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) 1999-2005 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, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2005 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{} 1999-2005 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., 51 Franklin Street, Fifth Floor, 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 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
483 process as well, since some of the source files are automatically
484 generated by Perl scripts. Last but not least, Bruno Haible's library
485 CLN is extensively used and needs to be installed on your system.
486 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
487 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
488 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
489 site} (it is covered by GPL) and install it prior to trying to install
490 GiNaC. The configure script checks if it can find it and if it cannot
491 it will refuse to continue.
494 @node Configuration, Building GiNaC, Prerequisites, Installation
495 @c node-name, next, previous, up
496 @section Configuration
497 @cindex configuration
500 To configure GiNaC means to prepare the source distribution for
501 building. It is done via a shell script called @command{configure} that
502 is shipped with the sources and was originally generated by GNU
503 Autoconf. Since a configure script generated by GNU Autoconf never
504 prompts, all customization must be done either via command line
505 parameters or environment variables. It accepts a list of parameters,
506 the complete set of which can be listed by calling it with the
507 @option{--help} option. The most important ones will be shortly
508 described in what follows:
513 @option{--disable-shared}: When given, this option switches off the
514 build of a shared library, i.e. a @file{.so} file. This may be convenient
515 when developing because it considerably speeds up compilation.
518 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
519 and headers are installed. It defaults to @file{/usr/local} which means
520 that the library is installed in the directory @file{/usr/local/lib},
521 the header files in @file{/usr/local/include/ginac} and the documentation
522 (like this one) into @file{/usr/local/share/doc/GiNaC}.
525 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
526 the library installed in some other directory than
527 @file{@var{PREFIX}/lib/}.
530 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
531 to have the header files installed in some other directory than
532 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
533 @option{--includedir=/usr/include} you will end up with the header files
534 sitting in the directory @file{/usr/include/ginac/}. Note that the
535 subdirectory @file{ginac} is enforced by this process in order to
536 keep the header files separated from others. This avoids some
537 clashes and allows for an easier deinstallation of GiNaC. This ought
538 to be considered A Good Thing (tm).
541 @option{--datadir=@var{DATADIR}}: This option may be given in case you
542 want to have the documentation installed in some other directory than
543 @file{@var{PREFIX}/share/doc/GiNaC/}.
547 In addition, you may specify some environment variables. @env{CXX}
548 holds the path and the name of the C++ compiler in case you want to
549 override the default in your path. (The @command{configure} script
550 searches your path for @command{c++}, @command{g++}, @command{gcc},
551 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
552 be very useful to define some compiler flags with the @env{CXXFLAGS}
553 environment variable, like optimization, debugging information and
554 warning levels. If omitted, it defaults to @option{-g
555 -O2}.@footnote{The @command{configure} script is itself generated from
556 the file @file{configure.ac}. It is only distributed in packaged
557 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
558 must generate @command{configure} along with the various
559 @file{Makefile.in} by using the @command{autogen.sh} script. This will
560 require a fair amount of support from your local toolchain, though.}
562 The whole process is illustrated in the following two
563 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
564 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
567 Here is a simple configuration for a site-wide GiNaC library assuming
568 everything is in default paths:
571 $ export CXXFLAGS="-Wall -O2"
575 And here is a configuration for a private static GiNaC library with
576 several components sitting in custom places (site-wide GCC and private
577 CLN). The compiler is persuaded to be picky and full assertions and
578 debugging information are switched on:
581 $ export CXX=/usr/local/gnu/bin/c++
582 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
583 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
584 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
585 $ ./configure --disable-shared --prefix=$(HOME)
589 @node Building GiNaC, Installing GiNaC, Configuration, Installation
590 @c node-name, next, previous, up
591 @section Building GiNaC
592 @cindex building GiNaC
594 After proper configuration you should just build the whole
599 at the command prompt and go for a cup of coffee. The exact time it
600 takes to compile GiNaC depends not only on the speed of your machines
601 but also on other parameters, for instance what value for @env{CXXFLAGS}
602 you entered. Optimization may be very time-consuming.
604 Just to make sure GiNaC works properly you may run a collection of
605 regression tests by typing
611 This will compile some sample programs, run them and check the output
612 for correctness. The regression tests fall in three categories. First,
613 the so called @emph{exams} are performed, simple tests where some
614 predefined input is evaluated (like a pupils' exam). Second, the
615 @emph{checks} test the coherence of results among each other with
616 possible random input. Third, some @emph{timings} are performed, which
617 benchmark some predefined problems with different sizes and display the
618 CPU time used in seconds. Each individual test should return a message
619 @samp{passed}. This is mostly intended to be a QA-check if something
620 was broken during development, not a sanity check of your system. Some
621 of the tests in sections @emph{checks} and @emph{timings} may require
622 insane amounts of memory and CPU time. Feel free to kill them if your
623 machine catches fire. Another quite important intent is to allow people
624 to fiddle around with optimization.
626 By default, the only documentation that will be built is this tutorial
627 in @file{.info} format. To build the GiNaC tutorial and reference manual
628 in HTML, DVI, PostScript, or PDF formats, use one of
637 Generally, the top-level Makefile runs recursively to the
638 subdirectories. It is therefore safe to go into any subdirectory
639 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
640 @var{target} there in case something went wrong.
643 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
644 @c node-name, next, previous, up
645 @section Installing GiNaC
648 To install GiNaC on your system, simply type
654 As described in the section about configuration the files will be
655 installed in the following directories (the directories will be created
656 if they don't already exist):
661 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
662 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
663 So will @file{libginac.so} unless the configure script was
664 given the option @option{--disable-shared}. The proper symlinks
665 will be established as well.
668 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
669 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
672 All documentation (info) will be stuffed into
673 @file{@var{PREFIX}/share/doc/GiNaC/} (or
674 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
678 For the sake of completeness we will list some other useful make
679 targets: @command{make clean} deletes all files generated by
680 @command{make}, i.e. all the object files. In addition @command{make
681 distclean} removes all files generated by the configuration and
682 @command{make maintainer-clean} goes one step further and deletes files
683 that may require special tools to rebuild (like the @command{libtool}
684 for instance). Finally @command{make uninstall} removes the installed
685 library, header files and documentation@footnote{Uninstallation does not
686 work after you have called @command{make distclean} since the
687 @file{Makefile} is itself generated by the configuration from
688 @file{Makefile.in} and hence deleted by @command{make distclean}. There
689 are two obvious ways out of this dilemma. First, you can run the
690 configuration again with the same @var{PREFIX} thus creating a
691 @file{Makefile} with a working @samp{uninstall} target. Second, you can
692 do it by hand since you now know where all the files went during
696 @node Basic Concepts, Expressions, Installing GiNaC, Top
697 @c node-name, next, previous, up
698 @chapter Basic Concepts
700 This chapter will describe the different fundamental objects that can be
701 handled by GiNaC. But before doing so, it is worthwhile introducing you
702 to the more commonly used class of expressions, representing a flexible
703 meta-class for storing all mathematical objects.
706 * Expressions:: The fundamental GiNaC class.
707 * Automatic evaluation:: Evaluation and canonicalization.
708 * Error handling:: How the library reports errors.
709 * The Class Hierarchy:: Overview of GiNaC's classes.
710 * Symbols:: Symbolic objects.
711 * Numbers:: Numerical objects.
712 * Constants:: Pre-defined constants.
713 * Fundamental containers:: Sums, products and powers.
714 * Lists:: Lists of expressions.
715 * Mathematical functions:: Mathematical functions.
716 * Relations:: Equality, Inequality and all that.
717 * Integrals:: Symbolic integrals.
718 * Matrices:: Matrices.
719 * Indexed objects:: Handling indexed quantities.
720 * Non-commutative objects:: Algebras with non-commutative products.
721 * Hash Maps:: A faster alternative to std::map<>.
725 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
726 @c node-name, next, previous, up
728 @cindex expression (class @code{ex})
731 The most common class of objects a user deals with is the expression
732 @code{ex}, representing a mathematical object like a variable, number,
733 function, sum, product, etc@dots{} Expressions may be put together to form
734 new expressions, passed as arguments to functions, and so on. Here is a
735 little collection of valid expressions:
738 ex MyEx1 = 5; // simple number
739 ex MyEx2 = x + 2*y; // polynomial in x and y
740 ex MyEx3 = (x + 1)/(x - 1); // rational expression
741 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
742 ex MyEx5 = MyEx4 + 1; // similar to above
745 Expressions are handles to other more fundamental objects, that often
746 contain other expressions thus creating a tree of expressions
747 (@xref{Internal Structures}, for particular examples). Most methods on
748 @code{ex} therefore run top-down through such an expression tree. For
749 example, the method @code{has()} scans recursively for occurrences of
750 something inside an expression. Thus, if you have declared @code{MyEx4}
751 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
752 the argument of @code{sin} and hence return @code{true}.
754 The next sections will outline the general picture of GiNaC's class
755 hierarchy and describe the classes of objects that are handled by
758 @subsection Note: Expressions and STL containers
760 GiNaC expressions (@code{ex} objects) have value semantics (they can be
761 assigned, reassigned and copied like integral types) but the operator
762 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
763 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
765 This implies that in order to use expressions in sorted containers such as
766 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
767 comparison predicate. GiNaC provides such a predicate, called
768 @code{ex_is_less}. For example, a set of expressions should be defined
769 as @code{std::set<ex, ex_is_less>}.
771 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
772 don't pose a problem. A @code{std::vector<ex>} works as expected.
774 @xref{Information About Expressions}, for more about comparing and ordering
778 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
779 @c node-name, next, previous, up
780 @section Automatic evaluation and canonicalization of expressions
783 GiNaC performs some automatic transformations on expressions, to simplify
784 them and put them into a canonical form. Some examples:
787 ex MyEx1 = 2*x - 1 + x; // 3*x-1
788 ex MyEx2 = x - x; // 0
789 ex MyEx3 = cos(2*Pi); // 1
790 ex MyEx4 = x*y/x; // y
793 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
794 evaluation}. GiNaC only performs transformations that are
798 at most of complexity
806 algebraically correct, possibly except for a set of measure zero (e.g.
807 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
810 There are two types of automatic transformations in GiNaC that may not
811 behave in an entirely obvious way at first glance:
815 The terms of sums and products (and some other things like the arguments of
816 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
817 into a canonical form that is deterministic, but not lexicographical or in
818 any other way easy to guess (it almost always depends on the number and
819 order of the symbols you define). However, constructing the same expression
820 twice, either implicitly or explicitly, will always result in the same
823 Expressions of the form 'number times sum' are automatically expanded (this
824 has to do with GiNaC's internal representation of sums and products). For
827 ex MyEx5 = 2*(x + y); // 2*x+2*y
828 ex MyEx6 = z*(x + y); // z*(x+y)
832 The general rule is that when you construct expressions, GiNaC automatically
833 creates them in canonical form, which might differ from the form you typed in
834 your program. This may create some awkward looking output (@samp{-y+x} instead
835 of @samp{x-y}) but allows for more efficient operation and usually yields
836 some immediate simplifications.
838 @cindex @code{eval()}
839 Internally, the anonymous evaluator in GiNaC is implemented by the methods
842 ex ex::eval(int level = 0) const;
843 ex basic::eval(int level = 0) const;
846 but unless you are extending GiNaC with your own classes or functions, there
847 should never be any reason to call them explicitly. All GiNaC methods that
848 transform expressions, like @code{subs()} or @code{normal()}, automatically
849 re-evaluate their results.
852 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
853 @c node-name, next, previous, up
854 @section Error handling
856 @cindex @code{pole_error} (class)
858 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
859 generated by GiNaC are subclassed from the standard @code{exception} class
860 defined in the @file{<stdexcept>} header. In addition to the predefined
861 @code{logic_error}, @code{domain_error}, @code{out_of_range},
862 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
863 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
864 exception that gets thrown when trying to evaluate a mathematical function
867 The @code{pole_error} class has a member function
870 int pole_error::degree() const;
873 that returns the order of the singularity (or 0 when the pole is
874 logarithmic or the order is undefined).
876 When using GiNaC it is useful to arrange for exceptions to be caught in
877 the main program even if you don't want to do any special error handling.
878 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
879 default exception handler of your C++ compiler's run-time system which
880 usually only aborts the program without giving any information what went
883 Here is an example for a @code{main()} function that catches and prints
884 exceptions generated by GiNaC:
889 #include <ginac/ginac.h>
891 using namespace GiNaC;
899 @} catch (exception &p) @{
900 cerr << p.what() << endl;
908 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
909 @c node-name, next, previous, up
910 @section The Class Hierarchy
912 GiNaC's class hierarchy consists of several classes representing
913 mathematical objects, all of which (except for @code{ex} and some
914 helpers) are internally derived from one abstract base class called
915 @code{basic}. You do not have to deal with objects of class
916 @code{basic}, instead you'll be dealing with symbols, numbers,
917 containers of expressions and so on.
921 To get an idea about what kinds of symbolic composites may be built we
922 have a look at the most important classes in the class hierarchy and
923 some of the relations among the classes:
925 @image{classhierarchy}
927 The abstract classes shown here (the ones without drop-shadow) are of no
928 interest for the user. They are used internally in order to avoid code
929 duplication if two or more classes derived from them share certain
930 features. An example is @code{expairseq}, a container for a sequence of
931 pairs each consisting of one expression and a number (@code{numeric}).
932 What @emph{is} visible to the user are the derived classes @code{add}
933 and @code{mul}, representing sums and products. @xref{Internal
934 Structures}, where these two classes are described in more detail. The
935 following table shortly summarizes what kinds of mathematical objects
936 are stored in the different classes:
939 @multitable @columnfractions .22 .78
940 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
941 @item @code{constant} @tab Constants like
948 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
949 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
950 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
951 @item @code{ncmul} @tab Products of non-commutative objects
952 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
957 @code{sqrt(}@math{2}@code{)}
960 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
961 @item @code{function} @tab A symbolic function like
968 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
969 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
970 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
971 @item @code{indexed} @tab Indexed object like @math{A_ij}
972 @item @code{tensor} @tab Special tensor like the delta and metric tensors
973 @item @code{idx} @tab Index of an indexed object
974 @item @code{varidx} @tab Index with variance
975 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
976 @item @code{wildcard} @tab Wildcard for pattern matching
977 @item @code{structure} @tab Template for user-defined classes
982 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
983 @c node-name, next, previous, up
985 @cindex @code{symbol} (class)
986 @cindex hierarchy of classes
989 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
990 manipulation what atoms are for chemistry.
992 A typical symbol definition looks like this:
997 This definition actually contains three very different things:
999 @item a C++ variable named @code{x}
1000 @item a @code{symbol} object stored in this C++ variable; this object
1001 represents the symbol in a GiNaC expression
1002 @item the string @code{"x"} which is the name of the symbol, used (almost)
1003 exclusively for printing expressions holding the symbol
1006 Symbols have an explicit name, supplied as a string during construction,
1007 because in C++, variable names can't be used as values, and the C++ compiler
1008 throws them away during compilation.
1010 It is possible to omit the symbol name in the definition:
1015 In this case, GiNaC will assign the symbol an internal, unique name of the
1016 form @code{symbolNNN}. This won't affect the usability of the symbol but
1017 the output of your calculations will become more readable if you give your
1018 symbols sensible names (for intermediate expressions that are only used
1019 internally such anonymous symbols can be quite useful, however).
1021 Now, here is one important property of GiNaC that differentiates it from
1022 other computer algebra programs you may have used: GiNaC does @emph{not} use
1023 the names of symbols to tell them apart, but a (hidden) serial number that
1024 is unique for each newly created @code{symbol} object. In you want to use
1025 one and the same symbol in different places in your program, you must only
1026 create one @code{symbol} object and pass that around. If you create another
1027 symbol, even if it has the same name, GiNaC will treat it as a different
1044 // prints "x^6" which looks right, but...
1046 cout << e.degree(x) << endl;
1047 // ...this doesn't work. The symbol "x" here is different from the one
1048 // in f() and in the expression returned by f(). Consequently, it
1053 One possibility to ensure that @code{f()} and @code{main()} use the same
1054 symbol is to pass the symbol as an argument to @code{f()}:
1056 ex f(int n, const ex & x)
1065 // Now, f() uses the same symbol.
1068 cout << e.degree(x) << endl;
1069 // prints "6", as expected
1073 Another possibility would be to define a global symbol @code{x} that is used
1074 by both @code{f()} and @code{main()}. If you are using global symbols and
1075 multiple compilation units you must take special care, however. Suppose
1076 that you have a header file @file{globals.h} in your program that defines
1077 a @code{symbol x("x");}. In this case, every unit that includes
1078 @file{globals.h} would also get its own definition of @code{x} (because
1079 header files are just inlined into the source code by the C++ preprocessor),
1080 and hence you would again end up with multiple equally-named, but different,
1081 symbols. Instead, the @file{globals.h} header should only contain a
1082 @emph{declaration} like @code{extern symbol x;}, with the definition of
1083 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1085 A different approach to ensuring that symbols used in different parts of
1086 your program are identical is to create them with a @emph{factory} function
1089 const symbol & get_symbol(const string & s)
1091 static map<string, symbol> directory;
1092 map<string, symbol>::iterator i = directory.find(s);
1093 if (i != directory.end())
1096 return directory.insert(make_pair(s, symbol(s))).first->second;
1100 This function returns one newly constructed symbol for each name that is
1101 passed in, and it returns the same symbol when called multiple times with
1102 the same name. Using this symbol factory, we can rewrite our example like
1107 return pow(get_symbol("x"), n);
1114 // Both calls of get_symbol("x") yield the same symbol.
1115 cout << e.degree(get_symbol("x")) << endl;
1120 Instead of creating symbols from strings we could also have
1121 @code{get_symbol()} take, for example, an integer number as its argument.
1122 In this case, we would probably want to give the generated symbols names
1123 that include this number, which can be accomplished with the help of an
1124 @code{ostringstream}.
1126 In general, if you're getting weird results from GiNaC such as an expression
1127 @samp{x-x} that is not simplified to zero, you should check your symbol
1130 As we said, the names of symbols primarily serve for purposes of expression
1131 output. But there are actually two instances where GiNaC uses the names for
1132 identifying symbols: When constructing an expression from a string, and when
1133 recreating an expression from an archive (@pxref{Input/Output}).
1135 In addition to its name, a symbol may contain a special string that is used
1138 symbol x("x", "\\Box");
1141 This creates a symbol that is printed as "@code{x}" in normal output, but
1142 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1143 information about the different output formats of expressions in GiNaC).
1144 GiNaC automatically creates proper LaTeX code for symbols having names of
1145 greek letters (@samp{alpha}, @samp{mu}, etc.).
1147 @cindex @code{subs()}
1148 Symbols in GiNaC can't be assigned values. If you need to store results of
1149 calculations and give them a name, use C++ variables of type @code{ex}.
1150 If you want to replace a symbol in an expression with something else, you
1151 can invoke the expression's @code{.subs()} method
1152 (@pxref{Substituting Expressions}).
1154 @cindex @code{realsymbol()}
1155 By default, symbols are expected to stand in for complex values, i.e. they live
1156 in the complex domain. As a consequence, operations like complex conjugation,
1157 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1158 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1159 because of the unknown imaginary part of @code{x}.
1160 On the other hand, if you are sure that your symbols will hold only real values, you
1161 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1162 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1163 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1166 @node Numbers, Constants, Symbols, Basic Concepts
1167 @c node-name, next, previous, up
1169 @cindex @code{numeric} (class)
1175 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1176 The classes therein serve as foundation classes for GiNaC. CLN stands
1177 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1178 In order to find out more about CLN's internals, the reader is referred to
1179 the documentation of that library. @inforef{Introduction, , cln}, for
1180 more information. Suffice to say that it is by itself build on top of
1181 another library, the GNU Multiple Precision library GMP, which is an
1182 extremely fast library for arbitrary long integers and rationals as well
1183 as arbitrary precision floating point numbers. It is very commonly used
1184 by several popular cryptographic applications. CLN extends GMP by
1185 several useful things: First, it introduces the complex number field
1186 over either reals (i.e. floating point numbers with arbitrary precision)
1187 or rationals. Second, it automatically converts rationals to integers
1188 if the denominator is unity and complex numbers to real numbers if the
1189 imaginary part vanishes and also correctly treats algebraic functions.
1190 Third it provides good implementations of state-of-the-art algorithms
1191 for all trigonometric and hyperbolic functions as well as for
1192 calculation of some useful constants.
1194 The user can construct an object of class @code{numeric} in several
1195 ways. The following example shows the four most important constructors.
1196 It uses construction from C-integer, construction of fractions from two
1197 integers, construction from C-float and construction from a string:
1201 #include <ginac/ginac.h>
1202 using namespace GiNaC;
1206 numeric two = 2; // exact integer 2
1207 numeric r(2,3); // exact fraction 2/3
1208 numeric e(2.71828); // floating point number
1209 numeric p = "3.14159265358979323846"; // constructor from string
1210 // Trott's constant in scientific notation:
1211 numeric trott("1.0841015122311136151E-2");
1213 std::cout << two*p << std::endl; // floating point 6.283...
1218 @cindex complex numbers
1219 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1224 numeric z1 = 2-3*I; // exact complex number 2-3i
1225 numeric z2 = 5.9+1.6*I; // complex floating point number
1229 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1230 This would, however, call C's built-in operator @code{/} for integers
1231 first and result in a numeric holding a plain integer 1. @strong{Never
1232 use the operator @code{/} on integers} unless you know exactly what you
1233 are doing! Use the constructor from two integers instead, as shown in
1234 the example above. Writing @code{numeric(1)/2} may look funny but works
1237 @cindex @code{Digits}
1239 We have seen now the distinction between exact numbers and floating
1240 point numbers. Clearly, the user should never have to worry about
1241 dynamically created exact numbers, since their `exactness' always
1242 determines how they ought to be handled, i.e. how `long' they are. The
1243 situation is different for floating point numbers. Their accuracy is
1244 controlled by one @emph{global} variable, called @code{Digits}. (For
1245 those readers who know about Maple: it behaves very much like Maple's
1246 @code{Digits}). All objects of class numeric that are constructed from
1247 then on will be stored with a precision matching that number of decimal
1252 #include <ginac/ginac.h>
1253 using namespace std;
1254 using namespace GiNaC;
1258 numeric three(3.0), one(1.0);
1259 numeric x = one/three;
1261 cout << "in " << Digits << " digits:" << endl;
1263 cout << Pi.evalf() << endl;
1275 The above example prints the following output to screen:
1279 0.33333333333333333334
1280 3.1415926535897932385
1282 0.33333333333333333333333333333333333333333333333333333333333333333334
1283 3.1415926535897932384626433832795028841971693993751058209749445923078
1287 Note that the last number is not necessarily rounded as you would
1288 naively expect it to be rounded in the decimal system. But note also,
1289 that in both cases you got a couple of extra digits. This is because
1290 numbers are internally stored by CLN as chunks of binary digits in order
1291 to match your machine's word size and to not waste precision. Thus, on
1292 architectures with different word size, the above output might even
1293 differ with regard to actually computed digits.
1295 It should be clear that objects of class @code{numeric} should be used
1296 for constructing numbers or for doing arithmetic with them. The objects
1297 one deals with most of the time are the polymorphic expressions @code{ex}.
1299 @subsection Tests on numbers
1301 Once you have declared some numbers, assigned them to expressions and
1302 done some arithmetic with them it is frequently desired to retrieve some
1303 kind of information from them like asking whether that number is
1304 integer, rational, real or complex. For those cases GiNaC provides
1305 several useful methods. (Internally, they fall back to invocations of
1306 certain CLN functions.)
1308 As an example, let's construct some rational number, multiply it with
1309 some multiple of its denominator and test what comes out:
1313 #include <ginac/ginac.h>
1314 using namespace std;
1315 using namespace GiNaC;
1317 // some very important constants:
1318 const numeric twentyone(21);
1319 const numeric ten(10);
1320 const numeric five(5);
1324 numeric answer = twentyone;
1327 cout << answer.is_integer() << endl; // false, it's 21/5
1329 cout << answer.is_integer() << endl; // true, it's 42 now!
1333 Note that the variable @code{answer} is constructed here as an integer
1334 by @code{numeric}'s copy constructor but in an intermediate step it
1335 holds a rational number represented as integer numerator and integer
1336 denominator. When multiplied by 10, the denominator becomes unity and
1337 the result is automatically converted to a pure integer again.
1338 Internally, the underlying CLN is responsible for this behavior and we
1339 refer the reader to CLN's documentation. Suffice to say that
1340 the same behavior applies to complex numbers as well as return values of
1341 certain functions. Complex numbers are automatically converted to real
1342 numbers if the imaginary part becomes zero. The full set of tests that
1343 can be applied is listed in the following table.
