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{simplify_indexed()}
2624 @subsection Simplifying indexed expressions
2626 In addition to the few automatic simplifications that GiNaC performs on
2627 indexed expressions (such as re-ordering the indices of symmetric tensors
2628 and calculating traces and convolutions of matrices and predefined tensors)
2632 ex ex::simplify_indexed();
2633 ex ex::simplify_indexed(const scalar_products & sp);
2636 that performs some more expensive operations:
2639 @item it checks the consistency of free indices in sums in the same way
2640 @code{get_free_indices()} does
2641 @item it tries to give dummy indices that appear in different terms of a sum
2642 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2643 @item it (symbolically) calculates all possible dummy index summations/contractions
2644 with the predefined tensors (this will be explained in more detail in the
2646 @item it detects contractions that vanish for symmetry reasons, for example
2647 the contraction of a symmetric and a totally antisymmetric tensor
2648 @item as a special case of dummy index summation, it can replace scalar products
2649 of two tensors with a user-defined value
2652 The last point is done with the help of the @code{scalar_products} class
2653 which is used to store scalar products with known values (this is not an
2654 arithmetic class, you just pass it to @code{simplify_indexed()}):
2658 symbol A("A"), B("B"), C("C"), i_sym("i");
2662 sp.add(A, B, 0); // A and B are orthogonal
2663 sp.add(A, C, 0); // A and C are orthogonal
2664 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2666 e = indexed(A + B, i) * indexed(A + C, i);
2668 // -> (B+A).i*(A+C).i
2670 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2676 The @code{scalar_products} object @code{sp} acts as a storage for the
2677 scalar products added to it with the @code{.add()} method. This method
2678 takes three arguments: the two expressions of which the scalar product is
2679 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2680 @code{simplify_indexed()} will replace all scalar products of indexed
2681 objects that have the symbols @code{A} and @code{B} as base expressions
2682 with the single value 0. The number, type and dimension of the indices
2683 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2685 @cindex @code{expand()}
2686 The example above also illustrates a feature of the @code{expand()} method:
2687 if passed the @code{expand_indexed} option it will distribute indices
2688 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2690 @cindex @code{tensor} (class)
2691 @subsection Predefined tensors
2693 Some frequently used special tensors such as the delta, epsilon and metric
2694 tensors are predefined in GiNaC. They have special properties when
2695 contracted with other tensor expressions and some of them have constant
2696 matrix representations (they will evaluate to a number when numeric
2697 indices are specified).
2699 @cindex @code{delta_tensor()}
2700 @subsubsection Delta tensor
2702 The delta tensor takes two indices, is symmetric and has the matrix
2703 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2704 @code{delta_tensor()}:
2708 symbol A("A"), B("B");
2710 idx i(symbol("i"), 3), j(symbol("j"), 3),
2711 k(symbol("k"), 3), l(symbol("l"), 3);
2713 ex e = indexed(A, i, j) * indexed(B, k, l)
2714 * delta_tensor(i, k) * delta_tensor(j, l);
2715 cout << e.simplify_indexed() << endl;
2718 cout << delta_tensor(i, i) << endl;
2723 @cindex @code{metric_tensor()}
2724 @subsubsection General metric tensor
2726 The function @code{metric_tensor()} creates a general symmetric metric
2727 tensor with two indices that can be used to raise/lower tensor indices. The
2728 metric tensor is denoted as @samp{g} in the output and if its indices are of
2729 mixed variance it is automatically replaced by a delta tensor:
2735 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2737 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2738 cout << e.simplify_indexed() << endl;
2741 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2742 cout << e.simplify_indexed() << endl;
2745 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2746 * metric_tensor(nu, rho);
2747 cout << e.simplify_indexed() << endl;
2750 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2751 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2752 + indexed(A, mu.toggle_variance(), rho));
2753 cout << e.simplify_indexed() << endl;
2758 @cindex @code{lorentz_g()}
2759 @subsubsection Minkowski metric tensor
2761 The Minkowski metric tensor is a special metric tensor with a constant
2762 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2763 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2764 It is created with the function @code{lorentz_g()} (although it is output as
2769 varidx mu(symbol("mu"), 4);
2771 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2772 * lorentz_g(mu, varidx(0, 4)); // negative signature
2773 cout << e.simplify_indexed() << endl;
2776 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2777 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2778 cout << e.simplify_indexed() << endl;
2783 @cindex @code{spinor_metric()}
2784 @subsubsection Spinor metric tensor
2786 The function @code{spinor_metric()} creates an antisymmetric tensor with
2787 two indices that is used to raise/lower indices of 2-component spinors.
2788 It is output as @samp{eps}:
2794 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2795 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2797 e = spinor_metric(A, B) * indexed(psi, B_co);
2798 cout << e.simplify_indexed() << endl;
2801 e = spinor_metric(A, B) * indexed(psi, A_co);
2802 cout << e.simplify_indexed() << endl;
2805 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2806 cout << e.simplify_indexed() << endl;
2809 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2810 cout << e.simplify_indexed() << endl;
2813 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2814 cout << e.simplify_indexed() << endl;
2817 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2818 cout << e.simplify_indexed() << endl;
2823 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2825 @cindex @code{epsilon_tensor()}
2826 @cindex @code{lorentz_eps()}
2827 @subsubsection Epsilon tensor
2829 The epsilon tensor is totally antisymmetric, its number of indices is equal
2830 to the dimension of the index space (the indices must all be of the same
2831 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2832 defined to be 1. Its behavior with indices that have a variance also
2833 depends on the signature of the metric. Epsilon tensors are output as
2836 There are three functions defined to create epsilon tensors in 2, 3 and 4
2840 ex epsilon_tensor(const ex & i1, const ex & i2);
2841 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2842 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2843 bool pos_sig = false);
2846 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2847 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2848 Minkowski space (the last @code{bool} argument specifies whether the metric
2849 has negative or positive signature, as in the case of the Minkowski metric
2854 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2855 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2856 e = lorentz_eps(mu, nu, rho, sig) *
2857 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2858 cout << simplify_indexed(e) << endl;
2859 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2861 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2862 symbol A("A"), B("B");
2863 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2864 cout << simplify_indexed(e) << endl;
2865 // -> -B.k*A.j*eps.i.k.j
2866 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2867 cout << simplify_indexed(e) << endl;
2872 @subsection Linear algebra
2874 The @code{matrix} class can be used with indices to do some simple linear
2875 algebra (linear combinations and products of vectors and matrices, traces
2876 and scalar products):
2880 idx i(symbol("i"), 2), j(symbol("j"), 2);
2881 symbol x("x"), y("y");
2883 // A is a 2x2 matrix, X is a 2x1 vector
2884 matrix A(2, 2), X(2, 1);
2889 cout << indexed(A, i, i) << endl;
2892 ex e = indexed(A, i, j) * indexed(X, j);
2893 cout << e.simplify_indexed() << endl;
2894 // -> [[2*y+x],[4*y+3*x]].i
2896 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2897 cout << e.simplify_indexed() << endl;
2898 // -> [[3*y+3*x,6*y+2*x]].j
2902 You can of course obtain the same results with the @code{matrix::add()},
2903 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2904 but with indices you don't have to worry about transposing matrices.
2906 Matrix indices always start at 0 and their dimension must match the number
2907 of rows/columns of the matrix. Matrices with one row or one column are
2908 vectors and can have one or two indices (it doesn't matter whether it's a
2909 row or a column vector). Other matrices must have two indices.
2911 You should be careful when using indices with variance on matrices. GiNaC
2912 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2913 @samp{F.mu.nu} are different matrices. In this case you should use only
2914 one form for @samp{F} and explicitly multiply it with a matrix representation
2915 of the metric tensor.
2918 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2919 @c node-name, next, previous, up
2920 @section Non-commutative objects
2922 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2923 non-commutative objects are built-in which are mostly of use in high energy
2927 @item Clifford (Dirac) algebra (class @code{clifford})
2928 @item su(3) Lie algebra (class @code{color})
2929 @item Matrices (unindexed) (class @code{matrix})
2932 The @code{clifford} and @code{color} classes are subclasses of
2933 @code{indexed} because the elements of these algebras usually carry
2934 indices. The @code{matrix} class is described in more detail in
2937 Unlike most computer algebra systems, GiNaC does not primarily provide an
2938 operator (often denoted @samp{&*}) for representing inert products of
2939 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2940 classes of objects involved, and non-commutative products are formed with
2941 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2942 figuring out by itself which objects commutate and will group the factors
2943 by their class. Consider this example:
2947 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2948 idx a(symbol("a"), 8), b(symbol("b"), 8);
2949 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2951 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2955 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2956 groups the non-commutative factors (the gammas and the su(3) generators)
2957 together while preserving the order of factors within each class (because
2958 Clifford objects commutate with color objects). The resulting expression is a
2959 @emph{commutative} product with two factors that are themselves non-commutative
2960 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2961 parentheses are placed around the non-commutative products in the output.
2963 @cindex @code{ncmul} (class)
2964 Non-commutative products are internally represented by objects of the class
2965 @code{ncmul}, as opposed to commutative products which are handled by the
2966 @code{mul} class. You will normally not have to worry about this distinction,
2969 The advantage of this approach is that you never have to worry about using
2970 (or forgetting to use) a special operator when constructing non-commutative
2971 expressions. Also, non-commutative products in GiNaC are more intelligent
2972 than in other computer algebra systems; they can, for example, automatically
2973 canonicalize themselves according to rules specified in the implementation
2974 of the non-commutative classes. The drawback is that to work with other than
2975 the built-in algebras you have to implement new classes yourself. Symbols
2976 always commutate and it's not possible to construct non-commutative products
2977 using symbols to represent the algebra elements or generators. User-defined
2978 functions can, however, be specified as being non-commutative.
2980 @cindex @code{return_type()}
2981 @cindex @code{return_type_tinfo()}
2982 Information about the commutativity of an object or expression can be
2983 obtained with the two member functions
2986 unsigned ex::return_type() const;
2987 unsigned ex::return_type_tinfo() const;
2990 The @code{return_type()} function returns one of three values (defined in
2991 the header file @file{flags.h}), corresponding to three categories of
2992 expressions in GiNaC:
2995 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2996 classes are of this kind.
2997 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2998 certain class of non-commutative objects which can be determined with the
2999 @code{return_type_tinfo()} method. Expressions of this category commutate
3000 with everything except @code{noncommutative} expressions of the same
3002 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3003 of non-commutative objects of different classes. Expressions of this
3004 category don't commutate with any other @code{noncommutative} or
3005 @code{noncommutative_composite} expressions.
3008 The value returned by the @code{return_type_tinfo()} method is valid only
3009 when the return type of the expression is @code{noncommutative}. It is a
3010 value that is unique to the class of the object and usually one of the
3011 constants in @file{tinfos.h}, or derived therefrom.
3013 Here are a couple of examples:
3016 @multitable @columnfractions 0.33 0.33 0.34
3017 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3018 @item @code{42} @tab @code{commutative} @tab -
3019 @item @code{2*x-y} @tab @code{commutative} @tab -
3020 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3021 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3022 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3023 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3027 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3028 @code{TINFO_clifford} for objects with a representation label of zero.
3029 Other representation labels yield a different @code{return_type_tinfo()},
3030 but it's the same for any two objects with the same label. This is also true
3033 A last note: With the exception of matrices, positive integer powers of
3034 non-commutative objects are automatically expanded in GiNaC. For example,
3035 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3036 non-commutative expressions).
3039 @cindex @code{clifford} (class)
3040 @subsection Clifford algebra
3043 Clifford algebras are supported in two flavours: Dirac gamma
3044 matrices (more physical) and generic Clifford algebras (more
3047 @cindex @code{dirac_gamma()}
3048 @subsubsection Dirac gamma matrices
3049 Dirac gamma matrices (note that GiNaC doesn't treat them
3050 as matrices) are designated as @samp{gamma~mu} and satisfy
3051 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3052 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3053 constructed by the function
3056 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3059 which takes two arguments: the index and a @dfn{representation label} in the
3060 range 0 to 255 which is used to distinguish elements of different Clifford
3061 algebras (this is also called a @dfn{spin line index}). Gammas with different
3062 labels commutate with each other. The dimension of the index can be 4 or (in
3063 the framework of dimensional regularization) any symbolic value. Spinor
3064 indices on Dirac gammas are not supported in GiNaC.
3066 @cindex @code{dirac_ONE()}
3067 The unity element of a Clifford algebra is constructed by
3070 ex dirac_ONE(unsigned char rl = 0);
3073 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3074 multiples of the unity element, even though it's customary to omit it.
3075 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3076 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3077 GiNaC will complain and/or produce incorrect results.
3079 @cindex @code{dirac_gamma5()}
3080 There is a special element @samp{gamma5} that commutates with all other
3081 gammas, has a unit square, and in 4 dimensions equals
3082 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3085 ex dirac_gamma5(unsigned char rl = 0);
3088 @cindex @code{dirac_gammaL()}
3089 @cindex @code{dirac_gammaR()}
3090 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3091 objects, constructed by
3094 ex dirac_gammaL(unsigned char rl = 0);
3095 ex dirac_gammaR(unsigned char rl = 0);
3098 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3099 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3101 @cindex @code{dirac_slash()}
3102 Finally, the function
3105 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3108 creates a term that represents a contraction of @samp{e} with the Dirac
3109 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3110 with a unique index whose dimension is given by the @code{dim} argument).
3111 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3113 In products of dirac gammas, superfluous unity elements are automatically
3114 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3115 and @samp{gammaR} are moved to the front.
3117 The @code{simplify_indexed()} function performs contractions in gamma strings,
3123 symbol a("a"), b("b"), D("D");
3124 varidx mu(symbol("mu"), D);
3125 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3126 * dirac_gamma(mu.toggle_variance());
3128 // -> gamma~mu*a\*gamma.mu
3129 e = e.simplify_indexed();
3132 cout << e.subs(D == 4) << endl;
3138 @cindex @code{dirac_trace()}
3139 To calculate the trace of an expression containing strings of Dirac gammas
3140 you use one of the functions
3143 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3144 const ex & trONE = 4);
3145 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3146 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3149 These functions take the trace over all gammas in the specified set @code{rls}
3150 or list @code{rll} of representation labels, or the single label @code{rl};
3151 gammas with other labels are left standing. The last argument to
3152 @code{dirac_trace()} is the value to be returned for the trace of the unity
3153 element, which defaults to 4.
3155 The @code{dirac_trace()} function is a linear functional that is equal to the
3156 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3157 functional is not cyclic in
3160 dimensions when acting on
3161 expressions containing @samp{gamma5}, so it's not a proper trace. This
3162 @samp{gamma5} scheme is described in greater detail in
3163 @cite{The Role of gamma5 in Dimensional Regularization}.
3165 The value of the trace itself is also usually different in 4 and in
3173 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3174 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3175 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3176 cout << dirac_trace(e).simplify_indexed() << endl;
3183 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3184 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3185 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3186 cout << dirac_trace(e).simplify_indexed() << endl;
3187 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3191 Here is an example for using @code{dirac_trace()} to compute a value that
3192 appears in the calculation of the one-loop vacuum polarization amplitude in
3197 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3198 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3201 sp.add(l, l, pow(l, 2));
3202 sp.add(l, q, ldotq);
3204 ex e = dirac_gamma(mu) *
3205 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3206 dirac_gamma(mu.toggle_variance()) *
3207 (dirac_slash(l, D) + m * dirac_ONE());
3208 e = dirac_trace(e).simplify_indexed(sp);
3209 e = e.collect(lst(l, ldotq, m));
3211 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3215 The @code{canonicalize_clifford()} function reorders all gamma products that
3216 appear in an expression to a canonical (but not necessarily simple) form.
3217 You can use this to compare two expressions or for further simplifications:
3221 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3222 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3224 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3226 e = canonicalize_clifford(e);
3228 // -> 2*ONE*eta~mu~nu
3232 @cindex @code{clifford_unit()}
3233 @subsubsection A generic Clifford algebra
3235 A generic Clifford algebra, i.e. a
3239 dimensional algebra with
3243 satisfying the identities
3245 $e_i e_j + e_j e_i = M(i, j) $
3248 e~i e~j + e~j e~i = M(i, j)
3250 for some matrix (@code{metric})
3251 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3252 entries. Such generators are created by the function
3255 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3258 where @code{mu} should be a @code{varidx} class object indexing the
3259 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3260 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3261 object, optional parameter @code{rl} allows to distinguish different
3262 Clifford algebras (which will commute with each other). Note that the call
3263 @code{clifford_unit(mu, minkmetric())} creates something very close to
3264 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3265 metric defining this Clifford number.
3267 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3268 the Clifford algebra units with a call like that
3271 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3274 since this may yield some further automatic simplifications.
3276 Individual generators of a Clifford algebra can be accessed in several
3282 varidx nu(symbol("nu"), 4);
3284 ex M = diag_matrix(lst(1, -1, 0, s));
3285 ex e = clifford_unit(nu, M);
3286 ex e0 = e.subs(nu == 0);
3287 ex e1 = e.subs(nu == 1);
3288 ex e2 = e.subs(nu == 2);
3289 ex e3 = e.subs(nu == 3);
3294 will produce four anti-commuting generators of a Clifford algebra with properties
3296 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3299 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3302 @cindex @code{lst_to_clifford()}
3303 A similar effect can be achieved from the function
3306 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3307 unsigned char rl = 0);
3308 ex lst_to_clifford(const ex & v, const ex & e);
3311 which converts a list or vector
3313 $v = (v^0, v^1, ..., v^n)$
3316 @samp{v = (v~0, v~1, ..., v~n)}
3321 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3324 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3327 directly supplied in the second form of the procedure. In the first form
3328 the Clifford unit @samp{e.k} is generated by the call of
3329 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3330 with the help of @code{lst_to_clifford()} as follows
3335 varidx nu(symbol("nu"), 4);
3337 ex M = diag_matrix(lst(1, -1, 0, s));
3338 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3339 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3340 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3341 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3346 @cindex @code{clifford_to_lst()}
3347 There is the inverse function
3350 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3353 which takes an expression @code{e} and tries to find a list
3355 $v = (v^0, v^1, ..., v^n)$
3358 @samp{v = (v~0, v~1, ..., v~n)}
3362 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3365 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3367 with respect to the given Clifford units @code{c} and with none of the
3368 @samp{v~k} containing Clifford units @code{c} (of course, this
3369 may be impossible). This function can use an @code{algebraic} method
3370 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3372 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3375 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3377 is zero or is not a @code{numeric} for some @samp{k}
3378 then the method will be automatically changed to symbolic. The same effect
3379 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3381 @cindex @code{clifford_prime()}
3382 @cindex @code{clifford_star()}
3383 @cindex @code{clifford_bar()}
3384 There are several functions for (anti-)automorphisms of Clifford algebras:
3387 ex clifford_prime(const ex & e)
3388 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3389 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3392 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3393 changes signs of all Clifford units in the expression. The reversion
3394 of a Clifford algebra @code{clifford_star()} coincides with the
3395 @code{conjugate()} method and effectively reverses the order of Clifford
3396 units in any product. Finally the main anti-automorphism
3397 of a Clifford algebra @code{clifford_bar()} is the composition of the
3398 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3399 in a product. These functions correspond to the notations
3414 used in Clifford algebra textbooks.
