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{x-y}) but allows for more efficient operation and usually yields
777 some immediate simplifications.
779 @cindex @code{eval()}
780 Internally, the anonymous evaluator in GiNaC is implemented by the methods
783 ex ex::eval(int level = 0) const;
784 ex basic::eval(int level = 0) const;
787 but unless you are extending GiNaC with your own classes or functions, there
788 should never be any reason to call them explicitly. All GiNaC methods that
789 transform expressions, like @code{subs()} or @code{normal()}, automatically
790 re-evaluate their results.
793 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
794 @c node-name, next, previous, up
795 @section Error handling
797 @cindex @code{pole_error} (class)
799 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
800 generated by GiNaC are subclassed from the standard @code{exception} class
801 defined in the @file{<stdexcept>} header. In addition to the predefined
802 @code{logic_error}, @code{domain_error}, @code{out_of_range},
803 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
804 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
805 exception that gets thrown when trying to evaluate a mathematical function
808 The @code{pole_error} class has a member function
811 int pole_error::degree() const;
814 that returns the order of the singularity (or 0 when the pole is
815 logarithmic or the order is undefined).
817 When using GiNaC it is useful to arrange for exceptions to be catched in
818 the main program even if you don't want to do any special error handling.
819 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
820 default exception handler of your C++ compiler's run-time system which
821 usually only aborts the program without giving any information what went
824 Here is an example for a @code{main()} function that catches and prints
825 exceptions generated by GiNaC:
830 #include <ginac/ginac.h>
832 using namespace GiNaC;
840 @} catch (exception &p) @{
841 cerr << p.what() << endl;
849 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
850 @c node-name, next, previous, up
851 @section The Class Hierarchy
853 GiNaC's class hierarchy consists of several classes representing
854 mathematical objects, all of which (except for @code{ex} and some
855 helpers) are internally derived from one abstract base class called
856 @code{basic}. You do not have to deal with objects of class
857 @code{basic}, instead you'll be dealing with symbols, numbers,
858 containers of expressions and so on.
862 To get an idea about what kinds of symbolic composites may be built we
863 have a look at the most important classes in the class hierarchy and
864 some of the relations among the classes:
866 @image{classhierarchy}
868 The abstract classes shown here (the ones without drop-shadow) are of no
869 interest for the user. They are used internally in order to avoid code
870 duplication if two or more classes derived from them share certain
871 features. An example is @code{expairseq}, a container for a sequence of
872 pairs each consisting of one expression and a number (@code{numeric}).
873 What @emph{is} visible to the user are the derived classes @code{add}
874 and @code{mul}, representing sums and products. @xref{Internal
875 Structures}, where these two classes are described in more detail. The
876 following table shortly summarizes what kinds of mathematical objects
877 are stored in the different classes:
880 @multitable @columnfractions .22 .78
881 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
882 @item @code{constant} @tab Constants like
889 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
890 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
891 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
892 @item @code{ncmul} @tab Products of non-commutative objects
893 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
898 @code{sqrt(}@math{2}@code{)}
901 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
902 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
903 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
904 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
905 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
906 @item @code{indexed} @tab Indexed object like @math{A_ij}
907 @item @code{tensor} @tab Special tensor like the delta and metric tensors
908 @item @code{idx} @tab Index of an indexed object
909 @item @code{varidx} @tab Index with variance
910 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
911 @item @code{wildcard} @tab Wildcard for pattern matching
912 @item @code{structure} @tab Template for user-defined classes
917 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
918 @c node-name, next, previous, up
920 @cindex @code{symbol} (class)
921 @cindex hierarchy of classes
924 Symbols are for symbolic manipulation what atoms are for chemistry. You
925 can declare objects of class @code{symbol} as any other object simply by
926 saying @code{symbol x,y;}. There is, however, a catch in here having to
927 do with the fact that C++ is a compiled language. The information about
928 the symbol's name is thrown away by the compiler but at a later stage
929 you may want to print expressions holding your symbols. In order to
930 avoid confusion GiNaC's symbols are able to know their own name. This
931 is accomplished by declaring its name for output at construction time in
932 the fashion @code{symbol x("x");}. If you declare a symbol using the
933 default constructor (i.e. without string argument) the system will deal
934 out a unique name. That name may not be suitable for printing but for
935 internal routines when no output is desired it is often enough. We'll
936 come across examples of such symbols later in this tutorial.
938 This implies that the strings passed to symbols at construction time may
939 not be used for comparing two of them. It is perfectly legitimate to
940 write @code{symbol x("x"),y("x");} but it is likely to lead into
941 trouble. Here, @code{x} and @code{y} are different symbols and
942 statements like @code{x-y} will not be simplified to zero although the
943 output @code{x-x} looks funny. Such output may also occur when there
944 are two different symbols in two scopes, for instance when you call a
945 function that declares a symbol with a name already existent in a symbol
946 in the calling function. Again, comparing them (using @code{operator==}
947 for instance) will always reveal their difference. Watch out, please.
949 @cindex @code{subs()}
950 Although symbols can be assigned expressions for internal reasons, you
951 should not do it (and we are not going to tell you how it is done). If
952 you want to replace a symbol with something else in an expression, you
953 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
956 @node Numbers, Constants, Symbols, Basic Concepts
957 @c node-name, next, previous, up
959 @cindex @code{numeric} (class)
965 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
966 The classes therein serve as foundation classes for GiNaC. CLN stands
967 for Class Library for Numbers or alternatively for Common Lisp Numbers.
968 In order to find out more about CLN's internals, the reader is referred to
969 the documentation of that library. @inforef{Introduction, , cln}, for
970 more information. Suffice to say that it is by itself build on top of
971 another library, the GNU Multiple Precision library GMP, which is an
972 extremely fast library for arbitrary long integers and rationals as well
973 as arbitrary precision floating point numbers. It is very commonly used
974 by several popular cryptographic applications. CLN extends GMP by
975 several useful things: First, it introduces the complex number field
976 over either reals (i.e. floating point numbers with arbitrary precision)
977 or rationals. Second, it automatically converts rationals to integers
978 if the denominator is unity and complex numbers to real numbers if the
979 imaginary part vanishes and also correctly treats algebraic functions.
980 Third it provides good implementations of state-of-the-art algorithms
981 for all trigonometric and hyperbolic functions as well as for
982 calculation of some useful constants.
984 The user can construct an object of class @code{numeric} in several
985 ways. The following example shows the four most important constructors.
986 It uses construction from C-integer, construction of fractions from two
987 integers, construction from C-float and construction from a string:
991 #include <ginac/ginac.h>
992 using namespace GiNaC;
996 numeric two = 2; // exact integer 2
997 numeric r(2,3); // exact fraction 2/3
998 numeric e(2.71828); // floating point number
999 numeric p = "3.14159265358979323846"; // constructor from string
1000 // Trott's constant in scientific notation:
1001 numeric trott("1.0841015122311136151E-2");
1003 std::cout << two*p << std::endl; // floating point 6.283...
1008 @cindex complex numbers
1009 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1014 numeric z1 = 2-3*I; // exact complex number 2-3i
1015 numeric z2 = 5.9+1.6*I; // complex floating point number
1019 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1020 This would, however, call C's built-in operator @code{/} for integers
1021 first and result in a numeric holding a plain integer 1. @strong{Never
1022 use the operator @code{/} on integers} unless you know exactly what you
1023 are doing! Use the constructor from two integers instead, as shown in
1024 the example above. Writing @code{numeric(1)/2} may look funny but works
1027 @cindex @code{Digits}
1029 We have seen now the distinction between exact numbers and floating
1030 point numbers. Clearly, the user should never have to worry about
1031 dynamically created exact numbers, since their `exactness' always
1032 determines how they ought to be handled, i.e. how `long' they are. The
1033 situation is different for floating point numbers. Their accuracy is
1034 controlled by one @emph{global} variable, called @code{Digits}. (For
1035 those readers who know about Maple: it behaves very much like Maple's
1036 @code{Digits}). All objects of class numeric that are constructed from
1037 then on will be stored with a precision matching that number of decimal
1042 #include <ginac/ginac.h>
1043 using namespace std;
1044 using namespace GiNaC;
1048 numeric three(3.0), one(1.0);
1049 numeric x = one/three;
1051 cout << "in " << Digits << " digits:" << endl;
1053 cout << Pi.evalf() << endl;
1065 The above example prints the following output to screen:
1069 0.33333333333333333334
1070 3.1415926535897932385
1072 0.33333333333333333333333333333333333333333333333333333333333333333334
1073 3.1415926535897932384626433832795028841971693993751058209749445923078
1077 Note that the last number is not necessarily rounded as you would
1078 naively expect it to be rounded in the decimal system. But note also,
1079 that in both cases you got a couple of extra digits. This is because
1080 numbers are internally stored by CLN as chunks of binary digits in order
1081 to match your machine's word size and to not waste precision. Thus, on
1082 architectures with different word size, the above output might even
1083 differ with regard to actually computed digits.
1085 It should be clear that objects of class @code{numeric} should be used
1086 for constructing numbers or for doing arithmetic with them. The objects
1087 one deals with most of the time are the polymorphic expressions @code{ex}.
1089 @subsection Tests on numbers
1091 Once you have declared some numbers, assigned them to expressions and
1092 done some arithmetic with them it is frequently desired to retrieve some
1093 kind of information from them like asking whether that number is
1094 integer, rational, real or complex. For those cases GiNaC provides
1095 several useful methods. (Internally, they fall back to invocations of
1096 certain CLN functions.)
1098 As an example, let's construct some rational number, multiply it with
1099 some multiple of its denominator and test what comes out:
1103 #include <ginac/ginac.h>
1104 using namespace std;
1105 using namespace GiNaC;
1107 // some very important constants:
1108 const numeric twentyone(21);
1109 const numeric ten(10);
1110 const numeric five(5);
1114 numeric answer = twentyone;
1117 cout << answer.is_integer() << endl; // false, it's 21/5
1119 cout << answer.is_integer() << endl; // true, it's 42 now!
1123 Note that the variable @code{answer} is constructed here as an integer
1124 by @code{numeric}'s copy constructor but in an intermediate step it
1125 holds a rational number represented as integer numerator and integer
1126 denominator. When multiplied by 10, the denominator becomes unity and
1127 the result is automatically converted to a pure integer again.
1128 Internally, the underlying CLN is responsible for this behavior and we
1129 refer the reader to CLN's documentation. Suffice to say that
1130 the same behavior applies to complex numbers as well as return values of
1131 certain functions. Complex numbers are automatically converted to real
1132 numbers if the imaginary part becomes zero. The full set of tests that
1133 can be applied is listed in the following table.
1136 @multitable @columnfractions .30 .70
1137 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1138 @item @code{.is_zero()}
1139 @tab @dots{}equal to zero
1140 @item @code{.is_positive()}
1141 @tab @dots{}not complex and greater than 0
1142 @item @code{.is_integer()}
1143 @tab @dots{}a (non-complex) integer
1144 @item @code{.is_pos_integer()}
1145 @tab @dots{}an integer and greater than 0
1146 @item @code{.is_nonneg_integer()}
1147 @tab @dots{}an integer and greater equal 0
1148 @item @code{.is_even()}
1149 @tab @dots{}an even integer
1150 @item @code{.is_odd()}
1151 @tab @dots{}an odd integer
1152 @item @code{.is_prime()}
1153 @tab @dots{}a prime integer (probabilistic primality test)
1154 @item @code{.is_rational()}
1155 @tab @dots{}an exact rational number (integers are rational, too)
1156 @item @code{.is_real()}
1157 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1158 @item @code{.is_cinteger()}
1159 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1160 @item @code{.is_crational()}
1161 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1165 @subsection Converting numbers
1167 Sometimes it is desirable to convert a @code{numeric} object back to a
1168 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1169 class provides a couple of methods for this purpose:
1171 @cindex @code{to_int()}
1172 @cindex @code{to_long()}
1173 @cindex @code{to_double()}
1174 @cindex @code{to_cl_N()}
1176 int numeric::to_int() const;
1177 long numeric::to_long() const;
1178 double numeric::to_double() const;
1179 cln::cl_N numeric::to_cl_N() const;
1182 @code{to_int()} and @code{to_long()} only work when the number they are
1183 applied on is an exact integer. Otherwise the program will halt with a
1184 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1185 rational number will return a floating-point approximation. Both
1186 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1187 part of complex numbers.
1190 @node Constants, Fundamental containers, Numbers, Basic Concepts
1191 @c node-name, next, previous, up
1193 @cindex @code{constant} (class)
1196 @cindex @code{Catalan}
1197 @cindex @code{Euler}
1198 @cindex @code{evalf()}
1199 Constants behave pretty much like symbols except that they return some
1200 specific number when the method @code{.evalf()} is called.
1202 The predefined known constants are:
1205 @multitable @columnfractions .14 .30 .56
1206 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1208 @tab Archimedes' constant
1209 @tab 3.14159265358979323846264338327950288
1210 @item @code{Catalan}
1211 @tab Catalan's constant
1212 @tab 0.91596559417721901505460351493238411
1214 @tab Euler's (or Euler-Mascheroni) constant
1215 @tab 0.57721566490153286060651209008240243
1220 @node Fundamental containers, Lists, Constants, Basic Concepts
1221 @c node-name, next, previous, up
1222 @section Sums, products and powers
1226 @cindex @code{power}
1228 Simple rational expressions are written down in GiNaC pretty much like
1229 in other CAS or like expressions involving numerical variables in C.
1230 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1231 been overloaded to achieve this goal. When you run the following
1232 code snippet, the constructor for an object of type @code{mul} is
1233 automatically called to hold the product of @code{a} and @code{b} and
1234 then the constructor for an object of type @code{add} is called to hold
1235 the sum of that @code{mul} object and the number one:
1239 symbol a("a"), b("b");
1244 @cindex @code{pow()}
1245 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1246 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1247 construction is necessary since we cannot safely overload the constructor
1248 @code{^} in C++ to construct a @code{power} object. If we did, it would
1249 have several counterintuitive and undesired effects:
1253 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1255 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1256 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1257 interpret this as @code{x^(a^b)}.
1259 Also, expressions involving integer exponents are very frequently used,
1260 which makes it even more dangerous to overload @code{^} since it is then
1261 hard to distinguish between the semantics as exponentiation and the one
1262 for exclusive or. (It would be embarrassing to return @code{1} where one
1263 has requested @code{2^3}.)
1266 @cindex @command{ginsh}
1267 All effects are contrary to mathematical notation and differ from the
1268 way most other CAS handle exponentiation, therefore overloading @code{^}
1269 is ruled out for GiNaC's C++ part. The situation is different in
1270 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1271 that the other frequently used exponentiation operator @code{**} does
1272 not exist at all in C++).
1274 To be somewhat more precise, objects of the three classes described
1275 here, are all containers for other expressions. An object of class
1276 @code{power} is best viewed as a container with two slots, one for the
1277 basis, one for the exponent. All valid GiNaC expressions can be
1278 inserted. However, basic transformations like simplifying
1279 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1280 when this is mathematically possible. If we replace the outer exponent
1281 three in the example by some symbols @code{a}, the simplification is not
1282 safe and will not be performed, since @code{a} might be @code{1/2} and
1285 Objects of type @code{add} and @code{mul} are containers with an
1286 arbitrary number of slots for expressions to be inserted. Again, simple
1287 and safe simplifications are carried out like transforming
1288 @code{3*x+4-x} to @code{2*x+4}.
1291 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1292 @c node-name, next, previous, up
1293 @section Lists of expressions
1294 @cindex @code{lst} (class)
1296 @cindex @code{nops()}
1298 @cindex @code{append()}
1299 @cindex @code{prepend()}
1300 @cindex @code{remove_first()}
1301 @cindex @code{remove_last()}
1302 @cindex @code{remove_all()}
1304 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1305 expressions. They are not as ubiquitous as in many other computer algebra
1306 packages, but are sometimes used to supply a variable number of arguments of
1307 the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
1308 so you should have a basic understanding of them.
1310 Lists of up to 16 expressions can be directly constructed from single
1315 symbol x("x"), y("y");
1316 lst l(x, 2, y, x+y);
1317 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1321 Use the @code{nops()} method to determine the size (number of expressions) of
1322 a list and the @code{op()} method or the @code{[]} operator to access
1323 individual elements:
1327 cout << l.nops() << endl; // prints '4'
1328 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1332 As with the standard @code{list<T>} container, accessing random elements of a
1333 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1334 sequential access to the elements of a list is possible with the
1335 iterator types provided by the @code{lst} class:
1338 typedef ... lst::const_iterator;
1339 typedef ... lst::const_reverse_iterator;
1340 lst::const_iterator lst::begin() const;
1341 lst::const_iterator lst::end() const;
1342 lst::const_reverse_iterator lst::rbegin() const;
1343 lst::const_reverse_iterator lst::rend() const;
1346 For example, to print the elements of a list individually you can use:
1351 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1356 which is one order faster than
1361 for (size_t i = 0; i < l.nops(); ++i)
1362 cout << l.op(i) << endl;
1366 These iterators also allow you to use some of the algorithms provided by
1367 the C++ standard library:
1371 // print the elements of the list (requires #include <iterator>)
1372 copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1374 // sum up the elements of the list (requires #include <numeric>)
1375 ex sum = accumulate(l.begin(), l.end(), ex(0));
1376 cout << sum << endl; // prints '2+2*x+2*y'
1380 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1381 (the only other one is @code{matrix}). You can modify single elements:
1385 l[1] = 42; // l is now @{x, 42, y, x+y@}
1386 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1390 You can append or prepend an expression to a list with the @code{append()}
1391 and @code{prepend()} methods:
1395 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1396 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1400 You can remove the first or last element of a list with @code{remove_first()}
1401 and @code{remove_last()}:
1405 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1406 l.remove_last(); // l is now @{x, 7, y, x+y@}
1410 You can remove all the elements of a list with @code{remove_all()}:
1414 l.remove_all(); // l is now empty
1418 You can bring the elements of a list into a canonical order with @code{sort()}:
1422 lst l1(x, 2, y, x+y);
1423 lst l2(2, x+y, x, y);
1426 // l1 and l2 are now equal
1430 Finally, you can remove all but the first element of consecutive groups of
1431 elements with @code{unique()}:
1435 lst l3(x, 2, 2, 2, y, x+y, y+x);
1436 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1441 @node Mathematical functions, Relations, Lists, Basic Concepts
1442 @c node-name, next, previous, up
1443 @section Mathematical functions
1444 @cindex @code{function} (class)
1445 @cindex trigonometric function
1446 @cindex hyperbolic function
1448 There are quite a number of useful functions hard-wired into GiNaC. For
1449 instance, all trigonometric and hyperbolic functions are implemented
1450 (@xref{Built-in Functions}, for a complete list).
1452 These functions (better called @emph{pseudofunctions}) are all objects
1453 of class @code{function}. They accept one or more expressions as
1454 arguments and return one expression. If the arguments are not
1455 numerical, the evaluation of the function may be halted, as it does in
1456 the next example, showing how a function returns itself twice and
1457 finally an expression that may be really useful:
1459 @cindex Gamma function
1460 @cindex @code{subs()}
1463 symbol x("x"), y("y");
1465 cout << tgamma(foo) << endl;
1466 // -> tgamma(x+(1/2)*y)
1467 ex bar = foo.subs(y==1);
1468 cout << tgamma(bar) << endl;
1470 ex foobar = bar.subs(x==7);
1471 cout << tgamma(foobar) << endl;
1472 // -> (135135/128)*Pi^(1/2)
1476 Besides evaluation most of these functions allow differentiation, series
1477 expansion and so on. Read the next chapter in order to learn more about
1480 It must be noted that these pseudofunctions are created by inline
1481 functions, where the argument list is templated. This means that
1482 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1483 @code{sin(ex(1))} and will therefore not result in a floating point
1484 number. Unless of course the function prototype is explicitly
1485 overridden -- which is the case for arguments of type @code{numeric}
1486 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1487 point number of class @code{numeric} you should call
1488 @code{sin(numeric(1))}. This is almost the same as calling
1489 @code{sin(1).evalf()} except that the latter will return a numeric
1490 wrapped inside an @code{ex}.
1493 @node Relations, Matrices, Mathematical functions, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{relational} (class)
1498 Sometimes, a relation holding between two expressions must be stored
1499 somehow. The class @code{relational} is a convenient container for such
1500 purposes. A relation is by definition a container for two @code{ex} and
1501 a relation between them that signals equality, inequality and so on.