1346 @multitable @columnfractions .30 .70
1347 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1348 @item @code{.is_zero()}
1349 @tab @dots{}equal to zero
1350 @item @code{.is_positive()}
1351 @tab @dots{}not complex and greater than 0
1352 @item @code{.is_integer()}
1353 @tab @dots{}a (non-complex) integer
1354 @item @code{.is_pos_integer()}
1355 @tab @dots{}an integer and greater than 0
1356 @item @code{.is_nonneg_integer()}
1357 @tab @dots{}an integer and greater equal 0
1358 @item @code{.is_even()}
1359 @tab @dots{}an even integer
1360 @item @code{.is_odd()}
1361 @tab @dots{}an odd integer
1362 @item @code{.is_prime()}
1363 @tab @dots{}a prime integer (probabilistic primality test)
1364 @item @code{.is_rational()}
1365 @tab @dots{}an exact rational number (integers are rational, too)
1366 @item @code{.is_real()}
1367 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1368 @item @code{.is_cinteger()}
1369 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1370 @item @code{.is_crational()}
1371 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1375 @subsection Numeric functions
1377 The following functions can be applied to @code{numeric} objects and will be
1378 evaluated immediately:
1381 @multitable @columnfractions .30 .70
1382 @item @strong{Name} @tab @strong{Function}
1383 @item @code{inverse(z)}
1384 @tab returns @math{1/z}
1385 @cindex @code{inverse()} (numeric)
1386 @item @code{pow(a, b)}
1387 @tab exponentiation @math{a^b}
1390 @item @code{real(z)}
1392 @cindex @code{real()}
1393 @item @code{imag(z)}
1395 @cindex @code{imag()}
1396 @item @code{csgn(z)}
1397 @tab complex sign (returns an @code{int})
1398 @item @code{numer(z)}
1399 @tab numerator of rational or complex rational number
1400 @item @code{denom(z)}
1401 @tab denominator of rational or complex rational number
1402 @item @code{sqrt(z)}
1404 @item @code{isqrt(n)}
1405 @tab integer square root
1406 @cindex @code{isqrt()}
1413 @item @code{asin(z)}
1415 @item @code{acos(z)}
1417 @item @code{atan(z)}
1418 @tab inverse tangent
1419 @item @code{atan(y, x)}
1420 @tab inverse tangent with two arguments
1421 @item @code{sinh(z)}
1422 @tab hyperbolic sine
1423 @item @code{cosh(z)}
1424 @tab hyperbolic cosine
1425 @item @code{tanh(z)}
1426 @tab hyperbolic tangent
1427 @item @code{asinh(z)}
1428 @tab inverse hyperbolic sine
1429 @item @code{acosh(z)}
1430 @tab inverse hyperbolic cosine
1431 @item @code{atanh(z)}
1432 @tab inverse hyperbolic tangent
1434 @tab exponential function
1436 @tab natural logarithm
1439 @item @code{zeta(z)}
1440 @tab Riemann's zeta function
1441 @item @code{tgamma(z)}
1443 @item @code{lgamma(z)}
1444 @tab logarithm of gamma function
1446 @tab psi (digamma) function
1447 @item @code{psi(n, z)}
1448 @tab derivatives of psi function (polygamma functions)
1449 @item @code{factorial(n)}
1450 @tab factorial function @math{n!}
1451 @item @code{doublefactorial(n)}
1452 @tab double factorial function @math{n!!}
1453 @cindex @code{doublefactorial()}
1454 @item @code{binomial(n, k)}
1455 @tab binomial coefficients
1456 @item @code{bernoulli(n)}
1457 @tab Bernoulli numbers
1458 @cindex @code{bernoulli()}
1459 @item @code{fibonacci(n)}
1460 @tab Fibonacci numbers
1461 @cindex @code{fibonacci()}
1462 @item @code{mod(a, b)}
1463 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1464 @cindex @code{mod()}
1465 @item @code{smod(a, b)}
1466 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1467 @cindex @code{smod()}
1468 @item @code{irem(a, b)}
1469 @tab integer remainder (has the sign of @math{a}, or is zero)
1470 @cindex @code{irem()}
1471 @item @code{irem(a, b, q)}
1472 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1473 @item @code{iquo(a, b)}
1474 @tab integer quotient
1475 @cindex @code{iquo()}
1476 @item @code{iquo(a, b, r)}
1477 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1478 @item @code{gcd(a, b)}
1479 @tab greatest common divisor
1480 @item @code{lcm(a, b)}
1481 @tab least common multiple
1485 Most of these functions are also available as symbolic functions that can be
1486 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1487 as polynomial algorithms.
1489 @subsection Converting numbers
1491 Sometimes it is desirable to convert a @code{numeric} object back to a
1492 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1493 class provides a couple of methods for this purpose:
1495 @cindex @code{to_int()}
1496 @cindex @code{to_long()}
1497 @cindex @code{to_double()}
1498 @cindex @code{to_cl_N()}
1500 int numeric::to_int() const;
1501 long numeric::to_long() const;
1502 double numeric::to_double() const;
1503 cln::cl_N numeric::to_cl_N() const;
1506 @code{to_int()} and @code{to_long()} only work when the number they are
1507 applied on is an exact integer. Otherwise the program will halt with a
1508 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1509 rational number will return a floating-point approximation. Both
1510 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1511 part of complex numbers.
1514 @node Constants, Fundamental containers, Numbers, Basic Concepts
1515 @c node-name, next, previous, up
1517 @cindex @code{constant} (class)
1520 @cindex @code{Catalan}
1521 @cindex @code{Euler}
1522 @cindex @code{evalf()}
1523 Constants behave pretty much like symbols except that they return some
1524 specific number when the method @code{.evalf()} is called.
1526 The predefined known constants are:
1529 @multitable @columnfractions .14 .30 .56
1530 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1532 @tab Archimedes' constant
1533 @tab 3.14159265358979323846264338327950288
1534 @item @code{Catalan}
1535 @tab Catalan's constant
1536 @tab 0.91596559417721901505460351493238411
1538 @tab Euler's (or Euler-Mascheroni) constant
1539 @tab 0.57721566490153286060651209008240243
1544 @node Fundamental containers, Lists, Constants, Basic Concepts
1545 @c node-name, next, previous, up
1546 @section Sums, products and powers
1550 @cindex @code{power}
1552 Simple rational expressions are written down in GiNaC pretty much like
1553 in other CAS or like expressions involving numerical variables in C.
1554 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1555 been overloaded to achieve this goal. When you run the following
1556 code snippet, the constructor for an object of type @code{mul} is
1557 automatically called to hold the product of @code{a} and @code{b} and
1558 then the constructor for an object of type @code{add} is called to hold
1559 the sum of that @code{mul} object and the number one:
1563 symbol a("a"), b("b");
1568 @cindex @code{pow()}
1569 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1570 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1571 construction is necessary since we cannot safely overload the constructor
1572 @code{^} in C++ to construct a @code{power} object. If we did, it would
1573 have several counterintuitive and undesired effects:
1577 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1579 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1580 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1581 interpret this as @code{x^(a^b)}.
1583 Also, expressions involving integer exponents are very frequently used,
1584 which makes it even more dangerous to overload @code{^} since it is then
1585 hard to distinguish between the semantics as exponentiation and the one
1586 for exclusive or. (It would be embarrassing to return @code{1} where one
1587 has requested @code{2^3}.)
1590 @cindex @command{ginsh}
1591 All effects are contrary to mathematical notation and differ from the
1592 way most other CAS handle exponentiation, therefore overloading @code{^}
1593 is ruled out for GiNaC's C++ part. The situation is different in
1594 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1595 that the other frequently used exponentiation operator @code{**} does
1596 not exist at all in C++).
1598 To be somewhat more precise, objects of the three classes described
1599 here, are all containers for other expressions. An object of class
1600 @code{power} is best viewed as a container with two slots, one for the
1601 basis, one for the exponent. All valid GiNaC expressions can be
1602 inserted. However, basic transformations like simplifying
1603 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1604 when this is mathematically possible. If we replace the outer exponent
1605 three in the example by some symbols @code{a}, the simplification is not
1606 safe and will not be performed, since @code{a} might be @code{1/2} and
1609 Objects of type @code{add} and @code{mul} are containers with an
1610 arbitrary number of slots for expressions to be inserted. Again, simple
1611 and safe simplifications are carried out like transforming
1612 @code{3*x+4-x} to @code{2*x+4}.
1615 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1616 @c node-name, next, previous, up
1617 @section Lists of expressions
1618 @cindex @code{lst} (class)
1620 @cindex @code{nops()}
1622 @cindex @code{append()}
1623 @cindex @code{prepend()}
1624 @cindex @code{remove_first()}
1625 @cindex @code{remove_last()}
1626 @cindex @code{remove_all()}
1628 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1629 expressions. They are not as ubiquitous as in many other computer algebra
1630 packages, but are sometimes used to supply a variable number of arguments of
1631 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1632 constructors, so you should have a basic understanding of them.
1634 Lists can be constructed by assigning a comma-separated sequence of
1639 symbol x("x"), y("y");
1642 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1647 There are also constructors that allow direct creation of lists of up to
1648 16 expressions, which is often more convenient but slightly less efficient:
1652 // This produces the same list 'l' as above:
1653 // lst l(x, 2, y, x+y);
1654 // lst l = lst(x, 2, y, x+y);
1658 Use the @code{nops()} method to determine the size (number of expressions) of
1659 a list and the @code{op()} method or the @code{[]} operator to access
1660 individual elements:
1664 cout << l.nops() << endl; // prints '4'
1665 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1669 As with the standard @code{list<T>} container, accessing random elements of a
1670 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1671 sequential access to the elements of a list is possible with the
1672 iterator types provided by the @code{lst} class:
1675 typedef ... lst::const_iterator;
1676 typedef ... lst::const_reverse_iterator;
1677 lst::const_iterator lst::begin() const;
1678 lst::const_iterator lst::end() const;
1679 lst::const_reverse_iterator lst::rbegin() const;
1680 lst::const_reverse_iterator lst::rend() const;
1683 For example, to print the elements of a list individually you can use:
1688 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1693 which is one order faster than
1698 for (size_t i = 0; i < l.nops(); ++i)
1699 cout << l.op(i) << endl;
1703 These iterators also allow you to use some of the algorithms provided by
1704 the C++ standard library:
1708 // print the elements of the list (requires #include <iterator>)
1709 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1711 // sum up the elements of the list (requires #include <numeric>)
1712 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1713 cout << sum << endl; // prints '2+2*x+2*y'
1717 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1718 (the only other one is @code{matrix}). You can modify single elements:
1722 l[1] = 42; // l is now @{x, 42, y, x+y@}
1723 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1727 You can append or prepend an expression to a list with the @code{append()}
1728 and @code{prepend()} methods:
1732 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1733 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1737 You can remove the first or last element of a list with @code{remove_first()}
1738 and @code{remove_last()}:
1742 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1743 l.remove_last(); // l is now @{x, 7, y, x+y@}
1747 You can remove all the elements of a list with @code{remove_all()}:
1751 l.remove_all(); // l is now empty
1755 You can bring the elements of a list into a canonical order with @code{sort()}:
1764 // l1 and l2 are now equal
1768 Finally, you can remove all but the first element of consecutive groups of
1769 elements with @code{unique()}:
1774 l3 = x, 2, 2, 2, y, x+y, y+x;
1775 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1780 @node Mathematical functions, Relations, Lists, Basic Concepts
1781 @c node-name, next, previous, up
1782 @section Mathematical functions
1783 @cindex @code{function} (class)
1784 @cindex trigonometric function
1785 @cindex hyperbolic function
1787 There are quite a number of useful functions hard-wired into GiNaC. For
1788 instance, all trigonometric and hyperbolic functions are implemented
1789 (@xref{Built-in Functions}, for a complete list).
1791 These functions (better called @emph{pseudofunctions}) are all objects
1792 of class @code{function}. They accept one or more expressions as
1793 arguments and return one expression. If the arguments are not
1794 numerical, the evaluation of the function may be halted, as it does in
1795 the next example, showing how a function returns itself twice and
1796 finally an expression that may be really useful:
1798 @cindex Gamma function
1799 @cindex @code{subs()}
1802 symbol x("x"), y("y");
1804 cout << tgamma(foo) << endl;
1805 // -> tgamma(x+(1/2)*y)
1806 ex bar = foo.subs(y==1);
1807 cout << tgamma(bar) << endl;
1809 ex foobar = bar.subs(x==7);
1810 cout << tgamma(foobar) << endl;
1811 // -> (135135/128)*Pi^(1/2)
1815 Besides evaluation most of these functions allow differentiation, series
1816 expansion and so on. Read the next chapter in order to learn more about
1819 It must be noted that these pseudofunctions are created by inline
1820 functions, where the argument list is templated. This means that
1821 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1822 @code{sin(ex(1))} and will therefore not result in a floating point
1823 number. Unless of course the function prototype is explicitly
1824 overridden -- which is the case for arguments of type @code{numeric}
1825 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1826 point number of class @code{numeric} you should call
1827 @code{sin(numeric(1))}. This is almost the same as calling
1828 @code{sin(1).evalf()} except that the latter will return a numeric
1829 wrapped inside an @code{ex}.
1832 @node Relations, Integrals, Mathematical functions, Basic Concepts
1833 @c node-name, next, previous, up
1835 @cindex @code{relational} (class)
1837 Sometimes, a relation holding between two expressions must be stored
1838 somehow. The class @code{relational} is a convenient container for such
1839 purposes. A relation is by definition a container for two @code{ex} and
1840 a relation between them that signals equality, inequality and so on.
1841 They are created by simply using the C++ operators @code{==}, @code{!=},
1842 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1844 @xref{Mathematical functions}, for examples where various applications
1845 of the @code{.subs()} method show how objects of class relational are
1846 used as arguments. There they provide an intuitive syntax for
1847 substitutions. They are also used as arguments to the @code{ex::series}
1848 method, where the left hand side of the relation specifies the variable
1849 to expand in and the right hand side the expansion point. They can also
1850 be used for creating systems of equations that are to be solved for
1851 unknown variables. But the most common usage of objects of this class
1852 is rather inconspicuous in statements of the form @code{if
1853 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1854 conversion from @code{relational} to @code{bool} takes place. Note,
1855 however, that @code{==} here does not perform any simplifications, hence
1856 @code{expand()} must be called explicitly.
1858 @node Integrals, Matrices, Relations, Basic Concepts
1859 @c node-name, next, previous, up
1861 @cindex @code{integral} (class)
1863 An object of class @dfn{integral} can be used to hold a symbolic integral.
1864 If you want to symbolically represent the integral of @code{x*x} from 0 to
1865 1, you would write this as
1867 integral(x, 0, 1, x*x)
1869 The first argument is the integration variable. It should be noted that
1870 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1871 fact, it can only integrate polynomials. An expression containing integrals
1872 can be evaluated symbolically by calling the
1876 method on it. Numerical evaluation is available by calling the
1880 method on an expression containing the integral. This will only evaluate
1881 integrals into a number if @code{subs}ing the integration variable by a
1882 number in the fourth argument of an integral and then @code{evalf}ing the
1883 result always results in a number. Of course, also the boundaries of the
1884 integration domain must @code{evalf} into numbers. It should be noted that
1885 trying to @code{evalf} a function with discontinuities in the integration
1886 domain is not recommended. The accuracy of the numeric evaluation of
1887 integrals is determined by the static member variable
1889 ex integral::relative_integration_error
1891 of the class @code{integral}. The default value of this is 10^-8.
1892 The integration works by halving the interval of integration, until numeric
1893 stability of the answer indicates that the requested accuracy has been
1894 reached. The maximum depth of the halving can be set via the static member
1897 int integral::max_integration_level
1899 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1900 return the integral unevaluated. The function that performs the numerical
1901 evaluation, is also available as
1903 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1906 This function will throw an exception if the maximum depth is exceeded. The
1907 last parameter of the function is optional and defaults to the
1908 @code{relative_integration_error}. To make sure that we do not do too
1909 much work if an expression contains the same integral multiple times,
1910 a lookup table is used.
1912 If you know that an expression holds an integral, you can get the
1913 integration variable, the left boundary, right boundary and integrand by
1914 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1915 @code{.op(3)}. Differentiating integrals with respect to variables works
1916 as expected. Note that it makes no sense to differentiate an integral
1917 with respect to the integration variable.
1919 @node Matrices, Indexed objects, Integrals, Basic Concepts
1920 @c node-name, next, previous, up
1922 @cindex @code{matrix} (class)
1924 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1925 matrix with @math{m} rows and @math{n} columns are accessed with two
1926 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1927 second one in the range 0@dots{}@math{n-1}.
1929 There are a couple of ways to construct matrices, with or without preset
1930 elements. The constructor
1933 matrix::matrix(unsigned r, unsigned c);
1936 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1939 The fastest way to create a matrix with preinitialized elements is to assign
1940 a list of comma-separated expressions to an empty matrix (see below for an
1941 example). But you can also specify the elements as a (flat) list with
1944 matrix::matrix(unsigned r, unsigned c, const lst & l);
1949 @cindex @code{lst_to_matrix()}
1951 ex lst_to_matrix(const lst & l);
1954 constructs a matrix from a list of lists, each list representing a matrix row.
1956 There is also a set of functions for creating some special types of
1959 @cindex @code{diag_matrix()}
1960 @cindex @code{unit_matrix()}
1961 @cindex @code{symbolic_matrix()}
1963 ex diag_matrix(const lst & l);
1964 ex unit_matrix(unsigned x);
1965 ex unit_matrix(unsigned r, unsigned c);
1966 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1967 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1968 const string & tex_base_name);
1971 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1972 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1973 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1974 matrix filled with newly generated symbols made of the specified base name
1975 and the position of each element in the matrix.
1977 Matrices often arise by omitting elements of another matrix. For
1978 instance, the submatrix @code{S} of a matrix @code{M} takes a
1979 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1980 by removing one row and one column from a matrix @code{M}. (The
1981 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1982 can be used for computing the inverse using Cramer's rule.)
1984 @cindex @code{sub_matrix()}
1985 @cindex @code{reduced_matrix()}
1987 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1988 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1991 The function @code{sub_matrix()} takes a row offset @code{r} and a
1992 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1993 columns. The function @code{reduced_matrix()} has two integer arguments
1994 that specify which row and column to remove:
2002 cout << reduced_matrix(m, 1, 1) << endl;
2003 // -> [[11,13],[31,33]]
2004 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2005 // -> [[22,23],[32,33]]
2009 Matrix elements can be accessed and set using the parenthesis (function call)
2013 const ex & matrix::operator()(unsigned r, unsigned c) const;
2014 ex & matrix::operator()(unsigned r, unsigned c);
2017 It is also possible to access the matrix elements in a linear fashion with
2018 the @code{op()} method. But C++-style subscripting with square brackets
2019 @samp{[]} is not available.
2021 Here are a couple of examples for constructing matrices:
2025 symbol a("a"), b("b");
2039 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2042 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2045 cout << diag_matrix(lst(a, b)) << endl;
2048 cout << unit_matrix(3) << endl;
2049 // -> [[1,0,0],[0,1,0],[0,0,1]]
2051 cout << symbolic_matrix(2, 3, "x") << endl;
2052 // -> [[x00,x01,x02],[x10,x11,x12]]
2056 @cindex @code{transpose()}
2057 There are three ways to do arithmetic with matrices. The first (and most
2058 direct one) is to use the methods provided by the @code{matrix} class:
2061 matrix matrix::add(const matrix & other) const;
2062 matrix matrix::sub(const matrix & other) const;
2063 matrix matrix::mul(const matrix & other) const;
2064 matrix matrix::mul_scalar(const ex & other) const;
2065 matrix matrix::pow(const ex & expn) const;
2066 matrix matrix::transpose() const;
2069 All of these methods return the result as a new matrix object. Here is an
2070 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2075 matrix A(2, 2), B(2, 2), C(2, 2);
2083 matrix result = A.mul(B).sub(C.mul_scalar(2));
2084 cout << result << endl;
2085 // -> [[-13,-6],[1,2]]
2090 @cindex @code{evalm()}
2091 The second (and probably the most natural) way is to construct an expression
2092 containing matrices with the usual arithmetic operators and @code{pow()}.
2093 For efficiency reasons, expressions with sums, products and powers of
2094 matrices are not automatically evaluated in GiNaC. You have to call the
2098 ex ex::evalm() const;
2101 to obtain the result:
2108 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2109 cout << e.evalm() << endl;
2110 // -> [[-13,-6],[1,2]]
2115 The non-commutativity of the product @code{A*B} in this example is
2116 automatically recognized by GiNaC. There is no need to use a special
2117 operator here. @xref{Non-commutative objects}, for more information about
2118 dealing with non-commutative expressions.
2120 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2121 to perform the arithmetic:
2126 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2127 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2129 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2130 cout << e.simplify_indexed() << endl;
2131 // -> [[-13,-6],[1,2]].i.j
2135 Using indices is most useful when working with rectangular matrices and
2136 one-dimensional vectors because you don't have to worry about having to
2137 transpose matrices before multiplying them. @xref{Indexed objects}, for
2138 more information about using matrices with indices, and about indices in
2141 The @code{matrix} class provides a couple of additional methods for
2142 computing determinants, traces, characteristic polynomials and ranks:
2144 @cindex @code{determinant()}
2145 @cindex @code{trace()}
2146 @cindex @code{charpoly()}
2147 @cindex @code{rank()}
2149 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2150 ex matrix::trace() const;
2151 ex matrix::charpoly(const ex & lambda) const;
2152 unsigned matrix::rank() const;
2155 The @samp{algo} argument of @code{determinant()} allows to select
2156 between different algorithms for calculating the determinant. The
2157 asymptotic speed (as parametrized by the matrix size) can greatly differ
2158 between those algorithms, depending on the nature of the matrix'
2159 entries. The possible values are defined in the @file{flags.h} header
2160 file. By default, GiNaC uses a heuristic to automatically select an
2161 algorithm that is likely (but not guaranteed) to give the result most
2164 @cindex @code{inverse()} (matrix)
2165 @cindex @code{solve()}
2166 Matrices may also be inverted using the @code{ex matrix::inverse()}
2167 method and linear systems may be solved with:
2170 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2171 unsigned algo=solve_algo::automatic) const;
2174 Assuming the matrix object this method is applied on is an @code{m}
2175 times @code{n} matrix, then @code{vars} must be a @code{n} times
2176 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2177 times @code{p} matrix. The returned matrix then has dimension @code{n}
2178 times @code{p} and in the case of an underdetermined system will still
2179 contain some of the indeterminates from @code{vars}. If the system is
2180 overdetermined, an exception is thrown.
2183 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2184 @c node-name, next, previous, up
2185 @section Indexed objects
2187 GiNaC allows you to handle expressions containing general indexed objects in
2188 arbitrary spaces. It is also able to canonicalize and simplify such
2189 expressions and perform symbolic dummy index summations. There are a number
2190 of predefined indexed objects provided, like delta and metric tensors.
2192 There are few restrictions placed on indexed objects and their indices and
2193 it is easy to construct nonsense expressions, but our intention is to
2194 provide a general framework that allows you to implement algorithms with
2195 indexed quantities, getting in the way as little as possible.
2197 @cindex @code{idx} (class)
2198 @cindex @code{indexed} (class)
2199 @subsection Indexed quantities and their indices
2201 Indexed expressions in GiNaC are constructed of two special types of objects,
2202 @dfn{index objects} and @dfn{indexed objects}.
2206 @cindex contravariant
2209 @item Index objects are of class @code{idx} or a subclass. Every index has
2210 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2211 the index lives in) which can both be arbitrary expressions but are usually
2212 a number or a simple symbol. In addition, indices of class @code{varidx} have
2213 a @dfn{variance} (they can be co- or contravariant), and indices of class
2214 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2216 @item Indexed objects are of class @code{indexed} or a subclass. They
2217 contain a @dfn{base expression} (which is the expression being indexed), and
2218 one or more indices.
2222 @strong{Please notice:} when printing expressions, covariant indices and indices
2223 without variance are denoted @samp{.i} while contravariant indices are
2224 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2225 value. In the following, we are going to use that notation in the text so
2226 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2227 not visible in the output.
2229 A simple example shall illustrate the concepts:
2233 #include <ginac/ginac.h>
2234 using namespace std;
2235 using namespace GiNaC;
2239 symbol i_sym("i"), j_sym("j");
2240 idx i(i_sym, 3), j(j_sym, 3);
2243 cout << indexed(A, i, j) << endl;
2245 cout << index_dimensions << indexed(A, i, j) << endl;
2247 cout << dflt; // reset cout to default output format (dimensions hidden)
2251 The @code{idx} constructor takes two arguments, the index value and the
2252 index dimension. First we define two index objects, @code{i} and @code{j},
2253 both with the numeric dimension 3. The value of the index @code{i} is the
2254 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2255 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2256 construct an expression containing one indexed object, @samp{A.i.j}. It has
2257 the symbol @code{A} as its base expression and the two indices @code{i} and
2260 The dimensions of indices are normally not visible in the output, but one
2261 can request them to be printed with the @code{index_dimensions} manipulator,
2264 Note the difference between the indices @code{i} and @code{j} which are of
2265 class @code{idx}, and the index values which are the symbols @code{i_sym}
2266 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2267 or numbers but must be index objects. For example, the following is not
2268 correct and will raise an exception:
2271 symbol i("i"), j("j");
2272 e = indexed(A, i, j); // ERROR: indices must be of type idx
2275 You can have multiple indexed objects in an expression, index values can
2276 be numeric, and index dimensions symbolic:
2280 symbol B("B"), dim("dim");
2281 cout << 4 * indexed(A, i)
2282 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2287 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2288 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2289 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2290 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2291 @code{simplify_indexed()} for that, see below).
2293 In fact, base expressions, index values and index dimensions can be
2294 arbitrary expressions:
2298 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2303 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2304 get an error message from this but you will probably not be able to do
2305 anything useful with it.
2307 @cindex @code{get_value()}
2308 @cindex @code{get_dimension()}
2312 ex idx::get_value();
2313 ex idx::get_dimension();
2316 return the value and dimension of an @code{idx} object. If you have an index
2317 in an expression, such as returned by calling @code{.op()} on an indexed
2318 object, you can get a reference to the @code{idx} object with the function
2319 @code{ex_to<idx>()} on the expression.
2321 There are also the methods
2324 bool idx::is_numeric();
2325 bool idx::is_symbolic();
2326 bool idx::is_dim_numeric();
2327 bool idx::is_dim_symbolic();
2330 for checking whether the value and dimension are numeric or symbolic
2331 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2332 About Expressions}) returns information about the index value.
2334 @cindex @code{varidx} (class)
2335 If you need co- and contravariant indices, use the @code{varidx} class:
2339 symbol mu_sym("mu"), nu_sym("nu");
2340 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2341 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2343 cout << indexed(A, mu, nu) << endl;
2345 cout << indexed(A, mu_co, nu) << endl;
2347 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2352 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2353 co- or contravariant. The default is a contravariant (upper) index, but
2354 this can be overridden by supplying a third argument to the @code{varidx}
2355 constructor. The two methods
2358 bool varidx::is_covariant();
2359 bool varidx::is_contravariant();
2362 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2363 to get the object reference from an expression). There's also the very useful
2367 ex varidx::toggle_variance();
2370 which makes a new index with the same value and dimension but the opposite
2371 variance. By using it you only have to define the index once.
2373 @cindex @code{spinidx} (class)
2374 The @code{spinidx} class provides dotted and undotted variant indices, as
2375 used in the Weyl-van-der-Waerden spinor formalism:
2379 symbol K("K"), C_sym("C"), D_sym("D");
2380 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2381 // contravariant, undotted
2382 spinidx C_co(C_sym, 2, true); // covariant index
2383 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2384 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2386 cout << indexed(K, C, D) << endl;
2388 cout << indexed(K, C_co, D_dot) << endl;
2390 cout << indexed(K, D_co_dot, D) << endl;
2395 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2396 dotted or undotted. The default is undotted but this can be overridden by
2397 supplying a fourth argument to the @code{spinidx} constructor. The two
2401 bool spinidx::is_dotted();
2402 bool spinidx::is_undotted();
2405 allow you to check whether or not a @code{spinidx} object is dotted (use
2406 @code{ex_to<spinidx>()} to get the object reference from an expression).
