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 two 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.
4289 @code{subs()} performs syntactic substitution of any complete algebraic
4290 object; it does not try to match sub-expressions as is demonstrated by the
4295 symbol x("x"), y("y"), z("z");
4297 ex e1 = pow(x+y, 2);
4298 cout << e1.subs(x+y == 4) << endl;
4301 ex e2 = sin(x)*sin(y)*cos(x);
4302 cout << e2.subs(sin(x) == cos(x)) << endl;
4303 // -> cos(x)^2*sin(y)
4306 cout << e3.subs(x+y == 4) << endl;
4308 // (and not 4+z as one might expect)
4312 A more powerful form of substitution using wildcards is described in the
4316 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4317 @c node-name, next, previous, up
4318 @section Pattern matching and advanced substitutions
4319 @cindex @code{wildcard} (class)
4320 @cindex Pattern matching
4322 GiNaC allows the use of patterns for checking whether an expression is of a
4323 certain form or contains subexpressions of a certain form, and for
4324 substituting expressions in a more general way.
4326 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4327 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4328 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4329 an unsigned integer number to allow having multiple different wildcards in a
4330 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4331 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4335 ex wild(unsigned label = 0);
4338 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4341 Some examples for patterns:
4343 @multitable @columnfractions .5 .5
4344 @item @strong{Constructed as} @tab @strong{Output as}
4345 @item @code{wild()} @tab @samp{$0}
4346 @item @code{pow(x,wild())} @tab @samp{x^$0}
4347 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4348 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4354 @item Wildcards behave like symbols and are subject to the same algebraic
4355 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4356 @item As shown in the last example, to use wildcards for indices you have to
4357 use them as the value of an @code{idx} object. This is because indices must
4358 always be of class @code{idx} (or a subclass).
4359 @item Wildcards only represent expressions or subexpressions. It is not
4360 possible to use them as placeholders for other properties like index
4361 dimension or variance, representation labels, symmetry of indexed objects
4363 @item Because wildcards are commutative, it is not possible to use wildcards
4364 as part of noncommutative products.
4365 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4366 are also valid patterns.
4369 @subsection Matching expressions
4370 @cindex @code{match()}
4371 The most basic application of patterns is to check whether an expression
4372 matches a given pattern. This is done by the function
4375 bool ex::match(const ex & pattern);
4376 bool ex::match(const ex & pattern, lst & repls);
4379 This function returns @code{true} when the expression matches the pattern
4380 and @code{false} if it doesn't. If used in the second form, the actual
4381 subexpressions matched by the wildcards get returned in the @code{repls}
4382 object as a list of relations of the form @samp{wildcard == expression}.
4383 If @code{match()} returns false, the state of @code{repls} is undefined.
4384 For reproducible results, the list should be empty when passed to
4385 @code{match()}, but it is also possible to find similarities in multiple
4386 expressions by passing in the result of a previous match.
4388 The matching algorithm works as follows:
4391 @item A single wildcard matches any expression. If one wildcard appears
4392 multiple times in a pattern, it must match the same expression in all
4393 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4394 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4395 @item If the expression is not of the same class as the pattern, the match
4396 fails (i.e. a sum only matches a sum, a function only matches a function,
4398 @item If the pattern is a function, it only matches the same function
4399 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4400 @item Except for sums and products, the match fails if the number of
4401 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4403 @item If there are no subexpressions, the expressions and the pattern must
4404 be equal (in the sense of @code{is_equal()}).
4405 @item Except for sums and products, each subexpression (@code{op()}) must
4406 match the corresponding subexpression of the pattern.
4409 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4410 account for their commutativity and associativity:
4413 @item If the pattern contains a term or factor that is a single wildcard,
4414 this one is used as the @dfn{global wildcard}. If there is more than one
4415 such wildcard, one of them is chosen as the global wildcard in a random
4417 @item Every term/factor of the pattern, except the global wildcard, is
4418 matched against every term of the expression in sequence. If no match is
4419 found, the whole match fails. Terms that did match are not considered in
4421 @item If there are no unmatched terms left, the match succeeds. Otherwise
4422 the match fails unless there is a global wildcard in the pattern, in
4423 which case this wildcard matches the remaining terms.
4426 In general, having more than one single wildcard as a term of a sum or a
4427 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4430 Here are some examples in @command{ginsh} to demonstrate how it works (the
4431 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4432 match fails, and the list of wildcard replacements otherwise):
4435 > match((x+y)^a,(x+y)^a);
4437 > match((x+y)^a,(x+y)^b);
4439 > match((x+y)^a,$1^$2);
4441 > match((x+y)^a,$1^$1);
4443 > match((x+y)^(x+y),$1^$1);
4445 > match((x+y)^(x+y),$1^$2);
4447 > match((a+b)*(a+c),($1+b)*($1+c));
4449 > match((a+b)*(a+c),(a+$1)*(a+$2));
4451 (Unpredictable. The result might also be [$1==c,$2==b].)
4452 > match((a+b)*(a+c),($1+$2)*($1+$3));
4453 (The result is undefined. Due to the sequential nature of the algorithm
4454 and the re-ordering of terms in GiNaC, the match for the first factor
4455 may be @{$1==a,$2==b@} in which case the match for the second factor
4456 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4458 > match(a*(x+y)+a*z+b,a*$1+$2);
4459 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4460 @{$1=x+y,$2=a*z+b@}.)
4461 > match(a+b+c+d+e+f,c);
4463 > match(a+b+c+d+e+f,c+$0);
4465 > match(a+b+c+d+e+f,c+e+$0);
4467 > match(a+b,a+b+$0);
4469 > match(a*b^2,a^$1*b^$2);
4471 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4472 even though a==a^1.)
4473 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4475 > match(atan2(y,x^2),atan2(y,$0));
4479 @subsection Matching parts of expressions
4480 @cindex @code{has()}
4481 A more general way to look for patterns in expressions is provided by the
4485 bool ex::has(const ex & pattern);
4488 This function checks whether a pattern is matched by an expression itself or
4489 by any of its subexpressions.
4491 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4492 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4495 > has(x*sin(x+y+2*a),y);
4497 > has(x*sin(x+y+2*a),x+y);
4499 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4500 has the subexpressions "x", "y" and "2*a".)
4501 > has(x*sin(x+y+2*a),x+y+$1);
4503 (But this is possible.)
4504 > has(x*sin(2*(x+y)+2*a),x+y);
4506 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4507 which "x+y" is not a subexpression.)
4510 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4512 > has(4*x^2-x+3,$1*x);
4514 > has(4*x^2+x+3,$1*x);
4516 (Another possible pitfall. The first expression matches because the term
4517 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4518 contains a linear term you should use the coeff() function instead.)
4521 @cindex @code{find()}
4525 bool ex::find(const ex & pattern, lst & found);
4528 works a bit like @code{has()} but it doesn't stop upon finding the first
4529 match. Instead, it appends all found matches to the specified list. If there
4530 are multiple occurrences of the same expression, it is entered only once to
4531 the list. @code{find()} returns false if no matches were found (in
4532 @command{ginsh}, it returns an empty list):
4535 > find(1+x+x^2+x^3,x);
4537 > find(1+x+x^2+x^3,y);
4539 > find(1+x+x^2+x^3,x^$1);
4541 (Note the absence of "x".)
4542 > expand((sin(x)+sin(y))*(a+b));
4543 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4548 @subsection Substituting expressions
4549 @cindex @code{subs()}
4550 Probably the most useful application of patterns is to use them for
4551 substituting expressions with the @code{subs()} method. Wildcards can be
4552 used in the search patterns as well as in the replacement expressions, where
4553 they get replaced by the expressions matched by them. @code{subs()} doesn't
4554 know anything about algebra; it performs purely syntactic substitutions.
4559 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4561 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4563 > subs((a+b+c)^2,a+b==x);
4565 > subs((a+b+c)^2,a+b+$1==x+$1);
4567 > subs(a+2*b,a+b==x);
4569 > subs(4*x^3-2*x^2+5*x-1,x==a);
4571 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4573 > subs(sin(1+sin(x)),sin($1)==cos($1));
4575 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4579 The last example would be written in C++ in this way:
4583 symbol a("a"), b("b"), x("x"), y("y");
4584 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4585 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4586 cout << e.expand() << endl;
4591 @subsection Algebraic substitutions
4592 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4593 enables smarter, algebraic substitutions in products and powers. If you want
4594 to substitute some factors of a product, you only need to list these factors
4595 in your pattern. Furthermore, if an (integer) power of some expression occurs
4596 in your pattern and in the expression that you want the substitution to occur
4597 in, it can be substituted as many times as possible, without getting negative
4600 An example clarifies it all (hopefully):
4603 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4604 subs_options::algebraic) << endl;
4605 // --> (y+x)^6+b^6+a^6
4607 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4609 // Powers and products are smart, but addition is just the same.
4611 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4614 // As I said: addition is just the same.
4616 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4617 // --> x^3*b*a^2+2*b
4619 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4621 // --> 2*b+x^3*b^(-1)*a^(-2)
4623 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4624 // --> -1-2*a^2+4*a^3+5*a
4626 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4627 subs_options::algebraic) << endl;
4628 // --> -1+5*x+4*x^3-2*x^2
4629 // You should not really need this kind of patterns very often now.
4630 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4632 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4633 subs_options::algebraic) << endl;
4634 // --> cos(1+cos(x))
4636 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4637 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4638 subs_options::algebraic)) << endl;
4643 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4644 @c node-name, next, previous, up
4645 @section Applying a Function on Subexpressions
4646 @cindex tree traversal
4647 @cindex @code{map()}
4649 Sometimes you may want to perform an operation on specific parts of an
4650 expression while leaving the general structure of it intact. An example
4651 of this would be a matrix trace operation: the trace of a sum is the sum
4652 of the traces of the individual terms. That is, the trace should @dfn{map}
4653 on the sum, by applying itself to each of the sum's operands. It is possible
4654 to do this manually which usually results in code like this:
4659 if (is_a<matrix>(e))
4660 return ex_to<matrix>(e).trace();
4661 else if (is_a<add>(e)) @{
4663 for (size_t i=0; i<e.nops(); i++)
4664 sum += calc_trace(e.op(i));
4666 @} else if (is_a<mul>)(e)) @{
4674 This is, however, slightly inefficient (if the sum is very large it can take
4675 a long time to add the terms one-by-one), and its applicability is limited to
4676 a rather small class of expressions. If @code{calc_trace()} is called with
4677 a relation or a list as its argument, you will probably want the trace to
4678 be taken on both sides of the relation or of all elements of the list.
4680 GiNaC offers the @code{map()} method to aid in the implementation of such
4684 ex ex::map(map_function & f) const;
4685 ex ex::map(ex (*f)(const ex & e)) const;
4688 In the first (preferred) form, @code{map()} takes a function object that
4689 is subclassed from the @code{map_function} class. In the second form, it
4690 takes a pointer to a function that accepts and returns an expression.
4691 @code{map()} constructs a new expression of the same type, applying the
4692 specified function on all subexpressions (in the sense of @code{op()}),
4695 The use of a function object makes it possible to supply more arguments to
4696 the function that is being mapped, or to keep local state information.
4697 The @code{map_function} class declares a virtual function call operator
4698 that you can overload. Here is a sample implementation of @code{calc_trace()}
4699 that uses @code{map()} in a recursive fashion:
4702 struct calc_trace : public map_function @{
4703 ex operator()(const ex &e)
4705 if (is_a<matrix>(e))
4706 return ex_to<matrix>(e).trace();
4707 else if (is_a<mul>(e)) @{
4710 return e.map(*this);
4715 This function object could then be used like this:
4719 ex M = ... // expression with matrices
4720 calc_trace do_trace;
4721 ex tr = do_trace(M);
4725 Here is another example for you to meditate over. It removes quadratic
4726 terms in a variable from an expanded polynomial:
4729 struct map_rem_quad : public map_function @{
4731 map_rem_quad(const ex & var_) : var(var_) @{@}
4733 ex operator()(const ex & e)
4735 if (is_a<add>(e) || is_a<mul>(e))
4736 return e.map(*this);
4737 else if (is_a<power>(e) &&
4738 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4748 symbol x("x"), y("y");
4751 for (int i=0; i<8; i++)
4752 e += pow(x, i) * pow(y, 8-i) * (i+1);
4754 // -> 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
4756 map_rem_quad rem_quad(x);
4757 cout << rem_quad(e) << endl;
4758 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4762 @command{ginsh} offers a slightly different implementation of @code{map()}
4763 that allows applying algebraic functions to operands. The second argument
4764 to @code{map()} is an expression containing the wildcard @samp{$0} which
4765 acts as the placeholder for the operands:
4770 > map(a+2*b,sin($0));
4772 > map(@{a,b,c@},$0^2+$0);
4773 @{a^2+a,b^2+b,c^2+c@}
4776 Note that it is only possible to use algebraic functions in the second
4777 argument. You can not use functions like @samp{diff()}, @samp{op()},
4778 @samp{subs()} etc. because these are evaluated immediately:
4781 > map(@{a,b,c@},diff($0,a));
4783 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4784 to "map(@{a,b,c@},0)".
4788 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4789 @c node-name, next, previous, up
4790 @section Visitors and Tree Traversal
4791 @cindex tree traversal
4792 @cindex @code{visitor} (class)
4793 @cindex @code{accept()}
4794 @cindex @code{visit()}
4795 @cindex @code{traverse()}
4796 @cindex @code{traverse_preorder()}
4797 @cindex @code{traverse_postorder()}
4799 Suppose that you need a function that returns a list of all indices appearing
4800 in an arbitrary expression. The indices can have any dimension, and for
4801 indices with variance you always want the covariant version returned.
4803 You can't use @code{get_free_indices()} because you also want to include
4804 dummy indices in the list, and you can't use @code{find()} as it needs
4805 specific index dimensions (and it would require two passes: one for indices
4806 with variance, one for plain ones).
