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 * Solving Linear Systems of Equations::
3812 * Input/Output:: Input and output of expressions.
3816 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3817 @c node-name, next, previous, up
3818 @section Getting information about expressions
3820 @subsection Checking expression types
3821 @cindex @code{is_a<@dots{}>()}
3822 @cindex @code{is_exactly_a<@dots{}>()}
3823 @cindex @code{ex_to<@dots{}>()}
3824 @cindex Converting @code{ex} to other classes
3825 @cindex @code{info()}
3826 @cindex @code{return_type()}
3827 @cindex @code{return_type_tinfo()}
3829 Sometimes it's useful to check whether a given expression is a plain number,
3830 a sum, a polynomial with integer coefficients, or of some other specific type.
3831 GiNaC provides a couple of functions for this:
3834 bool is_a<T>(const ex & e);
3835 bool is_exactly_a<T>(const ex & e);
3836 bool ex::info(unsigned flag);
3837 unsigned ex::return_type() const;
3838 unsigned ex::return_type_tinfo() const;
3841 When the test made by @code{is_a<T>()} returns true, it is safe to call
3842 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3843 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3844 example, assuming @code{e} is an @code{ex}:
3849 if (is_a<numeric>(e))
3850 numeric n = ex_to<numeric>(e);
3855 @code{is_a<T>(e)} allows you to check whether the top-level object of
3856 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3857 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3858 e.g., for checking whether an expression is a number, a sum, or a product:
3865 is_a<numeric>(e1); // true
3866 is_a<numeric>(e2); // false
3867 is_a<add>(e1); // false
3868 is_a<add>(e2); // true
3869 is_a<mul>(e1); // false
3870 is_a<mul>(e2); // false
3874 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3875 top-level object of an expression @samp{e} is an instance of the GiNaC
3876 class @samp{T}, not including parent classes.
3878 The @code{info()} method is used for checking certain attributes of
3879 expressions. The possible values for the @code{flag} argument are defined
3880 in @file{ginac/flags.h}, the most important being explained in the following
3884 @multitable @columnfractions .30 .70
3885 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3886 @item @code{numeric}
3887 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3889 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3890 @item @code{rational}
3891 @tab @dots{}an exact rational number (integers are rational, too)
3892 @item @code{integer}
3893 @tab @dots{}a (non-complex) integer
3894 @item @code{crational}
3895 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3896 @item @code{cinteger}
3897 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3898 @item @code{positive}
3899 @tab @dots{}not complex and greater than 0
3900 @item @code{negative}
3901 @tab @dots{}not complex and less than 0
3902 @item @code{nonnegative}
3903 @tab @dots{}not complex and greater than or equal to 0
3905 @tab @dots{}an integer greater than 0
3907 @tab @dots{}an integer less than 0
3908 @item @code{nonnegint}
3909 @tab @dots{}an integer greater than or equal to 0
3911 @tab @dots{}an even integer
3913 @tab @dots{}an odd integer
3915 @tab @dots{}a prime integer (probabilistic primality test)
3916 @item @code{relation}
3917 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3918 @item @code{relation_equal}
3919 @tab @dots{}a @code{==} relation
3920 @item @code{relation_not_equal}
3921 @tab @dots{}a @code{!=} relation
3922 @item @code{relation_less}
3923 @tab @dots{}a @code{<} relation
3924 @item @code{relation_less_or_equal}
3925 @tab @dots{}a @code{<=} relation
3926 @item @code{relation_greater}
3927 @tab @dots{}a @code{>} relation
3928 @item @code{relation_greater_or_equal}
3929 @tab @dots{}a @code{>=} relation
3931 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3933 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3934 @item @code{polynomial}
3935 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3936 @item @code{integer_polynomial}
3937 @tab @dots{}a polynomial with (non-complex) integer coefficients
3938 @item @code{cinteger_polynomial}
3939 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3940 @item @code{rational_polynomial}
3941 @tab @dots{}a polynomial with (non-complex) rational coefficients
3942 @item @code{crational_polynomial}
3943 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3944 @item @code{rational_function}
3945 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3946 @item @code{algebraic}
3947 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3951 To determine whether an expression is commutative or non-commutative and if
3952 so, with which other expressions it would commutate, you use the methods
3953 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3954 for an explanation of these.
3957 @subsection Accessing subexpressions
3960 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3961 @code{function}, act as containers for subexpressions. For example, the
3962 subexpressions of a sum (an @code{add} object) are the individual terms,
3963 and the subexpressions of a @code{function} are the function's arguments.
3965 @cindex @code{nops()}
3967 GiNaC provides several ways of accessing subexpressions. The first way is to
3972 ex ex::op(size_t i);
3975 @code{nops()} determines the number of subexpressions (operands) contained
3976 in the expression, while @code{op(i)} returns the @code{i}-th
3977 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3978 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3979 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3980 @math{i>0} are the indices.
3983 @cindex @code{const_iterator}
3984 The second way to access subexpressions is via the STL-style random-access
3985 iterator class @code{const_iterator} and the methods
3988 const_iterator ex::begin();
3989 const_iterator ex::end();
3992 @code{begin()} returns an iterator referring to the first subexpression;
3993 @code{end()} returns an iterator which is one-past the last subexpression.
3994 If the expression has no subexpressions, then @code{begin() == end()}. These
3995 iterators can also be used in conjunction with non-modifying STL algorithms.
3997 Here is an example that (non-recursively) prints the subexpressions of a
3998 given expression in three different ways:
4005 for (size_t i = 0; i != e.nops(); ++i)
4006 cout << e.op(i) << endl;
4009 for (const_iterator i = e.begin(); i != e.end(); ++i)
4012 // with iterators and STL copy()
4013 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4017 @cindex @code{const_preorder_iterator}
4018 @cindex @code{const_postorder_iterator}
4019 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4020 expression's immediate children. GiNaC provides two additional iterator
4021 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4022 that iterate over all objects in an expression tree, in preorder or postorder,
4023 respectively. They are STL-style forward iterators, and are created with the
4027 const_preorder_iterator ex::preorder_begin();
4028 const_preorder_iterator ex::preorder_end();
4029 const_postorder_iterator ex::postorder_begin();
4030 const_postorder_iterator ex::postorder_end();
4033 The following example illustrates the differences between
4034 @code{const_iterator}, @code{const_preorder_iterator}, and
4035 @code{const_postorder_iterator}:
4039 symbol A("A"), B("B"), C("C");
4040 ex e = lst(lst(A, B), C);
4042 std::copy(e.begin(), e.end(),
4043 std::ostream_iterator<ex>(cout, "\n"));
4047 std::copy(e.preorder_begin(), e.preorder_end(),
4048 std::ostream_iterator<ex>(cout, "\n"));
4055 std::copy(e.postorder_begin(), e.postorder_end(),
4056 std::ostream_iterator<ex>(cout, "\n"));
4065 @cindex @code{relational} (class)
4066 Finally, the left-hand side and right-hand side expressions of objects of
4067 class @code{relational} (and only of these) can also be accessed with the
4076 @subsection Comparing expressions
4077 @cindex @code{is_equal()}
4078 @cindex @code{is_zero()}
4080 Expressions can be compared with the usual C++ relational operators like
4081 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4082 the result is usually not determinable and the result will be @code{false},
4083 except in the case of the @code{!=} operator. You should also be aware that
4084 GiNaC will only do the most trivial test for equality (subtracting both
4085 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4088 Actually, if you construct an expression like @code{a == b}, this will be
4089 represented by an object of the @code{relational} class (@pxref{Relations})
4090 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4092 There are also two methods
4095 bool ex::is_equal(const ex & other);
4099 for checking whether one expression is equal to another, or equal to zero,
4103 @subsection Ordering expressions
4104 @cindex @code{ex_is_less} (class)
4105 @cindex @code{ex_is_equal} (class)
4106 @cindex @code{compare()}
4108 Sometimes it is necessary to establish a mathematically well-defined ordering
4109 on a set of arbitrary expressions, for example to use expressions as keys
4110 in a @code{std::map<>} container, or to bring a vector of expressions into
4111 a canonical order (which is done internally by GiNaC for sums and products).
4113 The operators @code{<}, @code{>} etc. described in the last section cannot
4114 be used for this, as they don't implement an ordering relation in the
4115 mathematical sense. In particular, they are not guaranteed to be
4116 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4117 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4120 By default, STL classes and algorithms use the @code{<} and @code{==}
4121 operators to compare objects, which are unsuitable for expressions, but GiNaC
4122 provides two functors that can be supplied as proper binary comparison
4123 predicates to the STL:
4126 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4128 bool operator()(const ex &lh, const ex &rh) const;
4131 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4133 bool operator()(const ex &lh, const ex &rh) const;
4137 For example, to define a @code{map} that maps expressions to strings you
4141 std::map<ex, std::string, ex_is_less> myMap;
4144 Omitting the @code{ex_is_less} template parameter will introduce spurious
4145 bugs because the map operates improperly.
4147 Other examples for the use of the functors:
4155 std::sort(v.begin(), v.end(), ex_is_less());
4157 // count the number of expressions equal to '1'
4158 unsigned num_ones = std::count_if(v.begin(), v.end(),
4159 std::bind2nd(ex_is_equal(), 1));
4162 The implementation of @code{ex_is_less} uses the member function
4165 int ex::compare(const ex & other) const;
4168 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4169 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4173 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4174 @c node-name, next, previous, up
4175 @section Numerical Evaluation
4176 @cindex @code{evalf()}
4178 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4179 To evaluate them using floating-point arithmetic you need to call
4182 ex ex::evalf(int level = 0) const;
4185 @cindex @code{Digits}
4186 The accuracy of the evaluation is controlled by the global object @code{Digits}
4187 which can be assigned an integer value. The default value of @code{Digits}
4188 is 17. @xref{Numbers}, for more information and examples.
4190 To evaluate an expression to a @code{double} floating-point number you can
4191 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4195 // Approximate sin(x/Pi)
4197 ex e = series(sin(x/Pi), x == 0, 6);
4199 // Evaluate numerically at x=0.1
4200 ex f = evalf(e.subs(x == 0.1));
4202 // ex_to<numeric> is an unsafe cast, so check the type first
4203 if (is_a<numeric>(f)) @{
4204 double d = ex_to<numeric>(f).to_double();
4213 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4214 @c node-name, next, previous, up
4215 @section Substituting expressions
4216 @cindex @code{subs()}
4218 Algebraic objects inside expressions can be replaced with arbitrary
4219 expressions via the @code{.subs()} method:
4222 ex ex::subs(const ex & e, unsigned options = 0);
4223 ex ex::subs(const exmap & m, unsigned options = 0);
4224 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4227 In the first form, @code{subs()} accepts a relational of the form
4228 @samp{object == expression} or a @code{lst} of such relationals:
4232 symbol x("x"), y("y");
4234 ex e1 = 2*x^2-4*x+3;
4235 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4239 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4244 If you specify multiple substitutions, they are performed in parallel, so e.g.
4245 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4247 The second form of @code{subs()} takes an @code{exmap} object which is a
4248 pair associative container that maps expressions to expressions (currently
4249 implemented as a @code{std::map}). This is the most efficient one of the
4250 three @code{subs()} forms and should be used when the number of objects to
4251 be substituted is large or unknown.
4253 Using this form, the second example from above would look like this:
4257 symbol x("x"), y("y");
4263 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4267 The third form of @code{subs()} takes two lists, one for the objects to be
4268 replaced and one for the expressions to be substituted (both lists must
4269 contain the same number of elements). Using this form, you would write
4273 symbol x("x"), y("y");
4276 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4280 The optional last argument to @code{subs()} is a combination of
4281 @code{subs_options} flags. There are three options available:
4282 @code{subs_options::no_pattern} disables pattern matching, which makes
4283 large @code{subs()} operations significantly faster if you are not using
4284 patterns. The second option, @code{subs_options::algebraic} enables
4285 algebraic substitutions in products and powers.
4286 @ref{Pattern Matching and Advanced Substitutions}, for more information
4287 about patterns and algebraic substitutions. The third option,
4288 @code{subs_options::no_index_renaming} disables the feature that dummy
4289 indices are renamed if the subsitution could give a result in which a
4290 dummy index occurs more than two times. This is sometimes necessary if
4291 you want to use @code{subs()} to rename your dummy indices.
4293 @code{subs()} performs syntactic substitution of any complete algebraic
4294 object; it does not try to match sub-expressions as is demonstrated by the
4299 symbol x("x"), y("y"), z("z");
4301 ex e1 = pow(x+y, 2);
4302 cout << e1.subs(x+y == 4) << endl;
4305 ex e2 = sin(x)*sin(y)*cos(x);
4306 cout << e2.subs(sin(x) == cos(x)) << endl;
4307 // -> cos(x)^2*sin(y)
4310 cout << e3.subs(x+y == 4) << endl;
4312 // (and not 4+z as one might expect)
4316 A more powerful form of substitution using wildcards is described in the
4320 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4321 @c node-name, next, previous, up
4322 @section Pattern matching and advanced substitutions
4323 @cindex @code{wildcard} (class)
4324 @cindex Pattern matching
4326 GiNaC allows the use of patterns for checking whether an expression is of a
4327 certain form or contains subexpressions of a certain form, and for
4328 substituting expressions in a more general way.
4330 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4331 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4332 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4333 an unsigned integer number to allow having multiple different wildcards in a
4334 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4335 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4339 ex wild(unsigned label = 0);
4342 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4345 Some examples for patterns:
4347 @multitable @columnfractions .5 .5
4348 @item @strong{Constructed as} @tab @strong{Output as}
4349 @item @code{wild()} @tab @samp{$0}
4350 @item @code{pow(x,wild())} @tab @samp{x^$0}
4351 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4352 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4358 @item Wildcards behave like symbols and are subject to the same algebraic
4359 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4360 @item As shown in the last example, to use wildcards for indices you have to
4361 use them as the value of an @code{idx} object. This is because indices must
4362 always be of class @code{idx} (or a subclass).
4363 @item Wildcards only represent expressions or subexpressions. It is not
4364 possible to use them as placeholders for other properties like index
4365 dimension or variance, representation labels, symmetry of indexed objects
4367 @item Because wildcards are commutative, it is not possible to use wildcards
4368 as part of noncommutative products.
4369 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4370 are also valid patterns.
4373 @subsection Matching expressions
4374 @cindex @code{match()}
4375 The most basic application of patterns is to check whether an expression
4376 matches a given pattern. This is done by the function
4379 bool ex::match(const ex & pattern);
4380 bool ex::match(const ex & pattern, lst & repls);
4383 This function returns @code{true} when the expression matches the pattern
4384 and @code{false} if it doesn't. If used in the second form, the actual
4385 subexpressions matched by the wildcards get returned in the @code{repls}
4386 object as a list of relations of the form @samp{wildcard == expression}.
4387 If @code{match()} returns false, the state of @code{repls} is undefined.
4388 For reproducible results, the list should be empty when passed to
4389 @code{match()}, but it is also possible to find similarities in multiple
4390 expressions by passing in the result of a previous match.
4392 The matching algorithm works as follows:
4395 @item A single wildcard matches any expression. If one wildcard appears
4396 multiple times in a pattern, it must match the same expression in all
4397 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4398 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4399 @item If the expression is not of the same class as the pattern, the match
4400 fails (i.e. a sum only matches a sum, a function only matches a function,
4402 @item If the pattern is a function, it only matches the same function
4403 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4404 @item Except for sums and products, the match fails if the number of
4405 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4407 @item If there are no subexpressions, the expressions and the pattern must
4408 be equal (in the sense of @code{is_equal()}).