2407 Finally, the two methods
2410 ex spinidx::toggle_dot();
2411 ex spinidx::toggle_variance_dot();
2414 create a new index with the same value and dimension but opposite dottedness
2415 and the same or opposite variance.
2417 @subsection Substituting indices
2419 @cindex @code{subs()}
2420 Sometimes you will want to substitute one symbolic index with another
2421 symbolic or numeric index, for example when calculating one specific element
2422 of a tensor expression. This is done with the @code{.subs()} method, as it
2423 is done for symbols (see @ref{Substituting Expressions}).
2425 You have two possibilities here. You can either substitute the whole index
2426 by another index or expression:
2430 ex e = indexed(A, mu_co);
2431 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2432 // -> A.mu becomes A~nu
2433 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2434 // -> A.mu becomes A~0
2435 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2436 // -> A.mu becomes A.0
2440 The third example shows that trying to replace an index with something that
2441 is not an index will substitute the index value instead.
2443 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2448 ex e = indexed(A, mu_co);
2449 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2450 // -> A.mu becomes A.nu
2451 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2452 // -> A.mu becomes A.0
2456 As you see, with the second method only the value of the index will get
2457 substituted. Its other properties, including its dimension, remain unchanged.
2458 If you want to change the dimension of an index you have to substitute the
2459 whole index by another one with the new dimension.
2461 Finally, substituting the base expression of an indexed object works as
2466 ex e = indexed(A, mu_co);
2467 cout << e << " becomes " << e.subs(A == A+B) << endl;
2468 // -> A.mu becomes (B+A).mu
2472 @subsection Symmetries
2473 @cindex @code{symmetry} (class)
2474 @cindex @code{sy_none()}
2475 @cindex @code{sy_symm()}
2476 @cindex @code{sy_anti()}
2477 @cindex @code{sy_cycl()}
2479 Indexed objects can have certain symmetry properties with respect to their
2480 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2481 that is constructed with the helper functions
2484 symmetry sy_none(...);
2485 symmetry sy_symm(...);
2486 symmetry sy_anti(...);
2487 symmetry sy_cycl(...);
2490 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2491 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2492 represents a cyclic symmetry. Each of these functions accepts up to four
2493 arguments which can be either symmetry objects themselves or unsigned integer
2494 numbers that represent an index position (counting from 0). A symmetry
2495 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2496 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2499 Here are some examples of symmetry definitions:
2504 e = indexed(A, i, j);
2505 e = indexed(A, sy_none(), i, j); // equivalent
2506 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2508 // Symmetric in all three indices:
2509 e = indexed(A, sy_symm(), i, j, k);
2510 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2511 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2512 // different canonical order
2514 // Symmetric in the first two indices only:
2515 e = indexed(A, sy_symm(0, 1), i, j, k);
2516 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2518 // Antisymmetric in the first and last index only (index ranges need not
2520 e = indexed(A, sy_anti(0, 2), i, j, k);
2521 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2523 // An example of a mixed symmetry: antisymmetric in the first two and
2524 // last two indices, symmetric when swapping the first and last index
2525 // pairs (like the Riemann curvature tensor):
2526 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2528 // Cyclic symmetry in all three indices:
2529 e = indexed(A, sy_cycl(), i, j, k);
2530 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2532 // The following examples are invalid constructions that will throw
2533 // an exception at run time.
2535 // An index may not appear multiple times:
2536 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2537 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2539 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2540 // same number of indices:
2541 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2543 // And of course, you cannot specify indices which are not there:
2544 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2548 If you need to specify more than four indices, you have to use the
2549 @code{.add()} method of the @code{symmetry} class. For example, to specify
2550 full symmetry in the first six indices you would write
2551 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2553 If an indexed object has a symmetry, GiNaC will automatically bring the
2554 indices into a canonical order which allows for some immediate simplifications:
2558 cout << indexed(A, sy_symm(), i, j)
2559 + indexed(A, sy_symm(), j, i) << endl;
2561 cout << indexed(B, sy_anti(), i, j)
2562 + indexed(B, sy_anti(), j, i) << endl;
2564 cout << indexed(B, sy_anti(), i, j, k)
2565 - indexed(B, sy_anti(), j, k, i) << endl;
2570 @cindex @code{get_free_indices()}
2572 @subsection Dummy indices
2574 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2575 that a summation over the index range is implied. Symbolic indices which are
2576 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2577 dummy nor free indices.
2579 To be recognized as a dummy index pair, the two indices must be of the same
2580 class and their value must be the same single symbol (an index like
2581 @samp{2*n+1} is never a dummy index). If the indices are of class
2582 @code{varidx} they must also be of opposite variance; if they are of class
2583 @code{spinidx} they must be both dotted or both undotted.
2585 The method @code{.get_free_indices()} returns a vector containing the free
2586 indices of an expression. It also checks that the free indices of the terms
2587 of a sum are consistent:
2591 symbol A("A"), B("B"), C("C");
2593 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2594 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2596 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2597 cout << exprseq(e.get_free_indices()) << endl;
2599 // 'j' and 'l' are dummy indices
2601 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2602 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2604 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2605 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2606 cout << exprseq(e.get_free_indices()) << endl;
2608 // 'nu' is a dummy index, but 'sigma' is not
2610 e = indexed(A, mu, mu);
2611 cout << exprseq(e.get_free_indices()) << endl;
2613 // 'mu' is not a dummy index because it appears twice with the same
2616 e = indexed(A, mu, nu) + 42;
2617 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2618 // this will throw an exception:
2619 // "add::get_free_indices: inconsistent indices in sum"
2623 @cindex @code{expand_dummy_sum()}
2624 A dummy index summation like
2631 can be expanded for indices with numeric
2632 dimensions (e.g. 3) into the explicit sum like
2634 $a_1b^1+a_2b^2+a_3b^3 $.
2637 a.1 b~1 + a.2 b~2 + a.3 b~3.
2639 This is performed by the function
2642 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2645 which takes an expression @code{e} and returns the expanded sum for all
2646 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2647 is set to @code{true} then all substitutions are made by @code{idx} class
2648 indices, i.e. without variance. In this case the above sum
2657 $a_1b_1+a_2b_2+a_3b_3 $.
2660 a.1 b.1 + a.2 b.2 + a.3 b.3.
2664 @cindex @code{simplify_indexed()}
2665 @subsection Simplifying indexed expressions
2667 In addition to the few automatic simplifications that GiNaC performs on
2668 indexed expressions (such as re-ordering the indices of symmetric tensors
2669 and calculating traces and convolutions of matrices and predefined tensors)
2673 ex ex::simplify_indexed();
2674 ex ex::simplify_indexed(const scalar_products & sp);
2677 that performs some more expensive operations:
2680 @item it checks the consistency of free indices in sums in the same way
2681 @code{get_free_indices()} does
2682 @item it tries to give dummy indices that appear in different terms of a sum
2683 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2684 @item it (symbolically) calculates all possible dummy index summations/contractions
2685 with the predefined tensors (this will be explained in more detail in the
2687 @item it detects contractions that vanish for symmetry reasons, for example
2688 the contraction of a symmetric and a totally antisymmetric tensor
2689 @item as a special case of dummy index summation, it can replace scalar products
2690 of two tensors with a user-defined value
2693 The last point is done with the help of the @code{scalar_products} class
2694 which is used to store scalar products with known values (this is not an
2695 arithmetic class, you just pass it to @code{simplify_indexed()}):
2699 symbol A("A"), B("B"), C("C"), i_sym("i");
2703 sp.add(A, B, 0); // A and B are orthogonal
2704 sp.add(A, C, 0); // A and C are orthogonal
2705 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2707 e = indexed(A + B, i) * indexed(A + C, i);
2709 // -> (B+A).i*(A+C).i
2711 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2717 The @code{scalar_products} object @code{sp} acts as a storage for the
2718 scalar products added to it with the @code{.add()} method. This method
2719 takes three arguments: the two expressions of which the scalar product is
2720 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2721 @code{simplify_indexed()} will replace all scalar products of indexed
2722 objects that have the symbols @code{A} and @code{B} as base expressions
2723 with the single value 0. The number, type and dimension of the indices
2724 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2726 @cindex @code{expand()}
2727 The example above also illustrates a feature of the @code{expand()} method:
2728 if passed the @code{expand_indexed} option it will distribute indices
2729 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2731 @cindex @code{tensor} (class)
2732 @subsection Predefined tensors
2734 Some frequently used special tensors such as the delta, epsilon and metric
2735 tensors are predefined in GiNaC. They have special properties when
2736 contracted with other tensor expressions and some of them have constant
2737 matrix representations (they will evaluate to a number when numeric
2738 indices are specified).
2740 @cindex @code{delta_tensor()}
2741 @subsubsection Delta tensor
2743 The delta tensor takes two indices, is symmetric and has the matrix
2744 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2745 @code{delta_tensor()}:
2749 symbol A("A"), B("B");
2751 idx i(symbol("i"), 3), j(symbol("j"), 3),
2752 k(symbol("k"), 3), l(symbol("l"), 3);
2754 ex e = indexed(A, i, j) * indexed(B, k, l)
2755 * delta_tensor(i, k) * delta_tensor(j, l);
2756 cout << e.simplify_indexed() << endl;
2759 cout << delta_tensor(i, i) << endl;
2764 @cindex @code{metric_tensor()}
2765 @subsubsection General metric tensor
2767 The function @code{metric_tensor()} creates a general symmetric metric
2768 tensor with two indices that can be used to raise/lower tensor indices. The
2769 metric tensor is denoted as @samp{g} in the output and if its indices are of
2770 mixed variance it is automatically replaced by a delta tensor:
2776 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2778 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2779 cout << e.simplify_indexed() << endl;
2782 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2783 cout << e.simplify_indexed() << endl;
2786 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2787 * metric_tensor(nu, rho);
2788 cout << e.simplify_indexed() << endl;
2791 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2792 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2793 + indexed(A, mu.toggle_variance(), rho));
2794 cout << e.simplify_indexed() << endl;
2799 @cindex @code{lorentz_g()}
2800 @subsubsection Minkowski metric tensor
2802 The Minkowski metric tensor is a special metric tensor with a constant
2803 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2804 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2805 It is created with the function @code{lorentz_g()} (although it is output as
2810 varidx mu(symbol("mu"), 4);
2812 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2813 * lorentz_g(mu, varidx(0, 4)); // negative signature
2814 cout << e.simplify_indexed() << endl;
2817 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2818 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2819 cout << e.simplify_indexed() << endl;
2824 @cindex @code{spinor_metric()}
2825 @subsubsection Spinor metric tensor
2827 The function @code{spinor_metric()} creates an antisymmetric tensor with
2828 two indices that is used to raise/lower indices of 2-component spinors.
2829 It is output as @samp{eps}:
2835 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2836 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2838 e = spinor_metric(A, B) * indexed(psi, B_co);
2839 cout << e.simplify_indexed() << endl;
2842 e = spinor_metric(A, B) * indexed(psi, A_co);
2843 cout << e.simplify_indexed() << endl;
2846 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2847 cout << e.simplify_indexed() << endl;
2850 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2851 cout << e.simplify_indexed() << endl;
2854 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2855 cout << e.simplify_indexed() << endl;
2858 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2859 cout << e.simplify_indexed() << endl;
2864 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2866 @cindex @code{epsilon_tensor()}
2867 @cindex @code{lorentz_eps()}
2868 @subsubsection Epsilon tensor
2870 The epsilon tensor is totally antisymmetric, its number of indices is equal
2871 to the dimension of the index space (the indices must all be of the same
2872 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2873 defined to be 1. Its behavior with indices that have a variance also
2874 depends on the signature of the metric. Epsilon tensors are output as
2877 There are three functions defined to create epsilon tensors in 2, 3 and 4
2881 ex epsilon_tensor(const ex & i1, const ex & i2);
2882 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2883 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2884 bool pos_sig = false);
2887 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2888 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2889 Minkowski space (the last @code{bool} argument specifies whether the metric
2890 has negative or positive signature, as in the case of the Minkowski metric
2895 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2896 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2897 e = lorentz_eps(mu, nu, rho, sig) *
2898 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2899 cout << simplify_indexed(e) << endl;
2900 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2902 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2903 symbol A("A"), B("B");
2904 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2905 cout << simplify_indexed(e) << endl;
2906 // -> -B.k*A.j*eps.i.k.j
2907 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2908 cout << simplify_indexed(e) << endl;
2913 @subsection Linear algebra
2915 The @code{matrix} class can be used with indices to do some simple linear
2916 algebra (linear combinations and products of vectors and matrices, traces
2917 and scalar products):
2921 idx i(symbol("i"), 2), j(symbol("j"), 2);
2922 symbol x("x"), y("y");
2924 // A is a 2x2 matrix, X is a 2x1 vector
2925 matrix A(2, 2), X(2, 1);
2930 cout << indexed(A, i, i) << endl;
2933 ex e = indexed(A, i, j) * indexed(X, j);
2934 cout << e.simplify_indexed() << endl;
2935 // -> [[2*y+x],[4*y+3*x]].i
2937 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2938 cout << e.simplify_indexed() << endl;
2939 // -> [[3*y+3*x,6*y+2*x]].j
2943 You can of course obtain the same results with the @code{matrix::add()},
2944 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2945 but with indices you don't have to worry about transposing matrices.
2947 Matrix indices always start at 0 and their dimension must match the number
2948 of rows/columns of the matrix. Matrices with one row or one column are
2949 vectors and can have one or two indices (it doesn't matter whether it's a
2950 row or a column vector). Other matrices must have two indices.
2952 You should be careful when using indices with variance on matrices. GiNaC
2953 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2954 @samp{F.mu.nu} are different matrices. In this case you should use only
2955 one form for @samp{F} and explicitly multiply it with a matrix representation
2956 of the metric tensor.
2959 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2960 @c node-name, next, previous, up
2961 @section Non-commutative objects
2963 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2964 non-commutative objects are built-in which are mostly of use in high energy
2968 @item Clifford (Dirac) algebra (class @code{clifford})
2969 @item su(3) Lie algebra (class @code{color})
2970 @item Matrices (unindexed) (class @code{matrix})
2973 The @code{clifford} and @code{color} classes are subclasses of
2974 @code{indexed} because the elements of these algebras usually carry
2975 indices. The @code{matrix} class is described in more detail in
2978 Unlike most computer algebra systems, GiNaC does not primarily provide an
2979 operator (often denoted @samp{&*}) for representing inert products of
2980 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2981 classes of objects involved, and non-commutative products are formed with
2982 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2983 figuring out by itself which objects commutate and will group the factors
2984 by their class. Consider this example:
2988 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2989 idx a(symbol("a"), 8), b(symbol("b"), 8);
2990 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2992 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2996 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2997 groups the non-commutative factors (the gammas and the su(3) generators)
2998 together while preserving the order of factors within each class (because
2999 Clifford objects commutate with color objects). The resulting expression is a
3000 @emph{commutative} product with two factors that are themselves non-commutative
3001 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3002 parentheses are placed around the non-commutative products in the output.
3004 @cindex @code{ncmul} (class)
3005 Non-commutative products are internally represented by objects of the class
3006 @code{ncmul}, as opposed to commutative products which are handled by the
3007 @code{mul} class. You will normally not have to worry about this distinction,
3010 The advantage of this approach is that you never have to worry about using
3011 (or forgetting to use) a special operator when constructing non-commutative
3012 expressions. Also, non-commutative products in GiNaC are more intelligent
3013 than in other computer algebra systems; they can, for example, automatically
3014 canonicalize themselves according to rules specified in the implementation
3015 of the non-commutative classes. The drawback is that to work with other than
3016 the built-in algebras you have to implement new classes yourself. Symbols
3017 always commutate and it's not possible to construct non-commutative products
3018 using symbols to represent the algebra elements or generators. User-defined
3019 functions can, however, be specified as being non-commutative.
3021 @cindex @code{return_type()}
3022 @cindex @code{return_type_tinfo()}
3023 Information about the commutativity of an object or expression can be
3024 obtained with the two member functions
3027 unsigned ex::return_type() const;
3028 unsigned ex::return_type_tinfo() const;
3031 The @code{return_type()} function returns one of three values (defined in
3032 the header file @file{flags.h}), corresponding to three categories of
3033 expressions in GiNaC:
3036 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3037 classes are of this kind.
3038 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3039 certain class of non-commutative objects which can be determined with the
3040 @code{return_type_tinfo()} method. Expressions of this category commutate
3041 with everything except @code{noncommutative} expressions of the same
3043 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3044 of non-commutative objects of different classes. Expressions of this
3045 category don't commutate with any other @code{noncommutative} or
3046 @code{noncommutative_composite} expressions.
3049 The value returned by the @code{return_type_tinfo()} method is valid only
3050 when the return type of the expression is @code{noncommutative}. It is a
3051 value that is unique to the class of the object and usually one of the
3052 constants in @file{tinfos.h}, or derived therefrom.
3054 Here are a couple of examples:
3057 @multitable @columnfractions 0.33 0.33 0.34
3058 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3059 @item @code{42} @tab @code{commutative} @tab -
3060 @item @code{2*x-y} @tab @code{commutative} @tab -
3061 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3062 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3063 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3064 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3068 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3069 @code{TINFO_clifford} for objects with a representation label of zero.
3070 Other representation labels yield a different @code{return_type_tinfo()},
3071 but it's the same for any two objects with the same label. This is also true
3074 A last note: With the exception of matrices, positive integer powers of
3075 non-commutative objects are automatically expanded in GiNaC. For example,
3076 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3077 non-commutative expressions).
3080 @cindex @code{clifford} (class)
3081 @subsection Clifford algebra
3084 Clifford algebras are supported in two flavours: Dirac gamma
3085 matrices (more physical) and generic Clifford algebras (more
3088 @cindex @code{dirac_gamma()}
3089 @subsubsection Dirac gamma matrices
3090 Dirac gamma matrices (note that GiNaC doesn't treat them
3091 as matrices) are designated as @samp{gamma~mu} and satisfy
3092 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3093 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3094 constructed by the function
3097 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3100 which takes two arguments: the index and a @dfn{representation label} in the
3101 range 0 to 255 which is used to distinguish elements of different Clifford
3102 algebras (this is also called a @dfn{spin line index}). Gammas with different
3103 labels commutate with each other. The dimension of the index can be 4 or (in
3104 the framework of dimensional regularization) any symbolic value. Spinor
3105 indices on Dirac gammas are not supported in GiNaC.
3107 @cindex @code{dirac_ONE()}
3108 The unity element of a Clifford algebra is constructed by
3111 ex dirac_ONE(unsigned char rl = 0);
3114 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3115 multiples of the unity element, even though it's customary to omit it.
3116 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3117 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3118 GiNaC will complain and/or produce incorrect results.
3120 @cindex @code{dirac_gamma5()}
3121 There is a special element @samp{gamma5} that commutates with all other
3122 gammas, has a unit square, and in 4 dimensions equals
3123 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3126 ex dirac_gamma5(unsigned char rl = 0);
3129 @cindex @code{dirac_gammaL()}
3130 @cindex @code{dirac_gammaR()}
3131 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3132 objects, constructed by
3135 ex dirac_gammaL(unsigned char rl = 0);
3136 ex dirac_gammaR(unsigned char rl = 0);
3139 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3140 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3142 @cindex @code{dirac_slash()}
3143 Finally, the function
3146 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3149 creates a term that represents a contraction of @samp{e} with the Dirac
3150 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3151 with a unique index whose dimension is given by the @code{dim} argument).
3152 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3154 In products of dirac gammas, superfluous unity elements are automatically
3155 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3156 and @samp{gammaR} are moved to the front.
3158 The @code{simplify_indexed()} function performs contractions in gamma strings,
3164 symbol a("a"), b("b"), D("D");
3165 varidx mu(symbol("mu"), D);
3166 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3167 * dirac_gamma(mu.toggle_variance());
3169 // -> gamma~mu*a\*gamma.mu
3170 e = e.simplify_indexed();
3173 cout << e.subs(D == 4) << endl;
3179 @cindex @code{dirac_trace()}
3180 To calculate the trace of an expression containing strings of Dirac gammas
3181 you use one of the functions
3184 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3185 const ex & trONE = 4);
3186 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3187 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3190 These functions take the trace over all gammas in the specified set @code{rls}
3191 or list @code{rll} of representation labels, or the single label @code{rl};
3192 gammas with other labels are left standing. The last argument to
3193 @code{dirac_trace()} is the value to be returned for the trace of the unity
3194 element, which defaults to 4.
3196 The @code{dirac_trace()} function is a linear functional that is equal to the
3197 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3198 functional is not cyclic in
3201 dimensions when acting on
3202 expressions containing @samp{gamma5}, so it's not a proper trace. This
3203 @samp{gamma5} scheme is described in greater detail in
3204 @cite{The Role of gamma5 in Dimensional Regularization}.
3206 The value of the trace itself is also usually different in 4 and in
3214 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3215 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3216 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3217 cout << dirac_trace(e).simplify_indexed() << endl;
3224 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3225 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3226 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3227 cout << dirac_trace(e).simplify_indexed() << endl;
3228 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3232 Here is an example for using @code{dirac_trace()} to compute a value that
3233 appears in the calculation of the one-loop vacuum polarization amplitude in
3238 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3239 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3242 sp.add(l, l, pow(l, 2));
3243 sp.add(l, q, ldotq);
3245 ex e = dirac_gamma(mu) *
3246 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3247 dirac_gamma(mu.toggle_variance()) *
3248 (dirac_slash(l, D) + m * dirac_ONE());
3249 e = dirac_trace(e).simplify_indexed(sp);
3250 e = e.collect(lst(l, ldotq, m));
3252 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3256 The @code{canonicalize_clifford()} function reorders all gamma products that
3257 appear in an expression to a canonical (but not necessarily simple) form.
3258 You can use this to compare two expressions or for further simplifications:
3262 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3263 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3265 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3267 e = canonicalize_clifford(e);
3269 // -> 2*ONE*eta~mu~nu
3273 @cindex @code{clifford_unit()}
3274 @subsubsection A generic Clifford algebra
3276 A generic Clifford algebra, i.e. a
3280 dimensional algebra with
3284 satisfying the identities
3286 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3289 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3291 for some bilinear form (@code{metric})
3292 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3293 and contain symbolic entries. Such generators are created by the
3297 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3298 bool anticommuting = false);
3301 where @code{mu} should be a @code{varidx} class object indexing the
3302 generators, an index @code{mu} with a numeric value may be of type
3304 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3305 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3306 object. Optional parameter @code{rl} allows to distinguish different
3307 Clifford algebras, which will commute with each other. The last
3308 optional parameter @code{anticommuting} defines if the anticommuting
3311 $e_i e_j + e_j e_i = 0$)
3314 e~i e~j + e~j e~i = 0)
3316 will be used for contraction of Clifford units. If the @code{metric} is
3317 supplied by a @code{matrix} object, then the value of
3318 @code{anticommuting} is calculated automatically and the supplied one
3319 will be ignored. One can overcome this by giving @code{metric} through
3320 matrix wrapped into an @code{indexed} object.
3322 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3323 something very close to @code{dirac_gamma(mu)}, although
3324 @code{dirac_gamma} have more efficient simplification mechanism.
3325 @cindex @code{clifford::get_metric()}
3326 The method @code{clifford::get_metric()} returns a metric defining this
3328 @cindex @code{clifford::is_anticommuting()}
3329 The method @code{clifford::is_anticommuting()} returns the
3330 @code{anticommuting} property of a unit.
3332 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3333 the Clifford algebra units with a call like that
3336 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3339 since this may yield some further automatic simplifications. Again, for a
3340 metric defined through a @code{matrix} such a symmetry is detected
3343 Individual generators of a Clifford algebra can be accessed in several
3349 varidx nu(symbol("nu"), 4);
3351 ex M = diag_matrix(lst(1, -1, 0, s));
3352 ex e = clifford_unit(nu, M);
3353 ex e0 = e.subs(nu == 0);
3354 ex e1 = e.subs(nu == 1);
3355 ex e2 = e.subs(nu == 2);
3356 ex e3 = e.subs(nu == 3);
3361 will produce four anti-commuting generators of a Clifford algebra with properties
3363 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3366 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3367 @code{pow(e3, 2) = s}.
3370 @cindex @code{lst_to_clifford()}
3371 A similar effect can be achieved from the function
3374 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3375 unsigned char rl = 0, bool anticommuting = false);
3376 ex lst_to_clifford(const ex & v, const ex & e);
3379 which converts a list or vector
3381 $v = (v^0, v^1, ..., v^n)$
3384 @samp{v = (v~0, v~1, ..., v~n)}
3389 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3392 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3395 directly supplied in the second form of the procedure. In the first form
3396 the Clifford unit @samp{e.k} is generated by the call of
3397 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3398 with the help of @code{lst_to_clifford()} as follows
3403 varidx nu(symbol("nu"), 4);
3405 ex M = diag_matrix(lst(1, -1, 0, s));
3406 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3407 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3408 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3409 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3414 @cindex @code{clifford_to_lst()}
3415 There is the inverse function
3418 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3421 which takes an expression @code{e} and tries to find a list
3423 $v = (v^0, v^1, ..., v^n)$
3426 @samp{v = (v~0, v~1, ..., v~n)}
3430 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3433 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3435 with respect to the given Clifford units @code{c} and with none of the
3436 @samp{v~k} containing Clifford units @code{c} (of course, this
3437 may be impossible). This function can use an @code{algebraic} method
3438 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3440 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3443 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3445 is zero or is not @code{numeric} for some @samp{k}
3446 then the method will be automatically changed to symbolic. The same effect
3447 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3449 @cindex @code{clifford_prime()}
3450 @cindex @code{clifford_star()}
3451 @cindex @code{clifford_bar()}
3452 There are several functions for (anti-)automorphisms of Clifford algebras:
3455 ex clifford_prime(const ex & e)
3456 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3457 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3460 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3461 changes signs of all Clifford units in the expression. The reversion
3462 of a Clifford algebra @code{clifford_star()} coincides with the
3463 @code{conjugate()} method and effectively reverses the order of Clifford
3464 units in any product. Finally the main anti-automorphism
3465 of a Clifford algebra @code{clifford_bar()} is the composition of the
3466 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3467 in a product. These functions correspond to the notations
3482 used in Clifford algebra textbooks.