3416 @cindex @code{clifford_norm()}
3420 ex clifford_norm(const ex & e);
3423 @cindex @code{clifford_inverse()}
3424 calculates the norm of a Clifford number from the expression
3426 $||e||^2 = e\overline{e}$.
3429 @code{||e||^2 = e \bar@{e@}}
3431 The inverse of a Clifford expression is returned by the function
3434 ex clifford_inverse(const ex & e);
3437 which calculates it as
3439 $e^{-1} = \overline{e}/||e||^2$.
3442 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3451 then an exception is raised.
3453 @cindex @code{remove_dirac_ONE()}
3454 If a Clifford number happens to be a factor of
3455 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3456 expression by the function
3459 ex remove_dirac_ONE(const ex & e);
3462 @cindex @code{canonicalize_clifford()}
3463 The function @code{canonicalize_clifford()} works for a
3464 generic Clifford algebra in a similar way as for Dirac gammas.
3466 The last provided function is
3468 @cindex @code{clifford_moebius_map()}
3470 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3471 const ex & d, const ex & v, const ex & G,
3472 unsigned char rl = 0);
3473 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3474 unsigned char rl = 0);
3477 It takes a list or vector @code{v} and makes the Moebius (conformal or
3478 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3479 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3480 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3481 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3482 ignored even if supplied. The returned value of this function is a list
3483 of components of the resulting vector.
3485 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3486 where @code{1} is the @code{representation_label} and @code{\nu} is the
3487 index of the corresponding unit. This provides a flexible typesetting
3488 with a suitable defintion of the @code{\clifford} command. For example, the
3491 \newcommand@{\clifford@}[1][]@{@}
3493 typesets all Clifford units identically, while the alternative definition
3495 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3497 prints units with @code{representation_label=0} as
3504 with @code{representation_label=1} as
3511 and with @code{representation_label=2} as
3519 @cindex @code{color} (class)
3520 @subsection Color algebra
3522 @cindex @code{color_T()}
3523 For computations in quantum chromodynamics, GiNaC implements the base elements
3524 and structure constants of the su(3) Lie algebra (color algebra). The base
3525 elements @math{T_a} are constructed by the function
3528 ex color_T(const ex & a, unsigned char rl = 0);
3531 which takes two arguments: the index and a @dfn{representation label} in the
3532 range 0 to 255 which is used to distinguish elements of different color
3533 algebras. Objects with different labels commutate with each other. The
3534 dimension of the index must be exactly 8 and it should be of class @code{idx},
3537 @cindex @code{color_ONE()}
3538 The unity element of a color algebra is constructed by
3541 ex color_ONE(unsigned char rl = 0);
3544 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3545 multiples of the unity element, even though it's customary to omit it.
3546 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3547 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3548 GiNaC may produce incorrect results.
3550 @cindex @code{color_d()}
3551 @cindex @code{color_f()}
3555 ex color_d(const ex & a, const ex & b, const ex & c);
3556 ex color_f(const ex & a, const ex & b, const ex & c);
3559 create the symmetric and antisymmetric structure constants @math{d_abc} and
3560 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3561 and @math{[T_a, T_b] = i f_abc T_c}.
3563 These functions evaluate to their numerical values,
3564 if you supply numeric indices to them. The index values should be in
3565 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3566 goes along better with the notations used in physical literature.
3568 @cindex @code{color_h()}
3569 There's an additional function
3572 ex color_h(const ex & a, const ex & b, const ex & c);
3575 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3577 The function @code{simplify_indexed()} performs some simplifications on
3578 expressions containing color objects:
3583 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3584 k(symbol("k"), 8), l(symbol("l"), 8);
3586 e = color_d(a, b, l) * color_f(a, b, k);
3587 cout << e.simplify_indexed() << endl;
3590 e = color_d(a, b, l) * color_d(a, b, k);
3591 cout << e.simplify_indexed() << endl;
3594 e = color_f(l, a, b) * color_f(a, b, k);
3595 cout << e.simplify_indexed() << endl;
3598 e = color_h(a, b, c) * color_h(a, b, c);
3599 cout << e.simplify_indexed() << endl;
3602 e = color_h(a, b, c) * color_T(b) * color_T(c);
3603 cout << e.simplify_indexed() << endl;
3606 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3607 cout << e.simplify_indexed() << endl;
3610 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3611 cout << e.simplify_indexed() << endl;
3612 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3616 @cindex @code{color_trace()}
3617 To calculate the trace of an expression containing color objects you use one
3621 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3622 ex color_trace(const ex & e, const lst & rll);
3623 ex color_trace(const ex & e, unsigned char rl = 0);
3626 These functions take the trace over all color @samp{T} objects in the
3627 specified set @code{rls} or list @code{rll} of representation labels, or the
3628 single label @code{rl}; @samp{T}s with other labels are left standing. For
3633 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3635 // -> -I*f.a.c.b+d.a.c.b
3640 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3641 @c node-name, next, previous, up
3644 @cindex @code{exhashmap} (class)
3646 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3647 that can be used as a drop-in replacement for the STL
3648 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3649 typically constant-time, element look-up than @code{map<>}.
3651 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3652 following differences:
3656 no @code{lower_bound()} and @code{upper_bound()} methods
3658 no reverse iterators, no @code{rbegin()}/@code{rend()}
3660 no @code{operator<(exhashmap, exhashmap)}
3662 the comparison function object @code{key_compare} is hardcoded to
3665 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3666 initial hash table size (the actual table size after construction may be
3667 larger than the specified value)
3669 the method @code{size_t bucket_count()} returns the current size of the hash
3672 @code{insert()} and @code{erase()} operations invalidate all iterators
3676 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3677 @c node-name, next, previous, up
3678 @chapter Methods and Functions
3681 In this chapter the most important algorithms provided by GiNaC will be
3682 described. Some of them are implemented as functions on expressions,
3683 others are implemented as methods provided by expression objects. If
3684 they are methods, there exists a wrapper function around it, so you can
3685 alternatively call it in a functional way as shown in the simple
3690 cout << "As method: " << sin(1).evalf() << endl;
3691 cout << "As function: " << evalf(sin(1)) << endl;
3695 @cindex @code{subs()}
3696 The general rule is that wherever methods accept one or more parameters
3697 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3698 wrapper accepts is the same but preceded by the object to act on
3699 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3700 most natural one in an OO model but it may lead to confusion for MapleV
3701 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3702 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3703 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3704 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3705 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3706 here. Also, users of MuPAD will in most cases feel more comfortable
3707 with GiNaC's convention. All function wrappers are implemented
3708 as simple inline functions which just call the corresponding method and
3709 are only provided for users uncomfortable with OO who are dead set to
3710 avoid method invocations. Generally, nested function wrappers are much
3711 harder to read than a sequence of methods and should therefore be
3712 avoided if possible. On the other hand, not everything in GiNaC is a
3713 method on class @code{ex} and sometimes calling a function cannot be
3717 * Information About Expressions::
3718 * Numerical Evaluation::
3719 * Substituting Expressions::
3720 * Pattern Matching and Advanced Substitutions::
3721 * Applying a Function on Subexpressions::
3722 * Visitors and Tree Traversal::
3723 * Polynomial Arithmetic:: Working with polynomials.
3724 * Rational Expressions:: Working with rational functions.
3725 * Symbolic Differentiation::
3726 * Series Expansion:: Taylor and Laurent expansion.
3728 * Built-in Functions:: List of predefined mathematical functions.
3729 * Multiple polylogarithms::
3730 * Complex Conjugation::
3731 * Built-in Functions:: List of predefined mathematical functions.
3732 * Solving Linear Systems of Equations::
3733 * Input/Output:: Input and output of expressions.
3737 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3738 @c node-name, next, previous, up
3739 @section Getting information about expressions
3741 @subsection Checking expression types
3742 @cindex @code{is_a<@dots{}>()}
3743 @cindex @code{is_exactly_a<@dots{}>()}
3744 @cindex @code{ex_to<@dots{}>()}
3745 @cindex Converting @code{ex} to other classes
3746 @cindex @code{info()}
3747 @cindex @code{return_type()}
3748 @cindex @code{return_type_tinfo()}
3750 Sometimes it's useful to check whether a given expression is a plain number,
3751 a sum, a polynomial with integer coefficients, or of some other specific type.
3752 GiNaC provides a couple of functions for this:
3755 bool is_a<T>(const ex & e);
3756 bool is_exactly_a<T>(const ex & e);
3757 bool ex::info(unsigned flag);
3758 unsigned ex::return_type() const;
3759 unsigned ex::return_type_tinfo() const;
3762 When the test made by @code{is_a<T>()} returns true, it is safe to call
3763 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3764 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3765 example, assuming @code{e} is an @code{ex}:
3770 if (is_a<numeric>(e))
3771 numeric n = ex_to<numeric>(e);
3776 @code{is_a<T>(e)} allows you to check whether the top-level object of
3777 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3778 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3779 e.g., for checking whether an expression is a number, a sum, or a product:
3786 is_a<numeric>(e1); // true
3787 is_a<numeric>(e2); // false
3788 is_a<add>(e1); // false
3789 is_a<add>(e2); // true
3790 is_a<mul>(e1); // false
3791 is_a<mul>(e2); // false
3795 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3796 top-level object of an expression @samp{e} is an instance of the GiNaC
3797 class @samp{T}, not including parent classes.
3799 The @code{info()} method is used for checking certain attributes of
3800 expressions. The possible values for the @code{flag} argument are defined
3801 in @file{ginac/flags.h}, the most important being explained in the following
3805 @multitable @columnfractions .30 .70
3806 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3807 @item @code{numeric}
3808 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3810 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3811 @item @code{rational}
3812 @tab @dots{}an exact rational number (integers are rational, too)
3813 @item @code{integer}
3814 @tab @dots{}a (non-complex) integer
3815 @item @code{crational}
3816 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3817 @item @code{cinteger}
3818 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3819 @item @code{positive}
3820 @tab @dots{}not complex and greater than 0
3821 @item @code{negative}
3822 @tab @dots{}not complex and less than 0
3823 @item @code{nonnegative}
3824 @tab @dots{}not complex and greater than or equal to 0
3826 @tab @dots{}an integer greater than 0
3828 @tab @dots{}an integer less than 0
3829 @item @code{nonnegint}
3830 @tab @dots{}an integer greater than or equal to 0
3832 @tab @dots{}an even integer
3834 @tab @dots{}an odd integer
3836 @tab @dots{}a prime integer (probabilistic primality test)
3837 @item @code{relation}
3838 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3839 @item @code{relation_equal}
3840 @tab @dots{}a @code{==} relation
3841 @item @code{relation_not_equal}
3842 @tab @dots{}a @code{!=} relation
3843 @item @code{relation_less}
3844 @tab @dots{}a @code{<} relation
3845 @item @code{relation_less_or_equal}
3846 @tab @dots{}a @code{<=} relation
3847 @item @code{relation_greater}
3848 @tab @dots{}a @code{>} relation
3849 @item @code{relation_greater_or_equal}
3850 @tab @dots{}a @code{>=} relation
3852 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3854 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3855 @item @code{polynomial}
3856 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3857 @item @code{integer_polynomial}
3858 @tab @dots{}a polynomial with (non-complex) integer coefficients
3859 @item @code{cinteger_polynomial}
3860 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3861 @item @code{rational_polynomial}
3862 @tab @dots{}a polynomial with (non-complex) rational coefficients
3863 @item @code{crational_polynomial}
3864 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3865 @item @code{rational_function}
3866 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3867 @item @code{algebraic}
3868 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3872 To determine whether an expression is commutative or non-commutative and if
3873 so, with which other expressions it would commutate, you use the methods
3874 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3875 for an explanation of these.
3878 @subsection Accessing subexpressions
3881 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3882 @code{function}, act as containers for subexpressions. For example, the
3883 subexpressions of a sum (an @code{add} object) are the individual terms,
3884 and the subexpressions of a @code{function} are the function's arguments.
3886 @cindex @code{nops()}
3888 GiNaC provides several ways of accessing subexpressions. The first way is to
3893 ex ex::op(size_t i);
3896 @code{nops()} determines the number of subexpressions (operands) contained
3897 in the expression, while @code{op(i)} returns the @code{i}-th
3898 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3899 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3900 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3901 @math{i>0} are the indices.
3904 @cindex @code{const_iterator}
3905 The second way to access subexpressions is via the STL-style random-access
3906 iterator class @code{const_iterator} and the methods
3909 const_iterator ex::begin();
3910 const_iterator ex::end();
3913 @code{begin()} returns an iterator referring to the first subexpression;
3914 @code{end()} returns an iterator which is one-past the last subexpression.
3915 If the expression has no subexpressions, then @code{begin() == end()}. These
3916 iterators can also be used in conjunction with non-modifying STL algorithms.
3918 Here is an example that (non-recursively) prints the subexpressions of a
3919 given expression in three different ways:
3926 for (size_t i = 0; i != e.nops(); ++i)
3927 cout << e.op(i) << endl;
3930 for (const_iterator i = e.begin(); i != e.end(); ++i)
3933 // with iterators and STL copy()
3934 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3938 @cindex @code{const_preorder_iterator}
3939 @cindex @code{const_postorder_iterator}
3940 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3941 expression's immediate children. GiNaC provides two additional iterator
3942 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3943 that iterate over all objects in an expression tree, in preorder or postorder,
3944 respectively. They are STL-style forward iterators, and are created with the
3948 const_preorder_iterator ex::preorder_begin();
3949 const_preorder_iterator ex::preorder_end();
3950 const_postorder_iterator ex::postorder_begin();
3951 const_postorder_iterator ex::postorder_end();
3954 The following example illustrates the differences between
3955 @code{const_iterator}, @code{const_preorder_iterator}, and
3956 @code{const_postorder_iterator}:
3960 symbol A("A"), B("B"), C("C");
3961 ex e = lst(lst(A, B), C);
3963 std::copy(e.begin(), e.end(),
3964 std::ostream_iterator<ex>(cout, "\n"));
3968 std::copy(e.preorder_begin(), e.preorder_end(),
3969 std::ostream_iterator<ex>(cout, "\n"));
3976 std::copy(e.postorder_begin(), e.postorder_end(),
3977 std::ostream_iterator<ex>(cout, "\n"));
3986 @cindex @code{relational} (class)
3987 Finally, the left-hand side and right-hand side expressions of objects of
3988 class @code{relational} (and only of these) can also be accessed with the
3997 @subsection Comparing expressions
3998 @cindex @code{is_equal()}
3999 @cindex @code{is_zero()}
4001 Expressions can be compared with the usual C++ relational operators like
4002 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4003 the result is usually not determinable and the result will be @code{false},
4004 except in the case of the @code{!=} operator. You should also be aware that
4005 GiNaC will only do the most trivial test for equality (subtracting both
4006 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4009 Actually, if you construct an expression like @code{a == b}, this will be
4010 represented by an object of the @code{relational} class (@pxref{Relations})
4011 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4013 There are also two methods
4016 bool ex::is_equal(const ex & other);
4020 for checking whether one expression is equal to another, or equal to zero,
4024 @subsection Ordering expressions
4025 @cindex @code{ex_is_less} (class)
4026 @cindex @code{ex_is_equal} (class)
4027 @cindex @code{compare()}
4029 Sometimes it is necessary to establish a mathematically well-defined ordering
4030 on a set of arbitrary expressions, for example to use expressions as keys
4031 in a @code{std::map<>} container, or to bring a vector of expressions into
4032 a canonical order (which is done internally by GiNaC for sums and products).
4034 The operators @code{<}, @code{>} etc. described in the last section cannot
4035 be used for this, as they don't implement an ordering relation in the
4036 mathematical sense. In particular, they are not guaranteed to be
4037 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4038 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4041 By default, STL classes and algorithms use the @code{<} and @code{==}
4042 operators to compare objects, which are unsuitable for expressions, but GiNaC
4043 provides two functors that can be supplied as proper binary comparison
4044 predicates to the STL:
4047 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4049 bool operator()(const ex &lh, const ex &rh) const;
4052 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4054 bool operator()(const ex &lh, const ex &rh) const;
4058 For example, to define a @code{map} that maps expressions to strings you
4062 std::map<ex, std::string, ex_is_less> myMap;
4065 Omitting the @code{ex_is_less} template parameter will introduce spurious
4066 bugs because the map operates improperly.
4068 Other examples for the use of the functors:
4076 std::sort(v.begin(), v.end(), ex_is_less());
4078 // count the number of expressions equal to '1'
4079 unsigned num_ones = std::count_if(v.begin(), v.end(),
4080 std::bind2nd(ex_is_equal(), 1));
4083 The implementation of @code{ex_is_less} uses the member function
4086 int ex::compare(const ex & other) const;
4089 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4090 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4094 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4095 @c node-name, next, previous, up
4096 @section Numerical Evaluation
4097 @cindex @code{evalf()}
4099 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4100 To evaluate them using floating-point arithmetic you need to call
4103 ex ex::evalf(int level = 0) const;
4106 @cindex @code{Digits}
4107 The accuracy of the evaluation is controlled by the global object @code{Digits}
4108 which can be assigned an integer value. The default value of @code{Digits}
4109 is 17. @xref{Numbers}, for more information and examples.
4111 To evaluate an expression to a @code{double} floating-point number you can
4112 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4116 // Approximate sin(x/Pi)
4118 ex e = series(sin(x/Pi), x == 0, 6);
4120 // Evaluate numerically at x=0.1
4121 ex f = evalf(e.subs(x == 0.1));
4123 // ex_to<numeric> is an unsafe cast, so check the type first
4124 if (is_a<numeric>(f)) @{
4125 double d = ex_to<numeric>(f).to_double();
4134 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4135 @c node-name, next, previous, up
4136 @section Substituting expressions
4137 @cindex @code{subs()}
4139 Algebraic objects inside expressions can be replaced with arbitrary
4140 expressions via the @code{.subs()} method:
4143 ex ex::subs(const ex & e, unsigned options = 0);
4144 ex ex::subs(const exmap & m, unsigned options = 0);
4145 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4148 In the first form, @code{subs()} accepts a relational of the form
4149 @samp{object == expression} or a @code{lst} of such relationals:
4153 symbol x("x"), y("y");
4155 ex e1 = 2*x^2-4*x+3;
4156 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4160 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4165 If you specify multiple substitutions, they are performed in parallel, so e.g.
4166 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4168 The second form of @code{subs()} takes an @code{exmap} object which is a
4169 pair associative container that maps expressions to expressions (currently
4170 implemented as a @code{std::map}). This is the most efficient one of the
4171 three @code{subs()} forms and should be used when the number of objects to
4172 be substituted is large or unknown.
