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 integrant 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 Matrix elements can be accessed and set using the parenthesis (function call)
1981 const ex & matrix::operator()(unsigned r, unsigned c) const;
1982 ex & matrix::operator()(unsigned r, unsigned c);
1985 It is also possible to access the matrix elements in a linear fashion with
1986 the @code{op()} method. But C++-style subscripting with square brackets
1987 @samp{[]} is not available.
1989 Here are a couple of examples for constructing matrices:
1993 symbol a("a"), b("b");
2007 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2010 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2013 cout << diag_matrix(lst(a, b)) << endl;
2016 cout << unit_matrix(3) << endl;
2017 // -> [[1,0,0],[0,1,0],[0,0,1]]
2019 cout << symbolic_matrix(2, 3, "x") << endl;
2020 // -> [[x00,x01,x02],[x10,x11,x12]]
2024 @cindex @code{transpose()}
2025 There are three ways to do arithmetic with matrices. The first (and most
2026 direct one) is to use the methods provided by the @code{matrix} class:
2029 matrix matrix::add(const matrix & other) const;
2030 matrix matrix::sub(const matrix & other) const;
2031 matrix matrix::mul(const matrix & other) const;
2032 matrix matrix::mul_scalar(const ex & other) const;
2033 matrix matrix::pow(const ex & expn) const;
2034 matrix matrix::transpose() const;
2037 All of these methods return the result as a new matrix object. Here is an
2038 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2043 matrix A(2, 2), B(2, 2), C(2, 2);
2051 matrix result = A.mul(B).sub(C.mul_scalar(2));
2052 cout << result << endl;
2053 // -> [[-13,-6],[1,2]]
2058 @cindex @code{evalm()}
2059 The second (and probably the most natural) way is to construct an expression
2060 containing matrices with the usual arithmetic operators and @code{pow()}.
2061 For efficiency reasons, expressions with sums, products and powers of
2062 matrices are not automatically evaluated in GiNaC. You have to call the
2066 ex ex::evalm() const;
2069 to obtain the result:
2076 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2077 cout << e.evalm() << endl;
2078 // -> [[-13,-6],[1,2]]
2083 The non-commutativity of the product @code{A*B} in this example is
2084 automatically recognized by GiNaC. There is no need to use a special
2085 operator here. @xref{Non-commutative objects}, for more information about
2086 dealing with non-commutative expressions.
2088 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2089 to perform the arithmetic:
2094 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2095 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2097 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2098 cout << e.simplify_indexed() << endl;
2099 // -> [[-13,-6],[1,2]].i.j
2103 Using indices is most useful when working with rectangular matrices and
2104 one-dimensional vectors because you don't have to worry about having to
2105 transpose matrices before multiplying them. @xref{Indexed objects}, for
2106 more information about using matrices with indices, and about indices in
2109 The @code{matrix} class provides a couple of additional methods for
2110 computing determinants, traces, characteristic polynomials and ranks:
2112 @cindex @code{determinant()}
2113 @cindex @code{trace()}
2114 @cindex @code{charpoly()}
2115 @cindex @code{rank()}
2117 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2118 ex matrix::trace() const;
2119 ex matrix::charpoly(const ex & lambda) const;
2120 unsigned matrix::rank() const;
2123 The @samp{algo} argument of @code{determinant()} allows to select
2124 between different algorithms for calculating the determinant. The
2125 asymptotic speed (as parametrized by the matrix size) can greatly differ
2126 between those algorithms, depending on the nature of the matrix'
2127 entries. The possible values are defined in the @file{flags.h} header
2128 file. By default, GiNaC uses a heuristic to automatically select an
2129 algorithm that is likely (but not guaranteed) to give the result most
2132 @cindex @code{inverse()} (matrix)
2133 @cindex @code{solve()}
2134 Matrices may also be inverted using the @code{ex matrix::inverse()}
2135 method and linear systems may be solved with:
2138 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2139 unsigned algo=solve_algo::automatic) const;
2142 Assuming the matrix object this method is applied on is an @code{m}
2143 times @code{n} matrix, then @code{vars} must be a @code{n} times
2144 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2145 times @code{p} matrix. The returned matrix then has dimension @code{n}
2146 times @code{p} and in the case of an underdetermined system will still
2147 contain some of the indeterminates from @code{vars}. If the system is
2148 overdetermined, an exception is thrown.
2151 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2152 @c node-name, next, previous, up
2153 @section Indexed objects
2155 GiNaC allows you to handle expressions containing general indexed objects in
2156 arbitrary spaces. It is also able to canonicalize and simplify such
2157 expressions and perform symbolic dummy index summations. There are a number
2158 of predefined indexed objects provided, like delta and metric tensors.
2160 There are few restrictions placed on indexed objects and their indices and
2161 it is easy to construct nonsense expressions, but our intention is to
2162 provide a general framework that allows you to implement algorithms with
2163 indexed quantities, getting in the way as little as possible.
2165 @cindex @code{idx} (class)
2166 @cindex @code{indexed} (class)
2167 @subsection Indexed quantities and their indices
2169 Indexed expressions in GiNaC are constructed of two special types of objects,
2170 @dfn{index objects} and @dfn{indexed objects}.
2174 @cindex contravariant
2177 @item Index objects are of class @code{idx} or a subclass. Every index has
2178 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2179 the index lives in) which can both be arbitrary expressions but are usually
2180 a number or a simple symbol. In addition, indices of class @code{varidx} have
2181 a @dfn{variance} (they can be co- or contravariant), and indices of class
2182 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2184 @item Indexed objects are of class @code{indexed} or a subclass. They
2185 contain a @dfn{base expression} (which is the expression being indexed), and
2186 one or more indices.
2190 @strong{Please notice:} when printing expressions, covariant indices and indices
2191 without variance are denoted @samp{.i} while contravariant indices are
2192 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2193 value. In the following, we are going to use that notation in the text so
2194 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2195 not visible in the output.
2197 A simple example shall illustrate the concepts:
2201 #include <ginac/ginac.h>
2202 using namespace std;
2203 using namespace GiNaC;
2207 symbol i_sym("i"), j_sym("j");
2208 idx i(i_sym, 3), j(j_sym, 3);
2211 cout << indexed(A, i, j) << endl;
2213 cout << index_dimensions << indexed(A, i, j) << endl;
2215 cout << dflt; // reset cout to default output format (dimensions hidden)
2219 The @code{idx} constructor takes two arguments, the index value and the
2220 index dimension. First we define two index objects, @code{i} and @code{j},
2221 both with the numeric dimension 3. The value of the index @code{i} is the
2222 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2223 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2224 construct an expression containing one indexed object, @samp{A.i.j}. It has
2225 the symbol @code{A} as its base expression and the two indices @code{i} and
2228 The dimensions of indices are normally not visible in the output, but one
2229 can request them to be printed with the @code{index_dimensions} manipulator,
2232 Note the difference between the indices @code{i} and @code{j} which are of
2233 class @code{idx}, and the index values which are the symbols @code{i_sym}
2234 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2235 or numbers but must be index objects. For example, the following is not
2236 correct and will raise an exception:
2239 symbol i("i"), j("j");
2240 e = indexed(A, i, j); // ERROR: indices must be of type idx
2243 You can have multiple indexed objects in an expression, index values can
2244 be numeric, and index dimensions symbolic:
2248 symbol B("B"), dim("dim");
2249 cout << 4 * indexed(A, i)
2250 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2255 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2256 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2257 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2258 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2259 @code{simplify_indexed()} for that, see below).
2261 In fact, base expressions, index values and index dimensions can be
2262 arbitrary expressions:
2266 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2271 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2272 get an error message from this but you will probably not be able to do
2273 anything useful with it.
2275 @cindex @code{get_value()}
2276 @cindex @code{get_dimension()}
2280 ex idx::get_value();
2281 ex idx::get_dimension();
2284 return the value and dimension of an @code{idx} object. If you have an index
2285 in an expression, such as returned by calling @code{.op()} on an indexed
2286 object, you can get a reference to the @code{idx} object with the function
2287 @code{ex_to<idx>()} on the expression.
2289 There are also the methods
2292 bool idx::is_numeric();
2293 bool idx::is_symbolic();
2294 bool idx::is_dim_numeric();
2295 bool idx::is_dim_symbolic();
2298 for checking whether the value and dimension are numeric or symbolic
2299 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2300 About Expressions}) returns information about the index value.
2302 @cindex @code{varidx} (class)
2303 If you need co- and contravariant indices, use the @code{varidx} class:
2307 symbol mu_sym("mu"), nu_sym("nu");
2308 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2309 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2311 cout << indexed(A, mu, nu) << endl;
2313 cout << indexed(A, mu_co, nu) << endl;
2315 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2320 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2321 co- or contravariant. The default is a contravariant (upper) index, but
2322 this can be overridden by supplying a third argument to the @code{varidx}
2323 constructor. The two methods
2326 bool varidx::is_covariant();
2327 bool varidx::is_contravariant();
2330 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2331 to get the object reference from an expression). There's also the very useful
2335 ex varidx::toggle_variance();
2338 which makes a new index with the same value and dimension but the opposite
2339 variance. By using it you only have to define the index once.
2341 @cindex @code{spinidx} (class)
2342 The @code{spinidx} class provides dotted and undotted variant indices, as
2343 used in the Weyl-van-der-Waerden spinor formalism:
2347 symbol K("K"), C_sym("C"), D_sym("D");
2348 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2349 // contravariant, undotted
2350 spinidx C_co(C_sym, 2, true); // covariant index
2351 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2352 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2354 cout << indexed(K, C, D) << endl;
2356 cout << indexed(K, C_co, D_dot) << endl;
2358 cout << indexed(K, D_co_dot, D) << endl;
2363 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2364 dotted or undotted. The default is undotted but this can be overridden by
2365 supplying a fourth argument to the @code{spinidx} constructor. The two
2369 bool spinidx::is_dotted();
2370 bool spinidx::is_undotted();
2373 allow you to check whether or not a @code{spinidx} object is dotted (use
2374 @code{ex_to<spinidx>()} to get the object reference from an expression).
2375 Finally, the two methods
2378 ex spinidx::toggle_dot();
2379 ex spinidx::toggle_variance_dot();
2382 create a new index with the same value and dimension but opposite dottedness
2383 and the same or opposite variance.
2385 @subsection Substituting indices
2387 @cindex @code{subs()}
2388 Sometimes you will want to substitute one symbolic index with another
2389 symbolic or numeric index, for example when calculating one specific element
2390 of a tensor expression. This is done with the @code{.subs()} method, as it
2391 is done for symbols (see @ref{Substituting Expressions}).
2393 You have two possibilities here. You can either substitute the whole index
2394 by another index or expression:
2398 ex e = indexed(A, mu_co);
2399 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2400 // -> A.mu becomes A~nu
2401 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2402 // -> A.mu becomes A~0
2403 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2404 // -> A.mu becomes A.0
2408 The third example shows that trying to replace an index with something that
2409 is not an index will substitute the index value instead.
2411 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2416 ex e = indexed(A, mu_co);
2417 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2418 // -> A.mu becomes A.nu
2419 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2420 // -> A.mu becomes A.0
2424 As you see, with the second method only the value of the index will get
2425 substituted. Its other properties, including its dimension, remain unchanged.
2426 If you want to change the dimension of an index you have to substitute the
2427 whole index by another one with the new dimension.
2429 Finally, substituting the base expression of an indexed object works as
2434 ex e = indexed(A, mu_co);
2435 cout << e << " becomes " << e.subs(A == A+B) << endl;
2436 // -> A.mu becomes (B+A).mu
2440 @subsection Symmetries
2441 @cindex @code{symmetry} (class)
2442 @cindex @code{sy_none()}
2443 @cindex @code{sy_symm()}
2444 @cindex @code{sy_anti()}
2445 @cindex @code{sy_cycl()}
2447 Indexed objects can have certain symmetry properties with respect to their
2448 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2449 that is constructed with the helper functions
2452 symmetry sy_none(...);
2453 symmetry sy_symm(...);
2454 symmetry sy_anti(...);
2455 symmetry sy_cycl(...);
2458 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2459 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2460 represents a cyclic symmetry. Each of these functions accepts up to four
2461 arguments which can be either symmetry objects themselves or unsigned integer
2462 numbers that represent an index position (counting from 0). A symmetry
2463 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2464 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2467 Here are some examples of symmetry definitions:
2472 e = indexed(A, i, j);
2473 e = indexed(A, sy_none(), i, j); // equivalent
2474 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2476 // Symmetric in all three indices:
2477 e = indexed(A, sy_symm(), i, j, k);
2478 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2479 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2480 // different canonical order
2482 // Symmetric in the first two indices only:
2483 e = indexed(A, sy_symm(0, 1), i, j, k);
2484 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2486 // Antisymmetric in the first and last index only (index ranges need not
2488 e = indexed(A, sy_anti(0, 2), i, j, k);
2489 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2491 // An example of a mixed symmetry: antisymmetric in the first two and
2492 // last two indices, symmetric when swapping the first and last index
2493 // pairs (like the Riemann curvature tensor):
2494 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2496 // Cyclic symmetry in all three indices:
2497 e = indexed(A, sy_cycl(), i, j, k);
2498 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2500 // The following examples are invalid constructions that will throw
2501 // an exception at run time.
2503 // An index may not appear multiple times:
2504 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2505 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2507 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2508 // same number of indices:
2509 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2511 // And of course, you cannot specify indices which are not there:
2512 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2516 If you need to specify more than four indices, you have to use the
2517 @code{.add()} method of the @code{symmetry} class. For example, to specify
2518 full symmetry in the first six indices you would write
2519 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2521 If an indexed object has a symmetry, GiNaC will automatically bring the
2522 indices into a canonical order which allows for some immediate simplifications:
2526 cout << indexed(A, sy_symm(), i, j)
2527 + indexed(A, sy_symm(), j, i) << endl;
2529 cout << indexed(B, sy_anti(), i, j)
2530 + indexed(B, sy_anti(), j, i) << endl;
2532 cout << indexed(B, sy_anti(), i, j, k)
2533 - indexed(B, sy_anti(), j, k, i) << endl;
2538 @cindex @code{get_free_indices()}
2540 @subsection Dummy indices
2542 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2543 that a summation over the index range is implied. Symbolic indices which are
2544 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2545 dummy nor free indices.
2547 To be recognized as a dummy index pair, the two indices must be of the same
2548 class and their value must be the same single symbol (an index like
2549 @samp{2*n+1} is never a dummy index). If the indices are of class
2550 @code{varidx} they must also be of opposite variance; if they are of class
2551 @code{spinidx} they must be both dotted or both undotted.
2553 The method @code{.get_free_indices()} returns a vector containing the free
2554 indices of an expression. It also checks that the free indices of the terms
2555 of a sum are consistent:
2559 symbol A("A"), B("B"), C("C");
2561 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2562 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2564 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2565 cout << exprseq(e.get_free_indices()) << endl;
2567 // 'j' and 'l' are dummy indices
2569 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2570 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2572 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2573 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2574 cout << exprseq(e.get_free_indices()) << endl;
2576 // 'nu' is a dummy index, but 'sigma' is not
2578 e = indexed(A, mu, mu);
2579 cout << exprseq(e.get_free_indices()) << endl;
2581 // 'mu' is not a dummy index because it appears twice with the same
2584 e = indexed(A, mu, nu) + 42;
2585 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2586 // this will throw an exception:
2587 // "add::get_free_indices: inconsistent indices in sum"
2591 @cindex @code{simplify_indexed()}
2592 @subsection Simplifying indexed expressions
2594 In addition to the few automatic simplifications that GiNaC performs on
2595 indexed expressions (such as re-ordering the indices of symmetric tensors
2596 and calculating traces and convolutions of matrices and predefined tensors)
2600 ex ex::simplify_indexed();
2601 ex ex::simplify_indexed(const scalar_products & sp);
2604 that performs some more expensive operations:
2607 @item it checks the consistency of free indices in sums in the same way
2608 @code{get_free_indices()} does
2609 @item it tries to give dummy indices that appear in different terms of a sum
2610 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2611 @item it (symbolically) calculates all possible dummy index summations/contractions
2612 with the predefined tensors (this will be explained in more detail in the
2614 @item it detects contractions that vanish for symmetry reasons, for example
2615 the contraction of a symmetric and a totally antisymmetric tensor
2616 @item as a special case of dummy index summation, it can replace scalar products
2617 of two tensors with a user-defined value
2620 The last point is done with the help of the @code{scalar_products} class
2621 which is used to store scalar products with known values (this is not an
2622 arithmetic class, you just pass it to @code{simplify_indexed()}):
2626 symbol A("A"), B("B"), C("C"), i_sym("i");
2630 sp.add(A, B, 0); // A and B are orthogonal
2631 sp.add(A, C, 0); // A and C are orthogonal
2632 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2634 e = indexed(A + B, i) * indexed(A + C, i);
2636 // -> (B+A).i*(A+C).i
2638 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2644 The @code{scalar_products} object @code{sp} acts as a storage for the
2645 scalar products added to it with the @code{.add()} method. This method
2646 takes three arguments: the two expressions of which the scalar product is
2647 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2648 @code{simplify_indexed()} will replace all scalar products of indexed
2649 objects that have the symbols @code{A} and @code{B} as base expressions
2650 with the single value 0. The number, type and dimension of the indices
2651 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2653 @cindex @code{expand()}
2654 The example above also illustrates a feature of the @code{expand()} method:
2655 if passed the @code{expand_indexed} option it will distribute indices
2656 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2658 @cindex @code{tensor} (class)
2659 @subsection Predefined tensors
2661 Some frequently used special tensors such as the delta, epsilon and metric
2662 tensors are predefined in GiNaC. They have special properties when
2663 contracted with other tensor expressions and some of them have constant
2664 matrix representations (they will evaluate to a number when numeric
2665 indices are specified).
2667 @cindex @code{delta_tensor()}
2668 @subsubsection Delta tensor
2670 The delta tensor takes two indices, is symmetric and has the matrix
2671 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2672 @code{delta_tensor()}:
2676 symbol A("A"), B("B");
2678 idx i(symbol("i"), 3), j(symbol("j"), 3),
2679 k(symbol("k"), 3), l(symbol("l"), 3);
2681 ex e = indexed(A, i, j) * indexed(B, k, l)
2682 * delta_tensor(i, k) * delta_tensor(j, l);
2683 cout << e.simplify_indexed() << endl;
2686 cout << delta_tensor(i, i) << endl;
2691 @cindex @code{metric_tensor()}
2692 @subsubsection General metric tensor
2694 The function @code{metric_tensor()} creates a general symmetric metric
2695 tensor with two indices that can be used to raise/lower tensor indices. The
2696 metric tensor is denoted as @samp{g} in the output and if its indices are of
2697 mixed variance it is automatically replaced by a delta tensor:
2703 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2705 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2706 cout << e.simplify_indexed() << endl;
2709 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2710 cout << e.simplify_indexed() << endl;
2713 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2714 * metric_tensor(nu, rho);
2715 cout << e.simplify_indexed() << endl;
2718 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2719 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2720 + indexed(A, mu.toggle_variance(), rho));
2721 cout << e.simplify_indexed() << endl;
2726 @cindex @code{lorentz_g()}
2727 @subsubsection Minkowski metric tensor
2729 The Minkowski metric tensor is a special metric tensor with a constant
2730 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2731 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2732 It is created with the function @code{lorentz_g()} (although it is output as
2737 varidx mu(symbol("mu"), 4);
2739 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2740 * lorentz_g(mu, varidx(0, 4)); // negative signature
2741 cout << e.simplify_indexed() << endl;
2744 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2745 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2746 cout << e.simplify_indexed() << endl;
2751 @cindex @code{spinor_metric()}
2752 @subsubsection Spinor metric tensor
2754 The function @code{spinor_metric()} creates an antisymmetric tensor with
2755 two indices that is used to raise/lower indices of 2-component spinors.