3484 @cindex @code{clifford_norm()}
3488 ex clifford_norm(const ex & e);
3491 @cindex @code{clifford_inverse()}
3492 calculates the norm of a Clifford number from the expression
3494 $||e||^2 = e\overline{e}$.
3497 @code{||e||^2 = e \bar@{e@}}
3499 The inverse of a Clifford expression is returned by the function
3502 ex clifford_inverse(const ex & e);
3505 which calculates it as
3507 $e^{-1} = \overline{e}/||e||^2$.
3510 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3519 then an exception is raised.
3521 @cindex @code{remove_dirac_ONE()}
3522 If a Clifford number happens to be a factor of
3523 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3524 expression by the function
3527 ex remove_dirac_ONE(const ex & e);
3530 @cindex @code{canonicalize_clifford()}
3531 The function @code{canonicalize_clifford()} works for a
3532 generic Clifford algebra in a similar way as for Dirac gammas.
3534 The next provided function is
3536 @cindex @code{clifford_moebius_map()}
3538 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3539 const ex & d, const ex & v, const ex & G,
3540 unsigned char rl = 0, bool anticommuting = false);
3541 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3542 unsigned char rl = 0, bool anticommuting = false);
3545 It takes a list or vector @code{v} and makes the Moebius (conformal or
3546 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3547 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3548 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3549 indexed object, tensormetric, matrix or a Clifford unit, in the later
3550 case the optional parameters @code{rl} and @code{anticommuting} are ignored
3551 even if supplied. The returned value of this function is a list of
3552 components of the resulting vector.
3554 @cindex @code{clifford_max_label()}
3555 Finally the function
3558 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3561 can detect a presence of Clifford objects in the expression @code{e}: if
3562 such objects are found it returns the maximal
3563 @code{representation_label} of them, otherwise @code{-1}. The optional
3564 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3565 be ignored during the search.
3567 LaTeX output for Clifford units looks like
3568 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3569 @code{representation_label} and @code{\nu} is the index of the
3570 corresponding unit. This provides a flexible typesetting with a suitable
3571 defintion of the @code{\clifford} command. For example, the definition
3573 \newcommand@{\clifford@}[1][]@{@}
3575 typesets all Clifford units identically, while the alternative definition
3577 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3579 prints units with @code{representation_label=0} as
3586 with @code{representation_label=1} as
3593 and with @code{representation_label=2} as
3601 @cindex @code{color} (class)
3602 @subsection Color algebra
3604 @cindex @code{color_T()}
3605 For computations in quantum chromodynamics, GiNaC implements the base elements
3606 and structure constants of the su(3) Lie algebra (color algebra). The base
3607 elements @math{T_a} are constructed by the function
3610 ex color_T(const ex & a, unsigned char rl = 0);
3613 which takes two arguments: the index and a @dfn{representation label} in the
3614 range 0 to 255 which is used to distinguish elements of different color
3615 algebras. Objects with different labels commutate with each other. The
3616 dimension of the index must be exactly 8 and it should be of class @code{idx},
3619 @cindex @code{color_ONE()}
3620 The unity element of a color algebra is constructed by
3623 ex color_ONE(unsigned char rl = 0);
3626 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3627 multiples of the unity element, even though it's customary to omit it.
3628 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3629 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3630 GiNaC may produce incorrect results.
3632 @cindex @code{color_d()}
3633 @cindex @code{color_f()}
3637 ex color_d(const ex & a, const ex & b, const ex & c);
3638 ex color_f(const ex & a, const ex & b, const ex & c);
3641 create the symmetric and antisymmetric structure constants @math{d_abc} and
3642 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3643 and @math{[T_a, T_b] = i f_abc T_c}.
3645 These functions evaluate to their numerical values,
3646 if you supply numeric indices to them. The index values should be in
3647 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3648 goes along better with the notations used in physical literature.
3650 @cindex @code{color_h()}
3651 There's an additional function
3654 ex color_h(const ex & a, const ex & b, const ex & c);
3657 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3659 The function @code{simplify_indexed()} performs some simplifications on
3660 expressions containing color objects:
3665 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3666 k(symbol("k"), 8), l(symbol("l"), 8);
3668 e = color_d(a, b, l) * color_f(a, b, k);
3669 cout << e.simplify_indexed() << endl;
3672 e = color_d(a, b, l) * color_d(a, b, k);
3673 cout << e.simplify_indexed() << endl;
3676 e = color_f(l, a, b) * color_f(a, b, k);
3677 cout << e.simplify_indexed() << endl;
3680 e = color_h(a, b, c) * color_h(a, b, c);
3681 cout << e.simplify_indexed() << endl;
3684 e = color_h(a, b, c) * color_T(b) * color_T(c);
3685 cout << e.simplify_indexed() << endl;
3688 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3689 cout << e.simplify_indexed() << endl;
3692 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3693 cout << e.simplify_indexed() << endl;
3694 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3698 @cindex @code{color_trace()}
3699 To calculate the trace of an expression containing color objects you use one
3703 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3704 ex color_trace(const ex & e, const lst & rll);
3705 ex color_trace(const ex & e, unsigned char rl = 0);
3708 These functions take the trace over all color @samp{T} objects in the
3709 specified set @code{rls} or list @code{rll} of representation labels, or the
3710 single label @code{rl}; @samp{T}s with other labels are left standing. For
3715 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3717 // -> -I*f.a.c.b+d.a.c.b
3722 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3723 @c node-name, next, previous, up
3726 @cindex @code{exhashmap} (class)
3728 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3729 that can be used as a drop-in replacement for the STL
3730 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3731 typically constant-time, element look-up than @code{map<>}.
3733 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3734 following differences:
3738 no @code{lower_bound()} and @code{upper_bound()} methods
3740 no reverse iterators, no @code{rbegin()}/@code{rend()}
3742 no @code{operator<(exhashmap, exhashmap)}
3744 the comparison function object @code{key_compare} is hardcoded to
3747 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3748 initial hash table size (the actual table size after construction may be
3749 larger than the specified value)
3751 the method @code{size_t bucket_count()} returns the current size of the hash
3754 @code{insert()} and @code{erase()} operations invalidate all iterators
3758 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3759 @c node-name, next, previous, up
3760 @chapter Methods and Functions
3763 In this chapter the most important algorithms provided by GiNaC will be
3764 described. Some of them are implemented as functions on expressions,
3765 others are implemented as methods provided by expression objects. If
3766 they are methods, there exists a wrapper function around it, so you can
3767 alternatively call it in a functional way as shown in the simple
3772 cout << "As method: " << sin(1).evalf() << endl;
3773 cout << "As function: " << evalf(sin(1)) << endl;
3777 @cindex @code{subs()}
3778 The general rule is that wherever methods accept one or more parameters
3779 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3780 wrapper accepts is the same but preceded by the object to act on
3781 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3782 most natural one in an OO model but it may lead to confusion for MapleV
3783 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3784 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3785 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3786 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3787 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3788 here. Also, users of MuPAD will in most cases feel more comfortable
3789 with GiNaC's convention. All function wrappers are implemented
3790 as simple inline functions which just call the corresponding method and
3791 are only provided for users uncomfortable with OO who are dead set to
3792 avoid method invocations. Generally, nested function wrappers are much
3793 harder to read than a sequence of methods and should therefore be
3794 avoided if possible. On the other hand, not everything in GiNaC is a
3795 method on class @code{ex} and sometimes calling a function cannot be
3799 * Information About Expressions::
3800 * Numerical Evaluation::
3801 * Substituting Expressions::
3802 * Pattern Matching and Advanced Substitutions::
3803 * Applying a Function on Subexpressions::
3804 * Visitors and Tree Traversal::
3805 * Polynomial Arithmetic:: Working with polynomials.
3806 * Rational Expressions:: Working with rational functions.
3807 * Symbolic Differentiation::
3808 * Series Expansion:: Taylor and Laurent expansion.
3810 * Built-in Functions:: List of predefined mathematical functions.
3811 * Multiple polylogarithms::
3812 * Complex Conjugation::
3813 * Built-in Functions:: List of predefined mathematical functions.
3814 * Solving Linear Systems of Equations::
3815 * Input/Output:: Input and output of expressions.
3819 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3820 @c node-name, next, previous, up
3821 @section Getting information about expressions
3823 @subsection Checking expression types
3824 @cindex @code{is_a<@dots{}>()}
3825 @cindex @code{is_exactly_a<@dots{}>()}
3826 @cindex @code{ex_to<@dots{}>()}
3827 @cindex Converting @code{ex} to other classes
3828 @cindex @code{info()}
3829 @cindex @code{return_type()}
3830 @cindex @code{return_type_tinfo()}
3832 Sometimes it's useful to check whether a given expression is a plain number,
3833 a sum, a polynomial with integer coefficients, or of some other specific type.
3834 GiNaC provides a couple of functions for this:
3837 bool is_a<T>(const ex & e);
3838 bool is_exactly_a<T>(const ex & e);
3839 bool ex::info(unsigned flag);
3840 unsigned ex::return_type() const;
3841 unsigned ex::return_type_tinfo() const;
3844 When the test made by @code{is_a<T>()} returns true, it is safe to call
3845 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3846 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3847 example, assuming @code{e} is an @code{ex}:
3852 if (is_a<numeric>(e))
3853 numeric n = ex_to<numeric>(e);
3858 @code{is_a<T>(e)} allows you to check whether the top-level object of
3859 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3860 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3861 e.g., for checking whether an expression is a number, a sum, or a product:
3868 is_a<numeric>(e1); // true
3869 is_a<numeric>(e2); // false
3870 is_a<add>(e1); // false
3871 is_a<add>(e2); // true
3872 is_a<mul>(e1); // false
3873 is_a<mul>(e2); // false
3877 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3878 top-level object of an expression @samp{e} is an instance of the GiNaC
3879 class @samp{T}, not including parent classes.
3881 The @code{info()} method is used for checking certain attributes of
3882 expressions. The possible values for the @code{flag} argument are defined
3883 in @file{ginac/flags.h}, the most important being explained in the following
3887 @multitable @columnfractions .30 .70
3888 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3889 @item @code{numeric}
3890 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3892 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3893 @item @code{rational}
3894 @tab @dots{}an exact rational number (integers are rational, too)
3895 @item @code{integer}
3896 @tab @dots{}a (non-complex) integer
3897 @item @code{crational}
3898 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3899 @item @code{cinteger}
3900 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3901 @item @code{positive}
3902 @tab @dots{}not complex and greater than 0
3903 @item @code{negative}
3904 @tab @dots{}not complex and less than 0
3905 @item @code{nonnegative}
3906 @tab @dots{}not complex and greater than or equal to 0
3908 @tab @dots{}an integer greater than 0
3910 @tab @dots{}an integer less than 0
3911 @item @code{nonnegint}
3912 @tab @dots{}an integer greater than or equal to 0
3914 @tab @dots{}an even integer
3916 @tab @dots{}an odd integer
3918 @tab @dots{}a prime integer (probabilistic primality test)
3919 @item @code{relation}
3920 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3921 @item @code{relation_equal}
3922 @tab @dots{}a @code{==} relation
3923 @item @code{relation_not_equal}
3924 @tab @dots{}a @code{!=} relation
3925 @item @code{relation_less}
3926 @tab @dots{}a @code{<} relation
3927 @item @code{relation_less_or_equal}
3928 @tab @dots{}a @code{<=} relation
3929 @item @code{relation_greater}
3930 @tab @dots{}a @code{>} relation
3931 @item @code{relation_greater_or_equal}
3932 @tab @dots{}a @code{>=} relation
3934 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3936 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3937 @item @code{polynomial}
3938 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3939 @item @code{integer_polynomial}
3940 @tab @dots{}a polynomial with (non-complex) integer coefficients
3941 @item @code{cinteger_polynomial}
3942 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3943 @item @code{rational_polynomial}
3944 @tab @dots{}a polynomial with (non-complex) rational coefficients
3945 @item @code{crational_polynomial}
3946 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3947 @item @code{rational_function}
3948 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3949 @item @code{algebraic}
3950 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3954 To determine whether an expression is commutative or non-commutative and if
3955 so, with which other expressions it would commutate, you use the methods
3956 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3957 for an explanation of these.
3960 @subsection Accessing subexpressions
3963 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3964 @code{function}, act as containers for subexpressions. For example, the
3965 subexpressions of a sum (an @code{add} object) are the individual terms,
3966 and the subexpressions of a @code{function} are the function's arguments.
3968 @cindex @code{nops()}
3970 GiNaC provides several ways of accessing subexpressions. The first way is to
3975 ex ex::op(size_t i);
3978 @code{nops()} determines the number of subexpressions (operands) contained
3979 in the expression, while @code{op(i)} returns the @code{i}-th
3980 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3981 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3982 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3983 @math{i>0} are the indices.
3986 @cindex @code{const_iterator}
3987 The second way to access subexpressions is via the STL-style random-access
3988 iterator class @code{const_iterator} and the methods
3991 const_iterator ex::begin();
3992 const_iterator ex::end();
3995 @code{begin()} returns an iterator referring to the first subexpression;
3996 @code{end()} returns an iterator which is one-past the last subexpression.
3997 If the expression has no subexpressions, then @code{begin() == end()}. These
3998 iterators can also be used in conjunction with non-modifying STL algorithms.
4000 Here is an example that (non-recursively) prints the subexpressions of a
4001 given expression in three different ways:
4008 for (size_t i = 0; i != e.nops(); ++i)
4009 cout << e.op(i) << endl;
4012 for (const_iterator i = e.begin(); i != e.end(); ++i)
4015 // with iterators and STL copy()
4016 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4020 @cindex @code{const_preorder_iterator}
4021 @cindex @code{const_postorder_iterator}
4022 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4023 expression's immediate children. GiNaC provides two additional iterator
4024 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4025 that iterate over all objects in an expression tree, in preorder or postorder,
4026 respectively. They are STL-style forward iterators, and are created with the
4030 const_preorder_iterator ex::preorder_begin();
4031 const_preorder_iterator ex::preorder_end();
4032 const_postorder_iterator ex::postorder_begin();
4033 const_postorder_iterator ex::postorder_end();
4036 The following example illustrates the differences between
4037 @code{const_iterator}, @code{const_preorder_iterator}, and
4038 @code{const_postorder_iterator}:
4042 symbol A("A"), B("B"), C("C");
4043 ex e = lst(lst(A, B), C);
4045 std::copy(e.begin(), e.end(),
4046 std::ostream_iterator<ex>(cout, "\n"));
4050 std::copy(e.preorder_begin(), e.preorder_end(),
4051 std::ostream_iterator<ex>(cout, "\n"));
4058 std::copy(e.postorder_begin(), e.postorder_end(),
4059 std::ostream_iterator<ex>(cout, "\n"));
4068 @cindex @code{relational} (class)
4069 Finally, the left-hand side and right-hand side expressions of objects of
4070 class @code{relational} (and only of these) can also be accessed with the
4079 @subsection Comparing expressions
4080 @cindex @code{is_equal()}
4081 @cindex @code{is_zero()}
4083 Expressions can be compared with the usual C++ relational operators like
4084 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4085 the result is usually not determinable and the result will be @code{false},
4086 except in the case of the @code{!=} operator. You should also be aware that
4087 GiNaC will only do the most trivial test for equality (subtracting both
4088 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4091 Actually, if you construct an expression like @code{a == b}, this will be
4092 represented by an object of the @code{relational} class (@pxref{Relations})
4093 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4095 There are also two methods
4098 bool ex::is_equal(const ex & other);
4102 for checking whether one expression is equal to another, or equal to zero,
4106 @subsection Ordering expressions
4107 @cindex @code{ex_is_less} (class)
4108 @cindex @code{ex_is_equal} (class)
4109 @cindex @code{compare()}
4111 Sometimes it is necessary to establish a mathematically well-defined ordering
4112 on a set of arbitrary expressions, for example to use expressions as keys
4113 in a @code{std::map<>} container, or to bring a vector of expressions into
4114 a canonical order (which is done internally by GiNaC for sums and products).
4116 The operators @code{<}, @code{>} etc. described in the last section cannot
4117 be used for this, as they don't implement an ordering relation in the
4118 mathematical sense. In particular, they are not guaranteed to be
4119 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4120 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4123 By default, STL classes and algorithms use the @code{<} and @code{==}
4124 operators to compare objects, which are unsuitable for expressions, but GiNaC
4125 provides two functors that can be supplied as proper binary comparison
4126 predicates to the STL:
4129 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4131 bool operator()(const ex &lh, const ex &rh) const;
4134 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4136 bool operator()(const ex &lh, const ex &rh) const;
4140 For example, to define a @code{map} that maps expressions to strings you
4144 std::map<ex, std::string, ex_is_less> myMap;
4147 Omitting the @code{ex_is_less} template parameter will introduce spurious
4148 bugs because the map operates improperly.
4150 Other examples for the use of the functors:
4158 std::sort(v.begin(), v.end(), ex_is_less());
4160 // count the number of expressions equal to '1'
4161 unsigned num_ones = std::count_if(v.begin(), v.end(),
4162 std::bind2nd(ex_is_equal(), 1));
4165 The implementation of @code{ex_is_less} uses the member function
4168 int ex::compare(const ex & other) const;
4171 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4172 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4176 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4177 @c node-name, next, previous, up
4178 @section Numerical Evaluation
4179 @cindex @code{evalf()}
4181 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4182 To evaluate them using floating-point arithmetic you need to call
4185 ex ex::evalf(int level = 0) const;
4188 @cindex @code{Digits}
4189 The accuracy of the evaluation is controlled by the global object @code{Digits}
4190 which can be assigned an integer value. The default value of @code{Digits}
4191 is 17. @xref{Numbers}, for more information and examples.
4193 To evaluate an expression to a @code{double} floating-point number you can
4194 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4198 // Approximate sin(x/Pi)
4200 ex e = series(sin(x/Pi), x == 0, 6);
4202 // Evaluate numerically at x=0.1
4203 ex f = evalf(e.subs(x == 0.1));
4205 // ex_to<numeric> is an unsafe cast, so check the type first
4206 if (is_a<numeric>(f)) @{
4207 double d = ex_to<numeric>(f).to_double();
4216 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4217 @c node-name, next, previous, up
4218 @section Substituting expressions
4219 @cindex @code{subs()}
4221 Algebraic objects inside expressions can be replaced with arbitrary
4222 expressions via the @code{.subs()} method:
4225 ex ex::subs(const ex & e, unsigned options = 0);
4226 ex ex::subs(const exmap & m, unsigned options = 0);
4227 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4230 In the first form, @code{subs()} accepts a relational of the form
4231 @samp{object == expression} or a @code{lst} of such relationals:
4235 symbol x("x"), y("y");
4237 ex e1 = 2*x^2-4*x+3;
4238 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4242 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4247 If you specify multiple substitutions, they are performed in parallel, so e.g.
4248 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4250 The second form of @code{subs()} takes an @code{exmap} object which is a
4251 pair associative container that maps expressions to expressions (currently
4252 implemented as a @code{std::map}). This is the most efficient one of the
4253 three @code{subs()} forms and should be used when the number of objects to
4254 be substituted is large or unknown.
4256 Using this form, the second example from above would look like this:
4260 symbol x("x"), y("y");
4266 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4270 The third form of @code{subs()} takes two lists, one for the objects to be
4271 replaced and one for the expressions to be substituted (both lists must
4272 contain the same number of elements). Using this form, you would write
4276 symbol x("x"), y("y");
4279 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4283 The optional last argument to @code{subs()} is a combination of
4284 @code{subs_options} flags. There are two options available:
4285 @code{subs_options::no_pattern} disables pattern matching, which makes
4286 large @code{subs()} operations significantly faster if you are not using
4287 patterns. The second option, @code{subs_options::algebraic} enables
4288 algebraic substitutions in products and powers.
4289 @ref{Pattern Matching and Advanced Substitutions}, for more information
4290 about patterns and algebraic substitutions.
4292 @code{subs()} performs syntactic substitution of any complete algebraic
4293 object; it does not try to match sub-expressions as is demonstrated by the
4298 symbol x("x"), y("y"), z("z");
4300 ex e1 = pow(x+y, 2);
4301 cout << e1.subs(x+y == 4) << endl;
4304 ex e2 = sin(x)*sin(y)*cos(x);
4305 cout << e2.subs(sin(x) == cos(x)) << endl;
4306 // -> cos(x)^2*sin(y)
4309 cout << e3.subs(x+y == 4) << endl;
4311 // (and not 4+z as one might expect)
4315 A more powerful form of substitution using wildcards is described in the
4319 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4320 @c node-name, next, previous, up
4321 @section Pattern matching and advanced substitutions
4322 @cindex @code{wildcard} (class)
4323 @cindex Pattern matching
4325 GiNaC allows the use of patterns for checking whether an expression is of a
4326 certain form or contains subexpressions of a certain form, and for
4327 substituting expressions in a more general way.
4329 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4330 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4331 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4332 an unsigned integer number to allow having multiple different wildcards in a
4333 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4334 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4338 ex wild(unsigned label = 0);
4341 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4344 Some examples for patterns:
4346 @multitable @columnfractions .5 .5
4347 @item @strong{Constructed as} @tab @strong{Output as}
4348 @item @code{wild()} @tab @samp{$0}
4349 @item @code{pow(x,wild())} @tab @samp{x^$0}
4350 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4351 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4357 @item Wildcards behave like symbols and are subject to the same algebraic
4358 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4359 @item As shown in the last example, to use wildcards for indices you have to
4360 use them as the value of an @code{idx} object. This is because indices must
4361 always be of class @code{idx} (or a subclass).
4362 @item Wildcards only represent expressions or subexpressions. It is not
4363 possible to use them as placeholders for other properties like index
4364 dimension or variance, representation labels, symmetry of indexed objects
4366 @item Because wildcards are commutative, it is not possible to use wildcards
4367 as part of noncommutative products.
4368 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4369 are also valid patterns.
4372 @subsection Matching expressions
4373 @cindex @code{match()}
4374 The most basic application of patterns is to check whether an expression
4375 matches a given pattern. This is done by the function
4378 bool ex::match(const ex & pattern);
4379 bool ex::match(const ex & pattern, lst & repls);
4382 This function returns @code{true} when the expression matches the pattern
4383 and @code{false} if it doesn't. If used in the second form, the actual
4384 subexpressions matched by the wildcards get returned in the @code{repls}
4385 object as a list of relations of the form @samp{wildcard == expression}.
4386 If @code{match()} returns false, the state of @code{repls} is undefined.
4387 For reproducible results, the list should be empty when passed to
4388 @code{match()}, but it is also possible to find similarities in multiple
4389 expressions by passing in the result of a previous match.
4391 The matching algorithm works as follows:
4394 @item A single wildcard matches any expression. If one wildcard appears
4395 multiple times in a pattern, it must match the same expression in all
4396 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4397 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4398 @item If the expression is not of the same class as the pattern, the match
4399 fails (i.e. a sum only matches a sum, a function only matches a function,
4401 @item If the pattern is a function, it only matches the same function
4402 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4403 @item Except for sums and products, the match fails if the number of
4404 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4406 @item If there are no subexpressions, the expressions and the pattern must
4407 be equal (in the sense of @code{is_equal()}).
4408 @item Except for sums and products, each subexpression (@code{op()}) must
4409 match the corresponding subexpression of the pattern.
4412 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4413 account for their commutativity and associativity:
4416 @item If the pattern contains a term or factor that is a single wildcard,
4417 this one is used as the @dfn{global wildcard}. If there is more than one
4418 such wildcard, one of them is chosen as the global wildcard in a random
4420 @item Every term/factor of the pattern, except the global wildcard, is
4421 matched against every term of the expression in sequence. If no match is
4422 found, the whole match fails. Terms that did match are not considered in
4424 @item If there are no unmatched terms left, the match succeeds. Otherwise
4425 the match fails unless there is a global wildcard in the pattern, in
4426 which case this wildcard matches the remaining terms.
4429 In general, having more than one single wildcard as a term of a sum or a
4430 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4433 Here are some examples in @command{ginsh} to demonstrate how it works (the
4434 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4435 match fails, and the list of wildcard replacements otherwise):
4438 > match((x+y)^a,(x+y)^a);
4440 > match((x+y)^a,(x+y)^b);
4442 > match((x+y)^a,$1^$2);
4444 > match((x+y)^a,$1^$1);
4446 > match((x+y)^(x+y),$1^$1);
4448 > match((x+y)^(x+y),$1^$2);
4450 > match((a+b)*(a+c),($1+b)*($1+c));
4452 > match((a+b)*(a+c),(a+$1)*(a+$2));
4454 (Unpredictable. The result might also be [$1==c,$2==b].)
4455 > match((a+b)*(a+c),($1+$2)*($1+$3));
4456 (The result is undefined. Due to the sequential nature of the algorithm
4457 and the re-ordering of terms in GiNaC, the match for the first factor
4458 may be @{$1==a,$2==b@} in which case the match for the second factor
4459 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4461 > match(a*(x+y)+a*z+b,a*$1+$2);
4462 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4463 @{$1=x+y,$2=a*z+b@}.)
4464 > match(a+b+c+d+e+f,c);
4466 > match(a+b+c+d+e+f,c+$0);
4468 > match(a+b+c+d+e+f,c+e+$0);
4470 > match(a+b,a+b+$0);
4472 > match(a*b^2,a^$1*b^$2);
4474 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4475 even though a==a^1.)
4476 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4478 > match(atan2(y,x^2),atan2(y,$0));
4482 @subsection Matching parts of expressions
4483 @cindex @code{has()}
4484 A more general way to look for patterns in expressions is provided by the
4488 bool ex::has(const ex & pattern);
4491 This function checks whether a pattern is matched by an expression itself or
4492 by any of its subexpressions.
4494 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4495 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4498 > has(x*sin(x+y+2*a),y);
4500 > has(x*sin(x+y+2*a),x+y);
4502 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4503 has the subexpressions "x", "y" and "2*a".)
4504 > has(x*sin(x+y+2*a),x+y+$1);
4506 (But this is possible.)
4507 > has(x*sin(2*(x+y)+2*a),x+y);
4509 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4510 which "x+y" is not a subexpression.)
4513 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4515 > has(4*x^2-x+3,$1*x);
4517 > has(4*x^2+x+3,$1*x);
4519 (Another possible pitfall. The first expression matches because the term
4520 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4521 contains a linear term you should use the coeff() function instead.)
