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, the CLN library
485 is used extensively and needs to be installed on your system.
486 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
487 (it is covered by GPL) and install it prior to trying to install
488 GiNaC. The configure script checks if it can find it and if it cannot
489 it will refuse to continue.
492 @node Configuration, Building GiNaC, Prerequisites, Installation
493 @c node-name, next, previous, up
494 @section Configuration
495 @cindex configuration
498 To configure GiNaC means to prepare the source distribution for
499 building. It is done via a shell script called @command{configure} that
500 is shipped with the sources and was originally generated by GNU
501 Autoconf. Since a configure script generated by GNU Autoconf never
502 prompts, all customization must be done either via command line
503 parameters or environment variables. It accepts a list of parameters,
504 the complete set of which can be listed by calling it with the
505 @option{--help} option. The most important ones will be shortly
506 described in what follows:
511 @option{--disable-shared}: When given, this option switches off the
512 build of a shared library, i.e. a @file{.so} file. This may be convenient
513 when developing because it considerably speeds up compilation.
516 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
517 and headers are installed. It defaults to @file{/usr/local} which means
518 that the library is installed in the directory @file{/usr/local/lib},
519 the header files in @file{/usr/local/include/ginac} and the documentation
520 (like this one) into @file{/usr/local/share/doc/GiNaC}.
523 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
524 the library installed in some other directory than
525 @file{@var{PREFIX}/lib/}.
528 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
529 to have the header files installed in some other directory than
530 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
531 @option{--includedir=/usr/include} you will end up with the header files
532 sitting in the directory @file{/usr/include/ginac/}. Note that the
533 subdirectory @file{ginac} is enforced by this process in order to
534 keep the header files separated from others. This avoids some
535 clashes and allows for an easier deinstallation of GiNaC. This ought
536 to be considered A Good Thing (tm).
539 @option{--datadir=@var{DATADIR}}: This option may be given in case you
540 want to have the documentation installed in some other directory than
541 @file{@var{PREFIX}/share/doc/GiNaC/}.
545 In addition, you may specify some environment variables. @env{CXX}
546 holds the path and the name of the C++ compiler in case you want to
547 override the default in your path. (The @command{configure} script
548 searches your path for @command{c++}, @command{g++}, @command{gcc},
549 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
550 be very useful to define some compiler flags with the @env{CXXFLAGS}
551 environment variable, like optimization, debugging information and
552 warning levels. If omitted, it defaults to @option{-g
553 -O2}.@footnote{The @command{configure} script is itself generated from
554 the file @file{configure.ac}. It is only distributed in packaged
555 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
556 must generate @command{configure} along with the various
557 @file{Makefile.in} by using the @command{autogen.sh} script. This will
558 require a fair amount of support from your local toolchain, though.}
560 The whole process is illustrated in the following two
561 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
562 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
565 Here is a simple configuration for a site-wide GiNaC library assuming
566 everything is in default paths:
569 $ export CXXFLAGS="-Wall -O2"
573 And here is a configuration for a private static GiNaC library with
574 several components sitting in custom places (site-wide GCC and private
575 CLN). The compiler is persuaded to be picky and full assertions and
576 debugging information are switched on:
579 $ export CXX=/usr/local/gnu/bin/c++
580 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
581 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
582 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
583 $ ./configure --disable-shared --prefix=$(HOME)
587 @node Building GiNaC, Installing GiNaC, Configuration, Installation
588 @c node-name, next, previous, up
589 @section Building GiNaC
590 @cindex building GiNaC
592 After proper configuration you should just build the whole
597 at the command prompt and go for a cup of coffee. The exact time it
598 takes to compile GiNaC depends not only on the speed of your machines
599 but also on other parameters, for instance what value for @env{CXXFLAGS}
600 you entered. Optimization may be very time-consuming.
602 Just to make sure GiNaC works properly you may run a collection of
603 regression tests by typing
609 This will compile some sample programs, run them and check the output
610 for correctness. The regression tests fall in three categories. First,
611 the so called @emph{exams} are performed, simple tests where some
612 predefined input is evaluated (like a pupils' exam). Second, the
613 @emph{checks} test the coherence of results among each other with
614 possible random input. Third, some @emph{timings} are performed, which
615 benchmark some predefined problems with different sizes and display the
616 CPU time used in seconds. Each individual test should return a message
617 @samp{passed}. This is mostly intended to be a QA-check if something
618 was broken during development, not a sanity check of your system. Some
619 of the tests in sections @emph{checks} and @emph{timings} may require
620 insane amounts of memory and CPU time. Feel free to kill them if your
621 machine catches fire. Another quite important intent is to allow people
622 to fiddle around with optimization.
624 By default, the only documentation that will be built is this tutorial
625 in @file{.info} format. To build the GiNaC tutorial and reference manual
626 in HTML, DVI, PostScript, or PDF formats, use one of
635 Generally, the top-level Makefile runs recursively to the
636 subdirectories. It is therefore safe to go into any subdirectory
637 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
638 @var{target} there in case something went wrong.
641 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
642 @c node-name, next, previous, up
643 @section Installing GiNaC
646 To install GiNaC on your system, simply type
652 As described in the section about configuration the files will be
653 installed in the following directories (the directories will be created
654 if they don't already exist):
659 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
660 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
661 So will @file{libginac.so} unless the configure script was
662 given the option @option{--disable-shared}. The proper symlinks
663 will be established as well.
666 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
667 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
670 All documentation (info) will be stuffed into
671 @file{@var{PREFIX}/share/doc/GiNaC/} (or
672 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
676 For the sake of completeness we will list some other useful make
677 targets: @command{make clean} deletes all files generated by
678 @command{make}, i.e. all the object files. In addition @command{make
679 distclean} removes all files generated by the configuration and
680 @command{make maintainer-clean} goes one step further and deletes files
681 that may require special tools to rebuild (like the @command{libtool}
682 for instance). Finally @command{make uninstall} removes the installed
683 library, header files and documentation@footnote{Uninstallation does not
684 work after you have called @command{make distclean} since the
685 @file{Makefile} is itself generated by the configuration from
686 @file{Makefile.in} and hence deleted by @command{make distclean}. There
687 are two obvious ways out of this dilemma. First, you can run the
688 configuration again with the same @var{PREFIX} thus creating a
689 @file{Makefile} with a working @samp{uninstall} target. Second, you can
690 do it by hand since you now know where all the files went during
694 @node Basic Concepts, Expressions, Installing GiNaC, Top
695 @c node-name, next, previous, up
696 @chapter Basic Concepts
698 This chapter will describe the different fundamental objects that can be
699 handled by GiNaC. But before doing so, it is worthwhile introducing you
700 to the more commonly used class of expressions, representing a flexible
701 meta-class for storing all mathematical objects.
704 * Expressions:: The fundamental GiNaC class.
705 * Automatic evaluation:: Evaluation and canonicalization.
706 * Error handling:: How the library reports errors.
707 * The Class Hierarchy:: Overview of GiNaC's classes.
708 * Symbols:: Symbolic objects.
709 * Numbers:: Numerical objects.
710 * Constants:: Pre-defined constants.
711 * Fundamental containers:: Sums, products and powers.
712 * Lists:: Lists of expressions.
713 * Mathematical functions:: Mathematical functions.
714 * Relations:: Equality, Inequality and all that.
715 * Integrals:: Symbolic integrals.
716 * Matrices:: Matrices.
717 * Indexed objects:: Handling indexed quantities.
718 * Non-commutative objects:: Algebras with non-commutative products.
719 * Hash Maps:: A faster alternative to std::map<>.
723 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
724 @c node-name, next, previous, up
726 @cindex expression (class @code{ex})
729 The most common class of objects a user deals with is the expression
730 @code{ex}, representing a mathematical object like a variable, number,
731 function, sum, product, etc@dots{} Expressions may be put together to form
732 new expressions, passed as arguments to functions, and so on. Here is a
733 little collection of valid expressions:
736 ex MyEx1 = 5; // simple number
737 ex MyEx2 = x + 2*y; // polynomial in x and y
738 ex MyEx3 = (x + 1)/(x - 1); // rational expression
739 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
740 ex MyEx5 = MyEx4 + 1; // similar to above
743 Expressions are handles to other more fundamental objects, that often
744 contain other expressions thus creating a tree of expressions
745 (@xref{Internal Structures}, for particular examples). Most methods on
746 @code{ex} therefore run top-down through such an expression tree. For
747 example, the method @code{has()} scans recursively for occurrences of
748 something inside an expression. Thus, if you have declared @code{MyEx4}
749 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
750 the argument of @code{sin} and hence return @code{true}.
752 The next sections will outline the general picture of GiNaC's class
753 hierarchy and describe the classes of objects that are handled by
756 @subsection Note: Expressions and STL containers
758 GiNaC expressions (@code{ex} objects) have value semantics (they can be
759 assigned, reassigned and copied like integral types) but the operator
760 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
761 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
763 This implies that in order to use expressions in sorted containers such as
764 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
765 comparison predicate. GiNaC provides such a predicate, called
766 @code{ex_is_less}. For example, a set of expressions should be defined
767 as @code{std::set<ex, ex_is_less>}.
769 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
770 don't pose a problem. A @code{std::vector<ex>} works as expected.
772 @xref{Information About Expressions}, for more about comparing and ordering
776 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
777 @c node-name, next, previous, up
778 @section Automatic evaluation and canonicalization of expressions
781 GiNaC performs some automatic transformations on expressions, to simplify
782 them and put them into a canonical form. Some examples:
785 ex MyEx1 = 2*x - 1 + x; // 3*x-1
786 ex MyEx2 = x - x; // 0
787 ex MyEx3 = cos(2*Pi); // 1
788 ex MyEx4 = x*y/x; // y
791 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
792 evaluation}. GiNaC only performs transformations that are
796 at most of complexity
804 algebraically correct, possibly except for a set of measure zero (e.g.
805 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
808 There are two types of automatic transformations in GiNaC that may not
809 behave in an entirely obvious way at first glance:
813 The terms of sums and products (and some other things like the arguments of
814 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
815 into a canonical form that is deterministic, but not lexicographical or in
816 any other way easy to guess (it almost always depends on the number and
817 order of the symbols you define). However, constructing the same expression
818 twice, either implicitly or explicitly, will always result in the same
821 Expressions of the form 'number times sum' are automatically expanded (this
822 has to do with GiNaC's internal representation of sums and products). For
825 ex MyEx5 = 2*(x + y); // 2*x+2*y
826 ex MyEx6 = z*(x + y); // z*(x+y)
830 The general rule is that when you construct expressions, GiNaC automatically
831 creates them in canonical form, which might differ from the form you typed in
832 your program. This may create some awkward looking output (@samp{-y+x} instead
833 of @samp{x-y}) but allows for more efficient operation and usually yields
834 some immediate simplifications.
836 @cindex @code{eval()}
837 Internally, the anonymous evaluator in GiNaC is implemented by the methods
840 ex ex::eval(int level = 0) const;
841 ex basic::eval(int level = 0) const;
844 but unless you are extending GiNaC with your own classes or functions, there
845 should never be any reason to call them explicitly. All GiNaC methods that
846 transform expressions, like @code{subs()} or @code{normal()}, automatically
847 re-evaluate their results.
850 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
851 @c node-name, next, previous, up
852 @section Error handling
854 @cindex @code{pole_error} (class)
856 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
857 generated by GiNaC are subclassed from the standard @code{exception} class
858 defined in the @file{<stdexcept>} header. In addition to the predefined
859 @code{logic_error}, @code{domain_error}, @code{out_of_range},
860 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
861 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
862 exception that gets thrown when trying to evaluate a mathematical function
865 The @code{pole_error} class has a member function
868 int pole_error::degree() const;
871 that returns the order of the singularity (or 0 when the pole is
872 logarithmic or the order is undefined).
874 When using GiNaC it is useful to arrange for exceptions to be caught in
875 the main program even if you don't want to do any special error handling.
876 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
877 default exception handler of your C++ compiler's run-time system which
878 usually only aborts the program without giving any information what went
881 Here is an example for a @code{main()} function that catches and prints
882 exceptions generated by GiNaC:
887 #include <ginac/ginac.h>
889 using namespace GiNaC;
897 @} catch (exception &p) @{
898 cerr << p.what() << endl;
906 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
907 @c node-name, next, previous, up
908 @section The Class Hierarchy
910 GiNaC's class hierarchy consists of several classes representing
911 mathematical objects, all of which (except for @code{ex} and some
912 helpers) are internally derived from one abstract base class called
913 @code{basic}. You do not have to deal with objects of class
914 @code{basic}, instead you'll be dealing with symbols, numbers,
915 containers of expressions and so on.
919 To get an idea about what kinds of symbolic composites may be built we
920 have a look at the most important classes in the class hierarchy and
921 some of the relations among the classes:
923 @image{classhierarchy}
925 The abstract classes shown here (the ones without drop-shadow) are of no
926 interest for the user. They are used internally in order to avoid code
927 duplication if two or more classes derived from them share certain
928 features. An example is @code{expairseq}, a container for a sequence of
929 pairs each consisting of one expression and a number (@code{numeric}).
930 What @emph{is} visible to the user are the derived classes @code{add}
931 and @code{mul}, representing sums and products. @xref{Internal
932 Structures}, where these two classes are described in more detail. The
933 following table shortly summarizes what kinds of mathematical objects
934 are stored in the different classes:
937 @multitable @columnfractions .22 .78
938 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
939 @item @code{constant} @tab Constants like
946 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
947 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
948 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
949 @item @code{ncmul} @tab Products of non-commutative objects
950 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
955 @code{sqrt(}@math{2}@code{)}
958 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
959 @item @code{function} @tab A symbolic function like
966 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
967 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
968 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
969 @item @code{indexed} @tab Indexed object like @math{A_ij}
970 @item @code{tensor} @tab Special tensor like the delta and metric tensors
971 @item @code{idx} @tab Index of an indexed object
972 @item @code{varidx} @tab Index with variance
973 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
974 @item @code{wildcard} @tab Wildcard for pattern matching
975 @item @code{structure} @tab Template for user-defined classes
980 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
981 @c node-name, next, previous, up
983 @cindex @code{symbol} (class)
984 @cindex hierarchy of classes
987 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
988 manipulation what atoms are for chemistry.
990 A typical symbol definition looks like this:
995 This definition actually contains three very different things:
997 @item a C++ variable named @code{x}
998 @item a @code{symbol} object stored in this C++ variable; this object
999 represents the symbol in a GiNaC expression
1000 @item the string @code{"x"} which is the name of the symbol, used (almost)
1001 exclusively for printing expressions holding the symbol
1004 Symbols have an explicit name, supplied as a string during construction,
1005 because in C++, variable names can't be used as values, and the C++ compiler
1006 throws them away during compilation.
1008 It is possible to omit the symbol name in the definition:
1013 In this case, GiNaC will assign the symbol an internal, unique name of the
1014 form @code{symbolNNN}. This won't affect the usability of the symbol but
1015 the output of your calculations will become more readable if you give your
1016 symbols sensible names (for intermediate expressions that are only used
1017 internally such anonymous symbols can be quite useful, however).
1019 Now, here is one important property of GiNaC that differentiates it from
1020 other computer algebra programs you may have used: GiNaC does @emph{not} use
1021 the names of symbols to tell them apart, but a (hidden) serial number that
1022 is unique for each newly created @code{symbol} object. In you want to use
1023 one and the same symbol in different places in your program, you must only
1024 create one @code{symbol} object and pass that around. If you create another
1025 symbol, even if it has the same name, GiNaC will treat it as a different
1042 // prints "x^6" which looks right, but...
1044 cout << e.degree(x) << endl;
1045 // ...this doesn't work. The symbol "x" here is different from the one
1046 // in f() and in the expression returned by f(). Consequently, it
1051 One possibility to ensure that @code{f()} and @code{main()} use the same
1052 symbol is to pass the symbol as an argument to @code{f()}:
1054 ex f(int n, const ex & x)
1063 // Now, f() uses the same symbol.
1066 cout << e.degree(x) << endl;
1067 // prints "6", as expected
1071 Another possibility would be to define a global symbol @code{x} that is used
1072 by both @code{f()} and @code{main()}. If you are using global symbols and
1073 multiple compilation units you must take special care, however. Suppose
1074 that you have a header file @file{globals.h} in your program that defines
1075 a @code{symbol x("x");}. In this case, every unit that includes
1076 @file{globals.h} would also get its own definition of @code{x} (because
1077 header files are just inlined into the source code by the C++ preprocessor),
1078 and hence you would again end up with multiple equally-named, but different,
1079 symbols. Instead, the @file{globals.h} header should only contain a
1080 @emph{declaration} like @code{extern symbol x;}, with the definition of
1081 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1083 A different approach to ensuring that symbols used in different parts of
1084 your program are identical is to create them with a @emph{factory} function
1087 const symbol & get_symbol(const string & s)
1089 static map<string, symbol> directory;
1090 map<string, symbol>::iterator i = directory.find(s);
1091 if (i != directory.end())
1094 return directory.insert(make_pair(s, symbol(s))).first->second;
1098 This function returns one newly constructed symbol for each name that is
1099 passed in, and it returns the same symbol when called multiple times with
1100 the same name. Using this symbol factory, we can rewrite our example like
1105 return pow(get_symbol("x"), n);
1112 // Both calls of get_symbol("x") yield the same symbol.
1113 cout << e.degree(get_symbol("x")) << endl;
1118 Instead of creating symbols from strings we could also have
1119 @code{get_symbol()} take, for example, an integer number as its argument.
1120 In this case, we would probably want to give the generated symbols names
1121 that include this number, which can be accomplished with the help of an
1122 @code{ostringstream}.
1124 In general, if you're getting weird results from GiNaC such as an expression
1125 @samp{x-x} that is not simplified to zero, you should check your symbol
1128 As we said, the names of symbols primarily serve for purposes of expression
1129 output. But there are actually two instances where GiNaC uses the names for
1130 identifying symbols: When constructing an expression from a string, and when
1131 recreating an expression from an archive (@pxref{Input/Output}).
1133 In addition to its name, a symbol may contain a special string that is used
1136 symbol x("x", "\\Box");
1139 This creates a symbol that is printed as "@code{x}" in normal output, but
1140 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1141 information about the different output formats of expressions in GiNaC).
1142 GiNaC automatically creates proper LaTeX code for symbols having names of
1143 greek letters (@samp{alpha}, @samp{mu}, etc.).
1145 @cindex @code{subs()}
1146 Symbols in GiNaC can't be assigned values. If you need to store results of
1147 calculations and give them a name, use C++ variables of type @code{ex}.
1148 If you want to replace a symbol in an expression with something else, you
1149 can invoke the expression's @code{.subs()} method
1150 (@pxref{Substituting Expressions}).
1152 @cindex @code{realsymbol()}
1153 By default, symbols are expected to stand in for complex values, i.e. they live
1154 in the complex domain. As a consequence, operations like complex conjugation,
1155 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1156 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1157 because of the unknown imaginary part of @code{x}.
1158 On the other hand, if you are sure that your symbols will hold only real values, you
1159 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1160 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1161 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @node Numbers, Constants, Symbols, Basic Concepts
1165 @c node-name, next, previous, up
1167 @cindex @code{numeric} (class)
1173 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1174 The classes therein serve as foundation classes for GiNaC. CLN stands
1175 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1176 In order to find out more about CLN's internals, the reader is referred to
1177 the documentation of that library. @inforef{Introduction, , cln}, for
1178 more information. Suffice to say that it is by itself build on top of
1179 another library, the GNU Multiple Precision library GMP, which is an
1180 extremely fast library for arbitrary long integers and rationals as well
1181 as arbitrary precision floating point numbers. It is very commonly used
1182 by several popular cryptographic applications. CLN extends GMP by
1183 several useful things: First, it introduces the complex number field
1184 over either reals (i.e. floating point numbers with arbitrary precision)
1185 or rationals. Second, it automatically converts rationals to integers
1186 if the denominator is unity and complex numbers to real numbers if the
1187 imaginary part vanishes and also correctly treats algebraic functions.
1188 Third it provides good implementations of state-of-the-art algorithms
1189 for all trigonometric and hyperbolic functions as well as for
1190 calculation of some useful constants.
1192 The user can construct an object of class @code{numeric} in several
1193 ways. The following example shows the four most important constructors.
1194 It uses construction from C-integer, construction of fractions from two
1195 integers, construction from C-float and construction from a string:
1199 #include <ginac/ginac.h>
1200 using namespace GiNaC;
1204 numeric two = 2; // exact integer 2
1205 numeric r(2,3); // exact fraction 2/3
1206 numeric e(2.71828); // floating point number
1207 numeric p = "3.14159265358979323846"; // constructor from string
1208 // Trott's constant in scientific notation:
1209 numeric trott("1.0841015122311136151E-2");
1211 std::cout << two*p << std::endl; // floating point 6.283...
1216 @cindex complex numbers
1217 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1222 numeric z1 = 2-3*I; // exact complex number 2-3i
1223 numeric z2 = 5.9+1.6*I; // complex floating point number
1227 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1228 This would, however, call C's built-in operator @code{/} for integers
1229 first and result in a numeric holding a plain integer 1. @strong{Never
1230 use the operator @code{/} on integers} unless you know exactly what you
1231 are doing! Use the constructor from two integers instead, as shown in
1232 the example above. Writing @code{numeric(1)/2} may look funny but works
1235 @cindex @code{Digits}
1237 We have seen now the distinction between exact numbers and floating
1238 point numbers. Clearly, the user should never have to worry about
1239 dynamically created exact numbers, since their `exactness' always
1240 determines how they ought to be handled, i.e. how `long' they are. The
1241 situation is different for floating point numbers. Their accuracy is
1242 controlled by one @emph{global} variable, called @code{Digits}. (For
1243 those readers who know about Maple: it behaves very much like Maple's
1244 @code{Digits}). All objects of class numeric that are constructed from
1245 then on will be stored with a precision matching that number of decimal
1250 #include <ginac/ginac.h>
1251 using namespace std;
1252 using namespace GiNaC;
1256 numeric three(3.0), one(1.0);
1257 numeric x = one/three;
1259 cout << "in " << Digits << " digits:" << endl;
1261 cout << Pi.evalf() << endl;
1273 The above example prints the following output to screen:
1277 0.33333333333333333334
1278 3.1415926535897932385
1280 0.33333333333333333333333333333333333333333333333333333333333333333334
1281 3.1415926535897932384626433832795028841971693993751058209749445923078
1285 Note that the last number is not necessarily rounded as you would
1286 naively expect it to be rounded in the decimal system. But note also,
1287 that in both cases you got a couple of extra digits. This is because
1288 numbers are internally stored by CLN as chunks of binary digits in order
1289 to match your machine's word size and to not waste precision. Thus, on
1290 architectures with different word size, the above output might even
1291 differ with regard to actually computed digits.
1293 It should be clear that objects of class @code{numeric} should be used
1294 for constructing numbers or for doing arithmetic with them. The objects
1295 one deals with most of the time are the polymorphic expressions @code{ex}.
1297 @subsection Tests on numbers
1299 Once you have declared some numbers, assigned them to expressions and
1300 done some arithmetic with them it is frequently desired to retrieve some
1301 kind of information from them like asking whether that number is
1302 integer, rational, real or complex. For those cases GiNaC provides
1303 several useful methods. (Internally, they fall back to invocations of
1304 certain CLN functions.)
1306 As an example, let's construct some rational number, multiply it with
1307 some multiple of its denominator and test what comes out:
1311 #include <ginac/ginac.h>
1312 using namespace std;
1313 using namespace GiNaC;
1315 // some very important constants:
1316 const numeric twentyone(21);
1317 const numeric ten(10);
1318 const numeric five(5);
1322 numeric answer = twentyone;
1325 cout << answer.is_integer() << endl; // false, it's 21/5
1327 cout << answer.is_integer() << endl; // true, it's 42 now!
1331 Note that the variable @code{answer} is constructed here as an integer
1332 by @code{numeric}'s copy constructor but in an intermediate step it
1333 holds a rational number represented as integer numerator and integer
1334 denominator. When multiplied by 10, the denominator becomes unity and
1335 the result is automatically converted to a pure integer again.
1336 Internally, the underlying CLN is responsible for this behavior and we
1337 refer the reader to CLN's documentation. Suffice to say that
1338 the same behavior applies to complex numbers as well as return values of
1339 certain functions. Complex numbers are automatically converted to real
1340 numbers if the imaginary part becomes zero. The full set of tests that
1341 can be applied is listed in the following table.