2756 It is output as @samp{eps}:
2762 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2763 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2765 e = spinor_metric(A, B) * indexed(psi, B_co);
2766 cout << e.simplify_indexed() << endl;
2769 e = spinor_metric(A, B) * indexed(psi, A_co);
2770 cout << e.simplify_indexed() << endl;
2773 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2774 cout << e.simplify_indexed() << endl;
2777 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2778 cout << e.simplify_indexed() << endl;
2781 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2782 cout << e.simplify_indexed() << endl;
2785 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2786 cout << e.simplify_indexed() << endl;
2791 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2793 @cindex @code{epsilon_tensor()}
2794 @cindex @code{lorentz_eps()}
2795 @subsubsection Epsilon tensor
2797 The epsilon tensor is totally antisymmetric, its number of indices is equal
2798 to the dimension of the index space (the indices must all be of the same
2799 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2800 defined to be 1. Its behavior with indices that have a variance also
2801 depends on the signature of the metric. Epsilon tensors are output as
2804 There are three functions defined to create epsilon tensors in 2, 3 and 4
2808 ex epsilon_tensor(const ex & i1, const ex & i2);
2809 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2810 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2811 bool pos_sig = false);
2814 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2815 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2816 Minkowski space (the last @code{bool} argument specifies whether the metric
2817 has negative or positive signature, as in the case of the Minkowski metric
2822 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2823 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2824 e = lorentz_eps(mu, nu, rho, sig) *
2825 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2826 cout << simplify_indexed(e) << endl;
2827 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2829 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2830 symbol A("A"), B("B");
2831 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2832 cout << simplify_indexed(e) << endl;
2833 // -> -B.k*A.j*eps.i.k.j
2834 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2835 cout << simplify_indexed(e) << endl;
2840 @subsection Linear algebra
2842 The @code{matrix} class can be used with indices to do some simple linear
2843 algebra (linear combinations and products of vectors and matrices, traces
2844 and scalar products):
2848 idx i(symbol("i"), 2), j(symbol("j"), 2);
2849 symbol x("x"), y("y");
2851 // A is a 2x2 matrix, X is a 2x1 vector
2852 matrix A(2, 2), X(2, 1);
2857 cout << indexed(A, i, i) << endl;
2860 ex e = indexed(A, i, j) * indexed(X, j);
2861 cout << e.simplify_indexed() << endl;
2862 // -> [[2*y+x],[4*y+3*x]].i
2864 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2865 cout << e.simplify_indexed() << endl;
2866 // -> [[3*y+3*x,6*y+2*x]].j
2870 You can of course obtain the same results with the @code{matrix::add()},
2871 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2872 but with indices you don't have to worry about transposing matrices.
2874 Matrix indices always start at 0 and their dimension must match the number
2875 of rows/columns of the matrix. Matrices with one row or one column are
2876 vectors and can have one or two indices (it doesn't matter whether it's a
2877 row or a column vector). Other matrices must have two indices.
2879 You should be careful when using indices with variance on matrices. GiNaC
2880 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2881 @samp{F.mu.nu} are different matrices. In this case you should use only
2882 one form for @samp{F} and explicitly multiply it with a matrix representation
2883 of the metric tensor.
2886 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2887 @c node-name, next, previous, up
2888 @section Non-commutative objects
2890 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2891 non-commutative objects are built-in which are mostly of use in high energy
2895 @item Clifford (Dirac) algebra (class @code{clifford})
2896 @item su(3) Lie algebra (class @code{color})
2897 @item Matrices (unindexed) (class @code{matrix})
2900 The @code{clifford} and @code{color} classes are subclasses of
2901 @code{indexed} because the elements of these algebras usually carry
2902 indices. The @code{matrix} class is described in more detail in
2905 Unlike most computer algebra systems, GiNaC does not primarily provide an
2906 operator (often denoted @samp{&*}) for representing inert products of
2907 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2908 classes of objects involved, and non-commutative products are formed with
2909 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2910 figuring out by itself which objects commutate and will group the factors
2911 by their class. Consider this example:
2915 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2916 idx a(symbol("a"), 8), b(symbol("b"), 8);
2917 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2919 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2923 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2924 groups the non-commutative factors (the gammas and the su(3) generators)
2925 together while preserving the order of factors within each class (because
2926 Clifford objects commutate with color objects). The resulting expression is a
2927 @emph{commutative} product with two factors that are themselves non-commutative
2928 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2929 parentheses are placed around the non-commutative products in the output.
2931 @cindex @code{ncmul} (class)
2932 Non-commutative products are internally represented by objects of the class
2933 @code{ncmul}, as opposed to commutative products which are handled by the
2934 @code{mul} class. You will normally not have to worry about this distinction,
2937 The advantage of this approach is that you never have to worry about using
2938 (or forgetting to use) a special operator when constructing non-commutative
2939 expressions. Also, non-commutative products in GiNaC are more intelligent
2940 than in other computer algebra systems; they can, for example, automatically
2941 canonicalize themselves according to rules specified in the implementation
2942 of the non-commutative classes. The drawback is that to work with other than
2943 the built-in algebras you have to implement new classes yourself. Symbols
2944 always commutate and it's not possible to construct non-commutative products
2945 using symbols to represent the algebra elements or generators. User-defined
2946 functions can, however, be specified as being non-commutative.
2948 @cindex @code{return_type()}
2949 @cindex @code{return_type_tinfo()}
2950 Information about the commutativity of an object or expression can be
2951 obtained with the two member functions
2954 unsigned ex::return_type() const;
2955 unsigned ex::return_type_tinfo() const;
2958 The @code{return_type()} function returns one of three values (defined in
2959 the header file @file{flags.h}), corresponding to three categories of
2960 expressions in GiNaC:
2963 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2964 classes are of this kind.
2965 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2966 certain class of non-commutative objects which can be determined with the
2967 @code{return_type_tinfo()} method. Expressions of this category commutate
2968 with everything except @code{noncommutative} expressions of the same
2970 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2971 of non-commutative objects of different classes. Expressions of this
2972 category don't commutate with any other @code{noncommutative} or
2973 @code{noncommutative_composite} expressions.
2976 The value returned by the @code{return_type_tinfo()} method is valid only
2977 when the return type of the expression is @code{noncommutative}. It is a
2978 value that is unique to the class of the object and usually one of the
2979 constants in @file{tinfos.h}, or derived therefrom.
2981 Here are a couple of examples:
2984 @multitable @columnfractions 0.33 0.33 0.34
2985 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2986 @item @code{42} @tab @code{commutative} @tab -
2987 @item @code{2*x-y} @tab @code{commutative} @tab -
2988 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2989 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2990 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2991 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2995 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2996 @code{TINFO_clifford} for objects with a representation label of zero.
2997 Other representation labels yield a different @code{return_type_tinfo()},
2998 but it's the same for any two objects with the same label. This is also true
3001 A last note: With the exception of matrices, positive integer powers of
3002 non-commutative objects are automatically expanded in GiNaC. For example,
3003 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3004 non-commutative expressions).
3007 @cindex @code{clifford} (class)
3008 @subsection Clifford algebra
3011 Clifford algebras are supported in two flavours: Dirac gamma
3012 matrices (more physical) and generic Clifford algebras (more
3015 @cindex @code{dirac_gamma()}
3016 @subsubsection Dirac gamma matrices
3017 Dirac gamma matrices (note that GiNaC doesn't treat them
3018 as matrices) are designated as @samp{gamma~mu} and satisfy
3019 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3020 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3021 constructed by the function
3024 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3027 which takes two arguments: the index and a @dfn{representation label} in the
3028 range 0 to 255 which is used to distinguish elements of different Clifford
3029 algebras (this is also called a @dfn{spin line index}). Gammas with different
3030 labels commutate with each other. The dimension of the index can be 4 or (in
3031 the framework of dimensional regularization) any symbolic value. Spinor
3032 indices on Dirac gammas are not supported in GiNaC.
3034 @cindex @code{dirac_ONE()}
3035 The unity element of a Clifford algebra is constructed by
3038 ex dirac_ONE(unsigned char rl = 0);
3041 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3042 multiples of the unity element, even though it's customary to omit it.
3043 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3044 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3045 GiNaC will complain and/or produce incorrect results.
3047 @cindex @code{dirac_gamma5()}
3048 There is a special element @samp{gamma5} that commutates with all other
3049 gammas, has a unit square, and in 4 dimensions equals
3050 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3053 ex dirac_gamma5(unsigned char rl = 0);
3056 @cindex @code{dirac_gammaL()}
3057 @cindex @code{dirac_gammaR()}
3058 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3059 objects, constructed by
3062 ex dirac_gammaL(unsigned char rl = 0);
3063 ex dirac_gammaR(unsigned char rl = 0);
3066 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3067 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3069 @cindex @code{dirac_slash()}
3070 Finally, the function
3073 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3076 creates a term that represents a contraction of @samp{e} with the Dirac
3077 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3078 with a unique index whose dimension is given by the @code{dim} argument).
3079 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3081 In products of dirac gammas, superfluous unity elements are automatically
3082 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3083 and @samp{gammaR} are moved to the front.
3085 The @code{simplify_indexed()} function performs contractions in gamma strings,
3091 symbol a("a"), b("b"), D("D");
3092 varidx mu(symbol("mu"), D);
3093 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3094 * dirac_gamma(mu.toggle_variance());
3096 // -> gamma~mu*a\*gamma.mu
3097 e = e.simplify_indexed();
3100 cout << e.subs(D == 4) << endl;
3106 @cindex @code{dirac_trace()}
3107 To calculate the trace of an expression containing strings of Dirac gammas
3108 you use one of the functions
3111 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3112 const ex & trONE = 4);
3113 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3114 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3117 These functions take the trace over all gammas in the specified set @code{rls}
3118 or list @code{rll} of representation labels, or the single label @code{rl};
3119 gammas with other labels are left standing. The last argument to
3120 @code{dirac_trace()} is the value to be returned for the trace of the unity
3121 element, which defaults to 4.
3123 The @code{dirac_trace()} function is a linear functional that is equal to the
3124 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3125 functional is not cyclic in
3128 dimensions when acting on
3129 expressions containing @samp{gamma5}, so it's not a proper trace. This
3130 @samp{gamma5} scheme is described in greater detail in
3131 @cite{The Role of gamma5 in Dimensional Regularization}.
3133 The value of the trace itself is also usually different in 4 and in
3141 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3142 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3143 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3144 cout << dirac_trace(e).simplify_indexed() << endl;
3151 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3152 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3153 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3154 cout << dirac_trace(e).simplify_indexed() << endl;
3155 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3159 Here is an example for using @code{dirac_trace()} to compute a value that
3160 appears in the calculation of the one-loop vacuum polarization amplitude in
3165 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3166 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3169 sp.add(l, l, pow(l, 2));
3170 sp.add(l, q, ldotq);
3172 ex e = dirac_gamma(mu) *
3173 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3174 dirac_gamma(mu.toggle_variance()) *
3175 (dirac_slash(l, D) + m * dirac_ONE());
3176 e = dirac_trace(e).simplify_indexed(sp);
3177 e = e.collect(lst(l, ldotq, m));
3179 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3183 The @code{canonicalize_clifford()} function reorders all gamma products that
3184 appear in an expression to a canonical (but not necessarily simple) form.
3185 You can use this to compare two expressions or for further simplifications:
3189 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3190 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3192 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3194 e = canonicalize_clifford(e);
3196 // -> 2*ONE*eta~mu~nu
3200 @cindex @code{clifford_unit()}
3201 @subsubsection A generic Clifford algebra
3203 A generic Clifford algebra, i.e. a
3207 dimensional algebra with
3211 satisfying the identities
3213 $e_i e_j + e_j e_i = M(i, j) $
3216 e~i e~j + e~j e~i = M(i, j)
3218 for some matrix (@code{metric})
3219 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3220 entries. Such generators are created by the function
3223 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3226 where @code{mu} should be a @code{varidx} class object indexing the
3227 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3228 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3229 object, optional parameter @code{rl} allows to distinguish different
3230 Clifford algebras (which will commute with each other). Note that the call
3231 @code{clifford_unit(mu, minkmetric())} creates something very close to
3232 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3233 metric defining this Clifford number.
3235 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3236 the Clifford algebra units with a call like that
3239 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3242 since this may yield some further automatic simplifications.
3244 Individual generators of a Clifford algebra can be accessed in several
3250 varidx nu(symbol("nu"), 4);
3252 ex M = diag_matrix(lst(1, -1, 0, s));
3253 ex e = clifford_unit(nu, M);
3254 ex e0 = e.subs(nu == 0);
3255 ex e1 = e.subs(nu == 1);
3256 ex e2 = e.subs(nu == 2);
3257 ex e3 = e.subs(nu == 3);
3262 will produce four anti-commuting generators of a Clifford algebra with properties
3264 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3267 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3270 @cindex @code{lst_to_clifford()}
3271 A similar effect can be achieved from the function
3274 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3275 unsigned char rl = 0);
3276 ex lst_to_clifford(const ex & v, const ex & e);
3279 which converts a list or vector
3281 $v = (v^0, v^1, ..., v^n)$
3284 @samp{v = (v~0, v~1, ..., v~n)}
3289 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3292 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3295 directly supplied in the second form of the procedure. In the first form
3296 the Clifford unit @samp{e.k} is generated by the call of
3297 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3298 with the help of @code{lst_to_clifford()} as follows
3303 varidx nu(symbol("nu"), 4);
3305 ex M = diag_matrix(lst(1, -1, 0, s));
3306 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3307 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3308 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3309 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3314 @cindex @code{clifford_to_lst()}
3315 There is the inverse function
3318 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3321 which takes an expression @code{e} and tries to find a list
3323 $v = (v^0, v^1, ..., v^n)$
3326 @samp{v = (v~0, v~1, ..., v~n)}
3330 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3333 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3335 with respect to the given Clifford units @code{c} and with none of the
3336 @samp{v~k} containing Clifford units @code{c} (of course, this
3337 may be impossible). This function can use an @code{algebraic} method
3338 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3340 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3343 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3345 is zero or is not a @code{numeric} for some @samp{k}
3346 then the method will be automatically changed to symbolic. The same effect
3347 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3349 @cindex @code{clifford_prime()}
3350 @cindex @code{clifford_star()}
3351 @cindex @code{clifford_bar()}
3352 There are several functions for (anti-)automorphisms of Clifford algebras:
3355 ex clifford_prime(const ex & e)
3356 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3357 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3360 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3361 changes signs of all Clifford units in the expression. The reversion
3362 of a Clifford algebra @code{clifford_star()} coincides with the
3363 @code{conjugate()} method and effectively reverses the order of Clifford
3364 units in any product. Finally the main anti-automorphism
3365 of a Clifford algebra @code{clifford_bar()} is the composition of the
3366 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3367 in a product. These functions correspond to the notations
3382 used in Clifford algebra textbooks.
3384 @cindex @code{clifford_norm()}
3388 ex clifford_norm(const ex & e);
3391 @cindex @code{clifford_inverse()}
3392 calculates the norm of a Clifford number from the expression
3394 $||e||^2 = e\overline{e}$.
3397 @code{||e||^2 = e \bar@{e@}}
3399 The inverse of a Clifford expression is returned by the function
3402 ex clifford_inverse(const ex & e);
3405 which calculates it as
3407 $e^{-1} = \overline{e}/||e||^2$.
3410 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3419 then an exception is raised.
3421 @cindex @code{remove_dirac_ONE()}
3422 If a Clifford number happens to be a factor of
3423 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3424 expression by the function
3427 ex remove_dirac_ONE(const ex & e);
3430 @cindex @code{canonicalize_clifford()}
3431 The function @code{canonicalize_clifford()} works for a
3432 generic Clifford algebra in a similar way as for Dirac gammas.
3434 The last provided function is
3436 @cindex @code{clifford_moebius_map()}
3438 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3439 const ex & d, const ex & v, const ex & G,
3440 unsigned char rl = 0);
3441 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3442 unsigned char rl = 0);
3445 It takes a list or vector @code{v} and makes the Moebius (conformal or
3446 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3447 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3448 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3449 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3450 ignored even if supplied. The returned value of this function is a list
3451 of components of the resulting vector.
3453 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3454 where @code{1} is the @code{representation_label} and @code{\nu} is the
3455 index of the corresponding unit. This provides a flexible typesetting
3456 with a suitable defintion of the @code{\clifford} command. For example, the
3459 \newcommand@{\clifford@}[1][]@{@}
3461 typesets all Clifford units identically, while the alternative definition
3463 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3465 prints units with @code{representation_label=0} as
3472 with @code{representation_label=1} as
3479 and with @code{representation_label=2} as
3487 @cindex @code{color} (class)
3488 @subsection Color algebra
3490 @cindex @code{color_T()}
3491 For computations in quantum chromodynamics, GiNaC implements the base elements
3492 and structure constants of the su(3) Lie algebra (color algebra). The base
3493 elements @math{T_a} are constructed by the function
3496 ex color_T(const ex & a, unsigned char rl = 0);
3499 which takes two arguments: the index and a @dfn{representation label} in the
3500 range 0 to 255 which is used to distinguish elements of different color
3501 algebras. Objects with different labels commutate with each other. The
3502 dimension of the index must be exactly 8 and it should be of class @code{idx},
3505 @cindex @code{color_ONE()}
3506 The unity element of a color algebra is constructed by
3509 ex color_ONE(unsigned char rl = 0);
3512 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3513 multiples of the unity element, even though it's customary to omit it.
3514 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3515 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3516 GiNaC may produce incorrect results.
3518 @cindex @code{color_d()}
3519 @cindex @code{color_f()}
3523 ex color_d(const ex & a, const ex & b, const ex & c);
3524 ex color_f(const ex & a, const ex & b, const ex & c);
3527 create the symmetric and antisymmetric structure constants @math{d_abc} and
3528 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3529 and @math{[T_a, T_b] = i f_abc T_c}.
3531 These functions evaluate to their numerical values,
3532 if you supply numeric indices to them. The index values should be in
3533 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3534 goes along better with the notations used in physical literature.
3536 @cindex @code{color_h()}
3537 There's an additional function
3540 ex color_h(const ex & a, const ex & b, const ex & c);
3543 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3545 The function @code{simplify_indexed()} performs some simplifications on
3546 expressions containing color objects:
3551 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3552 k(symbol("k"), 8), l(symbol("l"), 8);
3554 e = color_d(a, b, l) * color_f(a, b, k);
3555 cout << e.simplify_indexed() << endl;
3558 e = color_d(a, b, l) * color_d(a, b, k);
3559 cout << e.simplify_indexed() << endl;
3562 e = color_f(l, a, b) * color_f(a, b, k);
3563 cout << e.simplify_indexed() << endl;
3566 e = color_h(a, b, c) * color_h(a, b, c);
3567 cout << e.simplify_indexed() << endl;
3570 e = color_h(a, b, c) * color_T(b) * color_T(c);
3571 cout << e.simplify_indexed() << endl;
3574 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3575 cout << e.simplify_indexed() << endl;
3578 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3579 cout << e.simplify_indexed() << endl;
3580 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3584 @cindex @code{color_trace()}
3585 To calculate the trace of an expression containing color objects you use one
3589 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3590 ex color_trace(const ex & e, const lst & rll);
3591 ex color_trace(const ex & e, unsigned char rl = 0);
3594 These functions take the trace over all color @samp{T} objects in the
3595 specified set @code{rls} or list @code{rll} of representation labels, or the
3596 single label @code{rl}; @samp{T}s with other labels are left standing. For
3601 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3603 // -> -I*f.a.c.b+d.a.c.b
3608 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3609 @c node-name, next, previous, up
3612 @cindex @code{exhashmap} (class)
3614 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3615 that can be used as a drop-in replacement for the STL
3616 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3617 typically constant-time, element look-up than @code{map<>}.
3619 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3620 following differences:
3624 no @code{lower_bound()} and @code{upper_bound()} methods
3626 no reverse iterators, no @code{rbegin()}/@code{rend()}
3628 no @code{operator<(exhashmap, exhashmap)}
3630 the comparison function object @code{key_compare} is hardcoded to
3633 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3634 initial hash table size (the actual table size after construction may be
3635 larger than the specified value)
3637 the method @code{size_t bucket_count()} returns the current size of the hash
3640 @code{insert()} and @code{erase()} operations invalidate all iterators
3644 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3645 @c node-name, next, previous, up
3646 @chapter Methods and Functions
3649 In this chapter the most important algorithms provided by GiNaC will be
3650 described. Some of them are implemented as functions on expressions,
3651 others are implemented as methods provided by expression objects. If
3652 they are methods, there exists a wrapper function around it, so you can
3653 alternatively call it in a functional way as shown in the simple
3658 cout << "As method: " << sin(1).evalf() << endl;
3659 cout << "As function: " << evalf(sin(1)) << endl;
3663 @cindex @code{subs()}
3664 The general rule is that wherever methods accept one or more parameters
3665 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3666 wrapper accepts is the same but preceded by the object to act on
3667 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3668 most natural one in an OO model but it may lead to confusion for MapleV
3669 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3670 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3671 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3672 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3673 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3674 here. Also, users of MuPAD will in most cases feel more comfortable
3675 with GiNaC's convention. All function wrappers are implemented
3676 as simple inline functions which just call the corresponding method and
3677 are only provided for users uncomfortable with OO who are dead set to
3678 avoid method invocations. Generally, nested function wrappers are much
3679 harder to read than a sequence of methods and should therefore be
3680 avoided if possible. On the other hand, not everything in GiNaC is a
3681 method on class @code{ex} and sometimes calling a function cannot be
3685 * Information About Expressions::
3686 * Numerical Evaluation::
3687 * Substituting Expressions::
3688 * Pattern Matching and Advanced Substitutions::
3689 * Applying a Function on Subexpressions::
3690 * Visitors and Tree Traversal::
3691 * Polynomial Arithmetic:: Working with polynomials.