4524 @cindex @code{find()}
4528 bool ex::find(const ex & pattern, lst & found);
4531 works a bit like @code{has()} but it doesn't stop upon finding the first
4532 match. Instead, it appends all found matches to the specified list. If there
4533 are multiple occurrences of the same expression, it is entered only once to
4534 the list. @code{find()} returns false if no matches were found (in
4535 @command{ginsh}, it returns an empty list):
4538 > find(1+x+x^2+x^3,x);
4540 > find(1+x+x^2+x^3,y);
4542 > find(1+x+x^2+x^3,x^$1);
4544 (Note the absence of "x".)
4545 > expand((sin(x)+sin(y))*(a+b));
4546 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4551 @subsection Substituting expressions
4552 @cindex @code{subs()}
4553 Probably the most useful application of patterns is to use them for
4554 substituting expressions with the @code{subs()} method. Wildcards can be
4555 used in the search patterns as well as in the replacement expressions, where
4556 they get replaced by the expressions matched by them. @code{subs()} doesn't
4557 know anything about algebra; it performs purely syntactic substitutions.
4562 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4564 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4566 > subs((a+b+c)^2,a+b==x);
4568 > subs((a+b+c)^2,a+b+$1==x+$1);
4570 > subs(a+2*b,a+b==x);
4572 > subs(4*x^3-2*x^2+5*x-1,x==a);
4574 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4576 > subs(sin(1+sin(x)),sin($1)==cos($1));
4578 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4582 The last example would be written in C++ in this way:
4586 symbol a("a"), b("b"), x("x"), y("y");
4587 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4588 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4589 cout << e.expand() << endl;
4594 @subsection Algebraic substitutions
4595 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4596 enables smarter, algebraic substitutions in products and powers. If you want
4597 to substitute some factors of a product, you only need to list these factors
4598 in your pattern. Furthermore, if an (integer) power of some expression occurs
4599 in your pattern and in the expression that you want the substitution to occur
4600 in, it can be substituted as many times as possible, without getting negative
4603 An example clarifies it all (hopefully):
4606 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4607 subs_options::algebraic) << endl;
4608 // --> (y+x)^6+b^6+a^6
4610 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4612 // Powers and products are smart, but addition is just the same.
4614 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4617 // As I said: addition is just the same.
4619 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4620 // --> x^3*b*a^2+2*b
4622 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4624 // --> 2*b+x^3*b^(-1)*a^(-2)
4626 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4627 // --> -1-2*a^2+4*a^3+5*a
4629 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4630 subs_options::algebraic) << endl;
4631 // --> -1+5*x+4*x^3-2*x^2
4632 // You should not really need this kind of patterns very often now.
4633 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4635 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4636 subs_options::algebraic) << endl;
4637 // --> cos(1+cos(x))
4639 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4640 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4641 subs_options::algebraic)) << endl;
4646 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4647 @c node-name, next, previous, up
4648 @section Applying a Function on Subexpressions
4649 @cindex tree traversal
4650 @cindex @code{map()}
4652 Sometimes you may want to perform an operation on specific parts of an
4653 expression while leaving the general structure of it intact. An example
4654 of this would be a matrix trace operation: the trace of a sum is the sum
4655 of the traces of the individual terms. That is, the trace should @dfn{map}
4656 on the sum, by applying itself to each of the sum's operands. It is possible
4657 to do this manually which usually results in code like this:
4662 if (is_a<matrix>(e))
4663 return ex_to<matrix>(e).trace();
4664 else if (is_a<add>(e)) @{
4666 for (size_t i=0; i<e.nops(); i++)
4667 sum += calc_trace(e.op(i));
4669 @} else if (is_a<mul>)(e)) @{
4677 This is, however, slightly inefficient (if the sum is very large it can take
4678 a long time to add the terms one-by-one), and its applicability is limited to
4679 a rather small class of expressions. If @code{calc_trace()} is called with
4680 a relation or a list as its argument, you will probably want the trace to
4681 be taken on both sides of the relation or of all elements of the list.
4683 GiNaC offers the @code{map()} method to aid in the implementation of such
4687 ex ex::map(map_function & f) const;
4688 ex ex::map(ex (*f)(const ex & e)) const;
4691 In the first (preferred) form, @code{map()} takes a function object that
4692 is subclassed from the @code{map_function} class. In the second form, it
4693 takes a pointer to a function that accepts and returns an expression.
4694 @code{map()} constructs a new expression of the same type, applying the
4695 specified function on all subexpressions (in the sense of @code{op()}),
4698 The use of a function object makes it possible to supply more arguments to
4699 the function that is being mapped, or to keep local state information.
4700 The @code{map_function} class declares a virtual function call operator
4701 that you can overload. Here is a sample implementation of @code{calc_trace()}
4702 that uses @code{map()} in a recursive fashion:
4705 struct calc_trace : public map_function @{
4706 ex operator()(const ex &e)
4708 if (is_a<matrix>(e))
4709 return ex_to<matrix>(e).trace();
4710 else if (is_a<mul>(e)) @{
4713 return e.map(*this);
4718 This function object could then be used like this:
4722 ex M = ... // expression with matrices
4723 calc_trace do_trace;
4724 ex tr = do_trace(M);
4728 Here is another example for you to meditate over. It removes quadratic
4729 terms in a variable from an expanded polynomial:
4732 struct map_rem_quad : public map_function @{
4734 map_rem_quad(const ex & var_) : var(var_) @{@}
4736 ex operator()(const ex & e)
4738 if (is_a<add>(e) || is_a<mul>(e))
4739 return e.map(*this);
4740 else if (is_a<power>(e) &&
4741 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4751 symbol x("x"), y("y");
4754 for (int i=0; i<8; i++)
4755 e += pow(x, i) * pow(y, 8-i) * (i+1);
4757 // -> 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
4759 map_rem_quad rem_quad(x);
4760 cout << rem_quad(e) << endl;
4761 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4765 @command{ginsh} offers a slightly different implementation of @code{map()}
4766 that allows applying algebraic functions to operands. The second argument
4767 to @code{map()} is an expression containing the wildcard @samp{$0} which
4768 acts as the placeholder for the operands:
4773 > map(a+2*b,sin($0));
4775 > map(@{a,b,c@},$0^2+$0);
4776 @{a^2+a,b^2+b,c^2+c@}
4779 Note that it is only possible to use algebraic functions in the second
4780 argument. You can not use functions like @samp{diff()}, @samp{op()},
4781 @samp{subs()} etc. because these are evaluated immediately:
4784 > map(@{a,b,c@},diff($0,a));
4786 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4787 to "map(@{a,b,c@},0)".
4791 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4792 @c node-name, next, previous, up
4793 @section Visitors and Tree Traversal
4794 @cindex tree traversal
4795 @cindex @code{visitor} (class)
4796 @cindex @code{accept()}
4797 @cindex @code{visit()}
4798 @cindex @code{traverse()}
4799 @cindex @code{traverse_preorder()}
4800 @cindex @code{traverse_postorder()}
4802 Suppose that you need a function that returns a list of all indices appearing
4803 in an arbitrary expression. The indices can have any dimension, and for
4804 indices with variance you always want the covariant version returned.
4806 You can't use @code{get_free_indices()} because you also want to include
4807 dummy indices in the list, and you can't use @code{find()} as it needs
4808 specific index dimensions (and it would require two passes: one for indices
4809 with variance, one for plain ones).
4811 The obvious solution to this problem is a tree traversal with a type switch,
4812 such as the following:
4815 void gather_indices_helper(const ex & e, lst & l)
4817 if (is_a<varidx>(e)) @{
4818 const varidx & vi = ex_to<varidx>(e);
4819 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4820 @} else if (is_a<idx>(e)) @{
4823 size_t n = e.nops();
4824 for (size_t i = 0; i < n; ++i)
4825 gather_indices_helper(e.op(i), l);
4829 lst gather_indices(const ex & e)
4832 gather_indices_helper(e, l);
4839 This works fine but fans of object-oriented programming will feel
4840 uncomfortable with the type switch. One reason is that there is a possibility
4841 for subtle bugs regarding derived classes. If we had, for example, written
4844 if (is_a<idx>(e)) @{
4846 @} else if (is_a<varidx>(e)) @{
4850 in @code{gather_indices_helper}, the code wouldn't have worked because the
4851 first line "absorbs" all classes derived from @code{idx}, including
4852 @code{varidx}, so the special case for @code{varidx} would never have been
4855 Also, for a large number of classes, a type switch like the above can get
4856 unwieldy and inefficient (it's a linear search, after all).
4857 @code{gather_indices_helper} only checks for two classes, but if you had to
4858 write a function that required a different implementation for nearly
4859 every GiNaC class, the result would be very hard to maintain and extend.
4861 The cleanest approach to the problem would be to add a new virtual function
4862 to GiNaC's class hierarchy. In our example, there would be specializations
4863 for @code{idx} and @code{varidx} while the default implementation in
4864 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4865 impossible to add virtual member functions to existing classes without
4866 changing their source and recompiling everything. GiNaC comes with source,
4867 so you could actually do this, but for a small algorithm like the one
4868 presented this would be impractical.
4870 One solution to this dilemma is the @dfn{Visitor} design pattern,
4871 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4872 variation, described in detail in
4873 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4874 virtual functions to the class hierarchy to implement operations, GiNaC
4875 provides a single "bouncing" method @code{accept()} that takes an instance
4876 of a special @code{visitor} class and redirects execution to the one
4877 @code{visit()} virtual function of the visitor that matches the type of
4878 object that @code{accept()} was being invoked on.
4880 Visitors in GiNaC must derive from the global @code{visitor} class as well
4881 as from the class @code{T::visitor} of each class @code{T} they want to
4882 visit, and implement the member functions @code{void visit(const T &)} for
4888 void ex::accept(visitor & v) const;
4891 will then dispatch to the correct @code{visit()} member function of the
4892 specified visitor @code{v} for the type of GiNaC object at the root of the
4893 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4895 Here is an example of a visitor:
4899 : public visitor, // this is required
4900 public add::visitor, // visit add objects
4901 public numeric::visitor, // visit numeric objects
4902 public basic::visitor // visit basic objects
4904 void visit(const add & x)
4905 @{ cout << "called with an add object" << endl; @}
4907 void visit(const numeric & x)
4908 @{ cout << "called with a numeric object" << endl; @}
4910 void visit(const basic & x)
4911 @{ cout << "called with a basic object" << endl; @}
4915 which can be used as follows:
4926 // prints "called with a numeric object"
4928 // prints "called with an add object"
4930 // prints "called with a basic object"
4934 The @code{visit(const basic &)} method gets called for all objects that are
4935 not @code{numeric} or @code{add} and acts as an (optional) default.
4937 From a conceptual point of view, the @code{visit()} methods of the visitor
4938 behave like a newly added virtual function of the visited hierarchy.
4939 In addition, visitors can store state in member variables, and they can
4940 be extended by deriving a new visitor from an existing one, thus building
4941 hierarchies of visitors.
4943 We can now rewrite our index example from above with a visitor:
4946 class gather_indices_visitor
4947 : public visitor, public idx::visitor, public varidx::visitor
4951 void visit(const idx & i)
4956 void visit(const varidx & vi)
4958 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4962 const lst & get_result() // utility function
4971 What's missing is the tree traversal. We could implement it in
4972 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4975 void ex::traverse_preorder(visitor & v) const;
4976 void ex::traverse_postorder(visitor & v) const;
4977 void ex::traverse(visitor & v) const;
4980 @code{traverse_preorder()} visits a node @emph{before} visiting its
4981 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4982 visiting its subexpressions. @code{traverse()} is a synonym for
4983 @code{traverse_preorder()}.
4985 Here is a new implementation of @code{gather_indices()} that uses the visitor
4986 and @code{traverse()}:
4989 lst gather_indices(const ex & e)
4991 gather_indices_visitor v;
4993 return v.get_result();
4997 Alternatively, you could use pre- or postorder iterators for the tree
5001 lst gather_indices(const ex & e)
5003 gather_indices_visitor v;
5004 for (const_preorder_iterator i = e.preorder_begin();
5005 i != e.preorder_end(); ++i) @{
5008 return v.get_result();
5013 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
5014 @c node-name, next, previous, up
5015 @section Polynomial arithmetic
5017 @subsection Expanding and collecting
5018 @cindex @code{expand()}
5019 @cindex @code{collect()}
5020 @cindex @code{collect_common_factors()}
5022 A polynomial in one or more variables has many equivalent
5023 representations. Some useful ones serve a specific purpose. Consider
5024 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5025 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5026 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5027 representations are the recursive ones where one collects for exponents
5028 in one of the three variable. Since the factors are themselves
5029 polynomials in the remaining two variables the procedure can be
5030 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5031 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5034 To bring an expression into expanded form, its method
5037 ex ex::expand(unsigned options = 0);
5040 may be called. In our example above, this corresponds to @math{4*x*y +
5041 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5042 GiNaC is not easy to guess you should be prepared to see different
5043 orderings of terms in such sums!
5045 Another useful representation of multivariate polynomials is as a
5046 univariate polynomial in one of the variables with the coefficients
5047 being polynomials in the remaining variables. The method
5048 @code{collect()} accomplishes this task:
5051 ex ex::collect(const ex & s, bool distributed = false);
5054 The first argument to @code{collect()} can also be a list of objects in which
5055 case the result is either a recursively collected polynomial, or a polynomial
5056 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5057 by the @code{distributed} flag.
5059 Note that the original polynomial needs to be in expanded form (for the
5060 variables concerned) in order for @code{collect()} to be able to find the
5061 coefficients properly.
5063 The following @command{ginsh} transcript shows an application of @code{collect()}
5064 together with @code{find()}:
5067 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5068 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)
5069 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5070 > collect(a,@{p,q@});
5071 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5072 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5073 > collect(a,find(a,sin($1)));
5074 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5075 > collect(a,@{find(a,sin($1)),p,q@});
5076 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5077 > collect(a,@{find(a,sin($1)),d@});
5078 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5081 Polynomials can often be brought into a more compact form by collecting
5082 common factors from the terms of sums. This is accomplished by the function
5085 ex collect_common_factors(const ex & e);
5088 This function doesn't perform a full factorization but only looks for
5089 factors which are already explicitly present:
5092 > collect_common_factors(a*x+a*y);
5094 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5096 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5097 (c+a)*a*(x*y+y^2+x)*b
5100 @subsection Degree and coefficients
5101 @cindex @code{degree()}
5102 @cindex @code{ldegree()}
5103 @cindex @code{coeff()}
5105 The degree and low degree of a polynomial can be obtained using the two
5109 int ex::degree(const ex & s);
5110 int ex::ldegree(const ex & s);
5113 which also work reliably on non-expanded input polynomials (they even work
5114 on rational functions, returning the asymptotic degree). By definition, the
5115 degree of zero is zero. To extract a coefficient with a certain power from
5116 an expanded polynomial you use
5119 ex ex::coeff(const ex & s, int n);
5122 You can also obtain the leading and trailing coefficients with the methods
5125 ex ex::lcoeff(const ex & s);
5126 ex ex::tcoeff(const ex & s);
5129 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5132 An application is illustrated in the next example, where a multivariate
5133 polynomial is analyzed:
5137 symbol x("x"), y("y");
5138 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5139 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5140 ex Poly = PolyInp.expand();
5142 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5143 cout << "The x^" << i << "-coefficient is "
5144 << Poly.coeff(x,i) << endl;
5146 cout << "As polynomial in y: "
5147 << Poly.collect(y) << endl;
5151 When run, it returns an output in the following fashion:
5154 The x^0-coefficient is y^2+11*y
5155 The x^1-coefficient is 5*y^2-2*y
5156 The x^2-coefficient is -1
5157 The x^3-coefficient is 4*y
5158 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5161 As always, the exact output may vary between different versions of GiNaC
5162 or even from run to run since the internal canonical ordering is not
5163 within the user's sphere of influence.
5165 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5166 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5167 with non-polynomial expressions as they not only work with symbols but with
5168 constants, functions and indexed objects as well:
5172 symbol a("a"), b("b"), c("c"), x("x");
5173 idx i(symbol("i"), 3);
5175 ex e = pow(sin(x) - cos(x), 4);
5176 cout << e.degree(cos(x)) << endl;
5178 cout << e.expand().coeff(sin(x), 3) << endl;
5181 e = indexed(a+b, i) * indexed(b+c, i);
5182 e = e.expand(expand_options::expand_indexed);
5183 cout << e.collect(indexed(b, i)) << endl;
5184 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5189 @subsection Polynomial division
5190 @cindex polynomial division
5193 @cindex pseudo-remainder
5194 @cindex @code{quo()}
5195 @cindex @code{rem()}
5196 @cindex @code{prem()}
5197 @cindex @code{divide()}
5202 ex quo(const ex & a, const ex & b, const ex & x);
5203 ex rem(const ex & a, const ex & b, const ex & x);
5206 compute the quotient and remainder of univariate polynomials in the variable
5207 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5209 The additional function
5212 ex prem(const ex & a, const ex & b, const ex & x);
5215 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5216 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5218 Exact division of multivariate polynomials is performed by the function
5221 bool divide(const ex & a, const ex & b, ex & q);
5224 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5225 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5226 in which case the value of @code{q} is undefined.
5229 @subsection Unit, content and primitive part
5230 @cindex @code{unit()}
5231 @cindex @code{content()}
5232 @cindex @code{primpart()}
5233 @cindex @code{unitcontprim()}
5238 ex ex::unit(const ex & x);
5239 ex ex::content(const ex & x);
5240 ex ex::primpart(const ex & x);
5241 ex ex::primpart(const ex & x, const ex & c);
5244 return the unit part, content part, and primitive polynomial of a multivariate
5245 polynomial with respect to the variable @samp{x} (the unit part being the sign
5246 of the leading coefficient, the content part being the GCD of the coefficients,
5247 and the primitive polynomial being the input polynomial divided by the unit and
5248 content parts). The second variant of @code{primpart()} expects the previously
5249 calculated content part of the polynomial in @code{c}, which enables it to
5250 work faster in the case where the content part has already been computed. The
5251 product of unit, content, and primitive part is the original polynomial.
5253 Additionally, the method
5256 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5259 computes the unit, content, and primitive parts in one go, returning them
5260 in @code{u}, @code{c}, and @code{p}, respectively.
5263 @subsection GCD, LCM and resultant
5266 @cindex @code{gcd()}
5267 @cindex @code{lcm()}
5269 The functions for polynomial greatest common divisor and least common
5270 multiple have the synopsis
5273 ex gcd(const ex & a, const ex & b);
5274 ex lcm(const ex & a, const ex & b);
5277 The functions @code{gcd()} and @code{lcm()} accept two expressions
5278 @code{a} and @code{b} as arguments and return a new expression, their
5279 greatest common divisor or least common multiple, respectively. If the
5280 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5281 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5282 the coefficients must be rationals.
5285 #include <ginac/ginac.h>
5286 using namespace GiNaC;
5290 symbol x("x"), y("y"), z("z");
5291 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5292 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5294 ex P_gcd = gcd(P_a, P_b);
5296 ex P_lcm = lcm(P_a, P_b);
5297 // 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
5302 @cindex @code{resultant()}
5304 The resultant of two expressions only makes sense with polynomials.
5305 It is always computed with respect to a specific symbol within the
5306 expressions. The function has the interface
5309 ex resultant(const ex & a, const ex & b, const ex & s);
5312 Resultants are symmetric in @code{a} and @code{b}. The following example
5313 computes the resultant of two expressions with respect to @code{x} and
5314 @code{y}, respectively:
5317 #include <ginac/ginac.h>
5318 using namespace GiNaC;
5322 symbol x("x"), y("y");
5324 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5327 r = resultant(e1, e2, x);
5329 r = resultant(e1, e2, y);
5334 @subsection Square-free decomposition
5335 @cindex square-free decomposition
5336 @cindex factorization
5337 @cindex @code{sqrfree()}
5339 GiNaC still lacks proper factorization support. Some form of
5340 factorization is, however, easily implemented by noting that factors
5341 appearing in a polynomial with power two or more also appear in the
5342 derivative and hence can easily be found by computing the GCD of the
5343 original polynomial and its derivatives. Any decent system has an
5344 interface for this so called square-free factorization. So we provide
5347 ex sqrfree(const ex & a, const lst & l = lst());
5349 Here is an example that by the way illustrates how the exact form of the
5350 result may slightly depend on the order of differentiation, calling for
5351 some care with subsequent processing of the result:
5354 symbol x("x"), y("y");
5355 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5357 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5358 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5360 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5361 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5363 cout << sqrfree(BiVarPol) << endl;
5364 // -> depending on luck, any of the above
5367 Note also, how factors with the same exponents are not fully factorized
5371 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5372 @c node-name, next, previous, up
5373 @section Rational expressions
5375 @subsection The @code{normal} method
5376 @cindex @code{normal()}
5377 @cindex simplification
5378 @cindex temporary replacement
5380 Some basic form of simplification of expressions is called for frequently.
5381 GiNaC provides the method @code{.normal()}, which converts a rational function
5382 into an equivalent rational function of the form @samp{numerator/denominator}
5383 where numerator and denominator are coprime. If the input expression is already
5384 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5385 otherwise it performs fraction addition and multiplication.
5387 @code{.normal()} can also be used on expressions which are not rational functions
5388 as it will replace all non-rational objects (like functions or non-integer
5389 powers) by temporary symbols to bring the expression to the domain of rational
5390 functions before performing the normalization, and re-substituting these
5391 symbols afterwards. This algorithm is also available as a separate method
5392 @code{.to_rational()}, described below.
5394 This means that both expressions @code{t1} and @code{t2} are indeed
5395 simplified in this little code snippet:
5400 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5401 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5402 std::cout << "t1 is " << t1.normal() << std::endl;
5403 std::cout << "t2 is " << t2.normal() << std::endl;
5407 Of course this works for multivariate polynomials too, so the ratio of
5408 the sample-polynomials from the section about GCD and LCM above would be
5409 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5412 @subsection Numerator and denominator
5415 @cindex @code{numer()}
5416 @cindex @code{denom()}
5417 @cindex @code{numer_denom()}
5419 The numerator and denominator of an expression can be obtained with
5424 ex ex::numer_denom();
5427 These functions will first normalize the expression as described above and
5428 then return the numerator, denominator, or both as a list, respectively.
5429 If you need both numerator and denominator, calling @code{numer_denom()} is
5430 faster than using @code{numer()} and @code{denom()} separately.
5433 @subsection Converting to a polynomial or rational expression
5434 @cindex @code{to_polynomial()}
5435 @cindex @code{to_rational()}
5437 Some of the methods described so far only work on polynomials or rational
5438 functions. GiNaC provides a way to extend the domain of these functions to
5439 general expressions by using the temporary replacement algorithm described
5440 above. You do this by calling
5443 ex ex::to_polynomial(exmap & m);
5444 ex ex::to_polynomial(lst & l);
5448 ex ex::to_rational(exmap & m);
5449 ex ex::to_rational(lst & l);
5452 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5453 will be filled with the generated temporary symbols and their replacement
5454 expressions in a format that can be used directly for the @code{subs()}
5455 method. It can also already contain a list of replacements from an earlier
5456 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5457 possible to use it on multiple expressions and get consistent results.
5459 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5460 is probably best illustrated with an example:
5464 symbol x("x"), y("y");
5465 ex a = 2*x/sin(x) - y/(3*sin(x));
5469 ex p = a.to_polynomial(lp);
5470 cout << " = " << p << "\n with " << lp << endl;
5471 // = symbol3*symbol2*y+2*symbol2*x
5472 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5475 ex r = a.to_rational(lr);
5476 cout << " = " << r << "\n with " << lr << endl;
5477 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5478 // with @{symbol4==sin(x)@}
5482 The following more useful example will print @samp{sin(x)-cos(x)}:
5487 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5488 ex b = sin(x) + cos(x);
5491 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5492 cout << q.subs(m) << endl;
5497 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5498 @c node-name, next, previous, up
5499 @section Symbolic differentiation
5500 @cindex differentiation
5501 @cindex @code{diff()}
5503 @cindex product rule
5505 GiNaC's objects know how to differentiate themselves. Thus, a
5506 polynomial (class @code{add}) knows that its derivative is the sum of
5507 the derivatives of all the monomials:
5511 symbol x("x"), y("y"), z("z");
5512 ex P = pow(x, 5) + pow(x, 2) + y;
5514 cout << P.diff(x,2) << endl;
5516 cout << P.diff(y) << endl; // 1
5518 cout << P.diff(z) << endl; // 0
5523 If a second integer parameter @var{n} is given, the @code{diff} method
5524 returns the @var{n}th derivative.
5526 If @emph{every} object and every function is told what its derivative
5527 is, all derivatives of composed objects can be calculated using the
5528 chain rule and the product rule. Consider, for instance the expression
5529 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5530 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5531 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5532 out that the composition is the generating function for Euler Numbers,
5533 i.e. the so called @var{n}th Euler number is the coefficient of
5534 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5535 identity to code a function that generates Euler numbers in just three
5538 @cindex Euler numbers
5540 #include <ginac/ginac.h>
5541 using namespace GiNaC;
5543 ex EulerNumber(unsigned n)
5546 const ex generator = pow(cosh(x),-1);
5547 return generator.diff(x,n).subs(x==0);
5552 for (unsigned i=0; i<11; i+=2)
5553 std::cout << EulerNumber(i) << std::endl;
5558 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5559 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5560 @code{i} by two since all odd Euler numbers vanish anyways.
5563 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5564 @c node-name, next, previous, up
5565 @section Series expansion
5566 @cindex @code{series()}
5567 @cindex Taylor expansion
5568 @cindex Laurent expansion
5569 @cindex @code{pseries} (class)
5570 @cindex @code{Order()}
5572 Expressions know how to expand themselves as a Taylor series or (more
5573 generally) a Laurent series. As in most conventional Computer Algebra
5574 Systems, no distinction is made between those two. There is a class of
5575 its own for storing such series (@code{class pseries}) and a built-in
5576 function (called @code{Order}) for storing the order term of the series.