4409 @item Except for sums and products, each subexpression (@code{op()}) must
4410 match the corresponding subexpression of the pattern.
4413 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4414 account for their commutativity and associativity:
4417 @item If the pattern contains a term or factor that is a single wildcard,
4418 this one is used as the @dfn{global wildcard}. If there is more than one
4419 such wildcard, one of them is chosen as the global wildcard in a random
4421 @item Every term/factor of the pattern, except the global wildcard, is
4422 matched against every term of the expression in sequence. If no match is
4423 found, the whole match fails. Terms that did match are not considered in
4425 @item If there are no unmatched terms left, the match succeeds. Otherwise
4426 the match fails unless there is a global wildcard in the pattern, in
4427 which case this wildcard matches the remaining terms.
4430 In general, having more than one single wildcard as a term of a sum or a
4431 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4434 Here are some examples in @command{ginsh} to demonstrate how it works (the
4435 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4436 match fails, and the list of wildcard replacements otherwise):
4439 > match((x+y)^a,(x+y)^a);
4441 > match((x+y)^a,(x+y)^b);
4443 > match((x+y)^a,$1^$2);
4445 > match((x+y)^a,$1^$1);
4447 > match((x+y)^(x+y),$1^$1);
4449 > match((x+y)^(x+y),$1^$2);
4451 > match((a+b)*(a+c),($1+b)*($1+c));
4453 > match((a+b)*(a+c),(a+$1)*(a+$2));
4455 (Unpredictable. The result might also be [$1==c,$2==b].)
4456 > match((a+b)*(a+c),($1+$2)*($1+$3));
4457 (The result is undefined. Due to the sequential nature of the algorithm
4458 and the re-ordering of terms in GiNaC, the match for the first factor
4459 may be @{$1==a,$2==b@} in which case the match for the second factor
4460 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4462 > match(a*(x+y)+a*z+b,a*$1+$2);
4463 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4464 @{$1=x+y,$2=a*z+b@}.)
4465 > match(a+b+c+d+e+f,c);
4467 > match(a+b+c+d+e+f,c+$0);
4469 > match(a+b+c+d+e+f,c+e+$0);
4471 > match(a+b,a+b+$0);
4473 > match(a*b^2,a^$1*b^$2);
4475 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4476 even though a==a^1.)
4477 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4479 > match(atan2(y,x^2),atan2(y,$0));
4483 @subsection Matching parts of expressions
4484 @cindex @code{has()}
4485 A more general way to look for patterns in expressions is provided by the
4489 bool ex::has(const ex & pattern);
4492 This function checks whether a pattern is matched by an expression itself or
4493 by any of its subexpressions.
4495 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4496 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4499 > has(x*sin(x+y+2*a),y);
4501 > has(x*sin(x+y+2*a),x+y);
4503 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4504 has the subexpressions "x", "y" and "2*a".)
4505 > has(x*sin(x+y+2*a),x+y+$1);
4507 (But this is possible.)
4508 > has(x*sin(2*(x+y)+2*a),x+y);
4510 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4511 which "x+y" is not a subexpression.)
4514 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4516 > has(4*x^2-x+3,$1*x);
4518 > has(4*x^2+x+3,$1*x);
4520 (Another possible pitfall. The first expression matches because the term
4521 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4522 contains a linear term you should use the coeff() function instead.)
4525 @cindex @code{find()}
4529 bool ex::find(const ex & pattern, lst & found);
4532 works a bit like @code{has()} but it doesn't stop upon finding the first
4533 match. Instead, it appends all found matches to the specified list. If there
4534 are multiple occurrences of the same expression, it is entered only once to
4535 the list. @code{find()} returns false if no matches were found (in
4536 @command{ginsh}, it returns an empty list):
4539 > find(1+x+x^2+x^3,x);
4541 > find(1+x+x^2+x^3,y);
4543 > find(1+x+x^2+x^3,x^$1);
4545 (Note the absence of "x".)
4546 > expand((sin(x)+sin(y))*(a+b));
4547 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4552 @subsection Substituting expressions
4553 @cindex @code{subs()}
4554 Probably the most useful application of patterns is to use them for
4555 substituting expressions with the @code{subs()} method. Wildcards can be
4556 used in the search patterns as well as in the replacement expressions, where
4557 they get replaced by the expressions matched by them. @code{subs()} doesn't
4558 know anything about algebra; it performs purely syntactic substitutions.
4563 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4565 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4567 > subs((a+b+c)^2,a+b==x);
4569 > subs((a+b+c)^2,a+b+$1==x+$1);
4571 > subs(a+2*b,a+b==x);
4573 > subs(4*x^3-2*x^2+5*x-1,x==a);
4575 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4577 > subs(sin(1+sin(x)),sin($1)==cos($1));
4579 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4583 The last example would be written in C++ in this way:
4587 symbol a("a"), b("b"), x("x"), y("y");
4588 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4589 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4590 cout << e.expand() << endl;
4595 @subsection The option algebraic
4596 Both @code{has()} and @code{subs()} take an optional argument to pass them
4597 extra options. This section describes what happens if you give the former
4598 the option @code{has_options::algebraic} or the latter
4599 @code{subs:options::algebraic}. In that case the matching condition for
4600 powers and multiplications is changed in such a way that they become
4601 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4602 If you use these options you will find that
4603 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4604 Besides matching some of the factors of a product also powers match as
4605 often as is possible without getting negative exponents. For example
4606 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4607 @code{x*c^2*z}. This also works with negative powers:
4608 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4609 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4610 and not for locating @code{x+y} within @code{x+y+z}.
4613 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4614 @c node-name, next, previous, up
4615 @section Applying a Function on Subexpressions
4616 @cindex tree traversal
4617 @cindex @code{map()}
4619 Sometimes you may want to perform an operation on specific parts of an
4620 expression while leaving the general structure of it intact. An example
4621 of this would be a matrix trace operation: the trace of a sum is the sum
4622 of the traces of the individual terms. That is, the trace should @dfn{map}
4623 on the sum, by applying itself to each of the sum's operands. It is possible
4624 to do this manually which usually results in code like this:
4629 if (is_a<matrix>(e))
4630 return ex_to<matrix>(e).trace();
4631 else if (is_a<add>(e)) @{
4633 for (size_t i=0; i<e.nops(); i++)
4634 sum += calc_trace(e.op(i));
4636 @} else if (is_a<mul>)(e)) @{
4644 This is, however, slightly inefficient (if the sum is very large it can take
4645 a long time to add the terms one-by-one), and its applicability is limited to
4646 a rather small class of expressions. If @code{calc_trace()} is called with
4647 a relation or a list as its argument, you will probably want the trace to
4648 be taken on both sides of the relation or of all elements of the list.
4650 GiNaC offers the @code{map()} method to aid in the implementation of such
4654 ex ex::map(map_function & f) const;
4655 ex ex::map(ex (*f)(const ex & e)) const;
4658 In the first (preferred) form, @code{map()} takes a function object that
4659 is subclassed from the @code{map_function} class. In the second form, it
4660 takes a pointer to a function that accepts and returns an expression.
4661 @code{map()} constructs a new expression of the same type, applying the
4662 specified function on all subexpressions (in the sense of @code{op()}),
4665 The use of a function object makes it possible to supply more arguments to
4666 the function that is being mapped, or to keep local state information.
4667 The @code{map_function} class declares a virtual function call operator
4668 that you can overload. Here is a sample implementation of @code{calc_trace()}
4669 that uses @code{map()} in a recursive fashion:
4672 struct calc_trace : public map_function @{
4673 ex operator()(const ex &e)
4675 if (is_a<matrix>(e))
4676 return ex_to<matrix>(e).trace();
4677 else if (is_a<mul>(e)) @{
4680 return e.map(*this);
4685 This function object could then be used like this:
4689 ex M = ... // expression with matrices
4690 calc_trace do_trace;
4691 ex tr = do_trace(M);
4695 Here is another example for you to meditate over. It removes quadratic
4696 terms in a variable from an expanded polynomial:
4699 struct map_rem_quad : public map_function @{
4701 map_rem_quad(const ex & var_) : var(var_) @{@}
4703 ex operator()(const ex & e)
4705 if (is_a<add>(e) || is_a<mul>(e))
4706 return e.map(*this);
4707 else if (is_a<power>(e) &&
4708 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4718 symbol x("x"), y("y");
4721 for (int i=0; i<8; i++)
4722 e += pow(x, i) * pow(y, 8-i) * (i+1);
4724 // -> 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
4726 map_rem_quad rem_quad(x);
4727 cout << rem_quad(e) << endl;
4728 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4732 @command{ginsh} offers a slightly different implementation of @code{map()}
4733 that allows applying algebraic functions to operands. The second argument
4734 to @code{map()} is an expression containing the wildcard @samp{$0} which
4735 acts as the placeholder for the operands:
4740 > map(a+2*b,sin($0));
4742 > map(@{a,b,c@},$0^2+$0);
4743 @{a^2+a,b^2+b,c^2+c@}
4746 Note that it is only possible to use algebraic functions in the second
4747 argument. You can not use functions like @samp{diff()}, @samp{op()},
4748 @samp{subs()} etc. because these are evaluated immediately:
4751 > map(@{a,b,c@},diff($0,a));
4753 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4754 to "map(@{a,b,c@},0)".
4758 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4759 @c node-name, next, previous, up
4760 @section Visitors and Tree Traversal
4761 @cindex tree traversal
4762 @cindex @code{visitor} (class)
4763 @cindex @code{accept()}
4764 @cindex @code{visit()}
4765 @cindex @code{traverse()}
4766 @cindex @code{traverse_preorder()}
4767 @cindex @code{traverse_postorder()}
4769 Suppose that you need a function that returns a list of all indices appearing
4770 in an arbitrary expression. The indices can have any dimension, and for
4771 indices with variance you always want the covariant version returned.
4773 You can't use @code{get_free_indices()} because you also want to include
4774 dummy indices in the list, and you can't use @code{find()} as it needs
4775 specific index dimensions (and it would require two passes: one for indices
4776 with variance, one for plain ones).
4778 The obvious solution to this problem is a tree traversal with a type switch,
4779 such as the following:
4782 void gather_indices_helper(const ex & e, lst & l)
4784 if (is_a<varidx>(e)) @{
4785 const varidx & vi = ex_to<varidx>(e);
4786 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4787 @} else if (is_a<idx>(e)) @{
4790 size_t n = e.nops();
4791 for (size_t i = 0; i < n; ++i)
4792 gather_indices_helper(e.op(i), l);
4796 lst gather_indices(const ex & e)
4799 gather_indices_helper(e, l);
4806 This works fine but fans of object-oriented programming will feel
4807 uncomfortable with the type switch. One reason is that there is a possibility
4808 for subtle bugs regarding derived classes. If we had, for example, written
4811 if (is_a<idx>(e)) @{
4813 @} else if (is_a<varidx>(e)) @{
4817 in @code{gather_indices_helper}, the code wouldn't have worked because the
4818 first line "absorbs" all classes derived from @code{idx}, including
4819 @code{varidx}, so the special case for @code{varidx} would never have been
4822 Also, for a large number of classes, a type switch like the above can get
4823 unwieldy and inefficient (it's a linear search, after all).
4824 @code{gather_indices_helper} only checks for two classes, but if you had to
4825 write a function that required a different implementation for nearly
4826 every GiNaC class, the result would be very hard to maintain and extend.
4828 The cleanest approach to the problem would be to add a new virtual function
4829 to GiNaC's class hierarchy. In our example, there would be specializations
4830 for @code{idx} and @code{varidx} while the default implementation in
4831 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4832 impossible to add virtual member functions to existing classes without
4833 changing their source and recompiling everything. GiNaC comes with source,
4834 so you could actually do this, but for a small algorithm like the one
4835 presented this would be impractical.
4837 One solution to this dilemma is the @dfn{Visitor} design pattern,
4838 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4839 variation, described in detail in
4840 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4841 virtual functions to the class hierarchy to implement operations, GiNaC
4842 provides a single "bouncing" method @code{accept()} that takes an instance
4843 of a special @code{visitor} class and redirects execution to the one
4844 @code{visit()} virtual function of the visitor that matches the type of
4845 object that @code{accept()} was being invoked on.
4847 Visitors in GiNaC must derive from the global @code{visitor} class as well
4848 as from the class @code{T::visitor} of each class @code{T} they want to
4849 visit, and implement the member functions @code{void visit(const T &)} for
4855 void ex::accept(visitor & v) const;
4858 will then dispatch to the correct @code{visit()} member function of the
4859 specified visitor @code{v} for the type of GiNaC object at the root of the
4860 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4862 Here is an example of a visitor:
4866 : public visitor, // this is required
4867 public add::visitor, // visit add objects
4868 public numeric::visitor, // visit numeric objects
4869 public basic::visitor // visit basic objects
4871 void visit(const add & x)
4872 @{ cout << "called with an add object" << endl; @}
4874 void visit(const numeric & x)
4875 @{ cout << "called with a numeric object" << endl; @}
4877 void visit(const basic & x)
4878 @{ cout << "called with a basic object" << endl; @}
4882 which can be used as follows:
4893 // prints "called with a numeric object"
4895 // prints "called with an add object"
4897 // prints "called with a basic object"
4901 The @code{visit(const basic &)} method gets called for all objects that are
4902 not @code{numeric} or @code{add} and acts as an (optional) default.
4904 From a conceptual point of view, the @code{visit()} methods of the visitor
4905 behave like a newly added virtual function of the visited hierarchy.
4906 In addition, visitors can store state in member variables, and they can
4907 be extended by deriving a new visitor from an existing one, thus building
4908 hierarchies of visitors.
4910 We can now rewrite our index example from above with a visitor:
4913 class gather_indices_visitor
4914 : public visitor, public idx::visitor, public varidx::visitor
4918 void visit(const idx & i)
4923 void visit(const varidx & vi)
4925 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4929 const lst & get_result() // utility function
4938 What's missing is the tree traversal. We could implement it in
4939 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4942 void ex::traverse_preorder(visitor & v) const;
4943 void ex::traverse_postorder(visitor & v) const;
4944 void ex::traverse(visitor & v) const;
4947 @code{traverse_preorder()} visits a node @emph{before} visiting its
4948 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4949 visiting its subexpressions. @code{traverse()} is a synonym for
4950 @code{traverse_preorder()}.
4952 Here is a new implementation of @code{gather_indices()} that uses the visitor
4953 and @code{traverse()}:
4956 lst gather_indices(const ex & e)
4958 gather_indices_visitor v;
4960 return v.get_result();
4964 Alternatively, you could use pre- or postorder iterators for the tree
4968 lst gather_indices(const ex & e)
4970 gather_indices_visitor v;
4971 for (const_preorder_iterator i = e.preorder_begin();
4972 i != e.preorder_end(); ++i) @{
4975 return v.get_result();
4980 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4981 @c node-name, next, previous, up
4982 @section Polynomial arithmetic
4984 @subsection Expanding and collecting
4985 @cindex @code{expand()}
4986 @cindex @code{collect()}
4987 @cindex @code{collect_common_factors()}
4989 A polynomial in one or more variables has many equivalent
4990 representations. Some useful ones serve a specific purpose. Consider
4991 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4992 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4993 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4994 representations are the recursive ones where one collects for exponents
4995 in one of the three variable. Since the factors are themselves
4996 polynomials in the remaining two variables the procedure can be
4997 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4998 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5001 To bring an expression into expanded form, its method
5004 ex ex::expand(unsigned options = 0);
5007 may be called. In our example above, this corresponds to @math{4*x*y +
5008 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5009 GiNaC is not easy to guess you should be prepared to see different
5010 orderings of terms in such sums!
