y, x+y);
1292 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1296 Use the @code{nops()} method to determine the size (number of expressions) of
1297 a list and the @code{op()} method or the @code{[]} operator to access
1298 individual elements:
1302 cout << l.nops() << endl; // prints '4'
1303 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1307 As with the standard @code{list<T>} container, accessing random elements of a
1308 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1309 sequential access to the elements of a list is possible with the
1310 iterator types provided by the @code{lst} class:
1313 typedef ... lst::const_iterator;
1314 typedef ... lst::const_reverse_iterator;
1315 lst::const_iterator lst::begin() const;
1316 lst::const_iterator lst::end() const;
1317 lst::const_reverse_iterator lst::rbegin() const;
1318 lst::const_reverse_iterator lst::rend() const;
1321 For example, to print the elements of a list individually you can use:
1326 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1331 which is one order faster than
1336 for (size_t i = 0; i < l.nops(); ++i)
1337 cout << l.op(i) << endl;
1341 These iterators also allow you to use some of the algorithms provided by
1342 the C++ standard library:
1346 // print the elements of the list (requires #include <iterator>)
1347 copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1349 // sum up the elements of the list (requires #include <numeric>)
1350 ex sum = accumulate(l.begin(), l.end(), ex(0));
1351 cout << sum << endl; // prints '2+2*x+2*y'
1355 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1356 (the only other one is @code{matrix}). You can modify single elements:
1360 l[1] = 42; // l is now @{x, 42, y, x+y@}
1361 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1365 You can append or prepend an expression to a list with the @code{append()}
1366 and @code{prepend()} methods:
1370 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1371 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1375 You can remove the first or last element of a list with @code{remove_first()}
1376 and @code{remove_last()}:
1380 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1381 l.remove_last(); // l is now @{x, 7, y, x+y@}
1385 You can remove all the elements of a list with @code{remove_all()}:
1389 l.remove_all(); // l is now empty
1393 You can bring the elements of a list into a canonical order with @code{sort()}:
1397 lst l1(x, 2, y, x+y);
1398 lst l2(2, x+y, x, y);
1401 // l1 and l2 are now equal
1405 Finally, you can remove all but the first element of consecutive groups of
1406 elements with @code{unique()}:
1410 lst l3(x, 2, 2, 2, y, x+y, y+x);
1411 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1416 @node Mathematical functions, Relations, Lists, Basic Concepts
1417 @c node-name, next, previous, up
1418 @section Mathematical functions
1419 @cindex @code{function} (class)
1420 @cindex trigonometric function
1421 @cindex hyperbolic function
1423 There are quite a number of useful functions hard-wired into GiNaC. For
1424 instance, all trigonometric and hyperbolic functions are implemented
1425 (@xref{Built-in Functions}, for a complete list).
1427 These functions (better called @emph{pseudofunctions}) are all objects
1428 of class @code{function}. They accept one or more expressions as
1429 arguments and return one expression. If the arguments are not
1430 numerical, the evaluation of the function may be halted, as it does in
1431 the next example, showing how a function returns itself twice and
1432 finally an expression that may be really useful:
1434 @cindex Gamma function
1435 @cindex @code{subs()}
1438 symbol x("x"), y("y");
1440 cout << tgamma(foo) << endl;
1441 // -> tgamma(x+(1/2)*y)
1442 ex bar = foo.subs(y==1);
1443 cout << tgamma(bar) << endl;
1445 ex foobar = bar.subs(x==7);
1446 cout << tgamma(foobar) << endl;
1447 // -> (135135/128)*Pi^(1/2)
1451 Besides evaluation most of these functions allow differentiation, series
1452 expansion and so on. Read the next chapter in order to learn more about
1455 It must be noted that these pseudofunctions are created by inline
1456 functions, where the argument list is templated. This means that
1457 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1458 @code{sin(ex(1))} and will therefore not result in a floating point
1459 number. Unless of course the function prototype is explicitly
1460 overridden -- which is the case for arguments of type @code{numeric}
1461 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1462 point number of class @code{numeric} you should call
1463 @code{sin(numeric(1))}. This is almost the same as calling
1464 @code{sin(1).evalf()} except that the latter will return a numeric
1465 wrapped inside an @code{ex}.
1468 @node Relations, Matrices, Mathematical functions, Basic Concepts
1469 @c node-name, next, previous, up
1471 @cindex @code{relational} (class)
1473 Sometimes, a relation holding between two expressions must be stored
1474 somehow. The class @code{relational} is a convenient container for such
1475 purposes. A relation is by definition a container for two @code{ex} and
1476 a relation between them that signals equality, inequality and so on.
1477 They are created by simply using the C++ operators @code{==}, @code{!=},
1478 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1480 @xref{Mathematical functions}, for examples where various applications
1481 of the @code{.subs()} method show how objects of class relational are
1482 used as arguments. There they provide an intuitive syntax for
1483 substitutions. They are also used as arguments to the @code{ex::series}
1484 method, where the left hand side of the relation specifies the variable
1485 to expand in and the right hand side the expansion point. They can also
1486 be used for creating systems of equations that are to be solved for
1487 unknown variables. But the most common usage of objects of this class
1488 is rather inconspicuous in statements of the form @code{if
1489 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1490 conversion from @code{relational} to @code{bool} takes place. Note,
1491 however, that @code{==} here does not perform any simplifications, hence
1492 @code{expand()} must be called explicitly.
1495 @node Matrices, Indexed objects, Relations, Basic Concepts
1496 @c node-name, next, previous, up
1498 @cindex @code{matrix} (class)
1500 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1501 matrix with @math{m} rows and @math{n} columns are accessed with two
1502 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1503 second one in the range 0@dots{}@math{n-1}.
1505 There are a couple of ways to construct matrices, with or without preset
1508 @cindex @code{lst_to_matrix()}
1509 @cindex @code{diag_matrix()}
1510 @cindex @code{unit_matrix()}
1511 @cindex @code{symbolic_matrix()}
1513 matrix::matrix(unsigned r, unsigned c);
1514 matrix::matrix(unsigned r, unsigned c, const lst & l);
1515 ex lst_to_matrix(const lst & l);
1516 ex diag_matrix(const lst & l);
1517 ex unit_matrix(unsigned x);
1518 ex unit_matrix(unsigned r, unsigned c);
1519 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1520 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1523 The first two functions are @code{matrix} constructors which create a matrix
1524 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1525 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1526 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1527 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1528 constructs a diagonal matrix given the list of diagonal elements.
1529 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1530 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1531 with newly generated symbols made of the specified base name and the
1532 position of each element in the matrix.
1534 Matrix elements can be accessed and set using the parenthesis (function call)
1538 const ex & matrix::operator()(unsigned r, unsigned c) const;
1539 ex & matrix::operator()(unsigned r, unsigned c);
1542 It is also possible to access the matrix elements in a linear fashion with
1543 the @code{op()} method. But C++-style subscripting with square brackets
1544 @samp{[]} is not available.
1546 Here are a couple of examples of constructing matrices:
1550 symbol a("a"), b("b");
1558 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1561 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1564 cout << diag_matrix(lst(a, b)) << endl;
1567 cout << unit_matrix(3) << endl;
1568 // -> [[1,0,0],[0,1,0],[0,0,1]]
1570 cout << symbolic_matrix(2, 3, "x") << endl;
1571 // -> [[x00,x01,x02],[x10,x11,x12]]
1575 @cindex @code{transpose()}
1576 There are three ways to do arithmetic with matrices. The first (and most
1577 direct one) is to use the methods provided by the @code{matrix} class:
1580 matrix matrix::add(const matrix & other) const;
1581 matrix matrix::sub(const matrix & other) const;
1582 matrix matrix::mul(const matrix & other) const;
1583 matrix matrix::mul_scalar(const ex & other) const;
1584 matrix matrix::pow(const ex & expn) const;
1585 matrix matrix::transpose() const;
1588 All of these methods return the result as a new matrix object. Here is an
1589 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1594 matrix A(2, 2, lst(1, 2, 3, 4));
1595 matrix B(2, 2, lst(-1, 0, 2, 1));
1596 matrix C(2, 2, lst(8, 4, 2, 1));
1598 matrix result = A.mul(B).sub(C.mul_scalar(2));
1599 cout << result << endl;
1600 // -> [[-13,-6],[1,2]]
1605 @cindex @code{evalm()}
1606 The second (and probably the most natural) way is to construct an expression
1607 containing matrices with the usual arithmetic operators and @code{pow()}.
1608 For efficiency reasons, expressions with sums, products and powers of
1609 matrices are not automatically evaluated in GiNaC. You have to call the
1613 ex ex::evalm() const;
1616 to obtain the result:
1623 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1624 cout << e.evalm() << endl;
1625 // -> [[-13,-6],[1,2]]
1630 The non-commutativity of the product @code{A*B} in this example is
1631 automatically recognized by GiNaC. There is no need to use a special
1632 operator here. @xref{Non-commutative objects}, for more information about
1633 dealing with non-commutative expressions.
1635 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1636 to perform the arithmetic:
1641 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1642 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1644 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1645 cout << e.simplify_indexed() << endl;
1646 // -> [[-13,-6],[1,2]].i.j
1650 Using indices is most useful when working with rectangular matrices and
1651 one-dimensional vectors because you don't have to worry about having to
1652 transpose matrices before multiplying them. @xref{Indexed objects}, for
1653 more information about using matrices with indices, and about indices in
1656 The @code{matrix} class provides a couple of additional methods for
1657 computing determinants, traces, and characteristic polynomials:
1659 @cindex @code{determinant()}
1660 @cindex @code{trace()}
1661 @cindex @code{charpoly()}
1663 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1664 ex matrix::trace() const;
1665 ex matrix::charpoly(const symbol & lambda) const;
1668 The @samp{algo} argument of @code{determinant()} allows to select
1669 between different algorithms for calculating the determinant. The
1670 asymptotic speed (as parametrized by the matrix size) can greatly differ
1671 between those algorithms, depending on the nature of the matrix'
1672 entries. The possible values are defined in the @file{flags.h} header
1673 file. By default, GiNaC uses a heuristic to automatically select an
1674 algorithm that is likely (but not guaranteed) to give the result most
1677 @cindex @code{inverse()}
1678 @cindex @code{solve()}
1679 Matrices may also be inverted using the @code{ex matrix::inverse()}
1680 method and linear systems may be solved with:
1683 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1686 Assuming the matrix object this method is applied on is an @code{m}
1687 times @code{n} matrix, then @code{vars} must be a @code{n} times
1688 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1689 times @code{p} matrix. The returned matrix then has dimension @code{n}
1690 times @code{p} and in the case of an underdetermined system will still
1691 contain some of the indeterminates from @code{vars}. If the system is
1692 overdetermined, an exception is thrown.
1695 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1696 @c node-name, next, previous, up
1697 @section Indexed objects
1699 GiNaC allows you to handle expressions containing general indexed objects in
1700 arbitrary spaces. It is also able to canonicalize and simplify such
1701 expressions and perform symbolic dummy index summations. There are a number
1702 of predefined indexed objects provided, like delta and metric tensors.
1704 There are few restrictions placed on indexed objects and their indices and
1705 it is easy to construct nonsense expressions, but our intention is to
1706 provide a general framework that allows you to implement algorithms with
1707 indexed quantities, getting in the way as little as possible.
1709 @cindex @code{idx} (class)
1710 @cindex @code{indexed} (class)
1711 @subsection Indexed quantities and their indices
1713 Indexed expressions in GiNaC are constructed of two special types of objects,
1714 @dfn{index objects} and @dfn{indexed objects}.
1718 @cindex contravariant
1721 @item Index objects are of class @code{idx} or a subclass. Every index has
1722 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1723 the index lives in) which can both be arbitrary expressions but are usually
1724 a number or a simple symbol. In addition, indices of class @code{varidx} have
1725 a @dfn{variance} (they can be co- or contravariant), and indices of class
1726 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1728 @item Indexed objects are of class @code{indexed} or a subclass. They
1729 contain a @dfn{base expression} (which is the expression being indexed), and
1730 one or more indices.
1734 @strong{Note:} when printing expressions, covariant indices and indices
1735 without variance are denoted @samp{.i} while contravariant indices are
1736 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1737 value. In the following, we are going to use that notation in the text so
1738 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1739 not visible in the output.
1741 A simple example shall illustrate the concepts:
1745 #include <ginac/ginac.h>
1746 using namespace std;
1747 using namespace GiNaC;
1751 symbol i_sym("i"), j_sym("j");
1752 idx i(i_sym, 3), j(j_sym, 3);
1755 cout << indexed(A, i, j) << endl;
1757 cout << index_dimensions << indexed(A, i, j) << endl;
1759 cout << dflt; // reset cout to default output format (dimensions hidden)
1763 The @code{idx} constructor takes two arguments, the index value and the
1764 index dimension. First we define two index objects, @code{i} and @code{j},
1765 both with the numeric dimension 3. The value of the index @code{i} is the
1766 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1767 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1768 construct an expression containing one indexed object, @samp{A.i.j}. It has
1769 the symbol @code{A} as its base expression and the two indices @code{i} and
1772 The dimensions of indices are normally not visible in the output, but one
1773 can request them to be printed with the @code{index_dimensions} manipulator,
1776 Note the difference between the indices @code{i} and @code{j} which are of
1777 class @code{idx}, and the index values which are the symbols @code{i_sym}
1778 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1779 or numbers but must be index objects. For example, the following is not
1780 correct and will raise an exception:
1783 symbol i("i"), j("j");
1784 e = indexed(A, i, j); // ERROR: indices must be of type idx
1787 You can have multiple indexed objects in an expression, index values can
1788 be numeric, and index dimensions symbolic:
1792 symbol B("B"), dim("dim");
1793 cout << 4 * indexed(A, i)
1794 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1799 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1800 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1801 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1802 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1803 @code{simplify_indexed()} for that, see below).
1805 In fact, base expressions, index values and index dimensions can be
1806 arbitrary expressions:
1810 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1815 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1816 get an error message from this but you will probably not be able to do
1817 anything useful with it.
1819 @cindex @code{get_value()}
1820 @cindex @code{get_dimension()}
1824 ex idx::get_value();
1825 ex idx::get_dimension();
1828 return the value and dimension of an @code{idx} object. If you have an index
1829 in an expression, such as returned by calling @code{.op()} on an indexed
1830 object, you can get a reference to the @code{idx} object with the function
1831 @code{ex_to<idx>()} on the expression.
1833 There are also the methods
1836 bool idx::is_numeric();
1837 bool idx::is_symbolic();
1838 bool idx::is_dim_numeric();
1839 bool idx::is_dim_symbolic();
1842 for checking whether the value and dimension are numeric or symbolic
1843 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1844 About Expressions}) returns information about the index value.
1846 @cindex @code{varidx} (class)
1847 If you need co- and contravariant indices, use the @code{varidx} class:
1851 symbol mu_sym("mu"), nu_sym("nu");
1852 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1853 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1855 cout << indexed(A, mu, nu) << endl;
1857 cout << indexed(A, mu_co, nu) << endl;
1859 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1864 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1865 co- or contravariant. The default is a contravariant (upper) index, but
1866 this can be overridden by supplying a third argument to the @code{varidx}
1867 constructor. The two methods
1870 bool varidx::is_covariant();
1871 bool varidx::is_contravariant();
1874 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1875 to get the object reference from an expression). There's also the very useful
1879 ex varidx::toggle_variance();
1882 which makes a new index with the same value and dimension but the opposite
1883 variance. By using it you only have to define the index once.
1885 @cindex @code{spinidx} (class)
1886 The @code{spinidx} class provides dotted and undotted variant indices, as
1887 used in the Weyl-van-der-Waerden spinor formalism:
1891 symbol K("K"), C_sym("C"), D_sym("D");
1892 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1893 // contravariant, undotted
1894 spinidx C_co(C_sym, 2, true); // covariant index
1895 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1896 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1898 cout << indexed(K, C, D) << endl;
1900 cout << indexed(K, C_co, D_dot) << endl;
1902 cout << indexed(K, D_co_dot, D) << endl;
1907 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1908 dotted or undotted. The default is undotted but this can be overridden by
1909 supplying a fourth argument to the @code{spinidx} constructor. The two
1913 bool spinidx::is_dotted();
1914 bool spinidx::is_undotted();
1917 allow you to check whether or not a @code{spinidx} object is dotted (use
1918 @code{ex_to<spinidx>()} to get the object reference from an expression).
1919 Finally, the two methods
1922 ex spinidx::toggle_dot();
1923 ex spinidx::toggle_variance_dot();
1926 create a new index with the same value and dimension but opposite dottedness
1927 and the same or opposite variance.
1929 @subsection Substituting indices
1931 @cindex @code{subs()}
1932 Sometimes you will want to substitute one symbolic index with another
1933 symbolic or numeric index, for example when calculating one specific element
1934 of a tensor expression. This is done with the @code{.subs()} method, as it
1935 is done for symbols (see @ref{Substituting Expressions}).
1937 You have two possibilities here. You can either substitute the whole index
1938 by another index or expression:
1942 ex e = indexed(A, mu_co);
1943 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1944 // -> A.mu becomes A~nu
1945 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1946 // -> A.mu becomes A~0
1947 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1948 // -> A.mu becomes A.0
1952 The third example shows that trying to replace an index with something that
1953 is not an index will substitute the index value instead.
1955 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1960 ex e = indexed(A, mu_co);
1961 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1962 // -> A.mu becomes A.nu
1963 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1964 // -> A.mu becomes A.0
1968 As you see, with the second method only the value of the index will get
1969 substituted. Its other properties, including its dimension, remain unchanged.
1970 If you want to change the dimension of an index you have to substitute the
1971 whole index by another one with the new dimension.