5577 As a consequence, if you want to work with series, i.e. multiply two
5578 series, you need to call the method @code{ex::series} again to convert
5579 it to a series object with the usual structure (expansion plus order
5580 term). A sample application from special relativity could read:
5583 #include <ginac/ginac.h>
5584 using namespace std;
5585 using namespace GiNaC;
5589 symbol v("v"), c("c");
5591 ex gamma = 1/sqrt(1 - pow(v/c,2));
5592 ex mass_nonrel = gamma.series(v==0, 10);
5594 cout << "the relativistic mass increase with v is " << endl
5595 << mass_nonrel << endl;
5597 cout << "the inverse square of this series is " << endl
5598 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5602 Only calling the series method makes the last output simplify to
5603 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5604 series raised to the power @math{-2}.
5606 @cindex Machin's formula
5607 As another instructive application, let us calculate the numerical
5608 value of Archimedes' constant
5612 (for which there already exists the built-in constant @code{Pi})
5613 using John Machin's amazing formula
5615 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5618 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5620 This equation (and similar ones) were used for over 200 years for
5621 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5622 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5623 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5624 order term with it and the question arises what the system is supposed
5625 to do when the fractions are plugged into that order term. The solution
5626 is to use the function @code{series_to_poly()} to simply strip the order
5630 #include <ginac/ginac.h>
5631 using namespace GiNaC;
5633 ex machin_pi(int degr)
5636 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5637 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5638 -4*pi_expansion.subs(x==numeric(1,239));
5644 using std::cout; // just for fun, another way of...
5645 using std::endl; // ...dealing with this namespace std.
5647 for (int i=2; i<12; i+=2) @{
5648 pi_frac = machin_pi(i);
5649 cout << i << ":\t" << pi_frac << endl
5650 << "\t" << pi_frac.evalf() << endl;
5656 Note how we just called @code{.series(x,degr)} instead of
5657 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5658 method @code{series()}: if the first argument is a symbol the expression
5659 is expanded in that symbol around point @code{0}. When you run this
5660 program, it will type out:
5664 3.1832635983263598326
5665 4: 5359397032/1706489875
5666 3.1405970293260603143
5667 6: 38279241713339684/12184551018734375
5668 3.141621029325034425
5669 8: 76528487109180192540976/24359780855939418203125
5670 3.141591772182177295
5671 10: 327853873402258685803048818236/104359128170408663038552734375
5672 3.1415926824043995174
5676 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5677 @c node-name, next, previous, up
5678 @section Symmetrization
5679 @cindex @code{symmetrize()}
5680 @cindex @code{antisymmetrize()}
5681 @cindex @code{symmetrize_cyclic()}
5686 ex ex::symmetrize(const lst & l);
5687 ex ex::antisymmetrize(const lst & l);
5688 ex ex::symmetrize_cyclic(const lst & l);
5691 symmetrize an expression by returning the sum over all symmetric,
5692 antisymmetric or cyclic permutations of the specified list of objects,
5693 weighted by the number of permutations.
5695 The three additional methods
5698 ex ex::symmetrize();
5699 ex ex::antisymmetrize();
5700 ex ex::symmetrize_cyclic();
5703 symmetrize or antisymmetrize an expression over its free indices.
5705 Symmetrization is most useful with indexed expressions but can be used with
5706 almost any kind of object (anything that is @code{subs()}able):
5710 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5711 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5713 cout << indexed(A, i, j).symmetrize() << endl;
5714 // -> 1/2*A.j.i+1/2*A.i.j
5715 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5716 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5717 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5718 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5722 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5723 @c node-name, next, previous, up
5724 @section Predefined mathematical functions
5726 @subsection Overview
5728 GiNaC contains the following predefined mathematical functions:
5731 @multitable @columnfractions .30 .70
5732 @item @strong{Name} @tab @strong{Function}
5735 @cindex @code{abs()}
5736 @item @code{csgn(x)}
5738 @cindex @code{conjugate()}
5739 @item @code{conjugate(x)}
5740 @tab complex conjugation
5741 @cindex @code{csgn()}
5742 @item @code{sqrt(x)}
5743 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5744 @cindex @code{sqrt()}
5747 @cindex @code{sin()}
5750 @cindex @code{cos()}
5753 @cindex @code{tan()}
5754 @item @code{asin(x)}
5756 @cindex @code{asin()}
5757 @item @code{acos(x)}
5759 @cindex @code{acos()}
5760 @item @code{atan(x)}
5761 @tab inverse tangent
5762 @cindex @code{atan()}
5763 @item @code{atan2(y, x)}
5764 @tab inverse tangent with two arguments
5765 @item @code{sinh(x)}
5766 @tab hyperbolic sine
5767 @cindex @code{sinh()}
5768 @item @code{cosh(x)}
5769 @tab hyperbolic cosine
5770 @cindex @code{cosh()}
5771 @item @code{tanh(x)}
5772 @tab hyperbolic tangent
5773 @cindex @code{tanh()}
5774 @item @code{asinh(x)}
5775 @tab inverse hyperbolic sine
5776 @cindex @code{asinh()}
5777 @item @code{acosh(x)}
5778 @tab inverse hyperbolic cosine
5779 @cindex @code{acosh()}
5780 @item @code{atanh(x)}
5781 @tab inverse hyperbolic tangent
5782 @cindex @code{atanh()}
5784 @tab exponential function
5785 @cindex @code{exp()}
5787 @tab natural logarithm
5788 @cindex @code{log()}
5791 @cindex @code{Li2()}
5792 @item @code{Li(m, x)}
5793 @tab classical polylogarithm as well as multiple polylogarithm
5795 @item @code{G(a, y)}
5796 @tab multiple polylogarithm
5798 @item @code{G(a, s, y)}
5799 @tab multiple polylogarithm with explicit signs for the imaginary parts
5801 @item @code{S(n, p, x)}
5802 @tab Nielsen's generalized polylogarithm
5804 @item @code{H(m, x)}
5805 @tab harmonic polylogarithm
5807 @item @code{zeta(m)}
5808 @tab Riemann's zeta function as well as multiple zeta value
5809 @cindex @code{zeta()}
5810 @item @code{zeta(m, s)}
5811 @tab alternating Euler sum
5812 @cindex @code{zeta()}
5813 @item @code{zetaderiv(n, x)}
5814 @tab derivatives of Riemann's zeta function
5815 @item @code{tgamma(x)}
5817 @cindex @code{tgamma()}
5818 @cindex gamma function
5819 @item @code{lgamma(x)}
5820 @tab logarithm of gamma function
5821 @cindex @code{lgamma()}
5822 @item @code{beta(x, y)}
5823 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5824 @cindex @code{beta()}
5826 @tab psi (digamma) function
5827 @cindex @code{psi()}
5828 @item @code{psi(n, x)}
5829 @tab derivatives of psi function (polygamma functions)
5830 @item @code{factorial(n)}
5831 @tab factorial function @math{n!}
5832 @cindex @code{factorial()}
5833 @item @code{binomial(n, k)}
5834 @tab binomial coefficients
5835 @cindex @code{binomial()}
5836 @item @code{Order(x)}
5837 @tab order term function in truncated power series
5838 @cindex @code{Order()}
5843 For functions that have a branch cut in the complex plane GiNaC follows
5844 the conventions for C++ as defined in the ANSI standard as far as
5845 possible. In particular: the natural logarithm (@code{log}) and the
5846 square root (@code{sqrt}) both have their branch cuts running along the
5847 negative real axis where the points on the axis itself belong to the
5848 upper part (i.e. continuous with quadrant II). The inverse
5849 trigonometric and hyperbolic functions are not defined for complex
5850 arguments by the C++ standard, however. In GiNaC we follow the
5851 conventions used by CLN, which in turn follow the carefully designed
5852 definitions in the Common Lisp standard. It should be noted that this
5853 convention is identical to the one used by the C99 standard and by most
5854 serious CAS. It is to be expected that future revisions of the C++
5855 standard incorporate these functions in the complex domain in a manner
5856 compatible with C99.
5858 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5859 @c node-name, next, previous, up
5860 @subsection Multiple polylogarithms
5862 @cindex polylogarithm
5863 @cindex Nielsen's generalized polylogarithm
5864 @cindex harmonic polylogarithm
5865 @cindex multiple zeta value
5866 @cindex alternating Euler sum
5867 @cindex multiple polylogarithm
5869 The multiple polylogarithm is the most generic member of a family of functions,
5870 to which others like the harmonic polylogarithm, Nielsen's generalized
5871 polylogarithm and the multiple zeta value belong.
5872 Everyone of these functions can also be written as a multiple polylogarithm with specific
5873 parameters. This whole family of functions is therefore often referred to simply as
5874 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5875 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5876 @code{Li} and @code{G} in principle represent the same function, the different
5877 notations are more natural to the series representation or the integral
5878 representation, respectively.
5880 To facilitate the discussion of these functions we distinguish between indices and
5881 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5882 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5884 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5885 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5886 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5887 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5888 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5889 @code{s} is not given, the signs default to +1.
5890 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5891 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5892 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5893 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5894 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5896 The functions print in LaTeX format as
5898 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5904 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5907 $\zeta(m_1,m_2,\ldots,m_k)$.
5909 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5910 are printed with a line above, e.g.
5912 $\zeta(5,\overline{2})$.
5914 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5916 Definitions and analytical as well as numerical properties of multiple polylogarithms
5917 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5918 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5919 except for a few differences which will be explicitly stated in the following.
5921 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5922 that the indices and arguments are understood to be in the same order as in which they appear in
5923 the series representation. This means
5925 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5928 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5931 $\zeta(1,2)$ evaluates to infinity.
5933 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5936 The functions only evaluate if the indices are integers greater than zero, except for the indices
5937 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5938 will be interpreted as the sequence of signs for the corresponding indices
5939 @code{m} or the sign of the imaginary part for the
5940 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5941 @code{zeta(lst(3,4), lst(-1,1))} means
5943 $\zeta(\overline{3},4)$
5946 @code{G(lst(a,b), lst(-1,1), c)} means
5948 $G(a-0\epsilon,b+0\epsilon;c)$.
5950 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5951 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5952 e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
5953 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5954 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5955 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5956 evaluates also for negative integers and positive even integers. For example:
5959 > Li(@{3,1@},@{x,1@});
5962 -zeta(@{3,2@},@{-1,-1@})
5967 It is easy to tell for a given function into which other function it can be rewritten, may
5968 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5969 with negative indices or trailing zeros (the example above gives a hint). Signs can
5970 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5971 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5972 @code{Li} (@code{eval()} already cares for the possible downgrade):
5975 > convert_H_to_Li(@{0,-2,-1,3@},x);
5976 Li(@{3,1,3@},@{-x,1,-1@})
5977 > convert_H_to_Li(@{2,-1,0@},x);
5978 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5981 Every function can be numerically evaluated for
5982 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5983 global variable @code{Digits}:
5988 > evalf(zeta(@{3,1,3,1@}));
5989 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5992 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5993 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5995 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6000 In long expressions this helps a lot with debugging, because you can easily spot
6001 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6002 cancellations of divergencies happen.
6004 Useful publications:
6006 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6007 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6009 @cite{Harmonic Polylogarithms},
6010 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6012 @cite{Special Values of Multiple Polylogarithms},
6013 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6015 @cite{Numerical Evaluation of Multiple Polylogarithms},
6016 J.Vollinga, S.Weinzierl, hep-ph/0410259
6018 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
6019 @c node-name, next, previous, up
6020 @section Complex Conjugation
6022 @cindex @code{conjugate()}
6030 returns the complex conjugate of the expression. For all built-in functions and objects the
6031 conjugation gives the expected results:
6035 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6039 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6040 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6041 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6042 // -> -gamma5*gamma~b*gamma~a
6046 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
6047 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
6048 arguments. This is the default strategy. If you want to define your own functions and want to
6049 change this behavior, you have to supply a specialized conjugation method for your function
6050 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
6052 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
6053 @c node-name, next, previous, up
6054 @section Solving Linear Systems of Equations
6055 @cindex @code{lsolve()}
6057 The function @code{lsolve()} provides a convenient wrapper around some
6058 matrix operations that comes in handy when a system of linear equations
6062 ex lsolve(const ex & eqns, const ex & symbols,
6063 unsigned options = solve_algo::automatic);
6066 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6067 @code{relational}) while @code{symbols} is a @code{lst} of
6068 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
6071 It returns the @code{lst} of solutions as an expression. As an example,
6072 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6076 symbol a("a"), b("b"), x("x"), y("y");
6078 eqns = a*x+b*y==3, x-y==b;
6080 cout << lsolve(eqns, vars) << endl;
6081 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6084 When the linear equations @code{eqns} are underdetermined, the solution
6085 will contain one or more tautological entries like @code{x==x},
6086 depending on the rank of the system. When they are overdetermined, the
6087 solution will be an empty @code{lst}. Note the third optional parameter
6088 to @code{lsolve()}: it accepts the same parameters as
6089 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6093 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6094 @c node-name, next, previous, up
6095 @section Input and output of expressions
6098 @subsection Expression output
6100 @cindex output of expressions
6102 Expressions can simply be written to any stream:
6107 ex e = 4.5*I+pow(x,2)*3/2;
6108 cout << e << endl; // prints '4.5*I+3/2*x^2'
6112 The default output format is identical to the @command{ginsh} input syntax and
6113 to that used by most computer algebra systems, but not directly pastable
6114 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6115 is printed as @samp{x^2}).
6117 It is possible to print expressions in a number of different formats with
6118 a set of stream manipulators;
6121 std::ostream & dflt(std::ostream & os);
6122 std::ostream & latex(std::ostream & os);
6123 std::ostream & tree(std::ostream & os);
6124 std::ostream & csrc(std::ostream & os);
6125 std::ostream & csrc_float(std::ostream & os);
6126 std::ostream & csrc_double(std::ostream & os);
6127 std::ostream & csrc_cl_N(std::ostream & os);
6128 std::ostream & index_dimensions(std::ostream & os);
6129 std::ostream & no_index_dimensions(std::ostream & os);
6132 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6133 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6134 @code{print_csrc()} functions, respectively.
6137 All manipulators affect the stream state permanently. To reset the output
6138 format to the default, use the @code{dflt} manipulator:
6142 cout << latex; // all output to cout will be in LaTeX format from
6144 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6145 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6146 cout << dflt; // revert to default output format
6147 cout << e << endl; // prints '4.5*I+3/2*x^2'
6151 If you don't want to affect the format of the stream you're working with,
6152 you can output to a temporary @code{ostringstream} like this:
6157 s << latex << e; // format of cout remains unchanged
6158 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6163 @cindex @code{csrc_float}
6164 @cindex @code{csrc_double}
6165 @cindex @code{csrc_cl_N}
6166 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6167 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6168 format that can be directly used in a C or C++ program. The three possible
6169 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6170 classes provided by the CLN library):
6174 cout << "f = " << csrc_float << e << ";\n";
6175 cout << "d = " << csrc_double << e << ";\n";
6176 cout << "n = " << csrc_cl_N << e << ";\n";
6180 The above example will produce (note the @code{x^2} being converted to
6184 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6185 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6186 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6190 The @code{tree} manipulator allows dumping the internal structure of an
6191 expression for debugging purposes:
6202 add, hash=0x0, flags=0x3, nops=2
6203 power, hash=0x0, flags=0x3, nops=2
6204 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6205 2 (numeric), hash=0x6526b0fa, flags=0xf
6206 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6209 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6213 @cindex @code{latex}
6214 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6215 It is rather similar to the default format but provides some braces needed
6216 by LaTeX for delimiting boxes and also converts some common objects to
6217 conventional LaTeX names. It is possible to give symbols a special name for
6218 LaTeX output by supplying it as a second argument to the @code{symbol}
6221 For example, the code snippet
6225 symbol x("x", "\\circ");
6226 ex e = lgamma(x).series(x==0,3);
6227 cout << latex << e << endl;
6234 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6235 +\mathcal@{O@}(\circ^@{3@})
6238 @cindex @code{index_dimensions}
6239 @cindex @code{no_index_dimensions}
6240 Index dimensions are normally hidden in the output. To make them visible, use
6241 the @code{index_dimensions} manipulator. The dimensions will be written in
6242 square brackets behind each index value in the default and LaTeX output
6247 symbol x("x"), y("y");
6248 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6249 ex e = indexed(x, mu) * indexed(y, nu);
6252 // prints 'x~mu*y~nu'
6253 cout << index_dimensions << e << endl;
6254 // prints 'x~mu[4]*y~nu[4]'
6255 cout << no_index_dimensions << e << endl;
6256 // prints 'x~mu*y~nu'
6261 @cindex Tree traversal
6262 If you need any fancy special output format, e.g. for interfacing GiNaC
6263 with other algebra systems or for producing code for different
6264 programming languages, you can always traverse the expression tree yourself:
6267 static void my_print(const ex & e)
6269 if (is_a<function>(e))
6270 cout << ex_to<function>(e).get_name();
6272 cout << ex_to<basic>(e).class_name();
6274 size_t n = e.nops();
6276 for (size_t i=0; i<n; i++) @{
6288 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6296 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6297 symbol(y))),numeric(-2)))
6300 If you need an output format that makes it possible to accurately
6301 reconstruct an expression by feeding the output to a suitable parser or
6302 object factory, you should consider storing the expression in an
6303 @code{archive} object and reading the object properties from there.
6304 See the section on archiving for more information.
6307 @subsection Expression input
6308 @cindex input of expressions
6310 GiNaC provides no way to directly read an expression from a stream because
6311 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6312 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6313 @code{y} you defined in your program and there is no way to specify the
6314 desired symbols to the @code{>>} stream input operator.
6316 Instead, GiNaC lets you construct an expression from a string, specifying the
6317 list of symbols to be used:
6321 symbol x("x"), y("y");
6322 ex e("2*x+sin(y)", lst(x, y));
6326 The input syntax is the same as that used by @command{ginsh} and the stream
6327 output operator @code{<<}. The symbols in the string are matched by name to
6328 the symbols in the list and if GiNaC encounters a symbol not specified in
6329 the list it will throw an exception.
6331 With this constructor, it's also easy to implement interactive GiNaC programs:
6336 #include <stdexcept>
6337 #include <ginac/ginac.h>
6338 using namespace std;
6339 using namespace GiNaC;
6346 cout << "Enter an expression containing 'x': ";
6351 cout << "The derivative of " << e << " with respect to x is ";
6352 cout << e.diff(x) << ".\n";
6353 @} catch (exception &p) @{
6354 cerr << p.what() << endl;
6360 @subsection Archiving
6361 @cindex @code{archive} (class)
6364 GiNaC allows creating @dfn{archives} of expressions which can be stored
6365 to or retrieved from files. To create an archive, you declare an object
6366 of class @code{archive} and archive expressions in it, giving each
6367 expression a unique name:
6371 using namespace std;
6372 #include <ginac/ginac.h>
6373 using namespace GiNaC;
6377 symbol x("x"), y("y"), z("z");
6379 ex foo = sin(x + 2*y) + 3*z + 41;
6383 a.archive_ex(foo, "foo");
6384 a.archive_ex(bar, "the second one");
6388 The archive can then be written to a file:
6392 ofstream out("foobar.gar");
6398 The file @file{foobar.gar} contains all information that is needed to
6399 reconstruct the expressions @code{foo} and @code{bar}.
6401 @cindex @command{viewgar}
6402 The tool @command{viewgar} that comes with GiNaC can be used to view
6403 the contents of GiNaC archive files:
6406 $ viewgar foobar.gar
6407 foo = 41+sin(x+2*y)+3*z
6408 the second one = 42+sin(x+2*y)+3*z
6411 The point of writing archive files is of course that they can later be
6417 ifstream in("foobar.gar");
6422 And the stored expressions can be retrieved by their name:
6429 ex ex1 = a2.unarchive_ex(syms, "foo");
6430 ex ex2 = a2.unarchive_ex(syms, "the second one");
6432 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6433 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6434 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6438 Note that you have to supply a list of the symbols which are to be inserted
6439 in the expressions. Symbols in archives are stored by their name only and
6440 if you don't specify which symbols you have, unarchiving the expression will
6441 create new symbols with that name. E.g. if you hadn't included @code{x} in
6442 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6443 have had no effect because the @code{x} in @code{ex1} would have been a
6444 different symbol than the @code{x} which was defined at the beginning of
6445 the program, although both would appear as @samp{x} when printed.
6447 You can also use the information stored in an @code{archive} object to
6448 output expressions in a format suitable for exact reconstruction. The
6449 @code{archive} and @code{archive_node} classes have a couple of member
6450 functions that let you access the stored properties:
6453 static void my_print2(const archive_node & n)
6456 n.find_string("class", class_name);
6457 cout << class_name << "(";
6459 archive_node::propinfovector p;
6460 n.get_properties(p);
6462 size_t num = p.size();
6463 for (size_t i=0; i<num; i++) @{
6464 const string &name = p[i].name;
6465 if (name == "class")
6467 cout << name << "=";
6469 unsigned count = p[i].count;
6473 for (unsigned j=0; j<count; j++) @{
6474 switch (p[i].type) @{
6475 case archive_node::PTYPE_BOOL: @{
6477 n.find_bool(name, x, j);
6478 cout << (x ? "true" : "false");
6481 case archive_node::PTYPE_UNSIGNED: @{
6483 n.find_unsigned(name, x, j);
6487 case archive_node::PTYPE_STRING: @{
6489 n.find_string(name, x, j);
6490 cout << '\"' << x << '\"';
6493 case archive_node::PTYPE_NODE: @{
6494 const archive_node &x = n.find_ex_node(name, j);
6516 ex e = pow(2, x) - y;
6518 my_print2(ar.get_top_node(0)); cout << endl;
6526 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6527 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6528 overall_coeff=numeric(number="0"))
6531 Be warned, however, that the set of properties and their meaning for each
6532 class may change between GiNaC versions.
6535 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6536 @c node-name, next, previous, up
6537 @chapter Extending GiNaC
6539 By reading so far you should have gotten a fairly good understanding of
6540 GiNaC's design patterns. From here on you should start reading the
6541 sources. All we can do now is issue some recommendations how to tackle
6542 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6543 develop some useful extension please don't hesitate to contact the GiNaC
6544 authors---they will happily incorporate them into future versions.
6547 * What does not belong into GiNaC:: What to avoid.
6548 * Symbolic functions:: Implementing symbolic functions.
6549 * Printing:: Adding new output formats.
6550 * Structures:: Defining new algebraic classes (the easy way).
6551 * Adding classes:: Defining new algebraic classes (the hard way).
6555 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6556 @c node-name, next, previous, up
6557 @section What doesn't belong into GiNaC
6559 @cindex @command{ginsh}
6560 First of all, GiNaC's name must be read literally. It is designed to be
6561 a library for use within C++. The tiny @command{ginsh} accompanying
6562 GiNaC makes this even more clear: it doesn't even attempt to provide a
6563 language. There are no loops or conditional expressions in
6564 @command{ginsh}, it is merely a window into the library for the
6565 programmer to test stuff (or to show off). Still, the design of a
6566 complete CAS with a language of its own, graphical capabilities and all
6567 this on top of GiNaC is possible and is without doubt a nice project for
6570 There are many built-in functions in GiNaC that do not know how to
6571 evaluate themselves numerically to a precision declared at runtime
6572 (using @code{Digits}). Some may be evaluated at certain points, but not
6573 generally. This ought to be fixed. However, doing numerical
6574 computations with GiNaC's quite abstract classes is doomed to be
6575 inefficient. For this purpose, the underlying foundation classes
6576 provided by CLN are much better suited.
6579 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6580 @c node-name, next, previous, up
6581 @section Symbolic functions
6583 The easiest and most instructive way to start extending GiNaC is probably to
6584 create your own symbolic functions. These are implemented with the help of
6585 two preprocessor macros:
6587 @cindex @code{DECLARE_FUNCTION}
6588 @cindex @code{REGISTER_FUNCTION}
6590 DECLARE_FUNCTION_<n>P(<name>)
6591 REGISTER_FUNCTION(<name>, <options>)
6594 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6595 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6596 parameters of type @code{ex} and returns a newly constructed GiNaC
6597 @code{function} object that represents your function.
6599 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6600 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6601 set of options that associate the symbolic function with C++ functions you
6602 provide to implement the various methods such as evaluation, derivative,
6603 series expansion etc. They also describe additional attributes the function
6604 might have, such as symmetry and commutation properties, and a name for
6605 LaTeX output. Multiple options are separated by the member access operator
6606 @samp{.} and can be given in an arbitrary order.
6608 (By the way: in case you are worrying about all the macros above we can
6609 assure you that functions are GiNaC's most macro-intense classes. We have
6610 done our best to avoid macros where we can.)
6612 @subsection A minimal example
6614 Here is an example for the implementation of a function with two arguments
6615 that is not further evaluated:
6618 DECLARE_FUNCTION_2P(myfcn)
6620 REGISTER_FUNCTION(myfcn, dummy())
6623 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6624 in algebraic expressions:
6630 ex e = 2*myfcn(42, 1+3*x) - x;
6632 // prints '2*myfcn(42,1+3*x)-x'
6637 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6638 "no options". A function with no options specified merely acts as a kind of
6639 container for its arguments. It is a pure "dummy" function with no associated
6640 logic (which is, however, sometimes perfectly sufficient).
6642 Let's now have a look at the implementation of GiNaC's cosine function for an
6643 example of how to make an "intelligent" function.
6645 @subsection The cosine function
6647 The GiNaC header file @file{inifcns.h} contains the line
6650 DECLARE_FUNCTION_1P(cos)
6653 which declares to all programs using GiNaC that there is a function @samp{cos}
6654 that takes one @code{ex} as an argument. This is all they need to know to use
6655 this function in expressions.
6657 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6658 is its @code{REGISTER_FUNCTION} line:
6661 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6662 evalf_func(cos_evalf).
6663 derivative_func(cos_deriv).
6664 latex_name("\\cos"));
6667 There are four options defined for the cosine function. One of them
6668 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6669 other three indicate the C++ functions in which the "brains" of the cosine
6670 function are defined.
6672 @cindex @code{hold()}
6674 The @code{eval_func()} option specifies the C++ function that implements
6675 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6676 the same number of arguments as the associated symbolic function (one in this
6677 case) and returns the (possibly transformed or in some way simplified)
6678 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6679 of the automatic evaluation process). If no (further) evaluation is to take
6680 place, the @code{eval_func()} function must return the original function
6681 with @code{.hold()}, to avoid a potential infinite recursion. If your
6682 symbolic functions produce a segmentation fault or stack overflow when
6683 using them in expressions, you are probably missing a @code{.hold()}
6686 The @code{eval_func()} function for the cosine looks something like this
6687 (actually, it doesn't look like this at all, but it should give you an idea
6691 static ex cos_eval(const ex & x)
6693 if ("x is a multiple of 2*Pi")
6695 else if ("x is a multiple of Pi")
6697 else if ("x is a multiple of Pi/2")
6701 else if ("x has the form 'acos(y)'")
6703 else if ("x has the form 'asin(y)'")
6708 return cos(x).hold();
6712 This function is called every time the cosine is used in a symbolic expression:
6718 // this calls cos_eval(Pi), and inserts its return value into
6719 // the actual expression
6726 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6727 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6728 symbolic transformation can be done, the unmodified function is returned
6729 with @code{.hold()}.