4808 The obvious solution to this problem is a tree traversal with a type switch,
4809 such as the following:
4812 void gather_indices_helper(const ex & e, lst & l)
4814 if (is_a<varidx>(e)) @{
4815 const varidx & vi = ex_to<varidx>(e);
4816 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4817 @} else if (is_a<idx>(e)) @{
4820 size_t n = e.nops();
4821 for (size_t i = 0; i < n; ++i)
4822 gather_indices_helper(e.op(i), l);
4826 lst gather_indices(const ex & e)
4829 gather_indices_helper(e, l);
4836 This works fine but fans of object-oriented programming will feel
4837 uncomfortable with the type switch. One reason is that there is a possibility
4838 for subtle bugs regarding derived classes. If we had, for example, written
4841 if (is_a<idx>(e)) @{
4843 @} else if (is_a<varidx>(e)) @{
4847 in @code{gather_indices_helper}, the code wouldn't have worked because the
4848 first line "absorbs" all classes derived from @code{idx}, including
4849 @code{varidx}, so the special case for @code{varidx} would never have been
4852 Also, for a large number of classes, a type switch like the above can get
4853 unwieldy and inefficient (it's a linear search, after all).
4854 @code{gather_indices_helper} only checks for two classes, but if you had to
4855 write a function that required a different implementation for nearly
4856 every GiNaC class, the result would be very hard to maintain and extend.
4858 The cleanest approach to the problem would be to add a new virtual function
4859 to GiNaC's class hierarchy. In our example, there would be specializations
4860 for @code{idx} and @code{varidx} while the default implementation in
4861 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4862 impossible to add virtual member functions to existing classes without
4863 changing their source and recompiling everything. GiNaC comes with source,
4864 so you could actually do this, but for a small algorithm like the one
4865 presented this would be impractical.
4867 One solution to this dilemma is the @dfn{Visitor} design pattern,
4868 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4869 variation, described in detail in
4870 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4871 virtual functions to the class hierarchy to implement operations, GiNaC
4872 provides a single "bouncing" method @code{accept()} that takes an instance
4873 of a special @code{visitor} class and redirects execution to the one
4874 @code{visit()} virtual function of the visitor that matches the type of
4875 object that @code{accept()} was being invoked on.
4877 Visitors in GiNaC must derive from the global @code{visitor} class as well
4878 as from the class @code{T::visitor} of each class @code{T} they want to
4879 visit, and implement the member functions @code{void visit(const T &)} for
4885 void ex::accept(visitor & v) const;
4888 will then dispatch to the correct @code{visit()} member function of the
4889 specified visitor @code{v} for the type of GiNaC object at the root of the
4890 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4892 Here is an example of a visitor:
4896 : public visitor, // this is required
4897 public add::visitor, // visit add objects
4898 public numeric::visitor, // visit numeric objects
4899 public basic::visitor // visit basic objects
4901 void visit(const add & x)
4902 @{ cout << "called with an add object" << endl; @}
4904 void visit(const numeric & x)
4905 @{ cout << "called with a numeric object" << endl; @}
4907 void visit(const basic & x)
4908 @{ cout << "called with a basic object" << endl; @}
4912 which can be used as follows:
4923 // prints "called with a numeric object"
4925 // prints "called with an add object"
4927 // prints "called with a basic object"
4931 The @code{visit(const basic &)} method gets called for all objects that are
4932 not @code{numeric} or @code{add} and acts as an (optional) default.
4934 From a conceptual point of view, the @code{visit()} methods of the visitor
4935 behave like a newly added virtual function of the visited hierarchy.
4936 In addition, visitors can store state in member variables, and they can
4937 be extended by deriving a new visitor from an existing one, thus building
4938 hierarchies of visitors.
4940 We can now rewrite our index example from above with a visitor:
4943 class gather_indices_visitor
4944 : public visitor, public idx::visitor, public varidx::visitor
4948 void visit(const idx & i)
4953 void visit(const varidx & vi)
4955 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4959 const lst & get_result() // utility function
4968 What's missing is the tree traversal. We could implement it in
4969 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4972 void ex::traverse_preorder(visitor & v) const;
4973 void ex::traverse_postorder(visitor & v) const;
4974 void ex::traverse(visitor & v) const;
4977 @code{traverse_preorder()} visits a node @emph{before} visiting its
4978 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4979 visiting its subexpressions. @code{traverse()} is a synonym for
4980 @code{traverse_preorder()}.
4982 Here is a new implementation of @code{gather_indices()} that uses the visitor
4983 and @code{traverse()}:
4986 lst gather_indices(const ex & e)
4988 gather_indices_visitor v;
4990 return v.get_result();
4994 Alternatively, you could use pre- or postorder iterators for the tree
4998 lst gather_indices(const ex & e)
5000 gather_indices_visitor v;
5001 for (const_preorder_iterator i = e.preorder_begin();
5002 i != e.preorder_end(); ++i) @{
5005 return v.get_result();
5010 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
5011 @c node-name, next, previous, up
5012 @section Polynomial arithmetic
5014 @subsection Expanding and collecting
5015 @cindex @code{expand()}
5016 @cindex @code{collect()}
5017 @cindex @code{collect_common_factors()}
5019 A polynomial in one or more variables has many equivalent
5020 representations. Some useful ones serve a specific purpose. Consider
5021 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5022 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5023 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5024 representations are the recursive ones where one collects for exponents
5025 in one of the three variable. Since the factors are themselves
5026 polynomials in the remaining two variables the procedure can be
5027 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5028 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5031 To bring an expression into expanded form, its method
5034 ex ex::expand(unsigned options = 0);
5037 may be called. In our example above, this corresponds to @math{4*x*y +
5038 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5039 GiNaC is not easy to guess you should be prepared to see different
5040 orderings of terms in such sums!
5042 Another useful representation of multivariate polynomials is as a
5043 univariate polynomial in one of the variables with the coefficients
5044 being polynomials in the remaining variables. The method
5045 @code{collect()} accomplishes this task:
5048 ex ex::collect(const ex & s, bool distributed = false);
5051 The first argument to @code{collect()} can also be a list of objects in which
5052 case the result is either a recursively collected polynomial, or a polynomial
5053 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5054 by the @code{distributed} flag.
5056 Note that the original polynomial needs to be in expanded form (for the
5057 variables concerned) in order for @code{collect()} to be able to find the
5058 coefficients properly.
5060 The following @command{ginsh} transcript shows an application of @code{collect()}
5061 together with @code{find()}:
5064 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5065 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)
5066 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5067 > collect(a,@{p,q@});
5068 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5069 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5070 > collect(a,find(a,sin($1)));
5071 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5072 > collect(a,@{find(a,sin($1)),p,q@});
5073 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5074 > collect(a,@{find(a,sin($1)),d@});
5075 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5078 Polynomials can often be brought into a more compact form by collecting
5079 common factors from the terms of sums. This is accomplished by the function
5082 ex collect_common_factors(const ex & e);
5085 This function doesn't perform a full factorization but only looks for
5086 factors which are already explicitly present:
5089 > collect_common_factors(a*x+a*y);
5091 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5093 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5094 (c+a)*a*(x*y+y^2+x)*b
5097 @subsection Degree and coefficients
5098 @cindex @code{degree()}
5099 @cindex @code{ldegree()}
5100 @cindex @code{coeff()}
5102 The degree and low degree of a polynomial can be obtained using the two
5106 int ex::degree(const ex & s);
5107 int ex::ldegree(const ex & s);
5110 which also work reliably on non-expanded input polynomials (they even work
5111 on rational functions, returning the asymptotic degree). By definition, the
5112 degree of zero is zero. To extract a coefficient with a certain power from
5113 an expanded polynomial you use
5116 ex ex::coeff(const ex & s, int n);
5119 You can also obtain the leading and trailing coefficients with the methods
5122 ex ex::lcoeff(const ex & s);
5123 ex ex::tcoeff(const ex & s);
5126 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5129 An application is illustrated in the next example, where a multivariate
5130 polynomial is analyzed:
5134 symbol x("x"), y("y");
5135 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5136 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5137 ex Poly = PolyInp.expand();
5139 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5140 cout << "The x^" << i << "-coefficient is "
5141 << Poly.coeff(x,i) << endl;
5143 cout << "As polynomial in y: "
5144 << Poly.collect(y) << endl;
5148 When run, it returns an output in the following fashion:
5151 The x^0-coefficient is y^2+11*y
5152 The x^1-coefficient is 5*y^2-2*y
5153 The x^2-coefficient is -1
5154 The x^3-coefficient is 4*y
5155 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5158 As always, the exact output may vary between different versions of GiNaC
5159 or even from run to run since the internal canonical ordering is not
5160 within the user's sphere of influence.
5162 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5163 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5164 with non-polynomial expressions as they not only work with symbols but with
5165 constants, functions and indexed objects as well:
5169 symbol a("a"), b("b"), c("c"), x("x");
5170 idx i(symbol("i"), 3);
5172 ex e = pow(sin(x) - cos(x), 4);
5173 cout << e.degree(cos(x)) << endl;
5175 cout << e.expand().coeff(sin(x), 3) << endl;
5178 e = indexed(a+b, i) * indexed(b+c, i);
5179 e = e.expand(expand_options::expand_indexed);
5180 cout << e.collect(indexed(b, i)) << endl;
5181 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5186 @subsection Polynomial division
5187 @cindex polynomial division
5190 @cindex pseudo-remainder
5191 @cindex @code{quo()}
5192 @cindex @code{rem()}
5193 @cindex @code{prem()}
5194 @cindex @code{divide()}
5199 ex quo(const ex & a, const ex & b, const ex & x);
5200 ex rem(const ex & a, const ex & b, const ex & x);
5203 compute the quotient and remainder of univariate polynomials in the variable
5204 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5206 The additional function
5209 ex prem(const ex & a, const ex & b, const ex & x);
5212 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5213 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5215 Exact division of multivariate polynomials is performed by the function
5218 bool divide(const ex & a, const ex & b, ex & q);
5221 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5222 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5223 in which case the value of @code{q} is undefined.
5226 @subsection Unit, content and primitive part
5227 @cindex @code{unit()}
5228 @cindex @code{content()}
5229 @cindex @code{primpart()}
5230 @cindex @code{unitcontprim()}
5235 ex ex::unit(const ex & x);
5236 ex ex::content(const ex & x);
5237 ex ex::primpart(const ex & x);
5238 ex ex::primpart(const ex & x, const ex & c);
5241 return the unit part, content part, and primitive polynomial of a multivariate
5242 polynomial with respect to the variable @samp{x} (the unit part being the sign
5243 of the leading coefficient, the content part being the GCD of the coefficients,
5244 and the primitive polynomial being the input polynomial divided by the unit and
5245 content parts). The second variant of @code{primpart()} expects the previously
5246 calculated content part of the polynomial in @code{c}, which enables it to
5247 work faster in the case where the content part has already been computed. The
5248 product of unit, content, and primitive part is the original polynomial.
5250 Additionally, the method
5253 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5256 computes the unit, content, and primitive parts in one go, returning them
5257 in @code{u}, @code{c}, and @code{p}, respectively.
5260 @subsection GCD, LCM and resultant
5263 @cindex @code{gcd()}
5264 @cindex @code{lcm()}
5266 The functions for polynomial greatest common divisor and least common
5267 multiple have the synopsis
5270 ex gcd(const ex & a, const ex & b);
5271 ex lcm(const ex & a, const ex & b);
5274 The functions @code{gcd()} and @code{lcm()} accept two expressions
5275 @code{a} and @code{b} as arguments and return a new expression, their
5276 greatest common divisor or least common multiple, respectively. If the
5277 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5278 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5279 the coefficients must be rationals.
5282 #include <ginac/ginac.h>
5283 using namespace GiNaC;
5287 symbol x("x"), y("y"), z("z");
5288 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5289 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5291 ex P_gcd = gcd(P_a, P_b);
5293 ex P_lcm = lcm(P_a, P_b);
5294 // 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
5299 @cindex @code{resultant()}
5301 The resultant of two expressions only makes sense with polynomials.
5302 It is always computed with respect to a specific symbol within the
5303 expressions. The function has the interface
5306 ex resultant(const ex & a, const ex & b, const ex & s);
5309 Resultants are symmetric in @code{a} and @code{b}. The following example
5310 computes the resultant of two expressions with respect to @code{x} and
5311 @code{y}, respectively:
5314 #include <ginac/ginac.h>
5315 using namespace GiNaC;
5319 symbol x("x"), y("y");
5321 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5324 r = resultant(e1, e2, x);
5326 r = resultant(e1, e2, y);
5331 @subsection Square-free decomposition
5332 @cindex square-free decomposition
5333 @cindex factorization
5334 @cindex @code{sqrfree()}
5336 GiNaC still lacks proper factorization support. Some form of
5337 factorization is, however, easily implemented by noting that factors
5338 appearing in a polynomial with power two or more also appear in the
5339 derivative and hence can easily be found by computing the GCD of the
5340 original polynomial and its derivatives. Any decent system has an
5341 interface for this so called square-free factorization. So we provide
5344 ex sqrfree(const ex & a, const lst & l = lst());
5346 Here is an example that by the way illustrates how the exact form of the
5347 result may slightly depend on the order of differentiation, calling for
5348 some care with subsequent processing of the result:
5351 symbol x("x"), y("y");
5352 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5354 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5355 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5357 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5358 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5360 cout << sqrfree(BiVarPol) << endl;
5361 // -> depending on luck, any of the above
5364 Note also, how factors with the same exponents are not fully factorized
5368 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5369 @c node-name, next, previous, up
5370 @section Rational expressions
5372 @subsection The @code{normal} method
5373 @cindex @code{normal()}
5374 @cindex simplification
5375 @cindex temporary replacement
5377 Some basic form of simplification of expressions is called for frequently.
5378 GiNaC provides the method @code{.normal()}, which converts a rational function
5379 into an equivalent rational function of the form @samp{numerator/denominator}
5380 where numerator and denominator are coprime. If the input expression is already
5381 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5382 otherwise it performs fraction addition and multiplication.
5384 @code{.normal()} can also be used on expressions which are not rational functions
5385 as it will replace all non-rational objects (like functions or non-integer
5386 powers) by temporary symbols to bring the expression to the domain of rational
5387 functions before performing the normalization, and re-substituting these
5388 symbols afterwards. This algorithm is also available as a separate method
5389 @code{.to_rational()}, described below.