3692 * Rational Expressions:: Working with rational functions.
3693 * Symbolic Differentiation::
3694 * Series Expansion:: Taylor and Laurent expansion.
3696 * Built-in Functions:: List of predefined mathematical functions.
3697 * Multiple polylogarithms::
3698 * Complex Conjugation::
3699 * Built-in Functions:: List of predefined mathematical functions.
3700 * Solving Linear Systems of Equations::
3701 * Input/Output:: Input and output of expressions.
3705 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3706 @c node-name, next, previous, up
3707 @section Getting information about expressions
3709 @subsection Checking expression types
3710 @cindex @code{is_a<@dots{}>()}
3711 @cindex @code{is_exactly_a<@dots{}>()}
3712 @cindex @code{ex_to<@dots{}>()}
3713 @cindex Converting @code{ex} to other classes
3714 @cindex @code{info()}
3715 @cindex @code{return_type()}
3716 @cindex @code{return_type_tinfo()}
3718 Sometimes it's useful to check whether a given expression is a plain number,
3719 a sum, a polynomial with integer coefficients, or of some other specific type.
3720 GiNaC provides a couple of functions for this:
3723 bool is_a<T>(const ex & e);
3724 bool is_exactly_a<T>(const ex & e);
3725 bool ex::info(unsigned flag);
3726 unsigned ex::return_type() const;
3727 unsigned ex::return_type_tinfo() const;
3730 When the test made by @code{is_a<T>()} returns true, it is safe to call
3731 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3732 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3733 example, assuming @code{e} is an @code{ex}:
3738 if (is_a<numeric>(e))
3739 numeric n = ex_to<numeric>(e);
3744 @code{is_a<T>(e)} allows you to check whether the top-level object of
3745 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3746 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3747 e.g., for checking whether an expression is a number, a sum, or a product:
3754 is_a<numeric>(e1); // true
3755 is_a<numeric>(e2); // false
3756 is_a<add>(e1); // false
3757 is_a<add>(e2); // true
3758 is_a<mul>(e1); // false
3759 is_a<mul>(e2); // false
3763 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3764 top-level object of an expression @samp{e} is an instance of the GiNaC
3765 class @samp{T}, not including parent classes.
3767 The @code{info()} method is used for checking certain attributes of
3768 expressions. The possible values for the @code{flag} argument are defined
3769 in @file{ginac/flags.h}, the most important being explained in the following
3773 @multitable @columnfractions .30 .70
3774 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3775 @item @code{numeric}
3776 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3778 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3779 @item @code{rational}
3780 @tab @dots{}an exact rational number (integers are rational, too)
3781 @item @code{integer}
3782 @tab @dots{}a (non-complex) integer
3783 @item @code{crational}
3784 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3785 @item @code{cinteger}
3786 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3787 @item @code{positive}
3788 @tab @dots{}not complex and greater than 0
3789 @item @code{negative}
3790 @tab @dots{}not complex and less than 0
3791 @item @code{nonnegative}
3792 @tab @dots{}not complex and greater than or equal to 0
3794 @tab @dots{}an integer greater than 0
3796 @tab @dots{}an integer less than 0
3797 @item @code{nonnegint}
3798 @tab @dots{}an integer greater than or equal to 0
3800 @tab @dots{}an even integer
3802 @tab @dots{}an odd integer
3804 @tab @dots{}a prime integer (probabilistic primality test)
3805 @item @code{relation}
3806 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3807 @item @code{relation_equal}
3808 @tab @dots{}a @code{==} relation
3809 @item @code{relation_not_equal}
3810 @tab @dots{}a @code{!=} relation
3811 @item @code{relation_less}
3812 @tab @dots{}a @code{<} relation
3813 @item @code{relation_less_or_equal}
3814 @tab @dots{}a @code{<=} relation
3815 @item @code{relation_greater}
3816 @tab @dots{}a @code{>} relation
3817 @item @code{relation_greater_or_equal}
3818 @tab @dots{}a @code{>=} relation
3820 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3822 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3823 @item @code{polynomial}
3824 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3825 @item @code{integer_polynomial}
3826 @tab @dots{}a polynomial with (non-complex) integer coefficients
3827 @item @code{cinteger_polynomial}
3828 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3829 @item @code{rational_polynomial}
3830 @tab @dots{}a polynomial with (non-complex) rational coefficients
3831 @item @code{crational_polynomial}
3832 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3833 @item @code{rational_function}
3834 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3835 @item @code{algebraic}
3836 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3840 To determine whether an expression is commutative or non-commutative and if
3841 so, with which other expressions it would commutate, you use the methods
3842 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3843 for an explanation of these.
3846 @subsection Accessing subexpressions
3849 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3850 @code{function}, act as containers for subexpressions. For example, the
3851 subexpressions of a sum (an @code{add} object) are the individual terms,
3852 and the subexpressions of a @code{function} are the function's arguments.
3854 @cindex @code{nops()}
3856 GiNaC provides several ways of accessing subexpressions. The first way is to
3861 ex ex::op(size_t i);
3864 @code{nops()} determines the number of subexpressions (operands) contained
3865 in the expression, while @code{op(i)} returns the @code{i}-th
3866 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3867 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3868 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3869 @math{i>0} are the indices.
3872 @cindex @code{const_iterator}
3873 The second way to access subexpressions is via the STL-style random-access
3874 iterator class @code{const_iterator} and the methods
3877 const_iterator ex::begin();
3878 const_iterator ex::end();
3881 @code{begin()} returns an iterator referring to the first subexpression;
3882 @code{end()} returns an iterator which is one-past the last subexpression.
3883 If the expression has no subexpressions, then @code{begin() == end()}. These
3884 iterators can also be used in conjunction with non-modifying STL algorithms.
3886 Here is an example that (non-recursively) prints the subexpressions of a
3887 given expression in three different ways:
3894 for (size_t i = 0; i != e.nops(); ++i)
3895 cout << e.op(i) << endl;
3898 for (const_iterator i = e.begin(); i != e.end(); ++i)
3901 // with iterators and STL copy()
3902 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3906 @cindex @code{const_preorder_iterator}
3907 @cindex @code{const_postorder_iterator}
3908 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3909 expression's immediate children. GiNaC provides two additional iterator
3910 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3911 that iterate over all objects in an expression tree, in preorder or postorder,
3912 respectively. They are STL-style forward iterators, and are created with the
3916 const_preorder_iterator ex::preorder_begin();
3917 const_preorder_iterator ex::preorder_end();
3918 const_postorder_iterator ex::postorder_begin();
3919 const_postorder_iterator ex::postorder_end();
3922 The following example illustrates the differences between
3923 @code{const_iterator}, @code{const_preorder_iterator}, and
3924 @code{const_postorder_iterator}:
3928 symbol A("A"), B("B"), C("C");
3929 ex e = lst(lst(A, B), C);
3931 std::copy(e.begin(), e.end(),
3932 std::ostream_iterator<ex>(cout, "\n"));
3936 std::copy(e.preorder_begin(), e.preorder_end(),
3937 std::ostream_iterator<ex>(cout, "\n"));
3944 std::copy(e.postorder_begin(), e.postorder_end(),
3945 std::ostream_iterator<ex>(cout, "\n"));
3954 @cindex @code{relational} (class)
3955 Finally, the left-hand side and right-hand side expressions of objects of
3956 class @code{relational} (and only of these) can also be accessed with the
3965 @subsection Comparing expressions
3966 @cindex @code{is_equal()}
3967 @cindex @code{is_zero()}
3969 Expressions can be compared with the usual C++ relational operators like
3970 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3971 the result is usually not determinable and the result will be @code{false},
3972 except in the case of the @code{!=} operator. You should also be aware that
3973 GiNaC will only do the most trivial test for equality (subtracting both
3974 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3977 Actually, if you construct an expression like @code{a == b}, this will be
3978 represented by an object of the @code{relational} class (@pxref{Relations})
3979 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3981 There are also two methods
3984 bool ex::is_equal(const ex & other);
3988 for checking whether one expression is equal to another, or equal to zero,
3992 @subsection Ordering expressions
3993 @cindex @code{ex_is_less} (class)
3994 @cindex @code{ex_is_equal} (class)
3995 @cindex @code{compare()}
3997 Sometimes it is necessary to establish a mathematically well-defined ordering
3998 on a set of arbitrary expressions, for example to use expressions as keys
3999 in a @code{std::map<>} container, or to bring a vector of expressions into
4000 a canonical order (which is done internally by GiNaC for sums and products).
4002 The operators @code{<}, @code{>} etc. described in the last section cannot
4003 be used for this, as they don't implement an ordering relation in the
4004 mathematical sense. In particular, they are not guaranteed to be
4005 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4006 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4009 By default, STL classes and algorithms use the @code{<} and @code{==}
4010 operators to compare objects, which are unsuitable for expressions, but GiNaC
4011 provides two functors that can be supplied as proper binary comparison
4012 predicates to the STL:
4015 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4017 bool operator()(const ex &lh, const ex &rh) const;
4020 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4022 bool operator()(const ex &lh, const ex &rh) const;
4026 For example, to define a @code{map} that maps expressions to strings you
4030 std::map<ex, std::string, ex_is_less> myMap;
4033 Omitting the @code{ex_is_less} template parameter will introduce spurious
4034 bugs because the map operates improperly.
4036 Other examples for the use of the functors:
4044 std::sort(v.begin(), v.end(), ex_is_less());
4046 // count the number of expressions equal to '1'
4047 unsigned num_ones = std::count_if(v.begin(), v.end(),
4048 std::bind2nd(ex_is_equal(), 1));
4051 The implementation of @code{ex_is_less} uses the member function
4054 int ex::compare(const ex & other) const;
4057 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4058 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4062 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4063 @c node-name, next, previous, up
4064 @section Numerical Evaluation
4065 @cindex @code{evalf()}
4067 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4068 To evaluate them using floating-point arithmetic you need to call
4071 ex ex::evalf(int level = 0) const;
4074 @cindex @code{Digits}
4075 The accuracy of the evaluation is controlled by the global object @code{Digits}
4076 which can be assigned an integer value. The default value of @code{Digits}
4077 is 17. @xref{Numbers}, for more information and examples.
4079 To evaluate an expression to a @code{double} floating-point number you can
4080 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4084 // Approximate sin(x/Pi)
4086 ex e = series(sin(x/Pi), x == 0, 6);
4088 // Evaluate numerically at x=0.1
4089 ex f = evalf(e.subs(x == 0.1));
4091 // ex_to<numeric> is an unsafe cast, so check the type first
4092 if (is_a<numeric>(f)) @{
4093 double d = ex_to<numeric>(f).to_double();
4102 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4103 @c node-name, next, previous, up
4104 @section Substituting expressions
4105 @cindex @code{subs()}
4107 Algebraic objects inside expressions can be replaced with arbitrary
4108 expressions via the @code{.subs()} method:
4111 ex ex::subs(const ex & e, unsigned options = 0);
4112 ex ex::subs(const exmap & m, unsigned options = 0);
4113 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4116 In the first form, @code{subs()} accepts a relational of the form
4117 @samp{object == expression} or a @code{lst} of such relationals:
4121 symbol x("x"), y("y");
4123 ex e1 = 2*x^2-4*x+3;
4124 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4128 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4133 If you specify multiple substitutions, they are performed in parallel, so e.g.
4134 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4136 The second form of @code{subs()} takes an @code{exmap} object which is a
4137 pair associative container that maps expressions to expressions (currently
4138 implemented as a @code{std::map}). This is the most efficient one of the
4139 three @code{subs()} forms and should be used when the number of objects to
4140 be substituted is large or unknown.
4142 Using this form, the second example from above would look like this:
4146 symbol x("x"), y("y");
4152 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4156 The third form of @code{subs()} takes two lists, one for the objects to be
4157 replaced and one for the expressions to be substituted (both lists must
4158 contain the same number of elements). Using this form, you would write
4162 symbol x("x"), y("y");
4165 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4169 The optional last argument to @code{subs()} is a combination of
4170 @code{subs_options} flags. There are two options available:
4171 @code{subs_options::no_pattern} disables pattern matching, which makes
4172 large @code{subs()} operations significantly faster if you are not using
4173 patterns. The second option, @code{subs_options::algebraic} enables
4174 algebraic substitutions in products and powers.
4175 @ref{Pattern Matching and Advanced Substitutions}, for more information
4176 about patterns and algebraic substitutions.
4178 @code{subs()} performs syntactic substitution of any complete algebraic
4179 object; it does not try to match sub-expressions as is demonstrated by the
4184 symbol x("x"), y("y"), z("z");
4186 ex e1 = pow(x+y, 2);
4187 cout << e1.subs(x+y == 4) << endl;
4190 ex e2 = sin(x)*sin(y)*cos(x);
4191 cout << e2.subs(sin(x) == cos(x)) << endl;
4192 // -> cos(x)^2*sin(y)
4195 cout << e3.subs(x+y == 4) << endl;
4197 // (and not 4+z as one might expect)
4201 A more powerful form of substitution using wildcards is described in the
4205 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4206 @c node-name, next, previous, up
4207 @section Pattern matching and advanced substitutions
4208 @cindex @code{wildcard} (class)
4209 @cindex Pattern matching
4211 GiNaC allows the use of patterns for checking whether an expression is of a
4212 certain form or contains subexpressions of a certain form, and for
4213 substituting expressions in a more general way.
4215 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4216 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4217 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4218 an unsigned integer number to allow having multiple different wildcards in a
4219 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4220 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4224 ex wild(unsigned label = 0);
4227 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4230 Some examples for patterns:
4232 @multitable @columnfractions .5 .5
4233 @item @strong{Constructed as} @tab @strong{Output as}
4234 @item @code{wild()} @tab @samp{$0}
4235 @item @code{pow(x,wild())} @tab @samp{x^$0}
4236 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4237 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4243 @item Wildcards behave like symbols and are subject to the same algebraic
4244 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4245 @item As shown in the last example, to use wildcards for indices you have to
4246 use them as the value of an @code{idx} object. This is because indices must
4247 always be of class @code{idx} (or a subclass).
4248 @item Wildcards only represent expressions or subexpressions. It is not
4249 possible to use them as placeholders for other properties like index
4250 dimension or variance, representation labels, symmetry of indexed objects
4252 @item Because wildcards are commutative, it is not possible to use wildcards
4253 as part of noncommutative products.
4254 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4255 are also valid patterns.
4258 @subsection Matching expressions
4259 @cindex @code{match()}
4260 The most basic application of patterns is to check whether an expression
4261 matches a given pattern. This is done by the function
4264 bool ex::match(const ex & pattern);
4265 bool ex::match(const ex & pattern, lst & repls);
4268 This function returns @code{true} when the expression matches the pattern
4269 and @code{false} if it doesn't. If used in the second form, the actual
4270 subexpressions matched by the wildcards get returned in the @code{repls}
4271 object as a list of relations of the form @samp{wildcard == expression}.
4272 If @code{match()} returns false, the state of @code{repls} is undefined.
4273 For reproducible results, the list should be empty when passed to
4274 @code{match()}, but it is also possible to find similarities in multiple
4275 expressions by passing in the result of a previous match.
4277 The matching algorithm works as follows:
4280 @item A single wildcard matches any expression. If one wildcard appears
4281 multiple times in a pattern, it must match the same expression in all
4282 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4283 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4284 @item If the expression is not of the same class as the pattern, the match
4285 fails (i.e. a sum only matches a sum, a function only matches a function,
4287 @item If the pattern is a function, it only matches the same function
4288 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4289 @item Except for sums and products, the match fails if the number of
4290 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4292 @item If there are no subexpressions, the expressions and the pattern must
4293 be equal (in the sense of @code{is_equal()}).
4294 @item Except for sums and products, each subexpression (@code{op()}) must
4295 match the corresponding subexpression of the pattern.
4298 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4299 account for their commutativity and associativity:
4302 @item If the pattern contains a term or factor that is a single wildcard,
4303 this one is used as the @dfn{global wildcard}. If there is more than one
4304 such wildcard, one of them is chosen as the global wildcard in a random
4306 @item Every term/factor of the pattern, except the global wildcard, is
4307 matched against every term of the expression in sequence. If no match is
4308 found, the whole match fails. Terms that did match are not considered in
4310 @item If there are no unmatched terms left, the match succeeds. Otherwise
4311 the match fails unless there is a global wildcard in the pattern, in
4312 which case this wildcard matches the remaining terms.
4315 In general, having more than one single wildcard as a term of a sum or a
4316 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4319 Here are some examples in @command{ginsh} to demonstrate how it works (the
4320 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4321 match fails, and the list of wildcard replacements otherwise):
4324 > match((x+y)^a,(x+y)^a);
4326 > match((x+y)^a,(x+y)^b);
4328 > match((x+y)^a,$1^$2);
4330 > match((x+y)^a,$1^$1);
4332 > match((x+y)^(x+y),$1^$1);
4334 > match((x+y)^(x+y),$1^$2);
4336 > match((a+b)*(a+c),($1+b)*($1+c));
4338 > match((a+b)*(a+c),(a+$1)*(a+$2));
4340 (Unpredictable. The result might also be [$1==c,$2==b].)
4341 > match((a+b)*(a+c),($1+$2)*($1+$3));
4342 (The result is undefined. Due to the sequential nature of the algorithm
4343 and the re-ordering of terms in GiNaC, the match for the first factor
4344 may be @{$1==a,$2==b@} in which case the match for the second factor
4345 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4347 > match(a*(x+y)+a*z+b,a*$1+$2);
4348 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4349 @{$1=x+y,$2=a*z+b@}.)
4350 > match(a+b+c+d+e+f,c);
4352 > match(a+b+c+d+e+f,c+$0);
4354 > match(a+b+c+d+e+f,c+e+$0);
4356 > match(a+b,a+b+$0);
4358 > match(a*b^2,a^$1*b^$2);
4360 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4361 even though a==a^1.)
4362 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4364 > match(atan2(y,x^2),atan2(y,$0));
4368 @subsection Matching parts of expressions
4369 @cindex @code{has()}
4370 A more general way to look for patterns in expressions is provided by the
4374 bool ex::has(const ex & pattern);
4377 This function checks whether a pattern is matched by an expression itself or
4378 by any of its subexpressions.
4380 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4381 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4384 > has(x*sin(x+y+2*a),y);
4386 > has(x*sin(x+y+2*a),x+y);
4388 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4389 has the subexpressions "x", "y" and "2*a".)
4390 > has(x*sin(x+y+2*a),x+y+$1);
4392 (But this is possible.)
4393 > has(x*sin(2*(x+y)+2*a),x+y);
4395 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4396 which "x+y" is not a subexpression.)
4399 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4401 > has(4*x^2-x+3,$1*x);
4403 > has(4*x^2+x+3,$1*x);
4405 (Another possible pitfall. The first expression matches because the term
4406 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4407 contains a linear term you should use the coeff() function instead.)
4410 @cindex @code{find()}
4414 bool ex::find(const ex & pattern, lst & found);
4417 works a bit like @code{has()} but it doesn't stop upon finding the first
4418 match. Instead, it appends all found matches to the specified list. If there
4419 are multiple occurrences of the same expression, it is entered only once to
4420 the list. @code{find()} returns false if no matches were found (in
4421 @command{ginsh}, it returns an empty list):
4424 > find(1+x+x^2+x^3,x);
4426 > find(1+x+x^2+x^3,y);
4428 > find(1+x+x^2+x^3,x^$1);
4430 (Note the absence of "x".)
4431 > expand((sin(x)+sin(y))*(a+b));
4432 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4437 @subsection Substituting expressions
4438 @cindex @code{subs()}
4439 Probably the most useful application of patterns is to use them for
4440 substituting expressions with the @code{subs()} method. Wildcards can be
4441 used in the search patterns as well as in the replacement expressions, where
4442 they get replaced by the expressions matched by them. @code{subs()} doesn't
4443 know anything about algebra; it performs purely syntactic substitutions.
4448 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4450 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4452 > subs((a+b+c)^2,a+b==x);
4454 > subs((a+b+c)^2,a+b+$1==x+$1);
4456 > subs(a+2*b,a+b==x);
4458 > subs(4*x^3-2*x^2+5*x-1,x==a);
4460 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4462 > subs(sin(1+sin(x)),sin($1)==cos($1));
4464 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4468 The last example would be written in C++ in this way:
4472 symbol a("a"), b("b"), x("x"), y("y");
4473 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4474 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4475 cout << e.expand() << endl;
4480 @subsection Algebraic substitutions
4481 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4482 enables smarter, algebraic substitutions in products and powers. If you want
4483 to substitute some factors of a product, you only need to list these factors
4484 in your pattern. Furthermore, if an (integer) power of some expression occurs
4485 in your pattern and in the expression that you want the substitution to occur
4486 in, it can be substituted as many times as possible, without getting negative
4489 An example clarifies it all (hopefully):
4492 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4493 subs_options::algebraic) << endl;
4494 // --> (y+x)^6+b^6+a^6
4496 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4498 // Powers and products are smart, but addition is just the same.
