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(void) const;
814 that returns the order of the singularity (or 0 when the pole is
815 logarithmic or the order is undefined).
817 When using GiNaC it is useful to arrange for exceptions to be catched in
818 the main program even if you don't want to do any special error handling.
819 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
820 default exception handler of your C++ compiler's run-time system which
821 usually only aborts the program without giving any information what went
824 Here is an example for a @code{main()} function that catches and prints
825 exceptions generated by GiNaC:
830 #include <ginac/ginac.h>
832 using namespace GiNaC;
840 @} catch (exception &p) @{
841 cerr << p.what() << endl;
849 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
850 @c node-name, next, previous, up
851 @section The Class Hierarchy
853 GiNaC's class hierarchy consists of several classes representing
854 mathematical objects, all of which (except for @code{ex} and some
855 helpers) are internally derived from one abstract base class called
856 @code{basic}. You do not have to deal with objects of class
857 @code{basic}, instead you'll be dealing with symbols, numbers,
858 containers of expressions and so on.
862 To get an idea about what kinds of symbolic composites may be built we
863 have a look at the most important classes in the class hierarchy and
864 some of the relations among the classes:
866 @image{classhierarchy}
868 The abstract classes shown here (the ones without drop-shadow) are of no
869 interest for the user. They are used internally in order to avoid code
870 duplication if two or more classes derived from them share certain
871 features. An example is @code{expairseq}, a container for a sequence of
872 pairs each consisting of one expression and a number (@code{numeric}).
873 What @emph{is} visible to the user are the derived classes @code{add}
874 and @code{mul}, representing sums and products. @xref{Internal
875 Structures}, where these two classes are described in more detail. The
876 following table shortly summarizes what kinds of mathematical objects
877 are stored in the different classes:
880 @multitable @columnfractions .22 .78
881 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
882 @item @code{constant} @tab Constants like
889 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
890 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
891 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
892 @item @code{ncmul} @tab Products of non-commutative objects
893 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
898 @code{sqrt(}@math{2}@code{)}
901 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
902 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
903 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
904 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
905 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
906 @item @code{indexed} @tab Indexed object like @math{A_ij}
907 @item @code{tensor} @tab Special tensor like the delta and metric tensors
908 @item @code{idx} @tab Index of an indexed object
909 @item @code{varidx} @tab Index with variance
910 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
911 @item @code{wildcard} @tab Wildcard for pattern matching
916 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
917 @c node-name, next, previous, up
919 @cindex @code{symbol} (class)
920 @cindex hierarchy of classes
923 Symbols are for symbolic manipulation what atoms are for chemistry. You
924 can declare objects of class @code{symbol} as any other object simply by
925 saying @code{symbol x,y;}. There is, however, a catch in here having to
926 do with the fact that C++ is a compiled language. The information about
927 the symbol's name is thrown away by the compiler but at a later stage
928 you may want to print expressions holding your symbols. In order to
929 avoid confusion GiNaC's symbols are able to know their own name. This
930 is accomplished by declaring its name for output at construction time in
931 the fashion @code{symbol x("x");}. If you declare a symbol using the
932 default constructor (i.e. without string argument) the system will deal
933 out a unique name. That name may not be suitable for printing but for
934 internal routines when no output is desired it is often enough. We'll
935 come across examples of such symbols later in this tutorial.
937 This implies that the strings passed to symbols at construction time may
938 not be used for comparing two of them. It is perfectly legitimate to
939 write @code{symbol x("x"),y("x");} but it is likely to lead into
940 trouble. Here, @code{x} and @code{y} are different symbols and
941 statements like @code{x-y} will not be simplified to zero although the
942 output @code{x-x} looks funny. Such output may also occur when there
943 are two different symbols in two scopes, for instance when you call a
944 function that declares a symbol with a name already existent in a symbol
945 in the calling function. Again, comparing them (using @code{operator==}
946 for instance) will always reveal their difference. Watch out, please.
948 @cindex @code{subs()}
949 Although symbols can be assigned expressions for internal reasons, you
950 should not do it (and we are not going to tell you how it is done). If
951 you want to replace a symbol with something else in an expression, you
952 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
955 @node Numbers, Constants, Symbols, Basic Concepts
956 @c node-name, next, previous, up
958 @cindex @code{numeric} (class)
964 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
965 The classes therein serve as foundation classes for GiNaC. CLN stands
966 for Class Library for Numbers or alternatively for Common Lisp Numbers.
967 In order to find out more about CLN's internals, the reader is referred to
968 the documentation of that library. @inforef{Introduction, , cln}, for
969 more information. Suffice to say that it is by itself build on top of
970 another library, the GNU Multiple Precision library GMP, which is an
971 extremely fast library for arbitrary long integers and rationals as well
972 as arbitrary precision floating point numbers. It is very commonly used
973 by several popular cryptographic applications. CLN extends GMP by
974 several useful things: First, it introduces the complex number field
975 over either reals (i.e. floating point numbers with arbitrary precision)
976 or rationals. Second, it automatically converts rationals to integers
977 if the denominator is unity and complex numbers to real numbers if the
978 imaginary part vanishes and also correctly treats algebraic functions.
979 Third it provides good implementations of state-of-the-art algorithms
980 for all trigonometric and hyperbolic functions as well as for
981 calculation of some useful constants.
983 The user can construct an object of class @code{numeric} in several
984 ways. The following example shows the four most important constructors.
985 It uses construction from C-integer, construction of fractions from two
986 integers, construction from C-float and construction from a string:
990 #include <ginac/ginac.h>
991 using namespace GiNaC;
995 numeric two = 2; // exact integer 2
996 numeric r(2,3); // exact fraction 2/3
997 numeric e(2.71828); // floating point number
998 numeric p = "3.14159265358979323846"; // constructor from string
999 // Trott's constant in scientific notation:
1000 numeric trott("1.0841015122311136151E-2");
1002 std::cout << two*p << std::endl; // floating point 6.283...
1007 @cindex complex numbers
1008 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1013 numeric z1 = 2-3*I; // exact complex number 2-3i
1014 numeric z2 = 5.9+1.6*I; // complex floating point number
1018 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1019 This would, however, call C's built-in operator @code{/} for integers
1020 first and result in a numeric holding a plain integer 1. @strong{Never
1021 use the operator @code{/} on integers} unless you know exactly what you
1022 are doing! Use the constructor from two integers instead, as shown in
1023 the example above. Writing @code{numeric(1)/2} may look funny but works
1026 @cindex @code{Digits}
1028 We have seen now the distinction between exact numbers and floating
1029 point numbers. Clearly, the user should never have to worry about
1030 dynamically created exact numbers, since their `exactness' always
1031 determines how they ought to be handled, i.e. how `long' they are. The
1032 situation is different for floating point numbers. Their accuracy is
1033 controlled by one @emph{global} variable, called @code{Digits}. (For
1034 those readers who know about Maple: it behaves very much like Maple's
1035 @code{Digits}). All objects of class numeric that are constructed from
1036 then on will be stored with a precision matching that number of decimal
1041 #include <ginac/ginac.h>
1042 using namespace std;
1043 using namespace GiNaC;
1047 numeric three(3.0), one(1.0);
1048 numeric x = one/three;
1050 cout << "in " << Digits << " digits:" << endl;
1052 cout << Pi.evalf() << endl;
1064 The above example prints the following output to screen:
1068 0.33333333333333333334
1069 3.1415926535897932385
1071 0.33333333333333333333333333333333333333333333333333333333333333333334
1072 3.1415926535897932384626433832795028841971693993751058209749445923078
1076 Note that the last number is not necessarily rounded as you would
1077 naively expect it to be rounded in the decimal system. But note also,
1078 that in both cases you got a couple of extra digits. This is because
1079 numbers are internally stored by CLN as chunks of binary digits in order
1080 to match your machine's word size and to not waste precision. Thus, on
1081 architectures with different word size, the above output might even
1082 differ with regard to actually computed digits.
1084 It should be clear that objects of class @code{numeric} should be used
1085 for constructing numbers or for doing arithmetic with them. The objects
1086 one deals with most of the time are the polymorphic expressions @code{ex}.
1088 @subsection Tests on numbers
1090 Once you have declared some numbers, assigned them to expressions and
1091 done some arithmetic with them it is frequently desired to retrieve some
1092 kind of information from them like asking whether that number is
1093 integer, rational, real or complex. For those cases GiNaC provides
1094 several useful methods. (Internally, they fall back to invocations of
1095 certain CLN functions.)
1097 As an example, let's construct some rational number, multiply it with
1098 some multiple of its denominator and test what comes out:
1102 #include <ginac/ginac.h>
1103 using namespace std;
1104 using namespace GiNaC;
1106 // some very important constants:
1107 const numeric twentyone(21);
1108 const numeric ten(10);
1109 const numeric five(5);
1113 numeric answer = twentyone;
1116 cout << answer.is_integer() << endl; // false, it's 21/5
1118 cout << answer.is_integer() << endl; // true, it's 42 now!
1122 Note that the variable @code{answer} is constructed here as an integer
1123 by @code{numeric}'s copy constructor but in an intermediate step it
1124 holds a rational number represented as integer numerator and integer
1125 denominator. When multiplied by 10, the denominator becomes unity and
1126 the result is automatically converted to a pure integer again.
1127 Internally, the underlying CLN is responsible for this behavior and we
1128 refer the reader to CLN's documentation. Suffice to say that
1129 the same behavior applies to complex numbers as well as return values of
1130 certain functions. Complex numbers are automatically converted to real
1131 numbers if the imaginary part becomes zero. The full set of tests that
1132 can be applied is listed in the following table.
1135 @multitable @columnfractions .30 .70
1136 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1137 @item @code{.is_zero()}
1138 @tab @dots{}equal to zero
1139 @item @code{.is_positive()}
1140 @tab @dots{}not complex and greater than 0
1141 @item @code{.is_integer()}
1142 @tab @dots{}a (non-complex) integer
1143 @item @code{.is_pos_integer()}
1144 @tab @dots{}an integer and greater than 0
1145 @item @code{.is_nonneg_integer()}
1146 @tab @dots{}an integer and greater equal 0
1147 @item @code{.is_even()}
1148 @tab @dots{}an even integer
1149 @item @code{.is_odd()}
1150 @tab @dots{}an odd integer
1151 @item @code{.is_prime()}
1152 @tab @dots{}a prime integer (probabilistic primality test)
1153 @item @code{.is_rational()}
1154 @tab @dots{}an exact rational number (integers are rational, too)
1155 @item @code{.is_real()}
1156 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1157 @item @code{.is_cinteger()}
1158 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1159 @item @code{.is_crational()}
1160 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1165 @node Constants, Fundamental containers, Numbers, Basic Concepts
1166 @c node-name, next, previous, up
1168 @cindex @code{constant} (class)
1171 @cindex @code{Catalan}
1172 @cindex @code{Euler}
1173 @cindex @code{evalf()}
1174 Constants behave pretty much like symbols except that they return some
1175 specific number when the method @code{.evalf()} is called.
1177 The predefined known constants are:
1180 @multitable @columnfractions .14 .30 .56
1181 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1183 @tab Archimedes' constant
1184 @tab 3.14159265358979323846264338327950288
1185 @item @code{Catalan}
1186 @tab Catalan's constant
1187 @tab 0.91596559417721901505460351493238411
1189 @tab Euler's (or Euler-Mascheroni) constant
1190 @tab 0.57721566490153286060651209008240243
1195 @node Fundamental containers, Lists, Constants, Basic Concepts
1196 @c node-name, next, previous, up
1197 @section Sums, products and powers
1201 @cindex @code{power}
1203 Simple rational expressions are written down in GiNaC pretty much like
1204 in other CAS or like expressions involving numerical variables in C.
1205 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1206 been overloaded to achieve this goal. When you run the following
1207 code snippet, the constructor for an object of type @code{mul} is
1208 automatically called to hold the product of @code{a} and @code{b} and
1209 then the constructor for an object of type @code{add} is called to hold
1210 the sum of that @code{mul} object and the number one:
1214 symbol a("a"), b("b");
1219 @cindex @code{pow()}
1220 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1221 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1222 construction is necessary since we cannot safely overload the constructor
1223 @code{^} in C++ to construct a @code{power} object. If we did, it would
1224 have several counterintuitive and undesired effects:
1228 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1230 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1231 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1232 interpret this as @code{x^(a^b)}.
1234 Also, expressions involving integer exponents are very frequently used,
1235 which makes it even more dangerous to overload @code{^} since it is then
1236 hard to distinguish between the semantics as exponentiation and the one
1237 for exclusive or. (It would be embarrassing to return @code{1} where one
1238 has requested @code{2^3}.)
1241 @cindex @command{ginsh}
1242 All effects are contrary to mathematical notation and differ from the
1243 way most other CAS handle exponentiation, therefore overloading @code{^}
1244 is ruled out for GiNaC's C++ part. The situation is different in
1245 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1246 that the other frequently used exponentiation operator @code{**} does
1247 not exist at all in C++).
1249 To be somewhat more precise, objects of the three classes described
1250 here, are all containers for other expressions. An object of class
1251 @code{power} is best viewed as a container with two slots, one for the
1252 basis, one for the exponent. All valid GiNaC expressions can be
1253 inserted. However, basic transformations like simplifying
1254 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1255 when this is mathematically possible. If we replace the outer exponent
1256 three in the example by some symbols @code{a}, the simplification is not
1257 safe and will not be performed, since @code{a} might be @code{1/2} and
1260 Objects of type @code{add} and @code{mul} are containers with an
1261 arbitrary number of slots for expressions to be inserted. Again, simple
1262 and safe simplifications are carried out like transforming
1263 @code{3*x+4-x} to @code{2*x+4}.
1266 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1267 @c node-name, next, previous, up
1268 @section Lists of expressions
1269 @cindex @code{lst} (class)
1271 @cindex @code{nops()}
1273 @cindex @code{append()}
1274 @cindex @code{prepend()}
1275 @cindex @code{remove_first()}
1276 @cindex @code{remove_last()}
1278 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1279 expressions. They are not as ubiquitous as in many other computer algebra
1280 packages, but are sometimes used to supply a variable number of arguments of
1281 the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
1282 so you should have a basic understanding of them.
1284 Lists of up to 16 expressions can be directly constructed from single
1289 symbol x("x"), y("y");
1290 lst l(x, 2, y, x+y);
1291 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1295 Use the @code{nops()} method to determine the size (number of expressions) of
1296 a list and the @code{op()} method or the @code{[]} operator to access
1297 individual elements:
1301 cout << l.nops() << endl; // prints '4'
1302 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1306 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1307 (the only other one is @code{matrix}). You can modify single elements:
1311 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1315 You can append or prepend an expression to a list with the @code{append()}
1316 and @code{prepend()} methods:
1320 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1321 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1325 You can remove the first or last element of a list with @code{remove_first()}
1326 and @code{remove_last()}:
1330 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1331 l.remove_last(); // l is now @{x, 7, y, x+y@}
1335 You can bring the elements of a list into a canonical order with @code{sort()}:
1339 lst l1(x, 2, y, x+y);
1340 lst l2(2, x+y, x, y);
1343 // l1 and l2 are now equal
1347 Finally, you can remove all but the first element of consecutive groups of
1348 elements with @code{unique()}:
1352 lst l3(x, 2, 2, 2, y, x+y, y+x);
1353 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1358 @node Mathematical functions, Relations, Lists, Basic Concepts
1359 @c node-name, next, previous, up
1360 @section Mathematical functions
1361 @cindex @code{function} (class)
1362 @cindex trigonometric function
1363 @cindex hyperbolic function
1365 There are quite a number of useful functions hard-wired into GiNaC. For
1366 instance, all trigonometric and hyperbolic functions are implemented
1367 (@xref{Built-in Functions}, for a complete list).
1369 These functions (better called @emph{pseudofunctions}) are all objects
1370 of class @code{function}. They accept one or more expressions as
1371 arguments and return one expression. If the arguments are not
1372 numerical, the evaluation of the function may be halted, as it does in
1373 the next example, showing how a function returns itself twice and
1374 finally an expression that may be really useful:
1376 @cindex Gamma function
1377 @cindex @code{subs()}
1380 symbol x("x"), y("y");
1382 cout << tgamma(foo) << endl;
1383 // -> tgamma(x+(1/2)*y)
1384 ex bar = foo.subs(y==1);
1385 cout << tgamma(bar) << endl;
1387 ex foobar = bar.subs(x==7);
1388 cout << tgamma(foobar) << endl;
1389 // -> (135135/128)*Pi^(1/2)
1393 Besides evaluation most of these functions allow differentiation, series
1394 expansion and so on. Read the next chapter in order to learn more about
1397 It must be noted that these pseudofunctions are created by inline
1398 functions, where the argument list is templated. This means that
1399 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1400 @code{sin(ex(1))} and will therefore not result in a floating point
1401 number. Unless of course the function prototype is explicitly
1402 overridden -- which is the case for arguments of type @code{numeric}
1403 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1404 point number of class @code{numeric} you should call
1405 @code{sin(numeric(1))}. This is almost the same as calling
1406 @code{sin(1).evalf()} except that the latter will return a numeric
1407 wrapped inside an @code{ex}.
1410 @node Relations, Matrices, Mathematical functions, Basic Concepts
1411 @c node-name, next, previous, up
1413 @cindex @code{relational} (class)
1415 Sometimes, a relation holding between two expressions must be stored
1416 somehow. The class @code{relational} is a convenient container for such
1417 purposes. A relation is by definition a container for two @code{ex} and
1418 a relation between them that signals equality, inequality and so on.