5391 This means that both expressions @code{t1} and @code{t2} are indeed
5392 simplified in this little code snippet:
5397 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5398 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5399 std::cout << "t1 is " << t1.normal() << std::endl;
5400 std::cout << "t2 is " << t2.normal() << std::endl;
5404 Of course this works for multivariate polynomials too, so the ratio of
5405 the sample-polynomials from the section about GCD and LCM above would be
5406 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5409 @subsection Numerator and denominator
5412 @cindex @code{numer()}
5413 @cindex @code{denom()}
5414 @cindex @code{numer_denom()}
5416 The numerator and denominator of an expression can be obtained with
5421 ex ex::numer_denom();
5424 These functions will first normalize the expression as described above and
5425 then return the numerator, denominator, or both as a list, respectively.
5426 If you need both numerator and denominator, calling @code{numer_denom()} is
5427 faster than using @code{numer()} and @code{denom()} separately.
5430 @subsection Converting to a polynomial or rational expression
5431 @cindex @code{to_polynomial()}
5432 @cindex @code{to_rational()}
5434 Some of the methods described so far only work on polynomials or rational
5435 functions. GiNaC provides a way to extend the domain of these functions to
5436 general expressions by using the temporary replacement algorithm described
5437 above. You do this by calling
5440 ex ex::to_polynomial(exmap & m);
5441 ex ex::to_polynomial(lst & l);
5445 ex ex::to_rational(exmap & m);
5446 ex ex::to_rational(lst & l);
5449 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5450 will be filled with the generated temporary symbols and their replacement
5451 expressions in a format that can be used directly for the @code{subs()}
5452 method. It can also already contain a list of replacements from an earlier
5453 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5454 possible to use it on multiple expressions and get consistent results.
5456 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5457 is probably best illustrated with an example:
5461 symbol x("x"), y("y");
5462 ex a = 2*x/sin(x) - y/(3*sin(x));
5466 ex p = a.to_polynomial(lp);
5467 cout << " = " << p << "\n with " << lp << endl;
5468 // = symbol3*symbol2*y+2*symbol2*x
5469 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5472 ex r = a.to_rational(lr);
5473 cout << " = " << r << "\n with " << lr << endl;
5474 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5475 // with @{symbol4==sin(x)@}
5479 The following more useful example will print @samp{sin(x)-cos(x)}:
5484 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5485 ex b = sin(x) + cos(x);
5488 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5489 cout << q.subs(m) << endl;
5494 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5495 @c node-name, next, previous, up
5496 @section Symbolic differentiation
5497 @cindex differentiation
5498 @cindex @code{diff()}
5500 @cindex product rule
5502 GiNaC's objects know how to differentiate themselves. Thus, a
5503 polynomial (class @code{add}) knows that its derivative is the sum of
5504 the derivatives of all the monomials:
5508 symbol x("x"), y("y"), z("z");
5509 ex P = pow(x, 5) + pow(x, 2) + y;
5511 cout << P.diff(x,2) << endl;
5513 cout << P.diff(y) << endl; // 1
5515 cout << P.diff(z) << endl; // 0
5520 If a second integer parameter @var{n} is given, the @code{diff} method
5521 returns the @var{n}th derivative.
5523 If @emph{every} object and every function is told what its derivative
5524 is, all derivatives of composed objects can be calculated using the
5525 chain rule and the product rule. Consider, for instance the expression
5526 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5527 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5528 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5529 out that the composition is the generating function for Euler Numbers,
5530 i.e. the so called @var{n}th Euler number is the coefficient of
5531 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5532 identity to code a function that generates Euler numbers in just three
5535 @cindex Euler numbers
5537 #include <ginac/ginac.h>
5538 using namespace GiNaC;
5540 ex EulerNumber(unsigned n)
5543 const ex generator = pow(cosh(x),-1);
5544 return generator.diff(x,n).subs(x==0);
5549 for (unsigned i=0; i<11; i+=2)
5550 std::cout << EulerNumber(i) << std::endl;
5555 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5556 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5557 @code{i} by two since all odd Euler numbers vanish anyways.
5560 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5561 @c node-name, next, previous, up
5562 @section Series expansion
5563 @cindex @code{series()}
5564 @cindex Taylor expansion
5565 @cindex Laurent expansion
5566 @cindex @code{pseries} (class)
5567 @cindex @code{Order()}
5569 Expressions know how to expand themselves as a Taylor series or (more
5570 generally) a Laurent series. As in most conventional Computer Algebra
5571 Systems, no distinction is made between those two. There is a class of
5572 its own for storing such series (@code{class pseries}) and a built-in
5573 function (called @code{Order}) for storing the order term of the series.
5574 As a consequence, if you want to work with series, i.e. multiply two
5575 series, you need to call the method @code{ex::series} again to convert
5576 it to a series object with the usual structure (expansion plus order
5577 term). A sample application from special relativity could read:
5580 #include <ginac/ginac.h>
5581 using namespace std;
5582 using namespace GiNaC;
5586 symbol v("v"), c("c");
5588 ex gamma = 1/sqrt(1 - pow(v/c,2));
5589 ex mass_nonrel = gamma.series(v==0, 10);
5591 cout << "the relativistic mass increase with v is " << endl
5592 << mass_nonrel << endl;
5594 cout << "the inverse square of this series is " << endl
5595 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5599 Only calling the series method makes the last output simplify to
5600 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5601 series raised to the power @math{-2}.
5603 @cindex Machin's formula
5604 As another instructive application, let us calculate the numerical
5605 value of Archimedes' constant
5609 (for which there already exists the built-in constant @code{Pi})
5610 using John Machin's amazing formula
5612 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5615 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5617 This equation (and similar ones) were used for over 200 years for
5618 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5619 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5620 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5621 order term with it and the question arises what the system is supposed
5622 to do when the fractions are plugged into that order term. The solution
5623 is to use the function @code{series_to_poly()} to simply strip the order
5627 #include <ginac/ginac.h>
5628 using namespace GiNaC;
5630 ex machin_pi(int degr)
5633 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5634 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5635 -4*pi_expansion.subs(x==numeric(1,239));
5641 using std::cout; // just for fun, another way of...
5642 using std::endl; // ...dealing with this namespace std.
5644 for (int i=2; i<12; i+=2) @{
5645 pi_frac = machin_pi(i);
5646 cout << i << ":\t" << pi_frac << endl
5647 << "\t" << pi_frac.evalf() << endl;
5653 Note how we just called @code{.series(x,degr)} instead of
5654 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5655 method @code{series()}: if the first argument is a symbol the expression
5656 is expanded in that symbol around point @code{0}. When you run this
5657 program, it will type out:
5661 3.1832635983263598326
5662 4: 5359397032/1706489875
5663 3.1405970293260603143
5664 6: 38279241713339684/12184551018734375
5665 3.141621029325034425
5666 8: 76528487109180192540976/24359780855939418203125
5667 3.141591772182177295
5668 10: 327853873402258685803048818236/104359128170408663038552734375
5669 3.1415926824043995174
5673 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5674 @c node-name, next, previous, up
5675 @section Symmetrization
5676 @cindex @code{symmetrize()}
5677 @cindex @code{antisymmetrize()}
5678 @cindex @code{symmetrize_cyclic()}
5683 ex ex::symmetrize(const lst & l);
5684 ex ex::antisymmetrize(const lst & l);
5685 ex ex::symmetrize_cyclic(const lst & l);
5688 symmetrize an expression by returning the sum over all symmetric,
5689 antisymmetric or cyclic permutations of the specified list of objects,
5690 weighted by the number of permutations.
5692 The three additional methods
5695 ex ex::symmetrize();
5696 ex ex::antisymmetrize();
5697 ex ex::symmetrize_cyclic();
5700 symmetrize or antisymmetrize an expression over its free indices.
5702 Symmetrization is most useful with indexed expressions but can be used with
5703 almost any kind of object (anything that is @code{subs()}able):
5707 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5708 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5710 cout << indexed(A, i, j).symmetrize() << endl;
5711 // -> 1/2*A.j.i+1/2*A.i.j
5712 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5713 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5714 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5715 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5719 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5720 @c node-name, next, previous, up
5721 @section Predefined mathematical functions
5723 @subsection Overview
5725 GiNaC contains the following predefined mathematical functions:
5728 @multitable @columnfractions .30 .70
5729 @item @strong{Name} @tab @strong{Function}
5732 @cindex @code{abs()}
5733 @item @code{csgn(x)}
5735 @cindex @code{conjugate()}
5736 @item @code{conjugate(x)}
5737 @tab complex conjugation
5738 @cindex @code{csgn()}
5739 @item @code{sqrt(x)}
5740 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5741 @cindex @code{sqrt()}
5744 @cindex @code{sin()}
5747 @cindex @code{cos()}
5750 @cindex @code{tan()}
5751 @item @code{asin(x)}
5753 @cindex @code{asin()}
5754 @item @code{acos(x)}
5756 @cindex @code{acos()}
5757 @item @code{atan(x)}
5758 @tab inverse tangent
5759 @cindex @code{atan()}
5760 @item @code{atan2(y, x)}
5761 @tab inverse tangent with two arguments
5762 @item @code{sinh(x)}
5763 @tab hyperbolic sine
5764 @cindex @code{sinh()}
5765 @item @code{cosh(x)}
5766 @tab hyperbolic cosine
5767 @cindex @code{cosh()}
5768 @item @code{tanh(x)}
5769 @tab hyperbolic tangent
5770 @cindex @code{tanh()}
5771 @item @code{asinh(x)}
5772 @tab inverse hyperbolic sine
5773 @cindex @code{asinh()}
5774 @item @code{acosh(x)}
5775 @tab inverse hyperbolic cosine
5776 @cindex @code{acosh()}
5777 @item @code{atanh(x)}
5778 @tab inverse hyperbolic tangent
5779 @cindex @code{atanh()}
5781 @tab exponential function
5782 @cindex @code{exp()}
5784 @tab natural logarithm
5785 @cindex @code{log()}
5788 @cindex @code{Li2()}
5789 @item @code{Li(m, x)}
5790 @tab classical polylogarithm as well as multiple polylogarithm
5792 @item @code{G(a, y)}
5793 @tab multiple polylogarithm
5795 @item @code{G(a, s, y)}
5796 @tab multiple polylogarithm with explicit signs for the imaginary parts
5798 @item @code{S(n, p, x)}
5799 @tab Nielsen's generalized polylogarithm
5801 @item @code{H(m, x)}
5802 @tab harmonic polylogarithm
5804 @item @code{zeta(m)}
5805 @tab Riemann's zeta function as well as multiple zeta value
5806 @cindex @code{zeta()}
5807 @item @code{zeta(m, s)}
5808 @tab alternating Euler sum
5809 @cindex @code{zeta()}
5810 @item @code{zetaderiv(n, x)}
5811 @tab derivatives of Riemann's zeta function
5812 @item @code{tgamma(x)}
5814 @cindex @code{tgamma()}
5815 @cindex gamma function
5816 @item @code{lgamma(x)}
5817 @tab logarithm of gamma function
5818 @cindex @code{lgamma()}
5819 @item @code{beta(x, y)}
5820 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5821 @cindex @code{beta()}
5823 @tab psi (digamma) function
5824 @cindex @code{psi()}
5825 @item @code{psi(n, x)}
5826 @tab derivatives of psi function (polygamma functions)
5827 @item @code{factorial(n)}
5828 @tab factorial function @math{n!}
5829 @cindex @code{factorial()}
5830 @item @code{binomial(n, k)}
5831 @tab binomial coefficients
5832 @cindex @code{binomial()}
5833 @item @code{Order(x)}
5834 @tab order term function in truncated power series
5835 @cindex @code{Order()}
5840 For functions that have a branch cut in the complex plane GiNaC follows
5841 the conventions for C++ as defined in the ANSI standard as far as
5842 possible. In particular: the natural logarithm (@code{log}) and the
5843 square root (@code{sqrt}) both have their branch cuts running along the
5844 negative real axis where the points on the axis itself belong to the
5845 upper part (i.e. continuous with quadrant II). The inverse
5846 trigonometric and hyperbolic functions are not defined for complex
5847 arguments by the C++ standard, however. In GiNaC we follow the
5848 conventions used by CLN, which in turn follow the carefully designed
5849 definitions in the Common Lisp standard. It should be noted that this
5850 convention is identical to the one used by the C99 standard and by most
5851 serious CAS. It is to be expected that future revisions of the C++
5852 standard incorporate these functions in the complex domain in a manner
5853 compatible with C99.
5855 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5856 @c node-name, next, previous, up
5857 @subsection Multiple polylogarithms
5859 @cindex polylogarithm
5860 @cindex Nielsen's generalized polylogarithm
5861 @cindex harmonic polylogarithm
5862 @cindex multiple zeta value
5863 @cindex alternating Euler sum
5864 @cindex multiple polylogarithm
5866 The multiple polylogarithm is the most generic member of a family of functions,
5867 to which others like the harmonic polylogarithm, Nielsen's generalized
5868 polylogarithm and the multiple zeta value belong.
5869 Everyone of these functions can also be written as a multiple polylogarithm with specific
5870 parameters. This whole family of functions is therefore often referred to simply as
5871 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5872 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5873 @code{Li} and @code{G} in principle represent the same function, the different
5874 notations are more natural to the series representation or the integral
5875 representation, respectively.
5877 To facilitate the discussion of these functions we distinguish between indices and
5878 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5879 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5881 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5882 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5883 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5884 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5885 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5886 @code{s} is not given, the signs default to +1.
5887 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5888 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5889 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5890 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5891 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5893 The functions print in LaTeX format as
5895 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5901 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5904 $\zeta(m_1,m_2,\ldots,m_k)$.
5906 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5907 are printed with a line above, e.g.
5909 $\zeta(5,\overline{2})$.
5911 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5913 Definitions and analytical as well as numerical properties of multiple polylogarithms
5914 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5915 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5916 except for a few differences which will be explicitly stated in the following.
5918 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5919 that the indices and arguments are understood to be in the same order as in which they appear in
5920 the series representation. This means
5922 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5925 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5928 $\zeta(1,2)$ evaluates to infinity.