4500 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4503 // As I said: addition is just the same.
4505 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4506 // --> x^3*b*a^2+2*b
4508 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4510 // --> 2*b+x^3*b^(-1)*a^(-2)
4512 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4513 // --> -1-2*a^2+4*a^3+5*a
4515 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4516 subs_options::algebraic) << endl;
4517 // --> -1+5*x+4*x^3-2*x^2
4518 // You should not really need this kind of patterns very often now.
4519 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4521 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4522 subs_options::algebraic) << endl;
4523 // --> cos(1+cos(x))
4525 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4526 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4527 subs_options::algebraic)) << endl;
4532 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4533 @c node-name, next, previous, up
4534 @section Applying a Function on Subexpressions
4535 @cindex tree traversal
4536 @cindex @code{map()}
4538 Sometimes you may want to perform an operation on specific parts of an
4539 expression while leaving the general structure of it intact. An example
4540 of this would be a matrix trace operation: the trace of a sum is the sum
4541 of the traces of the individual terms. That is, the trace should @dfn{map}
4542 on the sum, by applying itself to each of the sum's operands. It is possible
4543 to do this manually which usually results in code like this:
4548 if (is_a<matrix>(e))
4549 return ex_to<matrix>(e).trace();
4550 else if (is_a<add>(e)) @{
4552 for (size_t i=0; i<e.nops(); i++)
4553 sum += calc_trace(e.op(i));
4555 @} else if (is_a<mul>)(e)) @{
4563 This is, however, slightly inefficient (if the sum is very large it can take
4564 a long time to add the terms one-by-one), and its applicability is limited to
4565 a rather small class of expressions. If @code{calc_trace()} is called with
4566 a relation or a list as its argument, you will probably want the trace to
4567 be taken on both sides of the relation or of all elements of the list.
4569 GiNaC offers the @code{map()} method to aid in the implementation of such
4573 ex ex::map(map_function & f) const;
4574 ex ex::map(ex (*f)(const ex & e)) const;
4577 In the first (preferred) form, @code{map()} takes a function object that
4578 is subclassed from the @code{map_function} class. In the second form, it
4579 takes a pointer to a function that accepts and returns an expression.
4580 @code{map()} constructs a new expression of the same type, applying the
4581 specified function on all subexpressions (in the sense of @code{op()}),
4584 The use of a function object makes it possible to supply more arguments to
4585 the function that is being mapped, or to keep local state information.
4586 The @code{map_function} class declares a virtual function call operator
4587 that you can overload. Here is a sample implementation of @code{calc_trace()}
4588 that uses @code{map()} in a recursive fashion:
4591 struct calc_trace : public map_function @{
4592 ex operator()(const ex &e)
4594 if (is_a<matrix>(e))
4595 return ex_to<matrix>(e).trace();
4596 else if (is_a<mul>(e)) @{
4599 return e.map(*this);
4604 This function object could then be used like this:
4608 ex M = ... // expression with matrices
4609 calc_trace do_trace;
4610 ex tr = do_trace(M);
4614 Here is another example for you to meditate over. It removes quadratic
4615 terms in a variable from an expanded polynomial:
4618 struct map_rem_quad : public map_function @{
4620 map_rem_quad(const ex & var_) : var(var_) @{@}
4622 ex operator()(const ex & e)
4624 if (is_a<add>(e) || is_a<mul>(e))
4625 return e.map(*this);
4626 else if (is_a<power>(e) &&
4627 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4637 symbol x("x"), y("y");
4640 for (int i=0; i<8; i++)
4641 e += pow(x, i) * pow(y, 8-i) * (i+1);
4643 // -> 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
4645 map_rem_quad rem_quad(x);
4646 cout << rem_quad(e) << endl;
4647 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4651 @command{ginsh} offers a slightly different implementation of @code{map()}
4652 that allows applying algebraic functions to operands. The second argument
4653 to @code{map()} is an expression containing the wildcard @samp{$0} which
4654 acts as the placeholder for the operands:
4659 > map(a+2*b,sin($0));
4661 > map(@{a,b,c@},$0^2+$0);
4662 @{a^2+a,b^2+b,c^2+c@}
4665 Note that it is only possible to use algebraic functions in the second
4666 argument. You can not use functions like @samp{diff()}, @samp{op()},
4667 @samp{subs()} etc. because these are evaluated immediately:
4670 > map(@{a,b,c@},diff($0,a));
4672 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4673 to "map(@{a,b,c@},0)".
4677 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4678 @c node-name, next, previous, up
4679 @section Visitors and Tree Traversal
4680 @cindex tree traversal
4681 @cindex @code{visitor} (class)
4682 @cindex @code{accept()}
4683 @cindex @code{visit()}
4684 @cindex @code{traverse()}
4685 @cindex @code{traverse_preorder()}
4686 @cindex @code{traverse_postorder()}
4688 Suppose that you need a function that returns a list of all indices appearing
4689 in an arbitrary expression. The indices can have any dimension, and for
4690 indices with variance you always want the covariant version returned.
4692 You can't use @code{get_free_indices()} because you also want to include
4693 dummy indices in the list, and you can't use @code{find()} as it needs
4694 specific index dimensions (and it would require two passes: one for indices
4695 with variance, one for plain ones).
4697 The obvious solution to this problem is a tree traversal with a type switch,
4698 such as the following:
4701 void gather_indices_helper(const ex & e, lst & l)
4703 if (is_a<varidx>(e)) @{
4704 const varidx & vi = ex_to<varidx>(e);
4705 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4706 @} else if (is_a<idx>(e)) @{
4709 size_t n = e.nops();
4710 for (size_t i = 0; i < n; ++i)
4711 gather_indices_helper(e.op(i), l);
4715 lst gather_indices(const ex & e)
4718 gather_indices_helper(e, l);
4725 This works fine but fans of object-oriented programming will feel
4726 uncomfortable with the type switch. One reason is that there is a possibility
4727 for subtle bugs regarding derived classes. If we had, for example, written
4730 if (is_a<idx>(e)) @{
4732 @} else if (is_a<varidx>(e)) @{
4736 in @code{gather_indices_helper}, the code wouldn't have worked because the
4737 first line "absorbs" all classes derived from @code{idx}, including
4738 @code{varidx}, so the special case for @code{varidx} would never have been
4741 Also, for a large number of classes, a type switch like the above can get
4742 unwieldy and inefficient (it's a linear search, after all).
4743 @code{gather_indices_helper} only checks for two classes, but if you had to
4744 write a function that required a different implementation for nearly
4745 every GiNaC class, the result would be very hard to maintain and extend.
4747 The cleanest approach to the problem would be to add a new virtual function
4748 to GiNaC's class hierarchy. In our example, there would be specializations
4749 for @code{idx} and @code{varidx} while the default implementation in
4750 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4751 impossible to add virtual member functions to existing classes without
4752 changing their source and recompiling everything. GiNaC comes with source,
4753 so you could actually do this, but for a small algorithm like the one
4754 presented this would be impractical.
4756 One solution to this dilemma is the @dfn{Visitor} design pattern,
4757 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4758 variation, described in detail in
4759 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4760 virtual functions to the class hierarchy to implement operations, GiNaC
4761 provides a single "bouncing" method @code{accept()} that takes an instance
4762 of a special @code{visitor} class and redirects execution to the one
4763 @code{visit()} virtual function of the visitor that matches the type of
4764 object that @code{accept()} was being invoked on.
4766 Visitors in GiNaC must derive from the global @code{visitor} class as well
4767 as from the class @code{T::visitor} of each class @code{T} they want to
4768 visit, and implement the member functions @code{void visit(const T &)} for
4774 void ex::accept(visitor & v) const;
4777 will then dispatch to the correct @code{visit()} member function of the
4778 specified visitor @code{v} for the type of GiNaC object at the root of the
4779 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4781 Here is an example of a visitor:
4785 : public visitor, // this is required
4786 public add::visitor, // visit add objects
4787 public numeric::visitor, // visit numeric objects
4788 public basic::visitor // visit basic objects
4790 void visit(const add & x)
4791 @{ cout << "called with an add object" << endl; @}
4793 void visit(const numeric & x)
4794 @{ cout << "called with a numeric object" << endl; @}
4796 void visit(const basic & x)
4797 @{ cout << "called with a basic object" << endl; @}
4801 which can be used as follows:
4812 // prints "called with a numeric object"
4814 // prints "called with an add object"
4816 // prints "called with a basic object"
4820 The @code{visit(const basic &)} method gets called for all objects that are
4821 not @code{numeric} or @code{add} and acts as an (optional) default.
4823 From a conceptual point of view, the @code{visit()} methods of the visitor
4824 behave like a newly added virtual function of the visited hierarchy.
4825 In addition, visitors can store state in member variables, and they can
4826 be extended by deriving a new visitor from an existing one, thus building
4827 hierarchies of visitors.
4829 We can now rewrite our index example from above with a visitor:
4832 class gather_indices_visitor
4833 : public visitor, public idx::visitor, public varidx::visitor
4837 void visit(const idx & i)
4842 void visit(const varidx & vi)
4844 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4848 const lst & get_result() // utility function
4857 What's missing is the tree traversal. We could implement it in
4858 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4861 void ex::traverse_preorder(visitor & v) const;
4862 void ex::traverse_postorder(visitor & v) const;
4863 void ex::traverse(visitor & v) const;
4866 @code{traverse_preorder()} visits a node @emph{before} visiting its
4867 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4868 visiting its subexpressions. @code{traverse()} is a synonym for
4869 @code{traverse_preorder()}.
4871 Here is a new implementation of @code{gather_indices()} that uses the visitor
4872 and @code{traverse()}:
4875 lst gather_indices(const ex & e)
4877 gather_indices_visitor v;
4879 return v.get_result();
4883 Alternatively, you could use pre- or postorder iterators for the tree
4887 lst gather_indices(const ex & e)
4889 gather_indices_visitor v;
4890 for (const_preorder_iterator i = e.preorder_begin();
4891 i != e.preorder_end(); ++i) @{
4894 return v.get_result();
4899 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4900 @c node-name, next, previous, up
4901 @section Polynomial arithmetic
4903 @subsection Expanding and collecting
4904 @cindex @code{expand()}
4905 @cindex @code{collect()}
4906 @cindex @code{collect_common_factors()}
4908 A polynomial in one or more variables has many equivalent
4909 representations. Some useful ones serve a specific purpose. Consider
4910 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4911 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4912 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4913 representations are the recursive ones where one collects for exponents
4914 in one of the three variable. Since the factors are themselves
4915 polynomials in the remaining two variables the procedure can be
4916 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4917 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4920 To bring an expression into expanded form, its method
4923 ex ex::expand(unsigned options = 0);
4926 may be called. In our example above, this corresponds to @math{4*x*y +
4927 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4928 GiNaC is not easy to guess you should be prepared to see different
4929 orderings of terms in such sums!
4931 Another useful representation of multivariate polynomials is as a
4932 univariate polynomial in one of the variables with the coefficients
4933 being polynomials in the remaining variables. The method
4934 @code{collect()} accomplishes this task:
4937 ex ex::collect(const ex & s, bool distributed = false);
4940 The first argument to @code{collect()} can also be a list of objects in which
4941 case the result is either a recursively collected polynomial, or a polynomial
4942 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4943 by the @code{distributed} flag.
4945 Note that the original polynomial needs to be in expanded form (for the
4946 variables concerned) in order for @code{collect()} to be able to find the
4947 coefficients properly.
4949 The following @command{ginsh} transcript shows an application of @code{collect()}
4950 together with @code{find()}:
4953 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4954 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)
4955 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4956 > collect(a,@{p,q@});
4957 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
4958 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4959 > collect(a,find(a,sin($1)));
4960 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4961 > collect(a,@{find(a,sin($1)),p,q@});
4962 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4963 > collect(a,@{find(a,sin($1)),d@});
4964 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4967 Polynomials can often be brought into a more compact form by collecting
4968 common factors from the terms of sums. This is accomplished by the function
4971 ex collect_common_factors(const ex & e);
4974 This function doesn't perform a full factorization but only looks for
4975 factors which are already explicitly present:
4978 > collect_common_factors(a*x+a*y);
4980 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4982 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4983 (c+a)*a*(x*y+y^2+x)*b
4986 @subsection Degree and coefficients
4987 @cindex @code{degree()}
4988 @cindex @code{ldegree()}
4989 @cindex @code{coeff()}
4991 The degree and low degree of a polynomial can be obtained using the two
4995 int ex::degree(const ex & s);
4996 int ex::ldegree(const ex & s);
4999 which also work reliably on non-expanded input polynomials (they even work
5000 on rational functions, returning the asymptotic degree). By definition, the
5001 degree of zero is zero. To extract a coefficient with a certain power from
5002 an expanded polynomial you use
5005 ex ex::coeff(const ex & s, int n);
5008 You can also obtain the leading and trailing coefficients with the methods
5011 ex ex::lcoeff(const ex & s);
5012 ex ex::tcoeff(const ex & s);
5015 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5018 An application is illustrated in the next example, where a multivariate
5019 polynomial is analyzed:
5023 symbol x("x"), y("y");
5024 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5025 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5026 ex Poly = PolyInp.expand();
5028 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5029 cout << "The x^" << i << "-coefficient is "
5030 << Poly.coeff(x,i) << endl;
5032 cout << "As polynomial in y: "
5033 << Poly.collect(y) << endl;
5037 When run, it returns an output in the following fashion:
5040 The x^0-coefficient is y^2+11*y
5041 The x^1-coefficient is 5*y^2-2*y
5042 The x^2-coefficient is -1
5043 The x^3-coefficient is 4*y
5044 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5047 As always, the exact output may vary between different versions of GiNaC
5048 or even from run to run since the internal canonical ordering is not
5049 within the user's sphere of influence.
5051 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5052 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5053 with non-polynomial expressions as they not only work with symbols but with
5054 constants, functions and indexed objects as well:
5058 symbol a("a"), b("b"), c("c"), x("x");
5059 idx i(symbol("i"), 3);
5061 ex e = pow(sin(x) - cos(x), 4);
5062 cout << e.degree(cos(x)) << endl;
5064 cout << e.expand().coeff(sin(x), 3) << endl;
5067 e = indexed(a+b, i) * indexed(b+c, i);
5068 e = e.expand(expand_options::expand_indexed);
5069 cout << e.collect(indexed(b, i)) << endl;
5070 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5075 @subsection Polynomial division
5076 @cindex polynomial division
5079 @cindex pseudo-remainder
5080 @cindex @code{quo()}
5081 @cindex @code{rem()}
5082 @cindex @code{prem()}
5083 @cindex @code{divide()}
5088 ex quo(const ex & a, const ex & b, const ex & x);
5089 ex rem(const ex & a, const ex & b, const ex & x);
5092 compute the quotient and remainder of univariate polynomials in the variable
5093 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5095 The additional function
5098 ex prem(const ex & a, const ex & b, const ex & x);
5101 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5102 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5104 Exact division of multivariate polynomials is performed by the function
5107 bool divide(const ex & a, const ex & b, ex & q);
5110 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5111 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5112 in which case the value of @code{q} is undefined.
5115 @subsection Unit, content and primitive part
5116 @cindex @code{unit()}
5117 @cindex @code{content()}
5118 @cindex @code{primpart()}
5119 @cindex @code{unitcontprim()}
5124 ex ex::unit(const ex & x);
5125 ex ex::content(const ex & x);
5126 ex ex::primpart(const ex & x);
5127 ex ex::primpart(const ex & x, const ex & c);
5130 return the unit part, content part, and primitive polynomial of a multivariate
5131 polynomial with respect to the variable @samp{x} (the unit part being the sign
5132 of the leading coefficient, the content part being the GCD of the coefficients,
5133 and the primitive polynomial being the input polynomial divided by the unit and
5134 content parts). The second variant of @code{primpart()} expects the previously
5135 calculated content part of the polynomial in @code{c}, which enables it to
5136 work faster in the case where the content part has already been computed. The
5137 product of unit, content, and primitive part is the original polynomial.
5139 Additionally, the method
5142 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5145 computes the unit, content, and primitive parts in one go, returning them
5146 in @code{u}, @code{c}, and @code{p}, respectively.
5149 @subsection GCD, LCM and resultant
5152 @cindex @code{gcd()}
5153 @cindex @code{lcm()}
5155 The functions for polynomial greatest common divisor and least common
5156 multiple have the synopsis
5159 ex gcd(const ex & a, const ex & b);
5160 ex lcm(const ex & a, const ex & b);
5163 The functions @code{gcd()} and @code{lcm()} accept two expressions
5164 @code{a} and @code{b} as arguments and return a new expression, their
5165 greatest common divisor or least common multiple, respectively. If the
5166 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5167 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5168 the coefficients must be rationals.
5171 #include <ginac/ginac.h>
5172 using namespace GiNaC;
5176 symbol x("x"), y("y"), z("z");
5177 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5178 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5180 ex P_gcd = gcd(P_a, P_b);
5182 ex P_lcm = lcm(P_a, P_b);
5183 // 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
5188 @cindex @code{resultant()}
5190 The resultant of two expressions only makes sense with polynomials.
5191 It is always computed with respect to a specific symbol within the
5192 expressions. The function has the interface
5195 ex resultant(const ex & a, const ex & b, const ex & s);
5198 Resultants are symmetric in @code{a} and @code{b}. The following example
5199 computes the resultant of two expressions with respect to @code{x} and
5200 @code{y}, respectively:
5203 #include <ginac/ginac.h>
5204 using namespace GiNaC;
5208 symbol x("x"), y("y");
5210 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5213 r = resultant(e1, e2, x);
5215 r = resultant(e1, e2, y);
5220 @subsection Square-free decomposition
5221 @cindex square-free decomposition
5222 @cindex factorization
5223 @cindex @code{sqrfree()}
5225 GiNaC still lacks proper factorization support. Some form of
5226 factorization is, however, easily implemented by noting that factors
5227 appearing in a polynomial with power two or more also appear in the
5228 derivative and hence can easily be found by computing the GCD of the
5229 original polynomial and its derivatives. Any decent system has an
5230 interface for this so called square-free factorization. So we provide
5233 ex sqrfree(const ex & a, const lst & l = lst());
5235 Here is an example that by the way illustrates how the exact form of the
5236 result may slightly depend on the order of differentiation, calling for
5237 some care with subsequent processing of the result:
5240 symbol x("x"), y("y");
5241 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5243 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5244 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5246 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5247 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5249 cout << sqrfree(BiVarPol) << endl;
5250 // -> depending on luck, any of the above
5253 Note also, how factors with the same exponents are not fully factorized
5257 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5258 @c node-name, next, previous, up
5259 @section Rational expressions
5261 @subsection The @code{normal} method
5262 @cindex @code{normal()}
5263 @cindex simplification
5264 @cindex temporary replacement
5266 Some basic form of simplification of expressions is called for frequently.
5267 GiNaC provides the method @code{.normal()}, which converts a rational function
5268 into an equivalent rational function of the form @samp{numerator/denominator}
5269 where numerator and denominator are coprime. If the input expression is already
5270 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5271 otherwise it performs fraction addition and multiplication.
5273 @code{.normal()} can also be used on expressions which are not rational functions
5274 as it will replace all non-rational objects (like functions or non-integer
5275 powers) by temporary symbols to bring the expression to the domain of rational
5276 functions before performing the normalization, and re-substituting these
5277 symbols afterwards. This algorithm is also available as a separate method
5278 @code{.to_rational()}, described below.
5280 This means that both expressions @code{t1} and @code{t2} are indeed
5281 simplified in this little code snippet:
5286 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5287 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5288 std::cout << "t1 is " << t1.normal() << std::endl;
5289 std::cout << "t2 is " << t2.normal() << std::endl;
5293 Of course this works for multivariate polynomials too, so the ratio of
5294 the sample-polynomials from the section about GCD and LCM above would be
5295 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5298 @subsection Numerator and denominator
5301 @cindex @code{numer()}
5302 @cindex @code{denom()}
5303 @cindex @code{numer_denom()}
5305 The numerator and denominator of an expression can be obtained with
5310 ex ex::numer_denom();
5313 These functions will first normalize the expression as described above and
5314 then return the numerator, denominator, or both as a list, respectively.
5315 If you need both numerator and denominator, calling @code{numer_denom()} is
5316 faster than using @code{numer()} and @code{denom()} separately.
