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
912 @item @code{structure} @tab Template for user-defined classes
917 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
918 @c node-name, next, previous, up
920 @cindex @code{symbol} (class)
921 @cindex hierarchy of classes
924 Symbols are for symbolic manipulation what atoms are for chemistry. You
925 can declare objects of class @code{symbol} as any other object simply by
926 saying @code{symbol x,y;}. There is, however, a catch in here having to
927 do with the fact that C++ is a compiled language. The information about
928 the symbol's name is thrown away by the compiler but at a later stage
929 you may want to print expressions holding your symbols. In order to
930 avoid confusion GiNaC's symbols are able to know their own name. This
931 is accomplished by declaring its name for output at construction time in
932 the fashion @code{symbol x("x");}. If you declare a symbol using the
933 default constructor (i.e. without string argument) the system will deal
934 out a unique name. That name may not be suitable for printing but for
935 internal routines when no output is desired it is often enough. We'll
936 come across examples of such symbols later in this tutorial.
938 This implies that the strings passed to symbols at construction time may
939 not be used for comparing two of them. It is perfectly legitimate to
940 write @code{symbol x("x"),y("x");} but it is likely to lead into
941 trouble. Here, @code{x} and @code{y} are different symbols and
942 statements like @code{x-y} will not be simplified to zero although the
943 output @code{x-x} looks funny. Such output may also occur when there
944 are two different symbols in two scopes, for instance when you call a
945 function that declares a symbol with a name already existent in a symbol
946 in the calling function. Again, comparing them (using @code{operator==}
947 for instance) will always reveal their difference. Watch out, please.
949 @cindex @code{subs()}
950 Although symbols can be assigned expressions for internal reasons, you
951 should not do it (and we are not going to tell you how it is done). If
952 you want to replace a symbol with something else in an expression, you
953 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
956 @node Numbers, Constants, Symbols, Basic Concepts
957 @c node-name, next, previous, up
959 @cindex @code{numeric} (class)
965 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
966 The classes therein serve as foundation classes for GiNaC. CLN stands
967 for Class Library for Numbers or alternatively for Common Lisp Numbers.
968 In order to find out more about CLN's internals, the reader is referred to
969 the documentation of that library. @inforef{Introduction, , cln}, for
970 more information. Suffice to say that it is by itself build on top of
971 another library, the GNU Multiple Precision library GMP, which is an
972 extremely fast library for arbitrary long integers and rationals as well
973 as arbitrary precision floating point numbers. It is very commonly used
974 by several popular cryptographic applications. CLN extends GMP by
975 several useful things: First, it introduces the complex number field
976 over either reals (i.e. floating point numbers with arbitrary precision)
977 or rationals. Second, it automatically converts rationals to integers
978 if the denominator is unity and complex numbers to real numbers if the
979 imaginary part vanishes and also correctly treats algebraic functions.
980 Third it provides good implementations of state-of-the-art algorithms
981 for all trigonometric and hyperbolic functions as well as for
982 calculation of some useful constants.
984 The user can construct an object of class @code{numeric} in several
985 ways. The following example shows the four most important constructors.
986 It uses construction from C-integer, construction of fractions from two
987 integers, construction from C-float and construction from a string:
991 #include <ginac/ginac.h>
992 using namespace GiNaC;
996 numeric two = 2; // exact integer 2
997 numeric r(2,3); // exact fraction 2/3
998 numeric e(2.71828); // floating point number
999 numeric p = "3.14159265358979323846"; // constructor from string
1000 // Trott's constant in scientific notation:
1001 numeric trott("1.0841015122311136151E-2");
1003 std::cout << two*p << std::endl; // floating point 6.283...
1008 @cindex complex numbers
1009 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1014 numeric z1 = 2-3*I; // exact complex number 2-3i
1015 numeric z2 = 5.9+1.6*I; // complex floating point number
1019 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1020 This would, however, call C's built-in operator @code{/} for integers
1021 first and result in a numeric holding a plain integer 1. @strong{Never
1022 use the operator @code{/} on integers} unless you know exactly what you
1023 are doing! Use the constructor from two integers instead, as shown in
1024 the example above. Writing @code{numeric(1)/2} may look funny but works
1027 @cindex @code{Digits}
1029 We have seen now the distinction between exact numbers and floating
1030 point numbers. Clearly, the user should never have to worry about
1031 dynamically created exact numbers, since their `exactness' always
1032 determines how they ought to be handled, i.e. how `long' they are. The
1033 situation is different for floating point numbers. Their accuracy is
1034 controlled by one @emph{global} variable, called @code{Digits}. (For
1035 those readers who know about Maple: it behaves very much like Maple's
1036 @code{Digits}). All objects of class numeric that are constructed from
1037 then on will be stored with a precision matching that number of decimal
1042 #include <ginac/ginac.h>
1043 using namespace std;
1044 using namespace GiNaC;
1048 numeric three(3.0), one(1.0);
1049 numeric x = one/three;
1051 cout << "in " << Digits << " digits:" << endl;
1053 cout << Pi.evalf() << endl;
1065 The above example prints the following output to screen:
1069 0.33333333333333333334
1070 3.1415926535897932385
1072 0.33333333333333333333333333333333333333333333333333333333333333333334
1073 3.1415926535897932384626433832795028841971693993751058209749445923078
1077 Note that the last number is not necessarily rounded as you would
1078 naively expect it to be rounded in the decimal system. But note also,
1079 that in both cases you got a couple of extra digits. This is because
1080 numbers are internally stored by CLN as chunks of binary digits in order
1081 to match your machine's word size and to not waste precision. Thus, on
1082 architectures with different word size, the above output might even
1083 differ with regard to actually computed digits.
1085 It should be clear that objects of class @code{numeric} should be used
1086 for constructing numbers or for doing arithmetic with them. The objects
1087 one deals with most of the time are the polymorphic expressions @code{ex}.
1089 @subsection Tests on numbers
1091 Once you have declared some numbers, assigned them to expressions and
1092 done some arithmetic with them it is frequently desired to retrieve some
1093 kind of information from them like asking whether that number is
1094 integer, rational, real or complex. For those cases GiNaC provides
1095 several useful methods. (Internally, they fall back to invocations of
1096 certain CLN functions.)
1098 As an example, let's construct some rational number, multiply it with
1099 some multiple of its denominator and test what comes out:
1103 #include <ginac/ginac.h>
1104 using namespace std;
1105 using namespace GiNaC;
1107 // some very important constants:
1108 const numeric twentyone(21);
1109 const numeric ten(10);
1110 const numeric five(5);
1114 numeric answer = twentyone;
1117 cout << answer.is_integer() << endl; // false, it's 21/5
1119 cout << answer.is_integer() << endl; // true, it's 42 now!
1123 Note that the variable @code{answer} is constructed here as an integer
1124 by @code{numeric}'s copy constructor but in an intermediate step it
1125 holds a rational number represented as integer numerator and integer
1126 denominator. When multiplied by 10, the denominator becomes unity and
1127 the result is automatically converted to a pure integer again.
1128 Internally, the underlying CLN is responsible for this behavior and we
1129 refer the reader to CLN's documentation. Suffice to say that
1130 the same behavior applies to complex numbers as well as return values of
1131 certain functions. Complex numbers are automatically converted to real
1132 numbers if the imaginary part becomes zero. The full set of tests that
1133 can be applied is listed in the following table.
1136 @multitable @columnfractions .30 .70
1137 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1138 @item @code{.is_zero()}
1139 @tab @dots{}equal to zero
1140 @item @code{.is_positive()}
1141 @tab @dots{}not complex and greater than 0
1142 @item @code{.is_integer()}
1143 @tab @dots{}a (non-complex) integer
1144 @item @code{.is_pos_integer()}
1145 @tab @dots{}an integer and greater than 0
1146 @item @code{.is_nonneg_integer()}
1147 @tab @dots{}an integer and greater equal 0
1148 @item @code{.is_even()}
1149 @tab @dots{}an even integer
1150 @item @code{.is_odd()}
1151 @tab @dots{}an odd integer
1152 @item @code{.is_prime()}
1153 @tab @dots{}a prime integer (probabilistic primality test)
1154 @item @code{.is_rational()}
1155 @tab @dots{}an exact rational number (integers are rational, too)
1156 @item @code{.is_real()}
1157 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1158 @item @code{.is_cinteger()}
1159 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1160 @item @code{.is_crational()}
1161 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1166 @node Constants, Fundamental containers, Numbers, Basic Concepts
1167 @c node-name, next, previous, up
1169 @cindex @code{constant} (class)
1172 @cindex @code{Catalan}
1173 @cindex @code{Euler}
1174 @cindex @code{evalf()}
1175 Constants behave pretty much like symbols except that they return some
1176 specific number when the method @code{.evalf()} is called.
1178 The predefined known constants are:
1181 @multitable @columnfractions .14 .30 .56
1182 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1184 @tab Archimedes' constant
1185 @tab 3.14159265358979323846264338327950288
1186 @item @code{Catalan}
1187 @tab Catalan's constant
1188 @tab 0.91596559417721901505460351493238411
1190 @tab Euler's (or Euler-Mascheroni) constant
1191 @tab 0.57721566490153286060651209008240243
1196 @node Fundamental containers, Lists, Constants, Basic Concepts
1197 @c node-name, next, previous, up
1198 @section Sums, products and powers
1202 @cindex @code{power}
1204 Simple rational expressions are written down in GiNaC pretty much like
1205 in other CAS or like expressions involving numerical variables in C.
1206 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1207 been overloaded to achieve this goal. When you run the following
1208 code snippet, the constructor for an object of type @code{mul} is
1209 automatically called to hold the product of @code{a} and @code{b} and
1210 then the constructor for an object of type @code{add} is called to hold
1211 the sum of that @code{mul} object and the number one:
1215 symbol a("a"), b("b");
1220 @cindex @code{pow()}
1221 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1222 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1223 construction is necessary since we cannot safely overload the constructor
1224 @code{^} in C++ to construct a @code{power} object. If we did, it would
1225 have several counterintuitive and undesired effects:
1229 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1231 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1232 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1233 interpret this as @code{x^(a^b)}.
1235 Also, expressions involving integer exponents are very frequently used,
1236 which makes it even more dangerous to overload @code{^} since it is then
1237 hard to distinguish between the semantics as exponentiation and the one
1238 for exclusive or. (It would be embarrassing to return @code{1} where one
1239 has requested @code{2^3}.)
1242 @cindex @command{ginsh}
1243 All effects are contrary to mathematical notation and differ from the
1244 way most other CAS handle exponentiation, therefore overloading @code{^}
1245 is ruled out for GiNaC's C++ part. The situation is different in
1246 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1247 that the other frequently used exponentiation operator @code{**} does
1248 not exist at all in C++).
1250 To be somewhat more precise, objects of the three classes described
1251 here, are all containers for other expressions. An object of class
1252 @code{power} is best viewed as a container with two slots, one for the
1253 basis, one for the exponent. All valid GiNaC expressions can be
1254 inserted. However, basic transformations like simplifying
1255 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1256 when this is mathematically possible. If we replace the outer exponent
1257 three in the example by some symbols @code{a}, the simplification is not
1258 safe and will not be performed, since @code{a} might be @code{1/2} and
1261 Objects of type @code{add} and @code{mul} are containers with an
1262 arbitrary number of slots for expressions to be inserted. Again, simple
1263 and safe simplifications are carried out like transforming
1264 @code{3*x+4-x} to @code{2*x+4}.
1267 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1268 @c node-name, next, previous, up
1269 @section Lists of expressions
1270 @cindex @code{lst} (class)
1272 @cindex @code{nops()}
1274 @cindex @code{append()}
1275 @cindex @code{prepend()}
1276 @cindex @code{remove_first()}
1277 @cindex @code{remove_last()}
1278 @cindex @code{remove_all()}
1280 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1281 expressions. They are not as ubiquitous as in many other computer algebra
1282 packages, but are sometimes used to supply a variable number of arguments of
1283 the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
1284 so you should have a basic understanding of them.
1286 Lists of up to 16 expressions can be directly constructed from single
1291 symbol x("x"), y("y");
1292 lst l(x, 2, y, x+y);
1293 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1297 Use the @code{nops()} method to determine the size (number of expressions) of
1298 a list and the @code{op()} method or the @code{[]} operator to access
1299 individual elements:
1303 cout << l.nops() << endl; // prints '4'
1304 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1308 As with the standard @code{list<T>} container, accessing random elements of a
1309 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1310 sequential access to the elements of a list is possible with the
1311 iterator types provided by the @code{lst} class:
1314 typedef ... lst::const_iterator;
1315 typedef ... lst::const_reverse_iterator;
1316 lst::const_iterator lst::begin() const;
1317 lst::const_iterator lst::end() const;
1318 lst::const_reverse_iterator lst::rbegin() const;
1319 lst::const_reverse_iterator lst::rend() const;
1322 For example, to print the elements of a list individually you can use:
1327 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1332 which is one order faster than
1337 for (size_t i = 0; i < l.nops(); ++i)
1338 cout << l.op(i) << endl;
1342 These iterators also allow you to use some of the algorithms provided by
1343 the C++ standard library:
1347 // print the elements of the list (requires #include <iterator>)
1348 copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1350 // sum up the elements of the list (requires #include <numeric>)
1351 ex sum = accumulate(l.begin(), l.end(), ex(0));
1352 cout << sum << endl; // prints '2+2*x+2*y'
1356 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1357 (the only other one is @code{matrix}). You can modify single elements:
1361 l[1] = 42; // l is now @{x, 42, y, x+y@}
1362 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1366 You can append or prepend an expression to a list with the @code{append()}
1367 and @code{prepend()} methods:
1371 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1372 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1376 You can remove the first or last element of a list with @code{remove_first()}
1377 and @code{remove_last()}:
1381 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1382 l.remove_last(); // l is now @{x, 7, y, x+y@}
1386 You can remove all the elements of a list with @code{remove_all()}:
1390 l.remove_all(); // l is now empty
1394 You can bring the elements of a list into a canonical order with @code{sort()}:
1398 lst l1(x, 2, y, x+y);
1399 lst l2(2, x+y, x, y);
1402 // l1 and l2 are now equal
1406 Finally, you can remove all but the first element of consecutive groups of
1407 elements with @code{unique()}:
1411 lst l3(x, 2, 2, 2, y, x+y, y+x);
1412 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1417 @node Mathematical functions, Relations, Lists, Basic Concepts
1418 @c node-name, next, previous, up
1419 @section Mathematical functions
1420 @cindex @code{function} (class)
1421 @cindex trigonometric function
1422 @cindex hyperbolic function
1424 There are quite a number of useful functions hard-wired into GiNaC. For
1425 instance, all trigonometric and hyperbolic functions are implemented
1426 (@xref{Built-in Functions}, for a complete list).
1428 These functions (better called @emph{pseudofunctions}) are all objects
1429 of class @code{function}. They accept one or more expressions as
1430 arguments and return one expression. If the arguments are not
1431 numerical, the evaluation of the function may be halted, as it does in
1432 the next example, showing how a function returns itself twice and
1433 finally an expression that may be really useful:
1435 @cindex Gamma function
1436 @cindex @code{subs()}
1439 symbol x("x"), y("y");
1441 cout << tgamma(foo) << endl;
1442 // -> tgamma(x+(1/2)*y)
1443 ex bar = foo.subs(y==1);
1444 cout << tgamma(bar) << endl;
1446 ex foobar = bar.subs(x==7);
1447 cout << tgamma(foobar) << endl;
1448 // -> (135135/128)*Pi^(1/2)
1452 Besides evaluation most of these functions allow differentiation, series
1453 expansion and so on. Read the next chapter in order to learn more about
1456 It must be noted that these pseudofunctions are created by inline
1457 functions, where the argument list is templated. This means that
1458 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1459 @code{sin(ex(1))} and will therefore not result in a floating point
1460 number. Unless of course the function prototype is explicitly
1461 overridden -- which is the case for arguments of type @code{numeric}
1462 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1463 point number of class @code{numeric} you should call
1464 @code{sin(numeric(1))}. This is almost the same as calling
1465 @code{sin(1).evalf()} except that the latter will return a numeric
1466 wrapped inside an @code{ex}.
1469 @node Relations, Matrices, Mathematical functions, Basic Concepts
1470 @c node-name, next, previous, up
1472 @cindex @code{relational} (class)
1474 Sometimes, a relation holding between two expressions must be stored
1475 somehow. The class @code{relational} is a convenient container for such
1476 purposes. A relation is by definition a container for two @code{ex} and
1477 a relation between them that signals equality, inequality and so on.
1478 They are created by simply using the C++ operators @code{==}, @code{!=},
1479 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1481 @xref{Mathematical functions}, for examples where various applications
1482 of the @code{.subs()} method show how objects of class relational are
1483 used as arguments. There they provide an intuitive syntax for
1484 substitutions. They are also used as arguments to the @code{ex::series}
1485 method, where the left hand side of the relation specifies the variable
1486 to expand in and the right hand side the expansion point. They can also
1487 be used for creating systems of equations that are to be solved for
1488 unknown variables. But the most common usage of objects of this class
1489 is rather inconspicuous in statements of the form @code{if
1490 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1491 conversion from @code{relational} to @code{bool} takes place. Note,
1492 however, that @code{==} here does not perform any simplifications, hence
1493 @code{expand()} must be called explicitly.
1496 @node Matrices, Indexed objects, Relations, Basic Concepts
1497 @c node-name, next, previous, up
1499 @cindex @code{matrix} (class)
1501 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1502 matrix with @math{m} rows and @math{n} columns are accessed with two
1503 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1504 second one in the range 0@dots{}@math{n-1}.
1506 There are a couple of ways to construct matrices, with or without preset
1509 @cindex @code{lst_to_matrix()}
1510 @cindex @code{diag_matrix()}
1511 @cindex @code{unit_matrix()}
1512 @cindex @code{symbolic_matrix()}
1514 matrix::matrix(unsigned r, unsigned c);
1515 matrix::matrix(unsigned r, unsigned c, const lst & l);
1516 ex lst_to_matrix(const lst & l);
1517 ex diag_matrix(const lst & l);
1518 ex unit_matrix(unsigned x);
1519 ex unit_matrix(unsigned r, unsigned c);
1520 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1521 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1524 The first two functions are @code{matrix} constructors which create a matrix
1525 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1526 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1527 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1528 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1529 constructs a diagonal matrix given the list of diagonal elements.
1530 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1531 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1532 with newly generated symbols made of the specified base name and the
1533 position of each element in the matrix.
1535 Matrix elements can be accessed and set using the parenthesis (function call)
1539 const ex & matrix::operator()(unsigned r, unsigned c) const;
1540 ex & matrix::operator()(unsigned r, unsigned c);
1543 It is also possible to access the matrix elements in a linear fashion with
1544 the @code{op()} method. But C++-style subscripting with square brackets
1545 @samp{[]} is not available.
1547 Here are a couple of examples of constructing matrices:
1551 symbol a("a"), b("b");
1559 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1562 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1565 cout << diag_matrix(lst(a, b)) << endl;
1568 cout << unit_matrix(3) << endl;
1569 // -> [[1,0,0],[0,1,0],[0,0,1]]
1571 cout << symbolic_matrix(2, 3, "x") << endl;
1572 // -> [[x00,x01,x02],[x10,x11,x12]]
1576 @cindex @code{transpose()}
1577 There are three ways to do arithmetic with matrices. The first (and most
1578 direct one) is to use the methods provided by the @code{matrix} class:
1581 matrix matrix::add(const matrix & other) const;
1582 matrix matrix::sub(const matrix & other) const;
1583 matrix matrix::mul(const matrix & other) const;
1584 matrix matrix::mul_scalar(const ex & other) const;
1585 matrix matrix::pow(const ex & expn) const;
1586 matrix matrix::transpose() const;
1589 All of these methods return the result as a new matrix object. Here is an
1590 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1595 matrix A(2, 2, lst(1, 2, 3, 4));
1596 matrix B(2, 2, lst(-1, 0, 2, 1));
1597 matrix C(2, 2, lst(8, 4, 2, 1));
1599 matrix result = A.mul(B).sub(C.mul_scalar(2));
1600 cout << result << endl;
1601 // -> [[-13,-6],[1,2]]
1606 @cindex @code{evalm()}
1607 The second (and probably the most natural) way is to construct an expression
1608 containing matrices with the usual arithmetic operators and @code{pow()}.