1973 Finally, substituting the base expression of an indexed object works as
1978 ex e = indexed(A, mu_co);
1979 cout << e << " becomes " << e.subs(A == A+B) << endl;
1980 // -> A.mu becomes (B+A).mu
1984 @subsection Symmetries
1985 @cindex @code{symmetry} (class)
1986 @cindex @code{sy_none()}
1987 @cindex @code{sy_symm()}
1988 @cindex @code{sy_anti()}
1989 @cindex @code{sy_cycl()}
1991 Indexed objects can have certain symmetry properties with respect to their
1992 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1993 that is constructed with the helper functions
1996 symmetry sy_none(...);
1997 symmetry sy_symm(...);
1998 symmetry sy_anti(...);
1999 symmetry sy_cycl(...);
2002 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2003 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2004 represents a cyclic symmetry. Each of these functions accepts up to four
2005 arguments which can be either symmetry objects themselves or unsigned integer
2006 numbers that represent an index position (counting from 0). A symmetry
2007 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2008 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2011 Here are some examples of symmetry definitions:
2016 e = indexed(A, i, j);
2017 e = indexed(A, sy_none(), i, j); // equivalent
2018 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2020 // Symmetric in all three indices:
2021 e = indexed(A, sy_symm(), i, j, k);
2022 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2023 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2024 // different canonical order
2026 // Symmetric in the first two indices only:
2027 e = indexed(A, sy_symm(0, 1), i, j, k);
2028 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2030 // Antisymmetric in the first and last index only (index ranges need not
2032 e = indexed(A, sy_anti(0, 2), i, j, k);
2033 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2035 // An example of a mixed symmetry: antisymmetric in the first two and
2036 // last two indices, symmetric when swapping the first and last index
2037 // pairs (like the Riemann curvature tensor):
2038 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2040 // Cyclic symmetry in all three indices:
2041 e = indexed(A, sy_cycl(), i, j, k);
2042 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2044 // The following examples are invalid constructions that will throw
2045 // an exception at run time.
2047 // An index may not appear multiple times:
2048 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2049 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2051 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2052 // same number of indices:
2053 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2055 // And of course, you cannot specify indices which are not there:
2056 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2060 If you need to specify more than four indices, you have to use the
2061 @code{.add()} method of the @code{symmetry} class. For example, to specify
2062 full symmetry in the first six indices you would write
2063 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2065 If an indexed object has a symmetry, GiNaC will automatically bring the
2066 indices into a canonical order which allows for some immediate simplifications:
2070 cout << indexed(A, sy_symm(), i, j)
2071 + indexed(A, sy_symm(), j, i) << endl;
2073 cout << indexed(B, sy_anti(), i, j)
2074 + indexed(B, sy_anti(), j, i) << endl;
2076 cout << indexed(B, sy_anti(), i, j, k)
2077 - indexed(B, sy_anti(), j, k, i) << endl;
2082 @cindex @code{get_free_indices()}
2084 @subsection Dummy indices
2086 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2087 that a summation over the index range is implied. Symbolic indices which are
2088 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2089 dummy nor free indices.
2091 To be recognized as a dummy index pair, the two indices must be of the same
2092 class and their value must be the same single symbol (an index like
2093 @samp{2*n+1} is never a dummy index). If the indices are of class
2094 @code{varidx} they must also be of opposite variance; if they are of class
2095 @code{spinidx} they must be both dotted or both undotted.
2097 The method @code{.get_free_indices()} returns a vector containing the free
2098 indices of an expression. It also checks that the free indices of the terms
2099 of a sum are consistent:
2103 symbol A("A"), B("B"), C("C");
2105 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2106 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2108 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2109 cout << exprseq(e.get_free_indices()) << endl;
2111 // 'j' and 'l' are dummy indices
2113 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2114 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2116 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2117 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2118 cout << exprseq(e.get_free_indices()) << endl;
2120 // 'nu' is a dummy index, but 'sigma' is not
2122 e = indexed(A, mu, mu);
2123 cout << exprseq(e.get_free_indices()) << endl;
2125 // 'mu' is not a dummy index because it appears twice with the same
2128 e = indexed(A, mu, nu) + 42;
2129 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2130 // this will throw an exception:
2131 // "add::get_free_indices: inconsistent indices in sum"
2135 @cindex @code{simplify_indexed()}
2136 @subsection Simplifying indexed expressions
2138 In addition to the few automatic simplifications that GiNaC performs on
2139 indexed expressions (such as re-ordering the indices of symmetric tensors
2140 and calculating traces and convolutions of matrices and predefined tensors)
2144 ex ex::simplify_indexed();
2145 ex ex::simplify_indexed(const scalar_products & sp);
2148 that performs some more expensive operations:
2151 @item it checks the consistency of free indices in sums in the same way
2152 @code{get_free_indices()} does
2153 @item it tries to give dummy indices that appear in different terms of a sum
2154 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2155 @item it (symbolically) calculates all possible dummy index summations/contractions
2156 with the predefined tensors (this will be explained in more detail in the
2158 @item it detects contractions that vanish for symmetry reasons, for example
2159 the contraction of a symmetric and a totally antisymmetric tensor
2160 @item as a special case of dummy index summation, it can replace scalar products
2161 of two tensors with a user-defined value
2164 The last point is done with the help of the @code{scalar_products} class
2165 which is used to store scalar products with known values (this is not an
2166 arithmetic class, you just pass it to @code{simplify_indexed()}):
2170 symbol A("A"), B("B"), C("C"), i_sym("i");
2174 sp.add(A, B, 0); // A and B are orthogonal
2175 sp.add(A, C, 0); // A and C are orthogonal
2176 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2178 e = indexed(A + B, i) * indexed(A + C, i);
2180 // -> (B+A).i*(A+C).i
2182 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2188 The @code{scalar_products} object @code{sp} acts as a storage for the
2189 scalar products added to it with the @code{.add()} method. This method
2190 takes three arguments: the two expressions of which the scalar product is
2191 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2192 @code{simplify_indexed()} will replace all scalar products of indexed
2193 objects that have the symbols @code{A} and @code{B} as base expressions
2194 with the single value 0. The number, type and dimension of the indices
2195 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2197 @cindex @code{expand()}
2198 The example above also illustrates a feature of the @code{expand()} method:
2199 if passed the @code{expand_indexed} option it will distribute indices
2200 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2202 @cindex @code{tensor} (class)
2203 @subsection Predefined tensors
2205 Some frequently used special tensors such as the delta, epsilon and metric
2206 tensors are predefined in GiNaC. They have special properties when
2207 contracted with other tensor expressions and some of them have constant
2208 matrix representations (they will evaluate to a number when numeric
2209 indices are specified).
2211 @cindex @code{delta_tensor()}
2212 @subsubsection Delta tensor
2214 The delta tensor takes two indices, is symmetric and has the matrix
2215 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2216 @code{delta_tensor()}:
2220 symbol A("A"), B("B");
2222 idx i(symbol("i"), 3), j(symbol("j"), 3),
2223 k(symbol("k"), 3), l(symbol("l"), 3);
2225 ex e = indexed(A, i, j) * indexed(B, k, l)
2226 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2227 cout << e.simplify_indexed() << endl;
2230 cout << delta_tensor(i, i) << endl;
2235 @cindex @code{metric_tensor()}
2236 @subsubsection General metric tensor
2238 The function @code{metric_tensor()} creates a general symmetric metric
2239 tensor with two indices that can be used to raise/lower tensor indices. The
2240 metric tensor is denoted as @samp{g} in the output and if its indices are of
2241 mixed variance it is automatically replaced by a delta tensor:
2247 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2249 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2250 cout << e.simplify_indexed() << endl;
2253 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2254 cout << e.simplify_indexed() << endl;
2257 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2258 * metric_tensor(nu, rho);
2259 cout << e.simplify_indexed() << endl;
2262 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2263 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2264 + indexed(A, mu.toggle_variance(), rho));
2265 cout << e.simplify_indexed() << endl;
2270 @cindex @code{lorentz_g()}
2271 @subsubsection Minkowski metric tensor
2273 The Minkowski metric tensor is a special metric tensor with a constant
2274 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2275 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2276 It is created with the function @code{lorentz_g()} (although it is output as
2281 varidx mu(symbol("mu"), 4);
2283 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2284 * lorentz_g(mu, varidx(0, 4)); // negative signature
2285 cout << e.simplify_indexed() << endl;
2288 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2289 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2290 cout << e.simplify_indexed() << endl;
2295 @cindex @code{spinor_metric()}
2296 @subsubsection Spinor metric tensor
2298 The function @code{spinor_metric()} creates an antisymmetric tensor with
2299 two indices that is used to raise/lower indices of 2-component spinors.
2300 It is output as @samp{eps}:
2306 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2307 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2309 e = spinor_metric(A, B) * indexed(psi, B_co);
2310 cout << e.simplify_indexed() << endl;
2313 e = spinor_metric(A, B) * indexed(psi, A_co);
2314 cout << e.simplify_indexed() << endl;
2317 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2318 cout << e.simplify_indexed() << endl;
2321 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2322 cout << e.simplify_indexed() << endl;
2325 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2326 cout << e.simplify_indexed() << endl;
2329 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2330 cout << e.simplify_indexed() << endl;
2335 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2337 @cindex @code{epsilon_tensor()}
2338 @cindex @code{lorentz_eps()}
2339 @subsubsection Epsilon tensor
2341 The epsilon tensor is totally antisymmetric, its number of indices is equal
2342 to the dimension of the index space (the indices must all be of the same
2343 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2344 defined to be 1. Its behavior with indices that have a variance also
2345 depends on the signature of the metric. Epsilon tensors are output as
2348 There are three functions defined to create epsilon tensors in 2, 3 and 4
2352 ex epsilon_tensor(const ex & i1, const ex & i2);
2353 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2354 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2357 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2358 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2359 Minkowski space (the last @code{bool} argument specifies whether the metric
2360 has negative or positive signature, as in the case of the Minkowski metric
2365 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2366 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2367 e = lorentz_eps(mu, nu, rho, sig) *
2368 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2369 cout << simplify_indexed(e) << endl;
2370 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2372 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2373 symbol A("A"), B("B");
2374 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2375 cout << simplify_indexed(e) << endl;
2376 // -> -B.k*A.j*eps.i.k.j
2377 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2378 cout << simplify_indexed(e) << endl;
2383 @subsection Linear algebra
2385 The @code{matrix} class can be used with indices to do some simple linear
2386 algebra (linear combinations and products of vectors and matrices, traces
2387 and scalar products):
2391 idx i(symbol("i"), 2), j(symbol("j"), 2);
2392 symbol x("x"), y("y");
2394 // A is a 2x2 matrix, X is a 2x1 vector
2395 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2397 cout << indexed(A, i, i) << endl;
2400 ex e = indexed(A, i, j) * indexed(X, j);
2401 cout << e.simplify_indexed() << endl;
2402 // -> [[2*y+x],[4*y+3*x]].i
2404 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2405 cout << e.simplify_indexed() << endl;
2406 // -> [[3*y+3*x,6*y+2*x]].j
2410 You can of course obtain the same results with the @code{matrix::add()},
2411 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2412 but with indices you don't have to worry about transposing matrices.
2414 Matrix indices always start at 0 and their dimension must match the number
2415 of rows/columns of the matrix. Matrices with one row or one column are
2416 vectors and can have one or two indices (it doesn't matter whether it's a
2417 row or a column vector). Other matrices must have two indices.
2419 You should be careful when using indices with variance on matrices. GiNaC
2420 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2421 @samp{F.mu.nu} are different matrices. In this case you should use only
2422 one form for @samp{F} and explicitly multiply it with a matrix representation
2423 of the metric tensor.
2426 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2427 @c node-name, next, previous, up
2428 @section Non-commutative objects
2430 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2431 non-commutative objects are built-in which are mostly of use in high energy
2435 @item Clifford (Dirac) algebra (class @code{clifford})
2436 @item su(3) Lie algebra (class @code{color})
2437 @item Matrices (unindexed) (class @code{matrix})
2440 The @code{clifford} and @code{color} classes are subclasses of
2441 @code{indexed} because the elements of these algebras usually carry
2442 indices. The @code{matrix} class is described in more detail in
2445 Unlike most computer algebra systems, GiNaC does not primarily provide an
2446 operator (often denoted @samp{&*}) for representing inert products of
2447 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2448 classes of objects involved, and non-commutative products are formed with
2449 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2450 figuring out by itself which objects commute and will group the factors
2451 by their class. Consider this example:
2455 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2456 idx a(symbol("a"), 8), b(symbol("b"), 8);
2457 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2459 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2463 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2464 groups the non-commutative factors (the gammas and the su(3) generators)
2465 together while preserving the order of factors within each class (because
2466 Clifford objects commute with color objects). The resulting expression is a
2467 @emph{commutative} product with two factors that are themselves non-commutative
2468 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2469 parentheses are placed around the non-commutative products in the output.
2471 @cindex @code{ncmul} (class)
2472 Non-commutative products are internally represented by objects of the class
2473 @code{ncmul}, as opposed to commutative products which are handled by the
2474 @code{mul} class. You will normally not have to worry about this distinction,
2477 The advantage of this approach is that you never have to worry about using
2478 (or forgetting to use) a special operator when constructing non-commutative
2479 expressions. Also, non-commutative products in GiNaC are more intelligent
2480 than in other computer algebra systems; they can, for example, automatically
2481 canonicalize themselves according to rules specified in the implementation
2482 of the non-commutative classes. The drawback is that to work with other than
2483 the built-in algebras you have to implement new classes yourself. Symbols
2484 always commute and it's not possible to construct non-commutative products
2485 using symbols to represent the algebra elements or generators. User-defined
2486 functions can, however, be specified as being non-commutative.
2488 @cindex @code{return_type()}
2489 @cindex @code{return_type_tinfo()}
2490 Information about the commutativity of an object or expression can be
2491 obtained with the two member functions
2494 unsigned ex::return_type() const;
2495 unsigned ex::return_type_tinfo() const;
2498 The @code{return_type()} function returns one of three values (defined in
2499 the header file @file{flags.h}), corresponding to three categories of
2500 expressions in GiNaC:
2503 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2504 classes are of this kind.
2505 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2506 certain class of non-commutative objects which can be determined with the
2507 @code{return_type_tinfo()} method. Expressions of this category commute
2508 with everything except @code{noncommutative} expressions of the same
2510 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2511 of non-commutative objects of different classes. Expressions of this
2512 category don't commute with any other @code{noncommutative} or
2513 @code{noncommutative_composite} expressions.
2516 The value returned by the @code{return_type_tinfo()} method is valid only
2517 when the return type of the expression is @code{noncommutative}. It is a
2518 value that is unique to the class of the object and usually one of the
2519 constants in @file{tinfos.h}, or derived therefrom.
2521 Here are a couple of examples:
2524 @multitable @columnfractions 0.33 0.33 0.34
2525 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2526 @item @code{42} @tab @code{commutative} @tab -
2527 @item @code{2*x-y} @tab @code{commutative} @tab -
2528 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2529 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2530 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2531 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2535 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2536 @code{TINFO_clifford} for objects with a representation label of zero.
2537 Other representation labels yield a different @code{return_type_tinfo()},
2538 but it's the same for any two objects with the same label. This is also true
2541 A last note: With the exception of matrices, positive integer powers of
2542 non-commutative objects are automatically expanded in GiNaC. For example,
2543 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2544 non-commutative expressions).
2547 @cindex @code{clifford} (class)
2548 @subsection Clifford algebra
2550 @cindex @code{dirac_gamma()}
2551 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2552 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2553 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2554 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2557 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2560 which takes two arguments: the index and a @dfn{representation label} in the
2561 range 0 to 255 which is used to distinguish elements of different Clifford
2562 algebras (this is also called a @dfn{spin line index}). Gammas with different
2563 labels commute with each other. The dimension of the index can be 4 or (in
2564 the framework of dimensional regularization) any symbolic value. Spinor
2565 indices on Dirac gammas are not supported in GiNaC.
2567 @cindex @code{dirac_ONE()}
2568 The unity element of a Clifford algebra is constructed by
2571 ex dirac_ONE(unsigned char rl = 0);
2574 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2575 multiples of the unity element, even though it's customary to omit it.
2576 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2577 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2578 GiNaC will complain and/or produce incorrect results.
2580 @cindex @code{dirac_gamma5()}
2581 There is a special element @samp{gamma5} that commutes with all other
2582 gammas, has a unit square, and in 4 dimensions equals
2583 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2586 ex dirac_gamma5(unsigned char rl = 0);
2589 @cindex @code{dirac_gammaL()}
2590 @cindex @code{dirac_gammaR()}
2591 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2592 objects, constructed by
2595 ex dirac_gammaL(unsigned char rl = 0);
2596 ex dirac_gammaR(unsigned char rl = 0);
2599 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2600 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2602 @cindex @code{dirac_slash()}
2603 Finally, the function
2606 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2609 creates a term that represents a contraction of @samp{e} with the Dirac
2610 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2611 with a unique index whose dimension is given by the @code{dim} argument).
2612 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2614 In products of dirac gammas, superfluous unity elements are automatically
2615 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2616 and @samp{gammaR} are moved to the front.
2618 The @code{simplify_indexed()} function performs contractions in gamma strings,
2624 symbol a("a"), b("b"), D("D");
2625 varidx mu(symbol("mu"), D);
2626 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2627 * dirac_gamma(mu.toggle_variance());
2629 // -> gamma~mu*a\*gamma.mu
2630 e = e.simplify_indexed();
2633 cout << e.subs(D == 4) << endl;
2639 @cindex @code{dirac_trace()}
2640 To calculate the trace of an expression containing strings of Dirac gammas
2641 you use the function
2644 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2647 This function takes the trace of all gammas with the specified representation
2648 label; gammas with other labels are left standing. The last argument to
2649 @code{dirac_trace()} is the value to be returned for the trace of the unity
2650 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2651 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2652 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2653 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2654 This @samp{gamma5} scheme is described in greater detail in
2655 @cite{The Role of gamma5 in Dimensional Regularization}.