1419 They are created by simply using the C++ operators @code{==}, @code{!=},
1420 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1422 @xref{Mathematical functions}, for examples where various applications
1423 of the @code{.subs()} method show how objects of class relational are
1424 used as arguments. There they provide an intuitive syntax for
1425 substitutions. They are also used as arguments to the @code{ex::series}
1426 method, where the left hand side of the relation specifies the variable
1427 to expand in and the right hand side the expansion point. They can also
1428 be used for creating systems of equations that are to be solved for
1429 unknown variables. But the most common usage of objects of this class
1430 is rather inconspicuous in statements of the form @code{if
1431 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1432 conversion from @code{relational} to @code{bool} takes place. Note,
1433 however, that @code{==} here does not perform any simplifications, hence
1434 @code{expand()} must be called explicitly.
1437 @node Matrices, Indexed objects, Relations, Basic Concepts
1438 @c node-name, next, previous, up
1440 @cindex @code{matrix} (class)
1442 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1443 matrix with @math{m} rows and @math{n} columns are accessed with two
1444 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1445 second one in the range 0@dots{}@math{n-1}.
1447 There are a couple of ways to construct matrices, with or without preset
1450 @cindex @code{lst_to_matrix()}
1451 @cindex @code{diag_matrix()}
1452 @cindex @code{unit_matrix()}
1453 @cindex @code{symbolic_matrix()}
1455 matrix::matrix(unsigned r, unsigned c);
1456 matrix::matrix(unsigned r, unsigned c, const lst & l);
1457 ex lst_to_matrix(const lst & l);
1458 ex diag_matrix(const lst & l);
1459 ex unit_matrix(unsigned x);
1460 ex unit_matrix(unsigned r, unsigned c);
1461 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1462 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1465 The first two functions are @code{matrix} constructors which create a matrix
1466 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1467 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1468 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1469 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1470 constructs a diagonal matrix given the list of diagonal elements.
1471 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1472 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1473 with newly generated symbols made of the specified base name and the
1474 position of each element in the matrix.
1476 Matrix elements can be accessed and set using the parenthesis (function call)
1480 const ex & matrix::operator()(unsigned r, unsigned c) const;
1481 ex & matrix::operator()(unsigned r, unsigned c);
1484 It is also possible to access the matrix elements in a linear fashion with
1485 the @code{op()} method. But C++-style subscripting with square brackets
1486 @samp{[]} is not available.
1488 Here are a couple of examples of constructing matrices:
1492 symbol a("a"), b("b");
1500 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1503 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1506 cout << diag_matrix(lst(a, b)) << endl;
1509 cout << unit_matrix(3) << endl;
1510 // -> [[1,0,0],[0,1,0],[0,0,1]]
1512 cout << symbolic_matrix(2, 3, "x") << endl;
1513 // -> [[x00,x01,x02],[x10,x11,x12]]
1517 @cindex @code{transpose()}
1518 @cindex @code{inverse()}
1519 There are three ways to do arithmetic with matrices. The first (and most
1520 efficient one) is to use the methods provided by the @code{matrix} class:
1523 matrix matrix::add(const matrix & other) const;
1524 matrix matrix::sub(const matrix & other) const;
1525 matrix matrix::mul(const matrix & other) const;
1526 matrix matrix::mul_scalar(const ex & other) const;
1527 matrix matrix::pow(const ex & expn) const;
1528 matrix matrix::transpose(void) const;
1529 matrix matrix::inverse(void) const;
1532 All of these methods return the result as a new matrix object. Here is an
1533 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1538 matrix A(2, 2, lst(1, 2, 3, 4));
1539 matrix B(2, 2, lst(-1, 0, 2, 1));
1540 matrix C(2, 2, lst(8, 4, 2, 1));
1542 matrix result = A.mul(B).sub(C.mul_scalar(2));
1543 cout << result << endl;
1544 // -> [[-13,-6],[1,2]]
1549 @cindex @code{evalm()}
1550 The second (and probably the most natural) way is to construct an expression
1551 containing matrices with the usual arithmetic operators and @code{pow()}.
1552 For efficiency reasons, expressions with sums, products and powers of
1553 matrices are not automatically evaluated in GiNaC. You have to call the
1557 ex ex::evalm() const;
1560 to obtain the result:
1567 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1568 cout << e.evalm() << endl;
1569 // -> [[-13,-6],[1,2]]
1574 The non-commutativity of the product @code{A*B} in this example is
1575 automatically recognized by GiNaC. There is no need to use a special
1576 operator here. @xref{Non-commutative objects}, for more information about
1577 dealing with non-commutative expressions.
1579 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1580 to perform the arithmetic:
1585 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1586 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1588 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1589 cout << e.simplify_indexed() << endl;
1590 // -> [[-13,-6],[1,2]].i.j
1594 Using indices is most useful when working with rectangular matrices and
1595 one-dimensional vectors because you don't have to worry about having to
1596 transpose matrices before multiplying them. @xref{Indexed objects}, for
1597 more information about using matrices with indices, and about indices in
1600 The @code{matrix} class provides a couple of additional methods for
1601 computing determinants, traces, and characteristic polynomials:
1603 @cindex @code{determinant()}
1604 @cindex @code{trace()}
1605 @cindex @code{charpoly()}
1607 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1608 ex matrix::trace(void) const;
1609 ex matrix::charpoly(const symbol & lambda) const;
1612 The @samp{algo} argument of @code{determinant()} allows to select between
1613 different algorithms for calculating the determinant. The possible values
1614 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1615 heuristic to automatically select an algorithm that is likely to give the
1616 result most quickly.
1619 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1620 @c node-name, next, previous, up
1621 @section Indexed objects
1623 GiNaC allows you to handle expressions containing general indexed objects in
1624 arbitrary spaces. It is also able to canonicalize and simplify such
1625 expressions and perform symbolic dummy index summations. There are a number
1626 of predefined indexed objects provided, like delta and metric tensors.
1628 There are few restrictions placed on indexed objects and their indices and
1629 it is easy to construct nonsense expressions, but our intention is to
1630 provide a general framework that allows you to implement algorithms with
1631 indexed quantities, getting in the way as little as possible.
1633 @cindex @code{idx} (class)
1634 @cindex @code{indexed} (class)
1635 @subsection Indexed quantities and their indices
1637 Indexed expressions in GiNaC are constructed of two special types of objects,
1638 @dfn{index objects} and @dfn{indexed objects}.
1642 @cindex contravariant
1645 @item Index objects are of class @code{idx} or a subclass. Every index has
1646 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1647 the index lives in) which can both be arbitrary expressions but are usually
1648 a number or a simple symbol. In addition, indices of class @code{varidx} have
1649 a @dfn{variance} (they can be co- or contravariant), and indices of class
1650 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1652 @item Indexed objects are of class @code{indexed} or a subclass. They
1653 contain a @dfn{base expression} (which is the expression being indexed), and
1654 one or more indices.
1658 @strong{Note:} when printing expressions, covariant indices and indices
1659 without variance are denoted @samp{.i} while contravariant indices are
1660 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1661 value. In the following, we are going to use that notation in the text so
1662 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1663 not visible in the output.
1665 A simple example shall illustrate the concepts:
1669 #include <ginac/ginac.h>
1670 using namespace std;
1671 using namespace GiNaC;
1675 symbol i_sym("i"), j_sym("j");
1676 idx i(i_sym, 3), j(j_sym, 3);
1679 cout << indexed(A, i, j) << endl;
1684 The @code{idx} constructor takes two arguments, the index value and the
1685 index dimension. First we define two index objects, @code{i} and @code{j},
1686 both with the numeric dimension 3. The value of the index @code{i} is the
1687 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1688 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1689 construct an expression containing one indexed object, @samp{A.i.j}. It has
1690 the symbol @code{A} as its base expression and the two indices @code{i} and
1693 Note the difference between the indices @code{i} and @code{j} which are of
1694 class @code{idx}, and the index values which are the symbols @code{i_sym}
1695 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1696 or numbers but must be index objects. For example, the following is not
1697 correct and will raise an exception:
1700 symbol i("i"), j("j");
1701 e = indexed(A, i, j); // ERROR: indices must be of type idx
1704 You can have multiple indexed objects in an expression, index values can
1705 be numeric, and index dimensions symbolic:
1709 symbol B("B"), dim("dim");
1710 cout << 4 * indexed(A, i)
1711 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1716 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1717 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1718 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1719 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1720 @code{simplify_indexed()} for that, see below).
1722 In fact, base expressions, index values and index dimensions can be
1723 arbitrary expressions:
1727 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1732 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1733 get an error message from this but you will probably not be able to do
1734 anything useful with it.
1736 @cindex @code{get_value()}
1737 @cindex @code{get_dimension()}
1741 ex idx::get_value(void);
1742 ex idx::get_dimension(void);
1745 return the value and dimension of an @code{idx} object. If you have an index
1746 in an expression, such as returned by calling @code{.op()} on an indexed
1747 object, you can get a reference to the @code{idx} object with the function
1748 @code{ex_to<idx>()} on the expression.
1750 There are also the methods
1753 bool idx::is_numeric(void);
1754 bool idx::is_symbolic(void);
1755 bool idx::is_dim_numeric(void);
1756 bool idx::is_dim_symbolic(void);
1759 for checking whether the value and dimension are numeric or symbolic
1760 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1761 About Expressions}) returns information about the index value.
1763 @cindex @code{varidx} (class)
1764 If you need co- and contravariant indices, use the @code{varidx} class:
1768 symbol mu_sym("mu"), nu_sym("nu");
1769 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1770 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1772 cout << indexed(A, mu, nu) << endl;
1774 cout << indexed(A, mu_co, nu) << endl;
1776 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1781 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1782 co- or contravariant. The default is a contravariant (upper) index, but
1783 this can be overridden by supplying a third argument to the @code{varidx}
1784 constructor. The two methods
1787 bool varidx::is_covariant(void);
1788 bool varidx::is_contravariant(void);
1791 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1792 to get the object reference from an expression). There's also the very useful
1796 ex varidx::toggle_variance(void);
1799 which makes a new index with the same value and dimension but the opposite
1800 variance. By using it you only have to define the index once.
1802 @cindex @code{spinidx} (class)
1803 The @code{spinidx} class provides dotted and undotted variant indices, as
1804 used in the Weyl-van-der-Waerden spinor formalism:
1808 symbol K("K"), C_sym("C"), D_sym("D");
1809 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1810 // contravariant, undotted
1811 spinidx C_co(C_sym, 2, true); // covariant index
1812 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1813 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1815 cout << indexed(K, C, D) << endl;
1817 cout << indexed(K, C_co, D_dot) << endl;
1819 cout << indexed(K, D_co_dot, D) << endl;
1824 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1825 dotted or undotted. The default is undotted but this can be overridden by
1826 supplying a fourth argument to the @code{spinidx} constructor. The two
1830 bool spinidx::is_dotted(void);
1831 bool spinidx::is_undotted(void);
1834 allow you to check whether or not a @code{spinidx} object is dotted (use
1835 @code{ex_to<spinidx>()} to get the object reference from an expression).
1836 Finally, the two methods
1839 ex spinidx::toggle_dot(void);
1840 ex spinidx::toggle_variance_dot(void);
1843 create a new index with the same value and dimension but opposite dottedness
1844 and the same or opposite variance.
1846 @subsection Substituting indices
1848 @cindex @code{subs()}
1849 Sometimes you will want to substitute one symbolic index with another
1850 symbolic or numeric index, for example when calculating one specific element
1851 of a tensor expression. This is done with the @code{.subs()} method, as it
1852 is done for symbols (see @ref{Substituting Expressions}).
1854 You have two possibilities here. You can either substitute the whole index
1855 by another index or expression:
1859 ex e = indexed(A, mu_co);
1860 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1861 // -> A.mu becomes A~nu
1862 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1863 // -> A.mu becomes A~0
1864 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1865 // -> A.mu becomes A.0
1869 The third example shows that trying to replace an index with something that
1870 is not an index will substitute the index value instead.
1872 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1877 ex e = indexed(A, mu_co);
1878 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1879 // -> A.mu becomes A.nu
1880 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1881 // -> A.mu becomes A.0
1885 As you see, with the second method only the value of the index will get
1886 substituted. Its other properties, including its dimension, remain unchanged.
1887 If you want to change the dimension of an index you have to substitute the
1888 whole index by another one with the new dimension.
1890 Finally, substituting the base expression of an indexed object works as
1895 ex e = indexed(A, mu_co);
1896 cout << e << " becomes " << e.subs(A == A+B) << endl;
1897 // -> A.mu becomes (B+A).mu
1901 @subsection Symmetries
1902 @cindex @code{symmetry} (class)
1903 @cindex @code{sy_none()}
1904 @cindex @code{sy_symm()}
1905 @cindex @code{sy_anti()}
1906 @cindex @code{sy_cycl()}
1908 Indexed objects can have certain symmetry properties with respect to their
1909 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1910 that is constructed with the helper functions
1913 symmetry sy_none(...);
1914 symmetry sy_symm(...);
1915 symmetry sy_anti(...);
1916 symmetry sy_cycl(...);
1919 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1920 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1921 represents a cyclic symmetry. Each of these functions accepts up to four
1922 arguments which can be either symmetry objects themselves or unsigned integer
1923 numbers that represent an index position (counting from 0). A symmetry
1924 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1925 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1928 Here are some examples of symmetry definitions:
1933 e = indexed(A, i, j);
1934 e = indexed(A, sy_none(), i, j); // equivalent
1935 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1937 // Symmetric in all three indices:
1938 e = indexed(A, sy_symm(), i, j, k);
1939 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1940 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1941 // different canonical order
1943 // Symmetric in the first two indices only:
1944 e = indexed(A, sy_symm(0, 1), i, j, k);
1945 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1947 // Antisymmetric in the first and last index only (index ranges need not
1949 e = indexed(A, sy_anti(0, 2), i, j, k);
1950 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1952 // An example of a mixed symmetry: antisymmetric in the first two and
1953 // last two indices, symmetric when swapping the first and last index
1954 // pairs (like the Riemann curvature tensor):
1955 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1957 // Cyclic symmetry in all three indices:
1958 e = indexed(A, sy_cycl(), i, j, k);
1959 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1961 // The following examples are invalid constructions that will throw
1962 // an exception at run time.
1964 // An index may not appear multiple times:
1965 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1966 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1968 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1969 // same number of indices:
1970 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1972 // And of course, you cannot specify indices which are not there:
1973 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1977 If you need to specify more than four indices, you have to use the
1978 @code{.add()} method of the @code{symmetry} class. For example, to specify
1979 full symmetry in the first six indices you would write
1980 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1982 If an indexed object has a symmetry, GiNaC will automatically bring the
1983 indices into a canonical order which allows for some immediate simplifications:
1987 cout << indexed(A, sy_symm(), i, j)
1988 + indexed(A, sy_symm(), j, i) << endl;
1990 cout << indexed(B, sy_anti(), i, j)
1991 + indexed(B, sy_anti(), j, i) << endl;
1993 cout << indexed(B, sy_anti(), i, j, k)
1994 - indexed(B, sy_anti(), j, k, i) << endl;
1999 @cindex @code{get_free_indices()}
2001 @subsection Dummy indices
2003 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2004 that a summation over the index range is implied. Symbolic indices which are
2005 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2006 dummy nor free indices.
2008 To be recognized as a dummy index pair, the two indices must be of the same
2009 class and their value must be the same single symbol (an index like
2010 @samp{2*n+1} is never a dummy index). If the indices are of class
2011 @code{varidx} they must also be of opposite variance; if they are of class
2012 @code{spinidx} they must be both dotted or both undotted.