5930 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5933 The functions only evaluate if the indices are integers greater than zero, except for the indices
5934 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5935 will be interpreted as the sequence of signs for the corresponding indices
5936 @code{m} or the sign of the imaginary part for the
5937 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5938 @code{zeta(lst(3,4), lst(-1,1))} means
5940 $\zeta(\overline{3},4)$
5943 @code{G(lst(a,b), lst(-1,1), c)} means
5945 $G(a-0\epsilon,b+0\epsilon;c)$.
5947 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5948 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5949 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
5950 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5951 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5952 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5953 evaluates also for negative integers and positive even integers. For example:
5956 > Li(@{3,1@},@{x,1@});
5959 -zeta(@{3,2@},@{-1,-1@})
5964 It is easy to tell for a given function into which other function it can be rewritten, may
5965 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5966 with negative indices or trailing zeros (the example above gives a hint). Signs can
5967 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5968 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5969 @code{Li} (@code{eval()} already cares for the possible downgrade):
5972 > convert_H_to_Li(@{0,-2,-1,3@},x);
5973 Li(@{3,1,3@},@{-x,1,-1@})
5974 > convert_H_to_Li(@{2,-1,0@},x);
5975 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5978 Every function can be numerically evaluated for
5979 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5980 global variable @code{Digits}:
5985 > evalf(zeta(@{3,1,3,1@}));
5986 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5989 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5990 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5992 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5997 In long expressions this helps a lot with debugging, because you can easily spot
5998 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5999 cancellations of divergencies happen.
6001 Useful publications:
6003 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6004 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6006 @cite{Harmonic Polylogarithms},
6007 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6009 @cite{Special Values of Multiple Polylogarithms},
6010 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6012 @cite{Numerical Evaluation of Multiple Polylogarithms},
6013 J.Vollinga, S.Weinzierl, hep-ph/0410259
6015 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
6016 @c node-name, next, previous, up
6017 @section Complex Conjugation
6019 @cindex @code{conjugate()}
6027 returns the complex conjugate of the expression. For all built-in functions and objects the
6028 conjugation gives the expected results:
6032 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6036 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6037 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6038 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6039 // -> -gamma5*gamma~b*gamma~a
6043 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
6044 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
6045 arguments. This is the default strategy. If you want to define your own functions and want to
6046 change this behavior, you have to supply a specialized conjugation method for your function
6047 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
6049 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
6050 @c node-name, next, previous, up
6051 @section Solving Linear Systems of Equations
6052 @cindex @code{lsolve()}
6054 The function @code{lsolve()} provides a convenient wrapper around some
6055 matrix operations that comes in handy when a system of linear equations
6059 ex lsolve(const ex & eqns, const ex & symbols,
6060 unsigned options = solve_algo::automatic);
6063 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6064 @code{relational}) while @code{symbols} is a @code{lst} of
6065 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
6068 It returns the @code{lst} of solutions as an expression. As an example,
6069 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6073 symbol a("a"), b("b"), x("x"), y("y");
6075 eqns = a*x+b*y==3, x-y==b;
6077 cout << lsolve(eqns, vars) << endl;
6078 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6081 When the linear equations @code{eqns} are underdetermined, the solution
6082 will contain one or more tautological entries like @code{x==x},
6083 depending on the rank of the system. When they are overdetermined, the
6084 solution will be an empty @code{lst}. Note the third optional parameter
6085 to @code{lsolve()}: it accepts the same parameters as
6086 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6090 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6091 @c node-name, next, previous, up
6092 @section Input and output of expressions
6095 @subsection Expression output
6097 @cindex output of expressions
6099 Expressions can simply be written to any stream:
6104 ex e = 4.5*I+pow(x,2)*3/2;
6105 cout << e << endl; // prints '4.5*I+3/2*x^2'
6109 The default output format is identical to the @command{ginsh} input syntax and
6110 to that used by most computer algebra systems, but not directly pastable
6111 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6112 is printed as @samp{x^2}).
6114 It is possible to print expressions in a number of different formats with
6115 a set of stream manipulators;
6118 std::ostream & dflt(std::ostream & os);
6119 std::ostream & latex(std::ostream & os);
6120 std::ostream & tree(std::ostream & os);
6121 std::ostream & csrc(std::ostream & os);
6122 std::ostream & csrc_float(std::ostream & os);
6123 std::ostream & csrc_double(std::ostream & os);
6124 std::ostream & csrc_cl_N(std::ostream & os);
6125 std::ostream & index_dimensions(std::ostream & os);
6126 std::ostream & no_index_dimensions(std::ostream & os);
6129 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6130 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6131 @code{print_csrc()} functions, respectively.
6134 All manipulators affect the stream state permanently. To reset the output
6135 format to the default, use the @code{dflt} manipulator:
6139 cout << latex; // all output to cout will be in LaTeX format from
6141 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6142 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6143 cout << dflt; // revert to default output format
6144 cout << e << endl; // prints '4.5*I+3/2*x^2'
6148 If you don't want to affect the format of the stream you're working with,
6149 you can output to a temporary @code{ostringstream} like this:
6154 s << latex << e; // format of cout remains unchanged
6155 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6160 @cindex @code{csrc_float}
6161 @cindex @code{csrc_double}
6162 @cindex @code{csrc_cl_N}
6163 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6164 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6165 format that can be directly used in a C or C++ program. The three possible
6166 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6167 classes provided by the CLN library):
6171 cout << "f = " << csrc_float << e << ";\n";
6172 cout << "d = " << csrc_double << e << ";\n";
6173 cout << "n = " << csrc_cl_N << e << ";\n";
6177 The above example will produce (note the @code{x^2} being converted to
6181 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6182 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6183 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6187 The @code{tree} manipulator allows dumping the internal structure of an
6188 expression for debugging purposes:
6199 add, hash=0x0, flags=0x3, nops=2
6200 power, hash=0x0, flags=0x3, nops=2
6201 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6202 2 (numeric), hash=0x6526b0fa, flags=0xf
6203 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6206 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6210 @cindex @code{latex}
6211 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6212 It is rather similar to the default format but provides some braces needed
6213 by LaTeX for delimiting boxes and also converts some common objects to
6214 conventional LaTeX names. It is possible to give symbols a special name for
6215 LaTeX output by supplying it as a second argument to the @code{symbol}
6218 For example, the code snippet
6222 symbol x("x", "\\circ");
6223 ex e = lgamma(x).series(x==0,3);
6224 cout << latex << e << endl;
6231 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6232 +\mathcal@{O@}(\circ^@{3@})
6235 @cindex @code{index_dimensions}
6236 @cindex @code{no_index_dimensions}
6237 Index dimensions are normally hidden in the output. To make them visible, use
6238 the @code{index_dimensions} manipulator. The dimensions will be written in
6239 square brackets behind each index value in the default and LaTeX output
6244 symbol x("x"), y("y");
6245 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6246 ex e = indexed(x, mu) * indexed(y, nu);
6249 // prints 'x~mu*y~nu'
6250 cout << index_dimensions << e << endl;
6251 // prints 'x~mu[4]*y~nu[4]'
6252 cout << no_index_dimensions << e << endl;
6253 // prints 'x~mu*y~nu'
6258 @cindex Tree traversal
6259 If you need any fancy special output format, e.g. for interfacing GiNaC
6260 with other algebra systems or for producing code for different
6261 programming languages, you can always traverse the expression tree yourself:
6264 static void my_print(const ex & e)
6266 if (is_a<function>(e))
6267 cout << ex_to<function>(e).get_name();
6269 cout << ex_to<basic>(e).class_name();
6271 size_t n = e.nops();
6273 for (size_t i=0; i<n; i++) @{
6285 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6293 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6294 symbol(y))),numeric(-2)))
6297 If you need an output format that makes it possible to accurately
6298 reconstruct an expression by feeding the output to a suitable parser or
6299 object factory, you should consider storing the expression in an
6300 @code{archive} object and reading the object properties from there.
6301 See the section on archiving for more information.
6304 @subsection Expression input
6305 @cindex input of expressions
6307 GiNaC provides no way to directly read an expression from a stream because
6308 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6309 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6310 @code{y} you defined in your program and there is no way to specify the
6311 desired symbols to the @code{>>} stream input operator.
6313 Instead, GiNaC lets you construct an expression from a string, specifying the
6314 list of symbols to be used:
6318 symbol x("x"), y("y");
6319 ex e("2*x+sin(y)", lst(x, y));
6323 The input syntax is the same as that used by @command{ginsh} and the stream
6324 output operator @code{<<}. The symbols in the string are matched by name to
6325 the symbols in the list and if GiNaC encounters a symbol not specified in
6326 the list it will throw an exception.
6328 With this constructor, it's also easy to implement interactive GiNaC programs:
6333 #include <stdexcept>
6334 #include <ginac/ginac.h>
6335 using namespace std;
6336 using namespace GiNaC;
6343 cout << "Enter an expression containing 'x': ";
6348 cout << "The derivative of " << e << " with respect to x is ";
6349 cout << e.diff(x) << ".\n";
6350 @} catch (exception &p) @{
6351 cerr << p.what() << endl;
6357 @subsection Archiving
6358 @cindex @code{archive} (class)
6361 GiNaC allows creating @dfn{archives} of expressions which can be stored
6362 to or retrieved from files. To create an archive, you declare an object
6363 of class @code{archive} and archive expressions in it, giving each
6364 expression a unique name:
6368 using namespace std;
6369 #include <ginac/ginac.h>
6370 using namespace GiNaC;
6374 symbol x("x"), y("y"), z("z");
6376 ex foo = sin(x + 2*y) + 3*z + 41;
6380 a.archive_ex(foo, "foo");
6381 a.archive_ex(bar, "the second one");
6385 The archive can then be written to a file:
6389 ofstream out("foobar.gar");
6395 The file @file{foobar.gar} contains all information that is needed to
6396 reconstruct the expressions @code{foo} and @code{bar}.
6398 @cindex @command{viewgar}
6399 The tool @command{viewgar} that comes with GiNaC can be used to view
6400 the contents of GiNaC archive files:
6403 $ viewgar foobar.gar
6404 foo = 41+sin(x+2*y)+3*z
6405 the second one = 42+sin(x+2*y)+3*z
6408 The point of writing archive files is of course that they can later be
6414 ifstream in("foobar.gar");
6419 And the stored expressions can be retrieved by their name:
6426 ex ex1 = a2.unarchive_ex(syms, "foo");
6427 ex ex2 = a2.unarchive_ex(syms, "the second one");
6429 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6430 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6431 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6435 Note that you have to supply a list of the symbols which are to be inserted
6436 in the expressions. Symbols in archives are stored by their name only and
6437 if you don't specify which symbols you have, unarchiving the expression will
6438 create new symbols with that name. E.g. if you hadn't included @code{x} in
6439 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6440 have had no effect because the @code{x} in @code{ex1} would have been a
6441 different symbol than the @code{x} which was defined at the beginning of
6442 the program, although both would appear as @samp{x} when printed.
6444 You can also use the information stored in an @code{archive} object to
6445 output expressions in a format suitable for exact reconstruction. The
6446 @code{archive} and @code{archive_node} classes have a couple of member
6447 functions that let you access the stored properties:
6450 static void my_print2(const archive_node & n)
6453 n.find_string("class", class_name);
6454 cout << class_name << "(";
6456 archive_node::propinfovector p;
6457 n.get_properties(p);
6459 size_t num = p.size();
6460 for (size_t i=0; i<num; i++) @{
6461 const string &name = p[i].name;
6462 if (name == "class")
6464 cout << name << "=";
6466 unsigned count = p[i].count;
6470 for (unsigned j=0; j<count; j++) @{
6471 switch (p[i].type) @{
6472 case archive_node::PTYPE_BOOL: @{
6474 n.find_bool(name, x, j);
6475 cout << (x ? "true" : "false");
6478 case archive_node::PTYPE_UNSIGNED: @{
6480 n.find_unsigned(name, x, j);
6484 case archive_node::PTYPE_STRING: @{
6486 n.find_string(name, x, j);
6487 cout << '\"' << x << '\"';
6490 case archive_node::PTYPE_NODE: @{
6491 const archive_node &x = n.find_ex_node(name, j);
6513 ex e = pow(2, x) - y;
6515 my_print2(ar.get_top_node(0)); cout << endl;
6523 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6524 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6525 overall_coeff=numeric(number="0"))
6528 Be warned, however, that the set of properties and their meaning for each
6529 class may change between GiNaC versions.
6532 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6533 @c node-name, next, previous, up
6534 @chapter Extending GiNaC
6536 By reading so far you should have gotten a fairly good understanding of
6537 GiNaC's design patterns. From here on you should start reading the
6538 sources. All we can do now is issue some recommendations how to tackle
6539 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6540 develop some useful extension please don't hesitate to contact the GiNaC
6541 authors---they will happily incorporate them into future versions.
6544 * What does not belong into GiNaC:: What to avoid.
6545 * Symbolic functions:: Implementing symbolic functions.
6546 * Printing:: Adding new output formats.
6547 * Structures:: Defining new algebraic classes (the easy way).
6548 * Adding classes:: Defining new algebraic classes (the hard way).
6552 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6553 @c node-name, next, previous, up
6554 @section What doesn't belong into GiNaC
6556 @cindex @command{ginsh}
6557 First of all, GiNaC's name must be read literally. It is designed to be
6558 a library for use within C++. The tiny @command{ginsh} accompanying
6559 GiNaC makes this even more clear: it doesn't even attempt to provide a
6560 language. There are no loops or conditional expressions in
6561 @command{ginsh}, it is merely a window into the library for the
6562 programmer to test stuff (or to show off). Still, the design of a
6563 complete CAS with a language of its own, graphical capabilities and all
6564 this on top of GiNaC is possible and is without doubt a nice project for
6567 There are many built-in functions in GiNaC that do not know how to
6568 evaluate themselves numerically to a precision declared at runtime
6569 (using @code{Digits}). Some may be evaluated at certain points, but not
6570 generally. This ought to be fixed. However, doing numerical
6571 computations with GiNaC's quite abstract classes is doomed to be
6572 inefficient. For this purpose, the underlying foundation classes
6573 provided by CLN are much better suited.