5319 @subsection Converting to a polynomial or rational expression
5320 @cindex @code{to_polynomial()}
5321 @cindex @code{to_rational()}
5323 Some of the methods described so far only work on polynomials or rational
5324 functions. GiNaC provides a way to extend the domain of these functions to
5325 general expressions by using the temporary replacement algorithm described
5326 above. You do this by calling
5329 ex ex::to_polynomial(exmap & m);
5330 ex ex::to_polynomial(lst & l);
5334 ex ex::to_rational(exmap & m);
5335 ex ex::to_rational(lst & l);
5338 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5339 will be filled with the generated temporary symbols and their replacement
5340 expressions in a format that can be used directly for the @code{subs()}
5341 method. It can also already contain a list of replacements from an earlier
5342 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5343 possible to use it on multiple expressions and get consistent results.
5345 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5346 is probably best illustrated with an example:
5350 symbol x("x"), y("y");
5351 ex a = 2*x/sin(x) - y/(3*sin(x));
5355 ex p = a.to_polynomial(lp);
5356 cout << " = " << p << "\n with " << lp << endl;
5357 // = symbol3*symbol2*y+2*symbol2*x
5358 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5361 ex r = a.to_rational(lr);
5362 cout << " = " << r << "\n with " << lr << endl;
5363 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5364 // with @{symbol4==sin(x)@}
5368 The following more useful example will print @samp{sin(x)-cos(x)}:
5373 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5374 ex b = sin(x) + cos(x);
5377 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5378 cout << q.subs(m) << endl;
5383 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5384 @c node-name, next, previous, up
5385 @section Symbolic differentiation
5386 @cindex differentiation
5387 @cindex @code{diff()}
5389 @cindex product rule
5391 GiNaC's objects know how to differentiate themselves. Thus, a
5392 polynomial (class @code{add}) knows that its derivative is the sum of
5393 the derivatives of all the monomials:
5397 symbol x("x"), y("y"), z("z");
5398 ex P = pow(x, 5) + pow(x, 2) + y;
5400 cout << P.diff(x,2) << endl;
5402 cout << P.diff(y) << endl; // 1
5404 cout << P.diff(z) << endl; // 0
5409 If a second integer parameter @var{n} is given, the @code{diff} method
5410 returns the @var{n}th derivative.
5412 If @emph{every} object and every function is told what its derivative
5413 is, all derivatives of composed objects can be calculated using the
5414 chain rule and the product rule. Consider, for instance the expression
5415 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5416 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5417 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5418 out that the composition is the generating function for Euler Numbers,
5419 i.e. the so called @var{n}th Euler number is the coefficient of
5420 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5421 identity to code a function that generates Euler numbers in just three
5424 @cindex Euler numbers
5426 #include <ginac/ginac.h>
5427 using namespace GiNaC;
5429 ex EulerNumber(unsigned n)
5432 const ex generator = pow(cosh(x),-1);
5433 return generator.diff(x,n).subs(x==0);
5438 for (unsigned i=0; i<11; i+=2)
5439 std::cout << EulerNumber(i) << std::endl;
5444 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5445 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5446 @code{i} by two since all odd Euler numbers vanish anyways.
5449 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5450 @c node-name, next, previous, up
5451 @section Series expansion
5452 @cindex @code{series()}
5453 @cindex Taylor expansion
5454 @cindex Laurent expansion
5455 @cindex @code{pseries} (class)
5456 @cindex @code{Order()}
5458 Expressions know how to expand themselves as a Taylor series or (more
5459 generally) a Laurent series. As in most conventional Computer Algebra
5460 Systems, no distinction is made between those two. There is a class of
5461 its own for storing such series (@code{class pseries}) and a built-in
5462 function (called @code{Order}) for storing the order term of the series.
5463 As a consequence, if you want to work with series, i.e. multiply two
5464 series, you need to call the method @code{ex::series} again to convert
5465 it to a series object with the usual structure (expansion plus order
5466 term). A sample application from special relativity could read:
5469 #include <ginac/ginac.h>
5470 using namespace std;
5471 using namespace GiNaC;
5475 symbol v("v"), c("c");
5477 ex gamma = 1/sqrt(1 - pow(v/c,2));
5478 ex mass_nonrel = gamma.series(v==0, 10);
5480 cout << "the relativistic mass increase with v is " << endl
5481 << mass_nonrel << endl;
5483 cout << "the inverse square of this series is " << endl
5484 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5488 Only calling the series method makes the last output simplify to
5489 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5490 series raised to the power @math{-2}.
5492 @cindex Machin's formula
5493 As another instructive application, let us calculate the numerical
5494 value of Archimedes' constant
5498 (for which there already exists the built-in constant @code{Pi})
5499 using John Machin's amazing formula
5501 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5504 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5506 This equation (and similar ones) were used for over 200 years for
5507 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5508 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5509 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5510 order term with it and the question arises what the system is supposed
5511 to do when the fractions are plugged into that order term. The solution
5512 is to use the function @code{series_to_poly()} to simply strip the order
5516 #include <ginac/ginac.h>
5517 using namespace GiNaC;
5519 ex machin_pi(int degr)
5522 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5523 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5524 -4*pi_expansion.subs(x==numeric(1,239));
5530 using std::cout; // just for fun, another way of...
5531 using std::endl; // ...dealing with this namespace std.
5533 for (int i=2; i<12; i+=2) @{
5534 pi_frac = machin_pi(i);
5535 cout << i << ":\t" << pi_frac << endl
5536 << "\t" << pi_frac.evalf() << endl;
5542 Note how we just called @code{.series(x,degr)} instead of
5543 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5544 method @code{series()}: if the first argument is a symbol the expression
5545 is expanded in that symbol around point @code{0}. When you run this
5546 program, it will type out:
5550 3.1832635983263598326
5551 4: 5359397032/1706489875
5552 3.1405970293260603143
5553 6: 38279241713339684/12184551018734375
5554 3.141621029325034425
5555 8: 76528487109180192540976/24359780855939418203125
5556 3.141591772182177295
5557 10: 327853873402258685803048818236/104359128170408663038552734375
5558 3.1415926824043995174
5562 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5563 @c node-name, next, previous, up
5564 @section Symmetrization
5565 @cindex @code{symmetrize()}
5566 @cindex @code{antisymmetrize()}
5567 @cindex @code{symmetrize_cyclic()}
5572 ex ex::symmetrize(const lst & l);
5573 ex ex::antisymmetrize(const lst & l);
5574 ex ex::symmetrize_cyclic(const lst & l);
5577 symmetrize an expression by returning the sum over all symmetric,
5578 antisymmetric or cyclic permutations of the specified list of objects,
5579 weighted by the number of permutations.
5581 The three additional methods
5584 ex ex::symmetrize();
5585 ex ex::antisymmetrize();
5586 ex ex::symmetrize_cyclic();
5589 symmetrize or antisymmetrize an expression over its free indices.
5591 Symmetrization is most useful with indexed expressions but can be used with
5592 almost any kind of object (anything that is @code{subs()}able):
5596 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5597 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5599 cout << indexed(A, i, j).symmetrize() << endl;
5600 // -> 1/2*A.j.i+1/2*A.i.j
5601 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5602 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5603 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5604 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5608 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5609 @c node-name, next, previous, up
5610 @section Predefined mathematical functions
5612 @subsection Overview
5614 GiNaC contains the following predefined mathematical functions:
5617 @multitable @columnfractions .30 .70
5618 @item @strong{Name} @tab @strong{Function}
5621 @cindex @code{abs()}
5622 @item @code{csgn(x)}
5624 @cindex @code{conjugate()}
5625 @item @code{conjugate(x)}
5626 @tab complex conjugation
5627 @cindex @code{csgn()}
5628 @item @code{sqrt(x)}
5629 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5630 @cindex @code{sqrt()}
5633 @cindex @code{sin()}
5636 @cindex @code{cos()}
5639 @cindex @code{tan()}
5640 @item @code{asin(x)}
5642 @cindex @code{asin()}
5643 @item @code{acos(x)}
5645 @cindex @code{acos()}
5646 @item @code{atan(x)}
5647 @tab inverse tangent
5648 @cindex @code{atan()}
5649 @item @code{atan2(y, x)}
5650 @tab inverse tangent with two arguments
5651 @item @code{sinh(x)}
5652 @tab hyperbolic sine
5653 @cindex @code{sinh()}
5654 @item @code{cosh(x)}
5655 @tab hyperbolic cosine
5656 @cindex @code{cosh()}
5657 @item @code{tanh(x)}
5658 @tab hyperbolic tangent
5659 @cindex @code{tanh()}
5660 @item @code{asinh(x)}
5661 @tab inverse hyperbolic sine
5662 @cindex @code{asinh()}
5663 @item @code{acosh(x)}
5664 @tab inverse hyperbolic cosine
5665 @cindex @code{acosh()}
5666 @item @code{atanh(x)}
5667 @tab inverse hyperbolic tangent
5668 @cindex @code{atanh()}
5670 @tab exponential function
5671 @cindex @code{exp()}
5673 @tab natural logarithm
5674 @cindex @code{log()}
5677 @cindex @code{Li2()}
5678 @item @code{Li(m, x)}
5679 @tab classical polylogarithm as well as multiple polylogarithm
5681 @item @code{G(a, y)}
5682 @tab multiple polylogarithm
5684 @item @code{G(a, s, y)}
5685 @tab multiple polylogarithm with explicit signs for the imaginary parts
5687 @item @code{S(n, p, x)}
5688 @tab Nielsen's generalized polylogarithm
5690 @item @code{H(m, x)}
5691 @tab harmonic polylogarithm
5693 @item @code{zeta(m)}
5694 @tab Riemann's zeta function as well as multiple zeta value
5695 @cindex @code{zeta()}
5696 @item @code{zeta(m, s)}
5697 @tab alternating Euler sum
5698 @cindex @code{zeta()}
5699 @item @code{zetaderiv(n, x)}
5700 @tab derivatives of Riemann's zeta function
5701 @item @code{tgamma(x)}
5703 @cindex @code{tgamma()}
5704 @cindex gamma function
5705 @item @code{lgamma(x)}
5706 @tab logarithm of gamma function
5707 @cindex @code{lgamma()}
5708 @item @code{beta(x, y)}
5709 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5710 @cindex @code{beta()}
5712 @tab psi (digamma) function
5713 @cindex @code{psi()}
5714 @item @code{psi(n, x)}
5715 @tab derivatives of psi function (polygamma functions)
5716 @item @code{factorial(n)}
5717 @tab factorial function @math{n!}
5718 @cindex @code{factorial()}
5719 @item @code{binomial(n, k)}
5720 @tab binomial coefficients
5721 @cindex @code{binomial()}
5722 @item @code{Order(x)}
5723 @tab order term function in truncated power series
5724 @cindex @code{Order()}
5729 For functions that have a branch cut in the complex plane GiNaC follows
5730 the conventions for C++ as defined in the ANSI standard as far as
5731 possible. In particular: the natural logarithm (@code{log}) and the
5732 square root (@code{sqrt}) both have their branch cuts running along the
5733 negative real axis where the points on the axis itself belong to the
5734 upper part (i.e. continuous with quadrant II). The inverse
5735 trigonometric and hyperbolic functions are not defined for complex
5736 arguments by the C++ standard, however. In GiNaC we follow the
5737 conventions used by CLN, which in turn follow the carefully designed
5738 definitions in the Common Lisp standard. It should be noted that this
5739 convention is identical to the one used by the C99 standard and by most
5740 serious CAS. It is to be expected that future revisions of the C++
5741 standard incorporate these functions in the complex domain in a manner
5742 compatible with C99.
5744 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5745 @c node-name, next, previous, up
5746 @subsection Multiple polylogarithms
5748 @cindex polylogarithm
5749 @cindex Nielsen's generalized polylogarithm
5750 @cindex harmonic polylogarithm
5751 @cindex multiple zeta value
5752 @cindex alternating Euler sum
5753 @cindex multiple polylogarithm
5755 The multiple polylogarithm is the most generic member of a family of functions,
5756 to which others like the harmonic polylogarithm, Nielsen's generalized
5757 polylogarithm and the multiple zeta value belong.
5758 Everyone of these functions can also be written as a multiple polylogarithm with specific
5759 parameters. This whole family of functions is therefore often referred to simply as
5760 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5761 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5762 @code{Li} and @code{G} in principle represent the same function, the different
5763 notations are more natural to the series representation or the integral
5764 representation, respectively.
5766 To facilitate the discussion of these functions we distinguish between indices and
5767 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5768 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5770 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5771 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5772 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5773 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5774 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5775 @code{s} is not given, the signs default to +1.
5776 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5777 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5778 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5779 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5780 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5782 The functions print in LaTeX format as
5784 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5790 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5793 $\zeta(m_1,m_2,\ldots,m_k)$.
5795 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5796 are printed with a line above, e.g.
5798 $\zeta(5,\overline{2})$.
5800 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5802 Definitions and analytical as well as numerical properties of multiple polylogarithms
5803 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5804 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5805 except for a few differences which will be explicitly stated in the following.
5807 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5808 that the indices and arguments are understood to be in the same order as in which they appear in
5809 the series representation. This means
5811 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5814 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5817 $\zeta(1,2)$ evaluates to infinity.
5819 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5822 The functions only evaluate if the indices are integers greater than zero, except for the indices
5823 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5824 will be interpreted as the sequence of signs for the corresponding indices
5825 @code{m} or the sign of the imaginary part for the
5826 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5827 @code{zeta(lst(3,4), lst(-1,1))} means
5829 $\zeta(\overline{3},4)$
5832 @code{G(lst(a,b), lst(-1,1), c)} means
5834 $G(a-0\epsilon,b+0\epsilon;c)$.
5836 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5837 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5838 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
5839 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5840 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5841 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5842 evaluates also for negative integers and positive even integers. For example:
5845 > Li(@{3,1@},@{x,1@});
5848 -zeta(@{3,2@},@{-1,-1@})
5853 It is easy to tell for a given function into which other function it can be rewritten, may
5854 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5855 with negative indices or trailing zeros (the example above gives a hint). Signs can
5856 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5857 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5858 @code{Li} (@code{eval()} already cares for the possible downgrade):
5861 > convert_H_to_Li(@{0,-2,-1,3@},x);
5862 Li(@{3,1,3@},@{-x,1,-1@})
5863 > convert_H_to_Li(@{2,-1,0@},x);
5864 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5867 Every function can be numerically evaluated for
5868 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5869 global variable @code{Digits}:
5874 > evalf(zeta(@{3,1,3,1@}));
5875 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5878 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5879 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5881 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5886 In long expressions this helps a lot with debugging, because you can easily spot
5887 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5888 cancellations of divergencies happen.
5890 Useful publications:
5892 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5893 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5895 @cite{Harmonic Polylogarithms},
5896 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5898 @cite{Special Values of Multiple Polylogarithms},
5899 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5901 @cite{Numerical Evaluation of Multiple Polylogarithms},
5902 J.Vollinga, S.Weinzierl, hep-ph/0410259
5904 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5905 @c node-name, next, previous, up
5906 @section Complex Conjugation
5908 @cindex @code{conjugate()}
5916 returns the complex conjugate of the expression. For all built-in functions and objects the
5917 conjugation gives the expected results:
5921 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5925 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5926 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5927 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5928 // -> -gamma5*gamma~b*gamma~a
5932 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5933 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5934 arguments. This is the default strategy. If you want to define your own functions and want to
5935 change this behavior, you have to supply a specialized conjugation method for your function
5936 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5938 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5939 @c node-name, next, previous, up
5940 @section Solving Linear Systems of Equations
5941 @cindex @code{lsolve()}
5943 The function @code{lsolve()} provides a convenient wrapper around some
5944 matrix operations that comes in handy when a system of linear equations
5948 ex lsolve(const ex & eqns, const ex & symbols,
5949 unsigned options = solve_algo::automatic);
5952 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5953 @code{relational}) while @code{symbols} is a @code{lst} of
5954 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5957 It returns the @code{lst} of solutions as an expression. As an example,
5958 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5962 symbol a("a"), b("b"), x("x"), y("y");
5964 eqns = a*x+b*y==3, x-y==b;
5966 cout << lsolve(eqns, vars) << endl;
5967 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5970 When the linear equations @code{eqns} are underdetermined, the solution
5971 will contain one or more tautological entries like @code{x==x},
5972 depending on the rank of the system. When they are overdetermined, the
5973 solution will be an empty @code{lst}. Note the third optional parameter
5974 to @code{lsolve()}: it accepts the same parameters as
5975 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5979 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5980 @c node-name, next, previous, up
5981 @section Input and output of expressions
5984 @subsection Expression output
5986 @cindex output of expressions
5988 Expressions can simply be written to any stream:
5993 ex e = 4.5*I+pow(x,2)*3/2;
5994 cout << e << endl; // prints '4.5*I+3/2*x^2'
5998 The default output format is identical to the @command{ginsh} input syntax and
5999 to that used by most computer algebra systems, but not directly pastable
6000 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6001 is printed as @samp{x^2}).
6003 It is possible to print expressions in a number of different formats with
6004 a set of stream manipulators;
6007 std::ostream & dflt(std::ostream & os);
6008 std::ostream & latex(std::ostream & os);
6009 std::ostream & tree(std::ostream & os);
6010 std::ostream & csrc(std::ostream & os);
6011 std::ostream & csrc_float(std::ostream & os);
6012 std::ostream & csrc_double(std::ostream & os);
6013 std::ostream & csrc_cl_N(std::ostream & os);
6014 std::ostream & index_dimensions(std::ostream & os);
6015 std::ostream & no_index_dimensions(std::ostream & os);
6018 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6019 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6020 @code{print_csrc()} functions, respectively.
6023 All manipulators affect the stream state permanently. To reset the output
6024 format to the default, use the @code{dflt} manipulator:
6028 cout << latex; // all output to cout will be in LaTeX format from
6030 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6031 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6032 cout << dflt; // revert to default output format
6033 cout << e << endl; // prints '4.5*I+3/2*x^2'
6037 If you don't want to affect the format of the stream you're working with,
6038 you can output to a temporary @code{ostringstream} like this:
6043 s << latex << e; // format of cout remains unchanged
6044 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6049 @cindex @code{csrc_float}
6050 @cindex @code{csrc_double}
6051 @cindex @code{csrc_cl_N}
6052 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6053 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6054 format that can be directly used in a C or C++ program. The three possible
6055 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6056 classes provided by the CLN library):
6060 cout << "f = " << csrc_float << e << ";\n";
6061 cout << "d = " << csrc_double << e << ";\n";
6062 cout << "n = " << csrc_cl_N << e << ";\n";
6066 The above example will produce (note the @code{x^2} being converted to
6070 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6071 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6072 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6076 The @code{tree} manipulator allows dumping the internal structure of an
6077 expression for debugging purposes:
6088 add, hash=0x0, flags=0x3, nops=2
6089 power, hash=0x0, flags=0x3, nops=2
6090 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6091 2 (numeric), hash=0x6526b0fa, flags=0xf
6092 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6095 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6099 @cindex @code{latex}
6100 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6101 It is rather similar to the default format but provides some braces needed
6102 by LaTeX for delimiting boxes and also converts some common objects to
6103 conventional LaTeX names. It is possible to give symbols a special name for
6104 LaTeX output by supplying it as a second argument to the @code{symbol}
6107 For example, the code snippet
6111 symbol x("x", "\\circ");
6112 ex e = lgamma(x).series(x==0,3);
6113 cout << latex << e << endl;
6120 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6121 +\mathcal@{O@}(\circ^@{3@})
6124 @cindex @code{index_dimensions}
6125 @cindex @code{no_index_dimensions}
6126 Index dimensions are normally hidden in the output. To make them visible, use
6127 the @code{index_dimensions} manipulator. The dimensions will be written in
6128 square brackets behind each index value in the default and LaTeX output
6133 symbol x("x"), y("y");
6134 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6135 ex e = indexed(x, mu) * indexed(y, nu);
6138 // prints 'x~mu*y~nu'
6139 cout << index_dimensions << e << endl;
6140 // prints 'x~mu[4]*y~nu[4]'
6141 cout << no_index_dimensions << e << endl;
6142 // prints 'x~mu*y~nu'
6147 @cindex Tree traversal
6148 If you need any fancy special output format, e.g. for interfacing GiNaC
6149 with other algebra systems or for producing code for different
6150 programming languages, you can always traverse the expression tree yourself:
6153 static void my_print(const ex & e)
6155 if (is_a<function>(e))
6156 cout << ex_to<function>(e).get_name();
6158 cout << ex_to<basic>(e).class_name();
6160 size_t n = e.nops();
6162 for (size_t i=0; i<n; i++) @{
6174 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6182 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6183 symbol(y))),numeric(-2)))
6186 If you need an output format that makes it possible to accurately
6187 reconstruct an expression by feeding the output to a suitable parser or
6188 object factory, you should consider storing the expression in an
6189 @code{archive} object and reading the object properties from there.