2014 The method @code{.get_free_indices()} returns a vector containing the free
2015 indices of an expression. It also checks that the free indices of the terms
2016 of a sum are consistent:
2020 symbol A("A"), B("B"), C("C");
2022 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2023 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2025 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2026 cout << exprseq(e.get_free_indices()) << endl;
2028 // 'j' and 'l' are dummy indices
2030 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2031 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2033 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2034 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2035 cout << exprseq(e.get_free_indices()) << endl;
2037 // 'nu' is a dummy index, but 'sigma' is not
2039 e = indexed(A, mu, mu);
2040 cout << exprseq(e.get_free_indices()) << endl;
2042 // 'mu' is not a dummy index because it appears twice with the same
2045 e = indexed(A, mu, nu) + 42;
2046 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2047 // this will throw an exception:
2048 // "add::get_free_indices: inconsistent indices in sum"
2052 @cindex @code{simplify_indexed()}
2053 @subsection Simplifying indexed expressions
2055 In addition to the few automatic simplifications that GiNaC performs on
2056 indexed expressions (such as re-ordering the indices of symmetric tensors
2057 and calculating traces and convolutions of matrices and predefined tensors)
2061 ex ex::simplify_indexed(void);
2062 ex ex::simplify_indexed(const scalar_products & sp);
2065 that performs some more expensive operations:
2068 @item it checks the consistency of free indices in sums in the same way
2069 @code{get_free_indices()} does
2070 @item it tries to give dummy indices that appear in different terms of a sum
2071 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2072 @item it (symbolically) calculates all possible dummy index summations/contractions
2073 with the predefined tensors (this will be explained in more detail in the
2075 @item it detects contractions that vanish for symmetry reasons, for example
2076 the contraction of a symmetric and a totally antisymmetric tensor
2077 @item as a special case of dummy index summation, it can replace scalar products
2078 of two tensors with a user-defined value
2081 The last point is done with the help of the @code{scalar_products} class
2082 which is used to store scalar products with known values (this is not an
2083 arithmetic class, you just pass it to @code{simplify_indexed()}):
2087 symbol A("A"), B("B"), C("C"), i_sym("i");
2091 sp.add(A, B, 0); // A and B are orthogonal
2092 sp.add(A, C, 0); // A and C are orthogonal
2093 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2095 e = indexed(A + B, i) * indexed(A + C, i);
2097 // -> (B+A).i*(A+C).i
2099 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2105 The @code{scalar_products} object @code{sp} acts as a storage for the
2106 scalar products added to it with the @code{.add()} method. This method
2107 takes three arguments: the two expressions of which the scalar product is
2108 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2109 @code{simplify_indexed()} will replace all scalar products of indexed
2110 objects that have the symbols @code{A} and @code{B} as base expressions
2111 with the single value 0. The number, type and dimension of the indices
2112 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2114 @cindex @code{expand()}
2115 The example above also illustrates a feature of the @code{expand()} method:
2116 if passed the @code{expand_indexed} option it will distribute indices
2117 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2119 @cindex @code{tensor} (class)
2120 @subsection Predefined tensors
2122 Some frequently used special tensors such as the delta, epsilon and metric
2123 tensors are predefined in GiNaC. They have special properties when
2124 contracted with other tensor expressions and some of them have constant
2125 matrix representations (they will evaluate to a number when numeric
2126 indices are specified).
2128 @cindex @code{delta_tensor()}
2129 @subsubsection Delta tensor
2131 The delta tensor takes two indices, is symmetric and has the matrix
2132 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2133 @code{delta_tensor()}:
2137 symbol A("A"), B("B");
2139 idx i(symbol("i"), 3), j(symbol("j"), 3),
2140 k(symbol("k"), 3), l(symbol("l"), 3);
2142 ex e = indexed(A, i, j) * indexed(B, k, l)
2143 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2144 cout << e.simplify_indexed() << endl;
2147 cout << delta_tensor(i, i) << endl;
2152 @cindex @code{metric_tensor()}
2153 @subsubsection General metric tensor
2155 The function @code{metric_tensor()} creates a general symmetric metric
2156 tensor with two indices that can be used to raise/lower tensor indices. The
2157 metric tensor is denoted as @samp{g} in the output and if its indices are of
2158 mixed variance it is automatically replaced by a delta tensor:
2164 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2166 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2167 cout << e.simplify_indexed() << endl;
2170 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2171 cout << e.simplify_indexed() << endl;
2174 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2175 * metric_tensor(nu, rho);
2176 cout << e.simplify_indexed() << endl;
2179 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2180 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2181 + indexed(A, mu.toggle_variance(), rho));
2182 cout << e.simplify_indexed() << endl;
2187 @cindex @code{lorentz_g()}
2188 @subsubsection Minkowski metric tensor
2190 The Minkowski metric tensor is a special metric tensor with a constant
2191 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2192 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2193 It is created with the function @code{lorentz_g()} (although it is output as
2198 varidx mu(symbol("mu"), 4);
2200 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2201 * lorentz_g(mu, varidx(0, 4)); // negative signature
2202 cout << e.simplify_indexed() << endl;
2205 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2206 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2207 cout << e.simplify_indexed() << endl;
2212 @cindex @code{spinor_metric()}
2213 @subsubsection Spinor metric tensor
2215 The function @code{spinor_metric()} creates an antisymmetric tensor with
2216 two indices that is used to raise/lower indices of 2-component spinors.
2217 It is output as @samp{eps}:
2223 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2224 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2226 e = spinor_metric(A, B) * indexed(psi, B_co);
2227 cout << e.simplify_indexed() << endl;
2230 e = spinor_metric(A, B) * indexed(psi, A_co);
2231 cout << e.simplify_indexed() << endl;
2234 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2235 cout << e.simplify_indexed() << endl;
2238 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2239 cout << e.simplify_indexed() << endl;
2242 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2243 cout << e.simplify_indexed() << endl;
2246 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2247 cout << e.simplify_indexed() << endl;
2252 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2254 @cindex @code{epsilon_tensor()}
2255 @cindex @code{lorentz_eps()}
2256 @subsubsection Epsilon tensor
2258 The epsilon tensor is totally antisymmetric, its number of indices is equal
2259 to the dimension of the index space (the indices must all be of the same
2260 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2261 defined to be 1. Its behavior with indices that have a variance also
2262 depends on the signature of the metric. Epsilon tensors are output as
2265 There are three functions defined to create epsilon tensors in 2, 3 and 4
2269 ex epsilon_tensor(const ex & i1, const ex & i2);
2270 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2271 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2274 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2275 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2276 Minkowski space (the last @code{bool} argument specifies whether the metric
2277 has negative or positive signature, as in the case of the Minkowski metric
2282 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2283 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2284 e = lorentz_eps(mu, nu, rho, sig) *
2285 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2286 cout << simplify_indexed(e) << endl;
2287 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2289 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2290 symbol A("A"), B("B");
2291 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2292 cout << simplify_indexed(e) << endl;
2293 // -> -B.k*A.j*eps.i.k.j
2294 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2295 cout << simplify_indexed(e) << endl;
2300 @subsection Linear algebra
2302 The @code{matrix} class can be used with indices to do some simple linear
2303 algebra (linear combinations and products of vectors and matrices, traces
2304 and scalar products):
2308 idx i(symbol("i"), 2), j(symbol("j"), 2);
2309 symbol x("x"), y("y");
2311 // A is a 2x2 matrix, X is a 2x1 vector
2312 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2314 cout << indexed(A, i, i) << endl;
2317 ex e = indexed(A, i, j) * indexed(X, j);
2318 cout << e.simplify_indexed() << endl;
2319 // -> [[2*y+x],[4*y+3*x]].i
2321 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2322 cout << e.simplify_indexed() << endl;
2323 // -> [[3*y+3*x,6*y+2*x]].j
2327 You can of course obtain the same results with the @code{matrix::add()},
2328 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2329 but with indices you don't have to worry about transposing matrices.
2331 Matrix indices always start at 0 and their dimension must match the number
2332 of rows/columns of the matrix. Matrices with one row or one column are
2333 vectors and can have one or two indices (it doesn't matter whether it's a
2334 row or a column vector). Other matrices must have two indices.
2336 You should be careful when using indices with variance on matrices. GiNaC
2337 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2338 @samp{F.mu.nu} are different matrices. In this case you should use only
2339 one form for @samp{F} and explicitly multiply it with a matrix representation
2340 of the metric tensor.
2343 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2344 @c node-name, next, previous, up
2345 @section Non-commutative objects
2347 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2348 non-commutative objects are built-in which are mostly of use in high energy
2352 @item Clifford (Dirac) algebra (class @code{clifford})
2353 @item su(3) Lie algebra (class @code{color})
2354 @item Matrices (unindexed) (class @code{matrix})
2357 The @code{clifford} and @code{color} classes are subclasses of
2358 @code{indexed} because the elements of these algebras usually carry
2359 indices. The @code{matrix} class is described in more detail in
2362 Unlike most computer algebra systems, GiNaC does not primarily provide an
2363 operator (often denoted @samp{&*}) for representing inert products of
2364 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2365 classes of objects involved, and non-commutative products are formed with
2366 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2367 figuring out by itself which objects commute and will group the factors
2368 by their class. Consider this example:
2372 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2373 idx a(symbol("a"), 8), b(symbol("b"), 8);
2374 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2376 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2380 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2381 groups the non-commutative factors (the gammas and the su(3) generators)
2382 together while preserving the order of factors within each class (because
2383 Clifford objects commute with color objects). The resulting expression is a
2384 @emph{commutative} product with two factors that are themselves non-commutative
2385 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2386 parentheses are placed around the non-commutative products in the output.
2388 @cindex @code{ncmul} (class)
2389 Non-commutative products are internally represented by objects of the class
2390 @code{ncmul}, as opposed to commutative products which are handled by the
2391 @code{mul} class. You will normally not have to worry about this distinction,
2394 The advantage of this approach is that you never have to worry about using
2395 (or forgetting to use) a special operator when constructing non-commutative
2396 expressions. Also, non-commutative products in GiNaC are more intelligent
2397 than in other computer algebra systems; they can, for example, automatically
2398 canonicalize themselves according to rules specified in the implementation
2399 of the non-commutative classes. The drawback is that to work with other than
2400 the built-in algebras you have to implement new classes yourself. Symbols
2401 always commute and it's not possible to construct non-commutative products
2402 using symbols to represent the algebra elements or generators. User-defined
2403 functions can, however, be specified as being non-commutative.
2405 @cindex @code{return_type()}
2406 @cindex @code{return_type_tinfo()}
2407 Information about the commutativity of an object or expression can be
2408 obtained with the two member functions
2411 unsigned ex::return_type(void) const;
2412 unsigned ex::return_type_tinfo(void) const;
2415 The @code{return_type()} function returns one of three values (defined in
2416 the header file @file{flags.h}), corresponding to three categories of
2417 expressions in GiNaC:
2420 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2421 classes are of this kind.
2422 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2423 certain class of non-commutative objects which can be determined with the
2424 @code{return_type_tinfo()} method. Expressions of this category commute
2425 with everything except @code{noncommutative} expressions of the same
2427 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2428 of non-commutative objects of different classes. Expressions of this
2429 category don't commute with any other @code{noncommutative} or
2430 @code{noncommutative_composite} expressions.
2433 The value returned by the @code{return_type_tinfo()} method is valid only
2434 when the return type of the expression is @code{noncommutative}. It is a
2435 value that is unique to the class of the object and usually one of the
2436 constants in @file{tinfos.h}, or derived therefrom.
2438 Here are a couple of examples:
2441 @multitable @columnfractions 0.33 0.33 0.34
2442 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2443 @item @code{42} @tab @code{commutative} @tab -
2444 @item @code{2*x-y} @tab @code{commutative} @tab -
2445 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2446 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2447 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2448 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2452 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2453 @code{TINFO_clifford} for objects with a representation label of zero.
2454 Other representation labels yield a different @code{return_type_tinfo()},
2455 but it's the same for any two objects with the same label. This is also true
2458 A last note: With the exception of matrices, positive integer powers of
2459 non-commutative objects are automatically expanded in GiNaC. For example,
2460 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2461 non-commutative expressions).
2464 @cindex @code{clifford} (class)
2465 @subsection Clifford algebra
2467 @cindex @code{dirac_gamma()}
2468 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2469 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2470 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2471 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2474 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2477 which takes two arguments: the index and a @dfn{representation label} in the
2478 range 0 to 255 which is used to distinguish elements of different Clifford
2479 algebras (this is also called a @dfn{spin line index}). Gammas with different
2480 labels commute with each other. The dimension of the index can be 4 or (in
2481 the framework of dimensional regularization) any symbolic value. Spinor
2482 indices on Dirac gammas are not supported in GiNaC.
2484 @cindex @code{dirac_ONE()}
2485 The unity element of a Clifford algebra is constructed by
2488 ex dirac_ONE(unsigned char rl = 0);
2491 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2492 multiples of the unity element, even though it's customary to omit it.
2493 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2494 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2495 GiNaC will complain and/or produce incorrect results.
2497 @cindex @code{dirac_gamma5()}
2498 There is a special element @samp{gamma5} that commutes with all other
2499 gammas, has a unit square, and in 4 dimensions equals
2500 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2503 ex dirac_gamma5(unsigned char rl = 0);
2506 @cindex @code{dirac_gammaL()}
2507 @cindex @code{dirac_gammaR()}
2508 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2509 objects, constructed by
2512 ex dirac_gammaL(unsigned char rl = 0);
2513 ex dirac_gammaR(unsigned char rl = 0);
2516 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2517 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2519 @cindex @code{dirac_slash()}
2520 Finally, the function
2523 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2526 creates a term that represents a contraction of @samp{e} with the Dirac
2527 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2528 with a unique index whose dimension is given by the @code{dim} argument).
2529 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2531 In products of dirac gammas, superfluous unity elements are automatically
2532 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2533 and @samp{gammaR} are moved to the front.
2535 The @code{simplify_indexed()} function performs contractions in gamma strings,
2541 symbol a("a"), b("b"), D("D");
2542 varidx mu(symbol("mu"), D);
2543 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2544 * dirac_gamma(mu.toggle_variance());
2546 // -> gamma~mu*a\*gamma.mu
2547 e = e.simplify_indexed();
2550 cout << e.subs(D == 4) << endl;
2556 @cindex @code{dirac_trace()}
2557 To calculate the trace of an expression containing strings of Dirac gammas
2558 you use the function
2561 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2564 This function takes the trace of all gammas with the specified representation
2565 label; gammas with other labels are left standing. The last argument to
2566 @code{dirac_trace()} is the value to be returned for the trace of the unity
2567 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2568 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2569 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2570 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2571 This @samp{gamma5} scheme is described in greater detail in
2572 @cite{The Role of gamma5 in Dimensional Regularization}.
2574 The value of the trace itself is also usually different in 4 and in
2575 @math{D != 4} dimensions:
2580 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2581 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2582 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2583 cout << dirac_trace(e).simplify_indexed() << endl;
2590 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2591 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2592 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2593 cout << dirac_trace(e).simplify_indexed() << endl;
2594 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2598 Here is an example for using @code{dirac_trace()} to compute a value that
2599 appears in the calculation of the one-loop vacuum polarization amplitude in
2604 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2605 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2608 sp.add(l, l, pow(l, 2));
2609 sp.add(l, q, ldotq);
2611 ex e = dirac_gamma(mu) *
2612 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2613 dirac_gamma(mu.toggle_variance()) *
2614 (dirac_slash(l, D) + m * dirac_ONE());
2615 e = dirac_trace(e).simplify_indexed(sp);
2616 e = e.collect(lst(l, ldotq, m));
2618 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2622 The @code{canonicalize_clifford()} function reorders all gamma products that
2623 appear in an expression to a canonical (but not necessarily simple) form.
2624 You can use this to compare two expressions or for further simplifications:
2628 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2629 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2631 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2633 e = canonicalize_clifford(e);
2640 @cindex @code{color} (class)
2641 @subsection Color algebra
2643 @cindex @code{color_T()}
2644 For computations in quantum chromodynamics, GiNaC implements the base elements
2645 and structure constants of the su(3) Lie algebra (color algebra). The base
2646 elements @math{T_a} are constructed by the function
2649 ex color_T(const ex & a, unsigned char rl = 0);
2652 which takes two arguments: the index and a @dfn{representation label} in the
2653 range 0 to 255 which is used to distinguish elements of different color
2654 algebras. Objects with different labels commute with each other. The
2655 dimension of the index must be exactly 8 and it should be of class @code{idx},
2658 @cindex @code{color_ONE()}
2659 The unity element of a color algebra is constructed by
2662 ex color_ONE(unsigned char rl = 0);
2665 @strong{Note:} You must always use @code{color_ONE()} when referring to
2666 multiples of the unity element, even though it's customary to omit it.
2667 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2668 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2669 GiNaC may produce incorrect results.
2671 @cindex @code{color_d()}
2672 @cindex @code{color_f()}
2676 ex color_d(const ex & a, const ex & b, const ex & c);
2677 ex color_f(const ex & a, const ex & b, const ex & c);
2680 create the symmetric and antisymmetric structure constants @math{d_abc} and
2681 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2682 and @math{[T_a, T_b] = i f_abc T_c}.
2684 @cindex @code{color_h()}
2685 There's an additional function
2688 ex color_h(const ex & a, const ex & b, const ex & c);
2691 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2693 The function @code{simplify_indexed()} performs some simplifications on
2694 expressions containing color objects:
2699 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2700 k(symbol("k"), 8), l(symbol("l"), 8);
2702 e = color_d(a, b, l) * color_f(a, b, k);
2703 cout << e.simplify_indexed() << endl;
2706 e = color_d(a, b, l) * color_d(a, b, k);
2707 cout << e.simplify_indexed() << endl;
2710 e = color_f(l, a, b) * color_f(a, b, k);
2711 cout << e.simplify_indexed() << endl;
2714 e = color_h(a, b, c) * color_h(a, b, c);
2715 cout << e.simplify_indexed() << endl;
2718 e = color_h(a, b, c) * color_T(b) * color_T(c);
2719 cout << e.simplify_indexed() << endl;
2722 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2723 cout << e.simplify_indexed() << endl;
2726 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2727 cout << e.simplify_indexed() << endl;
2728 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2732 @cindex @code{color_trace()}
2733 To calculate the trace of an expression containing color objects you use the
2737 ex color_trace(const ex & e, unsigned char rl = 0);
2740 This function takes the trace of all color @samp{T} objects with the
2741 specified representation label; @samp{T}s with other labels are left
2742 standing. For example:
2746 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2748 // -> -I*f.a.c.b+d.a.c.b
2753 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2754 @c node-name, next, previous, up
2755 @chapter Methods and Functions
2758 In this chapter the most important algorithms provided by GiNaC will be
2759 described. Some of them are implemented as functions on expressions,
2760 others are implemented as methods provided by expression objects. If
2761 they are methods, there exists a wrapper function around it, so you can
2762 alternatively call it in a functional way as shown in the simple
2767 cout << "As method: " << sin(1).evalf() << endl;
2768 cout << "As function: " << evalf(sin(1)) << endl;
2772 @cindex @code{subs()}
2773 The general rule is that wherever methods accept one or more parameters
2774 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2775 wrapper accepts is the same but preceded by the object to act on
2776 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2777 most natural one in an OO model but it may lead to confusion for MapleV
2778 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2779 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2780 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2781 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2782 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2783 here. Also, users of MuPAD will in most cases feel more comfortable
2784 with GiNaC's convention. All function wrappers are implemented
2785 as simple inline functions which just call the corresponding method and
2786 are only provided for users uncomfortable with OO who are dead set to
2787 avoid method invocations. Generally, nested function wrappers are much
2788 harder to read than a sequence of methods and should therefore be
2789 avoided if possible. On the other hand, not everything in GiNaC is a
2790 method on class @code{ex} and sometimes calling a function cannot be
2794 * Information About Expressions::
2795 * Substituting Expressions::
2796 * Pattern Matching and Advanced Substitutions::
2797 * Applying a Function on Subexpressions::
2798 * Polynomial Arithmetic:: Working with polynomials.