1609 For efficiency reasons, expressions with sums, products and powers of
1610 matrices are not automatically evaluated in GiNaC. You have to call the
1614 ex ex::evalm() const;
1617 to obtain the result:
1624 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1625 cout << e.evalm() << endl;
1626 // -> [[-13,-6],[1,2]]
1631 The non-commutativity of the product @code{A*B} in this example is
1632 automatically recognized by GiNaC. There is no need to use a special
1633 operator here. @xref{Non-commutative objects}, for more information about
1634 dealing with non-commutative expressions.
1636 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1637 to perform the arithmetic:
1642 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1643 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1645 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1646 cout << e.simplify_indexed() << endl;
1647 // -> [[-13,-6],[1,2]].i.j
1651 Using indices is most useful when working with rectangular matrices and
1652 one-dimensional vectors because you don't have to worry about having to
1653 transpose matrices before multiplying them. @xref{Indexed objects}, for
1654 more information about using matrices with indices, and about indices in
1657 The @code{matrix} class provides a couple of additional methods for
1658 computing determinants, traces, and characteristic polynomials:
1660 @cindex @code{determinant()}
1661 @cindex @code{trace()}
1662 @cindex @code{charpoly()}
1664 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1665 ex matrix::trace() const;
1666 ex matrix::charpoly(const symbol & lambda) const;
1669 The @samp{algo} argument of @code{determinant()} allows to select
1670 between different algorithms for calculating the determinant. The
1671 asymptotic speed (as parametrized by the matrix size) can greatly differ
1672 between those algorithms, depending on the nature of the matrix'
1673 entries. The possible values are defined in the @file{flags.h} header
1674 file. By default, GiNaC uses a heuristic to automatically select an
1675 algorithm that is likely (but not guaranteed) to give the result most
1678 @cindex @code{inverse()}
1679 @cindex @code{solve()}
1680 Matrices may also be inverted using the @code{ex matrix::inverse()}
1681 method and linear systems may be solved with:
1684 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1687 Assuming the matrix object this method is applied on is an @code{m}
1688 times @code{n} matrix, then @code{vars} must be a @code{n} times
1689 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1690 times @code{p} matrix. The returned matrix then has dimension @code{n}
1691 times @code{p} and in the case of an underdetermined system will still
1692 contain some of the indeterminates from @code{vars}. If the system is
1693 overdetermined, an exception is thrown.
1696 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1697 @c node-name, next, previous, up
1698 @section Indexed objects
1700 GiNaC allows you to handle expressions containing general indexed objects in
1701 arbitrary spaces. It is also able to canonicalize and simplify such
1702 expressions and perform symbolic dummy index summations. There are a number
1703 of predefined indexed objects provided, like delta and metric tensors.
1705 There are few restrictions placed on indexed objects and their indices and
1706 it is easy to construct nonsense expressions, but our intention is to
1707 provide a general framework that allows you to implement algorithms with
1708 indexed quantities, getting in the way as little as possible.
1710 @cindex @code{idx} (class)
1711 @cindex @code{indexed} (class)
1712 @subsection Indexed quantities and their indices
1714 Indexed expressions in GiNaC are constructed of two special types of objects,
1715 @dfn{index objects} and @dfn{indexed objects}.
1719 @cindex contravariant
1722 @item Index objects are of class @code{idx} or a subclass. Every index has
1723 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1724 the index lives in) which can both be arbitrary expressions but are usually
1725 a number or a simple symbol. In addition, indices of class @code{varidx} have
1726 a @dfn{variance} (they can be co- or contravariant), and indices of class
1727 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1729 @item Indexed objects are of class @code{indexed} or a subclass. They
1730 contain a @dfn{base expression} (which is the expression being indexed), and
1731 one or more indices.
1735 @strong{Note:} when printing expressions, covariant indices and indices
1736 without variance are denoted @samp{.i} while contravariant indices are
1737 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1738 value. In the following, we are going to use that notation in the text so
1739 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1740 not visible in the output.
1742 A simple example shall illustrate the concepts:
1746 #include <ginac/ginac.h>
1747 using namespace std;
1748 using namespace GiNaC;
1752 symbol i_sym("i"), j_sym("j");
1753 idx i(i_sym, 3), j(j_sym, 3);
1756 cout << indexed(A, i, j) << endl;
1758 cout << index_dimensions << indexed(A, i, j) << endl;
1760 cout << dflt; // reset cout to default output format (dimensions hidden)
1764 The @code{idx} constructor takes two arguments, the index value and the
1765 index dimension. First we define two index objects, @code{i} and @code{j},
1766 both with the numeric dimension 3. The value of the index @code{i} is the
1767 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1768 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1769 construct an expression containing one indexed object, @samp{A.i.j}. It has
1770 the symbol @code{A} as its base expression and the two indices @code{i} and
1773 The dimensions of indices are normally not visible in the output, but one
1774 can request them to be printed with the @code{index_dimensions} manipulator,
1777 Note the difference between the indices @code{i} and @code{j} which are of
1778 class @code{idx}, and the index values which are the symbols @code{i_sym}
1779 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1780 or numbers but must be index objects. For example, the following is not
1781 correct and will raise an exception:
1784 symbol i("i"), j("j");
1785 e = indexed(A, i, j); // ERROR: indices must be of type idx
1788 You can have multiple indexed objects in an expression, index values can
1789 be numeric, and index dimensions symbolic:
1793 symbol B("B"), dim("dim");
1794 cout << 4 * indexed(A, i)
1795 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1800 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1801 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1802 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1803 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1804 @code{simplify_indexed()} for that, see below).
1806 In fact, base expressions, index values and index dimensions can be
1807 arbitrary expressions:
1811 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1816 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1817 get an error message from this but you will probably not be able to do
1818 anything useful with it.
1820 @cindex @code{get_value()}
1821 @cindex @code{get_dimension()}
1825 ex idx::get_value();
1826 ex idx::get_dimension();
1829 return the value and dimension of an @code{idx} object. If you have an index
1830 in an expression, such as returned by calling @code{.op()} on an indexed
1831 object, you can get a reference to the @code{idx} object with the function
1832 @code{ex_to<idx>()} on the expression.
1834 There are also the methods
1837 bool idx::is_numeric();
1838 bool idx::is_symbolic();
1839 bool idx::is_dim_numeric();
1840 bool idx::is_dim_symbolic();
1843 for checking whether the value and dimension are numeric or symbolic
1844 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1845 About Expressions}) returns information about the index value.
1847 @cindex @code{varidx} (class)
1848 If you need co- and contravariant indices, use the @code{varidx} class:
1852 symbol mu_sym("mu"), nu_sym("nu");
1853 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1854 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1856 cout << indexed(A, mu, nu) << endl;
1858 cout << indexed(A, mu_co, nu) << endl;
1860 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1865 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1866 co- or contravariant. The default is a contravariant (upper) index, but
1867 this can be overridden by supplying a third argument to the @code{varidx}
1868 constructor. The two methods
1871 bool varidx::is_covariant();
1872 bool varidx::is_contravariant();
1875 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1876 to get the object reference from an expression). There's also the very useful
1880 ex varidx::toggle_variance();
1883 which makes a new index with the same value and dimension but the opposite
1884 variance. By using it you only have to define the index once.
1886 @cindex @code{spinidx} (class)
1887 The @code{spinidx} class provides dotted and undotted variant indices, as
1888 used in the Weyl-van-der-Waerden spinor formalism:
1892 symbol K("K"), C_sym("C"), D_sym("D");
1893 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1894 // contravariant, undotted
1895 spinidx C_co(C_sym, 2, true); // covariant index
1896 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1897 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1899 cout << indexed(K, C, D) << endl;
1901 cout << indexed(K, C_co, D_dot) << endl;
1903 cout << indexed(K, D_co_dot, D) << endl;
1908 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1909 dotted or undotted. The default is undotted but this can be overridden by
1910 supplying a fourth argument to the @code{spinidx} constructor. The two
1914 bool spinidx::is_dotted();
1915 bool spinidx::is_undotted();
1918 allow you to check whether or not a @code{spinidx} object is dotted (use
1919 @code{ex_to<spinidx>()} to get the object reference from an expression).
1920 Finally, the two methods
1923 ex spinidx::toggle_dot();
1924 ex spinidx::toggle_variance_dot();
1927 create a new index with the same value and dimension but opposite dottedness
1928 and the same or opposite variance.
1930 @subsection Substituting indices
1932 @cindex @code{subs()}
1933 Sometimes you will want to substitute one symbolic index with another
1934 symbolic or numeric index, for example when calculating one specific element
1935 of a tensor expression. This is done with the @code{.subs()} method, as it
1936 is done for symbols (see @ref{Substituting Expressions}).
1938 You have two possibilities here. You can either substitute the whole index
1939 by another index or expression:
1943 ex e = indexed(A, mu_co);
1944 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1945 // -> A.mu becomes A~nu
1946 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1947 // -> A.mu becomes A~0
1948 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1949 // -> A.mu becomes A.0
1953 The third example shows that trying to replace an index with something that
1954 is not an index will substitute the index value instead.
1956 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1961 ex e = indexed(A, mu_co);
1962 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1963 // -> A.mu becomes A.nu
1964 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1965 // -> A.mu becomes A.0
1969 As you see, with the second method only the value of the index will get
1970 substituted. Its other properties, including its dimension, remain unchanged.
1971 If you want to change the dimension of an index you have to substitute the
1972 whole index by another one with the new dimension.
1974 Finally, substituting the base expression of an indexed object works as
1979 ex e = indexed(A, mu_co);
1980 cout << e << " becomes " << e.subs(A == A+B) << endl;
1981 // -> A.mu becomes (B+A).mu
1985 @subsection Symmetries
1986 @cindex @code{symmetry} (class)
1987 @cindex @code{sy_none()}
1988 @cindex @code{sy_symm()}
1989 @cindex @code{sy_anti()}
1990 @cindex @code{sy_cycl()}
1992 Indexed objects can have certain symmetry properties with respect to their
1993 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1994 that is constructed with the helper functions
1997 symmetry sy_none(...);
1998 symmetry sy_symm(...);
1999 symmetry sy_anti(...);
2000 symmetry sy_cycl(...);
2003 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2004 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2005 represents a cyclic symmetry. Each of these functions accepts up to four
2006 arguments which can be either symmetry objects themselves or unsigned integer
2007 numbers that represent an index position (counting from 0). A symmetry
2008 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2009 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2012 Here are some examples of symmetry definitions:
2017 e = indexed(A, i, j);
2018 e = indexed(A, sy_none(), i, j); // equivalent
2019 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2021 // Symmetric in all three indices:
2022 e = indexed(A, sy_symm(), i, j, k);
2023 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2024 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2025 // different canonical order
2027 // Symmetric in the first two indices only:
2028 e = indexed(A, sy_symm(0, 1), i, j, k);
2029 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2031 // Antisymmetric in the first and last index only (index ranges need not
2033 e = indexed(A, sy_anti(0, 2), i, j, k);
2034 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2036 // An example of a mixed symmetry: antisymmetric in the first two and
2037 // last two indices, symmetric when swapping the first and last index
2038 // pairs (like the Riemann curvature tensor):
2039 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2041 // Cyclic symmetry in all three indices:
2042 e = indexed(A, sy_cycl(), i, j, k);
2043 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2045 // The following examples are invalid constructions that will throw
2046 // an exception at run time.
2048 // An index may not appear multiple times:
2049 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2050 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2052 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2053 // same number of indices:
2054 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2056 // And of course, you cannot specify indices which are not there:
2057 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2061 If you need to specify more than four indices, you have to use the
2062 @code{.add()} method of the @code{symmetry} class. For example, to specify
2063 full symmetry in the first six indices you would write
2064 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2066 If an indexed object has a symmetry, GiNaC will automatically bring the
2067 indices into a canonical order which allows for some immediate simplifications:
2071 cout << indexed(A, sy_symm(), i, j)
2072 + indexed(A, sy_symm(), j, i) << endl;
2074 cout << indexed(B, sy_anti(), i, j)
2075 + indexed(B, sy_anti(), j, i) << endl;
2077 cout << indexed(B, sy_anti(), i, j, k)
2078 - indexed(B, sy_anti(), j, k, i) << endl;
2083 @cindex @code{get_free_indices()}
2085 @subsection Dummy indices
2087 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2088 that a summation over the index range is implied. Symbolic indices which are
2089 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2090 dummy nor free indices.
2092 To be recognized as a dummy index pair, the two indices must be of the same
2093 class and their value must be the same single symbol (an index like
2094 @samp{2*n+1} is never a dummy index). If the indices are of class
2095 @code{varidx} they must also be of opposite variance; if they are of class
2096 @code{spinidx} they must be both dotted or both undotted.
2098 The method @code{.get_free_indices()} returns a vector containing the free
2099 indices of an expression. It also checks that the free indices of the terms
2100 of a sum are consistent:
2104 symbol A("A"), B("B"), C("C");
2106 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2107 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2109 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2110 cout << exprseq(e.get_free_indices()) << endl;
2112 // 'j' and 'l' are dummy indices
2114 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2115 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2117 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2118 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2119 cout << exprseq(e.get_free_indices()) << endl;
2121 // 'nu' is a dummy index, but 'sigma' is not
2123 e = indexed(A, mu, mu);
2124 cout << exprseq(e.get_free_indices()) << endl;
2126 // 'mu' is not a dummy index because it appears twice with the same
2129 e = indexed(A, mu, nu) + 42;
2130 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2131 // this will throw an exception:
2132 // "add::get_free_indices: inconsistent indices in sum"
2136 @cindex @code{simplify_indexed()}
2137 @subsection Simplifying indexed expressions
2139 In addition to the few automatic simplifications that GiNaC performs on
2140 indexed expressions (such as re-ordering the indices of symmetric tensors
2141 and calculating traces and convolutions of matrices and predefined tensors)
2145 ex ex::simplify_indexed();
2146 ex ex::simplify_indexed(const scalar_products & sp);
2149 that performs some more expensive operations:
2152 @item it checks the consistency of free indices in sums in the same way
2153 @code{get_free_indices()} does
2154 @item it tries to give dummy indices that appear in different terms of a sum
2155 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2156 @item it (symbolically) calculates all possible dummy index summations/contractions
2157 with the predefined tensors (this will be explained in more detail in the
2159 @item it detects contractions that vanish for symmetry reasons, for example
2160 the contraction of a symmetric and a totally antisymmetric tensor
2161 @item as a special case of dummy index summation, it can replace scalar products
2162 of two tensors with a user-defined value
2165 The last point is done with the help of the @code{scalar_products} class
2166 which is used to store scalar products with known values (this is not an
2167 arithmetic class, you just pass it to @code{simplify_indexed()}):
2171 symbol A("A"), B("B"), C("C"), i_sym("i");
2175 sp.add(A, B, 0); // A and B are orthogonal
2176 sp.add(A, C, 0); // A and C are orthogonal
2177 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2179 e = indexed(A + B, i) * indexed(A + C, i);
2181 // -> (B+A).i*(A+C).i
2183 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2189 The @code{scalar_products} object @code{sp} acts as a storage for the
2190 scalar products added to it with the @code{.add()} method. This method
2191 takes three arguments: the two expressions of which the scalar product is
2192 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2193 @code{simplify_indexed()} will replace all scalar products of indexed
2194 objects that have the symbols @code{A} and @code{B} as base expressions
2195 with the single value 0. The number, type and dimension of the indices
2196 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2198 @cindex @code{expand()}
2199 The example above also illustrates a feature of the @code{expand()} method:
2200 if passed the @code{expand_indexed} option it will distribute indices
2201 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2203 @cindex @code{tensor} (class)
2204 @subsection Predefined tensors
2206 Some frequently used special tensors such as the delta, epsilon and metric
2207 tensors are predefined in GiNaC. They have special properties when
2208 contracted with other tensor expressions and some of them have constant
2209 matrix representations (they will evaluate to a number when numeric
2210 indices are specified).
2212 @cindex @code{delta_tensor()}
2213 @subsubsection Delta tensor
2215 The delta tensor takes two indices, is symmetric and has the matrix
2216 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2217 @code{delta_tensor()}:
2221 symbol A("A"), B("B");
2223 idx i(symbol("i"), 3), j(symbol("j"), 3),
2224 k(symbol("k"), 3), l(symbol("l"), 3);
2226 ex e = indexed(A, i, j) * indexed(B, k, l)
2227 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2228 cout << e.simplify_indexed() << endl;
2231 cout << delta_tensor(i, i) << endl;
2236 @cindex @code{metric_tensor()}
2237 @subsubsection General metric tensor
2239 The function @code{metric_tensor()} creates a general symmetric metric
2240 tensor with two indices that can be used to raise/lower tensor indices. The
2241 metric tensor is denoted as @samp{g} in the output and if its indices are of
2242 mixed variance it is automatically replaced by a delta tensor:
2248 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2250 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2251 cout << e.simplify_indexed() << endl;
2254 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2255 cout << e.simplify_indexed() << endl;
2258 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2259 * metric_tensor(nu, rho);
2260 cout << e.simplify_indexed() << endl;
2263 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2264 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2265 + indexed(A, mu.toggle_variance(), rho));
2266 cout << e.simplify_indexed() << endl;
2271 @cindex @code{lorentz_g()}
2272 @subsubsection Minkowski metric tensor
2274 The Minkowski metric tensor is a special metric tensor with a constant
2275 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2276 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2277 It is created with the function @code{lorentz_g()} (although it is output as
2282 varidx mu(symbol("mu"), 4);
2284 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2285 * lorentz_g(mu, varidx(0, 4)); // negative signature
2286 cout << e.simplify_indexed() << endl;
2289 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2290 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2291 cout << e.simplify_indexed() << endl;
2296 @cindex @code{spinor_metric()}
2297 @subsubsection Spinor metric tensor
2299 The function @code{spinor_metric()} creates an antisymmetric tensor with
2300 two indices that is used to raise/lower indices of 2-component spinors.