6190 See the section on archiving for more information.
6193 @subsection Expression input
6194 @cindex input of expressions
6196 GiNaC provides no way to directly read an expression from a stream because
6197 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6198 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6199 @code{y} you defined in your program and there is no way to specify the
6200 desired symbols to the @code{>>} stream input operator.
6202 Instead, GiNaC lets you construct an expression from a string, specifying the
6203 list of symbols to be used:
6207 symbol x("x"), y("y");
6208 ex e("2*x+sin(y)", lst(x, y));
6212 The input syntax is the same as that used by @command{ginsh} and the stream
6213 output operator @code{<<}. The symbols in the string are matched by name to
6214 the symbols in the list and if GiNaC encounters a symbol not specified in
6215 the list it will throw an exception.
6217 With this constructor, it's also easy to implement interactive GiNaC programs:
6222 #include <stdexcept>
6223 #include <ginac/ginac.h>
6224 using namespace std;
6225 using namespace GiNaC;
6232 cout << "Enter an expression containing 'x': ";
6237 cout << "The derivative of " << e << " with respect to x is ";
6238 cout << e.diff(x) << ".\n";
6239 @} catch (exception &p) @{
6240 cerr << p.what() << endl;
6246 @subsection Archiving
6247 @cindex @code{archive} (class)
6250 GiNaC allows creating @dfn{archives} of expressions which can be stored
6251 to or retrieved from files. To create an archive, you declare an object
6252 of class @code{archive} and archive expressions in it, giving each
6253 expression a unique name:
6257 using namespace std;
6258 #include <ginac/ginac.h>
6259 using namespace GiNaC;
6263 symbol x("x"), y("y"), z("z");
6265 ex foo = sin(x + 2*y) + 3*z + 41;
6269 a.archive_ex(foo, "foo");
6270 a.archive_ex(bar, "the second one");
6274 The archive can then be written to a file:
6278 ofstream out("foobar.gar");
6284 The file @file{foobar.gar} contains all information that is needed to
6285 reconstruct the expressions @code{foo} and @code{bar}.
6287 @cindex @command{viewgar}
6288 The tool @command{viewgar} that comes with GiNaC can be used to view
6289 the contents of GiNaC archive files:
6292 $ viewgar foobar.gar
6293 foo = 41+sin(x+2*y)+3*z
6294 the second one = 42+sin(x+2*y)+3*z
6297 The point of writing archive files is of course that they can later be
6303 ifstream in("foobar.gar");
6308 And the stored expressions can be retrieved by their name:
6315 ex ex1 = a2.unarchive_ex(syms, "foo");
6316 ex ex2 = a2.unarchive_ex(syms, "the second one");
6318 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6319 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6320 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6324 Note that you have to supply a list of the symbols which are to be inserted
6325 in the expressions. Symbols in archives are stored by their name only and
6326 if you don't specify which symbols you have, unarchiving the expression will
6327 create new symbols with that name. E.g. if you hadn't included @code{x} in
6328 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6329 have had no effect because the @code{x} in @code{ex1} would have been a
6330 different symbol than the @code{x} which was defined at the beginning of
6331 the program, although both would appear as @samp{x} when printed.
6333 You can also use the information stored in an @code{archive} object to
6334 output expressions in a format suitable for exact reconstruction. The
6335 @code{archive} and @code{archive_node} classes have a couple of member
6336 functions that let you access the stored properties:
6339 static void my_print2(const archive_node & n)
6342 n.find_string("class", class_name);
6343 cout << class_name << "(";
6345 archive_node::propinfovector p;
6346 n.get_properties(p);
6348 size_t num = p.size();
6349 for (size_t i=0; i<num; i++) @{
6350 const string &name = p[i].name;
6351 if (name == "class")
6353 cout << name << "=";
6355 unsigned count = p[i].count;
6359 for (unsigned j=0; j<count; j++) @{
6360 switch (p[i].type) @{
6361 case archive_node::PTYPE_BOOL: @{
6363 n.find_bool(name, x, j);
6364 cout << (x ? "true" : "false");
6367 case archive_node::PTYPE_UNSIGNED: @{
6369 n.find_unsigned(name, x, j);
6373 case archive_node::PTYPE_STRING: @{
6375 n.find_string(name, x, j);
6376 cout << '\"' << x << '\"';
6379 case archive_node::PTYPE_NODE: @{
6380 const archive_node &x = n.find_ex_node(name, j);
6402 ex e = pow(2, x) - y;
6404 my_print2(ar.get_top_node(0)); cout << endl;
6412 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6413 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6414 overall_coeff=numeric(number="0"))
6417 Be warned, however, that the set of properties and their meaning for each
6418 class may change between GiNaC versions.
6421 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6422 @c node-name, next, previous, up
6423 @chapter Extending GiNaC
6425 By reading so far you should have gotten a fairly good understanding of
6426 GiNaC's design patterns. From here on you should start reading the
6427 sources. All we can do now is issue some recommendations how to tackle
6428 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6429 develop some useful extension please don't hesitate to contact the GiNaC
6430 authors---they will happily incorporate them into future versions.
6433 * What does not belong into GiNaC:: What to avoid.
6434 * Symbolic functions:: Implementing symbolic functions.
6435 * Printing:: Adding new output formats.
6436 * Structures:: Defining new algebraic classes (the easy way).
6437 * Adding classes:: Defining new algebraic classes (the hard way).
6441 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6442 @c node-name, next, previous, up
6443 @section What doesn't belong into GiNaC
6445 @cindex @command{ginsh}
6446 First of all, GiNaC's name must be read literally. It is designed to be
6447 a library for use within C++. The tiny @command{ginsh} accompanying
6448 GiNaC makes this even more clear: it doesn't even attempt to provide a
6449 language. There are no loops or conditional expressions in
6450 @command{ginsh}, it is merely a window into the library for the
6451 programmer to test stuff (or to show off). Still, the design of a
6452 complete CAS with a language of its own, graphical capabilities and all
6453 this on top of GiNaC is possible and is without doubt a nice project for
6456 There are many built-in functions in GiNaC that do not know how to
6457 evaluate themselves numerically to a precision declared at runtime
6458 (using @code{Digits}). Some may be evaluated at certain points, but not
6459 generally. This ought to be fixed. However, doing numerical
6460 computations with GiNaC's quite abstract classes is doomed to be
6461 inefficient. For this purpose, the underlying foundation classes
6462 provided by CLN are much better suited.
6465 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6466 @c node-name, next, previous, up
6467 @section Symbolic functions
6469 The easiest and most instructive way to start extending GiNaC is probably to
6470 create your own symbolic functions. These are implemented with the help of
6471 two preprocessor macros:
6473 @cindex @code{DECLARE_FUNCTION}
6474 @cindex @code{REGISTER_FUNCTION}
6476 DECLARE_FUNCTION_<n>P(<name>)
6477 REGISTER_FUNCTION(<name>, <options>)
6480 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6481 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6482 parameters of type @code{ex} and returns a newly constructed GiNaC
6483 @code{function} object that represents your function.
6485 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6486 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6487 set of options that associate the symbolic function with C++ functions you
6488 provide to implement the various methods such as evaluation, derivative,
6489 series expansion etc. They also describe additional attributes the function
6490 might have, such as symmetry and commutation properties, and a name for
6491 LaTeX output. Multiple options are separated by the member access operator
6492 @samp{.} and can be given in an arbitrary order.
6494 (By the way: in case you are worrying about all the macros above we can
6495 assure you that functions are GiNaC's most macro-intense classes. We have
6496 done our best to avoid macros where we can.)
6498 @subsection A minimal example
6500 Here is an example for the implementation of a function with two arguments
6501 that is not further evaluated:
6504 DECLARE_FUNCTION_2P(myfcn)
6506 REGISTER_FUNCTION(myfcn, dummy())
6509 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6510 in algebraic expressions:
6516 ex e = 2*myfcn(42, 1+3*x) - x;
6518 // prints '2*myfcn(42,1+3*x)-x'
6523 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6524 "no options". A function with no options specified merely acts as a kind of
6525 container for its arguments. It is a pure "dummy" function with no associated
6526 logic (which is, however, sometimes perfectly sufficient).
6528 Let's now have a look at the implementation of GiNaC's cosine function for an
6529 example of how to make an "intelligent" function.
6531 @subsection The cosine function
6533 The GiNaC header file @file{inifcns.h} contains the line
6536 DECLARE_FUNCTION_1P(cos)
6539 which declares to all programs using GiNaC that there is a function @samp{cos}
6540 that takes one @code{ex} as an argument. This is all they need to know to use
6541 this function in expressions.
6543 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6544 is its @code{REGISTER_FUNCTION} line:
6547 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6548 evalf_func(cos_evalf).
6549 derivative_func(cos_deriv).
6550 latex_name("\\cos"));
6553 There are four options defined for the cosine function. One of them
6554 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6555 other three indicate the C++ functions in which the "brains" of the cosine
6556 function are defined.
6558 @cindex @code{hold()}
6560 The @code{eval_func()} option specifies the C++ function that implements
6561 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6562 the same number of arguments as the associated symbolic function (one in this
6563 case) and returns the (possibly transformed or in some way simplified)
6564 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6565 of the automatic evaluation process). If no (further) evaluation is to take
6566 place, the @code{eval_func()} function must return the original function
6567 with @code{.hold()}, to avoid a potential infinite recursion. If your
6568 symbolic functions produce a segmentation fault or stack overflow when
6569 using them in expressions, you are probably missing a @code{.hold()}
6572 The @code{eval_func()} function for the cosine looks something like this
6573 (actually, it doesn't look like this at all, but it should give you an idea
6577 static ex cos_eval(const ex & x)
6579 if ("x is a multiple of 2*Pi")
6581 else if ("x is a multiple of Pi")
6583 else if ("x is a multiple of Pi/2")
6587 else if ("x has the form 'acos(y)'")
6589 else if ("x has the form 'asin(y)'")
6594 return cos(x).hold();
6598 This function is called every time the cosine is used in a symbolic expression:
6604 // this calls cos_eval(Pi), and inserts its return value into
6605 // the actual expression
6612 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6613 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6614 symbolic transformation can be done, the unmodified function is returned
6615 with @code{.hold()}.
6617 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6618 The user has to call @code{evalf()} for that. This is implemented in a
6622 static ex cos_evalf(const ex & x)
6624 if (is_a<numeric>(x))
6625 return cos(ex_to<numeric>(x));
6627 return cos(x).hold();
6631 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6632 in this case the @code{cos()} function for @code{numeric} objects, which in
6633 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6634 isn't really needed here, but reminds us that the corresponding @code{eval()}
6635 function would require it in this place.
6637 Differentiation will surely turn up and so we need to tell @code{cos}
6638 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6639 instance, are then handled automatically by @code{basic::diff} and
6643 static ex cos_deriv(const ex & x, unsigned diff_param)
6649 @cindex product rule
6650 The second parameter is obligatory but uninteresting at this point. It
6651 specifies which parameter to differentiate in a partial derivative in
6652 case the function has more than one parameter, and its main application
6653 is for correct handling of the chain rule.
6655 An implementation of the series expansion is not needed for @code{cos()} as
6656 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6657 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6658 the other hand, does have poles and may need to do Laurent expansion:
6661 static ex tan_series(const ex & x, const relational & rel,
6662 int order, unsigned options)
6664 // Find the actual expansion point
6665 const ex x_pt = x.subs(rel);
6667 if ("x_pt is not an odd multiple of Pi/2")
6668 throw do_taylor(); // tell function::series() to do Taylor expansion
6670 // On a pole, expand sin()/cos()
6671 return (sin(x)/cos(x)).series(rel, order+2, options);
6675 The @code{series()} implementation of a function @emph{must} return a
6676 @code{pseries} object, otherwise your code will crash.
6678 @subsection Function options
6680 GiNaC functions understand several more options which are always
6681 specified as @code{.option(params)}. None of them are required, but you
6682 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6683 is a do-nothing option called @code{dummy()} which you can use to define
6684 functions without any special options.
6687 eval_func(<C++ function>)
6688 evalf_func(<C++ function>)
6689 derivative_func(<C++ function>)
6690 series_func(<C++ function>)
6691 conjugate_func(<C++ function>)
6694 These specify the C++ functions that implement symbolic evaluation,
6695 numeric evaluation, partial derivatives, and series expansion, respectively.
6696 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6697 @code{diff()} and @code{series()}.
6699 The @code{eval_func()} function needs to use @code{.hold()} if no further
6700 automatic evaluation is desired or possible.
6702 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6703 expansion, which is correct if there are no poles involved. If the function
6704 has poles in the complex plane, the @code{series_func()} needs to check
6705 whether the expansion point is on a pole and fall back to Taylor expansion
6706 if it isn't. Otherwise, the pole usually needs to be regularized by some
6707 suitable transformation.
6710 latex_name(const string & n)
6713 specifies the LaTeX code that represents the name of the function in LaTeX
6714 output. The default is to put the function name in an @code{\mbox@{@}}.
6717 do_not_evalf_params()
6720 This tells @code{evalf()} to not recursively evaluate the parameters of the
6721 function before calling the @code{evalf_func()}.
6724 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6727 This allows you to explicitly specify the commutation properties of the
6728 function (@xref{Non-commutative objects}, for an explanation of
6729 (non)commutativity in GiNaC). For example, you can use
6730 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6731 GiNaC treat your function like a matrix. By default, functions inherit the
6732 commutation properties of their first argument.
6735 set_symmetry(const symmetry & s)
6738 specifies the symmetry properties of the function with respect to its
6739 arguments. @xref{Indexed objects}, for an explanation of symmetry
6740 specifications. GiNaC will automatically rearrange the arguments of
6741 symmetric functions into a canonical order.
6743 Sometimes you may want to have finer control over how functions are
6744 displayed in the output. For example, the @code{abs()} function prints
6745 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6746 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6750 print_func<C>(<C++ function>)
6753 option which is explained in the next section.
6755 @subsection Functions with a variable number of arguments
6757 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6758 functions with a fixed number of arguments. Sometimes, though, you may need
6759 to have a function that accepts a variable number of expressions. One way to
6760 accomplish this is to pass variable-length lists as arguments. The
6761 @code{Li()} function uses this method for multiple polylogarithms.
6763 It is also possible to define functions that accept a different number of
6764 parameters under the same function name, such as the @code{psi()} function
6765 which can be called either as @code{psi(z)} (the digamma function) or as
6766 @code{psi(n, z)} (polygamma functions). These are actually two different
6767 functions in GiNaC that, however, have the same name. Defining such
6768 functions is not possible with the macros but requires manually fiddling
6769 with GiNaC internals. If you are interested, please consult the GiNaC source
6770 code for the @code{psi()} function (@file{inifcns.h} and
6771 @file{inifcns_gamma.cpp}).
6774 @node Printing, Structures, Symbolic functions, Extending GiNaC
6775 @c node-name, next, previous, up
6776 @section GiNaC's expression output system
6778 GiNaC allows the output of expressions in a variety of different formats
6779 (@pxref{Input/Output}). This section will explain how expression output
6780 is implemented internally, and how to define your own output formats or
6781 change the output format of built-in algebraic objects. You will also want
6782 to read this section if you plan to write your own algebraic classes or
6785 @cindex @code{print_context} (class)
6786 @cindex @code{print_dflt} (class)
6787 @cindex @code{print_latex} (class)
6788 @cindex @code{print_tree} (class)
6789 @cindex @code{print_csrc} (class)
6790 All the different output formats are represented by a hierarchy of classes
6791 rooted in the @code{print_context} class, defined in the @file{print.h}
6796 the default output format
6798 output in LaTeX mathematical mode
6800 a dump of the internal expression structure (for debugging)
6802 the base class for C source output
6803 @item print_csrc_float
6804 C source output using the @code{float} type
6805 @item print_csrc_double
6806 C source output using the @code{double} type
6807 @item print_csrc_cl_N
6808 C source output using CLN types
6811 The @code{print_context} base class provides two public data members:
6823 @code{s} is a reference to the stream to output to, while @code{options}
6824 holds flags and modifiers. Currently, there is only one flag defined:
6825 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6826 to print the index dimension which is normally hidden.
6828 When you write something like @code{std::cout << e}, where @code{e} is
6829 an object of class @code{ex}, GiNaC will construct an appropriate
6830 @code{print_context} object (of a class depending on the selected output
6831 format), fill in the @code{s} and @code{options} members, and call
6833 @cindex @code{print()}
6835 void ex::print(const print_context & c, unsigned level = 0) const;
6838 which in turn forwards the call to the @code{print()} method of the
6839 top-level algebraic object contained in the expression.
6841 Unlike other methods, GiNaC classes don't usually override their
6842 @code{print()} method to implement expression output. Instead, the default
6843 implementation @code{basic::print(c, level)} performs a run-time double
6844 dispatch to a function selected by the dynamic type of the object and the
6845 passed @code{print_context}. To this end, GiNaC maintains a separate method
6846 table for each class, similar to the virtual function table used for ordinary
6847 (single) virtual function dispatch.
6849 The method table contains one slot for each possible @code{print_context}
6850 type, indexed by the (internally assigned) serial number of the type. Slots
6851 may be empty, in which case GiNaC will retry the method lookup with the
6852 @code{print_context} object's parent class, possibly repeating the process
6853 until it reaches the @code{print_context} base class. If there's still no
6854 method defined, the method table of the algebraic object's parent class
6855 is consulted, and so on, until a matching method is found (eventually it
6856 will reach the combination @code{basic/print_context}, which prints the
6857 object's class name enclosed in square brackets).
6859 You can think of the print methods of all the different classes and output
6860 formats as being arranged in a two-dimensional matrix with one axis listing
6861 the algebraic classes and the other axis listing the @code{print_context}
6864 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6865 to implement printing, but then they won't get any of the benefits of the
6866 double dispatch mechanism (such as the ability for derived classes to
6867 inherit only certain print methods from its parent, or the replacement of
6868 methods at run-time).
6870 @subsection Print methods for classes
6872 The method table for a class is set up either in the definition of the class,
6873 by passing the appropriate @code{print_func<C>()} option to
6874 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6875 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6876 can also be used to override existing methods dynamically.
6878 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6879 be a member function of the class (or one of its parent classes), a static
6880 member function, or an ordinary (global) C++ function. The @code{C} template
6881 parameter specifies the appropriate @code{print_context} type for which the
6882 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6883 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6884 the class is the one being implemented by
6885 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6887 For print methods that are member functions, their first argument must be of
6888 a type convertible to a @code{const C &}, and the second argument must be an
6891 For static members and global functions, the first argument must be of a type
6892 convertible to a @code{const T &}, the second argument must be of a type
6893 convertible to a @code{const C &}, and the third argument must be an
6894 @code{unsigned}. A global function will, of course, not have access to
6895 private and protected members of @code{T}.
6897 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6898 and @code{basic::print()}) is used for proper parenthesizing of the output
6899 (and by @code{print_tree} for proper indentation). It can be used for similar
6900 purposes if you write your own output formats.
6902 The explanations given above may seem complicated, but in practice it's
6903 really simple, as shown in the following example. Suppose that we want to
6904 display exponents in LaTeX output not as superscripts but with little
6905 upwards-pointing arrows. This can be achieved in the following way:
6908 void my_print_power_as_latex(const power & p,
6909 const print_latex & c,
6912 // get the precedence of the 'power' class
6913 unsigned power_prec = p.precedence();
6915 // if the parent operator has the same or a higher precedence
6916 // we need parentheses around the power
6917 if (level >= power_prec)
6920 // print the basis and exponent, each enclosed in braces, and
6921 // separated by an uparrow
6923 p.op(0).print(c, power_prec);
6924 c.s << "@}\\uparrow@{";
6925 p.op(1).print(c, power_prec);
6928 // don't forget the closing parenthesis
6929 if (level >= power_prec)
6935 // a sample expression
6936 symbol x("x"), y("y");
6937 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6939 // switch to LaTeX mode
6942 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6945 // now we replace the method for the LaTeX output of powers with
6947 set_print_func<power, print_latex>(my_print_power_as_latex);
6949 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
6960 The first argument of @code{my_print_power_as_latex} could also have been
6961 a @code{const basic &}, the second one a @code{const print_context &}.
6964 The above code depends on @code{mul} objects converting their operands to
6965 @code{power} objects for the purpose of printing.
6968 The output of products including negative powers as fractions is also
6969 controlled by the @code{mul} class.
6972 The @code{power/print_latex} method provided by GiNaC prints square roots
6973 using @code{\sqrt}, but the above code doesn't.
6977 It's not possible to restore a method table entry to its previous or default
6978 value. Once you have called @code{set_print_func()}, you can only override
6979 it with another call to @code{set_print_func()}, but you can't easily go back
6980 to the default behavior again (you can, of course, dig around in the GiNaC
6981 sources, find the method that is installed at startup
6982 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6983 one; that is, after you circumvent the C++ member access control@dots{}).