1502 They are created by simply using the C++ operators @code{==}, @code{!=},
1503 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1505 @xref{Mathematical functions}, for examples where various applications
1506 of the @code{.subs()} method show how objects of class relational are
1507 used as arguments. There they provide an intuitive syntax for
1508 substitutions. They are also used as arguments to the @code{ex::series}
1509 method, where the left hand side of the relation specifies the variable
1510 to expand in and the right hand side the expansion point. They can also
1511 be used for creating systems of equations that are to be solved for
1512 unknown variables. But the most common usage of objects of this class
1513 is rather inconspicuous in statements of the form @code{if
1514 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1515 conversion from @code{relational} to @code{bool} takes place. Note,
1516 however, that @code{==} here does not perform any simplifications, hence
1517 @code{expand()} must be called explicitly.
1520 @node Matrices, Indexed objects, Relations, Basic Concepts
1521 @c node-name, next, previous, up
1523 @cindex @code{matrix} (class)
1525 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1526 matrix with @math{m} rows and @math{n} columns are accessed with two
1527 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1528 second one in the range 0@dots{}@math{n-1}.
1530 There are a couple of ways to construct matrices, with or without preset
1533 @cindex @code{lst_to_matrix()}
1534 @cindex @code{diag_matrix()}
1535 @cindex @code{unit_matrix()}
1536 @cindex @code{symbolic_matrix()}
1538 matrix::matrix(unsigned r, unsigned c);
1539 matrix::matrix(unsigned r, unsigned c, const lst & l);
1540 ex lst_to_matrix(const lst & l);
1541 ex diag_matrix(const lst & l);
1542 ex unit_matrix(unsigned x);
1543 ex unit_matrix(unsigned r, unsigned c);
1544 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1545 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1548 The first two functions are @code{matrix} constructors which create a matrix
1549 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1550 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1551 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1552 from a list of lists, each list representing a matrix row. @code{diag_matrix()}
1553 constructs a diagonal matrix given the list of diagonal elements.
1554 @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
1555 unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
1556 with newly generated symbols made of the specified base name and the
1557 position of each element in the matrix.
1559 Matrix elements can be accessed and set using the parenthesis (function call)
1563 const ex & matrix::operator()(unsigned r, unsigned c) const;
1564 ex & matrix::operator()(unsigned r, unsigned c);
1567 It is also possible to access the matrix elements in a linear fashion with
1568 the @code{op()} method. But C++-style subscripting with square brackets
1569 @samp{[]} is not available.
1571 Here are a couple of examples of constructing matrices:
1575 symbol a("a"), b("b");
1583 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1586 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1589 cout << diag_matrix(lst(a, b)) << endl;
1592 cout << unit_matrix(3) << endl;
1593 // -> [[1,0,0],[0,1,0],[0,0,1]]
1595 cout << symbolic_matrix(2, 3, "x") << endl;
1596 // -> [[x00,x01,x02],[x10,x11,x12]]
1600 @cindex @code{transpose()}
1601 There are three ways to do arithmetic with matrices. The first (and most
1602 direct one) is to use the methods provided by the @code{matrix} class:
1605 matrix matrix::add(const matrix & other) const;
1606 matrix matrix::sub(const matrix & other) const;
1607 matrix matrix::mul(const matrix & other) const;
1608 matrix matrix::mul_scalar(const ex & other) const;
1609 matrix matrix::pow(const ex & expn) const;
1610 matrix matrix::transpose() const;
1613 All of these methods return the result as a new matrix object. Here is an
1614 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1619 matrix A(2, 2, lst(1, 2, 3, 4));
1620 matrix B(2, 2, lst(-1, 0, 2, 1));
1621 matrix C(2, 2, lst(8, 4, 2, 1));
1623 matrix result = A.mul(B).sub(C.mul_scalar(2));
1624 cout << result << endl;
1625 // -> [[-13,-6],[1,2]]
1630 @cindex @code{evalm()}
1631 The second (and probably the most natural) way is to construct an expression
1632 containing matrices with the usual arithmetic operators and @code{pow()}.
1633 For efficiency reasons, expressions with sums, products and powers of
1634 matrices are not automatically evaluated in GiNaC. You have to call the
1638 ex ex::evalm() const;
1641 to obtain the result:
1648 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1649 cout << e.evalm() << endl;
1650 // -> [[-13,-6],[1,2]]
1655 The non-commutativity of the product @code{A*B} in this example is
1656 automatically recognized by GiNaC. There is no need to use a special
1657 operator here. @xref{Non-commutative objects}, for more information about
1658 dealing with non-commutative expressions.
1660 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1661 to perform the arithmetic:
1666 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1667 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1669 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1670 cout << e.simplify_indexed() << endl;
1671 // -> [[-13,-6],[1,2]].i.j
1675 Using indices is most useful when working with rectangular matrices and
1676 one-dimensional vectors because you don't have to worry about having to
1677 transpose matrices before multiplying them. @xref{Indexed objects}, for
1678 more information about using matrices with indices, and about indices in
1681 The @code{matrix} class provides a couple of additional methods for
1682 computing determinants, traces, and characteristic polynomials:
1684 @cindex @code{determinant()}
1685 @cindex @code{trace()}
1686 @cindex @code{charpoly()}
1688 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1689 ex matrix::trace() const;
1690 ex matrix::charpoly(const ex & lambda) const;
1693 The @samp{algo} argument of @code{determinant()} allows to select
1694 between different algorithms for calculating the determinant. The
1695 asymptotic speed (as parametrized by the matrix size) can greatly differ
1696 between those algorithms, depending on the nature of the matrix'
1697 entries. The possible values are defined in the @file{flags.h} header
1698 file. By default, GiNaC uses a heuristic to automatically select an
1699 algorithm that is likely (but not guaranteed) to give the result most
1702 @cindex @code{inverse()}
1703 @cindex @code{solve()}
1704 Matrices may also be inverted using the @code{ex matrix::inverse()}
1705 method and linear systems may be solved with:
1708 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1711 Assuming the matrix object this method is applied on is an @code{m}
1712 times @code{n} matrix, then @code{vars} must be a @code{n} times
1713 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1714 times @code{p} matrix. The returned matrix then has dimension @code{n}
1715 times @code{p} and in the case of an underdetermined system will still
1716 contain some of the indeterminates from @code{vars}. If the system is
1717 overdetermined, an exception is thrown.
1720 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1721 @c node-name, next, previous, up
1722 @section Indexed objects
1724 GiNaC allows you to handle expressions containing general indexed objects in
1725 arbitrary spaces. It is also able to canonicalize and simplify such
1726 expressions and perform symbolic dummy index summations. There are a number
1727 of predefined indexed objects provided, like delta and metric tensors.
1729 There are few restrictions placed on indexed objects and their indices and
1730 it is easy to construct nonsense expressions, but our intention is to
1731 provide a general framework that allows you to implement algorithms with
1732 indexed quantities, getting in the way as little as possible.
1734 @cindex @code{idx} (class)
1735 @cindex @code{indexed} (class)
1736 @subsection Indexed quantities and their indices
1738 Indexed expressions in GiNaC are constructed of two special types of objects,
1739 @dfn{index objects} and @dfn{indexed objects}.
1743 @cindex contravariant
1746 @item Index objects are of class @code{idx} or a subclass. Every index has
1747 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1748 the index lives in) which can both be arbitrary expressions but are usually
1749 a number or a simple symbol. In addition, indices of class @code{varidx} have
1750 a @dfn{variance} (they can be co- or contravariant), and indices of class
1751 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1753 @item Indexed objects are of class @code{indexed} or a subclass. They
1754 contain a @dfn{base expression} (which is the expression being indexed), and
1755 one or more indices.
1759 @strong{Note:} when printing expressions, covariant indices and indices
1760 without variance are denoted @samp{.i} while contravariant indices are
1761 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1762 value. In the following, we are going to use that notation in the text so
1763 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1764 not visible in the output.
1766 A simple example shall illustrate the concepts:
1770 #include <ginac/ginac.h>
1771 using namespace std;
1772 using namespace GiNaC;
1776 symbol i_sym("i"), j_sym("j");
1777 idx i(i_sym, 3), j(j_sym, 3);
1780 cout << indexed(A, i, j) << endl;
1782 cout << index_dimensions << indexed(A, i, j) << endl;
1784 cout << dflt; // reset cout to default output format (dimensions hidden)
1788 The @code{idx} constructor takes two arguments, the index value and the
1789 index dimension. First we define two index objects, @code{i} and @code{j},
1790 both with the numeric dimension 3. The value of the index @code{i} is the
1791 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1792 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1793 construct an expression containing one indexed object, @samp{A.i.j}. It has
1794 the symbol @code{A} as its base expression and the two indices @code{i} and
1797 The dimensions of indices are normally not visible in the output, but one
1798 can request them to be printed with the @code{index_dimensions} manipulator,
1801 Note the difference between the indices @code{i} and @code{j} which are of
1802 class @code{idx}, and the index values which are the symbols @code{i_sym}
1803 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1804 or numbers but must be index objects. For example, the following is not
1805 correct and will raise an exception:
1808 symbol i("i"), j("j");
1809 e = indexed(A, i, j); // ERROR: indices must be of type idx
1812 You can have multiple indexed objects in an expression, index values can
1813 be numeric, and index dimensions symbolic:
1817 symbol B("B"), dim("dim");
1818 cout << 4 * indexed(A, i)
1819 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1824 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1825 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1826 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1827 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1828 @code{simplify_indexed()} for that, see below).
1830 In fact, base expressions, index values and index dimensions can be
1831 arbitrary expressions:
1835 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1840 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1841 get an error message from this but you will probably not be able to do
1842 anything useful with it.
1844 @cindex @code{get_value()}
1845 @cindex @code{get_dimension()}
1849 ex idx::get_value();
1850 ex idx::get_dimension();
1853 return the value and dimension of an @code{idx} object. If you have an index
1854 in an expression, such as returned by calling @code{.op()} on an indexed
1855 object, you can get a reference to the @code{idx} object with the function
1856 @code{ex_to<idx>()} on the expression.
1858 There are also the methods
1861 bool idx::is_numeric();
1862 bool idx::is_symbolic();
1863 bool idx::is_dim_numeric();
1864 bool idx::is_dim_symbolic();
1867 for checking whether the value and dimension are numeric or symbolic
1868 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1869 About Expressions}) returns information about the index value.
1871 @cindex @code{varidx} (class)
1872 If you need co- and contravariant indices, use the @code{varidx} class:
1876 symbol mu_sym("mu"), nu_sym("nu");
1877 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1878 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1880 cout << indexed(A, mu, nu) << endl;
1882 cout << indexed(A, mu_co, nu) << endl;
1884 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1889 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1890 co- or contravariant. The default is a contravariant (upper) index, but
1891 this can be overridden by supplying a third argument to the @code{varidx}
1892 constructor. The two methods
1895 bool varidx::is_covariant();
1896 bool varidx::is_contravariant();
1899 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1900 to get the object reference from an expression). There's also the very useful
1904 ex varidx::toggle_variance();
1907 which makes a new index with the same value and dimension but the opposite
1908 variance. By using it you only have to define the index once.
1910 @cindex @code{spinidx} (class)
1911 The @code{spinidx} class provides dotted and undotted variant indices, as
1912 used in the Weyl-van-der-Waerden spinor formalism:
1916 symbol K("K"), C_sym("C"), D_sym("D");
1917 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1918 // contravariant, undotted
1919 spinidx C_co(C_sym, 2, true); // covariant index
1920 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1921 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1923 cout << indexed(K, C, D) << endl;
1925 cout << indexed(K, C_co, D_dot) << endl;
1927 cout << indexed(K, D_co_dot, D) << endl;
1932 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1933 dotted or undotted. The default is undotted but this can be overridden by
1934 supplying a fourth argument to the @code{spinidx} constructor. The two
1938 bool spinidx::is_dotted();
1939 bool spinidx::is_undotted();
1942 allow you to check whether or not a @code{spinidx} object is dotted (use
1943 @code{ex_to<spinidx>()} to get the object reference from an expression).
1944 Finally, the two methods
1947 ex spinidx::toggle_dot();
1948 ex spinidx::toggle_variance_dot();
1951 create a new index with the same value and dimension but opposite dottedness
1952 and the same or opposite variance.
1954 @subsection Substituting indices
1956 @cindex @code{subs()}
1957 Sometimes you will want to substitute one symbolic index with another
1958 symbolic or numeric index, for example when calculating one specific element
1959 of a tensor expression. This is done with the @code{.subs()} method, as it
1960 is done for symbols (see @ref{Substituting Expressions}).
1962 You have two possibilities here. You can either substitute the whole index
1963 by another index or expression:
1967 ex e = indexed(A, mu_co);
1968 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1969 // -> A.mu becomes A~nu
1970 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1971 // -> A.mu becomes A~0
1972 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1973 // -> A.mu becomes A.0
1977 The third example shows that trying to replace an index with something that
1978 is not an index will substitute the index value instead.
1980 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1985 ex e = indexed(A, mu_co);
1986 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1987 // -> A.mu becomes A.nu
1988 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1989 // -> A.mu becomes A.0
1993 As you see, with the second method only the value of the index will get
1994 substituted. Its other properties, including its dimension, remain unchanged.
1995 If you want to change the dimension of an index you have to substitute the
1996 whole index by another one with the new dimension.
1998 Finally, substituting the base expression of an indexed object works as
2003 ex e = indexed(A, mu_co);
2004 cout << e << " becomes " << e.subs(A == A+B) << endl;
2005 // -> A.mu becomes (B+A).mu
2009 @subsection Symmetries
2010 @cindex @code{symmetry} (class)
2011 @cindex @code{sy_none()}
2012 @cindex @code{sy_symm()}
2013 @cindex @code{sy_anti()}
2014 @cindex @code{sy_cycl()}
2016 Indexed objects can have certain symmetry properties with respect to their
2017 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2018 that is constructed with the helper functions
2021 symmetry sy_none(...);
2022 symmetry sy_symm(...);
2023 symmetry sy_anti(...);
2024 symmetry sy_cycl(...);
2027 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2028 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2029 represents a cyclic symmetry. Each of these functions accepts up to four
2030 arguments which can be either symmetry objects themselves or unsigned integer
2031 numbers that represent an index position (counting from 0). A symmetry
2032 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2033 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2036 Here are some examples of symmetry definitions:
2041 e = indexed(A, i, j);
2042 e = indexed(A, sy_none(), i, j); // equivalent
2043 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2045 // Symmetric in all three indices:
2046 e = indexed(A, sy_symm(), i, j, k);
2047 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2048 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2049 // different canonical order
2051 // Symmetric in the first two indices only:
2052 e = indexed(A, sy_symm(0, 1), i, j, k);
2053 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2055 // Antisymmetric in the first and last index only (index ranges need not
2057 e = indexed(A, sy_anti(0, 2), i, j, k);
2058 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2060 // An example of a mixed symmetry: antisymmetric in the first two and
2061 // last two indices, symmetric when swapping the first and last index
2062 // pairs (like the Riemann curvature tensor):
2063 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2065 // Cyclic symmetry in all three indices:
2066 e = indexed(A, sy_cycl(), i, j, k);
2067 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2069 // The following examples are invalid constructions that will throw
2070 // an exception at run time.
2072 // An index may not appear multiple times:
2073 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2074 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2076 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2077 // same number of indices:
2078 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2080 // And of course, you cannot specify indices which are not there:
2081 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2085 If you need to specify more than four indices, you have to use the
2086 @code{.add()} method of the @code{symmetry} class. For example, to specify
2087 full symmetry in the first six indices you would write
2088 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2090 If an indexed object has a symmetry, GiNaC will automatically bring the
2091 indices into a canonical order which allows for some immediate simplifications:
2095 cout << indexed(A, sy_symm(), i, j)
2096 + indexed(A, sy_symm(), j, i) << endl;
2098 cout << indexed(B, sy_anti(), i, j)
2099 + indexed(B, sy_anti(), j, i) << endl;
2101 cout << indexed(B, sy_anti(), i, j, k)
2102 - indexed(B, sy_anti(), j, k, i) << endl;
2107 @cindex @code{get_free_indices()}
2109 @subsection Dummy indices
2111 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2112 that a summation over the index range is implied. Symbolic indices which are
2113 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2114 dummy nor free indices.
2116 To be recognized as a dummy index pair, the two indices must be of the same
2117 class and their value must be the same single symbol (an index like
2118 @samp{2*n+1} is never a dummy index). If the indices are of class
2119 @code{varidx} they must also be of opposite variance; if they are of class
2120 @code{spinidx} they must be both dotted or both undotted.
2122 The method @code{.get_free_indices()} returns a vector containing the free
2123 indices of an expression. It also checks that the free indices of the terms
2124 of a sum are consistent:
2128 symbol A("A"), B("B"), C("C");
2130 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2131 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2133 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2134 cout << exprseq(e.get_free_indices()) << endl;
2136 // 'j' and 'l' are dummy indices
2138 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2139 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2141 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2142 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2143 cout << exprseq(e.get_free_indices()) << endl;
2145 // 'nu' is a dummy index, but 'sigma' is not
2147 e = indexed(A, mu, mu);
2148 cout << exprseq(e.get_free_indices()) << endl;
2150 // 'mu' is not a dummy index because it appears twice with the same
2153 e = indexed(A, mu, nu) + 42;
2154 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2155 // this will throw an exception:
2156 // "add::get_free_indices: inconsistent indices in sum"
2160 @cindex @code{simplify_indexed()}
2161 @subsection Simplifying indexed expressions
2163 In addition to the few automatic simplifications that GiNaC performs on
2164 indexed expressions (such as re-ordering the indices of symmetric tensors
2165 and calculating traces and convolutions of matrices and predefined tensors)
2169 ex ex::simplify_indexed();
2170 ex ex::simplify_indexed(const scalar_products & sp);
2173 that performs some more expensive operations:
2176 @item it checks the consistency of free indices in sums in the same way
2177 @code{get_free_indices()} does
2178 @item it tries to give dummy indices that appear in different terms of a sum
2179 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2180 @item it (symbolically) calculates all possible dummy index summations/contractions
2181 with the predefined tensors (this will be explained in more detail in the
2183 @item it detects contractions that vanish for symmetry reasons, for example
2184 the contraction of a symmetric and a totally antisymmetric tensor
2185 @item as a special case of dummy index summation, it can replace scalar products
2186 of two tensors with a user-defined value
2189 The last point is done with the help of the @code{scalar_products} class
2190 which is used to store scalar products with known values (this is not an
2191 arithmetic class, you just pass it to @code{simplify_indexed()}):
2195 symbol A("A"), B("B"), C("C"), i_sym("i");
2199 sp.add(A, B, 0); // A and B are orthogonal
2200 sp.add(A, C, 0); // A and C are orthogonal
2201 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2203 e = indexed(A + B, i) * indexed(A + C, i);
2205 // -> (B+A).i*(A+C).i
2207 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2213 The @code{scalar_products} object @code{sp} acts as a storage for the
2214 scalar products added to it with the @code{.add()} method. This method
2215 takes three arguments: the two expressions of which the scalar product is
2216 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2217 @code{simplify_indexed()} will replace all scalar products of indexed
2218 objects that have the symbols @code{A} and @code{B} as base expressions
2219 with the single value 0. The number, type and dimension of the indices
2220 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2222 @cindex @code{expand()}
2223 The example above also illustrates a feature of the @code{expand()} method:
2224 if passed the @code{expand_indexed} option it will distribute indices
2225 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2227 @cindex @code{tensor} (class)
2228 @subsection Predefined tensors
2230 Some frequently used special tensors such as the delta, epsilon and metric
2231 tensors are predefined in GiNaC. They have special properties when
2232 contracted with other tensor expressions and some of them have constant
2233 matrix representations (they will evaluate to a number when numeric
2234 indices are specified).