1344 @multitable @columnfractions .30 .70
1345 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1346 @item @code{.is_zero()}
1347 @tab @dots{}equal to zero
1348 @item @code{.is_positive()}
1349 @tab @dots{}not complex and greater than 0
1350 @item @code{.is_integer()}
1351 @tab @dots{}a (non-complex) integer
1352 @item @code{.is_pos_integer()}
1353 @tab @dots{}an integer and greater than 0
1354 @item @code{.is_nonneg_integer()}
1355 @tab @dots{}an integer and greater equal 0
1356 @item @code{.is_even()}
1357 @tab @dots{}an even integer
1358 @item @code{.is_odd()}
1359 @tab @dots{}an odd integer
1360 @item @code{.is_prime()}
1361 @tab @dots{}a prime integer (probabilistic primality test)
1362 @item @code{.is_rational()}
1363 @tab @dots{}an exact rational number (integers are rational, too)
1364 @item @code{.is_real()}
1365 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1366 @item @code{.is_cinteger()}
1367 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1368 @item @code{.is_crational()}
1369 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1373 @subsection Numeric functions
1375 The following functions can be applied to @code{numeric} objects and will be
1376 evaluated immediately:
1379 @multitable @columnfractions .30 .70
1380 @item @strong{Name} @tab @strong{Function}
1381 @item @code{inverse(z)}
1382 @tab returns @math{1/z}
1383 @cindex @code{inverse()} (numeric)
1384 @item @code{pow(a, b)}
1385 @tab exponentiation @math{a^b}
1388 @item @code{real(z)}
1390 @cindex @code{real()}
1391 @item @code{imag(z)}
1393 @cindex @code{imag()}
1394 @item @code{csgn(z)}
1395 @tab complex sign (returns an @code{int})
1396 @item @code{numer(z)}
1397 @tab numerator of rational or complex rational number
1398 @item @code{denom(z)}
1399 @tab denominator of rational or complex rational number
1400 @item @code{sqrt(z)}
1402 @item @code{isqrt(n)}
1403 @tab integer square root
1404 @cindex @code{isqrt()}
1411 @item @code{asin(z)}
1413 @item @code{acos(z)}
1415 @item @code{atan(z)}
1416 @tab inverse tangent
1417 @item @code{atan(y, x)}
1418 @tab inverse tangent with two arguments
1419 @item @code{sinh(z)}
1420 @tab hyperbolic sine
1421 @item @code{cosh(z)}
1422 @tab hyperbolic cosine
1423 @item @code{tanh(z)}
1424 @tab hyperbolic tangent
1425 @item @code{asinh(z)}
1426 @tab inverse hyperbolic sine
1427 @item @code{acosh(z)}
1428 @tab inverse hyperbolic cosine
1429 @item @code{atanh(z)}
1430 @tab inverse hyperbolic tangent
1432 @tab exponential function
1434 @tab natural logarithm
1437 @item @code{zeta(z)}
1438 @tab Riemann's zeta function
1439 @item @code{tgamma(z)}
1441 @item @code{lgamma(z)}
1442 @tab logarithm of gamma function
1444 @tab psi (digamma) function
1445 @item @code{psi(n, z)}
1446 @tab derivatives of psi function (polygamma functions)
1447 @item @code{factorial(n)}
1448 @tab factorial function @math{n!}
1449 @item @code{doublefactorial(n)}
1450 @tab double factorial function @math{n!!}
1451 @cindex @code{doublefactorial()}
1452 @item @code{binomial(n, k)}
1453 @tab binomial coefficients
1454 @item @code{bernoulli(n)}
1455 @tab Bernoulli numbers
1456 @cindex @code{bernoulli()}
1457 @item @code{fibonacci(n)}
1458 @tab Fibonacci numbers
1459 @cindex @code{fibonacci()}
1460 @item @code{mod(a, b)}
1461 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1462 @cindex @code{mod()}
1463 @item @code{smod(a, b)}
1464 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1465 @cindex @code{smod()}
1466 @item @code{irem(a, b)}
1467 @tab integer remainder (has the sign of @math{a}, or is zero)
1468 @cindex @code{irem()}
1469 @item @code{irem(a, b, q)}
1470 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1471 @item @code{iquo(a, b)}
1472 @tab integer quotient
1473 @cindex @code{iquo()}
1474 @item @code{iquo(a, b, r)}
1475 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1476 @item @code{gcd(a, b)}
1477 @tab greatest common divisor
1478 @item @code{lcm(a, b)}
1479 @tab least common multiple
1483 Most of these functions are also available as symbolic functions that can be
1484 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1485 as polynomial algorithms.
1487 @subsection Converting numbers
1489 Sometimes it is desirable to convert a @code{numeric} object back to a
1490 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1491 class provides a couple of methods for this purpose:
1493 @cindex @code{to_int()}
1494 @cindex @code{to_long()}
1495 @cindex @code{to_double()}
1496 @cindex @code{to_cl_N()}
1498 int numeric::to_int() const;
1499 long numeric::to_long() const;
1500 double numeric::to_double() const;
1501 cln::cl_N numeric::to_cl_N() const;
1504 @code{to_int()} and @code{to_long()} only work when the number they are
1505 applied on is an exact integer. Otherwise the program will halt with a
1506 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1507 rational number will return a floating-point approximation. Both
1508 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1509 part of complex numbers.
1512 @node Constants, Fundamental containers, Numbers, Basic Concepts
1513 @c node-name, next, previous, up
1515 @cindex @code{constant} (class)
1518 @cindex @code{Catalan}
1519 @cindex @code{Euler}
1520 @cindex @code{evalf()}
1521 Constants behave pretty much like symbols except that they return some
1522 specific number when the method @code{.evalf()} is called.
1524 The predefined known constants are:
1527 @multitable @columnfractions .14 .30 .56
1528 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1530 @tab Archimedes' constant
1531 @tab 3.14159265358979323846264338327950288
1532 @item @code{Catalan}
1533 @tab Catalan's constant
1534 @tab 0.91596559417721901505460351493238411
1536 @tab Euler's (or Euler-Mascheroni) constant
1537 @tab 0.57721566490153286060651209008240243
1542 @node Fundamental containers, Lists, Constants, Basic Concepts
1543 @c node-name, next, previous, up
1544 @section Sums, products and powers
1548 @cindex @code{power}
1550 Simple rational expressions are written down in GiNaC pretty much like
1551 in other CAS or like expressions involving numerical variables in C.
1552 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1553 been overloaded to achieve this goal. When you run the following
1554 code snippet, the constructor for an object of type @code{mul} is
1555 automatically called to hold the product of @code{a} and @code{b} and
1556 then the constructor for an object of type @code{add} is called to hold
1557 the sum of that @code{mul} object and the number one:
1561 symbol a("a"), b("b");
1566 @cindex @code{pow()}
1567 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1568 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1569 construction is necessary since we cannot safely overload the constructor
1570 @code{^} in C++ to construct a @code{power} object. If we did, it would
1571 have several counterintuitive and undesired effects:
1575 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1577 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1578 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1579 interpret this as @code{x^(a^b)}.
1581 Also, expressions involving integer exponents are very frequently used,
1582 which makes it even more dangerous to overload @code{^} since it is then
1583 hard to distinguish between the semantics as exponentiation and the one
1584 for exclusive or. (It would be embarrassing to return @code{1} where one
1585 has requested @code{2^3}.)
1588 @cindex @command{ginsh}
1589 All effects are contrary to mathematical notation and differ from the
1590 way most other CAS handle exponentiation, therefore overloading @code{^}
1591 is ruled out for GiNaC's C++ part. The situation is different in
1592 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1593 that the other frequently used exponentiation operator @code{**} does
1594 not exist at all in C++).
1596 To be somewhat more precise, objects of the three classes described
1597 here, are all containers for other expressions. An object of class
1598 @code{power} is best viewed as a container with two slots, one for the
1599 basis, one for the exponent. All valid GiNaC expressions can be
1600 inserted. However, basic transformations like simplifying
1601 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1602 when this is mathematically possible. If we replace the outer exponent
1603 three in the example by some symbols @code{a}, the simplification is not
1604 safe and will not be performed, since @code{a} might be @code{1/2} and
1607 Objects of type @code{add} and @code{mul} are containers with an
1608 arbitrary number of slots for expressions to be inserted. Again, simple
1609 and safe simplifications are carried out like transforming
1610 @code{3*x+4-x} to @code{2*x+4}.
1613 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1614 @c node-name, next, previous, up
1615 @section Lists of expressions
1616 @cindex @code{lst} (class)
1618 @cindex @code{nops()}
1620 @cindex @code{append()}
1621 @cindex @code{prepend()}
1622 @cindex @code{remove_first()}
1623 @cindex @code{remove_last()}
1624 @cindex @code{remove_all()}
1626 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1627 expressions. They are not as ubiquitous as in many other computer algebra
1628 packages, but are sometimes used to supply a variable number of arguments of
1629 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1630 constructors, so you should have a basic understanding of them.
1632 Lists can be constructed by assigning a comma-separated sequence of
1637 symbol x("x"), y("y");
1640 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1645 There are also constructors that allow direct creation of lists of up to
1646 16 expressions, which is often more convenient but slightly less efficient:
1650 // This produces the same list 'l' as above:
1651 // lst l(x, 2, y, x+y);
1652 // lst l = lst(x, 2, y, x+y);
1656 Use the @code{nops()} method to determine the size (number of expressions) of
1657 a list and the @code{op()} method or the @code{[]} operator to access
1658 individual elements:
1662 cout << l.nops() << endl; // prints '4'
1663 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1667 As with the standard @code{list<T>} container, accessing random elements of a
1668 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1669 sequential access to the elements of a list is possible with the
1670 iterator types provided by the @code{lst} class:
1673 typedef ... lst::const_iterator;
1674 typedef ... lst::const_reverse_iterator;
1675 lst::const_iterator lst::begin() const;
1676 lst::const_iterator lst::end() const;
1677 lst::const_reverse_iterator lst::rbegin() const;
1678 lst::const_reverse_iterator lst::rend() const;
1681 For example, to print the elements of a list individually you can use:
1686 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1691 which is one order faster than
1696 for (size_t i = 0; i < l.nops(); ++i)
1697 cout << l.op(i) << endl;
1701 These iterators also allow you to use some of the algorithms provided by
1702 the C++ standard library:
1706 // print the elements of the list (requires #include <iterator>)
1707 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1709 // sum up the elements of the list (requires #include <numeric>)
1710 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1711 cout << sum << endl; // prints '2+2*x+2*y'
1715 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1716 (the only other one is @code{matrix}). You can modify single elements:
1720 l[1] = 42; // l is now @{x, 42, y, x+y@}
1721 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1725 You can append or prepend an expression to a list with the @code{append()}
1726 and @code{prepend()} methods:
1730 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1731 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1735 You can remove the first or last element of a list with @code{remove_first()}
1736 and @code{remove_last()}:
1740 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1741 l.remove_last(); // l is now @{x, 7, y, x+y@}
1745 You can remove all the elements of a list with @code{remove_all()}:
1749 l.remove_all(); // l is now empty
1753 You can bring the elements of a list into a canonical order with @code{sort()}:
1762 // l1 and l2 are now equal
1766 Finally, you can remove all but the first element of consecutive groups of
1767 elements with @code{unique()}:
1772 l3 = x, 2, 2, 2, y, x+y, y+x;
1773 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1778 @node Mathematical functions, Relations, Lists, Basic Concepts
1779 @c node-name, next, previous, up
1780 @section Mathematical functions
1781 @cindex @code{function} (class)
1782 @cindex trigonometric function
1783 @cindex hyperbolic function
1785 There are quite a number of useful functions hard-wired into GiNaC. For
1786 instance, all trigonometric and hyperbolic functions are implemented
1787 (@xref{Built-in Functions}, for a complete list).
1789 These functions (better called @emph{pseudofunctions}) are all objects
1790 of class @code{function}. They accept one or more expressions as
1791 arguments and return one expression. If the arguments are not
1792 numerical, the evaluation of the function may be halted, as it does in
1793 the next example, showing how a function returns itself twice and
1794 finally an expression that may be really useful:
1796 @cindex Gamma function
1797 @cindex @code{subs()}
1800 symbol x("x"), y("y");
1802 cout << tgamma(foo) << endl;
1803 // -> tgamma(x+(1/2)*y)
1804 ex bar = foo.subs(y==1);
1805 cout << tgamma(bar) << endl;
1807 ex foobar = bar.subs(x==7);
1808 cout << tgamma(foobar) << endl;
1809 // -> (135135/128)*Pi^(1/2)
1813 Besides evaluation most of these functions allow differentiation, series
1814 expansion and so on. Read the next chapter in order to learn more about
1817 It must be noted that these pseudofunctions are created by inline
1818 functions, where the argument list is templated. This means that
1819 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1820 @code{sin(ex(1))} and will therefore not result in a floating point
1821 number. Unless of course the function prototype is explicitly
1822 overridden -- which is the case for arguments of type @code{numeric}
1823 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1824 point number of class @code{numeric} you should call
1825 @code{sin(numeric(1))}. This is almost the same as calling
1826 @code{sin(1).evalf()} except that the latter will return a numeric
1827 wrapped inside an @code{ex}.
1830 @node Relations, Integrals, Mathematical functions, Basic Concepts
1831 @c node-name, next, previous, up
1833 @cindex @code{relational} (class)
1835 Sometimes, a relation holding between two expressions must be stored
1836 somehow. The class @code{relational} is a convenient container for such
1837 purposes. A relation is by definition a container for two @code{ex} and
1838 a relation between them that signals equality, inequality and so on.
1839 They are created by simply using the C++ operators @code{==}, @code{!=},
1840 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1842 @xref{Mathematical functions}, for examples where various applications
1843 of the @code{.subs()} method show how objects of class relational are
1844 used as arguments. There they provide an intuitive syntax for
1845 substitutions. They are also used as arguments to the @code{ex::series}
1846 method, where the left hand side of the relation specifies the variable
1847 to expand in and the right hand side the expansion point. They can also
1848 be used for creating systems of equations that are to be solved for
1849 unknown variables. But the most common usage of objects of this class
1850 is rather inconspicuous in statements of the form @code{if
1851 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1852 conversion from @code{relational} to @code{bool} takes place. Note,
1853 however, that @code{==} here does not perform any simplifications, hence
1854 @code{expand()} must be called explicitly.
1856 @node Integrals, Matrices, Relations, Basic Concepts
1857 @c node-name, next, previous, up
1859 @cindex @code{integral} (class)
1861 An object of class @dfn{integral} can be used to hold a symbolic integral.
1862 If you want to symbolically represent the integral of @code{x*x} from 0 to
1863 1, you would write this as
1865 integral(x, 0, 1, x*x)
1867 The first argument is the integration variable. It should be noted that
1868 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1869 fact, it can only integrate polynomials. An expression containing integrals
1870 can be evaluated symbolically by calling the
1874 method on it. Numerical evaluation is available by calling the
1878 method on an expression containing the integral. This will only evaluate
1879 integrals into a number if @code{subs}ing the integration variable by a
1880 number in the fourth argument of an integral and then @code{evalf}ing the
1881 result always results in a number. Of course, also the boundaries of the
1882 integration domain must @code{evalf} into numbers. It should be noted that
1883 trying to @code{evalf} a function with discontinuities in the integration
1884 domain is not recommended. The accuracy of the numeric evaluation of
1885 integrals is determined by the static member variable
1887 ex integral::relative_integration_error
1889 of the class @code{integral}. The default value of this is 10^-8.
1890 The integration works by halving the interval of integration, until numeric
1891 stability of the answer indicates that the requested accuracy has been
1892 reached. The maximum depth of the halving can be set via the static member
1895 int integral::max_integration_level
1897 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1898 return the integral unevaluated. The function that performs the numerical
1899 evaluation, is also available as
1901 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1904 This function will throw an exception if the maximum depth is exceeded. The
1905 last parameter of the function is optional and defaults to the
1906 @code{relative_integration_error}. To make sure that we do not do too
1907 much work if an expression contains the same integral multiple times,
1908 a lookup table is used.
1910 If you know that an expression holds an integral, you can get the
1911 integration variable, the left boundary, right boundary and integrand by
1912 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1913 @code{.op(3)}. Differentiating integrals with respect to variables works
1914 as expected. Note that it makes no sense to differentiate an integral
1915 with respect to the integration variable.
1917 @node Matrices, Indexed objects, Integrals, Basic Concepts
1918 @c node-name, next, previous, up
1920 @cindex @code{matrix} (class)
1922 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1923 matrix with @math{m} rows and @math{n} columns are accessed with two
1924 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1925 second one in the range 0@dots{}@math{n-1}.
1927 There are a couple of ways to construct matrices, with or without preset
1928 elements. The constructor
1931 matrix::matrix(unsigned r, unsigned c);
1934 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1937 The fastest way to create a matrix with preinitialized elements is to assign
1938 a list of comma-separated expressions to an empty matrix (see below for an
1939 example). But you can also specify the elements as a (flat) list with
1942 matrix::matrix(unsigned r, unsigned c, const lst & l);
1947 @cindex @code{lst_to_matrix()}
1949 ex lst_to_matrix(const lst & l);
1952 constructs a matrix from a list of lists, each list representing a matrix row.
1954 There is also a set of functions for creating some special types of
1957 @cindex @code{diag_matrix()}
1958 @cindex @code{unit_matrix()}
1959 @cindex @code{symbolic_matrix()}
1961 ex diag_matrix(const lst & l);
1962 ex unit_matrix(unsigned x);
1963 ex unit_matrix(unsigned r, unsigned c);
1964 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1965 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1966 const string & tex_base_name);
1969 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1970 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1971 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1972 matrix filled with newly generated symbols made of the specified base name
1973 and the position of each element in the matrix.
1975 Matrices often arise by omitting elements of another matrix. For
1976 instance, the submatrix @code{S} of a matrix @code{M} takes a
1977 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1978 by removing one row and one column from a matrix @code{M}. (The
1979 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1980 can be used for computing the inverse using Cramer's rule.)
1982 @cindex @code{sub_matrix()}
1983 @cindex @code{reduced_matrix()}
1985 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1986 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1989 The function @code{sub_matrix()} takes a row offset @code{r} and a
1990 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1991 columns. The function @code{reduced_matrix()} has two integer arguments
1992 that specify which row and column to remove:
2000 cout << reduced_matrix(m, 1, 1) << endl;
2001 // -> [[11,13],[31,33]]
2002 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2003 // -> [[22,23],[32,33]]
2007 Matrix elements can be accessed and set using the parenthesis (function call)
2011 const ex & matrix::operator()(unsigned r, unsigned c) const;
2012 ex & matrix::operator()(unsigned r, unsigned c);
2015 It is also possible to access the matrix elements in a linear fashion with
2016 the @code{op()} method. But C++-style subscripting with square brackets
2017 @samp{[]} is not available.
2019 Here are a couple of examples for constructing matrices:
2023 symbol a("a"), b("b");
2037 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2040 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2043 cout << diag_matrix(lst(a, b)) << endl;
2046 cout << unit_matrix(3) << endl;
2047 // -> [[1,0,0],[0,1,0],[0,0,1]]
2049 cout << symbolic_matrix(2, 3, "x") << endl;
2050 // -> [[x00,x01,x02],[x10,x11,x12]]
2054 @cindex @code{transpose()}
2055 There are three ways to do arithmetic with matrices. The first (and most
2056 direct one) is to use the methods provided by the @code{matrix} class:
2059 matrix matrix::add(const matrix & other) const;
2060 matrix matrix::sub(const matrix & other) const;
2061 matrix matrix::mul(const matrix & other) const;
2062 matrix matrix::mul_scalar(const ex & other) const;
2063 matrix matrix::pow(const ex & expn) const;
2064 matrix matrix::transpose() const;
2067 All of these methods return the result as a new matrix object. Here is an
2068 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2073 matrix A(2, 2), B(2, 2), C(2, 2);
2081 matrix result = A.mul(B).sub(C.mul_scalar(2));
2082 cout << result << endl;
2083 // -> [[-13,-6],[1,2]]
2088 @cindex @code{evalm()}
2089 The second (and probably the most natural) way is to construct an expression
2090 containing matrices with the usual arithmetic operators and @code{pow()}.
2091 For efficiency reasons, expressions with sums, products and powers of
2092 matrices are not automatically evaluated in GiNaC. You have to call the
2096 ex ex::evalm() const;
2099 to obtain the result:
2106 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2107 cout << e.evalm() << endl;
2108 // -> [[-13,-6],[1,2]]
2113 The non-commutativity of the product @code{A*B} in this example is
2114 automatically recognized by GiNaC. There is no need to use a special
2115 operator here. @xref{Non-commutative objects}, for more information about
2116 dealing with non-commutative expressions.
2118 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2119 to perform the arithmetic:
2124 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2125 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2127 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2128 cout << e.simplify_indexed() << endl;
2129 // -> [[-13,-6],[1,2]].i.j
2133 Using indices is most useful when working with rectangular matrices and
2134 one-dimensional vectors because you don't have to worry about having to
2135 transpose matrices before multiplying them. @xref{Indexed objects}, for
2136 more information about using matrices with indices, and about indices in
2139 The @code{matrix} class provides a couple of additional methods for
2140 computing determinants, traces, characteristic polynomials and ranks:
2142 @cindex @code{determinant()}
2143 @cindex @code{trace()}
2144 @cindex @code{charpoly()}
2145 @cindex @code{rank()}
2147 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2148 ex matrix::trace() const;
2149 ex matrix::charpoly(const ex & lambda) const;
2150 unsigned matrix::rank() const;
2153 The @samp{algo} argument of @code{determinant()} allows to select
2154 between different algorithms for calculating the determinant. The
2155 asymptotic speed (as parametrized by the matrix size) can greatly differ
2156 between those algorithms, depending on the nature of the matrix'
2157 entries. The possible values are defined in the @file{flags.h} header
2158 file. By default, GiNaC uses a heuristic to automatically select an
2159 algorithm that is likely (but not guaranteed) to give the result most
2162 @cindex @code{inverse()} (matrix)
2163 @cindex @code{solve()}
2164 Matrices may also be inverted using the @code{ex matrix::inverse()}
2165 method and linear systems may be solved with:
2168 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2169 unsigned algo=solve_algo::automatic) const;
2172 Assuming the matrix object this method is applied on is an @code{m}
2173 times @code{n} matrix, then @code{vars} must be a @code{n} times
2174 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2175 times @code{p} matrix. The returned matrix then has dimension @code{n}
2176 times @code{p} and in the case of an underdetermined system will still
2177 contain some of the indeterminates from @code{vars}. If the system is
2178 overdetermined, an exception is thrown.
2181 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2182 @c node-name, next, previous, up
2183 @section Indexed objects
2185 GiNaC allows you to handle expressions containing general indexed objects in
2186 arbitrary spaces. It is also able to canonicalize and simplify such
2187 expressions and perform symbolic dummy index summations. There are a number
2188 of predefined indexed objects provided, like delta and metric tensors.
2190 There are few restrictions placed on indexed objects and their indices and
2191 it is easy to construct nonsense expressions, but our intention is to
2192 provide a general framework that allows you to implement algorithms with
2193 indexed quantities, getting in the way as little as possible.
2195 @cindex @code{idx} (class)
2196 @cindex @code{indexed} (class)
2197 @subsection Indexed quantities and their indices
2199 Indexed expressions in GiNaC are constructed of two special types of objects,
2200 @dfn{index objects} and @dfn{indexed objects}.
2204 @cindex contravariant
2207 @item Index objects are of class @code{idx} or a subclass. Every index has
2208 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2209 the index lives in) which can both be arbitrary expressions but are usually
2210 a number or a simple symbol. In addition, indices of class @code{varidx} have
2211 a @dfn{variance} (they can be co- or contravariant), and indices of class
2212 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2214 @item Indexed objects are of class @code{indexed} or a subclass. They
2215 contain a @dfn{base expression} (which is the expression being indexed), and
2216 one or more indices.
2220 @strong{Please notice:} when printing expressions, covariant indices and indices
2221 without variance are denoted @samp{.i} while contravariant indices are
2222 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2223 value. In the following, we are going to use that notation in the text so
2224 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2225 not visible in the output.
2227 A simple example shall illustrate the concepts:
2231 #include <ginac/ginac.h>
2232 using namespace std;
2233 using namespace GiNaC;
2237 symbol i_sym("i"), j_sym("j");
2238 idx i(i_sym, 3), j(j_sym, 3);
2241 cout << indexed(A, i, j) << endl;
2243 cout << index_dimensions << indexed(A, i, j) << endl;
2245 cout << dflt; // reset cout to default output format (dimensions hidden)
2249 The @code{idx} constructor takes two arguments, the index value and the
2250 index dimension. First we define two index objects, @code{i} and @code{j},
2251 both with the numeric dimension 3. The value of the index @code{i} is the
2252 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2253 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2254 construct an expression containing one indexed object, @samp{A.i.j}. It has
2255 the symbol @code{A} as its base expression and the two indices @code{i} and
2258 The dimensions of indices are normally not visible in the output, but one
2259 can request them to be printed with the @code{index_dimensions} manipulator,
2262 Note the difference between the indices @code{i} and @code{j} which are of
2263 class @code{idx}, and the index values which are the symbols @code{i_sym}
2264 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2265 or numbers but must be index objects. For example, the following is not
2266 correct and will raise an exception:
2269 symbol i("i"), j("j");
2270 e = indexed(A, i, j); // ERROR: indices must be of type idx
2273 You can have multiple indexed objects in an expression, index values can
2274 be numeric, and index dimensions symbolic:
2278 symbol B("B"), dim("dim");
2279 cout << 4 * indexed(A, i)
2280 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2285 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2286 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2287 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2288 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2289 @code{simplify_indexed()} for that, see below).
2291 In fact, base expressions, index values and index dimensions can be
2292 arbitrary expressions:
2296 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2301 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2302 get an error message from this but you will probably not be able to do
2303 anything useful with it.
2305 @cindex @code{get_value()}
2306 @cindex @code{get_dimension()}
2310 ex idx::get_value();
2311 ex idx::get_dimension();
2314 return the value and dimension of an @code{idx} object. If you have an index
2315 in an expression, such as returned by calling @code{.op()} on an indexed
2316 object, you can get a reference to the @code{idx} object with the function
2317 @code{ex_to<idx>()} on the expression.
2319 There are also the methods
2322 bool idx::is_numeric();
2323 bool idx::is_symbolic();
2324 bool idx::is_dim_numeric();
2325 bool idx::is_dim_symbolic();
2328 for checking whether the value and dimension are numeric or symbolic
2329 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2330 About Expressions}) returns information about the index value.
2332 @cindex @code{varidx} (class)
2333 If you need co- and contravariant indices, use the @code{varidx} class:
2337 symbol mu_sym("mu"), nu_sym("nu");
2338 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2339 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2341 cout << indexed(A, mu, nu) << endl;
2343 cout << indexed(A, mu_co, nu) << endl;
2345 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2350 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2351 co- or contravariant. The default is a contravariant (upper) index, but
2352 this can be overridden by supplying a third argument to the @code{varidx}
2353 constructor. The two methods
2356 bool varidx::is_covariant();
2357 bool varidx::is_contravariant();
2360 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2361 to get the object reference from an expression). There's also the very useful
2365 ex varidx::toggle_variance();
2368 which makes a new index with the same value and dimension but the opposite
2369 variance. By using it you only have to define the index once.
2371 @cindex @code{spinidx} (class)
2372 The @code{spinidx} class provides dotted and undotted variant indices, as
2373 used in the Weyl-van-der-Waerden spinor formalism:
2377 symbol K("K"), C_sym("C"), D_sym("D");
2378 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2379 // contravariant, undotted
2380 spinidx C_co(C_sym, 2, true); // covariant index
2381 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2382 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2384 cout << indexed(K, C, D) << endl;
2386 cout << indexed(K, C_co, D_dot) << endl;
2388 cout << indexed(K, D_co_dot, D) << endl;
2393 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2394 dotted or undotted. The default is undotted but this can be overridden by
2395 supplying a fourth argument to the @code{spinidx} constructor. The two
2399 bool spinidx::is_dotted();
2400 bool spinidx::is_undotted();
2403 allow you to check whether or not a @code{spinidx} object is dotted (use
2404 @code{ex_to<spinidx>()} to get the object reference from an expression).