2657 The value of the trace itself is also usually different in 4 and in
2658 @math{D != 4} dimensions:
2663 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2664 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2665 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2666 cout << dirac_trace(e).simplify_indexed() << endl;
2673 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2674 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2675 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2676 cout << dirac_trace(e).simplify_indexed() << endl;
2677 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2681 Here is an example for using @code{dirac_trace()} to compute a value that
2682 appears in the calculation of the one-loop vacuum polarization amplitude in
2687 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2688 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2691 sp.add(l, l, pow(l, 2));
2692 sp.add(l, q, ldotq);
2694 ex e = dirac_gamma(mu) *
2695 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2696 dirac_gamma(mu.toggle_variance()) *
2697 (dirac_slash(l, D) + m * dirac_ONE());
2698 e = dirac_trace(e).simplify_indexed(sp);
2699 e = e.collect(lst(l, ldotq, m));
2701 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2705 The @code{canonicalize_clifford()} function reorders all gamma products that
2706 appear in an expression to a canonical (but not necessarily simple) form.
2707 You can use this to compare two expressions or for further simplifications:
2711 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2712 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2714 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2716 e = canonicalize_clifford(e);
2723 @cindex @code{color} (class)
2724 @subsection Color algebra
2726 @cindex @code{color_T()}
2727 For computations in quantum chromodynamics, GiNaC implements the base elements
2728 and structure constants of the su(3) Lie algebra (color algebra). The base
2729 elements @math{T_a} are constructed by the function
2732 ex color_T(const ex & a, unsigned char rl = 0);
2735 which takes two arguments: the index and a @dfn{representation label} in the
2736 range 0 to 255 which is used to distinguish elements of different color
2737 algebras. Objects with different labels commute with each other. The
2738 dimension of the index must be exactly 8 and it should be of class @code{idx},
2741 @cindex @code{color_ONE()}
2742 The unity element of a color algebra is constructed by
2745 ex color_ONE(unsigned char rl = 0);
2748 @strong{Note:} You must always use @code{color_ONE()} when referring to
2749 multiples of the unity element, even though it's customary to omit it.
2750 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2751 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2752 GiNaC may produce incorrect results.
2754 @cindex @code{color_d()}
2755 @cindex @code{color_f()}
2759 ex color_d(const ex & a, const ex & b, const ex & c);
2760 ex color_f(const ex & a, const ex & b, const ex & c);
2763 create the symmetric and antisymmetric structure constants @math{d_abc} and
2764 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2765 and @math{[T_a, T_b] = i f_abc T_c}.
2767 @cindex @code{color_h()}
2768 There's an additional function
2771 ex color_h(const ex & a, const ex & b, const ex & c);
2774 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2776 The function @code{simplify_indexed()} performs some simplifications on
2777 expressions containing color objects:
2782 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2783 k(symbol("k"), 8), l(symbol("l"), 8);
2785 e = color_d(a, b, l) * color_f(a, b, k);
2786 cout << e.simplify_indexed() << endl;
2789 e = color_d(a, b, l) * color_d(a, b, k);
2790 cout << e.simplify_indexed() << endl;
2793 e = color_f(l, a, b) * color_f(a, b, k);
2794 cout << e.simplify_indexed() << endl;
2797 e = color_h(a, b, c) * color_h(a, b, c);
2798 cout << e.simplify_indexed() << endl;
2801 e = color_h(a, b, c) * color_T(b) * color_T(c);
2802 cout << e.simplify_indexed() << endl;
2805 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2806 cout << e.simplify_indexed() << endl;
2809 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2810 cout << e.simplify_indexed() << endl;
2811 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2815 @cindex @code{color_trace()}
2816 To calculate the trace of an expression containing color objects you use the
2820 ex color_trace(const ex & e, unsigned char rl = 0);
2823 This function takes the trace of all color @samp{T} objects with the
2824 specified representation label; @samp{T}s with other labels are left
2825 standing. For example:
2829 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2831 // -> -I*f.a.c.b+d.a.c.b
2836 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2837 @c node-name, next, previous, up
2838 @chapter Methods and Functions
2841 In this chapter the most important algorithms provided by GiNaC will be
2842 described. Some of them are implemented as functions on expressions,
2843 others are implemented as methods provided by expression objects. If
2844 they are methods, there exists a wrapper function around it, so you can
2845 alternatively call it in a functional way as shown in the simple
2850 cout << "As method: " << sin(1).evalf() << endl;
2851 cout << "As function: " << evalf(sin(1)) << endl;
2855 @cindex @code{subs()}
2856 The general rule is that wherever methods accept one or more parameters
2857 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2858 wrapper accepts is the same but preceded by the object to act on
2859 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2860 most natural one in an OO model but it may lead to confusion for MapleV
2861 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2862 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2863 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2864 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2865 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2866 here. Also, users of MuPAD will in most cases feel more comfortable
2867 with GiNaC's convention. All function wrappers are implemented
2868 as simple inline functions which just call the corresponding method and
2869 are only provided for users uncomfortable with OO who are dead set to
2870 avoid method invocations. Generally, nested function wrappers are much
2871 harder to read than a sequence of methods and should therefore be
2872 avoided if possible. On the other hand, not everything in GiNaC is a
2873 method on class @code{ex} and sometimes calling a function cannot be
2877 * Information About Expressions::
2878 * Substituting Expressions::
2879 * Pattern Matching and Advanced Substitutions::
2880 * Applying a Function on Subexpressions::
2881 * Visitors and Tree Traversal::
2882 * Polynomial Arithmetic:: Working with polynomials.
2883 * Rational Expressions:: Working with rational functions.
2884 * Symbolic Differentiation::
2885 * Series Expansion:: Taylor and Laurent expansion.
2887 * Built-in Functions:: List of predefined mathematical functions.
2888 * Solving Linear Systems of Equations::
2889 * Input/Output:: Input and output of expressions.
2893 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2894 @c node-name, next, previous, up
2895 @section Getting information about expressions
2897 @subsection Checking expression types
2898 @cindex @code{is_a<@dots{}>()}
2899 @cindex @code{is_exactly_a<@dots{}>()}
2900 @cindex @code{ex_to<@dots{}>()}
2901 @cindex Converting @code{ex} to other classes
2902 @cindex @code{info()}
2903 @cindex @code{return_type()}
2904 @cindex @code{return_type_tinfo()}
2906 Sometimes it's useful to check whether a given expression is a plain number,
2907 a sum, a polynomial with integer coefficients, or of some other specific type.
2908 GiNaC provides a couple of functions for this:
2911 bool is_a<T>(const ex & e);
2912 bool is_exactly_a<T>(const ex & e);
2913 bool ex::info(unsigned flag);
2914 unsigned ex::return_type() const;
2915 unsigned ex::return_type_tinfo() const;
2918 When the test made by @code{is_a<T>()} returns true, it is safe to call
2919 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2920 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2921 example, assuming @code{e} is an @code{ex}:
2926 if (is_a<numeric>(e))
2927 numeric n = ex_to<numeric>(e);
2932 @code{is_a<T>(e)} allows you to check whether the top-level object of
2933 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2934 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2935 e.g., for checking whether an expression is a number, a sum, or a product:
2942 is_a<numeric>(e1); // true
2943 is_a<numeric>(e2); // false
2944 is_a<add>(e1); // false
2945 is_a<add>(e2); // true
2946 is_a<mul>(e1); // false
2947 is_a<mul>(e2); // false
2951 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2952 top-level object of an expression @samp{e} is an instance of the GiNaC
2953 class @samp{T}, not including parent classes.
2955 The @code{info()} method is used for checking certain attributes of
2956 expressions. The possible values for the @code{flag} argument are defined
2957 in @file{ginac/flags.h}, the most important being explained in the following
2961 @multitable @columnfractions .30 .70
2962 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2963 @item @code{numeric}
2964 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2966 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2967 @item @code{rational}
2968 @tab @dots{}an exact rational number (integers are rational, too)
2969 @item @code{integer}
2970 @tab @dots{}a (non-complex) integer
2971 @item @code{crational}
2972 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2973 @item @code{cinteger}
2974 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2975 @item @code{positive}
2976 @tab @dots{}not complex and greater than 0
2977 @item @code{negative}
2978 @tab @dots{}not complex and less than 0
2979 @item @code{nonnegative}
2980 @tab @dots{}not complex and greater than or equal to 0
2982 @tab @dots{}an integer greater than 0
2984 @tab @dots{}an integer less than 0
2985 @item @code{nonnegint}
2986 @tab @dots{}an integer greater than or equal to 0
2988 @tab @dots{}an even integer
2990 @tab @dots{}an odd integer
2992 @tab @dots{}a prime integer (probabilistic primality test)
2993 @item @code{relation}
2994 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2995 @item @code{relation_equal}
2996 @tab @dots{}a @code{==} relation
2997 @item @code{relation_not_equal}
2998 @tab @dots{}a @code{!=} relation
2999 @item @code{relation_less}
3000 @tab @dots{}a @code{<} relation
3001 @item @code{relation_less_or_equal}
3002 @tab @dots{}a @code{<=} relation
3003 @item @code{relation_greater}
3004 @tab @dots{}a @code{>} relation
3005 @item @code{relation_greater_or_equal}
3006 @tab @dots{}a @code{>=} relation
3008 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3010 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3011 @item @code{polynomial}
3012 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3013 @item @code{integer_polynomial}
3014 @tab @dots{}a polynomial with (non-complex) integer coefficients
3015 @item @code{cinteger_polynomial}
3016 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3017 @item @code{rational_polynomial}
3018 @tab @dots{}a polynomial with (non-complex) rational coefficients
3019 @item @code{crational_polynomial}
3020 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3021 @item @code{rational_function}
3022 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3023 @item @code{algebraic}
3024 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3028 To determine whether an expression is commutative or non-commutative and if
3029 so, with which other expressions it would commute, you use the methods
3030 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3031 for an explanation of these.
3034 @subsection Accessing subexpressions
3035 @cindex @code{nops()}
3038 @cindex @code{relational} (class)
3040 GiNaC provides the two methods
3044 ex ex::op(size_t i);
3047 for accessing the subexpressions in the container-like GiNaC classes like
3048 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3049 determines the number of subexpressions (@samp{operands}) contained, while
3050 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3051 In the case of a @code{power} object, @code{op(0)} will return the basis
3052 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3053 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3055 The left-hand and right-hand side expressions of objects of class
3056 @code{relational} (and only of these) can also be accessed with the methods
3064 @subsection Comparing expressions
3065 @cindex @code{is_equal()}
3066 @cindex @code{is_zero()}
3068 Expressions can be compared with the usual C++ relational operators like
3069 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3070 the result is usually not determinable and the result will be @code{false},
3071 except in the case of the @code{!=} operator. You should also be aware that
3072 GiNaC will only do the most trivial test for equality (subtracting both
3073 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3076 Actually, if you construct an expression like @code{a == b}, this will be
3077 represented by an object of the @code{relational} class (@pxref{Relations})
3078 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3080 There are also two methods
3083 bool ex::is_equal(const ex & other);
3087 for checking whether one expression is equal to another, or equal to zero,
3090 @strong{Warning:} You will also find an @code{ex::compare()} method in the
3091 GiNaC header files. This method is however only to be used internally by
3092 GiNaC to establish a canonical sort order for terms, and using it to compare
3093 expressions will give very surprising results.
3096 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
3097 @c node-name, next, previous, up
3098 @section Substituting expressions
3099 @cindex @code{subs()}
3101 Algebraic objects inside expressions can be replaced with arbitrary
3102 expressions via the @code{.subs()} method:
3105 ex ex::subs(const ex & e);
3106 ex ex::subs(const lst & syms, const lst & repls);
3109 In the first form, @code{subs()} accepts a relational of the form
3110 @samp{object == expression} or a @code{lst} of such relationals:
3114 symbol x("x"), y("y");
3116 ex e1 = 2*x^2-4*x+3;
3117 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3121 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3126 If you specify multiple substitutions, they are performed in parallel, so e.g.
3127 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3129 The second form of @code{subs()} takes two lists, one for the objects to be
3130 replaced and one for the expressions to be substituted (both lists must
3131 contain the same number of elements). Using this form, you would write
3132 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
3134 @code{subs()} performs syntactic substitution of any complete algebraic
3135 object; it does not try to match sub-expressions as is demonstrated by the
3140 symbol x("x"), y("y"), z("z");
3142 ex e1 = pow(x+y, 2);
3143 cout << e1.subs(x+y == 4) << endl;
3146 ex e2 = sin(x)*sin(y)*cos(x);
3147 cout << e2.subs(sin(x) == cos(x)) << endl;
3148 // -> cos(x)^2*sin(y)
3151 cout << e3.subs(x+y == 4) << endl;
3153 // (and not 4+z as one might expect)
3157 A more powerful form of substitution using wildcards is described in the
3161 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3162 @c node-name, next, previous, up
3163 @section Pattern matching and advanced substitutions
3164 @cindex @code{wildcard} (class)
3165 @cindex Pattern matching
3167 GiNaC allows the use of patterns for checking whether an expression is of a
3168 certain form or contains subexpressions of a certain form, and for
3169 substituting expressions in a more general way.
3171 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3172 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3173 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3174 an unsigned integer number to allow having multiple different wildcards in a
3175 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3176 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3180 ex wild(unsigned label = 0);
3183 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3186 Some examples for patterns:
3188 @multitable @columnfractions .5 .5
3189 @item @strong{Constructed as} @tab @strong{Output as}
3190 @item @code{wild()} @tab @samp{$0}
3191 @item @code{pow(x,wild())} @tab @samp{x^$0}
3192 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3193 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3199 @item Wildcards behave like symbols and are subject to the same algebraic
3200 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3201 @item As shown in the last example, to use wildcards for indices you have to
3202 use them as the value of an @code{idx} object. This is because indices must
3203 always be of class @code{idx} (or a subclass).
3204 @item Wildcards only represent expressions or subexpressions. It is not
3205 possible to use them as placeholders for other properties like index
3206 dimension or variance, representation labels, symmetry of indexed objects
3208 @item Because wildcards are commutative, it is not possible to use wildcards
3209 as part of noncommutative products.
3210 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3211 are also valid patterns.
3214 @subsection Matching expressions
3215 @cindex @code{match()}
3216 The most basic application of patterns is to check whether an expression
3217 matches a given pattern. This is done by the function
3220 bool ex::match(const ex & pattern);
3221 bool ex::match(const ex & pattern, lst & repls);
3224 This function returns @code{true} when the expression matches the pattern
3225 and @code{false} if it doesn't. If used in the second form, the actual
3226 subexpressions matched by the wildcards get returned in the @code{repls}
3227 object as a list of relations of the form @samp{wildcard == expression}.
3228 If @code{match()} returns false, the state of @code{repls} is undefined.
3229 For reproducible results, the list should be empty when passed to
3230 @code{match()}, but it is also possible to find similarities in multiple
3231 expressions by passing in the result of a previous match.
3233 The matching algorithm works as follows:
3236 @item A single wildcard matches any expression. If one wildcard appears
3237 multiple times in a pattern, it must match the same expression in all
3238 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3239 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3240 @item If the expression is not of the same class as the pattern, the match
3241 fails (i.e. a sum only matches a sum, a function only matches a function,
3243 @item If the pattern is a function, it only matches the same function
3244 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3245 @item Except for sums and products, the match fails if the number of
3246 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3248 @item If there are no subexpressions, the expressions and the pattern must
3249 be equal (in the sense of @code{is_equal()}).
3250 @item Except for sums and products, each subexpression (@code{op()}) must
3251 match the corresponding subexpression of the pattern.
3254 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3255 account for their commutativity and associativity:
3258 @item If the pattern contains a term or factor that is a single wildcard,
3259 this one is used as the @dfn{global wildcard}. If there is more than one
3260 such wildcard, one of them is chosen as the global wildcard in a random
3262 @item Every term/factor of the pattern, except the global wildcard, is
3263 matched against every term of the expression in sequence. If no match is
3264 found, the whole match fails. Terms that did match are not considered in
3266 @item If there are no unmatched terms left, the match succeeds. Otherwise
3267 the match fails unless there is a global wildcard in the pattern, in
3268 which case this wildcard matches the remaining terms.
3271 In general, having more than one single wildcard as a term of a sum or a
3272 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3275 Here are some examples in @command{ginsh} to demonstrate how it works (the
3276 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3277 match fails, and the list of wildcard replacements otherwise):
3280 > match((x+y)^a,(x+y)^a);
3282 > match((x+y)^a,(x+y)^b);
3284 > match((x+y)^a,$1^$2);
3286 > match((x+y)^a,$1^$1);
3288 > match((x+y)^(x+y),$1^$1);
3290 > match((x+y)^(x+y),$1^$2);
3292 > match((a+b)*(a+c),($1+b)*($1+c));
3294 > match((a+b)*(a+c),(a+$1)*(a+$2));
3296 (Unpredictable. The result might also be [$1==c,$2==b].)
3297 > match((a+b)*(a+c),($1+$2)*($1+$3));
3298 (The result is undefined. Due to the sequential nature of the algorithm
3299 and the re-ordering of terms in GiNaC, the match for the first factor
3300 may be @{$1==a,$2==b@} in which case the match for the second factor
3301 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3303 > match(a*(x+y)+a*z+b,a*$1+$2);
3304 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3305 @{$1=x+y,$2=a*z+b@}.)
3306 > match(a+b+c+d+e+f,c);
3308 > match(a+b+c+d+e+f,c+$0);
3310 > match(a+b+c+d+e+f,c+e+$0);
3312 > match(a+b,a+b+$0);
3314 > match(a*b^2,a^$1*b^$2);
3316 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3317 even though a==a^1.)
3318 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3320 > match(atan2(y,x^2),atan2(y,$0));
3324 @subsection Matching parts of expressions
3325 @cindex @code{has()}
3326 A more general way to look for patterns in expressions is provided by the
3330 bool ex::has(const ex & pattern);
3333 This function checks whether a pattern is matched by an expression itself or
3334 by any of its subexpressions.
3336 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3337 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3340 > has(x*sin(x+y+2*a),y);
3342 > has(x*sin(x+y+2*a),x+y);
3344 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3345 has the subexpressions "x", "y" and "2*a".)
3346 > has(x*sin(x+y+2*a),x+y+$1);
3348 (But this is possible.)
3349 > has(x*sin(2*(x+y)+2*a),x+y);
3351 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3352 which "x+y" is not a subexpression.)
3355 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3357 > has(4*x^2-x+3,$1*x);
3359 > has(4*x^2+x+3,$1*x);
3361 (Another possible pitfall. The first expression matches because the term
3362 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3363 contains a linear term you should use the coeff() function instead.)