2236 @cindex @code{delta_tensor()}
2237 @subsubsection Delta tensor
2239 The delta tensor takes two indices, is symmetric and has the matrix
2240 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2241 @code{delta_tensor()}:
2245 symbol A("A"), B("B");
2247 idx i(symbol("i"), 3), j(symbol("j"), 3),
2248 k(symbol("k"), 3), l(symbol("l"), 3);
2250 ex e = indexed(A, i, j) * indexed(B, k, l)
2251 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2252 cout << e.simplify_indexed() << endl;
2255 cout << delta_tensor(i, i) << endl;
2260 @cindex @code{metric_tensor()}
2261 @subsubsection General metric tensor
2263 The function @code{metric_tensor()} creates a general symmetric metric
2264 tensor with two indices that can be used to raise/lower tensor indices. The
2265 metric tensor is denoted as @samp{g} in the output and if its indices are of
2266 mixed variance it is automatically replaced by a delta tensor:
2272 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2274 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2275 cout << e.simplify_indexed() << endl;
2278 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2279 cout << e.simplify_indexed() << endl;
2282 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2283 * metric_tensor(nu, rho);
2284 cout << e.simplify_indexed() << endl;
2287 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2288 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2289 + indexed(A, mu.toggle_variance(), rho));
2290 cout << e.simplify_indexed() << endl;
2295 @cindex @code{lorentz_g()}
2296 @subsubsection Minkowski metric tensor
2298 The Minkowski metric tensor is a special metric tensor with a constant
2299 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2300 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2301 It is created with the function @code{lorentz_g()} (although it is output as
2306 varidx mu(symbol("mu"), 4);
2308 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2309 * lorentz_g(mu, varidx(0, 4)); // negative signature
2310 cout << e.simplify_indexed() << endl;
2313 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2314 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2315 cout << e.simplify_indexed() << endl;
2320 @cindex @code{spinor_metric()}
2321 @subsubsection Spinor metric tensor
2323 The function @code{spinor_metric()} creates an antisymmetric tensor with
2324 two indices that is used to raise/lower indices of 2-component spinors.
2325 It is output as @samp{eps}:
2331 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2332 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2334 e = spinor_metric(A, B) * indexed(psi, B_co);
2335 cout << e.simplify_indexed() << endl;
2338 e = spinor_metric(A, B) * indexed(psi, A_co);
2339 cout << e.simplify_indexed() << endl;
2342 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2343 cout << e.simplify_indexed() << endl;
2346 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2347 cout << e.simplify_indexed() << endl;
2350 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2351 cout << e.simplify_indexed() << endl;
2354 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2355 cout << e.simplify_indexed() << endl;
2360 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2362 @cindex @code{epsilon_tensor()}
2363 @cindex @code{lorentz_eps()}
2364 @subsubsection Epsilon tensor
2366 The epsilon tensor is totally antisymmetric, its number of indices is equal
2367 to the dimension of the index space (the indices must all be of the same
2368 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2369 defined to be 1. Its behavior with indices that have a variance also
2370 depends on the signature of the metric. Epsilon tensors are output as
2373 There are three functions defined to create epsilon tensors in 2, 3 and 4
2377 ex epsilon_tensor(const ex & i1, const ex & i2);
2378 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2379 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2382 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2383 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2384 Minkowski space (the last @code{bool} argument specifies whether the metric
2385 has negative or positive signature, as in the case of the Minkowski metric
2390 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2391 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2392 e = lorentz_eps(mu, nu, rho, sig) *
2393 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2394 cout << simplify_indexed(e) << endl;
2395 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2397 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2398 symbol A("A"), B("B");
2399 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2400 cout << simplify_indexed(e) << endl;
2401 // -> -B.k*A.j*eps.i.k.j
2402 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2403 cout << simplify_indexed(e) << endl;
2408 @subsection Linear algebra
2410 The @code{matrix} class can be used with indices to do some simple linear
2411 algebra (linear combinations and products of vectors and matrices, traces
2412 and scalar products):
2416 idx i(symbol("i"), 2), j(symbol("j"), 2);
2417 symbol x("x"), y("y");
2419 // A is a 2x2 matrix, X is a 2x1 vector
2420 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2422 cout << indexed(A, i, i) << endl;
2425 ex e = indexed(A, i, j) * indexed(X, j);
2426 cout << e.simplify_indexed() << endl;
2427 // -> [[2*y+x],[4*y+3*x]].i
2429 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2430 cout << e.simplify_indexed() << endl;
2431 // -> [[3*y+3*x,6*y+2*x]].j
2435 You can of course obtain the same results with the @code{matrix::add()},
2436 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2437 but with indices you don't have to worry about transposing matrices.
2439 Matrix indices always start at 0 and their dimension must match the number
2440 of rows/columns of the matrix. Matrices with one row or one column are
2441 vectors and can have one or two indices (it doesn't matter whether it's a
2442 row or a column vector). Other matrices must have two indices.
2444 You should be careful when using indices with variance on matrices. GiNaC
2445 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2446 @samp{F.mu.nu} are different matrices. In this case you should use only
2447 one form for @samp{F} and explicitly multiply it with a matrix representation
2448 of the metric tensor.
2451 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2452 @c node-name, next, previous, up
2453 @section Non-commutative objects
2455 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2456 non-commutative objects are built-in which are mostly of use in high energy
2460 @item Clifford (Dirac) algebra (class @code{clifford})
2461 @item su(3) Lie algebra (class @code{color})
2462 @item Matrices (unindexed) (class @code{matrix})
2465 The @code{clifford} and @code{color} classes are subclasses of
2466 @code{indexed} because the elements of these algebras usually carry
2467 indices. The @code{matrix} class is described in more detail in
2470 Unlike most computer algebra systems, GiNaC does not primarily provide an
2471 operator (often denoted @samp{&*}) for representing inert products of
2472 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2473 classes of objects involved, and non-commutative products are formed with
2474 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2475 figuring out by itself which objects commute and will group the factors
2476 by their class. Consider this example:
2480 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2481 idx a(symbol("a"), 8), b(symbol("b"), 8);
2482 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2484 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2488 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2489 groups the non-commutative factors (the gammas and the su(3) generators)
2490 together while preserving the order of factors within each class (because
2491 Clifford objects commute with color objects). The resulting expression is a
2492 @emph{commutative} product with two factors that are themselves non-commutative
2493 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2494 parentheses are placed around the non-commutative products in the output.
2496 @cindex @code{ncmul} (class)
2497 Non-commutative products are internally represented by objects of the class
2498 @code{ncmul}, as opposed to commutative products which are handled by the
2499 @code{mul} class. You will normally not have to worry about this distinction,
2502 The advantage of this approach is that you never have to worry about using
2503 (or forgetting to use) a special operator when constructing non-commutative
2504 expressions. Also, non-commutative products in GiNaC are more intelligent
2505 than in other computer algebra systems; they can, for example, automatically
2506 canonicalize themselves according to rules specified in the implementation
2507 of the non-commutative classes. The drawback is that to work with other than
2508 the built-in algebras you have to implement new classes yourself. Symbols
2509 always commute and it's not possible to construct non-commutative products
2510 using symbols to represent the algebra elements or generators. User-defined
2511 functions can, however, be specified as being non-commutative.
2513 @cindex @code{return_type()}
2514 @cindex @code{return_type_tinfo()}
2515 Information about the commutativity of an object or expression can be
2516 obtained with the two member functions
2519 unsigned ex::return_type() const;
2520 unsigned ex::return_type_tinfo() const;
2523 The @code{return_type()} function returns one of three values (defined in
2524 the header file @file{flags.h}), corresponding to three categories of
2525 expressions in GiNaC:
2528 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2529 classes are of this kind.
2530 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2531 certain class of non-commutative objects which can be determined with the
2532 @code{return_type_tinfo()} method. Expressions of this category commute
2533 with everything except @code{noncommutative} expressions of the same
2535 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2536 of non-commutative objects of different classes. Expressions of this
2537 category don't commute with any other @code{noncommutative} or
2538 @code{noncommutative_composite} expressions.
2541 The value returned by the @code{return_type_tinfo()} method is valid only
2542 when the return type of the expression is @code{noncommutative}. It is a
2543 value that is unique to the class of the object and usually one of the
2544 constants in @file{tinfos.h}, or derived therefrom.
2546 Here are a couple of examples:
2549 @multitable @columnfractions 0.33 0.33 0.34
2550 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2551 @item @code{42} @tab @code{commutative} @tab -
2552 @item @code{2*x-y} @tab @code{commutative} @tab -
2553 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2554 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2555 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2556 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2560 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2561 @code{TINFO_clifford} for objects with a representation label of zero.
2562 Other representation labels yield a different @code{return_type_tinfo()},
2563 but it's the same for any two objects with the same label. This is also true
2566 A last note: With the exception of matrices, positive integer powers of
2567 non-commutative objects are automatically expanded in GiNaC. For example,
2568 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2569 non-commutative expressions).
2572 @cindex @code{clifford} (class)
2573 @subsection Clifford algebra
2575 @cindex @code{dirac_gamma()}
2576 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2577 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2578 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2579 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2582 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2585 which takes two arguments: the index and a @dfn{representation label} in the
2586 range 0 to 255 which is used to distinguish elements of different Clifford
2587 algebras (this is also called a @dfn{spin line index}). Gammas with different
2588 labels commute with each other. The dimension of the index can be 4 or (in
2589 the framework of dimensional regularization) any symbolic value. Spinor
2590 indices on Dirac gammas are not supported in GiNaC.
2592 @cindex @code{dirac_ONE()}
2593 The unity element of a Clifford algebra is constructed by
2596 ex dirac_ONE(unsigned char rl = 0);
2599 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2600 multiples of the unity element, even though it's customary to omit it.
2601 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2602 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2603 GiNaC will complain and/or produce incorrect results.
2605 @cindex @code{dirac_gamma5()}
2606 There is a special element @samp{gamma5} that commutes with all other
2607 gammas, has a unit square, and in 4 dimensions equals
2608 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2611 ex dirac_gamma5(unsigned char rl = 0);
2614 @cindex @code{dirac_gammaL()}
2615 @cindex @code{dirac_gammaR()}
2616 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2617 objects, constructed by
2620 ex dirac_gammaL(unsigned char rl = 0);
2621 ex dirac_gammaR(unsigned char rl = 0);
2624 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2625 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2627 @cindex @code{dirac_slash()}
2628 Finally, the function
2631 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2634 creates a term that represents a contraction of @samp{e} with the Dirac
2635 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2636 with a unique index whose dimension is given by the @code{dim} argument).
2637 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2639 In products of dirac gammas, superfluous unity elements are automatically
2640 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2641 and @samp{gammaR} are moved to the front.
2643 The @code{simplify_indexed()} function performs contractions in gamma strings,
2649 symbol a("a"), b("b"), D("D");
2650 varidx mu(symbol("mu"), D);
2651 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2652 * dirac_gamma(mu.toggle_variance());
2654 // -> gamma~mu*a\*gamma.mu
2655 e = e.simplify_indexed();
2658 cout << e.subs(D == 4) << endl;
2664 @cindex @code{dirac_trace()}
2665 To calculate the trace of an expression containing strings of Dirac gammas
2666 you use the function
2669 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2672 This function takes the trace of all gammas with the specified representation
2673 label; gammas with other labels are left standing. The last argument to
2674 @code{dirac_trace()} is the value to be returned for the trace of the unity
2675 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2676 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2677 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2678 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2679 This @samp{gamma5} scheme is described in greater detail in
2680 @cite{The Role of gamma5 in Dimensional Regularization}.
2682 The value of the trace itself is also usually different in 4 and in
2683 @math{D != 4} dimensions:
2688 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2689 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2690 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2691 cout << dirac_trace(e).simplify_indexed() << endl;
2698 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2699 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2700 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2701 cout << dirac_trace(e).simplify_indexed() << endl;
2702 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2706 Here is an example for using @code{dirac_trace()} to compute a value that
2707 appears in the calculation of the one-loop vacuum polarization amplitude in
2712 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2713 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2716 sp.add(l, l, pow(l, 2));
2717 sp.add(l, q, ldotq);
2719 ex e = dirac_gamma(mu) *
2720 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2721 dirac_gamma(mu.toggle_variance()) *
2722 (dirac_slash(l, D) + m * dirac_ONE());
2723 e = dirac_trace(e).simplify_indexed(sp);
2724 e = e.collect(lst(l, ldotq, m));
2726 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2730 The @code{canonicalize_clifford()} function reorders all gamma products that
2731 appear in an expression to a canonical (but not necessarily simple) form.
2732 You can use this to compare two expressions or for further simplifications:
2736 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2737 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2739 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2741 e = canonicalize_clifford(e);
2748 @cindex @code{color} (class)
2749 @subsection Color algebra
2751 @cindex @code{color_T()}
2752 For computations in quantum chromodynamics, GiNaC implements the base elements
2753 and structure constants of the su(3) Lie algebra (color algebra). The base
2754 elements @math{T_a} are constructed by the function
2757 ex color_T(const ex & a, unsigned char rl = 0);
2760 which takes two arguments: the index and a @dfn{representation label} in the
2761 range 0 to 255 which is used to distinguish elements of different color
2762 algebras. Objects with different labels commute with each other. The
2763 dimension of the index must be exactly 8 and it should be of class @code{idx},
2766 @cindex @code{color_ONE()}
2767 The unity element of a color algebra is constructed by
2770 ex color_ONE(unsigned char rl = 0);
2773 @strong{Note:} You must always use @code{color_ONE()} when referring to
2774 multiples of the unity element, even though it's customary to omit it.
2775 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2776 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2777 GiNaC may produce incorrect results.
2779 @cindex @code{color_d()}
2780 @cindex @code{color_f()}
2784 ex color_d(const ex & a, const ex & b, const ex & c);
2785 ex color_f(const ex & a, const ex & b, const ex & c);
2788 create the symmetric and antisymmetric structure constants @math{d_abc} and
2789 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2790 and @math{[T_a, T_b] = i f_abc T_c}.
2792 @cindex @code{color_h()}
2793 There's an additional function
2796 ex color_h(const ex & a, const ex & b, const ex & c);
2799 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2801 The function @code{simplify_indexed()} performs some simplifications on
2802 expressions containing color objects:
2807 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2808 k(symbol("k"), 8), l(symbol("l"), 8);
2810 e = color_d(a, b, l) * color_f(a, b, k);
2811 cout << e.simplify_indexed() << endl;
2814 e = color_d(a, b, l) * color_d(a, b, k);
2815 cout << e.simplify_indexed() << endl;
2818 e = color_f(l, a, b) * color_f(a, b, k);
2819 cout << e.simplify_indexed() << endl;
2822 e = color_h(a, b, c) * color_h(a, b, c);
2823 cout << e.simplify_indexed() << endl;
2826 e = color_h(a, b, c) * color_T(b) * color_T(c);
2827 cout << e.simplify_indexed() << endl;
2830 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2831 cout << e.simplify_indexed() << endl;
2834 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2835 cout << e.simplify_indexed() << endl;
2836 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2840 @cindex @code{color_trace()}
2841 To calculate the trace of an expression containing color objects you use the
2845 ex color_trace(const ex & e, unsigned char rl = 0);
2848 This function takes the trace of all color @samp{T} objects with the
2849 specified representation label; @samp{T}s with other labels are left
2850 standing. For example:
2854 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2856 // -> -I*f.a.c.b+d.a.c.b
2861 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2862 @c node-name, next, previous, up
2863 @chapter Methods and Functions
2866 In this chapter the most important algorithms provided by GiNaC will be
2867 described. Some of them are implemented as functions on expressions,
2868 others are implemented as methods provided by expression objects. If
2869 they are methods, there exists a wrapper function around it, so you can
2870 alternatively call it in a functional way as shown in the simple
2875 cout << "As method: " << sin(1).evalf() << endl;
2876 cout << "As function: " << evalf(sin(1)) << endl;
2880 @cindex @code{subs()}
2881 The general rule is that wherever methods accept one or more parameters
2882 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2883 wrapper accepts is the same but preceded by the object to act on
2884 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2885 most natural one in an OO model but it may lead to confusion for MapleV
2886 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2887 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2888 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2889 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2890 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2891 here. Also, users of MuPAD will in most cases feel more comfortable
2892 with GiNaC's convention. All function wrappers are implemented
2893 as simple inline functions which just call the corresponding method and
2894 are only provided for users uncomfortable with OO who are dead set to
2895 avoid method invocations. Generally, nested function wrappers are much
2896 harder to read than a sequence of methods and should therefore be
2897 avoided if possible. On the other hand, not everything in GiNaC is a
2898 method on class @code{ex} and sometimes calling a function cannot be
2902 * Information About Expressions::
2903 * Numerical Evaluation::
2904 * Substituting Expressions::
2905 * Pattern Matching and Advanced Substitutions::
2906 * Applying a Function on Subexpressions::
2907 * Visitors and Tree Traversal::
2908 * Polynomial Arithmetic:: Working with polynomials.
2909 * Rational Expressions:: Working with rational functions.
2910 * Symbolic Differentiation::
2911 * Series Expansion:: Taylor and Laurent expansion.
2913 * Built-in Functions:: List of predefined mathematical functions.
2914 * Solving Linear Systems of Equations::
2915 * Input/Output:: Input and output of expressions.
2919 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
2920 @c node-name, next, previous, up
2921 @section Getting information about expressions
2923 @subsection Checking expression types
2924 @cindex @code{is_a<@dots{}>()}
2925 @cindex @code{is_exactly_a<@dots{}>()}
2926 @cindex @code{ex_to<@dots{}>()}
2927 @cindex Converting @code{ex} to other classes
2928 @cindex @code{info()}
2929 @cindex @code{return_type()}
2930 @cindex @code{return_type_tinfo()}
2932 Sometimes it's useful to check whether a given expression is a plain number,
2933 a sum, a polynomial with integer coefficients, or of some other specific type.
2934 GiNaC provides a couple of functions for this:
2937 bool is_a<T>(const ex & e);
2938 bool is_exactly_a<T>(const ex & e);
2939 bool ex::info(unsigned flag);
2940 unsigned ex::return_type() const;
2941 unsigned ex::return_type_tinfo() const;
2944 When the test made by @code{is_a<T>()} returns true, it is safe to call
2945 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2946 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2947 example, assuming @code{e} is an @code{ex}:
2952 if (is_a<numeric>(e))
2953 numeric n = ex_to<numeric>(e);
2958 @code{is_a<T>(e)} allows you to check whether the top-level object of
2959 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2960 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2961 e.g., for checking whether an expression is a number, a sum, or a product:
2968 is_a<numeric>(e1); // true
2969 is_a<numeric>(e2); // false
2970 is_a<add>(e1); // false
2971 is_a<add>(e2); // true
2972 is_a<mul>(e1); // false
2973 is_a<mul>(e2); // false
2977 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2978 top-level object of an expression @samp{e} is an instance of the GiNaC
2979 class @samp{T}, not including parent classes.
2981 The @code{info()} method is used for checking certain attributes of
2982 expressions. The possible values for the @code{flag} argument are defined
2983 in @file{ginac/flags.h}, the most important being explained in the following
2987 @multitable @columnfractions .30 .70
2988 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2989 @item @code{numeric}
2990 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2992 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2993 @item @code{rational}
2994 @tab @dots{}an exact rational number (integers are rational, too)
2995 @item @code{integer}
2996 @tab @dots{}a (non-complex) integer
2997 @item @code{crational}
2998 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2999 @item @code{cinteger}
3000 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3001 @item @code{positive}
3002 @tab @dots{}not complex and greater than 0
3003 @item @code{negative}
3004 @tab @dots{}not complex and less than 0
3005 @item @code{nonnegative}
3006 @tab @dots{}not complex and greater than or equal to 0
3008 @tab @dots{}an integer greater than 0
3010 @tab @dots{}an integer less than 0
3011 @item @code{nonnegint}
3012 @tab @dots{}an integer greater than or equal to 0
3014 @tab @dots{}an even integer
3016 @tab @dots{}an odd integer
3018 @tab @dots{}a prime integer (probabilistic primality test)
3019 @item @code{relation}
3020 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3021 @item @code{relation_equal}
3022 @tab @dots{}a @code{==} relation
3023 @item @code{relation_not_equal}
3024 @tab @dots{}a @code{!=} relation
3025 @item @code{relation_less}
3026 @tab @dots{}a @code{<} relation
3027 @item @code{relation_less_or_equal}
3028 @tab @dots{}a @code{<=} relation
3029 @item @code{relation_greater}
3030 @tab @dots{}a @code{>} relation
3031 @item @code{relation_greater_or_equal}
3032 @tab @dots{}a @code{>=} relation
3034 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3036 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3037 @item @code{polynomial}
3038 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3039 @item @code{integer_polynomial}
3040 @tab @dots{}a polynomial with (non-complex) integer coefficients
3041 @item @code{cinteger_polynomial}
3042 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3043 @item @code{rational_polynomial}
3044 @tab @dots{}a polynomial with (non-complex) rational coefficients
3045 @item @code{crational_polynomial}
3046 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3047 @item @code{rational_function}
3048 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3049 @item @code{algebraic}
3050 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3054 To determine whether an expression is commutative or non-commutative and if
3055 so, with which other expressions it would commute, you use the methods
3056 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3057 for an explanation of these.