2301 It is output as @samp{eps}:
2307 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2308 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2310 e = spinor_metric(A, B) * indexed(psi, B_co);
2311 cout << e.simplify_indexed() << endl;
2314 e = spinor_metric(A, B) * indexed(psi, A_co);
2315 cout << e.simplify_indexed() << endl;
2318 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2319 cout << e.simplify_indexed() << endl;
2322 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2323 cout << e.simplify_indexed() << endl;
2326 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2327 cout << e.simplify_indexed() << endl;
2330 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2331 cout << e.simplify_indexed() << endl;
2336 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2338 @cindex @code{epsilon_tensor()}
2339 @cindex @code{lorentz_eps()}
2340 @subsubsection Epsilon tensor
2342 The epsilon tensor is totally antisymmetric, its number of indices is equal
2343 to the dimension of the index space (the indices must all be of the same
2344 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2345 defined to be 1. Its behavior with indices that have a variance also
2346 depends on the signature of the metric. Epsilon tensors are output as
2349 There are three functions defined to create epsilon tensors in 2, 3 and 4
2353 ex epsilon_tensor(const ex & i1, const ex & i2);
2354 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2355 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2358 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2359 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2360 Minkowski space (the last @code{bool} argument specifies whether the metric
2361 has negative or positive signature, as in the case of the Minkowski metric
2366 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2367 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2368 e = lorentz_eps(mu, nu, rho, sig) *
2369 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2370 cout << simplify_indexed(e) << endl;
2371 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2373 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2374 symbol A("A"), B("B");
2375 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2376 cout << simplify_indexed(e) << endl;
2377 // -> -B.k*A.j*eps.i.k.j
2378 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2379 cout << simplify_indexed(e) << endl;
2384 @subsection Linear algebra
2386 The @code{matrix} class can be used with indices to do some simple linear
2387 algebra (linear combinations and products of vectors and matrices, traces
2388 and scalar products):
2392 idx i(symbol("i"), 2), j(symbol("j"), 2);
2393 symbol x("x"), y("y");
2395 // A is a 2x2 matrix, X is a 2x1 vector
2396 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2398 cout << indexed(A, i, i) << endl;
2401 ex e = indexed(A, i, j) * indexed(X, j);
2402 cout << e.simplify_indexed() << endl;
2403 // -> [[2*y+x],[4*y+3*x]].i
2405 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2406 cout << e.simplify_indexed() << endl;
2407 // -> [[3*y+3*x,6*y+2*x]].j
2411 You can of course obtain the same results with the @code{matrix::add()},
2412 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2413 but with indices you don't have to worry about transposing matrices.
2415 Matrix indices always start at 0 and their dimension must match the number
2416 of rows/columns of the matrix. Matrices with one row or one column are
2417 vectors and can have one or two indices (it doesn't matter whether it's a
2418 row or a column vector). Other matrices must have two indices.
2420 You should be careful when using indices with variance on matrices. GiNaC
2421 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2422 @samp{F.mu.nu} are different matrices. In this case you should use only
2423 one form for @samp{F} and explicitly multiply it with a matrix representation
2424 of the metric tensor.
2427 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2428 @c node-name, next, previous, up
2429 @section Non-commutative objects
2431 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2432 non-commutative objects are built-in which are mostly of use in high energy
2436 @item Clifford (Dirac) algebra (class @code{clifford})
2437 @item su(3) Lie algebra (class @code{color})
2438 @item Matrices (unindexed) (class @code{matrix})
2441 The @code{clifford} and @code{color} classes are subclasses of
2442 @code{indexed} because the elements of these algebras usually carry
2443 indices. The @code{matrix} class is described in more detail in
2446 Unlike most computer algebra systems, GiNaC does not primarily provide an
2447 operator (often denoted @samp{&*}) for representing inert products of
2448 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2449 classes of objects involved, and non-commutative products are formed with
2450 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2451 figuring out by itself which objects commute and will group the factors
2452 by their class. Consider this example:
2456 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2457 idx a(symbol("a"), 8), b(symbol("b"), 8);
2458 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2460 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2464 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2465 groups the non-commutative factors (the gammas and the su(3) generators)
2466 together while preserving the order of factors within each class (because
2467 Clifford objects commute with color objects). The resulting expression is a
2468 @emph{commutative} product with two factors that are themselves non-commutative
2469 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2470 parentheses are placed around the non-commutative products in the output.
2472 @cindex @code{ncmul} (class)
2473 Non-commutative products are internally represented by objects of the class
2474 @code{ncmul}, as opposed to commutative products which are handled by the
2475 @code{mul} class. You will normally not have to worry about this distinction,
2478 The advantage of this approach is that you never have to worry about using
2479 (or forgetting to use) a special operator when constructing non-commutative
2480 expressions. Also, non-commutative products in GiNaC are more intelligent
2481 than in other computer algebra systems; they can, for example, automatically
2482 canonicalize themselves according to rules specified in the implementation
2483 of the non-commutative classes. The drawback is that to work with other than
2484 the built-in algebras you have to implement new classes yourself. Symbols
2485 always commute and it's not possible to construct non-commutative products
2486 using symbols to represent the algebra elements or generators. User-defined
2487 functions can, however, be specified as being non-commutative.
2489 @cindex @code{return_type()}
2490 @cindex @code{return_type_tinfo()}
2491 Information about the commutativity of an object or expression can be
2492 obtained with the two member functions
2495 unsigned ex::return_type() const;
2496 unsigned ex::return_type_tinfo() const;
2499 The @code{return_type()} function returns one of three values (defined in
2500 the header file @file{flags.h}), corresponding to three categories of
2501 expressions in GiNaC:
2504 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2505 classes are of this kind.
2506 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2507 certain class of non-commutative objects which can be determined with the
2508 @code{return_type_tinfo()} method. Expressions of this category commute
2509 with everything except @code{noncommutative} expressions of the same
2511 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2512 of non-commutative objects of different classes. Expressions of this
2513 category don't commute with any other @code{noncommutative} or
2514 @code{noncommutative_composite} expressions.
2517 The value returned by the @code{return_type_tinfo()} method is valid only
2518 when the return type of the expression is @code{noncommutative}. It is a
2519 value that is unique to the class of the object and usually one of the
2520 constants in @file{tinfos.h}, or derived therefrom.
2522 Here are a couple of examples:
2525 @multitable @columnfractions 0.33 0.33 0.34
2526 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2527 @item @code{42} @tab @code{commutative} @tab -
2528 @item @code{2*x-y} @tab @code{commutative} @tab -
2529 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2530 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2531 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2532 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2536 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2537 @code{TINFO_clifford} for objects with a representation label of zero.
2538 Other representation labels yield a different @code{return_type_tinfo()},
2539 but it's the same for any two objects with the same label. This is also true
2542 A last note: With the exception of matrices, positive integer powers of
2543 non-commutative objects are automatically expanded in GiNaC. For example,
2544 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2545 non-commutative expressions).
2548 @cindex @code{clifford} (class)
2549 @subsection Clifford algebra
2551 @cindex @code{dirac_gamma()}
2552 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2553 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2554 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2555 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2558 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2561 which takes two arguments: the index and a @dfn{representation label} in the
2562 range 0 to 255 which is used to distinguish elements of different Clifford
2563 algebras (this is also called a @dfn{spin line index}). Gammas with different
2564 labels commute with each other. The dimension of the index can be 4 or (in
2565 the framework of dimensional regularization) any symbolic value. Spinor
2566 indices on Dirac gammas are not supported in GiNaC.
2568 @cindex @code{dirac_ONE()}
2569 The unity element of a Clifford algebra is constructed by
2572 ex dirac_ONE(unsigned char rl = 0);
2575 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2576 multiples of the unity element, even though it's customary to omit it.
2577 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2578 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2579 GiNaC will complain and/or produce incorrect results.
2581 @cindex @code{dirac_gamma5()}
2582 There is a special element @samp{gamma5} that commutes with all other
2583 gammas, has a unit square, and in 4 dimensions equals
2584 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2587 ex dirac_gamma5(unsigned char rl = 0);
2590 @cindex @code{dirac_gammaL()}
2591 @cindex @code{dirac_gammaR()}
2592 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2593 objects, constructed by
2596 ex dirac_gammaL(unsigned char rl = 0);
2597 ex dirac_gammaR(unsigned char rl = 0);
2600 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2601 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2603 @cindex @code{dirac_slash()}
2604 Finally, the function
2607 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2610 creates a term that represents a contraction of @samp{e} with the Dirac
2611 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2612 with a unique index whose dimension is given by the @code{dim} argument).
2613 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2615 In products of dirac gammas, superfluous unity elements are automatically
2616 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2617 and @samp{gammaR} are moved to the front.
2619 The @code{simplify_indexed()} function performs contractions in gamma strings,
2625 symbol a("a"), b("b"), D("D");
2626 varidx mu(symbol("mu"), D);
2627 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2628 * dirac_gamma(mu.toggle_variance());
2630 // -> gamma~mu*a\*gamma.mu
2631 e = e.simplify_indexed();
2634 cout << e.subs(D == 4) << endl;
2640 @cindex @code{dirac_trace()}
2641 To calculate the trace of an expression containing strings of Dirac gammas
2642 you use the function
2645 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2648 This function takes the trace of all gammas with the specified representation
2649 label; gammas with other labels are left standing. The last argument to
2650 @code{dirac_trace()} is the value to be returned for the trace of the unity
2651 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2652 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2653 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2654 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2655 This @samp{gamma5} scheme is described in greater detail in
2656 @cite{The Role of gamma5 in Dimensional Regularization}.
2658 The value of the trace itself is also usually different in 4 and in
2659 @math{D != 4} dimensions:
2664 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2665 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2666 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2667 cout << dirac_trace(e).simplify_indexed() << endl;
2674 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2675 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2676 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2677 cout << dirac_trace(e).simplify_indexed() << endl;
2678 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2682 Here is an example for using @code{dirac_trace()} to compute a value that
2683 appears in the calculation of the one-loop vacuum polarization amplitude in
2688 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2689 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2692 sp.add(l, l, pow(l, 2));
2693 sp.add(l, q, ldotq);
2695 ex e = dirac_gamma(mu) *
2696 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2697 dirac_gamma(mu.toggle_variance()) *
2698 (dirac_slash(l, D) + m * dirac_ONE());
2699 e = dirac_trace(e).simplify_indexed(sp);
2700 e = e.collect(lst(l, ldotq, m));
2702 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2706 The @code{canonicalize_clifford()} function reorders all gamma products that
2707 appear in an expression to a canonical (but not necessarily simple) form.
2708 You can use this to compare two expressions or for further simplifications:
2712 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2713 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2715 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2717 e = canonicalize_clifford(e);
2724 @cindex @code{color} (class)
2725 @subsection Color algebra
2727 @cindex @code{color_T()}
2728 For computations in quantum chromodynamics, GiNaC implements the base elements
2729 and structure constants of the su(3) Lie algebra (color algebra). The base
2730 elements @math{T_a} are constructed by the function
2733 ex color_T(const ex & a, unsigned char rl = 0);
2736 which takes two arguments: the index and a @dfn{representation label} in the
2737 range 0 to 255 which is used to distinguish elements of different color
2738 algebras. Objects with different labels commute with each other. The
2739 dimension of the index must be exactly 8 and it should be of class @code{idx},
2742 @cindex @code{color_ONE()}
2743 The unity element of a color algebra is constructed by
2746 ex color_ONE(unsigned char rl = 0);
2749 @strong{Note:} You must always use @code{color_ONE()} when referring to
2750 multiples of the unity element, even though it's customary to omit it.
2751 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2752 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2753 GiNaC may produce incorrect results.
2755 @cindex @code{color_d()}
2756 @cindex @code{color_f()}
2760 ex color_d(const ex & a, const ex & b, const ex & c);
2761 ex color_f(const ex & a, const ex & b, const ex & c);
2764 create the symmetric and antisymmetric structure constants @math{d_abc} and
2765 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2766 and @math{[T_a, T_b] = i f_abc T_c}.
2768 @cindex @code{color_h()}
2769 There's an additional function
2772 ex color_h(const ex & a, const ex & b, const ex & c);
2775 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2777 The function @code{simplify_indexed()} performs some simplifications on
2778 expressions containing color objects:
2783 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2784 k(symbol("k"), 8), l(symbol("l"), 8);
2786 e = color_d(a, b, l) * color_f(a, b, k);
2787 cout << e.simplify_indexed() << endl;
2790 e = color_d(a, b, l) * color_d(a, b, k);
2791 cout << e.simplify_indexed() << endl;
2794 e = color_f(l, a, b) * color_f(a, b, k);
2795 cout << e.simplify_indexed() << endl;
2798 e = color_h(a, b, c) * color_h(a, b, c);
2799 cout << e.simplify_indexed() << endl;
2802 e = color_h(a, b, c) * color_T(b) * color_T(c);
2803 cout << e.simplify_indexed() << endl;
2806 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2807 cout << e.simplify_indexed() << endl;
2810 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2811 cout << e.simplify_indexed() << endl;
2812 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2816 @cindex @code{color_trace()}
2817 To calculate the trace of an expression containing color objects you use the
2821 ex color_trace(const ex & e, unsigned char rl = 0);
2824 This function takes the trace of all color @samp{T} objects with the
2825 specified representation label; @samp{T}s with other labels are left
2826 standing. For example:
2830 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2832 // -> -I*f.a.c.b+d.a.c.b
2837 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2838 @c node-name, next, previous, up
2839 @chapter Methods and Functions
2842 In this chapter the most important algorithms provided by GiNaC will be
2843 described. Some of them are implemented as functions on expressions,
2844 others are implemented as methods provided by expression objects. If
2845 they are methods, there exists a wrapper function around it, so you can
2846 alternatively call it in a functional way as shown in the simple
2851 cout << "As method: " << sin(1).evalf() << endl;
2852 cout << "As function: " << evalf(sin(1)) << endl;
2856 @cindex @code{subs()}
2857 The general rule is that wherever methods accept one or more parameters
2858 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2859 wrapper accepts is the same but preceded by the object to act on
2860 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2861 most natural one in an OO model but it may lead to confusion for MapleV
2862 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2863 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2864 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2865 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2866 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2867 here. Also, users of MuPAD will in most cases feel more comfortable
2868 with GiNaC's convention. All function wrappers are implemented
2869 as simple inline functions which just call the corresponding method and
2870 are only provided for users uncomfortable with OO who are dead set to
2871 avoid method invocations. Generally, nested function wrappers are much
2872 harder to read than a sequence of methods and should therefore be
2873 avoided if possible. On the other hand, not everything in GiNaC is a
2874 method on class @code{ex} and sometimes calling a function cannot be
2878 * Information About Expressions::
2879 * Substituting Expressions::
2880 * Pattern Matching and Advanced Substitutions::
2881 * Applying a Function on Subexpressions::
2882 * Visitors and Tree Traversal::
2883 * Polynomial Arithmetic:: Working with polynomials.
2884 * Rational Expressions:: Working with rational functions.
2885 * Symbolic Differentiation::
2886 * Series Expansion:: Taylor and Laurent expansion.
2888 * Built-in Functions:: List of predefined mathematical functions.
2889 * Solving Linear Systems of Equations::
2890 * Input/Output:: Input and output of expressions.
2894 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2895 @c node-name, next, previous, up
2896 @section Getting information about expressions
2898 @subsection Checking expression types
2899 @cindex @code{is_a<@dots{}>()}
2900 @cindex @code{is_exactly_a<@dots{}>()}
2901 @cindex @code{ex_to<@dots{}>()}
2902 @cindex Converting @code{ex} to other classes
2903 @cindex @code{info()}
2904 @cindex @code{return_type()}
2905 @cindex @code{return_type_tinfo()}
2907 Sometimes it's useful to check whether a given expression is a plain number,
2908 a sum, a polynomial with integer coefficients, or of some other specific type.
2909 GiNaC provides a couple of functions for this:
2912 bool is_a<T>(const ex & e);
2913 bool is_exactly_a<T>(const ex & e);
2914 bool ex::info(unsigned flag);
2915 unsigned ex::return_type() const;
2916 unsigned ex::return_type_tinfo() const;
2919 When the test made by @code{is_a<T>()} returns true, it is safe to call
2920 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2921 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2922 example, assuming @code{e} is an @code{ex}:
2927 if (is_a<numeric>(e))
2928 numeric n = ex_to<numeric>(e);
2933 @code{is_a<T>(e)} allows you to check whether the top-level object of
2934 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2935 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2936 e.g., for checking whether an expression is a number, a sum, or a product:
2943 is_a<numeric>(e1); // true
2944 is_a<numeric>(e2); // false
2945 is_a<add>(e1); // false
2946 is_a<add>(e2); // true
2947 is_a<mul>(e1); // false
2948 is_a<mul>(e2); // false
2952 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2953 top-level object of an expression @samp{e} is an instance of the GiNaC
2954 class @samp{T}, not including parent classes.
2956 The @code{info()} method is used for checking certain attributes of
2957 expressions. The possible values for the @code{flag} argument are defined
2958 in @file{ginac/flags.h}, the most important being explained in the following
2962 @multitable @columnfractions .30 .70
2963 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2964 @item @code{numeric}
2965 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2967 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2968 @item @code{rational}
2969 @tab @dots{}an exact rational number (integers are rational, too)
2970 @item @code{integer}
2971 @tab @dots{}a (non-complex) integer
2972 @item @code{crational}
2973 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2974 @item @code{cinteger}
2975 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2976 @item @code{positive}
2977 @tab @dots{}not complex and greater than 0
2978 @item @code{negative}
2979 @tab @dots{}not complex and less than 0
2980 @item @code{nonnegative}
2981 @tab @dots{}not complex and greater than or equal to 0
2983 @tab @dots{}an integer greater than 0
2985 @tab @dots{}an integer less than 0
2986 @item @code{nonnegint}
2987 @tab @dots{}an integer greater than or equal to 0
2989 @tab @dots{}an even integer
2991 @tab @dots{}an odd integer
2993 @tab @dots{}a prime integer (probabilistic primality test)
2994 @item @code{relation}
2995 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2996 @item @code{relation_equal}
2997 @tab @dots{}a @code{==} relation
2998 @item @code{relation_not_equal}
2999 @tab @dots{}a @code{!=} relation
3000 @item @code{relation_less}
3001 @tab @dots{}a @code{<} relation
3002 @item @code{relation_less_or_equal}
3003 @tab @dots{}a @code{<=} relation
3004 @item @code{relation_greater}
3005 @tab @dots{}a @code{>} relation
3006 @item @code{relation_greater_or_equal}
3007 @tab @dots{}a @code{>=} relation
3009 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3011 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3012 @item @code{polynomial}
3013 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3014 @item @code{integer_polynomial}
3015 @tab @dots{}a polynomial with (non-complex) integer coefficients
3016 @item @code{cinteger_polynomial}
3017 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3018 @item @code{rational_polynomial}
3019 @tab @dots{}a polynomial with (non-complex) rational coefficients
3020 @item @code{crational_polynomial}
3021 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3022 @item @code{rational_function}
3023 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3024 @item @code{algebraic}
3025 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3029 To determine whether an expression is commutative or non-commutative and if
3030 so, with which other expressions it would commute, you use the methods
3031 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3032 for an explanation of these.
3035 @subsection Accessing subexpressions
3036 @cindex @code{nops()}
3039 @cindex @code{relational} (class)
3041 GiNaC provides the two methods
3045 ex ex::op(size_t i);
3048 for accessing the subexpressions in the container-like GiNaC classes like
3049 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3050 determines the number of subexpressions (@samp{operands}) contained, while
3051 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3052 In the case of a @code{power} object, @code{op(0)} will return the basis
3053 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3054 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3056 The left-hand and right-hand side expressions of objects of class
3057 @code{relational} (and only of these) can also be accessed with the methods
3065 @subsection Comparing expressions
3066 @cindex @code{is_equal()}
3067 @cindex @code{is_zero()}
3069 Expressions can be compared with the usual C++ relational operators like
3070 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3071 the result is usually not determinable and the result will be @code{false},
3072 except in the case of the @code{!=} operator. You should also be aware that
3073 GiNaC will only do the most trivial test for equality (subtracting both
3074 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3077 Actually, if you construct an expression like @code{a == b}, this will be
3078 represented by an object of the @code{relational} class (@pxref{Relations})
3079 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3081 There are also two methods
3084 bool ex::is_equal(const ex & other);
3088 for checking whether one expression is equal to another, or equal to zero,
3091 @strong{Warning:} You will also find an @code{ex::compare()} method in the
3092 GiNaC header files. This method is however only to be used internally by
3093 GiNaC to establish a canonical sort order for terms, and using it to compare
3094 expressions will give very surprising results.