2405 Finally, the two methods
2408 ex spinidx::toggle_dot();
2409 ex spinidx::toggle_variance_dot();
2412 create a new index with the same value and dimension but opposite dottedness
2413 and the same or opposite variance.
2415 @subsection Substituting indices
2417 @cindex @code{subs()}
2418 Sometimes you will want to substitute one symbolic index with another
2419 symbolic or numeric index, for example when calculating one specific element
2420 of a tensor expression. This is done with the @code{.subs()} method, as it
2421 is done for symbols (see @ref{Substituting Expressions}).
2423 You have two possibilities here. You can either substitute the whole index
2424 by another index or expression:
2428 ex e = indexed(A, mu_co);
2429 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2430 // -> A.mu becomes A~nu
2431 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2432 // -> A.mu becomes A~0
2433 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2434 // -> A.mu becomes A.0
2438 The third example shows that trying to replace an index with something that
2439 is not an index will substitute the index value instead.
2441 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2446 ex e = indexed(A, mu_co);
2447 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2448 // -> A.mu becomes A.nu
2449 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2450 // -> A.mu becomes A.0
2454 As you see, with the second method only the value of the index will get
2455 substituted. Its other properties, including its dimension, remain unchanged.
2456 If you want to change the dimension of an index you have to substitute the
2457 whole index by another one with the new dimension.
2459 Finally, substituting the base expression of an indexed object works as
2464 ex e = indexed(A, mu_co);
2465 cout << e << " becomes " << e.subs(A == A+B) << endl;
2466 // -> A.mu becomes (B+A).mu
2470 @subsection Symmetries
2471 @cindex @code{symmetry} (class)
2472 @cindex @code{sy_none()}
2473 @cindex @code{sy_symm()}
2474 @cindex @code{sy_anti()}
2475 @cindex @code{sy_cycl()}
2477 Indexed objects can have certain symmetry properties with respect to their
2478 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2479 that is constructed with the helper functions
2482 symmetry sy_none(...);
2483 symmetry sy_symm(...);
2484 symmetry sy_anti(...);
2485 symmetry sy_cycl(...);
2488 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2489 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2490 represents a cyclic symmetry. Each of these functions accepts up to four
2491 arguments which can be either symmetry objects themselves or unsigned integer
2492 numbers that represent an index position (counting from 0). A symmetry
2493 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2494 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2497 Here are some examples of symmetry definitions:
2502 e = indexed(A, i, j);
2503 e = indexed(A, sy_none(), i, j); // equivalent
2504 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2506 // Symmetric in all three indices:
2507 e = indexed(A, sy_symm(), i, j, k);
2508 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2509 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2510 // different canonical order
2512 // Symmetric in the first two indices only:
2513 e = indexed(A, sy_symm(0, 1), i, j, k);
2514 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2516 // Antisymmetric in the first and last index only (index ranges need not
2518 e = indexed(A, sy_anti(0, 2), i, j, k);
2519 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2521 // An example of a mixed symmetry: antisymmetric in the first two and
2522 // last two indices, symmetric when swapping the first and last index
2523 // pairs (like the Riemann curvature tensor):
2524 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2526 // Cyclic symmetry in all three indices:
2527 e = indexed(A, sy_cycl(), i, j, k);
2528 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2530 // The following examples are invalid constructions that will throw
2531 // an exception at run time.
2533 // An index may not appear multiple times:
2534 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2535 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2537 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2538 // same number of indices:
2539 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2541 // And of course, you cannot specify indices which are not there:
2542 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2546 If you need to specify more than four indices, you have to use the
2547 @code{.add()} method of the @code{symmetry} class. For example, to specify
2548 full symmetry in the first six indices you would write
2549 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2551 If an indexed object has a symmetry, GiNaC will automatically bring the
2552 indices into a canonical order which allows for some immediate simplifications:
2556 cout << indexed(A, sy_symm(), i, j)
2557 + indexed(A, sy_symm(), j, i) << endl;
2559 cout << indexed(B, sy_anti(), i, j)
2560 + indexed(B, sy_anti(), j, i) << endl;
2562 cout << indexed(B, sy_anti(), i, j, k)
2563 - indexed(B, sy_anti(), j, k, i) << endl;
2568 @cindex @code{get_free_indices()}
2570 @subsection Dummy indices
2572 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2573 that a summation over the index range is implied. Symbolic indices which are
2574 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2575 dummy nor free indices.
2577 To be recognized as a dummy index pair, the two indices must be of the same
2578 class and their value must be the same single symbol (an index like
2579 @samp{2*n+1} is never a dummy index). If the indices are of class
2580 @code{varidx} they must also be of opposite variance; if they are of class
2581 @code{spinidx} they must be both dotted or both undotted.
2583 The method @code{.get_free_indices()} returns a vector containing the free
2584 indices of an expression. It also checks that the free indices of the terms
2585 of a sum are consistent:
2589 symbol A("A"), B("B"), C("C");
2591 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2592 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2594 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2595 cout << exprseq(e.get_free_indices()) << endl;
2597 // 'j' and 'l' are dummy indices
2599 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2600 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2602 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2603 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2604 cout << exprseq(e.get_free_indices()) << endl;
2606 // 'nu' is a dummy index, but 'sigma' is not
2608 e = indexed(A, mu, mu);
2609 cout << exprseq(e.get_free_indices()) << endl;
2611 // 'mu' is not a dummy index because it appears twice with the same
2614 e = indexed(A, mu, nu) + 42;
2615 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2616 // this will throw an exception:
2617 // "add::get_free_indices: inconsistent indices in sum"
2621 @cindex @code{expand_dummy_sum()}
2622 A dummy index summation like
2629 can be expanded for indices with numeric
2630 dimensions (e.g. 3) into the explicit sum like
2632 $a_1b^1+a_2b^2+a_3b^3 $.
2635 a.1 b~1 + a.2 b~2 + a.3 b~3.
2637 This is performed by the function
2640 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2643 which takes an expression @code{e} and returns the expanded sum for all
2644 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2645 is set to @code{true} then all substitutions are made by @code{idx} class
2646 indices, i.e. without variance. In this case the above sum
2655 $a_1b_1+a_2b_2+a_3b_3 $.
2658 a.1 b.1 + a.2 b.2 + a.3 b.3.
2662 @cindex @code{simplify_indexed()}
2663 @subsection Simplifying indexed expressions
2665 In addition to the few automatic simplifications that GiNaC performs on
2666 indexed expressions (such as re-ordering the indices of symmetric tensors
2667 and calculating traces and convolutions of matrices and predefined tensors)
2671 ex ex::simplify_indexed();
2672 ex ex::simplify_indexed(const scalar_products & sp);
2675 that performs some more expensive operations:
2678 @item it checks the consistency of free indices in sums in the same way
2679 @code{get_free_indices()} does
2680 @item it tries to give dummy indices that appear in different terms of a sum
2681 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2682 @item it (symbolically) calculates all possible dummy index summations/contractions
2683 with the predefined tensors (this will be explained in more detail in the
2685 @item it detects contractions that vanish for symmetry reasons, for example
2686 the contraction of a symmetric and a totally antisymmetric tensor
2687 @item as a special case of dummy index summation, it can replace scalar products
2688 of two tensors with a user-defined value
2691 The last point is done with the help of the @code{scalar_products} class
2692 which is used to store scalar products with known values (this is not an
2693 arithmetic class, you just pass it to @code{simplify_indexed()}):
2697 symbol A("A"), B("B"), C("C"), i_sym("i");
2701 sp.add(A, B, 0); // A and B are orthogonal
2702 sp.add(A, C, 0); // A and C are orthogonal
2703 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2705 e = indexed(A + B, i) * indexed(A + C, i);
2707 // -> (B+A).i*(A+C).i
2709 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2715 The @code{scalar_products} object @code{sp} acts as a storage for the
2716 scalar products added to it with the @code{.add()} method. This method
2717 takes three arguments: the two expressions of which the scalar product is
2718 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2719 @code{simplify_indexed()} will replace all scalar products of indexed
2720 objects that have the symbols @code{A} and @code{B} as base expressions
2721 with the single value 0. The number, type and dimension of the indices
2722 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2724 @cindex @code{expand()}
2725 The example above also illustrates a feature of the @code{expand()} method:
2726 if passed the @code{expand_indexed} option it will distribute indices
2727 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2729 @cindex @code{tensor} (class)
2730 @subsection Predefined tensors
2732 Some frequently used special tensors such as the delta, epsilon and metric
2733 tensors are predefined in GiNaC. They have special properties when
2734 contracted with other tensor expressions and some of them have constant
2735 matrix representations (they will evaluate to a number when numeric
2736 indices are specified).
2738 @cindex @code{delta_tensor()}
2739 @subsubsection Delta tensor
2741 The delta tensor takes two indices, is symmetric and has the matrix
2742 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2743 @code{delta_tensor()}:
2747 symbol A("A"), B("B");
2749 idx i(symbol("i"), 3), j(symbol("j"), 3),
2750 k(symbol("k"), 3), l(symbol("l"), 3);
2752 ex e = indexed(A, i, j) * indexed(B, k, l)
2753 * delta_tensor(i, k) * delta_tensor(j, l);
2754 cout << e.simplify_indexed() << endl;
2757 cout << delta_tensor(i, i) << endl;
2762 @cindex @code{metric_tensor()}
2763 @subsubsection General metric tensor
2765 The function @code{metric_tensor()} creates a general symmetric metric
2766 tensor with two indices that can be used to raise/lower tensor indices. The
2767 metric tensor is denoted as @samp{g} in the output and if its indices are of
2768 mixed variance it is automatically replaced by a delta tensor:
2774 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2776 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2777 cout << e.simplify_indexed() << endl;
2780 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2781 cout << e.simplify_indexed() << endl;
2784 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2785 * metric_tensor(nu, rho);
2786 cout << e.simplify_indexed() << endl;
2789 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2790 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2791 + indexed(A, mu.toggle_variance(), rho));
2792 cout << e.simplify_indexed() << endl;
2797 @cindex @code{lorentz_g()}
2798 @subsubsection Minkowski metric tensor
2800 The Minkowski metric tensor is a special metric tensor with a constant
2801 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2802 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2803 It is created with the function @code{lorentz_g()} (although it is output as
2808 varidx mu(symbol("mu"), 4);
2810 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2811 * lorentz_g(mu, varidx(0, 4)); // negative signature
2812 cout << e.simplify_indexed() << endl;
2815 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2816 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2817 cout << e.simplify_indexed() << endl;
2822 @cindex @code{spinor_metric()}
2823 @subsubsection Spinor metric tensor
2825 The function @code{spinor_metric()} creates an antisymmetric tensor with
2826 two indices that is used to raise/lower indices of 2-component spinors.
2827 It is output as @samp{eps}:
2833 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2834 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2836 e = spinor_metric(A, B) * indexed(psi, B_co);
2837 cout << e.simplify_indexed() << endl;
2840 e = spinor_metric(A, B) * indexed(psi, A_co);
2841 cout << e.simplify_indexed() << endl;
2844 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2845 cout << e.simplify_indexed() << endl;
2848 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2849 cout << e.simplify_indexed() << endl;
2852 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2853 cout << e.simplify_indexed() << endl;
2856 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2857 cout << e.simplify_indexed() << endl;
2862 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2864 @cindex @code{epsilon_tensor()}
2865 @cindex @code{lorentz_eps()}
2866 @subsubsection Epsilon tensor
2868 The epsilon tensor is totally antisymmetric, its number of indices is equal
2869 to the dimension of the index space (the indices must all be of the same
2870 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2871 defined to be 1. Its behavior with indices that have a variance also
2872 depends on the signature of the metric. Epsilon tensors are output as
2875 There are three functions defined to create epsilon tensors in 2, 3 and 4
2879 ex epsilon_tensor(const ex & i1, const ex & i2);
2880 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2881 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2882 bool pos_sig = false);
2885 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2886 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2887 Minkowski space (the last @code{bool} argument specifies whether the metric
2888 has negative or positive signature, as in the case of the Minkowski metric
2893 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2894 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2895 e = lorentz_eps(mu, nu, rho, sig) *
2896 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2897 cout << simplify_indexed(e) << endl;
2898 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2900 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2901 symbol A("A"), B("B");
2902 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2903 cout << simplify_indexed(e) << endl;
2904 // -> -B.k*A.j*eps.i.k.j
2905 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2906 cout << simplify_indexed(e) << endl;
2911 @subsection Linear algebra
2913 The @code{matrix} class can be used with indices to do some simple linear
2914 algebra (linear combinations and products of vectors and matrices, traces
2915 and scalar products):
2919 idx i(symbol("i"), 2), j(symbol("j"), 2);
2920 symbol x("x"), y("y");
2922 // A is a 2x2 matrix, X is a 2x1 vector
2923 matrix A(2, 2), X(2, 1);
2928 cout << indexed(A, i, i) << endl;
2931 ex e = indexed(A, i, j) * indexed(X, j);
2932 cout << e.simplify_indexed() << endl;
2933 // -> [[2*y+x],[4*y+3*x]].i
2935 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2936 cout << e.simplify_indexed() << endl;
2937 // -> [[3*y+3*x,6*y+2*x]].j
2941 You can of course obtain the same results with the @code{matrix::add()},
2942 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2943 but with indices you don't have to worry about transposing matrices.
2945 Matrix indices always start at 0 and their dimension must match the number
2946 of rows/columns of the matrix. Matrices with one row or one column are
2947 vectors and can have one or two indices (it doesn't matter whether it's a
2948 row or a column vector). Other matrices must have two indices.
2950 You should be careful when using indices with variance on matrices. GiNaC
2951 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2952 @samp{F.mu.nu} are different matrices. In this case you should use only
2953 one form for @samp{F} and explicitly multiply it with a matrix representation
2954 of the metric tensor.
2957 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2958 @c node-name, next, previous, up
2959 @section Non-commutative objects
2961 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2962 non-commutative objects are built-in which are mostly of use in high energy
2966 @item Clifford (Dirac) algebra (class @code{clifford})
2967 @item su(3) Lie algebra (class @code{color})
2968 @item Matrices (unindexed) (class @code{matrix})
2971 The @code{clifford} and @code{color} classes are subclasses of
2972 @code{indexed} because the elements of these algebras usually carry
2973 indices. The @code{matrix} class is described in more detail in
2976 Unlike most computer algebra systems, GiNaC does not primarily provide an
2977 operator (often denoted @samp{&*}) for representing inert products of
2978 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2979 classes of objects involved, and non-commutative products are formed with
2980 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2981 figuring out by itself which objects commutate and will group the factors
2982 by their class. Consider this example:
2986 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2987 idx a(symbol("a"), 8), b(symbol("b"), 8);
2988 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2990 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2994 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2995 groups the non-commutative factors (the gammas and the su(3) generators)
2996 together while preserving the order of factors within each class (because
2997 Clifford objects commutate with color objects). The resulting expression is a
2998 @emph{commutative} product with two factors that are themselves non-commutative
2999 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3000 parentheses are placed around the non-commutative products in the output.
3002 @cindex @code{ncmul} (class)
3003 Non-commutative products are internally represented by objects of the class
3004 @code{ncmul}, as opposed to commutative products which are handled by the
3005 @code{mul} class. You will normally not have to worry about this distinction,
3008 The advantage of this approach is that you never have to worry about using
3009 (or forgetting to use) a special operator when constructing non-commutative
3010 expressions. Also, non-commutative products in GiNaC are more intelligent
3011 than in other computer algebra systems; they can, for example, automatically
3012 canonicalize themselves according to rules specified in the implementation
3013 of the non-commutative classes. The drawback is that to work with other than
3014 the built-in algebras you have to implement new classes yourself. Symbols
3015 always commutate and it's not possible to construct non-commutative products
3016 using symbols to represent the algebra elements or generators. User-defined
3017 functions can, however, be specified as being non-commutative.
3019 @cindex @code{return_type()}
3020 @cindex @code{return_type_tinfo()}
3021 Information about the commutativity of an object or expression can be
3022 obtained with the two member functions
3025 unsigned ex::return_type() const;
3026 unsigned ex::return_type_tinfo() const;
3029 The @code{return_type()} function returns one of three values (defined in
3030 the header file @file{flags.h}), corresponding to three categories of
3031 expressions in GiNaC:
3034 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3035 classes are of this kind.
3036 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3037 certain class of non-commutative objects which can be determined with the
3038 @code{return_type_tinfo()} method. Expressions of this category commutate
3039 with everything except @code{noncommutative} expressions of the same
3041 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3042 of non-commutative objects of different classes. Expressions of this
3043 category don't commutate with any other @code{noncommutative} or
3044 @code{noncommutative_composite} expressions.
3047 The value returned by the @code{return_type_tinfo()} method is valid only
3048 when the return type of the expression is @code{noncommutative}. It is a
3049 value that is unique to the class of the object and usually one of the
3050 constants in @file{tinfos.h}, or derived therefrom.
3052 Here are a couple of examples:
3055 @multitable @columnfractions 0.33 0.33 0.34
3056 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3057 @item @code{42} @tab @code{commutative} @tab -
3058 @item @code{2*x-y} @tab @code{commutative} @tab -
3059 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3060 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3061 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3062 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3066 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3067 @code{TINFO_clifford} for objects with a representation label of zero.
3068 Other representation labels yield a different @code{return_type_tinfo()},
3069 but it's the same for any two objects with the same label. This is also true
3072 A last note: With the exception of matrices, positive integer powers of
3073 non-commutative objects are automatically expanded in GiNaC. For example,
3074 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3075 non-commutative expressions).
3078 @cindex @code{clifford} (class)
3079 @subsection Clifford algebra
3082 Clifford algebras are supported in two flavours: Dirac gamma
3083 matrices (more physical) and generic Clifford algebras (more
3086 @cindex @code{dirac_gamma()}
3087 @subsubsection Dirac gamma matrices
3088 Dirac gamma matrices (note that GiNaC doesn't treat them
3089 as matrices) are designated as @samp{gamma~mu} and satisfy
3090 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3091 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3092 constructed by the function
3095 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3098 which takes two arguments: the index and a @dfn{representation label} in the
3099 range 0 to 255 which is used to distinguish elements of different Clifford
3100 algebras (this is also called a @dfn{spin line index}). Gammas with different
3101 labels commutate with each other. The dimension of the index can be 4 or (in
3102 the framework of dimensional regularization) any symbolic value. Spinor
3103 indices on Dirac gammas are not supported in GiNaC.
3105 @cindex @code{dirac_ONE()}
3106 The unity element of a Clifford algebra is constructed by
3109 ex dirac_ONE(unsigned char rl = 0);
3112 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3113 multiples of the unity element, even though it's customary to omit it.
3114 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3115 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3116 GiNaC will complain and/or produce incorrect results.
3118 @cindex @code{dirac_gamma5()}
3119 There is a special element @samp{gamma5} that commutates with all other
3120 gammas, has a unit square, and in 4 dimensions equals
3121 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3124 ex dirac_gamma5(unsigned char rl = 0);
3127 @cindex @code{dirac_gammaL()}
3128 @cindex @code{dirac_gammaR()}
3129 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3130 objects, constructed by
3133 ex dirac_gammaL(unsigned char rl = 0);
3134 ex dirac_gammaR(unsigned char rl = 0);
3137 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3138 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3140 @cindex @code{dirac_slash()}
3141 Finally, the function
3144 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3147 creates a term that represents a contraction of @samp{e} with the Dirac
3148 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3149 with a unique index whose dimension is given by the @code{dim} argument).
3150 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3152 In products of dirac gammas, superfluous unity elements are automatically
3153 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3154 and @samp{gammaR} are moved to the front.
3156 The @code{simplify_indexed()} function performs contractions in gamma strings,
3162 symbol a("a"), b("b"), D("D");
3163 varidx mu(symbol("mu"), D);
3164 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3165 * dirac_gamma(mu.toggle_variance());
3167 // -> gamma~mu*a\*gamma.mu
3168 e = e.simplify_indexed();
3171 cout << e.subs(D == 4) << endl;
3177 @cindex @code{dirac_trace()}
3178 To calculate the trace of an expression containing strings of Dirac gammas
3179 you use one of the functions
3182 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3183 const ex & trONE = 4);
3184 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3185 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3188 These functions take the trace over all gammas in the specified set @code{rls}
3189 or list @code{rll} of representation labels, or the single label @code{rl};
3190 gammas with other labels are left standing. The last argument to
3191 @code{dirac_trace()} is the value to be returned for the trace of the unity
3192 element, which defaults to 4.
3194 The @code{dirac_trace()} function is a linear functional that is equal to the
3195 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3196 functional is not cyclic in
3199 dimensions when acting on
3200 expressions containing @samp{gamma5}, so it's not a proper trace. This
3201 @samp{gamma5} scheme is described in greater detail in
3202 @cite{The Role of gamma5 in Dimensional Regularization}.
3204 The value of the trace itself is also usually different in 4 and in
3212 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3213 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3214 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3215 cout << dirac_trace(e).simplify_indexed() << endl;
3222 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3223 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3224 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3225 cout << dirac_trace(e).simplify_indexed() << endl;
3226 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3230 Here is an example for using @code{dirac_trace()} to compute a value that
3231 appears in the calculation of the one-loop vacuum polarization amplitude in
3236 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3237 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3240 sp.add(l, l, pow(l, 2));
3241 sp.add(l, q, ldotq);
3243 ex e = dirac_gamma(mu) *
3244 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3245 dirac_gamma(mu.toggle_variance()) *
3246 (dirac_slash(l, D) + m * dirac_ONE());
3247 e = dirac_trace(e).simplify_indexed(sp);
3248 e = e.collect(lst(l, ldotq, m));
3250 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3254 The @code{canonicalize_clifford()} function reorders all gamma products that
3255 appear in an expression to a canonical (but not necessarily simple) form.
3256 You can use this to compare two expressions or for further simplifications:
3260 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3261 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3263 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3265 e = canonicalize_clifford(e);
3267 // -> 2*ONE*eta~mu~nu
3271 @cindex @code{clifford_unit()}
3272 @subsubsection A generic Clifford algebra
3274 A generic Clifford algebra, i.e. a
3278 dimensional algebra with
3282 satisfying the identities
3284 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3287 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3289 for some bilinear form (@code{metric})
3290 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3291 and contain symbolic entries. Such generators are created by the
3295 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3296 bool anticommuting = false);
3299 where @code{mu} should be a @code{varidx} class object indexing the
3300 generators, an index @code{mu} with a numeric value may be of type
3302 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3303 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3304 object. Optional parameter @code{rl} allows to distinguish different
3305 Clifford algebras, which will commute with each other. The last
3306 optional parameter @code{anticommuting} defines if the anticommuting
3309 $e_i e_j + e_j e_i = 0$)
3312 e~i e~j + e~j e~i = 0)
3314 will be used for contraction of Clifford units. If the @code{metric} is
3315 supplied by a @code{matrix} object, then the value of
3316 @code{anticommuting} is calculated automatically and the supplied one
3317 will be ignored. One can overcome this by giving @code{metric} through
3318 matrix wrapped into an @code{indexed} object.
3320 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3321 something very close to @code{dirac_gamma(mu)}, although
3322 @code{dirac_gamma} have more efficient simplification mechanism.
3323 @cindex @code{clifford::get_metric()}
3324 The method @code{clifford::get_metric()} returns a metric defining this
3326 @cindex @code{clifford::is_anticommuting()}
3327 The method @code{clifford::is_anticommuting()} returns the
3328 @code{anticommuting} property of a unit.
3330 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3331 the Clifford algebra units with a call like that
3334 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3337 since this may yield some further automatic simplifications. Again, for a
3338 metric defined through a @code{matrix} such a symmetry is detected
3341 Individual generators of a Clifford algebra can be accessed in several
3347 varidx nu(symbol("nu"), 4);
3349 ex M = diag_matrix(lst(1, -1, 0, s));
3350 ex e = clifford_unit(nu, M);
3351 ex e0 = e.subs(nu == 0);
3352 ex e1 = e.subs(nu == 1);
3353 ex e2 = e.subs(nu == 2);
3354 ex e3 = e.subs(nu == 3);
3359 will produce four anti-commuting generators of a Clifford algebra with properties
3361 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3364 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3365 @code{pow(e3, 2) = s}.
3368 @cindex @code{lst_to_clifford()}
3369 A similar effect can be achieved from the function
3372 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3373 unsigned char rl = 0, bool anticommuting = false);
3374 ex lst_to_clifford(const ex & v, const ex & e);
3377 which converts a list or vector
3379 $v = (v^0, v^1, ..., v^n)$
3382 @samp{v = (v~0, v~1, ..., v~n)}
3387 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3390 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3393 directly supplied in the second form of the procedure. In the first form
3394 the Clifford unit @samp{e.k} is generated by the call of
3395 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3396 with the help of @code{lst_to_clifford()} as follows
3401 varidx nu(symbol("nu"), 4);
3403 ex M = diag_matrix(lst(1, -1, 0, s));
3404 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3405 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3406 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3407 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3412 @cindex @code{clifford_to_lst()}
3413 There is the inverse function
3416 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3419 which takes an expression @code{e} and tries to find a list
3421 $v = (v^0, v^1, ..., v^n)$
3424 @samp{v = (v~0, v~1, ..., v~n)}
3428 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3431 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3433 with respect to the given Clifford units @code{c} and with none of the
3434 @samp{v~k} containing Clifford units @code{c} (of course, this
3435 may be impossible). This function can use an @code{algebraic} method
3436 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3438 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3441 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3443 is zero or is not @code{numeric} for some @samp{k}
3444 then the method will be automatically changed to symbolic. The same effect
3445 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3447 @cindex @code{clifford_prime()}
3448 @cindex @code{clifford_star()}
3449 @cindex @code{clifford_bar()}
3450 There are several functions for (anti-)automorphisms of Clifford algebras:
3453 ex clifford_prime(const ex & e)
3454 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3455 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3458 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3459 changes signs of all Clifford units in the expression. The reversion
3460 of a Clifford algebra @code{clifford_star()} coincides with the
3461 @code{conjugate()} method and effectively reverses the order of Clifford
3462 units in any product. Finally the main anti-automorphism
3463 of a Clifford algebra @code{clifford_bar()} is the composition of the
3464 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3465 in a product. These functions correspond to the notations
3480 used in Clifford algebra textbooks.