6731 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6732 The user has to call @code{evalf()} for that. This is implemented in a
6736 static ex cos_evalf(const ex & x)
6738 if (is_a<numeric>(x))
6739 return cos(ex_to<numeric>(x));
6741 return cos(x).hold();
6745 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6746 in this case the @code{cos()} function for @code{numeric} objects, which in
6747 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6748 isn't really needed here, but reminds us that the corresponding @code{eval()}
6749 function would require it in this place.
6751 Differentiation will surely turn up and so we need to tell @code{cos}
6752 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6753 instance, are then handled automatically by @code{basic::diff} and
6757 static ex cos_deriv(const ex & x, unsigned diff_param)
6763 @cindex product rule
6764 The second parameter is obligatory but uninteresting at this point. It
6765 specifies which parameter to differentiate in a partial derivative in
6766 case the function has more than one parameter, and its main application
6767 is for correct handling of the chain rule.
6769 An implementation of the series expansion is not needed for @code{cos()} as
6770 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6771 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6772 the other hand, does have poles and may need to do Laurent expansion:
6775 static ex tan_series(const ex & x, const relational & rel,
6776 int order, unsigned options)
6778 // Find the actual expansion point
6779 const ex x_pt = x.subs(rel);
6781 if ("x_pt is not an odd multiple of Pi/2")
6782 throw do_taylor(); // tell function::series() to do Taylor expansion
6784 // On a pole, expand sin()/cos()
6785 return (sin(x)/cos(x)).series(rel, order+2, options);
6789 The @code{series()} implementation of a function @emph{must} return a
6790 @code{pseries} object, otherwise your code will crash.
6792 @subsection Function options
6794 GiNaC functions understand several more options which are always
6795 specified as @code{.option(params)}. None of them are required, but you
6796 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6797 is a do-nothing option called @code{dummy()} which you can use to define
6798 functions without any special options.
6801 eval_func(<C++ function>)
6802 evalf_func(<C++ function>)
6803 derivative_func(<C++ function>)
6804 series_func(<C++ function>)
6805 conjugate_func(<C++ function>)
6808 These specify the C++ functions that implement symbolic evaluation,
6809 numeric evaluation, partial derivatives, and series expansion, respectively.
6810 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6811 @code{diff()} and @code{series()}.
6813 The @code{eval_func()} function needs to use @code{.hold()} if no further
6814 automatic evaluation is desired or possible.
6816 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6817 expansion, which is correct if there are no poles involved. If the function
6818 has poles in the complex plane, the @code{series_func()} needs to check
6819 whether the expansion point is on a pole and fall back to Taylor expansion
6820 if it isn't. Otherwise, the pole usually needs to be regularized by some
6821 suitable transformation.
6824 latex_name(const string & n)
6827 specifies the LaTeX code that represents the name of the function in LaTeX
6828 output. The default is to put the function name in an @code{\mbox@{@}}.
6831 do_not_evalf_params()
6834 This tells @code{evalf()} to not recursively evaluate the parameters of the
6835 function before calling the @code{evalf_func()}.
6838 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6841 This allows you to explicitly specify the commutation properties of the
6842 function (@xref{Non-commutative objects}, for an explanation of
6843 (non)commutativity in GiNaC). For example, you can use
6844 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6845 GiNaC treat your function like a matrix. By default, functions inherit the
6846 commutation properties of their first argument.
6849 set_symmetry(const symmetry & s)
6852 specifies the symmetry properties of the function with respect to its
6853 arguments. @xref{Indexed objects}, for an explanation of symmetry
6854 specifications. GiNaC will automatically rearrange the arguments of
6855 symmetric functions into a canonical order.
6857 Sometimes you may want to have finer control over how functions are
6858 displayed in the output. For example, the @code{abs()} function prints
6859 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6860 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6864 print_func<C>(<C++ function>)
6867 option which is explained in the next section.
6869 @subsection Functions with a variable number of arguments
6871 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6872 functions with a fixed number of arguments. Sometimes, though, you may need
6873 to have a function that accepts a variable number of expressions. One way to
6874 accomplish this is to pass variable-length lists as arguments. The
6875 @code{Li()} function uses this method for multiple polylogarithms.
6877 It is also possible to define functions that accept a different number of
6878 parameters under the same function name, such as the @code{psi()} function
6879 which can be called either as @code{psi(z)} (the digamma function) or as
6880 @code{psi(n, z)} (polygamma functions). These are actually two different
6881 functions in GiNaC that, however, have the same name. Defining such
6882 functions is not possible with the macros but requires manually fiddling
6883 with GiNaC internals. If you are interested, please consult the GiNaC source
6884 code for the @code{psi()} function (@file{inifcns.h} and
6885 @file{inifcns_gamma.cpp}).
6888 @node Printing, Structures, Symbolic functions, Extending GiNaC
6889 @c node-name, next, previous, up
6890 @section GiNaC's expression output system
6892 GiNaC allows the output of expressions in a variety of different formats
6893 (@pxref{Input/Output}). This section will explain how expression output
6894 is implemented internally, and how to define your own output formats or
6895 change the output format of built-in algebraic objects. You will also want
6896 to read this section if you plan to write your own algebraic classes or
6899 @cindex @code{print_context} (class)
6900 @cindex @code{print_dflt} (class)
6901 @cindex @code{print_latex} (class)
6902 @cindex @code{print_tree} (class)
6903 @cindex @code{print_csrc} (class)
6904 All the different output formats are represented by a hierarchy of classes
6905 rooted in the @code{print_context} class, defined in the @file{print.h}
6910 the default output format
6912 output in LaTeX mathematical mode
6914 a dump of the internal expression structure (for debugging)
6916 the base class for C source output
6917 @item print_csrc_float
6918 C source output using the @code{float} type
6919 @item print_csrc_double
6920 C source output using the @code{double} type
6921 @item print_csrc_cl_N
6922 C source output using CLN types
6925 The @code{print_context} base class provides two public data members:
6937 @code{s} is a reference to the stream to output to, while @code{options}
6938 holds flags and modifiers. Currently, there is only one flag defined:
6939 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6940 to print the index dimension which is normally hidden.
6942 When you write something like @code{std::cout << e}, where @code{e} is
6943 an object of class @code{ex}, GiNaC will construct an appropriate
6944 @code{print_context} object (of a class depending on the selected output
6945 format), fill in the @code{s} and @code{options} members, and call
6947 @cindex @code{print()}
6949 void ex::print(const print_context & c, unsigned level = 0) const;
6952 which in turn forwards the call to the @code{print()} method of the
6953 top-level algebraic object contained in the expression.
6955 Unlike other methods, GiNaC classes don't usually override their
6956 @code{print()} method to implement expression output. Instead, the default
6957 implementation @code{basic::print(c, level)} performs a run-time double
6958 dispatch to a function selected by the dynamic type of the object and the
6959 passed @code{print_context}. To this end, GiNaC maintains a separate method
6960 table for each class, similar to the virtual function table used for ordinary
6961 (single) virtual function dispatch.
6963 The method table contains one slot for each possible @code{print_context}
6964 type, indexed by the (internally assigned) serial number of the type. Slots
6965 may be empty, in which case GiNaC will retry the method lookup with the
6966 @code{print_context} object's parent class, possibly repeating the process
6967 until it reaches the @code{print_context} base class. If there's still no
6968 method defined, the method table of the algebraic object's parent class
6969 is consulted, and so on, until a matching method is found (eventually it
6970 will reach the combination @code{basic/print_context}, which prints the
6971 object's class name enclosed in square brackets).
6973 You can think of the print methods of all the different classes and output
6974 formats as being arranged in a two-dimensional matrix with one axis listing
6975 the algebraic classes and the other axis listing the @code{print_context}
6978 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6979 to implement printing, but then they won't get any of the benefits of the
6980 double dispatch mechanism (such as the ability for derived classes to
6981 inherit only certain print methods from its parent, or the replacement of
6982 methods at run-time).
6984 @subsection Print methods for classes
6986 The method table for a class is set up either in the definition of the class,
6987 by passing the appropriate @code{print_func<C>()} option to
6988 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6989 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6990 can also be used to override existing methods dynamically.
6992 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6993 be a member function of the class (or one of its parent classes), a static
6994 member function, or an ordinary (global) C++ function. The @code{C} template
6995 parameter specifies the appropriate @code{print_context} type for which the
6996 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6997 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6998 the class is the one being implemented by
6999 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7001 For print methods that are member functions, their first argument must be of
7002 a type convertible to a @code{const C &}, and the second argument must be an
7005 For static members and global functions, the first argument must be of a type
7006 convertible to a @code{const T &}, the second argument must be of a type
7007 convertible to a @code{const C &}, and the third argument must be an
7008 @code{unsigned}. A global function will, of course, not have access to
7009 private and protected members of @code{T}.
7011 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7012 and @code{basic::print()}) is used for proper parenthesizing of the output
7013 (and by @code{print_tree} for proper indentation). It can be used for similar
7014 purposes if you write your own output formats.
7016 The explanations given above may seem complicated, but in practice it's
7017 really simple, as shown in the following example. Suppose that we want to
7018 display exponents in LaTeX output not as superscripts but with little
7019 upwards-pointing arrows. This can be achieved in the following way:
7022 void my_print_power_as_latex(const power & p,
7023 const print_latex & c,
7026 // get the precedence of the 'power' class
7027 unsigned power_prec = p.precedence();
7029 // if the parent operator has the same or a higher precedence
7030 // we need parentheses around the power
7031 if (level >= power_prec)
7034 // print the basis and exponent, each enclosed in braces, and
7035 // separated by an uparrow
7037 p.op(0).print(c, power_prec);
7038 c.s << "@}\\uparrow@{";
7039 p.op(1).print(c, power_prec);
7042 // don't forget the closing parenthesis
7043 if (level >= power_prec)
7049 // a sample expression
7050 symbol x("x"), y("y");
7051 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7053 // switch to LaTeX mode
7056 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7059 // now we replace the method for the LaTeX output of powers with
7061 set_print_func<power, print_latex>(my_print_power_as_latex);
7063 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7074 The first argument of @code{my_print_power_as_latex} could also have been
7075 a @code{const basic &}, the second one a @code{const print_context &}.
7078 The above code depends on @code{mul} objects converting their operands to
7079 @code{power} objects for the purpose of printing.
7082 The output of products including negative powers as fractions is also
7083 controlled by the @code{mul} class.
7086 The @code{power/print_latex} method provided by GiNaC prints square roots
7087 using @code{\sqrt}, but the above code doesn't.
7091 It's not possible to restore a method table entry to its previous or default
7092 value. Once you have called @code{set_print_func()}, you can only override
7093 it with another call to @code{set_print_func()}, but you can't easily go back
7094 to the default behavior again (you can, of course, dig around in the GiNaC
7095 sources, find the method that is installed at startup
7096 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7097 one; that is, after you circumvent the C++ member access control@dots{}).
7099 @subsection Print methods for functions
7101 Symbolic functions employ a print method dispatch mechanism similar to the
7102 one used for classes. The methods are specified with @code{print_func<C>()}
7103 function options. If you don't specify any special print methods, the function
7104 will be printed with its name (or LaTeX name, if supplied), followed by a
7105 comma-separated list of arguments enclosed in parentheses.
7107 For example, this is what GiNaC's @samp{abs()} function is defined like:
7110 static ex abs_eval(const ex & arg) @{ ... @}
7111 static ex abs_evalf(const ex & arg) @{ ... @}
7113 static void abs_print_latex(const ex & arg, const print_context & c)
7115 c.s << "@{|"; arg.print(c); c.s << "|@}";
7118 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7120 c.s << "fabs("; arg.print(c); c.s << ")";
7123 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7124 evalf_func(abs_evalf).
7125 print_func<print_latex>(abs_print_latex).
7126 print_func<print_csrc_float>(abs_print_csrc_float).
7127 print_func<print_csrc_double>(abs_print_csrc_float));
7130 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7131 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7133 There is currently no equivalent of @code{set_print_func()} for functions.
7135 @subsection Adding new output formats
7137 Creating a new output format involves subclassing @code{print_context},
7138 which is somewhat similar to adding a new algebraic class
7139 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7140 that needs to go into the class definition, and a corresponding macro
7141 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7142 Every @code{print_context} class needs to provide a default constructor
7143 and a constructor from an @code{std::ostream} and an @code{unsigned}
7146 Here is an example for a user-defined @code{print_context} class:
7149 class print_myformat : public print_dflt
7151 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7153 print_myformat(std::ostream & os, unsigned opt = 0)
7154 : print_dflt(os, opt) @{@}
7157 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7159 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7162 That's all there is to it. None of the actual expression output logic is
7163 implemented in this class. It merely serves as a selector for choosing
7164 a particular format. The algorithms for printing expressions in the new
7165 format are implemented as print methods, as described above.
7167 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7168 exactly like GiNaC's default output format:
7173 ex e = pow(x, 2) + 1;
7175 // this prints "1+x^2"
7178 // this also prints "1+x^2"
7179 e.print(print_myformat()); cout << endl;
7185 To fill @code{print_myformat} with life, we need to supply appropriate
7186 print methods with @code{set_print_func()}, like this:
7189 // This prints powers with '**' instead of '^'. See the LaTeX output
7190 // example above for explanations.
7191 void print_power_as_myformat(const power & p,
7192 const print_myformat & c,
7195 unsigned power_prec = p.precedence();
7196 if (level >= power_prec)
7198 p.op(0).print(c, power_prec);
7200 p.op(1).print(c, power_prec);
7201 if (level >= power_prec)
7207 // install a new print method for power objects
7208 set_print_func<power, print_myformat>(print_power_as_myformat);
7210 // now this prints "1+x**2"
7211 e.print(print_myformat()); cout << endl;
7213 // but the default format is still "1+x^2"
7219 @node Structures, Adding classes, Printing, Extending GiNaC
7220 @c node-name, next, previous, up
7223 If you are doing some very specialized things with GiNaC, or if you just
7224 need some more organized way to store data in your expressions instead of
7225 anonymous lists, you may want to implement your own algebraic classes.
7226 ('algebraic class' means any class directly or indirectly derived from
7227 @code{basic} that can be used in GiNaC expressions).
7229 GiNaC offers two ways of accomplishing this: either by using the
7230 @code{structure<T>} template class, or by rolling your own class from
7231 scratch. This section will discuss the @code{structure<T>} template which
7232 is easier to use but more limited, while the implementation of custom
7233 GiNaC classes is the topic of the next section. However, you may want to
7234 read both sections because many common concepts and member functions are
7235 shared by both concepts, and it will also allow you to decide which approach
7236 is most suited to your needs.
7238 The @code{structure<T>} template, defined in the GiNaC header file
7239 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7240 or @code{class}) into a GiNaC object that can be used in expressions.
7242 @subsection Example: scalar products
7244 Let's suppose that we need a way to handle some kind of abstract scalar
7245 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7246 product class have to store their left and right operands, which can in turn
7247 be arbitrary expressions. Here is a possible way to represent such a
7248 product in a C++ @code{struct}:
7252 using namespace std;
7254 #include <ginac/ginac.h>
7255 using namespace GiNaC;
7261 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7265 The default constructor is required. Now, to make a GiNaC class out of this
7266 data structure, we need only one line:
7269 typedef structure<sprod_s> sprod;
7272 That's it. This line constructs an algebraic class @code{sprod} which
7273 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7274 expressions like any other GiNaC class:
7278 symbol a("a"), b("b");
7279 ex e = sprod(sprod_s(a, b));
7283 Note the difference between @code{sprod} which is the algebraic class, and
7284 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7285 and @code{right} data members. As shown above, an @code{sprod} can be
7286 constructed from an @code{sprod_s} object.
7288 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7289 you could define a little wrapper function like this:
7292 inline ex make_sprod(ex left, ex right)
7294 return sprod(sprod_s(left, right));
7298 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7299 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7300 @code{get_struct()}:
7304 cout << ex_to<sprod>(e)->left << endl;
7306 cout << ex_to<sprod>(e).get_struct().right << endl;
7311 You only have read access to the members of @code{sprod_s}.
7313 The type definition of @code{sprod} is enough to write your own algorithms
7314 that deal with scalar products, for example:
7319 if (is_a<sprod>(p)) @{
7320 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7321 return make_sprod(sp.right, sp.left);
7332 @subsection Structure output
7334 While the @code{sprod} type is useable it still leaves something to be
7335 desired, most notably proper output:
7340 // -> [structure object]
7344 By default, any structure types you define will be printed as
7345 @samp{[structure object]}. To override this you can either specialize the
7346 template's @code{print()} member function, or specify print methods with
7347 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7348 it's not possible to supply class options like @code{print_func<>()} to
7349 structures, so for a self-contained structure type you need to resort to
7350 overriding the @code{print()} function, which is also what we will do here.
7352 The member functions of GiNaC classes are described in more detail in the
7353 next section, but it shouldn't be hard to figure out what's going on here:
7356 void sprod::print(const print_context & c, unsigned level) const
7358 // tree debug output handled by superclass
7359 if (is_a<print_tree>(c))
7360 inherited::print(c, level);
7362 // get the contained sprod_s object
7363 const sprod_s & sp = get_struct();
7365 // print_context::s is a reference to an ostream
7366 c.s << "<" << sp.left << "|" << sp.right << ">";
7370 Now we can print expressions containing scalar products:
7376 cout << swap_sprod(e) << endl;
7381 @subsection Comparing structures
7383 The @code{sprod} class defined so far still has one important drawback: all
7384 scalar products are treated as being equal because GiNaC doesn't know how to
7385 compare objects of type @code{sprod_s}. This can lead to some confusing
7386 and undesired behavior:
7390 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7392 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7393 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7397 To remedy this, we first need to define the operators @code{==} and @code{<}
7398 for objects of type @code{sprod_s}:
7401 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7403 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7406 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7408 return lhs.left.compare(rhs.left) < 0
7409 ? true : lhs.right.compare(rhs.right) < 0;
7413 The ordering established by the @code{<} operator doesn't have to make any
7414 algebraic sense, but it needs to be well defined. Note that we can't use
7415 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7416 in the implementation of these operators because they would construct
7417 GiNaC @code{relational} objects which in the case of @code{<} do not
7418 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7419 decide which one is algebraically 'less').
7421 Next, we need to change our definition of the @code{sprod} type to let
7422 GiNaC know that an ordering relation exists for the embedded objects:
7425 typedef structure<sprod_s, compare_std_less> sprod;
7428 @code{sprod} objects then behave as expected:
7432 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7433 // -> <a|b>-<a^2|b^2>
7434 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7435 // -> <a|b>+<a^2|b^2>
7436 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7438 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7443 The @code{compare_std_less} policy parameter tells GiNaC to use the
7444 @code{std::less} and @code{std::equal_to} functors to compare objects of
7445 type @code{sprod_s}. By default, these functors forward their work to the
7446 standard @code{<} and @code{==} operators, which we have overloaded.
7447 Alternatively, we could have specialized @code{std::less} and
7448 @code{std::equal_to} for class @code{sprod_s}.
7450 GiNaC provides two other comparison policies for @code{structure<T>}
7451 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7452 which does a bit-wise comparison of the contained @code{T} objects.
7453 This should be used with extreme care because it only works reliably with
7454 built-in integral types, and it also compares any padding (filler bytes of
7455 undefined value) that the @code{T} class might have.
7457 @subsection Subexpressions
7459 Our scalar product class has two subexpressions: the left and right
7460 operands. It might be a good idea to make them accessible via the standard
7461 @code{nops()} and @code{op()} methods:
7464 size_t sprod::nops() const
7469 ex sprod::op(size_t i) const
7473 return get_struct().left;
7475 return get_struct().right;
7477 throw std::range_error("sprod::op(): no such operand");
7482 Implementing @code{nops()} and @code{op()} for container types such as
7483 @code{sprod} has two other nice side effects:
7487 @code{has()} works as expected
7489 GiNaC generates better hash keys for the objects (the default implementation
7490 of @code{calchash()} takes subexpressions into account)
7493 @cindex @code{let_op()}
7494 There is a non-const variant of @code{op()} called @code{let_op()} that
7495 allows replacing subexpressions:
7498 ex & sprod::let_op(size_t i)
7500 // every non-const member function must call this
7501 ensure_if_modifiable();
7505 return get_struct().left;
7507 return get_struct().right;
7509 throw std::range_error("sprod::let_op(): no such operand");
7514 Once we have provided @code{let_op()} we also get @code{subs()} and
7515 @code{map()} for free. In fact, every container class that returns a non-null
7516 @code{nops()} value must either implement @code{let_op()} or provide custom
7517 implementations of @code{subs()} and @code{map()}.
7519 In turn, the availability of @code{map()} enables the recursive behavior of a
7520 couple of other default method implementations, in particular @code{evalf()},
7521 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7522 we probably want to provide our own version of @code{expand()} for scalar
7523 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7524 This is left as an exercise for the reader.
7526 The @code{structure<T>} template defines many more member functions that
7527 you can override by specialization to customize the behavior of your
7528 structures. You are referred to the next section for a description of
7529 some of these (especially @code{eval()}). There is, however, one topic
7530 that shall be addressed here, as it demonstrates one peculiarity of the
7531 @code{structure<T>} template: archiving.
7533 @subsection Archiving structures
7535 If you don't know how the archiving of GiNaC objects is implemented, you
7536 should first read the next section and then come back here. You're back?
7539 To implement archiving for structures it is not enough to provide
7540 specializations for the @code{archive()} member function and the
7541 unarchiving constructor (the @code{unarchive()} function has a default
7542 implementation). You also need to provide a unique name (as a string literal)
7543 for each structure type you define. This is because in GiNaC archives,
7544 the class of an object is stored as a string, the class name.
7546 By default, this class name (as returned by the @code{class_name()} member
7547 function) is @samp{structure} for all structure classes. This works as long
7548 as you have only defined one structure type, but if you use two or more you
7549 need to provide a different name for each by specializing the
7550 @code{get_class_name()} member function. Here is a sample implementation
7551 for enabling archiving of the scalar product type defined above:
7554 const char *sprod::get_class_name() @{ return "sprod"; @}
7556 void sprod::archive(archive_node & n) const
7558 inherited::archive(n);
7559 n.add_ex("left", get_struct().left);
7560 n.add_ex("right", get_struct().right);
7563 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7565 n.find_ex("left", get_struct().left, sym_lst);
7566 n.find_ex("right", get_struct().right, sym_lst);
7570 Note that the unarchiving constructor is @code{sprod::structure} and not
7571 @code{sprod::sprod}, and that we don't need to supply an
7572 @code{sprod::unarchive()} function.
7575 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7576 @c node-name, next, previous, up
7577 @section Adding classes
7579 The @code{structure<T>} template provides an way to extend GiNaC with custom
7580 algebraic classes that is easy to use but has its limitations, the most
7581 severe of which being that you can't add any new member functions to
7582 structures. To be able to do this, you need to write a new class definition
7585 This section will explain how to implement new algebraic classes in GiNaC by
7586 giving the example of a simple 'string' class. After reading this section
7587 you will know how to properly declare a GiNaC class and what the minimum
7588 required member functions are that you have to implement. We only cover the
7589 implementation of a 'leaf' class here (i.e. one that doesn't contain
7590 subexpressions). Creating a container class like, for example, a class
7591 representing tensor products is more involved but this section should give
7592 you enough information so you can consult the source to GiNaC's predefined
7593 classes if you want to implement something more complicated.
7595 @subsection GiNaC's run-time type information system
7597 @cindex hierarchy of classes
7599 All algebraic classes (that is, all classes that can appear in expressions)
7600 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7601 @code{basic *} (which is essentially what an @code{ex} is) represents a
7602 generic pointer to an algebraic class. Occasionally it is necessary to find
7603 out what the class of an object pointed to by a @code{basic *} really is.
7604 Also, for the unarchiving of expressions it must be possible to find the
7605 @code{unarchive()} function of a class given the class name (as a string). A
7606 system that provides this kind of information is called a run-time type
7607 information (RTTI) system. The C++ language provides such a thing (see the
7608 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7609 implements its own, simpler RTTI.
7611 The RTTI in GiNaC is based on two mechanisms:
7616 The @code{basic} class declares a member variable @code{tinfo_key} which
7617 holds an unsigned integer that identifies the object's class. These numbers
7618 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7619 classes. They all start with @code{TINFO_}.
7622 By means of some clever tricks with static members, GiNaC maintains a list
7623 of information for all classes derived from @code{basic}. The information
7624 available includes the class names, the @code{tinfo_key}s, and pointers
7625 to the unarchiving functions. This class registry is defined in the
7626 @file{registrar.h} header file.
7630 The disadvantage of this proprietary RTTI implementation is that there's
7631 a little more to do when implementing new classes (C++'s RTTI works more
7632 or less automatically) but don't worry, most of the work is simplified by
7635 @subsection A minimalistic example
7637 Now we will start implementing a new class @code{mystring} that allows
7638 placing character strings in algebraic expressions (this is not very useful,
7639 but it's just an example). This class will be a direct subclass of
7640 @code{basic}. You can use this sample implementation as a starting point
7641 for your own classes.
7643 The code snippets given here assume that you have included some header files
7649 #include <stdexcept>
7650 using namespace std;
7652 #include <ginac/ginac.h>
7653 using namespace GiNaC;
7656 The first thing we have to do is to define a @code{tinfo_key} for our new
7657 class. This can be any arbitrary unsigned number that is not already taken
7658 by one of the existing classes but it's better to come up with something
7659 that is unlikely to clash with keys that might be added in the future. The
7660 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7661 which is not a requirement but we are going to stick with this scheme:
7664 const unsigned TINFO_mystring = 0x42420001U;
7667 Now we can write down the class declaration. The class stores a C++
7668 @code{string} and the user shall be able to construct a @code{mystring}
7669 object from a C or C++ string:
7672 class mystring : public basic
7674 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7677 mystring(const string &s);
7678 mystring(const char *s);
7684 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7687 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7688 macros are defined in @file{registrar.h}. They take the name of the class
7689 and its direct superclass as arguments and insert all required declarations
7690 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7691 the first line after the opening brace of the class definition. The
7692 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7693 source (at global scope, of course, not inside a function).