6985 @subsection Print methods for functions
6987 Symbolic functions employ a print method dispatch mechanism similar to the
6988 one used for classes. The methods are specified with @code{print_func<C>()}
6989 function options. If you don't specify any special print methods, the function
6990 will be printed with its name (or LaTeX name, if supplied), followed by a
6991 comma-separated list of arguments enclosed in parentheses.
6993 For example, this is what GiNaC's @samp{abs()} function is defined like:
6996 static ex abs_eval(const ex & arg) @{ ... @}
6997 static ex abs_evalf(const ex & arg) @{ ... @}
6999 static void abs_print_latex(const ex & arg, const print_context & c)
7001 c.s << "@{|"; arg.print(c); c.s << "|@}";
7004 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7006 c.s << "fabs("; arg.print(c); c.s << ")";
7009 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7010 evalf_func(abs_evalf).
7011 print_func<print_latex>(abs_print_latex).
7012 print_func<print_csrc_float>(abs_print_csrc_float).
7013 print_func<print_csrc_double>(abs_print_csrc_float));
7016 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7017 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7019 There is currently no equivalent of @code{set_print_func()} for functions.
7021 @subsection Adding new output formats
7023 Creating a new output format involves subclassing @code{print_context},
7024 which is somewhat similar to adding a new algebraic class
7025 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7026 that needs to go into the class definition, and a corresponding macro
7027 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7028 Every @code{print_context} class needs to provide a default constructor
7029 and a constructor from an @code{std::ostream} and an @code{unsigned}
7032 Here is an example for a user-defined @code{print_context} class:
7035 class print_myformat : public print_dflt
7037 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7039 print_myformat(std::ostream & os, unsigned opt = 0)
7040 : print_dflt(os, opt) @{@}
7043 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7045 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7048 That's all there is to it. None of the actual expression output logic is
7049 implemented in this class. It merely serves as a selector for choosing
7050 a particular format. The algorithms for printing expressions in the new
7051 format are implemented as print methods, as described above.
7053 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7054 exactly like GiNaC's default output format:
7059 ex e = pow(x, 2) + 1;
7061 // this prints "1+x^2"
7064 // this also prints "1+x^2"
7065 e.print(print_myformat()); cout << endl;
7071 To fill @code{print_myformat} with life, we need to supply appropriate
7072 print methods with @code{set_print_func()}, like this:
7075 // This prints powers with '**' instead of '^'. See the LaTeX output
7076 // example above for explanations.
7077 void print_power_as_myformat(const power & p,
7078 const print_myformat & c,
7081 unsigned power_prec = p.precedence();
7082 if (level >= power_prec)
7084 p.op(0).print(c, power_prec);
7086 p.op(1).print(c, power_prec);
7087 if (level >= power_prec)
7093 // install a new print method for power objects
7094 set_print_func<power, print_myformat>(print_power_as_myformat);
7096 // now this prints "1+x**2"
7097 e.print(print_myformat()); cout << endl;
7099 // but the default format is still "1+x^2"
7105 @node Structures, Adding classes, Printing, Extending GiNaC
7106 @c node-name, next, previous, up
7109 If you are doing some very specialized things with GiNaC, or if you just
7110 need some more organized way to store data in your expressions instead of
7111 anonymous lists, you may want to implement your own algebraic classes.
7112 ('algebraic class' means any class directly or indirectly derived from
7113 @code{basic} that can be used in GiNaC expressions).
7115 GiNaC offers two ways of accomplishing this: either by using the
7116 @code{structure<T>} template class, or by rolling your own class from
7117 scratch. This section will discuss the @code{structure<T>} template which
7118 is easier to use but more limited, while the implementation of custom
7119 GiNaC classes is the topic of the next section. However, you may want to
7120 read both sections because many common concepts and member functions are
7121 shared by both concepts, and it will also allow you to decide which approach
7122 is most suited to your needs.
7124 The @code{structure<T>} template, defined in the GiNaC header file
7125 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7126 or @code{class}) into a GiNaC object that can be used in expressions.
7128 @subsection Example: scalar products
7130 Let's suppose that we need a way to handle some kind of abstract scalar
7131 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7132 product class have to store their left and right operands, which can in turn
7133 be arbitrary expressions. Here is a possible way to represent such a
7134 product in a C++ @code{struct}:
7138 using namespace std;
7140 #include <ginac/ginac.h>
7141 using namespace GiNaC;
7147 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7151 The default constructor is required. Now, to make a GiNaC class out of this
7152 data structure, we need only one line:
7155 typedef structure<sprod_s> sprod;
7158 That's it. This line constructs an algebraic class @code{sprod} which
7159 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7160 expressions like any other GiNaC class:
7164 symbol a("a"), b("b");
7165 ex e = sprod(sprod_s(a, b));
7169 Note the difference between @code{sprod} which is the algebraic class, and
7170 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7171 and @code{right} data members. As shown above, an @code{sprod} can be
7172 constructed from an @code{sprod_s} object.
7174 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7175 you could define a little wrapper function like this:
7178 inline ex make_sprod(ex left, ex right)
7180 return sprod(sprod_s(left, right));
7184 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7185 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7186 @code{get_struct()}:
7190 cout << ex_to<sprod>(e)->left << endl;
7192 cout << ex_to<sprod>(e).get_struct().right << endl;
7197 You only have read access to the members of @code{sprod_s}.
7199 The type definition of @code{sprod} is enough to write your own algorithms
7200 that deal with scalar products, for example:
7205 if (is_a<sprod>(p)) @{
7206 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7207 return make_sprod(sp.right, sp.left);
7218 @subsection Structure output
7220 While the @code{sprod} type is useable it still leaves something to be
7221 desired, most notably proper output:
7226 // -> [structure object]
7230 By default, any structure types you define will be printed as
7231 @samp{[structure object]}. To override this you can either specialize the
7232 template's @code{print()} member function, or specify print methods with
7233 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7234 it's not possible to supply class options like @code{print_func<>()} to
7235 structures, so for a self-contained structure type you need to resort to
7236 overriding the @code{print()} function, which is also what we will do here.
7238 The member functions of GiNaC classes are described in more detail in the
7239 next section, but it shouldn't be hard to figure out what's going on here:
7242 void sprod::print(const print_context & c, unsigned level) const
7244 // tree debug output handled by superclass
7245 if (is_a<print_tree>(c))
7246 inherited::print(c, level);
7248 // get the contained sprod_s object
7249 const sprod_s & sp = get_struct();
7251 // print_context::s is a reference to an ostream
7252 c.s << "<" << sp.left << "|" << sp.right << ">";
7256 Now we can print expressions containing scalar products:
7262 cout << swap_sprod(e) << endl;
7267 @subsection Comparing structures
7269 The @code{sprod} class defined so far still has one important drawback: all
7270 scalar products are treated as being equal because GiNaC doesn't know how to
7271 compare objects of type @code{sprod_s}. This can lead to some confusing
7272 and undesired behavior:
7276 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7278 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7279 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7283 To remedy this, we first need to define the operators @code{==} and @code{<}
7284 for objects of type @code{sprod_s}:
7287 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7289 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7292 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7294 return lhs.left.compare(rhs.left) < 0
7295 ? true : lhs.right.compare(rhs.right) < 0;
7299 The ordering established by the @code{<} operator doesn't have to make any
7300 algebraic sense, but it needs to be well defined. Note that we can't use
7301 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7302 in the implementation of these operators because they would construct
7303 GiNaC @code{relational} objects which in the case of @code{<} do not
7304 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7305 decide which one is algebraically 'less').
7307 Next, we need to change our definition of the @code{sprod} type to let
7308 GiNaC know that an ordering relation exists for the embedded objects:
7311 typedef structure<sprod_s, compare_std_less> sprod;
7314 @code{sprod} objects then behave as expected:
7318 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7319 // -> <a|b>-<a^2|b^2>
7320 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7321 // -> <a|b>+<a^2|b^2>
7322 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7324 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7329 The @code{compare_std_less} policy parameter tells GiNaC to use the
7330 @code{std::less} and @code{std::equal_to} functors to compare objects of
7331 type @code{sprod_s}. By default, these functors forward their work to the
7332 standard @code{<} and @code{==} operators, which we have overloaded.
7333 Alternatively, we could have specialized @code{std::less} and
7334 @code{std::equal_to} for class @code{sprod_s}.
7336 GiNaC provides two other comparison policies for @code{structure<T>}
7337 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7338 which does a bit-wise comparison of the contained @code{T} objects.
7339 This should be used with extreme care because it only works reliably with
7340 built-in integral types, and it also compares any padding (filler bytes of
7341 undefined value) that the @code{T} class might have.
7343 @subsection Subexpressions
7345 Our scalar product class has two subexpressions: the left and right
7346 operands. It might be a good idea to make them accessible via the standard
7347 @code{nops()} and @code{op()} methods:
7350 size_t sprod::nops() const
7355 ex sprod::op(size_t i) const
7359 return get_struct().left;
7361 return get_struct().right;
7363 throw std::range_error("sprod::op(): no such operand");
7368 Implementing @code{nops()} and @code{op()} for container types such as
7369 @code{sprod} has two other nice side effects:
7373 @code{has()} works as expected
7375 GiNaC generates better hash keys for the objects (the default implementation
7376 of @code{calchash()} takes subexpressions into account)
7379 @cindex @code{let_op()}
7380 There is a non-const variant of @code{op()} called @code{let_op()} that
7381 allows replacing subexpressions:
7384 ex & sprod::let_op(size_t i)
7386 // every non-const member function must call this
7387 ensure_if_modifiable();
7391 return get_struct().left;
7393 return get_struct().right;
7395 throw std::range_error("sprod::let_op(): no such operand");
7400 Once we have provided @code{let_op()} we also get @code{subs()} and
7401 @code{map()} for free. In fact, every container class that returns a non-null
7402 @code{nops()} value must either implement @code{let_op()} or provide custom
7403 implementations of @code{subs()} and @code{map()}.
7405 In turn, the availability of @code{map()} enables the recursive behavior of a
7406 couple of other default method implementations, in particular @code{evalf()},
7407 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7408 we probably want to provide our own version of @code{expand()} for scalar
7409 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7410 This is left as an exercise for the reader.
7412 The @code{structure<T>} template defines many more member functions that
7413 you can override by specialization to customize the behavior of your
7414 structures. You are referred to the next section for a description of
7415 some of these (especially @code{eval()}). There is, however, one topic
7416 that shall be addressed here, as it demonstrates one peculiarity of the
7417 @code{structure<T>} template: archiving.
7419 @subsection Archiving structures
7421 If you don't know how the archiving of GiNaC objects is implemented, you
7422 should first read the next section and then come back here. You're back?
7425 To implement archiving for structures it is not enough to provide
7426 specializations for the @code{archive()} member function and the
7427 unarchiving constructor (the @code{unarchive()} function has a default
7428 implementation). You also need to provide a unique name (as a string literal)
7429 for each structure type you define. This is because in GiNaC archives,
7430 the class of an object is stored as a string, the class name.
7432 By default, this class name (as returned by the @code{class_name()} member
7433 function) is @samp{structure} for all structure classes. This works as long
7434 as you have only defined one structure type, but if you use two or more you
7435 need to provide a different name for each by specializing the
7436 @code{get_class_name()} member function. Here is a sample implementation
7437 for enabling archiving of the scalar product type defined above:
7440 const char *sprod::get_class_name() @{ return "sprod"; @}
7442 void sprod::archive(archive_node & n) const
7444 inherited::archive(n);
7445 n.add_ex("left", get_struct().left);
7446 n.add_ex("right", get_struct().right);
7449 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7451 n.find_ex("left", get_struct().left, sym_lst);
7452 n.find_ex("right", get_struct().right, sym_lst);
7456 Note that the unarchiving constructor is @code{sprod::structure} and not
7457 @code{sprod::sprod}, and that we don't need to supply an
7458 @code{sprod::unarchive()} function.
7461 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7462 @c node-name, next, previous, up
7463 @section Adding classes
7465 The @code{structure<T>} template provides an way to extend GiNaC with custom
7466 algebraic classes that is easy to use but has its limitations, the most
7467 severe of which being that you can't add any new member functions to
7468 structures. To be able to do this, you need to write a new class definition
7471 This section will explain how to implement new algebraic classes in GiNaC by
7472 giving the example of a simple 'string' class. After reading this section
7473 you will know how to properly declare a GiNaC class and what the minimum
7474 required member functions are that you have to implement. We only cover the
7475 implementation of a 'leaf' class here (i.e. one that doesn't contain
7476 subexpressions). Creating a container class like, for example, a class
7477 representing tensor products is more involved but this section should give
7478 you enough information so you can consult the source to GiNaC's predefined
7479 classes if you want to implement something more complicated.
7481 @subsection GiNaC's run-time type information system
7483 @cindex hierarchy of classes
7485 All algebraic classes (that is, all classes that can appear in expressions)
7486 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7487 @code{basic *} (which is essentially what an @code{ex} is) represents a
7488 generic pointer to an algebraic class. Occasionally it is necessary to find
7489 out what the class of an object pointed to by a @code{basic *} really is.
7490 Also, for the unarchiving of expressions it must be possible to find the
7491 @code{unarchive()} function of a class given the class name (as a string). A
7492 system that provides this kind of information is called a run-time type
7493 information (RTTI) system. The C++ language provides such a thing (see the
7494 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7495 implements its own, simpler RTTI.
7497 The RTTI in GiNaC is based on two mechanisms:
7502 The @code{basic} class declares a member variable @code{tinfo_key} which
7503 holds an unsigned integer that identifies the object's class. These numbers
7504 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7505 classes. They all start with @code{TINFO_}.
7508 By means of some clever tricks with static members, GiNaC maintains a list
7509 of information for all classes derived from @code{basic}. The information
7510 available includes the class names, the @code{tinfo_key}s, and pointers
7511 to the unarchiving functions. This class registry is defined in the
7512 @file{registrar.h} header file.
7516 The disadvantage of this proprietary RTTI implementation is that there's
7517 a little more to do when implementing new classes (C++'s RTTI works more
7518 or less automatically) but don't worry, most of the work is simplified by
7521 @subsection A minimalistic example
7523 Now we will start implementing a new class @code{mystring} that allows
7524 placing character strings in algebraic expressions (this is not very useful,
7525 but it's just an example). This class will be a direct subclass of
7526 @code{basic}. You can use this sample implementation as a starting point
7527 for your own classes.
7529 The code snippets given here assume that you have included some header files
7535 #include <stdexcept>
7536 using namespace std;
7538 #include <ginac/ginac.h>
7539 using namespace GiNaC;
7542 The first thing we have to do is to define a @code{tinfo_key} for our new
7543 class. This can be any arbitrary unsigned number that is not already taken
7544 by one of the existing classes but it's better to come up with something
7545 that is unlikely to clash with keys that might be added in the future. The
7546 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7547 which is not a requirement but we are going to stick with this scheme:
7550 const unsigned TINFO_mystring = 0x42420001U;
7553 Now we can write down the class declaration. The class stores a C++
7554 @code{string} and the user shall be able to construct a @code{mystring}
7555 object from a C or C++ string:
7558 class mystring : public basic
7560 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7563 mystring(const string &s);
7564 mystring(const char *s);
7570 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7573 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7574 macros are defined in @file{registrar.h}. They take the name of the class
7575 and its direct superclass as arguments and insert all required declarations
7576 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7577 the first line after the opening brace of the class definition. The
7578 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7579 source (at global scope, of course, not inside a function).
7581 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7582 declarations of the default constructor and a couple of other functions that
7583 are required. It also defines a type @code{inherited} which refers to the
7584 superclass so you don't have to modify your code every time you shuffle around
7585 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7586 class with the GiNaC RTTI (there is also a
7587 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7588 options for the class, and which we will be using instead in a few minutes).
7590 Now there are seven member functions we have to implement to get a working
7596 @code{mystring()}, the default constructor.
7599 @code{void archive(archive_node &n)}, the archiving function. This stores all
7600 information needed to reconstruct an object of this class inside an
7601 @code{archive_node}.
7604 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7605 constructor. This constructs an instance of the class from the information
7606 found in an @code{archive_node}.
7609 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7610 unarchiving function. It constructs a new instance by calling the unarchiving
7614 @cindex @code{compare_same_type()}
7615 @code{int compare_same_type(const basic &other)}, which is used internally
7616 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7617 -1, depending on the relative order of this object and the @code{other}
7618 object. If it returns 0, the objects are considered equal.
7619 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7620 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7621 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7622 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7623 must provide a @code{compare_same_type()} function, even those representing
7624 objects for which no reasonable algebraic ordering relationship can be
7628 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7629 which are the two constructors we declared.
7633 Let's proceed step-by-step. The default constructor looks like this:
7636 mystring::mystring() : inherited(TINFO_mystring) @{@}
7639 The golden rule is that in all constructors you have to set the
7640 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7641 it will be set by the constructor of the superclass and all hell will break
7642 loose in the RTTI. For your convenience, the @code{basic} class provides
7643 a constructor that takes a @code{tinfo_key} value, which we are using here
7644 (remember that in our case @code{inherited == basic}). If the superclass
7645 didn't have such a constructor, we would have to set the @code{tinfo_key}
7646 to the right value manually.
7648 In the default constructor you should set all other member variables to
7649 reasonable default values (we don't need that here since our @code{str}
7650 member gets set to an empty string automatically).
7652 Next are the three functions for archiving. You have to implement them even
7653 if you don't plan to use archives, but the minimum required implementation
7654 is really simple. First, the archiving function:
7657 void mystring::archive(archive_node &n) const
7659 inherited::archive(n);
7660 n.add_string("string", str);
7664 The only thing that is really required is calling the @code{archive()}
7665 function of the superclass. Optionally, you can store all information you
7666 deem necessary for representing the object into the passed
7667 @code{archive_node}. We are just storing our string here. For more
7668 information on how the archiving works, consult the @file{archive.h} header
7671 The unarchiving constructor is basically the inverse of the archiving
7675 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7677 n.find_string("string", str);
7681 If you don't need archiving, just leave this function empty (but you must
7682 invoke the unarchiving constructor of the superclass). Note that we don't
7683 have to set the @code{tinfo_key} here because it is done automatically
7684 by the unarchiving constructor of the @code{basic} class.
7686 Finally, the unarchiving function:
7689 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7691 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7695 You don't have to understand how exactly this works. Just copy these
7696 four lines into your code literally (replacing the class name, of
7697 course). It calls the unarchiving constructor of the class and unless
7698 you are doing something very special (like matching @code{archive_node}s
7699 to global objects) you don't need a different implementation. For those
7700 who are interested: setting the @code{dynallocated} flag puts the object
7701 under the control of GiNaC's garbage collection. It will get deleted
7702 automatically once it is no longer referenced.
7704 Our @code{compare_same_type()} function uses a provided function to compare
7708 int mystring::compare_same_type(const basic &other) const
7710 const mystring &o = static_cast<const mystring &>(other);
7711 int cmpval = str.compare(o.str);
7714 else if (cmpval < 0)
7721 Although this function takes a @code{basic &}, it will always be a reference
7722 to an object of exactly the same class (objects of different classes are not
7723 comparable), so the cast is safe. If this function returns 0, the two objects
7724 are considered equal (in the sense that @math{A-B=0}), so you should compare
7725 all relevant member variables.
7727 Now the only thing missing is our two new constructors:
7730 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7731 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7734 No surprises here. We set the @code{str} member from the argument and
7735 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7737 That's it! We now have a minimal working GiNaC class that can store
7738 strings in algebraic expressions. Let's confirm that the RTTI works:
7741 ex e = mystring("Hello, world!");
7742 cout << is_a<mystring>(e) << endl;
7745 cout << e.bp->class_name() << endl;
7749 Obviously it does. Let's see what the expression @code{e} looks like:
7753 // -> [mystring object]
7756 Hm, not exactly what we expect, but of course the @code{mystring} class
7757 doesn't yet know how to print itself. This can be done either by implementing
7758 the @code{print()} member function, or, preferably, by specifying a
7759 @code{print_func<>()} class option. Let's say that we want to print the string
7760 surrounded by double quotes:
7763 class mystring : public basic
7767 void do_print(const print_context &c, unsigned level = 0) const;
7771 void mystring::do_print(const print_context &c, unsigned level) const
7773 // print_context::s is a reference to an ostream
7774 c.s << '\"' << str << '\"';
7778 The @code{level} argument is only required for container classes to
7779 correctly parenthesize the output.
7781 Now we need to tell GiNaC that @code{mystring} objects should use the
7782 @code{do_print()} member function for printing themselves. For this, we
7786 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7792 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7793 print_func<print_context>(&mystring::do_print))
7796 Let's try again to print the expression:
7800 // -> "Hello, world!"
