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) 1999-2006 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{} 1999-2006 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{} 1999-2006 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 expressions}), 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
1159 values, you would like to have such functions evaluated. Therefore GiNaC
1160 allows you to specify
1161 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1162 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @cindex @code{possymbol()}
1165 Furthermore, it is also possible to declare a symbol as positive. This will,
1166 for instance, enable the automatic simplification of @code{abs(x)} into
1167 @code{x}. This is done by declaying the symbol as @code{possymbol x("x");}.
1170 @node Numbers, Constants, Symbols, Basic concepts
1171 @c node-name, next, previous, up
1173 @cindex @code{numeric} (class)
1179 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1180 The classes therein serve as foundation classes for GiNaC. CLN stands
1181 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1182 In order to find out more about CLN's internals, the reader is referred to
1183 the documentation of that library. @inforef{Introduction, , cln}, for
1184 more information. Suffice to say that it is by itself build on top of
1185 another library, the GNU Multiple Precision library GMP, which is an
1186 extremely fast library for arbitrary long integers and rationals as well
1187 as arbitrary precision floating point numbers. It is very commonly used
1188 by several popular cryptographic applications. CLN extends GMP by
1189 several useful things: First, it introduces the complex number field
1190 over either reals (i.e. floating point numbers with arbitrary precision)
1191 or rationals. Second, it automatically converts rationals to integers
1192 if the denominator is unity and complex numbers to real numbers if the
1193 imaginary part vanishes and also correctly treats algebraic functions.
1194 Third it provides good implementations of state-of-the-art algorithms
1195 for all trigonometric and hyperbolic functions as well as for
1196 calculation of some useful constants.
1198 The user can construct an object of class @code{numeric} in several
1199 ways. The following example shows the four most important constructors.
1200 It uses construction from C-integer, construction of fractions from two
1201 integers, construction from C-float and construction from a string:
1205 #include <ginac/ginac.h>
1206 using namespace GiNaC;
1210 numeric two = 2; // exact integer 2
1211 numeric r(2,3); // exact fraction 2/3
1212 numeric e(2.71828); // floating point number
1213 numeric p = "3.14159265358979323846"; // constructor from string
1214 // Trott's constant in scientific notation:
1215 numeric trott("1.0841015122311136151E-2");
1217 std::cout << two*p << std::endl; // floating point 6.283...
1222 @cindex complex numbers
1223 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1228 numeric z1 = 2-3*I; // exact complex number 2-3i
1229 numeric z2 = 5.9+1.6*I; // complex floating point number
1233 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1234 This would, however, call C's built-in operator @code{/} for integers
1235 first and result in a numeric holding a plain integer 1. @strong{Never
1236 use the operator @code{/} on integers} unless you know exactly what you
1237 are doing! Use the constructor from two integers instead, as shown in
1238 the example above. Writing @code{numeric(1)/2} may look funny but works
1241 @cindex @code{Digits}
1243 We have seen now the distinction between exact numbers and floating
1244 point numbers. Clearly, the user should never have to worry about
1245 dynamically created exact numbers, since their `exactness' always
1246 determines how they ought to be handled, i.e. how `long' they are. The
1247 situation is different for floating point numbers. Their accuracy is
1248 controlled by one @emph{global} variable, called @code{Digits}. (For
1249 those readers who know about Maple: it behaves very much like Maple's
1250 @code{Digits}). All objects of class numeric that are constructed from
1251 then on will be stored with a precision matching that number of decimal
1256 #include <ginac/ginac.h>
1257 using namespace std;
1258 using namespace GiNaC;
1262 numeric three(3.0), one(1.0);
1263 numeric x = one/three;
1265 cout << "in " << Digits << " digits:" << endl;
1267 cout << Pi.evalf() << endl;
1279 The above example prints the following output to screen:
1283 0.33333333333333333334
1284 3.1415926535897932385
1286 0.33333333333333333333333333333333333333333333333333333333333333333334
1287 3.1415926535897932384626433832795028841971693993751058209749445923078
1291 Note that the last number is not necessarily rounded as you would
1292 naively expect it to be rounded in the decimal system. But note also,
1293 that in both cases you got a couple of extra digits. This is because
1294 numbers are internally stored by CLN as chunks of binary digits in order
1295 to match your machine's word size and to not waste precision. Thus, on
1296 architectures with different word size, the above output might even
1297 differ with regard to actually computed digits.
1299 It should be clear that objects of class @code{numeric} should be used
1300 for constructing numbers or for doing arithmetic with them. The objects
1301 one deals with most of the time are the polymorphic expressions @code{ex}.
1303 @subsection Tests on numbers
1305 Once you have declared some numbers, assigned them to expressions and
1306 done some arithmetic with them it is frequently desired to retrieve some
1307 kind of information from them like asking whether that number is
1308 integer, rational, real or complex. For those cases GiNaC provides
1309 several useful methods. (Internally, they fall back to invocations of
1310 certain CLN functions.)