5012 Another useful representation of multivariate polynomials is as a
5013 univariate polynomial in one of the variables with the coefficients
5014 being polynomials in the remaining variables. The method
5015 @code{collect()} accomplishes this task:
5018 ex ex::collect(const ex & s, bool distributed = false);
5021 The first argument to @code{collect()} can also be a list of objects in which
5022 case the result is either a recursively collected polynomial, or a polynomial
5023 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5024 by the @code{distributed} flag.
5026 Note that the original polynomial needs to be in expanded form (for the
5027 variables concerned) in order for @code{collect()} to be able to find the
5028 coefficients properly.
5030 The following @command{ginsh} transcript shows an application of @code{collect()}
5031 together with @code{find()}:
5034 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5035 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)
5036 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5037 > collect(a,@{p,q@});
5038 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5039 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5040 > collect(a,find(a,sin($1)));
5041 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5042 > collect(a,@{find(a,sin($1)),p,q@});
5043 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5044 > collect(a,@{find(a,sin($1)),d@});
5045 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5048 Polynomials can often be brought into a more compact form by collecting
5049 common factors from the terms of sums. This is accomplished by the function
5052 ex collect_common_factors(const ex & e);
5055 This function doesn't perform a full factorization but only looks for
5056 factors which are already explicitly present:
5059 > collect_common_factors(a*x+a*y);
5061 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5063 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5064 (c+a)*a*(x*y+y^2+x)*b
5067 @subsection Degree and coefficients
5068 @cindex @code{degree()}
5069 @cindex @code{ldegree()}
5070 @cindex @code{coeff()}
5072 The degree and low degree of a polynomial can be obtained using the two
5076 int ex::degree(const ex & s);
5077 int ex::ldegree(const ex & s);
5080 which also work reliably on non-expanded input polynomials (they even work
5081 on rational functions, returning the asymptotic degree). By definition, the
5082 degree of zero is zero. To extract a coefficient with a certain power from
5083 an expanded polynomial you use
5086 ex ex::coeff(const ex & s, int n);
5089 You can also obtain the leading and trailing coefficients with the methods
5092 ex ex::lcoeff(const ex & s);
5093 ex ex::tcoeff(const ex & s);
5096 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5099 An application is illustrated in the next example, where a multivariate
5100 polynomial is analyzed:
5104 symbol x("x"), y("y");
5105 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5106 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5107 ex Poly = PolyInp.expand();
5109 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5110 cout << "The x^" << i << "-coefficient is "
5111 << Poly.coeff(x,i) << endl;
5113 cout << "As polynomial in y: "
5114 << Poly.collect(y) << endl;
5118 When run, it returns an output in the following fashion:
5121 The x^0-coefficient is y^2+11*y
5122 The x^1-coefficient is 5*y^2-2*y
5123 The x^2-coefficient is -1
5124 The x^3-coefficient is 4*y
5125 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5128 As always, the exact output may vary between different versions of GiNaC
5129 or even from run to run since the internal canonical ordering is not
5130 within the user's sphere of influence.
5132 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5133 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5134 with non-polynomial expressions as they not only work with symbols but with
5135 constants, functions and indexed objects as well:
5139 symbol a("a"), b("b"), c("c"), x("x");
5140 idx i(symbol("i"), 3);
5142 ex e = pow(sin(x) - cos(x), 4);
5143 cout << e.degree(cos(x)) << endl;
5145 cout << e.expand().coeff(sin(x), 3) << endl;
5148 e = indexed(a+b, i) * indexed(b+c, i);
5149 e = e.expand(expand_options::expand_indexed);
5150 cout << e.collect(indexed(b, i)) << endl;
5151 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5156 @subsection Polynomial division
5157 @cindex polynomial division
5160 @cindex pseudo-remainder
5161 @cindex @code{quo()}
5162 @cindex @code{rem()}
5163 @cindex @code{prem()}
5164 @cindex @code{divide()}
5169 ex quo(const ex & a, const ex & b, const ex & x);
5170 ex rem(const ex & a, const ex & b, const ex & x);
5173 compute the quotient and remainder of univariate polynomials in the variable
5174 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5176 The additional function
5179 ex prem(const ex & a, const ex & b, const ex & x);
5182 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5183 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5185 Exact division of multivariate polynomials is performed by the function
5188 bool divide(const ex & a, const ex & b, ex & q);
5191 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5192 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5193 in which case the value of @code{q} is undefined.
5196 @subsection Unit, content and primitive part
5197 @cindex @code{unit()}
5198 @cindex @code{content()}
5199 @cindex @code{primpart()}
5200 @cindex @code{unitcontprim()}
5205 ex ex::unit(const ex & x);
5206 ex ex::content(const ex & x);
5207 ex ex::primpart(const ex & x);
5208 ex ex::primpart(const ex & x, const ex & c);
5211 return the unit part, content part, and primitive polynomial of a multivariate
5212 polynomial with respect to the variable @samp{x} (the unit part being the sign
5213 of the leading coefficient, the content part being the GCD of the coefficients,
5214 and the primitive polynomial being the input polynomial divided by the unit and
5215 content parts). The second variant of @code{primpart()} expects the previously
5216 calculated content part of the polynomial in @code{c}, which enables it to
5217 work faster in the case where the content part has already been computed. The
5218 product of unit, content, and primitive part is the original polynomial.
5220 Additionally, the method
5223 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5226 computes the unit, content, and primitive parts in one go, returning them
5227 in @code{u}, @code{c}, and @code{p}, respectively.
5230 @subsection GCD, LCM and resultant
5233 @cindex @code{gcd()}
5234 @cindex @code{lcm()}
5236 The functions for polynomial greatest common divisor and least common
5237 multiple have the synopsis
5240 ex gcd(const ex & a, const ex & b);
5241 ex lcm(const ex & a, const ex & b);
5244 The functions @code{gcd()} and @code{lcm()} accept two expressions
5245 @code{a} and @code{b} as arguments and return a new expression, their
5246 greatest common divisor or least common multiple, respectively. If the
5247 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5248 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5249 the coefficients must be rationals.
5252 #include <ginac/ginac.h>
5253 using namespace GiNaC;
5257 symbol x("x"), y("y"), z("z");
5258 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5259 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5261 ex P_gcd = gcd(P_a, P_b);
5263 ex P_lcm = lcm(P_a, P_b);
5264 // 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
5269 @cindex @code{resultant()}
5271 The resultant of two expressions only makes sense with polynomials.
5272 It is always computed with respect to a specific symbol within the
5273 expressions. The function has the interface
5276 ex resultant(const ex & a, const ex & b, const ex & s);
5279 Resultants are symmetric in @code{a} and @code{b}. The following example
5280 computes the resultant of two expressions with respect to @code{x} and
5281 @code{y}, respectively:
5284 #include <ginac/ginac.h>
5285 using namespace GiNaC;
5289 symbol x("x"), y("y");
5291 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5294 r = resultant(e1, e2, x);
5296 r = resultant(e1, e2, y);
5301 @subsection Square-free decomposition
5302 @cindex square-free decomposition
5303 @cindex factorization
5304 @cindex @code{sqrfree()}
5306 GiNaC still lacks proper factorization support. Some form of
5307 factorization is, however, easily implemented by noting that factors
5308 appearing in a polynomial with power two or more also appear in the
5309 derivative and hence can easily be found by computing the GCD of the
5310 original polynomial and its derivatives. Any decent system has an
5311 interface for this so called square-free factorization. So we provide
5314 ex sqrfree(const ex & a, const lst & l = lst());
5316 Here is an example that by the way illustrates how the exact form of the
5317 result may slightly depend on the order of differentiation, calling for
5318 some care with subsequent processing of the result:
5321 symbol x("x"), y("y");
5322 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5324 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5325 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5327 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5328 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5330 cout << sqrfree(BiVarPol) << endl;
5331 // -> depending on luck, any of the above
5334 Note also, how factors with the same exponents are not fully factorized
5338 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5339 @c node-name, next, previous, up
5340 @section Rational expressions
5342 @subsection The @code{normal} method
5343 @cindex @code{normal()}
5344 @cindex simplification
5345 @cindex temporary replacement
5347 Some basic form of simplification of expressions is called for frequently.
5348 GiNaC provides the method @code{.normal()}, which converts a rational function
5349 into an equivalent rational function of the form @samp{numerator/denominator}
5350 where numerator and denominator are coprime. If the input expression is already
5351 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5352 otherwise it performs fraction addition and multiplication.
5354 @code{.normal()} can also be used on expressions which are not rational functions
5355 as it will replace all non-rational objects (like functions or non-integer
5356 powers) by temporary symbols to bring the expression to the domain of rational
5357 functions before performing the normalization, and re-substituting these
5358 symbols afterwards. This algorithm is also available as a separate method
5359 @code{.to_rational()}, described below.
5361 This means that both expressions @code{t1} and @code{t2} are indeed
5362 simplified in this little code snippet:
5367 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5368 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5369 std::cout << "t1 is " << t1.normal() << std::endl;
5370 std::cout << "t2 is " << t2.normal() << std::endl;
5374 Of course this works for multivariate polynomials too, so the ratio of
5375 the sample-polynomials from the section about GCD and LCM above would be
5376 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5379 @subsection Numerator and denominator
5382 @cindex @code{numer()}
5383 @cindex @code{denom()}
5384 @cindex @code{numer_denom()}
5386 The numerator and denominator of an expression can be obtained with
5391 ex ex::numer_denom();
5394 These functions will first normalize the expression as described above and
5395 then return the numerator, denominator, or both as a list, respectively.
5396 If you need both numerator and denominator, calling @code{numer_denom()} is
5397 faster than using @code{numer()} and @code{denom()} separately.
5400 @subsection Converting to a polynomial or rational expression
5401 @cindex @code{to_polynomial()}
5402 @cindex @code{to_rational()}
5404 Some of the methods described so far only work on polynomials or rational
5405 functions. GiNaC provides a way to extend the domain of these functions to
5406 general expressions by using the temporary replacement algorithm described
5407 above. You do this by calling
5410 ex ex::to_polynomial(exmap & m);
5411 ex ex::to_polynomial(lst & l);
5415 ex ex::to_rational(exmap & m);
5416 ex ex::to_rational(lst & l);
5419 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5420 will be filled with the generated temporary symbols and their replacement
5421 expressions in a format that can be used directly for the @code{subs()}
5422 method. It can also already contain a list of replacements from an earlier
5423 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5424 possible to use it on multiple expressions and get consistent results.
5426 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5427 is probably best illustrated with an example:
5431 symbol x("x"), y("y");
5432 ex a = 2*x/sin(x) - y/(3*sin(x));
5436 ex p = a.to_polynomial(lp);
5437 cout << " = " << p << "\n with " << lp << endl;
5438 // = symbol3*symbol2*y+2*symbol2*x
5439 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5442 ex r = a.to_rational(lr);
5443 cout << " = " << r << "\n with " << lr << endl;
5444 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5445 // with @{symbol4==sin(x)@}
5449 The following more useful example will print @samp{sin(x)-cos(x)}:
5454 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5455 ex b = sin(x) + cos(x);
5458 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5459 cout << q.subs(m) << endl;
5464 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5465 @c node-name, next, previous, up
5466 @section Symbolic differentiation
5467 @cindex differentiation
5468 @cindex @code{diff()}
5470 @cindex product rule
5472 GiNaC's objects know how to differentiate themselves. Thus, a
5473 polynomial (class @code{add}) knows that its derivative is the sum of
5474 the derivatives of all the monomials:
5478 symbol x("x"), y("y"), z("z");
5479 ex P = pow(x, 5) + pow(x, 2) + y;
5481 cout << P.diff(x,2) << endl;
5483 cout << P.diff(y) << endl; // 1
5485 cout << P.diff(z) << endl; // 0
5490 If a second integer parameter @var{n} is given, the @code{diff} method
5491 returns the @var{n}th derivative.
5493 If @emph{every} object and every function is told what its derivative
5494 is, all derivatives of composed objects can be calculated using the
5495 chain rule and the product rule. Consider, for instance the expression
5496 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5497 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5498 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5499 out that the composition is the generating function for Euler Numbers,
5500 i.e. the so called @var{n}th Euler number is the coefficient of
5501 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5502 identity to code a function that generates Euler numbers in just three
5505 @cindex Euler numbers
5507 #include <ginac/ginac.h>
5508 using namespace GiNaC;
5510 ex EulerNumber(unsigned n)
5513 const ex generator = pow(cosh(x),-1);
5514 return generator.diff(x,n).subs(x==0);
5519 for (unsigned i=0; i<11; i+=2)
5520 std::cout << EulerNumber(i) << std::endl;
5525 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5526 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5527 @code{i} by two since all odd Euler numbers vanish anyways.
5530 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5531 @c node-name, next, previous, up
5532 @section Series expansion
5533 @cindex @code{series()}
5534 @cindex Taylor expansion
5535 @cindex Laurent expansion
5536 @cindex @code{pseries} (class)
5537 @cindex @code{Order()}
5539 Expressions know how to expand themselves as a Taylor series or (more
5540 generally) a Laurent series. As in most conventional Computer Algebra
5541 Systems, no distinction is made between those two. There is a class of
5542 its own for storing such series (@code{class pseries}) and a built-in
5543 function (called @code{Order}) for storing the order term of the series.
5544 As a consequence, if you want to work with series, i.e. multiply two
5545 series, you need to call the method @code{ex::series} again to convert
5546 it to a series object with the usual structure (expansion plus order
5547 term). A sample application from special relativity could read:
5550 #include <ginac/ginac.h>
5551 using namespace std;
5552 using namespace GiNaC;
5556 symbol v("v"), c("c");
5558 ex gamma = 1/sqrt(1 - pow(v/c,2));
5559 ex mass_nonrel = gamma.series(v==0, 10);
5561 cout << "the relativistic mass increase with v is " << endl
5562 << mass_nonrel << endl;
5564 cout << "the inverse square of this series is " << endl
5565 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5569 Only calling the series method makes the last output simplify to
5570 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5571 series raised to the power @math{-2}.
5573 @cindex Machin's formula
5574 As another instructive application, let us calculate the numerical
5575 value of Archimedes' constant
5579 (for which there already exists the built-in constant @code{Pi})
5580 using John Machin's amazing formula
5582 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5585 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5587 This equation (and similar ones) were used for over 200 years for
5588 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5589 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5590 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5591 order term with it and the question arises what the system is supposed
5592 to do when the fractions are plugged into that order term. The solution
5593 is to use the function @code{series_to_poly()} to simply strip the order
5597 #include <ginac/ginac.h>
5598 using namespace GiNaC;
5600 ex machin_pi(int degr)
5603 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5604 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5605 -4*pi_expansion.subs(x==numeric(1,239));
5611 using std::cout; // just for fun, another way of...
5612 using std::endl; // ...dealing with this namespace std.
5614 for (int i=2; i<12; i+=2) @{
5615 pi_frac = machin_pi(i);
5616 cout << i << ":\t" << pi_frac << endl
5617 << "\t" << pi_frac.evalf() << endl;
5623 Note how we just called @code{.series(x,degr)} instead of
5624 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5625 method @code{series()}: if the first argument is a symbol the expression
5626 is expanded in that symbol around point @code{0}. When you run this
5627 program, it will type out:
5631 3.1832635983263598326
5632 4: 5359397032/1706489875
5633 3.1405970293260603143
5634 6: 38279241713339684/12184551018734375
5635 3.141621029325034425
5636 8: 76528487109180192540976/24359780855939418203125
5637 3.141591772182177295
5638 10: 327853873402258685803048818236/104359128170408663038552734375
5639 3.1415926824043995174
5643 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5644 @c node-name, next, previous, up
5645 @section Symmetrization
5646 @cindex @code{symmetrize()}
5647 @cindex @code{antisymmetrize()}
5648 @cindex @code{symmetrize_cyclic()}
5653 ex ex::symmetrize(const lst & l);
5654 ex ex::antisymmetrize(const lst & l);
5655 ex ex::symmetrize_cyclic(const lst & l);
5658 symmetrize an expression by returning the sum over all symmetric,
5659 antisymmetric or cyclic permutations of the specified list of objects,
5660 weighted by the number of permutations.