7695 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7696 declarations of the default constructor and a couple of other functions that
7697 are required. It also defines a type @code{inherited} which refers to the
7698 superclass so you don't have to modify your code every time you shuffle around
7699 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7700 class with the GiNaC RTTI (there is also a
7701 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7702 options for the class, and which we will be using instead in a few minutes).
7704 Now there are seven member functions we have to implement to get a working
7710 @code{mystring()}, the default constructor.
7713 @code{void archive(archive_node &n)}, the archiving function. This stores all
7714 information needed to reconstruct an object of this class inside an
7715 @code{archive_node}.
7718 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7719 constructor. This constructs an instance of the class from the information
7720 found in an @code{archive_node}.
7723 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7724 unarchiving function. It constructs a new instance by calling the unarchiving
7728 @cindex @code{compare_same_type()}
7729 @code{int compare_same_type(const basic &other)}, which is used internally
7730 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7731 -1, depending on the relative order of this object and the @code{other}
7732 object. If it returns 0, the objects are considered equal.
7733 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7734 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7735 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7736 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7737 must provide a @code{compare_same_type()} function, even those representing
7738 objects for which no reasonable algebraic ordering relationship can be
7742 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7743 which are the two constructors we declared.
7747 Let's proceed step-by-step. The default constructor looks like this:
7750 mystring::mystring() : inherited(TINFO_mystring) @{@}
7753 The golden rule is that in all constructors you have to set the
7754 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7755 it will be set by the constructor of the superclass and all hell will break
7756 loose in the RTTI. For your convenience, the @code{basic} class provides
7757 a constructor that takes a @code{tinfo_key} value, which we are using here
7758 (remember that in our case @code{inherited == basic}). If the superclass
7759 didn't have such a constructor, we would have to set the @code{tinfo_key}
7760 to the right value manually.
7762 In the default constructor you should set all other member variables to
7763 reasonable default values (we don't need that here since our @code{str}
7764 member gets set to an empty string automatically).
7766 Next are the three functions for archiving. You have to implement them even
7767 if you don't plan to use archives, but the minimum required implementation
7768 is really simple. First, the archiving function:
7771 void mystring::archive(archive_node &n) const
7773 inherited::archive(n);
7774 n.add_string("string", str);
7778 The only thing that is really required is calling the @code{archive()}
7779 function of the superclass. Optionally, you can store all information you
7780 deem necessary for representing the object into the passed
7781 @code{archive_node}. We are just storing our string here. For more
7782 information on how the archiving works, consult the @file{archive.h} header
7785 The unarchiving constructor is basically the inverse of the archiving
7789 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7791 n.find_string("string", str);
7795 If you don't need archiving, just leave this function empty (but you must
7796 invoke the unarchiving constructor of the superclass). Note that we don't
7797 have to set the @code{tinfo_key} here because it is done automatically
7798 by the unarchiving constructor of the @code{basic} class.
7800 Finally, the unarchiving function:
7803 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7805 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7809 You don't have to understand how exactly this works. Just copy these
7810 four lines into your code literally (replacing the class name, of
7811 course). It calls the unarchiving constructor of the class and unless
7812 you are doing something very special (like matching @code{archive_node}s
7813 to global objects) you don't need a different implementation. For those
7814 who are interested: setting the @code{dynallocated} flag puts the object
7815 under the control of GiNaC's garbage collection. It will get deleted
7816 automatically once it is no longer referenced.
7818 Our @code{compare_same_type()} function uses a provided function to compare
7822 int mystring::compare_same_type(const basic &other) const
7824 const mystring &o = static_cast<const mystring &>(other);
7825 int cmpval = str.compare(o.str);
7828 else if (cmpval < 0)
7835 Although this function takes a @code{basic &}, it will always be a reference
7836 to an object of exactly the same class (objects of different classes are not
7837 comparable), so the cast is safe. If this function returns 0, the two objects
7838 are considered equal (in the sense that @math{A-B=0}), so you should compare
7839 all relevant member variables.
7841 Now the only thing missing is our two new constructors:
7844 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7845 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7848 No surprises here. We set the @code{str} member from the argument and
7849 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7851 That's it! We now have a minimal working GiNaC class that can store
7852 strings in algebraic expressions. Let's confirm that the RTTI works:
7855 ex e = mystring("Hello, world!");
7856 cout << is_a<mystring>(e) << endl;
7859 cout << e.bp->class_name() << endl;
7863 Obviously it does. Let's see what the expression @code{e} looks like:
7867 // -> [mystring object]
7870 Hm, not exactly what we expect, but of course the @code{mystring} class
7871 doesn't yet know how to print itself. This can be done either by implementing
7872 the @code{print()} member function, or, preferably, by specifying a
7873 @code{print_func<>()} class option. Let's say that we want to print the string
7874 surrounded by double quotes:
7877 class mystring : public basic
7881 void do_print(const print_context &c, unsigned level = 0) const;
7885 void mystring::do_print(const print_context &c, unsigned level) const
7887 // print_context::s is a reference to an ostream
7888 c.s << '\"' << str << '\"';
7892 The @code{level} argument is only required for container classes to
7893 correctly parenthesize the output.
7895 Now we need to tell GiNaC that @code{mystring} objects should use the
7896 @code{do_print()} member function for printing themselves. For this, we
7900 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7906 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7907 print_func<print_context>(&mystring::do_print))
7910 Let's try again to print the expression:
7914 // -> "Hello, world!"
7917 Much better. If we wanted to have @code{mystring} objects displayed in a
7918 different way depending on the output format (default, LaTeX, etc.), we
7919 would have supplied multiple @code{print_func<>()} options with different
7920 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7921 separated by dots. This is similar to the way options are specified for
7922 symbolic functions. @xref{Printing}, for a more in-depth description of the
7923 way expression output is implemented in GiNaC.
7925 The @code{mystring} class can be used in arbitrary expressions:
7928 e += mystring("GiNaC rulez");
7930 // -> "GiNaC rulez"+"Hello, world!"
7933 (GiNaC's automatic term reordering is in effect here), or even
7936 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7938 // -> "One string"^(2*sin(-"Another string"+Pi))
7941 Whether this makes sense is debatable but remember that this is only an
7942 example. At least it allows you to implement your own symbolic algorithms
7945 Note that GiNaC's algebraic rules remain unchanged:
7948 e = mystring("Wow") * mystring("Wow");
7952 e = pow(mystring("First")-mystring("Second"), 2);
7953 cout << e.expand() << endl;
7954 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7957 There's no way to, for example, make GiNaC's @code{add} class perform string
7958 concatenation. You would have to implement this yourself.
7960 @subsection Automatic evaluation
7963 @cindex @code{eval()}
7964 @cindex @code{hold()}
7965 When dealing with objects that are just a little more complicated than the
7966 simple string objects we have implemented, chances are that you will want to
7967 have some automatic simplifications or canonicalizations performed on them.
7968 This is done in the evaluation member function @code{eval()}. Let's say that
7969 we wanted all strings automatically converted to lowercase with
7970 non-alphabetic characters stripped, and empty strings removed:
7973 class mystring : public basic
7977 ex eval(int level = 0) const;
7981 ex mystring::eval(int level) const
7984 for (int i=0; i<str.length(); i++) @{
7986 if (c >= 'A' && c <= 'Z')
7987 new_str += tolower(c);
7988 else if (c >= 'a' && c <= 'z')
7992 if (new_str.length() == 0)
7995 return mystring(new_str).hold();
7999 The @code{level} argument is used to limit the recursion depth of the
8000 evaluation. We don't have any subexpressions in the @code{mystring}
8001 class so we are not concerned with this. If we had, we would call the
8002 @code{eval()} functions of the subexpressions with @code{level - 1} as
8003 the argument if @code{level != 1}. The @code{hold()} member function
8004 sets a flag in the object that prevents further evaluation. Otherwise
8005 we might end up in an endless loop. When you want to return the object
8006 unmodified, use @code{return this->hold();}.
8008 Let's confirm that it works:
8011 ex e = mystring("Hello, world!") + mystring("!?#");
8015 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8020 @subsection Optional member functions
8022 We have implemented only a small set of member functions to make the class
8023 work in the GiNaC framework. There are two functions that are not strictly
8024 required but will make operations with objects of the class more efficient:
8026 @cindex @code{calchash()}
8027 @cindex @code{is_equal_same_type()}
8029 unsigned calchash() const;
8030 bool is_equal_same_type(const basic &other) const;
8033 The @code{calchash()} method returns an @code{unsigned} hash value for the
8034 object which will allow GiNaC to compare and canonicalize expressions much
8035 more efficiently. You should consult the implementation of some of the built-in
8036 GiNaC classes for examples of hash functions. The default implementation of
8037 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8038 class and all subexpressions that are accessible via @code{op()}.
8040 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8041 tests for equality without establishing an ordering relation, which is often
8042 faster. The default implementation of @code{is_equal_same_type()} just calls
8043 @code{compare_same_type()} and tests its result for zero.
8045 @subsection Other member functions
8047 For a real algebraic class, there are probably some more functions that you
8048 might want to provide:
8051 bool info(unsigned inf) const;
8052 ex evalf(int level = 0) const;
8053 ex series(const relational & r, int order, unsigned options = 0) const;
8054 ex derivative(const symbol & s) const;
8057 If your class stores sub-expressions (see the scalar product example in the
8058 previous section) you will probably want to override
8060 @cindex @code{let_op()}
8063 ex op(size_t i) const;
8064 ex & let_op(size_t i);
8065 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8066 ex map(map_function & f) const;
8069 @code{let_op()} is a variant of @code{op()} that allows write access. The
8070 default implementations of @code{subs()} and @code{map()} use it, so you have
8071 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8073 You can, of course, also add your own new member functions. Remember
8074 that the RTTI may be used to get information about what kinds of objects
8075 you are dealing with (the position in the class hierarchy) and that you
8076 can always extract the bare object from an @code{ex} by stripping the
8077 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8078 should become a need.
8080 That's it. May the source be with you!
8083 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8084 @c node-name, next, previous, up
8085 @chapter A Comparison With Other CAS
8088 This chapter will give you some information on how GiNaC compares to
8089 other, traditional Computer Algebra Systems, like @emph{Maple},
8090 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8091 disadvantages over these systems.
8094 * Advantages:: Strengths of the GiNaC approach.
8095 * Disadvantages:: Weaknesses of the GiNaC approach.
8096 * Why C++?:: Attractiveness of C++.
8099 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8100 @c node-name, next, previous, up
8103 GiNaC has several advantages over traditional Computer
8104 Algebra Systems, like
8109 familiar language: all common CAS implement their own proprietary
8110 grammar which you have to learn first (and maybe learn again when your
8111 vendor decides to `enhance' it). With GiNaC you can write your program
8112 in common C++, which is standardized.
8116 structured data types: you can build up structured data types using
8117 @code{struct}s or @code{class}es together with STL features instead of
8118 using unnamed lists of lists of lists.
8121 strongly typed: in CAS, you usually have only one kind of variables
8122 which can hold contents of an arbitrary type. This 4GL like feature is
8123 nice for novice programmers, but dangerous.
8126 development tools: powerful development tools exist for C++, like fancy
8127 editors (e.g. with automatic indentation and syntax highlighting),
8128 debuggers, visualization tools, documentation generators@dots{}
8131 modularization: C++ programs can easily be split into modules by
8132 separating interface and implementation.
8135 price: GiNaC is distributed under the GNU Public License which means
8136 that it is free and available with source code. And there are excellent
8137 C++-compilers for free, too.
8140 extendable: you can add your own classes to GiNaC, thus extending it on
8141 a very low level. Compare this to a traditional CAS that you can
8142 usually only extend on a high level by writing in the language defined
8143 by the parser. In particular, it turns out to be almost impossible to
8144 fix bugs in a traditional system.
8147 multiple interfaces: Though real GiNaC programs have to be written in
8148 some editor, then be compiled, linked and executed, there are more ways
8149 to work with the GiNaC engine. Many people want to play with
8150 expressions interactively, as in traditional CASs. Currently, two such
8151 windows into GiNaC have been implemented and many more are possible: the
8152 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8153 types to a command line and second, as a more consistent approach, an
8154 interactive interface to the Cint C++ interpreter has been put together
8155 (called GiNaC-cint) that allows an interactive scripting interface
8156 consistent with the C++ language. It is available from the usual GiNaC
8160 seamless integration: it is somewhere between difficult and impossible
8161 to call CAS functions from within a program written in C++ or any other
8162 programming language and vice versa. With GiNaC, your symbolic routines
8163 are part of your program. You can easily call third party libraries,
8164 e.g. for numerical evaluation or graphical interaction. All other
8165 approaches are much more cumbersome: they range from simply ignoring the
8166 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8167 system (i.e. @emph{Yacas}).
8170 efficiency: often large parts of a program do not need symbolic
8171 calculations at all. Why use large integers for loop variables or
8172 arbitrary precision arithmetics where @code{int} and @code{double} are
8173 sufficient? For pure symbolic applications, GiNaC is comparable in
8174 speed with other CAS.
8179 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8180 @c node-name, next, previous, up
8181 @section Disadvantages
8183 Of course it also has some disadvantages:
8188 advanced features: GiNaC cannot compete with a program like
8189 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8190 which grows since 1981 by the work of dozens of programmers, with
8191 respect to mathematical features. Integration, factorization,
8192 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8193 not planned for the near future).
8196 portability: While the GiNaC library itself is designed to avoid any
8197 platform dependent features (it should compile on any ANSI compliant C++
8198 compiler), the currently used version of the CLN library (fast large
8199 integer and arbitrary precision arithmetics) can only by compiled
8200 without hassle on systems with the C++ compiler from the GNU Compiler
8201 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8202 macros to let the compiler gather all static initializations, which
8203 works for GNU C++ only. Feel free to contact the authors in case you
8204 really believe that you need to use a different compiler. We have
8205 occasionally used other compilers and may be able to give you advice.}
8206 GiNaC uses recent language features like explicit constructors, mutable
8207 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8208 literally. Recent GCC versions starting at 2.95.3, although itself not
8209 yet ANSI compliant, support all needed features.
8214 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8215 @c node-name, next, previous, up
8218 Why did we choose to implement GiNaC in C++ instead of Java or any other
8219 language? C++ is not perfect: type checking is not strict (casting is
8220 possible), separation between interface and implementation is not
8221 complete, object oriented design is not enforced. The main reason is
8222 the often scolded feature of operator overloading in C++. While it may
8223 be true that operating on classes with a @code{+} operator is rarely
8224 meaningful, it is perfectly suited for algebraic expressions. Writing
8225 @math{3x+5y} as @code{3*x+5*y} instead of
8226 @code{x.times(3).plus(y.times(5))} looks much more natural.
8227 Furthermore, the main developers are more familiar with C++ than with
8228 any other programming language.
8231 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8232 @c node-name, next, previous, up
8233 @appendix Internal Structures
8236 * Expressions are reference counted::
8237 * Internal representation of products and sums::
8240 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8241 @c node-name, next, previous, up
8242 @appendixsection Expressions are reference counted
8244 @cindex reference counting
8245 @cindex copy-on-write
8246 @cindex garbage collection
8247 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8248 where the counter belongs to the algebraic objects derived from class
8249 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8250 which @code{ex} contains an instance. If you understood that, you can safely
8251 skip the rest of this passage.
8253 Expressions are extremely light-weight since internally they work like
8254 handles to the actual representation. They really hold nothing more
8255 than a pointer to some other object. What this means in practice is
8256 that whenever you create two @code{ex} and set the second equal to the
8257 first no copying process is involved. Instead, the copying takes place
8258 as soon as you try to change the second. Consider the simple sequence
8263 #include <ginac/ginac.h>
8264 using namespace std;
8265 using namespace GiNaC;
8269 symbol x("x"), y("y"), z("z");
8272 e1 = sin(x + 2*y) + 3*z + 41;
8273 e2 = e1; // e2 points to same object as e1
8274 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8275 e2 += 1; // e2 is copied into a new object
8276 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8280 The line @code{e2 = e1;} creates a second expression pointing to the
8281 object held already by @code{e1}. The time involved for this operation
8282 is therefore constant, no matter how large @code{e1} was. Actual
8283 copying, however, must take place in the line @code{e2 += 1;} because
8284 @code{e1} and @code{e2} are not handles for the same object any more.
8285 This concept is called @dfn{copy-on-write semantics}. It increases
8286 performance considerably whenever one object occurs multiple times and
8287 represents a simple garbage collection scheme because when an @code{ex}
8288 runs out of scope its destructor checks whether other expressions handle
8289 the object it points to too and deletes the object from memory if that
8290 turns out not to be the case. A slightly less trivial example of
8291 differentiation using the chain-rule should make clear how powerful this
8296 symbol x("x"), y("y");
8300 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8301 cout << e1 << endl // prints x+3*y
8302 << e2 << endl // prints (x+3*y)^3
8303 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8307 Here, @code{e1} will actually be referenced three times while @code{e2}
8308 will be referenced two times. When the power of an expression is built,
8309 that expression needs not be copied. Likewise, since the derivative of
8310 a power of an expression can be easily expressed in terms of that
8311 expression, no copying of @code{e1} is involved when @code{e3} is
8312 constructed. So, when @code{e3} is constructed it will print as
8313 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8314 holds a reference to @code{e2} and the factor in front is just
8317 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8318 semantics. When you insert an expression into a second expression, the
8319 result behaves exactly as if the contents of the first expression were
8320 inserted. But it may be useful to remember that this is not what
8321 happens. Knowing this will enable you to write much more efficient
8322 code. If you still have an uncertain feeling with copy-on-write
8323 semantics, we recommend you have a look at the
8324 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8325 Marshall Cline. Chapter 16 covers this issue and presents an
8326 implementation which is pretty close to the one in GiNaC.
8329 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8330 @c node-name, next, previous, up
8331 @appendixsection Internal representation of products and sums
8333 @cindex representation
8336 @cindex @code{power}
8337 Although it should be completely transparent for the user of
8338 GiNaC a short discussion of this topic helps to understand the sources
8339 and also explain performance to a large degree. Consider the
8340 unexpanded symbolic expression
8342 $2d^3 \left( 4a + 5b - 3 \right)$
8345 @math{2*d^3*(4*a+5*b-3)}
8347 which could naively be represented by a tree of linear containers for
8348 addition and multiplication, one container for exponentiation with base
8349 and exponent and some atomic leaves of symbols and numbers in this
8354 @cindex pair-wise representation
8355 However, doing so results in a rather deeply nested tree which will
8356 quickly become inefficient to manipulate. We can improve on this by
8357 representing the sum as a sequence of terms, each one being a pair of a
8358 purely numeric multiplicative coefficient and its rest. In the same
8359 spirit we can store the multiplication as a sequence of terms, each
8360 having a numeric exponent and a possibly complicated base, the tree
8361 becomes much more flat:
8365 The number @code{3} above the symbol @code{d} shows that @code{mul}
8366 objects are treated similarly where the coefficients are interpreted as
8367 @emph{exponents} now. Addition of sums of terms or multiplication of
8368 products with numerical exponents can be coded to be very efficient with
8369 such a pair-wise representation. Internally, this handling is performed
8370 by most CAS in this way. It typically speeds up manipulations by an
8371 order of magnitude. The overall multiplicative factor @code{2} and the
8372 additive term @code{-3} look somewhat out of place in this
8373 representation, however, since they are still carrying a trivial
8374 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8375 this is avoided by adding a field that carries an overall numeric
8376 coefficient. This results in the realistic picture of internal
8379 $2d^3 \left( 4a + 5b - 3 \right)$:
8382 @math{2*d^3*(4*a+5*b-3)}:
8388 This also allows for a better handling of numeric radicals, since
8389 @code{sqrt(2)} can now be carried along calculations. Now it should be
8390 clear, why both classes @code{add} and @code{mul} are derived from the
8391 same abstract class: the data representation is the same, only the
8392 semantics differs. In the class hierarchy, methods for polynomial
8393 expansion and the like are reimplemented for @code{add} and @code{mul},
8394 but the data structure is inherited from @code{expairseq}.
8397 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8398 @c node-name, next, previous, up
8399 @appendix Package Tools
8401 If you are creating a software package that uses the GiNaC library,
8402 setting the correct command line options for the compiler and linker
8403 can be difficult. GiNaC includes two tools to make this process easier.
8406 * ginac-config:: A shell script to detect compiler and linker flags.
8407 * AM_PATH_GINAC:: Macro for GNU automake.
8411 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8412 @c node-name, next, previous, up
8413 @section @command{ginac-config}
8414 @cindex ginac-config
8416 @command{ginac-config} is a shell script that you can use to determine
8417 the compiler and linker command line options required to compile and
8418 link a program with the GiNaC library.
8420 @command{ginac-config} takes the following flags:
8424 Prints out the version of GiNaC installed.
8426 Prints '-I' flags pointing to the installed header files.
8428 Prints out the linker flags necessary to link a program against GiNaC.
8429 @item --prefix[=@var{PREFIX}]
8430 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8431 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8432 Otherwise, prints out the configured value of @env{$prefix}.
8433 @item --exec-prefix[=@var{PREFIX}]
8434 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8435 Otherwise, prints out the configured value of @env{$exec_prefix}.
8438 Typically, @command{ginac-config} will be used within a configure
8439 script, as described below. It, however, can also be used directly from
8440 the command line using backquotes to compile a simple program. For
8444 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8447 This command line might expand to (for example):
8450 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8451 -lginac -lcln -lstdc++
8454 Not only is the form using @command{ginac-config} easier to type, it will
8455 work on any system, no matter how GiNaC was configured.
8458 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8459 @c node-name, next, previous, up
8460 @section @samp{AM_PATH_GINAC}
8461 @cindex AM_PATH_GINAC
8463 For packages configured using GNU automake, GiNaC also provides
8464 a macro to automate the process of checking for GiNaC.
8467 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8468 [, @var{ACTION-IF-NOT-FOUND}]]])
8476 Determines the location of GiNaC using @command{ginac-config}, which is
8477 either found in the user's path, or from the environment variable
8478 @env{GINACLIB_CONFIG}.
8481 Tests the installed libraries to make sure that their version
8482 is later than @var{MINIMUM-VERSION}. (A default version will be used
8486 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8487 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8488 variable to the output of @command{ginac-config --libs}, and calls
8489 @samp{AC_SUBST()} for these variables so they can be used in generated
8490 makefiles, and then executes @var{ACTION-IF-FOUND}.
8493 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8494 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8498 This macro is in file @file{ginac.m4} which is installed in
8499 @file{$datadir/aclocal}. Note that if automake was installed with a
8500 different @samp{--prefix} than GiNaC, you will either have to manually
8501 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8502 aclocal the @samp{-I} option when running it.
8505 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8506 * Example package:: Example of a package using AM_PATH_GINAC.
8510 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8511 @c node-name, next, previous, up
8512 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8514 Simply make sure that @command{ginac-config} is in your path, and run
8515 the configure script.
8522 The directory where the GiNaC libraries are installed needs
8523 to be found by your system's dynamic linker.
8525 This is generally done by
8528 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8534 setting the environment variable @env{LD_LIBRARY_PATH},
8537 or, as a last resort,
8540 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8541 running configure, for instance:
8544 LDFLAGS=-R/home/cbauer/lib ./configure
8549 You can also specify a @command{ginac-config} not in your path by
8550 setting the @env{GINACLIB_CONFIG} environment variable to the
8551 name of the executable
8554 If you move the GiNaC package from its installed location,
8555 you will either need to modify @command{ginac-config} script
8556 manually to point to the new location or rebuild GiNaC.
8567 --with-ginac-prefix=@var{PREFIX}
8568 --with-ginac-exec-prefix=@var{PREFIX}
8571 are provided to override the prefix and exec-prefix that were stored
8572 in the @command{ginac-config} shell script by GiNaC's configure. You are
8573 generally better off configuring GiNaC with the right path to begin with.
8577 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8578 @c node-name, next, previous, up
8579 @subsection Example of a package using @samp{AM_PATH_GINAC}
8581 The following shows how to build a simple package using automake
8582 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8586 #include <ginac/ginac.h>
8590 GiNaC::symbol x("x");
8591 GiNaC::ex a = GiNaC::sin(x);
8592 std::cout << "Derivative of " << a
8593 << " is " << a.diff(x) << std::endl;
8598 You should first read the introductory portions of the automake
8599 Manual, if you are not already familiar with it.
8601 Two files are needed, @file{configure.in}, which is used to build the
8605 dnl Process this file with autoconf to produce a configure script.
8607 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8613 AM_PATH_GINAC(0.9.0, [
8614 LIBS="$LIBS $GINACLIB_LIBS"
8615 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8616 ], AC_MSG_ERROR([need to have GiNaC installed]))
8621 The only command in this which is not standard for automake
8622 is the @samp{AM_PATH_GINAC} macro.
8624 That command does the following: If a GiNaC version greater or equal
8625 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8626 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8627 the error message `need to have GiNaC installed'
8629 And the @file{Makefile.am}, which will be used to build the Makefile.
8632 ## Process this file with automake to produce Makefile.in
8633 bin_PROGRAMS = simple
8634 simple_SOURCES = simple.cpp
8637 This @file{Makefile.am}, says that we are building a single executable,
8638 from a single source file @file{simple.cpp}. Since every program
8639 we are building uses GiNaC we simply added the GiNaC options
8640 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8641 want to specify them on a per-program basis: for instance by
8645 simple_LDADD = $(GINACLIB_LIBS)
8646 INCLUDES = $(GINACLIB_CPPFLAGS)
8649 to the @file{Makefile.am}.
8651 To try this example out, create a new directory and add the three
8654 Now execute the following commands:
8657 $ automake --add-missing
8662 You now have a package that can be built in the normal fashion
8671 @node Bibliography, Concept Index, Example package, Top
8672 @c node-name, next, previous, up
8673 @appendix Bibliography
8678 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8681 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8684 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8687 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8690 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8691 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8694 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8695 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8696 Academic Press, London
8699 @cite{Computer Algebra Systems - A Practical Guide},
8700 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8703 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8704 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8707 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8708 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8711 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8716 @node Concept Index, , Bibliography, Top
8717 @c node-name, next, previous, up
8718 @unnumbered Concept Index