3366 @cindex @code{find()}
3370 bool ex::find(const ex & pattern, lst & found);
3373 works a bit like @code{has()} but it doesn't stop upon finding the first
3374 match. Instead, it appends all found matches to the specified list. If there
3375 are multiple occurrences of the same expression, it is entered only once to
3376 the list. @code{find()} returns false if no matches were found (in
3377 @command{ginsh}, it returns an empty list):
3380 > find(1+x+x^2+x^3,x);
3382 > find(1+x+x^2+x^3,y);
3384 > find(1+x+x^2+x^3,x^$1);
3386 (Note the absence of "x".)
3387 > expand((sin(x)+sin(y))*(a+b));
3388 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3393 @subsection Substituting expressions
3394 @cindex @code{subs()}
3395 Probably the most useful application of patterns is to use them for
3396 substituting expressions with the @code{subs()} method. Wildcards can be
3397 used in the search patterns as well as in the replacement expressions, where
3398 they get replaced by the expressions matched by them. @code{subs()} doesn't
3399 know anything about algebra; it performs purely syntactic substitutions.
3404 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3406 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3408 > subs((a+b+c)^2,a+b==x);
3410 > subs((a+b+c)^2,a+b+$1==x+$1);
3412 > subs(a+2*b,a+b==x);
3414 > subs(4*x^3-2*x^2+5*x-1,x==a);
3416 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3418 > subs(sin(1+sin(x)),sin($1)==cos($1));
3420 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3424 The last example would be written in C++ in this way:
3428 symbol a("a"), b("b"), x("x"), y("y");
3429 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3430 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3431 cout << e.expand() << endl;
3436 @subsection Algebraic substitutions
3437 The @code{subs()} method has an extra, optional, argument. This argument can
3438 be used to pass one of the @code{subs_options} to it. The only option that is
3439 currently available is the @code{subs_algebraic} option which affects
3440 products and powers. If you want to substitute some factors of a product, you
3441 only need to list these factors in your pattern. Furthermore, if an (integer)
3442 power of some expression occurs in your pattern and in the expression that you
3443 want the substitution to occur in, it can be substituted as many times as
3444 possible, without getting negative powers.
3446 An example clarifies it all (hopefully):
3449 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3450 subs_options::subs_algebraic) << endl;
3451 // --> (y+x)^6+b^6+a^6
3453 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::subs_algebraic) << endl;
3455 // Powers and products are smart, but addition is just the same.
3457 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::subs_algebraic)
3460 // As I said: addition is just the same.
3462 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::subs_algebraic) << endl;
3463 // --> x^3*b*a^2+2*b
3465 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::subs_algebraic)
3467 // --> 2*b+x^3*b^(-1)*a^(-2)
3469 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::subs_algebraic) << endl;
3470 // --> -1-2*a^2+4*a^3+5*a
3472 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3473 subs_options::subs_algebraic) << endl;
3474 // --> -1+5*x+4*x^3-2*x^2
3475 // You should not really need this kind of patterns very often now.
3476 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3478 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3479 subs_options::subs_algebraic) << endl;
3480 // --> cos(1+cos(x))
3482 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3483 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3484 subs_options::subs_algebraic)) << endl;
3489 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3490 @c node-name, next, previous, up
3491 @section Applying a Function on Subexpressions
3492 @cindex tree traversal
3493 @cindex @code{map()}
3495 Sometimes you may want to perform an operation on specific parts of an
3496 expression while leaving the general structure of it intact. An example
3497 of this would be a matrix trace operation: the trace of a sum is the sum
3498 of the traces of the individual terms. That is, the trace should @dfn{map}
3499 on the sum, by applying itself to each of the sum's operands. It is possible
3500 to do this manually which usually results in code like this:
3505 if (is_a<matrix>(e))
3506 return ex_to<matrix>(e).trace();
3507 else if (is_a<add>(e)) @{
3509 for (size_t i=0; i<e.nops(); i++)
3510 sum += calc_trace(e.op(i));
3512 @} else if (is_a<mul>)(e)) @{
3520 This is, however, slightly inefficient (if the sum is very large it can take
3521 a long time to add the terms one-by-one), and its applicability is limited to
3522 a rather small class of expressions. If @code{calc_trace()} is called with
3523 a relation or a list as its argument, you will probably want the trace to
3524 be taken on both sides of the relation or of all elements of the list.
3526 GiNaC offers the @code{map()} method to aid in the implementation of such
3530 ex ex::map(map_function & f) const;
3531 ex ex::map(ex (*f)(const ex & e)) const;
3534 In the first (preferred) form, @code{map()} takes a function object that
3535 is subclassed from the @code{map_function} class. In the second form, it
3536 takes a pointer to a function that accepts and returns an expression.
3537 @code{map()} constructs a new expression of the same type, applying the
3538 specified function on all subexpressions (in the sense of @code{op()}),
3541 The use of a function object makes it possible to supply more arguments to
3542 the function that is being mapped, or to keep local state information.
3543 The @code{map_function} class declares a virtual function call operator
3544 that you can overload. Here is a sample implementation of @code{calc_trace()}
3545 that uses @code{map()} in a recursive fashion:
3548 struct calc_trace : public map_function @{
3549 ex operator()(const ex &e)
3551 if (is_a<matrix>(e))
3552 return ex_to<matrix>(e).trace();
3553 else if (is_a<mul>(e)) @{
3556 return e.map(*this);
3561 This function object could then be used like this:
3565 ex M = ... // expression with matrices
3566 calc_trace do_trace;
3567 ex tr = do_trace(M);
3571 Here is another example for you to meditate over. It removes quadratic
3572 terms in a variable from an expanded polynomial:
3575 struct map_rem_quad : public map_function @{
3577 map_rem_quad(const ex & var_) : var(var_) @{@}
3579 ex operator()(const ex & e)
3581 if (is_a<add>(e) || is_a<mul>(e))
3582 return e.map(*this);
3583 else if (is_a<power>(e) &&
3584 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3594 symbol x("x"), y("y");
3597 for (int i=0; i<8; i++)
3598 e += pow(x, i) * pow(y, 8-i) * (i+1);
3600 // -> 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
3602 map_rem_quad rem_quad(x);
3603 cout << rem_quad(e) << endl;
3604 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3608 @command{ginsh} offers a slightly different implementation of @code{map()}
3609 that allows applying algebraic functions to operands. The second argument
3610 to @code{map()} is an expression containing the wildcard @samp{$0} which
3611 acts as the placeholder for the operands:
3616 > map(a+2*b,sin($0));
3618 > map(@{a,b,c@},$0^2+$0);
3619 @{a^2+a,b^2+b,c^2+c@}
3622 Note that it is only possible to use algebraic functions in the second
3623 argument. You can not use functions like @samp{diff()}, @samp{op()},
3624 @samp{subs()} etc. because these are evaluated immediately:
3627 > map(@{a,b,c@},diff($0,a));
3629 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3630 to "map(@{a,b,c@},0)".
3634 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3635 @c node-name, next, previous, up
3636 @section Visitors and Tree Traversal
3637 @cindex tree traversal
3638 @cindex @code{visitor} (class)
3639 @cindex @code{accept()}
3640 @cindex @code{visit()}
3641 @cindex @code{traverse()}
3642 @cindex @code{traverse_preorder()}
3643 @cindex @code{traverse_postorder()}
3645 Suppose that you need a function that returns a list of all indices appearing
3646 in an arbitrary expression. The indices can have any dimension, and for
3647 indices with variance you always want the covariant version returned.
3649 You can't use @code{get_free_indices()} because you also want to include
3650 dummy indices in the list, and you can't use @code{find()} as it needs
3651 specific index dimensions (and it would require two passes: one for indices
3652 with variance, one for plain ones).
3654 The obvious solution to this problem is a tree traversal with a type switch,
3655 such as the following:
3658 void gather_indices_helper(const ex & e, lst & l)
3660 if (is_a<varidx>(e)) @{
3661 const varidx & vi = ex_to<varidx>(e);
3662 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3663 @} else if (is_a<idx>(e)) @{
3666 size_t n = e.nops();
3667 for (size_t i = 0; i < n; ++i)
3668 gather_indices_helper(e.op(i), l);
3672 lst gather_indices(const ex & e)
3675 gather_indices_helper(e, l);
3682 This works fine but fans of object-oriented programming will feel
3683 uncomfortable with the type switch. One reason is that there is a possibility
3684 for subtle bugs regarding derived classes. If we had, for example, written
3687 if (is_a<idx>(e)) @{
3689 @} else if (is_a<varidx>(e)) @{
3693 in @code{gather_indices_helper}, the code wouldn't have worked because the
3694 first line "absorbs" all classes derived from @code{idx}, including
3695 @code{varidx}, so the special case for @code{varidx} would never have been
3698 Also, for a large number of classes, a type switch like the above can get
3699 unwieldy and inefficient (it's a linear search, after all).
3700 @code{gather_indices_helper} only checks for two classes, but if you had to
3701 write a function that required a different implementation for nearly
3702 every GiNaC class, the result would be very hard to maintain and extend.
3704 The cleanest approach to the problem would be to add a new virtual function
3705 to GiNaC's class hierarchy. In our example, there would be specializations
3706 for @code{idx} and @code{varidx} while the default implementation in
3707 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
3708 impossible to add virtual member functions to existing classes without
3709 changing their source and recompiling everything. GiNaC comes with source,
3710 so you could actually do this, but for a small algorithm like the one
3711 presented this would be impractical.
3713 One solution to this dilemma is the @dfn{Visitor} design pattern,
3714 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
3715 variation, described in detail in
3716 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
3717 virtual functions to the class hierarchy to implement operations, GiNaC
3718 provides a single "bouncing" method @code{accept()} that takes an instance
3719 of a special @code{visitor} class and redirects execution to the one
3720 @code{visit()} virtual function of the visitor that matches the type of
3721 object that @code{accept()} was being invoked on.
3723 Visitors in GiNaC must derive from the global @code{visitor} class as well
3724 as from the class @code{T::visitor} of each class @code{T} they want to
3725 visit, and implement the member functions @code{void visit(const T &)} for
3731 void ex::accept(visitor & v) const;
3734 will then dispatch to the correct @code{visit()} member function of the
3735 specified visitor @code{v} for the type of GiNaC object at the root of the
3736 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
3738 Here is an example of a visitor:
3742 : public visitor, // this is required
3743 public add::visitor, // visit add objects
3744 public numeric::visitor, // visit numeric objects
3745 public basic::visitor // visit basic objects
3747 void visit(const add & x)
3748 @{ cout << "called with an add object" << endl; @}
3750 void visit(const numeric & x)
3751 @{ cout << "called with a numeric object" << endl; @}
3753 void visit(const basic & x)
3754 @{ cout << "called with a basic object" << endl; @}
3758 which can be used as follows:
3769 // prints "called with a numeric object"
3771 // prints "called with an add object"
3773 // prints "called with a basic object"
3777 The @code{visit(const basic &)} method gets called for all objects that are
3778 not @code{numeric} or @code{add} and acts as an (optional) default.
3780 From a conceptual point of view, the @code{visit()} methods of the visitor
3781 behave like a newly added virtual function of the visited hierarchy.
3782 In addition, visitors can store state in member variables, and they can
3783 be extended by deriving a new visitor from an existing one, thus building
3784 hierarchies of visitors.
3786 We can now rewrite our index example from above with a visitor:
3789 class gather_indices_visitor
3790 : public visitor, public idx::visitor, public varidx::visitor
3794 void visit(const idx & i)
3799 void visit(const varidx & vi)
3801 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3805 const lst & get_result() // utility function
3814 What's missing is the tree traversal. We could implement it in
3815 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
3818 void ex::traverse_preorder(visitor & v) const;
3819 void ex::traverse_postorder(visitor & v) const;
3820 void ex::traverse(visitor & v) const;
3823 @code{traverse_preorder()} visits a node @emph{before} visiting its
3824 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
3825 visiting its subexpressions. @code{traverse()} is a synonym for
3826 @code{traverse_preorder()}.
3828 Here is a new implementation of @code{gather_indices()} that uses the visitor
3829 and @code{traverse()}:
3832 lst gather_indices(const ex & e)
3834 gather_indices_visitor v;
3836 return v.get_result();
3841 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
3842 @c node-name, next, previous, up
3843 @section Polynomial arithmetic
3845 @subsection Expanding and collecting
3846 @cindex @code{expand()}
3847 @cindex @code{collect()}
3848 @cindex @code{collect_common_factors()}
3850 A polynomial in one or more variables has many equivalent
3851 representations. Some useful ones serve a specific purpose. Consider
3852 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3853 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3854 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3855 representations are the recursive ones where one collects for exponents
3856 in one of the three variable. Since the factors are themselves
3857 polynomials in the remaining two variables the procedure can be
3858 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3859 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3862 To bring an expression into expanded form, its method
3868 may be called. In our example above, this corresponds to @math{4*x*y +
3869 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3870 GiNaC is not easily guessable you should be prepared to see different
3871 orderings of terms in such sums!
3873 Another useful representation of multivariate polynomials is as a
3874 univariate polynomial in one of the variables with the coefficients
3875 being polynomials in the remaining variables. The method
3876 @code{collect()} accomplishes this task:
3879 ex ex::collect(const ex & s, bool distributed = false);
3882 The first argument to @code{collect()} can also be a list of objects in which
3883 case the result is either a recursively collected polynomial, or a polynomial
3884 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3885 by the @code{distributed} flag.
3887 Note that the original polynomial needs to be in expanded form (for the
3888 variables concerned) in order for @code{collect()} to be able to find the
3889 coefficients properly.
3891 The following @command{ginsh} transcript shows an application of @code{collect()}
3892 together with @code{find()}:
3895 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3896 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
3897 > collect(a,@{p,q@});
3898 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)
3899 > collect(a,find(a,sin($1)));
3900 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3901 > collect(a,@{find(a,sin($1)),p,q@});
3902 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3903 > collect(a,@{find(a,sin($1)),d@});
3904 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3907 Polynomials can often be brought into a more compact form by collecting
3908 common factors from the terms of sums. This is accomplished by the function
3911 ex collect_common_factors(const ex & e);
3914 This function doesn't perform a full factorization but only looks for
3915 factors which are already explicitly present:
3918 > collect_common_factors(a*x+a*y);
3920 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
3922 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
3923 (c+a)*a*(x*y+y^2+x)*b
3926 @subsection Degree and coefficients
3927 @cindex @code{degree()}
3928 @cindex @code{ldegree()}
3929 @cindex @code{coeff()}
3931 The degree and low degree of a polynomial can be obtained using the two
3935 int ex::degree(const ex & s);
3936 int ex::ldegree(const ex & s);
3939 which also work reliably on non-expanded input polynomials (they even work
3940 on rational functions, returning the asymptotic degree). To extract
3941 a coefficient with a certain power from an expanded polynomial you use
3944 ex ex::coeff(const ex & s, int n);
3947 You can also obtain the leading and trailing coefficients with the methods
3950 ex ex::lcoeff(const ex & s);
3951 ex ex::tcoeff(const ex & s);
3954 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3957 An application is illustrated in the next example, where a multivariate
3958 polynomial is analyzed:
3962 symbol x("x"), y("y");
3963 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3964 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3965 ex Poly = PolyInp.expand();
3967 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3968 cout << "The x^" << i << "-coefficient is "
3969 << Poly.coeff(x,i) << endl;
3971 cout << "As polynomial in y: "
3972 << Poly.collect(y) << endl;
3976 When run, it returns an output in the following fashion:
3979 The x^0-coefficient is y^2+11*y
3980 The x^1-coefficient is 5*y^2-2*y
3981 The x^2-coefficient is -1
3982 The x^3-coefficient is 4*y
3983 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3986 As always, the exact output may vary between different versions of GiNaC
3987 or even from run to run since the internal canonical ordering is not
3988 within the user's sphere of influence.
3990 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3991 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3992 with non-polynomial expressions as they not only work with symbols but with
3993 constants, functions and indexed objects as well:
3997 symbol a("a"), b("b"), c("c");
3998 idx i(symbol("i"), 3);
4000 ex e = pow(sin(x) - cos(x), 4);
4001 cout << e.degree(cos(x)) << endl;
4003 cout << e.expand().coeff(sin(x), 3) << endl;
4006 e = indexed(a+b, i) * indexed(b+c, i);
4007 e = e.expand(expand_options::expand_indexed);
4008 cout << e.collect(indexed(b, i)) << endl;
4009 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4014 @subsection Polynomial division
4015 @cindex polynomial division
4018 @cindex pseudo-remainder
4019 @cindex @code{quo()}
4020 @cindex @code{rem()}
4021 @cindex @code{prem()}
4022 @cindex @code{divide()}
4027 ex quo(const ex & a, const ex & b, const symbol & x);
4028 ex rem(const ex & a, const ex & b, const symbol & x);
4031 compute the quotient and remainder of univariate polynomials in the variable
4032 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4034 The additional function
4037 ex prem(const ex & a, const ex & b, const symbol & x);
4040 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4041 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4043 Exact division of multivariate polynomials is performed by the function
4046 bool divide(const ex & a, const ex & b, ex & q);
4049 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4050 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4051 in which case the value of @code{q} is undefined.
4054 @subsection Unit, content and primitive part
4055 @cindex @code{unit()}
4056 @cindex @code{content()}
4057 @cindex @code{primpart()}
4062 ex ex::unit(const symbol & x);
4063 ex ex::content(const symbol & x);
4064 ex ex::primpart(const symbol & x);
4067 return the unit part, content part, and primitive polynomial of a multivariate
4068 polynomial with respect to the variable @samp{x} (the unit part being the sign
4069 of the leading coefficient, the content part being the GCD of the coefficients,
4070 and the primitive polynomial being the input polynomial divided by the unit and
4071 content parts). The product of unit, content, and primitive part is the
4072 original polynomial.