4174 Using this form, the second example from above would look like this:
4178 symbol x("x"), y("y");
4184 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4188 The third form of @code{subs()} takes two lists, one for the objects to be
4189 replaced and one for the expressions to be substituted (both lists must
4190 contain the same number of elements). Using this form, you would write
4194 symbol x("x"), y("y");
4197 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4201 The optional last argument to @code{subs()} is a combination of
4202 @code{subs_options} flags. There are two options available:
4203 @code{subs_options::no_pattern} disables pattern matching, which makes
4204 large @code{subs()} operations significantly faster if you are not using
4205 patterns. The second option, @code{subs_options::algebraic} enables
4206 algebraic substitutions in products and powers.
4207 @ref{Pattern Matching and Advanced Substitutions}, for more information
4208 about patterns and algebraic substitutions.
4210 @code{subs()} performs syntactic substitution of any complete algebraic
4211 object; it does not try to match sub-expressions as is demonstrated by the
4216 symbol x("x"), y("y"), z("z");
4218 ex e1 = pow(x+y, 2);
4219 cout << e1.subs(x+y == 4) << endl;
4222 ex e2 = sin(x)*sin(y)*cos(x);
4223 cout << e2.subs(sin(x) == cos(x)) << endl;
4224 // -> cos(x)^2*sin(y)
4227 cout << e3.subs(x+y == 4) << endl;
4229 // (and not 4+z as one might expect)
4233 A more powerful form of substitution using wildcards is described in the
4237 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4238 @c node-name, next, previous, up
4239 @section Pattern matching and advanced substitutions
4240 @cindex @code{wildcard} (class)
4241 @cindex Pattern matching
4243 GiNaC allows the use of patterns for checking whether an expression is of a
4244 certain form or contains subexpressions of a certain form, and for
4245 substituting expressions in a more general way.
4247 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4248 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4249 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4250 an unsigned integer number to allow having multiple different wildcards in a
4251 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4252 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4256 ex wild(unsigned label = 0);
4259 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4262 Some examples for patterns:
4264 @multitable @columnfractions .5 .5
4265 @item @strong{Constructed as} @tab @strong{Output as}
4266 @item @code{wild()} @tab @samp{$0}
4267 @item @code{pow(x,wild())} @tab @samp{x^$0}
4268 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4269 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4275 @item Wildcards behave like symbols and are subject to the same algebraic
4276 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4277 @item As shown in the last example, to use wildcards for indices you have to
4278 use them as the value of an @code{idx} object. This is because indices must
4279 always be of class @code{idx} (or a subclass).
4280 @item Wildcards only represent expressions or subexpressions. It is not
4281 possible to use them as placeholders for other properties like index
4282 dimension or variance, representation labels, symmetry of indexed objects
4284 @item Because wildcards are commutative, it is not possible to use wildcards
4285 as part of noncommutative products.
4286 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4287 are also valid patterns.
4290 @subsection Matching expressions
4291 @cindex @code{match()}
4292 The most basic application of patterns is to check whether an expression
4293 matches a given pattern. This is done by the function
4296 bool ex::match(const ex & pattern);
4297 bool ex::match(const ex & pattern, lst & repls);
4300 This function returns @code{true} when the expression matches the pattern
4301 and @code{false} if it doesn't. If used in the second form, the actual
4302 subexpressions matched by the wildcards get returned in the @code{repls}
4303 object as a list of relations of the form @samp{wildcard == expression}.
4304 If @code{match()} returns false, the state of @code{repls} is undefined.
4305 For reproducible results, the list should be empty when passed to
4306 @code{match()}, but it is also possible to find similarities in multiple
4307 expressions by passing in the result of a previous match.
4309 The matching algorithm works as follows:
4312 @item A single wildcard matches any expression. If one wildcard appears
4313 multiple times in a pattern, it must match the same expression in all
4314 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4315 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4316 @item If the expression is not of the same class as the pattern, the match
4317 fails (i.e. a sum only matches a sum, a function only matches a function,
4319 @item If the pattern is a function, it only matches the same function
4320 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4321 @item Except for sums and products, the match fails if the number of
4322 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4324 @item If there are no subexpressions, the expressions and the pattern must
4325 be equal (in the sense of @code{is_equal()}).
4326 @item Except for sums and products, each subexpression (@code{op()}) must
4327 match the corresponding subexpression of the pattern.
4330 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4331 account for their commutativity and associativity:
4334 @item If the pattern contains a term or factor that is a single wildcard,
4335 this one is used as the @dfn{global wildcard}. If there is more than one
4336 such wildcard, one of them is chosen as the global wildcard in a random
4338 @item Every term/factor of the pattern, except the global wildcard, is
4339 matched against every term of the expression in sequence. If no match is
4340 found, the whole match fails. Terms that did match are not considered in
4342 @item If there are no unmatched terms left, the match succeeds. Otherwise
4343 the match fails unless there is a global wildcard in the pattern, in
4344 which case this wildcard matches the remaining terms.
4347 In general, having more than one single wildcard as a term of a sum or a
4348 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4351 Here are some examples in @command{ginsh} to demonstrate how it works (the
4352 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4353 match fails, and the list of wildcard replacements otherwise):
4356 > match((x+y)^a,(x+y)^a);
4358 > match((x+y)^a,(x+y)^b);
4360 > match((x+y)^a,$1^$2);
4362 > match((x+y)^a,$1^$1);
4364 > match((x+y)^(x+y),$1^$1);
4366 > match((x+y)^(x+y),$1^$2);
4368 > match((a+b)*(a+c),($1+b)*($1+c));
4370 > match((a+b)*(a+c),(a+$1)*(a+$2));
4372 (Unpredictable. The result might also be [$1==c,$2==b].)
4373 > match((a+b)*(a+c),($1+$2)*($1+$3));
4374 (The result is undefined. Due to the sequential nature of the algorithm
4375 and the re-ordering of terms in GiNaC, the match for the first factor
4376 may be @{$1==a,$2==b@} in which case the match for the second factor
4377 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4379 > match(a*(x+y)+a*z+b,a*$1+$2);
4380 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4381 @{$1=x+y,$2=a*z+b@}.)
4382 > match(a+b+c+d+e+f,c);
4384 > match(a+b+c+d+e+f,c+$0);
4386 > match(a+b+c+d+e+f,c+e+$0);
4388 > match(a+b,a+b+$0);
4390 > match(a*b^2,a^$1*b^$2);
4392 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4393 even though a==a^1.)
4394 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4396 > match(atan2(y,x^2),atan2(y,$0));
4400 @subsection Matching parts of expressions
4401 @cindex @code{has()}
4402 A more general way to look for patterns in expressions is provided by the
4406 bool ex::has(const ex & pattern);
4409 This function checks whether a pattern is matched by an expression itself or
4410 by any of its subexpressions.
4412 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4413 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4416 > has(x*sin(x+y+2*a),y);
4418 > has(x*sin(x+y+2*a),x+y);
4420 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4421 has the subexpressions "x", "y" and "2*a".)
4422 > has(x*sin(x+y+2*a),x+y+$1);
4424 (But this is possible.)
4425 > has(x*sin(2*(x+y)+2*a),x+y);
4427 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4428 which "x+y" is not a subexpression.)
4431 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4433 > has(4*x^2-x+3,$1*x);
4435 > has(4*x^2+x+3,$1*x);
4437 (Another possible pitfall. The first expression matches because the term
4438 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4439 contains a linear term you should use the coeff() function instead.)
4442 @cindex @code{find()}
4446 bool ex::find(const ex & pattern, lst & found);
4449 works a bit like @code{has()} but it doesn't stop upon finding the first
4450 match. Instead, it appends all found matches to the specified list. If there
4451 are multiple occurrences of the same expression, it is entered only once to
4452 the list. @code{find()} returns false if no matches were found (in
4453 @command{ginsh}, it returns an empty list):
4456 > find(1+x+x^2+x^3,x);
4458 > find(1+x+x^2+x^3,y);
4460 > find(1+x+x^2+x^3,x^$1);
4462 (Note the absence of "x".)
4463 > expand((sin(x)+sin(y))*(a+b));
4464 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4469 @subsection Substituting expressions
4470 @cindex @code{subs()}
4471 Probably the most useful application of patterns is to use them for
4472 substituting expressions with the @code{subs()} method. Wildcards can be
4473 used in the search patterns as well as in the replacement expressions, where
4474 they get replaced by the expressions matched by them. @code{subs()} doesn't
4475 know anything about algebra; it performs purely syntactic substitutions.
4480 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4482 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4484 > subs((a+b+c)^2,a+b==x);
4486 > subs((a+b+c)^2,a+b+$1==x+$1);
4488 > subs(a+2*b,a+b==x);
4490 > subs(4*x^3-2*x^2+5*x-1,x==a);
4492 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4494 > subs(sin(1+sin(x)),sin($1)==cos($1));
4496 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4500 The last example would be written in C++ in this way:
4504 symbol a("a"), b("b"), x("x"), y("y");
4505 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4506 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4507 cout << e.expand() << endl;
4512 @subsection Algebraic substitutions
4513 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4514 enables smarter, algebraic substitutions in products and powers. If you want
4515 to substitute some factors of a product, you only need to list these factors
4516 in your pattern. Furthermore, if an (integer) power of some expression occurs
4517 in your pattern and in the expression that you want the substitution to occur
4518 in, it can be substituted as many times as possible, without getting negative
4521 An example clarifies it all (hopefully):
4524 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4525 subs_options::algebraic) << endl;
4526 // --> (y+x)^6+b^6+a^6
4528 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4530 // Powers and products are smart, but addition is just the same.
4532 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4535 // As I said: addition is just the same.
4537 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4538 // --> x^3*b*a^2+2*b
4540 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4542 // --> 2*b+x^3*b^(-1)*a^(-2)
4544 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4545 // --> -1-2*a^2+4*a^3+5*a
4547 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4548 subs_options::algebraic) << endl;
4549 // --> -1+5*x+4*x^3-2*x^2
4550 // You should not really need this kind of patterns very often now.
4551 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4553 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4554 subs_options::algebraic) << endl;
4555 // --> cos(1+cos(x))
4557 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4558 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4559 subs_options::algebraic)) << endl;
4564 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4565 @c node-name, next, previous, up
4566 @section Applying a Function on Subexpressions
4567 @cindex tree traversal
4568 @cindex @code{map()}
4570 Sometimes you may want to perform an operation on specific parts of an
4571 expression while leaving the general structure of it intact. An example
4572 of this would be a matrix trace operation: the trace of a sum is the sum
4573 of the traces of the individual terms. That is, the trace should @dfn{map}
4574 on the sum, by applying itself to each of the sum's operands. It is possible
4575 to do this manually which usually results in code like this:
4580 if (is_a<matrix>(e))
4581 return ex_to<matrix>(e).trace();
4582 else if (is_a<add>(e)) @{
4584 for (size_t i=0; i<e.nops(); i++)
4585 sum += calc_trace(e.op(i));
4587 @} else if (is_a<mul>)(e)) @{
4595 This is, however, slightly inefficient (if the sum is very large it can take
4596 a long time to add the terms one-by-one), and its applicability is limited to
4597 a rather small class of expressions. If @code{calc_trace()} is called with
4598 a relation or a list as its argument, you will probably want the trace to
4599 be taken on both sides of the relation or of all elements of the list.
4601 GiNaC offers the @code{map()} method to aid in the implementation of such
4605 ex ex::map(map_function & f) const;
4606 ex ex::map(ex (*f)(const ex & e)) const;
4609 In the first (preferred) form, @code{map()} takes a function object that
4610 is subclassed from the @code{map_function} class. In the second form, it
4611 takes a pointer to a function that accepts and returns an expression.
4612 @code{map()} constructs a new expression of the same type, applying the
4613 specified function on all subexpressions (in the sense of @code{op()}),
4616 The use of a function object makes it possible to supply more arguments to
4617 the function that is being mapped, or to keep local state information.
4618 The @code{map_function} class declares a virtual function call operator
4619 that you can overload. Here is a sample implementation of @code{calc_trace()}
4620 that uses @code{map()} in a recursive fashion:
4623 struct calc_trace : public map_function @{
4624 ex operator()(const ex &e)
4626 if (is_a<matrix>(e))
4627 return ex_to<matrix>(e).trace();
4628 else if (is_a<mul>(e)) @{
4631 return e.map(*this);
4636 This function object could then be used like this:
4640 ex M = ... // expression with matrices
4641 calc_trace do_trace;
4642 ex tr = do_trace(M);
4646 Here is another example for you to meditate over. It removes quadratic
4647 terms in a variable from an expanded polynomial:
4650 struct map_rem_quad : public map_function @{
4652 map_rem_quad(const ex & var_) : var(var_) @{@}
4654 ex operator()(const ex & e)
4656 if (is_a<add>(e) || is_a<mul>(e))
4657 return e.map(*this);
4658 else if (is_a<power>(e) &&
4659 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4669 symbol x("x"), y("y");
4672 for (int i=0; i<8; i++)
4673 e += pow(x, i) * pow(y, 8-i) * (i+1);
4675 // -> 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
4677 map_rem_quad rem_quad(x);
4678 cout << rem_quad(e) << endl;
4679 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4683 @command{ginsh} offers a slightly different implementation of @code{map()}
4684 that allows applying algebraic functions to operands. The second argument
4685 to @code{map()} is an expression containing the wildcard @samp{$0} which
4686 acts as the placeholder for the operands:
4691 > map(a+2*b,sin($0));
4693 > map(@{a,b,c@},$0^2+$0);
4694 @{a^2+a,b^2+b,c^2+c@}
4697 Note that it is only possible to use algebraic functions in the second
4698 argument. You can not use functions like @samp{diff()}, @samp{op()},
4699 @samp{subs()} etc. because these are evaluated immediately:
4702 > map(@{a,b,c@},diff($0,a));
4704 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4705 to "map(@{a,b,c@},0)".
4709 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4710 @c node-name, next, previous, up
4711 @section Visitors and Tree Traversal
4712 @cindex tree traversal
4713 @cindex @code{visitor} (class)
4714 @cindex @code{accept()}
4715 @cindex @code{visit()}
4716 @cindex @code{traverse()}
4717 @cindex @code{traverse_preorder()}
4718 @cindex @code{traverse_postorder()}
4720 Suppose that you need a function that returns a list of all indices appearing
4721 in an arbitrary expression. The indices can have any dimension, and for
4722 indices with variance you always want the covariant version returned.
4724 You can't use @code{get_free_indices()} because you also want to include
4725 dummy indices in the list, and you can't use @code{find()} as it needs
4726 specific index dimensions (and it would require two passes: one for indices
4727 with variance, one for plain ones).
4729 The obvious solution to this problem is a tree traversal with a type switch,
4730 such as the following:
4733 void gather_indices_helper(const ex & e, lst & l)
4735 if (is_a<varidx>(e)) @{
4736 const varidx & vi = ex_to<varidx>(e);
4737 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4738 @} else if (is_a<idx>(e)) @{
4741 size_t n = e.nops();
4742 for (size_t i = 0; i < n; ++i)
4743 gather_indices_helper(e.op(i), l);
4747 lst gather_indices(const ex & e)
4750 gather_indices_helper(e, l);
4757 This works fine but fans of object-oriented programming will feel
4758 uncomfortable with the type switch. One reason is that there is a possibility
4759 for subtle bugs regarding derived classes. If we had, for example, written
4762 if (is_a<idx>(e)) @{
4764 @} else if (is_a<varidx>(e)) @{
4768 in @code{gather_indices_helper}, the code wouldn't have worked because the
4769 first line "absorbs" all classes derived from @code{idx}, including
4770 @code{varidx}, so the special case for @code{varidx} would never have been
4773 Also, for a large number of classes, a type switch like the above can get
4774 unwieldy and inefficient (it's a linear search, after all).
4775 @code{gather_indices_helper} only checks for two classes, but if you had to
4776 write a function that required a different implementation for nearly
4777 every GiNaC class, the result would be very hard to maintain and extend.
4779 The cleanest approach to the problem would be to add a new virtual function
4780 to GiNaC's class hierarchy. In our example, there would be specializations
4781 for @code{idx} and @code{varidx} while the default implementation in
4782 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4783 impossible to add virtual member functions to existing classes without
4784 changing their source and recompiling everything. GiNaC comes with source,
4785 so you could actually do this, but for a small algorithm like the one
4786 presented this would be impractical.
4788 One solution to this dilemma is the @dfn{Visitor} design pattern,
4789 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4790 variation, described in detail in
4791 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4792 virtual functions to the class hierarchy to implement operations, GiNaC
4793 provides a single "bouncing" method @code{accept()} that takes an instance
4794 of a special @code{visitor} class and redirects execution to the one
4795 @code{visit()} virtual function of the visitor that matches the type of
4796 object that @code{accept()} was being invoked on.
4798 Visitors in GiNaC must derive from the global @code{visitor} class as well
4799 as from the class @code{T::visitor} of each class @code{T} they want to
4800 visit, and implement the member functions @code{void visit(const T &)} for
4806 void ex::accept(visitor & v) const;
4809 will then dispatch to the correct @code{visit()} member function of the
4810 specified visitor @code{v} for the type of GiNaC object at the root of the
4811 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4813 Here is an example of a visitor:
4817 : public visitor, // this is required
4818 public add::visitor, // visit add objects
4819 public numeric::visitor, // visit numeric objects
4820 public basic::visitor // visit basic objects
4822 void visit(const add & x)
4823 @{ cout << "called with an add object" << endl; @}
4825 void visit(const numeric & x)
4826 @{ cout << "called with a numeric object" << endl; @}
4828 void visit(const basic & x)
4829 @{ cout << "called with a basic object" << endl; @}
4833 which can be used as follows:
4844 // prints "called with a numeric object"
4846 // prints "called with an add object"
4848 // prints "called with a basic object"
4852 The @code{visit(const basic &)} method gets called for all objects that are
4853 not @code{numeric} or @code{add} and acts as an (optional) default.
4855 From a conceptual point of view, the @code{visit()} methods of the visitor
4856 behave like a newly added virtual function of the visited hierarchy.
4857 In addition, visitors can store state in member variables, and they can
4858 be extended by deriving a new visitor from an existing one, thus building
4859 hierarchies of visitors.
4861 We can now rewrite our index example from above with a visitor:
4864 class gather_indices_visitor
4865 : public visitor, public idx::visitor, public varidx::visitor
4869 void visit(const idx & i)
4874 void visit(const varidx & vi)
4876 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4880 const lst & get_result() // utility function
4889 What's missing is the tree traversal. We could implement it in
4890 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4893 void ex::traverse_preorder(visitor & v) const;
4894 void ex::traverse_postorder(visitor & v) const;
4895 void ex::traverse(visitor & v) const;
4898 @code{traverse_preorder()} visits a node @emph{before} visiting its
4899 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4900 visiting its subexpressions. @code{traverse()} is a synonym for
4901 @code{traverse_preorder()}.