6576 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6577 @c node-name, next, previous, up
6578 @section Symbolic functions
6580 The easiest and most instructive way to start extending GiNaC is probably to
6581 create your own symbolic functions. These are implemented with the help of
6582 two preprocessor macros:
6584 @cindex @code{DECLARE_FUNCTION}
6585 @cindex @code{REGISTER_FUNCTION}
6587 DECLARE_FUNCTION_<n>P(<name>)
6588 REGISTER_FUNCTION(<name>, <options>)
6591 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6592 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6593 parameters of type @code{ex} and returns a newly constructed GiNaC
6594 @code{function} object that represents your function.
6596 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6597 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6598 set of options that associate the symbolic function with C++ functions you
6599 provide to implement the various methods such as evaluation, derivative,
6600 series expansion etc. They also describe additional attributes the function
6601 might have, such as symmetry and commutation properties, and a name for
6602 LaTeX output. Multiple options are separated by the member access operator
6603 @samp{.} and can be given in an arbitrary order.
6605 (By the way: in case you are worrying about all the macros above we can
6606 assure you that functions are GiNaC's most macro-intense classes. We have
6607 done our best to avoid macros where we can.)
6609 @subsection A minimal example
6611 Here is an example for the implementation of a function with two arguments
6612 that is not further evaluated:
6615 DECLARE_FUNCTION_2P(myfcn)
6617 REGISTER_FUNCTION(myfcn, dummy())
6620 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6621 in algebraic expressions:
6627 ex e = 2*myfcn(42, 1+3*x) - x;
6629 // prints '2*myfcn(42,1+3*x)-x'
6634 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6635 "no options". A function with no options specified merely acts as a kind of
6636 container for its arguments. It is a pure "dummy" function with no associated
6637 logic (which is, however, sometimes perfectly sufficient).
6639 Let's now have a look at the implementation of GiNaC's cosine function for an
6640 example of how to make an "intelligent" function.
6642 @subsection The cosine function
6644 The GiNaC header file @file{inifcns.h} contains the line
6647 DECLARE_FUNCTION_1P(cos)
6650 which declares to all programs using GiNaC that there is a function @samp{cos}
6651 that takes one @code{ex} as an argument. This is all they need to know to use
6652 this function in expressions.
6654 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6655 is its @code{REGISTER_FUNCTION} line:
6658 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6659 evalf_func(cos_evalf).
6660 derivative_func(cos_deriv).
6661 latex_name("\\cos"));
6664 There are four options defined for the cosine function. One of them
6665 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6666 other three indicate the C++ functions in which the "brains" of the cosine
6667 function are defined.
6669 @cindex @code{hold()}
6671 The @code{eval_func()} option specifies the C++ function that implements
6672 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6673 the same number of arguments as the associated symbolic function (one in this
6674 case) and returns the (possibly transformed or in some way simplified)
6675 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6676 of the automatic evaluation process). If no (further) evaluation is to take
6677 place, the @code{eval_func()} function must return the original function
6678 with @code{.hold()}, to avoid a potential infinite recursion. If your
6679 symbolic functions produce a segmentation fault or stack overflow when
6680 using them in expressions, you are probably missing a @code{.hold()}
6683 The @code{eval_func()} function for the cosine looks something like this
6684 (actually, it doesn't look like this at all, but it should give you an idea
6688 static ex cos_eval(const ex & x)
6690 if ("x is a multiple of 2*Pi")
6692 else if ("x is a multiple of Pi")
6694 else if ("x is a multiple of Pi/2")
6698 else if ("x has the form 'acos(y)'")
6700 else if ("x has the form 'asin(y)'")
6705 return cos(x).hold();
6709 This function is called every time the cosine is used in a symbolic expression:
6715 // this calls cos_eval(Pi), and inserts its return value into
6716 // the actual expression
6723 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6724 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6725 symbolic transformation can be done, the unmodified function is returned
6726 with @code{.hold()}.
6728 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6729 The user has to call @code{evalf()} for that. This is implemented in a
6733 static ex cos_evalf(const ex & x)
6735 if (is_a<numeric>(x))
6736 return cos(ex_to<numeric>(x));
6738 return cos(x).hold();
6742 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6743 in this case the @code{cos()} function for @code{numeric} objects, which in
6744 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6745 isn't really needed here, but reminds us that the corresponding @code{eval()}
6746 function would require it in this place.
6748 Differentiation will surely turn up and so we need to tell @code{cos}
6749 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6750 instance, are then handled automatically by @code{basic::diff} and
6754 static ex cos_deriv(const ex & x, unsigned diff_param)
6760 @cindex product rule
6761 The second parameter is obligatory but uninteresting at this point. It
6762 specifies which parameter to differentiate in a partial derivative in
6763 case the function has more than one parameter, and its main application
6764 is for correct handling of the chain rule.
6766 An implementation of the series expansion is not needed for @code{cos()} as
6767 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6768 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6769 the other hand, does have poles and may need to do Laurent expansion:
6772 static ex tan_series(const ex & x, const relational & rel,
6773 int order, unsigned options)
6775 // Find the actual expansion point
6776 const ex x_pt = x.subs(rel);
6778 if ("x_pt is not an odd multiple of Pi/2")
6779 throw do_taylor(); // tell function::series() to do Taylor expansion
6781 // On a pole, expand sin()/cos()
6782 return (sin(x)/cos(x)).series(rel, order+2, options);
6786 The @code{series()} implementation of a function @emph{must} return a
6787 @code{pseries} object, otherwise your code will crash.
6789 @subsection Function options
6791 GiNaC functions understand several more options which are always
6792 specified as @code{.option(params)}. None of them are required, but you
6793 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6794 is a do-nothing option called @code{dummy()} which you can use to define
6795 functions without any special options.
6798 eval_func(<C++ function>)
6799 evalf_func(<C++ function>)
6800 derivative_func(<C++ function>)
6801 series_func(<C++ function>)
6802 conjugate_func(<C++ function>)
6805 These specify the C++ functions that implement symbolic evaluation,
6806 numeric evaluation, partial derivatives, and series expansion, respectively.
6807 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6808 @code{diff()} and @code{series()}.
6810 The @code{eval_func()} function needs to use @code{.hold()} if no further
6811 automatic evaluation is desired or possible.
6813 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6814 expansion, which is correct if there are no poles involved. If the function
6815 has poles in the complex plane, the @code{series_func()} needs to check
6816 whether the expansion point is on a pole and fall back to Taylor expansion
6817 if it isn't. Otherwise, the pole usually needs to be regularized by some
6818 suitable transformation.
6821 latex_name(const string & n)
6824 specifies the LaTeX code that represents the name of the function in LaTeX
6825 output. The default is to put the function name in an @code{\mbox@{@}}.
6828 do_not_evalf_params()
6831 This tells @code{evalf()} to not recursively evaluate the parameters of the
6832 function before calling the @code{evalf_func()}.
6835 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6838 This allows you to explicitly specify the commutation properties of the
6839 function (@xref{Non-commutative objects}, for an explanation of
6840 (non)commutativity in GiNaC). For example, you can use
6841 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6842 GiNaC treat your function like a matrix. By default, functions inherit the
6843 commutation properties of their first argument.
6846 set_symmetry(const symmetry & s)
6849 specifies the symmetry properties of the function with respect to its
6850 arguments. @xref{Indexed objects}, for an explanation of symmetry
6851 specifications. GiNaC will automatically rearrange the arguments of
6852 symmetric functions into a canonical order.
6854 Sometimes you may want to have finer control over how functions are
6855 displayed in the output. For example, the @code{abs()} function prints
6856 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6857 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6861 print_func<C>(<C++ function>)
6864 option which is explained in the next section.
6866 @subsection Functions with a variable number of arguments
6868 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6869 functions with a fixed number of arguments. Sometimes, though, you may need
6870 to have a function that accepts a variable number of expressions. One way to
6871 accomplish this is to pass variable-length lists as arguments. The
6872 @code{Li()} function uses this method for multiple polylogarithms.
6874 It is also possible to define functions that accept a different number of
6875 parameters under the same function name, such as the @code{psi()} function
6876 which can be called either as @code{psi(z)} (the digamma function) or as
6877 @code{psi(n, z)} (polygamma functions). These are actually two different
6878 functions in GiNaC that, however, have the same name. Defining such
6879 functions is not possible with the macros but requires manually fiddling
6880 with GiNaC internals. If you are interested, please consult the GiNaC source
6881 code for the @code{psi()} function (@file{inifcns.h} and
6882 @file{inifcns_gamma.cpp}).
6885 @node Printing, Structures, Symbolic functions, Extending GiNaC
6886 @c node-name, next, previous, up
6887 @section GiNaC's expression output system
6889 GiNaC allows the output of expressions in a variety of different formats
6890 (@pxref{Input/Output}). This section will explain how expression output
6891 is implemented internally, and how to define your own output formats or
6892 change the output format of built-in algebraic objects. You will also want
6893 to read this section if you plan to write your own algebraic classes or
6896 @cindex @code{print_context} (class)
6897 @cindex @code{print_dflt} (class)
6898 @cindex @code{print_latex} (class)
6899 @cindex @code{print_tree} (class)
6900 @cindex @code{print_csrc} (class)
6901 All the different output formats are represented by a hierarchy of classes
6902 rooted in the @code{print_context} class, defined in the @file{print.h}
6907 the default output format
6909 output in LaTeX mathematical mode
6911 a dump of the internal expression structure (for debugging)
6913 the base class for C source output
6914 @item print_csrc_float
6915 C source output using the @code{float} type
6916 @item print_csrc_double
6917 C source output using the @code{double} type
6918 @item print_csrc_cl_N
6919 C source output using CLN types
6922 The @code{print_context} base class provides two public data members:
6934 @code{s} is a reference to the stream to output to, while @code{options}
6935 holds flags and modifiers. Currently, there is only one flag defined:
6936 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6937 to print the index dimension which is normally hidden.
6939 When you write something like @code{std::cout << e}, where @code{e} is
6940 an object of class @code{ex}, GiNaC will construct an appropriate
6941 @code{print_context} object (of a class depending on the selected output
6942 format), fill in the @code{s} and @code{options} members, and call
6944 @cindex @code{print()}
6946 void ex::print(const print_context & c, unsigned level = 0) const;
6949 which in turn forwards the call to the @code{print()} method of the
6950 top-level algebraic object contained in the expression.
6952 Unlike other methods, GiNaC classes don't usually override their
6953 @code{print()} method to implement expression output. Instead, the default
6954 implementation @code{basic::print(c, level)} performs a run-time double
6955 dispatch to a function selected by the dynamic type of the object and the
6956 passed @code{print_context}. To this end, GiNaC maintains a separate method
6957 table for each class, similar to the virtual function table used for ordinary
6958 (single) virtual function dispatch.
6960 The method table contains one slot for each possible @code{print_context}
6961 type, indexed by the (internally assigned) serial number of the type. Slots
6962 may be empty, in which case GiNaC will retry the method lookup with the
6963 @code{print_context} object's parent class, possibly repeating the process
6964 until it reaches the @code{print_context} base class. If there's still no
6965 method defined, the method table of the algebraic object's parent class
6966 is consulted, and so on, until a matching method is found (eventually it
6967 will reach the combination @code{basic/print_context}, which prints the
6968 object's class name enclosed in square brackets).
6970 You can think of the print methods of all the different classes and output
6971 formats as being arranged in a two-dimensional matrix with one axis listing
6972 the algebraic classes and the other axis listing the @code{print_context}
6975 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6976 to implement printing, but then they won't get any of the benefits of the
6977 double dispatch mechanism (such as the ability for derived classes to
6978 inherit only certain print methods from its parent, or the replacement of
6979 methods at run-time).
6981 @subsection Print methods for classes
6983 The method table for a class is set up either in the definition of the class,
6984 by passing the appropriate @code{print_func<C>()} option to
6985 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6986 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6987 can also be used to override existing methods dynamically.
6989 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6990 be a member function of the class (or one of its parent classes), a static
6991 member function, or an ordinary (global) C++ function. The @code{C} template
6992 parameter specifies the appropriate @code{print_context} type for which the
6993 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6994 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6995 the class is the one being implemented by
6996 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6998 For print methods that are member functions, their first argument must be of
6999 a type convertible to a @code{const C &}, and the second argument must be an
7002 For static members and global functions, the first argument must be of a type
7003 convertible to a @code{const T &}, the second argument must be of a type
7004 convertible to a @code{const C &}, and the third argument must be an
7005 @code{unsigned}. A global function will, of course, not have access to
7006 private and protected members of @code{T}.
7008 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7009 and @code{basic::print()}) is used for proper parenthesizing of the output
7010 (and by @code{print_tree} for proper indentation). It can be used for similar
7011 purposes if you write your own output formats.
7013 The explanations given above may seem complicated, but in practice it's
7014 really simple, as shown in the following example. Suppose that we want to
7015 display exponents in LaTeX output not as superscripts but with little
7016 upwards-pointing arrows. This can be achieved in the following way:
7019 void my_print_power_as_latex(const power & p,
7020 const print_latex & c,
7023 // get the precedence of the 'power' class
7024 unsigned power_prec = p.precedence();
7026 // if the parent operator has the same or a higher precedence
7027 // we need parentheses around the power
7028 if (level >= power_prec)
7031 // print the basis and exponent, each enclosed in braces, and
7032 // separated by an uparrow
7034 p.op(0).print(c, power_prec);
7035 c.s << "@}\\uparrow@{";
7036 p.op(1).print(c, power_prec);
7039 // don't forget the closing parenthesis
7040 if (level >= power_prec)
7046 // a sample expression
7047 symbol x("x"), y("y");
7048 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7050 // switch to LaTeX mode
7053 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7056 // now we replace the method for the LaTeX output of powers with
7058 set_print_func<power, print_latex>(my_print_power_as_latex);
7060 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7071 The first argument of @code{my_print_power_as_latex} could also have been
7072 a @code{const basic &}, the second one a @code{const print_context &}.
7075 The above code depends on @code{mul} objects converting their operands to
7076 @code{power} objects for the purpose of printing.
7079 The output of products including negative powers as fractions is also
7080 controlled by the @code{mul} class.