3097 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
3098 @c node-name, next, previous, up
3099 @section Substituting expressions
3100 @cindex @code{subs()}
3102 Algebraic objects inside expressions can be replaced with arbitrary
3103 expressions via the @code{.subs()} method:
3106 ex ex::subs(const ex & e, unsigned options = 0);
3107 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3110 In the first form, @code{subs()} accepts a relational of the form
3111 @samp{object == expression} or a @code{lst} of such relationals:
3115 symbol x("x"), y("y");
3117 ex e1 = 2*x^2-4*x+3;
3118 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3122 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3127 If you specify multiple substitutions, they are performed in parallel, so e.g.
3128 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3130 The second form of @code{subs()} takes two lists, one for the objects to be
3131 replaced and one for the expressions to be substituted (both lists must
3132 contain the same number of elements). Using this form, you would write
3133 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
3135 The optional last argument to @code{subs()} is a combination of
3136 @code{subs_options} flags. There are two options available:
3137 @code{subs_options::subs_no_pattern} disables pattern matching, which makes
3138 large @code{subs()} operations significantly faster if you are not using
3139 patterns. The second option, @code{subs_options::subs_algebraic} enables
3140 algebraic substitutions in products and powers.
3141 @ref{Pattern Matching and Advanced Substitutions}, for more information
3142 about patterns and algebraic substitutions.
3144 @code{subs()} performs syntactic substitution of any complete algebraic
3145 object; it does not try to match sub-expressions as is demonstrated by the
3150 symbol x("x"), y("y"), z("z");
3152 ex e1 = pow(x+y, 2);
3153 cout << e1.subs(x+y == 4) << endl;
3156 ex e2 = sin(x)*sin(y)*cos(x);
3157 cout << e2.subs(sin(x) == cos(x)) << endl;
3158 // -> cos(x)^2*sin(y)
3161 cout << e3.subs(x+y == 4) << endl;
3163 // (and not 4+z as one might expect)
3167 A more powerful form of substitution using wildcards is described in the
3171 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3172 @c node-name, next, previous, up
3173 @section Pattern matching and advanced substitutions
3174 @cindex @code{wildcard} (class)
3175 @cindex Pattern matching
3177 GiNaC allows the use of patterns for checking whether an expression is of a
3178 certain form or contains subexpressions of a certain form, and for
3179 substituting expressions in a more general way.
3181 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3182 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3183 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3184 an unsigned integer number to allow having multiple different wildcards in a
3185 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3186 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3190 ex wild(unsigned label = 0);
3193 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3196 Some examples for patterns:
3198 @multitable @columnfractions .5 .5
3199 @item @strong{Constructed as} @tab @strong{Output as}
3200 @item @code{wild()} @tab @samp{$0}
3201 @item @code{pow(x,wild())} @tab @samp{x^$0}
3202 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3203 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3209 @item Wildcards behave like symbols and are subject to the same algebraic
3210 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3211 @item As shown in the last example, to use wildcards for indices you have to
3212 use them as the value of an @code{idx} object. This is because indices must
3213 always be of class @code{idx} (or a subclass).
3214 @item Wildcards only represent expressions or subexpressions. It is not
3215 possible to use them as placeholders for other properties like index
3216 dimension or variance, representation labels, symmetry of indexed objects
3218 @item Because wildcards are commutative, it is not possible to use wildcards
3219 as part of noncommutative products.
3220 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3221 are also valid patterns.
3224 @subsection Matching expressions
3225 @cindex @code{match()}
3226 The most basic application of patterns is to check whether an expression
3227 matches a given pattern. This is done by the function
3230 bool ex::match(const ex & pattern);
3231 bool ex::match(const ex & pattern, lst & repls);
3234 This function returns @code{true} when the expression matches the pattern
3235 and @code{false} if it doesn't. If used in the second form, the actual
3236 subexpressions matched by the wildcards get returned in the @code{repls}
3237 object as a list of relations of the form @samp{wildcard == expression}.
3238 If @code{match()} returns false, the state of @code{repls} is undefined.
3239 For reproducible results, the list should be empty when passed to
3240 @code{match()}, but it is also possible to find similarities in multiple
3241 expressions by passing in the result of a previous match.
3243 The matching algorithm works as follows:
3246 @item A single wildcard matches any expression. If one wildcard appears
3247 multiple times in a pattern, it must match the same expression in all
3248 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3249 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3250 @item If the expression is not of the same class as the pattern, the match
3251 fails (i.e. a sum only matches a sum, a function only matches a function,
3253 @item If the pattern is a function, it only matches the same function
3254 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3255 @item Except for sums and products, the match fails if the number of
3256 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3258 @item If there are no subexpressions, the expressions and the pattern must
3259 be equal (in the sense of @code{is_equal()}).
3260 @item Except for sums and products, each subexpression (@code{op()}) must
3261 match the corresponding subexpression of the pattern.
3264 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3265 account for their commutativity and associativity:
3268 @item If the pattern contains a term or factor that is a single wildcard,
3269 this one is used as the @dfn{global wildcard}. If there is more than one
3270 such wildcard, one of them is chosen as the global wildcard in a random
3272 @item Every term/factor of the pattern, except the global wildcard, is
3273 matched against every term of the expression in sequence. If no match is
3274 found, the whole match fails. Terms that did match are not considered in
3276 @item If there are no unmatched terms left, the match succeeds. Otherwise
3277 the match fails unless there is a global wildcard in the pattern, in
3278 which case this wildcard matches the remaining terms.
3281 In general, having more than one single wildcard as a term of a sum or a
3282 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3285 Here are some examples in @command{ginsh} to demonstrate how it works (the
3286 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3287 match fails, and the list of wildcard replacements otherwise):
3290 > match((x+y)^a,(x+y)^a);
3292 > match((x+y)^a,(x+y)^b);
3294 > match((x+y)^a,$1^$2);
3296 > match((x+y)^a,$1^$1);
3298 > match((x+y)^(x+y),$1^$1);
3300 > match((x+y)^(x+y),$1^$2);
3302 > match((a+b)*(a+c),($1+b)*($1+c));
3304 > match((a+b)*(a+c),(a+$1)*(a+$2));
3306 (Unpredictable. The result might also be [$1==c,$2==b].)
3307 > match((a+b)*(a+c),($1+$2)*($1+$3));
3308 (The result is undefined. Due to the sequential nature of the algorithm
3309 and the re-ordering of terms in GiNaC, the match for the first factor
3310 may be @{$1==a,$2==b@} in which case the match for the second factor
3311 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3313 > match(a*(x+y)+a*z+b,a*$1+$2);
3314 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3315 @{$1=x+y,$2=a*z+b@}.)
3316 > match(a+b+c+d+e+f,c);
3318 > match(a+b+c+d+e+f,c+$0);
3320 > match(a+b+c+d+e+f,c+e+$0);
3322 > match(a+b,a+b+$0);
3324 > match(a*b^2,a^$1*b^$2);
3326 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3327 even though a==a^1.)
3328 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3330 > match(atan2(y,x^2),atan2(y,$0));
3334 @subsection Matching parts of expressions
3335 @cindex @code{has()}
3336 A more general way to look for patterns in expressions is provided by the
3340 bool ex::has(const ex & pattern);
3343 This function checks whether a pattern is matched by an expression itself or
3344 by any of its subexpressions.
3346 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3347 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3350 > has(x*sin(x+y+2*a),y);
3352 > has(x*sin(x+y+2*a),x+y);
3354 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3355 has the subexpressions "x", "y" and "2*a".)
3356 > has(x*sin(x+y+2*a),x+y+$1);
3358 (But this is possible.)
3359 > has(x*sin(2*(x+y)+2*a),x+y);
3361 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3362 which "x+y" is not a subexpression.)
3365 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3367 > has(4*x^2-x+3,$1*x);
3369 > has(4*x^2+x+3,$1*x);
3371 (Another possible pitfall. The first expression matches because the term
3372 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3373 contains a linear term you should use the coeff() function instead.)
3376 @cindex @code{find()}
3380 bool ex::find(const ex & pattern, lst & found);
3383 works a bit like @code{has()} but it doesn't stop upon finding the first
3384 match. Instead, it appends all found matches to the specified list. If there
3385 are multiple occurrences of the same expression, it is entered only once to
3386 the list. @code{find()} returns false if no matches were found (in
3387 @command{ginsh}, it returns an empty list):
3390 > find(1+x+x^2+x^3,x);
3392 > find(1+x+x^2+x^3,y);
3394 > find(1+x+x^2+x^3,x^$1);
3396 (Note the absence of "x".)
3397 > expand((sin(x)+sin(y))*(a+b));
3398 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3403 @subsection Substituting expressions
3404 @cindex @code{subs()}
3405 Probably the most useful application of patterns is to use them for
3406 substituting expressions with the @code{subs()} method. Wildcards can be
3407 used in the search patterns as well as in the replacement expressions, where
3408 they get replaced by the expressions matched by them. @code{subs()} doesn't
3409 know anything about algebra; it performs purely syntactic substitutions.
3414 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3416 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3418 > subs((a+b+c)^2,a+b==x);
3420 > subs((a+b+c)^2,a+b+$1==x+$1);
3422 > subs(a+2*b,a+b==x);
3424 > subs(4*x^3-2*x^2+5*x-1,x==a);
3426 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3428 > subs(sin(1+sin(x)),sin($1)==cos($1));
3430 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3434 The last example would be written in C++ in this way:
3438 symbol a("a"), b("b"), x("x"), y("y");
3439 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3440 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3441 cout << e.expand() << endl;
3446 @subsection Algebraic substitutions
3447 Supplying the @code{subs_options::subs_algebraic} option to @code{subs()}
3448 enables smarter, algebraic substitutions in products and powers. If you want
3449 to substitute some factors of a product, you only need to list these factors
3450 in your pattern. Furthermore, if an (integer) power of some expression occurs
3451 in your pattern and in the expression that you want the substitution to occur
3452 in, it can be substituted as many times as possible, without getting negative
3455 An example clarifies it all (hopefully):
3458 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3459 subs_options::subs_algebraic) << endl;
3460 // --> (y+x)^6+b^6+a^6
3462 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::subs_algebraic) << endl;
3464 // Powers and products are smart, but addition is just the same.
3466 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::subs_algebraic)
3469 // As I said: addition is just the same.
3471 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::subs_algebraic) << endl;
3472 // --> x^3*b*a^2+2*b
3474 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::subs_algebraic)
3476 // --> 2*b+x^3*b^(-1)*a^(-2)
3478 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::subs_algebraic) << endl;
3479 // --> -1-2*a^2+4*a^3+5*a
3481 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3482 subs_options::subs_algebraic) << endl;
3483 // --> -1+5*x+4*x^3-2*x^2
3484 // You should not really need this kind of patterns very often now.
3485 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3487 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3488 subs_options::subs_algebraic) << endl;
3489 // --> cos(1+cos(x))
3491 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3492 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3493 subs_options::subs_algebraic)) << endl;
3498 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3499 @c node-name, next, previous, up
3500 @section Applying a Function on Subexpressions
3501 @cindex tree traversal
3502 @cindex @code{map()}
3504 Sometimes you may want to perform an operation on specific parts of an
3505 expression while leaving the general structure of it intact. An example
3506 of this would be a matrix trace operation: the trace of a sum is the sum
3507 of the traces of the individual terms. That is, the trace should @dfn{map}
3508 on the sum, by applying itself to each of the sum's operands. It is possible
3509 to do this manually which usually results in code like this:
3514 if (is_a<matrix>(e))
3515 return ex_to<matrix>(e).trace();
3516 else if (is_a<add>(e)) @{
3518 for (size_t i=0; i<e.nops(); i++)
3519 sum += calc_trace(e.op(i));
3521 @} else if (is_a<mul>)(e)) @{
3529 This is, however, slightly inefficient (if the sum is very large it can take
3530 a long time to add the terms one-by-one), and its applicability is limited to
3531 a rather small class of expressions. If @code{calc_trace()} is called with
3532 a relation or a list as its argument, you will probably want the trace to
3533 be taken on both sides of the relation or of all elements of the list.
3535 GiNaC offers the @code{map()} method to aid in the implementation of such
3539 ex ex::map(map_function & f) const;
3540 ex ex::map(ex (*f)(const ex & e)) const;
3543 In the first (preferred) form, @code{map()} takes a function object that
3544 is subclassed from the @code{map_function} class. In the second form, it
3545 takes a pointer to a function that accepts and returns an expression.
3546 @code{map()} constructs a new expression of the same type, applying the
3547 specified function on all subexpressions (in the sense of @code{op()}),
3550 The use of a function object makes it possible to supply more arguments to
3551 the function that is being mapped, or to keep local state information.
3552 The @code{map_function} class declares a virtual function call operator
3553 that you can overload. Here is a sample implementation of @code{calc_trace()}
3554 that uses @code{map()} in a recursive fashion:
3557 struct calc_trace : public map_function @{
3558 ex operator()(const ex &e)
3560 if (is_a<matrix>(e))
3561 return ex_to<matrix>(e).trace();
3562 else if (is_a<mul>(e)) @{
3565 return e.map(*this);
3570 This function object could then be used like this:
3574 ex M = ... // expression with matrices
3575 calc_trace do_trace;
3576 ex tr = do_trace(M);
3580 Here is another example for you to meditate over. It removes quadratic
3581 terms in a variable from an expanded polynomial:
3584 struct map_rem_quad : public map_function @{
3586 map_rem_quad(const ex & var_) : var(var_) @{@}
3588 ex operator()(const ex & e)
3590 if (is_a<add>(e) || is_a<mul>(e))
3591 return e.map(*this);
3592 else if (is_a<power>(e) &&
3593 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3603 symbol x("x"), y("y");
3606 for (int i=0; i<8; i++)
3607 e += pow(x, i) * pow(y, 8-i) * (i+1);
3609 // -> 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
3611 map_rem_quad rem_quad(x);
3612 cout << rem_quad(e) << endl;
3613 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3617 @command{ginsh} offers a slightly different implementation of @code{map()}
3618 that allows applying algebraic functions to operands. The second argument
3619 to @code{map()} is an expression containing the wildcard @samp{$0} which
3620 acts as the placeholder for the operands:
3625 > map(a+2*b,sin($0));
3627 > map(@{a,b,c@},$0^2+$0);
3628 @{a^2+a,b^2+b,c^2+c@}
3631 Note that it is only possible to use algebraic functions in the second
3632 argument. You can not use functions like @samp{diff()}, @samp{op()},
3633 @samp{subs()} etc. because these are evaluated immediately:
3636 > map(@{a,b,c@},diff($0,a));
3638 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3639 to "map(@{a,b,c@},0)".
3643 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3644 @c node-name, next, previous, up
3645 @section Visitors and Tree Traversal
3646 @cindex tree traversal
3647 @cindex @code{visitor} (class)
3648 @cindex @code{accept()}
3649 @cindex @code{visit()}
3650 @cindex @code{traverse()}
3651 @cindex @code{traverse_preorder()}
3652 @cindex @code{traverse_postorder()}
3654 Suppose that you need a function that returns a list of all indices appearing
3655 in an arbitrary expression. The indices can have any dimension, and for
3656 indices with variance you always want the covariant version returned.
3658 You can't use @code{get_free_indices()} because you also want to include
3659 dummy indices in the list, and you can't use @code{find()} as it needs
3660 specific index dimensions (and it would require two passes: one for indices
3661 with variance, one for plain ones).
3663 The obvious solution to this problem is a tree traversal with a type switch,
3664 such as the following:
3667 void gather_indices_helper(const ex & e, lst & l)
3669 if (is_a<varidx>(e)) @{
3670 const varidx & vi = ex_to<varidx>(e);
3671 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3672 @} else if (is_a<idx>(e)) @{
3675 size_t n = e.nops();
3676 for (size_t i = 0; i < n; ++i)
3677 gather_indices_helper(e.op(i), l);
3681 lst gather_indices(const ex & e)
3684 gather_indices_helper(e, l);
3691 This works fine but fans of object-oriented programming will feel
3692 uncomfortable with the type switch. One reason is that there is a possibility
3693 for subtle bugs regarding derived classes. If we had, for example, written
3696 if (is_a<idx>(e)) @{
3698 @} else if (is_a<varidx>(e)) @{
3702 in @code{gather_indices_helper}, the code wouldn't have worked because the
3703 first line "absorbs" all classes derived from @code{idx}, including
3704 @code{varidx}, so the special case for @code{varidx} would never have been
3707 Also, for a large number of classes, a type switch like the above can get
3708 unwieldy and inefficient (it's a linear search, after all).
3709 @code{gather_indices_helper} only checks for two classes, but if you had to
3710 write a function that required a different implementation for nearly
3711 every GiNaC class, the result would be very hard to maintain and extend.
3713 The cleanest approach to the problem would be to add a new virtual function
3714 to GiNaC's class hierarchy. In our example, there would be specializations
3715 for @code{idx} and @code{varidx} while the default implementation in
3716 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
3717 impossible to add virtual member functions to existing classes without
3718 changing their source and recompiling everything. GiNaC comes with source,
3719 so you could actually do this, but for a small algorithm like the one
3720 presented this would be impractical.
3722 One solution to this dilemma is the @dfn{Visitor} design pattern,
3723 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
3724 variation, described in detail in
3725 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
3726 virtual functions to the class hierarchy to implement operations, GiNaC
3727 provides a single "bouncing" method @code{accept()} that takes an instance
3728 of a special @code{visitor} class and redirects execution to the one
3729 @code{visit()} virtual function of the visitor that matches the type of
3730 object that @code{accept()} was being invoked on.
3732 Visitors in GiNaC must derive from the global @code{visitor} class as well
3733 as from the class @code{T::visitor} of each class @code{T} they want to
3734 visit, and implement the member functions @code{void visit(const T &)} for
3740 void ex::accept(visitor & v) const;
3743 will then dispatch to the correct @code{visit()} member function of the
3744 specified visitor @code{v} for the type of GiNaC object at the root of the
3745 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
3747 Here is an example of a visitor:
3751 : public visitor, // this is required
3752 public add::visitor, // visit add objects
3753 public numeric::visitor, // visit numeric objects
3754 public basic::visitor // visit basic objects
3756 void visit(const add & x)
3757 @{ cout << "called with an add object" << endl; @}
3759 void visit(const numeric & x)
3760 @{ cout << "called with a numeric object" << endl; @}
3762 void visit(const basic & x)
3763 @{ cout << "called with a basic object" << endl; @}
3767 which can be used as follows:
3778 // prints "called with a numeric object"
3780 // prints "called with an add object"
3782 // prints "called with a basic object"
3786 The @code{visit(const basic &)} method gets called for all objects that are
3787 not @code{numeric} or @code{add} and acts as an (optional) default.
3789 From a conceptual point of view, the @code{visit()} methods of the visitor
3790 behave like a newly added virtual function of the visited hierarchy.
3791 In addition, visitors can store state in member variables, and they can
3792 be extended by deriving a new visitor from an existing one, thus building
3793 hierarchies of visitors.
3795 We can now rewrite our index example from above with a visitor:
3798 class gather_indices_visitor
3799 : public visitor, public idx::visitor, public varidx::visitor
3803 void visit(const idx & i)
3808 void visit(const varidx & vi)
3810 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3814 const lst & get_result() // utility function
3823 What's missing is the tree traversal. We could implement it in
3824 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
3827 void ex::traverse_preorder(visitor & v) const;
3828 void ex::traverse_postorder(visitor & v) const;
3829 void ex::traverse(visitor & v) const;
3832 @code{traverse_preorder()} visits a node @emph{before} visiting its
3833 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
3834 visiting its subexpressions. @code{traverse()} is a synonym for
3835 @code{traverse_preorder()}.