4903 Here is a new implementation of @code{gather_indices()} that uses the visitor
4904 and @code{traverse()}:
4907 lst gather_indices(const ex & e)
4909 gather_indices_visitor v;
4911 return v.get_result();
4915 Alternatively, you could use pre- or postorder iterators for the tree
4919 lst gather_indices(const ex & e)
4921 gather_indices_visitor v;
4922 for (const_preorder_iterator i = e.preorder_begin();
4923 i != e.preorder_end(); ++i) @{
4926 return v.get_result();
4931 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4932 @c node-name, next, previous, up
4933 @section Polynomial arithmetic
4935 @subsection Expanding and collecting
4936 @cindex @code{expand()}
4937 @cindex @code{collect()}
4938 @cindex @code{collect_common_factors()}
4940 A polynomial in one or more variables has many equivalent
4941 representations. Some useful ones serve a specific purpose. Consider
4942 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4943 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4944 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4945 representations are the recursive ones where one collects for exponents
4946 in one of the three variable. Since the factors are themselves
4947 polynomials in the remaining two variables the procedure can be
4948 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4949 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4952 To bring an expression into expanded form, its method
4955 ex ex::expand(unsigned options = 0);
4958 may be called. In our example above, this corresponds to @math{4*x*y +
4959 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4960 GiNaC is not easy to guess you should be prepared to see different
4961 orderings of terms in such sums!
4963 Another useful representation of multivariate polynomials is as a
4964 univariate polynomial in one of the variables with the coefficients
4965 being polynomials in the remaining variables. The method
4966 @code{collect()} accomplishes this task:
4969 ex ex::collect(const ex & s, bool distributed = false);
4972 The first argument to @code{collect()} can also be a list of objects in which
4973 case the result is either a recursively collected polynomial, or a polynomial
4974 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4975 by the @code{distributed} flag.
4977 Note that the original polynomial needs to be in expanded form (for the
4978 variables concerned) in order for @code{collect()} to be able to find the
4979 coefficients properly.
4981 The following @command{ginsh} transcript shows an application of @code{collect()}
4982 together with @code{find()}:
4985 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4986 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)
4987 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4988 > collect(a,@{p,q@});
4989 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
4990 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4991 > collect(a,find(a,sin($1)));
4992 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4993 > collect(a,@{find(a,sin($1)),p,q@});
4994 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4995 > collect(a,@{find(a,sin($1)),d@});
4996 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4999 Polynomials can often be brought into a more compact form by collecting
5000 common factors from the terms of sums. This is accomplished by the function
5003 ex collect_common_factors(const ex & e);
5006 This function doesn't perform a full factorization but only looks for
5007 factors which are already explicitly present:
5010 > collect_common_factors(a*x+a*y);
5012 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5014 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5015 (c+a)*a*(x*y+y^2+x)*b
5018 @subsection Degree and coefficients
5019 @cindex @code{degree()}
5020 @cindex @code{ldegree()}
5021 @cindex @code{coeff()}
5023 The degree and low degree of a polynomial can be obtained using the two
5027 int ex::degree(const ex & s);
5028 int ex::ldegree(const ex & s);
5031 which also work reliably on non-expanded input polynomials (they even work
5032 on rational functions, returning the asymptotic degree). By definition, the
5033 degree of zero is zero. To extract a coefficient with a certain power from
5034 an expanded polynomial you use
5037 ex ex::coeff(const ex & s, int n);
5040 You can also obtain the leading and trailing coefficients with the methods
5043 ex ex::lcoeff(const ex & s);
5044 ex ex::tcoeff(const ex & s);
5047 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5050 An application is illustrated in the next example, where a multivariate
5051 polynomial is analyzed:
5055 symbol x("x"), y("y");
5056 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5057 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5058 ex Poly = PolyInp.expand();
5060 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5061 cout << "The x^" << i << "-coefficient is "
5062 << Poly.coeff(x,i) << endl;
5064 cout << "As polynomial in y: "
5065 << Poly.collect(y) << endl;
5069 When run, it returns an output in the following fashion:
5072 The x^0-coefficient is y^2+11*y
5073 The x^1-coefficient is 5*y^2-2*y
5074 The x^2-coefficient is -1
5075 The x^3-coefficient is 4*y
5076 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5079 As always, the exact output may vary between different versions of GiNaC
5080 or even from run to run since the internal canonical ordering is not
5081 within the user's sphere of influence.
5083 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5084 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5085 with non-polynomial expressions as they not only work with symbols but with
5086 constants, functions and indexed objects as well:
5090 symbol a("a"), b("b"), c("c"), x("x");
5091 idx i(symbol("i"), 3);
5093 ex e = pow(sin(x) - cos(x), 4);
5094 cout << e.degree(cos(x)) << endl;
5096 cout << e.expand().coeff(sin(x), 3) << endl;
5099 e = indexed(a+b, i) * indexed(b+c, i);
5100 e = e.expand(expand_options::expand_indexed);
5101 cout << e.collect(indexed(b, i)) << endl;
5102 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5107 @subsection Polynomial division
5108 @cindex polynomial division
5111 @cindex pseudo-remainder
5112 @cindex @code{quo()}
5113 @cindex @code{rem()}
5114 @cindex @code{prem()}
5115 @cindex @code{divide()}
5120 ex quo(const ex & a, const ex & b, const ex & x);
5121 ex rem(const ex & a, const ex & b, const ex & x);
5124 compute the quotient and remainder of univariate polynomials in the variable
5125 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5127 The additional function
5130 ex prem(const ex & a, const ex & b, const ex & x);
5133 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5134 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5136 Exact division of multivariate polynomials is performed by the function
5139 bool divide(const ex & a, const ex & b, ex & q);
5142 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5143 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5144 in which case the value of @code{q} is undefined.
5147 @subsection Unit, content and primitive part
5148 @cindex @code{unit()}
5149 @cindex @code{content()}
5150 @cindex @code{primpart()}
5151 @cindex @code{unitcontprim()}
5156 ex ex::unit(const ex & x);
5157 ex ex::content(const ex & x);
5158 ex ex::primpart(const ex & x);
5159 ex ex::primpart(const ex & x, const ex & c);
5162 return the unit part, content part, and primitive polynomial of a multivariate
5163 polynomial with respect to the variable @samp{x} (the unit part being the sign
5164 of the leading coefficient, the content part being the GCD of the coefficients,
5165 and the primitive polynomial being the input polynomial divided by the unit and
5166 content parts). The second variant of @code{primpart()} expects the previously
5167 calculated content part of the polynomial in @code{c}, which enables it to
5168 work faster in the case where the content part has already been computed. The
5169 product of unit, content, and primitive part is the original polynomial.
5171 Additionally, the method
5174 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5177 computes the unit, content, and primitive parts in one go, returning them
5178 in @code{u}, @code{c}, and @code{p}, respectively.
5181 @subsection GCD, LCM and resultant
5184 @cindex @code{gcd()}
5185 @cindex @code{lcm()}
5187 The functions for polynomial greatest common divisor and least common
5188 multiple have the synopsis
5191 ex gcd(const ex & a, const ex & b);
5192 ex lcm(const ex & a, const ex & b);
5195 The functions @code{gcd()} and @code{lcm()} accept two expressions
5196 @code{a} and @code{b} as arguments and return a new expression, their
5197 greatest common divisor or least common multiple, respectively. If the
5198 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5199 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5200 the coefficients must be rationals.
5203 #include <ginac/ginac.h>
5204 using namespace GiNaC;
5208 symbol x("x"), y("y"), z("z");
5209 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5210 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5212 ex P_gcd = gcd(P_a, P_b);
5214 ex P_lcm = lcm(P_a, P_b);
5215 // 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
5220 @cindex @code{resultant()}
5222 The resultant of two expressions only makes sense with polynomials.
5223 It is always computed with respect to a specific symbol within the
5224 expressions. The function has the interface
5227 ex resultant(const ex & a, const ex & b, const ex & s);
5230 Resultants are symmetric in @code{a} and @code{b}. The following example
5231 computes the resultant of two expressions with respect to @code{x} and
5232 @code{y}, respectively:
5235 #include <ginac/ginac.h>
5236 using namespace GiNaC;
5240 symbol x("x"), y("y");
5242 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5245 r = resultant(e1, e2, x);
5247 r = resultant(e1, e2, y);
5252 @subsection Square-free decomposition
5253 @cindex square-free decomposition
5254 @cindex factorization
5255 @cindex @code{sqrfree()}
5257 GiNaC still lacks proper factorization support. Some form of
5258 factorization is, however, easily implemented by noting that factors
5259 appearing in a polynomial with power two or more also appear in the
5260 derivative and hence can easily be found by computing the GCD of the
5261 original polynomial and its derivatives. Any decent system has an
5262 interface for this so called square-free factorization. So we provide
5265 ex sqrfree(const ex & a, const lst & l = lst());
5267 Here is an example that by the way illustrates how the exact form of the
5268 result may slightly depend on the order of differentiation, calling for
5269 some care with subsequent processing of the result:
5272 symbol x("x"), y("y");
5273 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5275 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5276 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5278 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5279 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5281 cout << sqrfree(BiVarPol) << endl;
5282 // -> depending on luck, any of the above
5285 Note also, how factors with the same exponents are not fully factorized
5289 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5290 @c node-name, next, previous, up
5291 @section Rational expressions
5293 @subsection The @code{normal} method
5294 @cindex @code{normal()}
5295 @cindex simplification
5296 @cindex temporary replacement
5298 Some basic form of simplification of expressions is called for frequently.
5299 GiNaC provides the method @code{.normal()}, which converts a rational function
5300 into an equivalent rational function of the form @samp{numerator/denominator}
5301 where numerator and denominator are coprime. If the input expression is already
5302 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5303 otherwise it performs fraction addition and multiplication.
5305 @code{.normal()} can also be used on expressions which are not rational functions
5306 as it will replace all non-rational objects (like functions or non-integer
5307 powers) by temporary symbols to bring the expression to the domain of rational
5308 functions before performing the normalization, and re-substituting these
5309 symbols afterwards. This algorithm is also available as a separate method
5310 @code{.to_rational()}, described below.
5312 This means that both expressions @code{t1} and @code{t2} are indeed
5313 simplified in this little code snippet:
5318 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5319 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5320 std::cout << "t1 is " << t1.normal() << std::endl;
5321 std::cout << "t2 is " << t2.normal() << std::endl;
5325 Of course this works for multivariate polynomials too, so the ratio of
5326 the sample-polynomials from the section about GCD and LCM above would be
5327 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5330 @subsection Numerator and denominator
5333 @cindex @code{numer()}
5334 @cindex @code{denom()}
5335 @cindex @code{numer_denom()}
5337 The numerator and denominator of an expression can be obtained with
5342 ex ex::numer_denom();
5345 These functions will first normalize the expression as described above and
5346 then return the numerator, denominator, or both as a list, respectively.
5347 If you need both numerator and denominator, calling @code{numer_denom()} is
5348 faster than using @code{numer()} and @code{denom()} separately.
5351 @subsection Converting to a polynomial or rational expression
5352 @cindex @code{to_polynomial()}
5353 @cindex @code{to_rational()}
5355 Some of the methods described so far only work on polynomials or rational
5356 functions. GiNaC provides a way to extend the domain of these functions to
5357 general expressions by using the temporary replacement algorithm described
5358 above. You do this by calling
5361 ex ex::to_polynomial(exmap & m);
5362 ex ex::to_polynomial(lst & l);
5366 ex ex::to_rational(exmap & m);
5367 ex ex::to_rational(lst & l);
5370 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5371 will be filled with the generated temporary symbols and their replacement
5372 expressions in a format that can be used directly for the @code{subs()}
5373 method. It can also already contain a list of replacements from an earlier
5374 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5375 possible to use it on multiple expressions and get consistent results.
5377 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5378 is probably best illustrated with an example:
5382 symbol x("x"), y("y");
5383 ex a = 2*x/sin(x) - y/(3*sin(x));
5387 ex p = a.to_polynomial(lp);
5388 cout << " = " << p << "\n with " << lp << endl;
5389 // = symbol3*symbol2*y+2*symbol2*x
5390 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5393 ex r = a.to_rational(lr);
5394 cout << " = " << r << "\n with " << lr << endl;
5395 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5396 // with @{symbol4==sin(x)@}
5400 The following more useful example will print @samp{sin(x)-cos(x)}:
5405 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5406 ex b = sin(x) + cos(x);
5409 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5410 cout << q.subs(m) << endl;
5415 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5416 @c node-name, next, previous, up
5417 @section Symbolic differentiation
5418 @cindex differentiation
5419 @cindex @code{diff()}
5421 @cindex product rule
5423 GiNaC's objects know how to differentiate themselves. Thus, a
5424 polynomial (class @code{add}) knows that its derivative is the sum of
5425 the derivatives of all the monomials:
5429 symbol x("x"), y("y"), z("z");
5430 ex P = pow(x, 5) + pow(x, 2) + y;
5432 cout << P.diff(x,2) << endl;
5434 cout << P.diff(y) << endl; // 1
5436 cout << P.diff(z) << endl; // 0
5441 If a second integer parameter @var{n} is given, the @code{diff} method
5442 returns the @var{n}th derivative.
5444 If @emph{every} object and every function is told what its derivative
5445 is, all derivatives of composed objects can be calculated using the
5446 chain rule and the product rule. Consider, for instance the expression
5447 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5448 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5449 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5450 out that the composition is the generating function for Euler Numbers,
5451 i.e. the so called @var{n}th Euler number is the coefficient of
5452 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5453 identity to code a function that generates Euler numbers in just three
5456 @cindex Euler numbers
5458 #include <ginac/ginac.h>
5459 using namespace GiNaC;
5461 ex EulerNumber(unsigned n)
5464 const ex generator = pow(cosh(x),-1);
5465 return generator.diff(x,n).subs(x==0);
5470 for (unsigned i=0; i<11; i+=2)
5471 std::cout << EulerNumber(i) << std::endl;
5476 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5477 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5478 @code{i} by two since all odd Euler numbers vanish anyways.
5481 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5482 @c node-name, next, previous, up
5483 @section Series expansion
5484 @cindex @code{series()}
5485 @cindex Taylor expansion
5486 @cindex Laurent expansion
5487 @cindex @code{pseries} (class)
5488 @cindex @code{Order()}
5490 Expressions know how to expand themselves as a Taylor series or (more
5491 generally) a Laurent series. As in most conventional Computer Algebra
5492 Systems, no distinction is made between those two. There is a class of
5493 its own for storing such series (@code{class pseries}) and a built-in
5494 function (called @code{Order}) for storing the order term of the series.
5495 As a consequence, if you want to work with series, i.e. multiply two
5496 series, you need to call the method @code{ex::series} again to convert
5497 it to a series object with the usual structure (expansion plus order
5498 term). A sample application from special relativity could read:
5501 #include <ginac/ginac.h>
5502 using namespace std;
5503 using namespace GiNaC;
5507 symbol v("v"), c("c");
5509 ex gamma = 1/sqrt(1 - pow(v/c,2));
5510 ex mass_nonrel = gamma.series(v==0, 10);
5512 cout << "the relativistic mass increase with v is " << endl
5513 << mass_nonrel << endl;
5515 cout << "the inverse square of this series is " << endl
5516 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5520 Only calling the series method makes the last output simplify to
5521 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5522 series raised to the power @math{-2}.
5524 @cindex Machin's formula
5525 As another instructive application, let us calculate the numerical
5526 value of Archimedes' constant
5530 (for which there already exists the built-in constant @code{Pi})
5531 using John Machin's amazing formula
5533 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5536 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5538 This equation (and similar ones) were used for over 200 years for
5539 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5540 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5541 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5542 order term with it and the question arises what the system is supposed
5543 to do when the fractions are plugged into that order term. The solution
5544 is to use the function @code{series_to_poly()} to simply strip the order
5548 #include <ginac/ginac.h>
5549 using namespace GiNaC;
5551 ex machin_pi(int degr)
5554 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5555 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5556 -4*pi_expansion.subs(x==numeric(1,239));
5562 using std::cout; // just for fun, another way of...
5563 using std::endl; // ...dealing with this namespace std.
5565 for (int i=2; i<12; i+=2) @{
5566 pi_frac = machin_pi(i);
5567 cout << i << ":\t" << pi_frac << endl
5568 << "\t" << pi_frac.evalf() << endl;
5574 Note how we just called @code{.series(x,degr)} instead of
5575 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5576 method @code{series()}: if the first argument is a symbol the expression
5577 is expanded in that symbol around point @code{0}. When you run this
5578 program, it will type out:
5582 3.1832635983263598326
5583 4: 5359397032/1706489875
5584 3.1405970293260603143
5585 6: 38279241713339684/12184551018734375
5586 3.141621029325034425
5587 8: 76528487109180192540976/24359780855939418203125
5588 3.141591772182177295
5589 10: 327853873402258685803048818236/104359128170408663038552734375
5590 3.1415926824043995174
5594 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5595 @c node-name, next, previous, up
5596 @section Symmetrization
5597 @cindex @code{symmetrize()}
5598 @cindex @code{antisymmetrize()}
5599 @cindex @code{symmetrize_cyclic()}
5604 ex ex::symmetrize(const lst & l);
5605 ex ex::antisymmetrize(const lst & l);
5606 ex ex::symmetrize_cyclic(const lst & l);
5609 symmetrize an expression by returning the sum over all symmetric,
5610 antisymmetric or cyclic permutations of the specified list of objects,
5611 weighted by the number of permutations.
5613 The three additional methods
5616 ex ex::symmetrize();
5617 ex ex::antisymmetrize();
5618 ex ex::symmetrize_cyclic();
5621 symmetrize or antisymmetrize an expression over its free indices.