4075 @subsection GCD and LCM
4078 @cindex @code{gcd()}
4079 @cindex @code{lcm()}
4081 The functions for polynomial greatest common divisor and least common
4082 multiple have the synopsis
4085 ex gcd(const ex & a, const ex & b);
4086 ex lcm(const ex & a, const ex & b);
4089 The functions @code{gcd()} and @code{lcm()} accept two expressions
4090 @code{a} and @code{b} as arguments and return a new expression, their
4091 greatest common divisor or least common multiple, respectively. If the
4092 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4093 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4096 #include <ginac/ginac.h>
4097 using namespace GiNaC;
4101 symbol x("x"), y("y"), z("z");
4102 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4103 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4105 ex P_gcd = gcd(P_a, P_b);
4107 ex P_lcm = lcm(P_a, P_b);
4108 // 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
4113 @subsection Square-free decomposition
4114 @cindex square-free decomposition
4115 @cindex factorization
4116 @cindex @code{sqrfree()}
4118 GiNaC still lacks proper factorization support. Some form of
4119 factorization is, however, easily implemented by noting that factors
4120 appearing in a polynomial with power two or more also appear in the
4121 derivative and hence can easily be found by computing the GCD of the
4122 original polynomial and its derivatives. Any decent system has an
4123 interface for this so called square-free factorization. So we provide
4126 ex sqrfree(const ex & a, const lst & l = lst());
4128 Here is an example that by the way illustrates how the exact form of the
4129 result may slightly depend on the order of differentiation, calling for
4130 some care with subsequent processing of the result:
4133 symbol x("x"), y("y");
4134 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4136 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4137 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4139 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4140 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4142 cout << sqrfree(BiVarPol) << endl;
4143 // -> depending on luck, any of the above
4146 Note also, how factors with the same exponents are not fully factorized
4150 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4151 @c node-name, next, previous, up
4152 @section Rational expressions
4154 @subsection The @code{normal} method
4155 @cindex @code{normal()}
4156 @cindex simplification
4157 @cindex temporary replacement
4159 Some basic form of simplification of expressions is called for frequently.
4160 GiNaC provides the method @code{.normal()}, which converts a rational function
4161 into an equivalent rational function of the form @samp{numerator/denominator}
4162 where numerator and denominator are coprime. If the input expression is already
4163 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4164 otherwise it performs fraction addition and multiplication.
4166 @code{.normal()} can also be used on expressions which are not rational functions
4167 as it will replace all non-rational objects (like functions or non-integer
4168 powers) by temporary symbols to bring the expression to the domain of rational
4169 functions before performing the normalization, and re-substituting these
4170 symbols afterwards. This algorithm is also available as a separate method
4171 @code{.to_rational()}, described below.
4173 This means that both expressions @code{t1} and @code{t2} are indeed
4174 simplified in this little code snippet:
4179 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4180 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4181 std::cout << "t1 is " << t1.normal() << std::endl;
4182 std::cout << "t2 is " << t2.normal() << std::endl;
4186 Of course this works for multivariate polynomials too, so the ratio of
4187 the sample-polynomials from the section about GCD and LCM above would be
4188 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4191 @subsection Numerator and denominator
4194 @cindex @code{numer()}
4195 @cindex @code{denom()}
4196 @cindex @code{numer_denom()}
4198 The numerator and denominator of an expression can be obtained with
4203 ex ex::numer_denom();
4206 These functions will first normalize the expression as described above and
4207 then return the numerator, denominator, or both as a list, respectively.
4208 If you need both numerator and denominator, calling @code{numer_denom()} is
4209 faster than using @code{numer()} and @code{denom()} separately.
4212 @subsection Converting to a polynomial or rational expression
4213 @cindex @code{to_polynomial()}
4214 @cindex @code{to_rational()}
4216 Some of the methods described so far only work on polynomials or rational
4217 functions. GiNaC provides a way to extend the domain of these functions to
4218 general expressions by using the temporary replacement algorithm described
4219 above. You do this by calling
4222 ex ex::to_polynomial(lst &l);
4226 ex ex::to_rational(lst &l);
4229 on the expression to be converted. The supplied @code{lst} will be filled
4230 with the generated temporary symbols and their replacement expressions in
4231 a format that can be used directly for the @code{subs()} method. It can also
4232 already contain a list of replacements from an earlier application of
4233 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
4234 it on multiple expressions and get consistent results.
4236 The difference betwerrn @code{.to_polynomial()} and @code{.to_rational()}
4237 is probably best illustrated with an example:
4241 symbol x("x"), y("y");
4242 ex a = 2*x/sin(x) - y/(3*sin(x));
4246 ex p = a.to_polynomial(lp);
4247 cout << " = " << p << "\n with " << lp << endl;
4248 // = symbol3*symbol2*y+2*symbol2*x
4249 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4252 ex r = a.to_rational(lr);
4253 cout << " = " << r << "\n with " << lr << endl;
4254 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4255 // with @{symbol4==sin(x)@}
4259 The following more useful example will print @samp{sin(x)-cos(x)}:
4264 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4265 ex b = sin(x) + cos(x);
4268 divide(a.to_polynomial(l), b.to_polynomial(l), q);
4269 cout << q.subs(l) << endl;
4274 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4275 @c node-name, next, previous, up
4276 @section Symbolic differentiation
4277 @cindex differentiation
4278 @cindex @code{diff()}
4280 @cindex product rule
4282 GiNaC's objects know how to differentiate themselves. Thus, a
4283 polynomial (class @code{add}) knows that its derivative is the sum of
4284 the derivatives of all the monomials:
4288 symbol x("x"), y("y"), z("z");
4289 ex P = pow(x, 5) + pow(x, 2) + y;
4291 cout << P.diff(x,2) << endl;
4293 cout << P.diff(y) << endl; // 1
4295 cout << P.diff(z) << endl; // 0
4300 If a second integer parameter @var{n} is given, the @code{diff} method
4301 returns the @var{n}th derivative.
4303 If @emph{every} object and every function is told what its derivative
4304 is, all derivatives of composed objects can be calculated using the
4305 chain rule and the product rule. Consider, for instance the expression
4306 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4307 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4308 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4309 out that the composition is the generating function for Euler Numbers,
4310 i.e. the so called @var{n}th Euler number is the coefficient of
4311 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4312 identity to code a function that generates Euler numbers in just three
4315 @cindex Euler numbers
4317 #include <ginac/ginac.h>
4318 using namespace GiNaC;
4320 ex EulerNumber(unsigned n)
4323 const ex generator = pow(cosh(x),-1);
4324 return generator.diff(x,n).subs(x==0);
4329 for (unsigned i=0; i<11; i+=2)
4330 std::cout << EulerNumber(i) << std::endl;
4335 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4336 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4337 @code{i} by two since all odd Euler numbers vanish anyways.
4340 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4341 @c node-name, next, previous, up
4342 @section Series expansion
4343 @cindex @code{series()}
4344 @cindex Taylor expansion
4345 @cindex Laurent expansion
4346 @cindex @code{pseries} (class)
4347 @cindex @code{Order()}
4349 Expressions know how to expand themselves as a Taylor series or (more
4350 generally) a Laurent series. As in most conventional Computer Algebra
4351 Systems, no distinction is made between those two. There is a class of
4352 its own for storing such series (@code{class pseries}) and a built-in
4353 function (called @code{Order}) for storing the order term of the series.
4354 As a consequence, if you want to work with series, i.e. multiply two
4355 series, you need to call the method @code{ex::series} again to convert
4356 it to a series object with the usual structure (expansion plus order
4357 term). A sample application from special relativity could read:
4360 #include <ginac/ginac.h>
4361 using namespace std;
4362 using namespace GiNaC;
4366 symbol v("v"), c("c");
4368 ex gamma = 1/sqrt(1 - pow(v/c,2));
4369 ex mass_nonrel = gamma.series(v==0, 10);
4371 cout << "the relativistic mass increase with v is " << endl
4372 << mass_nonrel << endl;
4374 cout << "the inverse square of this series is " << endl
4375 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4379 Only calling the series method makes the last output simplify to
4380 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4381 series raised to the power @math{-2}.
4383 @cindex Machin's formula
4384 As another instructive application, let us calculate the numerical
4385 value of Archimedes' constant
4389 (for which there already exists the built-in constant @code{Pi})
4390 using John Machin's amazing formula
4392 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4395 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4397 This equation (and similar ones) were used for over 200 years for
4398 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4399 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4400 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4401 order term with it and the question arises what the system is supposed
4402 to do when the fractions are plugged into that order term. The solution
4403 is to use the function @code{series_to_poly()} to simply strip the order
4407 #include <ginac/ginac.h>
4408 using namespace GiNaC;
4410 ex machin_pi(int degr)
4413 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4414 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4415 -4*pi_expansion.subs(x==numeric(1,239));
4421 using std::cout; // just for fun, another way of...
4422 using std::endl; // ...dealing with this namespace std.
4424 for (int i=2; i<12; i+=2) @{
4425 pi_frac = machin_pi(i);
4426 cout << i << ":\t" << pi_frac << endl
4427 << "\t" << pi_frac.evalf() << endl;
4433 Note how we just called @code{.series(x,degr)} instead of
4434 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4435 method @code{series()}: if the first argument is a symbol the expression
4436 is expanded in that symbol around point @code{0}. When you run this
4437 program, it will type out:
4441 3.1832635983263598326
4442 4: 5359397032/1706489875
4443 3.1405970293260603143
4444 6: 38279241713339684/12184551018734375
4445 3.141621029325034425
4446 8: 76528487109180192540976/24359780855939418203125
4447 3.141591772182177295
4448 10: 327853873402258685803048818236/104359128170408663038552734375
4449 3.1415926824043995174
4453 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4454 @c node-name, next, previous, up
4455 @section Symmetrization
4456 @cindex @code{symmetrize()}
4457 @cindex @code{antisymmetrize()}
4458 @cindex @code{symmetrize_cyclic()}
4463 ex ex::symmetrize(const lst & l);
4464 ex ex::antisymmetrize(const lst & l);
4465 ex ex::symmetrize_cyclic(const lst & l);
4468 symmetrize an expression by returning the sum over all symmetric,
4469 antisymmetric or cyclic permutations of the specified list of objects,
4470 weighted by the number of permutations.
4472 The three additional methods
4475 ex ex::symmetrize();
4476 ex ex::antisymmetrize();
4477 ex ex::symmetrize_cyclic();
4480 symmetrize or antisymmetrize an expression over its free indices.
4482 Symmetrization is most useful with indexed expressions but can be used with
4483 almost any kind of object (anything that is @code{subs()}able):
4487 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4488 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4490 cout << indexed(A, i, j).symmetrize() << endl;
4491 // -> 1/2*A.j.i+1/2*A.i.j
4492 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4493 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4494 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4495 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4500 @node Built-in Functions, Solving Linear Systems of Equations, Symmetrization, Methods and Functions
4501 @c node-name, next, previous, up
4502 @section Predefined mathematical functions
4504 GiNaC contains the following predefined mathematical functions:
4507 @multitable @columnfractions .30 .70
4508 @item @strong{Name} @tab @strong{Function}
4511 @cindex @code{abs()}
4512 @item @code{csgn(x)}
4514 @cindex @code{csgn()}
4515 @item @code{sqrt(x)}
4516 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4517 @cindex @code{sqrt()}
4520 @cindex @code{sin()}
4523 @cindex @code{cos()}
4526 @cindex @code{tan()}
4527 @item @code{asin(x)}
4529 @cindex @code{asin()}
4530 @item @code{acos(x)}
4532 @cindex @code{acos()}
4533 @item @code{atan(x)}
4534 @tab inverse tangent
4535 @cindex @code{atan()}
4536 @item @code{atan2(y, x)}
4537 @tab inverse tangent with two arguments
4538 @item @code{sinh(x)}
4539 @tab hyperbolic sine
4540 @cindex @code{sinh()}
4541 @item @code{cosh(x)}
4542 @tab hyperbolic cosine
4543 @cindex @code{cosh()}
4544 @item @code{tanh(x)}
4545 @tab hyperbolic tangent
4546 @cindex @code{tanh()}
4547 @item @code{asinh(x)}
4548 @tab inverse hyperbolic sine
4549 @cindex @code{asinh()}
4550 @item @code{acosh(x)}
4551 @tab inverse hyperbolic cosine
4552 @cindex @code{acosh()}
4553 @item @code{atanh(x)}
4554 @tab inverse hyperbolic tangent
4555 @cindex @code{atanh()}
4557 @tab exponential function
4558 @cindex @code{exp()}
4560 @tab natural logarithm
4561 @cindex @code{log()}
4564 @cindex @code{Li2()}
4565 @item @code{zeta(x)}
4566 @tab Riemann's zeta function
4567 @cindex @code{zeta()}
4568 @item @code{zeta(n, x)}
4569 @tab derivatives of Riemann's zeta function
4570 @item @code{tgamma(x)}
4572 @cindex @code{tgamma()}
4573 @cindex Gamma function
4574 @item @code{lgamma(x)}
4575 @tab logarithm of Gamma function
4576 @cindex @code{lgamma()}
4577 @item @code{beta(x, y)}
4578 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4579 @cindex @code{beta()}
4581 @tab psi (digamma) function
4582 @cindex @code{psi()}
4583 @item @code{psi(n, x)}
4584 @tab derivatives of psi function (polygamma functions)
4585 @item @code{factorial(n)}
4586 @tab factorial function
4587 @cindex @code{factorial()}
4588 @item @code{binomial(n, m)}
4589 @tab binomial coefficients
4590 @cindex @code{binomial()}
4591 @item @code{Order(x)}
4592 @tab order term function in truncated power series
4593 @cindex @code{Order()}
4598 For functions that have a branch cut in the complex plane GiNaC follows
4599 the conventions for C++ as defined in the ANSI standard as far as
4600 possible. In particular: the natural logarithm (@code{log}) and the
4601 square root (@code{sqrt}) both have their branch cuts running along the
4602 negative real axis where the points on the axis itself belong to the
4603 upper part (i.e. continuous with quadrant II). The inverse
4604 trigonometric and hyperbolic functions are not defined for complex
4605 arguments by the C++ standard, however. In GiNaC we follow the
4606 conventions used by CLN, which in turn follow the carefully designed
4607 definitions in the Common Lisp standard. It should be noted that this
4608 convention is identical to the one used by the C99 standard and by most
4609 serious CAS. It is to be expected that future revisions of the C++
4610 standard incorporate these functions in the complex domain in a manner
4611 compatible with C99.
4614 @node Solving Linear Systems of Equations, Input/Output, Built-in Functions, Methods and Functions
4615 @c node-name, next, previous, up
4616 @section Solving Linear Systems of Equations
4617 @cindex @code{lsolve()}
4619 The function @code{lsolve()} provides a convenient wrapper around some
4620 matrix operations that comes in handy when a system of linear equations
4624 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
4627 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
4628 @code{relational}) while @code{symbols} is a @code{lst} of
4629 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
4632 It returns the @code{lst} of solutions as an expression. As an example,
4633 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
4637 symbol a("a"), b("b"), x("x"), y("y");
4639 eqns.append(a*x+b*y==3).append(x-y==b);
4641 vars.append(x).append(y);
4642 cout << lsolve(eqns, vars) << endl;
4643 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
4646 When the linear equations @code{eqns} are underdetermined, the solution
4647 will contain one or more tautological entries like @code{x==x},
4648 depending on the rank of the system. When they are overdetermined, the
4649 solution will be an empty @code{lst}. Note the third optional parameter
4650 to @code{lsolve()}: it accepts the same parameters as
4651 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
4655 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
4656 @c node-name, next, previous, up
4657 @section Input and output of expressions
4660 @subsection Expression output
4662 @cindex output of expressions
4664 Expressions can simply be written to any stream:
4669 ex e = 4.5*I+pow(x,2)*3/2;
4670 cout << e << endl; // prints '4.5*I+3/2*x^2'
4674 The default output format is identical to the @command{ginsh} input syntax and
4675 to that used by most computer algebra systems, but not directly pastable
4676 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4677 is printed as @samp{x^2}).
4679 It is possible to print expressions in a number of different formats with
4680 a set of stream manipulators;
4683 std::ostream & dflt(std::ostream & os);
4684 std::ostream & latex(std::ostream & os);
4685 std::ostream & tree(std::ostream & os);
4686 std::ostream & csrc(std::ostream & os);
4687 std::ostream & csrc_float(std::ostream & os);
4688 std::ostream & csrc_double(std::ostream & os);
4689 std::ostream & csrc_cl_N(std::ostream & os);
4690 std::ostream & index_dimensions(std::ostream & os);
4691 std::ostream & no_index_dimensions(std::ostream & os);
4694 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
4695 @command{ginsh} via the @code{print()}, @code{print_latex()} and
4696 @code{print_csrc()} functions, respectively.