3482 @cindex @code{clifford_norm()}
3486 ex clifford_norm(const ex & e);
3489 @cindex @code{clifford_inverse()}
3490 calculates the norm of a Clifford number from the expression
3492 $||e||^2 = e\overline{e}$.
3495 @code{||e||^2 = e \bar@{e@}}
3497 The inverse of a Clifford expression is returned by the function
3500 ex clifford_inverse(const ex & e);
3503 which calculates it as
3505 $e^{-1} = \overline{e}/||e||^2$.
3508 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3517 then an exception is raised.
3519 @cindex @code{remove_dirac_ONE()}
3520 If a Clifford number happens to be a factor of
3521 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3522 expression by the function
3525 ex remove_dirac_ONE(const ex & e);
3528 @cindex @code{canonicalize_clifford()}
3529 The function @code{canonicalize_clifford()} works for a
3530 generic Clifford algebra in a similar way as for Dirac gammas.
3532 The next provided function is
3534 @cindex @code{clifford_moebius_map()}
3536 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3537 const ex & d, const ex & v, const ex & G,
3538 unsigned char rl = 0, bool anticommuting = false);
3539 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3540 unsigned char rl = 0, bool anticommuting = false);
3543 It takes a list or vector @code{v} and makes the Moebius (conformal or
3544 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3545 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3546 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3547 indexed object, tensormetric, matrix or a Clifford unit, in the later
3548 case the optional parameters @code{rl} and @code{anticommuting} are ignored
3549 even if supplied. The returned value of this function is a list of
3550 components of the resulting vector.
3552 @cindex @code{clifford_max_label()}
3553 Finally the function
3556 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3559 can detect a presence of Clifford objects in the expression @code{e}: if
3560 such objects are found it returns the maximal
3561 @code{representation_label} of them, otherwise @code{-1}. The optional
3562 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3563 be ignored during the search.
3565 LaTeX output for Clifford units looks like
3566 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3567 @code{representation_label} and @code{\nu} is the index of the
3568 corresponding unit. This provides a flexible typesetting with a suitable
3569 defintion of the @code{\clifford} command. For example, the definition
3571 \newcommand@{\clifford@}[1][]@{@}
3573 typesets all Clifford units identically, while the alternative definition
3575 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3577 prints units with @code{representation_label=0} as
3584 with @code{representation_label=1} as
3591 and with @code{representation_label=2} as
3599 @cindex @code{color} (class)
3600 @subsection Color algebra
3602 @cindex @code{color_T()}
3603 For computations in quantum chromodynamics, GiNaC implements the base elements
3604 and structure constants of the su(3) Lie algebra (color algebra). The base
3605 elements @math{T_a} are constructed by the function
3608 ex color_T(const ex & a, unsigned char rl = 0);
3611 which takes two arguments: the index and a @dfn{representation label} in the
3612 range 0 to 255 which is used to distinguish elements of different color
3613 algebras. Objects with different labels commutate with each other. The
3614 dimension of the index must be exactly 8 and it should be of class @code{idx},
3617 @cindex @code{color_ONE()}
3618 The unity element of a color algebra is constructed by
3621 ex color_ONE(unsigned char rl = 0);
3624 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3625 multiples of the unity element, even though it's customary to omit it.
3626 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3627 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3628 GiNaC may produce incorrect results.
3630 @cindex @code{color_d()}
3631 @cindex @code{color_f()}
3635 ex color_d(const ex & a, const ex & b, const ex & c);
3636 ex color_f(const ex & a, const ex & b, const ex & c);
3639 create the symmetric and antisymmetric structure constants @math{d_abc} and
3640 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3641 and @math{[T_a, T_b] = i f_abc T_c}.
3643 These functions evaluate to their numerical values,
3644 if you supply numeric indices to them. The index values should be in
3645 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3646 goes along better with the notations used in physical literature.
3648 @cindex @code{color_h()}
3649 There's an additional function
3652 ex color_h(const ex & a, const ex & b, const ex & c);
3655 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3657 The function @code{simplify_indexed()} performs some simplifications on
3658 expressions containing color objects:
3663 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3664 k(symbol("k"), 8), l(symbol("l"), 8);
3666 e = color_d(a, b, l) * color_f(a, b, k);
3667 cout << e.simplify_indexed() << endl;
3670 e = color_d(a, b, l) * color_d(a, b, k);
3671 cout << e.simplify_indexed() << endl;
3674 e = color_f(l, a, b) * color_f(a, b, k);
3675 cout << e.simplify_indexed() << endl;
3678 e = color_h(a, b, c) * color_h(a, b, c);
3679 cout << e.simplify_indexed() << endl;
3682 e = color_h(a, b, c) * color_T(b) * color_T(c);
3683 cout << e.simplify_indexed() << endl;
3686 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3687 cout << e.simplify_indexed() << endl;
3690 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3691 cout << e.simplify_indexed() << endl;
3692 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3696 @cindex @code{color_trace()}
3697 To calculate the trace of an expression containing color objects you use one
3701 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3702 ex color_trace(const ex & e, const lst & rll);
3703 ex color_trace(const ex & e, unsigned char rl = 0);
3706 These functions take the trace over all color @samp{T} objects in the
3707 specified set @code{rls} or list @code{rll} of representation labels, or the
3708 single label @code{rl}; @samp{T}s with other labels are left standing. For
3713 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3715 // -> -I*f.a.c.b+d.a.c.b
3720 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3721 @c node-name, next, previous, up
3724 @cindex @code{exhashmap} (class)
3726 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3727 that can be used as a drop-in replacement for the STL
3728 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3729 typically constant-time, element look-up than @code{map<>}.
3731 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3732 following differences:
3736 no @code{lower_bound()} and @code{upper_bound()} methods
3738 no reverse iterators, no @code{rbegin()}/@code{rend()}
3740 no @code{operator<(exhashmap, exhashmap)}
3742 the comparison function object @code{key_compare} is hardcoded to
3745 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3746 initial hash table size (the actual table size after construction may be
3747 larger than the specified value)
3749 the method @code{size_t bucket_count()} returns the current size of the hash
3752 @code{insert()} and @code{erase()} operations invalidate all iterators
3756 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3757 @c node-name, next, previous, up
3758 @chapter Methods and Functions
3761 In this chapter the most important algorithms provided by GiNaC will be
3762 described. Some of them are implemented as functions on expressions,
3763 others are implemented as methods provided by expression objects. If
3764 they are methods, there exists a wrapper function around it, so you can
3765 alternatively call it in a functional way as shown in the simple
3770 cout << "As method: " << sin(1).evalf() << endl;
3771 cout << "As function: " << evalf(sin(1)) << endl;
3775 @cindex @code{subs()}
3776 The general rule is that wherever methods accept one or more parameters
3777 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3778 wrapper accepts is the same but preceded by the object to act on
3779 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3780 most natural one in an OO model but it may lead to confusion for MapleV
3781 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3782 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3783 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3784 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3785 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3786 here. Also, users of MuPAD will in most cases feel more comfortable
3787 with GiNaC's convention. All function wrappers are implemented
3788 as simple inline functions which just call the corresponding method and
3789 are only provided for users uncomfortable with OO who are dead set to
3790 avoid method invocations. Generally, nested function wrappers are much
3791 harder to read than a sequence of methods and should therefore be
3792 avoided if possible. On the other hand, not everything in GiNaC is a
3793 method on class @code{ex} and sometimes calling a function cannot be
3797 * Information About Expressions::
3798 * Numerical Evaluation::
3799 * Substituting Expressions::
3800 * Pattern Matching and Advanced Substitutions::
3801 * Applying a Function on Subexpressions::
3802 * Visitors and Tree Traversal::
3803 * Polynomial Arithmetic:: Working with polynomials.
3804 * Rational Expressions:: Working with rational functions.
3805 * Symbolic Differentiation::
3806 * Series Expansion:: Taylor and Laurent expansion.
3808 * Built-in Functions:: List of predefined mathematical functions.
3809 * Multiple polylogarithms::
3810 * Complex Conjugation::
3811 * Built-in Functions:: List of predefined mathematical functions.
3812 * Solving Linear Systems of Equations::
3813 * Input/Output:: Input and output of expressions.
3817 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3818 @c node-name, next, previous, up
3819 @section Getting information about expressions
3821 @subsection Checking expression types
3822 @cindex @code{is_a<@dots{}>()}
3823 @cindex @code{is_exactly_a<@dots{}>()}
3824 @cindex @code{ex_to<@dots{}>()}
3825 @cindex Converting @code{ex} to other classes
3826 @cindex @code{info()}
3827 @cindex @code{return_type()}
3828 @cindex @code{return_type_tinfo()}
3830 Sometimes it's useful to check whether a given expression is a plain number,
3831 a sum, a polynomial with integer coefficients, or of some other specific type.
3832 GiNaC provides a couple of functions for this:
3835 bool is_a<T>(const ex & e);
3836 bool is_exactly_a<T>(const ex & e);
3837 bool ex::info(unsigned flag);
3838 unsigned ex::return_type() const;
3839 unsigned ex::return_type_tinfo() const;
3842 When the test made by @code{is_a<T>()} returns true, it is safe to call
3843 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3844 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3845 example, assuming @code{e} is an @code{ex}:
3850 if (is_a<numeric>(e))
3851 numeric n = ex_to<numeric>(e);
3856 @code{is_a<T>(e)} allows you to check whether the top-level object of
3857 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3858 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3859 e.g., for checking whether an expression is a number, a sum, or a product:
3866 is_a<numeric>(e1); // true
3867 is_a<numeric>(e2); // false
3868 is_a<add>(e1); // false
3869 is_a<add>(e2); // true
3870 is_a<mul>(e1); // false
3871 is_a<mul>(e2); // false
3875 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3876 top-level object of an expression @samp{e} is an instance of the GiNaC
3877 class @samp{T}, not including parent classes.
3879 The @code{info()} method is used for checking certain attributes of
3880 expressions. The possible values for the @code{flag} argument are defined
3881 in @file{ginac/flags.h}, the most important being explained in the following
3885 @multitable @columnfractions .30 .70
3886 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3887 @item @code{numeric}
3888 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3890 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3891 @item @code{rational}
3892 @tab @dots{}an exact rational number (integers are rational, too)
3893 @item @code{integer}
3894 @tab @dots{}a (non-complex) integer
3895 @item @code{crational}
3896 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3897 @item @code{cinteger}
3898 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3899 @item @code{positive}
3900 @tab @dots{}not complex and greater than 0
3901 @item @code{negative}
3902 @tab @dots{}not complex and less than 0
3903 @item @code{nonnegative}
3904 @tab @dots{}not complex and greater than or equal to 0
3906 @tab @dots{}an integer greater than 0
3908 @tab @dots{}an integer less than 0
3909 @item @code{nonnegint}
3910 @tab @dots{}an integer greater than or equal to 0
3912 @tab @dots{}an even integer
3914 @tab @dots{}an odd integer
3916 @tab @dots{}a prime integer (probabilistic primality test)
3917 @item @code{relation}
3918 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3919 @item @code{relation_equal}
3920 @tab @dots{}a @code{==} relation
3921 @item @code{relation_not_equal}
3922 @tab @dots{}a @code{!=} relation
3923 @item @code{relation_less}
3924 @tab @dots{}a @code{<} relation
3925 @item @code{relation_less_or_equal}
3926 @tab @dots{}a @code{<=} relation
3927 @item @code{relation_greater}
3928 @tab @dots{}a @code{>} relation
3929 @item @code{relation_greater_or_equal}
3930 @tab @dots{}a @code{>=} relation
3932 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3934 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3935 @item @code{polynomial}
3936 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3937 @item @code{integer_polynomial}
3938 @tab @dots{}a polynomial with (non-complex) integer coefficients
3939 @item @code{cinteger_polynomial}
3940 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3941 @item @code{rational_polynomial}
3942 @tab @dots{}a polynomial with (non-complex) rational coefficients
3943 @item @code{crational_polynomial}
3944 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3945 @item @code{rational_function}
3946 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3947 @item @code{algebraic}
3948 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3952 To determine whether an expression is commutative or non-commutative and if
3953 so, with which other expressions it would commutate, you use the methods
3954 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3955 for an explanation of these.
3958 @subsection Accessing subexpressions
3961 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3962 @code{function}, act as containers for subexpressions. For example, the
3963 subexpressions of a sum (an @code{add} object) are the individual terms,
3964 and the subexpressions of a @code{function} are the function's arguments.
3966 @cindex @code{nops()}
3968 GiNaC provides several ways of accessing subexpressions. The first way is to
3973 ex ex::op(size_t i);
3976 @code{nops()} determines the number of subexpressions (operands) contained
3977 in the expression, while @code{op(i)} returns the @code{i}-th
3978 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3979 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3980 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3981 @math{i>0} are the indices.
3984 @cindex @code{const_iterator}
3985 The second way to access subexpressions is via the STL-style random-access
3986 iterator class @code{const_iterator} and the methods
3989 const_iterator ex::begin();
3990 const_iterator ex::end();
3993 @code{begin()} returns an iterator referring to the first subexpression;
3994 @code{end()} returns an iterator which is one-past the last subexpression.
3995 If the expression has no subexpressions, then @code{begin() == end()}. These
3996 iterators can also be used in conjunction with non-modifying STL algorithms.
3998 Here is an example that (non-recursively) prints the subexpressions of a
3999 given expression in three different ways:
4006 for (size_t i = 0; i != e.nops(); ++i)
4007 cout << e.op(i) << endl;
4010 for (const_iterator i = e.begin(); i != e.end(); ++i)
4013 // with iterators and STL copy()
4014 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4018 @cindex @code{const_preorder_iterator}
4019 @cindex @code{const_postorder_iterator}
4020 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4021 expression's immediate children. GiNaC provides two additional iterator
4022 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4023 that iterate over all objects in an expression tree, in preorder or postorder,
4024 respectively. They are STL-style forward iterators, and are created with the
4028 const_preorder_iterator ex::preorder_begin();
4029 const_preorder_iterator ex::preorder_end();
4030 const_postorder_iterator ex::postorder_begin();
4031 const_postorder_iterator ex::postorder_end();
4034 The following example illustrates the differences between
4035 @code{const_iterator}, @code{const_preorder_iterator}, and
4036 @code{const_postorder_iterator}:
4040 symbol A("A"), B("B"), C("C");
4041 ex e = lst(lst(A, B), C);
4043 std::copy(e.begin(), e.end(),
4044 std::ostream_iterator<ex>(cout, "\n"));
4048 std::copy(e.preorder_begin(), e.preorder_end(),
4049 std::ostream_iterator<ex>(cout, "\n"));
4056 std::copy(e.postorder_begin(), e.postorder_end(),
4057 std::ostream_iterator<ex>(cout, "\n"));
4066 @cindex @code{relational} (class)
4067 Finally, the left-hand side and right-hand side expressions of objects of
4068 class @code{relational} (and only of these) can also be accessed with the
4077 @subsection Comparing expressions
4078 @cindex @code{is_equal()}
4079 @cindex @code{is_zero()}
4081 Expressions can be compared with the usual C++ relational operators like
4082 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4083 the result is usually not determinable and the result will be @code{false},
4084 except in the case of the @code{!=} operator. You should also be aware that
4085 GiNaC will only do the most trivial test for equality (subtracting both
4086 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4089 Actually, if you construct an expression like @code{a == b}, this will be
4090 represented by an object of the @code{relational} class (@pxref{Relations})
4091 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4093 There are also two methods
4096 bool ex::is_equal(const ex & other);
4100 for checking whether one expression is equal to another, or equal to zero,
4104 @subsection Ordering expressions
4105 @cindex @code{ex_is_less} (class)
4106 @cindex @code{ex_is_equal} (class)
4107 @cindex @code{compare()}
4109 Sometimes it is necessary to establish a mathematically well-defined ordering
4110 on a set of arbitrary expressions, for example to use expressions as keys
4111 in a @code{std::map<>} container, or to bring a vector of expressions into
4112 a canonical order (which is done internally by GiNaC for sums and products).
4114 The operators @code{<}, @code{>} etc. described in the last section cannot
4115 be used for this, as they don't implement an ordering relation in the
4116 mathematical sense. In particular, they are not guaranteed to be
4117 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4118 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4121 By default, STL classes and algorithms use the @code{<} and @code{==}
4122 operators to compare objects, which are unsuitable for expressions, but GiNaC
4123 provides two functors that can be supplied as proper binary comparison
4124 predicates to the STL:
4127 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4129 bool operator()(const ex &lh, const ex &rh) const;
4132 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4134 bool operator()(const ex &lh, const ex &rh) const;
4138 For example, to define a @code{map} that maps expressions to strings you
4142 std::map<ex, std::string, ex_is_less> myMap;
4145 Omitting the @code{ex_is_less} template parameter will introduce spurious
4146 bugs because the map operates improperly.
4148 Other examples for the use of the functors:
4156 std::sort(v.begin(), v.end(), ex_is_less());
4158 // count the number of expressions equal to '1'
4159 unsigned num_ones = std::count_if(v.begin(), v.end(),
4160 std::bind2nd(ex_is_equal(), 1));
4163 The implementation of @code{ex_is_less} uses the member function
4166 int ex::compare(const ex & other) const;
4169 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4170 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4174 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4175 @c node-name, next, previous, up
4176 @section Numerical Evaluation
4177 @cindex @code{evalf()}
4179 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4180 To evaluate them using floating-point arithmetic you need to call
4183 ex ex::evalf(int level = 0) const;
4186 @cindex @code{Digits}
4187 The accuracy of the evaluation is controlled by the global object @code{Digits}
4188 which can be assigned an integer value. The default value of @code{Digits}
4189 is 17. @xref{Numbers}, for more information and examples.
4191 To evaluate an expression to a @code{double} floating-point number you can
4192 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4196 // Approximate sin(x/Pi)
4198 ex e = series(sin(x/Pi), x == 0, 6);
4200 // Evaluate numerically at x=0.1
4201 ex f = evalf(e.subs(x == 0.1));
4203 // ex_to<numeric> is an unsafe cast, so check the type first
4204 if (is_a<numeric>(f)) @{
4205 double d = ex_to<numeric>(f).to_double();
4214 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4215 @c node-name, next, previous, up
4216 @section Substituting expressions
4217 @cindex @code{subs()}
4219 Algebraic objects inside expressions can be replaced with arbitrary
4220 expressions via the @code{.subs()} method:
4223 ex ex::subs(const ex & e, unsigned options = 0);
4224 ex ex::subs(const exmap & m, unsigned options = 0);
4225 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4228 In the first form, @code{subs()} accepts a relational of the form
4229 @samp{object == expression} or a @code{lst} of such relationals:
4233 symbol x("x"), y("y");
4235 ex e1 = 2*x^2-4*x+3;
4236 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4240 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4245 If you specify multiple substitutions, they are performed in parallel, so e.g.
4246 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4248 The second form of @code{subs()} takes an @code{exmap} object which is a
4249 pair associative container that maps expressions to expressions (currently
4250 implemented as a @code{std::map}). This is the most efficient one of the
4251 three @code{subs()} forms and should be used when the number of objects to
4252 be substituted is large or unknown.
4254 Using this form, the second example from above would look like this:
4258 symbol x("x"), y("y");
4264 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4268 The third form of @code{subs()} takes two lists, one for the objects to be
4269 replaced and one for the expressions to be substituted (both lists must
4270 contain the same number of elements). Using this form, you would write
4274 symbol x("x"), y("y");
4277 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4281 The optional last argument to @code{subs()} is a combination of
4282 @code{subs_options} flags. There are two options available:
4283 @code{subs_options::no_pattern} disables pattern matching, which makes
4284 large @code{subs()} operations significantly faster if you are not using
4285 patterns. The second option, @code{subs_options::algebraic} enables
4286 algebraic substitutions in products and powers.
4287 @ref{Pattern Matching and Advanced Substitutions}, for more information
4288 about patterns and algebraic substitutions.
4290 @code{subs()} performs syntactic substitution of any complete algebraic
4291 object; it does not try to match sub-expressions as is demonstrated by the
4296 symbol x("x"), y("y"), z("z");
4298 ex e1 = pow(x+y, 2);
4299 cout << e1.subs(x+y == 4) << endl;
4302 ex e2 = sin(x)*sin(y)*cos(x);
4303 cout << e2.subs(sin(x) == cos(x)) << endl;
4304 // -> cos(x)^2*sin(y)
4307 cout << e3.subs(x+y == 4) << endl;
4309 // (and not 4+z as one might expect)
4313 A more powerful form of substitution using wildcards is described in the
4317 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4318 @c node-name, next, previous, up
4319 @section Pattern matching and advanced substitutions
4320 @cindex @code{wildcard} (class)
4321 @cindex Pattern matching
4323 GiNaC allows the use of patterns for checking whether an expression is of a
4324 certain form or contains subexpressions of a certain form, and for
4325 substituting expressions in a more general way.
4327 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4328 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4329 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4330 an unsigned integer number to allow having multiple different wildcards in a
4331 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4332 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4336 ex wild(unsigned label = 0);
4339 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4342 Some examples for patterns:
4344 @multitable @columnfractions .5 .5
4345 @item @strong{Constructed as} @tab @strong{Output as}
4346 @item @code{wild()} @tab @samp{$0}
4347 @item @code{pow(x,wild())} @tab @samp{x^$0}
4348 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4349 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4355 @item Wildcards behave like symbols and are subject to the same algebraic
4356 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4357 @item As shown in the last example, to use wildcards for indices you have to
4358 use them as the value of an @code{idx} object. This is because indices must
4359 always be of class @code{idx} (or a subclass).
4360 @item Wildcards only represent expressions or subexpressions. It is not
4361 possible to use them as placeholders for other properties like index
4362 dimension or variance, representation labels, symmetry of indexed objects
4364 @item Because wildcards are commutative, it is not possible to use wildcards
4365 as part of noncommutative products.
4366 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4367 are also valid patterns.
4370 @subsection Matching expressions
4371 @cindex @code{match()}
4372 The most basic application of patterns is to check whether an expression
4373 matches a given pattern. This is done by the function
4376 bool ex::match(const ex & pattern);
4377 bool ex::match(const ex & pattern, lst & repls);
4380 This function returns @code{true} when the expression matches the pattern
4381 and @code{false} if it doesn't. If used in the second form, the actual
4382 subexpressions matched by the wildcards get returned in the @code{repls}
4383 object as a list of relations of the form @samp{wildcard == expression}.
4384 If @code{match()} returns false, the state of @code{repls} is undefined.
4385 For reproducible results, the list should be empty when passed to
4386 @code{match()}, but it is also possible to find similarities in multiple
4387 expressions by passing in the result of a previous match.
4389 The matching algorithm works as follows:
4392 @item A single wildcard matches any expression. If one wildcard appears
4393 multiple times in a pattern, it must match the same expression in all
4394 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4395 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4396 @item If the expression is not of the same class as the pattern, the match
4397 fails (i.e. a sum only matches a sum, a function only matches a function,
4399 @item If the pattern is a function, it only matches the same function
4400 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4401 @item Except for sums and products, the match fails if the number of
4402 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4404 @item If there are no subexpressions, the expressions and the pattern must
4405 be equal (in the sense of @code{is_equal()}).
4406 @item Except for sums and products, each subexpression (@code{op()}) must
4407 match the corresponding subexpression of the pattern.
4410 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4411 account for their commutativity and associativity:
4414 @item If the pattern contains a term or factor that is a single wildcard,
4415 this one is used as the @dfn{global wildcard}. If there is more than one
4416 such wildcard, one of them is chosen as the global wildcard in a random
4418 @item Every term/factor of the pattern, except the global wildcard, is
4419 matched against every term of the expression in sequence. If no match is
4420 found, the whole match fails. Terms that did match are not considered in
4422 @item If there are no unmatched terms left, the match succeeds. Otherwise
4423 the match fails unless there is a global wildcard in the pattern, in
4424 which case this wildcard matches the remaining terms.
4427 In general, having more than one single wildcard as a term of a sum or a
4428 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4431 Here are some examples in @command{ginsh} to demonstrate how it works (the
4432 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4433 match fails, and the list of wildcard replacements otherwise):
4436 > match((x+y)^a,(x+y)^a);
4438 > match((x+y)^a,(x+y)^b);
4440 > match((x+y)^a,$1^$2);
4442 > match((x+y)^a,$1^$1);
4444 > match((x+y)^(x+y),$1^$1);
4446 > match((x+y)^(x+y),$1^$2);
4448 > match((a+b)*(a+c),($1+b)*($1+c));
4450 > match((a+b)*(a+c),(a+$1)*(a+$2));
4452 (Unpredictable. The result might also be [$1==c,$2==b].)
4453 > match((a+b)*(a+c),($1+$2)*($1+$3));
4454 (The result is undefined. Due to the sequential nature of the algorithm
4455 and the re-ordering of terms in GiNaC, the match for the first factor
4456 may be @{$1==a,$2==b@} in which case the match for the second factor
4457 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4459 > match(a*(x+y)+a*z+b,a*$1+$2);
4460 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4461 @{$1=x+y,$2=a*z+b@}.)
4462 > match(a+b+c+d+e+f,c);
4464 > match(a+b+c+d+e+f,c+$0);
4466 > match(a+b+c+d+e+f,c+e+$0);
4468 > match(a+b,a+b+$0);
4470 > match(a*b^2,a^$1*b^$2);
4472 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4473 even though a==a^1.)
4474 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4476 > match(atan2(y,x^2),atan2(y,$0));
4480 @subsection Matching parts of expressions
4481 @cindex @code{has()}
4482 A more general way to look for patterns in expressions is provided by the
4486 bool ex::has(const ex & pattern);
4489 This function checks whether a pattern is matched by an expression itself or
4490 by any of its subexpressions.
4492 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4493 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4496 > has(x*sin(x+y+2*a),y);
4498 > has(x*sin(x+y+2*a),x+y);
4500 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4501 has the subexpressions "x", "y" and "2*a".)
4502 > has(x*sin(x+y+2*a),x+y+$1);
4504 (But this is possible.)
4505 > has(x*sin(2*(x+y)+2*a),x+y);
4507 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4508 which "x+y" is not a subexpression.)
4511 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4513 > has(4*x^2-x+3,$1*x);
4515 > has(4*x^2+x+3,$1*x);
4517 (Another possible pitfall. The first expression matches because the term
4518 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4519 contains a linear term you should use the coeff() function instead.)