3060 @subsection Accessing subexpressions
3061 @cindex @code{nops()}
3064 @cindex @code{relational} (class)
3066 GiNaC provides the two methods
3070 ex ex::op(size_t i);
3073 for accessing the subexpressions in the container-like GiNaC classes like
3074 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3075 determines the number of subexpressions (@samp{operands}) contained, while
3076 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3077 In the case of a @code{power} object, @code{op(0)} will return the basis
3078 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3079 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3081 The left-hand and right-hand side expressions of objects of class
3082 @code{relational} (and only of these) can also be accessed with the methods
3090 @subsection Comparing expressions
3091 @cindex @code{is_equal()}
3092 @cindex @code{is_zero()}
3094 Expressions can be compared with the usual C++ relational operators like
3095 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3096 the result is usually not determinable and the result will be @code{false},
3097 except in the case of the @code{!=} operator. You should also be aware that
3098 GiNaC will only do the most trivial test for equality (subtracting both
3099 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3102 Actually, if you construct an expression like @code{a == b}, this will be
3103 represented by an object of the @code{relational} class (@pxref{Relations})
3104 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3106 There are also two methods
3109 bool ex::is_equal(const ex & other);
3113 for checking whether one expression is equal to another, or equal to zero,
3117 @subsection Ordering expressions
3118 @cindex @code{ex_is_less} (class)
3119 @cindex @code{ex_is_equal} (class)
3120 @cindex @code{compare()}
3122 Sometimes it is necessary to establish a mathematically well-defined ordering
3123 on a set of arbitrary expressions, for example to use expressions as keys
3124 in a @code{std::map<>} container, or to bring a vector of expressions into
3125 a canonical order (which is done internally by GiNaC for sums and products).
3127 The operators @code{<}, @code{>} etc. described in the last section cannot
3128 be used for this, as they don't implement an ordering relation in the
3129 mathematical sense. In particular, they are not guaranteed to be
3130 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3131 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3134 By default, STL classes and algorithms use the @code{<} and @code{==}
3135 operators to compare objects, which are unsuitable for expressions, but GiNaC
3136 provides two functors that can be supplied as proper binary comparison
3137 predicates to the STL:
3140 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3142 bool operator()(const ex &lh, const ex &rh) const;
3145 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3147 bool operator()(const ex &lh, const ex &rh) const;
3151 For example, to define a @code{map} that maps expressions to strings you
3155 std::map<ex, std::string, ex_is_less> myMap;
3158 Omitting the @code{ex_is_less} template parameter will introduce spurious
3159 bugs because the map operates improperly.
3161 Other examples for the use of the functors:
3169 std::sort(v.begin(), v.end(), ex_is_less());
3171 // count the number of expressions equal to '1'
3172 unsigned num_ones = std::count_if(v.begin(), v.end(),
3173 std::bind2nd(ex_is_equal(), 1));
3176 The implementation of @code{ex_is_less} uses the member function
3179 int ex::compare(const ex & other) const;
3182 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3183 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3187 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3188 @c node-name, next, previous, up
3189 @section Numercial Evaluation
3190 @cindex @code{evalf()}
3192 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3193 To evaluate them using floating-point arithmetic you need to call
3196 ex ex::evalf(int level = 0) const;
3199 @cindex @code{Digits}
3200 The accuracy of the evaluation is controlled by the global object @code{Digits}
3201 which can be assigned an integer value. The default value of @code{Digits}
3202 is 17. @xref{Numbers}, for more information and examples.
3204 To evaluate an expression to a @code{double} floating-point number you can
3205 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3209 // Approximate sin(x/Pi)
3211 ex e = series(sin(x/Pi), x == 0, 6);
3213 // Evaluate numerically at x=0.1
3214 ex f = evalf(e.subs(x == 0.1));
3216 // ex_to<numeric> is an unsafe cast, so check the type first
3217 if (is_a<numeric>(f)) @{
3218 double d = ex_to<numeric>(f).to_double();
3227 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3228 @c node-name, next, previous, up
3229 @section Substituting expressions
3230 @cindex @code{subs()}
3232 Algebraic objects inside expressions can be replaced with arbitrary
3233 expressions via the @code{.subs()} method:
3236 ex ex::subs(const ex & e, unsigned options = 0);
3237 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3240 In the first form, @code{subs()} accepts a relational of the form
3241 @samp{object == expression} or a @code{lst} of such relationals:
3245 symbol x("x"), y("y");
3247 ex e1 = 2*x^2-4*x+3;
3248 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3252 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3257 If you specify multiple substitutions, they are performed in parallel, so e.g.
3258 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3260 The second form of @code{subs()} takes two lists, one for the objects to be
3261 replaced and one for the expressions to be substituted (both lists must
3262 contain the same number of elements). Using this form, you would write
3263 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
3265 The optional last argument to @code{subs()} is a combination of
3266 @code{subs_options} flags. There are two options available:
3267 @code{subs_options::no_pattern} disables pattern matching, which makes
3268 large @code{subs()} operations significantly faster if you are not using
3269 patterns. The second option, @code{subs_options::algebraic} enables
3270 algebraic substitutions in products and powers.
3271 @ref{Pattern Matching and Advanced Substitutions}, for more information
3272 about patterns and algebraic substitutions.
3274 @code{subs()} performs syntactic substitution of any complete algebraic
3275 object; it does not try to match sub-expressions as is demonstrated by the
3280 symbol x("x"), y("y"), z("z");
3282 ex e1 = pow(x+y, 2);
3283 cout << e1.subs(x+y == 4) << endl;
3286 ex e2 = sin(x)*sin(y)*cos(x);
3287 cout << e2.subs(sin(x) == cos(x)) << endl;
3288 // -> cos(x)^2*sin(y)
3291 cout << e3.subs(x+y == 4) << endl;
3293 // (and not 4+z as one might expect)
3297 A more powerful form of substitution using wildcards is described in the
3301 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3302 @c node-name, next, previous, up
3303 @section Pattern matching and advanced substitutions
3304 @cindex @code{wildcard} (class)
3305 @cindex Pattern matching
3307 GiNaC allows the use of patterns for checking whether an expression is of a
3308 certain form or contains subexpressions of a certain form, and for
3309 substituting expressions in a more general way.
3311 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3312 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3313 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3314 an unsigned integer number to allow having multiple different wildcards in a
3315 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3316 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3320 ex wild(unsigned label = 0);
3323 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3326 Some examples for patterns:
3328 @multitable @columnfractions .5 .5
3329 @item @strong{Constructed as} @tab @strong{Output as}
3330 @item @code{wild()} @tab @samp{$0}
3331 @item @code{pow(x,wild())} @tab @samp{x^$0}
3332 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3333 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3339 @item Wildcards behave like symbols and are subject to the same algebraic
3340 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3341 @item As shown in the last example, to use wildcards for indices you have to
3342 use them as the value of an @code{idx} object. This is because indices must
3343 always be of class @code{idx} (or a subclass).
3344 @item Wildcards only represent expressions or subexpressions. It is not
3345 possible to use them as placeholders for other properties like index
3346 dimension or variance, representation labels, symmetry of indexed objects
3348 @item Because wildcards are commutative, it is not possible to use wildcards
3349 as part of noncommutative products.
3350 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3351 are also valid patterns.
3354 @subsection Matching expressions
3355 @cindex @code{match()}
3356 The most basic application of patterns is to check whether an expression
3357 matches a given pattern. This is done by the function
3360 bool ex::match(const ex & pattern);
3361 bool ex::match(const ex & pattern, lst & repls);
3364 This function returns @code{true} when the expression matches the pattern
3365 and @code{false} if it doesn't. If used in the second form, the actual
3366 subexpressions matched by the wildcards get returned in the @code{repls}
3367 object as a list of relations of the form @samp{wildcard == expression}.
3368 If @code{match()} returns false, the state of @code{repls} is undefined.
3369 For reproducible results, the list should be empty when passed to
3370 @code{match()}, but it is also possible to find similarities in multiple
3371 expressions by passing in the result of a previous match.
3373 The matching algorithm works as follows:
3376 @item A single wildcard matches any expression. If one wildcard appears
3377 multiple times in a pattern, it must match the same expression in all
3378 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3379 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3380 @item If the expression is not of the same class as the pattern, the match
3381 fails (i.e. a sum only matches a sum, a function only matches a function,
3383 @item If the pattern is a function, it only matches the same function
3384 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3385 @item Except for sums and products, the match fails if the number of
3386 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3388 @item If there are no subexpressions, the expressions and the pattern must
3389 be equal (in the sense of @code{is_equal()}).
3390 @item Except for sums and products, each subexpression (@code{op()}) must
3391 match the corresponding subexpression of the pattern.
3394 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3395 account for their commutativity and associativity:
3398 @item If the pattern contains a term or factor that is a single wildcard,
3399 this one is used as the @dfn{global wildcard}. If there is more than one
3400 such wildcard, one of them is chosen as the global wildcard in a random
3402 @item Every term/factor of the pattern, except the global wildcard, is
3403 matched against every term of the expression in sequence. If no match is
3404 found, the whole match fails. Terms that did match are not considered in
3406 @item If there are no unmatched terms left, the match succeeds. Otherwise
3407 the match fails unless there is a global wildcard in the pattern, in
3408 which case this wildcard matches the remaining terms.
3411 In general, having more than one single wildcard as a term of a sum or a
3412 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3415 Here are some examples in @command{ginsh} to demonstrate how it works (the
3416 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3417 match fails, and the list of wildcard replacements otherwise):
3420 > match((x+y)^a,(x+y)^a);
3422 > match((x+y)^a,(x+y)^b);
3424 > match((x+y)^a,$1^$2);
3426 > match((x+y)^a,$1^$1);
3428 > match((x+y)^(x+y),$1^$1);
3430 > match((x+y)^(x+y),$1^$2);
3432 > match((a+b)*(a+c),($1+b)*($1+c));
3434 > match((a+b)*(a+c),(a+$1)*(a+$2));
3436 (Unpredictable. The result might also be [$1==c,$2==b].)
3437 > match((a+b)*(a+c),($1+$2)*($1+$3));
3438 (The result is undefined. Due to the sequential nature of the algorithm
3439 and the re-ordering of terms in GiNaC, the match for the first factor
3440 may be @{$1==a,$2==b@} in which case the match for the second factor
3441 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3443 > match(a*(x+y)+a*z+b,a*$1+$2);
3444 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3445 @{$1=x+y,$2=a*z+b@}.)
3446 > match(a+b+c+d+e+f,c);
3448 > match(a+b+c+d+e+f,c+$0);
3450 > match(a+b+c+d+e+f,c+e+$0);
3452 > match(a+b,a+b+$0);
3454 > match(a*b^2,a^$1*b^$2);
3456 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3457 even though a==a^1.)
3458 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3460 > match(atan2(y,x^2),atan2(y,$0));
3464 @subsection Matching parts of expressions
3465 @cindex @code{has()}
3466 A more general way to look for patterns in expressions is provided by the
3470 bool ex::has(const ex & pattern);
3473 This function checks whether a pattern is matched by an expression itself or
3474 by any of its subexpressions.
3476 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3477 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3480 > has(x*sin(x+y+2*a),y);
3482 > has(x*sin(x+y+2*a),x+y);
3484 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3485 has the subexpressions "x", "y" and "2*a".)
3486 > has(x*sin(x+y+2*a),x+y+$1);
3488 (But this is possible.)
3489 > has(x*sin(2*(x+y)+2*a),x+y);
3491 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3492 which "x+y" is not a subexpression.)
3495 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3497 > has(4*x^2-x+3,$1*x);
3499 > has(4*x^2+x+3,$1*x);
3501 (Another possible pitfall. The first expression matches because the term
3502 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3503 contains a linear term you should use the coeff() function instead.)
3506 @cindex @code{find()}
3510 bool ex::find(const ex & pattern, lst & found);
3513 works a bit like @code{has()} but it doesn't stop upon finding the first
3514 match. Instead, it appends all found matches to the specified list. If there
3515 are multiple occurrences of the same expression, it is entered only once to
3516 the list. @code{find()} returns false if no matches were found (in
3517 @command{ginsh}, it returns an empty list):
3520 > find(1+x+x^2+x^3,x);
3522 > find(1+x+x^2+x^3,y);
3524 > find(1+x+x^2+x^3,x^$1);
3526 (Note the absence of "x".)
3527 > expand((sin(x)+sin(y))*(a+b));
3528 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3533 @subsection Substituting expressions
3534 @cindex @code{subs()}
3535 Probably the most useful application of patterns is to use them for
3536 substituting expressions with the @code{subs()} method. Wildcards can be
3537 used in the search patterns as well as in the replacement expressions, where
3538 they get replaced by the expressions matched by them. @code{subs()} doesn't
3539 know anything about algebra; it performs purely syntactic substitutions.
3544 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3546 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3548 > subs((a+b+c)^2,a+b==x);
3550 > subs((a+b+c)^2,a+b+$1==x+$1);
3552 > subs(a+2*b,a+b==x);
3554 > subs(4*x^3-2*x^2+5*x-1,x==a);
3556 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3558 > subs(sin(1+sin(x)),sin($1)==cos($1));
3560 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3564 The last example would be written in C++ in this way:
3568 symbol a("a"), b("b"), x("x"), y("y");
3569 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3570 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3571 cout << e.expand() << endl;
3576 @subsection Algebraic substitutions
3577 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3578 enables smarter, algebraic substitutions in products and powers. If you want
3579 to substitute some factors of a product, you only need to list these factors
3580 in your pattern. Furthermore, if an (integer) power of some expression occurs
3581 in your pattern and in the expression that you want the substitution to occur
3582 in, it can be substituted as many times as possible, without getting negative
3585 An example clarifies it all (hopefully):
3588 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3589 subs_options::algebraic) << endl;
3590 // --> (y+x)^6+b^6+a^6
3592 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3594 // Powers and products are smart, but addition is just the same.
3596 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3599 // As I said: addition is just the same.
3601 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3602 // --> x^3*b*a^2+2*b
3604 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3606 // --> 2*b+x^3*b^(-1)*a^(-2)
3608 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3609 // --> -1-2*a^2+4*a^3+5*a
3611 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3612 subs_options::algebraic) << endl;
3613 // --> -1+5*x+4*x^3-2*x^2
3614 // You should not really need this kind of patterns very often now.
3615 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3617 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3618 subs_options::algebraic) << endl;
3619 // --> cos(1+cos(x))
3621 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3622 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3623 subs_options::algebraic)) << endl;
3628 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3629 @c node-name, next, previous, up
3630 @section Applying a Function on Subexpressions
3631 @cindex tree traversal
3632 @cindex @code{map()}
3634 Sometimes you may want to perform an operation on specific parts of an
3635 expression while leaving the general structure of it intact. An example
3636 of this would be a matrix trace operation: the trace of a sum is the sum
3637 of the traces of the individual terms. That is, the trace should @dfn{map}
3638 on the sum, by applying itself to each of the sum's operands. It is possible
3639 to do this manually which usually results in code like this:
3644 if (is_a<matrix>(e))
3645 return ex_to<matrix>(e).trace();
3646 else if (is_a<add>(e)) @{
3648 for (size_t i=0; i<e.nops(); i++)
3649 sum += calc_trace(e.op(i));
3651 @} else if (is_a<mul>)(e)) @{
3659 This is, however, slightly inefficient (if the sum is very large it can take
3660 a long time to add the terms one-by-one), and its applicability is limited to
3661 a rather small class of expressions. If @code{calc_trace()} is called with
3662 a relation or a list as its argument, you will probably want the trace to
3663 be taken on both sides of the relation or of all elements of the list.
3665 GiNaC offers the @code{map()} method to aid in the implementation of such
3669 ex ex::map(map_function & f) const;
3670 ex ex::map(ex (*f)(const ex & e)) const;
3673 In the first (preferred) form, @code{map()} takes a function object that
3674 is subclassed from the @code{map_function} class. In the second form, it
3675 takes a pointer to a function that accepts and returns an expression.
3676 @code{map()} constructs a new expression of the same type, applying the
3677 specified function on all subexpressions (in the sense of @code{op()}),
3680 The use of a function object makes it possible to supply more arguments to
3681 the function that is being mapped, or to keep local state information.
3682 The @code{map_function} class declares a virtual function call operator
3683 that you can overload. Here is a sample implementation of @code{calc_trace()}
3684 that uses @code{map()} in a recursive fashion:
3687 struct calc_trace : public map_function @{
3688 ex operator()(const ex &e)
3690 if (is_a<matrix>(e))
3691 return ex_to<matrix>(e).trace();
3692 else if (is_a<mul>(e)) @{
3695 return e.map(*this);
3700 This function object could then be used like this:
3704 ex M = ... // expression with matrices
3705 calc_trace do_trace;
3706 ex tr = do_trace(M);
3710 Here is another example for you to meditate over. It removes quadratic
3711 terms in a variable from an expanded polynomial:
3714 struct map_rem_quad : public map_function @{
3716 map_rem_quad(const ex & var_) : var(var_) @{@}
3718 ex operator()(const ex & e)
3720 if (is_a<add>(e) || is_a<mul>(e))
3721 return e.map(*this);
3722 else if (is_a<power>(e) &&
3723 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3733 symbol x("x"), y("y");
3736 for (int i=0; i<8; i++)
3737 e += pow(x, i) * pow(y, 8-i) * (i+1);
3739 // -> 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
3741 map_rem_quad rem_quad(x);
3742 cout << rem_quad(e) << endl;
3743 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3747 @command{ginsh} offers a slightly different implementation of @code{map()}
3748 that allows applying algebraic functions to operands. The second argument
3749 to @code{map()} is an expression containing the wildcard @samp{$0} which
3750 acts as the placeholder for the operands:
3755 > map(a+2*b,sin($0));
3757 > map(@{a,b,c@},$0^2+$0);
3758 @{a^2+a,b^2+b,c^2+c@}
3761 Note that it is only possible to use algebraic functions in the second
3762 argument. You can not use functions like @samp{diff()}, @samp{op()},
3763 @samp{subs()} etc. because these are evaluated immediately:
3766 > map(@{a,b,c@},diff($0,a));
3768 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3769 to "map(@{a,b,c@},0)".
3773 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3774 @c node-name, next, previous, up
3775 @section Visitors and Tree Traversal
3776 @cindex tree traversal
3777 @cindex @code{visitor} (class)
3778 @cindex @code{accept()}
3779 @cindex @code{visit()}
3780 @cindex @code{traverse()}
3781 @cindex @code{traverse_preorder()}
3782 @cindex @code{traverse_postorder()}
3784 Suppose that you need a function that returns a list of all indices appearing
3785 in an arbitrary expression. The indices can have any dimension, and for
3786 indices with variance you always want the covariant version returned.
3788 You can't use @code{get_free_indices()} because you also want to include
3789 dummy indices in the list, and you can't use @code{find()} as it needs
3790 specific index dimensions (and it would require two passes: one for indices
3791 with variance, one for plain ones).
3793 The obvious solution to this problem is a tree traversal with a type switch,
3794 such as the following:
3797 void gather_indices_helper(const ex & e, lst & l)
3799 if (is_a<varidx>(e)) @{
3800 const varidx & vi = ex_to<varidx>(e);
3801 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3802 @} else if (is_a<idx>(e)) @{
3805 size_t n = e.nops();
3806 for (size_t i = 0; i < n; ++i)
3807 gather_indices_helper(e.op(i), l);
3811 lst gather_indices(const ex & e)
3814 gather_indices_helper(e, l);
3821 This works fine but fans of object-oriented programming will feel
3822 uncomfortable with the type switch. One reason is that there is a possibility
3823 for subtle bugs regarding derived classes. If we had, for example, written
3826 if (is_a<idx>(e)) @{
3828 @} else if (is_a<varidx>(e)) @{
3832 in @code{gather_indices_helper}, the code wouldn't have worked because the
3833 first line "absorbs" all classes derived from @code{idx}, including
3834 @code{varidx}, so the special case for @code{varidx} would never have been
3837 Also, for a large number of classes, a type switch like the above can get
3838 unwieldy and inefficient (it's a linear search, after all).
3839 @code{gather_indices_helper} only checks for two classes, but if you had to
3840 write a function that required a different implementation for nearly
3841 every GiNaC class, the result would be very hard to maintain and extend.