1312 As an example, let's construct some rational number, multiply it with
1313 some multiple of its denominator and test what comes out:
1317 #include <ginac/ginac.h>
1318 using namespace std;
1319 using namespace GiNaC;
1321 // some very important constants:
1322 const numeric twentyone(21);
1323 const numeric ten(10);
1324 const numeric five(5);
1328 numeric answer = twentyone;
1331 cout << answer.is_integer() << endl; // false, it's 21/5
1333 cout << answer.is_integer() << endl; // true, it's 42 now!
1337 Note that the variable @code{answer} is constructed here as an integer
1338 by @code{numeric}'s copy constructor but in an intermediate step it
1339 holds a rational number represented as integer numerator and integer
1340 denominator. When multiplied by 10, the denominator becomes unity and
1341 the result is automatically converted to a pure integer again.
1342 Internally, the underlying CLN is responsible for this behavior and we
1343 refer the reader to CLN's documentation. Suffice to say that
1344 the same behavior applies to complex numbers as well as return values of
1345 certain functions. Complex numbers are automatically converted to real
1346 numbers if the imaginary part becomes zero. The full set of tests that
1347 can be applied is listed in the following table.
1350 @multitable @columnfractions .30 .70
1351 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1352 @item @code{.is_zero()}
1353 @tab @dots{}equal to zero
1354 @item @code{.is_positive()}
1355 @tab @dots{}not complex and greater than 0
1356 @item @code{.is_integer()}
1357 @tab @dots{}a (non-complex) integer
1358 @item @code{.is_pos_integer()}
1359 @tab @dots{}an integer and greater than 0
1360 @item @code{.is_nonneg_integer()}
1361 @tab @dots{}an integer and greater equal 0
1362 @item @code{.is_even()}
1363 @tab @dots{}an even integer
1364 @item @code{.is_odd()}
1365 @tab @dots{}an odd integer
1366 @item @code{.is_prime()}
1367 @tab @dots{}a prime integer (probabilistic primality test)
1368 @item @code{.is_rational()}
1369 @tab @dots{}an exact rational number (integers are rational, too)
1370 @item @code{.is_real()}
1371 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1372 @item @code{.is_cinteger()}
1373 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1374 @item @code{.is_crational()}
1375 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1379 @subsection Numeric functions
1381 The following functions can be applied to @code{numeric} objects and will be
1382 evaluated immediately:
1385 @multitable @columnfractions .30 .70
1386 @item @strong{Name} @tab @strong{Function}
1387 @item @code{inverse(z)}
1388 @tab returns @math{1/z}
1389 @cindex @code{inverse()} (numeric)
1390 @item @code{pow(a, b)}
1391 @tab exponentiation @math{a^b}
1394 @item @code{real(z)}
1396 @cindex @code{real()}
1397 @item @code{imag(z)}
1399 @cindex @code{imag()}
1400 @item @code{csgn(z)}
1401 @tab complex sign (returns an @code{int})
1402 @item @code{step(x)}
1403 @tab step function (returns an @code{numeric})
1404 @item @code{numer(z)}
1405 @tab numerator of rational or complex rational number
1406 @item @code{denom(z)}
1407 @tab denominator of rational or complex rational number
1408 @item @code{sqrt(z)}
1410 @item @code{isqrt(n)}
1411 @tab integer square root
1412 @cindex @code{isqrt()}
1419 @item @code{asin(z)}
1421 @item @code{acos(z)}
1423 @item @code{atan(z)}
1424 @tab inverse tangent
1425 @item @code{atan(y, x)}
1426 @tab inverse tangent with two arguments
1427 @item @code{sinh(z)}
1428 @tab hyperbolic sine
1429 @item @code{cosh(z)}
1430 @tab hyperbolic cosine
1431 @item @code{tanh(z)}
1432 @tab hyperbolic tangent
1433 @item @code{asinh(z)}
1434 @tab inverse hyperbolic sine
1435 @item @code{acosh(z)}
1436 @tab inverse hyperbolic cosine
1437 @item @code{atanh(z)}
1438 @tab inverse hyperbolic tangent
1440 @tab exponential function
1442 @tab natural logarithm
1445 @item @code{zeta(z)}
1446 @tab Riemann's zeta function
1447 @item @code{tgamma(z)}
1449 @item @code{lgamma(z)}
1450 @tab logarithm of gamma function
1452 @tab psi (digamma) function
1453 @item @code{psi(n, z)}
1454 @tab derivatives of psi function (polygamma functions)
1455 @item @code{factorial(n)}
1456 @tab factorial function @math{n!}
1457 @item @code{doublefactorial(n)}
1458 @tab double factorial function @math{n!!}
1459 @cindex @code{doublefactorial()}
1460 @item @code{binomial(n, k)}
1461 @tab binomial coefficients
1462 @item @code{bernoulli(n)}
1463 @tab Bernoulli numbers
1464 @cindex @code{bernoulli()}
1465 @item @code{fibonacci(n)}
1466 @tab Fibonacci numbers
1467 @cindex @code{fibonacci()}
1468 @item @code{mod(a, b)}
1469 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1470 @cindex @code{mod()}
1471 @item @code{smod(a, b)}
1472 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1473 @cindex @code{smod()}
1474 @item @code{irem(a, b)}
1475 @tab integer remainder (has the sign of @math{a}, or is zero)
1476 @cindex @code{irem()}
1477 @item @code{irem(a, b, q)}
1478 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1479 @item @code{iquo(a, b)}
1480 @tab integer quotient
1481 @cindex @code{iquo()}
1482 @item @code{iquo(a, b, r)}
1483 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1484 @item @code{gcd(a, b)}
1485 @tab greatest common divisor
1486 @item @code{lcm(a, b)}
1487 @tab least common multiple
1491 Most of these functions are also available as symbolic functions that can be
1492 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1493 as polynomial algorithms.