4522 @cindex @code{find()}
4526 bool ex::find(const ex & pattern, lst & found);
4529 works a bit like @code{has()} but it doesn't stop upon finding the first
4530 match. Instead, it appends all found matches to the specified list. If there
4531 are multiple occurrences of the same expression, it is entered only once to
4532 the list. @code{find()} returns false if no matches were found (in
4533 @command{ginsh}, it returns an empty list):
4536 > find(1+x+x^2+x^3,x);
4538 > find(1+x+x^2+x^3,y);
4540 > find(1+x+x^2+x^3,x^$1);
4542 (Note the absence of "x".)
4543 > expand((sin(x)+sin(y))*(a+b));
4544 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4549 @subsection Substituting expressions
4550 @cindex @code{subs()}
4551 Probably the most useful application of patterns is to use them for
4552 substituting expressions with the @code{subs()} method. Wildcards can be
4553 used in the search patterns as well as in the replacement expressions, where
4554 they get replaced by the expressions matched by them. @code{subs()} doesn't
4555 know anything about algebra; it performs purely syntactic substitutions.
4560 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4562 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4564 > subs((a+b+c)^2,a+b==x);
4566 > subs((a+b+c)^2,a+b+$1==x+$1);
4568 > subs(a+2*b,a+b==x);
4570 > subs(4*x^3-2*x^2+5*x-1,x==a);
4572 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4574 > subs(sin(1+sin(x)),sin($1)==cos($1));
4576 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4580 The last example would be written in C++ in this way:
4584 symbol a("a"), b("b"), x("x"), y("y");
4585 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4586 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4587 cout << e.expand() << endl;
4592 @subsection Algebraic substitutions
4593 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4594 enables smarter, algebraic substitutions in products and powers. If you want
4595 to substitute some factors of a product, you only need to list these factors
4596 in your pattern. Furthermore, if an (integer) power of some expression occurs
4597 in your pattern and in the expression that you want the substitution to occur
4598 in, it can be substituted as many times as possible, without getting negative
4601 An example clarifies it all (hopefully):
4604 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4605 subs_options::algebraic) << endl;
4606 // --> (y+x)^6+b^6+a^6
4608 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4610 // Powers and products are smart, but addition is just the same.
4612 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4615 // As I said: addition is just the same.
4617 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4618 // --> x^3*b*a^2+2*b
4620 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4622 // --> 2*b+x^3*b^(-1)*a^(-2)
4624 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4625 // --> -1-2*a^2+4*a^3+5*a
4627 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4628 subs_options::algebraic) << endl;
4629 // --> -1+5*x+4*x^3-2*x^2
4630 // You should not really need this kind of patterns very often now.
4631 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4633 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4634 subs_options::algebraic) << endl;
4635 // --> cos(1+cos(x))
4637 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4638 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4639 subs_options::algebraic)) << endl;
4644 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4645 @c node-name, next, previous, up
4646 @section Applying a Function on Subexpressions
4647 @cindex tree traversal
4648 @cindex @code{map()}
4650 Sometimes you may want to perform an operation on specific parts of an
4651 expression while leaving the general structure of it intact. An example
4652 of this would be a matrix trace operation: the trace of a sum is the sum
4653 of the traces of the individual terms. That is, the trace should @dfn{map}
4654 on the sum, by applying itself to each of the sum's operands. It is possible
4655 to do this manually which usually results in code like this:
4660 if (is_a<matrix>(e))
4661 return ex_to<matrix>(e).trace();
4662 else if (is_a<add>(e)) @{
4664 for (size_t i=0; i<e.nops(); i++)
4665 sum += calc_trace(e.op(i));
4667 @} else if (is_a<mul>)(e)) @{
4675 This is, however, slightly inefficient (if the sum is very large it can take
4676 a long time to add the terms one-by-one), and its applicability is limited to
4677 a rather small class of expressions. If @code{calc_trace()} is called with
4678 a relation or a list as its argument, you will probably want the trace to
4679 be taken on both sides of the relation or of all elements of the list.
4681 GiNaC offers the @code{map()} method to aid in the implementation of such
4685 ex ex::map(map_function & f) const;
4686 ex ex::map(ex (*f)(const ex & e)) const;
4689 In the first (preferred) form, @code{map()} takes a function object that
4690 is subclassed from the @code{map_function} class. In the second form, it
4691 takes a pointer to a function that accepts and returns an expression.
4692 @code{map()} constructs a new expression of the same type, applying the
4693 specified function on all subexpressions (in the sense of @code{op()}),
4696 The use of a function object makes it possible to supply more arguments to
4697 the function that is being mapped, or to keep local state information.
4698 The @code{map_function} class declares a virtual function call operator
4699 that you can overload. Here is a sample implementation of @code{calc_trace()}
4700 that uses @code{map()} in a recursive fashion:
4703 struct calc_trace : public map_function @{
4704 ex operator()(const ex &e)
4706 if (is_a<matrix>(e))
4707 return ex_to<matrix>(e).trace();
4708 else if (is_a<mul>(e)) @{
4711 return e.map(*this);
4716 This function object could then be used like this:
4720 ex M = ... // expression with matrices
4721 calc_trace do_trace;
4722 ex tr = do_trace(M);
4726 Here is another example for you to meditate over. It removes quadratic
4727 terms in a variable from an expanded polynomial:
4730 struct map_rem_quad : public map_function @{
4732 map_rem_quad(const ex & var_) : var(var_) @{@}
4734 ex operator()(const ex & e)
4736 if (is_a<add>(e) || is_a<mul>(e))
4737 return e.map(*this);
4738 else if (is_a<power>(e) &&
4739 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4749 symbol x("x"), y("y");
4752 for (int i=0; i<8; i++)
4753 e += pow(x, i) * pow(y, 8-i) * (i+1);
4755 // -> 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
4757 map_rem_quad rem_quad(x);
4758 cout << rem_quad(e) << endl;
4759 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4763 @command{ginsh} offers a slightly different implementation of @code{map()}
4764 that allows applying algebraic functions to operands. The second argument
4765 to @code{map()} is an expression containing the wildcard @samp{$0} which
4766 acts as the placeholder for the operands:
4771 > map(a+2*b,sin($0));
4773 > map(@{a,b,c@},$0^2+$0);
4774 @{a^2+a,b^2+b,c^2+c@}
4777 Note that it is only possible to use algebraic functions in the second
4778 argument. You can not use functions like @samp{diff()}, @samp{op()},
4779 @samp{subs()} etc. because these are evaluated immediately:
4782 > map(@{a,b,c@},diff($0,a));
4784 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4785 to "map(@{a,b,c@},0)".
4789 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4790 @c node-name, next, previous, up
4791 @section Visitors and Tree Traversal
4792 @cindex tree traversal
4793 @cindex @code{visitor} (class)
4794 @cindex @code{accept()}
4795 @cindex @code{visit()}
4796 @cindex @code{traverse()}
4797 @cindex @code{traverse_preorder()}
4798 @cindex @code{traverse_postorder()}
4800 Suppose that you need a function that returns a list of all indices appearing
4801 in an arbitrary expression. The indices can have any dimension, and for
4802 indices with variance you always want the covariant version returned.
4804 You can't use @code{get_free_indices()} because you also want to include
4805 dummy indices in the list, and you can't use @code{find()} as it needs
4806 specific index dimensions (and it would require two passes: one for indices
4807 with variance, one for plain ones).
4809 The obvious solution to this problem is a tree traversal with a type switch,
4810 such as the following:
4813 void gather_indices_helper(const ex & e, lst & l)
4815 if (is_a<varidx>(e)) @{
4816 const varidx & vi = ex_to<varidx>(e);
4817 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4818 @} else if (is_a<idx>(e)) @{
4821 size_t n = e.nops();
4822 for (size_t i = 0; i < n; ++i)
4823 gather_indices_helper(e.op(i), l);
4827 lst gather_indices(const ex & e)
4830 gather_indices_helper(e, l);
4837 This works fine but fans of object-oriented programming will feel
4838 uncomfortable with the type switch. One reason is that there is a possibility
4839 for subtle bugs regarding derived classes. If we had, for example, written
4842 if (is_a<idx>(e)) @{
4844 @} else if (is_a<varidx>(e)) @{
4848 in @code{gather_indices_helper}, the code wouldn't have worked because the
4849 first line "absorbs" all classes derived from @code{idx}, including
4850 @code{varidx}, so the special case for @code{varidx} would never have been
4853 Also, for a large number of classes, a type switch like the above can get
4854 unwieldy and inefficient (it's a linear search, after all).
4855 @code{gather_indices_helper} only checks for two classes, but if you had to
4856 write a function that required a different implementation for nearly
4857 every GiNaC class, the result would be very hard to maintain and extend.
4859 The cleanest approach to the problem would be to add a new virtual function
4860 to GiNaC's class hierarchy. In our example, there would be specializations
4861 for @code{idx} and @code{varidx} while the default implementation in
4862 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4863 impossible to add virtual member functions to existing classes without
4864 changing their source and recompiling everything. GiNaC comes with source,
4865 so you could actually do this, but for a small algorithm like the one
4866 presented this would be impractical.
4868 One solution to this dilemma is the @dfn{Visitor} design pattern,
4869 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4870 variation, described in detail in
4871 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4872 virtual functions to the class hierarchy to implement operations, GiNaC
4873 provides a single "bouncing" method @code{accept()} that takes an instance
4874 of a special @code{visitor} class and redirects execution to the one
4875 @code{visit()} virtual function of the visitor that matches the type of
4876 object that @code{accept()} was being invoked on.
4878 Visitors in GiNaC must derive from the global @code{visitor} class as well
4879 as from the class @code{T::visitor} of each class @code{T} they want to
4880 visit, and implement the member functions @code{void visit(const T &)} for
4886 void ex::accept(visitor & v) const;
4889 will then dispatch to the correct @code{visit()} member function of the
4890 specified visitor @code{v} for the type of GiNaC object at the root of the
4891 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4893 Here is an example of a visitor:
4897 : public visitor, // this is required
4898 public add::visitor, // visit add objects
4899 public numeric::visitor, // visit numeric objects
4900 public basic::visitor // visit basic objects
4902 void visit(const add & x)
4903 @{ cout << "called with an add object" << endl; @}
4905 void visit(const numeric & x)
4906 @{ cout << "called with a numeric object" << endl; @}
4908 void visit(const basic & x)
4909 @{ cout << "called with a basic object" << endl; @}
4913 which can be used as follows:
4924 // prints "called with a numeric object"
4926 // prints "called with an add object"
4928 // prints "called with a basic object"
4932 The @code{visit(const basic &)} method gets called for all objects that are
4933 not @code{numeric} or @code{add} and acts as an (optional) default.
4935 From a conceptual point of view, the @code{visit()} methods of the visitor
4936 behave like a newly added virtual function of the visited hierarchy.
4937 In addition, visitors can store state in member variables, and they can
4938 be extended by deriving a new visitor from an existing one, thus building
4939 hierarchies of visitors.
4941 We can now rewrite our index example from above with a visitor:
4944 class gather_indices_visitor
4945 : public visitor, public idx::visitor, public varidx::visitor
4949 void visit(const idx & i)
4954 void visit(const varidx & vi)
4956 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4960 const lst & get_result() // utility function
4969 What's missing is the tree traversal. We could implement it in
4970 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4973 void ex::traverse_preorder(visitor & v) const;
4974 void ex::traverse_postorder(visitor & v) const;
4975 void ex::traverse(visitor & v) const;
4978 @code{traverse_preorder()} visits a node @emph{before} visiting its
4979 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4980 visiting its subexpressions. @code{traverse()} is a synonym for
4981 @code{traverse_preorder()}.
4983 Here is a new implementation of @code{gather_indices()} that uses the visitor
4984 and @code{traverse()}:
4987 lst gather_indices(const ex & e)
4989 gather_indices_visitor v;
4991 return v.get_result();
4995 Alternatively, you could use pre- or postorder iterators for the tree
4999 lst gather_indices(const ex & e)
5001 gather_indices_visitor v;
5002 for (const_preorder_iterator i = e.preorder_begin();
5003 i != e.preorder_end(); ++i) @{
5006 return v.get_result();
5011 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
5012 @c node-name, next, previous, up
5013 @section Polynomial arithmetic
5015 @subsection Expanding and collecting
5016 @cindex @code{expand()}
5017 @cindex @code{collect()}
5018 @cindex @code{collect_common_factors()}
5020 A polynomial in one or more variables has many equivalent
5021 representations. Some useful ones serve a specific purpose. Consider
5022 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5023 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5024 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5025 representations are the recursive ones where one collects for exponents
5026 in one of the three variable. Since the factors are themselves
5027 polynomials in the remaining two variables the procedure can be
5028 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5029 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5032 To bring an expression into expanded form, its method
5035 ex ex::expand(unsigned options = 0);
5038 may be called. In our example above, this corresponds to @math{4*x*y +
5039 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5040 GiNaC is not easy to guess you should be prepared to see different
5041 orderings of terms in such sums!
5043 Another useful representation of multivariate polynomials is as a
5044 univariate polynomial in one of the variables with the coefficients
5045 being polynomials in the remaining variables. The method
5046 @code{collect()} accomplishes this task:
5049 ex ex::collect(const ex & s, bool distributed = false);
5052 The first argument to @code{collect()} can also be a list of objects in which
5053 case the result is either a recursively collected polynomial, or a polynomial
5054 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5055 by the @code{distributed} flag.
5057 Note that the original polynomial needs to be in expanded form (for the
5058 variables concerned) in order for @code{collect()} to be able to find the
5059 coefficients properly.
5061 The following @command{ginsh} transcript shows an application of @code{collect()}
5062 together with @code{find()}:
5065 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5066 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)
5067 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5068 > collect(a,@{p,q@});
5069 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5070 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5071 > collect(a,find(a,sin($1)));
5072 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5073 > collect(a,@{find(a,sin($1)),p,q@});
5074 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5075 > collect(a,@{find(a,sin($1)),d@});
5076 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5079 Polynomials can often be brought into a more compact form by collecting
5080 common factors from the terms of sums. This is accomplished by the function
5083 ex collect_common_factors(const ex & e);
5086 This function doesn't perform a full factorization but only looks for
5087 factors which are already explicitly present:
5090 > collect_common_factors(a*x+a*y);
5092 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5094 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5095 (c+a)*a*(x*y+y^2+x)*b
5098 @subsection Degree and coefficients
5099 @cindex @code{degree()}
5100 @cindex @code{ldegree()}
5101 @cindex @code{coeff()}
5103 The degree and low degree of a polynomial can be obtained using the two
5107 int ex::degree(const ex & s);
5108 int ex::ldegree(const ex & s);
5111 which also work reliably on non-expanded input polynomials (they even work
5112 on rational functions, returning the asymptotic degree). By definition, the
5113 degree of zero is zero. To extract a coefficient with a certain power from
5114 an expanded polynomial you use
5117 ex ex::coeff(const ex & s, int n);
5120 You can also obtain the leading and trailing coefficients with the methods
5123 ex ex::lcoeff(const ex & s);
5124 ex ex::tcoeff(const ex & s);
5127 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5130 An application is illustrated in the next example, where a multivariate
5131 polynomial is analyzed:
5135 symbol x("x"), y("y");
5136 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5137 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5138 ex Poly = PolyInp.expand();
5140 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5141 cout << "The x^" << i << "-coefficient is "
5142 << Poly.coeff(x,i) << endl;
5144 cout << "As polynomial in y: "
5145 << Poly.collect(y) << endl;
5149 When run, it returns an output in the following fashion:
5152 The x^0-coefficient is y^2+11*y
5153 The x^1-coefficient is 5*y^2-2*y
5154 The x^2-coefficient is -1
5155 The x^3-coefficient is 4*y
5156 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5159 As always, the exact output may vary between different versions of GiNaC
5160 or even from run to run since the internal canonical ordering is not
5161 within the user's sphere of influence.
5163 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5164 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5165 with non-polynomial expressions as they not only work with symbols but with
5166 constants, functions and indexed objects as well:
5170 symbol a("a"), b("b"), c("c"), x("x");
5171 idx i(symbol("i"), 3);
5173 ex e = pow(sin(x) - cos(x), 4);
5174 cout << e.degree(cos(x)) << endl;
5176 cout << e.expand().coeff(sin(x), 3) << endl;
5179 e = indexed(a+b, i) * indexed(b+c, i);
5180 e = e.expand(expand_options::expand_indexed);
5181 cout << e.collect(indexed(b, i)) << endl;
5182 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5187 @subsection Polynomial division
5188 @cindex polynomial division
5191 @cindex pseudo-remainder
5192 @cindex @code{quo()}
5193 @cindex @code{rem()}
5194 @cindex @code{prem()}
5195 @cindex @code{divide()}
5200 ex quo(const ex & a, const ex & b, const ex & x);
5201 ex rem(const ex & a, const ex & b, const ex & x);
5204 compute the quotient and remainder of univariate polynomials in the variable
5205 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5207 The additional function
5210 ex prem(const ex & a, const ex & b, const ex & x);
5213 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5214 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5216 Exact division of multivariate polynomials is performed by the function
5219 bool divide(const ex & a, const ex & b, ex & q);
5222 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5223 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5224 in which case the value of @code{q} is undefined.
5227 @subsection Unit, content and primitive part
5228 @cindex @code{unit()}
5229 @cindex @code{content()}
5230 @cindex @code{primpart()}
5231 @cindex @code{unitcontprim()}
5236 ex ex::unit(const ex & x);
5237 ex ex::content(const ex & x);
5238 ex ex::primpart(const ex & x);
5239 ex ex::primpart(const ex & x, const ex & c);
5242 return the unit part, content part, and primitive polynomial of a multivariate
5243 polynomial with respect to the variable @samp{x} (the unit part being the sign
5244 of the leading coefficient, the content part being the GCD of the coefficients,
5245 and the primitive polynomial being the input polynomial divided by the unit and
5246 content parts). The second variant of @code{primpart()} expects the previously
5247 calculated content part of the polynomial in @code{c}, which enables it to
5248 work faster in the case where the content part has already been computed. The
5249 product of unit, content, and primitive part is the original polynomial.
5251 Additionally, the method
5254 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5257 computes the unit, content, and primitive parts in one go, returning them
5258 in @code{u}, @code{c}, and @code{p}, respectively.
5261 @subsection GCD, LCM and resultant
5264 @cindex @code{gcd()}
5265 @cindex @code{lcm()}
5267 The functions for polynomial greatest common divisor and least common
5268 multiple have the synopsis
5271 ex gcd(const ex & a, const ex & b);
5272 ex lcm(const ex & a, const ex & b);
5275 The functions @code{gcd()} and @code{lcm()} accept two expressions
5276 @code{a} and @code{b} as arguments and return a new expression, their
5277 greatest common divisor or least common multiple, respectively. If the
5278 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5279 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5280 the coefficients must be rationals.
5283 #include <ginac/ginac.h>
5284 using namespace GiNaC;
5288 symbol x("x"), y("y"), z("z");
5289 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5290 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5292 ex P_gcd = gcd(P_a, P_b);
5294 ex P_lcm = lcm(P_a, P_b);
5295 // 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
5300 @cindex @code{resultant()}
5302 The resultant of two expressions only makes sense with polynomials.
5303 It is always computed with respect to a specific symbol within the
5304 expressions. The function has the interface
5307 ex resultant(const ex & a, const ex & b, const ex & s);
5310 Resultants are symmetric in @code{a} and @code{b}. The following example
5311 computes the resultant of two expressions with respect to @code{x} and
5312 @code{y}, respectively:
5315 #include <ginac/ginac.h>
5316 using namespace GiNaC;
5320 symbol x("x"), y("y");
5322 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5325 r = resultant(e1, e2, x);
5327 r = resultant(e1, e2, y);
5332 @subsection Square-free decomposition
5333 @cindex square-free decomposition
5334 @cindex factorization
5335 @cindex @code{sqrfree()}
5337 GiNaC still lacks proper factorization support. Some form of
5338 factorization is, however, easily implemented by noting that factors
5339 appearing in a polynomial with power two or more also appear in the
5340 derivative and hence can easily be found by computing the GCD of the
5341 original polynomial and its derivatives. Any decent system has an
5342 interface for this so called square-free factorization. So we provide
5345 ex sqrfree(const ex & a, const lst & l = lst());
5347 Here is an example that by the way illustrates how the exact form of the
5348 result may slightly depend on the order of differentiation, calling for
5349 some care with subsequent processing of the result:
5352 symbol x("x"), y("y");
5353 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5355 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5356 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5358 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5359 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5361 cout << sqrfree(BiVarPol) << endl;
5362 // -> depending on luck, any of the above
5365 Note also, how factors with the same exponents are not fully factorized
5369 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5370 @c node-name, next, previous, up
5371 @section Rational expressions
5373 @subsection The @code{normal} method
5374 @cindex @code{normal()}
5375 @cindex simplification
5376 @cindex temporary replacement
5378 Some basic form of simplification of expressions is called for frequently.
5379 GiNaC provides the method @code{.normal()}, which converts a rational function
5380 into an equivalent rational function of the form @samp{numerator/denominator}
5381 where numerator and denominator are coprime. If the input expression is already
5382 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5383 otherwise it performs fraction addition and multiplication.
5385 @code{.normal()} can also be used on expressions which are not rational functions
5386 as it will replace all non-rational objects (like functions or non-integer
5387 powers) by temporary symbols to bring the expression to the domain of rational
5388 functions before performing the normalization, and re-substituting these
5389 symbols afterwards. This algorithm is also available as a separate method
5390 @code{.to_rational()}, described below.
5392 This means that both expressions @code{t1} and @code{t2} are indeed
5393 simplified in this little code snippet:
5398 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5399 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5400 std::cout << "t1 is " << t1.normal() << std::endl;
5401 std::cout << "t2 is " << t2.normal() << std::endl;
5405 Of course this works for multivariate polynomials too, so the ratio of
5406 the sample-polynomials from the section about GCD and LCM above would be
5407 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5410 @subsection Numerator and denominator
5413 @cindex @code{numer()}
5414 @cindex @code{denom()}
5415 @cindex @code{numer_denom()}
5417 The numerator and denominator of an expression can be obtained with
5422 ex ex::numer_denom();
5425 These functions will first normalize the expression as described above and
5426 then return the numerator, denominator, or both as a list, respectively.
5427 If you need both numerator and denominator, calling @code{numer_denom()} is
5428 faster than using @code{numer()} and @code{denom()} separately.
5431 @subsection Converting to a polynomial or rational expression
5432 @cindex @code{to_polynomial()}
5433 @cindex @code{to_rational()}
5435 Some of the methods described so far only work on polynomials or rational
5436 functions. GiNaC provides a way to extend the domain of these functions to
5437 general expressions by using the temporary replacement algorithm described
5438 above. You do this by calling
5441 ex ex::to_polynomial(exmap & m);
5442 ex ex::to_polynomial(lst & l);
5446 ex ex::to_rational(exmap & m);
5447 ex ex::to_rational(lst & l);
5450 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5451 will be filled with the generated temporary symbols and their replacement
5452 expressions in a format that can be used directly for the @code{subs()}
5453 method. It can also already contain a list of replacements from an earlier
5454 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5455 possible to use it on multiple expressions and get consistent results.
5457 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5458 is probably best illustrated with an example:
5462 symbol x("x"), y("y");
5463 ex a = 2*x/sin(x) - y/(3*sin(x));
5467 ex p = a.to_polynomial(lp);
5468 cout << " = " << p << "\n with " << lp << endl;
5469 // = symbol3*symbol2*y+2*symbol2*x
5470 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5473 ex r = a.to_rational(lr);
5474 cout << " = " << r << "\n with " << lr << endl;
5475 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5476 // with @{symbol4==sin(x)@}
5480 The following more useful example will print @samp{sin(x)-cos(x)}:
5485 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5486 ex b = sin(x) + cos(x);
5489 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5490 cout << q.subs(m) << endl;
5495 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5496 @c node-name, next, previous, up
5497 @section Symbolic differentiation
5498 @cindex differentiation
5499 @cindex @code{diff()}
5501 @cindex product rule
5503 GiNaC's objects know how to differentiate themselves. Thus, a
5504 polynomial (class @code{add}) knows that its derivative is the sum of
5505 the derivatives of all the monomials:
5509 symbol x("x"), y("y"), z("z");
5510 ex P = pow(x, 5) + pow(x, 2) + y;
5512 cout << P.diff(x,2) << endl;
5514 cout << P.diff(y) << endl; // 1
5516 cout << P.diff(z) << endl; // 0
5521 If a second integer parameter @var{n} is given, the @code{diff} method
5522 returns the @var{n}th derivative.
5524 If @emph{every} object and every function is told what its derivative
5525 is, all derivatives of composed objects can be calculated using the
5526 chain rule and the product rule. Consider, for instance the expression
5527 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5528 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5529 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5530 out that the composition is the generating function for Euler Numbers,
5531 i.e. the so called @var{n}th Euler number is the coefficient of
5532 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5533 identity to code a function that generates Euler numbers in just three
5536 @cindex Euler numbers
5538 #include <ginac/ginac.h>
5539 using namespace GiNaC;
5541 ex EulerNumber(unsigned n)
5544 const ex generator = pow(cosh(x),-1);
5545 return generator.diff(x,n).subs(x==0);
5550 for (unsigned i=0; i<11; i+=2)
5551 std::cout << EulerNumber(i) << std::endl;
5556 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5557 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5558 @code{i} by two since all odd Euler numbers vanish anyways.
5561 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5562 @c node-name, next, previous, up
5563 @section Series expansion
5564 @cindex @code{series()}
5565 @cindex Taylor expansion
5566 @cindex Laurent expansion
5567 @cindex @code{pseries} (class)
5568 @cindex @code{Order()}
5570 Expressions know how to expand themselves as a Taylor series or (more
5571 generally) a Laurent series. As in most conventional Computer Algebra
5572 Systems, no distinction is made between those two. There is a class of
5573 its own for storing such series (@code{class pseries}) and a built-in
5574 function (called @code{Order}) for storing the order term of the series.
5575 As a consequence, if you want to work with series, i.e. multiply two
5576 series, you need to call the method @code{ex::series} again to convert
5577 it to a series object with the usual structure (expansion plus order
5578 term). A sample application from special relativity could read:
5581 #include <ginac/ginac.h>
5582 using namespace std;
5583 using namespace GiNaC;
5587 symbol v("v"), c("c");
5589 ex gamma = 1/sqrt(1 - pow(v/c,2));
5590 ex mass_nonrel = gamma.series(v==0, 10);
5592 cout << "the relativistic mass increase with v is " << endl
5593 << mass_nonrel << endl;
5595 cout << "the inverse square of this series is " << endl
5596 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5600 Only calling the series method makes the last output simplify to
5601 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5602 series raised to the power @math{-2}.