5662 The three additional methods
5665 ex ex::symmetrize();
5666 ex ex::antisymmetrize();
5667 ex ex::symmetrize_cyclic();
5670 symmetrize or antisymmetrize an expression over its free indices.
5672 Symmetrization is most useful with indexed expressions but can be used with
5673 almost any kind of object (anything that is @code{subs()}able):
5677 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5678 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5680 cout << indexed(A, i, j).symmetrize() << endl;
5681 // -> 1/2*A.j.i+1/2*A.i.j
5682 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5683 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5684 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5685 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5689 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5690 @c node-name, next, previous, up
5691 @section Predefined mathematical functions
5693 @subsection Overview
5695 GiNaC contains the following predefined mathematical functions:
5698 @multitable @columnfractions .30 .70
5699 @item @strong{Name} @tab @strong{Function}
5702 @cindex @code{abs()}
5703 @item @code{csgn(x)}
5705 @cindex @code{conjugate()}
5706 @item @code{conjugate(x)}
5707 @tab complex conjugation
5708 @cindex @code{csgn()}
5709 @item @code{sqrt(x)}
5710 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5711 @cindex @code{sqrt()}
5714 @cindex @code{sin()}
5717 @cindex @code{cos()}
5720 @cindex @code{tan()}
5721 @item @code{asin(x)}
5723 @cindex @code{asin()}
5724 @item @code{acos(x)}
5726 @cindex @code{acos()}
5727 @item @code{atan(x)}
5728 @tab inverse tangent
5729 @cindex @code{atan()}
5730 @item @code{atan2(y, x)}
5731 @tab inverse tangent with two arguments
5732 @item @code{sinh(x)}
5733 @tab hyperbolic sine
5734 @cindex @code{sinh()}
5735 @item @code{cosh(x)}
5736 @tab hyperbolic cosine
5737 @cindex @code{cosh()}
5738 @item @code{tanh(x)}
5739 @tab hyperbolic tangent
5740 @cindex @code{tanh()}
5741 @item @code{asinh(x)}
5742 @tab inverse hyperbolic sine
5743 @cindex @code{asinh()}
5744 @item @code{acosh(x)}
5745 @tab inverse hyperbolic cosine
5746 @cindex @code{acosh()}
5747 @item @code{atanh(x)}
5748 @tab inverse hyperbolic tangent
5749 @cindex @code{atanh()}
5751 @tab exponential function
5752 @cindex @code{exp()}
5754 @tab natural logarithm
5755 @cindex @code{log()}
5758 @cindex @code{Li2()}
5759 @item @code{Li(m, x)}
5760 @tab classical polylogarithm as well as multiple polylogarithm
5762 @item @code{G(a, y)}
5763 @tab multiple polylogarithm
5765 @item @code{G(a, s, y)}
5766 @tab multiple polylogarithm with explicit signs for the imaginary parts
5768 @item @code{S(n, p, x)}
5769 @tab Nielsen's generalized polylogarithm
5771 @item @code{H(m, x)}
5772 @tab harmonic polylogarithm
5774 @item @code{zeta(m)}
5775 @tab Riemann's zeta function as well as multiple zeta value
5776 @cindex @code{zeta()}
5777 @item @code{zeta(m, s)}
5778 @tab alternating Euler sum
5779 @cindex @code{zeta()}
5780 @item @code{zetaderiv(n, x)}
5781 @tab derivatives of Riemann's zeta function
5782 @item @code{tgamma(x)}
5784 @cindex @code{tgamma()}
5785 @cindex gamma function
5786 @item @code{lgamma(x)}
5787 @tab logarithm of gamma function
5788 @cindex @code{lgamma()}
5789 @item @code{beta(x, y)}
5790 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5791 @cindex @code{beta()}
5793 @tab psi (digamma) function
5794 @cindex @code{psi()}
5795 @item @code{psi(n, x)}
5796 @tab derivatives of psi function (polygamma functions)
5797 @item @code{factorial(n)}
5798 @tab factorial function @math{n!}
5799 @cindex @code{factorial()}
5800 @item @code{binomial(n, k)}
5801 @tab binomial coefficients
5802 @cindex @code{binomial()}
5803 @item @code{Order(x)}
5804 @tab order term function in truncated power series
5805 @cindex @code{Order()}
5810 For functions that have a branch cut in the complex plane GiNaC follows
5811 the conventions for C++ as defined in the ANSI standard as far as
5812 possible. In particular: the natural logarithm (@code{log}) and the
5813 square root (@code{sqrt}) both have their branch cuts running along the
5814 negative real axis where the points on the axis itself belong to the
5815 upper part (i.e. continuous with quadrant II). The inverse
5816 trigonometric and hyperbolic functions are not defined for complex
5817 arguments by the C++ standard, however. In GiNaC we follow the
5818 conventions used by CLN, which in turn follow the carefully designed
5819 definitions in the Common Lisp standard. It should be noted that this
5820 convention is identical to the one used by the C99 standard and by most
5821 serious CAS. It is to be expected that future revisions of the C++
5822 standard incorporate these functions in the complex domain in a manner
5823 compatible with C99.
5825 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5826 @c node-name, next, previous, up
5827 @subsection Multiple polylogarithms
5829 @cindex polylogarithm
5830 @cindex Nielsen's generalized polylogarithm
5831 @cindex harmonic polylogarithm
5832 @cindex multiple zeta value
5833 @cindex alternating Euler sum
5834 @cindex multiple polylogarithm
5836 The multiple polylogarithm is the most generic member of a family of functions,
5837 to which others like the harmonic polylogarithm, Nielsen's generalized
5838 polylogarithm and the multiple zeta value belong.
5839 Everyone of these functions can also be written as a multiple polylogarithm with specific
5840 parameters. This whole family of functions is therefore often referred to simply as
5841 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5842 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5843 @code{Li} and @code{G} in principle represent the same function, the different
5844 notations are more natural to the series representation or the integral
5845 representation, respectively.
5847 To facilitate the discussion of these functions we distinguish between indices and
5848 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5849 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5851 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5852 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5853 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5854 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5855 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5856 @code{s} is not given, the signs default to +1.
5857 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5858 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5859 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5860 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5861 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5863 The functions print in LaTeX format as
5865 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5871 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5874 $\zeta(m_1,m_2,\ldots,m_k)$.
5876 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5877 are printed with a line above, e.g.
5879 $\zeta(5,\overline{2})$.
5881 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5883 Definitions and analytical as well as numerical properties of multiple polylogarithms
5884 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5885 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5886 except for a few differences which will be explicitly stated in the following.
5888 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5889 that the indices and arguments are understood to be in the same order as in which they appear in
5890 the series representation. This means
5892 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5895 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5898 $\zeta(1,2)$ evaluates to infinity.
5900 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5903 The functions only evaluate if the indices are integers greater than zero, except for the indices
5904 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5905 will be interpreted as the sequence of signs for the corresponding indices
5906 @code{m} or the sign of the imaginary part for the
5907 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5908 @code{zeta(lst(3,4), lst(-1,1))} means
5910 $\zeta(\overline{3},4)$
5913 @code{G(lst(a,b), lst(-1,1), c)} means
5915 $G(a-0\epsilon,b+0\epsilon;c)$.
5917 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5918 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5919 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
5920 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5921 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5922 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5923 evaluates also for negative integers and positive even integers. For example:
5926 > Li(@{3,1@},@{x,1@});
5929 -zeta(@{3,2@},@{-1,-1@})
5934 It is easy to tell for a given function into which other function it can be rewritten, may
5935 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5936 with negative indices or trailing zeros (the example above gives a hint). Signs can
5937 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5938 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5939 @code{Li} (@code{eval()} already cares for the possible downgrade):
5942 > convert_H_to_Li(@{0,-2,-1,3@},x);
5943 Li(@{3,1,3@},@{-x,1,-1@})
5944 > convert_H_to_Li(@{2,-1,0@},x);
5945 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5948 Every function can be numerically evaluated for
5949 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5950 global variable @code{Digits}:
5955 > evalf(zeta(@{3,1,3,1@}));
5956 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5959 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5960 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5962 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5967 In long expressions this helps a lot with debugging, because you can easily spot
5968 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5969 cancellations of divergencies happen.
5971 Useful publications:
5973 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5974 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5976 @cite{Harmonic Polylogarithms},
5977 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5979 @cite{Special Values of Multiple Polylogarithms},
5980 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5982 @cite{Numerical Evaluation of Multiple Polylogarithms},
5983 J.Vollinga, S.Weinzierl, hep-ph/0410259
5985 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5986 @c node-name, next, previous, up
5987 @section Complex Conjugation
5989 @cindex @code{conjugate()}
5997 returns the complex conjugate of the expression. For all built-in functions and objects the
5998 conjugation gives the expected results:
6002 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6006 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6007 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6008 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6009 // -> -gamma5*gamma~b*gamma~a
6013 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
6014 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
6015 arguments. This is the default strategy. If you want to define your own functions and want to
6016 change this behavior, you have to supply a specialized conjugation method for your function
6017 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
6019 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
6020 @c node-name, next, previous, up
6021 @section Solving Linear Systems of Equations
6022 @cindex @code{lsolve()}
6024 The function @code{lsolve()} provides a convenient wrapper around some
6025 matrix operations that comes in handy when a system of linear equations
6029 ex lsolve(const ex & eqns, const ex & symbols,
6030 unsigned options = solve_algo::automatic);
6033 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6034 @code{relational}) while @code{symbols} is a @code{lst} of
6035 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
6038 It returns the @code{lst} of solutions as an expression. As an example,
6039 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6043 symbol a("a"), b("b"), x("x"), y("y");
6045 eqns = a*x+b*y==3, x-y==b;
6047 cout << lsolve(eqns, vars) << endl;
6048 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6051 When the linear equations @code{eqns} are underdetermined, the solution
6052 will contain one or more tautological entries like @code{x==x},
6053 depending on the rank of the system. When they are overdetermined, the
6054 solution will be an empty @code{lst}. Note the third optional parameter
6055 to @code{lsolve()}: it accepts the same parameters as
6056 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6060 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6061 @c node-name, next, previous, up
6062 @section Input and output of expressions
6065 @subsection Expression output
6067 @cindex output of expressions
6069 Expressions can simply be written to any stream:
6074 ex e = 4.5*I+pow(x,2)*3/2;
6075 cout << e << endl; // prints '4.5*I+3/2*x^2'
6079 The default output format is identical to the @command{ginsh} input syntax and
6080 to that used by most computer algebra systems, but not directly pastable
6081 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6082 is printed as @samp{x^2}).
6084 It is possible to print expressions in a number of different formats with
6085 a set of stream manipulators;
6088 std::ostream & dflt(std::ostream & os);
6089 std::ostream & latex(std::ostream & os);
6090 std::ostream & tree(std::ostream & os);
6091 std::ostream & csrc(std::ostream & os);
6092 std::ostream & csrc_float(std::ostream & os);
6093 std::ostream & csrc_double(std::ostream & os);
6094 std::ostream & csrc_cl_N(std::ostream & os);
6095 std::ostream & index_dimensions(std::ostream & os);
6096 std::ostream & no_index_dimensions(std::ostream & os);
6099 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6100 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6101 @code{print_csrc()} functions, respectively.
6104 All manipulators affect the stream state permanently. To reset the output
6105 format to the default, use the @code{dflt} manipulator:
6109 cout << latex; // all output to cout will be in LaTeX format from
6111 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6112 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6113 cout << dflt; // revert to default output format
6114 cout << e << endl; // prints '4.5*I+3/2*x^2'
6118 If you don't want to affect the format of the stream you're working with,
6119 you can output to a temporary @code{ostringstream} like this:
6124 s << latex << e; // format of cout remains unchanged
6125 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6130 @cindex @code{csrc_float}
6131 @cindex @code{csrc_double}
6132 @cindex @code{csrc_cl_N}
6133 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6134 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6135 format that can be directly used in a C or C++ program. The three possible
6136 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6137 classes provided by the CLN library):
6141 cout << "f = " << csrc_float << e << ";\n";
6142 cout << "d = " << csrc_double << e << ";\n";
6143 cout << "n = " << csrc_cl_N << e << ";\n";
6147 The above example will produce (note the @code{x^2} being converted to
6151 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6152 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6153 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6157 The @code{tree} manipulator allows dumping the internal structure of an
6158 expression for debugging purposes:
6169 add, hash=0x0, flags=0x3, nops=2
6170 power, hash=0x0, flags=0x3, nops=2
6171 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6172 2 (numeric), hash=0x6526b0fa, flags=0xf
6173 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6176 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6180 @cindex @code{latex}
6181 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6182 It is rather similar to the default format but provides some braces needed
6183 by LaTeX for delimiting boxes and also converts some common objects to
6184 conventional LaTeX names. It is possible to give symbols a special name for
6185 LaTeX output by supplying it as a second argument to the @code{symbol}
6188 For example, the code snippet
6192 symbol x("x", "\\circ");
6193 ex e = lgamma(x).series(x==0,3);
6194 cout << latex << e << endl;
6201 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6202 +\mathcal@{O@}(\circ^@{3@})
6205 @cindex @code{index_dimensions}
6206 @cindex @code{no_index_dimensions}
6207 Index dimensions are normally hidden in the output. To make them visible, use
6208 the @code{index_dimensions} manipulator. The dimensions will be written in
6209 square brackets behind each index value in the default and LaTeX output
6214 symbol x("x"), y("y");
6215 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6216 ex e = indexed(x, mu) * indexed(y, nu);
6219 // prints 'x~mu*y~nu'
6220 cout << index_dimensions << e << endl;
6221 // prints 'x~mu[4]*y~nu[4]'
6222 cout << no_index_dimensions << e << endl;
6223 // prints 'x~mu*y~nu'
6228 @cindex Tree traversal
6229 If you need any fancy special output format, e.g. for interfacing GiNaC
6230 with other algebra systems or for producing code for different
6231 programming languages, you can always traverse the expression tree yourself:
6234 static void my_print(const ex & e)
6236 if (is_a<function>(e))
6237 cout << ex_to<function>(e).get_name();
6239 cout << ex_to<basic>(e).class_name();
6241 size_t n = e.nops();
6243 for (size_t i=0; i<n; i++) @{
6255 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6263 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6264 symbol(y))),numeric(-2)))
6267 If you need an output format that makes it possible to accurately
6268 reconstruct an expression by feeding the output to a suitable parser or
6269 object factory, you should consider storing the expression in an
6270 @code{archive} object and reading the object properties from there.
6271 See the section on archiving for more information.
6274 @subsection Expression input
6275 @cindex input of expressions
6277 GiNaC provides no way to directly read an expression from a stream because
6278 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6279 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6280 @code{y} you defined in your program and there is no way to specify the
6281 desired symbols to the @code{>>} stream input operator.