3837 Here is a new implementation of @code{gather_indices()} that uses the visitor
3838 and @code{traverse()}:
3841 lst gather_indices(const ex & e)
3843 gather_indices_visitor v;
3845 return v.get_result();
3850 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
3851 @c node-name, next, previous, up
3852 @section Polynomial arithmetic
3854 @subsection Expanding and collecting
3855 @cindex @code{expand()}
3856 @cindex @code{collect()}
3857 @cindex @code{collect_common_factors()}
3859 A polynomial in one or more variables has many equivalent
3860 representations. Some useful ones serve a specific purpose. Consider
3861 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3862 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3863 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3864 representations are the recursive ones where one collects for exponents
3865 in one of the three variable. Since the factors are themselves
3866 polynomials in the remaining two variables the procedure can be
3867 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3868 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3871 To bring an expression into expanded form, its method
3874 ex ex::expand(unsigned options = 0);
3877 may be called. In our example above, this corresponds to @math{4*x*y +
3878 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3879 GiNaC is not easily guessable you should be prepared to see different
3880 orderings of terms in such sums!
3882 Another useful representation of multivariate polynomials is as a
3883 univariate polynomial in one of the variables with the coefficients
3884 being polynomials in the remaining variables. The method
3885 @code{collect()} accomplishes this task:
3888 ex ex::collect(const ex & s, bool distributed = false);
3891 The first argument to @code{collect()} can also be a list of objects in which
3892 case the result is either a recursively collected polynomial, or a polynomial
3893 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3894 by the @code{distributed} flag.
3896 Note that the original polynomial needs to be in expanded form (for the
3897 variables concerned) in order for @code{collect()} to be able to find the
3898 coefficients properly.
3900 The following @command{ginsh} transcript shows an application of @code{collect()}
3901 together with @code{find()}:
3904 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3905 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
3906 > collect(a,@{p,q@});
3907 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)
3908 > collect(a,find(a,sin($1)));
3909 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3910 > collect(a,@{find(a,sin($1)),p,q@});
3911 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3912 > collect(a,@{find(a,sin($1)),d@});
3913 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3916 Polynomials can often be brought into a more compact form by collecting
3917 common factors from the terms of sums. This is accomplished by the function
3920 ex collect_common_factors(const ex & e);
3923 This function doesn't perform a full factorization but only looks for
3924 factors which are already explicitly present:
3927 > collect_common_factors(a*x+a*y);
3929 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
3931 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
3932 (c+a)*a*(x*y+y^2+x)*b
3935 @subsection Degree and coefficients
3936 @cindex @code{degree()}
3937 @cindex @code{ldegree()}
3938 @cindex @code{coeff()}
3940 The degree and low degree of a polynomial can be obtained using the two
3944 int ex::degree(const ex & s);
3945 int ex::ldegree(const ex & s);
3948 which also work reliably on non-expanded input polynomials (they even work
3949 on rational functions, returning the asymptotic degree). To extract
3950 a coefficient with a certain power from an expanded polynomial you use
3953 ex ex::coeff(const ex & s, int n);
3956 You can also obtain the leading and trailing coefficients with the methods
3959 ex ex::lcoeff(const ex & s);
3960 ex ex::tcoeff(const ex & s);
3963 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3966 An application is illustrated in the next example, where a multivariate
3967 polynomial is analyzed:
3971 symbol x("x"), y("y");
3972 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3973 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3974 ex Poly = PolyInp.expand();
3976 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3977 cout << "The x^" << i << "-coefficient is "
3978 << Poly.coeff(x,i) << endl;
3980 cout << "As polynomial in y: "
3981 << Poly.collect(y) << endl;
3985 When run, it returns an output in the following fashion:
3988 The x^0-coefficient is y^2+11*y
3989 The x^1-coefficient is 5*y^2-2*y
3990 The x^2-coefficient is -1
3991 The x^3-coefficient is 4*y
3992 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3995 As always, the exact output may vary between different versions of GiNaC
3996 or even from run to run since the internal canonical ordering is not
3997 within the user's sphere of influence.
3999 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4000 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4001 with non-polynomial expressions as they not only work with symbols but with
4002 constants, functions and indexed objects as well:
4006 symbol a("a"), b("b"), c("c");
4007 idx i(symbol("i"), 3);
4009 ex e = pow(sin(x) - cos(x), 4);
4010 cout << e.degree(cos(x)) << endl;
4012 cout << e.expand().coeff(sin(x), 3) << endl;
4015 e = indexed(a+b, i) * indexed(b+c, i);
4016 e = e.expand(expand_options::expand_indexed);
4017 cout << e.collect(indexed(b, i)) << endl;
4018 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4023 @subsection Polynomial division
4024 @cindex polynomial division
4027 @cindex pseudo-remainder
4028 @cindex @code{quo()}
4029 @cindex @code{rem()}
4030 @cindex @code{prem()}
4031 @cindex @code{divide()}
4036 ex quo(const ex & a, const ex & b, const symbol & x);
4037 ex rem(const ex & a, const ex & b, const symbol & x);
4040 compute the quotient and remainder of univariate polynomials in the variable
4041 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4043 The additional function
4046 ex prem(const ex & a, const ex & b, const symbol & x);
4049 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4050 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4052 Exact division of multivariate polynomials is performed by the function
4055 bool divide(const ex & a, const ex & b, ex & q);
4058 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4059 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4060 in which case the value of @code{q} is undefined.
4063 @subsection Unit, content and primitive part
4064 @cindex @code{unit()}
4065 @cindex @code{content()}
4066 @cindex @code{primpart()}
4071 ex ex::unit(const symbol & x);
4072 ex ex::content(const symbol & x);
4073 ex ex::primpart(const symbol & x);
4076 return the unit part, content part, and primitive polynomial of a multivariate
4077 polynomial with respect to the variable @samp{x} (the unit part being the sign
4078 of the leading coefficient, the content part being the GCD of the coefficients,
4079 and the primitive polynomial being the input polynomial divided by the unit and
4080 content parts). The product of unit, content, and primitive part is the
4081 original polynomial.
4084 @subsection GCD and LCM
4087 @cindex @code{gcd()}
4088 @cindex @code{lcm()}
4090 The functions for polynomial greatest common divisor and least common
4091 multiple have the synopsis
4094 ex gcd(const ex & a, const ex & b);
4095 ex lcm(const ex & a, const ex & b);
4098 The functions @code{gcd()} and @code{lcm()} accept two expressions
4099 @code{a} and @code{b} as arguments and return a new expression, their
4100 greatest common divisor or least common multiple, respectively. If the
4101 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4102 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4105 #include <ginac/ginac.h>
4106 using namespace GiNaC;
4110 symbol x("x"), y("y"), z("z");
4111 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4112 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4114 ex P_gcd = gcd(P_a, P_b);
4116 ex P_lcm = lcm(P_a, P_b);
4117 // 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
4122 @subsection Square-free decomposition
4123 @cindex square-free decomposition
4124 @cindex factorization
4125 @cindex @code{sqrfree()}
4127 GiNaC still lacks proper factorization support. Some form of
4128 factorization is, however, easily implemented by noting that factors
4129 appearing in a polynomial with power two or more also appear in the
4130 derivative and hence can easily be found by computing the GCD of the
4131 original polynomial and its derivatives. Any decent system has an
4132 interface for this so called square-free factorization. So we provide
4135 ex sqrfree(const ex & a, const lst & l = lst());
4137 Here is an example that by the way illustrates how the exact form of the
4138 result may slightly depend on the order of differentiation, calling for
4139 some care with subsequent processing of the result:
4142 symbol x("x"), y("y");
4143 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4145 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4146 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4148 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4149 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4151 cout << sqrfree(BiVarPol) << endl;
4152 // -> depending on luck, any of the above
4155 Note also, how factors with the same exponents are not fully factorized
4159 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4160 @c node-name, next, previous, up
4161 @section Rational expressions
4163 @subsection The @code{normal} method
4164 @cindex @code{normal()}
4165 @cindex simplification
4166 @cindex temporary replacement
4168 Some basic form of simplification of expressions is called for frequently.
4169 GiNaC provides the method @code{.normal()}, which converts a rational function
4170 into an equivalent rational function of the form @samp{numerator/denominator}
4171 where numerator and denominator are coprime. If the input expression is already
4172 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4173 otherwise it performs fraction addition and multiplication.
4175 @code{.normal()} can also be used on expressions which are not rational functions
4176 as it will replace all non-rational objects (like functions or non-integer
4177 powers) by temporary symbols to bring the expression to the domain of rational
4178 functions before performing the normalization, and re-substituting these
4179 symbols afterwards. This algorithm is also available as a separate method
4180 @code{.to_rational()}, described below.
4182 This means that both expressions @code{t1} and @code{t2} are indeed
4183 simplified in this little code snippet:
4188 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4189 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4190 std::cout << "t1 is " << t1.normal() << std::endl;
4191 std::cout << "t2 is " << t2.normal() << std::endl;
4195 Of course this works for multivariate polynomials too, so the ratio of
4196 the sample-polynomials from the section about GCD and LCM above would be
4197 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4200 @subsection Numerator and denominator
4203 @cindex @code{numer()}
4204 @cindex @code{denom()}
4205 @cindex @code{numer_denom()}
4207 The numerator and denominator of an expression can be obtained with
4212 ex ex::numer_denom();
4215 These functions will first normalize the expression as described above and
4216 then return the numerator, denominator, or both as a list, respectively.
4217 If you need both numerator and denominator, calling @code{numer_denom()} is
4218 faster than using @code{numer()} and @code{denom()} separately.
4221 @subsection Converting to a polynomial or rational expression
4222 @cindex @code{to_polynomial()}
4223 @cindex @code{to_rational()}
4225 Some of the methods described so far only work on polynomials or rational
4226 functions. GiNaC provides a way to extend the domain of these functions to
4227 general expressions by using the temporary replacement algorithm described
4228 above. You do this by calling
4231 ex ex::to_polynomial(lst &l);
4235 ex ex::to_rational(lst &l);
4238 on the expression to be converted. The supplied @code{lst} will be filled
4239 with the generated temporary symbols and their replacement expressions in
4240 a format that can be used directly for the @code{subs()} method. It can also
4241 already contain a list of replacements from an earlier application of
4242 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
4243 it on multiple expressions and get consistent results.
4245 The difference betwerrn @code{.to_polynomial()} and @code{.to_rational()}
4246 is probably best illustrated with an example:
4250 symbol x("x"), y("y");
4251 ex a = 2*x/sin(x) - y/(3*sin(x));
4255 ex p = a.to_polynomial(lp);
4256 cout << " = " << p << "\n with " << lp << endl;
4257 // = symbol3*symbol2*y+2*symbol2*x
4258 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4261 ex r = a.to_rational(lr);
4262 cout << " = " << r << "\n with " << lr << endl;
4263 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4264 // with @{symbol4==sin(x)@}
4268 The following more useful example will print @samp{sin(x)-cos(x)}:
4273 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4274 ex b = sin(x) + cos(x);
4277 divide(a.to_polynomial(l), b.to_polynomial(l), q);
4278 cout << q.subs(l) << endl;
4283 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4284 @c node-name, next, previous, up
4285 @section Symbolic differentiation
4286 @cindex differentiation
4287 @cindex @code{diff()}
4289 @cindex product rule
4291 GiNaC's objects know how to differentiate themselves. Thus, a
4292 polynomial (class @code{add}) knows that its derivative is the sum of
4293 the derivatives of all the monomials:
4297 symbol x("x"), y("y"), z("z");
4298 ex P = pow(x, 5) + pow(x, 2) + y;
4300 cout << P.diff(x,2) << endl;
4302 cout << P.diff(y) << endl; // 1
4304 cout << P.diff(z) << endl; // 0
4309 If a second integer parameter @var{n} is given, the @code{diff} method
4310 returns the @var{n}th derivative.
4312 If @emph{every} object and every function is told what its derivative
4313 is, all derivatives of composed objects can be calculated using the
4314 chain rule and the product rule. Consider, for instance the expression
4315 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4316 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4317 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4318 out that the composition is the generating function for Euler Numbers,
4319 i.e. the so called @var{n}th Euler number is the coefficient of
4320 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4321 identity to code a function that generates Euler numbers in just three
4324 @cindex Euler numbers
4326 #include <ginac/ginac.h>
4327 using namespace GiNaC;
4329 ex EulerNumber(unsigned n)
4332 const ex generator = pow(cosh(x),-1);
4333 return generator.diff(x,n).subs(x==0);
4338 for (unsigned i=0; i<11; i+=2)
4339 std::cout << EulerNumber(i) << std::endl;
4344 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4345 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4346 @code{i} by two since all odd Euler numbers vanish anyways.
4349 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4350 @c node-name, next, previous, up
4351 @section Series expansion
4352 @cindex @code{series()}
4353 @cindex Taylor expansion
4354 @cindex Laurent expansion
4355 @cindex @code{pseries} (class)
4356 @cindex @code{Order()}
4358 Expressions know how to expand themselves as a Taylor series or (more
4359 generally) a Laurent series. As in most conventional Computer Algebra
4360 Systems, no distinction is made between those two. There is a class of
4361 its own for storing such series (@code{class pseries}) and a built-in
4362 function (called @code{Order}) for storing the order term of the series.
4363 As a consequence, if you want to work with series, i.e. multiply two
4364 series, you need to call the method @code{ex::series} again to convert
4365 it to a series object with the usual structure (expansion plus order
4366 term). A sample application from special relativity could read:
4369 #include <ginac/ginac.h>
4370 using namespace std;
4371 using namespace GiNaC;
4375 symbol v("v"), c("c");
4377 ex gamma = 1/sqrt(1 - pow(v/c,2));
4378 ex mass_nonrel = gamma.series(v==0, 10);
4380 cout << "the relativistic mass increase with v is " << endl
4381 << mass_nonrel << endl;
4383 cout << "the inverse square of this series is " << endl
4384 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4388 Only calling the series method makes the last output simplify to
4389 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4390 series raised to the power @math{-2}.
4392 @cindex Machin's formula
4393 As another instructive application, let us calculate the numerical
4394 value of Archimedes' constant
4398 (for which there already exists the built-in constant @code{Pi})
4399 using John Machin's amazing formula
4401 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4404 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4406 This equation (and similar ones) were used for over 200 years for
4407 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4408 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4409 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4410 order term with it and the question arises what the system is supposed
4411 to do when the fractions are plugged into that order term. The solution
4412 is to use the function @code{series_to_poly()} to simply strip the order
4416 #include <ginac/ginac.h>
4417 using namespace GiNaC;
4419 ex machin_pi(int degr)
4422 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4423 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4424 -4*pi_expansion.subs(x==numeric(1,239));
4430 using std::cout; // just for fun, another way of...
4431 using std::endl; // ...dealing with this namespace std.
4433 for (int i=2; i<12; i+=2) @{
4434 pi_frac = machin_pi(i);
4435 cout << i << ":\t" << pi_frac << endl
4436 << "\t" << pi_frac.evalf() << endl;
4442 Note how we just called @code{.series(x,degr)} instead of
4443 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4444 method @code{series()}: if the first argument is a symbol the expression
4445 is expanded in that symbol around point @code{0}. When you run this
4446 program, it will type out:
4450 3.1832635983263598326
4451 4: 5359397032/1706489875
4452 3.1405970293260603143
4453 6: 38279241713339684/12184551018734375
4454 3.141621029325034425
4455 8: 76528487109180192540976/24359780855939418203125
4456 3.141591772182177295
4457 10: 327853873402258685803048818236/104359128170408663038552734375
4458 3.1415926824043995174
4462 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4463 @c node-name, next, previous, up
4464 @section Symmetrization
4465 @cindex @code{symmetrize()}
4466 @cindex @code{antisymmetrize()}
4467 @cindex @code{symmetrize_cyclic()}
4472 ex ex::symmetrize(const lst & l);
4473 ex ex::antisymmetrize(const lst & l);
4474 ex ex::symmetrize_cyclic(const lst & l);
4477 symmetrize an expression by returning the sum over all symmetric,
4478 antisymmetric or cyclic permutations of the specified list of objects,
4479 weighted by the number of permutations.
4481 The three additional methods
4484 ex ex::symmetrize();
4485 ex ex::antisymmetrize();
4486 ex ex::symmetrize_cyclic();
4489 symmetrize or antisymmetrize an expression over its free indices.
4491 Symmetrization is most useful with indexed expressions but can be used with
4492 almost any kind of object (anything that is @code{subs()}able):
4496 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4497 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4499 cout << indexed(A, i, j).symmetrize() << endl;
4500 // -> 1/2*A.j.i+1/2*A.i.j
4501 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4502 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4503 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4504 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4509 @node Built-in Functions, Solving Linear Systems of Equations, Symmetrization, Methods and Functions
4510 @c node-name, next, previous, up
4511 @section Predefined mathematical functions
4513 GiNaC contains the following predefined mathematical functions:
4516 @multitable @columnfractions .30 .70
4517 @item @strong{Name} @tab @strong{Function}
4520 @cindex @code{abs()}
4521 @item @code{csgn(x)}
4523 @cindex @code{csgn()}
4524 @item @code{sqrt(x)}
4525 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4526 @cindex @code{sqrt()}
4529 @cindex @code{sin()}
4532 @cindex @code{cos()}
4535 @cindex @code{tan()}
4536 @item @code{asin(x)}
4538 @cindex @code{asin()}
4539 @item @code{acos(x)}
4541 @cindex @code{acos()}
4542 @item @code{atan(x)}
4543 @tab inverse tangent
4544 @cindex @code{atan()}
4545 @item @code{atan2(y, x)}
4546 @tab inverse tangent with two arguments
4547 @item @code{sinh(x)}
4548 @tab hyperbolic sine
4549 @cindex @code{sinh()}
4550 @item @code{cosh(x)}
4551 @tab hyperbolic cosine
4552 @cindex @code{cosh()}
4553 @item @code{tanh(x)}
4554 @tab hyperbolic tangent
4555 @cindex @code{tanh()}
4556 @item @code{asinh(x)}
4557 @tab inverse hyperbolic sine
4558 @cindex @code{asinh()}
4559 @item @code{acosh(x)}
4560 @tab inverse hyperbolic cosine
4561 @cindex @code{acosh()}
4562 @item @code{atanh(x)}
4563 @tab inverse hyperbolic tangent
4564 @cindex @code{atanh()}
4566 @tab exponential function
4567 @cindex @code{exp()}
4569 @tab natural logarithm
4570 @cindex @code{log()}
4573 @cindex @code{Li2()}
4574 @item @code{zeta(x)}
4575 @tab Riemann's zeta function
4576 @cindex @code{zeta()}
4577 @item @code{zeta(n, x)}
4578 @tab derivatives of Riemann's zeta function
4579 @item @code{tgamma(x)}
4581 @cindex @code{tgamma()}
4582 @cindex Gamma function
4583 @item @code{lgamma(x)}
4584 @tab logarithm of Gamma function
4585 @cindex @code{lgamma()}
4586 @item @code{beta(x, y)}
4587 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4588 @cindex @code{beta()}
4590 @tab psi (digamma) function
4591 @cindex @code{psi()}
4592 @item @code{psi(n, x)}
4593 @tab derivatives of psi function (polygamma functions)
4594 @item @code{factorial(n)}
4595 @tab factorial function
4596 @cindex @code{factorial()}
4597 @item @code{binomial(n, m)}
4598 @tab binomial coefficients
4599 @cindex @code{binomial()}
4600 @item @code{Order(x)}
4601 @tab order term function in truncated power series
4602 @cindex @code{Order()}
4607 For functions that have a branch cut in the complex plane GiNaC follows
4608 the conventions for C++ as defined in the ANSI standard as far as
4609 possible. In particular: the natural logarithm (@code{log}) and the
4610 square root (@code{sqrt}) both have their branch cuts running along the
4611 negative real axis where the points on the axis itself belong to the
4612 upper part (i.e. continuous with quadrant II). The inverse
4613 trigonometric and hyperbolic functions are not defined for complex
4614 arguments by the C++ standard, however. In GiNaC we follow the
4615 conventions used by CLN, which in turn follow the carefully designed
4616 definitions in the Common Lisp standard. It should be noted that this
4617 convention is identical to the one used by the C99 standard and by most
4618 serious CAS. It is to be expected that future revisions of the C++
4619 standard incorporate these functions in the complex domain in a manner
4620 compatible with C99.