5604 @cindex Machin's formula
5605 As another instructive application, let us calculate the numerical
5606 value of Archimedes' constant
5610 (for which there already exists the built-in constant @code{Pi})
5611 using John Machin's amazing formula
5613 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5616 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5618 This equation (and similar ones) were used for over 200 years for
5619 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5620 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5621 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5622 order term with it and the question arises what the system is supposed
5623 to do when the fractions are plugged into that order term. The solution
5624 is to use the function @code{series_to_poly()} to simply strip the order
5628 #include <ginac/ginac.h>
5629 using namespace GiNaC;
5631 ex machin_pi(int degr)
5634 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5635 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5636 -4*pi_expansion.subs(x==numeric(1,239));
5642 using std::cout; // just for fun, another way of...
5643 using std::endl; // ...dealing with this namespace std.
5645 for (int i=2; i<12; i+=2) @{
5646 pi_frac = machin_pi(i);
5647 cout << i << ":\t" << pi_frac << endl
5648 << "\t" << pi_frac.evalf() << endl;
5654 Note how we just called @code{.series(x,degr)} instead of
5655 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5656 method @code{series()}: if the first argument is a symbol the expression
5657 is expanded in that symbol around point @code{0}. When you run this
5658 program, it will type out:
5662 3.1832635983263598326
5663 4: 5359397032/1706489875
5664 3.1405970293260603143
5665 6: 38279241713339684/12184551018734375
5666 3.141621029325034425
5667 8: 76528487109180192540976/24359780855939418203125
5668 3.141591772182177295
5669 10: 327853873402258685803048818236/104359128170408663038552734375
5670 3.1415926824043995174
5674 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5675 @c node-name, next, previous, up
5676 @section Symmetrization
5677 @cindex @code{symmetrize()}
5678 @cindex @code{antisymmetrize()}
5679 @cindex @code{symmetrize_cyclic()}
5684 ex ex::symmetrize(const lst & l);
5685 ex ex::antisymmetrize(const lst & l);
5686 ex ex::symmetrize_cyclic(const lst & l);
5689 symmetrize an expression by returning the sum over all symmetric,
5690 antisymmetric or cyclic permutations of the specified list of objects,
5691 weighted by the number of permutations.
5693 The three additional methods
5696 ex ex::symmetrize();
5697 ex ex::antisymmetrize();
5698 ex ex::symmetrize_cyclic();
5701 symmetrize or antisymmetrize an expression over its free indices.
5703 Symmetrization is most useful with indexed expressions but can be used with
5704 almost any kind of object (anything that is @code{subs()}able):
5708 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5709 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5711 cout << indexed(A, i, j).symmetrize() << endl;
5712 // -> 1/2*A.j.i+1/2*A.i.j
5713 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5714 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5715 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5716 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5720 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5721 @c node-name, next, previous, up
5722 @section Predefined mathematical functions
5724 @subsection Overview
5726 GiNaC contains the following predefined mathematical functions:
5729 @multitable @columnfractions .30 .70
5730 @item @strong{Name} @tab @strong{Function}
5733 @cindex @code{abs()}
5734 @item @code{csgn(x)}
5736 @cindex @code{conjugate()}
5737 @item @code{conjugate(x)}
5738 @tab complex conjugation
5739 @cindex @code{csgn()}
5740 @item @code{sqrt(x)}
5741 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5742 @cindex @code{sqrt()}
5745 @cindex @code{sin()}
5748 @cindex @code{cos()}
5751 @cindex @code{tan()}
5752 @item @code{asin(x)}
5754 @cindex @code{asin()}
5755 @item @code{acos(x)}
5757 @cindex @code{acos()}
5758 @item @code{atan(x)}
5759 @tab inverse tangent
5760 @cindex @code{atan()}
5761 @item @code{atan2(y, x)}
5762 @tab inverse tangent with two arguments
5763 @item @code{sinh(x)}
5764 @tab hyperbolic sine
5765 @cindex @code{sinh()}
5766 @item @code{cosh(x)}
5767 @tab hyperbolic cosine
5768 @cindex @code{cosh()}
5769 @item @code{tanh(x)}
5770 @tab hyperbolic tangent
5771 @cindex @code{tanh()}
5772 @item @code{asinh(x)}
5773 @tab inverse hyperbolic sine
5774 @cindex @code{asinh()}
5775 @item @code{acosh(x)}
5776 @tab inverse hyperbolic cosine
5777 @cindex @code{acosh()}
5778 @item @code{atanh(x)}
5779 @tab inverse hyperbolic tangent
5780 @cindex @code{atanh()}
5782 @tab exponential function
5783 @cindex @code{exp()}
5785 @tab natural logarithm
5786 @cindex @code{log()}
5789 @cindex @code{Li2()}
5790 @item @code{Li(m, x)}
5791 @tab classical polylogarithm as well as multiple polylogarithm
5793 @item @code{G(a, y)}
5794 @tab multiple polylogarithm
5796 @item @code{G(a, s, y)}
5797 @tab multiple polylogarithm with explicit signs for the imaginary parts
5799 @item @code{S(n, p, x)}
5800 @tab Nielsen's generalized polylogarithm
5802 @item @code{H(m, x)}
5803 @tab harmonic polylogarithm
5805 @item @code{zeta(m)}
5806 @tab Riemann's zeta function as well as multiple zeta value
5807 @cindex @code{zeta()}
5808 @item @code{zeta(m, s)}
5809 @tab alternating Euler sum
5810 @cindex @code{zeta()}
5811 @item @code{zetaderiv(n, x)}
5812 @tab derivatives of Riemann's zeta function
5813 @item @code{tgamma(x)}
5815 @cindex @code{tgamma()}
5816 @cindex gamma function
5817 @item @code{lgamma(x)}
5818 @tab logarithm of gamma function
5819 @cindex @code{lgamma()}
5820 @item @code{beta(x, y)}
5821 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5822 @cindex @code{beta()}
5824 @tab psi (digamma) function
5825 @cindex @code{psi()}
5826 @item @code{psi(n, x)}
5827 @tab derivatives of psi function (polygamma functions)
5828 @item @code{factorial(n)}
5829 @tab factorial function @math{n!}
5830 @cindex @code{factorial()}
5831 @item @code{binomial(n, k)}
5832 @tab binomial coefficients
5833 @cindex @code{binomial()}
5834 @item @code{Order(x)}
5835 @tab order term function in truncated power series
5836 @cindex @code{Order()}
5841 For functions that have a branch cut in the complex plane GiNaC follows
5842 the conventions for C++ as defined in the ANSI standard as far as
5843 possible. In particular: the natural logarithm (@code{log}) and the
5844 square root (@code{sqrt}) both have their branch cuts running along the
5845 negative real axis where the points on the axis itself belong to the
5846 upper part (i.e. continuous with quadrant II). The inverse
5847 trigonometric and hyperbolic functions are not defined for complex
5848 arguments by the C++ standard, however. In GiNaC we follow the
5849 conventions used by CLN, which in turn follow the carefully designed
5850 definitions in the Common Lisp standard. It should be noted that this
5851 convention is identical to the one used by the C99 standard and by most
5852 serious CAS. It is to be expected that future revisions of the C++
5853 standard incorporate these functions in the complex domain in a manner
5854 compatible with C99.
5856 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5857 @c node-name, next, previous, up
5858 @subsection Multiple polylogarithms
5860 @cindex polylogarithm
5861 @cindex Nielsen's generalized polylogarithm
5862 @cindex harmonic polylogarithm
5863 @cindex multiple zeta value
5864 @cindex alternating Euler sum
5865 @cindex multiple polylogarithm
5867 The multiple polylogarithm is the most generic member of a family of functions,
5868 to which others like the harmonic polylogarithm, Nielsen's generalized
5869 polylogarithm and the multiple zeta value belong.
5870 Everyone of these functions can also be written as a multiple polylogarithm with specific
5871 parameters. This whole family of functions is therefore often referred to simply as
5872 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5873 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5874 @code{Li} and @code{G} in principle represent the same function, the different
5875 notations are more natural to the series representation or the integral
5876 representation, respectively.
5878 To facilitate the discussion of these functions we distinguish between indices and
5879 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5880 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5882 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5883 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5884 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5885 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5886 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5887 @code{s} is not given, the signs default to +1.
5888 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5889 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5890 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5891 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5892 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5894 The functions print in LaTeX format as
5896 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5902 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5905 $\zeta(m_1,m_2,\ldots,m_k)$.
5907 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5908 are printed with a line above, e.g.
5910 $\zeta(5,\overline{2})$.
5912 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5914 Definitions and analytical as well as numerical properties of multiple polylogarithms
5915 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5916 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5917 except for a few differences which will be explicitly stated in the following.
5919 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5920 that the indices and arguments are understood to be in the same order as in which they appear in
5921 the series representation. This means
5923 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5926 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5929 $\zeta(1,2)$ evaluates to infinity.
5931 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5934 The functions only evaluate if the indices are integers greater than zero, except for the indices
5935 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5936 will be interpreted as the sequence of signs for the corresponding indices
5937 @code{m} or the sign of the imaginary part for the
5938 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5939 @code{zeta(lst(3,4), lst(-1,1))} means
5941 $\zeta(\overline{3},4)$
5944 @code{G(lst(a,b), lst(-1,1), c)} means
5946 $G(a-0\epsilon,b+0\epsilon;c)$.
5948 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5949 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5950 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
5951 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5952 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5953 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5954 evaluates also for negative integers and positive even integers. For example:
5957 > Li(@{3,1@},@{x,1@});
5960 -zeta(@{3,2@},@{-1,-1@})
5965 It is easy to tell for a given function into which other function it can be rewritten, may
5966 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5967 with negative indices or trailing zeros (the example above gives a hint). Signs can
5968 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5969 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5970 @code{Li} (@code{eval()} already cares for the possible downgrade):
5973 > convert_H_to_Li(@{0,-2,-1,3@},x);
5974 Li(@{3,1,3@},@{-x,1,-1@})
5975 > convert_H_to_Li(@{2,-1,0@},x);
5976 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5979 Every function can be numerically evaluated for
5980 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5981 global variable @code{Digits}:
5986 > evalf(zeta(@{3,1,3,1@}));
5987 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5990 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5991 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5993 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5998 In long expressions this helps a lot with debugging, because you can easily spot
5999 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6000 cancellations of divergencies happen.
6002 Useful publications:
6004 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6005 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6007 @cite{Harmonic Polylogarithms},
6008 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6010 @cite{Special Values of Multiple Polylogarithms},
6011 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6013 @cite{Numerical Evaluation of Multiple Polylogarithms},
6014 J.Vollinga, S.Weinzierl, hep-ph/0410259
6016 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
6017 @c node-name, next, previous, up
6018 @section Complex Conjugation
6020 @cindex @code{conjugate()}
6028 returns the complex conjugate of the expression. For all built-in functions and objects the
6029 conjugation gives the expected results:
6033 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6037 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6038 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6039 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6040 // -> -gamma5*gamma~b*gamma~a
6044 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
6045 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
6046 arguments. This is the default strategy. If you want to define your own functions and want to
6047 change this behavior, you have to supply a specialized conjugation method for your function
6048 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
6050 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
6051 @c node-name, next, previous, up
6052 @section Solving Linear Systems of Equations
6053 @cindex @code{lsolve()}
6055 The function @code{lsolve()} provides a convenient wrapper around some
6056 matrix operations that comes in handy when a system of linear equations
6060 ex lsolve(const ex & eqns, const ex & symbols,
6061 unsigned options = solve_algo::automatic);
6064 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6065 @code{relational}) while @code{symbols} is a @code{lst} of
6066 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
6069 It returns the @code{lst} of solutions as an expression. As an example,
6070 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6074 symbol a("a"), b("b"), x("x"), y("y");
6076 eqns = a*x+b*y==3, x-y==b;
6078 cout << lsolve(eqns, vars) << endl;
6079 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6082 When the linear equations @code{eqns} are underdetermined, the solution
6083 will contain one or more tautological entries like @code{x==x},
6084 depending on the rank of the system. When they are overdetermined, the
6085 solution will be an empty @code{lst}. Note the third optional parameter
6086 to @code{lsolve()}: it accepts the same parameters as
6087 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6091 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6092 @c node-name, next, previous, up
6093 @section Input and output of expressions
6096 @subsection Expression output
6098 @cindex output of expressions
6100 Expressions can simply be written to any stream:
6105 ex e = 4.5*I+pow(x,2)*3/2;
6106 cout << e << endl; // prints '4.5*I+3/2*x^2'
6110 The default output format is identical to the @command{ginsh} input syntax and
6111 to that used by most computer algebra systems, but not directly pastable
6112 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6113 is printed as @samp{x^2}).
6115 It is possible to print expressions in a number of different formats with
6116 a set of stream manipulators;
6119 std::ostream & dflt(std::ostream & os);
6120 std::ostream & latex(std::ostream & os);
6121 std::ostream & tree(std::ostream & os);
6122 std::ostream & csrc(std::ostream & os);
6123 std::ostream & csrc_float(std::ostream & os);
6124 std::ostream & csrc_double(std::ostream & os);
6125 std::ostream & csrc_cl_N(std::ostream & os);
6126 std::ostream & index_dimensions(std::ostream & os);
6127 std::ostream & no_index_dimensions(std::ostream & os);
6130 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6131 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6132 @code{print_csrc()} functions, respectively.
6135 All manipulators affect the stream state permanently. To reset the output
6136 format to the default, use the @code{dflt} manipulator:
6140 cout << latex; // all output to cout will be in LaTeX format from
6142 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6143 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6144 cout << dflt; // revert to default output format
6145 cout << e << endl; // prints '4.5*I+3/2*x^2'
6149 If you don't want to affect the format of the stream you're working with,
6150 you can output to a temporary @code{ostringstream} like this:
6155 s << latex << e; // format of cout remains unchanged
6156 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6161 @cindex @code{csrc_float}
6162 @cindex @code{csrc_double}
6163 @cindex @code{csrc_cl_N}
6164 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6165 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6166 format that can be directly used in a C or C++ program. The three possible
6167 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6168 classes provided by the CLN library):
6172 cout << "f = " << csrc_float << e << ";\n";
6173 cout << "d = " << csrc_double << e << ";\n";
6174 cout << "n = " << csrc_cl_N << e << ";\n";
6178 The above example will produce (note the @code{x^2} being converted to
6182 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6183 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6184 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6188 The @code{tree} manipulator allows dumping the internal structure of an
6189 expression for debugging purposes:
6200 add, hash=0x0, flags=0x3, nops=2
6201 power, hash=0x0, flags=0x3, nops=2
6202 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6203 2 (numeric), hash=0x6526b0fa, flags=0xf
6204 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6207 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6211 @cindex @code{latex}
6212 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6213 It is rather similar to the default format but provides some braces needed
6214 by LaTeX for delimiting boxes and also converts some common objects to
6215 conventional LaTeX names. It is possible to give symbols a special name for
6216 LaTeX output by supplying it as a second argument to the @code{symbol}
6219 For example, the code snippet
6223 symbol x("x", "\\circ");
6224 ex e = lgamma(x).series(x==0,3);
6225 cout << latex << e << endl;
6232 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6233 +\mathcal@{O@}(\circ^@{3@})
6236 @cindex @code{index_dimensions}
6237 @cindex @code{no_index_dimensions}
6238 Index dimensions are normally hidden in the output. To make them visible, use
6239 the @code{index_dimensions} manipulator. The dimensions will be written in
6240 square brackets behind each index value in the default and LaTeX output
6245 symbol x("x"), y("y");
6246 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6247 ex e = indexed(x, mu) * indexed(y, nu);
6250 // prints 'x~mu*y~nu'
6251 cout << index_dimensions << e << endl;
6252 // prints 'x~mu[4]*y~nu[4]'
6253 cout << no_index_dimensions << e << endl;
6254 // prints 'x~mu*y~nu'
6259 @cindex Tree traversal
6260 If you need any fancy special output format, e.g. for interfacing GiNaC
6261 with other algebra systems or for producing code for different
6262 programming languages, you can always traverse the expression tree yourself:
6265 static void my_print(const ex & e)
6267 if (is_a<function>(e))
6268 cout << ex_to<function>(e).get_name();
6270 cout << ex_to<basic>(e).class_name();
6272 size_t n = e.nops();
6274 for (size_t i=0; i<n; i++) @{
6286 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6294 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6295 symbol(y))),numeric(-2)))
6298 If you need an output format that makes it possible to accurately
6299 reconstruct an expression by feeding the output to a suitable parser or
6300 object factory, you should consider storing the expression in an
6301 @code{archive} object and reading the object properties from there.
6302 See the section on archiving for more information.
6305 @subsection Expression input
6306 @cindex input of expressions
6308 GiNaC provides no way to directly read an expression from a stream because
6309 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6310 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6311 @code{y} you defined in your program and there is no way to specify the
6312 desired symbols to the @code{>>} stream input operator.
6314 Instead, GiNaC lets you construct an expression from a string, specifying the
6315 list of symbols to be used:
6319 symbol x("x"), y("y");
6320 ex e("2*x+sin(y)", lst(x, y));
6324 The input syntax is the same as that used by @command{ginsh} and the stream
6325 output operator @code{<<}. The symbols in the string are matched by name to
6326 the symbols in the list and if GiNaC encounters a symbol not specified in
6327 the list it will throw an exception.
6329 With this constructor, it's also easy to implement interactive GiNaC programs:
6334 #include <stdexcept>
6335 #include <ginac/ginac.h>
6336 using namespace std;
6337 using namespace GiNaC;
6344 cout << "Enter an expression containing 'x': ";
6349 cout << "The derivative of " << e << " with respect to x is ";
6350 cout << e.diff(x) << ".\n";
6351 @} catch (exception &p) @{
6352 cerr << p.what() << endl;
6358 @subsection Archiving
6359 @cindex @code{archive} (class)
6362 GiNaC allows creating @dfn{archives} of expressions which can be stored
6363 to or retrieved from files. To create an archive, you declare an object
6364 of class @code{archive} and archive expressions in it, giving each
6365 expression a unique name:
6369 using namespace std;
6370 #include <ginac/ginac.h>
6371 using namespace GiNaC;
6375 symbol x("x"), y("y"), z("z");
6377 ex foo = sin(x + 2*y) + 3*z + 41;
6381 a.archive_ex(foo, "foo");
6382 a.archive_ex(bar, "the second one");
6386 The archive can then be written to a file:
6390 ofstream out("foobar.gar");
6396 The file @file{foobar.gar} contains all information that is needed to
6397 reconstruct the expressions @code{foo} and @code{bar}.
6399 @cindex @command{viewgar}
6400 The tool @command{viewgar} that comes with GiNaC can be used to view
6401 the contents of GiNaC archive files:
6404 $ viewgar foobar.gar
6405 foo = 41+sin(x+2*y)+3*z
6406 the second one = 42+sin(x+2*y)+3*z
6409 The point of writing archive files is of course that they can later be
6415 ifstream in("foobar.gar");
6420 And the stored expressions can be retrieved by their name:
6427 ex ex1 = a2.unarchive_ex(syms, "foo");
6428 ex ex2 = a2.unarchive_ex(syms, "the second one");
6430 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6431 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6432 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6436 Note that you have to supply a list of the symbols which are to be inserted
6437 in the expressions. Symbols in archives are stored by their name only and
6438 if you don't specify which symbols you have, unarchiving the expression will
6439 create new symbols with that name. E.g. if you hadn't included @code{x} in
6440 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6441 have had no effect because the @code{x} in @code{ex1} would have been a
6442 different symbol than the @code{x} which was defined at the beginning of
6443 the program, although both would appear as @samp{x} when printed.
6445 You can also use the information stored in an @code{archive} object to
6446 output expressions in a format suitable for exact reconstruction. The
6447 @code{archive} and @code{archive_node} classes have a couple of member
6448 functions that let you access the stored properties:
6451 static void my_print2(const archive_node & n)
6454 n.find_string("class", class_name);
6455 cout << class_name << "(";
6457 archive_node::propinfovector p;
6458 n.get_properties(p);
6460 size_t num = p.size();
6461 for (size_t i=0; i<num; i++) @{
6462 const string &name = p[i].name;
6463 if (name == "class")
6465 cout << name << "=";
6467 unsigned count = p[i].count;
6471 for (unsigned j=0; j<count; j++) @{
6472 switch (p[i].type) @{
6473 case archive_node::PTYPE_BOOL: @{
6475 n.find_bool(name, x, j);
6476 cout << (x ? "true" : "false");
6479 case archive_node::PTYPE_UNSIGNED: @{
6481 n.find_unsigned(name, x, j);
6485 case archive_node::PTYPE_STRING: @{
6487 n.find_string(name, x, j);
6488 cout << '\"' << x << '\"';
6491 case archive_node::PTYPE_NODE: @{
6492 const archive_node &x = n.find_ex_node(name, j);
6514 ex e = pow(2, x) - y;
6516 my_print2(ar.get_top_node(0)); cout << endl;
6524 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6525 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6526 overall_coeff=numeric(number="0"))
6529 Be warned, however, that the set of properties and their meaning for each
6530 class may change between GiNaC versions.
6533 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6534 @c node-name, next, previous, up
6535 @chapter Extending GiNaC
6537 By reading so far you should have gotten a fairly good understanding of
6538 GiNaC's design patterns. From here on you should start reading the
6539 sources. All we can do now is issue some recommendations how to tackle
6540 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6541 develop some useful extension please don't hesitate to contact the GiNaC
6542 authors---they will happily incorporate them into future versions.
6545 * What does not belong into GiNaC:: What to avoid.
6546 * Symbolic functions:: Implementing symbolic functions.
6547 * Printing:: Adding new output formats.
6548 * Structures:: Defining new algebraic classes (the easy way).
6549 * Adding classes:: Defining new algebraic classes (the hard way).
6553 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6554 @c node-name, next, previous, up
6555 @section What doesn't belong into GiNaC
6557 @cindex @command{ginsh}
6558 First of all, GiNaC's name must be read literally. It is designed to be
6559 a library for use within C++. The tiny @command{ginsh} accompanying
6560 GiNaC makes this even more clear: it doesn't even attempt to provide a
6561 language. There are no loops or conditional expressions in
6562 @command{ginsh}, it is merely a window into the library for the
6563 programmer to test stuff (or to show off). Still, the design of a
6564 complete CAS with a language of its own, graphical capabilities and all
6565 this on top of GiNaC is possible and is without doubt a nice project for
6568 There are many built-in functions in GiNaC that do not know how to
6569 evaluate themselves numerically to a precision declared at runtime
6570 (using @code{Digits}). Some may be evaluated at certain points, but not
6571 generally. This ought to be fixed. However, doing numerical
6572 computations with GiNaC's quite abstract classes is doomed to be
6573 inefficient. For this purpose, the underlying foundation classes
6574 provided by CLN are much better suited.
6577 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6578 @c node-name, next, previous, up
6579 @section Symbolic functions
6581 The easiest and most instructive way to start extending GiNaC is probably to
6582 create your own symbolic functions. These are implemented with the help of
6583 two preprocessor macros:
6585 @cindex @code{DECLARE_FUNCTION}
6586 @cindex @code{REGISTER_FUNCTION}
6588 DECLARE_FUNCTION_<n>P(<name>)
6589 REGISTER_FUNCTION(<name>, <options>)
6592 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6593 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6594 parameters of type @code{ex} and returns a newly constructed GiNaC
6595 @code{function} object that represents your function.
6597 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6598 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6599 set of options that associate the symbolic function with C++ functions you
6600 provide to implement the various methods such as evaluation, derivative,
6601 series expansion etc. They also describe additional attributes the function
6602 might have, such as symmetry and commutation properties, and a name for
6603 LaTeX output. Multiple options are separated by the member access operator
6604 @samp{.} and can be given in an arbitrary order.
6606 (By the way: in case you are worrying about all the macros above we can
6607 assure you that functions are GiNaC's most macro-intense classes. We have
6608 done our best to avoid macros where we can.)
6610 @subsection A minimal example
6612 Here is an example for the implementation of a function with two arguments
6613 that is not further evaluated:
6616 DECLARE_FUNCTION_2P(myfcn)
6618 REGISTER_FUNCTION(myfcn, dummy())
6621 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6622 in algebraic expressions:
6628 ex e = 2*myfcn(42, 1+3*x) - x;
6630 // prints '2*myfcn(42,1+3*x)-x'
6635 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6636 "no options". A function with no options specified merely acts as a kind of
6637 container for its arguments. It is a pure "dummy" function with no associated
6638 logic (which is, however, sometimes perfectly sufficient).
6640 Let's now have a look at the implementation of GiNaC's cosine function for an
6641 example of how to make an "intelligent" function.
6643 @subsection The cosine function
6645 The GiNaC header file @file{inifcns.h} contains the line
6648 DECLARE_FUNCTION_1P(cos)
6651 which declares to all programs using GiNaC that there is a function @samp{cos}
6652 that takes one @code{ex} as an argument. This is all they need to know to use
6653 this function in expressions.
6655 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6656 is its @code{REGISTER_FUNCTION} line:
6659 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6660 evalf_func(cos_evalf).
6661 derivative_func(cos_deriv).
6662 latex_name("\\cos"));
6665 There are four options defined for the cosine function. One of them
6666 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6667 other three indicate the C++ functions in which the "brains" of the cosine
6668 function are defined.
6670 @cindex @code{hold()}
6672 The @code{eval_func()} option specifies the C++ function that implements
6673 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6674 the same number of arguments as the associated symbolic function (one in this
6675 case) and returns the (possibly transformed or in some way simplified)
6676 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6677 of the automatic evaluation process). If no (further) evaluation is to take
6678 place, the @code{eval_func()} function must return the original function
6679 with @code{.hold()}, to avoid a potential infinite recursion. If your
6680 symbolic functions produce a segmentation fault or stack overflow when
6681 using them in expressions, you are probably missing a @code{.hold()}
6684 The @code{eval_func()} function for the cosine looks something like this
6685 (actually, it doesn't look like this at all, but it should give you an idea
6689 static ex cos_eval(const ex & x)
6691 if ("x is a multiple of 2*Pi")
6693 else if ("x is a multiple of Pi")
6695 else if ("x is a multiple of Pi/2")
6699 else if ("x has the form 'acos(y)'")
6701 else if ("x has the form 'asin(y)'")
6706 return cos(x).hold();
6710 This function is called every time the cosine is used in a symbolic expression:
6716 // this calls cos_eval(Pi), and inserts its return value into
6717 // the actual expression
6724 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6725 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6726 symbolic transformation can be done, the unmodified function is returned
6727 with @code{.hold()}.