7083 The @code{power/print_latex} method provided by GiNaC prints square roots
7084 using @code{\sqrt}, but the above code doesn't.
7088 It's not possible to restore a method table entry to its previous or default
7089 value. Once you have called @code{set_print_func()}, you can only override
7090 it with another call to @code{set_print_func()}, but you can't easily go back
7091 to the default behavior again (you can, of course, dig around in the GiNaC
7092 sources, find the method that is installed at startup
7093 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7094 one; that is, after you circumvent the C++ member access control@dots{}).
7096 @subsection Print methods for functions
7098 Symbolic functions employ a print method dispatch mechanism similar to the
7099 one used for classes. The methods are specified with @code{print_func<C>()}
7100 function options. If you don't specify any special print methods, the function
7101 will be printed with its name (or LaTeX name, if supplied), followed by a
7102 comma-separated list of arguments enclosed in parentheses.
7104 For example, this is what GiNaC's @samp{abs()} function is defined like:
7107 static ex abs_eval(const ex & arg) @{ ... @}
7108 static ex abs_evalf(const ex & arg) @{ ... @}
7110 static void abs_print_latex(const ex & arg, const print_context & c)
7112 c.s << "@{|"; arg.print(c); c.s << "|@}";
7115 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7117 c.s << "fabs("; arg.print(c); c.s << ")";
7120 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7121 evalf_func(abs_evalf).
7122 print_func<print_latex>(abs_print_latex).
7123 print_func<print_csrc_float>(abs_print_csrc_float).
7124 print_func<print_csrc_double>(abs_print_csrc_float));
7127 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7128 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7130 There is currently no equivalent of @code{set_print_func()} for functions.
7132 @subsection Adding new output formats
7134 Creating a new output format involves subclassing @code{print_context},
7135 which is somewhat similar to adding a new algebraic class
7136 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7137 that needs to go into the class definition, and a corresponding macro
7138 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7139 Every @code{print_context} class needs to provide a default constructor
7140 and a constructor from an @code{std::ostream} and an @code{unsigned}
7143 Here is an example for a user-defined @code{print_context} class:
7146 class print_myformat : public print_dflt
7148 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7150 print_myformat(std::ostream & os, unsigned opt = 0)
7151 : print_dflt(os, opt) @{@}
7154 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7156 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7159 That's all there is to it. None of the actual expression output logic is
7160 implemented in this class. It merely serves as a selector for choosing
7161 a particular format. The algorithms for printing expressions in the new
7162 format are implemented as print methods, as described above.
7164 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7165 exactly like GiNaC's default output format:
7170 ex e = pow(x, 2) + 1;
7172 // this prints "1+x^2"
7175 // this also prints "1+x^2"
7176 e.print(print_myformat()); cout << endl;
7182 To fill @code{print_myformat} with life, we need to supply appropriate
7183 print methods with @code{set_print_func()}, like this:
7186 // This prints powers with '**' instead of '^'. See the LaTeX output
7187 // example above for explanations.
7188 void print_power_as_myformat(const power & p,
7189 const print_myformat & c,
7192 unsigned power_prec = p.precedence();
7193 if (level >= power_prec)
7195 p.op(0).print(c, power_prec);
7197 p.op(1).print(c, power_prec);
7198 if (level >= power_prec)
7204 // install a new print method for power objects
7205 set_print_func<power, print_myformat>(print_power_as_myformat);
7207 // now this prints "1+x**2"
7208 e.print(print_myformat()); cout << endl;
7210 // but the default format is still "1+x^2"
7216 @node Structures, Adding classes, Printing, Extending GiNaC
7217 @c node-name, next, previous, up
7220 If you are doing some very specialized things with GiNaC, or if you just
7221 need some more organized way to store data in your expressions instead of
7222 anonymous lists, you may want to implement your own algebraic classes.
7223 ('algebraic class' means any class directly or indirectly derived from
7224 @code{basic} that can be used in GiNaC expressions).
7226 GiNaC offers two ways of accomplishing this: either by using the
7227 @code{structure<T>} template class, or by rolling your own class from
7228 scratch. This section will discuss the @code{structure<T>} template which
7229 is easier to use but more limited, while the implementation of custom
7230 GiNaC classes is the topic of the next section. However, you may want to
7231 read both sections because many common concepts and member functions are
7232 shared by both concepts, and it will also allow you to decide which approach
7233 is most suited to your needs.
7235 The @code{structure<T>} template, defined in the GiNaC header file
7236 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7237 or @code{class}) into a GiNaC object that can be used in expressions.
7239 @subsection Example: scalar products
7241 Let's suppose that we need a way to handle some kind of abstract scalar
7242 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7243 product class have to store their left and right operands, which can in turn
7244 be arbitrary expressions. Here is a possible way to represent such a
7245 product in a C++ @code{struct}:
7249 using namespace std;
7251 #include <ginac/ginac.h>
7252 using namespace GiNaC;
7258 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7262 The default constructor is required. Now, to make a GiNaC class out of this
7263 data structure, we need only one line:
7266 typedef structure<sprod_s> sprod;
7269 That's it. This line constructs an algebraic class @code{sprod} which
7270 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7271 expressions like any other GiNaC class:
7275 symbol a("a"), b("b");
7276 ex e = sprod(sprod_s(a, b));
7280 Note the difference between @code{sprod} which is the algebraic class, and
7281 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7282 and @code{right} data members. As shown above, an @code{sprod} can be
7283 constructed from an @code{sprod_s} object.
7285 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7286 you could define a little wrapper function like this:
7289 inline ex make_sprod(ex left, ex right)
7291 return sprod(sprod_s(left, right));
7295 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7296 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7297 @code{get_struct()}:
7301 cout << ex_to<sprod>(e)->left << endl;
7303 cout << ex_to<sprod>(e).get_struct().right << endl;
7308 You only have read access to the members of @code{sprod_s}.
7310 The type definition of @code{sprod} is enough to write your own algorithms
7311 that deal with scalar products, for example:
7316 if (is_a<sprod>(p)) @{
7317 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7318 return make_sprod(sp.right, sp.left);
7329 @subsection Structure output
7331 While the @code{sprod} type is useable it still leaves something to be
7332 desired, most notably proper output:
7337 // -> [structure object]
7341 By default, any structure types you define will be printed as
7342 @samp{[structure object]}. To override this you can either specialize the
7343 template's @code{print()} member function, or specify print methods with
7344 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7345 it's not possible to supply class options like @code{print_func<>()} to
7346 structures, so for a self-contained structure type you need to resort to
7347 overriding the @code{print()} function, which is also what we will do here.
7349 The member functions of GiNaC classes are described in more detail in the
7350 next section, but it shouldn't be hard to figure out what's going on here:
7353 void sprod::print(const print_context & c, unsigned level) const
7355 // tree debug output handled by superclass
7356 if (is_a<print_tree>(c))
7357 inherited::print(c, level);
7359 // get the contained sprod_s object
7360 const sprod_s & sp = get_struct();
7362 // print_context::s is a reference to an ostream
7363 c.s << "<" << sp.left << "|" << sp.right << ">";
7367 Now we can print expressions containing scalar products:
7373 cout << swap_sprod(e) << endl;
7378 @subsection Comparing structures
7380 The @code{sprod} class defined so far still has one important drawback: all
7381 scalar products are treated as being equal because GiNaC doesn't know how to
7382 compare objects of type @code{sprod_s}. This can lead to some confusing
7383 and undesired behavior:
7387 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7389 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7390 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7394 To remedy this, we first need to define the operators @code{==} and @code{<}
7395 for objects of type @code{sprod_s}:
7398 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7400 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7403 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7405 return lhs.left.compare(rhs.left) < 0
7406 ? true : lhs.right.compare(rhs.right) < 0;
7410 The ordering established by the @code{<} operator doesn't have to make any
7411 algebraic sense, but it needs to be well defined. Note that we can't use
7412 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7413 in the implementation of these operators because they would construct
7414 GiNaC @code{relational} objects which in the case of @code{<} do not
7415 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7416 decide which one is algebraically 'less').
7418 Next, we need to change our definition of the @code{sprod} type to let
7419 GiNaC know that an ordering relation exists for the embedded objects:
7422 typedef structure<sprod_s, compare_std_less> sprod;
7425 @code{sprod} objects then behave as expected:
7429 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7430 // -> <a|b>-<a^2|b^2>
7431 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7432 // -> <a|b>+<a^2|b^2>
7433 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7435 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7440 The @code{compare_std_less} policy parameter tells GiNaC to use the
7441 @code{std::less} and @code{std::equal_to} functors to compare objects of
7442 type @code{sprod_s}. By default, these functors forward their work to the
7443 standard @code{<} and @code{==} operators, which we have overloaded.
7444 Alternatively, we could have specialized @code{std::less} and
7445 @code{std::equal_to} for class @code{sprod_s}.
7447 GiNaC provides two other comparison policies for @code{structure<T>}
7448 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7449 which does a bit-wise comparison of the contained @code{T} objects.
7450 This should be used with extreme care because it only works reliably with
7451 built-in integral types, and it also compares any padding (filler bytes of
7452 undefined value) that the @code{T} class might have.
7454 @subsection Subexpressions
7456 Our scalar product class has two subexpressions: the left and right
7457 operands. It might be a good idea to make them accessible via the standard
7458 @code{nops()} and @code{op()} methods:
7461 size_t sprod::nops() const
7466 ex sprod::op(size_t i) const
7470 return get_struct().left;
7472 return get_struct().right;
7474 throw std::range_error("sprod::op(): no such operand");
7479 Implementing @code{nops()} and @code{op()} for container types such as
7480 @code{sprod} has two other nice side effects:
7484 @code{has()} works as expected
7486 GiNaC generates better hash keys for the objects (the default implementation
7487 of @code{calchash()} takes subexpressions into account)
7490 @cindex @code{let_op()}
7491 There is a non-const variant of @code{op()} called @code{let_op()} that
7492 allows replacing subexpressions:
7495 ex & sprod::let_op(size_t i)
7497 // every non-const member function must call this
7498 ensure_if_modifiable();
7502 return get_struct().left;
7504 return get_struct().right;
7506 throw std::range_error("sprod::let_op(): no such operand");
7511 Once we have provided @code{let_op()} we also get @code{subs()} and
7512 @code{map()} for free. In fact, every container class that returns a non-null
7513 @code{nops()} value must either implement @code{let_op()} or provide custom
7514 implementations of @code{subs()} and @code{map()}.
7516 In turn, the availability of @code{map()} enables the recursive behavior of a
7517 couple of other default method implementations, in particular @code{evalf()},
7518 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7519 we probably want to provide our own version of @code{expand()} for scalar
7520 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7521 This is left as an exercise for the reader.
7523 The @code{structure<T>} template defines many more member functions that
7524 you can override by specialization to customize the behavior of your
7525 structures. You are referred to the next section for a description of
7526 some of these (especially @code{eval()}). There is, however, one topic
7527 that shall be addressed here, as it demonstrates one peculiarity of the
7528 @code{structure<T>} template: archiving.
7530 @subsection Archiving structures
7532 If you don't know how the archiving of GiNaC objects is implemented, you
7533 should first read the next section and then come back here. You're back?
7536 To implement archiving for structures it is not enough to provide
7537 specializations for the @code{archive()} member function and the
7538 unarchiving constructor (the @code{unarchive()} function has a default
7539 implementation). You also need to provide a unique name (as a string literal)
7540 for each structure type you define. This is because in GiNaC archives,
7541 the class of an object is stored as a string, the class name.
7543 By default, this class name (as returned by the @code{class_name()} member
7544 function) is @samp{structure} for all structure classes. This works as long
7545 as you have only defined one structure type, but if you use two or more you
7546 need to provide a different name for each by specializing the
7547 @code{get_class_name()} member function. Here is a sample implementation
7548 for enabling archiving of the scalar product type defined above:
7551 const char *sprod::get_class_name() @{ return "sprod"; @}
7553 void sprod::archive(archive_node & n) const
7555 inherited::archive(n);
7556 n.add_ex("left", get_struct().left);
7557 n.add_ex("right", get_struct().right);
7560 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7562 n.find_ex("left", get_struct().left, sym_lst);
7563 n.find_ex("right", get_struct().right, sym_lst);
7567 Note that the unarchiving constructor is @code{sprod::structure} and not
7568 @code{sprod::sprod}, and that we don't need to supply an
7569 @code{sprod::unarchive()} function.
7572 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7573 @c node-name, next, previous, up
7574 @section Adding classes
7576 The @code{structure<T>} template provides an way to extend GiNaC with custom
7577 algebraic classes that is easy to use but has its limitations, the most
7578 severe of which being that you can't add any new member functions to
7579 structures. To be able to do this, you need to write a new class definition
7582 This section will explain how to implement new algebraic classes in GiNaC by
7583 giving the example of a simple 'string' class. After reading this section
7584 you will know how to properly declare a GiNaC class and what the minimum
7585 required member functions are that you have to implement. We only cover the
7586 implementation of a 'leaf' class here (i.e. one that doesn't contain
7587 subexpressions). Creating a container class like, for example, a class
7588 representing tensor products is more involved but this section should give
7589 you enough information so you can consult the source to GiNaC's predefined
7590 classes if you want to implement something more complicated.
7592 @subsection GiNaC's run-time type information system
7594 @cindex hierarchy of classes
7596 All algebraic classes (that is, all classes that can appear in expressions)
7597 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7598 @code{basic *} (which is essentially what an @code{ex} is) represents a
7599 generic pointer to an algebraic class. Occasionally it is necessary to find
7600 out what the class of an object pointed to by a @code{basic *} really is.
7601 Also, for the unarchiving of expressions it must be possible to find the
7602 @code{unarchive()} function of a class given the class name (as a string). A
7603 system that provides this kind of information is called a run-time type
7604 information (RTTI) system. The C++ language provides such a thing (see the
7605 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7606 implements its own, simpler RTTI.
7608 The RTTI in GiNaC is based on two mechanisms:
7613 The @code{basic} class declares a member variable @code{tinfo_key} which
7614 holds an unsigned integer that identifies the object's class. These numbers
7615 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7616 classes. They all start with @code{TINFO_}.