3843 The cleanest approach to the problem would be to add a new virtual function
3844 to GiNaC's class hierarchy. In our example, there would be specializations
3845 for @code{idx} and @code{varidx} while the default implementation in
3846 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
3847 impossible to add virtual member functions to existing classes without
3848 changing their source and recompiling everything. GiNaC comes with source,
3849 so you could actually do this, but for a small algorithm like the one
3850 presented this would be impractical.
3852 One solution to this dilemma is the @dfn{Visitor} design pattern,
3853 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
3854 variation, described in detail in
3855 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
3856 virtual functions to the class hierarchy to implement operations, GiNaC
3857 provides a single "bouncing" method @code{accept()} that takes an instance
3858 of a special @code{visitor} class and redirects execution to the one
3859 @code{visit()} virtual function of the visitor that matches the type of
3860 object that @code{accept()} was being invoked on.
3862 Visitors in GiNaC must derive from the global @code{visitor} class as well
3863 as from the class @code{T::visitor} of each class @code{T} they want to
3864 visit, and implement the member functions @code{void visit(const T &)} for
3870 void ex::accept(visitor & v) const;
3873 will then dispatch to the correct @code{visit()} member function of the
3874 specified visitor @code{v} for the type of GiNaC object at the root of the
3875 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
3877 Here is an example of a visitor:
3881 : public visitor, // this is required
3882 public add::visitor, // visit add objects
3883 public numeric::visitor, // visit numeric objects
3884 public basic::visitor // visit basic objects
3886 void visit(const add & x)
3887 @{ cout << "called with an add object" << endl; @}
3889 void visit(const numeric & x)
3890 @{ cout << "called with a numeric object" << endl; @}
3892 void visit(const basic & x)
3893 @{ cout << "called with a basic object" << endl; @}
3897 which can be used as follows:
3908 // prints "called with a numeric object"
3910 // prints "called with an add object"
3912 // prints "called with a basic object"
3916 The @code{visit(const basic &)} method gets called for all objects that are
3917 not @code{numeric} or @code{add} and acts as an (optional) default.
3919 From a conceptual point of view, the @code{visit()} methods of the visitor
3920 behave like a newly added virtual function of the visited hierarchy.
3921 In addition, visitors can store state in member variables, and they can
3922 be extended by deriving a new visitor from an existing one, thus building
3923 hierarchies of visitors.
3925 We can now rewrite our index example from above with a visitor:
3928 class gather_indices_visitor
3929 : public visitor, public idx::visitor, public varidx::visitor
3933 void visit(const idx & i)
3938 void visit(const varidx & vi)
3940 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3944 const lst & get_result() // utility function
3953 What's missing is the tree traversal. We could implement it in
3954 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
3957 void ex::traverse_preorder(visitor & v) const;
3958 void ex::traverse_postorder(visitor & v) const;
3959 void ex::traverse(visitor & v) const;
3962 @code{traverse_preorder()} visits a node @emph{before} visiting its
3963 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
3964 visiting its subexpressions. @code{traverse()} is a synonym for
3965 @code{traverse_preorder()}.
3967 Here is a new implementation of @code{gather_indices()} that uses the visitor
3968 and @code{traverse()}:
3971 lst gather_indices(const ex & e)
3973 gather_indices_visitor v;
3975 return v.get_result();
3980 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
3981 @c node-name, next, previous, up
3982 @section Polynomial arithmetic
3984 @subsection Expanding and collecting
3985 @cindex @code{expand()}
3986 @cindex @code{collect()}
3987 @cindex @code{collect_common_factors()}
3989 A polynomial in one or more variables has many equivalent
3990 representations. Some useful ones serve a specific purpose. Consider
3991 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3992 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3993 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3994 representations are the recursive ones where one collects for exponents
3995 in one of the three variable. Since the factors are themselves
3996 polynomials in the remaining two variables the procedure can be
3997 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3998 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4001 To bring an expression into expanded form, its method
4004 ex ex::expand(unsigned options = 0);
4007 may be called. In our example above, this corresponds to @math{4*x*y +
4008 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4009 GiNaC is not easily guessable you should be prepared to see different
4010 orderings of terms in such sums!
4012 Another useful representation of multivariate polynomials is as a
4013 univariate polynomial in one of the variables with the coefficients
4014 being polynomials in the remaining variables. The method
4015 @code{collect()} accomplishes this task:
4018 ex ex::collect(const ex & s, bool distributed = false);
4021 The first argument to @code{collect()} can also be a list of objects in which
4022 case the result is either a recursively collected polynomial, or a polynomial
4023 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4024 by the @code{distributed} flag.
4026 Note that the original polynomial needs to be in expanded form (for the
4027 variables concerned) in order for @code{collect()} to be able to find the
4028 coefficients properly.
4030 The following @command{ginsh} transcript shows an application of @code{collect()}
4031 together with @code{find()}:
4034 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4035 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
4036 > collect(a,@{p,q@});
4037 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)
4038 > collect(a,find(a,sin($1)));
4039 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4040 > collect(a,@{find(a,sin($1)),p,q@});
4041 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4042 > collect(a,@{find(a,sin($1)),d@});
4043 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4046 Polynomials can often be brought into a more compact form by collecting
4047 common factors from the terms of sums. This is accomplished by the function
4050 ex collect_common_factors(const ex & e);
4053 This function doesn't perform a full factorization but only looks for
4054 factors which are already explicitly present:
4057 > collect_common_factors(a*x+a*y);
4059 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4061 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4062 (c+a)*a*(x*y+y^2+x)*b
4065 @subsection Degree and coefficients
4066 @cindex @code{degree()}
4067 @cindex @code{ldegree()}
4068 @cindex @code{coeff()}
4070 The degree and low degree of a polynomial can be obtained using the two
4074 int ex::degree(const ex & s);
4075 int ex::ldegree(const ex & s);
4078 which also work reliably on non-expanded input polynomials (they even work
4079 on rational functions, returning the asymptotic degree). To extract
4080 a coefficient with a certain power from an expanded polynomial you use
4083 ex ex::coeff(const ex & s, int n);
4086 You can also obtain the leading and trailing coefficients with the methods
4089 ex ex::lcoeff(const ex & s);
4090 ex ex::tcoeff(const ex & s);
4093 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4096 An application is illustrated in the next example, where a multivariate
4097 polynomial is analyzed:
4101 symbol x("x"), y("y");
4102 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4103 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4104 ex Poly = PolyInp.expand();
4106 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4107 cout << "The x^" << i << "-coefficient is "
4108 << Poly.coeff(x,i) << endl;
4110 cout << "As polynomial in y: "
4111 << Poly.collect(y) << endl;
4115 When run, it returns an output in the following fashion:
4118 The x^0-coefficient is y^2+11*y
4119 The x^1-coefficient is 5*y^2-2*y
4120 The x^2-coefficient is -1
4121 The x^3-coefficient is 4*y
4122 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4125 As always, the exact output may vary between different versions of GiNaC
4126 or even from run to run since the internal canonical ordering is not
4127 within the user's sphere of influence.
4129 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4130 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4131 with non-polynomial expressions as they not only work with symbols but with
4132 constants, functions and indexed objects as well:
4136 symbol a("a"), b("b"), c("c");
4137 idx i(symbol("i"), 3);
4139 ex e = pow(sin(x) - cos(x), 4);
4140 cout << e.degree(cos(x)) << endl;
4142 cout << e.expand().coeff(sin(x), 3) << endl;
4145 e = indexed(a+b, i) * indexed(b+c, i);
4146 e = e.expand(expand_options::expand_indexed);
4147 cout << e.collect(indexed(b, i)) << endl;
4148 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4153 @subsection Polynomial division
4154 @cindex polynomial division
4157 @cindex pseudo-remainder
4158 @cindex @code{quo()}
4159 @cindex @code{rem()}
4160 @cindex @code{prem()}
4161 @cindex @code{divide()}
4166 ex quo(const ex & a, const ex & b, const ex & x);
4167 ex rem(const ex & a, const ex & b, const ex & x);
4170 compute the quotient and remainder of univariate polynomials in the variable
4171 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4173 The additional function
4176 ex prem(const ex & a, const ex & b, const ex & x);
4179 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4180 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4182 Exact division of multivariate polynomials is performed by the function
4185 bool divide(const ex & a, const ex & b, ex & q);
4188 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4189 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4190 in which case the value of @code{q} is undefined.
4193 @subsection Unit, content and primitive part
4194 @cindex @code{unit()}
4195 @cindex @code{content()}
4196 @cindex @code{primpart()}
4201 ex ex::unit(const ex & x);
4202 ex ex::content(const ex & x);
4203 ex ex::primpart(const ex & x);
4206 return the unit part, content part, and primitive polynomial of a multivariate
4207 polynomial with respect to the variable @samp{x} (the unit part being the sign
4208 of the leading coefficient, the content part being the GCD of the coefficients,
4209 and the primitive polynomial being the input polynomial divided by the unit and
4210 content parts). The product of unit, content, and primitive part is the
4211 original polynomial.
4214 @subsection GCD and LCM
4217 @cindex @code{gcd()}
4218 @cindex @code{lcm()}
4220 The functions for polynomial greatest common divisor and least common
4221 multiple have the synopsis
4224 ex gcd(const ex & a, const ex & b);
4225 ex lcm(const ex & a, const ex & b);
4228 The functions @code{gcd()} and @code{lcm()} accept two expressions
4229 @code{a} and @code{b} as arguments and return a new expression, their
4230 greatest common divisor or least common multiple, respectively. If the
4231 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4232 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4235 #include <ginac/ginac.h>
4236 using namespace GiNaC;
4240 symbol x("x"), y("y"), z("z");
4241 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4242 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4244 ex P_gcd = gcd(P_a, P_b);
4246 ex P_lcm = lcm(P_a, P_b);
4247 // 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
4252 @subsection Square-free decomposition
4253 @cindex square-free decomposition
4254 @cindex factorization
4255 @cindex @code{sqrfree()}
4257 GiNaC still lacks proper factorization support. Some form of
4258 factorization is, however, easily implemented by noting that factors
4259 appearing in a polynomial with power two or more also appear in the
4260 derivative and hence can easily be found by computing the GCD of the
4261 original polynomial and its derivatives. Any decent system has an
4262 interface for this so called square-free factorization. So we provide
4265 ex sqrfree(const ex & a, const lst & l = lst());
4267 Here is an example that by the way illustrates how the exact form of the
4268 result may slightly depend on the order of differentiation, calling for
4269 some care with subsequent processing of the result:
4272 symbol x("x"), y("y");
4273 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4275 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4276 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4278 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4279 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4281 cout << sqrfree(BiVarPol) << endl;
4282 // -> depending on luck, any of the above
4285 Note also, how factors with the same exponents are not fully factorized
4289 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4290 @c node-name, next, previous, up
4291 @section Rational expressions
4293 @subsection The @code{normal} method
4294 @cindex @code{normal()}
4295 @cindex simplification
4296 @cindex temporary replacement
4298 Some basic form of simplification of expressions is called for frequently.
4299 GiNaC provides the method @code{.normal()}, which converts a rational function
4300 into an equivalent rational function of the form @samp{numerator/denominator}
4301 where numerator and denominator are coprime. If the input expression is already
4302 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4303 otherwise it performs fraction addition and multiplication.
4305 @code{.normal()} can also be used on expressions which are not rational functions
4306 as it will replace all non-rational objects (like functions or non-integer
4307 powers) by temporary symbols to bring the expression to the domain of rational
4308 functions before performing the normalization, and re-substituting these
4309 symbols afterwards. This algorithm is also available as a separate method
4310 @code{.to_rational()}, described below.
4312 This means that both expressions @code{t1} and @code{t2} are indeed
4313 simplified in this little code snippet:
4318 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4319 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4320 std::cout << "t1 is " << t1.normal() << std::endl;
4321 std::cout << "t2 is " << t2.normal() << std::endl;
4325 Of course this works for multivariate polynomials too, so the ratio of
4326 the sample-polynomials from the section about GCD and LCM above would be
4327 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4330 @subsection Numerator and denominator
4333 @cindex @code{numer()}
4334 @cindex @code{denom()}
4335 @cindex @code{numer_denom()}
4337 The numerator and denominator of an expression can be obtained with
4342 ex ex::numer_denom();
4345 These functions will first normalize the expression as described above and
4346 then return the numerator, denominator, or both as a list, respectively.
4347 If you need both numerator and denominator, calling @code{numer_denom()} is
4348 faster than using @code{numer()} and @code{denom()} separately.
4351 @subsection Converting to a polynomial or rational expression
4352 @cindex @code{to_polynomial()}
4353 @cindex @code{to_rational()}
4355 Some of the methods described so far only work on polynomials or rational
4356 functions. GiNaC provides a way to extend the domain of these functions to
4357 general expressions by using the temporary replacement algorithm described
4358 above. You do this by calling
4361 ex ex::to_polynomial(lst &l);
4365 ex ex::to_rational(lst &l);
4368 on the expression to be converted. The supplied @code{lst} will be filled
4369 with the generated temporary symbols and their replacement expressions in
4370 a format that can be used directly for the @code{subs()} method. It can also
4371 already contain a list of replacements from an earlier application of
4372 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
4373 it on multiple expressions and get consistent results.
4375 The difference betwerrn @code{.to_polynomial()} and @code{.to_rational()}
4376 is probably best illustrated with an example:
4380 symbol x("x"), y("y");
4381 ex a = 2*x/sin(x) - y/(3*sin(x));
4385 ex p = a.to_polynomial(lp);
4386 cout << " = " << p << "\n with " << lp << endl;
4387 // = symbol3*symbol2*y+2*symbol2*x
4388 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4391 ex r = a.to_rational(lr);
4392 cout << " = " << r << "\n with " << lr << endl;
4393 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4394 // with @{symbol4==sin(x)@}
4398 The following more useful example will print @samp{sin(x)-cos(x)}:
4403 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4404 ex b = sin(x) + cos(x);
4407 divide(a.to_polynomial(l), b.to_polynomial(l), q);
4408 cout << q.subs(l) << endl;
4413 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4414 @c node-name, next, previous, up
4415 @section Symbolic differentiation
4416 @cindex differentiation
4417 @cindex @code{diff()}
4419 @cindex product rule
4421 GiNaC's objects know how to differentiate themselves. Thus, a
4422 polynomial (class @code{add}) knows that its derivative is the sum of
4423 the derivatives of all the monomials:
4427 symbol x("x"), y("y"), z("z");
4428 ex P = pow(x, 5) + pow(x, 2) + y;
4430 cout << P.diff(x,2) << endl;
4432 cout << P.diff(y) << endl; // 1
4434 cout << P.diff(z) << endl; // 0
4439 If a second integer parameter @var{n} is given, the @code{diff} method
4440 returns the @var{n}th derivative.
4442 If @emph{every} object and every function is told what its derivative
4443 is, all derivatives of composed objects can be calculated using the
4444 chain rule and the product rule. Consider, for instance the expression
4445 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4446 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4447 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4448 out that the composition is the generating function for Euler Numbers,
4449 i.e. the so called @var{n}th Euler number is the coefficient of
4450 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4451 identity to code a function that generates Euler numbers in just three
4454 @cindex Euler numbers
4456 #include <ginac/ginac.h>
4457 using namespace GiNaC;
4459 ex EulerNumber(unsigned n)
4462 const ex generator = pow(cosh(x),-1);
4463 return generator.diff(x,n).subs(x==0);
4468 for (unsigned i=0; i<11; i+=2)
4469 std::cout << EulerNumber(i) << std::endl;
4474 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4475 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4476 @code{i} by two since all odd Euler numbers vanish anyways.
4479 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4480 @c node-name, next, previous, up
4481 @section Series expansion
4482 @cindex @code{series()}
4483 @cindex Taylor expansion
4484 @cindex Laurent expansion
4485 @cindex @code{pseries} (class)
4486 @cindex @code{Order()}
4488 Expressions know how to expand themselves as a Taylor series or (more
4489 generally) a Laurent series. As in most conventional Computer Algebra
4490 Systems, no distinction is made between those two. There is a class of
4491 its own for storing such series (@code{class pseries}) and a built-in
4492 function (called @code{Order}) for storing the order term of the series.
4493 As a consequence, if you want to work with series, i.e. multiply two
4494 series, you need to call the method @code{ex::series} again to convert
4495 it to a series object with the usual structure (expansion plus order
4496 term). A sample application from special relativity could read:
4499 #include <ginac/ginac.h>
4500 using namespace std;
4501 using namespace GiNaC;
4505 symbol v("v"), c("c");
4507 ex gamma = 1/sqrt(1 - pow(v/c,2));
4508 ex mass_nonrel = gamma.series(v==0, 10);
4510 cout << "the relativistic mass increase with v is " << endl
4511 << mass_nonrel << endl;
4513 cout << "the inverse square of this series is " << endl
4514 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4518 Only calling the series method makes the last output simplify to
4519 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4520 series raised to the power @math{-2}.
4522 @cindex Machin's formula
4523 As another instructive application, let us calculate the numerical
4524 value of Archimedes' constant
4528 (for which there already exists the built-in constant @code{Pi})
4529 using John Machin's amazing formula
4531 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4534 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4536 This equation (and similar ones) were used for over 200 years for
4537 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4538 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4539 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4540 order term with it and the question arises what the system is supposed
4541 to do when the fractions are plugged into that order term. The solution
4542 is to use the function @code{series_to_poly()} to simply strip the order
4546 #include <ginac/ginac.h>
4547 using namespace GiNaC;
4549 ex machin_pi(int degr)
4552 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4553 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4554 -4*pi_expansion.subs(x==numeric(1,239));
4560 using std::cout; // just for fun, another way of...
4561 using std::endl; // ...dealing with this namespace std.
4563 for (int i=2; i<12; i+=2) @{
4564 pi_frac = machin_pi(i);
4565 cout << i << ":\t" << pi_frac << endl
4566 << "\t" << pi_frac.evalf() << endl;
4572 Note how we just called @code{.series(x,degr)} instead of
4573 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4574 method @code{series()}: if the first argument is a symbol the expression
4575 is expanded in that symbol around point @code{0}. When you run this
4576 program, it will type out:
4580 3.1832635983263598326
4581 4: 5359397032/1706489875
4582 3.1405970293260603143
4583 6: 38279241713339684/12184551018734375
4584 3.141621029325034425
4585 8: 76528487109180192540976/24359780855939418203125
4586 3.141591772182177295
4587 10: 327853873402258685803048818236/104359128170408663038552734375
4588 3.1415926824043995174
4592 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4593 @c node-name, next, previous, up
4594 @section Symmetrization
4595 @cindex @code{symmetrize()}
4596 @cindex @code{antisymmetrize()}
4597 @cindex @code{symmetrize_cyclic()}
4602 ex ex::symmetrize(const lst & l);
4603 ex ex::antisymmetrize(const lst & l);
4604 ex ex::symmetrize_cyclic(const lst & l);
4607 symmetrize an expression by returning the sum over all symmetric,
4608 antisymmetric or cyclic permutations of the specified list of objects,
4609 weighted by the number of permutations.
4611 The three additional methods
4614 ex ex::symmetrize();
4615 ex ex::antisymmetrize();
4616 ex ex::symmetrize_cyclic();
4619 symmetrize or antisymmetrize an expression over its free indices.