6729 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6730 The user has to call @code{evalf()} for that. This is implemented in a
6734 static ex cos_evalf(const ex & x)
6736 if (is_a<numeric>(x))
6737 return cos(ex_to<numeric>(x));
6739 return cos(x).hold();
6743 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6744 in this case the @code{cos()} function for @code{numeric} objects, which in
6745 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6746 isn't really needed here, but reminds us that the corresponding @code{eval()}
6747 function would require it in this place.
6749 Differentiation will surely turn up and so we need to tell @code{cos}
6750 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6751 instance, are then handled automatically by @code{basic::diff} and
6755 static ex cos_deriv(const ex & x, unsigned diff_param)
6761 @cindex product rule
6762 The second parameter is obligatory but uninteresting at this point. It
6763 specifies which parameter to differentiate in a partial derivative in
6764 case the function has more than one parameter, and its main application
6765 is for correct handling of the chain rule.
6767 An implementation of the series expansion is not needed for @code{cos()} as
6768 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6769 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6770 the other hand, does have poles and may need to do Laurent expansion:
6773 static ex tan_series(const ex & x, const relational & rel,
6774 int order, unsigned options)
6776 // Find the actual expansion point
6777 const ex x_pt = x.subs(rel);
6779 if ("x_pt is not an odd multiple of Pi/2")
6780 throw do_taylor(); // tell function::series() to do Taylor expansion
6782 // On a pole, expand sin()/cos()
6783 return (sin(x)/cos(x)).series(rel, order+2, options);
6787 The @code{series()} implementation of a function @emph{must} return a
6788 @code{pseries} object, otherwise your code will crash.
6790 @subsection Function options
6792 GiNaC functions understand several more options which are always
6793 specified as @code{.option(params)}. None of them are required, but you
6794 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6795 is a do-nothing option called @code{dummy()} which you can use to define
6796 functions without any special options.
6799 eval_func(<C++ function>)
6800 evalf_func(<C++ function>)
6801 derivative_func(<C++ function>)
6802 series_func(<C++ function>)
6803 conjugate_func(<C++ function>)
6806 These specify the C++ functions that implement symbolic evaluation,
6807 numeric evaluation, partial derivatives, and series expansion, respectively.
6808 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6809 @code{diff()} and @code{series()}.
6811 The @code{eval_func()} function needs to use @code{.hold()} if no further
6812 automatic evaluation is desired or possible.
6814 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6815 expansion, which is correct if there are no poles involved. If the function
6816 has poles in the complex plane, the @code{series_func()} needs to check
6817 whether the expansion point is on a pole and fall back to Taylor expansion
6818 if it isn't. Otherwise, the pole usually needs to be regularized by some
6819 suitable transformation.
6822 latex_name(const string & n)
6825 specifies the LaTeX code that represents the name of the function in LaTeX
6826 output. The default is to put the function name in an @code{\mbox@{@}}.
6829 do_not_evalf_params()
6832 This tells @code{evalf()} to not recursively evaluate the parameters of the
6833 function before calling the @code{evalf_func()}.
6836 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6839 This allows you to explicitly specify the commutation properties of the
6840 function (@xref{Non-commutative objects}, for an explanation of
6841 (non)commutativity in GiNaC). For example, you can use
6842 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6843 GiNaC treat your function like a matrix. By default, functions inherit the
6844 commutation properties of their first argument.
6847 set_symmetry(const symmetry & s)
6850 specifies the symmetry properties of the function with respect to its
6851 arguments. @xref{Indexed objects}, for an explanation of symmetry
6852 specifications. GiNaC will automatically rearrange the arguments of
6853 symmetric functions into a canonical order.
6855 Sometimes you may want to have finer control over how functions are
6856 displayed in the output. For example, the @code{abs()} function prints
6857 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6858 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6862 print_func<C>(<C++ function>)
6865 option which is explained in the next section.
6867 @subsection Functions with a variable number of arguments
6869 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6870 functions with a fixed number of arguments. Sometimes, though, you may need
6871 to have a function that accepts a variable number of expressions. One way to
6872 accomplish this is to pass variable-length lists as arguments. The
6873 @code{Li()} function uses this method for multiple polylogarithms.
6875 It is also possible to define functions that accept a different number of
6876 parameters under the same function name, such as the @code{psi()} function
6877 which can be called either as @code{psi(z)} (the digamma function) or as
6878 @code{psi(n, z)} (polygamma functions). These are actually two different
6879 functions in GiNaC that, however, have the same name. Defining such
6880 functions is not possible with the macros but requires manually fiddling
6881 with GiNaC internals. If you are interested, please consult the GiNaC source
6882 code for the @code{psi()} function (@file{inifcns.h} and
6883 @file{inifcns_gamma.cpp}).
6886 @node Printing, Structures, Symbolic functions, Extending GiNaC
6887 @c node-name, next, previous, up
6888 @section GiNaC's expression output system
6890 GiNaC allows the output of expressions in a variety of different formats
6891 (@pxref{Input/Output}). This section will explain how expression output
6892 is implemented internally, and how to define your own output formats or
6893 change the output format of built-in algebraic objects. You will also want
6894 to read this section if you plan to write your own algebraic classes or
6897 @cindex @code{print_context} (class)
6898 @cindex @code{print_dflt} (class)
6899 @cindex @code{print_latex} (class)
6900 @cindex @code{print_tree} (class)
6901 @cindex @code{print_csrc} (class)
6902 All the different output formats are represented by a hierarchy of classes
6903 rooted in the @code{print_context} class, defined in the @file{print.h}
6908 the default output format
6910 output in LaTeX mathematical mode
6912 a dump of the internal expression structure (for debugging)
6914 the base class for C source output
6915 @item print_csrc_float
6916 C source output using the @code{float} type
6917 @item print_csrc_double
6918 C source output using the @code{double} type
6919 @item print_csrc_cl_N
6920 C source output using CLN types
6923 The @code{print_context} base class provides two public data members:
6935 @code{s} is a reference to the stream to output to, while @code{options}
6936 holds flags and modifiers. Currently, there is only one flag defined:
6937 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6938 to print the index dimension which is normally hidden.
6940 When you write something like @code{std::cout << e}, where @code{e} is
6941 an object of class @code{ex}, GiNaC will construct an appropriate
6942 @code{print_context} object (of a class depending on the selected output
6943 format), fill in the @code{s} and @code{options} members, and call
6945 @cindex @code{print()}
6947 void ex::print(const print_context & c, unsigned level = 0) const;
6950 which in turn forwards the call to the @code{print()} method of the
6951 top-level algebraic object contained in the expression.
6953 Unlike other methods, GiNaC classes don't usually override their
6954 @code{print()} method to implement expression output. Instead, the default
6955 implementation @code{basic::print(c, level)} performs a run-time double
6956 dispatch to a function selected by the dynamic type of the object and the
6957 passed @code{print_context}. To this end, GiNaC maintains a separate method
6958 table for each class, similar to the virtual function table used for ordinary
6959 (single) virtual function dispatch.
6961 The method table contains one slot for each possible @code{print_context}
6962 type, indexed by the (internally assigned) serial number of the type. Slots
6963 may be empty, in which case GiNaC will retry the method lookup with the
6964 @code{print_context} object's parent class, possibly repeating the process
6965 until it reaches the @code{print_context} base class. If there's still no
6966 method defined, the method table of the algebraic object's parent class
6967 is consulted, and so on, until a matching method is found (eventually it
6968 will reach the combination @code{basic/print_context}, which prints the
6969 object's class name enclosed in square brackets).
6971 You can think of the print methods of all the different classes and output
6972 formats as being arranged in a two-dimensional matrix with one axis listing
6973 the algebraic classes and the other axis listing the @code{print_context}
6976 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6977 to implement printing, but then they won't get any of the benefits of the
6978 double dispatch mechanism (such as the ability for derived classes to
6979 inherit only certain print methods from its parent, or the replacement of
6980 methods at run-time).
6982 @subsection Print methods for classes
6984 The method table for a class is set up either in the definition of the class,
6985 by passing the appropriate @code{print_func<C>()} option to
6986 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6987 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6988 can also be used to override existing methods dynamically.
6990 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6991 be a member function of the class (or one of its parent classes), a static
6992 member function, or an ordinary (global) C++ function. The @code{C} template
6993 parameter specifies the appropriate @code{print_context} type for which the
6994 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6995 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6996 the class is the one being implemented by
6997 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6999 For print methods that are member functions, their first argument must be of
7000 a type convertible to a @code{const C &}, and the second argument must be an
7003 For static members and global functions, the first argument must be of a type
7004 convertible to a @code{const T &}, the second argument must be of a type
7005 convertible to a @code{const C &}, and the third argument must be an
7006 @code{unsigned}. A global function will, of course, not have access to
7007 private and protected members of @code{T}.
7009 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7010 and @code{basic::print()}) is used for proper parenthesizing of the output
7011 (and by @code{print_tree} for proper indentation). It can be used for similar
7012 purposes if you write your own output formats.
7014 The explanations given above may seem complicated, but in practice it's
7015 really simple, as shown in the following example. Suppose that we want to
7016 display exponents in LaTeX output not as superscripts but with little
7017 upwards-pointing arrows. This can be achieved in the following way:
7020 void my_print_power_as_latex(const power & p,
7021 const print_latex & c,
7024 // get the precedence of the 'power' class
7025 unsigned power_prec = p.precedence();
7027 // if the parent operator has the same or a higher precedence
7028 // we need parentheses around the power
7029 if (level >= power_prec)
7032 // print the basis and exponent, each enclosed in braces, and
7033 // separated by an uparrow
7035 p.op(0).print(c, power_prec);
7036 c.s << "@}\\uparrow@{";
7037 p.op(1).print(c, power_prec);
7040 // don't forget the closing parenthesis
7041 if (level >= power_prec)
7047 // a sample expression
7048 symbol x("x"), y("y");
7049 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7051 // switch to LaTeX mode
7054 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7057 // now we replace the method for the LaTeX output of powers with
7059 set_print_func<power, print_latex>(my_print_power_as_latex);
7061 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7072 The first argument of @code{my_print_power_as_latex} could also have been
7073 a @code{const basic &}, the second one a @code{const print_context &}.
7076 The above code depends on @code{mul} objects converting their operands to
7077 @code{power} objects for the purpose of printing.
7080 The output of products including negative powers as fractions is also
7081 controlled by the @code{mul} class.
7084 The @code{power/print_latex} method provided by GiNaC prints square roots
7085 using @code{\sqrt}, but the above code doesn't.
7089 It's not possible to restore a method table entry to its previous or default
7090 value. Once you have called @code{set_print_func()}, you can only override
7091 it with another call to @code{set_print_func()}, but you can't easily go back
7092 to the default behavior again (you can, of course, dig around in the GiNaC
7093 sources, find the method that is installed at startup
7094 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7095 one; that is, after you circumvent the C++ member access control@dots{}).
7097 @subsection Print methods for functions
7099 Symbolic functions employ a print method dispatch mechanism similar to the
7100 one used for classes. The methods are specified with @code{print_func<C>()}
7101 function options. If you don't specify any special print methods, the function
7102 will be printed with its name (or LaTeX name, if supplied), followed by a
7103 comma-separated list of arguments enclosed in parentheses.
7105 For example, this is what GiNaC's @samp{abs()} function is defined like:
7108 static ex abs_eval(const ex & arg) @{ ... @}
7109 static ex abs_evalf(const ex & arg) @{ ... @}
7111 static void abs_print_latex(const ex & arg, const print_context & c)
7113 c.s << "@{|"; arg.print(c); c.s << "|@}";
7116 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7118 c.s << "fabs("; arg.print(c); c.s << ")";
7121 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7122 evalf_func(abs_evalf).
7123 print_func<print_latex>(abs_print_latex).
7124 print_func<print_csrc_float>(abs_print_csrc_float).
7125 print_func<print_csrc_double>(abs_print_csrc_float));
7128 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7129 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7131 There is currently no equivalent of @code{set_print_func()} for functions.
7133 @subsection Adding new output formats
7135 Creating a new output format involves subclassing @code{print_context},
7136 which is somewhat similar to adding a new algebraic class
7137 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7138 that needs to go into the class definition, and a corresponding macro
7139 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7140 Every @code{print_context} class needs to provide a default constructor
7141 and a constructor from an @code{std::ostream} and an @code{unsigned}
7144 Here is an example for a user-defined @code{print_context} class:
7147 class print_myformat : public print_dflt
7149 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7151 print_myformat(std::ostream & os, unsigned opt = 0)
7152 : print_dflt(os, opt) @{@}
7155 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7157 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7160 That's all there is to it. None of the actual expression output logic is
7161 implemented in this class. It merely serves as a selector for choosing
7162 a particular format. The algorithms for printing expressions in the new
7163 format are implemented as print methods, as described above.
7165 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7166 exactly like GiNaC's default output format:
7171 ex e = pow(x, 2) + 1;
7173 // this prints "1+x^2"
7176 // this also prints "1+x^2"
7177 e.print(print_myformat()); cout << endl;
7183 To fill @code{print_myformat} with life, we need to supply appropriate
7184 print methods with @code{set_print_func()}, like this:
7187 // This prints powers with '**' instead of '^'. See the LaTeX output
7188 // example above for explanations.
7189 void print_power_as_myformat(const power & p,
7190 const print_myformat & c,
7193 unsigned power_prec = p.precedence();
7194 if (level >= power_prec)
7196 p.op(0).print(c, power_prec);
7198 p.op(1).print(c, power_prec);
7199 if (level >= power_prec)
7205 // install a new print method for power objects
7206 set_print_func<power, print_myformat>(print_power_as_myformat);
7208 // now this prints "1+x**2"
7209 e.print(print_myformat()); cout << endl;
7211 // but the default format is still "1+x^2"
7217 @node Structures, Adding classes, Printing, Extending GiNaC
7218 @c node-name, next, previous, up
7221 If you are doing some very specialized things with GiNaC, or if you just
7222 need some more organized way to store data in your expressions instead of
7223 anonymous lists, you may want to implement your own algebraic classes.
7224 ('algebraic class' means any class directly or indirectly derived from
7225 @code{basic} that can be used in GiNaC expressions).
7227 GiNaC offers two ways of accomplishing this: either by using the
7228 @code{structure<T>} template class, or by rolling your own class from
7229 scratch. This section will discuss the @code{structure<T>} template which
7230 is easier to use but more limited, while the implementation of custom
7231 GiNaC classes is the topic of the next section. However, you may want to
7232 read both sections because many common concepts and member functions are
7233 shared by both concepts, and it will also allow you to decide which approach
7234 is most suited to your needs.
7236 The @code{structure<T>} template, defined in the GiNaC header file
7237 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7238 or @code{class}) into a GiNaC object that can be used in expressions.
7240 @subsection Example: scalar products
7242 Let's suppose that we need a way to handle some kind of abstract scalar
7243 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7244 product class have to store their left and right operands, which can in turn
7245 be arbitrary expressions. Here is a possible way to represent such a
7246 product in a C++ @code{struct}:
7250 using namespace std;
7252 #include <ginac/ginac.h>
7253 using namespace GiNaC;
7259 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7263 The default constructor is required. Now, to make a GiNaC class out of this
7264 data structure, we need only one line:
7267 typedef structure<sprod_s> sprod;
7270 That's it. This line constructs an algebraic class @code{sprod} which
7271 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7272 expressions like any other GiNaC class:
7276 symbol a("a"), b("b");
7277 ex e = sprod(sprod_s(a, b));
7281 Note the difference between @code{sprod} which is the algebraic class, and
7282 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7283 and @code{right} data members. As shown above, an @code{sprod} can be
7284 constructed from an @code{sprod_s} object.
7286 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7287 you could define a little wrapper function like this:
7290 inline ex make_sprod(ex left, ex right)
7292 return sprod(sprod_s(left, right));
7296 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7297 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7298 @code{get_struct()}:
7302 cout << ex_to<sprod>(e)->left << endl;
7304 cout << ex_to<sprod>(e).get_struct().right << endl;
7309 You only have read access to the members of @code{sprod_s}.
7311 The type definition of @code{sprod} is enough to write your own algorithms
7312 that deal with scalar products, for example:
7317 if (is_a<sprod>(p)) @{
7318 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7319 return make_sprod(sp.right, sp.left);
7330 @subsection Structure output
7332 While the @code{sprod} type is useable it still leaves something to be
7333 desired, most notably proper output:
7338 // -> [structure object]
7342 By default, any structure types you define will be printed as
7343 @samp{[structure object]}. To override this you can either specialize the
7344 template's @code{print()} member function, or specify print methods with
7345 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7346 it's not possible to supply class options like @code{print_func<>()} to
7347 structures, so for a self-contained structure type you need to resort to
7348 overriding the @code{print()} function, which is also what we will do here.
7350 The member functions of GiNaC classes are described in more detail in the
7351 next section, but it shouldn't be hard to figure out what's going on here:
7354 void sprod::print(const print_context & c, unsigned level) const
7356 // tree debug output handled by superclass
7357 if (is_a<print_tree>(c))
7358 inherited::print(c, level);
7360 // get the contained sprod_s object
7361 const sprod_s & sp = get_struct();
7363 // print_context::s is a reference to an ostream
7364 c.s << "<" << sp.left << "|" << sp.right << ">";
7368 Now we can print expressions containing scalar products:
7374 cout << swap_sprod(e) << endl;
7379 @subsection Comparing structures
7381 The @code{sprod} class defined so far still has one important drawback: all
7382 scalar products are treated as being equal because GiNaC doesn't know how to
7383 compare objects of type @code{sprod_s}. This can lead to some confusing
7384 and undesired behavior:
7388 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7390 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7391 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7395 To remedy this, we first need to define the operators @code{==} and @code{<}
7396 for objects of type @code{sprod_s}:
7399 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7401 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7404 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7406 return lhs.left.compare(rhs.left) < 0
7407 ? true : lhs.right.compare(rhs.right) < 0;
7411 The ordering established by the @code{<} operator doesn't have to make any
7412 algebraic sense, but it needs to be well defined. Note that we can't use
7413 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7414 in the implementation of these operators because they would construct
7415 GiNaC @code{relational} objects which in the case of @code{<} do not
7416 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7417 decide which one is algebraically 'less').
7419 Next, we need to change our definition of the @code{sprod} type to let
7420 GiNaC know that an ordering relation exists for the embedded objects:
7423 typedef structure<sprod_s, compare_std_less> sprod;
7426 @code{sprod} objects then behave as expected:
7430 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7431 // -> <a|b>-<a^2|b^2>
7432 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7433 // -> <a|b>+<a^2|b^2>
7434 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7436 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7441 The @code{compare_std_less} policy parameter tells GiNaC to use the
7442 @code{std::less} and @code{std::equal_to} functors to compare objects of
7443 type @code{sprod_s}. By default, these functors forward their work to the
7444 standard @code{<} and @code{==} operators, which we have overloaded.
7445 Alternatively, we could have specialized @code{std::less} and
7446 @code{std::equal_to} for class @code{sprod_s}.
7448 GiNaC provides two other comparison policies for @code{structure<T>}
7449 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7450 which does a bit-wise comparison of the contained @code{T} objects.
7451 This should be used with extreme care because it only works reliably with
7452 built-in integral types, and it also compares any padding (filler bytes of
7453 undefined value) that the @code{T} class might have.
7455 @subsection Subexpressions
7457 Our scalar product class has two subexpressions: the left and right
7458 operands. It might be a good idea to make them accessible via the standard
7459 @code{nops()} and @code{op()} methods:
7462 size_t sprod::nops() const
7467 ex sprod::op(size_t i) const
7471 return get_struct().left;
7473 return get_struct().right;
7475 throw std::range_error("sprod::op(): no such operand");
7480 Implementing @code{nops()} and @code{op()} for container types such as
7481 @code{sprod} has two other nice side effects:
7485 @code{has()} works as expected
7487 GiNaC generates better hash keys for the objects (the default implementation
7488 of @code{calchash()} takes subexpressions into account)
7491 @cindex @code{let_op()}
7492 There is a non-const variant of @code{op()} called @code{let_op()} that
7493 allows replacing subexpressions:
7496 ex & sprod::let_op(size_t i)
7498 // every non-const member function must call this
7499 ensure_if_modifiable();
7503 return get_struct().left;
7505 return get_struct().right;
7507 throw std::range_error("sprod::let_op(): no such operand");
7512 Once we have provided @code{let_op()} we also get @code{subs()} and
7513 @code{map()} for free. In fact, every container class that returns a non-null
7514 @code{nops()} value must either implement @code{let_op()} or provide custom
7515 implementations of @code{subs()} and @code{map()}.
7517 In turn, the availability of @code{map()} enables the recursive behavior of a
7518 couple of other default method implementations, in particular @code{evalf()},
7519 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7520 we probably want to provide our own version of @code{expand()} for scalar
7521 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7522 This is left as an exercise for the reader.
7524 The @code{structure<T>} template defines many more member functions that
7525 you can override by specialization to customize the behavior of your
7526 structures. You are referred to the next section for a description of
7527 some of these (especially @code{eval()}). There is, however, one topic
7528 that shall be addressed here, as it demonstrates one peculiarity of the
7529 @code{structure<T>} template: archiving.
7531 @subsection Archiving structures
7533 If you don't know how the archiving of GiNaC objects is implemented, you
7534 should first read the next section and then come back here. You're back?
7537 To implement archiving for structures it is not enough to provide
7538 specializations for the @code{archive()} member function and the
7539 unarchiving constructor (the @code{unarchive()} function has a default
7540 implementation). You also need to provide a unique name (as a string literal)
7541 for each structure type you define. This is because in GiNaC archives,
7542 the class of an object is stored as a string, the class name.
7544 By default, this class name (as returned by the @code{class_name()} member
7545 function) is @samp{structure} for all structure classes. This works as long
7546 as you have only defined one structure type, but if you use two or more you
7547 need to provide a different name for each by specializing the
7548 @code{get_class_name()} member function. Here is a sample implementation
7549 for enabling archiving of the scalar product type defined above:
7552 const char *sprod::get_class_name() @{ return "sprod"; @}
7554 void sprod::archive(archive_node & n) const
7556 inherited::archive(n);
7557 n.add_ex("left", get_struct().left);
7558 n.add_ex("right", get_struct().right);
7561 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7563 n.find_ex("left", get_struct().left, sym_lst);
7564 n.find_ex("right", get_struct().right, sym_lst);
7568 Note that the unarchiving constructor is @code{sprod::structure} and not
7569 @code{sprod::sprod}, and that we don't need to supply an
7570 @code{sprod::unarchive()} function.
7573 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7574 @c node-name, next, previous, up
7575 @section Adding classes
7577 The @code{structure<T>} template provides an way to extend GiNaC with custom
7578 algebraic classes that is easy to use but has its limitations, the most
7579 severe of which being that you can't add any new member functions to
7580 structures. To be able to do this, you need to write a new class definition
7583 This section will explain how to implement new algebraic classes in GiNaC by
7584 giving the example of a simple 'string' class. After reading this section
7585 you will know how to properly declare a GiNaC class and what the minimum
7586 required member functions are that you have to implement. We only cover the
7587 implementation of a 'leaf' class here (i.e. one that doesn't contain
7588 subexpressions). Creating a container class like, for example, a class
7589 representing tensor products is more involved but this section should give
7590 you enough information so you can consult the source to GiNaC's predefined
7591 classes if you want to implement something more complicated.
7593 @subsection GiNaC's run-time type information system
7595 @cindex hierarchy of classes
7597 All algebraic classes (that is, all classes that can appear in expressions)
7598 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7599 @code{basic *} (which is essentially what an @code{ex} is) represents a
7600 generic pointer to an algebraic class. Occasionally it is necessary to find
7601 out what the class of an object pointed to by a @code{basic *} really is.
7602 Also, for the unarchiving of expressions it must be possible to find the
7603 @code{unarchive()} function of a class given the class name (as a string). A
7604 system that provides this kind of information is called a run-time type
7605 information (RTTI) system. The C++ language provides such a thing (see the
7606 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7607 implements its own, simpler RTTI.
7609 The RTTI in GiNaC is based on two mechanisms:
7614 The @code{basic} class declares a member variable @code{tinfo_key} which
7615 holds an unsigned integer that identifies the object's class. These numbers
7616 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7617 classes. They all start with @code{TINFO_}.
7620 By means of some clever tricks with static members, GiNaC maintains a list
7621 of information for all classes derived from @code{basic}. The information
7622 available includes the class names, the @code{tinfo_key}s, and pointers
7623 to the unarchiving functions. This class registry is defined in the
7624 @file{registrar.h} header file.
7628 The disadvantage of this proprietary RTTI implementation is that there's
7629 a little more to do when implementing new classes (C++'s RTTI works more
7630 or less automatically) but don't worry, most of the work is simplified by
7633 @subsection A minimalistic example
7635 Now we will start implementing a new class @code{mystring} that allows
7636 placing character strings in algebraic expressions (this is not very useful,
7637 but it's just an example). This class will be a direct subclass of
7638 @code{basic}. You can use this sample implementation as a starting point
7639 for your own classes.
7641 The code snippets given here assume that you have included some header files
7647 #include <stdexcept>
7648 using namespace std;
7650 #include <ginac/ginac.h>
7651 using namespace GiNaC;
7654 The first thing we have to do is to define a @code{tinfo_key} for our new
7655 class. This can be any arbitrary unsigned number that is not already taken
7656 by one of the existing classes but it's better to come up with something
7657 that is unlikely to clash with keys that might be added in the future. The
7658 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7659 which is not a requirement but we are going to stick with this scheme:
7662 const unsigned TINFO_mystring = 0x42420001U;
7665 Now we can write down the class declaration. The class stores a C++
7666 @code{string} and the user shall be able to construct a @code{mystring}
7667 object from a C or C++ string:
7670 class mystring : public basic
7672 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7675 mystring(const string &s);
7676 mystring(const char *s);
7682 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7685 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7686 macros are defined in @file{registrar.h}. They take the name of the class
7687 and its direct superclass as arguments and insert all required declarations
7688 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7689 the first line after the opening brace of the class definition. The
7690 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7691 source (at global scope, of course, not inside a function).