4699 All manipulators affect the stream state permanently. To reset the output
4700 format to the default, use the @code{dflt} manipulator:
4704 cout << latex; // all output to cout will be in LaTeX format from now on
4705 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4706 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
4707 cout << dflt; // revert to default output format
4708 cout << e << endl; // prints '4.5*I+3/2*x^2'
4712 If you don't want to affect the format of the stream you're working with,
4713 you can output to a temporary @code{ostringstream} like this:
4718 s << latex << e; // format of cout remains unchanged
4719 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4724 @cindex @code{csrc_float}
4725 @cindex @code{csrc_double}
4726 @cindex @code{csrc_cl_N}
4727 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
4728 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
4729 format that can be directly used in a C or C++ program. The three possible
4730 formats select the data types used for numbers (@code{csrc_cl_N} uses the
4731 classes provided by the CLN library):
4735 cout << "f = " << csrc_float << e << ";\n";
4736 cout << "d = " << csrc_double << e << ";\n";
4737 cout << "n = " << csrc_cl_N << e << ";\n";
4741 The above example will produce (note the @code{x^2} being converted to
4745 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
4746 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
4747 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
4751 The @code{tree} manipulator allows dumping the internal structure of an
4752 expression for debugging purposes:
4763 add, hash=0x0, flags=0x3, nops=2
4764 power, hash=0x0, flags=0x3, nops=2
4765 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
4766 2 (numeric), hash=0x6526b0fa, flags=0xf
4767 3/2 (numeric), hash=0xf9828fbd, flags=0xf
4770 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
4774 @cindex @code{latex}
4775 The @code{latex} output format is for LaTeX parsing in mathematical mode.
4776 It is rather similar to the default format but provides some braces needed
4777 by LaTeX for delimiting boxes and also converts some common objects to
4778 conventional LaTeX names. It is possible to give symbols a special name for
4779 LaTeX output by supplying it as a second argument to the @code{symbol}
4782 For example, the code snippet
4786 symbol x("x", "\\circ");
4787 ex e = lgamma(x).series(x==0,3);
4788 cout << latex << e << endl;
4795 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
4798 @cindex @code{index_dimensions}
4799 @cindex @code{no_index_dimensions}
4800 Index dimensions are normally hidden in the output. To make them visible, use
4801 the @code{index_dimensions} manipulator. The dimensions will be written in
4802 square brackets behind each index value in the default and LaTeX output
4807 symbol x("x"), y("y");
4808 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
4809 ex e = indexed(x, mu) * indexed(y, nu);
4812 // prints 'x~mu*y~nu'
4813 cout << index_dimensions << e << endl;
4814 // prints 'x~mu[4]*y~nu[4]'
4815 cout << no_index_dimensions << e << endl;
4816 // prints 'x~mu*y~nu'
4821 @cindex Tree traversal
4822 If you need any fancy special output format, e.g. for interfacing GiNaC
4823 with other algebra systems or for producing code for different
4824 programming languages, you can always traverse the expression tree yourself:
4827 static void my_print(const ex & e)
4829 if (is_a<function>(e))
4830 cout << ex_to<function>(e).get_name();
4832 cout << e.bp->class_name();
4834 size_t n = e.nops();
4836 for (size_t i=0; i<n; i++) @{
4848 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4856 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4857 symbol(y))),numeric(-2)))
4860 If you need an output format that makes it possible to accurately
4861 reconstruct an expression by feeding the output to a suitable parser or
4862 object factory, you should consider storing the expression in an
4863 @code{archive} object and reading the object properties from there.
4864 See the section on archiving for more information.
4867 @subsection Expression input
4868 @cindex input of expressions
4870 GiNaC provides no way to directly read an expression from a stream because
4871 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4872 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4873 @code{y} you defined in your program and there is no way to specify the
4874 desired symbols to the @code{>>} stream input operator.
4876 Instead, GiNaC lets you construct an expression from a string, specifying the
4877 list of symbols to be used:
4881 symbol x("x"), y("y");
4882 ex e("2*x+sin(y)", lst(x, y));
4886 The input syntax is the same as that used by @command{ginsh} and the stream
4887 output operator @code{<<}. The symbols in the string are matched by name to
4888 the symbols in the list and if GiNaC encounters a symbol not specified in
4889 the list it will throw an exception.
4891 With this constructor, it's also easy to implement interactive GiNaC programs:
4896 #include <stdexcept>
4897 #include <ginac/ginac.h>
4898 using namespace std;
4899 using namespace GiNaC;
4906 cout << "Enter an expression containing 'x': ";
4911 cout << "The derivative of " << e << " with respect to x is ";
4912 cout << e.diff(x) << ".\n";
4913 @} catch (exception &p) @{
4914 cerr << p.what() << endl;
4920 @subsection Archiving
4921 @cindex @code{archive} (class)
4924 GiNaC allows creating @dfn{archives} of expressions which can be stored
4925 to or retrieved from files. To create an archive, you declare an object
4926 of class @code{archive} and archive expressions in it, giving each
4927 expression a unique name:
4931 using namespace std;
4932 #include <ginac/ginac.h>
4933 using namespace GiNaC;
4937 symbol x("x"), y("y"), z("z");
4939 ex foo = sin(x + 2*y) + 3*z + 41;
4943 a.archive_ex(foo, "foo");
4944 a.archive_ex(bar, "the second one");
4948 The archive can then be written to a file:
4952 ofstream out("foobar.gar");
4958 The file @file{foobar.gar} contains all information that is needed to
4959 reconstruct the expressions @code{foo} and @code{bar}.
4961 @cindex @command{viewgar}
4962 The tool @command{viewgar} that comes with GiNaC can be used to view
4963 the contents of GiNaC archive files:
4966 $ viewgar foobar.gar
4967 foo = 41+sin(x+2*y)+3*z
4968 the second one = 42+sin(x+2*y)+3*z
4971 The point of writing archive files is of course that they can later be
4977 ifstream in("foobar.gar");
4982 And the stored expressions can be retrieved by their name:
4988 ex ex1 = a2.unarchive_ex(syms, "foo");
4989 ex ex2 = a2.unarchive_ex(syms, "the second one");
4991 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4992 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4993 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4997 Note that you have to supply a list of the symbols which are to be inserted
4998 in the expressions. Symbols in archives are stored by their name only and
4999 if you don't specify which symbols you have, unarchiving the expression will
5000 create new symbols with that name. E.g. if you hadn't included @code{x} in
5001 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5002 have had no effect because the @code{x} in @code{ex1} would have been a
5003 different symbol than the @code{x} which was defined at the beginning of
5004 the program, although both would appear as @samp{x} when printed.
5006 You can also use the information stored in an @code{archive} object to
5007 output expressions in a format suitable for exact reconstruction. The
5008 @code{archive} and @code{archive_node} classes have a couple of member
5009 functions that let you access the stored properties:
5012 static void my_print2(const archive_node & n)
5015 n.find_string("class", class_name);
5016 cout << class_name << "(";
5018 archive_node::propinfovector p;
5019 n.get_properties(p);
5021 size_t num = p.size();
5022 for (size_t i=0; i<num; i++) @{
5023 const string &name = p[i].name;
5024 if (name == "class")
5026 cout << name << "=";
5028 unsigned count = p[i].count;
5032 for (unsigned j=0; j<count; j++) @{
5033 switch (p[i].type) @{
5034 case archive_node::PTYPE_BOOL: @{
5036 n.find_bool(name, x, j);
5037 cout << (x ? "true" : "false");
5040 case archive_node::PTYPE_UNSIGNED: @{
5042 n.find_unsigned(name, x, j);
5046 case archive_node::PTYPE_STRING: @{
5048 n.find_string(name, x, j);
5049 cout << '\"' << x << '\"';
5052 case archive_node::PTYPE_NODE: @{
5053 const archive_node &x = n.find_ex_node(name, j);
5075 ex e = pow(2, x) - y;
5077 my_print2(ar.get_top_node(0)); cout << endl;
5085 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5086 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5087 overall_coeff=numeric(number="0"))
5090 Be warned, however, that the set of properties and their meaning for each
5091 class may change between GiNaC versions.
5094 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5095 @c node-name, next, previous, up
5096 @chapter Extending GiNaC
5098 By reading so far you should have gotten a fairly good understanding of
5099 GiNaC's design-patterns. From here on you should start reading the
5100 sources. All we can do now is issue some recommendations how to tackle
5101 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5102 develop some useful extension please don't hesitate to contact the GiNaC
5103 authors---they will happily incorporate them into future versions.
5106 * What does not belong into GiNaC:: What to avoid.
5107 * Symbolic functions:: Implementing symbolic functions.
5108 * Structures:: Defining new algebraic classes (the easy way).
5109 * Adding classes:: Defining new algebraic classes (the hard way).
5113 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5114 @c node-name, next, previous, up
5115 @section What doesn't belong into GiNaC
5117 @cindex @command{ginsh}
5118 First of all, GiNaC's name must be read literally. It is designed to be
5119 a library for use within C++. The tiny @command{ginsh} accompanying
5120 GiNaC makes this even more clear: it doesn't even attempt to provide a
5121 language. There are no loops or conditional expressions in
5122 @command{ginsh}, it is merely a window into the library for the
5123 programmer to test stuff (or to show off). Still, the design of a
5124 complete CAS with a language of its own, graphical capabilities and all
5125 this on top of GiNaC is possible and is without doubt a nice project for
5128 There are many built-in functions in GiNaC that do not know how to
5129 evaluate themselves numerically to a precision declared at runtime
5130 (using @code{Digits}). Some may be evaluated at certain points, but not
5131 generally. This ought to be fixed. However, doing numerical
5132 computations with GiNaC's quite abstract classes is doomed to be
5133 inefficient. For this purpose, the underlying foundation classes
5134 provided by CLN are much better suited.
5137 @node Symbolic functions, Structures, What does not belong into GiNaC, Extending GiNaC
5138 @c node-name, next, previous, up
5139 @section Symbolic functions
5141 The easiest and most instructive way to start extending GiNaC is probably to
5142 create your own symbolic functions. These are implemented with the help of
5143 two preprocessor macros:
5145 @cindex @code{DECLARE_FUNCTION}
5146 @cindex @code{REGISTER_FUNCTION}
5148 DECLARE_FUNCTION_<n>P(<name>)
5149 REGISTER_FUNCTION(<name>, <options>)
5152 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5153 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5154 parameters of type @code{ex} and returns a newly constructed GiNaC
5155 @code{function} object that represents your function.
5157 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5158 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5159 set of options that associate the symbolic function with C++ functions you
5160 provide to implement the various methods such as evaluation, derivative,
5161 series expansion etc. They also describe additional attributes the function
5162 might have, such as symmetry and commutation properties, and a name for
5163 LaTeX output. Multiple options are separated by the member access operator
5164 @samp{.} and can be given in an arbitrary order.
5166 (By the way: in case you are worrying about all the macros above we can
5167 assure you that functions are GiNaC's most macro-intense classes. We have
5168 done our best to avoid macros where we can.)
5170 @subsection A minimal example
5172 Here is an example for the implementation of a function with two arguments
5173 that is not further evaluated:
5176 DECLARE_FUNCTION_2P(myfcn)
5178 static ex myfcn_eval(const ex & x, const ex & y)
5180 return myfcn(x, y).hold();
5183 REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
5186 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5187 in algebraic expressions:
5193 ex e = 2*myfcn(42, 3*x+1) - x;
5194 // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
5195 // the actual expression
5197 // prints '2*myfcn(42,1+3*x)-x'
5202 @cindex @code{hold()}
5204 The @code{eval_func()} option specifies the C++ function that implements
5205 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5206 the same number of arguments as the associated symbolic function (two in this
5207 case) and returns the (possibly transformed or in some way simplified)
5208 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5209 of the automatic evaluation process). If no (further) evaluation is to take
5210 place, the @code{eval_func()} function must return the original function
5211 with @code{.hold()}, to avoid a potential infinite recursion. If your
5212 symbolic functions produce a segmentation fault or stack overflow when
5213 using them in expressions, you are probably missing a @code{.hold()}
5216 There is not much you can do with the @code{myfcn} function. It merely acts
5217 as a kind of container for its arguments (which is, however, sometimes
5218 perfectly sufficient). Let's have a look at the implementation of GiNaC's
5221 @subsection The cosine function
5223 The GiNaC header file @file{inifcns.h} contains the line
5226 DECLARE_FUNCTION_1P(cos)
5229 which declares to all programs using GiNaC that there is a function @samp{cos}
5230 that takes one @code{ex} as an argument. This is all they need to know to use
5231 this function in expressions.
5233 The implementation of the cosine function is in @file{inifcns_trans.cpp}. The
5234 @code{eval_func()} function looks something like this (actually, it doesn't
5235 look like this at all, but it should give you an idea what is going on):
5238 static ex cos_eval(const ex & x)
5240 if (<x is a multiple of 2*Pi>)
5242 else if (<x is a multiple of Pi>)
5244 else if (<x is a multiple of Pi/2>)
5248 else if (<x has the form 'acos(y)'>)
5250 else if (<x has the form 'asin(y)'>)
5255 return cos(x).hold();
5259 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5260 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5261 symbolic transformation can be done, the unmodified function is returned
5262 with @code{.hold()}.
5264 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5265 The user has to call @code{evalf()} for that. This is implemented in a
5269 static ex cos_evalf(const ex & x)
5271 if (is_a<numeric>(x))
5272 return cos(ex_to<numeric>(x));
5274 return cos(x).hold();
5278 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5279 in this case the @code{cos()} function for @code{numeric} objects, which in
5280 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5281 isn't really needed here, but reminds us that the corresponding @code{eval()}
5282 function would require it in this place.
5284 Differentiation will surely turn up and so we need to tell @code{cos}
5285 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5286 instance, are then handled automatically by @code{basic::diff} and
5290 static ex cos_deriv(const ex & x, unsigned diff_param)
5296 @cindex product rule
5297 The second parameter is obligatory but uninteresting at this point. It
5298 specifies which parameter to differentiate in a partial derivative in
5299 case the function has more than one parameter, and its main application
5300 is for correct handling of the chain rule.
5302 An implementation of the series expansion is not needed for @code{cos()} as
5303 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5304 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5305 the other hand, does have poles and may need to do Laurent expansion:
5308 static ex tan_series(const ex & x, const relational & rel,
5309 int order, unsigned options)
5311 // Find the actual expansion point
5312 const ex x_pt = x.subs(rel);
5314 if (<x_pt is not an odd multiple of Pi/2>)
5315 throw do_taylor(); // tell function::series() to do Taylor expansion
5317 // On a pole, expand sin()/cos()
5318 return (sin(x)/cos(x)).series(rel, order+2, options);
5322 The @code{series()} implementation of a function @emph{must} return a
5323 @code{pseries} object, otherwise your code will crash.
5325 Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
5326 macro is used to tell the system how the @code{cos()} function behaves:
5329 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5330 evalf_func(cos_evalf).
5331 derivative_func(cos_deriv).
5332 latex_name("\\cos"));
5335 This registers the @code{cos_eval()}, @code{cos_evalf()} and
5336 @code{cos_deriv()} C++ functions with the @code{cos()} function, and also
5337 gives it a proper LaTeX name.
5339 @subsection Function options
5341 GiNaC functions understand several more options which are always
5342 specified as @code{.option(params)}. None of them are required, but you
5343 need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
5344 the @code{eval()} method).
5347 eval_func(<C++ function>)
5348 evalf_func(<C++ function>)
5349 derivative_func(<C++ function>)
5350 series_func(<C++ function>)
5353 These specify the C++ functions that implement symbolic evaluation,
5354 numeric evaluation, partial derivatives, and series expansion, respectively.
5355 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5356 @code{diff()} and @code{series()}.
5358 The @code{eval_func()} function needs to use @code{.hold()} if no further
5359 automatic evaluation is desired or possible.
5361 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5362 expansion, which is correct if there are no poles involved. If the function
5363 has poles in the complex plane, the @code{series_func()} needs to check
5364 whether the expansion point is on a pole and fall back to Taylor expansion
5365 if it isn't. Otherwise, the pole usually needs to be regularized by some
5366 suitable transformation.
5369 latex_name(const string & n)
5372 specifies the LaTeX code that represents the name of the function in LaTeX
5373 output. The default is to put the function name in an @code{\mbox@{@}}.
5376 do_not_evalf_params()
5379 This tells @code{evalf()} to not recursively evaluate the parameters of the
5380 function before calling the @code{evalf_func()}.
5383 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5386 This allows you to explicitly specify the commutation properties of the
5387 function (@xref{Non-commutative objects}, for an explanation of
5388 (non)commutativity in GiNaC). For example, you can use
5389 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5390 GiNaC treat your function like a matrix. By default, functions inherit the
5391 commutation properties of their first argument.
5394 set_symmetry(const symmetry & s)
5397 specifies the symmetry properties of the function with respect to its
5398 arguments. @xref{Indexed objects}, for an explanation of symmetry
5399 specifications. GiNaC will automatically rearrange the arguments of
5400 symmetric functions into a canonical order.
5403 @node Structures, Adding classes, Symbolic functions, Extending GiNaC
5404 @c node-name, next, previous, up
5407 If you are doing some very specialized things with GiNaC, or if you just
5408 need some more organized way to store data in your expressions instead of
5409 anonymous lists, you may want to implement your own algebraic classes.
5410 ('algebraic class' means any class directly or indirectly derived from
5411 @code{basic} that can be used in GiNaC expressions).
5413 GiNaC offers two ways of accomplishing this: either by using the
5414 @code{structure<T>} template class, or by rolling your own class from
5415 scratch. This section will discuss the @code{structure<T>} template which
5416 is easier to use but more limited, while the implementation of custom
5417 GiNaC classes is the topic of the next section. However, you may want to
5418 read both sections because many common concepts and member functions are
5419 shared by both concepts, and it will also allow you to decide which approach
5420 is most suited to your needs.
5422 The @code{structure<T>} template, defined in the GiNaC header file
5423 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
5424 or @code{class}) into a GiNaC object that can be used in expressions.
5426 @subsection Example: scalar products
5428 Let's suppose that we need a way to handle some kind of abstract scalar
5429 product of the form @samp{<x|y>} in expressions. Objects of the scalar
5430 product class have to store their left and right operands, which can in turn
5431 be arbitrary expressions. Here is a possible way to represent such a
5432 product in a C++ @code{struct}:
5436 using namespace std;
5438 #include <ginac/ginac.h>
5439 using namespace GiNaC;
5445 sprod_s(ex l, ex r) : left(l), right(r) @{@}
5449 The default constructor is required. Now, to make a GiNaC class out of this
5450 data structure, we need only one line:
5453 typedef structure<sprod_s> sprod;
5456 That's it. This line constructs an algebraic class @code{sprod} which
5457 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
5458 expressions like any other GiNaC class:
5462 symbol a("a"), b("b");
5463 ex e = sprod(sprod_s(a, b));
5467 Note the difference between @code{sprod} which is the algebraic class, and
5468 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
5469 and @code{right} data members. As shown above, an @code{sprod} can be
5470 constructed from an @code{sprod_s} object.