1495 @subsection Converting numbers
1497 Sometimes it is desirable to convert a @code{numeric} object back to a
1498 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1499 class provides a couple of methods for this purpose:
1501 @cindex @code{to_int()}
1502 @cindex @code{to_long()}
1503 @cindex @code{to_double()}
1504 @cindex @code{to_cl_N()}
1506 int numeric::to_int() const;
1507 long numeric::to_long() const;
1508 double numeric::to_double() const;
1509 cln::cl_N numeric::to_cl_N() const;
1512 @code{to_int()} and @code{to_long()} only work when the number they are
1513 applied on is an exact integer. Otherwise the program will halt with a
1514 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1515 rational number will return a floating-point approximation. Both
1516 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1517 part of complex numbers.
1520 @node Constants, Fundamental containers, Numbers, Basic concepts
1521 @c node-name, next, previous, up
1523 @cindex @code{constant} (class)
1526 @cindex @code{Catalan}
1527 @cindex @code{Euler}
1528 @cindex @code{evalf()}
1529 Constants behave pretty much like symbols except that they return some
1530 specific number when the method @code{.evalf()} is called.
1532 The predefined known constants are:
1535 @multitable @columnfractions .14 .30 .56
1536 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1538 @tab Archimedes' constant
1539 @tab 3.14159265358979323846264338327950288
1540 @item @code{Catalan}
1541 @tab Catalan's constant
1542 @tab 0.91596559417721901505460351493238411
1544 @tab Euler's (or Euler-Mascheroni) constant
1545 @tab 0.57721566490153286060651209008240243
1550 @node Fundamental containers, Lists, Constants, Basic concepts
1551 @c node-name, next, previous, up
1552 @section Sums, products and powers
1556 @cindex @code{power}
1558 Simple rational expressions are written down in GiNaC pretty much like
1559 in other CAS or like expressions involving numerical variables in C.
1560 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1561 been overloaded to achieve this goal. When you run the following
1562 code snippet, the constructor for an object of type @code{mul} is
1563 automatically called to hold the product of @code{a} and @code{b} and
1564 then the constructor for an object of type @code{add} is called to hold
1565 the sum of that @code{mul} object and the number one:
1569 symbol a("a"), b("b");
1574 @cindex @code{pow()}
1575 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1576 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1577 construction is necessary since we cannot safely overload the constructor
1578 @code{^} in C++ to construct a @code{power} object. If we did, it would
1579 have several counterintuitive and undesired effects:
1583 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1585 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1586 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1587 interpret this as @code{x^(a^b)}.
1589 Also, expressions involving integer exponents are very frequently used,
1590 which makes it even more dangerous to overload @code{^} since it is then
1591 hard to distinguish between the semantics as exponentiation and the one
1592 for exclusive or. (It would be embarrassing to return @code{1} where one
1593 has requested @code{2^3}.)
1596 @cindex @command{ginsh}
1597 All effects are contrary to mathematical notation and differ from the
1598 way most other CAS handle exponentiation, therefore overloading @code{^}
1599 is ruled out for GiNaC's C++ part. The situation is different in
1600 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1601 that the other frequently used exponentiation operator @code{**} does
1602 not exist at all in C++).