6283 Instead, GiNaC lets you construct an expression from a string, specifying the
6284 list of symbols to be used:
6288 symbol x("x"), y("y");
6289 ex e("2*x+sin(y)", lst(x, y));
6293 The input syntax is the same as that used by @command{ginsh} and the stream
6294 output operator @code{<<}. The symbols in the string are matched by name to
6295 the symbols in the list and if GiNaC encounters a symbol not specified in
6296 the list it will throw an exception.
6298 With this constructor, it's also easy to implement interactive GiNaC programs:
6303 #include <stdexcept>
6304 #include <ginac/ginac.h>
6305 using namespace std;
6306 using namespace GiNaC;
6313 cout << "Enter an expression containing 'x': ";
6318 cout << "The derivative of " << e << " with respect to x is ";
6319 cout << e.diff(x) << ".\n";
6320 @} catch (exception &p) @{
6321 cerr << p.what() << endl;
6327 @subsection Archiving
6328 @cindex @code{archive} (class)
6331 GiNaC allows creating @dfn{archives} of expressions which can be stored
6332 to or retrieved from files. To create an archive, you declare an object
6333 of class @code{archive} and archive expressions in it, giving each
6334 expression a unique name:
6338 using namespace std;
6339 #include <ginac/ginac.h>
6340 using namespace GiNaC;
6344 symbol x("x"), y("y"), z("z");
6346 ex foo = sin(x + 2*y) + 3*z + 41;
6350 a.archive_ex(foo, "foo");
6351 a.archive_ex(bar, "the second one");
6355 The archive can then be written to a file:
6359 ofstream out("foobar.gar");
6365 The file @file{foobar.gar} contains all information that is needed to
6366 reconstruct the expressions @code{foo} and @code{bar}.
6368 @cindex @command{viewgar}
6369 The tool @command{viewgar} that comes with GiNaC can be used to view
6370 the contents of GiNaC archive files:
6373 $ viewgar foobar.gar
6374 foo = 41+sin(x+2*y)+3*z
6375 the second one = 42+sin(x+2*y)+3*z
6378 The point of writing archive files is of course that they can later be
6384 ifstream in("foobar.gar");
6389 And the stored expressions can be retrieved by their name:
6396 ex ex1 = a2.unarchive_ex(syms, "foo");
6397 ex ex2 = a2.unarchive_ex(syms, "the second one");
6399 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6400 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6401 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6405 Note that you have to supply a list of the symbols which are to be inserted
6406 in the expressions. Symbols in archives are stored by their name only and
6407 if you don't specify which symbols you have, unarchiving the expression will
6408 create new symbols with that name. E.g. if you hadn't included @code{x} in
6409 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6410 have had no effect because the @code{x} in @code{ex1} would have been a
6411 different symbol than the @code{x} which was defined at the beginning of
6412 the program, although both would appear as @samp{x} when printed.
6414 You can also use the information stored in an @code{archive} object to
6415 output expressions in a format suitable for exact reconstruction. The
6416 @code{archive} and @code{archive_node} classes have a couple of member
6417 functions that let you access the stored properties:
6420 static void my_print2(const archive_node & n)
6423 n.find_string("class", class_name);
6424 cout << class_name << "(";
6426 archive_node::propinfovector p;
6427 n.get_properties(p);
6429 size_t num = p.size();
6430 for (size_t i=0; i<num; i++) @{
6431 const string &name = p[i].name;
6432 if (name == "class")
6434 cout << name << "=";
6436 unsigned count = p[i].count;
6440 for (unsigned j=0; j<count; j++) @{
6441 switch (p[i].type) @{
6442 case archive_node::PTYPE_BOOL: @{
6444 n.find_bool(name, x, j);
6445 cout << (x ? "true" : "false");
6448 case archive_node::PTYPE_UNSIGNED: @{
6450 n.find_unsigned(name, x, j);
6454 case archive_node::PTYPE_STRING: @{
6456 n.find_string(name, x, j);
6457 cout << '\"' << x << '\"';
6460 case archive_node::PTYPE_NODE: @{
6461 const archive_node &x = n.find_ex_node(name, j);
6483 ex e = pow(2, x) - y;
6485 my_print2(ar.get_top_node(0)); cout << endl;
6493 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6494 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6495 overall_coeff=numeric(number="0"))
6498 Be warned, however, that the set of properties and their meaning for each
6499 class may change between GiNaC versions.
6502 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6503 @c node-name, next, previous, up
6504 @chapter Extending GiNaC
6506 By reading so far you should have gotten a fairly good understanding of
6507 GiNaC's design patterns. From here on you should start reading the
6508 sources. All we can do now is issue some recommendations how to tackle
6509 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6510 develop some useful extension please don't hesitate to contact the GiNaC
6511 authors---they will happily incorporate them into future versions.
6514 * What does not belong into GiNaC:: What to avoid.
6515 * Symbolic functions:: Implementing symbolic functions.
6516 * Printing:: Adding new output formats.
6517 * Structures:: Defining new algebraic classes (the easy way).
6518 * Adding classes:: Defining new algebraic classes (the hard way).
6522 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6523 @c node-name, next, previous, up
6524 @section What doesn't belong into GiNaC
6526 @cindex @command{ginsh}
6527 First of all, GiNaC's name must be read literally. It is designed to be
6528 a library for use within C++. The tiny @command{ginsh} accompanying
6529 GiNaC makes this even more clear: it doesn't even attempt to provide a
6530 language. There are no loops or conditional expressions in
6531 @command{ginsh}, it is merely a window into the library for the
6532 programmer to test stuff (or to show off). Still, the design of a
6533 complete CAS with a language of its own, graphical capabilities and all
6534 this on top of GiNaC is possible and is without doubt a nice project for
6537 There are many built-in functions in GiNaC that do not know how to
6538 evaluate themselves numerically to a precision declared at runtime
6539 (using @code{Digits}). Some may be evaluated at certain points, but not
6540 generally. This ought to be fixed. However, doing numerical
6541 computations with GiNaC's quite abstract classes is doomed to be
6542 inefficient. For this purpose, the underlying foundation classes
6543 provided by CLN are much better suited.
6546 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6547 @c node-name, next, previous, up
6548 @section Symbolic functions
6550 The easiest and most instructive way to start extending GiNaC is probably to
6551 create your own symbolic functions. These are implemented with the help of
6552 two preprocessor macros:
6554 @cindex @code{DECLARE_FUNCTION}
6555 @cindex @code{REGISTER_FUNCTION}
6557 DECLARE_FUNCTION_<n>P(<name>)
6558 REGISTER_FUNCTION(<name>, <options>)
6561 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6562 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6563 parameters of type @code{ex} and returns a newly constructed GiNaC
6564 @code{function} object that represents your function.
6566 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6567 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6568 set of options that associate the symbolic function with C++ functions you
6569 provide to implement the various methods such as evaluation, derivative,
6570 series expansion etc. They also describe additional attributes the function
6571 might have, such as symmetry and commutation properties, and a name for
6572 LaTeX output. Multiple options are separated by the member access operator
6573 @samp{.} and can be given in an arbitrary order.
6575 (By the way: in case you are worrying about all the macros above we can
6576 assure you that functions are GiNaC's most macro-intense classes. We have
6577 done our best to avoid macros where we can.)
6579 @subsection A minimal example
6581 Here is an example for the implementation of a function with two arguments
6582 that is not further evaluated:
6585 DECLARE_FUNCTION_2P(myfcn)
6587 REGISTER_FUNCTION(myfcn, dummy())
6590 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6591 in algebraic expressions:
6597 ex e = 2*myfcn(42, 1+3*x) - x;
6599 // prints '2*myfcn(42,1+3*x)-x'
6604 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6605 "no options". A function with no options specified merely acts as a kind of
6606 container for its arguments. It is a pure "dummy" function with no associated
6607 logic (which is, however, sometimes perfectly sufficient).
6609 Let's now have a look at the implementation of GiNaC's cosine function for an
6610 example of how to make an "intelligent" function.
6612 @subsection The cosine function
6614 The GiNaC header file @file{inifcns.h} contains the line
6617 DECLARE_FUNCTION_1P(cos)
6620 which declares to all programs using GiNaC that there is a function @samp{cos}
6621 that takes one @code{ex} as an argument. This is all they need to know to use
6622 this function in expressions.
6624 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6625 is its @code{REGISTER_FUNCTION} line:
6628 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6629 evalf_func(cos_evalf).
6630 derivative_func(cos_deriv).
6631 latex_name("\\cos"));
6634 There are four options defined for the cosine function. One of them
6635 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6636 other three indicate the C++ functions in which the "brains" of the cosine
6637 function are defined.
6639 @cindex @code{hold()}
6641 The @code{eval_func()} option specifies the C++ function that implements
6642 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6643 the same number of arguments as the associated symbolic function (one in this
6644 case) and returns the (possibly transformed or in some way simplified)
6645 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6646 of the automatic evaluation process). If no (further) evaluation is to take
6647 place, the @code{eval_func()} function must return the original function
6648 with @code{.hold()}, to avoid a potential infinite recursion. If your
6649 symbolic functions produce a segmentation fault or stack overflow when
6650 using them in expressions, you are probably missing a @code{.hold()}
6653 The @code{eval_func()} function for the cosine looks something like this
6654 (actually, it doesn't look like this at all, but it should give you an idea
6658 static ex cos_eval(const ex & x)
6660 if ("x is a multiple of 2*Pi")
6662 else if ("x is a multiple of Pi")
6664 else if ("x is a multiple of Pi/2")
6668 else if ("x has the form 'acos(y)'")
6670 else if ("x has the form 'asin(y)'")
6675 return cos(x).hold();
6679 This function is called every time the cosine is used in a symbolic expression:
6685 // this calls cos_eval(Pi), and inserts its return value into
6686 // the actual expression
6693 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6694 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6695 symbolic transformation can be done, the unmodified function is returned
6696 with @code{.hold()}.
6698 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6699 The user has to call @code{evalf()} for that. This is implemented in a
6703 static ex cos_evalf(const ex & x)
6705 if (is_a<numeric>(x))
6706 return cos(ex_to<numeric>(x));
6708 return cos(x).hold();
6712 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6713 in this case the @code{cos()} function for @code{numeric} objects, which in
6714 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6715 isn't really needed here, but reminds us that the corresponding @code{eval()}
6716 function would require it in this place.
6718 Differentiation will surely turn up and so we need to tell @code{cos}
6719 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6720 instance, are then handled automatically by @code{basic::diff} and
6724 static ex cos_deriv(const ex & x, unsigned diff_param)
6730 @cindex product rule
6731 The second parameter is obligatory but uninteresting at this point. It
6732 specifies which parameter to differentiate in a partial derivative in
6733 case the function has more than one parameter, and its main application
6734 is for correct handling of the chain rule.
6736 An implementation of the series expansion is not needed for @code{cos()} as
6737 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6738 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6739 the other hand, does have poles and may need to do Laurent expansion:
6742 static ex tan_series(const ex & x, const relational & rel,
6743 int order, unsigned options)
6745 // Find the actual expansion point
6746 const ex x_pt = x.subs(rel);
6748 if ("x_pt is not an odd multiple of Pi/2")
6749 throw do_taylor(); // tell function::series() to do Taylor expansion
6751 // On a pole, expand sin()/cos()
6752 return (sin(x)/cos(x)).series(rel, order+2, options);
6756 The @code{series()} implementation of a function @emph{must} return a
6757 @code{pseries} object, otherwise your code will crash.
6759 @subsection Function options
6761 GiNaC functions understand several more options which are always
6762 specified as @code{.option(params)}. None of them are required, but you
6763 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6764 is a do-nothing option called @code{dummy()} which you can use to define
6765 functions without any special options.
6768 eval_func(<C++ function>)
6769 evalf_func(<C++ function>)
6770 derivative_func(<C++ function>)
6771 series_func(<C++ function>)
6772 conjugate_func(<C++ function>)
6775 These specify the C++ functions that implement symbolic evaluation,
6776 numeric evaluation, partial derivatives, and series expansion, respectively.
6777 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6778 @code{diff()} and @code{series()}.
6780 The @code{eval_func()} function needs to use @code{.hold()} if no further
6781 automatic evaluation is desired or possible.
6783 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6784 expansion, which is correct if there are no poles involved. If the function
6785 has poles in the complex plane, the @code{series_func()} needs to check
6786 whether the expansion point is on a pole and fall back to Taylor expansion
6787 if it isn't. Otherwise, the pole usually needs to be regularized by some
6788 suitable transformation.
6791 latex_name(const string & n)
6794 specifies the LaTeX code that represents the name of the function in LaTeX
6795 output. The default is to put the function name in an @code{\mbox@{@}}.
6798 do_not_evalf_params()
6801 This tells @code{evalf()} to not recursively evaluate the parameters of the
6802 function before calling the @code{evalf_func()}.
6805 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6808 This allows you to explicitly specify the commutation properties of the
6809 function (@xref{Non-commutative objects}, for an explanation of
6810 (non)commutativity in GiNaC). For example, you can use
6811 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6812 GiNaC treat your function like a matrix. By default, functions inherit the
6813 commutation properties of their first argument.
6816 set_symmetry(const symmetry & s)
6819 specifies the symmetry properties of the function with respect to its
6820 arguments. @xref{Indexed objects}, for an explanation of symmetry
6821 specifications. GiNaC will automatically rearrange the arguments of
6822 symmetric functions into a canonical order.
6824 Sometimes you may want to have finer control over how functions are
6825 displayed in the output. For example, the @code{abs()} function prints
6826 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6827 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6831 print_func<C>(<C++ function>)
6834 option which is explained in the next section.
6836 @subsection Functions with a variable number of arguments
6838 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6839 functions with a fixed number of arguments. Sometimes, though, you may need
6840 to have a function that accepts a variable number of expressions. One way to
6841 accomplish this is to pass variable-length lists as arguments. The
6842 @code{Li()} function uses this method for multiple polylogarithms.
6844 It is also possible to define functions that accept a different number of
6845 parameters under the same function name, such as the @code{psi()} function
6846 which can be called either as @code{psi(z)} (the digamma function) or as
6847 @code{psi(n, z)} (polygamma functions). These are actually two different
6848 functions in GiNaC that, however, have the same name. Defining such
6849 functions is not possible with the macros but requires manually fiddling
6850 with GiNaC internals. If you are interested, please consult the GiNaC source
6851 code for the @code{psi()} function (@file{inifcns.h} and
6852 @file{inifcns_gamma.cpp}).
6855 @node Printing, Structures, Symbolic functions, Extending GiNaC
6856 @c node-name, next, previous, up
6857 @section GiNaC's expression output system
6859 GiNaC allows the output of expressions in a variety of different formats
6860 (@pxref{Input/Output}). This section will explain how expression output
6861 is implemented internally, and how to define your own output formats or
6862 change the output format of built-in algebraic objects. You will also want
6863 to read this section if you plan to write your own algebraic classes or
6866 @cindex @code{print_context} (class)
6867 @cindex @code{print_dflt} (class)
6868 @cindex @code{print_latex} (class)
6869 @cindex @code{print_tree} (class)
6870 @cindex @code{print_csrc} (class)
6871 All the different output formats are represented by a hierarchy of classes
6872 rooted in the @code{print_context} class, defined in the @file{print.h}
6877 the default output format
6879 output in LaTeX mathematical mode
6881 a dump of the internal expression structure (for debugging)
6883 the base class for C source output
6884 @item print_csrc_float
6885 C source output using the @code{float} type
6886 @item print_csrc_double
6887 C source output using the @code{double} type
6888 @item print_csrc_cl_N
6889 C source output using CLN types
6892 The @code{print_context} base class provides two public data members:
6904 @code{s} is a reference to the stream to output to, while @code{options}
6905 holds flags and modifiers. Currently, there is only one flag defined:
6906 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6907 to print the index dimension which is normally hidden.