2799 * Rational Expressions:: Working with rational functions.
2800 * Symbolic Differentiation::
2801 * Series Expansion:: Taylor and Laurent expansion.
2803 * Built-in Functions:: List of predefined mathematical functions.
2804 * Input/Output:: Input and output of expressions.
2808 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2809 @c node-name, next, previous, up
2810 @section Getting information about expressions
2812 @subsection Checking expression types
2813 @cindex @code{is_a<@dots{}>()}
2814 @cindex @code{is_exactly_a<@dots{}>()}
2815 @cindex @code{ex_to<@dots{}>()}
2816 @cindex Converting @code{ex} to other classes
2817 @cindex @code{info()}
2818 @cindex @code{return_type()}
2819 @cindex @code{return_type_tinfo()}
2821 Sometimes it's useful to check whether a given expression is a plain number,
2822 a sum, a polynomial with integer coefficients, or of some other specific type.
2823 GiNaC provides a couple of functions for this:
2826 bool is_a<T>(const ex & e);
2827 bool is_exactly_a<T>(const ex & e);
2828 bool ex::info(unsigned flag);
2829 unsigned ex::return_type(void) const;
2830 unsigned ex::return_type_tinfo(void) const;
2833 When the test made by @code{is_a<T>()} returns true, it is safe to call
2834 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2835 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2836 example, assuming @code{e} is an @code{ex}:
2841 if (is_a<numeric>(e))
2842 numeric n = ex_to<numeric>(e);
2847 @code{is_a<T>(e)} allows you to check whether the top-level object of
2848 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2849 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2850 e.g., for checking whether an expression is a number, a sum, or a product:
2857 is_a<numeric>(e1); // true
2858 is_a<numeric>(e2); // false
2859 is_a<add>(e1); // false
2860 is_a<add>(e2); // true
2861 is_a<mul>(e1); // false
2862 is_a<mul>(e2); // false
2866 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2867 top-level object of an expression @samp{e} is an instance of the GiNaC
2868 class @samp{T}, not including parent classes.
2870 The @code{info()} method is used for checking certain attributes of
2871 expressions. The possible values for the @code{flag} argument are defined
2872 in @file{ginac/flags.h}, the most important being explained in the following
2876 @multitable @columnfractions .30 .70
2877 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2878 @item @code{numeric}
2879 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2881 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2882 @item @code{rational}
2883 @tab @dots{}an exact rational number (integers are rational, too)
2884 @item @code{integer}
2885 @tab @dots{}a (non-complex) integer
2886 @item @code{crational}
2887 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2888 @item @code{cinteger}
2889 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2890 @item @code{positive}
2891 @tab @dots{}not complex and greater than 0
2892 @item @code{negative}
2893 @tab @dots{}not complex and less than 0
2894 @item @code{nonnegative}
2895 @tab @dots{}not complex and greater than or equal to 0
2897 @tab @dots{}an integer greater than 0
2899 @tab @dots{}an integer less than 0
2900 @item @code{nonnegint}
2901 @tab @dots{}an integer greater than or equal to 0
2903 @tab @dots{}an even integer
2905 @tab @dots{}an odd integer
2907 @tab @dots{}a prime integer (probabilistic primality test)
2908 @item @code{relation}
2909 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2910 @item @code{relation_equal}
2911 @tab @dots{}a @code{==} relation
2912 @item @code{relation_not_equal}
2913 @tab @dots{}a @code{!=} relation
2914 @item @code{relation_less}
2915 @tab @dots{}a @code{<} relation
2916 @item @code{relation_less_or_equal}
2917 @tab @dots{}a @code{<=} relation
2918 @item @code{relation_greater}
2919 @tab @dots{}a @code{>} relation
2920 @item @code{relation_greater_or_equal}
2921 @tab @dots{}a @code{>=} relation
2923 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2925 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2926 @item @code{polynomial}
2927 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2928 @item @code{integer_polynomial}
2929 @tab @dots{}a polynomial with (non-complex) integer coefficients
2930 @item @code{cinteger_polynomial}
2931 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2932 @item @code{rational_polynomial}
2933 @tab @dots{}a polynomial with (non-complex) rational coefficients
2934 @item @code{crational_polynomial}
2935 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2936 @item @code{rational_function}
2937 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2938 @item @code{algebraic}
2939 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2943 To determine whether an expression is commutative or non-commutative and if
2944 so, with which other expressions it would commute, you use the methods
2945 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2946 for an explanation of these.
2949 @subsection Accessing subexpressions
2950 @cindex @code{nops()}
2953 @cindex @code{relational} (class)
2955 GiNaC provides the two methods
2958 unsigned ex::nops();
2959 ex ex::op(unsigned i);
2962 for accessing the subexpressions in the container-like GiNaC classes like
2963 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2964 determines the number of subexpressions (@samp{operands}) contained, while
2965 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2966 In the case of a @code{power} object, @code{op(0)} will return the basis
2967 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2968 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2970 The left-hand and right-hand side expressions of objects of class
2971 @code{relational} (and only of these) can also be accessed with the methods
2979 @subsection Comparing expressions
2980 @cindex @code{is_equal()}
2981 @cindex @code{is_zero()}
2983 Expressions can be compared with the usual C++ relational operators like
2984 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2985 the result is usually not determinable and the result will be @code{false},
2986 except in the case of the @code{!=} operator. You should also be aware that
2987 GiNaC will only do the most trivial test for equality (subtracting both
2988 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2991 Actually, if you construct an expression like @code{a == b}, this will be
2992 represented by an object of the @code{relational} class (@pxref{Relations})
2993 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
2995 There are also two methods
2998 bool ex::is_equal(const ex & other);
3002 for checking whether one expression is equal to another, or equal to zero,
3005 @strong{Warning:} You will also find an @code{ex::compare()} method in the
3006 GiNaC header files. This method is however only to be used internally by
3007 GiNaC to establish a canonical sort order for terms, and using it to compare
3008 expressions will give very surprising results.
3011 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
3012 @c node-name, next, previous, up
3013 @section Substituting expressions
3014 @cindex @code{subs()}
3016 Algebraic objects inside expressions can be replaced with arbitrary
3017 expressions via the @code{.subs()} method:
3020 ex ex::subs(const ex & e);
3021 ex ex::subs(const lst & syms, const lst & repls);
3024 In the first form, @code{subs()} accepts a relational of the form
3025 @samp{object == expression} or a @code{lst} of such relationals:
3029 symbol x("x"), y("y");
3031 ex e1 = 2*x^2-4*x+3;
3032 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3036 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3041 If you specify multiple substitutions, they are performed in parallel, so e.g.
3042 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3044 The second form of @code{subs()} takes two lists, one for the objects to be
3045 replaced and one for the expressions to be substituted (both lists must
3046 contain the same number of elements). Using this form, you would write
3047 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
3049 @code{subs()} performs syntactic substitution of any complete algebraic
3050 object; it does not try to match sub-expressions as is demonstrated by the
3055 symbol x("x"), y("y"), z("z");
3057 ex e1 = pow(x+y, 2);
3058 cout << e1.subs(x+y == 4) << endl;
3061 ex e2 = sin(x)*sin(y)*cos(x);
3062 cout << e2.subs(sin(x) == cos(x)) << endl;
3063 // -> cos(x)^2*sin(y)
3066 cout << e3.subs(x+y == 4) << endl;
3068 // (and not 4+z as one might expect)
3072 A more powerful form of substitution using wildcards is described in the
3076 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3077 @c node-name, next, previous, up
3078 @section Pattern matching and advanced substitutions
3079 @cindex @code{wildcard} (class)
3080 @cindex Pattern matching
3082 GiNaC allows the use of patterns for checking whether an expression is of a
3083 certain form or contains subexpressions of a certain form, and for
3084 substituting expressions in a more general way.
3086 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3087 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3088 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3089 an unsigned integer number to allow having multiple different wildcards in a
3090 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3091 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3095 ex wild(unsigned label = 0);
3098 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3101 Some examples for patterns:
3103 @multitable @columnfractions .5 .5
3104 @item @strong{Constructed as} @tab @strong{Output as}
3105 @item @code{wild()} @tab @samp{$0}
3106 @item @code{pow(x,wild())} @tab @samp{x^$0}
3107 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3108 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3114 @item Wildcards behave like symbols and are subject to the same algebraic
3115 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3116 @item As shown in the last example, to use wildcards for indices you have to
3117 use them as the value of an @code{idx} object. This is because indices must
3118 always be of class @code{idx} (or a subclass).
3119 @item Wildcards only represent expressions or subexpressions. It is not
3120 possible to use them as placeholders for other properties like index
3121 dimension or variance, representation labels, symmetry of indexed objects
3123 @item Because wildcards are commutative, it is not possible to use wildcards
3124 as part of noncommutative products.
3125 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3126 are also valid patterns.
3129 @subsection Matching expressions
3130 @cindex @code{match()}
3131 The most basic application of patterns is to check whether an expression
3132 matches a given pattern. This is done by the function
3135 bool ex::match(const ex & pattern);
3136 bool ex::match(const ex & pattern, lst & repls);
3139 This function returns @code{true} when the expression matches the pattern
3140 and @code{false} if it doesn't. If used in the second form, the actual
3141 subexpressions matched by the wildcards get returned in the @code{repls}
3142 object as a list of relations of the form @samp{wildcard == expression}.
3143 If @code{match()} returns false, the state of @code{repls} is undefined.
3144 For reproducible results, the list should be empty when passed to
3145 @code{match()}, but it is also possible to find similarities in multiple
3146 expressions by passing in the result of a previous match.
3148 The matching algorithm works as follows:
3151 @item A single wildcard matches any expression. If one wildcard appears
3152 multiple times in a pattern, it must match the same expression in all
3153 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3154 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3155 @item If the expression is not of the same class as the pattern, the match
3156 fails (i.e. a sum only matches a sum, a function only matches a function,
3158 @item If the pattern is a function, it only matches the same function
3159 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3160 @item Except for sums and products, the match fails if the number of
3161 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3163 @item If there are no subexpressions, the expressions and the pattern must
3164 be equal (in the sense of @code{is_equal()}).
3165 @item Except for sums and products, each subexpression (@code{op()}) must
3166 match the corresponding subexpression of the pattern.
3169 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3170 account for their commutativity and associativity:
3173 @item If the pattern contains a term or factor that is a single wildcard,
3174 this one is used as the @dfn{global wildcard}. If there is more than one
3175 such wildcard, one of them is chosen as the global wildcard in a random
3177 @item Every term/factor of the pattern, except the global wildcard, is
3178 matched against every term of the expression in sequence. If no match is
3179 found, the whole match fails. Terms that did match are not considered in
3181 @item If there are no unmatched terms left, the match succeeds. Otherwise
3182 the match fails unless there is a global wildcard in the pattern, in
3183 which case this wildcard matches the remaining terms.
3186 In general, having more than one single wildcard as a term of a sum or a
3187 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3190 Here are some examples in @command{ginsh} to demonstrate how it works (the
3191 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3192 match fails, and the list of wildcard replacements otherwise):
3195 > match((x+y)^a,(x+y)^a);
3197 > match((x+y)^a,(x+y)^b);
3199 > match((x+y)^a,$1^$2);
3201 > match((x+y)^a,$1^$1);
3203 > match((x+y)^(x+y),$1^$1);
3205 > match((x+y)^(x+y),$1^$2);
3207 > match((a+b)*(a+c),($1+b)*($1+c));
3209 > match((a+b)*(a+c),(a+$1)*(a+$2));
3211 (Unpredictable. The result might also be [$1==c,$2==b].)
3212 > match((a+b)*(a+c),($1+$2)*($1+$3));
3213 (The result is undefined. Due to the sequential nature of the algorithm
3214 and the re-ordering of terms in GiNaC, the match for the first factor
3215 may be @{$1==a,$2==b@} in which case the match for the second factor
3216 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3218 > match(a*(x+y)+a*z+b,a*$1+$2);
3219 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3220 @{$1=x+y,$2=a*z+b@}.)
3221 > match(a+b+c+d+e+f,c);
3223 > match(a+b+c+d+e+f,c+$0);
3225 > match(a+b+c+d+e+f,c+e+$0);
3227 > match(a+b,a+b+$0);
3229 > match(a*b^2,a^$1*b^$2);
3231 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3232 even though a==a^1.)
3233 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3235 > match(atan2(y,x^2),atan2(y,$0));
3239 @subsection Matching parts of expressions
3240 @cindex @code{has()}
3241 A more general way to look for patterns in expressions is provided by the
3245 bool ex::has(const ex & pattern);
3248 This function checks whether a pattern is matched by an expression itself or
3249 by any of its subexpressions.
3251 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3252 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3255 > has(x*sin(x+y+2*a),y);
3257 > has(x*sin(x+y+2*a),x+y);
3259 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3260 has the subexpressions "x", "y" and "2*a".)
3261 > has(x*sin(x+y+2*a),x+y+$1);
3263 (But this is possible.)
3264 > has(x*sin(2*(x+y)+2*a),x+y);
3266 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3267 which "x+y" is not a subexpression.)
3270 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3272 > has(4*x^2-x+3,$1*x);
3274 > has(4*x^2+x+3,$1*x);
3276 (Another possible pitfall. The first expression matches because the term
3277 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3278 contains a linear term you should use the coeff() function instead.)
3281 @cindex @code{find()}
3285 bool ex::find(const ex & pattern, lst & found);
3288 works a bit like @code{has()} but it doesn't stop upon finding the first
3289 match. Instead, it inserts all found matches into the specified list. If
3290 there are multiple occurrences of the same expression, it is entered only
3291 once to the list. @code{find()} returns false if no matches were found (in
3292 @command{ginsh}, it returns an empty list):
3295 > find(1+x+x^2+x^3,x);
3297 > find(1+x+x^2+x^3,y);
3299 > find(1+x+x^2+x^3,x^$1);
3301 (Note the absence of "x".)
3302 > expand((sin(x)+sin(y))*(a+b));
3303 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3308 @subsection Substituting expressions
3309 @cindex @code{subs()}
3310 Probably the most useful application of patterns is to use them for
3311 substituting expressions with the @code{subs()} method. Wildcards can be
3312 used in the search patterns as well as in the replacement expressions, where
3313 they get replaced by the expressions matched by them. @code{subs()} doesn't
3314 know anything about algebra; it performs purely syntactic substitutions.
3319 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3321 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3323 > subs((a+b+c)^2,a+b==x);
3325 > subs((a+b+c)^2,a+b+$1==x+$1);
3327 > subs(a+2*b,a+b==x);
3329 > subs(4*x^3-2*x^2+5*x-1,x==a);
3331 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3333 > subs(sin(1+sin(x)),sin($1)==cos($1));
3335 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3339 The last example would be written in C++ in this way:
3343 symbol a("a"), b("b"), x("x"), y("y");
3344 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3345 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3346 cout << e.expand() << endl;
3352 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3353 @c node-name, next, previous, up
3354 @section Applying a Function on Subexpressions
3355 @cindex Tree traversal
3356 @cindex @code{map()}
3358 Sometimes you may want to perform an operation on specific parts of an
3359 expression while leaving the general structure of it intact. An example
3360 of this would be a matrix trace operation: the trace of a sum is the sum
3361 of the traces of the individual terms. That is, the trace should @dfn{map}
3362 on the sum, by applying itself to each of the sum's operands. It is possible
3363 to do this manually which usually results in code like this:
3368 if (is_a<matrix>(e))
3369 return ex_to<matrix>(e).trace();
3370 else if (is_a<add>(e)) @{
3372 for (unsigned i=0; i<e.nops(); i++)
3373 sum += calc_trace(e.op(i));
3375 @} else if (is_a<mul>)(e)) @{
3383 This is, however, slightly inefficient (if the sum is very large it can take
3384 a long time to add the terms one-by-one), and its applicability is limited to
3385 a rather small class of expressions. If @code{calc_trace()} is called with
3386 a relation or a list as its argument, you will probably want the trace to
3387 be taken on both sides of the relation or of all elements of the list.
3389 GiNaC offers the @code{map()} method to aid in the implementation of such
3393 ex ex::map(map_function & f) const;
3394 ex ex::map(ex (*f)(const ex & e)) const;
3397 In the first (preferred) form, @code{map()} takes a function object that
3398 is subclassed from the @code{map_function} class. In the second form, it
3399 takes a pointer to a function that accepts and returns an expression.