7619 By means of some clever tricks with static members, GiNaC maintains a list
7620 of information for all classes derived from @code{basic}. The information
7621 available includes the class names, the @code{tinfo_key}s, and pointers
7622 to the unarchiving functions. This class registry is defined in the
7623 @file{registrar.h} header file.
7627 The disadvantage of this proprietary RTTI implementation is that there's
7628 a little more to do when implementing new classes (C++'s RTTI works more
7629 or less automatically) but don't worry, most of the work is simplified by
7632 @subsection A minimalistic example
7634 Now we will start implementing a new class @code{mystring} that allows
7635 placing character strings in algebraic expressions (this is not very useful,
7636 but it's just an example). This class will be a direct subclass of
7637 @code{basic}. You can use this sample implementation as a starting point
7638 for your own classes.
7640 The code snippets given here assume that you have included some header files
7646 #include <stdexcept>
7647 using namespace std;
7649 #include <ginac/ginac.h>
7650 using namespace GiNaC;
7653 The first thing we have to do is to define a @code{tinfo_key} for our new
7654 class. This can be any arbitrary unsigned number that is not already taken
7655 by one of the existing classes but it's better to come up with something
7656 that is unlikely to clash with keys that might be added in the future. The
7657 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7658 which is not a requirement but we are going to stick with this scheme:
7661 const unsigned TINFO_mystring = 0x42420001U;
7664 Now we can write down the class declaration. The class stores a C++
7665 @code{string} and the user shall be able to construct a @code{mystring}
7666 object from a C or C++ string:
7669 class mystring : public basic
7671 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7674 mystring(const string &s);
7675 mystring(const char *s);
7681 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7684 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7685 macros are defined in @file{registrar.h}. They take the name of the class
7686 and its direct superclass as arguments and insert all required declarations
7687 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7688 the first line after the opening brace of the class definition. The
7689 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7690 source (at global scope, of course, not inside a function).
7692 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7693 declarations of the default constructor and a couple of other functions that
7694 are required. It also defines a type @code{inherited} which refers to the
7695 superclass so you don't have to modify your code every time you shuffle around
7696 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7697 class with the GiNaC RTTI (there is also a
7698 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7699 options for the class, and which we will be using instead in a few minutes).
7701 Now there are seven member functions we have to implement to get a working
7707 @code{mystring()}, the default constructor.
7710 @code{void archive(archive_node &n)}, the archiving function. This stores all
7711 information needed to reconstruct an object of this class inside an
7712 @code{archive_node}.
7715 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7716 constructor. This constructs an instance of the class from the information
7717 found in an @code{archive_node}.
7720 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7721 unarchiving function. It constructs a new instance by calling the unarchiving
7725 @cindex @code{compare_same_type()}
7726 @code{int compare_same_type(const basic &other)}, which is used internally
7727 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7728 -1, depending on the relative order of this object and the @code{other}
7729 object. If it returns 0, the objects are considered equal.
7730 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7731 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7732 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7733 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7734 must provide a @code{compare_same_type()} function, even those representing
7735 objects for which no reasonable algebraic ordering relationship can be
7739 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7740 which are the two constructors we declared.
7744 Let's proceed step-by-step. The default constructor looks like this:
7747 mystring::mystring() : inherited(TINFO_mystring) @{@}
7750 The golden rule is that in all constructors you have to set the
7751 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7752 it will be set by the constructor of the superclass and all hell will break
7753 loose in the RTTI. For your convenience, the @code{basic} class provides
7754 a constructor that takes a @code{tinfo_key} value, which we are using here
7755 (remember that in our case @code{inherited == basic}). If the superclass
7756 didn't have such a constructor, we would have to set the @code{tinfo_key}
7757 to the right value manually.
7759 In the default constructor you should set all other member variables to
7760 reasonable default values (we don't need that here since our @code{str}
7761 member gets set to an empty string automatically).
7763 Next are the three functions for archiving. You have to implement them even
7764 if you don't plan to use archives, but the minimum required implementation
7765 is really simple. First, the archiving function:
7768 void mystring::archive(archive_node &n) const
7770 inherited::archive(n);
7771 n.add_string("string", str);
7775 The only thing that is really required is calling the @code{archive()}
7776 function of the superclass. Optionally, you can store all information you
7777 deem necessary for representing the object into the passed
7778 @code{archive_node}. We are just storing our string here. For more
7779 information on how the archiving works, consult the @file{archive.h} header
7782 The unarchiving constructor is basically the inverse of the archiving
7786 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7788 n.find_string("string", str);
7792 If you don't need archiving, just leave this function empty (but you must
7793 invoke the unarchiving constructor of the superclass). Note that we don't
7794 have to set the @code{tinfo_key} here because it is done automatically
7795 by the unarchiving constructor of the @code{basic} class.
7797 Finally, the unarchiving function:
7800 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7802 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7806 You don't have to understand how exactly this works. Just copy these
7807 four lines into your code literally (replacing the class name, of
7808 course). It calls the unarchiving constructor of the class and unless
7809 you are doing something very special (like matching @code{archive_node}s
7810 to global objects) you don't need a different implementation. For those
7811 who are interested: setting the @code{dynallocated} flag puts the object
7812 under the control of GiNaC's garbage collection. It will get deleted
7813 automatically once it is no longer referenced.
7815 Our @code{compare_same_type()} function uses a provided function to compare
7819 int mystring::compare_same_type(const basic &other) const
7821 const mystring &o = static_cast<const mystring &>(other);
7822 int cmpval = str.compare(o.str);
7825 else if (cmpval < 0)
7832 Although this function takes a @code{basic &}, it will always be a reference
7833 to an object of exactly the same class (objects of different classes are not
7834 comparable), so the cast is safe. If this function returns 0, the two objects
7835 are considered equal (in the sense that @math{A-B=0}), so you should compare
7836 all relevant member variables.
7838 Now the only thing missing is our two new constructors:
7841 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7842 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7845 No surprises here. We set the @code{str} member from the argument and
7846 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7848 That's it! We now have a minimal working GiNaC class that can store
7849 strings in algebraic expressions. Let's confirm that the RTTI works:
7852 ex e = mystring("Hello, world!");
7853 cout << is_a<mystring>(e) << endl;
7856 cout << e.bp->class_name() << endl;
7860 Obviously it does. Let's see what the expression @code{e} looks like:
7864 // -> [mystring object]
7867 Hm, not exactly what we expect, but of course the @code{mystring} class
7868 doesn't yet know how to print itself. This can be done either by implementing
7869 the @code{print()} member function, or, preferably, by specifying a
7870 @code{print_func<>()} class option. Let's say that we want to print the string
7871 surrounded by double quotes:
7874 class mystring : public basic
7878 void do_print(const print_context &c, unsigned level = 0) const;
7882 void mystring::do_print(const print_context &c, unsigned level) const
7884 // print_context::s is a reference to an ostream
7885 c.s << '\"' << str << '\"';
7889 The @code{level} argument is only required for container classes to
7890 correctly parenthesize the output.
7892 Now we need to tell GiNaC that @code{mystring} objects should use the
7893 @code{do_print()} member function for printing themselves. For this, we
7897 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7903 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7904 print_func<print_context>(&mystring::do_print))
7907 Let's try again to print the expression:
7911 // -> "Hello, world!"
7914 Much better. If we wanted to have @code{mystring} objects displayed in a
7915 different way depending on the output format (default, LaTeX, etc.), we
7916 would have supplied multiple @code{print_func<>()} options with different
7917 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7918 separated by dots. This is similar to the way options are specified for
7919 symbolic functions. @xref{Printing}, for a more in-depth description of the
7920 way expression output is implemented in GiNaC.
7922 The @code{mystring} class can be used in arbitrary expressions:
7925 e += mystring("GiNaC rulez");
7927 // -> "GiNaC rulez"+"Hello, world!"
7930 (GiNaC's automatic term reordering is in effect here), or even
7933 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7935 // -> "One string"^(2*sin(-"Another string"+Pi))
7938 Whether this makes sense is debatable but remember that this is only an
7939 example. At least it allows you to implement your own symbolic algorithms
7942 Note that GiNaC's algebraic rules remain unchanged:
7945 e = mystring("Wow") * mystring("Wow");
7949 e = pow(mystring("First")-mystring("Second"), 2);
7950 cout << e.expand() << endl;
7951 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7954 There's no way to, for example, make GiNaC's @code{add} class perform string
7955 concatenation. You would have to implement this yourself.
7957 @subsection Automatic evaluation
7960 @cindex @code{eval()}
7961 @cindex @code{hold()}
7962 When dealing with objects that are just a little more complicated than the
7963 simple string objects we have implemented, chances are that you will want to
7964 have some automatic simplifications or canonicalizations performed on them.
7965 This is done in the evaluation member function @code{eval()}. Let's say that
7966 we wanted all strings automatically converted to lowercase with
7967 non-alphabetic characters stripped, and empty strings removed:
7970 class mystring : public basic
7974 ex eval(int level = 0) const;
7978 ex mystring::eval(int level) const
7981 for (int i=0; i<str.length(); i++) @{
7983 if (c >= 'A' && c <= 'Z')
7984 new_str += tolower(c);
7985 else if (c >= 'a' && c <= 'z')
7989 if (new_str.length() == 0)
7992 return mystring(new_str).hold();
7996 The @code{level} argument is used to limit the recursion depth of the
7997 evaluation. We don't have any subexpressions in the @code{mystring}
7998 class so we are not concerned with this. If we had, we would call the
7999 @code{eval()} functions of the subexpressions with @code{level - 1} as
8000 the argument if @code{level != 1}. The @code{hold()} member function
8001 sets a flag in the object that prevents further evaluation. Otherwise
8002 we might end up in an endless loop. When you want to return the object
8003 unmodified, use @code{return this->hold();}.
8005 Let's confirm that it works:
8008 ex e = mystring("Hello, world!") + mystring("!?#");
8012 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8017 @subsection Optional member functions
8019 We have implemented only a small set of member functions to make the class
8020 work in the GiNaC framework. There are two functions that are not strictly
8021 required but will make operations with objects of the class more efficient:
8023 @cindex @code{calchash()}
8024 @cindex @code{is_equal_same_type()}
8026 unsigned calchash() const;
8027 bool is_equal_same_type(const basic &other) const;
8030 The @code{calchash()} method returns an @code{unsigned} hash value for the
8031 object which will allow GiNaC to compare and canonicalize expressions much
8032 more efficiently. You should consult the implementation of some of the built-in
8033 GiNaC classes for examples of hash functions. The default implementation of
8034 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8035 class and all subexpressions that are accessible via @code{op()}.
8037 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8038 tests for equality without establishing an ordering relation, which is often
8039 faster. The default implementation of @code{is_equal_same_type()} just calls
8040 @code{compare_same_type()} and tests its result for zero.
8042 @subsection Other member functions
8044 For a real algebraic class, there are probably some more functions that you
8045 might want to provide:
8048 bool info(unsigned inf) const;
8049 ex evalf(int level = 0) const;
8050 ex series(const relational & r, int order, unsigned options = 0) const;
8051 ex derivative(const symbol & s) const;
8054 If your class stores sub-expressions (see the scalar product example in the
8055 previous section) you will probably want to override
8057 @cindex @code{let_op()}
8060 ex op(size_t i) const;
8061 ex & let_op(size_t i);
8062 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8063 ex map(map_function & f) const;
8066 @code{let_op()} is a variant of @code{op()} that allows write access. The
8067 default implementations of @code{subs()} and @code{map()} use it, so you have
8068 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8070 You can, of course, also add your own new member functions. Remember
8071 that the RTTI may be used to get information about what kinds of objects
8072 you are dealing with (the position in the class hierarchy) and that you
8073 can always extract the bare object from an @code{ex} by stripping the
8074 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8075 should become a need.
8077 That's it. May the source be with you!
8080 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8081 @c node-name, next, previous, up
8082 @chapter A Comparison With Other CAS
8085 This chapter will give you some information on how GiNaC compares to
8086 other, traditional Computer Algebra Systems, like @emph{Maple},
8087 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8088 disadvantages over these systems.
8091 * Advantages:: Strengths of the GiNaC approach.
8092 * Disadvantages:: Weaknesses of the GiNaC approach.
8093 * Why C++?:: Attractiveness of C++.
8096 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8097 @c node-name, next, previous, up
8100 GiNaC has several advantages over traditional Computer
8101 Algebra Systems, like
8106 familiar language: all common CAS implement their own proprietary
8107 grammar which you have to learn first (and maybe learn again when your
8108 vendor decides to `enhance' it). With GiNaC you can write your program
8109 in common C++, which is standardized.
8113 structured data types: you can build up structured data types using
8114 @code{struct}s or @code{class}es together with STL features instead of
8115 using unnamed lists of lists of lists.
8118 strongly typed: in CAS, you usually have only one kind of variables
8119 which can hold contents of an arbitrary type. This 4GL like feature is
8120 nice for novice programmers, but dangerous.
8123 development tools: powerful development tools exist for C++, like fancy
8124 editors (e.g. with automatic indentation and syntax highlighting),
8125 debuggers, visualization tools, documentation generators@dots{}
8128 modularization: C++ programs can easily be split into modules by
8129 separating interface and implementation.
8132 price: GiNaC is distributed under the GNU Public License which means
8133 that it is free and available with source code. And there are excellent
8134 C++-compilers for free, too.
8137 extendable: you can add your own classes to GiNaC, thus extending it on
8138 a very low level. Compare this to a traditional CAS that you can
8139 usually only extend on a high level by writing in the language defined
8140 by the parser. In particular, it turns out to be almost impossible to
8141 fix bugs in a traditional system.
8144 multiple interfaces: Though real GiNaC programs have to be written in
8145 some editor, then be compiled, linked and executed, there are more ways
8146 to work with the GiNaC engine. Many people want to play with
8147 expressions interactively, as in traditional CASs. Currently, two such
8148 windows into GiNaC have been implemented and many more are possible: the
8149 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8150 types to a command line and second, as a more consistent approach, an
8151 interactive interface to the Cint C++ interpreter has been put together
8152 (called GiNaC-cint) that allows an interactive scripting interface
8153 consistent with the C++ language. It is available from the usual GiNaC
8157 seamless integration: it is somewhere between difficult and impossible
8158 to call CAS functions from within a program written in C++ or any other
8159 programming language and vice versa. With GiNaC, your symbolic routines
8160 are part of your program. You can easily call third party libraries,
8161 e.g. for numerical evaluation or graphical interaction. All other
8162 approaches are much more cumbersome: they range from simply ignoring the
8163 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8164 system (i.e. @emph{Yacas}).