5623 Symmetrization is most useful with indexed expressions but can be used with
5624 almost any kind of object (anything that is @code{subs()}able):
5628 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5629 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5631 cout << indexed(A, i, j).symmetrize() << endl;
5632 // -> 1/2*A.j.i+1/2*A.i.j
5633 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5634 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5635 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5636 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5640 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5641 @c node-name, next, previous, up
5642 @section Predefined mathematical functions
5644 @subsection Overview
5646 GiNaC contains the following predefined mathematical functions:
5649 @multitable @columnfractions .30 .70
5650 @item @strong{Name} @tab @strong{Function}
5653 @cindex @code{abs()}
5654 @item @code{csgn(x)}
5656 @cindex @code{conjugate()}
5657 @item @code{conjugate(x)}
5658 @tab complex conjugation
5659 @cindex @code{csgn()}
5660 @item @code{sqrt(x)}
5661 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5662 @cindex @code{sqrt()}
5665 @cindex @code{sin()}
5668 @cindex @code{cos()}
5671 @cindex @code{tan()}
5672 @item @code{asin(x)}
5674 @cindex @code{asin()}
5675 @item @code{acos(x)}
5677 @cindex @code{acos()}
5678 @item @code{atan(x)}
5679 @tab inverse tangent
5680 @cindex @code{atan()}
5681 @item @code{atan2(y, x)}
5682 @tab inverse tangent with two arguments
5683 @item @code{sinh(x)}
5684 @tab hyperbolic sine
5685 @cindex @code{sinh()}
5686 @item @code{cosh(x)}
5687 @tab hyperbolic cosine
5688 @cindex @code{cosh()}
5689 @item @code{tanh(x)}
5690 @tab hyperbolic tangent
5691 @cindex @code{tanh()}
5692 @item @code{asinh(x)}
5693 @tab inverse hyperbolic sine
5694 @cindex @code{asinh()}
5695 @item @code{acosh(x)}
5696 @tab inverse hyperbolic cosine
5697 @cindex @code{acosh()}
5698 @item @code{atanh(x)}
5699 @tab inverse hyperbolic tangent
5700 @cindex @code{atanh()}
5702 @tab exponential function
5703 @cindex @code{exp()}
5705 @tab natural logarithm
5706 @cindex @code{log()}
5709 @cindex @code{Li2()}
5710 @item @code{Li(m, x)}
5711 @tab classical polylogarithm as well as multiple polylogarithm
5713 @item @code{G(a, y)}
5714 @tab multiple polylogarithm
5716 @item @code{G(a, s, y)}
5717 @tab multiple polylogarithm with explicit signs for the imaginary parts
5719 @item @code{S(n, p, x)}
5720 @tab Nielsen's generalized polylogarithm
5722 @item @code{H(m, x)}
5723 @tab harmonic polylogarithm
5725 @item @code{zeta(m)}
5726 @tab Riemann's zeta function as well as multiple zeta value
5727 @cindex @code{zeta()}
5728 @item @code{zeta(m, s)}
5729 @tab alternating Euler sum
5730 @cindex @code{zeta()}
5731 @item @code{zetaderiv(n, x)}
5732 @tab derivatives of Riemann's zeta function
5733 @item @code{tgamma(x)}
5735 @cindex @code{tgamma()}
5736 @cindex gamma function
5737 @item @code{lgamma(x)}
5738 @tab logarithm of gamma function
5739 @cindex @code{lgamma()}
5740 @item @code{beta(x, y)}
5741 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5742 @cindex @code{beta()}
5744 @tab psi (digamma) function
5745 @cindex @code{psi()}
5746 @item @code{psi(n, x)}
5747 @tab derivatives of psi function (polygamma functions)
5748 @item @code{factorial(n)}
5749 @tab factorial function @math{n!}
5750 @cindex @code{factorial()}
5751 @item @code{binomial(n, k)}
5752 @tab binomial coefficients
5753 @cindex @code{binomial()}
5754 @item @code{Order(x)}
5755 @tab order term function in truncated power series
5756 @cindex @code{Order()}
5761 For functions that have a branch cut in the complex plane GiNaC follows
5762 the conventions for C++ as defined in the ANSI standard as far as
5763 possible. In particular: the natural logarithm (@code{log}) and the
5764 square root (@code{sqrt}) both have their branch cuts running along the
5765 negative real axis where the points on the axis itself belong to the
5766 upper part (i.e. continuous with quadrant II). The inverse
5767 trigonometric and hyperbolic functions are not defined for complex
5768 arguments by the C++ standard, however. In GiNaC we follow the
5769 conventions used by CLN, which in turn follow the carefully designed
5770 definitions in the Common Lisp standard. It should be noted that this
5771 convention is identical to the one used by the C99 standard and by most
5772 serious CAS. It is to be expected that future revisions of the C++
5773 standard incorporate these functions in the complex domain in a manner
5774 compatible with C99.
5776 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5777 @c node-name, next, previous, up
5778 @subsection Multiple polylogarithms
5780 @cindex polylogarithm
5781 @cindex Nielsen's generalized polylogarithm
5782 @cindex harmonic polylogarithm
5783 @cindex multiple zeta value
5784 @cindex alternating Euler sum
5785 @cindex multiple polylogarithm
5787 The multiple polylogarithm is the most generic member of a family of functions,
5788 to which others like the harmonic polylogarithm, Nielsen's generalized
5789 polylogarithm and the multiple zeta value belong.
5790 Everyone of these functions can also be written as a multiple polylogarithm with specific
5791 parameters. This whole family of functions is therefore often referred to simply as
5792 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5793 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5794 @code{Li} and @code{G} in principle represent the same function, the different
5795 notations are more natural to the series representation or the integral
5796 representation, respectively.
5798 To facilitate the discussion of these functions we distinguish between indices and
5799 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5800 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5802 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5803 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5804 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5805 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5806 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5807 @code{s} is not given, the signs default to +1.
5808 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5809 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5810 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5811 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5812 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5814 The functions print in LaTeX format as
5816 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5822 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5825 $\zeta(m_1,m_2,\ldots,m_k)$.
5827 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5828 are printed with a line above, e.g.
5830 $\zeta(5,\overline{2})$.
5832 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5834 Definitions and analytical as well as numerical properties of multiple polylogarithms
5835 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5836 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5837 except for a few differences which will be explicitly stated in the following.
5839 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5840 that the indices and arguments are understood to be in the same order as in which they appear in
5841 the series representation. This means
5843 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5846 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5849 $\zeta(1,2)$ evaluates to infinity.
5851 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5854 The functions only evaluate if the indices are integers greater than zero, except for the indices
5855 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5856 will be interpreted as the sequence of signs for the corresponding indices
5857 @code{m} or the sign of the imaginary part for the
5858 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5859 @code{zeta(lst(3,4), lst(-1,1))} means
5861 $\zeta(\overline{3},4)$
5864 @code{G(lst(a,b), lst(-1,1), c)} means
5866 $G(a-0\epsilon,b+0\epsilon;c)$.
5868 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5869 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5870 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
5871 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5872 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5873 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5874 evaluates also for negative integers and positive even integers. For example:
5877 > Li(@{3,1@},@{x,1@});
5880 -zeta(@{3,2@},@{-1,-1@})
5885 It is easy to tell for a given function into which other function it can be rewritten, may
5886 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5887 with negative indices or trailing zeros (the example above gives a hint). Signs can
5888 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5889 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5890 @code{Li} (@code{eval()} already cares for the possible downgrade):
5893 > convert_H_to_Li(@{0,-2,-1,3@},x);
5894 Li(@{3,1,3@},@{-x,1,-1@})
5895 > convert_H_to_Li(@{2,-1,0@},x);
5896 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5899 Every function can be numerically evaluated for
5900 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5901 global variable @code{Digits}:
5906 > evalf(zeta(@{3,1,3,1@}));
5907 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5910 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5911 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5913 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5918 In long expressions this helps a lot with debugging, because you can easily spot
5919 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5920 cancellations of divergencies happen.
5922 Useful publications:
5924 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5925 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5927 @cite{Harmonic Polylogarithms},
5928 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5930 @cite{Special Values of Multiple Polylogarithms},
5931 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5933 @cite{Numerical Evaluation of Multiple Polylogarithms},
5934 J.Vollinga, S.Weinzierl, hep-ph/0410259
5936 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5937 @c node-name, next, previous, up
5938 @section Complex Conjugation
5940 @cindex @code{conjugate()}
5948 returns the complex conjugate of the expression. For all built-in functions and objects the
5949 conjugation gives the expected results:
5953 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5957 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5958 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5959 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5960 // -> -gamma5*gamma~b*gamma~a
5964 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5965 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5966 arguments. This is the default strategy. If you want to define your own functions and want to
5967 change this behavior, you have to supply a specialized conjugation method for your function
5968 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5970 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5971 @c node-name, next, previous, up
5972 @section Solving Linear Systems of Equations
5973 @cindex @code{lsolve()}
5975 The function @code{lsolve()} provides a convenient wrapper around some
5976 matrix operations that comes in handy when a system of linear equations
5980 ex lsolve(const ex & eqns, const ex & symbols,
5981 unsigned options = solve_algo::automatic);
5984 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5985 @code{relational}) while @code{symbols} is a @code{lst} of
5986 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5989 It returns the @code{lst} of solutions as an expression. As an example,
5990 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5994 symbol a("a"), b("b"), x("x"), y("y");
5996 eqns = a*x+b*y==3, x-y==b;
5998 cout << lsolve(eqns, vars) << endl;
5999 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6002 When the linear equations @code{eqns} are underdetermined, the solution
6003 will contain one or more tautological entries like @code{x==x},
6004 depending on the rank of the system. When they are overdetermined, the
6005 solution will be an empty @code{lst}. Note the third optional parameter
6006 to @code{lsolve()}: it accepts the same parameters as
6007 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6011 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6012 @c node-name, next, previous, up
6013 @section Input and output of expressions
6016 @subsection Expression output
6018 @cindex output of expressions
6020 Expressions can simply be written to any stream:
6025 ex e = 4.5*I+pow(x,2)*3/2;
6026 cout << e << endl; // prints '4.5*I+3/2*x^2'
6030 The default output format is identical to the @command{ginsh} input syntax and
6031 to that used by most computer algebra systems, but not directly pastable
6032 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6033 is printed as @samp{x^2}).
6035 It is possible to print expressions in a number of different formats with
6036 a set of stream manipulators;
6039 std::ostream & dflt(std::ostream & os);
6040 std::ostream & latex(std::ostream & os);
6041 std::ostream & tree(std::ostream & os);
6042 std::ostream & csrc(std::ostream & os);
6043 std::ostream & csrc_float(std::ostream & os);
6044 std::ostream & csrc_double(std::ostream & os);
6045 std::ostream & csrc_cl_N(std::ostream & os);
6046 std::ostream & index_dimensions(std::ostream & os);
6047 std::ostream & no_index_dimensions(std::ostream & os);
6050 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6051 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6052 @code{print_csrc()} functions, respectively.
6055 All manipulators affect the stream state permanently. To reset the output
6056 format to the default, use the @code{dflt} manipulator:
6060 cout << latex; // all output to cout will be in LaTeX format from
6062 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6063 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6064 cout << dflt; // revert to default output format
6065 cout << e << endl; // prints '4.5*I+3/2*x^2'
6069 If you don't want to affect the format of the stream you're working with,
6070 you can output to a temporary @code{ostringstream} like this:
6075 s << latex << e; // format of cout remains unchanged
6076 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6081 @cindex @code{csrc_float}
6082 @cindex @code{csrc_double}
6083 @cindex @code{csrc_cl_N}
6084 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6085 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6086 format that can be directly used in a C or C++ program. The three possible
6087 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6088 classes provided by the CLN library):
6092 cout << "f = " << csrc_float << e << ";\n";
6093 cout << "d = " << csrc_double << e << ";\n";
6094 cout << "n = " << csrc_cl_N << e << ";\n";
6098 The above example will produce (note the @code{x^2} being converted to
6102 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6103 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6104 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6108 The @code{tree} manipulator allows dumping the internal structure of an
6109 expression for debugging purposes:
6120 add, hash=0x0, flags=0x3, nops=2
6121 power, hash=0x0, flags=0x3, nops=2
6122 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6123 2 (numeric), hash=0x6526b0fa, flags=0xf
6124 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6127 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6131 @cindex @code{latex}
6132 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6133 It is rather similar to the default format but provides some braces needed
6134 by LaTeX for delimiting boxes and also converts some common objects to
6135 conventional LaTeX names. It is possible to give symbols a special name for
6136 LaTeX output by supplying it as a second argument to the @code{symbol}
6139 For example, the code snippet
6143 symbol x("x", "\\circ");
6144 ex e = lgamma(x).series(x==0,3);
6145 cout << latex << e << endl;
6152 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6153 +\mathcal@{O@}(\circ^@{3@})
6156 @cindex @code{index_dimensions}
6157 @cindex @code{no_index_dimensions}
6158 Index dimensions are normally hidden in the output. To make them visible, use
6159 the @code{index_dimensions} manipulator. The dimensions will be written in
6160 square brackets behind each index value in the default and LaTeX output
6165 symbol x("x"), y("y");
6166 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6167 ex e = indexed(x, mu) * indexed(y, nu);
6170 // prints 'x~mu*y~nu'
6171 cout << index_dimensions << e << endl;
6172 // prints 'x~mu[4]*y~nu[4]'
6173 cout << no_index_dimensions << e << endl;
6174 // prints 'x~mu*y~nu'
6179 @cindex Tree traversal
6180 If you need any fancy special output format, e.g. for interfacing GiNaC
6181 with other algebra systems or for producing code for different
6182 programming languages, you can always traverse the expression tree yourself:
6185 static void my_print(const ex & e)
6187 if (is_a<function>(e))
6188 cout << ex_to<function>(e).get_name();
6190 cout << ex_to<basic>(e).class_name();
6192 size_t n = e.nops();
6194 for (size_t i=0; i<n; i++) @{
6206 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6214 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6215 symbol(y))),numeric(-2)))
6218 If you need an output format that makes it possible to accurately
6219 reconstruct an expression by feeding the output to a suitable parser or
6220 object factory, you should consider storing the expression in an
6221 @code{archive} object and reading the object properties from there.
6222 See the section on archiving for more information.
6225 @subsection Expression input
6226 @cindex input of expressions
6228 GiNaC provides no way to directly read an expression from a stream because
6229 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6230 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6231 @code{y} you defined in your program and there is no way to specify the
6232 desired symbols to the @code{>>} stream input operator.
6234 Instead, GiNaC lets you construct an expression from a string, specifying the
6235 list of symbols to be used:
6239 symbol x("x"), y("y");
6240 ex e("2*x+sin(y)", lst(x, y));
6244 The input syntax is the same as that used by @command{ginsh} and the stream
6245 output operator @code{<<}. The symbols in the string are matched by name to
6246 the symbols in the list and if GiNaC encounters a symbol not specified in
6247 the list it will throw an exception.
6249 With this constructor, it's also easy to implement interactive GiNaC programs:
6254 #include <stdexcept>
6255 #include <ginac/ginac.h>
6256 using namespace std;
6257 using namespace GiNaC;
6264 cout << "Enter an expression containing 'x': ";
6269 cout << "The derivative of " << e << " with respect to x is ";
6270 cout << e.diff(x) << ".\n";
6271 @} catch (exception &p) @{
6272 cerr << p.what() << endl;
6278 @subsection Archiving
6279 @cindex @code{archive} (class)
6282 GiNaC allows creating @dfn{archives} of expressions which can be stored
6283 to or retrieved from files. To create an archive, you declare an object
6284 of class @code{archive} and archive expressions in it, giving each
6285 expression a unique name:
6289 using namespace std;
6290 #include <ginac/ginac.h>
6291 using namespace GiNaC;
6295 symbol x("x"), y("y"), z("z");
6297 ex foo = sin(x + 2*y) + 3*z + 41;
6301 a.archive_ex(foo, "foo");
6302 a.archive_ex(bar, "the second one");
6306 The archive can then be written to a file:
6310 ofstream out("foobar.gar");
6316 The file @file{foobar.gar} contains all information that is needed to
6317 reconstruct the expressions @code{foo} and @code{bar}.
6319 @cindex @command{viewgar}
6320 The tool @command{viewgar} that comes with GiNaC can be used to view
6321 the contents of GiNaC archive files:
6324 $ viewgar foobar.gar
6325 foo = 41+sin(x+2*y)+3*z
6326 the second one = 42+sin(x+2*y)+3*z
6329 The point of writing archive files is of course that they can later be
6335 ifstream in("foobar.gar");
6340 And the stored expressions can be retrieved by their name:
6347 ex ex1 = a2.unarchive_ex(syms, "foo");
6348 ex ex2 = a2.unarchive_ex(syms, "the second one");
6350 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6351 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6352 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6356 Note that you have to supply a list of the symbols which are to be inserted
6357 in the expressions. Symbols in archives are stored by their name only and
6358 if you don't specify which symbols you have, unarchiving the expression will
6359 create new symbols with that name. E.g. if you hadn't included @code{x} in
6360 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6361 have had no effect because the @code{x} in @code{ex1} would have been a
6362 different symbol than the @code{x} which was defined at the beginning of
6363 the program, although both would appear as @samp{x} when printed.
6365 You can also use the information stored in an @code{archive} object to
6366 output expressions in a format suitable for exact reconstruction. The
6367 @code{archive} and @code{archive_node} classes have a couple of member
6368 functions that let you access the stored properties:
6371 static void my_print2(const archive_node & n)
6374 n.find_string("class", class_name);
6375 cout << class_name << "(";
6377 archive_node::propinfovector p;
6378 n.get_properties(p);
6380 size_t num = p.size();
6381 for (size_t i=0; i<num; i++) @{
6382 const string &name = p[i].name;
6383 if (name == "class")
6385 cout << name << "=";
6387 unsigned count = p[i].count;
6391 for (unsigned j=0; j<count; j++) @{
6392 switch (p[i].type) @{
6393 case archive_node::PTYPE_BOOL: @{
6395 n.find_bool(name, x, j);
6396 cout << (x ? "true" : "false");
6399 case archive_node::PTYPE_UNSIGNED: @{
6401 n.find_unsigned(name, x, j);
6405 case archive_node::PTYPE_STRING: @{
6407 n.find_string(name, x, j);
6408 cout << '\"' << x << '\"';
6411 case archive_node::PTYPE_NODE: @{
6412 const archive_node &x = n.find_ex_node(name, j);
6434 ex e = pow(2, x) - y;
6436 my_print2(ar.get_top_node(0)); cout << endl;
6444 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6445 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6446 overall_coeff=numeric(number="0"))
6449 Be warned, however, that the set of properties and their meaning for each
6450 class may change between GiNaC versions.
6453 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6454 @c node-name, next, previous, up
6455 @chapter Extending GiNaC
6457 By reading so far you should have gotten a fairly good understanding of
6458 GiNaC's design patterns. From here on you should start reading the
6459 sources. All we can do now is issue some recommendations how to tackle
6460 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6461 develop some useful extension please don't hesitate to contact the GiNaC
6462 authors---they will happily incorporate them into future versions.
6465 * What does not belong into GiNaC:: What to avoid.
6466 * Symbolic functions:: Implementing symbolic functions.
6467 * Printing:: Adding new output formats.
6468 * Structures:: Defining new algebraic classes (the easy way).
6469 * Adding classes:: Defining new algebraic classes (the hard way).
6473 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6474 @c node-name, next, previous, up
6475 @section What doesn't belong into GiNaC
6477 @cindex @command{ginsh}
6478 First of all, GiNaC's name must be read literally. It is designed to be
6479 a library for use within C++. The tiny @command{ginsh} accompanying
6480 GiNaC makes this even more clear: it doesn't even attempt to provide a
6481 language. There are no loops or conditional expressions in
6482 @command{ginsh}, it is merely a window into the library for the
6483 programmer to test stuff (or to show off). Still, the design of a
6484 complete CAS with a language of its own, graphical capabilities and all
6485 this on top of GiNaC is possible and is without doubt a nice project for
6488 There are many built-in functions in GiNaC that do not know how to
6489 evaluate themselves numerically to a precision declared at runtime
6490 (using @code{Digits}). Some may be evaluated at certain points, but not
6491 generally. This ought to be fixed. However, doing numerical
6492 computations with GiNaC's quite abstract classes is doomed to be
6493 inefficient. For this purpose, the underlying foundation classes
6494 provided by CLN are much better suited.