4623 @node Solving Linear Systems of Equations, Input/Output, Built-in Functions, Methods and Functions
4624 @c node-name, next, previous, up
4625 @section Solving Linear Systems of Equations
4626 @cindex @code{lsolve()}
4628 The function @code{lsolve()} provides a convenient wrapper around some
4629 matrix operations that comes in handy when a system of linear equations
4633 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
4636 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
4637 @code{relational}) while @code{symbols} is a @code{lst} of
4638 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
4641 It returns the @code{lst} of solutions as an expression. As an example,
4642 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
4646 symbol a("a"), b("b"), x("x"), y("y");
4648 eqns.append(a*x+b*y==3).append(x-y==b);
4650 vars.append(x).append(y);
4651 cout << lsolve(eqns, vars) << endl;
4652 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
4655 When the linear equations @code{eqns} are underdetermined, the solution
4656 will contain one or more tautological entries like @code{x==x},
4657 depending on the rank of the system. When they are overdetermined, the
4658 solution will be an empty @code{lst}. Note the third optional parameter
4659 to @code{lsolve()}: it accepts the same parameters as
4660 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
4664 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
4665 @c node-name, next, previous, up
4666 @section Input and output of expressions
4669 @subsection Expression output
4671 @cindex output of expressions
4673 Expressions can simply be written to any stream:
4678 ex e = 4.5*I+pow(x,2)*3/2;
4679 cout << e << endl; // prints '4.5*I+3/2*x^2'
4683 The default output format is identical to the @command{ginsh} input syntax and
4684 to that used by most computer algebra systems, but not directly pastable
4685 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4686 is printed as @samp{x^2}).
4688 It is possible to print expressions in a number of different formats with
4689 a set of stream manipulators;
4692 std::ostream & dflt(std::ostream & os);
4693 std::ostream & latex(std::ostream & os);
4694 std::ostream & tree(std::ostream & os);
4695 std::ostream & csrc(std::ostream & os);
4696 std::ostream & csrc_float(std::ostream & os);
4697 std::ostream & csrc_double(std::ostream & os);
4698 std::ostream & csrc_cl_N(std::ostream & os);
4699 std::ostream & index_dimensions(std::ostream & os);
4700 std::ostream & no_index_dimensions(std::ostream & os);
4703 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
4704 @command{ginsh} via the @code{print()}, @code{print_latex()} and
4705 @code{print_csrc()} functions, respectively.
4708 All manipulators affect the stream state permanently. To reset the output
4709 format to the default, use the @code{dflt} manipulator:
4713 cout << latex; // all output to cout will be in LaTeX format from now on
4714 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4715 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
4716 cout << dflt; // revert to default output format
4717 cout << e << endl; // prints '4.5*I+3/2*x^2'
4721 If you don't want to affect the format of the stream you're working with,
4722 you can output to a temporary @code{ostringstream} like this:
4727 s << latex << e; // format of cout remains unchanged
4728 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4733 @cindex @code{csrc_float}
4734 @cindex @code{csrc_double}
4735 @cindex @code{csrc_cl_N}
4736 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
4737 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
4738 format that can be directly used in a C or C++ program. The three possible
4739 formats select the data types used for numbers (@code{csrc_cl_N} uses the
4740 classes provided by the CLN library):
4744 cout << "f = " << csrc_float << e << ";\n";
4745 cout << "d = " << csrc_double << e << ";\n";
4746 cout << "n = " << csrc_cl_N << e << ";\n";
4750 The above example will produce (note the @code{x^2} being converted to
4754 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
4755 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
4756 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
4760 The @code{tree} manipulator allows dumping the internal structure of an
4761 expression for debugging purposes:
4772 add, hash=0x0, flags=0x3, nops=2
4773 power, hash=0x0, flags=0x3, nops=2
4774 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
4775 2 (numeric), hash=0x6526b0fa, flags=0xf
4776 3/2 (numeric), hash=0xf9828fbd, flags=0xf
4779 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
4783 @cindex @code{latex}
4784 The @code{latex} output format is for LaTeX parsing in mathematical mode.
4785 It is rather similar to the default format but provides some braces needed
4786 by LaTeX for delimiting boxes and also converts some common objects to
4787 conventional LaTeX names. It is possible to give symbols a special name for
4788 LaTeX output by supplying it as a second argument to the @code{symbol}
4791 For example, the code snippet
4795 symbol x("x", "\\circ");
4796 ex e = lgamma(x).series(x==0,3);
4797 cout << latex << e << endl;
4804 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
4807 @cindex @code{index_dimensions}
4808 @cindex @code{no_index_dimensions}
4809 Index dimensions are normally hidden in the output. To make them visible, use
4810 the @code{index_dimensions} manipulator. The dimensions will be written in
4811 square brackets behind each index value in the default and LaTeX output
4816 symbol x("x"), y("y");
4817 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
4818 ex e = indexed(x, mu) * indexed(y, nu);
4821 // prints 'x~mu*y~nu'
4822 cout << index_dimensions << e << endl;
4823 // prints 'x~mu[4]*y~nu[4]'
4824 cout << no_index_dimensions << e << endl;
4825 // prints 'x~mu*y~nu'
4830 @cindex Tree traversal
4831 If you need any fancy special output format, e.g. for interfacing GiNaC
4832 with other algebra systems or for producing code for different
4833 programming languages, you can always traverse the expression tree yourself:
4836 static void my_print(const ex & e)
4838 if (is_a<function>(e))
4839 cout << ex_to<function>(e).get_name();
4841 cout << e.bp->class_name();
4843 size_t n = e.nops();
4845 for (size_t i=0; i<n; i++) @{
4857 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4865 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4866 symbol(y))),numeric(-2)))
4869 If you need an output format that makes it possible to accurately
4870 reconstruct an expression by feeding the output to a suitable parser or
4871 object factory, you should consider storing the expression in an
4872 @code{archive} object and reading the object properties from there.
4873 See the section on archiving for more information.
4876 @subsection Expression input
4877 @cindex input of expressions
4879 GiNaC provides no way to directly read an expression from a stream because
4880 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4881 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4882 @code{y} you defined in your program and there is no way to specify the
4883 desired symbols to the @code{>>} stream input operator.
4885 Instead, GiNaC lets you construct an expression from a string, specifying the
4886 list of symbols to be used:
4890 symbol x("x"), y("y");
4891 ex e("2*x+sin(y)", lst(x, y));
4895 The input syntax is the same as that used by @command{ginsh} and the stream
4896 output operator @code{<<}. The symbols in the string are matched by name to
4897 the symbols in the list and if GiNaC encounters a symbol not specified in
4898 the list it will throw an exception.
4900 With this constructor, it's also easy to implement interactive GiNaC programs:
4905 #include <stdexcept>
4906 #include <ginac/ginac.h>
4907 using namespace std;
4908 using namespace GiNaC;
4915 cout << "Enter an expression containing 'x': ";
4920 cout << "The derivative of " << e << " with respect to x is ";
4921 cout << e.diff(x) << ".\n";
4922 @} catch (exception &p) @{
4923 cerr << p.what() << endl;
4929 @subsection Archiving
4930 @cindex @code{archive} (class)
4933 GiNaC allows creating @dfn{archives} of expressions which can be stored
4934 to or retrieved from files. To create an archive, you declare an object
4935 of class @code{archive} and archive expressions in it, giving each
4936 expression a unique name:
4940 using namespace std;
4941 #include <ginac/ginac.h>
4942 using namespace GiNaC;
4946 symbol x("x"), y("y"), z("z");
4948 ex foo = sin(x + 2*y) + 3*z + 41;
4952 a.archive_ex(foo, "foo");
4953 a.archive_ex(bar, "the second one");
4957 The archive can then be written to a file:
4961 ofstream out("foobar.gar");
4967 The file @file{foobar.gar} contains all information that is needed to
4968 reconstruct the expressions @code{foo} and @code{bar}.
4970 @cindex @command{viewgar}
4971 The tool @command{viewgar} that comes with GiNaC can be used to view
4972 the contents of GiNaC archive files:
4975 $ viewgar foobar.gar
4976 foo = 41+sin(x+2*y)+3*z
4977 the second one = 42+sin(x+2*y)+3*z
4980 The point of writing archive files is of course that they can later be
4986 ifstream in("foobar.gar");
4991 And the stored expressions can be retrieved by their name:
4997 ex ex1 = a2.unarchive_ex(syms, "foo");
4998 ex ex2 = a2.unarchive_ex(syms, "the second one");
5000 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5001 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5002 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5006 Note that you have to supply a list of the symbols which are to be inserted
5007 in the expressions. Symbols in archives are stored by their name only and
5008 if you don't specify which symbols you have, unarchiving the expression will
5009 create new symbols with that name. E.g. if you hadn't included @code{x} in
5010 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5011 have had no effect because the @code{x} in @code{ex1} would have been a
5012 different symbol than the @code{x} which was defined at the beginning of
5013 the program, although both would appear as @samp{x} when printed.
5015 You can also use the information stored in an @code{archive} object to
5016 output expressions in a format suitable for exact reconstruction. The
5017 @code{archive} and @code{archive_node} classes have a couple of member
5018 functions that let you access the stored properties:
5021 static void my_print2(const archive_node & n)
5024 n.find_string("class", class_name);
5025 cout << class_name << "(";
5027 archive_node::propinfovector p;
5028 n.get_properties(p);
5030 size_t num = p.size();
5031 for (size_t i=0; i<num; i++) @{
5032 const string &name = p[i].name;
5033 if (name == "class")
5035 cout << name << "=";
5037 unsigned count = p[i].count;
5041 for (unsigned j=0; j<count; j++) @{
5042 switch (p[i].type) @{
5043 case archive_node::PTYPE_BOOL: @{
5045 n.find_bool(name, x, j);
5046 cout << (x ? "true" : "false");
5049 case archive_node::PTYPE_UNSIGNED: @{
5051 n.find_unsigned(name, x, j);
5055 case archive_node::PTYPE_STRING: @{
5057 n.find_string(name, x, j);
5058 cout << '\"' << x << '\"';
5061 case archive_node::PTYPE_NODE: @{
5062 const archive_node &x = n.find_ex_node(name, j);
5084 ex e = pow(2, x) - y;
5086 my_print2(ar.get_top_node(0)); cout << endl;
5094 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5095 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5096 overall_coeff=numeric(number="0"))
5099 Be warned, however, that the set of properties and their meaning for each
5100 class may change between GiNaC versions.
5103 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5104 @c node-name, next, previous, up
5105 @chapter Extending GiNaC
5107 By reading so far you should have gotten a fairly good understanding of
5108 GiNaC's design-patterns. From here on you should start reading the
5109 sources. All we can do now is issue some recommendations how to tackle
5110 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5111 develop some useful extension please don't hesitate to contact the GiNaC
5112 authors---they will happily incorporate them into future versions.
5115 * What does not belong into GiNaC:: What to avoid.
5116 * Symbolic functions:: Implementing symbolic functions.
5117 * Structures:: Defining new algebraic classes (the easy way).
5118 * Adding classes:: Defining new algebraic classes (the hard way).
5122 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5123 @c node-name, next, previous, up
5124 @section What doesn't belong into GiNaC
5126 @cindex @command{ginsh}
5127 First of all, GiNaC's name must be read literally. It is designed to be
5128 a library for use within C++. The tiny @command{ginsh} accompanying
5129 GiNaC makes this even more clear: it doesn't even attempt to provide a
5130 language. There are no loops or conditional expressions in
5131 @command{ginsh}, it is merely a window into the library for the
5132 programmer to test stuff (or to show off). Still, the design of a
5133 complete CAS with a language of its own, graphical capabilities and all
5134 this on top of GiNaC is possible and is without doubt a nice project for
5137 There are many built-in functions in GiNaC that do not know how to
5138 evaluate themselves numerically to a precision declared at runtime
5139 (using @code{Digits}). Some may be evaluated at certain points, but not
5140 generally. This ought to be fixed. However, doing numerical
5141 computations with GiNaC's quite abstract classes is doomed to be
5142 inefficient. For this purpose, the underlying foundation classes
5143 provided by CLN are much better suited.
5146 @node Symbolic functions, Structures, What does not belong into GiNaC, Extending GiNaC
5147 @c node-name, next, previous, up
5148 @section Symbolic functions
5150 The easiest and most instructive way to start extending GiNaC is probably to
5151 create your own symbolic functions. These are implemented with the help of
5152 two preprocessor macros:
5154 @cindex @code{DECLARE_FUNCTION}
5155 @cindex @code{REGISTER_FUNCTION}
5157 DECLARE_FUNCTION_<n>P(<name>)
5158 REGISTER_FUNCTION(<name>, <options>)
5161 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5162 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5163 parameters of type @code{ex} and returns a newly constructed GiNaC
5164 @code{function} object that represents your function.
5166 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5167 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5168 set of options that associate the symbolic function with C++ functions you
5169 provide to implement the various methods such as evaluation, derivative,
5170 series expansion etc. They also describe additional attributes the function
5171 might have, such as symmetry and commutation properties, and a name for
5172 LaTeX output. Multiple options are separated by the member access operator
5173 @samp{.} and can be given in an arbitrary order.
5175 (By the way: in case you are worrying about all the macros above we can
5176 assure you that functions are GiNaC's most macro-intense classes. We have
5177 done our best to avoid macros where we can.)
5179 @subsection A minimal example
5181 Here is an example for the implementation of a function with two arguments
5182 that is not further evaluated:
5185 DECLARE_FUNCTION_2P(myfcn)
5187 static ex myfcn_eval(const ex & x, const ex & y)
5189 return myfcn(x, y).hold();
5192 REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
5195 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5196 in algebraic expressions:
5202 ex e = 2*myfcn(42, 3*x+1) - x;
5203 // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
5204 // the actual expression
5206 // prints '2*myfcn(42,1+3*x)-x'
5211 @cindex @code{hold()}
5213 The @code{eval_func()} option specifies the C++ function that implements
5214 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5215 the same number of arguments as the associated symbolic function (two in this
5216 case) and returns the (possibly transformed or in some way simplified)
5217 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5218 of the automatic evaluation process). If no (further) evaluation is to take
5219 place, the @code{eval_func()} function must return the original function
5220 with @code{.hold()}, to avoid a potential infinite recursion. If your
5221 symbolic functions produce a segmentation fault or stack overflow when
5222 using them in expressions, you are probably missing a @code{.hold()}
5225 There is not much you can do with the @code{myfcn} function. It merely acts
5226 as a kind of container for its arguments (which is, however, sometimes
5227 perfectly sufficient). Let's have a look at the implementation of GiNaC's
5230 @subsection The cosine function
5232 The GiNaC header file @file{inifcns.h} contains the line
5235 DECLARE_FUNCTION_1P(cos)
5238 which declares to all programs using GiNaC that there is a function @samp{cos}
5239 that takes one @code{ex} as an argument. This is all they need to know to use
5240 this function in expressions.
5242 The implementation of the cosine function is in @file{inifcns_trans.cpp}. The
5243 @code{eval_func()} function looks something like this (actually, it doesn't
5244 look like this at all, but it should give you an idea what is going on):
5247 static ex cos_eval(const ex & x)
5249 if (<x is a multiple of 2*Pi>)
5251 else if (<x is a multiple of Pi>)
5253 else if (<x is a multiple of Pi/2>)
5257 else if (<x has the form 'acos(y)'>)
5259 else if (<x has the form 'asin(y)'>)
5264 return cos(x).hold();
5268 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5269 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5270 symbolic transformation can be done, the unmodified function is returned
5271 with @code{.hold()}.
5273 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5274 The user has to call @code{evalf()} for that. This is implemented in a
5278 static ex cos_evalf(const ex & x)
5280 if (is_a<numeric>(x))
5281 return cos(ex_to<numeric>(x));
5283 return cos(x).hold();
5287 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5288 in this case the @code{cos()} function for @code{numeric} objects, which in
5289 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5290 isn't really needed here, but reminds us that the corresponding @code{eval()}
5291 function would require it in this place.
5293 Differentiation will surely turn up and so we need to tell @code{cos}
5294 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5295 instance, are then handled automatically by @code{basic::diff} and
5299 static ex cos_deriv(const ex & x, unsigned diff_param)
5305 @cindex product rule
5306 The second parameter is obligatory but uninteresting at this point. It
5307 specifies which parameter to differentiate in a partial derivative in
5308 case the function has more than one parameter, and its main application
5309 is for correct handling of the chain rule.
5311 An implementation of the series expansion is not needed for @code{cos()} as
5312 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5313 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5314 the other hand, does have poles and may need to do Laurent expansion:
5317 static ex tan_series(const ex & x, const relational & rel,
5318 int order, unsigned options)
5320 // Find the actual expansion point
5321 const ex x_pt = x.subs(rel);
5323 if (<x_pt is not an odd multiple of Pi/2>)
5324 throw do_taylor(); // tell function::series() to do Taylor expansion
5326 // On a pole, expand sin()/cos()
5327 return (sin(x)/cos(x)).series(rel, order+2, options);
5331 The @code{series()} implementation of a function @emph{must} return a
5332 @code{pseries} object, otherwise your code will crash.
5334 Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
5335 macro is used to tell the system how the @code{cos()} function behaves:
5338 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5339 evalf_func(cos_evalf).
5340 derivative_func(cos_deriv).
5341 latex_name("\\cos"));
5344 This registers the @code{cos_eval()}, @code{cos_evalf()} and
5345 @code{cos_deriv()} C++ functions with the @code{cos()} function, and also
5346 gives it a proper LaTeX name.
5348 @subsection Function options
5350 GiNaC functions understand several more options which are always
5351 specified as @code{.option(params)}. None of them are required, but you
5352 need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
5353 the @code{eval()} method).
5356 eval_func(<C++ function>)
5357 evalf_func(<C++ function>)
5358 derivative_func(<C++ function>)
5359 series_func(<C++ function>)
5362 These specify the C++ functions that implement symbolic evaluation,
5363 numeric evaluation, partial derivatives, and series expansion, respectively.
5364 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5365 @code{diff()} and @code{series()}.
5367 The @code{eval_func()} function needs to use @code{.hold()} if no further
5368 automatic evaluation is desired or possible.
5370 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5371 expansion, which is correct if there are no poles involved. If the function
5372 has poles in the complex plane, the @code{series_func()} needs to check
5373 whether the expansion point is on a pole and fall back to Taylor expansion
5374 if it isn't. Otherwise, the pole usually needs to be regularized by some
5375 suitable transformation.
5378 latex_name(const string & n)
5381 specifies the LaTeX code that represents the name of the function in LaTeX
5382 output. The default is to put the function name in an @code{\mbox@{@}}.
5385 do_not_evalf_params()
5388 This tells @code{evalf()} to not recursively evaluate the parameters of the
5389 function before calling the @code{evalf_func()}.