3400 @code{map()} constructs a new expression of the same type, applying the
3401 specified function on all subexpressions (in the sense of @code{op()}),
3404 The use of a function object makes it possible to supply more arguments to
3405 the function that is being mapped, or to keep local state information.
3406 The @code{map_function} class declares a virtual function call operator
3407 that you can overload. Here is a sample implementation of @code{calc_trace()}
3408 that uses @code{map()} in a recursive fashion:
3411 struct calc_trace : public map_function @{
3412 ex operator()(const ex &e)
3414 if (is_a<matrix>(e))
3415 return ex_to<matrix>(e).trace();
3416 else if (is_a<mul>(e)) @{
3419 return e.map(*this);
3424 This function object could then be used like this:
3428 ex M = ... // expression with matrices
3429 calc_trace do_trace;
3430 ex tr = do_trace(M);
3434 Here is another example for you to meditate over. It removes quadratic
3435 terms in a variable from an expanded polynomial:
3438 struct map_rem_quad : public map_function @{
3440 map_rem_quad(const ex & var_) : var(var_) @{@}
3442 ex operator()(const ex & e)
3444 if (is_a<add>(e) || is_a<mul>(e))
3445 return e.map(*this);
3446 else if (is_a<power>(e) &&
3447 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3457 symbol x("x"), y("y");
3460 for (int i=0; i<8; i++)
3461 e += pow(x, i) * pow(y, 8-i) * (i+1);
3463 // -> 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
3465 map_rem_quad rem_quad(x);
3466 cout << rem_quad(e) << endl;
3467 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3471 @command{ginsh} offers a slightly different implementation of @code{map()}
3472 that allows applying algebraic functions to operands. The second argument
3473 to @code{map()} is an expression containing the wildcard @samp{$0} which
3474 acts as the placeholder for the operands:
3479 > map(a+2*b,sin($0));
3481 > map(@{a,b,c@},$0^2+$0);
3482 @{a^2+a,b^2+b,c^2+c@}
3485 Note that it is only possible to use algebraic functions in the second
3486 argument. You can not use functions like @samp{diff()}, @samp{op()},
3487 @samp{subs()} etc. because these are evaluated immediately:
3490 > map(@{a,b,c@},diff($0,a));
3492 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3493 to "map(@{a,b,c@},0)".
3497 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3498 @c node-name, next, previous, up
3499 @section Polynomial arithmetic
3501 @subsection Expanding and collecting
3502 @cindex @code{expand()}
3503 @cindex @code{collect()}
3504 @cindex @code{collect_common_factors()}
3506 A polynomial in one or more variables has many equivalent
3507 representations. Some useful ones serve a specific purpose. Consider
3508 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3509 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3510 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3511 representations are the recursive ones where one collects for exponents
3512 in one of the three variable. Since the factors are themselves
3513 polynomials in the remaining two variables the procedure can be
3514 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3515 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3518 To bring an expression into expanded form, its method
3524 may be called. In our example above, this corresponds to @math{4*x*y +
3525 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3526 GiNaC is not easily guessable you should be prepared to see different
3527 orderings of terms in such sums!
3529 Another useful representation of multivariate polynomials is as a
3530 univariate polynomial in one of the variables with the coefficients
3531 being polynomials in the remaining variables. The method
3532 @code{collect()} accomplishes this task:
3535 ex ex::collect(const ex & s, bool distributed = false);
3538 The first argument to @code{collect()} can also be a list of objects in which
3539 case the result is either a recursively collected polynomial, or a polynomial
3540 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3541 by the @code{distributed} flag.
3543 Note that the original polynomial needs to be in expanded form (for the
3544 variables concerned) in order for @code{collect()} to be able to find the
3545 coefficients properly.
3547 The following @command{ginsh} transcript shows an application of @code{collect()}
3548 together with @code{find()}:
3551 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3552 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
3553 > collect(a,@{p,q@});
3554 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)
3555 > collect(a,find(a,sin($1)));
3556 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3557 > collect(a,@{find(a,sin($1)),p,q@});
3558 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3559 > collect(a,@{find(a,sin($1)),d@});
3560 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3563 Polynomials can often be brought into a more compact form by collecting
3564 common factors from the terms of sums. This is accomplished by the function
3567 ex collect_common_factors(const ex & e);
3570 This function doesn't perform a full factorization but only looks for
3571 factors which are already explicitly present:
3574 > collect_common_factors(a*x+a*y);
3576 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
3578 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
3579 (c+a)*a*(x*y+y^2+x)*b
3582 @subsection Degree and coefficients
3583 @cindex @code{degree()}
3584 @cindex @code{ldegree()}
3585 @cindex @code{coeff()}
3587 The degree and low degree of a polynomial can be obtained using the two
3591 int ex::degree(const ex & s);
3592 int ex::ldegree(const ex & s);
3595 These functions only work reliably if the input polynomial is collected in
3596 terms of the object @samp{s}. Otherwise, they are only guaranteed to return
3597 the upper/lower bounds of the exponents. If you need accurate results, you
3598 have to call @code{expand()} and/or @code{collect()} on the input polynomial.
3606 > degree(expand(a),x);
3610 @code{degree()} also works on rational functions, returning the asymptotic
3614 > degree((x+1)/(x^3+1),x);
3618 If the input is not a polynomial or rational function in the variable @samp{s},
3619 the behavior of @code{degree()} and @code{ldegree()} is undefined.
3621 To extract a coefficient with a certain power from an expanded
3625 ex ex::coeff(const ex & s, int n);
3628 You can also obtain the leading and trailing coefficients with the methods
3631 ex ex::lcoeff(const ex & s);
3632 ex ex::tcoeff(const ex & s);
3635 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3638 An application is illustrated in the next example, where a multivariate
3639 polynomial is analyzed:
3643 symbol x("x"), y("y");
3644 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3645 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3646 ex Poly = PolyInp.expand();
3648 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3649 cout << "The x^" << i << "-coefficient is "
3650 << Poly.coeff(x,i) << endl;
3652 cout << "As polynomial in y: "
3653 << Poly.collect(y) << endl;
3657 When run, it returns an output in the following fashion:
3660 The x^0-coefficient is y^2+11*y
3661 The x^1-coefficient is 5*y^2-2*y
3662 The x^2-coefficient is -1
3663 The x^3-coefficient is 4*y
3664 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3667 As always, the exact output may vary between different versions of GiNaC
3668 or even from run to run since the internal canonical ordering is not
3669 within the user's sphere of influence.
3671 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3672 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3673 with non-polynomial expressions as they not only work with symbols but with
3674 constants, functions and indexed objects as well:
3678 symbol a("a"), b("b"), c("c");
3679 idx i(symbol("i"), 3);
3681 ex e = pow(sin(x) - cos(x), 4);
3682 cout << e.degree(cos(x)) << endl;
3684 cout << e.expand().coeff(sin(x), 3) << endl;
3687 e = indexed(a+b, i) * indexed(b+c, i);
3688 e = e.expand(expand_options::expand_indexed);
3689 cout << e.collect(indexed(b, i)) << endl;
3690 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3695 @subsection Polynomial division
3696 @cindex polynomial division
3699 @cindex pseudo-remainder
3700 @cindex @code{quo()}
3701 @cindex @code{rem()}
3702 @cindex @code{prem()}
3703 @cindex @code{divide()}
3708 ex quo(const ex & a, const ex & b, const symbol & x);
3709 ex rem(const ex & a, const ex & b, const symbol & x);
3712 compute the quotient and remainder of univariate polynomials in the variable
3713 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3715 The additional function
3718 ex prem(const ex & a, const ex & b, const symbol & x);
3721 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3722 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3724 Exact division of multivariate polynomials is performed by the function
3727 bool divide(const ex & a, const ex & b, ex & q);
3730 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3731 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3732 in which case the value of @code{q} is undefined.
3735 @subsection Unit, content and primitive part
3736 @cindex @code{unit()}
3737 @cindex @code{content()}
3738 @cindex @code{primpart()}
3743 ex ex::unit(const symbol & x);
3744 ex ex::content(const symbol & x);
3745 ex ex::primpart(const symbol & x);
3748 return the unit part, content part, and primitive polynomial of a multivariate
3749 polynomial with respect to the variable @samp{x} (the unit part being the sign
3750 of the leading coefficient, the content part being the GCD of the coefficients,
3751 and the primitive polynomial being the input polynomial divided by the unit and
3752 content parts). The product of unit, content, and primitive part is the
3753 original polynomial.
3756 @subsection GCD and LCM
3759 @cindex @code{gcd()}
3760 @cindex @code{lcm()}
3762 The functions for polynomial greatest common divisor and least common
3763 multiple have the synopsis
3766 ex gcd(const ex & a, const ex & b);
3767 ex lcm(const ex & a, const ex & b);
3770 The functions @code{gcd()} and @code{lcm()} accept two expressions
3771 @code{a} and @code{b} as arguments and return a new expression, their
3772 greatest common divisor or least common multiple, respectively. If the
3773 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3774 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3777 #include <ginac/ginac.h>
3778 using namespace GiNaC;
3782 symbol x("x"), y("y"), z("z");
3783 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3784 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3786 ex P_gcd = gcd(P_a, P_b);
3788 ex P_lcm = lcm(P_a, P_b);
3789 // 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
3794 @subsection Square-free decomposition
3795 @cindex square-free decomposition
3796 @cindex factorization
3797 @cindex @code{sqrfree()}
3799 GiNaC still lacks proper factorization support. Some form of
3800 factorization is, however, easily implemented by noting that factors
3801 appearing in a polynomial with power two or more also appear in the
3802 derivative and hence can easily be found by computing the GCD of the
3803 original polynomial and its derivatives. Any decent system has an
3804 interface for this so called square-free factorization. So we provide
3807 ex sqrfree(const ex & a, const lst & l = lst());
3809 Here is an example that by the way illustrates how the exact form of the
3810 result may slightly depend on the order of differentiation, calling for
3811 some care with subsequent processing of the result:
3814 symbol x("x"), y("y");
3815 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
3817 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3818 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
3820 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3821 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
3823 cout << sqrfree(BiVarPol) << endl;
3824 // -> depending on luck, any of the above
3827 Note also, how factors with the same exponents are not fully factorized
3831 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3832 @c node-name, next, previous, up
3833 @section Rational expressions
3835 @subsection The @code{normal} method
3836 @cindex @code{normal()}
3837 @cindex simplification
3838 @cindex temporary replacement
3840 Some basic form of simplification of expressions is called for frequently.
3841 GiNaC provides the method @code{.normal()}, which converts a rational function
3842 into an equivalent rational function of the form @samp{numerator/denominator}
3843 where numerator and denominator are coprime. If the input expression is already
3844 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3845 otherwise it performs fraction addition and multiplication.
3847 @code{.normal()} can also be used on expressions which are not rational functions
3848 as it will replace all non-rational objects (like functions or non-integer
3849 powers) by temporary symbols to bring the expression to the domain of rational
3850 functions before performing the normalization, and re-substituting these
3851 symbols afterwards. This algorithm is also available as a separate method
3852 @code{.to_rational()}, described below.
3854 This means that both expressions @code{t1} and @code{t2} are indeed
3855 simplified in this little code snippet:
3860 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3861 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3862 std::cout << "t1 is " << t1.normal() << std::endl;
3863 std::cout << "t2 is " << t2.normal() << std::endl;
3867 Of course this works for multivariate polynomials too, so the ratio of
3868 the sample-polynomials from the section about GCD and LCM above would be
3869 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3872 @subsection Numerator and denominator
3875 @cindex @code{numer()}
3876 @cindex @code{denom()}
3877 @cindex @code{numer_denom()}
3879 The numerator and denominator of an expression can be obtained with
3884 ex ex::numer_denom();
3887 These functions will first normalize the expression as described above and
3888 then return the numerator, denominator, or both as a list, respectively.
3889 If you need both numerator and denominator, calling @code{numer_denom()} is
3890 faster than using @code{numer()} and @code{denom()} separately.
3893 @subsection Converting to a rational expression
3894 @cindex @code{to_rational()}
3896 Some of the methods described so far only work on polynomials or rational
3897 functions. GiNaC provides a way to extend the domain of these functions to
3898 general expressions by using the temporary replacement algorithm described
3899 above. You do this by calling
3902 ex ex::to_rational(lst &l);
3905 on the expression to be converted. The supplied @code{lst} will be filled
3906 with the generated temporary symbols and their replacement expressions in
3907 a format that can be used directly for the @code{subs()} method. It can also
3908 already contain a list of replacements from an earlier application of
3909 @code{.to_rational()}, so it's possible to use it on multiple expressions
3910 and get consistent results.
3917 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3918 ex b = sin(x) + cos(x);
3921 divide(a.to_rational(l), b.to_rational(l), q);
3922 cout << q.subs(l) << endl;
3926 will print @samp{sin(x)-cos(x)}.
3929 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3930 @c node-name, next, previous, up
3931 @section Symbolic differentiation
3932 @cindex differentiation
3933 @cindex @code{diff()}
3935 @cindex product rule
3937 GiNaC's objects know how to differentiate themselves. Thus, a
3938 polynomial (class @code{add}) knows that its derivative is the sum of
3939 the derivatives of all the monomials:
3943 symbol x("x"), y("y"), z("z");
3944 ex P = pow(x, 5) + pow(x, 2) + y;
3946 cout << P.diff(x,2) << endl;
3948 cout << P.diff(y) << endl; // 1
3950 cout << P.diff(z) << endl; // 0
3955 If a second integer parameter @var{n} is given, the @code{diff} method
3956 returns the @var{n}th derivative.
3958 If @emph{every} object and every function is told what its derivative
3959 is, all derivatives of composed objects can be calculated using the
3960 chain rule and the product rule. Consider, for instance the expression
3961 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3962 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3963 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3964 out that the composition is the generating function for Euler Numbers,
3965 i.e. the so called @var{n}th Euler number is the coefficient of
3966 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3967 identity to code a function that generates Euler numbers in just three
3970 @cindex Euler numbers
3972 #include <ginac/ginac.h>
3973 using namespace GiNaC;
3975 ex EulerNumber(unsigned n)
3978 const ex generator = pow(cosh(x),-1);
3979 return generator.diff(x,n).subs(x==0);
3984 for (unsigned i=0; i<11; i+=2)
3985 std::cout << EulerNumber(i) << std::endl;
3990 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3991 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3992 @code{i} by two since all odd Euler numbers vanish anyways.
3995 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3996 @c node-name, next, previous, up
3997 @section Series expansion
3998 @cindex @code{series()}
3999 @cindex Taylor expansion
4000 @cindex Laurent expansion
4001 @cindex @code{pseries} (class)
4002 @cindex @code{Order()}
4004 Expressions know how to expand themselves as a Taylor series or (more
4005 generally) a Laurent series. As in most conventional Computer Algebra
4006 Systems, no distinction is made between those two. There is a class of
4007 its own for storing such series (@code{class pseries}) and a built-in
4008 function (called @code{Order}) for storing the order term of the series.
4009 As a consequence, if you want to work with series, i.e. multiply two
4010 series, you need to call the method @code{ex::series} again to convert
4011 it to a series object with the usual structure (expansion plus order
4012 term). A sample application from special relativity could read:
4015 #include <ginac/ginac.h>
4016 using namespace std;
4017 using namespace GiNaC;
4021 symbol v("v"), c("c");
4023 ex gamma = 1/sqrt(1 - pow(v/c,2));
4024 ex mass_nonrel = gamma.series(v==0, 10);
4026 cout << "the relativistic mass increase with v is " << endl
4027 << mass_nonrel << endl;
4029 cout << "the inverse square of this series is " << endl
4030 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4034 Only calling the series method makes the last output simplify to
4035 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4036 series raised to the power @math{-2}.
4038 @cindex Machin's formula
4039 As another instructive application, let us calculate the numerical
4040 value of Archimedes' constant
4044 (for which there already exists the built-in constant @code{Pi})
4045 using Machin's amazing formula
4047 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4050 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4052 We may expand the arcus tangent around @code{0} and insert the fractions
4053 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
4054 carries an order term with it and the question arises what the system is
4055 supposed to do when the fractions are plugged into that order term. The
4056 solution is to use the function @code{series_to_poly()} to simply strip
4060 #include <ginac/ginac.h>
4061 using namespace GiNaC;
4063 ex machin_pi(int degr)
4066 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4067 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4068 -4*pi_expansion.subs(x==numeric(1,239));
4074 using std::cout; // just for fun, another way of...
4075 using std::endl; // ...dealing with this namespace std.
4077 for (int i=2; i<12; i+=2) @{
4078 pi_frac = machin_pi(i);
4079 cout << i << ":\t" << pi_frac << endl
4080 << "\t" << pi_frac.evalf() << endl;
4086 Note how we just called @code{.series(x,degr)} instead of
4087 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4088 method @code{series()}: if the first argument is a symbol the expression
4089 is expanded in that symbol around point @code{0}. When you run this
4090 program, it will type out:
4094 3.1832635983263598326
4095 4: 5359397032/1706489875
4096 3.1405970293260603143
4097 6: 38279241713339684/12184551018734375
4098 3.141621029325034425
4099 8: 76528487109180192540976/24359780855939418203125
4100 3.141591772182177295
4101 10: 327853873402258685803048818236/104359128170408663038552734375
4102 3.1415926824043995174
4106 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4107 @c node-name, next, previous, up
4108 @section Symmetrization
4109 @cindex @code{symmetrize()}
4110 @cindex @code{antisymmetrize()}
4111 @cindex @code{symmetrize_cyclic()}
4116 ex ex::symmetrize(const lst & l);
4117 ex ex::antisymmetrize(const lst & l);
4118 ex ex::symmetrize_cyclic(const lst & l);
4121 symmetrize an expression by returning the sum over all symmetric,
4122 antisymmetric or cyclic permutations of the specified list of objects,
4123 weighted by the number of permutations.