4621 Symmetrization is most useful with indexed expressions but can be used with
4622 almost any kind of object (anything that is @code{subs()}able):
4626 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4627 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4629 cout << indexed(A, i, j).symmetrize() << endl;
4630 // -> 1/2*A.j.i+1/2*A.i.j
4631 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4632 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4633 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4634 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4639 @node Built-in Functions, Solving Linear Systems of Equations, Symmetrization, Methods and Functions
4640 @c node-name, next, previous, up
4641 @section Predefined mathematical functions
4643 GiNaC contains the following predefined mathematical functions:
4646 @multitable @columnfractions .30 .70
4647 @item @strong{Name} @tab @strong{Function}
4650 @cindex @code{abs()}
4651 @item @code{csgn(x)}
4653 @cindex @code{csgn()}
4654 @item @code{sqrt(x)}
4655 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4656 @cindex @code{sqrt()}
4659 @cindex @code{sin()}
4662 @cindex @code{cos()}
4665 @cindex @code{tan()}
4666 @item @code{asin(x)}
4668 @cindex @code{asin()}
4669 @item @code{acos(x)}
4671 @cindex @code{acos()}
4672 @item @code{atan(x)}
4673 @tab inverse tangent
4674 @cindex @code{atan()}
4675 @item @code{atan2(y, x)}
4676 @tab inverse tangent with two arguments
4677 @item @code{sinh(x)}
4678 @tab hyperbolic sine
4679 @cindex @code{sinh()}
4680 @item @code{cosh(x)}
4681 @tab hyperbolic cosine
4682 @cindex @code{cosh()}
4683 @item @code{tanh(x)}
4684 @tab hyperbolic tangent
4685 @cindex @code{tanh()}
4686 @item @code{asinh(x)}
4687 @tab inverse hyperbolic sine
4688 @cindex @code{asinh()}
4689 @item @code{acosh(x)}
4690 @tab inverse hyperbolic cosine
4691 @cindex @code{acosh()}
4692 @item @code{atanh(x)}
4693 @tab inverse hyperbolic tangent
4694 @cindex @code{atanh()}
4696 @tab exponential function
4697 @cindex @code{exp()}
4699 @tab natural logarithm
4700 @cindex @code{log()}
4703 @cindex @code{Li2()}
4704 @item @code{zeta(x)}
4705 @tab Riemann's zeta function
4706 @cindex @code{zeta()}
4707 @item @code{zeta(n, x)}
4708 @tab derivatives of Riemann's zeta function
4709 @item @code{tgamma(x)}
4711 @cindex @code{tgamma()}
4712 @cindex Gamma function
4713 @item @code{lgamma(x)}
4714 @tab logarithm of Gamma function
4715 @cindex @code{lgamma()}
4716 @item @code{beta(x, y)}
4717 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4718 @cindex @code{beta()}
4720 @tab psi (digamma) function
4721 @cindex @code{psi()}
4722 @item @code{psi(n, x)}
4723 @tab derivatives of psi function (polygamma functions)
4724 @item @code{factorial(n)}
4725 @tab factorial function
4726 @cindex @code{factorial()}
4727 @item @code{binomial(n, m)}
4728 @tab binomial coefficients
4729 @cindex @code{binomial()}
4730 @item @code{Order(x)}
4731 @tab order term function in truncated power series
4732 @cindex @code{Order()}
4733 @item @code{Li(n,x)}
4736 @item @code{S(n,p,x)}
4737 @tab Nielsen's generalized polylogarithm
4739 @item @code{H(m_lst,x)}
4740 @tab harmonic polylogarithm
4742 @item @code{Li(m_lst,x_lst)}
4743 @tab multiple polylogarithm
4745 @item @code{mZeta(m_lst)}
4746 @tab multiple zeta value
4747 @cindex @code{mZeta()}
4752 For functions that have a branch cut in the complex plane GiNaC follows
4753 the conventions for C++ as defined in the ANSI standard as far as
4754 possible. In particular: the natural logarithm (@code{log}) and the
4755 square root (@code{sqrt}) both have their branch cuts running along the
4756 negative real axis where the points on the axis itself belong to the
4757 upper part (i.e. continuous with quadrant II). The inverse
4758 trigonometric and hyperbolic functions are not defined for complex
4759 arguments by the C++ standard, however. In GiNaC we follow the
4760 conventions used by CLN, which in turn follow the carefully designed
4761 definitions in the Common Lisp standard. It should be noted that this
4762 convention is identical to the one used by the C99 standard and by most
4763 serious CAS. It is to be expected that future revisions of the C++
4764 standard incorporate these functions in the complex domain in a manner
4765 compatible with C99.
4768 @node Solving Linear Systems of Equations, Input/Output, Built-in Functions, Methods and Functions
4769 @c node-name, next, previous, up
4770 @section Solving Linear Systems of Equations
4771 @cindex @code{lsolve()}
4773 The function @code{lsolve()} provides a convenient wrapper around some
4774 matrix operations that comes in handy when a system of linear equations
4778 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
4781 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
4782 @code{relational}) while @code{symbols} is a @code{lst} of
4783 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
4786 It returns the @code{lst} of solutions as an expression. As an example,
4787 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
4791 symbol a("a"), b("b"), x("x"), y("y");
4793 eqns.append(a*x+b*y==3).append(x-y==b);
4795 vars.append(x).append(y);
4796 cout << lsolve(eqns, vars) << endl;
4797 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
4800 When the linear equations @code{eqns} are underdetermined, the solution
4801 will contain one or more tautological entries like @code{x==x},
4802 depending on the rank of the system. When they are overdetermined, the
4803 solution will be an empty @code{lst}. Note the third optional parameter
4804 to @code{lsolve()}: it accepts the same parameters as
4805 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
4809 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
4810 @c node-name, next, previous, up
4811 @section Input and output of expressions
4814 @subsection Expression output
4816 @cindex output of expressions
4818 Expressions can simply be written to any stream:
4823 ex e = 4.5*I+pow(x,2)*3/2;
4824 cout << e << endl; // prints '4.5*I+3/2*x^2'
4828 The default output format is identical to the @command{ginsh} input syntax and
4829 to that used by most computer algebra systems, but not directly pastable
4830 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4831 is printed as @samp{x^2}).
4833 It is possible to print expressions in a number of different formats with
4834 a set of stream manipulators;
4837 std::ostream & dflt(std::ostream & os);
4838 std::ostream & latex(std::ostream & os);
4839 std::ostream & tree(std::ostream & os);
4840 std::ostream & csrc(std::ostream & os);
4841 std::ostream & csrc_float(std::ostream & os);
4842 std::ostream & csrc_double(std::ostream & os);
4843 std::ostream & csrc_cl_N(std::ostream & os);
4844 std::ostream & index_dimensions(std::ostream & os);
4845 std::ostream & no_index_dimensions(std::ostream & os);
4848 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
4849 @command{ginsh} via the @code{print()}, @code{print_latex()} and
4850 @code{print_csrc()} functions, respectively.
4853 All manipulators affect the stream state permanently. To reset the output
4854 format to the default, use the @code{dflt} manipulator:
4858 cout << latex; // all output to cout will be in LaTeX format from now on
4859 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4860 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
4861 cout << dflt; // revert to default output format
4862 cout << e << endl; // prints '4.5*I+3/2*x^2'
4866 If you don't want to affect the format of the stream you're working with,
4867 you can output to a temporary @code{ostringstream} like this:
4872 s << latex << e; // format of cout remains unchanged
4873 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4878 @cindex @code{csrc_float}
4879 @cindex @code{csrc_double}
4880 @cindex @code{csrc_cl_N}
4881 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
4882 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
4883 format that can be directly used in a C or C++ program. The three possible
4884 formats select the data types used for numbers (@code{csrc_cl_N} uses the
4885 classes provided by the CLN library):
4889 cout << "f = " << csrc_float << e << ";\n";
4890 cout << "d = " << csrc_double << e << ";\n";
4891 cout << "n = " << csrc_cl_N << e << ";\n";
4895 The above example will produce (note the @code{x^2} being converted to
4899 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
4900 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
4901 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
4905 The @code{tree} manipulator allows dumping the internal structure of an
4906 expression for debugging purposes:
4917 add, hash=0x0, flags=0x3, nops=2
4918 power, hash=0x0, flags=0x3, nops=2
4919 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
4920 2 (numeric), hash=0x6526b0fa, flags=0xf
4921 3/2 (numeric), hash=0xf9828fbd, flags=0xf
4924 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
4928 @cindex @code{latex}
4929 The @code{latex} output format is for LaTeX parsing in mathematical mode.
4930 It is rather similar to the default format but provides some braces needed
4931 by LaTeX for delimiting boxes and also converts some common objects to
4932 conventional LaTeX names. It is possible to give symbols a special name for
4933 LaTeX output by supplying it as a second argument to the @code{symbol}
4936 For example, the code snippet
4940 symbol x("x", "\\circ");
4941 ex e = lgamma(x).series(x==0,3);
4942 cout << latex << e << endl;
4949 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
4952 @cindex @code{index_dimensions}
4953 @cindex @code{no_index_dimensions}
4954 Index dimensions are normally hidden in the output. To make them visible, use
4955 the @code{index_dimensions} manipulator. The dimensions will be written in
4956 square brackets behind each index value in the default and LaTeX output
4961 symbol x("x"), y("y");
4962 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
4963 ex e = indexed(x, mu) * indexed(y, nu);
4966 // prints 'x~mu*y~nu'
4967 cout << index_dimensions << e << endl;
4968 // prints 'x~mu[4]*y~nu[4]'
4969 cout << no_index_dimensions << e << endl;
4970 // prints 'x~mu*y~nu'
4975 @cindex Tree traversal
4976 If you need any fancy special output format, e.g. for interfacing GiNaC
4977 with other algebra systems or for producing code for different
4978 programming languages, you can always traverse the expression tree yourself:
4981 static void my_print(const ex & e)
4983 if (is_a<function>(e))
4984 cout << ex_to<function>(e).get_name();
4986 cout << e.bp->class_name();
4988 size_t n = e.nops();
4990 for (size_t i=0; i<n; i++) @{
5002 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5010 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5011 symbol(y))),numeric(-2)))
5014 If you need an output format that makes it possible to accurately
5015 reconstruct an expression by feeding the output to a suitable parser or
5016 object factory, you should consider storing the expression in an
5017 @code{archive} object and reading the object properties from there.
5018 See the section on archiving for more information.
5021 @subsection Expression input
5022 @cindex input of expressions
5024 GiNaC provides no way to directly read an expression from a stream because
5025 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5026 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5027 @code{y} you defined in your program and there is no way to specify the
5028 desired symbols to the @code{>>} stream input operator.
5030 Instead, GiNaC lets you construct an expression from a string, specifying the
5031 list of symbols to be used:
5035 symbol x("x"), y("y");
5036 ex e("2*x+sin(y)", lst(x, y));
5040 The input syntax is the same as that used by @command{ginsh} and the stream
5041 output operator @code{<<}. The symbols in the string are matched by name to
5042 the symbols in the list and if GiNaC encounters a symbol not specified in
5043 the list it will throw an exception.
5045 With this constructor, it's also easy to implement interactive GiNaC programs:
5050 #include <stdexcept>
5051 #include <ginac/ginac.h>
5052 using namespace std;
5053 using namespace GiNaC;
5060 cout << "Enter an expression containing 'x': ";
5065 cout << "The derivative of " << e << " with respect to x is ";
5066 cout << e.diff(x) << ".\n";
5067 @} catch (exception &p) @{
5068 cerr << p.what() << endl;
5074 @subsection Archiving
5075 @cindex @code{archive} (class)
5078 GiNaC allows creating @dfn{archives} of expressions which can be stored
5079 to or retrieved from files. To create an archive, you declare an object
5080 of class @code{archive} and archive expressions in it, giving each
5081 expression a unique name:
5085 using namespace std;
5086 #include <ginac/ginac.h>
5087 using namespace GiNaC;
5091 symbol x("x"), y("y"), z("z");
5093 ex foo = sin(x + 2*y) + 3*z + 41;
5097 a.archive_ex(foo, "foo");
5098 a.archive_ex(bar, "the second one");
5102 The archive can then be written to a file:
5106 ofstream out("foobar.gar");
5112 The file @file{foobar.gar} contains all information that is needed to
5113 reconstruct the expressions @code{foo} and @code{bar}.
5115 @cindex @command{viewgar}
5116 The tool @command{viewgar} that comes with GiNaC can be used to view
5117 the contents of GiNaC archive files:
5120 $ viewgar foobar.gar
5121 foo = 41+sin(x+2*y)+3*z
5122 the second one = 42+sin(x+2*y)+3*z
5125 The point of writing archive files is of course that they can later be
5131 ifstream in("foobar.gar");
5136 And the stored expressions can be retrieved by their name:
5142 ex ex1 = a2.unarchive_ex(syms, "foo");
5143 ex ex2 = a2.unarchive_ex(syms, "the second one");
5145 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5146 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5147 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5151 Note that you have to supply a list of the symbols which are to be inserted
5152 in the expressions. Symbols in archives are stored by their name only and
5153 if you don't specify which symbols you have, unarchiving the expression will
5154 create new symbols with that name. E.g. if you hadn't included @code{x} in
5155 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5156 have had no effect because the @code{x} in @code{ex1} would have been a
5157 different symbol than the @code{x} which was defined at the beginning of
5158 the program, although both would appear as @samp{x} when printed.
5160 You can also use the information stored in an @code{archive} object to
5161 output expressions in a format suitable for exact reconstruction. The
5162 @code{archive} and @code{archive_node} classes have a couple of member
5163 functions that let you access the stored properties:
5166 static void my_print2(const archive_node & n)
5169 n.find_string("class", class_name);
5170 cout << class_name << "(";
5172 archive_node::propinfovector p;
5173 n.get_properties(p);
5175 size_t num = p.size();
5176 for (size_t i=0; i<num; i++) @{
5177 const string &name = p[i].name;
5178 if (name == "class")
5180 cout << name << "=";
5182 unsigned count = p[i].count;
5186 for (unsigned j=0; j<count; j++) @{
5187 switch (p[i].type) @{
5188 case archive_node::PTYPE_BOOL: @{
5190 n.find_bool(name, x, j);
5191 cout << (x ? "true" : "false");
5194 case archive_node::PTYPE_UNSIGNED: @{
5196 n.find_unsigned(name, x, j);
5200 case archive_node::PTYPE_STRING: @{
5202 n.find_string(name, x, j);
5203 cout << '\"' << x << '\"';
5206 case archive_node::PTYPE_NODE: @{
5207 const archive_node &x = n.find_ex_node(name, j);
5229 ex e = pow(2, x) - y;
5231 my_print2(ar.get_top_node(0)); cout << endl;
5239 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5240 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5241 overall_coeff=numeric(number="0"))
5244 Be warned, however, that the set of properties and their meaning for each
5245 class may change between GiNaC versions.
5248 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5249 @c node-name, next, previous, up
5250 @chapter Extending GiNaC
5252 By reading so far you should have gotten a fairly good understanding of
5253 GiNaC's design-patterns. From here on you should start reading the
5254 sources. All we can do now is issue some recommendations how to tackle
5255 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5256 develop some useful extension please don't hesitate to contact the GiNaC
5257 authors---they will happily incorporate them into future versions.
5260 * What does not belong into GiNaC:: What to avoid.
5261 * Symbolic functions:: Implementing symbolic functions.
5262 * Structures:: Defining new algebraic classes (the easy way).
5263 * Adding classes:: Defining new algebraic classes (the hard way).
5267 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5268 @c node-name, next, previous, up
5269 @section What doesn't belong into GiNaC
5271 @cindex @command{ginsh}
5272 First of all, GiNaC's name must be read literally. It is designed to be
5273 a library for use within C++. The tiny @command{ginsh} accompanying
5274 GiNaC makes this even more clear: it doesn't even attempt to provide a
5275 language. There are no loops or conditional expressions in
5276 @command{ginsh}, it is merely a window into the library for the
5277 programmer to test stuff (or to show off). Still, the design of a
5278 complete CAS with a language of its own, graphical capabilities and all
5279 this on top of GiNaC is possible and is without doubt a nice project for
5282 There are many built-in functions in GiNaC that do not know how to
5283 evaluate themselves numerically to a precision declared at runtime
5284 (using @code{Digits}). Some may be evaluated at certain points, but not
5285 generally. This ought to be fixed. However, doing numerical
5286 computations with GiNaC's quite abstract classes is doomed to be
5287 inefficient. For this purpose, the underlying foundation classes
5288 provided by CLN are much better suited.
5291 @node Symbolic functions, Structures, What does not belong into GiNaC, Extending GiNaC
5292 @c node-name, next, previous, up
5293 @section Symbolic functions
5295 The easiest and most instructive way to start extending GiNaC is probably to
5296 create your own symbolic functions. These are implemented with the help of
5297 two preprocessor macros:
5299 @cindex @code{DECLARE_FUNCTION}
5300 @cindex @code{REGISTER_FUNCTION}
5302 DECLARE_FUNCTION_<n>P(<name>)
5303 REGISTER_FUNCTION(<name>, <options>)
5306 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5307 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5308 parameters of type @code{ex} and returns a newly constructed GiNaC
5309 @code{function} object that represents your function.
5311 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5312 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5313 set of options that associate the symbolic function with C++ functions you
5314 provide to implement the various methods such as evaluation, derivative,
5315 series expansion etc. They also describe additional attributes the function
5316 might have, such as symmetry and commutation properties, and a name for
5317 LaTeX output. Multiple options are separated by the member access operator
5318 @samp{.} and can be given in an arbitrary order.
5320 (By the way: in case you are worrying about all the macros above we can
5321 assure you that functions are GiNaC's most macro-intense classes. We have
5322 done our best to avoid macros where we can.)
5324 @subsection A minimal example
5326 Here is an example for the implementation of a function with two arguments
5327 that is not further evaluated:
5330 DECLARE_FUNCTION_2P(myfcn)
5332 static ex myfcn_eval(const ex & x, const ex & y)
5334 return myfcn(x, y).hold();
5337 REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
5340 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5341 in algebraic expressions:
5347 ex e = 2*myfcn(42, 3*x+1) - x;
5348 // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
5349 // the actual expression
5351 // prints '2*myfcn(42,1+3*x)-x'
5356 @cindex @code{hold()}
5358 The @code{eval_func()} option specifies the C++ function that implements
5359 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5360 the same number of arguments as the associated symbolic function (two in this
5361 case) and returns the (possibly transformed or in some way simplified)
5362 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5363 of the automatic evaluation process). If no (further) evaluation is to take
5364 place, the @code{eval_func()} function must return the original function
5365 with @code{.hold()}, to avoid a potential infinite recursion. If your
5366 symbolic functions produce a segmentation fault or stack overflow when
5367 using them in expressions, you are probably missing a @code{.hold()}
5370 There is not much you can do with the @code{myfcn} function. It merely acts
5371 as a kind of container for its arguments (which is, however, sometimes
5372 perfectly sufficient). Let's have a look at the implementation of GiNaC's
5375 @subsection The cosine function
5377 The GiNaC header file @file{inifcns.h} contains the line
5380 DECLARE_FUNCTION_1P(cos)
5383 which declares to all programs using GiNaC that there is a function @samp{cos}
5384 that takes one @code{ex} as an argument. This is all they need to know to use
5385 this function in expressions.
5387 The implementation of the cosine function is in @file{inifcns_trans.cpp}. The
5388 @code{eval_func()} function looks something like this (actually, it doesn't
5389 look like this at all, but it should give you an idea what is going on):
5392 static ex cos_eval(const ex & x)
5394 if (<x is a multiple of 2*Pi>)
5396 else if (<x is a multiple of Pi>)
5398 else if (<x is a multiple of Pi/2>)
5402 else if (<x has the form 'acos(y)'>)
5404 else if (<x has the form 'asin(y)'>)
5409 return cos(x).hold();
5413 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5414 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5415 symbolic transformation can be done, the unmodified function is returned
5416 with @code{.hold()}.
5418 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5419 The user has to call @code{evalf()} for that. This is implemented in a
5423 static ex cos_evalf(const ex & x)
5425 if (is_a<numeric>(x))
5426 return cos(ex_to<numeric>(x));
5428 return cos(x).hold();
5432 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5433 in this case the @code{cos()} function for @code{numeric} objects, which in
5434 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5435 isn't really needed here, but reminds us that the corresponding @code{eval()}
5436 function would require it in this place.
5438 Differentiation will surely turn up and so we need to tell @code{cos}
5439 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5440 instance, are then handled automatically by @code{basic::diff} and
5444 static ex cos_deriv(const ex & x, unsigned diff_param)
5450 @cindex product rule
5451 The second parameter is obligatory but uninteresting at this point. It
5452 specifies which parameter to differentiate in a partial derivative in
5453 case the function has more than one parameter, and its main application
5454 is for correct handling of the chain rule.
5456 An implementation of the series expansion is not needed for @code{cos()} as
5457 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5458 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5459 the other hand, does have poles and may need to do Laurent expansion:
5462 static ex tan_series(const ex & x, const relational & rel,
5463 int order, unsigned options)
5465 // Find the actual expansion point
5466 const ex x_pt = x.subs(rel);
5468 if (<x_pt is not an odd multiple of Pi/2>)
5469 throw do_taylor(); // tell function::series() to do Taylor expansion
5471 // On a pole, expand sin()/cos()
5472 return (sin(x)/cos(x)).series(rel, order+2, options);
5476 The @code{series()} implementation of a function @emph{must} return a
5477 @code{pseries} object, otherwise your code will crash.
5479 Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
5480 macro is used to tell the system how the @code{cos()} function behaves:
5483 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5484 evalf_func(cos_evalf).