5392 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5395 This allows you to explicitly specify the commutation properties of the
5396 function (@xref{Non-commutative objects}, for an explanation of
5397 (non)commutativity in GiNaC). For example, you can use
5398 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5399 GiNaC treat your function like a matrix. By default, functions inherit the
5400 commutation properties of their first argument.
5403 set_symmetry(const symmetry & s)
5406 specifies the symmetry properties of the function with respect to its
5407 arguments. @xref{Indexed objects}, for an explanation of symmetry
5408 specifications. GiNaC will automatically rearrange the arguments of
5409 symmetric functions into a canonical order.
5412 @node Structures, Adding classes, Symbolic functions, Extending GiNaC
5413 @c node-name, next, previous, up
5416 If you are doing some very specialized things with GiNaC, or if you just
5417 need some more organized way to store data in your expressions instead of
5418 anonymous lists, you may want to implement your own algebraic classes.
5419 ('algebraic class' means any class directly or indirectly derived from
5420 @code{basic} that can be used in GiNaC expressions).
5422 GiNaC offers two ways of accomplishing this: either by using the
5423 @code{structure<T>} template class, or by rolling your own class from
5424 scratch. This section will discuss the @code{structure<T>} template which
5425 is easier to use but more limited, while the implementation of custom
5426 GiNaC classes is the topic of the next section. However, you may want to
5427 read both sections because many common concepts and member functions are
5428 shared by both concepts, and it will also allow you to decide which approach
5429 is most suited to your needs.
5431 The @code{structure<T>} template, defined in the GiNaC header file
5432 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
5433 or @code{class}) into a GiNaC object that can be used in expressions.
5435 @subsection Example: scalar products
5437 Let's suppose that we need a way to handle some kind of abstract scalar
5438 product of the form @samp{<x|y>} in expressions. Objects of the scalar
5439 product class have to store their left and right operands, which can in turn
5440 be arbitrary expressions. Here is a possible way to represent such a
5441 product in a C++ @code{struct}:
5445 using namespace std;
5447 #include <ginac/ginac.h>
5448 using namespace GiNaC;
5454 sprod_s(ex l, ex r) : left(l), right(r) @{@}
5458 The default constructor is required. Now, to make a GiNaC class out of this
5459 data structure, we need only one line:
5462 typedef structure<sprod_s> sprod;
5465 That's it. This line constructs an algebraic class @code{sprod} which
5466 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
5467 expressions like any other GiNaC class:
5471 symbol a("a"), b("b");
5472 ex e = sprod(sprod_s(a, b));
5476 Note the difference between @code{sprod} which is the algebraic class, and
5477 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
5478 and @code{right} data members. As shown above, an @code{sprod} can be
5479 constructed from an @code{sprod_s} object.
5481 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
5482 you could define a little wrapper function like this:
5485 inline ex make_sprod(ex left, ex right)
5487 return sprod(sprod_s(left, right));
5491 The @code{sprod_s} object contained in @code{sprod} can be accessed with
5492 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
5493 @code{get_struct()}:
5497 cout << ex_to<sprod>(e)->left << endl;
5499 cout << ex_to<sprod>(e).get_struct().right << endl;
5504 You only have read access to the members of @code{sprod_s}.
5506 The type definition of @code{sprod} is enough to write your own algorithms
5507 that deal with scalar products, for example:
5512 if (is_a<sprod>(p)) @{
5513 const sprod_s & sp = ex_to<sprod>(p).get_struct();
5514 return make_sprod(sp.right, sp.left);
5525 @subsection Structure output
5527 While the @code{sprod} type is useable it still leaves something to be
5528 desired, most notably proper output:
5533 // -> [structure object]
5537 By default, any structure types you define will be printed as
5538 @samp{[structure object]}. To override this, you can specialize the
5539 template's @code{print()} member function. The member functions of
5540 GiNaC classes are described in more detail in the next section, but
5541 it shouldn't be hard to figure out what's going on here:
5544 void sprod::print(const print_context & c, unsigned level) const
5546 // tree debug output handled by superclass
5547 if (is_a<print_tree>(c))
5548 inherited::print(c, level);
5550 // get the contained sprod_s object
5551 const sprod_s & sp = get_struct();
5553 // print_context::s is a reference to an ostream
5554 c.s << "<" << sp.left << "|" << sp.right << ">";
5558 Now we can print expressions containing scalar products:
5564 cout << swap_sprod(e) << endl;
5569 @subsection Comparing structures
5571 The @code{sprod} class defined so far still has one important drawback: all
5572 scalar products are treated as being equal because GiNaC doesn't know how to
5573 compare objects of type @code{sprod_s}. This can lead to some confusing
5574 and undesired behavior:
5578 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
5580 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
5581 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
5585 To remedy this, we first need to define the operators @code{==} and @code{<}
5586 for objects of type @code{sprod_s}:
5589 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
5591 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
5594 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
5596 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
5600 The ordering established by the @code{<} operator doesn't have to make any
5601 algebraic sense, but it needs to be well defined. Note that we can't use
5602 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
5603 in the implementation of these operators because they would construct
5604 GiNaC @code{relational} objects which in the case of @code{<} do not
5605 establish a well defined ordering (for arbitrary expressions, GiNaC can't
5606 decide which one is algebraically 'less').
5608 Next, we need to change our definition of the @code{sprod} type to let
5609 GiNaC know that an ordering relation exists for the embedded objects:
5612 typedef structure<sprod_s, compare_std_less> sprod;
5615 @code{sprod} objects then behave as expected:
5619 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
5620 // -> <a|b>-<a^2|b^2>
5621 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
5622 // -> <a|b>+<a^2|b^2>
5623 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
5625 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
5630 The @code{compare_std_less} policy parameter tells GiNaC to use the
5631 @code{std::less} and @code{std::equal_to} functors to compare objects of
5632 type @code{sprod_s}. By default, these functors forward their work to the
5633 standard @code{<} and @code{==} operators, which we have overloaded.
5634 Alternatively, we could have specialized @code{std::less} and
5635 @code{std::equal_to} for class @code{sprod_s}.
5637 GiNaC provides two other comparison policies for @code{structure<T>}
5638 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
5639 which does a bit-wise comparison of the contained @code{T} objects.
5640 This should be used with extreme care because it only works reliably with
5641 built-in integral types, and it also compares any padding (filler bytes of
5642 undefined value) that the @code{T} class might have.
5644 @subsection Subexpressions
5646 Our scalar product class has two subexpressions: the left and right
5647 operands. It might be a good idea to make them accessible via the standard
5648 @code{nops()} and @code{op()} methods:
5651 size_t sprod::nops() const
5656 ex sprod::op(size_t i) const
5660 return get_struct().left;
5662 return get_struct().right;
5664 throw std::range_error("sprod::op(): no such operand");
5669 Implementing @code{nops()} and @code{op()} for container types such as
5670 @code{sprod} has two other nice side effects:
5674 @code{has()} works as expected
5676 GiNaC generates better hash keys for the objects (the default implementation
5677 of @code{calchash()} takes subexpressions into account)
5680 @cindex @code{let_op()}
5681 There is a non-const variant of @code{op()} called @code{let_op()} that
5682 allows replacing subexpressions:
5685 ex & sprod::let_op(size_t i)
5687 // every non-const member function must call this
5688 ensure_if_modifiable();
5692 return get_struct().left;
5694 return get_struct().right;
5696 throw std::range_error("sprod::let_op(): no such operand");
5701 Once we have provided @code{let_op()} we also get @code{subs()} and
5702 @code{map()} for free. In fact, every container class that returns a non-null
5703 @code{nops()} value must either implement @code{let_op()} or provide custom
5704 implementations of @code{subs()} and @code{map()}.
5706 In turn, the availability of @code{map()} enables the recursive behavior of a
5707 couple of other default method implementations, in particular @code{evalf()},
5708 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
5709 we probably want to provide our own version of @code{expand()} for scalar
5710 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
5711 This is left as an exercise for the reader.
5713 The @code{structure<T>} template defines many more member functions that
5714 you can override by specialization to customize the behavior of your
5715 structures. You are referred to the next section for a description of
5716 some of these (especially @code{eval()}). There is, however, one topic
5717 that shall be addressed here, as it demonstrates one peculiarity of the
5718 @code{structure<T>} template: archiving.
5720 @subsection Archiving structures
5722 If you don't know how the archiving of GiNaC objects is implemented, you
5723 should first read the next section and then come back here. You're back?
5726 To implement archiving for structures it is not enough to provide
5727 specializations for the @code{archive()} member function and the
5728 unarchiving constructor (the @code{unarchive()} function has a default
5729 implementation). You also need to provide a unique name (as a string literal)
5730 for each structure type you define. This is because in GiNaC archives,
5731 the class of an object is stored as a string, the class name.
5733 By default, this class name (as returned by the @code{class_name()} member
5734 function) is @samp{structure} for all structure classes. This works as long
5735 as you have only defined one structure type, but if you use two or more you
5736 need to provide a different name for each by specializing the
5737 @code{get_class_name()} member function. Here is a sample implementation
5738 for enabling archiving of the scalar product type defined above:
5741 const char *sprod::get_class_name() @{ return "sprod"; @}
5743 void sprod::archive(archive_node & n) const
5745 inherited::archive(n);
5746 n.add_ex("left", get_struct().left);
5747 n.add_ex("right", get_struct().right);
5750 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
5752 n.find_ex("left", get_struct().left, sym_lst);
5753 n.find_ex("right", get_struct().right, sym_lst);
5757 Note that the unarchiving constructor is @code{sprod::structure} and not
5758 @code{sprod::sprod}, and that we don't need to supply an
5759 @code{sprod::unarchive()} function.
5762 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
5763 @c node-name, next, previous, up
5764 @section Adding classes
5766 The @code{structure<T>} template provides an way to extend GiNaC with custom
5767 algebraic classes that is easy to use but has its limitations, the most
5768 severe of which being that you can't add any new member functions to
5769 structures. To be able to do this, you need to write a new class definition
5772 This section will explain how to implement new algebraic classes in GiNaC by
5773 giving the example of a simple 'string' class. After reading this section
5774 you will know how to properly declare a GiNaC class and what the minimum
5775 required member functions are that you have to implement. We only cover the
5776 implementation of a 'leaf' class here (i.e. one that doesn't contain
5777 subexpressions). Creating a container class like, for example, a class
5778 representing tensor products is more involved but this section should give
5779 you enough information so you can consult the source to GiNaC's predefined
5780 classes if you want to implement something more complicated.
5782 @subsection GiNaC's run-time type information system
5784 @cindex hierarchy of classes
5786 All algebraic classes (that is, all classes that can appear in expressions)
5787 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
5788 @code{basic *} (which is essentially what an @code{ex} is) represents a
5789 generic pointer to an algebraic class. Occasionally it is necessary to find
5790 out what the class of an object pointed to by a @code{basic *} really is.
5791 Also, for the unarchiving of expressions it must be possible to find the
5792 @code{unarchive()} function of a class given the class name (as a string). A
5793 system that provides this kind of information is called a run-time type
5794 information (RTTI) system. The C++ language provides such a thing (see the
5795 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
5796 implements its own, simpler RTTI.
5798 The RTTI in GiNaC is based on two mechanisms:
5803 The @code{basic} class declares a member variable @code{tinfo_key} which
5804 holds an unsigned integer that identifies the object's class. These numbers
5805 are defined in the @file{tinfos.h} header file for the built-in GiNaC
5806 classes. They all start with @code{TINFO_}.
5809 By means of some clever tricks with static members, GiNaC maintains a list
5810 of information for all classes derived from @code{basic}. The information
5811 available includes the class names, the @code{tinfo_key}s, and pointers
5812 to the unarchiving functions. This class registry is defined in the
5813 @file{registrar.h} header file.
5817 The disadvantage of this proprietary RTTI implementation is that there's
5818 a little more to do when implementing new classes (C++'s RTTI works more
5819 or less automatic) but don't worry, most of the work is simplified by
5822 @subsection A minimalistic example
5824 Now we will start implementing a new class @code{mystring} that allows
5825 placing character strings in algebraic expressions (this is not very useful,
5826 but it's just an example). This class will be a direct subclass of
5827 @code{basic}. You can use this sample implementation as a starting point
5828 for your own classes.
5830 The code snippets given here assume that you have included some header files
5836 #include <stdexcept>
5837 using namespace std;
5839 #include <ginac/ginac.h>
5840 using namespace GiNaC;
5843 The first thing we have to do is to define a @code{tinfo_key} for our new
5844 class. This can be any arbitrary unsigned number that is not already taken
5845 by one of the existing classes but it's better to come up with something
5846 that is unlikely to clash with keys that might be added in the future. The
5847 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
5848 which is not a requirement but we are going to stick with this scheme:
5851 const unsigned TINFO_mystring = 0x42420001U;
5854 Now we can write down the class declaration. The class stores a C++
5855 @code{string} and the user shall be able to construct a @code{mystring}
5856 object from a C or C++ string:
5859 class mystring : public basic
5861 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
5864 mystring(const string &s);
5865 mystring(const char *s);
5871 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
5874 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
5875 macros are defined in @file{registrar.h}. They take the name of the class
5876 and its direct superclass as arguments and insert all required declarations
5877 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
5878 the first line after the opening brace of the class definition. The
5879 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
5880 source (at global scope, of course, not inside a function).
5882 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
5883 declarations of the default constructor and a couple of other functions that
5884 are required. It also defines a type @code{inherited} which refers to the
5885 superclass so you don't have to modify your code every time you shuffle around
5886 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
5887 class with the GiNaC RTTI.
5889 Now there are seven member functions we have to implement to get a working
5895 @code{mystring()}, the default constructor.
5898 @code{void archive(archive_node &n)}, the archiving function. This stores all
5899 information needed to reconstruct an object of this class inside an
5900 @code{archive_node}.
5903 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
5904 constructor. This constructs an instance of the class from the information
5905 found in an @code{archive_node}.
5908 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
5909 unarchiving function. It constructs a new instance by calling the unarchiving
5913 @cindex @code{compare_same_type()}
5914 @code{int compare_same_type(const basic &other)}, which is used internally
5915 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
5916 -1, depending on the relative order of this object and the @code{other}
5917 object. If it returns 0, the objects are considered equal.
5918 @strong{Note:} This has nothing to do with the (numeric) ordering
5919 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
5920 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
5921 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
5922 must provide a @code{compare_same_type()} function, even those representing
5923 objects for which no reasonable algebraic ordering relationship can be
5927 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
5928 which are the two constructors we declared.
5932 Let's proceed step-by-step. The default constructor looks like this:
5935 mystring::mystring() : inherited(TINFO_mystring) @{@}
5938 The golden rule is that in all constructors you have to set the
5939 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
5940 it will be set by the constructor of the superclass and all hell will break
5941 loose in the RTTI. For your convenience, the @code{basic} class provides
5942 a constructor that takes a @code{tinfo_key} value, which we are using here
5943 (remember that in our case @code{inherited == basic}). If the superclass
5944 didn't have such a constructor, we would have to set the @code{tinfo_key}
5945 to the right value manually.
5947 In the default constructor you should set all other member variables to
5948 reasonable default values (we don't need that here since our @code{str}
5949 member gets set to an empty string automatically).
5951 Next are the three functions for archiving. You have to implement them even
5952 if you don't plan to use archives, but the minimum required implementation
5953 is really simple. First, the archiving function:
5956 void mystring::archive(archive_node &n) const
5958 inherited::archive(n);
5959 n.add_string("string", str);
5963 The only thing that is really required is calling the @code{archive()}
5964 function of the superclass. Optionally, you can store all information you
5965 deem necessary for representing the object into the passed
5966 @code{archive_node}. We are just storing our string here. For more
5967 information on how the archiving works, consult the @file{archive.h} header
5970 The unarchiving constructor is basically the inverse of the archiving
5974 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
5976 n.find_string("string", str);
5980 If you don't need archiving, just leave this function empty (but you must
5981 invoke the unarchiving constructor of the superclass). Note that we don't
5982 have to set the @code{tinfo_key} here because it is done automatically
5983 by the unarchiving constructor of the @code{basic} class.
5985 Finally, the unarchiving function:
5988 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
5990 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
5994 You don't have to understand how exactly this works. Just copy these
5995 four lines into your code literally (replacing the class name, of
5996 course). It calls the unarchiving constructor of the class and unless
5997 you are doing something very special (like matching @code{archive_node}s
5998 to global objects) you don't need a different implementation. For those
5999 who are interested: setting the @code{dynallocated} flag puts the object
6000 under the control of GiNaC's garbage collection. It will get deleted
6001 automatically once it is no longer referenced.
6003 Our @code{compare_same_type()} function uses a provided function to compare
6007 int mystring::compare_same_type(const basic &other) const
6009 const mystring &o = static_cast<const mystring &>(other);
6010 int cmpval = str.compare(o.str);
6013 else if (cmpval < 0)
6020 Although this function takes a @code{basic &}, it will always be a reference
6021 to an object of exactly the same class (objects of different classes are not
6022 comparable), so the cast is safe. If this function returns 0, the two objects
6023 are considered equal (in the sense that @math{A-B=0}), so you should compare
6024 all relevant member variables.
6026 Now the only thing missing is our two new constructors:
6029 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6030 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6033 No surprises here. We set the @code{str} member from the argument and
6034 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6036 That's it! We now have a minimal working GiNaC class that can store
6037 strings in algebraic expressions. Let's confirm that the RTTI works:
6040 ex e = mystring("Hello, world!");
6041 cout << is_a<mystring>(e) << endl;
6044 cout << e.bp->class_name() << endl;
6048 Obviously it does. Let's see what the expression @code{e} looks like:
6052 // -> [mystring object]
6055 Hm, not exactly what we expect, but of course the @code{mystring} class
6056 doesn't yet know how to print itself. This is done in the @code{print()}
6057 member function. Let's say that we wanted to print the string surrounded
6061 class mystring : public basic
6065 void print(const print_context &c, unsigned level = 0) const;
6069 void mystring::print(const print_context &c, unsigned level) const
6071 // print_context::s is a reference to an ostream
6072 c.s << '\"' << str << '\"';
6076 The @code{level} argument is only required for container classes to
6077 correctly parenthesize the output. Let's try again to print the expression:
6081 // -> "Hello, world!"
6084 Much better. The @code{mystring} class can be used in arbitrary expressions:
6087 e += mystring("GiNaC rulez");
6089 // -> "GiNaC rulez"+"Hello, world!"
6092 (GiNaC's automatic term reordering is in effect here), or even
6095 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
6097 // -> "One string"^(2*sin(-"Another string"+Pi))
6100 Whether this makes sense is debatable but remember that this is only an
6101 example. At least it allows you to implement your own symbolic algorithms
6104 Note that GiNaC's algebraic rules remain unchanged:
6107 e = mystring("Wow") * mystring("Wow");
6111 e = pow(mystring("First")-mystring("Second"), 2);
6112 cout << e.expand() << endl;
6113 // -> -2*"First"*"Second"+"First"^2+"Second"^2
6116 There's no way to, for example, make GiNaC's @code{add} class perform string
6117 concatenation. You would have to implement this yourself.
6119 @subsection Automatic evaluation
6122 @cindex @code{eval()}
6123 @cindex @code{hold()}
6124 When dealing with objects that are just a little more complicated than the
6125 simple string objects we have implemented, chances are that you will want to
6126 have some automatic simplifications or canonicalizations performed on them.