7693 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7694 declarations of the default constructor and a couple of other functions that
7695 are required. It also defines a type @code{inherited} which refers to the
7696 superclass so you don't have to modify your code every time you shuffle around
7697 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7698 class with the GiNaC RTTI (there is also a
7699 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7700 options for the class, and which we will be using instead in a few minutes).
7702 Now there are seven member functions we have to implement to get a working
7708 @code{mystring()}, the default constructor.
7711 @code{void archive(archive_node &n)}, the archiving function. This stores all
7712 information needed to reconstruct an object of this class inside an
7713 @code{archive_node}.
7716 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7717 constructor. This constructs an instance of the class from the information
7718 found in an @code{archive_node}.
7721 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7722 unarchiving function. It constructs a new instance by calling the unarchiving
7726 @cindex @code{compare_same_type()}
7727 @code{int compare_same_type(const basic &other)}, which is used internally
7728 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7729 -1, depending on the relative order of this object and the @code{other}
7730 object. If it returns 0, the objects are considered equal.
7731 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7732 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7733 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7734 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7735 must provide a @code{compare_same_type()} function, even those representing
7736 objects for which no reasonable algebraic ordering relationship can be
7740 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7741 which are the two constructors we declared.
7745 Let's proceed step-by-step. The default constructor looks like this:
7748 mystring::mystring() : inherited(TINFO_mystring) @{@}
7751 The golden rule is that in all constructors you have to set the
7752 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7753 it will be set by the constructor of the superclass and all hell will break
7754 loose in the RTTI. For your convenience, the @code{basic} class provides
7755 a constructor that takes a @code{tinfo_key} value, which we are using here
7756 (remember that in our case @code{inherited == basic}). If the superclass
7757 didn't have such a constructor, we would have to set the @code{tinfo_key}
7758 to the right value manually.
7760 In the default constructor you should set all other member variables to
7761 reasonable default values (we don't need that here since our @code{str}
7762 member gets set to an empty string automatically).
7764 Next are the three functions for archiving. You have to implement them even
7765 if you don't plan to use archives, but the minimum required implementation
7766 is really simple. First, the archiving function:
7769 void mystring::archive(archive_node &n) const
7771 inherited::archive(n);
7772 n.add_string("string", str);
7776 The only thing that is really required is calling the @code{archive()}
7777 function of the superclass. Optionally, you can store all information you
7778 deem necessary for representing the object into the passed
7779 @code{archive_node}. We are just storing our string here. For more
7780 information on how the archiving works, consult the @file{archive.h} header
7783 The unarchiving constructor is basically the inverse of the archiving
7787 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7789 n.find_string("string", str);
7793 If you don't need archiving, just leave this function empty (but you must
7794 invoke the unarchiving constructor of the superclass). Note that we don't
7795 have to set the @code{tinfo_key} here because it is done automatically
7796 by the unarchiving constructor of the @code{basic} class.
7798 Finally, the unarchiving function:
7801 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7803 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7807 You don't have to understand how exactly this works. Just copy these
7808 four lines into your code literally (replacing the class name, of
7809 course). It calls the unarchiving constructor of the class and unless
7810 you are doing something very special (like matching @code{archive_node}s
7811 to global objects) you don't need a different implementation. For those
7812 who are interested: setting the @code{dynallocated} flag puts the object
7813 under the control of GiNaC's garbage collection. It will get deleted
7814 automatically once it is no longer referenced.
7816 Our @code{compare_same_type()} function uses a provided function to compare
7820 int mystring::compare_same_type(const basic &other) const
7822 const mystring &o = static_cast<const mystring &>(other);
7823 int cmpval = str.compare(o.str);
7826 else if (cmpval < 0)
7833 Although this function takes a @code{basic &}, it will always be a reference
7834 to an object of exactly the same class (objects of different classes are not
7835 comparable), so the cast is safe. If this function returns 0, the two objects
7836 are considered equal (in the sense that @math{A-B=0}), so you should compare
7837 all relevant member variables.
7839 Now the only thing missing is our two new constructors:
7842 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7843 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7846 No surprises here. We set the @code{str} member from the argument and
7847 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7849 That's it! We now have a minimal working GiNaC class that can store
7850 strings in algebraic expressions. Let's confirm that the RTTI works:
7853 ex e = mystring("Hello, world!");
7854 cout << is_a<mystring>(e) << endl;
7857 cout << e.bp->class_name() << endl;
7861 Obviously it does. Let's see what the expression @code{e} looks like:
7865 // -> [mystring object]
7868 Hm, not exactly what we expect, but of course the @code{mystring} class
7869 doesn't yet know how to print itself. This can be done either by implementing
7870 the @code{print()} member function, or, preferably, by specifying a
7871 @code{print_func<>()} class option. Let's say that we want to print the string
7872 surrounded by double quotes:
7875 class mystring : public basic
7879 void do_print(const print_context &c, unsigned level = 0) const;
7883 void mystring::do_print(const print_context &c, unsigned level) const
7885 // print_context::s is a reference to an ostream
7886 c.s << '\"' << str << '\"';
7890 The @code{level} argument is only required for container classes to
7891 correctly parenthesize the output.
7893 Now we need to tell GiNaC that @code{mystring} objects should use the
7894 @code{do_print()} member function for printing themselves. For this, we
7898 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7904 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7905 print_func<print_context>(&mystring::do_print))
7908 Let's try again to print the expression:
7912 // -> "Hello, world!"
7915 Much better. If we wanted to have @code{mystring} objects displayed in a
7916 different way depending on the output format (default, LaTeX, etc.), we
7917 would have supplied multiple @code{print_func<>()} options with different
7918 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7919 separated by dots. This is similar to the way options are specified for
7920 symbolic functions. @xref{Printing}, for a more in-depth description of the
7921 way expression output is implemented in GiNaC.
7923 The @code{mystring} class can be used in arbitrary expressions:
7926 e += mystring("GiNaC rulez");
7928 // -> "GiNaC rulez"+"Hello, world!"
7931 (GiNaC's automatic term reordering is in effect here), or even
7934 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7936 // -> "One string"^(2*sin(-"Another string"+Pi))
7939 Whether this makes sense is debatable but remember that this is only an
7940 example. At least it allows you to implement your own symbolic algorithms
7943 Note that GiNaC's algebraic rules remain unchanged:
7946 e = mystring("Wow") * mystring("Wow");
7950 e = pow(mystring("First")-mystring("Second"), 2);
7951 cout << e.expand() << endl;
7952 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7955 There's no way to, for example, make GiNaC's @code{add} class perform string
7956 concatenation. You would have to implement this yourself.
7958 @subsection Automatic evaluation
7961 @cindex @code{eval()}
7962 @cindex @code{hold()}
7963 When dealing with objects that are just a little more complicated than the
7964 simple string objects we have implemented, chances are that you will want to
7965 have some automatic simplifications or canonicalizations performed on them.
7966 This is done in the evaluation member function @code{eval()}. Let's say that
7967 we wanted all strings automatically converted to lowercase with
7968 non-alphabetic characters stripped, and empty strings removed:
7971 class mystring : public basic
7975 ex eval(int level = 0) const;
7979 ex mystring::eval(int level) const
7982 for (int i=0; i<str.length(); i++) @{
7984 if (c >= 'A' && c <= 'Z')
7985 new_str += tolower(c);
7986 else if (c >= 'a' && c <= 'z')
7990 if (new_str.length() == 0)
7993 return mystring(new_str).hold();
7997 The @code{level} argument is used to limit the recursion depth of the
7998 evaluation. We don't have any subexpressions in the @code{mystring}
7999 class so we are not concerned with this. If we had, we would call the
8000 @code{eval()} functions of the subexpressions with @code{level - 1} as
8001 the argument if @code{level != 1}. The @code{hold()} member function
8002 sets a flag in the object that prevents further evaluation. Otherwise
8003 we might end up in an endless loop. When you want to return the object
8004 unmodified, use @code{return this->hold();}.
8006 Let's confirm that it works:
8009 ex e = mystring("Hello, world!") + mystring("!?#");
8013 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8018 @subsection Optional member functions
8020 We have implemented only a small set of member functions to make the class
8021 work in the GiNaC framework. There are two functions that are not strictly
8022 required but will make operations with objects of the class more efficient:
8024 @cindex @code{calchash()}
8025 @cindex @code{is_equal_same_type()}
8027 unsigned calchash() const;
8028 bool is_equal_same_type(const basic &other) const;
8031 The @code{calchash()} method returns an @code{unsigned} hash value for the
8032 object which will allow GiNaC to compare and canonicalize expressions much
8033 more efficiently. You should consult the implementation of some of the built-in
8034 GiNaC classes for examples of hash functions. The default implementation of
8035 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8036 class and all subexpressions that are accessible via @code{op()}.
8038 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8039 tests for equality without establishing an ordering relation, which is often
8040 faster. The default implementation of @code{is_equal_same_type()} just calls
8041 @code{compare_same_type()} and tests its result for zero.
8043 @subsection Other member functions
8045 For a real algebraic class, there are probably some more functions that you
8046 might want to provide:
8049 bool info(unsigned inf) const;
8050 ex evalf(int level = 0) const;
8051 ex series(const relational & r, int order, unsigned options = 0) const;
8052 ex derivative(const symbol & s) const;
8055 If your class stores sub-expressions (see the scalar product example in the
8056 previous section) you will probably want to override
8058 @cindex @code{let_op()}
8061 ex op(size_t i) const;
8062 ex & let_op(size_t i);
8063 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8064 ex map(map_function & f) const;
8067 @code{let_op()} is a variant of @code{op()} that allows write access. The
8068 default implementations of @code{subs()} and @code{map()} use it, so you have
8069 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8071 You can, of course, also add your own new member functions. Remember
8072 that the RTTI may be used to get information about what kinds of objects
8073 you are dealing with (the position in the class hierarchy) and that you
8074 can always extract the bare object from an @code{ex} by stripping the
8075 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8076 should become a need.
8078 That's it. May the source be with you!
8081 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8082 @c node-name, next, previous, up
8083 @chapter A Comparison With Other CAS
8086 This chapter will give you some information on how GiNaC compares to
8087 other, traditional Computer Algebra Systems, like @emph{Maple},
8088 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8089 disadvantages over these systems.
8092 * Advantages:: Strengths of the GiNaC approach.
8093 * Disadvantages:: Weaknesses of the GiNaC approach.
8094 * Why C++?:: Attractiveness of C++.
8097 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8098 @c node-name, next, previous, up
8101 GiNaC has several advantages over traditional Computer
8102 Algebra Systems, like
8107 familiar language: all common CAS implement their own proprietary
8108 grammar which you have to learn first (and maybe learn again when your
8109 vendor decides to `enhance' it). With GiNaC you can write your program
8110 in common C++, which is standardized.
8114 structured data types: you can build up structured data types using
8115 @code{struct}s or @code{class}es together with STL features instead of
8116 using unnamed lists of lists of lists.
8119 strongly typed: in CAS, you usually have only one kind of variables
8120 which can hold contents of an arbitrary type. This 4GL like feature is
8121 nice for novice programmers, but dangerous.
8124 development tools: powerful development tools exist for C++, like fancy
8125 editors (e.g. with automatic indentation and syntax highlighting),
8126 debuggers, visualization tools, documentation generators@dots{}
8129 modularization: C++ programs can easily be split into modules by
8130 separating interface and implementation.
8133 price: GiNaC is distributed under the GNU Public License which means
8134 that it is free and available with source code. And there are excellent
8135 C++-compilers for free, too.
8138 extendable: you can add your own classes to GiNaC, thus extending it on
8139 a very low level. Compare this to a traditional CAS that you can
8140 usually only extend on a high level by writing in the language defined
8141 by the parser. In particular, it turns out to be almost impossible to
8142 fix bugs in a traditional system.
8145 multiple interfaces: Though real GiNaC programs have to be written in
8146 some editor, then be compiled, linked and executed, there are more ways
8147 to work with the GiNaC engine. Many people want to play with
8148 expressions interactively, as in traditional CASs. Currently, two such
8149 windows into GiNaC have been implemented and many more are possible: the
8150 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8151 types to a command line and second, as a more consistent approach, an
8152 interactive interface to the Cint C++ interpreter has been put together
8153 (called GiNaC-cint) that allows an interactive scripting interface
8154 consistent with the C++ language. It is available from the usual GiNaC
8158 seamless integration: it is somewhere between difficult and impossible
8159 to call CAS functions from within a program written in C++ or any other
8160 programming language and vice versa. With GiNaC, your symbolic routines
8161 are part of your program. You can easily call third party libraries,
8162 e.g. for numerical evaluation or graphical interaction. All other
8163 approaches are much more cumbersome: they range from simply ignoring the
8164 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8165 system (i.e. @emph{Yacas}).
8168 efficiency: often large parts of a program do not need symbolic
8169 calculations at all. Why use large integers for loop variables or
8170 arbitrary precision arithmetics where @code{int} and @code{double} are
8171 sufficient? For pure symbolic applications, GiNaC is comparable in
8172 speed with other CAS.
8177 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8178 @c node-name, next, previous, up
8179 @section Disadvantages
8181 Of course it also has some disadvantages:
8186 advanced features: GiNaC cannot compete with a program like
8187 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8188 which grows since 1981 by the work of dozens of programmers, with
8189 respect to mathematical features. Integration, factorization,
8190 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8191 not planned for the near future).
8194 portability: While the GiNaC library itself is designed to avoid any
8195 platform dependent features (it should compile on any ANSI compliant C++
8196 compiler), the currently used version of the CLN library (fast large
8197 integer and arbitrary precision arithmetics) can only by compiled
8198 without hassle on systems with the C++ compiler from the GNU Compiler
8199 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8200 macros to let the compiler gather all static initializations, which
8201 works for GNU C++ only. Feel free to contact the authors in case you
8202 really believe that you need to use a different compiler. We have
8203 occasionally used other compilers and may be able to give you advice.}
8204 GiNaC uses recent language features like explicit constructors, mutable
8205 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8206 literally. Recent GCC versions starting at 2.95.3, although itself not
8207 yet ANSI compliant, support all needed features.
8212 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8213 @c node-name, next, previous, up
8216 Why did we choose to implement GiNaC in C++ instead of Java or any other
8217 language? C++ is not perfect: type checking is not strict (casting is
8218 possible), separation between interface and implementation is not
8219 complete, object oriented design is not enforced. The main reason is
8220 the often scolded feature of operator overloading in C++. While it may
8221 be true that operating on classes with a @code{+} operator is rarely
8222 meaningful, it is perfectly suited for algebraic expressions. Writing
8223 @math{3x+5y} as @code{3*x+5*y} instead of
8224 @code{x.times(3).plus(y.times(5))} looks much more natural.
8225 Furthermore, the main developers are more familiar with C++ than with
8226 any other programming language.
8229 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8230 @c node-name, next, previous, up
8231 @appendix Internal Structures
8234 * Expressions are reference counted::
8235 * Internal representation of products and sums::
8238 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8239 @c node-name, next, previous, up
8240 @appendixsection Expressions are reference counted
8242 @cindex reference counting
8243 @cindex copy-on-write
8244 @cindex garbage collection
8245 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8246 where the counter belongs to the algebraic objects derived from class
8247 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8248 which @code{ex} contains an instance. If you understood that, you can safely
8249 skip the rest of this passage.
8251 Expressions are extremely light-weight since internally they work like
8252 handles to the actual representation. They really hold nothing more
8253 than a pointer to some other object. What this means in practice is
8254 that whenever you create two @code{ex} and set the second equal to the
8255 first no copying process is involved. Instead, the copying takes place
8256 as soon as you try to change the second. Consider the simple sequence
8261 #include <ginac/ginac.h>
8262 using namespace std;
8263 using namespace GiNaC;
8267 symbol x("x"), y("y"), z("z");
8270 e1 = sin(x + 2*y) + 3*z + 41;
8271 e2 = e1; // e2 points to same object as e1
8272 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8273 e2 += 1; // e2 is copied into a new object
8274 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8278 The line @code{e2 = e1;} creates a second expression pointing to the
8279 object held already by @code{e1}. The time involved for this operation
8280 is therefore constant, no matter how large @code{e1} was. Actual
8281 copying, however, must take place in the line @code{e2 += 1;} because
8282 @code{e1} and @code{e2} are not handles for the same object any more.
8283 This concept is called @dfn{copy-on-write semantics}. It increases
8284 performance considerably whenever one object occurs multiple times and
8285 represents a simple garbage collection scheme because when an @code{ex}
8286 runs out of scope its destructor checks whether other expressions handle
8287 the object it points to too and deletes the object from memory if that
8288 turns out not to be the case. A slightly less trivial example of
8289 differentiation using the chain-rule should make clear how powerful this
8294 symbol x("x"), y("y");
8298 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8299 cout << e1 << endl // prints x+3*y
8300 << e2 << endl // prints (x+3*y)^3
8301 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8305 Here, @code{e1} will actually be referenced three times while @code{e2}
8306 will be referenced two times. When the power of an expression is built,
8307 that expression needs not be copied. Likewise, since the derivative of
8308 a power of an expression can be easily expressed in terms of that
8309 expression, no copying of @code{e1} is involved when @code{e3} is
8310 constructed. So, when @code{e3} is constructed it will print as
8311 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8312 holds a reference to @code{e2} and the factor in front is just
8315 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8316 semantics. When you insert an expression into a second expression, the
8317 result behaves exactly as if the contents of the first expression were
8318 inserted. But it may be useful to remember that this is not what
8319 happens. Knowing this will enable you to write much more efficient
8320 code. If you still have an uncertain feeling with copy-on-write
8321 semantics, we recommend you have a look at the
8322 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8323 Marshall Cline. Chapter 16 covers this issue and presents an
8324 implementation which is pretty close to the one in GiNaC.
8327 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8328 @c node-name, next, previous, up
8329 @appendixsection Internal representation of products and sums
8331 @cindex representation
8334 @cindex @code{power}
8335 Although it should be completely transparent for the user of
8336 GiNaC a short discussion of this topic helps to understand the sources
8337 and also explain performance to a large degree. Consider the
8338 unexpanded symbolic expression
8340 $2d^3 \left( 4a + 5b - 3 \right)$
8343 @math{2*d^3*(4*a+5*b-3)}
8345 which could naively be represented by a tree of linear containers for
8346 addition and multiplication, one container for exponentiation with base
8347 and exponent and some atomic leaves of symbols and numbers in this
8352 @cindex pair-wise representation
8353 However, doing so results in a rather deeply nested tree which will
8354 quickly become inefficient to manipulate. We can improve on this by
8355 representing the sum as a sequence of terms, each one being a pair of a
8356 purely numeric multiplicative coefficient and its rest. In the same
8357 spirit we can store the multiplication as a sequence of terms, each
8358 having a numeric exponent and a possibly complicated base, the tree
8359 becomes much more flat:
8363 The number @code{3} above the symbol @code{d} shows that @code{mul}
8364 objects are treated similarly where the coefficients are interpreted as
8365 @emph{exponents} now. Addition of sums of terms or multiplication of
8366 products with numerical exponents can be coded to be very efficient with
8367 such a pair-wise representation. Internally, this handling is performed
8368 by most CAS in this way. It typically speeds up manipulations by an
8369 order of magnitude. The overall multiplicative factor @code{2} and the
8370 additive term @code{-3} look somewhat out of place in this
8371 representation, however, since they are still carrying a trivial
8372 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8373 this is avoided by adding a field that carries an overall numeric
8374 coefficient. This results in the realistic picture of internal
8377 $2d^3 \left( 4a + 5b - 3 \right)$:
8380 @math{2*d^3*(4*a+5*b-3)}:
8386 This also allows for a better handling of numeric radicals, since
8387 @code{sqrt(2)} can now be carried along calculations. Now it should be
8388 clear, why both classes @code{add} and @code{mul} are derived from the
8389 same abstract class: the data representation is the same, only the
8390 semantics differs. In the class hierarchy, methods for polynomial
8391 expansion and the like are reimplemented for @code{add} and @code{mul},
8392 but the data structure is inherited from @code{expairseq}.
8395 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8396 @c node-name, next, previous, up
8397 @appendix Package Tools
8399 If you are creating a software package that uses the GiNaC library,
8400 setting the correct command line options for the compiler and linker
8401 can be difficult. GiNaC includes two tools to make this process easier.
8404 * ginac-config:: A shell script to detect compiler and linker flags.
8405 * AM_PATH_GINAC:: Macro for GNU automake.
8409 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8410 @c node-name, next, previous, up
8411 @section @command{ginac-config}
8412 @cindex ginac-config
8414 @command{ginac-config} is a shell script that you can use to determine
8415 the compiler and linker command line options required to compile and
8416 link a program with the GiNaC library.
8418 @command{ginac-config} takes the following flags:
8422 Prints out the version of GiNaC installed.
8424 Prints '-I' flags pointing to the installed header files.
8426 Prints out the linker flags necessary to link a program against GiNaC.
8427 @item --prefix[=@var{PREFIX}]
8428 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8429 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8430 Otherwise, prints out the configured value of @env{$prefix}.
8431 @item --exec-prefix[=@var{PREFIX}]
8432 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8433 Otherwise, prints out the configured value of @env{$exec_prefix}.
8436 Typically, @command{ginac-config} will be used within a configure
8437 script, as described below. It, however, can also be used directly from
8438 the command line using backquotes to compile a simple program. For
8442 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8445 This command line might expand to (for example):
8448 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8449 -lginac -lcln -lstdc++
8452 Not only is the form using @command{ginac-config} easier to type, it will
8453 work on any system, no matter how GiNaC was configured.
8456 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8457 @c node-name, next, previous, up
8458 @section @samp{AM_PATH_GINAC}
8459 @cindex AM_PATH_GINAC
8461 For packages configured using GNU automake, GiNaC also provides
8462 a macro to automate the process of checking for GiNaC.
8465 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8466 [, @var{ACTION-IF-NOT-FOUND}]]])
8474 Determines the location of GiNaC using @command{ginac-config}, which is
8475 either found in the user's path, or from the environment variable
8476 @env{GINACLIB_CONFIG}.
8479 Tests the installed libraries to make sure that their version
8480 is later than @var{MINIMUM-VERSION}. (A default version will be used
8484 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8485 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8486 variable to the output of @command{ginac-config --libs}, and calls
8487 @samp{AC_SUBST()} for these variables so they can be used in generated
8488 makefiles, and then executes @var{ACTION-IF-FOUND}.
8491 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8492 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8496 This macro is in file @file{ginac.m4} which is installed in
8497 @file{$datadir/aclocal}. Note that if automake was installed with a
8498 different @samp{--prefix} than GiNaC, you will either have to manually
8499 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8500 aclocal the @samp{-I} option when running it.
8503 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8504 * Example package:: Example of a package using AM_PATH_GINAC.
8508 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8509 @c node-name, next, previous, up
8510 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8512 Simply make sure that @command{ginac-config} is in your path, and run
8513 the configure script.
8520 The directory where the GiNaC libraries are installed needs
8521 to be found by your system's dynamic linker.
8523 This is generally done by
8526 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8532 setting the environment variable @env{LD_LIBRARY_PATH},
8535 or, as a last resort,
8538 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8539 running configure, for instance:
8542 LDFLAGS=-R/home/cbauer/lib ./configure
8547 You can also specify a @command{ginac-config} not in your path by
8548 setting the @env{GINACLIB_CONFIG} environment variable to the
8549 name of the executable
8552 If you move the GiNaC package from its installed location,
8553 you will either need to modify @command{ginac-config} script
8554 manually to point to the new location or rebuild GiNaC.
8565 --with-ginac-prefix=@var{PREFIX}
8566 --with-ginac-exec-prefix=@var{PREFIX}
8569 are provided to override the prefix and exec-prefix that were stored
8570 in the @command{ginac-config} shell script by GiNaC's configure. You are
8571 generally better off configuring GiNaC with the right path to begin with.
8575 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8576 @c node-name, next, previous, up
8577 @subsection Example of a package using @samp{AM_PATH_GINAC}
8579 The following shows how to build a simple package using automake
8580 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8584 #include <ginac/ginac.h>
8588 GiNaC::symbol x("x");
8589 GiNaC::ex a = GiNaC::sin(x);
8590 std::cout << "Derivative of " << a
8591 << " is " << a.diff(x) << std::endl;
8596 You should first read the introductory portions of the automake
8597 Manual, if you are not already familiar with it.
8599 Two files are needed, @file{configure.in}, which is used to build the
8603 dnl Process this file with autoconf to produce a configure script.
8605 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8611 AM_PATH_GINAC(0.9.0, [
8612 LIBS="$LIBS $GINACLIB_LIBS"
8613 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8614 ], AC_MSG_ERROR([need to have GiNaC installed]))
8619 The only command in this which is not standard for automake
8620 is the @samp{AM_PATH_GINAC} macro.
8622 That command does the following: If a GiNaC version greater or equal
8623 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8624 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8625 the error message `need to have GiNaC installed'
8627 And the @file{Makefile.am}, which will be used to build the Makefile.
8630 ## Process this file with automake to produce Makefile.in
8631 bin_PROGRAMS = simple
8632 simple_SOURCES = simple.cpp
8635 This @file{Makefile.am}, says that we are building a single executable,
8636 from a single source file @file{simple.cpp}. Since every program
8637 we are building uses GiNaC we simply added the GiNaC options
8638 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8639 want to specify them on a per-program basis: for instance by
8643 simple_LDADD = $(GINACLIB_LIBS)
8644 INCLUDES = $(GINACLIB_CPPFLAGS)
8647 to the @file{Makefile.am}.
8649 To try this example out, create a new directory and add the three
8652 Now execute the following commands:
8655 $ automake --add-missing
8660 You now have a package that can be built in the normal fashion
8669 @node Bibliography, Concept Index, Example package, Top
8670 @c node-name, next, previous, up
8671 @appendix Bibliography
8676 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8679 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8682 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8685 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8688 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8689 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8692 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8693 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8694 Academic Press, London
8697 @cite{Computer Algebra Systems - A Practical Guide},
8698 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8701 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8702 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8705 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8706 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8709 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8714 @node Concept Index, , Bibliography, Top
8715 @c node-name, next, previous, up
8716 @unnumbered Concept Index