5485 derivative_func(cos_deriv).
5486 latex_name("\\cos"));
5489 This registers the @code{cos_eval()}, @code{cos_evalf()} and
5490 @code{cos_deriv()} C++ functions with the @code{cos()} function, and also
5491 gives it a proper LaTeX name.
5493 @subsection Function options
5495 GiNaC functions understand several more options which are always
5496 specified as @code{.option(params)}. None of them are required, but you
5497 need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
5498 the @code{eval()} method).
5501 eval_func(<C++ function>)
5502 evalf_func(<C++ function>)
5503 derivative_func(<C++ function>)
5504 series_func(<C++ function>)
5507 These specify the C++ functions that implement symbolic evaluation,
5508 numeric evaluation, partial derivatives, and series expansion, respectively.
5509 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5510 @code{diff()} and @code{series()}.
5512 The @code{eval_func()} function needs to use @code{.hold()} if no further
5513 automatic evaluation is desired or possible.
5515 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5516 expansion, which is correct if there are no poles involved. If the function
5517 has poles in the complex plane, the @code{series_func()} needs to check
5518 whether the expansion point is on a pole and fall back to Taylor expansion
5519 if it isn't. Otherwise, the pole usually needs to be regularized by some
5520 suitable transformation.
5523 latex_name(const string & n)
5526 specifies the LaTeX code that represents the name of the function in LaTeX
5527 output. The default is to put the function name in an @code{\mbox@{@}}.
5530 do_not_evalf_params()
5533 This tells @code{evalf()} to not recursively evaluate the parameters of the
5534 function before calling the @code{evalf_func()}.
5537 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5540 This allows you to explicitly specify the commutation properties of the
5541 function (@xref{Non-commutative objects}, for an explanation of
5542 (non)commutativity in GiNaC). For example, you can use
5543 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5544 GiNaC treat your function like a matrix. By default, functions inherit the
5545 commutation properties of their first argument.
5548 set_symmetry(const symmetry & s)
5551 specifies the symmetry properties of the function with respect to its
5552 arguments. @xref{Indexed objects}, for an explanation of symmetry
5553 specifications. GiNaC will automatically rearrange the arguments of
5554 symmetric functions into a canonical order.
5557 @node Structures, Adding classes, Symbolic functions, Extending GiNaC
5558 @c node-name, next, previous, up
5561 If you are doing some very specialized things with GiNaC, or if you just
5562 need some more organized way to store data in your expressions instead of
5563 anonymous lists, you may want to implement your own algebraic classes.
5564 ('algebraic class' means any class directly or indirectly derived from
5565 @code{basic} that can be used in GiNaC expressions).
5567 GiNaC offers two ways of accomplishing this: either by using the
5568 @code{structure<T>} template class, or by rolling your own class from
5569 scratch. This section will discuss the @code{structure<T>} template which
5570 is easier to use but more limited, while the implementation of custom
5571 GiNaC classes is the topic of the next section. However, you may want to
5572 read both sections because many common concepts and member functions are
5573 shared by both concepts, and it will also allow you to decide which approach
5574 is most suited to your needs.
5576 The @code{structure<T>} template, defined in the GiNaC header file
5577 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
5578 or @code{class}) into a GiNaC object that can be used in expressions.
5580 @subsection Example: scalar products
5582 Let's suppose that we need a way to handle some kind of abstract scalar
5583 product of the form @samp{<x|y>} in expressions. Objects of the scalar
5584 product class have to store their left and right operands, which can in turn
5585 be arbitrary expressions. Here is a possible way to represent such a
5586 product in a C++ @code{struct}:
5590 using namespace std;
5592 #include <ginac/ginac.h>
5593 using namespace GiNaC;
5599 sprod_s(ex l, ex r) : left(l), right(r) @{@}
5603 The default constructor is required. Now, to make a GiNaC class out of this
5604 data structure, we need only one line:
5607 typedef structure<sprod_s> sprod;
5610 That's it. This line constructs an algebraic class @code{sprod} which
5611 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
5612 expressions like any other GiNaC class:
5616 symbol a("a"), b("b");
5617 ex e = sprod(sprod_s(a, b));
5621 Note the difference between @code{sprod} which is the algebraic class, and
5622 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
5623 and @code{right} data members. As shown above, an @code{sprod} can be
5624 constructed from an @code{sprod_s} object.
5626 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
5627 you could define a little wrapper function like this:
5630 inline ex make_sprod(ex left, ex right)
5632 return sprod(sprod_s(left, right));
5636 The @code{sprod_s} object contained in @code{sprod} can be accessed with
5637 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
5638 @code{get_struct()}:
5642 cout << ex_to<sprod>(e)->left << endl;
5644 cout << ex_to<sprod>(e).get_struct().right << endl;
5649 You only have read access to the members of @code{sprod_s}.
5651 The type definition of @code{sprod} is enough to write your own algorithms
5652 that deal with scalar products, for example:
5657 if (is_a<sprod>(p)) @{
5658 const sprod_s & sp = ex_to<sprod>(p).get_struct();
5659 return make_sprod(sp.right, sp.left);
5670 @subsection Structure output
5672 While the @code{sprod} type is useable it still leaves something to be
5673 desired, most notably proper output:
5678 // -> [structure object]
5682 By default, any structure types you define will be printed as
5683 @samp{[structure object]}. To override this, you can specialize the
5684 template's @code{print()} member function. The member functions of
5685 GiNaC classes are described in more detail in the next section, but
5686 it shouldn't be hard to figure out what's going on here:
5689 void sprod::print(const print_context & c, unsigned level) const
5691 // tree debug output handled by superclass
5692 if (is_a<print_tree>(c))
5693 inherited::print(c, level);
5695 // get the contained sprod_s object
5696 const sprod_s & sp = get_struct();
5698 // print_context::s is a reference to an ostream
5699 c.s << "<" << sp.left << "|" << sp.right << ">";
5703 Now we can print expressions containing scalar products:
5709 cout << swap_sprod(e) << endl;
5714 @subsection Comparing structures
5716 The @code{sprod} class defined so far still has one important drawback: all
5717 scalar products are treated as being equal because GiNaC doesn't know how to
5718 compare objects of type @code{sprod_s}. This can lead to some confusing
5719 and undesired behavior:
5723 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
5725 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
5726 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
5730 To remedy this, we first need to define the operators @code{==} and @code{<}
5731 for objects of type @code{sprod_s}:
5734 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
5736 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
5739 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
5741 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
5745 The ordering established by the @code{<} operator doesn't have to make any
5746 algebraic sense, but it needs to be well defined. Note that we can't use
5747 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
5748 in the implementation of these operators because they would construct
5749 GiNaC @code{relational} objects which in the case of @code{<} do not
5750 establish a well defined ordering (for arbitrary expressions, GiNaC can't
5751 decide which one is algebraically 'less').
5753 Next, we need to change our definition of the @code{sprod} type to let
5754 GiNaC know that an ordering relation exists for the embedded objects:
5757 typedef structure<sprod_s, compare_std_less> sprod;
5760 @code{sprod} objects then behave as expected:
5764 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
5765 // -> <a|b>-<a^2|b^2>
5766 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
5767 // -> <a|b>+<a^2|b^2>
5768 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
5770 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
5775 The @code{compare_std_less} policy parameter tells GiNaC to use the
5776 @code{std::less} and @code{std::equal_to} functors to compare objects of
5777 type @code{sprod_s}. By default, these functors forward their work to the
5778 standard @code{<} and @code{==} operators, which we have overloaded.
5779 Alternatively, we could have specialized @code{std::less} and
5780 @code{std::equal_to} for class @code{sprod_s}.
5782 GiNaC provides two other comparison policies for @code{structure<T>}
5783 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
5784 which does a bit-wise comparison of the contained @code{T} objects.
5785 This should be used with extreme care because it only works reliably with
5786 built-in integral types, and it also compares any padding (filler bytes of
5787 undefined value) that the @code{T} class might have.
5789 @subsection Subexpressions
5791 Our scalar product class has two subexpressions: the left and right
5792 operands. It might be a good idea to make them accessible via the standard
5793 @code{nops()} and @code{op()} methods:
5796 size_t sprod::nops() const
5801 ex sprod::op(size_t i) const
5805 return get_struct().left;
5807 return get_struct().right;
5809 throw std::range_error("sprod::op(): no such operand");
5814 Implementing @code{nops()} and @code{op()} for container types such as
5815 @code{sprod} has two other nice side effects:
5819 @code{has()} works as expected
5821 GiNaC generates better hash keys for the objects (the default implementation
5822 of @code{calchash()} takes subexpressions into account)
5825 @cindex @code{let_op()}
5826 There is a non-const variant of @code{op()} called @code{let_op()} that
5827 allows replacing subexpressions:
5830 ex & sprod::let_op(size_t i)
5832 // every non-const member function must call this
5833 ensure_if_modifiable();
5837 return get_struct().left;
5839 return get_struct().right;
5841 throw std::range_error("sprod::let_op(): no such operand");
5846 Once we have provided @code{let_op()} we also get @code{subs()} and
5847 @code{map()} for free. In fact, every container class that returns a non-null
5848 @code{nops()} value must either implement @code{let_op()} or provide custom
5849 implementations of @code{subs()} and @code{map()}.
5851 In turn, the availability of @code{map()} enables the recursive behavior of a
5852 couple of other default method implementations, in particular @code{evalf()},
5853 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
5854 we probably want to provide our own version of @code{expand()} for scalar
5855 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
5856 This is left as an exercise for the reader.
5858 The @code{structure<T>} template defines many more member functions that
5859 you can override by specialization to customize the behavior of your
5860 structures. You are referred to the next section for a description of
5861 some of these (especially @code{eval()}). There is, however, one topic
5862 that shall be addressed here, as it demonstrates one peculiarity of the
5863 @code{structure<T>} template: archiving.
5865 @subsection Archiving structures
5867 If you don't know how the archiving of GiNaC objects is implemented, you
5868 should first read the next section and then come back here. You're back?
5871 To implement archiving for structures it is not enough to provide
5872 specializations for the @code{archive()} member function and the
5873 unarchiving constructor (the @code{unarchive()} function has a default
5874 implementation). You also need to provide a unique name (as a string literal)
5875 for each structure type you define. This is because in GiNaC archives,
5876 the class of an object is stored as a string, the class name.
5878 By default, this class name (as returned by the @code{class_name()} member
5879 function) is @samp{structure} for all structure classes. This works as long
5880 as you have only defined one structure type, but if you use two or more you
5881 need to provide a different name for each by specializing the
5882 @code{get_class_name()} member function. Here is a sample implementation
5883 for enabling archiving of the scalar product type defined above:
5886 const char *sprod::get_class_name() @{ return "sprod"; @}
5888 void sprod::archive(archive_node & n) const
5890 inherited::archive(n);
5891 n.add_ex("left", get_struct().left);
5892 n.add_ex("right", get_struct().right);
5895 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
5897 n.find_ex("left", get_struct().left, sym_lst);
5898 n.find_ex("right", get_struct().right, sym_lst);
5902 Note that the unarchiving constructor is @code{sprod::structure} and not
5903 @code{sprod::sprod}, and that we don't need to supply an
5904 @code{sprod::unarchive()} function.
5907 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
5908 @c node-name, next, previous, up
5909 @section Adding classes
5911 The @code{structure<T>} template provides an way to extend GiNaC with custom
5912 algebraic classes that is easy to use but has its limitations, the most
5913 severe of which being that you can't add any new member functions to
5914 structures. To be able to do this, you need to write a new class definition
5917 This section will explain how to implement new algebraic classes in GiNaC by
5918 giving the example of a simple 'string' class. After reading this section
5919 you will know how to properly declare a GiNaC class and what the minimum
5920 required member functions are that you have to implement. We only cover the
5921 implementation of a 'leaf' class here (i.e. one that doesn't contain
5922 subexpressions). Creating a container class like, for example, a class
5923 representing tensor products is more involved but this section should give
5924 you enough information so you can consult the source to GiNaC's predefined
5925 classes if you want to implement something more complicated.
5927 @subsection GiNaC's run-time type information system
5929 @cindex hierarchy of classes
5931 All algebraic classes (that is, all classes that can appear in expressions)
5932 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
5933 @code{basic *} (which is essentially what an @code{ex} is) represents a
5934 generic pointer to an algebraic class. Occasionally it is necessary to find
5935 out what the class of an object pointed to by a @code{basic *} really is.
5936 Also, for the unarchiving of expressions it must be possible to find the
5937 @code{unarchive()} function of a class given the class name (as a string). A
5938 system that provides this kind of information is called a run-time type
5939 information (RTTI) system. The C++ language provides such a thing (see the
5940 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
5941 implements its own, simpler RTTI.
5943 The RTTI in GiNaC is based on two mechanisms:
5948 The @code{basic} class declares a member variable @code{tinfo_key} which
5949 holds an unsigned integer that identifies the object's class. These numbers
5950 are defined in the @file{tinfos.h} header file for the built-in GiNaC
5951 classes. They all start with @code{TINFO_}.
5954 By means of some clever tricks with static members, GiNaC maintains a list
5955 of information for all classes derived from @code{basic}. The information
5956 available includes the class names, the @code{tinfo_key}s, and pointers
5957 to the unarchiving functions. This class registry is defined in the
5958 @file{registrar.h} header file.
5962 The disadvantage of this proprietary RTTI implementation is that there's
5963 a little more to do when implementing new classes (C++'s RTTI works more
5964 or less automatic) but don't worry, most of the work is simplified by
5967 @subsection A minimalistic example
5969 Now we will start implementing a new class @code{mystring} that allows
5970 placing character strings in algebraic expressions (this is not very useful,
5971 but it's just an example). This class will be a direct subclass of
5972 @code{basic}. You can use this sample implementation as a starting point
5973 for your own classes.
5975 The code snippets given here assume that you have included some header files
5981 #include <stdexcept>
5982 using namespace std;
5984 #include <ginac/ginac.h>
5985 using namespace GiNaC;
5988 The first thing we have to do is to define a @code{tinfo_key} for our new
5989 class. This can be any arbitrary unsigned number that is not already taken
5990 by one of the existing classes but it's better to come up with something
5991 that is unlikely to clash with keys that might be added in the future. The
5992 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
5993 which is not a requirement but we are going to stick with this scheme:
5996 const unsigned TINFO_mystring = 0x42420001U;
5999 Now we can write down the class declaration. The class stores a C++
6000 @code{string} and the user shall be able to construct a @code{mystring}
6001 object from a C or C++ string:
6004 class mystring : public basic
6006 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
6009 mystring(const string &s);
6010 mystring(const char *s);
6016 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6019 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
6020 macros are defined in @file{registrar.h}. They take the name of the class
6021 and its direct superclass as arguments and insert all required declarations
6022 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
6023 the first line after the opening brace of the class definition. The
6024 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
6025 source (at global scope, of course, not inside a function).
6027 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
6028 declarations of the default constructor and a couple of other functions that
6029 are required. It also defines a type @code{inherited} which refers to the
6030 superclass so you don't have to modify your code every time you shuffle around
6031 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
6032 class with the GiNaC RTTI.
6034 Now there are seven member functions we have to implement to get a working
6040 @code{mystring()}, the default constructor.
6043 @code{void archive(archive_node &n)}, the archiving function. This stores all
6044 information needed to reconstruct an object of this class inside an
6045 @code{archive_node}.
6048 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
6049 constructor. This constructs an instance of the class from the information
6050 found in an @code{archive_node}.
6053 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
6054 unarchiving function. It constructs a new instance by calling the unarchiving
6058 @cindex @code{compare_same_type()}
6059 @code{int compare_same_type(const basic &other)}, which is used internally
6060 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
6061 -1, depending on the relative order of this object and the @code{other}
6062 object. If it returns 0, the objects are considered equal.
6063 @strong{Note:} This has nothing to do with the (numeric) ordering
6064 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
6065 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
6066 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
6067 must provide a @code{compare_same_type()} function, even those representing
6068 objects for which no reasonable algebraic ordering relationship can be
6072 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
6073 which are the two constructors we declared.
6077 Let's proceed step-by-step. The default constructor looks like this:
6080 mystring::mystring() : inherited(TINFO_mystring) @{@}
6083 The golden rule is that in all constructors you have to set the
6084 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
6085 it will be set by the constructor of the superclass and all hell will break
6086 loose in the RTTI. For your convenience, the @code{basic} class provides
6087 a constructor that takes a @code{tinfo_key} value, which we are using here
6088 (remember that in our case @code{inherited == basic}). If the superclass
6089 didn't have such a constructor, we would have to set the @code{tinfo_key}
6090 to the right value manually.
6092 In the default constructor you should set all other member variables to
6093 reasonable default values (we don't need that here since our @code{str}
6094 member gets set to an empty string automatically).
6096 Next are the three functions for archiving. You have to implement them even
6097 if you don't plan to use archives, but the minimum required implementation
6098 is really simple. First, the archiving function:
6101 void mystring::archive(archive_node &n) const
6103 inherited::archive(n);
6104 n.add_string("string", str);
6108 The only thing that is really required is calling the @code{archive()}
6109 function of the superclass. Optionally, you can store all information you
6110 deem necessary for representing the object into the passed
6111 @code{archive_node}. We are just storing our string here. For more
6112 information on how the archiving works, consult the @file{archive.h} header
6115 The unarchiving constructor is basically the inverse of the archiving
6119 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
6121 n.find_string("string", str);
6125 If you don't need archiving, just leave this function empty (but you must
6126 invoke the unarchiving constructor of the superclass). Note that we don't
6127 have to set the @code{tinfo_key} here because it is done automatically
6128 by the unarchiving constructor of the @code{basic} class.
6130 Finally, the unarchiving function:
6133 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
6135 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
6139 You don't have to understand how exactly this works. Just copy these
6140 four lines into your code literally (replacing the class name, of
6141 course). It calls the unarchiving constructor of the class and unless
6142 you are doing something very special (like matching @code{archive_node}s
6143 to global objects) you don't need a different implementation. For those
6144 who are interested: setting the @code{dynallocated} flag puts the object
6145 under the control of GiNaC's garbage collection. It will get deleted
6146 automatically once it is no longer referenced.
6148 Our @code{compare_same_type()} function uses a provided function to compare
6152 int mystring::compare_same_type(const basic &other) const
6154 const mystring &o = static_cast<const mystring &>(other);
6155 int cmpval = str.compare(o.str);
6158 else if (cmpval < 0)
6165 Although this function takes a @code{basic &}, it will always be a reference
6166 to an object of exactly the same class (objects of different classes are not
6167 comparable), so the cast is safe. If this function returns 0, the two objects
6168 are considered equal (in the sense that @math{A-B=0}), so you should compare
6169 all relevant member variables.
6171 Now the only thing missing is our two new constructors:
6174 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6175 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6178 No surprises here. We set the @code{str} member from the argument and
6179 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6181 That's it! We now have a minimal working GiNaC class that can store
6182 strings in algebraic expressions. Let's confirm that the RTTI works:
6185 ex e = mystring("Hello, world!");
6186 cout << is_a<mystring>(e) << endl;
6189 cout << e.bp->class_name() << endl;
6193 Obviously it does. Let's see what the expression @code{e} looks like:
6197 // -> [mystring object]
6200 Hm, not exactly what we expect, but of course the @code{mystring} class
6201 doesn't yet know how to print itself. This is done in the @code{print()}
6202 member function. Let's say that we wanted to print the string surrounded
6206 class mystring : public basic
6210 void print(const print_context &c, unsigned level = 0) const;
6214 void mystring::print(const print_context &c, unsigned level) const
6216 // print_context::s is a reference to an ostream
6217 c.s << '\"' << str << '\"';
6221 The @code{level} argument is only required for container classes to
6222 correctly parenthesize the output. Let's try again to print the expression:
6226 // -> "Hello, world!"
6229 Much better. The @code{mystring} class can be used in arbitrary expressions:
6232 e += mystring("GiNaC rulez");
6234 // -> "GiNaC rulez"+"Hello, world!"
6237 (GiNaC's automatic term reordering is in effect here), or even
6240 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
6242 // -> "One string"^(2*sin(-"Another string"+Pi))
6245 Whether this makes sense is debatable but remember that this is only an
6246 example. At least it allows you to implement your own symbolic algorithms
6249 Note that GiNaC's algebraic rules remain unchanged:
6252 e = mystring("Wow") * mystring("Wow");
6256 e = pow(mystring("First")-mystring("Second"), 2);
6257 cout << e.expand() << endl;
6258 // -> -2*"First"*"Second"+"First"^2+"Second"^2
6261 There's no way to, for example, make GiNaC's @code{add} class perform string
6262 concatenation. You would have to implement this yourself.