6909 When you write something like @code{std::cout << e}, where @code{e} is
6910 an object of class @code{ex}, GiNaC will construct an appropriate
6911 @code{print_context} object (of a class depending on the selected output
6912 format), fill in the @code{s} and @code{options} members, and call
6914 @cindex @code{print()}
6916 void ex::print(const print_context & c, unsigned level = 0) const;
6919 which in turn forwards the call to the @code{print()} method of the
6920 top-level algebraic object contained in the expression.
6922 Unlike other methods, GiNaC classes don't usually override their
6923 @code{print()} method to implement expression output. Instead, the default
6924 implementation @code{basic::print(c, level)} performs a run-time double
6925 dispatch to a function selected by the dynamic type of the object and the
6926 passed @code{print_context}. To this end, GiNaC maintains a separate method
6927 table for each class, similar to the virtual function table used for ordinary
6928 (single) virtual function dispatch.
6930 The method table contains one slot for each possible @code{print_context}
6931 type, indexed by the (internally assigned) serial number of the type. Slots
6932 may be empty, in which case GiNaC will retry the method lookup with the
6933 @code{print_context} object's parent class, possibly repeating the process
6934 until it reaches the @code{print_context} base class. If there's still no
6935 method defined, the method table of the algebraic object's parent class
6936 is consulted, and so on, until a matching method is found (eventually it
6937 will reach the combination @code{basic/print_context}, which prints the
6938 object's class name enclosed in square brackets).
6940 You can think of the print methods of all the different classes and output
6941 formats as being arranged in a two-dimensional matrix with one axis listing
6942 the algebraic classes and the other axis listing the @code{print_context}
6945 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6946 to implement printing, but then they won't get any of the benefits of the
6947 double dispatch mechanism (such as the ability for derived classes to
6948 inherit only certain print methods from its parent, or the replacement of
6949 methods at run-time).
6951 @subsection Print methods for classes
6953 The method table for a class is set up either in the definition of the class,
6954 by passing the appropriate @code{print_func<C>()} option to
6955 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6956 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6957 can also be used to override existing methods dynamically.
6959 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6960 be a member function of the class (or one of its parent classes), a static
6961 member function, or an ordinary (global) C++ function. The @code{C} template
6962 parameter specifies the appropriate @code{print_context} type for which the
6963 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6964 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6965 the class is the one being implemented by
6966 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6968 For print methods that are member functions, their first argument must be of
6969 a type convertible to a @code{const C &}, and the second argument must be an
6972 For static members and global functions, the first argument must be of a type
6973 convertible to a @code{const T &}, the second argument must be of a type
6974 convertible to a @code{const C &}, and the third argument must be an
6975 @code{unsigned}. A global function will, of course, not have access to
6976 private and protected members of @code{T}.
6978 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6979 and @code{basic::print()}) is used for proper parenthesizing of the output
6980 (and by @code{print_tree} for proper indentation). It can be used for similar
6981 purposes if you write your own output formats.
6983 The explanations given above may seem complicated, but in practice it's
6984 really simple, as shown in the following example. Suppose that we want to
6985 display exponents in LaTeX output not as superscripts but with little
6986 upwards-pointing arrows. This can be achieved in the following way:
6989 void my_print_power_as_latex(const power & p,
6990 const print_latex & c,
6993 // get the precedence of the 'power' class
6994 unsigned power_prec = p.precedence();
6996 // if the parent operator has the same or a higher precedence
6997 // we need parentheses around the power
6998 if (level >= power_prec)
7001 // print the basis and exponent, each enclosed in braces, and
7002 // separated by an uparrow
7004 p.op(0).print(c, power_prec);
7005 c.s << "@}\\uparrow@{";
7006 p.op(1).print(c, power_prec);
7009 // don't forget the closing parenthesis
7010 if (level >= power_prec)
7016 // a sample expression
7017 symbol x("x"), y("y");
7018 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7020 // switch to LaTeX mode
7023 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7026 // now we replace the method for the LaTeX output of powers with
7028 set_print_func<power, print_latex>(my_print_power_as_latex);
7030 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7041 The first argument of @code{my_print_power_as_latex} could also have been
7042 a @code{const basic &}, the second one a @code{const print_context &}.
7045 The above code depends on @code{mul} objects converting their operands to
7046 @code{power} objects for the purpose of printing.
7049 The output of products including negative powers as fractions is also
7050 controlled by the @code{mul} class.
7053 The @code{power/print_latex} method provided by GiNaC prints square roots
7054 using @code{\sqrt}, but the above code doesn't.
7058 It's not possible to restore a method table entry to its previous or default
7059 value. Once you have called @code{set_print_func()}, you can only override
7060 it with another call to @code{set_print_func()}, but you can't easily go back
7061 to the default behavior again (you can, of course, dig around in the GiNaC
7062 sources, find the method that is installed at startup
7063 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7064 one; that is, after you circumvent the C++ member access control@dots{}).
7066 @subsection Print methods for functions
7068 Symbolic functions employ a print method dispatch mechanism similar to the
7069 one used for classes. The methods are specified with @code{print_func<C>()}
7070 function options. If you don't specify any special print methods, the function
7071 will be printed with its name (or LaTeX name, if supplied), followed by a
7072 comma-separated list of arguments enclosed in parentheses.
7074 For example, this is what GiNaC's @samp{abs()} function is defined like:
7077 static ex abs_eval(const ex & arg) @{ ... @}
7078 static ex abs_evalf(const ex & arg) @{ ... @}
7080 static void abs_print_latex(const ex & arg, const print_context & c)
7082 c.s << "@{|"; arg.print(c); c.s << "|@}";
7085 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7087 c.s << "fabs("; arg.print(c); c.s << ")";
7090 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7091 evalf_func(abs_evalf).
7092 print_func<print_latex>(abs_print_latex).
7093 print_func<print_csrc_float>(abs_print_csrc_float).
7094 print_func<print_csrc_double>(abs_print_csrc_float));
7097 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7098 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7100 There is currently no equivalent of @code{set_print_func()} for functions.
7102 @subsection Adding new output formats
7104 Creating a new output format involves subclassing @code{print_context},
7105 which is somewhat similar to adding a new algebraic class
7106 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7107 that needs to go into the class definition, and a corresponding macro
7108 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7109 Every @code{print_context} class needs to provide a default constructor
7110 and a constructor from an @code{std::ostream} and an @code{unsigned}
7113 Here is an example for a user-defined @code{print_context} class:
7116 class print_myformat : public print_dflt
7118 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7120 print_myformat(std::ostream & os, unsigned opt = 0)
7121 : print_dflt(os, opt) @{@}
7124 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7126 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7129 That's all there is to it. None of the actual expression output logic is
7130 implemented in this class. It merely serves as a selector for choosing
7131 a particular format. The algorithms for printing expressions in the new
7132 format are implemented as print methods, as described above.
7134 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7135 exactly like GiNaC's default output format:
7140 ex e = pow(x, 2) + 1;
7142 // this prints "1+x^2"
7145 // this also prints "1+x^2"
7146 e.print(print_myformat()); cout << endl;
7152 To fill @code{print_myformat} with life, we need to supply appropriate
7153 print methods with @code{set_print_func()}, like this:
7156 // This prints powers with '**' instead of '^'. See the LaTeX output
7157 // example above for explanations.
7158 void print_power_as_myformat(const power & p,
7159 const print_myformat & c,
7162 unsigned power_prec = p.precedence();
7163 if (level >= power_prec)
7165 p.op(0).print(c, power_prec);
7167 p.op(1).print(c, power_prec);
7168 if (level >= power_prec)
7174 // install a new print method for power objects
7175 set_print_func<power, print_myformat>(print_power_as_myformat);
7177 // now this prints "1+x**2"
7178 e.print(print_myformat()); cout << endl;
7180 // but the default format is still "1+x^2"
7186 @node Structures, Adding classes, Printing, Extending GiNaC
7187 @c node-name, next, previous, up
7190 If you are doing some very specialized things with GiNaC, or if you just
7191 need some more organized way to store data in your expressions instead of
7192 anonymous lists, you may want to implement your own algebraic classes.
7193 ('algebraic class' means any class directly or indirectly derived from
7194 @code{basic} that can be used in GiNaC expressions).
7196 GiNaC offers two ways of accomplishing this: either by using the
7197 @code{structure<T>} template class, or by rolling your own class from
7198 scratch. This section will discuss the @code{structure<T>} template which
7199 is easier to use but more limited, while the implementation of custom
7200 GiNaC classes is the topic of the next section. However, you may want to
7201 read both sections because many common concepts and member functions are
7202 shared by both concepts, and it will also allow you to decide which approach
7203 is most suited to your needs.
7205 The @code{structure<T>} template, defined in the GiNaC header file
7206 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7207 or @code{class}) into a GiNaC object that can be used in expressions.
7209 @subsection Example: scalar products
7211 Let's suppose that we need a way to handle some kind of abstract scalar
7212 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7213 product class have to store their left and right operands, which can in turn
7214 be arbitrary expressions. Here is a possible way to represent such a
7215 product in a C++ @code{struct}:
7219 using namespace std;
7221 #include <ginac/ginac.h>
7222 using namespace GiNaC;
7228 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7232 The default constructor is required. Now, to make a GiNaC class out of this
7233 data structure, we need only one line:
7236 typedef structure<sprod_s> sprod;
7239 That's it. This line constructs an algebraic class @code{sprod} which
7240 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7241 expressions like any other GiNaC class:
7245 symbol a("a"), b("b");
7246 ex e = sprod(sprod_s(a, b));
7250 Note the difference between @code{sprod} which is the algebraic class, and
7251 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7252 and @code{right} data members. As shown above, an @code{sprod} can be
7253 constructed from an @code{sprod_s} object.
7255 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7256 you could define a little wrapper function like this:
7259 inline ex make_sprod(ex left, ex right)
7261 return sprod(sprod_s(left, right));
7265 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7266 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7267 @code{get_struct()}:
7271 cout << ex_to<sprod>(e)->left << endl;
7273 cout << ex_to<sprod>(e).get_struct().right << endl;
7278 You only have read access to the members of @code{sprod_s}.
7280 The type definition of @code{sprod} is enough to write your own algorithms
7281 that deal with scalar products, for example:
7286 if (is_a<sprod>(p)) @{
7287 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7288 return make_sprod(sp.right, sp.left);
7299 @subsection Structure output
7301 While the @code{sprod} type is useable it still leaves something to be
7302 desired, most notably proper output:
7307 // -> [structure object]
7311 By default, any structure types you define will be printed as
7312 @samp{[structure object]}. To override this you can either specialize the
7313 template's @code{print()} member function, or specify print methods with
7314 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7315 it's not possible to supply class options like @code{print_func<>()} to
7316 structures, so for a self-contained structure type you need to resort to
7317 overriding the @code{print()} function, which is also what we will do here.
7319 The member functions of GiNaC classes are described in more detail in the
7320 next section, but it shouldn't be hard to figure out what's going on here:
7323 void sprod::print(const print_context & c, unsigned level) const
7325 // tree debug output handled by superclass
7326 if (is_a<print_tree>(c))
7327 inherited::print(c, level);
7329 // get the contained sprod_s object
7330 const sprod_s & sp = get_struct();
7332 // print_context::s is a reference to an ostream
7333 c.s << "<" << sp.left << "|" << sp.right << ">";
7337 Now we can print expressions containing scalar products:
7343 cout << swap_sprod(e) << endl;
7348 @subsection Comparing structures
7350 The @code{sprod} class defined so far still has one important drawback: all
7351 scalar products are treated as being equal because GiNaC doesn't know how to
7352 compare objects of type @code{sprod_s}. This can lead to some confusing
7353 and undesired behavior:
7357 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7359 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7360 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7364 To remedy this, we first need to define the operators @code{==} and @code{<}
7365 for objects of type @code{sprod_s}:
7368 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7370 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7373 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7375 return lhs.left.compare(rhs.left) < 0
7376 ? true : lhs.right.compare(rhs.right) < 0;
7380 The ordering established by the @code{<} operator doesn't have to make any
7381 algebraic sense, but it needs to be well defined. Note that we can't use
7382 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7383 in the implementation of these operators because they would construct
7384 GiNaC @code{relational} objects which in the case of @code{<} do not
7385 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7386 decide which one is algebraically 'less').
7388 Next, we need to change our definition of the @code{sprod} type to let
7389 GiNaC know that an ordering relation exists for the embedded objects:
7392 typedef structure<sprod_s, compare_std_less> sprod;
7395 @code{sprod} objects then behave as expected:
7399 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7400 // -> <a|b>-<a^2|b^2>
7401 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7402 // -> <a|b>+<a^2|b^2>
7403 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7405 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7410 The @code{compare_std_less} policy parameter tells GiNaC to use the
7411 @code{std::less} and @code{std::equal_to} functors to compare objects of
7412 type @code{sprod_s}. By default, these functors forward their work to the
7413 standard @code{<} and @code{==} operators, which we have overloaded.
7414 Alternatively, we could have specialized @code{std::less} and
7415 @code{std::equal_to} for class @code{sprod_s}.
7417 GiNaC provides two other comparison policies for @code{structure<T>}
7418 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7419 which does a bit-wise comparison of the contained @code{T} objects.
7420 This should be used with extreme care because it only works reliably with
7421 built-in integral types, and it also compares any padding (filler bytes of
7422 undefined value) that the @code{T} class might have.
7424 @subsection Subexpressions
7426 Our scalar product class has two subexpressions: the left and right
7427 operands. It might be a good idea to make them accessible via the standard
7428 @code{nops()} and @code{op()} methods:
7431 size_t sprod::nops() const
7436 ex sprod::op(size_t i) const
7440 return get_struct().left;
7442 return get_struct().right;
7444 throw std::range_error("sprod::op(): no such operand");
7449 Implementing @code{nops()} and @code{op()} for container types such as
7450 @code{sprod} has two other nice side effects:
7454 @code{has()} works as expected
7456 GiNaC generates better hash keys for the objects (the default implementation
7457 of @code{calchash()} takes subexpressions into account)
7460 @cindex @code{let_op()}
7461 There is a non-const variant of @code{op()} called @code{let_op()} that
7462 allows replacing subexpressions:
7465 ex & sprod::let_op(size_t i)
7467 // every non-const member function must call this
7468 ensure_if_modifiable();
7472 return get_struct().left;
7474 return get_struct().right;
7476 throw std::range_error("sprod::let_op(): no such operand");
7481 Once we have provided @code{let_op()} we also get @code{subs()} and
7482 @code{map()} for free. In fact, every container class that returns a non-null
7483 @code{nops()} value must either implement @code{let_op()} or provide custom
7484 implementations of @code{subs()} and @code{map()}.
7486 In turn, the availability of @code{map()} enables the recursive behavior of a
7487 couple of other default method implementations, in particular @code{evalf()},
7488 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7489 we probably want to provide our own version of @code{expand()} for scalar
7490 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7491 This is left as an exercise for the reader.
7493 The @code{structure<T>} template defines many more member functions that
7494 you can override by specialization to customize the behavior of your
7495 structures. You are referred to the next section for a description of
7496 some of these (especially @code{eval()}). There is, however, one topic
7497 that shall be addressed here, as it demonstrates one peculiarity of the
7498 @code{structure<T>} template: archiving.
7500 @subsection Archiving structures
7502 If you don't know how the archiving of GiNaC objects is implemented, you
7503 should first read the next section and then come back here. You're back?