5472 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
5473 you could define a little wrapper function like this:
5476 inline ex make_sprod(ex left, ex right)
5478 return sprod(sprod_s(left, right));
5482 The @code{sprod_s} object contained in @code{sprod} can be accessed with
5483 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
5484 @code{get_struct()}:
5488 cout << ex_to<sprod>(e)->left << endl;
5490 cout << ex_to<sprod>(e).get_struct().right << endl;
5495 You only have read access to the members of @code{sprod_s}.
5497 The type definition of @code{sprod} is enough to write your own algorithms
5498 that deal with scalar products, for example:
5503 if (is_a<sprod>(p)) @{
5504 const sprod_s & sp = ex_to<sprod>(p).get_struct();
5505 return make_sprod(sp.right, sp.left);
5516 @subsection Structure output
5518 While the @code{sprod} type is useable it still leaves something to be
5519 desired, most notably proper output:
5524 // -> [structure object]
5528 By default, any structure types you define will be printed as
5529 @samp{[structure object]}. To override this, you can specialize the
5530 template's @code{print()} member function. The member functions of
5531 GiNaC classes are described in more detail in the next section, but
5532 it shouldn't be hard to figure out what's going on here:
5535 void sprod::print(const print_context & c, unsigned level) const
5537 // tree debug output handled by superclass
5538 if (is_a<print_tree>(c))
5539 inherited::print(c, level);
5541 // get the contained sprod_s object
5542 const sprod_s & sp = get_struct();
5544 // print_context::s is a reference to an ostream
5545 c.s << "<" << sp.left << "|" << sp.right << ">";
5549 Now we can print expressions containing scalar products:
5555 cout << swap_sprod(e) << endl;
5560 @subsection Comparing structures
5562 The @code{sprod} class defined so far still has one important drawback: all
5563 scalar products are treated as being equal because GiNaC doesn't know how to
5564 compare objects of type @code{sprod_s}. This can lead to some confusing
5565 and undesired behavior:
5569 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
5571 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
5572 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
5576 To remedy this, we first need to define the operators @code{==} and @code{<}
5577 for objects of type @code{sprod_s}:
5580 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
5582 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
5585 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
5587 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
5591 The ordering established by the @code{<} operator doesn't have to make any
5592 algebraic sense, but it needs to be well defined. Note that we can't use
5593 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
5594 in the implementation of these operators because they would construct
5595 GiNaC @code{relational} objects which in the case of @code{<} do not
5596 establish a well defined ordering (for arbitrary expressions, GiNaC can't
5597 decide which one is algebraically 'less').
5599 Next, we need to change our definition of the @code{sprod} type to let
5600 GiNaC know that an ordering relation exists for the embedded objects:
5603 typedef structure<sprod_s, compare_std_less> sprod;
5606 @code{sprod} objects then behave as expected:
5610 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
5611 // -> <a|b>-<a^2|b^2>
5612 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
5613 // -> <a|b>+<a^2|b^2>
5614 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
5616 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
5621 The @code{compare_std_less} policy parameter tells GiNaC to use the
5622 @code{std::less} and @code{std::equal_to} functors to compare objects of
5623 type @code{sprod_s}. By default, these functors forward their work to the
5624 standard @code{<} and @code{==} operators, which we have overloaded.
5625 Alternatively, we could have specialized @code{std::less} and
5626 @code{std::equal_to} for class @code{sprod_s}.
5628 GiNaC provides two other comparison policies for @code{structure<T>}
5629 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
5630 which does a bit-wise comparison of the contained @code{T} objects.
5631 This should be used with extreme care because it only works reliably with
5632 built-in integral types, and it also compares any padding (filler bytes of
5633 undefined value) that the @code{T} class might have.
5635 @subsection Subexpressions
5637 Our scalar product class has two subexpressions: the left and right
5638 operands. It might be a good idea to make them accessible via the standard
5639 @code{nops()} and @code{op()} methods:
5642 size_t sprod::nops() const
5647 ex sprod::op(size_t i) const
5651 return get_struct().left;
5653 return get_struct().right;
5655 throw std::range_error("sprod::op(): no such operand");
5660 Implementing @code{nops()} and @code{op()} for container types such as
5661 @code{sprod} has two other nice side effects:
5665 @code{has()} works as expected
5667 GiNaC generates better hash keys for the objects (the default implementation
5668 of @code{calchash()} takes subexpressions into account)
5671 @cindex @code{let_op()}
5672 There is a non-const variant of @code{op()} called @code{let_op()} that
5673 allows replacing subexpressions:
5676 ex & sprod::let_op(size_t i)
5678 // every non-const member function must call this
5679 ensure_if_modifiable();
5683 return get_struct().left;
5685 return get_struct().right;
5687 throw std::range_error("sprod::let_op(): no such operand");
5692 Once we have provided @code{let_op()} we also get @code{subs()} and
5693 @code{map()} for free. In fact, every container class that returns a non-null
5694 @code{nops()} value must either implement @code{let_op()} or provide custom
5695 implementations of @code{subs()} and @code{map()}.
5697 In turn, the availability of @code{map()} enables the recursive behavior of a
5698 couple of other default method implementations, in particular @code{evalf()},
5699 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
5700 we probably want to provide our own version of @code{expand()} for scalar
5701 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
5702 This is left as an exercise for the reader.
5704 The @code{structure<T>} template defines many more member functions that
5705 you can override by specialization to customize the behavior of your
5706 structures. You are referred to the next section for a description of
5707 some of these (especially @code{eval()}). There is, however, one topic
5708 that shall be addressed here, as it demonstrates one peculiarity of the
5709 @code{structure<T>} template: archiving.
5711 @subsection Archiving structures
5713 If you don't know how the archiving of GiNaC objects is implemented, you
5714 should first read the next section and then come back here. You're back?
5717 To implement archiving for structures it is not enough to provide
5718 specializations for the @code{archive()} member function and the
5719 unarchiving constructor (the @code{unarchive()} function has a default
5720 implementation). You also need to provide a unique name (as a string literal)
5721 for each structure type you define. This is because in GiNaC archives,
5722 the class of an object is stored as a string, the class name.
5724 By default, this class name (as returned by the @code{class_name()} member
5725 function) is @samp{structure} for all structure classes. This works as long
5726 as you have only defined one structure type, but if you use two or more you
5727 need to provide a different name for each by specializing the
5728 @code{get_class_name()} member function. Here is a sample implementation
5729 for enabling archiving of the scalar product type defined above:
5732 const char *sprod::get_class_name() @{ return "sprod"; @}
5734 void sprod::archive(archive_node & n) const
5736 inherited::archive(n);
5737 n.add_ex("left", get_struct().left);
5738 n.add_ex("right", get_struct().right);
5741 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
5743 n.find_ex("left", get_struct().left, sym_lst);
5744 n.find_ex("right", get_struct().right, sym_lst);
5748 Note that the unarchiving constructor is @code{sprod::structure} and not
5749 @code{sprod::sprod}, and that we don't need to supply an
5750 @code{sprod::unarchive()} function.
5753 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
5754 @c node-name, next, previous, up
5755 @section Adding classes
5757 The @code{structure<T>} template provides an way to extend GiNaC with custom
5758 algebraic classes that is easy to use but has its limitations, the most
5759 severe of which being that you can't add any new member functions to
5760 structures. To be able to do this, you need to write a new class definition
5763 This section will explain how to implement new algebraic classes in GiNaC by
5764 giving the example of a simple 'string' class. After reading this section
5765 you will know how to properly declare a GiNaC class and what the minimum
5766 required member functions are that you have to implement. We only cover the
5767 implementation of a 'leaf' class here (i.e. one that doesn't contain
5768 subexpressions). Creating a container class like, for example, a class
5769 representing tensor products is more involved but this section should give
5770 you enough information so you can consult the source to GiNaC's predefined
5771 classes if you want to implement something more complicated.
5773 @subsection GiNaC's run-time type information system
5775 @cindex hierarchy of classes
5777 All algebraic classes (that is, all classes that can appear in expressions)
5778 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
5779 @code{basic *} (which is essentially what an @code{ex} is) represents a
5780 generic pointer to an algebraic class. Occasionally it is necessary to find
5781 out what the class of an object pointed to by a @code{basic *} really is.
5782 Also, for the unarchiving of expressions it must be possible to find the
5783 @code{unarchive()} function of a class given the class name (as a string). A
5784 system that provides this kind of information is called a run-time type
5785 information (RTTI) system. The C++ language provides such a thing (see the
5786 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
5787 implements its own, simpler RTTI.
5789 The RTTI in GiNaC is based on two mechanisms:
5794 The @code{basic} class declares a member variable @code{tinfo_key} which
5795 holds an unsigned integer that identifies the object's class. These numbers
5796 are defined in the @file{tinfos.h} header file for the built-in GiNaC
5797 classes. They all start with @code{TINFO_}.
5800 By means of some clever tricks with static members, GiNaC maintains a list
5801 of information for all classes derived from @code{basic}. The information
5802 available includes the class names, the @code{tinfo_key}s, and pointers
5803 to the unarchiving functions. This class registry is defined in the
5804 @file{registrar.h} header file.
5808 The disadvantage of this proprietary RTTI implementation is that there's
5809 a little more to do when implementing new classes (C++'s RTTI works more
5810 or less automatic) but don't worry, most of the work is simplified by
5813 @subsection A minimalistic example
5815 Now we will start implementing a new class @code{mystring} that allows
5816 placing character strings in algebraic expressions (this is not very useful,
5817 but it's just an example). This class will be a direct subclass of
5818 @code{basic}. You can use this sample implementation as a starting point
5819 for your own classes.
5821 The code snippets given here assume that you have included some header files
5827 #include <stdexcept>
5828 using namespace std;
5830 #include <ginac/ginac.h>
5831 using namespace GiNaC;
5834 The first thing we have to do is to define a @code{tinfo_key} for our new
5835 class. This can be any arbitrary unsigned number that is not already taken
5836 by one of the existing classes but it's better to come up with something
5837 that is unlikely to clash with keys that might be added in the future. The
5838 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
5839 which is not a requirement but we are going to stick with this scheme:
5842 const unsigned TINFO_mystring = 0x42420001U;
5845 Now we can write down the class declaration. The class stores a C++
5846 @code{string} and the user shall be able to construct a @code{mystring}
5847 object from a C or C++ string:
5850 class mystring : public basic
5852 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
5855 mystring(const string &s);
5856 mystring(const char *s);
5862 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
5865 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
5866 macros are defined in @file{registrar.h}. They take the name of the class
5867 and its direct superclass as arguments and insert all required declarations
5868 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
5869 the first line after the opening brace of the class definition. The
5870 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
5871 source (at global scope, of course, not inside a function).
5873 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
5874 declarations of the default constructor and a couple of other functions that
5875 are required. It also defines a type @code{inherited} which refers to the
5876 superclass so you don't have to modify your code every time you shuffle around
5877 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
5878 class with the GiNaC RTTI.
5880 Now there are seven member functions we have to implement to get a working
5886 @code{mystring()}, the default constructor.
5889 @code{void archive(archive_node &n)}, the archiving function. This stores all
5890 information needed to reconstruct an object of this class inside an
5891 @code{archive_node}.
5894 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
5895 constructor. This constructs an instance of the class from the information
5896 found in an @code{archive_node}.
5899 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
5900 unarchiving function. It constructs a new instance by calling the unarchiving
5904 @cindex @code{compare_same_type()}
5905 @code{int compare_same_type(const basic &other)}, which is used internally
5906 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
5907 -1, depending on the relative order of this object and the @code{other}
5908 object. If it returns 0, the objects are considered equal.
5909 @strong{Note:} This has nothing to do with the (numeric) ordering
5910 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
5911 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
5912 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
5913 must provide a @code{compare_same_type()} function, even those representing
5914 objects for which no reasonable algebraic ordering relationship can be
5918 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
5919 which are the two constructors we declared.
5923 Let's proceed step-by-step. The default constructor looks like this:
5926 mystring::mystring() : inherited(TINFO_mystring) @{@}
5929 The golden rule is that in all constructors you have to set the
5930 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
5931 it will be set by the constructor of the superclass and all hell will break
5932 loose in the RTTI. For your convenience, the @code{basic} class provides
5933 a constructor that takes a @code{tinfo_key} value, which we are using here
5934 (remember that in our case @code{inherited == basic}). If the superclass
5935 didn't have such a constructor, we would have to set the @code{tinfo_key}
5936 to the right value manually.
5938 In the default constructor you should set all other member variables to
5939 reasonable default values (we don't need that here since our @code{str}
5940 member gets set to an empty string automatically).
5942 Next are the three functions for archiving. You have to implement them even
5943 if you don't plan to use archives, but the minimum required implementation
5944 is really simple. First, the archiving function:
5947 void mystring::archive(archive_node &n) const
5949 inherited::archive(n);
5950 n.add_string("string", str);
5954 The only thing that is really required is calling the @code{archive()}
5955 function of the superclass. Optionally, you can store all information you
5956 deem necessary for representing the object into the passed
5957 @code{archive_node}. We are just storing our string here. For more
5958 information on how the archiving works, consult the @file{archive.h} header
5961 The unarchiving constructor is basically the inverse of the archiving
5965 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
5967 n.find_string("string", str);
5971 If you don't need archiving, just leave this function empty (but you must
5972 invoke the unarchiving constructor of the superclass). Note that we don't
5973 have to set the @code{tinfo_key} here because it is done automatically
5974 by the unarchiving constructor of the @code{basic} class.
5976 Finally, the unarchiving function:
5979 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
5981 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
5985 You don't have to understand how exactly this works. Just copy these
5986 four lines into your code literally (replacing the class name, of
5987 course). It calls the unarchiving constructor of the class and unless
5988 you are doing something very special (like matching @code{archive_node}s
5989 to global objects) you don't need a different implementation. For those
5990 who are interested: setting the @code{dynallocated} flag puts the object
5991 under the control of GiNaC's garbage collection. It will get deleted
5992 automatically once it is no longer referenced.
5994 Our @code{compare_same_type()} function uses a provided function to compare
5998 int mystring::compare_same_type(const basic &other) const
6000 const mystring &o = static_cast<const mystring &>(other);
6001 int cmpval = str.compare(o.str);
6004 else if (cmpval < 0)
6011 Although this function takes a @code{basic &}, it will always be a reference
6012 to an object of exactly the same class (objects of different classes are not
6013 comparable), so the cast is safe. If this function returns 0, the two objects
6014 are considered equal (in the sense that @math{A-B=0}), so you should compare
6015 all relevant member variables.
6017 Now the only thing missing is our two new constructors:
6020 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6021 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6024 No surprises here. We set the @code{str} member from the argument and
6025 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6027 That's it! We now have a minimal working GiNaC class that can store
6028 strings in algebraic expressions. Let's confirm that the RTTI works:
6031 ex e = mystring("Hello, world!");
6032 cout << is_a<mystring>(e) << endl;
6035 cout << e.bp->class_name() << endl;
6039 Obviously it does. Let's see what the expression @code{e} looks like:
6043 // -> [mystring object]
6046 Hm, not exactly what we expect, but of course the @code{mystring} class
6047 doesn't yet know how to print itself. This is done in the @code{print()}
6048 member function. Let's say that we wanted to print the string surrounded
6052 class mystring : public basic
6056 void print(const print_context &c, unsigned level = 0) const;
6060 void mystring::print(const print_context &c, unsigned level) const
6062 // print_context::s is a reference to an ostream
6063 c.s << '\"' << str << '\"';
6067 The @code{level} argument is only required for container classes to
6068 correctly parenthesize the output. Let's try again to print the expression:
6072 // -> "Hello, world!"
6075 Much better. The @code{mystring} class can be used in arbitrary expressions:
6078 e += mystring("GiNaC rulez");
6080 // -> "GiNaC rulez"+"Hello, world!"
6083 (GiNaC's automatic term reordering is in effect here), or even
6086 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
6088 // -> "One string"^(2*sin(-"Another string"+Pi))
6091 Whether this makes sense is debatable but remember that this is only an
6092 example. At least it allows you to implement your own symbolic algorithms
6095 Note that GiNaC's algebraic rules remain unchanged:
6098 e = mystring("Wow") * mystring("Wow");
6102 e = pow(mystring("First")-mystring("Second"), 2);
6103 cout << e.expand() << endl;
6104 // -> -2*"First"*"Second"+"First"^2+"Second"^2
6107 There's no way to, for example, make GiNaC's @code{add} class perform string
6108 concatenation. You would have to implement this yourself.
6110 @subsection Automatic evaluation
6113 @cindex @code{eval()}
6114 @cindex @code{hold()}
6115 When dealing with objects that are just a little more complicated than the
6116 simple string objects we have implemented, chances are that you will want to
6117 have some automatic simplifications or canonicalizations performed on them.
6118 This is done in the evaluation member function @code{eval()}. Let's say that
6119 we wanted all strings automatically converted to lowercase with
6120 non-alphabetic characters stripped, and empty strings removed:
6123 class mystring : public basic
6127 ex eval(int level = 0) const;
6131 ex mystring::eval(int level) const
6134 for (int i=0; i<str.length(); i++) @{
6136 if (c >= 'A' && c <= 'Z')
6137 new_str += tolower(c);
6138 else if (c >= 'a' && c <= 'z')
6142 if (new_str.length() == 0)
6145 return mystring(new_str).hold();
6149 The @code{level} argument is used to limit the recursion depth of the
6150 evaluation. We don't have any subexpressions in the @code{mystring}
6151 class so we are not concerned with this. If we had, we would call the
6152 @code{eval()} functions of the subexpressions with @code{level - 1} as
6153 the argument if @code{level != 1}. The @code{hold()} member function
6154 sets a flag in the object that prevents further evaluation. Otherwise
6155 we might end up in an endless loop. When you want to return the object
6156 unmodified, use @code{return this->hold();}.
6158 Let's confirm that it works:
6161 ex e = mystring("Hello, world!") + mystring("!?#");
6165 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
6170 @subsection Optional member functions
6172 We have implemented only a small set of member functions to make the class
6173 work in the GiNaC framework. There are two functions that are not strictly
6174 required but will make operations with objects of the class more efficient:
6176 @cindex @code{calchash()}
6177 @cindex @code{is_equal_same_type()}
6179 unsigned calchash() const;
6180 bool is_equal_same_type(const basic &other) const;
6183 The @code{calchash()} method returns an @code{unsigned} hash value for the
6184 object which will allow GiNaC to compare and canonicalize expressions much
6185 more efficiently. You should consult the implementation of some of the built-in
6186 GiNaC classes for examples of hash functions. The default implementation of
6187 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
6188 class and all subexpressions that are accessible via @code{op()}.