6497 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6498 @c node-name, next, previous, up
6499 @section Symbolic functions
6501 The easiest and most instructive way to start extending GiNaC is probably to
6502 create your own symbolic functions. These are implemented with the help of
6503 two preprocessor macros:
6505 @cindex @code{DECLARE_FUNCTION}
6506 @cindex @code{REGISTER_FUNCTION}
6508 DECLARE_FUNCTION_<n>P(<name>)
6509 REGISTER_FUNCTION(<name>, <options>)
6512 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6513 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6514 parameters of type @code{ex} and returns a newly constructed GiNaC
6515 @code{function} object that represents your function.
6517 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6518 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6519 set of options that associate the symbolic function with C++ functions you
6520 provide to implement the various methods such as evaluation, derivative,
6521 series expansion etc. They also describe additional attributes the function
6522 might have, such as symmetry and commutation properties, and a name for
6523 LaTeX output. Multiple options are separated by the member access operator
6524 @samp{.} and can be given in an arbitrary order.
6526 (By the way: in case you are worrying about all the macros above we can
6527 assure you that functions are GiNaC's most macro-intense classes. We have
6528 done our best to avoid macros where we can.)
6530 @subsection A minimal example
6532 Here is an example for the implementation of a function with two arguments
6533 that is not further evaluated:
6536 DECLARE_FUNCTION_2P(myfcn)
6538 REGISTER_FUNCTION(myfcn, dummy())
6541 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6542 in algebraic expressions:
6548 ex e = 2*myfcn(42, 1+3*x) - x;
6550 // prints '2*myfcn(42,1+3*x)-x'
6555 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6556 "no options". A function with no options specified merely acts as a kind of
6557 container for its arguments. It is a pure "dummy" function with no associated
6558 logic (which is, however, sometimes perfectly sufficient).
6560 Let's now have a look at the implementation of GiNaC's cosine function for an
6561 example of how to make an "intelligent" function.
6563 @subsection The cosine function
6565 The GiNaC header file @file{inifcns.h} contains the line
6568 DECLARE_FUNCTION_1P(cos)
6571 which declares to all programs using GiNaC that there is a function @samp{cos}
6572 that takes one @code{ex} as an argument. This is all they need to know to use
6573 this function in expressions.
6575 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6576 is its @code{REGISTER_FUNCTION} line:
6579 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6580 evalf_func(cos_evalf).
6581 derivative_func(cos_deriv).
6582 latex_name("\\cos"));
6585 There are four options defined for the cosine function. One of them
6586 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6587 other three indicate the C++ functions in which the "brains" of the cosine
6588 function are defined.
6590 @cindex @code{hold()}
6592 The @code{eval_func()} option specifies the C++ function that implements
6593 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6594 the same number of arguments as the associated symbolic function (one in this
6595 case) and returns the (possibly transformed or in some way simplified)
6596 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6597 of the automatic evaluation process). If no (further) evaluation is to take
6598 place, the @code{eval_func()} function must return the original function
6599 with @code{.hold()}, to avoid a potential infinite recursion. If your
6600 symbolic functions produce a segmentation fault or stack overflow when
6601 using them in expressions, you are probably missing a @code{.hold()}
6604 The @code{eval_func()} function for the cosine looks something like this
6605 (actually, it doesn't look like this at all, but it should give you an idea
6609 static ex cos_eval(const ex & x)
6611 if ("x is a multiple of 2*Pi")
6613 else if ("x is a multiple of Pi")
6615 else if ("x is a multiple of Pi/2")
6619 else if ("x has the form 'acos(y)'")
6621 else if ("x has the form 'asin(y)'")
6626 return cos(x).hold();
6630 This function is called every time the cosine is used in a symbolic expression:
6636 // this calls cos_eval(Pi), and inserts its return value into
6637 // the actual expression
6644 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6645 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6646 symbolic transformation can be done, the unmodified function is returned
6647 with @code{.hold()}.
6649 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6650 The user has to call @code{evalf()} for that. This is implemented in a
6654 static ex cos_evalf(const ex & x)
6656 if (is_a<numeric>(x))
6657 return cos(ex_to<numeric>(x));
6659 return cos(x).hold();
6663 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6664 in this case the @code{cos()} function for @code{numeric} objects, which in
6665 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6666 isn't really needed here, but reminds us that the corresponding @code{eval()}
6667 function would require it in this place.
6669 Differentiation will surely turn up and so we need to tell @code{cos}
6670 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6671 instance, are then handled automatically by @code{basic::diff} and
6675 static ex cos_deriv(const ex & x, unsigned diff_param)
6681 @cindex product rule
6682 The second parameter is obligatory but uninteresting at this point. It
6683 specifies which parameter to differentiate in a partial derivative in
6684 case the function has more than one parameter, and its main application
6685 is for correct handling of the chain rule.
6687 An implementation of the series expansion is not needed for @code{cos()} as
6688 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6689 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6690 the other hand, does have poles and may need to do Laurent expansion:
6693 static ex tan_series(const ex & x, const relational & rel,
6694 int order, unsigned options)
6696 // Find the actual expansion point
6697 const ex x_pt = x.subs(rel);
6699 if ("x_pt is not an odd multiple of Pi/2")
6700 throw do_taylor(); // tell function::series() to do Taylor expansion
6702 // On a pole, expand sin()/cos()
6703 return (sin(x)/cos(x)).series(rel, order+2, options);
6707 The @code{series()} implementation of a function @emph{must} return a
6708 @code{pseries} object, otherwise your code will crash.
6710 @subsection Function options
6712 GiNaC functions understand several more options which are always
6713 specified as @code{.option(params)}. None of them are required, but you
6714 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6715 is a do-nothing option called @code{dummy()} which you can use to define
6716 functions without any special options.
6719 eval_func(<C++ function>)
6720 evalf_func(<C++ function>)
6721 derivative_func(<C++ function>)
6722 series_func(<C++ function>)
6723 conjugate_func(<C++ function>)
6726 These specify the C++ functions that implement symbolic evaluation,
6727 numeric evaluation, partial derivatives, and series expansion, respectively.
6728 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6729 @code{diff()} and @code{series()}.
6731 The @code{eval_func()} function needs to use @code{.hold()} if no further
6732 automatic evaluation is desired or possible.
6734 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6735 expansion, which is correct if there are no poles involved. If the function
6736 has poles in the complex plane, the @code{series_func()} needs to check
6737 whether the expansion point is on a pole and fall back to Taylor expansion
6738 if it isn't. Otherwise, the pole usually needs to be regularized by some
6739 suitable transformation.
6742 latex_name(const string & n)
6745 specifies the LaTeX code that represents the name of the function in LaTeX
6746 output. The default is to put the function name in an @code{\mbox@{@}}.
6749 do_not_evalf_params()
6752 This tells @code{evalf()} to not recursively evaluate the parameters of the
6753 function before calling the @code{evalf_func()}.
6756 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6759 This allows you to explicitly specify the commutation properties of the
6760 function (@xref{Non-commutative objects}, for an explanation of
6761 (non)commutativity in GiNaC). For example, you can use
6762 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6763 GiNaC treat your function like a matrix. By default, functions inherit the
6764 commutation properties of their first argument.
6767 set_symmetry(const symmetry & s)
6770 specifies the symmetry properties of the function with respect to its
6771 arguments. @xref{Indexed objects}, for an explanation of symmetry
6772 specifications. GiNaC will automatically rearrange the arguments of
6773 symmetric functions into a canonical order.
6775 Sometimes you may want to have finer control over how functions are
6776 displayed in the output. For example, the @code{abs()} function prints
6777 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6778 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6782 print_func<C>(<C++ function>)
6785 option which is explained in the next section.
6787 @subsection Functions with a variable number of arguments
6789 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6790 functions with a fixed number of arguments. Sometimes, though, you may need
6791 to have a function that accepts a variable number of expressions. One way to
6792 accomplish this is to pass variable-length lists as arguments. The
6793 @code{Li()} function uses this method for multiple polylogarithms.
6795 It is also possible to define functions that accept a different number of
6796 parameters under the same function name, such as the @code{psi()} function
6797 which can be called either as @code{psi(z)} (the digamma function) or as
6798 @code{psi(n, z)} (polygamma functions). These are actually two different
6799 functions in GiNaC that, however, have the same name. Defining such
6800 functions is not possible with the macros but requires manually fiddling
6801 with GiNaC internals. If you are interested, please consult the GiNaC source
6802 code for the @code{psi()} function (@file{inifcns.h} and
6803 @file{inifcns_gamma.cpp}).
6806 @node Printing, Structures, Symbolic functions, Extending GiNaC
6807 @c node-name, next, previous, up
6808 @section GiNaC's expression output system
6810 GiNaC allows the output of expressions in a variety of different formats
6811 (@pxref{Input/Output}). This section will explain how expression output
6812 is implemented internally, and how to define your own output formats or
6813 change the output format of built-in algebraic objects. You will also want
6814 to read this section if you plan to write your own algebraic classes or
6817 @cindex @code{print_context} (class)
6818 @cindex @code{print_dflt} (class)
6819 @cindex @code{print_latex} (class)
6820 @cindex @code{print_tree} (class)
6821 @cindex @code{print_csrc} (class)
6822 All the different output formats are represented by a hierarchy of classes
6823 rooted in the @code{print_context} class, defined in the @file{print.h}
6828 the default output format
6830 output in LaTeX mathematical mode
6832 a dump of the internal expression structure (for debugging)
6834 the base class for C source output
6835 @item print_csrc_float
6836 C source output using the @code{float} type
6837 @item print_csrc_double
6838 C source output using the @code{double} type
6839 @item print_csrc_cl_N
6840 C source output using CLN types
6843 The @code{print_context} base class provides two public data members:
6855 @code{s} is a reference to the stream to output to, while @code{options}
6856 holds flags and modifiers. Currently, there is only one flag defined:
6857 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6858 to print the index dimension which is normally hidden.
6860 When you write something like @code{std::cout << e}, where @code{e} is
6861 an object of class @code{ex}, GiNaC will construct an appropriate
6862 @code{print_context} object (of a class depending on the selected output
6863 format), fill in the @code{s} and @code{options} members, and call
6865 @cindex @code{print()}
6867 void ex::print(const print_context & c, unsigned level = 0) const;
6870 which in turn forwards the call to the @code{print()} method of the
6871 top-level algebraic object contained in the expression.
6873 Unlike other methods, GiNaC classes don't usually override their
6874 @code{print()} method to implement expression output. Instead, the default
6875 implementation @code{basic::print(c, level)} performs a run-time double
6876 dispatch to a function selected by the dynamic type of the object and the
6877 passed @code{print_context}. To this end, GiNaC maintains a separate method
6878 table for each class, similar to the virtual function table used for ordinary
6879 (single) virtual function dispatch.
6881 The method table contains one slot for each possible @code{print_context}
6882 type, indexed by the (internally assigned) serial number of the type. Slots
6883 may be empty, in which case GiNaC will retry the method lookup with the
6884 @code{print_context} object's parent class, possibly repeating the process
6885 until it reaches the @code{print_context} base class. If there's still no
6886 method defined, the method table of the algebraic object's parent class
6887 is consulted, and so on, until a matching method is found (eventually it
6888 will reach the combination @code{basic/print_context}, which prints the
6889 object's class name enclosed in square brackets).
6891 You can think of the print methods of all the different classes and output
6892 formats as being arranged in a two-dimensional matrix with one axis listing
6893 the algebraic classes and the other axis listing the @code{print_context}
6896 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6897 to implement printing, but then they won't get any of the benefits of the
6898 double dispatch mechanism (such as the ability for derived classes to
6899 inherit only certain print methods from its parent, or the replacement of
6900 methods at run-time).
6902 @subsection Print methods for classes
6904 The method table for a class is set up either in the definition of the class,
6905 by passing the appropriate @code{print_func<C>()} option to
6906 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6907 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6908 can also be used to override existing methods dynamically.
6910 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6911 be a member function of the class (or one of its parent classes), a static
6912 member function, or an ordinary (global) C++ function. The @code{C} template
6913 parameter specifies the appropriate @code{print_context} type for which the
6914 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6915 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6916 the class is the one being implemented by
6917 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6919 For print methods that are member functions, their first argument must be of
6920 a type convertible to a @code{const C &}, and the second argument must be an
6923 For static members and global functions, the first argument must be of a type
6924 convertible to a @code{const T &}, the second argument must be of a type
6925 convertible to a @code{const C &}, and the third argument must be an
6926 @code{unsigned}. A global function will, of course, not have access to
6927 private and protected members of @code{T}.
6929 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6930 and @code{basic::print()}) is used for proper parenthesizing of the output
6931 (and by @code{print_tree} for proper indentation). It can be used for similar
6932 purposes if you write your own output formats.
6934 The explanations given above may seem complicated, but in practice it's
6935 really simple, as shown in the following example. Suppose that we want to
6936 display exponents in LaTeX output not as superscripts but with little
6937 upwards-pointing arrows. This can be achieved in the following way:
6940 void my_print_power_as_latex(const power & p,
6941 const print_latex & c,
6944 // get the precedence of the 'power' class
6945 unsigned power_prec = p.precedence();
6947 // if the parent operator has the same or a higher precedence
6948 // we need parentheses around the power
6949 if (level >= power_prec)
6952 // print the basis and exponent, each enclosed in braces, and
6953 // separated by an uparrow
6955 p.op(0).print(c, power_prec);
6956 c.s << "@}\\uparrow@{";
6957 p.op(1).print(c, power_prec);
6960 // don't forget the closing parenthesis
6961 if (level >= power_prec)
6967 // a sample expression
6968 symbol x("x"), y("y");
6969 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6971 // switch to LaTeX mode
6974 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6977 // now we replace the method for the LaTeX output of powers with
6979 set_print_func<power, print_latex>(my_print_power_as_latex);
6981 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
6992 The first argument of @code{my_print_power_as_latex} could also have been
6993 a @code{const basic &}, the second one a @code{const print_context &}.
6996 The above code depends on @code{mul} objects converting their operands to
6997 @code{power} objects for the purpose of printing.
7000 The output of products including negative powers as fractions is also
7001 controlled by the @code{mul} class.
7004 The @code{power/print_latex} method provided by GiNaC prints square roots
7005 using @code{\sqrt}, but the above code doesn't.
7009 It's not possible to restore a method table entry to its previous or default
7010 value. Once you have called @code{set_print_func()}, you can only override
7011 it with another call to @code{set_print_func()}, but you can't easily go back
7012 to the default behavior again (you can, of course, dig around in the GiNaC
7013 sources, find the method that is installed at startup
7014 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7015 one; that is, after you circumvent the C++ member access control@dots{}).
7017 @subsection Print methods for functions
7019 Symbolic functions employ a print method dispatch mechanism similar to the
7020 one used for classes. The methods are specified with @code{print_func<C>()}
7021 function options. If you don't specify any special print methods, the function
7022 will be printed with its name (or LaTeX name, if supplied), followed by a
7023 comma-separated list of arguments enclosed in parentheses.
7025 For example, this is what GiNaC's @samp{abs()} function is defined like:
7028 static ex abs_eval(const ex & arg) @{ ... @}
7029 static ex abs_evalf(const ex & arg) @{ ... @}
7031 static void abs_print_latex(const ex & arg, const print_context & c)
7033 c.s << "@{|"; arg.print(c); c.s << "|@}";
7036 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7038 c.s << "fabs("; arg.print(c); c.s << ")";
7041 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7042 evalf_func(abs_evalf).
7043 print_func<print_latex>(abs_print_latex).
7044 print_func<print_csrc_float>(abs_print_csrc_float).
7045 print_func<print_csrc_double>(abs_print_csrc_float));
7048 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7049 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7051 There is currently no equivalent of @code{set_print_func()} for functions.
7053 @subsection Adding new output formats
7055 Creating a new output format involves subclassing @code{print_context},
7056 which is somewhat similar to adding a new algebraic class
7057 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7058 that needs to go into the class definition, and a corresponding macro
7059 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7060 Every @code{print_context} class needs to provide a default constructor
7061 and a constructor from an @code{std::ostream} and an @code{unsigned}
7064 Here is an example for a user-defined @code{print_context} class:
7067 class print_myformat : public print_dflt
7069 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7071 print_myformat(std::ostream & os, unsigned opt = 0)
7072 : print_dflt(os, opt) @{@}
7075 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7077 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7080 That's all there is to it. None of the actual expression output logic is
7081 implemented in this class. It merely serves as a selector for choosing
7082 a particular format. The algorithms for printing expressions in the new
7083 format are implemented as print methods, as described above.
7085 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7086 exactly like GiNaC's default output format:
7091 ex e = pow(x, 2) + 1;
7093 // this prints "1+x^2"
7096 // this also prints "1+x^2"
7097 e.print(print_myformat()); cout << endl;
7103 To fill @code{print_myformat} with life, we need to supply appropriate
7104 print methods with @code{set_print_func()}, like this:
7107 // This prints powers with '**' instead of '^'. See the LaTeX output
7108 // example above for explanations.
7109 void print_power_as_myformat(const power & p,
7110 const print_myformat & c,
7113 unsigned power_prec = p.precedence();
7114 if (level >= power_prec)
7116 p.op(0).print(c, power_prec);
7118 p.op(1).print(c, power_prec);
7119 if (level >= power_prec)
7125 // install a new print method for power objects
7126 set_print_func<power, print_myformat>(print_power_as_myformat);
7128 // now this prints "1+x**2"
7129 e.print(print_myformat()); cout << endl;
7131 // but the default format is still "1+x^2"
7137 @node Structures, Adding classes, Printing, Extending GiNaC
7138 @c node-name, next, previous, up
7141 If you are doing some very specialized things with GiNaC, or if you just
7142 need some more organized way to store data in your expressions instead of
7143 anonymous lists, you may want to implement your own algebraic classes.
7144 ('algebraic class' means any class directly or indirectly derived from
7145 @code{basic} that can be used in GiNaC expressions).
7147 GiNaC offers two ways of accomplishing this: either by using the
7148 @code{structure<T>} template class, or by rolling your own class from
7149 scratch. This section will discuss the @code{structure<T>} template which
7150 is easier to use but more limited, while the implementation of custom
7151 GiNaC classes is the topic of the next section. However, you may want to
7152 read both sections because many common concepts and member functions are
7153 shared by both concepts, and it will also allow you to decide which approach
7154 is most suited to your needs.
7156 The @code{structure<T>} template, defined in the GiNaC header file
7157 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7158 or @code{class}) into a GiNaC object that can be used in expressions.