6127 This is done in the evaluation member function @code{eval()}. Let's say that
6128 we wanted all strings automatically converted to lowercase with
6129 non-alphabetic characters stripped, and empty strings removed:
6132 class mystring : public basic
6136 ex eval(int level = 0) const;
6140 ex mystring::eval(int level) const
6143 for (int i=0; i<str.length(); i++) @{
6145 if (c >= 'A' && c <= 'Z')
6146 new_str += tolower(c);
6147 else if (c >= 'a' && c <= 'z')
6151 if (new_str.length() == 0)
6154 return mystring(new_str).hold();
6158 The @code{level} argument is used to limit the recursion depth of the
6159 evaluation. We don't have any subexpressions in the @code{mystring}
6160 class so we are not concerned with this. If we had, we would call the
6161 @code{eval()} functions of the subexpressions with @code{level - 1} as
6162 the argument if @code{level != 1}. The @code{hold()} member function
6163 sets a flag in the object that prevents further evaluation. Otherwise
6164 we might end up in an endless loop. When you want to return the object
6165 unmodified, use @code{return this->hold();}.
6167 Let's confirm that it works:
6170 ex e = mystring("Hello, world!") + mystring("!?#");
6174 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
6179 @subsection Optional member functions
6181 We have implemented only a small set of member functions to make the class
6182 work in the GiNaC framework. There are two functions that are not strictly
6183 required but will make operations with objects of the class more efficient:
6185 @cindex @code{calchash()}
6186 @cindex @code{is_equal_same_type()}
6188 unsigned calchash() const;
6189 bool is_equal_same_type(const basic &other) const;
6192 The @code{calchash()} method returns an @code{unsigned} hash value for the
6193 object which will allow GiNaC to compare and canonicalize expressions much
6194 more efficiently. You should consult the implementation of some of the built-in
6195 GiNaC classes for examples of hash functions. The default implementation of
6196 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
6197 class and all subexpressions that are accessible via @code{op()}.
6199 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
6200 tests for equality without establishing an ordering relation, which is often
6201 faster. The default implementation of @code{is_equal_same_type()} just calls
6202 @code{compare_same_type()} and tests its result for zero.
6204 @subsection Other member functions
6206 For a real algebraic class, there are probably some more functions that you
6207 might want to provide:
6210 bool info(unsigned inf) const;
6211 ex evalf(int level = 0) const;
6212 ex series(const relational & r, int order, unsigned options = 0) const;
6213 ex derivative(const symbol & s) const;
6216 If your class stores sub-expressions (see the scalar product example in the
6217 previous section) you will probably want to override
6219 @cindex @code{let_op()}
6222 ex op(size_t i) const;
6223 ex & let_op(size_t i);
6224 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
6225 ex map(map_function & f) const;
6228 @code{let_op()} is a variant of @code{op()} that allows write access. The
6229 default implementations of @code{subs()} and @code{map()} use it, so you have
6230 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
6232 You can, of course, also add your own new member functions. Remember
6233 that the RTTI may be used to get information about what kinds of objects
6234 you are dealing with (the position in the class hierarchy) and that you
6235 can always extract the bare object from an @code{ex} by stripping the
6236 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
6237 should become a need.
6239 That's it. May the source be with you!
6242 @node A Comparison With Other CAS, Advantages, Adding classes, Top
6243 @c node-name, next, previous, up
6244 @chapter A Comparison With Other CAS
6247 This chapter will give you some information on how GiNaC compares to
6248 other, traditional Computer Algebra Systems, like @emph{Maple},
6249 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
6250 disadvantages over these systems.
6253 * Advantages:: Strengths of the GiNaC approach.
6254 * Disadvantages:: Weaknesses of the GiNaC approach.
6255 * Why C++?:: Attractiveness of C++.
6258 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
6259 @c node-name, next, previous, up
6262 GiNaC has several advantages over traditional Computer
6263 Algebra Systems, like
6268 familiar language: all common CAS implement their own proprietary
6269 grammar which you have to learn first (and maybe learn again when your
6270 vendor decides to `enhance' it). With GiNaC you can write your program
6271 in common C++, which is standardized.
6275 structured data types: you can build up structured data types using
6276 @code{struct}s or @code{class}es together with STL features instead of
6277 using unnamed lists of lists of lists.
6280 strongly typed: in CAS, you usually have only one kind of variables
6281 which can hold contents of an arbitrary type. This 4GL like feature is
6282 nice for novice programmers, but dangerous.
6285 development tools: powerful development tools exist for C++, like fancy
6286 editors (e.g. with automatic indentation and syntax highlighting),
6287 debuggers, visualization tools, documentation generators@dots{}
6290 modularization: C++ programs can easily be split into modules by
6291 separating interface and implementation.
6294 price: GiNaC is distributed under the GNU Public License which means
6295 that it is free and available with source code. And there are excellent
6296 C++-compilers for free, too.
6299 extendable: you can add your own classes to GiNaC, thus extending it on
6300 a very low level. Compare this to a traditional CAS that you can
6301 usually only extend on a high level by writing in the language defined
6302 by the parser. In particular, it turns out to be almost impossible to
6303 fix bugs in a traditional system.
6306 multiple interfaces: Though real GiNaC programs have to be written in
6307 some editor, then be compiled, linked and executed, there are more ways
6308 to work with the GiNaC engine. Many people want to play with
6309 expressions interactively, as in traditional CASs. Currently, two such
6310 windows into GiNaC have been implemented and many more are possible: the
6311 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
6312 types to a command line and second, as a more consistent approach, an
6313 interactive interface to the Cint C++ interpreter has been put together
6314 (called GiNaC-cint) that allows an interactive scripting interface
6315 consistent with the C++ language. It is available from the usual GiNaC
6319 seamless integration: it is somewhere between difficult and impossible
6320 to call CAS functions from within a program written in C++ or any other
6321 programming language and vice versa. With GiNaC, your symbolic routines
6322 are part of your program. You can easily call third party libraries,
6323 e.g. for numerical evaluation or graphical interaction. All other
6324 approaches are much more cumbersome: they range from simply ignoring the
6325 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
6326 system (i.e. @emph{Yacas}).
6329 efficiency: often large parts of a program do not need symbolic
6330 calculations at all. Why use large integers for loop variables or
6331 arbitrary precision arithmetics where @code{int} and @code{double} are
6332 sufficient? For pure symbolic applications, GiNaC is comparable in
6333 speed with other CAS.
6338 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
6339 @c node-name, next, previous, up
6340 @section Disadvantages
6342 Of course it also has some disadvantages:
6347 advanced features: GiNaC cannot compete with a program like
6348 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
6349 which grows since 1981 by the work of dozens of programmers, with
6350 respect to mathematical features. Integration, factorization,
6351 non-trivial simplifications, limits etc. are missing in GiNaC (and are
6352 not planned for the near future).
6355 portability: While the GiNaC library itself is designed to avoid any
6356 platform dependent features (it should compile on any ANSI compliant C++
6357 compiler), the currently used version of the CLN library (fast large
6358 integer and arbitrary precision arithmetics) can only by compiled
6359 without hassle on systems with the C++ compiler from the GNU Compiler
6360 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
6361 macros to let the compiler gather all static initializations, which
6362 works for GNU C++ only. Feel free to contact the authors in case you
6363 really believe that you need to use a different compiler. We have
6364 occasionally used other compilers and may be able to give you advice.}
6365 GiNaC uses recent language features like explicit constructors, mutable
6366 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
6367 literally. Recent GCC versions starting at 2.95.3, although itself not
6368 yet ANSI compliant, support all needed features.
6373 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
6374 @c node-name, next, previous, up
6377 Why did we choose to implement GiNaC in C++ instead of Java or any other
6378 language? C++ is not perfect: type checking is not strict (casting is
6379 possible), separation between interface and implementation is not
6380 complete, object oriented design is not enforced. The main reason is
6381 the often scolded feature of operator overloading in C++. While it may
6382 be true that operating on classes with a @code{+} operator is rarely
6383 meaningful, it is perfectly suited for algebraic expressions. Writing
6384 @math{3x+5y} as @code{3*x+5*y} instead of
6385 @code{x.times(3).plus(y.times(5))} looks much more natural.
6386 Furthermore, the main developers are more familiar with C++ than with
6387 any other programming language.
6390 @node Internal Structures, Expressions are reference counted, Why C++? , Top
6391 @c node-name, next, previous, up
6392 @appendix Internal Structures
6395 * Expressions are reference counted::
6396 * Internal representation of products and sums::
6399 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
6400 @c node-name, next, previous, up
6401 @appendixsection Expressions are reference counted
6403 @cindex reference counting
6404 @cindex copy-on-write
6405 @cindex garbage collection
6406 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
6407 where the counter belongs to the algebraic objects derived from class
6408 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
6409 which @code{ex} contains an instance. If you understood that, you can safely
6410 skip the rest of this passage.
6412 Expressions are extremely light-weight since internally they work like
6413 handles to the actual representation. They really hold nothing more
6414 than a pointer to some other object. What this means in practice is
6415 that whenever you create two @code{ex} and set the second equal to the
6416 first no copying process is involved. Instead, the copying takes place
6417 as soon as you try to change the second. Consider the simple sequence
6422 #include <ginac/ginac.h>
6423 using namespace std;
6424 using namespace GiNaC;
6428 symbol x("x"), y("y"), z("z");
6431 e1 = sin(x + 2*y) + 3*z + 41;
6432 e2 = e1; // e2 points to same object as e1
6433 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
6434 e2 += 1; // e2 is copied into a new object
6435 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
6439 The line @code{e2 = e1;} creates a second expression pointing to the
6440 object held already by @code{e1}. The time involved for this operation
6441 is therefore constant, no matter how large @code{e1} was. Actual
6442 copying, however, must take place in the line @code{e2 += 1;} because
6443 @code{e1} and @code{e2} are not handles for the same object any more.
6444 This concept is called @dfn{copy-on-write semantics}. It increases
6445 performance considerably whenever one object occurs multiple times and
6446 represents a simple garbage collection scheme because when an @code{ex}
6447 runs out of scope its destructor checks whether other expressions handle
6448 the object it points to too and deletes the object from memory if that
6449 turns out not to be the case. A slightly less trivial example of
6450 differentiation using the chain-rule should make clear how powerful this
6455 symbol x("x"), y("y");
6459 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
6460 cout << e1 << endl // prints x+3*y
6461 << e2 << endl // prints (x+3*y)^3
6462 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
6466 Here, @code{e1} will actually be referenced three times while @code{e2}
6467 will be referenced two times. When the power of an expression is built,
6468 that expression needs not be copied. Likewise, since the derivative of
6469 a power of an expression can be easily expressed in terms of that
6470 expression, no copying of @code{e1} is involved when @code{e3} is
6471 constructed. So, when @code{e3} is constructed it will print as
6472 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
6473 holds a reference to @code{e2} and the factor in front is just
6476 As a user of GiNaC, you cannot see this mechanism of copy-on-write
6477 semantics. When you insert an expression into a second expression, the
6478 result behaves exactly as if the contents of the first expression were
6479 inserted. But it may be useful to remember that this is not what
6480 happens. Knowing this will enable you to write much more efficient
6481 code. If you still have an uncertain feeling with copy-on-write
6482 semantics, we recommend you have a look at the
6483 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
6484 Marshall Cline. Chapter 16 covers this issue and presents an
6485 implementation which is pretty close to the one in GiNaC.
6488 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
6489 @c node-name, next, previous, up
6490 @appendixsection Internal representation of products and sums
6492 @cindex representation
6495 @cindex @code{power}
6496 Although it should be completely transparent for the user of
6497 GiNaC a short discussion of this topic helps to understand the sources
6498 and also explain performance to a large degree. Consider the
6499 unexpanded symbolic expression
6501 $2d^3 \left( 4a + 5b - 3 \right)$
6504 @math{2*d^3*(4*a+5*b-3)}
6506 which could naively be represented by a tree of linear containers for
6507 addition and multiplication, one container for exponentiation with base
6508 and exponent and some atomic leaves of symbols and numbers in this
6513 @cindex pair-wise representation
6514 However, doing so results in a rather deeply nested tree which will
6515 quickly become inefficient to manipulate. We can improve on this by
6516 representing the sum as a sequence of terms, each one being a pair of a
6517 purely numeric multiplicative coefficient and its rest. In the same
6518 spirit we can store the multiplication as a sequence of terms, each
6519 having a numeric exponent and a possibly complicated base, the tree
6520 becomes much more flat:
6524 The number @code{3} above the symbol @code{d} shows that @code{mul}
6525 objects are treated similarly where the coefficients are interpreted as
6526 @emph{exponents} now. Addition of sums of terms or multiplication of
6527 products with numerical exponents can be coded to be very efficient with
6528 such a pair-wise representation. Internally, this handling is performed
6529 by most CAS in this way. It typically speeds up manipulations by an
6530 order of magnitude. The overall multiplicative factor @code{2} and the
6531 additive term @code{-3} look somewhat out of place in this
6532 representation, however, since they are still carrying a trivial
6533 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
6534 this is avoided by adding a field that carries an overall numeric
6535 coefficient. This results in the realistic picture of internal
6538 $2d^3 \left( 4a + 5b - 3 \right)$:
6541 @math{2*d^3*(4*a+5*b-3)}:
6547 This also allows for a better handling of numeric radicals, since
6548 @code{sqrt(2)} can now be carried along calculations. Now it should be
6549 clear, why both classes @code{add} and @code{mul} are derived from the
6550 same abstract class: the data representation is the same, only the
6551 semantics differs. In the class hierarchy, methods for polynomial
6552 expansion and the like are reimplemented for @code{add} and @code{mul},
6553 but the data structure is inherited from @code{expairseq}.
6556 @node Package Tools, ginac-config, Internal representation of products and sums, Top
6557 @c node-name, next, previous, up
6558 @appendix Package Tools
6560 If you are creating a software package that uses the GiNaC library,
6561 setting the correct command line options for the compiler and linker
6562 can be difficult. GiNaC includes two tools to make this process easier.
6565 * ginac-config:: A shell script to detect compiler and linker flags.
6566 * AM_PATH_GINAC:: Macro for GNU automake.
6570 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
6571 @c node-name, next, previous, up
6572 @section @command{ginac-config}
6573 @cindex ginac-config
6575 @command{ginac-config} is a shell script that you can use to determine
6576 the compiler and linker command line options required to compile and
6577 link a program with the GiNaC library.
6579 @command{ginac-config} takes the following flags:
6583 Prints out the version of GiNaC installed.
6585 Prints '-I' flags pointing to the installed header files.
6587 Prints out the linker flags necessary to link a program against GiNaC.
6588 @item --prefix[=@var{PREFIX}]
6589 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
6590 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
6591 Otherwise, prints out the configured value of @env{$prefix}.
6592 @item --exec-prefix[=@var{PREFIX}]
6593 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
6594 Otherwise, prints out the configured value of @env{$exec_prefix}.
6597 Typically, @command{ginac-config} will be used within a configure
6598 script, as described below. It, however, can also be used directly from
6599 the command line using backquotes to compile a simple program. For
6603 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
6606 This command line might expand to (for example):
6609 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
6610 -lginac -lcln -lstdc++
6613 Not only is the form using @command{ginac-config} easier to type, it will
6614 work on any system, no matter how GiNaC was configured.
6617 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
6618 @c node-name, next, previous, up
6619 @section @samp{AM_PATH_GINAC}
6620 @cindex AM_PATH_GINAC
6622 For packages configured using GNU automake, GiNaC also provides
6623 a macro to automate the process of checking for GiNaC.
6626 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
6634 Determines the location of GiNaC using @command{ginac-config}, which is
6635 either found in the user's path, or from the environment variable
6636 @env{GINACLIB_CONFIG}.
6639 Tests the installed libraries to make sure that their version
6640 is later than @var{MINIMUM-VERSION}. (A default version will be used
6644 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
6645 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
6646 variable to the output of @command{ginac-config --libs}, and calls
6647 @samp{AC_SUBST()} for these variables so they can be used in generated
6648 makefiles, and then executes @var{ACTION-IF-FOUND}.
6651 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
6652 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
6656 This macro is in file @file{ginac.m4} which is installed in
6657 @file{$datadir/aclocal}. Note that if automake was installed with a
6658 different @samp{--prefix} than GiNaC, you will either have to manually
6659 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
6660 aclocal the @samp{-I} option when running it.
6663 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
6664 * Example package:: Example of a package using AM_PATH_GINAC.
6668 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
6669 @c node-name, next, previous, up
6670 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
6672 Simply make sure that @command{ginac-config} is in your path, and run
6673 the configure script.
6680 The directory where the GiNaC libraries are installed needs
6681 to be found by your system's dynamic linker.
6683 This is generally done by
6686 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
6692 setting the environment variable @env{LD_LIBRARY_PATH},
6695 or, as a last resort,
6698 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
6699 running configure, for instance:
6702 LDFLAGS=-R/home/cbauer/lib ./configure
6707 You can also specify a @command{ginac-config} not in your path by
6708 setting the @env{GINACLIB_CONFIG} environment variable to the
6709 name of the executable
6712 If you move the GiNaC package from its installed location,
6713 you will either need to modify @command{ginac-config} script
6714 manually to point to the new location or rebuild GiNaC.
6725 --with-ginac-prefix=@var{PREFIX}
6726 --with-ginac-exec-prefix=@var{PREFIX}
6729 are provided to override the prefix and exec-prefix that were stored
6730 in the @command{ginac-config} shell script by GiNaC's configure. You are
6731 generally better off configuring GiNaC with the right path to begin with.
6735 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
6736 @c node-name, next, previous, up
6737 @subsection Example of a package using @samp{AM_PATH_GINAC}
6739 The following shows how to build a simple package using automake
6740 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
6743 #include <ginac/ginac.h>
6747 GiNaC::symbol x("x");
6748 GiNaC::ex a = GiNaC::sin(x);
6749 std::cout << "Derivative of " << a
6750 << " is " << a.diff(x) << std::endl;
6755 You should first read the introductory portions of the automake
6756 Manual, if you are not already familiar with it.
6758 Two files are needed, @file{configure.in}, which is used to build the
6762 dnl Process this file with autoconf to produce a configure script.
6764 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
6770 AM_PATH_GINAC(0.9.0, [
6771 LIBS="$LIBS $GINACLIB_LIBS"
6772 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
6773 ], AC_MSG_ERROR([need to have GiNaC installed]))
6778 The only command in this which is not standard for automake
6779 is the @samp{AM_PATH_GINAC} macro.
6781 That command does the following: If a GiNaC version greater or equal
6782 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
6783 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
6784 the error message `need to have GiNaC installed'
6786 And the @file{Makefile.am}, which will be used to build the Makefile.
6789 ## Process this file with automake to produce Makefile.in
6790 bin_PROGRAMS = simple
6791 simple_SOURCES = simple.cpp
6794 This @file{Makefile.am}, says that we are building a single executable,
6795 from a single source file @file{simple.cpp}. Since every program
6796 we are building uses GiNaC we simply added the GiNaC options
6797 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
6798 want to specify them on a per-program basis: for instance by
6802 simple_LDADD = $(GINACLIB_LIBS)
6803 INCLUDES = $(GINACLIB_CPPFLAGS)
6806 to the @file{Makefile.am}.
6808 To try this example out, create a new directory and add the three
6811 Now execute the following commands:
6814 $ automake --add-missing
6819 You now have a package that can be built in the normal fashion
6828 @node Bibliography, Concept Index, Example package, Top
6829 @c node-name, next, previous, up
6830 @appendix Bibliography
6835 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
6838 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
6841 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
6844 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
6847 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
6848 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
6851 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
6852 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
6853 Academic Press, London
6856 @cite{Computer Algebra Systems - A Practical Guide},
6857 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
6860 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
6861 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
6864 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
6865 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
6868 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
6873 @node Concept Index, , Bibliography, Top
6874 @c node-name, next, previous, up
6875 @unnumbered Concept Index