4125 The three additional methods
4128 ex ex::symmetrize();
4129 ex ex::antisymmetrize();
4130 ex ex::symmetrize_cyclic();
4133 symmetrize or antisymmetrize an expression over its free indices.
4135 Symmetrization is most useful with indexed expressions but can be used with
4136 almost any kind of object (anything that is @code{subs()}able):
4140 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4141 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4143 cout << indexed(A, i, j).symmetrize() << endl;
4144 // -> 1/2*A.j.i+1/2*A.i.j
4145 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4146 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4147 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4148 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4153 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
4154 @c node-name, next, previous, up
4155 @section Predefined mathematical functions
4157 GiNaC contains the following predefined mathematical functions:
4160 @multitable @columnfractions .30 .70
4161 @item @strong{Name} @tab @strong{Function}
4164 @cindex @code{abs()}
4165 @item @code{csgn(x)}
4167 @cindex @code{csgn()}
4168 @item @code{sqrt(x)}
4169 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4170 @cindex @code{sqrt()}
4173 @cindex @code{sin()}
4176 @cindex @code{cos()}
4179 @cindex @code{tan()}
4180 @item @code{asin(x)}
4182 @cindex @code{asin()}
4183 @item @code{acos(x)}
4185 @cindex @code{acos()}
4186 @item @code{atan(x)}
4187 @tab inverse tangent
4188 @cindex @code{atan()}
4189 @item @code{atan2(y, x)}
4190 @tab inverse tangent with two arguments
4191 @item @code{sinh(x)}
4192 @tab hyperbolic sine
4193 @cindex @code{sinh()}
4194 @item @code{cosh(x)}
4195 @tab hyperbolic cosine
4196 @cindex @code{cosh()}
4197 @item @code{tanh(x)}
4198 @tab hyperbolic tangent
4199 @cindex @code{tanh()}
4200 @item @code{asinh(x)}
4201 @tab inverse hyperbolic sine
4202 @cindex @code{asinh()}
4203 @item @code{acosh(x)}
4204 @tab inverse hyperbolic cosine
4205 @cindex @code{acosh()}
4206 @item @code{atanh(x)}
4207 @tab inverse hyperbolic tangent
4208 @cindex @code{atanh()}
4210 @tab exponential function
4211 @cindex @code{exp()}
4213 @tab natural logarithm
4214 @cindex @code{log()}
4217 @cindex @code{Li2()}
4218 @item @code{zeta(x)}
4219 @tab Riemann's zeta function
4220 @cindex @code{zeta()}
4221 @item @code{zeta(n, x)}
4222 @tab derivatives of Riemann's zeta function
4223 @item @code{tgamma(x)}
4225 @cindex @code{tgamma()}
4226 @cindex Gamma function
4227 @item @code{lgamma(x)}
4228 @tab logarithm of Gamma function
4229 @cindex @code{lgamma()}
4230 @item @code{beta(x, y)}
4231 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4232 @cindex @code{beta()}
4234 @tab psi (digamma) function
4235 @cindex @code{psi()}
4236 @item @code{psi(n, x)}
4237 @tab derivatives of psi function (polygamma functions)
4238 @item @code{factorial(n)}
4239 @tab factorial function
4240 @cindex @code{factorial()}
4241 @item @code{binomial(n, m)}
4242 @tab binomial coefficients
4243 @cindex @code{binomial()}
4244 @item @code{Order(x)}
4245 @tab order term function in truncated power series
4246 @cindex @code{Order()}
4251 For functions that have a branch cut in the complex plane GiNaC follows
4252 the conventions for C++ as defined in the ANSI standard as far as
4253 possible. In particular: the natural logarithm (@code{log}) and the
4254 square root (@code{sqrt}) both have their branch cuts running along the
4255 negative real axis where the points on the axis itself belong to the
4256 upper part (i.e. continuous with quadrant II). The inverse
4257 trigonometric and hyperbolic functions are not defined for complex
4258 arguments by the C++ standard, however. In GiNaC we follow the
4259 conventions used by CLN, which in turn follow the carefully designed
4260 definitions in the Common Lisp standard. It should be noted that this
4261 convention is identical to the one used by the C99 standard and by most
4262 serious CAS. It is to be expected that future revisions of the C++
4263 standard incorporate these functions in the complex domain in a manner
4264 compatible with C99.
4267 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
4268 @c node-name, next, previous, up
4269 @section Input and output of expressions
4272 @subsection Expression output
4274 @cindex output of expressions
4276 The easiest way to print an expression is to write it to a stream:
4281 ex e = 4.5+pow(x,2)*3/2;
4282 cout << e << endl; // prints '(4.5)+3/2*x^2'
4286 The output format is identical to the @command{ginsh} input syntax and
4287 to that used by most computer algebra systems, but not directly pastable
4288 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4289 is printed as @samp{x^2}).
4291 It is possible to print expressions in a number of different formats with
4295 void ex::print(const print_context & c, unsigned level = 0);
4298 @cindex @code{print_context} (class)
4299 The type of @code{print_context} object passed in determines the format
4300 of the output. The possible types are defined in @file{ginac/print.h}.
4301 All constructors of @code{print_context} and derived classes take an
4302 @code{ostream &} as their first argument.
4304 To print an expression in a way that can be directly used in a C or C++
4305 program, you pass a @code{print_csrc} object like this:
4309 cout << "float f = ";
4310 e.print(print_csrc_float(cout));
4313 cout << "double d = ";
4314 e.print(print_csrc_double(cout));
4317 cout << "cl_N n = ";
4318 e.print(print_csrc_cl_N(cout));
4323 The three possible types mostly affect the way in which floating point
4324 numbers are written.
4326 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4329 float f = (3.0/2.0)*(x*x)+4.500000e+00;
4330 double d = (3.0/2.0)*(x*x)+4.5000000000000000e+00;
4331 cl_N n = cln::cl_RA("3/2")*(x*x)+cln::cl_F("4.5_17");
4334 The @code{print_context} type @code{print_tree} provides a dump of the
4335 internal structure of an expression for debugging purposes:
4339 e.print(print_tree(cout));
4346 add, hash=0x0, flags=0x3, nops=2
4347 power, hash=0x9, flags=0x3, nops=2
4348 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4349 2 (numeric), hash=0x80000042, flags=0xf
4350 3/2 (numeric), hash=0x80000061, flags=0xf
4353 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4357 This kind of output is also available in @command{ginsh} as the @code{print()}
4360 Another useful output format is for LaTeX parsing in mathematical mode.
4361 It is rather similar to the default @code{print_context} but provides
4362 some braces needed by LaTeX for delimiting boxes and also converts some
4363 common objects to conventional LaTeX names. It is possible to give symbols
4364 a special name for LaTeX output by supplying it as a second argument to
4365 the @code{symbol} constructor.
4367 For example, the code snippet
4372 ex foo = lgamma(x).series(x==0,3);
4373 foo.print(print_latex(std::cout));
4379 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4382 @cindex Tree traversal
4383 If you need any fancy special output format, e.g. for interfacing GiNaC
4384 with other algebra systems or for producing code for different
4385 programming languages, you can always traverse the expression tree yourself:
4388 static void my_print(const ex & e)
4390 if (is_a<function>(e))
4391 cout << ex_to<function>(e).get_name();
4393 cout << e.bp->class_name();
4395 unsigned n = e.nops();
4397 for (unsigned i=0; i<n; i++) @{
4409 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4417 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4418 symbol(y))),numeric(-2)))
4421 If you need an output format that makes it possible to accurately
4422 reconstruct an expression by feeding the output to a suitable parser or
4423 object factory, you should consider storing the expression in an
4424 @code{archive} object and reading the object properties from there.
4425 See the section on archiving for more information.
4428 @subsection Expression input
4429 @cindex input of expressions
4431 GiNaC provides no way to directly read an expression from a stream because
4432 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4433 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4434 @code{y} you defined in your program and there is no way to specify the
4435 desired symbols to the @code{>>} stream input operator.
4437 Instead, GiNaC lets you construct an expression from a string, specifying the
4438 list of symbols and indices to be used:
4442 symbol x("x"), y("y"), p("p");
4443 idx i(symbol("i"), 3);
4444 ex e("2*x+sin(y)+p.i", lst(x, y, p, i));
4448 The input syntax is the same as that used by @command{ginsh} and the stream
4449 output operator @code{<<}. The symbols and indices in the string are matched
4450 by name to the symbols and indices in the list and if GiNaC encounters a
4451 symbol or index not specified in the list it will throw an exception. Only
4452 indices whose values are single symbols can be used (i.e. numeric indices
4453 or compound indices as in "A.(2*n+1)" are not allowed).
4455 With this constructor, it's also easy to implement interactive GiNaC programs:
4460 #include <stdexcept>
4461 #include <ginac/ginac.h>
4462 using namespace std;
4463 using namespace GiNaC;
4470 cout << "Enter an expression containing 'x': ";
4475 cout << "The derivative of " << e << " with respect to x is ";
4476 cout << e.diff(x) << ".\n";
4477 @} catch (exception &p) @{
4478 cerr << p.what() << endl;
4484 @subsection Archiving
4485 @cindex @code{archive} (class)
4488 GiNaC allows creating @dfn{archives} of expressions which can be stored
4489 to or retrieved from files. To create an archive, you declare an object
4490 of class @code{archive} and archive expressions in it, giving each
4491 expression a unique name:
4495 using namespace std;
4496 #include <ginac/ginac.h>
4497 using namespace GiNaC;
4501 symbol x("x"), y("y"), z("z");
4503 ex foo = sin(x + 2*y) + 3*z + 41;
4507 a.archive_ex(foo, "foo");
4508 a.archive_ex(bar, "the second one");
4512 The archive can then be written to a file:
4516 ofstream out("foobar.gar");
4522 The file @file{foobar.gar} contains all information that is needed to
4523 reconstruct the expressions @code{foo} and @code{bar}.
4525 @cindex @command{viewgar}
4526 The tool @command{viewgar} that comes with GiNaC can be used to view
4527 the contents of GiNaC archive files:
4530 $ viewgar foobar.gar
4531 foo = 41+sin(x+2*y)+3*z
4532 the second one = 42+sin(x+2*y)+3*z
4535 The point of writing archive files is of course that they can later be
4541 ifstream in("foobar.gar");
4546 And the stored expressions can be retrieved by their name:
4552 ex ex1 = a2.unarchive_ex(syms, "foo");
4553 ex ex2 = a2.unarchive_ex(syms, "the second one");
4555 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4556 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4557 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4561 Note that you have to supply a list of the symbols which are to be inserted
4562 in the expressions. Symbols in archives are stored by their name only and
4563 if you don't specify which symbols you have, unarchiving the expression will
4564 create new symbols with that name. E.g. if you hadn't included @code{x} in
4565 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4566 have had no effect because the @code{x} in @code{ex1} would have been a
4567 different symbol than the @code{x} which was defined at the beginning of
4568 the program, although both would appear as @samp{x} when printed.
4570 You can also use the information stored in an @code{archive} object to
4571 output expressions in a format suitable for exact reconstruction. The
4572 @code{archive} and @code{archive_node} classes have a couple of member
4573 functions that let you access the stored properties:
4576 static void my_print2(const archive_node & n)
4579 n.find_string("class", class_name);
4580 cout << class_name << "(";
4582 archive_node::propinfovector p;
4583 n.get_properties(p);
4585 unsigned num = p.size();
4586 for (unsigned i=0; i<num; i++) @{
4587 const string &name = p[i].name;
4588 if (name == "class")
4590 cout << name << "=";
4592 unsigned count = p[i].count;
4596 for (unsigned j=0; j<count; j++) @{
4597 switch (p[i].type) @{
4598 case archive_node::PTYPE_BOOL: @{
4600 n.find_bool(name, x, j);
4601 cout << (x ? "true" : "false");
4604 case archive_node::PTYPE_UNSIGNED: @{
4606 n.find_unsigned(name, x, j);
4610 case archive_node::PTYPE_STRING: @{
4612 n.find_string(name, x, j);
4613 cout << '\"' << x << '\"';
4616 case archive_node::PTYPE_NODE: @{
4617 const archive_node &x = n.find_ex_node(name, j);
4639 ex e = pow(2, x) - y;
4641 my_print2(ar.get_top_node(0)); cout << endl;
4649 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4650 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4651 overall_coeff=numeric(number="0"))
4654 Be warned, however, that the set of properties and their meaning for each
4655 class may change between GiNaC versions.
4658 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4659 @c node-name, next, previous, up
4660 @chapter Extending GiNaC
4662 By reading so far you should have gotten a fairly good understanding of
4663 GiNaC's design-patterns. From here on you should start reading the
4664 sources. All we can do now is issue some recommendations how to tackle
4665 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4666 develop some useful extension please don't hesitate to contact the GiNaC
4667 authors---they will happily incorporate them into future versions.
4670 * What does not belong into GiNaC:: What to avoid.
4671 * Symbolic functions:: Implementing symbolic functions.
4672 * Adding classes:: Defining new algebraic classes.
4676 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4677 @c node-name, next, previous, up
4678 @section What doesn't belong into GiNaC
4680 @cindex @command{ginsh}
4681 First of all, GiNaC's name must be read literally. It is designed to be
4682 a library for use within C++. The tiny @command{ginsh} accompanying
4683 GiNaC makes this even more clear: it doesn't even attempt to provide a
4684 language. There are no loops or conditional expressions in
4685 @command{ginsh}, it is merely a window into the library for the
4686 programmer to test stuff (or to show off). Still, the design of a
4687 complete CAS with a language of its own, graphical capabilities and all
4688 this on top of GiNaC is possible and is without doubt a nice project for
4691 There are many built-in functions in GiNaC that do not know how to
4692 evaluate themselves numerically to a precision declared at runtime
4693 (using @code{Digits}). Some may be evaluated at certain points, but not
4694 generally. This ought to be fixed. However, doing numerical
4695 computations with GiNaC's quite abstract classes is doomed to be
4696 inefficient. For this purpose, the underlying foundation classes
4697 provided by CLN are much better suited.
4700 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4701 @c node-name, next, previous, up
4702 @section Symbolic functions
4704 The easiest and most instructive way to start extending GiNaC is probably to
4705 create your own symbolic functions. These are implemented with the help of
4706 two preprocessor macros:
4708 @cindex @code{DECLARE_FUNCTION}
4709 @cindex @code{REGISTER_FUNCTION}
4711 DECLARE_FUNCTION_<n>P(<name>)
4712 REGISTER_FUNCTION(<name>, <options>)
4715 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
4716 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
4717 parameters of type @code{ex} and returns a newly constructed GiNaC
4718 @code{function} object that represents your function.
4720 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
4721 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
4722 set of options that associate the symbolic function with C++ functions you
4723 provide to implement the various methods such as evaluation, derivative,
4724 series expansion etc. They also describe additional attributes the function
4725 might have, such as symmetry and commutation properties, and a name for
4726 LaTeX output. Multiple options are separated by the member access operator
4727 @samp{.} and can be given in an arbitrary order.
4729 (By the way: in case you are worrying about all the macros above we can
4730 assure you that functions are GiNaC's most macro-intense classes. We have
4731 done our best to avoid macros where we can.)
4733 @subsection A minimal example
4735 Here is an example for the implementation of a function with two arguments
4736 that is not further evaluated:
4739 DECLARE_FUNCTION_2P(myfcn)
4741 static ex myfcn_eval(const ex & x, const ex & y)
4743 return myfcn(x, y).hold();
4746 REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
4749 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
4750 in algebraic expressions:
4756 ex e = 2*myfcn(42, 3*x+1) - x;
4757 // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
4758 // the actual expression
4760 // prints '2*myfcn(42,1+3*x)-x'
4765 @cindex @code{hold()}
4767 The @code{eval_func()} option specifies the C++ function that implements
4768 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
4769 the same number of arguments as the associated symbolic function (two in this
4770 case) and returns the (possibly transformed or in some way simplified)
4771 symbolically evaluated function (@xref{Automatic evaluation}, for a description
4772 of the automatic evaluation process). If no (further) evaluation is to take
4773 place, the @code{eval_func()} function must return the original function
4774 with @code{.hold()}, to avoid a potential infinite recursion. If your
4775 symbolic functions produce a segmentation fault or stack overflow when
4776 using them in expressions, you are probably missing a @code{.hold()}
4779 There is not much you can do with the @code{myfcn} function. It merely acts
4780 as a kind of container for its arguments (which is, however, sometimes
4781 perfectly sufficient). Let's have a look at the implementation of GiNaC's
4784 @subsection The cosine function
4786 The GiNaC header file @file{inifcns.h} contains the line
4789 DECLARE_FUNCTION_1P(cos)
4792 which declares to all programs using GiNaC that there is a function @samp{cos}
4793 that takes one @code{ex} as an argument. This is all they need to know to use
4794 this function in expressions.