7160 @subsection Example: scalar products
7162 Let's suppose that we need a way to handle some kind of abstract scalar
7163 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7164 product class have to store their left and right operands, which can in turn
7165 be arbitrary expressions. Here is a possible way to represent such a
7166 product in a C++ @code{struct}:
7170 using namespace std;
7172 #include <ginac/ginac.h>
7173 using namespace GiNaC;
7179 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7183 The default constructor is required. Now, to make a GiNaC class out of this
7184 data structure, we need only one line:
7187 typedef structure<sprod_s> sprod;
7190 That's it. This line constructs an algebraic class @code{sprod} which
7191 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7192 expressions like any other GiNaC class:
7196 symbol a("a"), b("b");
7197 ex e = sprod(sprod_s(a, b));
7201 Note the difference between @code{sprod} which is the algebraic class, and
7202 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7203 and @code{right} data members. As shown above, an @code{sprod} can be
7204 constructed from an @code{sprod_s} object.
7206 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7207 you could define a little wrapper function like this:
7210 inline ex make_sprod(ex left, ex right)
7212 return sprod(sprod_s(left, right));
7216 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7217 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7218 @code{get_struct()}:
7222 cout << ex_to<sprod>(e)->left << endl;
7224 cout << ex_to<sprod>(e).get_struct().right << endl;
7229 You only have read access to the members of @code{sprod_s}.
7231 The type definition of @code{sprod} is enough to write your own algorithms
7232 that deal with scalar products, for example:
7237 if (is_a<sprod>(p)) @{
7238 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7239 return make_sprod(sp.right, sp.left);
7250 @subsection Structure output
7252 While the @code{sprod} type is useable it still leaves something to be
7253 desired, most notably proper output:
7258 // -> [structure object]
7262 By default, any structure types you define will be printed as
7263 @samp{[structure object]}. To override this you can either specialize the
7264 template's @code{print()} member function, or specify print methods with
7265 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7266 it's not possible to supply class options like @code{print_func<>()} to
7267 structures, so for a self-contained structure type you need to resort to
7268 overriding the @code{print()} function, which is also what we will do here.
7270 The member functions of GiNaC classes are described in more detail in the
7271 next section, but it shouldn't be hard to figure out what's going on here:
7274 void sprod::print(const print_context & c, unsigned level) const
7276 // tree debug output handled by superclass
7277 if (is_a<print_tree>(c))
7278 inherited::print(c, level);
7280 // get the contained sprod_s object
7281 const sprod_s & sp = get_struct();
7283 // print_context::s is a reference to an ostream
7284 c.s << "<" << sp.left << "|" << sp.right << ">";
7288 Now we can print expressions containing scalar products:
7294 cout << swap_sprod(e) << endl;
7299 @subsection Comparing structures
7301 The @code{sprod} class defined so far still has one important drawback: all
7302 scalar products are treated as being equal because GiNaC doesn't know how to
7303 compare objects of type @code{sprod_s}. This can lead to some confusing
7304 and undesired behavior:
7308 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7310 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7311 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7315 To remedy this, we first need to define the operators @code{==} and @code{<}
7316 for objects of type @code{sprod_s}:
7319 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7321 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7324 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7326 return lhs.left.compare(rhs.left) < 0
7327 ? true : lhs.right.compare(rhs.right) < 0;
7331 The ordering established by the @code{<} operator doesn't have to make any
7332 algebraic sense, but it needs to be well defined. Note that we can't use
7333 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7334 in the implementation of these operators because they would construct
7335 GiNaC @code{relational} objects which in the case of @code{<} do not
7336 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7337 decide which one is algebraically 'less').
7339 Next, we need to change our definition of the @code{sprod} type to let
7340 GiNaC know that an ordering relation exists for the embedded objects:
7343 typedef structure<sprod_s, compare_std_less> sprod;
7346 @code{sprod} objects then behave as expected:
7350 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7351 // -> <a|b>-<a^2|b^2>
7352 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7353 // -> <a|b>+<a^2|b^2>
7354 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7356 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7361 The @code{compare_std_less} policy parameter tells GiNaC to use the
7362 @code{std::less} and @code{std::equal_to} functors to compare objects of
7363 type @code{sprod_s}. By default, these functors forward their work to the
7364 standard @code{<} and @code{==} operators, which we have overloaded.
7365 Alternatively, we could have specialized @code{std::less} and
7366 @code{std::equal_to} for class @code{sprod_s}.
7368 GiNaC provides two other comparison policies for @code{structure<T>}
7369 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7370 which does a bit-wise comparison of the contained @code{T} objects.
7371 This should be used with extreme care because it only works reliably with
7372 built-in integral types, and it also compares any padding (filler bytes of
7373 undefined value) that the @code{T} class might have.
7375 @subsection Subexpressions
7377 Our scalar product class has two subexpressions: the left and right
7378 operands. It might be a good idea to make them accessible via the standard
7379 @code{nops()} and @code{op()} methods:
7382 size_t sprod::nops() const
7387 ex sprod::op(size_t i) const
7391 return get_struct().left;
7393 return get_struct().right;
7395 throw std::range_error("sprod::op(): no such operand");
7400 Implementing @code{nops()} and @code{op()} for container types such as
7401 @code{sprod} has two other nice side effects:
7405 @code{has()} works as expected
7407 GiNaC generates better hash keys for the objects (the default implementation
7408 of @code{calchash()} takes subexpressions into account)
7411 @cindex @code{let_op()}
7412 There is a non-const variant of @code{op()} called @code{let_op()} that
7413 allows replacing subexpressions:
7416 ex & sprod::let_op(size_t i)
7418 // every non-const member function must call this
7419 ensure_if_modifiable();
7423 return get_struct().left;
7425 return get_struct().right;
7427 throw std::range_error("sprod::let_op(): no such operand");
7432 Once we have provided @code{let_op()} we also get @code{subs()} and
7433 @code{map()} for free. In fact, every container class that returns a non-null
7434 @code{nops()} value must either implement @code{let_op()} or provide custom
7435 implementations of @code{subs()} and @code{map()}.
7437 In turn, the availability of @code{map()} enables the recursive behavior of a
7438 couple of other default method implementations, in particular @code{evalf()},
7439 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7440 we probably want to provide our own version of @code{expand()} for scalar
7441 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7442 This is left as an exercise for the reader.
7444 The @code{structure<T>} template defines many more member functions that
7445 you can override by specialization to customize the behavior of your
7446 structures. You are referred to the next section for a description of
7447 some of these (especially @code{eval()}). There is, however, one topic
7448 that shall be addressed here, as it demonstrates one peculiarity of the
7449 @code{structure<T>} template: archiving.
7451 @subsection Archiving structures
7453 If you don't know how the archiving of GiNaC objects is implemented, you
7454 should first read the next section and then come back here. You're back?
7457 To implement archiving for structures it is not enough to provide
7458 specializations for the @code{archive()} member function and the
7459 unarchiving constructor (the @code{unarchive()} function has a default
7460 implementation). You also need to provide a unique name (as a string literal)
7461 for each structure type you define. This is because in GiNaC archives,
7462 the class of an object is stored as a string, the class name.
7464 By default, this class name (as returned by the @code{class_name()} member
7465 function) is @samp{structure} for all structure classes. This works as long
7466 as you have only defined one structure type, but if you use two or more you
7467 need to provide a different name for each by specializing the
7468 @code{get_class_name()} member function. Here is a sample implementation
7469 for enabling archiving of the scalar product type defined above:
7472 const char *sprod::get_class_name() @{ return "sprod"; @}
7474 void sprod::archive(archive_node & n) const
7476 inherited::archive(n);
7477 n.add_ex("left", get_struct().left);
7478 n.add_ex("right", get_struct().right);
7481 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7483 n.find_ex("left", get_struct().left, sym_lst);
7484 n.find_ex("right", get_struct().right, sym_lst);
7488 Note that the unarchiving constructor is @code{sprod::structure} and not
7489 @code{sprod::sprod}, and that we don't need to supply an
7490 @code{sprod::unarchive()} function.
7493 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7494 @c node-name, next, previous, up
7495 @section Adding classes
7497 The @code{structure<T>} template provides an way to extend GiNaC with custom
7498 algebraic classes that is easy to use but has its limitations, the most
7499 severe of which being that you can't add any new member functions to
7500 structures. To be able to do this, you need to write a new class definition
7503 This section will explain how to implement new algebraic classes in GiNaC by
7504 giving the example of a simple 'string' class. After reading this section
7505 you will know how to properly declare a GiNaC class and what the minimum
7506 required member functions are that you have to implement. We only cover the
7507 implementation of a 'leaf' class here (i.e. one that doesn't contain
7508 subexpressions). Creating a container class like, for example, a class
7509 representing tensor products is more involved but this section should give
7510 you enough information so you can consult the source to GiNaC's predefined
7511 classes if you want to implement something more complicated.
7513 @subsection GiNaC's run-time type information system
7515 @cindex hierarchy of classes
7517 All algebraic classes (that is, all classes that can appear in expressions)
7518 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7519 @code{basic *} (which is essentially what an @code{ex} is) represents a
7520 generic pointer to an algebraic class. Occasionally it is necessary to find
7521 out what the class of an object pointed to by a @code{basic *} really is.
7522 Also, for the unarchiving of expressions it must be possible to find the
7523 @code{unarchive()} function of a class given the class name (as a string). A
7524 system that provides this kind of information is called a run-time type
7525 information (RTTI) system. The C++ language provides such a thing (see the
7526 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7527 implements its own, simpler RTTI.
7529 The RTTI in GiNaC is based on two mechanisms:
7534 The @code{basic} class declares a member variable @code{tinfo_key} which
7535 holds an unsigned integer that identifies the object's class. These numbers
7536 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7537 classes. They all start with @code{TINFO_}.
7540 By means of some clever tricks with static members, GiNaC maintains a list
7541 of information for all classes derived from @code{basic}. The information
7542 available includes the class names, the @code{tinfo_key}s, and pointers
7543 to the unarchiving functions. This class registry is defined in the
7544 @file{registrar.h} header file.
7548 The disadvantage of this proprietary RTTI implementation is that there's
7549 a little more to do when implementing new classes (C++'s RTTI works more
7550 or less automatically) but don't worry, most of the work is simplified by
7553 @subsection A minimalistic example
7555 Now we will start implementing a new class @code{mystring} that allows
7556 placing character strings in algebraic expressions (this is not very useful,
7557 but it's just an example). This class will be a direct subclass of
7558 @code{basic}. You can use this sample implementation as a starting point
7559 for your own classes.
7561 The code snippets given here assume that you have included some header files
7567 #include <stdexcept>
7568 using namespace std;
7570 #include <ginac/ginac.h>
7571 using namespace GiNaC;
7574 The first thing we have to do is to define a @code{tinfo_key} for our new
7575 class. This can be any arbitrary unsigned number that is not already taken
7576 by one of the existing classes but it's better to come up with something
7577 that is unlikely to clash with keys that might be added in the future. The
7578 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7579 which is not a requirement but we are going to stick with this scheme:
7582 const unsigned TINFO_mystring = 0x42420001U;
7585 Now we can write down the class declaration. The class stores a C++
7586 @code{string} and the user shall be able to construct a @code{mystring}
7587 object from a C or C++ string:
7590 class mystring : public basic
7592 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7595 mystring(const string &s);
7596 mystring(const char *s);
7602 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7605 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7606 macros are defined in @file{registrar.h}. They take the name of the class
7607 and its direct superclass as arguments and insert all required declarations
7608 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7609 the first line after the opening brace of the class definition. The
7610 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7611 source (at global scope, of course, not inside a function).
7613 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7614 declarations of the default constructor and a couple of other functions that
7615 are required. It also defines a type @code{inherited} which refers to the
7616 superclass so you don't have to modify your code every time you shuffle around
7617 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7618 class with the GiNaC RTTI (there is also a
7619 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7620 options for the class, and which we will be using instead in a few minutes).
7622 Now there are seven member functions we have to implement to get a working
7628 @code{mystring()}, the default constructor.
7631 @code{void archive(archive_node &n)}, the archiving function. This stores all
7632 information needed to reconstruct an object of this class inside an
7633 @code{archive_node}.
7636 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7637 constructor. This constructs an instance of the class from the information
7638 found in an @code{archive_node}.
7641 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7642 unarchiving function. It constructs a new instance by calling the unarchiving
7646 @cindex @code{compare_same_type()}
7647 @code{int compare_same_type(const basic &other)}, which is used internally
7648 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7649 -1, depending on the relative order of this object and the @code{other}
7650 object. If it returns 0, the objects are considered equal.
7651 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7652 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7653 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7654 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7655 must provide a @code{compare_same_type()} function, even those representing
7656 objects for which no reasonable algebraic ordering relationship can be
7660 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7661 which are the two constructors we declared.
7665 Let's proceed step-by-step. The default constructor looks like this:
7668 mystring::mystring() : inherited(TINFO_mystring) @{@}
7671 The golden rule is that in all constructors you have to set the
7672 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7673 it will be set by the constructor of the superclass and all hell will break
7674 loose in the RTTI. For your convenience, the @code{basic} class provides
7675 a constructor that takes a @code{tinfo_key} value, which we are using here
7676 (remember that in our case @code{inherited == basic}). If the superclass
7677 didn't have such a constructor, we would have to set the @code{tinfo_key}
7678 to the right value manually.
7680 In the default constructor you should set all other member variables to
7681 reasonable default values (we don't need that here since our @code{str}
7682 member gets set to an empty string automatically).
7684 Next are the three functions for archiving. You have to implement them even
7685 if you don't plan to use archives, but the minimum required implementation
7686 is really simple. First, the archiving function:
7689 void mystring::archive(archive_node &n) const
7691 inherited::archive(n);
7692 n.add_string("string", str);
7696 The only thing that is really required is calling the @code{archive()}
7697 function of the superclass. Optionally, you can store all information you
7698 deem necessary for representing the object into the passed
7699 @code{archive_node}. We are just storing our string here. For more
7700 information on how the archiving works, consult the @file{archive.h} header
7703 The unarchiving constructor is basically the inverse of the archiving
7707 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7709 n.find_string("string", str);
7713 If you don't need archiving, just leave this function empty (but you must
7714 invoke the unarchiving constructor of the superclass). Note that we don't
7715 have to set the @code{tinfo_key} here because it is done automatically
7716 by the unarchiving constructor of the @code{basic} class.
7718 Finally, the unarchiving function:
7721 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7723 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7727 You don't have to understand how exactly this works. Just copy these
7728 four lines into your code literally (replacing the class name, of
7729 course). It calls the unarchiving constructor of the class and unless
7730 you are doing something very special (like matching @code{archive_node}s
7731 to global objects) you don't need a different implementation. For those
7732 who are interested: setting the @code{dynallocated} flag puts the object
7733 under the control of GiNaC's garbage collection. It will get deleted
7734 automatically once it is no longer referenced.
7736 Our @code{compare_same_type()} function uses a provided function to compare
7740 int mystring::compare_same_type(const basic &other) const
7742 const mystring &o = static_cast<const mystring &>(other);
7743 int cmpval = str.compare(o.str);
7746 else if (cmpval < 0)
7753 Although this function takes a @code{basic &}, it will always be a reference
7754 to an object of exactly the same class (objects of different classes are not
7755 comparable), so the cast is safe. If this function returns 0, the two objects
7756 are considered equal (in the sense that @math{A-B=0}), so you should compare
7757 all relevant member variables.
7759 Now the only thing missing is our two new constructors:
7762 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7763 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7766 No surprises here. We set the @code{str} member from the argument and
7767 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7769 That's it! We now have a minimal working GiNaC class that can store
7770 strings in algebraic expressions. Let's confirm that the RTTI works:
7773 ex e = mystring("Hello, world!");
7774 cout << is_a<mystring>(e) << endl;
7777 cout << e.bp->class_name() << endl;
7781 Obviously it does. Let's see what the expression @code{e} looks like:
7785 // -> [mystring object]
7788 Hm, not exactly what we expect, but of course the @code{mystring} class
7789 doesn't yet know how to print itself. This can be done either by implementing
7790 the @code{print()} member function, or, preferably, by specifying a
7791 @code{print_func<>()} class option. Let's say that we want to print the string
7792 surrounded by double quotes:
7795 class mystring : public basic
7799 void do_print(const print_context &c, unsigned level = 0) const;
7803 void mystring::do_print(const print_context &c, unsigned level) const
7805 // print_context::s is a reference to an ostream
7806 c.s << '\"' << str << '\"';
7810 The @code{level} argument is only required for container classes to
7811 correctly parenthesize the output.
7813 Now we need to tell GiNaC that @code{mystring} objects should use the
7814 @code{do_print()} member function for printing themselves. For this, we
7818 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7824 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7825 print_func<print_context>(&mystring::do_print))
7828 Let's try again to print the expression:
7832 // -> "Hello, world!"
7835 Much better. If we wanted to have @code{mystring} objects displayed in a
7836 different way depending on the output format (default, LaTeX, etc.), we
7837 would have supplied multiple @code{print_func<>()} options with different
7838 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7839 separated by dots. This is similar to the way options are specified for
7840 symbolic functions. @xref{Printing}, for a more in-depth description of the
7841 way expression output is implemented in GiNaC.
7843 The @code{mystring} class can be used in arbitrary expressions:
7846 e += mystring("GiNaC rulez");
7848 // -> "GiNaC rulez"+"Hello, world!"
7851 (GiNaC's automatic term reordering is in effect here), or even
7854 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7856 // -> "One string"^(2*sin(-"Another string"+Pi))
7859 Whether this makes sense is debatable but remember that this is only an
7860 example. At least it allows you to implement your own symbolic algorithms
7863 Note that GiNaC's algebraic rules remain unchanged:
7866 e = mystring("Wow") * mystring("Wow");
7870 e = pow(mystring("First")-mystring("Second"), 2);
7871 cout << e.expand() << endl;
7872 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7875 There's no way to, for example, make GiNaC's @code{add} class perform string
7876 concatenation. You would have to implement this yourself.
7878 @subsection Automatic evaluation
7881 @cindex @code{eval()}
7882 @cindex @code{hold()}
7883 When dealing with objects that are just a little more complicated than the
7884 simple string objects we have implemented, chances are that you will want to
7885 have some automatic simplifications or canonicalizations performed on them.
7886 This is done in the evaluation member function @code{eval()}. Let's say that
7887 we wanted all strings automatically converted to lowercase with
7888 non-alphabetic characters stripped, and empty strings removed:
7891 class mystring : public basic
7895 ex eval(int level = 0) const;
7899 ex mystring::eval(int level) const
7902 for (int i=0; i<str.length(); i++) @{
7904 if (c >= 'A' && c <= 'Z')
7905 new_str += tolower(c);
7906 else if (c >= 'a' && c <= 'z')
7910 if (new_str.length() == 0)
7913 return mystring(new_str).hold();
7917 The @code{level} argument is used to limit the recursion depth of the
7918 evaluation. We don't have any subexpressions in the @code{mystring}
7919 class so we are not concerned with this. If we had, we would call the
7920 @code{eval()} functions of the subexpressions with @code{level - 1} as
7921 the argument if @code{level != 1}. The @code{hold()} member function
7922 sets a flag in the object that prevents further evaluation. Otherwise
7923 we might end up in an endless loop. When you want to return the object
7924 unmodified, use @code{return this->hold();}.