8167 efficiency: often large parts of a program do not need symbolic
8168 calculations at all. Why use large integers for loop variables or
8169 arbitrary precision arithmetics where @code{int} and @code{double} are
8170 sufficient? For pure symbolic applications, GiNaC is comparable in
8171 speed with other CAS.
8176 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8177 @c node-name, next, previous, up
8178 @section Disadvantages
8180 Of course it also has some disadvantages:
8185 advanced features: GiNaC cannot compete with a program like
8186 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8187 which grows since 1981 by the work of dozens of programmers, with
8188 respect to mathematical features. Integration, factorization,
8189 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8190 not planned for the near future).
8193 portability: While the GiNaC library itself is designed to avoid any
8194 platform dependent features (it should compile on any ANSI compliant C++
8195 compiler), the currently used version of the CLN library (fast large
8196 integer and arbitrary precision arithmetics) can only by compiled
8197 without hassle on systems with the C++ compiler from the GNU Compiler
8198 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8199 macros to let the compiler gather all static initializations, which
8200 works for GNU C++ only. Feel free to contact the authors in case you
8201 really believe that you need to use a different compiler. We have
8202 occasionally used other compilers and may be able to give you advice.}
8203 GiNaC uses recent language features like explicit constructors, mutable
8204 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8205 literally. Recent GCC versions starting at 2.95.3, although itself not
8206 yet ANSI compliant, support all needed features.
8211 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8212 @c node-name, next, previous, up
8215 Why did we choose to implement GiNaC in C++ instead of Java or any other
8216 language? C++ is not perfect: type checking is not strict (casting is
8217 possible), separation between interface and implementation is not
8218 complete, object oriented design is not enforced. The main reason is
8219 the often scolded feature of operator overloading in C++. While it may
8220 be true that operating on classes with a @code{+} operator is rarely
8221 meaningful, it is perfectly suited for algebraic expressions. Writing
8222 @math{3x+5y} as @code{3*x+5*y} instead of
8223 @code{x.times(3).plus(y.times(5))} looks much more natural.
8224 Furthermore, the main developers are more familiar with C++ than with
8225 any other programming language.
8228 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8229 @c node-name, next, previous, up
8230 @appendix Internal Structures
8233 * Expressions are reference counted::
8234 * Internal representation of products and sums::
8237 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8238 @c node-name, next, previous, up
8239 @appendixsection Expressions are reference counted
8241 @cindex reference counting
8242 @cindex copy-on-write
8243 @cindex garbage collection
8244 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8245 where the counter belongs to the algebraic objects derived from class
8246 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8247 which @code{ex} contains an instance. If you understood that, you can safely
8248 skip the rest of this passage.
8250 Expressions are extremely light-weight since internally they work like
8251 handles to the actual representation. They really hold nothing more
8252 than a pointer to some other object. What this means in practice is
8253 that whenever you create two @code{ex} and set the second equal to the
8254 first no copying process is involved. Instead, the copying takes place
8255 as soon as you try to change the second. Consider the simple sequence
8260 #include <ginac/ginac.h>
8261 using namespace std;
8262 using namespace GiNaC;
8266 symbol x("x"), y("y"), z("z");
8269 e1 = sin(x + 2*y) + 3*z + 41;
8270 e2 = e1; // e2 points to same object as e1
8271 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8272 e2 += 1; // e2 is copied into a new object
8273 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8277 The line @code{e2 = e1;} creates a second expression pointing to the
8278 object held already by @code{e1}. The time involved for this operation
8279 is therefore constant, no matter how large @code{e1} was. Actual
8280 copying, however, must take place in the line @code{e2 += 1;} because
8281 @code{e1} and @code{e2} are not handles for the same object any more.
8282 This concept is called @dfn{copy-on-write semantics}. It increases
8283 performance considerably whenever one object occurs multiple times and
8284 represents a simple garbage collection scheme because when an @code{ex}
8285 runs out of scope its destructor checks whether other expressions handle
8286 the object it points to too and deletes the object from memory if that
8287 turns out not to be the case. A slightly less trivial example of
8288 differentiation using the chain-rule should make clear how powerful this
8293 symbol x("x"), y("y");
8297 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8298 cout << e1 << endl // prints x+3*y
8299 << e2 << endl // prints (x+3*y)^3
8300 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8304 Here, @code{e1} will actually be referenced three times while @code{e2}
8305 will be referenced two times. When the power of an expression is built,
8306 that expression needs not be copied. Likewise, since the derivative of
8307 a power of an expression can be easily expressed in terms of that
8308 expression, no copying of @code{e1} is involved when @code{e3} is
8309 constructed. So, when @code{e3} is constructed it will print as
8310 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8311 holds a reference to @code{e2} and the factor in front is just
8314 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8315 semantics. When you insert an expression into a second expression, the
8316 result behaves exactly as if the contents of the first expression were
8317 inserted. But it may be useful to remember that this is not what
8318 happens. Knowing this will enable you to write much more efficient
8319 code. If you still have an uncertain feeling with copy-on-write
8320 semantics, we recommend you have a look at the
8321 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8322 Marshall Cline. Chapter 16 covers this issue and presents an
8323 implementation which is pretty close to the one in GiNaC.
8326 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8327 @c node-name, next, previous, up
8328 @appendixsection Internal representation of products and sums
8330 @cindex representation
8333 @cindex @code{power}
8334 Although it should be completely transparent for the user of
8335 GiNaC a short discussion of this topic helps to understand the sources
8336 and also explain performance to a large degree. Consider the
8337 unexpanded symbolic expression
8339 $2d^3 \left( 4a + 5b - 3 \right)$
8342 @math{2*d^3*(4*a+5*b-3)}
8344 which could naively be represented by a tree of linear containers for
8345 addition and multiplication, one container for exponentiation with base
8346 and exponent and some atomic leaves of symbols and numbers in this
8351 @cindex pair-wise representation
8352 However, doing so results in a rather deeply nested tree which will
8353 quickly become inefficient to manipulate. We can improve on this by
8354 representing the sum as a sequence of terms, each one being a pair of a
8355 purely numeric multiplicative coefficient and its rest. In the same
8356 spirit we can store the multiplication as a sequence of terms, each
8357 having a numeric exponent and a possibly complicated base, the tree
8358 becomes much more flat:
8362 The number @code{3} above the symbol @code{d} shows that @code{mul}
8363 objects are treated similarly where the coefficients are interpreted as
8364 @emph{exponents} now. Addition of sums of terms or multiplication of
8365 products with numerical exponents can be coded to be very efficient with
8366 such a pair-wise representation. Internally, this handling is performed
8367 by most CAS in this way. It typically speeds up manipulations by an
8368 order of magnitude. The overall multiplicative factor @code{2} and the
8369 additive term @code{-3} look somewhat out of place in this
8370 representation, however, since they are still carrying a trivial
8371 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8372 this is avoided by adding a field that carries an overall numeric
8373 coefficient. This results in the realistic picture of internal
8376 $2d^3 \left( 4a + 5b - 3 \right)$:
8379 @math{2*d^3*(4*a+5*b-3)}:
8385 This also allows for a better handling of numeric radicals, since
8386 @code{sqrt(2)} can now be carried along calculations. Now it should be
8387 clear, why both classes @code{add} and @code{mul} are derived from the
8388 same abstract class: the data representation is the same, only the
8389 semantics differs. In the class hierarchy, methods for polynomial
8390 expansion and the like are reimplemented for @code{add} and @code{mul},
8391 but the data structure is inherited from @code{expairseq}.
8394 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8395 @c node-name, next, previous, up
8396 @appendix Package Tools
8398 If you are creating a software package that uses the GiNaC library,
8399 setting the correct command line options for the compiler and linker
8400 can be difficult. GiNaC includes two tools to make this process easier.
8403 * ginac-config:: A shell script to detect compiler and linker flags.
8404 * AM_PATH_GINAC:: Macro for GNU automake.
8408 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8409 @c node-name, next, previous, up
8410 @section @command{ginac-config}
8411 @cindex ginac-config
8413 @command{ginac-config} is a shell script that you can use to determine
8414 the compiler and linker command line options required to compile and
8415 link a program with the GiNaC library.
8417 @command{ginac-config} takes the following flags:
8421 Prints out the version of GiNaC installed.
8423 Prints '-I' flags pointing to the installed header files.
8425 Prints out the linker flags necessary to link a program against GiNaC.
8426 @item --prefix[=@var{PREFIX}]
8427 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8428 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8429 Otherwise, prints out the configured value of @env{$prefix}.
8430 @item --exec-prefix[=@var{PREFIX}]
8431 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8432 Otherwise, prints out the configured value of @env{$exec_prefix}.
8435 Typically, @command{ginac-config} will be used within a configure
8436 script, as described below. It, however, can also be used directly from
8437 the command line using backquotes to compile a simple program. For
8441 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8444 This command line might expand to (for example):
8447 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8448 -lginac -lcln -lstdc++
8451 Not only is the form using @command{ginac-config} easier to type, it will
8452 work on any system, no matter how GiNaC was configured.
8455 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8456 @c node-name, next, previous, up
8457 @section @samp{AM_PATH_GINAC}
8458 @cindex AM_PATH_GINAC
8460 For packages configured using GNU automake, GiNaC also provides
8461 a macro to automate the process of checking for GiNaC.
8464 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8465 [, @var{ACTION-IF-NOT-FOUND}]]])
8473 Determines the location of GiNaC using @command{ginac-config}, which is
8474 either found in the user's path, or from the environment variable
8475 @env{GINACLIB_CONFIG}.
8478 Tests the installed libraries to make sure that their version
8479 is later than @var{MINIMUM-VERSION}. (A default version will be used
8483 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8484 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8485 variable to the output of @command{ginac-config --libs}, and calls
8486 @samp{AC_SUBST()} for these variables so they can be used in generated
8487 makefiles, and then executes @var{ACTION-IF-FOUND}.
8490 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8491 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8495 This macro is in file @file{ginac.m4} which is installed in
8496 @file{$datadir/aclocal}. Note that if automake was installed with a
8497 different @samp{--prefix} than GiNaC, you will either have to manually
8498 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8499 aclocal the @samp{-I} option when running it.
8502 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8503 * Example package:: Example of a package using AM_PATH_GINAC.
8507 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8508 @c node-name, next, previous, up
8509 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8511 Simply make sure that @command{ginac-config} is in your path, and run
8512 the configure script.
8519 The directory where the GiNaC libraries are installed needs
8520 to be found by your system's dynamic linker.
8522 This is generally done by
8525 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8531 setting the environment variable @env{LD_LIBRARY_PATH},
8534 or, as a last resort,
8537 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8538 running configure, for instance:
8541 LDFLAGS=-R/home/cbauer/lib ./configure
8546 You can also specify a @command{ginac-config} not in your path by
8547 setting the @env{GINACLIB_CONFIG} environment variable to the
8548 name of the executable
8551 If you move the GiNaC package from its installed location,
8552 you will either need to modify @command{ginac-config} script
8553 manually to point to the new location or rebuild GiNaC.
8564 --with-ginac-prefix=@var{PREFIX}
8565 --with-ginac-exec-prefix=@var{PREFIX}
8568 are provided to override the prefix and exec-prefix that were stored
8569 in the @command{ginac-config} shell script by GiNaC's configure. You are
8570 generally better off configuring GiNaC with the right path to begin with.
8574 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8575 @c node-name, next, previous, up
8576 @subsection Example of a package using @samp{AM_PATH_GINAC}
8578 The following shows how to build a simple package using automake
8579 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8583 #include <ginac/ginac.h>
8587 GiNaC::symbol x("x");
8588 GiNaC::ex a = GiNaC::sin(x);
8589 std::cout << "Derivative of " << a
8590 << " is " << a.diff(x) << std::endl;
8595 You should first read the introductory portions of the automake
8596 Manual, if you are not already familiar with it.
8598 Two files are needed, @file{configure.in}, which is used to build the
8602 dnl Process this file with autoconf to produce a configure script.
8604 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8610 AM_PATH_GINAC(0.9.0, [
8611 LIBS="$LIBS $GINACLIB_LIBS"
8612 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8613 ], AC_MSG_ERROR([need to have GiNaC installed]))
8618 The only command in this which is not standard for automake
8619 is the @samp{AM_PATH_GINAC} macro.
8621 That command does the following: If a GiNaC version greater or equal
8622 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8623 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8624 the error message `need to have GiNaC installed'
8626 And the @file{Makefile.am}, which will be used to build the Makefile.
8629 ## Process this file with automake to produce Makefile.in
8630 bin_PROGRAMS = simple
8631 simple_SOURCES = simple.cpp
8634 This @file{Makefile.am}, says that we are building a single executable,
8635 from a single source file @file{simple.cpp}. Since every program
8636 we are building uses GiNaC we simply added the GiNaC options
8637 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8638 want to specify them on a per-program basis: for instance by
8642 simple_LDADD = $(GINACLIB_LIBS)
8643 INCLUDES = $(GINACLIB_CPPFLAGS)
8646 to the @file{Makefile.am}.
8648 To try this example out, create a new directory and add the three
8651 Now execute the following commands:
8654 $ automake --add-missing
8659 You now have a package that can be built in the normal fashion
8668 @node Bibliography, Concept Index, Example package, Top
8669 @c node-name, next, previous, up
8670 @appendix Bibliography
8675 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8678 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8681 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8684 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8687 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8688 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8691 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8692 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8693 Academic Press, London
8696 @cite{Computer Algebra Systems - A Practical Guide},
8697 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8700 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8701 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8704 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8705 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8708 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8713 @node Concept Index, , Bibliography, Top
8714 @c node-name, next, previous, up
8715 @unnumbered Concept Index