4796 The implementation of the cosine function is in @file{inifcns_trans.cpp}. The
4797 @code{eval_func()} function looks something like this (actually, it doesn't
4798 look like this at all, but it should give you an idea what is going on):
4801 static ex cos_eval(const ex & x)
4803 if (<x is a multiple of 2*Pi>)
4805 else if (<x is a multiple of Pi>)
4807 else if (<x is a multiple of Pi/2>)
4811 else if (<x has the form 'acos(y)'>)
4813 else if (<x has the form 'asin(y)'>)
4818 return cos(x).hold();
4822 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
4823 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
4824 symbolic transformation can be done, the unmodified function is returned
4825 with @code{.hold()}.
4827 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
4828 The user has to call @code{evalf()} for that. This is implemented in a
4832 static ex cos_evalf(const ex & x)
4834 if (is_a<numeric>(x))
4835 return cos(ex_to<numeric>(x));
4837 return cos(x).hold();
4841 Since we are lazy we defer the problem of numeric evaluation to somebody else,
4842 in this case the @code{cos()} function for @code{numeric} objects, which in
4843 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
4844 isn't really needed here, but reminds us that the corresponding @code{eval()}
4845 function would require it in this place.
4847 Differentiation will surely turn up and so we need to tell @code{cos}
4848 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
4849 instance, are then handled automatically by @code{basic::diff} and
4853 static ex cos_deriv(const ex & x, unsigned diff_param)
4859 @cindex product rule
4860 The second parameter is obligatory but uninteresting at this point. It
4861 specifies which parameter to differentiate in a partial derivative in
4862 case the function has more than one parameter, and its main application
4863 is for correct handling of the chain rule.
4865 An implementation of the series expansion is not needed for @code{cos()} as
4866 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
4867 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
4868 the other hand, does have poles and may need to do Laurent expansion:
4871 static ex tan_series(const ex & x, const relational & rel,
4872 int order, unsigned options)
4874 // Find the actual expansion point
4875 const ex x_pt = x.subs(rel);
4877 if (<x_pt is not an odd multiple of Pi/2>)
4878 throw do_taylor(); // tell function::series() to do Taylor expansion
4880 // On a pole, expand sin()/cos()
4881 return (sin(x)/cos(x)).series(rel, order+2, options);
4885 The @code{series()} implementation of a function @emph{must} return a
4886 @code{pseries} object, otherwise your code will crash.
4888 Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
4889 macro is used to tell the system how the @code{cos()} function behaves:
4892 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4893 evalf_func(cos_evalf).
4894 derivative_func(cos_deriv).
4895 latex_name("\\cos"));
4898 This registers the @code{cos_eval()}, @code{cos_evalf()} and
4899 @code{cos_deriv()} C++ functions with the @code{cos()} function, and also
4900 gives it a proper LaTeX name.
4902 @subsection Function options
4904 GiNaC functions understand several more options which are always
4905 specified as @code{.option(params)}. None of them are required, but you
4906 need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
4907 the @code{eval()} method).
4910 eval_func(<C++ function>)
4911 evalf_func(<C++ function>)
4912 derivative_func(<C++ function>)
4913 series_func(<C++ function>)
4916 These specify the C++ functions that implement symbolic evaluation,
4917 numeric evaluation, partial derivatives, and series expansion, respectively.
4918 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
4919 @code{diff()} and @code{series()}.
4921 The @code{eval_func()} function needs to use @code{.hold()} if no further
4922 automatic evaluation is desired or possible.
4924 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
4925 expansion, which is correct if there are no poles involved. If the function
4926 has poles in the complex plane, the @code{series_func()} needs to check
4927 whether the expansion point is on a pole and fall back to Taylor expansion
4928 if it isn't. Otherwise, the pole usually needs to be regularized by some
4929 suitable transformation.
4932 latex_name(const string & n)
4935 specifies the LaTeX code that represents the name of the function in LaTeX
4936 output. The default is to put the function name in an @code{\mbox@{@}}.
4939 do_not_evalf_params()
4942 This tells @code{evalf()} to not recursively evaluate the parameters of the
4943 function before calling the @code{evalf_func()}.
4946 set_return_type(unsigned return_type, unsigned return_type_tinfo)
4949 This allows you to explicitly specify the commutation properties of the
4950 function (@xref{Non-commutative objects}, for an explanation of
4951 (non)commutativity in GiNaC). For example, you can use
4952 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
4953 GiNaC treat your function like a matrix. By default, functions inherit the
4954 commutation properties of their first argument.
4957 set_symmetry(const symmetry & s)
4960 specifies the symmetry properties of the function with respect to its
4961 arguments. @xref{Indexed objects}, for an explanation of symmetry
4962 specifications. GiNaC will automatically rearrange the arguments of
4963 symmetric functions into a canonical order.
4966 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4967 @c node-name, next, previous, up
4968 @section Adding classes
4970 If you are doing some very specialized things with GiNaC you may find that
4971 you have to implement your own algebraic classes to fit your needs. This
4972 section will explain how to do this by giving the example of a simple
4973 'string' class. After reading this section you will know how to properly
4974 declare a GiNaC class and what the minimum required member functions are
4975 that you have to implement. We only cover the implementation of a 'leaf'
4976 class here (i.e. one that doesn't contain subexpressions). Creating a
4977 container class like, for example, a class representing tensor products is
4978 more involved but this section should give you enough information so you can
4979 consult the source to GiNaC's predefined classes if you want to implement
4980 something more complicated.
4982 @subsection GiNaC's run-time type information system
4984 @cindex hierarchy of classes
4986 All algebraic classes (that is, all classes that can appear in expressions)
4987 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4988 @code{basic *} (which is essentially what an @code{ex} is) represents a
4989 generic pointer to an algebraic class. Occasionally it is necessary to find
4990 out what the class of an object pointed to by a @code{basic *} really is.
4991 Also, for the unarchiving of expressions it must be possible to find the
4992 @code{unarchive()} function of a class given the class name (as a string). A
4993 system that provides this kind of information is called a run-time type
4994 information (RTTI) system. The C++ language provides such a thing (see the
4995 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4996 implements its own, simpler RTTI.
4998 The RTTI in GiNaC is based on two mechanisms:
5003 The @code{basic} class declares a member variable @code{tinfo_key} which
5004 holds an unsigned integer that identifies the object's class. These numbers
5005 are defined in the @file{tinfos.h} header file for the built-in GiNaC
5006 classes. They all start with @code{TINFO_}.
5009 By means of some clever tricks with static members, GiNaC maintains a list
5010 of information for all classes derived from @code{basic}. The information
5011 available includes the class names, the @code{tinfo_key}s, and pointers
5012 to the unarchiving functions. This class registry is defined in the
5013 @file{registrar.h} header file.
5017 The disadvantage of this proprietary RTTI implementation is that there's
5018 a little more to do when implementing new classes (C++'s RTTI works more
5019 or less automatic) but don't worry, most of the work is simplified by
5022 @subsection A minimalistic example
5024 Now we will start implementing a new class @code{mystring} that allows
5025 placing character strings in algebraic expressions (this is not very useful,
5026 but it's just an example). This class will be a direct subclass of
5027 @code{basic}. You can use this sample implementation as a starting point
5028 for your own classes.
5030 The code snippets given here assume that you have included some header files
5036 #include <stdexcept>
5037 using namespace std;
5039 #include <ginac/ginac.h>
5040 using namespace GiNaC;
5043 The first thing we have to do is to define a @code{tinfo_key} for our new
5044 class. This can be any arbitrary unsigned number that is not already taken
5045 by one of the existing classes but it's better to come up with something
5046 that is unlikely to clash with keys that might be added in the future. The
5047 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
5048 which is not a requirement but we are going to stick with this scheme:
5051 const unsigned TINFO_mystring = 0x42420001U;
5054 Now we can write down the class declaration. The class stores a C++
5055 @code{string} and the user shall be able to construct a @code{mystring}
5056 object from a C or C++ string:
5059 class mystring : public basic
5061 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
5064 mystring(const string &s);
5065 mystring(const char *s);
5071 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
5074 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
5075 macros are defined in @file{registrar.h}. They take the name of the class
5076 and its direct superclass as arguments and insert all required declarations
5077 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
5078 the first line after the opening brace of the class definition. The
5079 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
5080 source (at global scope, of course, not inside a function).
5082 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
5083 declarations of the default and copy constructor, the destructor, the
5084 assignment operator and a couple of other functions that are required. It
5085 also defines a type @code{inherited} which refers to the superclass so you
5086 don't have to modify your code every time you shuffle around the class
5087 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
5088 constructor, the destructor and the assignment operator.
5090 Now there are nine member functions we have to implement to get a working
5096 @code{mystring()}, the default constructor.
5099 @code{void destroy(bool call_parent)}, which is used in the destructor and the
5100 assignment operator to free dynamically allocated members. The @code{call_parent}
5101 specifies whether the @code{destroy()} function of the superclass is to be
5105 @code{void copy(const mystring &other)}, which is used in the copy constructor
5106 and assignment operator to copy the member variables over from another
5107 object of the same class.
5110 @code{void archive(archive_node &n)}, the archiving function. This stores all
5111 information needed to reconstruct an object of this class inside an
5112 @code{archive_node}.
5115 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
5116 constructor. This constructs an instance of the class from the information
5117 found in an @code{archive_node}.
5120 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
5121 unarchiving function. It constructs a new instance by calling the unarchiving
5125 @code{int compare_same_type(const basic &other)}, which is used internally
5126 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
5127 -1, depending on the relative order of this object and the @code{other}
5128 object. If it returns 0, the objects are considered equal.
5129 @strong{Note:} This has nothing to do with the (numeric) ordering
5130 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
5131 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
5132 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
5133 must provide a @code{compare_same_type()} function, even those representing
5134 objects for which no reasonable algebraic ordering relationship can be
5138 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
5139 which are the two constructors we declared.
5143 Let's proceed step-by-step. The default constructor looks like this:
5146 mystring::mystring() : inherited(TINFO_mystring)
5148 // dynamically allocate resources here if required
5152 The golden rule is that in all constructors you have to set the
5153 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
5154 it will be set by the constructor of the superclass and all hell will break
5155 loose in the RTTI. For your convenience, the @code{basic} class provides
5156 a constructor that takes a @code{tinfo_key} value, which we are using here
5157 (remember that in our case @code{inherited = basic}). If the superclass
5158 didn't have such a constructor, we would have to set the @code{tinfo_key}
5159 to the right value manually.
5161 In the default constructor you should set all other member variables to
5162 reasonable default values (we don't need that here since our @code{str}
5163 member gets set to an empty string automatically). The constructor(s) are of
5164 course also the right place to allocate any dynamic resources you require.
5166 Next, the @code{destroy()} function:
5169 void mystring::destroy(bool call_parent)
5171 // free dynamically allocated resources here if required
5173 inherited::destroy(call_parent);
5177 This function is where we free all dynamically allocated resources. We
5178 don't have any so we're not doing anything here, but if we had, for
5179 example, used a C-style @code{char *} to store our string, this would be
5180 the place to @code{delete[]} the string storage. If @code{call_parent}
5181 is true, we have to call the @code{destroy()} function of the superclass
5182 after we're done (to mimic C++'s automatic invocation of superclass
5183 destructors where @code{destroy()} is called from outside a destructor).
5185 The @code{copy()} function just copies over the member variables from
5189 void mystring::copy(const mystring &other)
5191 inherited::copy(other);
5196 We can simply overwrite the member variables here. There's no need to worry
5197 about dynamically allocated storage. The assignment operator (which is
5198 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
5199 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
5200 explicitly call the @code{copy()} function of the superclass here so
5201 all the member variables will get copied.
5203 Next are the three functions for archiving. You have to implement them even
5204 if you don't plan to use archives, but the minimum required implementation
5205 is really simple. First, the archiving function:
5208 void mystring::archive(archive_node &n) const
5210 inherited::archive(n);
5211 n.add_string("string", str);
5215 The only thing that is really required is calling the @code{archive()}
5216 function of the superclass. Optionally, you can store all information you
5217 deem necessary for representing the object into the passed
5218 @code{archive_node}. We are just storing our string here. For more
5219 information on how the archiving works, consult the @file{archive.h} header
5222 The unarchiving constructor is basically the inverse of the archiving
5226 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
5228 n.find_string("string", str);
5232 If you don't need archiving, just leave this function empty (but you must
5233 invoke the unarchiving constructor of the superclass). Note that we don't
5234 have to set the @code{tinfo_key} here because it is done automatically
5235 by the unarchiving constructor of the @code{basic} class.
5237 Finally, the unarchiving function:
5240 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
5242 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
5246 You don't have to understand how exactly this works. Just copy these
5247 four lines into your code literally (replacing the class name, of
5248 course). It calls the unarchiving constructor of the class and unless
5249 you are doing something very special (like matching @code{archive_node}s
5250 to global objects) you don't need a different implementation. For those
5251 who are interested: setting the @code{dynallocated} flag puts the object
5252 under the control of GiNaC's garbage collection. It will get deleted
5253 automatically once it is no longer referenced.
5255 Our @code{compare_same_type()} function uses a provided function to compare
5259 int mystring::compare_same_type(const basic &other) const
5261 const mystring &o = static_cast<const mystring &>(other);
5262 int cmpval = str.compare(o.str);
5265 else if (cmpval < 0)
5272 Although this function takes a @code{basic &}, it will always be a reference
5273 to an object of exactly the same class (objects of different classes are not
5274 comparable), so the cast is safe. If this function returns 0, the two objects
5275 are considered equal (in the sense that @math{A-B=0}), so you should compare
5276 all relevant member variables.
5278 Now the only thing missing is our two new constructors:
5281 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
5283 // dynamically allocate resources here if required
5286 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
5288 // dynamically allocate resources here if required
5292 No surprises here. We set the @code{str} member from the argument and
5293 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
5295 That's it! We now have a minimal working GiNaC class that can store
5296 strings in algebraic expressions. Let's confirm that the RTTI works:
5299 ex e = mystring("Hello, world!");
5300 cout << is_a<mystring>(e) << endl;
5303 cout << e.bp->class_name() << endl;
5307 Obviously it does. Let's see what the expression @code{e} looks like:
5311 // -> [mystring object]
5314 Hm, not exactly what we expect, but of course the @code{mystring} class
5315 doesn't yet know how to print itself. This is done in the @code{print()}
5316 member function. Let's say that we wanted to print the string surrounded
5320 class mystring : public basic
5324 void print(const print_context &c, unsigned level = 0) const;
5328 void mystring::print(const print_context &c, unsigned level) const
5330 // print_context::s is a reference to an ostream
5331 c.s << '\"' << str << '\"';
5335 The @code{level} argument is only required for container classes to
5336 correctly parenthesize the output. Let's try again to print the expression:
5340 // -> "Hello, world!"
5343 Much better. The @code{mystring} class can be used in arbitrary expressions:
5346 e += mystring("GiNaC rulez");
5348 // -> "GiNaC rulez"+"Hello, world!"
5351 (GiNaC's automatic term reordering is in effect here), or even
5354 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
5356 // -> "One string"^(2*sin(-"Another string"+Pi))
5359 Whether this makes sense is debatable but remember that this is only an
5360 example. At least it allows you to implement your own symbolic algorithms
5363 Note that GiNaC's algebraic rules remain unchanged:
5366 e = mystring("Wow") * mystring("Wow");
5370 e = pow(mystring("First")-mystring("Second"), 2);
5371 cout << e.expand() << endl;
5372 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5375 There's no way to, for example, make GiNaC's @code{add} class perform string
5376 concatenation. You would have to implement this yourself.
5378 @subsection Automatic evaluation
5381 @cindex @code{eval()}
5382 @cindex @code{hold()}
5383 When dealing with objects that are just a little more complicated than the
5384 simple string objects we have implemented, chances are that you will want to
5385 have some automatic simplifications or canonicalizations performed on them.
5386 This is done in the evaluation member function @code{eval()}. Let's say that
5387 we wanted all strings automatically converted to lowercase with
5388 non-alphabetic characters stripped, and empty strings removed:
5391 class mystring : public basic
5395 ex eval(int level = 0) const;
5399 ex mystring::eval(int level) const
5402 for (int i=0; i<str.length(); i++) @{
5404 if (c >= 'A' && c <= 'Z')
5405 new_str += tolower(c);
5406 else if (c >= 'a' && c <= 'z')
5410 if (new_str.length() == 0)
5413 return mystring(new_str).hold();
5417 The @code{level} argument is used to limit the recursion depth of the
5418 evaluation. We don't have any subexpressions in the @code{mystring}
5419 class so we are not concerned with this. If we had, we would call the
5420 @code{eval()} functions of the subexpressions with @code{level - 1} as
5421 the argument if @code{level != 1}. The @code{hold()} member function
5422 sets a flag in the object that prevents further evaluation. Otherwise
5423 we might end up in an endless loop. When you want to return the object
5424 unmodified, use @code{return this->hold();}.
5426 Let's confirm that it works:
5429 ex e = mystring("Hello, world!") + mystring("!?#");
5433 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5438 @subsection Other member functions
5440 We have implemented only a small set of member functions to make the class
5441 work in the GiNaC framework. For a real algebraic class, there are probably
5442 some more functions that you might want to re-implement:
5445 bool info(unsigned inf) const;
5446 ex evalf(int level = 0) const;
5447 ex series(const relational & r, int order, unsigned options = 0) const;
5448 ex derivative(const symbol & s) const;
5451 If your class stores sub-expressions you will probably want to override
5453 @cindex @code{let_op()}
5455 unsigned nops() cont;
5458 ex map(map_function & f) const;
5459 ex subs(const lst & ls, const lst & lr, bool no_pattern = false) const;
5462 @code{let_op()} is a variant of @code{op()} that allows write access. The
5463 default implementation of @code{map()} uses it, so you have to implement
5464 either @code{let_op()} or @code{map()}.