6190 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
6191 tests for equality without establishing an ordering relation, which is often
6192 faster. The default implementation of @code{is_equal_same_type()} just calls
6193 @code{compare_same_type()} and tests its result for zero.
6195 @subsection Other member functions
6197 For a real algebraic class, there are probably some more functions that you
6198 might want to provide:
6201 bool info(unsigned inf) const;
6202 ex evalf(int level = 0) const;
6203 ex series(const relational & r, int order, unsigned options = 0) const;
6204 ex derivative(const symbol & s) const;
6207 If your class stores sub-expressions (see the scalar product example in the
6208 previous section) you will probably want to override
6210 @cindex @code{let_op()}
6213 ex op(size_t i) const;
6214 ex & let_op(size_t i);
6215 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
6216 ex map(map_function & f) const;
6219 @code{let_op()} is a variant of @code{op()} that allows write access. The
6220 default implementations of @code{subs()} and @code{map()} use it, so you have
6221 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
6223 You can, of course, also add your own new member functions. Remember
6224 that the RTTI may be used to get information about what kinds of objects
6225 you are dealing with (the position in the class hierarchy) and that you
6226 can always extract the bare object from an @code{ex} by stripping the
6227 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
6228 should become a need.
6230 That's it. May the source be with you!
6233 @node A Comparison With Other CAS, Advantages, Adding classes, Top
6234 @c node-name, next, previous, up
6235 @chapter A Comparison With Other CAS
6238 This chapter will give you some information on how GiNaC compares to
6239 other, traditional Computer Algebra Systems, like @emph{Maple},
6240 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
6241 disadvantages over these systems.
6244 * Advantages:: Strengths of the GiNaC approach.
6245 * Disadvantages:: Weaknesses of the GiNaC approach.
6246 * Why C++?:: Attractiveness of C++.
6249 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
6250 @c node-name, next, previous, up
6253 GiNaC has several advantages over traditional Computer
6254 Algebra Systems, like
6259 familiar language: all common CAS implement their own proprietary
6260 grammar which you have to learn first (and maybe learn again when your
6261 vendor decides to `enhance' it). With GiNaC you can write your program
6262 in common C++, which is standardized.
6266 structured data types: you can build up structured data types using
6267 @code{struct}s or @code{class}es together with STL features instead of
6268 using unnamed lists of lists of lists.
6271 strongly typed: in CAS, you usually have only one kind of variables
6272 which can hold contents of an arbitrary type. This 4GL like feature is
6273 nice for novice programmers, but dangerous.
6276 development tools: powerful development tools exist for C++, like fancy
6277 editors (e.g. with automatic indentation and syntax highlighting),
6278 debuggers, visualization tools, documentation generators@dots{}
6281 modularization: C++ programs can easily be split into modules by
6282 separating interface and implementation.
6285 price: GiNaC is distributed under the GNU Public License which means
6286 that it is free and available with source code. And there are excellent
6287 C++-compilers for free, too.
6290 extendable: you can add your own classes to GiNaC, thus extending it on
6291 a very low level. Compare this to a traditional CAS that you can
6292 usually only extend on a high level by writing in the language defined
6293 by the parser. In particular, it turns out to be almost impossible to
6294 fix bugs in a traditional system.
6297 multiple interfaces: Though real GiNaC programs have to be written in
6298 some editor, then be compiled, linked and executed, there are more ways
6299 to work with the GiNaC engine. Many people want to play with
6300 expressions interactively, as in traditional CASs. Currently, two such
6301 windows into GiNaC have been implemented and many more are possible: the
6302 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
6303 types to a command line and second, as a more consistent approach, an
6304 interactive interface to the Cint C++ interpreter has been put together
6305 (called GiNaC-cint) that allows an interactive scripting interface
6306 consistent with the C++ language. It is available from the usual GiNaC
6310 seamless integration: it is somewhere between difficult and impossible
6311 to call CAS functions from within a program written in C++ or any other
6312 programming language and vice versa. With GiNaC, your symbolic routines
6313 are part of your program. You can easily call third party libraries,
6314 e.g. for numerical evaluation or graphical interaction. All other
6315 approaches are much more cumbersome: they range from simply ignoring the
6316 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
6317 system (i.e. @emph{Yacas}).
6320 efficiency: often large parts of a program do not need symbolic
6321 calculations at all. Why use large integers for loop variables or
6322 arbitrary precision arithmetics where @code{int} and @code{double} are
6323 sufficient? For pure symbolic applications, GiNaC is comparable in
6324 speed with other CAS.
6329 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
6330 @c node-name, next, previous, up
6331 @section Disadvantages
6333 Of course it also has some disadvantages:
6338 advanced features: GiNaC cannot compete with a program like
6339 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
6340 which grows since 1981 by the work of dozens of programmers, with
6341 respect to mathematical features. Integration, factorization,
6342 non-trivial simplifications, limits etc. are missing in GiNaC (and are
6343 not planned for the near future).
6346 portability: While the GiNaC library itself is designed to avoid any
6347 platform dependent features (it should compile on any ANSI compliant C++
6348 compiler), the currently used version of the CLN library (fast large
6349 integer and arbitrary precision arithmetics) can only by compiled
6350 without hassle on systems with the C++ compiler from the GNU Compiler
6351 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
6352 macros to let the compiler gather all static initializations, which
6353 works for GNU C++ only. Feel free to contact the authors in case you
6354 really believe that you need to use a different compiler. We have
6355 occasionally used other compilers and may be able to give you advice.}
6356 GiNaC uses recent language features like explicit constructors, mutable
6357 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
6358 literally. Recent GCC versions starting at 2.95.3, although itself not
6359 yet ANSI compliant, support all needed features.
6364 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
6365 @c node-name, next, previous, up
6368 Why did we choose to implement GiNaC in C++ instead of Java or any other
6369 language? C++ is not perfect: type checking is not strict (casting is
6370 possible), separation between interface and implementation is not
6371 complete, object oriented design is not enforced. The main reason is
6372 the often scolded feature of operator overloading in C++. While it may
6373 be true that operating on classes with a @code{+} operator is rarely
6374 meaningful, it is perfectly suited for algebraic expressions. Writing
6375 @math{3x+5y} as @code{3*x+5*y} instead of
6376 @code{x.times(3).plus(y.times(5))} looks much more natural.
6377 Furthermore, the main developers are more familiar with C++ than with
6378 any other programming language.
6381 @node Internal Structures, Expressions are reference counted, Why C++? , Top
6382 @c node-name, next, previous, up
6383 @appendix Internal Structures
6386 * Expressions are reference counted::
6387 * Internal representation of products and sums::
6390 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
6391 @c node-name, next, previous, up
6392 @appendixsection Expressions are reference counted
6394 @cindex reference counting
6395 @cindex copy-on-write
6396 @cindex garbage collection
6397 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
6398 where the counter belongs to the algebraic objects derived from class
6399 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
6400 which @code{ex} contains an instance. If you understood that, you can safely
6401 skip the rest of this passage.
6403 Expressions are extremely light-weight since internally they work like
6404 handles to the actual representation. They really hold nothing more
6405 than a pointer to some other object. What this means in practice is
6406 that whenever you create two @code{ex} and set the second equal to the
6407 first no copying process is involved. Instead, the copying takes place
6408 as soon as you try to change the second. Consider the simple sequence
6413 #include <ginac/ginac.h>
6414 using namespace std;
6415 using namespace GiNaC;
6419 symbol x("x"), y("y"), z("z");
6422 e1 = sin(x + 2*y) + 3*z + 41;
6423 e2 = e1; // e2 points to same object as e1
6424 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
6425 e2 += 1; // e2 is copied into a new object
6426 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
6430 The line @code{e2 = e1;} creates a second expression pointing to the
6431 object held already by @code{e1}. The time involved for this operation
6432 is therefore constant, no matter how large @code{e1} was. Actual
6433 copying, however, must take place in the line @code{e2 += 1;} because
6434 @code{e1} and @code{e2} are not handles for the same object any more.
6435 This concept is called @dfn{copy-on-write semantics}. It increases
6436 performance considerably whenever one object occurs multiple times and
6437 represents a simple garbage collection scheme because when an @code{ex}
6438 runs out of scope its destructor checks whether other expressions handle
6439 the object it points to too and deletes the object from memory if that
6440 turns out not to be the case. A slightly less trivial example of
6441 differentiation using the chain-rule should make clear how powerful this
6446 symbol x("x"), y("y");
6450 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
6451 cout << e1 << endl // prints x+3*y
6452 << e2 << endl // prints (x+3*y)^3
6453 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
6457 Here, @code{e1} will actually be referenced three times while @code{e2}
6458 will be referenced two times. When the power of an expression is built,
6459 that expression needs not be copied. Likewise, since the derivative of
6460 a power of an expression can be easily expressed in terms of that
6461 expression, no copying of @code{e1} is involved when @code{e3} is
6462 constructed. So, when @code{e3} is constructed it will print as
6463 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
6464 holds a reference to @code{e2} and the factor in front is just
6467 As a user of GiNaC, you cannot see this mechanism of copy-on-write
6468 semantics. When you insert an expression into a second expression, the
6469 result behaves exactly as if the contents of the first expression were
6470 inserted. But it may be useful to remember that this is not what
6471 happens. Knowing this will enable you to write much more efficient
6472 code. If you still have an uncertain feeling with copy-on-write
6473 semantics, we recommend you have a look at the
6474 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
6475 Marshall Cline. Chapter 16 covers this issue and presents an
6476 implementation which is pretty close to the one in GiNaC.
6479 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
6480 @c node-name, next, previous, up
6481 @appendixsection Internal representation of products and sums
6483 @cindex representation
6486 @cindex @code{power}
6487 Although it should be completely transparent for the user of
6488 GiNaC a short discussion of this topic helps to understand the sources
6489 and also explain performance to a large degree. Consider the
6490 unexpanded symbolic expression
6492 $2d^3 \left( 4a + 5b - 3 \right)$
6495 @math{2*d^3*(4*a+5*b-3)}
6497 which could naively be represented by a tree of linear containers for
6498 addition and multiplication, one container for exponentiation with base
6499 and exponent and some atomic leaves of symbols and numbers in this
6504 @cindex pair-wise representation
6505 However, doing so results in a rather deeply nested tree which will
6506 quickly become inefficient to manipulate. We can improve on this by
6507 representing the sum as a sequence of terms, each one being a pair of a
6508 purely numeric multiplicative coefficient and its rest. In the same
6509 spirit we can store the multiplication as a sequence of terms, each
6510 having a numeric exponent and a possibly complicated base, the tree
6511 becomes much more flat:
6515 The number @code{3} above the symbol @code{d} shows that @code{mul}
6516 objects are treated similarly where the coefficients are interpreted as
6517 @emph{exponents} now. Addition of sums of terms or multiplication of
6518 products with numerical exponents can be coded to be very efficient with
6519 such a pair-wise representation. Internally, this handling is performed
6520 by most CAS in this way. It typically speeds up manipulations by an
6521 order of magnitude. The overall multiplicative factor @code{2} and the
6522 additive term @code{-3} look somewhat out of place in this
6523 representation, however, since they are still carrying a trivial
6524 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
6525 this is avoided by adding a field that carries an overall numeric
6526 coefficient. This results in the realistic picture of internal
6529 $2d^3 \left( 4a + 5b - 3 \right)$:
6532 @math{2*d^3*(4*a+5*b-3)}:
6538 This also allows for a better handling of numeric radicals, since
6539 @code{sqrt(2)} can now be carried along calculations. Now it should be
6540 clear, why both classes @code{add} and @code{mul} are derived from the
6541 same abstract class: the data representation is the same, only the
6542 semantics differs. In the class hierarchy, methods for polynomial
6543 expansion and the like are reimplemented for @code{add} and @code{mul},
6544 but the data structure is inherited from @code{expairseq}.
6547 @node Package Tools, ginac-config, Internal representation of products and sums, Top
6548 @c node-name, next, previous, up
6549 @appendix Package Tools
6551 If you are creating a software package that uses the GiNaC library,
6552 setting the correct command line options for the compiler and linker
6553 can be difficult. GiNaC includes two tools to make this process easier.
6556 * ginac-config:: A shell script to detect compiler and linker flags.
6557 * AM_PATH_GINAC:: Macro for GNU automake.
6561 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
6562 @c node-name, next, previous, up
6563 @section @command{ginac-config}
6564 @cindex ginac-config
6566 @command{ginac-config} is a shell script that you can use to determine
6567 the compiler and linker command line options required to compile and
6568 link a program with the GiNaC library.
6570 @command{ginac-config} takes the following flags:
6574 Prints out the version of GiNaC installed.
6576 Prints '-I' flags pointing to the installed header files.
6578 Prints out the linker flags necessary to link a program against GiNaC.
6579 @item --prefix[=@var{PREFIX}]
6580 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
6581 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
6582 Otherwise, prints out the configured value of @env{$prefix}.
6583 @item --exec-prefix[=@var{PREFIX}]
6584 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
6585 Otherwise, prints out the configured value of @env{$exec_prefix}.
6588 Typically, @command{ginac-config} will be used within a configure
6589 script, as described below. It, however, can also be used directly from
6590 the command line using backquotes to compile a simple program. For
6594 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
6597 This command line might expand to (for example):
6600 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
6601 -lginac -lcln -lstdc++
6604 Not only is the form using @command{ginac-config} easier to type, it will
6605 work on any system, no matter how GiNaC was configured.
6608 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
6609 @c node-name, next, previous, up
6610 @section @samp{AM_PATH_GINAC}
6611 @cindex AM_PATH_GINAC
6613 For packages configured using GNU automake, GiNaC also provides
6614 a macro to automate the process of checking for GiNaC.
6617 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
6625 Determines the location of GiNaC using @command{ginac-config}, which is
6626 either found in the user's path, or from the environment variable
6627 @env{GINACLIB_CONFIG}.
6630 Tests the installed libraries to make sure that their version
6631 is later than @var{MINIMUM-VERSION}. (A default version will be used
6635 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
6636 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
6637 variable to the output of @command{ginac-config --libs}, and calls
6638 @samp{AC_SUBST()} for these variables so they can be used in generated
6639 makefiles, and then executes @var{ACTION-IF-FOUND}.
6642 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
6643 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
6647 This macro is in file @file{ginac.m4} which is installed in
6648 @file{$datadir/aclocal}. Note that if automake was installed with a
6649 different @samp{--prefix} than GiNaC, you will either have to manually
6650 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
6651 aclocal the @samp{-I} option when running it.
6654 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
6655 * Example package:: Example of a package using AM_PATH_GINAC.
6659 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
6660 @c node-name, next, previous, up
6661 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
6663 Simply make sure that @command{ginac-config} is in your path, and run
6664 the configure script.
6671 The directory where the GiNaC libraries are installed needs
6672 to be found by your system's dynamic linker.
6674 This is generally done by
6677 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
6683 setting the environment variable @env{LD_LIBRARY_PATH},
6686 or, as a last resort,
6689 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
6690 running configure, for instance:
6693 LDFLAGS=-R/home/cbauer/lib ./configure
6698 You can also specify a @command{ginac-config} not in your path by
6699 setting the @env{GINACLIB_CONFIG} environment variable to the
6700 name of the executable
6703 If you move the GiNaC package from its installed location,
6704 you will either need to modify @command{ginac-config} script
6705 manually to point to the new location or rebuild GiNaC.
6716 --with-ginac-prefix=@var{PREFIX}
6717 --with-ginac-exec-prefix=@var{PREFIX}
6720 are provided to override the prefix and exec-prefix that were stored
6721 in the @command{ginac-config} shell script by GiNaC's configure. You are
6722 generally better off configuring GiNaC with the right path to begin with.
6726 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
6727 @c node-name, next, previous, up
6728 @subsection Example of a package using @samp{AM_PATH_GINAC}
6730 The following shows how to build a simple package using automake
6731 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
6734 #include <ginac/ginac.h>
6738 GiNaC::symbol x("x");
6739 GiNaC::ex a = GiNaC::sin(x);
6740 std::cout << "Derivative of " << a
6741 << " is " << a.diff(x) << std::endl;
6746 You should first read the introductory portions of the automake
6747 Manual, if you are not already familiar with it.
6749 Two files are needed, @file{configure.in}, which is used to build the
6753 dnl Process this file with autoconf to produce a configure script.
6755 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
6761 AM_PATH_GINAC(0.9.0, [
6762 LIBS="$LIBS $GINACLIB_LIBS"
6763 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
6764 ], AC_MSG_ERROR([need to have GiNaC installed]))
6769 The only command in this which is not standard for automake
6770 is the @samp{AM_PATH_GINAC} macro.
6772 That command does the following: If a GiNaC version greater or equal
6773 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
6774 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
6775 the error message `need to have GiNaC installed'
6777 And the @file{Makefile.am}, which will be used to build the Makefile.
6780 ## Process this file with automake to produce Makefile.in
6781 bin_PROGRAMS = simple
6782 simple_SOURCES = simple.cpp
6785 This @file{Makefile.am}, says that we are building a single executable,
6786 from a single source file @file{simple.cpp}. Since every program
6787 we are building uses GiNaC we simply added the GiNaC options
6788 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
6789 want to specify them on a per-program basis: for instance by
6793 simple_LDADD = $(GINACLIB_LIBS)
6794 INCLUDES = $(GINACLIB_CPPFLAGS)
6797 to the @file{Makefile.am}.
6799 To try this example out, create a new directory and add the three
6802 Now execute the following commands:
6805 $ automake --add-missing
6810 You now have a package that can be built in the normal fashion
6819 @node Bibliography, Concept Index, Example package, Top
6820 @c node-name, next, previous, up
6821 @appendix Bibliography
6826 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
6829 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
6832 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
6835 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
6838 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
6839 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
6842 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
6843 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
6844 Academic Press, London
6847 @cite{Computer Algebra Systems - A Practical Guide},
6848 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
6851 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
6852 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
6855 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
6856 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
6859 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
6864 @node Concept Index, , Bibliography, Top
6865 @c node-name, next, previous, up
6866 @unnumbered Concept Index