6264 @subsection Automatic evaluation
6267 @cindex @code{eval()}
6268 @cindex @code{hold()}
6269 When dealing with objects that are just a little more complicated than the
6270 simple string objects we have implemented, chances are that you will want to
6271 have some automatic simplifications or canonicalizations performed on them.
6272 This is done in the evaluation member function @code{eval()}. Let's say that
6273 we wanted all strings automatically converted to lowercase with
6274 non-alphabetic characters stripped, and empty strings removed:
6277 class mystring : public basic
6281 ex eval(int level = 0) const;
6285 ex mystring::eval(int level) const
6288 for (int i=0; i<str.length(); i++) @{
6290 if (c >= 'A' && c <= 'Z')
6291 new_str += tolower(c);
6292 else if (c >= 'a' && c <= 'z')
6296 if (new_str.length() == 0)
6299 return mystring(new_str).hold();
6303 The @code{level} argument is used to limit the recursion depth of the
6304 evaluation. We don't have any subexpressions in the @code{mystring}
6305 class so we are not concerned with this. If we had, we would call the
6306 @code{eval()} functions of the subexpressions with @code{level - 1} as
6307 the argument if @code{level != 1}. The @code{hold()} member function
6308 sets a flag in the object that prevents further evaluation. Otherwise
6309 we might end up in an endless loop. When you want to return the object
6310 unmodified, use @code{return this->hold();}.
6312 Let's confirm that it works:
6315 ex e = mystring("Hello, world!") + mystring("!?#");
6319 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
6324 @subsection Optional member functions
6326 We have implemented only a small set of member functions to make the class
6327 work in the GiNaC framework. There are two functions that are not strictly
6328 required but will make operations with objects of the class more efficient:
6330 @cindex @code{calchash()}
6331 @cindex @code{is_equal_same_type()}
6333 unsigned calchash() const;
6334 bool is_equal_same_type(const basic &other) const;
6337 The @code{calchash()} method returns an @code{unsigned} hash value for the
6338 object which will allow GiNaC to compare and canonicalize expressions much
6339 more efficiently. You should consult the implementation of some of the built-in
6340 GiNaC classes for examples of hash functions. The default implementation of
6341 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
6342 class and all subexpressions that are accessible via @code{op()}.
6344 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
6345 tests for equality without establishing an ordering relation, which is often
6346 faster. The default implementation of @code{is_equal_same_type()} just calls
6347 @code{compare_same_type()} and tests its result for zero.
6349 @subsection Other member functions
6351 For a real algebraic class, there are probably some more functions that you
6352 might want to provide:
6355 bool info(unsigned inf) const;
6356 ex evalf(int level = 0) const;
6357 ex series(const relational & r, int order, unsigned options = 0) const;
6358 ex derivative(const symbol & s) const;
6361 If your class stores sub-expressions (see the scalar product example in the
6362 previous section) you will probably want to override
6364 @cindex @code{let_op()}
6367 ex op(size_t i) const;
6368 ex & let_op(size_t i);
6369 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
6370 ex map(map_function & f) const;
6373 @code{let_op()} is a variant of @code{op()} that allows write access. The
6374 default implementations of @code{subs()} and @code{map()} use it, so you have
6375 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
6377 You can, of course, also add your own new member functions. Remember
6378 that the RTTI may be used to get information about what kinds of objects
6379 you are dealing with (the position in the class hierarchy) and that you
6380 can always extract the bare object from an @code{ex} by stripping the
6381 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
6382 should become a need.
6384 That's it. May the source be with you!
6387 @node A Comparison With Other CAS, Advantages, Adding classes, Top
6388 @c node-name, next, previous, up
6389 @chapter A Comparison With Other CAS
6392 This chapter will give you some information on how GiNaC compares to
6393 other, traditional Computer Algebra Systems, like @emph{Maple},
6394 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
6395 disadvantages over these systems.
6398 * Advantages:: Strengths of the GiNaC approach.
6399 * Disadvantages:: Weaknesses of the GiNaC approach.
6400 * Why C++?:: Attractiveness of C++.
6403 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
6404 @c node-name, next, previous, up
6407 GiNaC has several advantages over traditional Computer
6408 Algebra Systems, like
6413 familiar language: all common CAS implement their own proprietary
6414 grammar which you have to learn first (and maybe learn again when your
6415 vendor decides to `enhance' it). With GiNaC you can write your program
6416 in common C++, which is standardized.
6420 structured data types: you can build up structured data types using
6421 @code{struct}s or @code{class}es together with STL features instead of
6422 using unnamed lists of lists of lists.
6425 strongly typed: in CAS, you usually have only one kind of variables
6426 which can hold contents of an arbitrary type. This 4GL like feature is
6427 nice for novice programmers, but dangerous.
6430 development tools: powerful development tools exist for C++, like fancy
6431 editors (e.g. with automatic indentation and syntax highlighting),
6432 debuggers, visualization tools, documentation generators@dots{}
6435 modularization: C++ programs can easily be split into modules by
6436 separating interface and implementation.
6439 price: GiNaC is distributed under the GNU Public License which means
6440 that it is free and available with source code. And there are excellent
6441 C++-compilers for free, too.
6444 extendable: you can add your own classes to GiNaC, thus extending it on
6445 a very low level. Compare this to a traditional CAS that you can
6446 usually only extend on a high level by writing in the language defined
6447 by the parser. In particular, it turns out to be almost impossible to
6448 fix bugs in a traditional system.
6451 multiple interfaces: Though real GiNaC programs have to be written in
6452 some editor, then be compiled, linked and executed, there are more ways
6453 to work with the GiNaC engine. Many people want to play with
6454 expressions interactively, as in traditional CASs. Currently, two such
6455 windows into GiNaC have been implemented and many more are possible: the
6456 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
6457 types to a command line and second, as a more consistent approach, an
6458 interactive interface to the Cint C++ interpreter has been put together
6459 (called GiNaC-cint) that allows an interactive scripting interface
6460 consistent with the C++ language. It is available from the usual GiNaC
6464 seamless integration: it is somewhere between difficult and impossible
6465 to call CAS functions from within a program written in C++ or any other
6466 programming language and vice versa. With GiNaC, your symbolic routines
6467 are part of your program. You can easily call third party libraries,
6468 e.g. for numerical evaluation or graphical interaction. All other
6469 approaches are much more cumbersome: they range from simply ignoring the
6470 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
6471 system (i.e. @emph{Yacas}).
6474 efficiency: often large parts of a program do not need symbolic
6475 calculations at all. Why use large integers for loop variables or
6476 arbitrary precision arithmetics where @code{int} and @code{double} are
6477 sufficient? For pure symbolic applications, GiNaC is comparable in
6478 speed with other CAS.
6483 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
6484 @c node-name, next, previous, up
6485 @section Disadvantages
6487 Of course it also has some disadvantages:
6492 advanced features: GiNaC cannot compete with a program like
6493 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
6494 which grows since 1981 by the work of dozens of programmers, with
6495 respect to mathematical features. Integration, factorization,
6496 non-trivial simplifications, limits etc. are missing in GiNaC (and are
6497 not planned for the near future).
6500 portability: While the GiNaC library itself is designed to avoid any
6501 platform dependent features (it should compile on any ANSI compliant C++
6502 compiler), the currently used version of the CLN library (fast large
6503 integer and arbitrary precision arithmetics) can only by compiled
6504 without hassle on systems with the C++ compiler from the GNU Compiler
6505 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
6506 macros to let the compiler gather all static initializations, which
6507 works for GNU C++ only. Feel free to contact the authors in case you
6508 really believe that you need to use a different compiler. We have
6509 occasionally used other compilers and may be able to give you advice.}
6510 GiNaC uses recent language features like explicit constructors, mutable
6511 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
6512 literally. Recent GCC versions starting at 2.95.3, although itself not
6513 yet ANSI compliant, support all needed features.
6518 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
6519 @c node-name, next, previous, up
6522 Why did we choose to implement GiNaC in C++ instead of Java or any other
6523 language? C++ is not perfect: type checking is not strict (casting is
6524 possible), separation between interface and implementation is not
6525 complete, object oriented design is not enforced. The main reason is
6526 the often scolded feature of operator overloading in C++. While it may
6527 be true that operating on classes with a @code{+} operator is rarely
6528 meaningful, it is perfectly suited for algebraic expressions. Writing
6529 @math{3x+5y} as @code{3*x+5*y} instead of
6530 @code{x.times(3).plus(y.times(5))} looks much more natural.
6531 Furthermore, the main developers are more familiar with C++ than with
6532 any other programming language.
6535 @node Internal Structures, Expressions are reference counted, Why C++? , Top
6536 @c node-name, next, previous, up
6537 @appendix Internal Structures
6540 * Expressions are reference counted::
6541 * Internal representation of products and sums::
6544 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
6545 @c node-name, next, previous, up
6546 @appendixsection Expressions are reference counted
6548 @cindex reference counting
6549 @cindex copy-on-write
6550 @cindex garbage collection
6551 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
6552 where the counter belongs to the algebraic objects derived from class
6553 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
6554 which @code{ex} contains an instance. If you understood that, you can safely
6555 skip the rest of this passage.
6557 Expressions are extremely light-weight since internally they work like
6558 handles to the actual representation. They really hold nothing more
6559 than a pointer to some other object. What this means in practice is
6560 that whenever you create two @code{ex} and set the second equal to the
6561 first no copying process is involved. Instead, the copying takes place
6562 as soon as you try to change the second. Consider the simple sequence
6567 #include <ginac/ginac.h>
6568 using namespace std;
6569 using namespace GiNaC;
6573 symbol x("x"), y("y"), z("z");
6576 e1 = sin(x + 2*y) + 3*z + 41;
6577 e2 = e1; // e2 points to same object as e1
6578 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
6579 e2 += 1; // e2 is copied into a new object
6580 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
6584 The line @code{e2 = e1;} creates a second expression pointing to the
6585 object held already by @code{e1}. The time involved for this operation
6586 is therefore constant, no matter how large @code{e1} was. Actual
6587 copying, however, must take place in the line @code{e2 += 1;} because
6588 @code{e1} and @code{e2} are not handles for the same object any more.
6589 This concept is called @dfn{copy-on-write semantics}. It increases
6590 performance considerably whenever one object occurs multiple times and
6591 represents a simple garbage collection scheme because when an @code{ex}
6592 runs out of scope its destructor checks whether other expressions handle
6593 the object it points to too and deletes the object from memory if that
6594 turns out not to be the case. A slightly less trivial example of
6595 differentiation using the chain-rule should make clear how powerful this
6600 symbol x("x"), y("y");
6604 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
6605 cout << e1 << endl // prints x+3*y
6606 << e2 << endl // prints (x+3*y)^3
6607 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
6611 Here, @code{e1} will actually be referenced three times while @code{e2}
6612 will be referenced two times. When the power of an expression is built,
6613 that expression needs not be copied. Likewise, since the derivative of
6614 a power of an expression can be easily expressed in terms of that
6615 expression, no copying of @code{e1} is involved when @code{e3} is
6616 constructed. So, when @code{e3} is constructed it will print as
6617 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
6618 holds a reference to @code{e2} and the factor in front is just
6621 As a user of GiNaC, you cannot see this mechanism of copy-on-write
6622 semantics. When you insert an expression into a second expression, the
6623 result behaves exactly as if the contents of the first expression were
6624 inserted. But it may be useful to remember that this is not what
6625 happens. Knowing this will enable you to write much more efficient
6626 code. If you still have an uncertain feeling with copy-on-write
6627 semantics, we recommend you have a look at the
6628 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
6629 Marshall Cline. Chapter 16 covers this issue and presents an
6630 implementation which is pretty close to the one in GiNaC.
6633 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
6634 @c node-name, next, previous, up
6635 @appendixsection Internal representation of products and sums
6637 @cindex representation
6640 @cindex @code{power}
6641 Although it should be completely transparent for the user of
6642 GiNaC a short discussion of this topic helps to understand the sources
6643 and also explain performance to a large degree. Consider the
6644 unexpanded symbolic expression
6646 $2d^3 \left( 4a + 5b - 3 \right)$
6649 @math{2*d^3*(4*a+5*b-3)}
6651 which could naively be represented by a tree of linear containers for
6652 addition and multiplication, one container for exponentiation with base
6653 and exponent and some atomic leaves of symbols and numbers in this
6658 @cindex pair-wise representation
6659 However, doing so results in a rather deeply nested tree which will
6660 quickly become inefficient to manipulate. We can improve on this by
6661 representing the sum as a sequence of terms, each one being a pair of a
6662 purely numeric multiplicative coefficient and its rest. In the same
6663 spirit we can store the multiplication as a sequence of terms, each
6664 having a numeric exponent and a possibly complicated base, the tree
6665 becomes much more flat:
6669 The number @code{3} above the symbol @code{d} shows that @code{mul}
6670 objects are treated similarly where the coefficients are interpreted as
6671 @emph{exponents} now. Addition of sums of terms or multiplication of
6672 products with numerical exponents can be coded to be very efficient with
6673 such a pair-wise representation. Internally, this handling is performed
6674 by most CAS in this way. It typically speeds up manipulations by an
6675 order of magnitude. The overall multiplicative factor @code{2} and the
6676 additive term @code{-3} look somewhat out of place in this
6677 representation, however, since they are still carrying a trivial
6678 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
6679 this is avoided by adding a field that carries an overall numeric
6680 coefficient. This results in the realistic picture of internal
6683 $2d^3 \left( 4a + 5b - 3 \right)$:
6686 @math{2*d^3*(4*a+5*b-3)}:
6692 This also allows for a better handling of numeric radicals, since
6693 @code{sqrt(2)} can now be carried along calculations. Now it should be
6694 clear, why both classes @code{add} and @code{mul} are derived from the
6695 same abstract class: the data representation is the same, only the
6696 semantics differs. In the class hierarchy, methods for polynomial
6697 expansion and the like are reimplemented for @code{add} and @code{mul},
6698 but the data structure is inherited from @code{expairseq}.
6701 @node Package Tools, ginac-config, Internal representation of products and sums, Top
6702 @c node-name, next, previous, up
6703 @appendix Package Tools
6705 If you are creating a software package that uses the GiNaC library,
6706 setting the correct command line options for the compiler and linker
6707 can be difficult. GiNaC includes two tools to make this process easier.
6710 * ginac-config:: A shell script to detect compiler and linker flags.
6711 * AM_PATH_GINAC:: Macro for GNU automake.
6715 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
6716 @c node-name, next, previous, up
6717 @section @command{ginac-config}
6718 @cindex ginac-config
6720 @command{ginac-config} is a shell script that you can use to determine
6721 the compiler and linker command line options required to compile and
6722 link a program with the GiNaC library.
6724 @command{ginac-config} takes the following flags:
6728 Prints out the version of GiNaC installed.
6730 Prints '-I' flags pointing to the installed header files.
6732 Prints out the linker flags necessary to link a program against GiNaC.
6733 @item --prefix[=@var{PREFIX}]
6734 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
6735 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
6736 Otherwise, prints out the configured value of @env{$prefix}.
6737 @item --exec-prefix[=@var{PREFIX}]
6738 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
6739 Otherwise, prints out the configured value of @env{$exec_prefix}.
6742 Typically, @command{ginac-config} will be used within a configure
6743 script, as described below. It, however, can also be used directly from
6744 the command line using backquotes to compile a simple program. For
6748 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
6751 This command line might expand to (for example):
6754 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
6755 -lginac -lcln -lstdc++
6758 Not only is the form using @command{ginac-config} easier to type, it will
6759 work on any system, no matter how GiNaC was configured.
6762 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
6763 @c node-name, next, previous, up
6764 @section @samp{AM_PATH_GINAC}
6765 @cindex AM_PATH_GINAC
6767 For packages configured using GNU automake, GiNaC also provides
6768 a macro to automate the process of checking for GiNaC.
6771 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
6779 Determines the location of GiNaC using @command{ginac-config}, which is
6780 either found in the user's path, or from the environment variable
6781 @env{GINACLIB_CONFIG}.
6784 Tests the installed libraries to make sure that their version
6785 is later than @var{MINIMUM-VERSION}. (A default version will be used
6789 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
6790 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
6791 variable to the output of @command{ginac-config --libs}, and calls
6792 @samp{AC_SUBST()} for these variables so they can be used in generated
6793 makefiles, and then executes @var{ACTION-IF-FOUND}.
6796 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
6797 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
6801 This macro is in file @file{ginac.m4} which is installed in
6802 @file{$datadir/aclocal}. Note that if automake was installed with a
6803 different @samp{--prefix} than GiNaC, you will either have to manually
6804 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
6805 aclocal the @samp{-I} option when running it.
6808 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
6809 * Example package:: Example of a package using AM_PATH_GINAC.
6813 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
6814 @c node-name, next, previous, up
6815 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
6817 Simply make sure that @command{ginac-config} is in your path, and run
6818 the configure script.
6825 The directory where the GiNaC libraries are installed needs
6826 to be found by your system's dynamic linker.
6828 This is generally done by
6831 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
6837 setting the environment variable @env{LD_LIBRARY_PATH},
6840 or, as a last resort,
6843 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
6844 running configure, for instance:
6847 LDFLAGS=-R/home/cbauer/lib ./configure
6852 You can also specify a @command{ginac-config} not in your path by
6853 setting the @env{GINACLIB_CONFIG} environment variable to the
6854 name of the executable
6857 If you move the GiNaC package from its installed location,
6858 you will either need to modify @command{ginac-config} script
6859 manually to point to the new location or rebuild GiNaC.
6870 --with-ginac-prefix=@var{PREFIX}
6871 --with-ginac-exec-prefix=@var{PREFIX}
6874 are provided to override the prefix and exec-prefix that were stored
6875 in the @command{ginac-config} shell script by GiNaC's configure. You are
6876 generally better off configuring GiNaC with the right path to begin with.
6880 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
6881 @c node-name, next, previous, up
6882 @subsection Example of a package using @samp{AM_PATH_GINAC}
6884 The following shows how to build a simple package using automake
6885 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
6888 #include <ginac/ginac.h>
6892 GiNaC::symbol x("x");
6893 GiNaC::ex a = GiNaC::sin(x);
6894 std::cout << "Derivative of " << a
6895 << " is " << a.diff(x) << std::endl;
6900 You should first read the introductory portions of the automake
6901 Manual, if you are not already familiar with it.
6903 Two files are needed, @file{configure.in}, which is used to build the
6907 dnl Process this file with autoconf to produce a configure script.
6909 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
6915 AM_PATH_GINAC(0.9.0, [
6916 LIBS="$LIBS $GINACLIB_LIBS"
6917 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
6918 ], AC_MSG_ERROR([need to have GiNaC installed]))
6923 The only command in this which is not standard for automake
6924 is the @samp{AM_PATH_GINAC} macro.
6926 That command does the following: If a GiNaC version greater or equal
6927 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
6928 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
6929 the error message `need to have GiNaC installed'
6931 And the @file{Makefile.am}, which will be used to build the Makefile.
6934 ## Process this file with automake to produce Makefile.in
6935 bin_PROGRAMS = simple
6936 simple_SOURCES = simple.cpp
6939 This @file{Makefile.am}, says that we are building a single executable,
6940 from a single source file @file{simple.cpp}. Since every program
6941 we are building uses GiNaC we simply added the GiNaC options
6942 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
6943 want to specify them on a per-program basis: for instance by
6947 simple_LDADD = $(GINACLIB_LIBS)
6948 INCLUDES = $(GINACLIB_CPPFLAGS)
6951 to the @file{Makefile.am}.
6953 To try this example out, create a new directory and add the three
6956 Now execute the following commands:
6959 $ automake --add-missing
6964 You now have a package that can be built in the normal fashion
6973 @node Bibliography, Concept Index, Example package, Top
6974 @c node-name, next, previous, up
6975 @appendix Bibliography
6980 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
6983 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
6986 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
6989 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
6992 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
6993 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
6996 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
6997 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
6998 Academic Press, London
7001 @cite{Computer Algebra Systems - A Practical Guide},
7002 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
7005 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
7006 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
7009 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
7010 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
7013 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
7018 @node Concept Index, , Bibliography, Top
7019 @c node-name, next, previous, up
7020 @unnumbered Concept Index