5466 If your class stores any data that is not accessible via @code{op()}, you
5467 should also implement
5469 @cindex @code{calchash()}
5471 unsigned calchash(void) const;
5474 This function returns an @code{unsigned} hash value for the object which
5475 will allow GiNaC to compare and canonicalize expressions much more
5476 efficiently. You should consult the implementation of some of the built-in
5477 GiNaC classes for examples of hash functions.
5479 You can, of course, also add your own new member functions. Remember
5480 that the RTTI may be used to get information about what kinds of objects
5481 you are dealing with (the position in the class hierarchy) and that you
5482 can always extract the bare object from an @code{ex} by stripping the
5483 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5484 should become a need.
5486 That's it. May the source be with you!
5489 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5490 @c node-name, next, previous, up
5491 @chapter A Comparison With Other CAS
5494 This chapter will give you some information on how GiNaC compares to
5495 other, traditional Computer Algebra Systems, like @emph{Maple},
5496 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5497 disadvantages over these systems.
5500 * Advantages:: Strengths of the GiNaC approach.
5501 * Disadvantages:: Weaknesses of the GiNaC approach.
5502 * Why C++?:: Attractiveness of C++.
5505 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5506 @c node-name, next, previous, up
5509 GiNaC has several advantages over traditional Computer
5510 Algebra Systems, like
5515 familiar language: all common CAS implement their own proprietary
5516 grammar which you have to learn first (and maybe learn again when your
5517 vendor decides to `enhance' it). With GiNaC you can write your program
5518 in common C++, which is standardized.
5522 structured data types: you can build up structured data types using
5523 @code{struct}s or @code{class}es together with STL features instead of
5524 using unnamed lists of lists of lists.
5527 strongly typed: in CAS, you usually have only one kind of variables
5528 which can hold contents of an arbitrary type. This 4GL like feature is
5529 nice for novice programmers, but dangerous.
5532 development tools: powerful development tools exist for C++, like fancy
5533 editors (e.g. with automatic indentation and syntax highlighting),
5534 debuggers, visualization tools, documentation generators@dots{}
5537 modularization: C++ programs can easily be split into modules by
5538 separating interface and implementation.
5541 price: GiNaC is distributed under the GNU Public License which means
5542 that it is free and available with source code. And there are excellent
5543 C++-compilers for free, too.
5546 extendable: you can add your own classes to GiNaC, thus extending it on
5547 a very low level. Compare this to a traditional CAS that you can
5548 usually only extend on a high level by writing in the language defined
5549 by the parser. In particular, it turns out to be almost impossible to
5550 fix bugs in a traditional system.
5553 multiple interfaces: Though real GiNaC programs have to be written in
5554 some editor, then be compiled, linked and executed, there are more ways
5555 to work with the GiNaC engine. Many people want to play with
5556 expressions interactively, as in traditional CASs. Currently, two such
5557 windows into GiNaC have been implemented and many more are possible: the
5558 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5559 types to a command line and second, as a more consistent approach, an
5560 interactive interface to the Cint C++ interpreter has been put together
5561 (called GiNaC-cint) that allows an interactive scripting interface
5562 consistent with the C++ language. It is available from the usual GiNaC
5566 seamless integration: it is somewhere between difficult and impossible
5567 to call CAS functions from within a program written in C++ or any other
5568 programming language and vice versa. With GiNaC, your symbolic routines
5569 are part of your program. You can easily call third party libraries,
5570 e.g. for numerical evaluation or graphical interaction. All other
5571 approaches are much more cumbersome: they range from simply ignoring the
5572 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5573 system (i.e. @emph{Yacas}).
5576 efficiency: often large parts of a program do not need symbolic
5577 calculations at all. Why use large integers for loop variables or
5578 arbitrary precision arithmetics where @code{int} and @code{double} are
5579 sufficient? For pure symbolic applications, GiNaC is comparable in
5580 speed with other CAS.
5585 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5586 @c node-name, next, previous, up
5587 @section Disadvantages
5589 Of course it also has some disadvantages:
5594 advanced features: GiNaC cannot compete with a program like
5595 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5596 which grows since 1981 by the work of dozens of programmers, with
5597 respect to mathematical features. Integration, factorization,
5598 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5599 not planned for the near future).
5602 portability: While the GiNaC library itself is designed to avoid any
5603 platform dependent features (it should compile on any ANSI compliant C++
5604 compiler), the currently used version of the CLN library (fast large
5605 integer and arbitrary precision arithmetics) can only by compiled
5606 without hassle on systems with the C++ compiler from the GNU Compiler
5607 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
5608 macros to let the compiler gather all static initializations, which
5609 works for GNU C++ only. Feel free to contact the authors in case you
5610 really believe that you need to use a different compiler. We have
5611 occasionally used other compilers and may be able to give you advice.}
5612 GiNaC uses recent language features like explicit constructors, mutable
5613 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
5614 literally. Recent GCC versions starting at 2.95.3, although itself not
5615 yet ANSI compliant, support all needed features.
5620 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5621 @c node-name, next, previous, up
5624 Why did we choose to implement GiNaC in C++ instead of Java or any other
5625 language? C++ is not perfect: type checking is not strict (casting is
5626 possible), separation between interface and implementation is not
5627 complete, object oriented design is not enforced. The main reason is
5628 the often scolded feature of operator overloading in C++. While it may
5629 be true that operating on classes with a @code{+} operator is rarely
5630 meaningful, it is perfectly suited for algebraic expressions. Writing
5631 @math{3x+5y} as @code{3*x+5*y} instead of
5632 @code{x.times(3).plus(y.times(5))} looks much more natural.
5633 Furthermore, the main developers are more familiar with C++ than with
5634 any other programming language.
5637 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5638 @c node-name, next, previous, up
5639 @appendix Internal Structures
5642 * Expressions are reference counted::
5643 * Internal representation of products and sums::
5646 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5647 @c node-name, next, previous, up
5648 @appendixsection Expressions are reference counted
5650 @cindex reference counting
5651 @cindex copy-on-write
5652 @cindex garbage collection
5653 An expression is extremely light-weight since internally it works like a
5654 handle to the actual representation and really holds nothing more than a
5655 pointer to some other object. What this means in practice is that
5656 whenever you create two @code{ex} and set the second equal to the first
5657 no copying process is involved. Instead, the copying takes place as soon
5658 as you try to change the second. Consider the simple sequence of code:
5662 #include <ginac/ginac.h>
5663 using namespace std;
5664 using namespace GiNaC;
5668 symbol x("x"), y("y"), z("z");
5671 e1 = sin(x + 2*y) + 3*z + 41;
5672 e2 = e1; // e2 points to same object as e1
5673 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5674 e2 += 1; // e2 is copied into a new object
5675 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5679 The line @code{e2 = e1;} creates a second expression pointing to the
5680 object held already by @code{e1}. The time involved for this operation
5681 is therefore constant, no matter how large @code{e1} was. Actual
5682 copying, however, must take place in the line @code{e2 += 1;} because
5683 @code{e1} and @code{e2} are not handles for the same object any more.
5684 This concept is called @dfn{copy-on-write semantics}. It increases
5685 performance considerably whenever one object occurs multiple times and
5686 represents a simple garbage collection scheme because when an @code{ex}
5687 runs out of scope its destructor checks whether other expressions handle
5688 the object it points to too and deletes the object from memory if that
5689 turns out not to be the case. A slightly less trivial example of
5690 differentiation using the chain-rule should make clear how powerful this
5695 symbol x("x"), y("y");
5699 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5700 cout << e1 << endl // prints x+3*y
5701 << e2 << endl // prints (x+3*y)^3
5702 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5706 Here, @code{e1} will actually be referenced three times while @code{e2}
5707 will be referenced two times. When the power of an expression is built,
5708 that expression needs not be copied. Likewise, since the derivative of
5709 a power of an expression can be easily expressed in terms of that
5710 expression, no copying of @code{e1} is involved when @code{e3} is
5711 constructed. So, when @code{e3} is constructed it will print as
5712 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5713 holds a reference to @code{e2} and the factor in front is just
5716 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5717 semantics. When you insert an expression into a second expression, the
5718 result behaves exactly as if the contents of the first expression were
5719 inserted. But it may be useful to remember that this is not what
5720 happens. Knowing this will enable you to write much more efficient
5721 code. If you still have an uncertain feeling with copy-on-write
5722 semantics, we recommend you have a look at the
5723 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
5724 Marshall Cline. Chapter 16 covers this issue and presents an
5725 implementation which is pretty close to the one in GiNaC.
5728 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5729 @c node-name, next, previous, up
5730 @appendixsection Internal representation of products and sums
5732 @cindex representation
5735 @cindex @code{power}
5736 Although it should be completely transparent for the user of
5737 GiNaC a short discussion of this topic helps to understand the sources
5738 and also explain performance to a large degree. Consider the
5739 unexpanded symbolic expression
5741 $2d^3 \left( 4a + 5b - 3 \right)$
5744 @math{2*d^3*(4*a+5*b-3)}
5746 which could naively be represented by a tree of linear containers for
5747 addition and multiplication, one container for exponentiation with base
5748 and exponent and some atomic leaves of symbols and numbers in this
5753 @cindex pair-wise representation
5754 However, doing so results in a rather deeply nested tree which will
5755 quickly become inefficient to manipulate. We can improve on this by
5756 representing the sum as a sequence of terms, each one being a pair of a
5757 purely numeric multiplicative coefficient and its rest. In the same
5758 spirit we can store the multiplication as a sequence of terms, each
5759 having a numeric exponent and a possibly complicated base, the tree
5760 becomes much more flat:
5764 The number @code{3} above the symbol @code{d} shows that @code{mul}
5765 objects are treated similarly where the coefficients are interpreted as
5766 @emph{exponents} now. Addition of sums of terms or multiplication of
5767 products with numerical exponents can be coded to be very efficient with
5768 such a pair-wise representation. Internally, this handling is performed
5769 by most CAS in this way. It typically speeds up manipulations by an
5770 order of magnitude. The overall multiplicative factor @code{2} and the
5771 additive term @code{-3} look somewhat out of place in this
5772 representation, however, since they are still carrying a trivial
5773 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5774 this is avoided by adding a field that carries an overall numeric
5775 coefficient. This results in the realistic picture of internal
5778 $2d^3 \left( 4a + 5b - 3 \right)$:
5781 @math{2*d^3*(4*a+5*b-3)}:
5787 This also allows for a better handling of numeric radicals, since
5788 @code{sqrt(2)} can now be carried along calculations. Now it should be
5789 clear, why both classes @code{add} and @code{mul} are derived from the
5790 same abstract class: the data representation is the same, only the
5791 semantics differs. In the class hierarchy, methods for polynomial
5792 expansion and the like are reimplemented for @code{add} and @code{mul},
5793 but the data structure is inherited from @code{expairseq}.
5796 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5797 @c node-name, next, previous, up
5798 @appendix Package Tools
5800 If you are creating a software package that uses the GiNaC library,
5801 setting the correct command line options for the compiler and linker
5802 can be difficult. GiNaC includes two tools to make this process easier.
5805 * ginac-config:: A shell script to detect compiler and linker flags.
5806 * AM_PATH_GINAC:: Macro for GNU automake.
5810 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5811 @c node-name, next, previous, up
5812 @section @command{ginac-config}
5813 @cindex ginac-config
5815 @command{ginac-config} is a shell script that you can use to determine
5816 the compiler and linker command line options required to compile and
5817 link a program with the GiNaC library.
5819 @command{ginac-config} takes the following flags:
5823 Prints out the version of GiNaC installed.
5825 Prints '-I' flags pointing to the installed header files.
5827 Prints out the linker flags necessary to link a program against GiNaC.
5828 @item --prefix[=@var{PREFIX}]
5829 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5830 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5831 Otherwise, prints out the configured value of @env{$prefix}.
5832 @item --exec-prefix[=@var{PREFIX}]
5833 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5834 Otherwise, prints out the configured value of @env{$exec_prefix}.
5837 Typically, @command{ginac-config} will be used within a configure
5838 script, as described below. It, however, can also be used directly from
5839 the command line using backquotes to compile a simple program. For
5843 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5846 This command line might expand to (for example):
5849 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5850 -lginac -lcln -lstdc++
5853 Not only is the form using @command{ginac-config} easier to type, it will
5854 work on any system, no matter how GiNaC was configured.
5857 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5858 @c node-name, next, previous, up
5859 @section @samp{AM_PATH_GINAC}
5860 @cindex AM_PATH_GINAC
5862 For packages configured using GNU automake, GiNaC also provides
5863 a macro to automate the process of checking for GiNaC.
5866 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5874 Determines the location of GiNaC using @command{ginac-config}, which is
5875 either found in the user's path, or from the environment variable
5876 @env{GINACLIB_CONFIG}.
5879 Tests the installed libraries to make sure that their version
5880 is later than @var{MINIMUM-VERSION}. (A default version will be used
5884 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5885 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5886 variable to the output of @command{ginac-config --libs}, and calls
5887 @samp{AC_SUBST()} for these variables so they can be used in generated
5888 makefiles, and then executes @var{ACTION-IF-FOUND}.
5891 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5892 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5896 This macro is in file @file{ginac.m4} which is installed in
5897 @file{$datadir/aclocal}. Note that if automake was installed with a
5898 different @samp{--prefix} than GiNaC, you will either have to manually
5899 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5900 aclocal the @samp{-I} option when running it.
5903 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5904 * Example package:: Example of a package using AM_PATH_GINAC.
5908 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5909 @c node-name, next, previous, up
5910 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5912 Simply make sure that @command{ginac-config} is in your path, and run
5913 the configure script.
5920 The directory where the GiNaC libraries are installed needs
5921 to be found by your system's dynamic linker.
5923 This is generally done by
5926 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5932 setting the environment variable @env{LD_LIBRARY_PATH},
5935 or, as a last resort,
5938 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5939 running configure, for instance:
5942 LDFLAGS=-R/home/cbauer/lib ./configure
5947 You can also specify a @command{ginac-config} not in your path by
5948 setting the @env{GINACLIB_CONFIG} environment variable to the
5949 name of the executable
5952 If you move the GiNaC package from its installed location,
5953 you will either need to modify @command{ginac-config} script
5954 manually to point to the new location or rebuild GiNaC.
5965 --with-ginac-prefix=@var{PREFIX}
5966 --with-ginac-exec-prefix=@var{PREFIX}
5969 are provided to override the prefix and exec-prefix that were stored
5970 in the @command{ginac-config} shell script by GiNaC's configure. You are
5971 generally better off configuring GiNaC with the right path to begin with.
5975 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5976 @c node-name, next, previous, up
5977 @subsection Example of a package using @samp{AM_PATH_GINAC}
5979 The following shows how to build a simple package using automake
5980 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5983 #include <ginac/ginac.h>
5987 GiNaC::symbol x("x");
5988 GiNaC::ex a = GiNaC::sin(x);
5989 std::cout << "Derivative of " << a
5990 << " is " << a.diff(x) << std::endl;
5995 You should first read the introductory portions of the automake
5996 Manual, if you are not already familiar with it.
5998 Two files are needed, @file{configure.in}, which is used to build the
6002 dnl Process this file with autoconf to produce a configure script.
6004 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
6010 AM_PATH_GINAC(0.9.0, [
6011 LIBS="$LIBS $GINACLIB_LIBS"
6012 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
6013 ], AC_MSG_ERROR([need to have GiNaC installed]))
6018 The only command in this which is not standard for automake
6019 is the @samp{AM_PATH_GINAC} macro.
6021 That command does the following: If a GiNaC version greater or equal
6022 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
6023 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
6024 the error message `need to have GiNaC installed'
6026 And the @file{Makefile.am}, which will be used to build the Makefile.
6029 ## Process this file with automake to produce Makefile.in
6030 bin_PROGRAMS = simple
6031 simple_SOURCES = simple.cpp
6034 This @file{Makefile.am}, says that we are building a single executable,
6035 from a single source file @file{simple.cpp}. Since every program
6036 we are building uses GiNaC we simply added the GiNaC options
6037 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
6038 want to specify them on a per-program basis: for instance by
6042 simple_LDADD = $(GINACLIB_LIBS)
6043 INCLUDES = $(GINACLIB_CPPFLAGS)
6046 to the @file{Makefile.am}.
6048 To try this example out, create a new directory and add the three
6051 Now execute the following commands:
6054 $ automake --add-missing
6059 You now have a package that can be built in the normal fashion
6068 @node Bibliography, Concept Index, Example package, Top
6069 @c node-name, next, previous, up
6070 @appendix Bibliography
6075 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
6078 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
6081 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
6084 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
6087 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
6088 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
6091 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
6092 James H. Davenport, Yvon Siret, and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
6093 Academic Press, London
6096 @cite{Computer Algebra Systems - A Practical Guide},
6097 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
6100 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
6101 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
6104 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
6109 @node Concept Index, , Bibliography, Top
6110 @c node-name, next, previous, up
6111 @unnumbered Concept Index