7926 Let's confirm that it works:
7929 ex e = mystring("Hello, world!") + mystring("!?#");
7933 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7938 @subsection Optional member functions
7940 We have implemented only a small set of member functions to make the class
7941 work in the GiNaC framework. There are two functions that are not strictly
7942 required but will make operations with objects of the class more efficient:
7944 @cindex @code{calchash()}
7945 @cindex @code{is_equal_same_type()}
7947 unsigned calchash() const;
7948 bool is_equal_same_type(const basic &other) const;
7951 The @code{calchash()} method returns an @code{unsigned} hash value for the
7952 object which will allow GiNaC to compare and canonicalize expressions much
7953 more efficiently. You should consult the implementation of some of the built-in
7954 GiNaC classes for examples of hash functions. The default implementation of
7955 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7956 class and all subexpressions that are accessible via @code{op()}.
7958 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7959 tests for equality without establishing an ordering relation, which is often
7960 faster. The default implementation of @code{is_equal_same_type()} just calls
7961 @code{compare_same_type()} and tests its result for zero.
7963 @subsection Other member functions
7965 For a real algebraic class, there are probably some more functions that you
7966 might want to provide:
7969 bool info(unsigned inf) const;
7970 ex evalf(int level = 0) const;
7971 ex series(const relational & r, int order, unsigned options = 0) const;
7972 ex derivative(const symbol & s) const;
7975 If your class stores sub-expressions (see the scalar product example in the
7976 previous section) you will probably want to override
7978 @cindex @code{let_op()}
7981 ex op(size_t i) const;
7982 ex & let_op(size_t i);
7983 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7984 ex map(map_function & f) const;
7987 @code{let_op()} is a variant of @code{op()} that allows write access. The
7988 default implementations of @code{subs()} and @code{map()} use it, so you have
7989 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7991 You can, of course, also add your own new member functions. Remember
7992 that the RTTI may be used to get information about what kinds of objects
7993 you are dealing with (the position in the class hierarchy) and that you
7994 can always extract the bare object from an @code{ex} by stripping the
7995 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7996 should become a need.
7998 That's it. May the source be with you!
8001 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8002 @c node-name, next, previous, up
8003 @chapter A Comparison With Other CAS
8006 This chapter will give you some information on how GiNaC compares to
8007 other, traditional Computer Algebra Systems, like @emph{Maple},
8008 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8009 disadvantages over these systems.
8012 * Advantages:: Strengths of the GiNaC approach.
8013 * Disadvantages:: Weaknesses of the GiNaC approach.
8014 * Why C++?:: Attractiveness of C++.
8017 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8018 @c node-name, next, previous, up
8021 GiNaC has several advantages over traditional Computer
8022 Algebra Systems, like
8027 familiar language: all common CAS implement their own proprietary
8028 grammar which you have to learn first (and maybe learn again when your
8029 vendor decides to `enhance' it). With GiNaC you can write your program
8030 in common C++, which is standardized.
8034 structured data types: you can build up structured data types using
8035 @code{struct}s or @code{class}es together with STL features instead of
8036 using unnamed lists of lists of lists.
8039 strongly typed: in CAS, you usually have only one kind of variables
8040 which can hold contents of an arbitrary type. This 4GL like feature is
8041 nice for novice programmers, but dangerous.
8044 development tools: powerful development tools exist for C++, like fancy
8045 editors (e.g. with automatic indentation and syntax highlighting),
8046 debuggers, visualization tools, documentation generators@dots{}
8049 modularization: C++ programs can easily be split into modules by
8050 separating interface and implementation.
8053 price: GiNaC is distributed under the GNU Public License which means
8054 that it is free and available with source code. And there are excellent
8055 C++-compilers for free, too.
8058 extendable: you can add your own classes to GiNaC, thus extending it on
8059 a very low level. Compare this to a traditional CAS that you can
8060 usually only extend on a high level by writing in the language defined
8061 by the parser. In particular, it turns out to be almost impossible to
8062 fix bugs in a traditional system.
8065 multiple interfaces: Though real GiNaC programs have to be written in
8066 some editor, then be compiled, linked and executed, there are more ways
8067 to work with the GiNaC engine. Many people want to play with
8068 expressions interactively, as in traditional CASs. Currently, two such
8069 windows into GiNaC have been implemented and many more are possible: the
8070 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8071 types to a command line and second, as a more consistent approach, an
8072 interactive interface to the Cint C++ interpreter has been put together
8073 (called GiNaC-cint) that allows an interactive scripting interface
8074 consistent with the C++ language. It is available from the usual GiNaC
8078 seamless integration: it is somewhere between difficult and impossible
8079 to call CAS functions from within a program written in C++ or any other
8080 programming language and vice versa. With GiNaC, your symbolic routines
8081 are part of your program. You can easily call third party libraries,
8082 e.g. for numerical evaluation or graphical interaction. All other
8083 approaches are much more cumbersome: they range from simply ignoring the
8084 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8085 system (i.e. @emph{Yacas}).
8088 efficiency: often large parts of a program do not need symbolic
8089 calculations at all. Why use large integers for loop variables or
8090 arbitrary precision arithmetics where @code{int} and @code{double} are
8091 sufficient? For pure symbolic applications, GiNaC is comparable in
8092 speed with other CAS.
8097 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8098 @c node-name, next, previous, up
8099 @section Disadvantages
8101 Of course it also has some disadvantages:
8106 advanced features: GiNaC cannot compete with a program like
8107 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8108 which grows since 1981 by the work of dozens of programmers, with
8109 respect to mathematical features. Integration, factorization,
8110 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8111 not planned for the near future).
8114 portability: While the GiNaC library itself is designed to avoid any
8115 platform dependent features (it should compile on any ANSI compliant C++
8116 compiler), the currently used version of the CLN library (fast large
8117 integer and arbitrary precision arithmetics) can only by compiled
8118 without hassle on systems with the C++ compiler from the GNU Compiler
8119 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8120 macros to let the compiler gather all static initializations, which
8121 works for GNU C++ only. Feel free to contact the authors in case you
8122 really believe that you need to use a different compiler. We have
8123 occasionally used other compilers and may be able to give you advice.}
8124 GiNaC uses recent language features like explicit constructors, mutable
8125 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8126 literally. Recent GCC versions starting at 2.95.3, although itself not
8127 yet ANSI compliant, support all needed features.
8132 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8133 @c node-name, next, previous, up
8136 Why did we choose to implement GiNaC in C++ instead of Java or any other
8137 language? C++ is not perfect: type checking is not strict (casting is
8138 possible), separation between interface and implementation is not
8139 complete, object oriented design is not enforced. The main reason is
8140 the often scolded feature of operator overloading in C++. While it may
8141 be true that operating on classes with a @code{+} operator is rarely
8142 meaningful, it is perfectly suited for algebraic expressions. Writing
8143 @math{3x+5y} as @code{3*x+5*y} instead of
8144 @code{x.times(3).plus(y.times(5))} looks much more natural.
8145 Furthermore, the main developers are more familiar with C++ than with
8146 any other programming language.
8149 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8150 @c node-name, next, previous, up
8151 @appendix Internal Structures
8154 * Expressions are reference counted::
8155 * Internal representation of products and sums::
8158 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8159 @c node-name, next, previous, up
8160 @appendixsection Expressions are reference counted
8162 @cindex reference counting
8163 @cindex copy-on-write
8164 @cindex garbage collection
8165 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8166 where the counter belongs to the algebraic objects derived from class
8167 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8168 which @code{ex} contains an instance. If you understood that, you can safely
8169 skip the rest of this passage.
8171 Expressions are extremely light-weight since internally they work like
8172 handles to the actual representation. They really hold nothing more
8173 than a pointer to some other object. What this means in practice is
8174 that whenever you create two @code{ex} and set the second equal to the
8175 first no copying process is involved. Instead, the copying takes place
8176 as soon as you try to change the second. Consider the simple sequence
8181 #include <ginac/ginac.h>
8182 using namespace std;
8183 using namespace GiNaC;
8187 symbol x("x"), y("y"), z("z");
8190 e1 = sin(x + 2*y) + 3*z + 41;
8191 e2 = e1; // e2 points to same object as e1
8192 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8193 e2 += 1; // e2 is copied into a new object
8194 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8198 The line @code{e2 = e1;} creates a second expression pointing to the
8199 object held already by @code{e1}. The time involved for this operation
8200 is therefore constant, no matter how large @code{e1} was. Actual
8201 copying, however, must take place in the line @code{e2 += 1;} because
8202 @code{e1} and @code{e2} are not handles for the same object any more.
8203 This concept is called @dfn{copy-on-write semantics}. It increases
8204 performance considerably whenever one object occurs multiple times and
8205 represents a simple garbage collection scheme because when an @code{ex}
8206 runs out of scope its destructor checks whether other expressions handle
8207 the object it points to too and deletes the object from memory if that
8208 turns out not to be the case. A slightly less trivial example of
8209 differentiation using the chain-rule should make clear how powerful this
8214 symbol x("x"), y("y");
8218 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8219 cout << e1 << endl // prints x+3*y
8220 << e2 << endl // prints (x+3*y)^3
8221 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8225 Here, @code{e1} will actually be referenced three times while @code{e2}
8226 will be referenced two times. When the power of an expression is built,
8227 that expression needs not be copied. Likewise, since the derivative of
8228 a power of an expression can be easily expressed in terms of that
8229 expression, no copying of @code{e1} is involved when @code{e3} is
8230 constructed. So, when @code{e3} is constructed it will print as
8231 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8232 holds a reference to @code{e2} and the factor in front is just
8235 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8236 semantics. When you insert an expression into a second expression, the
8237 result behaves exactly as if the contents of the first expression were
8238 inserted. But it may be useful to remember that this is not what
8239 happens. Knowing this will enable you to write much more efficient
8240 code. If you still have an uncertain feeling with copy-on-write
8241 semantics, we recommend you have a look at the
8242 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8243 Marshall Cline. Chapter 16 covers this issue and presents an
8244 implementation which is pretty close to the one in GiNaC.
8247 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8248 @c node-name, next, previous, up
8249 @appendixsection Internal representation of products and sums
8251 @cindex representation
8254 @cindex @code{power}
8255 Although it should be completely transparent for the user of
8256 GiNaC a short discussion of this topic helps to understand the sources
8257 and also explain performance to a large degree. Consider the
8258 unexpanded symbolic expression
8260 $2d^3 \left( 4a + 5b - 3 \right)$
8263 @math{2*d^3*(4*a+5*b-3)}
8265 which could naively be represented by a tree of linear containers for
8266 addition and multiplication, one container for exponentiation with base
8267 and exponent and some atomic leaves of symbols and numbers in this
8272 @cindex pair-wise representation
8273 However, doing so results in a rather deeply nested tree which will
8274 quickly become inefficient to manipulate. We can improve on this by
8275 representing the sum as a sequence of terms, each one being a pair of a
8276 purely numeric multiplicative coefficient and its rest. In the same
8277 spirit we can store the multiplication as a sequence of terms, each
8278 having a numeric exponent and a possibly complicated base, the tree
8279 becomes much more flat:
8283 The number @code{3} above the symbol @code{d} shows that @code{mul}
8284 objects are treated similarly where the coefficients are interpreted as
8285 @emph{exponents} now. Addition of sums of terms or multiplication of
8286 products with numerical exponents can be coded to be very efficient with
8287 such a pair-wise representation. Internally, this handling is performed
8288 by most CAS in this way. It typically speeds up manipulations by an
8289 order of magnitude. The overall multiplicative factor @code{2} and the
8290 additive term @code{-3} look somewhat out of place in this
8291 representation, however, since they are still carrying a trivial
8292 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8293 this is avoided by adding a field that carries an overall numeric
8294 coefficient. This results in the realistic picture of internal
8297 $2d^3 \left( 4a + 5b - 3 \right)$:
8300 @math{2*d^3*(4*a+5*b-3)}:
8306 This also allows for a better handling of numeric radicals, since
8307 @code{sqrt(2)} can now be carried along calculations. Now it should be
8308 clear, why both classes @code{add} and @code{mul} are derived from the
8309 same abstract class: the data representation is the same, only the
8310 semantics differs. In the class hierarchy, methods for polynomial
8311 expansion and the like are reimplemented for @code{add} and @code{mul},
8312 but the data structure is inherited from @code{expairseq}.
8315 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8316 @c node-name, next, previous, up
8317 @appendix Package Tools
8319 If you are creating a software package that uses the GiNaC library,
8320 setting the correct command line options for the compiler and linker
8321 can be difficult. GiNaC includes two tools to make this process easier.
8324 * ginac-config:: A shell script to detect compiler and linker flags.
8325 * AM_PATH_GINAC:: Macro for GNU automake.
8329 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8330 @c node-name, next, previous, up
8331 @section @command{ginac-config}
8332 @cindex ginac-config
8334 @command{ginac-config} is a shell script that you can use to determine
8335 the compiler and linker command line options required to compile and
8336 link a program with the GiNaC library.
8338 @command{ginac-config} takes the following flags:
8342 Prints out the version of GiNaC installed.
8344 Prints '-I' flags pointing to the installed header files.
8346 Prints out the linker flags necessary to link a program against GiNaC.
8347 @item --prefix[=@var{PREFIX}]
8348 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8349 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8350 Otherwise, prints out the configured value of @env{$prefix}.
8351 @item --exec-prefix[=@var{PREFIX}]
8352 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8353 Otherwise, prints out the configured value of @env{$exec_prefix}.
8356 Typically, @command{ginac-config} will be used within a configure
8357 script, as described below. It, however, can also be used directly from
8358 the command line using backquotes to compile a simple program. For
8362 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8365 This command line might expand to (for example):
8368 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8369 -lginac -lcln -lstdc++
8372 Not only is the form using @command{ginac-config} easier to type, it will
8373 work on any system, no matter how GiNaC was configured.
8376 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8377 @c node-name, next, previous, up
8378 @section @samp{AM_PATH_GINAC}
8379 @cindex AM_PATH_GINAC
8381 For packages configured using GNU automake, GiNaC also provides
8382 a macro to automate the process of checking for GiNaC.
8385 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8386 [, @var{ACTION-IF-NOT-FOUND}]]])
8394 Determines the location of GiNaC using @command{ginac-config}, which is
8395 either found in the user's path, or from the environment variable
8396 @env{GINACLIB_CONFIG}.
8399 Tests the installed libraries to make sure that their version
8400 is later than @var{MINIMUM-VERSION}. (A default version will be used
8404 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8405 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8406 variable to the output of @command{ginac-config --libs}, and calls
8407 @samp{AC_SUBST()} for these variables so they can be used in generated
8408 makefiles, and then executes @var{ACTION-IF-FOUND}.
8411 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8412 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8416 This macro is in file @file{ginac.m4} which is installed in
8417 @file{$datadir/aclocal}. Note that if automake was installed with a
8418 different @samp{--prefix} than GiNaC, you will either have to manually
8419 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8420 aclocal the @samp{-I} option when running it.
8423 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8424 * Example package:: Example of a package using AM_PATH_GINAC.
8428 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8429 @c node-name, next, previous, up
8430 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8432 Simply make sure that @command{ginac-config} is in your path, and run
8433 the configure script.
8440 The directory where the GiNaC libraries are installed needs
8441 to be found by your system's dynamic linker.
8443 This is generally done by
8446 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8452 setting the environment variable @env{LD_LIBRARY_PATH},
8455 or, as a last resort,
8458 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8459 running configure, for instance:
8462 LDFLAGS=-R/home/cbauer/lib ./configure
8467 You can also specify a @command{ginac-config} not in your path by
8468 setting the @env{GINACLIB_CONFIG} environment variable to the
8469 name of the executable
8472 If you move the GiNaC package from its installed location,
8473 you will either need to modify @command{ginac-config} script
8474 manually to point to the new location or rebuild GiNaC.
8485 --with-ginac-prefix=@var{PREFIX}
8486 --with-ginac-exec-prefix=@var{PREFIX}
8489 are provided to override the prefix and exec-prefix that were stored
8490 in the @command{ginac-config} shell script by GiNaC's configure. You are
8491 generally better off configuring GiNaC with the right path to begin with.
8495 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8496 @c node-name, next, previous, up
8497 @subsection Example of a package using @samp{AM_PATH_GINAC}
8499 The following shows how to build a simple package using automake
8500 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8504 #include <ginac/ginac.h>
8508 GiNaC::symbol x("x");
8509 GiNaC::ex a = GiNaC::sin(x);
8510 std::cout << "Derivative of " << a
8511 << " is " << a.diff(x) << std::endl;
8516 You should first read the introductory portions of the automake
8517 Manual, if you are not already familiar with it.
8519 Two files are needed, @file{configure.in}, which is used to build the
8523 dnl Process this file with autoconf to produce a configure script.
8525 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8531 AM_PATH_GINAC(0.9.0, [
8532 LIBS="$LIBS $GINACLIB_LIBS"
8533 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8534 ], AC_MSG_ERROR([need to have GiNaC installed]))
8539 The only command in this which is not standard for automake
8540 is the @samp{AM_PATH_GINAC} macro.
8542 That command does the following: If a GiNaC version greater or equal
8543 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8544 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8545 the error message `need to have GiNaC installed'
8547 And the @file{Makefile.am}, which will be used to build the Makefile.
8550 ## Process this file with automake to produce Makefile.in
8551 bin_PROGRAMS = simple
8552 simple_SOURCES = simple.cpp
8555 This @file{Makefile.am}, says that we are building a single executable,
8556 from a single source file @file{simple.cpp}. Since every program
8557 we are building uses GiNaC we simply added the GiNaC options
8558 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8559 want to specify them on a per-program basis: for instance by
8563 simple_LDADD = $(GINACLIB_LIBS)
8564 INCLUDES = $(GINACLIB_CPPFLAGS)
8567 to the @file{Makefile.am}.
8569 To try this example out, create a new directory and add the three
8572 Now execute the following commands:
8575 $ automake --add-missing
8580 You now have a package that can be built in the normal fashion
8589 @node Bibliography, Concept Index, Example package, Top
8590 @c node-name, next, previous, up
8591 @appendix Bibliography
8596 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8599 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8602 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8605 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8608 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8609 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8612 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8613 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8614 Academic Press, London
8617 @cite{Computer Algebra Systems - A Practical Guide},
8618 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8621 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8622 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8625 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8626 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8629 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8634 @node Concept Index, , Bibliography, Top
8635 @c node-name, next, previous, up
8636 @unnumbered Concept Index