7803 Much better. If we wanted to have @code{mystring} objects displayed in a
7804 different way depending on the output format (default, LaTeX, etc.), we
7805 would have supplied multiple @code{print_func<>()} options with different
7806 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7807 separated by dots. This is similar to the way options are specified for
7808 symbolic functions. @xref{Printing}, for a more in-depth description of the
7809 way expression output is implemented in GiNaC.
7811 The @code{mystring} class can be used in arbitrary expressions:
7814 e += mystring("GiNaC rulez");
7816 // -> "GiNaC rulez"+"Hello, world!"
7819 (GiNaC's automatic term reordering is in effect here), or even
7822 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7824 // -> "One string"^(2*sin(-"Another string"+Pi))
7827 Whether this makes sense is debatable but remember that this is only an
7828 example. At least it allows you to implement your own symbolic algorithms
7831 Note that GiNaC's algebraic rules remain unchanged:
7834 e = mystring("Wow") * mystring("Wow");
7838 e = pow(mystring("First")-mystring("Second"), 2);
7839 cout << e.expand() << endl;
7840 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7843 There's no way to, for example, make GiNaC's @code{add} class perform string
7844 concatenation. You would have to implement this yourself.
7846 @subsection Automatic evaluation
7849 @cindex @code{eval()}
7850 @cindex @code{hold()}
7851 When dealing with objects that are just a little more complicated than the
7852 simple string objects we have implemented, chances are that you will want to
7853 have some automatic simplifications or canonicalizations performed on them.
7854 This is done in the evaluation member function @code{eval()}. Let's say that
7855 we wanted all strings automatically converted to lowercase with
7856 non-alphabetic characters stripped, and empty strings removed:
7859 class mystring : public basic
7863 ex eval(int level = 0) const;
7867 ex mystring::eval(int level) const
7870 for (int i=0; i<str.length(); i++) @{
7872 if (c >= 'A' && c <= 'Z')
7873 new_str += tolower(c);
7874 else if (c >= 'a' && c <= 'z')
7878 if (new_str.length() == 0)
7881 return mystring(new_str).hold();
7885 The @code{level} argument is used to limit the recursion depth of the
7886 evaluation. We don't have any subexpressions in the @code{mystring}
7887 class so we are not concerned with this. If we had, we would call the
7888 @code{eval()} functions of the subexpressions with @code{level - 1} as
7889 the argument if @code{level != 1}. The @code{hold()} member function
7890 sets a flag in the object that prevents further evaluation. Otherwise
7891 we might end up in an endless loop. When you want to return the object
7892 unmodified, use @code{return this->hold();}.
7894 Let's confirm that it works:
7897 ex e = mystring("Hello, world!") + mystring("!?#");
7901 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7906 @subsection Optional member functions
7908 We have implemented only a small set of member functions to make the class
7909 work in the GiNaC framework. There are two functions that are not strictly
7910 required but will make operations with objects of the class more efficient:
7912 @cindex @code{calchash()}
7913 @cindex @code{is_equal_same_type()}
7915 unsigned calchash() const;
7916 bool is_equal_same_type(const basic &other) const;
7919 The @code{calchash()} method returns an @code{unsigned} hash value for the
7920 object which will allow GiNaC to compare and canonicalize expressions much
7921 more efficiently. You should consult the implementation of some of the built-in
7922 GiNaC classes for examples of hash functions. The default implementation of
7923 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7924 class and all subexpressions that are accessible via @code{op()}.
7926 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7927 tests for equality without establishing an ordering relation, which is often
7928 faster. The default implementation of @code{is_equal_same_type()} just calls
7929 @code{compare_same_type()} and tests its result for zero.
7931 @subsection Other member functions
7933 For a real algebraic class, there are probably some more functions that you
7934 might want to provide:
7937 bool info(unsigned inf) const;
7938 ex evalf(int level = 0) const;
7939 ex series(const relational & r, int order, unsigned options = 0) const;
7940 ex derivative(const symbol & s) const;
7943 If your class stores sub-expressions (see the scalar product example in the
7944 previous section) you will probably want to override
7946 @cindex @code{let_op()}
7949 ex op(size_t i) const;
7950 ex & let_op(size_t i);
7951 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7952 ex map(map_function & f) const;
7955 @code{let_op()} is a variant of @code{op()} that allows write access. The
7956 default implementations of @code{subs()} and @code{map()} use it, so you have
7957 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7959 You can, of course, also add your own new member functions. Remember
7960 that the RTTI may be used to get information about what kinds of objects
7961 you are dealing with (the position in the class hierarchy) and that you
7962 can always extract the bare object from an @code{ex} by stripping the
7963 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7964 should become a need.
7966 That's it. May the source be with you!
7969 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7970 @c node-name, next, previous, up
7971 @chapter A Comparison With Other CAS
7974 This chapter will give you some information on how GiNaC compares to
7975 other, traditional Computer Algebra Systems, like @emph{Maple},
7976 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7977 disadvantages over these systems.
7980 * Advantages:: Strengths of the GiNaC approach.
7981 * Disadvantages:: Weaknesses of the GiNaC approach.
7982 * Why C++?:: Attractiveness of C++.
7985 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7986 @c node-name, next, previous, up
7989 GiNaC has several advantages over traditional Computer
7990 Algebra Systems, like
7995 familiar language: all common CAS implement their own proprietary
7996 grammar which you have to learn first (and maybe learn again when your
7997 vendor decides to `enhance' it). With GiNaC you can write your program
7998 in common C++, which is standardized.
8002 structured data types: you can build up structured data types using
8003 @code{struct}s or @code{class}es together with STL features instead of
8004 using unnamed lists of lists of lists.
8007 strongly typed: in CAS, you usually have only one kind of variables
8008 which can hold contents of an arbitrary type. This 4GL like feature is
8009 nice for novice programmers, but dangerous.
8012 development tools: powerful development tools exist for C++, like fancy
8013 editors (e.g. with automatic indentation and syntax highlighting),
8014 debuggers, visualization tools, documentation generators@dots{}
8017 modularization: C++ programs can easily be split into modules by
8018 separating interface and implementation.
8021 price: GiNaC is distributed under the GNU Public License which means
8022 that it is free and available with source code. And there are excellent
8023 C++-compilers for free, too.
8026 extendable: you can add your own classes to GiNaC, thus extending it on
8027 a very low level. Compare this to a traditional CAS that you can
8028 usually only extend on a high level by writing in the language defined
8029 by the parser. In particular, it turns out to be almost impossible to
8030 fix bugs in a traditional system.
8033 multiple interfaces: Though real GiNaC programs have to be written in
8034 some editor, then be compiled, linked and executed, there are more ways
8035 to work with the GiNaC engine. Many people want to play with
8036 expressions interactively, as in traditional CASs. Currently, two such
8037 windows into GiNaC have been implemented and many more are possible: the
8038 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8039 types to a command line and second, as a more consistent approach, an
8040 interactive interface to the Cint C++ interpreter has been put together
8041 (called GiNaC-cint) that allows an interactive scripting interface
8042 consistent with the C++ language. It is available from the usual GiNaC
8046 seamless integration: it is somewhere between difficult and impossible
8047 to call CAS functions from within a program written in C++ or any other
8048 programming language and vice versa. With GiNaC, your symbolic routines
8049 are part of your program. You can easily call third party libraries,
8050 e.g. for numerical evaluation or graphical interaction. All other
8051 approaches are much more cumbersome: they range from simply ignoring the
8052 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8053 system (i.e. @emph{Yacas}).
8056 efficiency: often large parts of a program do not need symbolic
8057 calculations at all. Why use large integers for loop variables or
8058 arbitrary precision arithmetics where @code{int} and @code{double} are
8059 sufficient? For pure symbolic applications, GiNaC is comparable in
8060 speed with other CAS.
8065 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8066 @c node-name, next, previous, up
8067 @section Disadvantages
8069 Of course it also has some disadvantages:
8074 advanced features: GiNaC cannot compete with a program like
8075 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8076 which grows since 1981 by the work of dozens of programmers, with
8077 respect to mathematical features. Integration, factorization,
8078 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8079 not planned for the near future).
8082 portability: While the GiNaC library itself is designed to avoid any
8083 platform dependent features (it should compile on any ANSI compliant C++
8084 compiler), the currently used version of the CLN library (fast large
8085 integer and arbitrary precision arithmetics) can only by compiled
8086 without hassle on systems with the C++ compiler from the GNU Compiler
8087 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8088 macros to let the compiler gather all static initializations, which
8089 works for GNU C++ only. Feel free to contact the authors in case you
8090 really believe that you need to use a different compiler. We have
8091 occasionally used other compilers and may be able to give you advice.}
8092 GiNaC uses recent language features like explicit constructors, mutable
8093 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8094 literally. Recent GCC versions starting at 2.95.3, although itself not
8095 yet ANSI compliant, support all needed features.
8100 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8101 @c node-name, next, previous, up
8104 Why did we choose to implement GiNaC in C++ instead of Java or any other
8105 language? C++ is not perfect: type checking is not strict (casting is
8106 possible), separation between interface and implementation is not
8107 complete, object oriented design is not enforced. The main reason is
8108 the often scolded feature of operator overloading in C++. While it may
8109 be true that operating on classes with a @code{+} operator is rarely
8110 meaningful, it is perfectly suited for algebraic expressions. Writing
8111 @math{3x+5y} as @code{3*x+5*y} instead of
8112 @code{x.times(3).plus(y.times(5))} looks much more natural.
8113 Furthermore, the main developers are more familiar with C++ than with
8114 any other programming language.
8117 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8118 @c node-name, next, previous, up
8119 @appendix Internal Structures
8122 * Expressions are reference counted::
8123 * Internal representation of products and sums::
8126 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8127 @c node-name, next, previous, up
8128 @appendixsection Expressions are reference counted
8130 @cindex reference counting
8131 @cindex copy-on-write
8132 @cindex garbage collection
8133 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8134 where the counter belongs to the algebraic objects derived from class
8135 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8136 which @code{ex} contains an instance. If you understood that, you can safely
8137 skip the rest of this passage.
8139 Expressions are extremely light-weight since internally they work like
8140 handles to the actual representation. They really hold nothing more
8141 than a pointer to some other object. What this means in practice is
8142 that whenever you create two @code{ex} and set the second equal to the
8143 first no copying process is involved. Instead, the copying takes place
8144 as soon as you try to change the second. Consider the simple sequence
8149 #include <ginac/ginac.h>
8150 using namespace std;
8151 using namespace GiNaC;
8155 symbol x("x"), y("y"), z("z");
8158 e1 = sin(x + 2*y) + 3*z + 41;
8159 e2 = e1; // e2 points to same object as e1
8160 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8161 e2 += 1; // e2 is copied into a new object
8162 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8166 The line @code{e2 = e1;} creates a second expression pointing to the
8167 object held already by @code{e1}. The time involved for this operation
8168 is therefore constant, no matter how large @code{e1} was. Actual
8169 copying, however, must take place in the line @code{e2 += 1;} because
8170 @code{e1} and @code{e2} are not handles for the same object any more.
8171 This concept is called @dfn{copy-on-write semantics}. It increases
8172 performance considerably whenever one object occurs multiple times and
8173 represents a simple garbage collection scheme because when an @code{ex}
8174 runs out of scope its destructor checks whether other expressions handle
8175 the object it points to too and deletes the object from memory if that
8176 turns out not to be the case. A slightly less trivial example of
8177 differentiation using the chain-rule should make clear how powerful this
8182 symbol x("x"), y("y");
8186 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8187 cout << e1 << endl // prints x+3*y
8188 << e2 << endl // prints (x+3*y)^3
8189 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8193 Here, @code{e1} will actually be referenced three times while @code{e2}
8194 will be referenced two times. When the power of an expression is built,
8195 that expression needs not be copied. Likewise, since the derivative of
8196 a power of an expression can be easily expressed in terms of that
8197 expression, no copying of @code{e1} is involved when @code{e3} is
8198 constructed. So, when @code{e3} is constructed it will print as
8199 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8200 holds a reference to @code{e2} and the factor in front is just
8203 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8204 semantics. When you insert an expression into a second expression, the
8205 result behaves exactly as if the contents of the first expression were
8206 inserted. But it may be useful to remember that this is not what
8207 happens. Knowing this will enable you to write much more efficient
8208 code. If you still have an uncertain feeling with copy-on-write
8209 semantics, we recommend you have a look at the
8210 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8211 Marshall Cline. Chapter 16 covers this issue and presents an
8212 implementation which is pretty close to the one in GiNaC.
8215 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8216 @c node-name, next, previous, up
8217 @appendixsection Internal representation of products and sums
8219 @cindex representation
8222 @cindex @code{power}
8223 Although it should be completely transparent for the user of
8224 GiNaC a short discussion of this topic helps to understand the sources
8225 and also explain performance to a large degree. Consider the
8226 unexpanded symbolic expression
8228 $2d^3 \left( 4a + 5b - 3 \right)$
8231 @math{2*d^3*(4*a+5*b-3)}
8233 which could naively be represented by a tree of linear containers for
8234 addition and multiplication, one container for exponentiation with base
8235 and exponent and some atomic leaves of symbols and numbers in this
8240 @cindex pair-wise representation
8241 However, doing so results in a rather deeply nested tree which will
8242 quickly become inefficient to manipulate. We can improve on this by
8243 representing the sum as a sequence of terms, each one being a pair of a
8244 purely numeric multiplicative coefficient and its rest. In the same
8245 spirit we can store the multiplication as a sequence of terms, each
8246 having a numeric exponent and a possibly complicated base, the tree
8247 becomes much more flat:
8251 The number @code{3} above the symbol @code{d} shows that @code{mul}
8252 objects are treated similarly where the coefficients are interpreted as
8253 @emph{exponents} now. Addition of sums of terms or multiplication of
8254 products with numerical exponents can be coded to be very efficient with
8255 such a pair-wise representation. Internally, this handling is performed
8256 by most CAS in this way. It typically speeds up manipulations by an
8257 order of magnitude. The overall multiplicative factor @code{2} and the
8258 additive term @code{-3} look somewhat out of place in this
8259 representation, however, since they are still carrying a trivial
8260 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8261 this is avoided by adding a field that carries an overall numeric
8262 coefficient. This results in the realistic picture of internal
8265 $2d^3 \left( 4a + 5b - 3 \right)$:
8268 @math{2*d^3*(4*a+5*b-3)}:
8274 This also allows for a better handling of numeric radicals, since
8275 @code{sqrt(2)} can now be carried along calculations. Now it should be
8276 clear, why both classes @code{add} and @code{mul} are derived from the
8277 same abstract class: the data representation is the same, only the
8278 semantics differs. In the class hierarchy, methods for polynomial
8279 expansion and the like are reimplemented for @code{add} and @code{mul},
8280 but the data structure is inherited from @code{expairseq}.
8283 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8284 @c node-name, next, previous, up
8285 @appendix Package Tools
8287 If you are creating a software package that uses the GiNaC library,
8288 setting the correct command line options for the compiler and linker
8289 can be difficult. GiNaC includes two tools to make this process easier.
8292 * ginac-config:: A shell script to detect compiler and linker flags.
8293 * AM_PATH_GINAC:: Macro for GNU automake.
8297 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8298 @c node-name, next, previous, up
8299 @section @command{ginac-config}
8300 @cindex ginac-config
8302 @command{ginac-config} is a shell script that you can use to determine
8303 the compiler and linker command line options required to compile and
8304 link a program with the GiNaC library.
8306 @command{ginac-config} takes the following flags:
8310 Prints out the version of GiNaC installed.
8312 Prints '-I' flags pointing to the installed header files.
8314 Prints out the linker flags necessary to link a program against GiNaC.
8315 @item --prefix[=@var{PREFIX}]
8316 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8317 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8318 Otherwise, prints out the configured value of @env{$prefix}.
8319 @item --exec-prefix[=@var{PREFIX}]
8320 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8321 Otherwise, prints out the configured value of @env{$exec_prefix}.
8324 Typically, @command{ginac-config} will be used within a configure
8325 script, as described below. It, however, can also be used directly from
8326 the command line using backquotes to compile a simple program. For
8330 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8333 This command line might expand to (for example):
8336 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8337 -lginac -lcln -lstdc++
8340 Not only is the form using @command{ginac-config} easier to type, it will
8341 work on any system, no matter how GiNaC was configured.
8344 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8345 @c node-name, next, previous, up
8346 @section @samp{AM_PATH_GINAC}
8347 @cindex AM_PATH_GINAC
8349 For packages configured using GNU automake, GiNaC also provides
8350 a macro to automate the process of checking for GiNaC.
8353 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8354 [, @var{ACTION-IF-NOT-FOUND}]]])
8362 Determines the location of GiNaC using @command{ginac-config}, which is
8363 either found in the user's path, or from the environment variable
8364 @env{GINACLIB_CONFIG}.
8367 Tests the installed libraries to make sure that their version
8368 is later than @var{MINIMUM-VERSION}. (A default version will be used
8372 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8373 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8374 variable to the output of @command{ginac-config --libs}, and calls
8375 @samp{AC_SUBST()} for these variables so they can be used in generated
8376 makefiles, and then executes @var{ACTION-IF-FOUND}.
8379 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8380 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8384 This macro is in file @file{ginac.m4} which is installed in
8385 @file{$datadir/aclocal}. Note that if automake was installed with a
8386 different @samp{--prefix} than GiNaC, you will either have to manually
8387 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8388 aclocal the @samp{-I} option when running it.
8391 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8392 * Example package:: Example of a package using AM_PATH_GINAC.
8396 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8397 @c node-name, next, previous, up
8398 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8400 Simply make sure that @command{ginac-config} is in your path, and run
8401 the configure script.
8408 The directory where the GiNaC libraries are installed needs
8409 to be found by your system's dynamic linker.
8411 This is generally done by
8414 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8420 setting the environment variable @env{LD_LIBRARY_PATH},
8423 or, as a last resort,
8426 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8427 running configure, for instance:
8430 LDFLAGS=-R/home/cbauer/lib ./configure
8435 You can also specify a @command{ginac-config} not in your path by
8436 setting the @env{GINACLIB_CONFIG} environment variable to the
8437 name of the executable
8440 If you move the GiNaC package from its installed location,
8441 you will either need to modify @command{ginac-config} script
8442 manually to point to the new location or rebuild GiNaC.
8453 --with-ginac-prefix=@var{PREFIX}
8454 --with-ginac-exec-prefix=@var{PREFIX}
8457 are provided to override the prefix and exec-prefix that were stored
8458 in the @command{ginac-config} shell script by GiNaC's configure. You are
8459 generally better off configuring GiNaC with the right path to begin with.
8463 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8464 @c node-name, next, previous, up
8465 @subsection Example of a package using @samp{AM_PATH_GINAC}
8467 The following shows how to build a simple package using automake
8468 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8472 #include <ginac/ginac.h>
8476 GiNaC::symbol x("x");
8477 GiNaC::ex a = GiNaC::sin(x);
8478 std::cout << "Derivative of " << a
8479 << " is " << a.diff(x) << std::endl;
8484 You should first read the introductory portions of the automake
8485 Manual, if you are not already familiar with it.
8487 Two files are needed, @file{configure.in}, which is used to build the
8491 dnl Process this file with autoconf to produce a configure script.
8493 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8499 AM_PATH_GINAC(0.9.0, [
8500 LIBS="$LIBS $GINACLIB_LIBS"
8501 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8502 ], AC_MSG_ERROR([need to have GiNaC installed]))
8507 The only command in this which is not standard for automake
8508 is the @samp{AM_PATH_GINAC} macro.
8510 That command does the following: If a GiNaC version greater or equal
8511 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8512 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8513 the error message `need to have GiNaC installed'
8515 And the @file{Makefile.am}, which will be used to build the Makefile.
8518 ## Process this file with automake to produce Makefile.in
8519 bin_PROGRAMS = simple
8520 simple_SOURCES = simple.cpp
8523 This @file{Makefile.am}, says that we are building a single executable,
8524 from a single source file @file{simple.cpp}. Since every program
8525 we are building uses GiNaC we simply added the GiNaC options
8526 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8527 want to specify them on a per-program basis: for instance by
8531 simple_LDADD = $(GINACLIB_LIBS)
8532 INCLUDES = $(GINACLIB_CPPFLAGS)
8535 to the @file{Makefile.am}.
8537 To try this example out, create a new directory and add the three
8540 Now execute the following commands:
8543 $ automake --add-missing
8548 You now have a package that can be built in the normal fashion
8557 @node Bibliography, Concept Index, Example package, Top
8558 @c node-name, next, previous, up
8559 @appendix Bibliography
8564 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8567 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8570 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8573 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8576 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8577 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8580 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8581 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8582 Academic Press, London
8585 @cite{Computer Algebra Systems - A Practical Guide},
8586 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8589 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8590 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8593 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8594 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8597 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8602 @node Concept Index, , Bibliography, Top
8603 @c node-name, next, previous, up
8604 @unnumbered Concept Index