7506 To implement archiving for structures it is not enough to provide
7507 specializations for the @code{archive()} member function and the
7508 unarchiving constructor (the @code{unarchive()} function has a default
7509 implementation). You also need to provide a unique name (as a string literal)
7510 for each structure type you define. This is because in GiNaC archives,
7511 the class of an object is stored as a string, the class name.
7513 By default, this class name (as returned by the @code{class_name()} member
7514 function) is @samp{structure} for all structure classes. This works as long
7515 as you have only defined one structure type, but if you use two or more you
7516 need to provide a different name for each by specializing the
7517 @code{get_class_name()} member function. Here is a sample implementation
7518 for enabling archiving of the scalar product type defined above:
7521 const char *sprod::get_class_name() @{ return "sprod"; @}
7523 void sprod::archive(archive_node & n) const
7525 inherited::archive(n);
7526 n.add_ex("left", get_struct().left);
7527 n.add_ex("right", get_struct().right);
7530 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7532 n.find_ex("left", get_struct().left, sym_lst);
7533 n.find_ex("right", get_struct().right, sym_lst);
7537 Note that the unarchiving constructor is @code{sprod::structure} and not
7538 @code{sprod::sprod}, and that we don't need to supply an
7539 @code{sprod::unarchive()} function.
7542 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7543 @c node-name, next, previous, up
7544 @section Adding classes
7546 The @code{structure<T>} template provides an way to extend GiNaC with custom
7547 algebraic classes that is easy to use but has its limitations, the most
7548 severe of which being that you can't add any new member functions to
7549 structures. To be able to do this, you need to write a new class definition
7552 This section will explain how to implement new algebraic classes in GiNaC by
7553 giving the example of a simple 'string' class. After reading this section
7554 you will know how to properly declare a GiNaC class and what the minimum
7555 required member functions are that you have to implement. We only cover the
7556 implementation of a 'leaf' class here (i.e. one that doesn't contain
7557 subexpressions). Creating a container class like, for example, a class
7558 representing tensor products is more involved but this section should give
7559 you enough information so you can consult the source to GiNaC's predefined
7560 classes if you want to implement something more complicated.
7562 @subsection GiNaC's run-time type information system
7564 @cindex hierarchy of classes
7566 All algebraic classes (that is, all classes that can appear in expressions)
7567 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7568 @code{basic *} (which is essentially what an @code{ex} is) represents a
7569 generic pointer to an algebraic class. Occasionally it is necessary to find
7570 out what the class of an object pointed to by a @code{basic *} really is.
7571 Also, for the unarchiving of expressions it must be possible to find the
7572 @code{unarchive()} function of a class given the class name (as a string). A
7573 system that provides this kind of information is called a run-time type
7574 information (RTTI) system. The C++ language provides such a thing (see the
7575 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7576 implements its own, simpler RTTI.
7578 The RTTI in GiNaC is based on two mechanisms:
7583 The @code{basic} class declares a member variable @code{tinfo_key} which
7584 holds an unsigned integer that identifies the object's class. These numbers
7585 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7586 classes. They all start with @code{TINFO_}.
7589 By means of some clever tricks with static members, GiNaC maintains a list
7590 of information for all classes derived from @code{basic}. The information
7591 available includes the class names, the @code{tinfo_key}s, and pointers
7592 to the unarchiving functions. This class registry is defined in the
7593 @file{registrar.h} header file.
7597 The disadvantage of this proprietary RTTI implementation is that there's
7598 a little more to do when implementing new classes (C++'s RTTI works more
7599 or less automatically) but don't worry, most of the work is simplified by
7602 @subsection A minimalistic example
7604 Now we will start implementing a new class @code{mystring} that allows
7605 placing character strings in algebraic expressions (this is not very useful,
7606 but it's just an example). This class will be a direct subclass of
7607 @code{basic}. You can use this sample implementation as a starting point
7608 for your own classes.
7610 The code snippets given here assume that you have included some header files
7616 #include <stdexcept>
7617 using namespace std;
7619 #include <ginac/ginac.h>
7620 using namespace GiNaC;
7623 The first thing we have to do is to define a @code{tinfo_key} for our new
7624 class. This can be any arbitrary unsigned number that is not already taken
7625 by one of the existing classes but it's better to come up with something
7626 that is unlikely to clash with keys that might be added in the future. The
7627 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7628 which is not a requirement but we are going to stick with this scheme:
7631 const unsigned TINFO_mystring = 0x42420001U;
7634 Now we can write down the class declaration. The class stores a C++
7635 @code{string} and the user shall be able to construct a @code{mystring}
7636 object from a C or C++ string:
7639 class mystring : public basic
7641 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7644 mystring(const string &s);
7645 mystring(const char *s);
7651 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7654 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7655 macros are defined in @file{registrar.h}. They take the name of the class
7656 and its direct superclass as arguments and insert all required declarations
7657 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7658 the first line after the opening brace of the class definition. The
7659 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7660 source (at global scope, of course, not inside a function).
7662 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7663 declarations of the default constructor and a couple of other functions that
7664 are required. It also defines a type @code{inherited} which refers to the
7665 superclass so you don't have to modify your code every time you shuffle around
7666 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7667 class with the GiNaC RTTI (there is also a
7668 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7669 options for the class, and which we will be using instead in a few minutes).
7671 Now there are seven member functions we have to implement to get a working
7677 @code{mystring()}, the default constructor.
7680 @code{void archive(archive_node &n)}, the archiving function. This stores all
7681 information needed to reconstruct an object of this class inside an
7682 @code{archive_node}.
7685 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7686 constructor. This constructs an instance of the class from the information
7687 found in an @code{archive_node}.
7690 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7691 unarchiving function. It constructs a new instance by calling the unarchiving
7695 @cindex @code{compare_same_type()}
7696 @code{int compare_same_type(const basic &other)}, which is used internally
7697 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7698 -1, depending on the relative order of this object and the @code{other}
7699 object. If it returns 0, the objects are considered equal.
7700 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7701 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7702 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7703 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7704 must provide a @code{compare_same_type()} function, even those representing
7705 objects for which no reasonable algebraic ordering relationship can be
7709 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7710 which are the two constructors we declared.
7714 Let's proceed step-by-step. The default constructor looks like this:
7717 mystring::mystring() : inherited(TINFO_mystring) @{@}
7720 The golden rule is that in all constructors you have to set the
7721 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7722 it will be set by the constructor of the superclass and all hell will break
7723 loose in the RTTI. For your convenience, the @code{basic} class provides
7724 a constructor that takes a @code{tinfo_key} value, which we are using here
7725 (remember that in our case @code{inherited == basic}). If the superclass
7726 didn't have such a constructor, we would have to set the @code{tinfo_key}
7727 to the right value manually.
7729 In the default constructor you should set all other member variables to
7730 reasonable default values (we don't need that here since our @code{str}
7731 member gets set to an empty string automatically).
7733 Next are the three functions for archiving. You have to implement them even
7734 if you don't plan to use archives, but the minimum required implementation
7735 is really simple. First, the archiving function:
7738 void mystring::archive(archive_node &n) const
7740 inherited::archive(n);
7741 n.add_string("string", str);
7745 The only thing that is really required is calling the @code{archive()}
7746 function of the superclass. Optionally, you can store all information you
7747 deem necessary for representing the object into the passed
7748 @code{archive_node}. We are just storing our string here. For more
7749 information on how the archiving works, consult the @file{archive.h} header
7752 The unarchiving constructor is basically the inverse of the archiving
7756 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7758 n.find_string("string", str);
7762 If you don't need archiving, just leave this function empty (but you must
7763 invoke the unarchiving constructor of the superclass). Note that we don't
7764 have to set the @code{tinfo_key} here because it is done automatically
7765 by the unarchiving constructor of the @code{basic} class.
7767 Finally, the unarchiving function:
7770 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7772 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7776 You don't have to understand how exactly this works. Just copy these
7777 four lines into your code literally (replacing the class name, of
7778 course). It calls the unarchiving constructor of the class and unless
7779 you are doing something very special (like matching @code{archive_node}s
7780 to global objects) you don't need a different implementation. For those
7781 who are interested: setting the @code{dynallocated} flag puts the object
7782 under the control of GiNaC's garbage collection. It will get deleted
7783 automatically once it is no longer referenced.
7785 Our @code{compare_same_type()} function uses a provided function to compare
7789 int mystring::compare_same_type(const basic &other) const
7791 const mystring &o = static_cast<const mystring &>(other);
7792 int cmpval = str.compare(o.str);
7795 else if (cmpval < 0)
7802 Although this function takes a @code{basic &}, it will always be a reference
7803 to an object of exactly the same class (objects of different classes are not
7804 comparable), so the cast is safe. If this function returns 0, the two objects
7805 are considered equal (in the sense that @math{A-B=0}), so you should compare
7806 all relevant member variables.
7808 Now the only thing missing is our two new constructors:
7811 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7812 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7815 No surprises here. We set the @code{str} member from the argument and
7816 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7818 That's it! We now have a minimal working GiNaC class that can store
7819 strings in algebraic expressions. Let's confirm that the RTTI works:
7822 ex e = mystring("Hello, world!");
7823 cout << is_a<mystring>(e) << endl;
7826 cout << ex_to<basic>(e).class_name() << endl;
7830 Obviously it does. Let's see what the expression @code{e} looks like:
7834 // -> [mystring object]
7837 Hm, not exactly what we expect, but of course the @code{mystring} class
7838 doesn't yet know how to print itself. This can be done either by implementing
7839 the @code{print()} member function, or, preferably, by specifying a
7840 @code{print_func<>()} class option. Let's say that we want to print the string
7841 surrounded by double quotes:
7844 class mystring : public basic
7848 void do_print(const print_context &c, unsigned level = 0) const;
7852 void mystring::do_print(const print_context &c, unsigned level) const
7854 // print_context::s is a reference to an ostream
7855 c.s << '\"' << str << '\"';
7859 The @code{level} argument is only required for container classes to
7860 correctly parenthesize the output.
7862 Now we need to tell GiNaC that @code{mystring} objects should use the
7863 @code{do_print()} member function for printing themselves. For this, we
7867 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7873 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7874 print_func<print_context>(&mystring::do_print))
7877 Let's try again to print the expression:
7881 // -> "Hello, world!"
7884 Much better. If we wanted to have @code{mystring} objects displayed in a
7885 different way depending on the output format (default, LaTeX, etc.), we
7886 would have supplied multiple @code{print_func<>()} options with different
7887 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7888 separated by dots. This is similar to the way options are specified for
7889 symbolic functions. @xref{Printing}, for a more in-depth description of the
7890 way expression output is implemented in GiNaC.
7892 The @code{mystring} class can be used in arbitrary expressions:
7895 e += mystring("GiNaC rulez");
7897 // -> "GiNaC rulez"+"Hello, world!"
7900 (GiNaC's automatic term reordering is in effect here), or even
7903 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7905 // -> "One string"^(2*sin(-"Another string"+Pi))
7908 Whether this makes sense is debatable but remember that this is only an
7909 example. At least it allows you to implement your own symbolic algorithms
7912 Note that GiNaC's algebraic rules remain unchanged:
7915 e = mystring("Wow") * mystring("Wow");
7919 e = pow(mystring("First")-mystring("Second"), 2);
7920 cout << e.expand() << endl;
7921 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7924 There's no way to, for example, make GiNaC's @code{add} class perform string
7925 concatenation. You would have to implement this yourself.
7927 @subsection Automatic evaluation
7930 @cindex @code{eval()}
7931 @cindex @code{hold()}
7932 When dealing with objects that are just a little more complicated than the
7933 simple string objects we have implemented, chances are that you will want to
7934 have some automatic simplifications or canonicalizations performed on them.
7935 This is done in the evaluation member function @code{eval()}. Let's say that
7936 we wanted all strings automatically converted to lowercase with
7937 non-alphabetic characters stripped, and empty strings removed:
7940 class mystring : public basic
7944 ex eval(int level = 0) const;
7948 ex mystring::eval(int level) const
7951 for (int i=0; i<str.length(); i++) @{
7953 if (c >= 'A' && c <= 'Z')
7954 new_str += tolower(c);
7955 else if (c >= 'a' && c <= 'z')
7959 if (new_str.length() == 0)
7962 return mystring(new_str).hold();
7966 The @code{level} argument is used to limit the recursion depth of the
7967 evaluation. We don't have any subexpressions in the @code{mystring}
7968 class so we are not concerned with this. If we had, we would call the
7969 @code{eval()} functions of the subexpressions with @code{level - 1} as
7970 the argument if @code{level != 1}. The @code{hold()} member function
7971 sets a flag in the object that prevents further evaluation. Otherwise
7972 we might end up in an endless loop. When you want to return the object
7973 unmodified, use @code{return this->hold();}.
7975 Let's confirm that it works:
7978 ex e = mystring("Hello, world!") + mystring("!?#");
7982 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7987 @subsection Optional member functions
7989 We have implemented only a small set of member functions to make the class
7990 work in the GiNaC framework. There are two functions that are not strictly
7991 required but will make operations with objects of the class more efficient:
7993 @cindex @code{calchash()}
7994 @cindex @code{is_equal_same_type()}
7996 unsigned calchash() const;
7997 bool is_equal_same_type(const basic &other) const;
8000 The @code{calchash()} method returns an @code{unsigned} hash value for the
8001 object which will allow GiNaC to compare and canonicalize expressions much
8002 more efficiently. You should consult the implementation of some of the built-in
8003 GiNaC classes for examples of hash functions. The default implementation of
8004 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8005 class and all subexpressions that are accessible via @code{op()}.
8007 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8008 tests for equality without establishing an ordering relation, which is often
8009 faster. The default implementation of @code{is_equal_same_type()} just calls
8010 @code{compare_same_type()} and tests its result for zero.
8012 @subsection Other member functions
8014 For a real algebraic class, there are probably some more functions that you
8015 might want to provide:
8018 bool info(unsigned inf) const;
8019 ex evalf(int level = 0) const;
8020 ex series(const relational & r, int order, unsigned options = 0) const;
8021 ex derivative(const symbol & s) const;
8024 If your class stores sub-expressions (see the scalar product example in the
8025 previous section) you will probably want to override
8027 @cindex @code{let_op()}
8030 ex op(size_t i) const;
8031 ex & let_op(size_t i);
8032 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8033 ex map(map_function & f) const;
8036 @code{let_op()} is a variant of @code{op()} that allows write access. The
8037 default implementations of @code{subs()} and @code{map()} use it, so you have
8038 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8040 You can, of course, also add your own new member functions. Remember
8041 that the RTTI may be used to get information about what kinds of objects
8042 you are dealing with (the position in the class hierarchy) and that you
8043 can always extract the bare object from an @code{ex} by stripping the
8044 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8045 should become a need.
8047 That's it. May the source be with you!
8050 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8051 @c node-name, next, previous, up
8052 @chapter A Comparison With Other CAS
8055 This chapter will give you some information on how GiNaC compares to
8056 other, traditional Computer Algebra Systems, like @emph{Maple},
8057 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8058 disadvantages over these systems.
8061 * Advantages:: Strengths of the GiNaC approach.
8062 * Disadvantages:: Weaknesses of the GiNaC approach.
8063 * Why C++?:: Attractiveness of C++.
8066 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8067 @c node-name, next, previous, up
8070 GiNaC has several advantages over traditional Computer
8071 Algebra Systems, like
8076 familiar language: all common CAS implement their own proprietary
8077 grammar which you have to learn first (and maybe learn again when your
8078 vendor decides to `enhance' it). With GiNaC you can write your program
8079 in common C++, which is standardized.
8083 structured data types: you can build up structured data types using
8084 @code{struct}s or @code{class}es together with STL features instead of
8085 using unnamed lists of lists of lists.
8088 strongly typed: in CAS, you usually have only one kind of variables
8089 which can hold contents of an arbitrary type. This 4GL like feature is
8090 nice for novice programmers, but dangerous.
8093 development tools: powerful development tools exist for C++, like fancy
8094 editors (e.g. with automatic indentation and syntax highlighting),
8095 debuggers, visualization tools, documentation generators@dots{}
8098 modularization: C++ programs can easily be split into modules by
8099 separating interface and implementation.
8102 price: GiNaC is distributed under the GNU Public License which means
8103 that it is free and available with source code. And there are excellent
8104 C++-compilers for free, too.
8107 extendable: you can add your own classes to GiNaC, thus extending it on
8108 a very low level. Compare this to a traditional CAS that you can
8109 usually only extend on a high level by writing in the language defined
8110 by the parser. In particular, it turns out to be almost impossible to
8111 fix bugs in a traditional system.
8114 multiple interfaces: Though real GiNaC programs have to be written in
8115 some editor, then be compiled, linked and executed, there are more ways
8116 to work with the GiNaC engine. Many people want to play with
8117 expressions interactively, as in traditional CASs. Currently, two such
8118 windows into GiNaC have been implemented and many more are possible: the
8119 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8120 types to a command line and second, as a more consistent approach, an
8121 interactive interface to the Cint C++ interpreter has been put together
8122 (called GiNaC-cint) that allows an interactive scripting interface
8123 consistent with the C++ language. It is available from the usual GiNaC
8127 seamless integration: it is somewhere between difficult and impossible
8128 to call CAS functions from within a program written in C++ or any other
8129 programming language and vice versa. With GiNaC, your symbolic routines
8130 are part of your program. You can easily call third party libraries,
8131 e.g. for numerical evaluation or graphical interaction. All other
8132 approaches are much more cumbersome: they range from simply ignoring the
8133 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8134 system (i.e. @emph{Yacas}).
8137 efficiency: often large parts of a program do not need symbolic
8138 calculations at all. Why use large integers for loop variables or
8139 arbitrary precision arithmetics where @code{int} and @code{double} are
8140 sufficient? For pure symbolic applications, GiNaC is comparable in
8141 speed with other CAS.
8146 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8147 @c node-name, next, previous, up
8148 @section Disadvantages
8150 Of course it also has some disadvantages:
8155 advanced features: GiNaC cannot compete with a program like
8156 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8157 which grows since 1981 by the work of dozens of programmers, with
8158 respect to mathematical features. Integration, factorization,
8159 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8160 not planned for the near future).
8163 portability: While the GiNaC library itself is designed to avoid any
8164 platform dependent features (it should compile on any ANSI compliant C++
8165 compiler), the currently used version of the CLN library (fast large
8166 integer and arbitrary precision arithmetics) can only by compiled
8167 without hassle on systems with the C++ compiler from the GNU Compiler
8168 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8169 macros to let the compiler gather all static initializations, which
8170 works for GNU C++ only. Feel free to contact the authors in case you
8171 really believe that you need to use a different compiler. We have
8172 occasionally used other compilers and may be able to give you advice.}
8173 GiNaC uses recent language features like explicit constructors, mutable
8174 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8175 literally. Recent GCC versions starting at 2.95.3, although itself not
8176 yet ANSI compliant, support all needed features.
8181 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8182 @c node-name, next, previous, up
8185 Why did we choose to implement GiNaC in C++ instead of Java or any other
8186 language? C++ is not perfect: type checking is not strict (casting is
8187 possible), separation between interface and implementation is not
8188 complete, object oriented design is not enforced. The main reason is
8189 the often scolded feature of operator overloading in C++. While it may
8190 be true that operating on classes with a @code{+} operator is rarely
8191 meaningful, it is perfectly suited for algebraic expressions. Writing
8192 @math{3x+5y} as @code{3*x+5*y} instead of
8193 @code{x.times(3).plus(y.times(5))} looks much more natural.
8194 Furthermore, the main developers are more familiar with C++ than with
8195 any other programming language.
8198 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8199 @c node-name, next, previous, up
8200 @appendix Internal Structures
8203 * Expressions are reference counted::
8204 * Internal representation of products and sums::
8207 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8208 @c node-name, next, previous, up
8209 @appendixsection Expressions are reference counted
8211 @cindex reference counting
8212 @cindex copy-on-write
8213 @cindex garbage collection
8214 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8215 where the counter belongs to the algebraic objects derived from class
8216 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8217 which @code{ex} contains an instance. If you understood that, you can safely
8218 skip the rest of this passage.
8220 Expressions are extremely light-weight since internally they work like
8221 handles to the actual representation. They really hold nothing more
8222 than a pointer to some other object. What this means in practice is
8223 that whenever you create two @code{ex} and set the second equal to the
8224 first no copying process is involved. Instead, the copying takes place
8225 as soon as you try to change the second. Consider the simple sequence
8230 #include <ginac/ginac.h>
8231 using namespace std;
8232 using namespace GiNaC;
8236 symbol x("x"), y("y"), z("z");
8239 e1 = sin(x + 2*y) + 3*z + 41;
8240 e2 = e1; // e2 points to same object as e1
8241 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8242 e2 += 1; // e2 is copied into a new object
8243 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8247 The line @code{e2 = e1;} creates a second expression pointing to the
8248 object held already by @code{e1}. The time involved for this operation
8249 is therefore constant, no matter how large @code{e1} was. Actual
8250 copying, however, must take place in the line @code{e2 += 1;} because
8251 @code{e1} and @code{e2} are not handles for the same object any more.
8252 This concept is called @dfn{copy-on-write semantics}. It increases
8253 performance considerably whenever one object occurs multiple times and
8254 represents a simple garbage collection scheme because when an @code{ex}
8255 runs out of scope its destructor checks whether other expressions handle
8256 the object it points to too and deletes the object from memory if that
8257 turns out not to be the case. A slightly less trivial example of
8258 differentiation using the chain-rule should make clear how powerful this
8263 symbol x("x"), y("y");
8267 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8268 cout << e1 << endl // prints x+3*y
8269 << e2 << endl // prints (x+3*y)^3
8270 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8274 Here, @code{e1} will actually be referenced three times while @code{e2}
8275 will be referenced two times. When the power of an expression is built,
8276 that expression needs not be copied. Likewise, since the derivative of
8277 a power of an expression can be easily expressed in terms of that
8278 expression, no copying of @code{e1} is involved when @code{e3} is
8279 constructed. So, when @code{e3} is constructed it will print as
8280 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8281 holds a reference to @code{e2} and the factor in front is just
8284 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8285 semantics. When you insert an expression into a second expression, the
8286 result behaves exactly as if the contents of the first expression were
8287 inserted. But it may be useful to remember that this is not what
8288 happens. Knowing this will enable you to write much more efficient
8289 code. If you still have an uncertain feeling with copy-on-write
8290 semantics, we recommend you have a look at the
8291 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8292 Marshall Cline. Chapter 16 covers this issue and presents an
8293 implementation which is pretty close to the one in GiNaC.
8296 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8297 @c node-name, next, previous, up
8298 @appendixsection Internal representation of products and sums
8300 @cindex representation
8303 @cindex @code{power}
8304 Although it should be completely transparent for the user of
8305 GiNaC a short discussion of this topic helps to understand the sources
8306 and also explain performance to a large degree. Consider the
8307 unexpanded symbolic expression
8309 $2d^3 \left( 4a + 5b - 3 \right)$
8312 @math{2*d^3*(4*a+5*b-3)}
8314 which could naively be represented by a tree of linear containers for
8315 addition and multiplication, one container for exponentiation with base
8316 and exponent and some atomic leaves of symbols and numbers in this
8321 @cindex pair-wise representation
8322 However, doing so results in a rather deeply nested tree which will
8323 quickly become inefficient to manipulate. We can improve on this by
8324 representing the sum as a sequence of terms, each one being a pair of a
8325 purely numeric multiplicative coefficient and its rest. In the same
8326 spirit we can store the multiplication as a sequence of terms, each
8327 having a numeric exponent and a possibly complicated base, the tree
8328 becomes much more flat:
8332 The number @code{3} above the symbol @code{d} shows that @code{mul}
8333 objects are treated similarly where the coefficients are interpreted as
8334 @emph{exponents} now. Addition of sums of terms or multiplication of
8335 products with numerical exponents can be coded to be very efficient with
8336 such a pair-wise representation. Internally, this handling is performed
8337 by most CAS in this way. It typically speeds up manipulations by an
8338 order of magnitude. The overall multiplicative factor @code{2} and the
8339 additive term @code{-3} look somewhat out of place in this
8340 representation, however, since they are still carrying a trivial
8341 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8342 this is avoided by adding a field that carries an overall numeric
8343 coefficient. This results in the realistic picture of internal
8346 $2d^3 \left( 4a + 5b - 3 \right)$:
8349 @math{2*d^3*(4*a+5*b-3)}:
8355 This also allows for a better handling of numeric radicals, since
8356 @code{sqrt(2)} can now be carried along calculations. Now it should be
8357 clear, why both classes @code{add} and @code{mul} are derived from the
8358 same abstract class: the data representation is the same, only the
8359 semantics differs. In the class hierarchy, methods for polynomial
8360 expansion and the like are reimplemented for @code{add} and @code{mul},
8361 but the data structure is inherited from @code{expairseq}.
8364 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8365 @c node-name, next, previous, up
8366 @appendix Package Tools
8368 If you are creating a software package that uses the GiNaC library,
8369 setting the correct command line options for the compiler and linker
8370 can be difficult. GiNaC includes two tools to make this process easier.
8373 * ginac-config:: A shell script to detect compiler and linker flags.
8374 * AM_PATH_GINAC:: Macro for GNU automake.
8378 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8379 @c node-name, next, previous, up
8380 @section @command{ginac-config}
8381 @cindex ginac-config
8383 @command{ginac-config} is a shell script that you can use to determine
8384 the compiler and linker command line options required to compile and
8385 link a program with the GiNaC library.
8387 @command{ginac-config} takes the following flags:
8391 Prints out the version of GiNaC installed.
8393 Prints '-I' flags pointing to the installed header files.
8395 Prints out the linker flags necessary to link a program against GiNaC.
8396 @item --prefix[=@var{PREFIX}]
8397 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8398 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8399 Otherwise, prints out the configured value of @env{$prefix}.
8400 @item --exec-prefix[=@var{PREFIX}]
8401 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8402 Otherwise, prints out the configured value of @env{$exec_prefix}.
8405 Typically, @command{ginac-config} will be used within a configure
8406 script, as described below. It, however, can also be used directly from
8407 the command line using backquotes to compile a simple program. For
8411 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8414 This command line might expand to (for example):
8417 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8418 -lginac -lcln -lstdc++
8421 Not only is the form using @command{ginac-config} easier to type, it will
8422 work on any system, no matter how GiNaC was configured.
8425 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8426 @c node-name, next, previous, up
8427 @section @samp{AM_PATH_GINAC}
8428 @cindex AM_PATH_GINAC
8430 For packages configured using GNU automake, GiNaC also provides
8431 a macro to automate the process of checking for GiNaC.
8434 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8435 [, @var{ACTION-IF-NOT-FOUND}]]])
8443 Determines the location of GiNaC using @command{ginac-config}, which is
8444 either found in the user's path, or from the environment variable
8445 @env{GINACLIB_CONFIG}.
8448 Tests the installed libraries to make sure that their version
8449 is later than @var{MINIMUM-VERSION}. (A default version will be used
8453 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8454 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8455 variable to the output of @command{ginac-config --libs}, and calls
8456 @samp{AC_SUBST()} for these variables so they can be used in generated
8457 makefiles, and then executes @var{ACTION-IF-FOUND}.
8460 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8461 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8465 This macro is in file @file{ginac.m4} which is installed in
8466 @file{$datadir/aclocal}. Note that if automake was installed with a
8467 different @samp{--prefix} than GiNaC, you will either have to manually
8468 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8469 aclocal the @samp{-I} option when running it.
8472 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8473 * Example package:: Example of a package using AM_PATH_GINAC.
8477 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8478 @c node-name, next, previous, up
8479 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8481 Simply make sure that @command{ginac-config} is in your path, and run
8482 the configure script.
8489 The directory where the GiNaC libraries are installed needs
8490 to be found by your system's dynamic linker.
8492 This is generally done by
8495 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8501 setting the environment variable @env{LD_LIBRARY_PATH},
8504 or, as a last resort,
8507 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8508 running configure, for instance:
8511 LDFLAGS=-R/home/cbauer/lib ./configure
8516 You can also specify a @command{ginac-config} not in your path by
8517 setting the @env{GINACLIB_CONFIG} environment variable to the
8518 name of the executable
8521 If you move the GiNaC package from its installed location,
8522 you will either need to modify @command{ginac-config} script
8523 manually to point to the new location or rebuild GiNaC.
8534 --with-ginac-prefix=@var{PREFIX}
8535 --with-ginac-exec-prefix=@var{PREFIX}
8538 are provided to override the prefix and exec-prefix that were stored
8539 in the @command{ginac-config} shell script by GiNaC's configure. You are
8540 generally better off configuring GiNaC with the right path to begin with.
8544 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8545 @c node-name, next, previous, up
8546 @subsection Example of a package using @samp{AM_PATH_GINAC}
8548 The following shows how to build a simple package using automake
8549 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8553 #include <ginac/ginac.h>
8557 GiNaC::symbol x("x");
8558 GiNaC::ex a = GiNaC::sin(x);
8559 std::cout << "Derivative of " << a
8560 << " is " << a.diff(x) << std::endl;
8565 You should first read the introductory portions of the automake
8566 Manual, if you are not already familiar with it.
8568 Two files are needed, @file{configure.in}, which is used to build the
8572 dnl Process this file with autoconf to produce a configure script.
8574 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8580 AM_PATH_GINAC(0.9.0, [
8581 LIBS="$LIBS $GINACLIB_LIBS"
8582 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8583 ], AC_MSG_ERROR([need to have GiNaC installed]))
8588 The only command in this which is not standard for automake
8589 is the @samp{AM_PATH_GINAC} macro.
8591 That command does the following: If a GiNaC version greater or equal
8592 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8593 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8594 the error message `need to have GiNaC installed'
8596 And the @file{Makefile.am}, which will be used to build the Makefile.
8599 ## Process this file with automake to produce Makefile.in
8600 bin_PROGRAMS = simple
8601 simple_SOURCES = simple.cpp
8604 This @file{Makefile.am}, says that we are building a single executable,
8605 from a single source file @file{simple.cpp}. Since every program
8606 we are building uses GiNaC we simply added the GiNaC options
8607 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8608 want to specify them on a per-program basis: for instance by
8612 simple_LDADD = $(GINACLIB_LIBS)
8613 INCLUDES = $(GINACLIB_CPPFLAGS)
8616 to the @file{Makefile.am}.
8618 To try this example out, create a new directory and add the three
8621 Now execute the following commands:
8624 $ automake --add-missing
8629 You now have a package that can be built in the normal fashion
8638 @node Bibliography, Concept Index, Example package, Top
8639 @c node-name, next, previous, up
8640 @appendix Bibliography
8645 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8648 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8651 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8654 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8657 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8658 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8661 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8662 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8663 Academic Press, London
8666 @cite{Computer Algebra Systems - A Practical Guide},
8667 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8670 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8671 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8674 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8675 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8678 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8683 @node Concept Index, , Bibliography, Top
8684 @c node-name, next, previous, up
8685 @unnumbered Concept Index