1604 To be somewhat more precise, objects of the three classes described
1605 here, are all containers for other expressions. An object of class
1606 @code{power} is best viewed as a container with two slots, one for the
1607 basis, one for the exponent. All valid GiNaC expressions can be
1608 inserted. However, basic transformations like simplifying
1609 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1610 when this is mathematically possible. If we replace the outer exponent
1611 three in the example by some symbols @code{a}, the simplification is not
1612 safe and will not be performed, since @code{a} might be @code{1/2} and
1615 Objects of type @code{add} and @code{mul} are containers with an
1616 arbitrary number of slots for expressions to be inserted. Again, simple
1617 and safe simplifications are carried out like transforming
1618 @code{3*x+4-x} to @code{2*x+4}.
1621 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1622 @c node-name, next, previous, up
1623 @section Lists of expressions
1624 @cindex @code{lst} (class)
1626 @cindex @code{nops()}
1628 @cindex @code{append()}
1629 @cindex @code{prepend()}
1630 @cindex @code{remove_first()}
1631 @cindex @code{remove_last()}
1632 @cindex @code{remove_all()}
1634 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1635 expressions. They are not as ubiquitous as in many other computer algebra
1636 packages, but are sometimes used to supply a variable number of arguments of
1637 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1638 constructors, so you should have a basic understanding of them.
1640 Lists can be constructed by assigning a comma-separated sequence of
1645 symbol x("x"), y("y");
1648 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1653 There are also constructors that allow direct creation of lists of up to
1654 16 expressions, which is often more convenient but slightly less efficient:
1658 // This produces the same list 'l' as above:
1659 // lst l(x, 2, y, x+y);
1660 // lst l = lst(x, 2, y, x+y);
1664 Use the @code{nops()} method to determine the size (number of expressions) of
1665 a list and the @code{op()} method or the @code{[]} operator to access
1666 individual elements:
1670 cout << l.nops() << endl; // prints '4'
1671 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1675 As with the standard @code{list<T>} container, accessing random elements of a
1676 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1677 sequential access to the elements of a list is possible with the
1678 iterator types provided by the @code{lst} class:
1681 typedef ... lst::const_iterator;
1682 typedef ... lst::const_reverse_iterator;
1683 lst::const_iterator lst::begin() const;
1684 lst::const_iterator lst::end() const;
1685 lst::const_reverse_iterator lst::rbegin() const;
1686 lst::const_reverse_iterator lst::rend() const;
1689 For example, to print the elements of a list individually you can use:
1694 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1699 which is one order faster than
1704 for (size_t i = 0; i < l.nops(); ++i)
1705 cout << l.op(i) << endl;
1709 These iterators also allow you to use some of the algorithms provided by
1710 the C++ standard library:
1714 // print the elements of the list (requires #include <iterator>)
1715 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1717 // sum up the elements of the list (requires #include <numeric>)
1718 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1719 cout << sum << endl; // prints '2+2*x+2*y'
1723 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1724 (the only other one is @code{matrix}). You can modify single elements:
1728 l[1] = 42; // l is now @{x, 42, y, x+y@}
1729 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1733 You can append or prepend an expression to a list with the @code{append()}
1734 and @code{prepend()} methods:
1738 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1739 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1743 You can remove the first or last element of a list with @code{remove_first()}
1744 and @code{remove_last()}:
1748 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1749 l.remove_last(); // l is now @{x, 7, y, x+y@}
1753 You can remove all the elements of a list with @code{remove_all()}:
1757 l.remove_all(); // l is now empty
1761 You can bring the elements of a list into a canonical order with @code{sort()}:
1770 // l1 and l2 are now equal
1774 Finally, you can remove all but the first element of consecutive groups of
1775 elements with @code{unique()}:
1780 l3 = x, 2, 2, 2, y, x+y, y+x;
1781 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1786 @node Mathematical functions, Relations, Lists, Basic concepts
1787 @c node-name, next, previous, up
1788 @section Mathematical functions
1789 @cindex @code{function} (class)
1790 @cindex trigonometric function
1791 @cindex hyperbolic function
1793 There are quite a number of useful functions hard-wired into GiNaC. For
1794 instance, all trigonometric and hyperbolic functions are implemented
1795 (@xref{Built-in functions}, for a complete list).
1797 These functions (better called @emph{pseudofunctions}) are all objects
1798 of class @code{function}. They accept one or more expressions as
1799 arguments and return one expression. If the arguments are not
1800 numerical, the evaluation of the function may be halted, as it does in
1801 the next example, showing how a function returns itself twice and
1802 finally an expression that may be really useful:
1804 @cindex Gamma function
1805 @cindex @code{subs()}
1808 symbol x("x"), y("y");
1810 cout << tgamma(foo) << endl;
1811 // -> tgamma(x+(1/2)*y)
1812 ex bar = foo.subs(y==1);
1813 cout << tgamma(bar) << endl;
1815 ex foobar = bar.subs(x==7);
1816 cout << tgamma(foobar) << endl;
1817 // -> (135135/128)*Pi^(1/2)
1821 Besides evaluation most of these functions allow differentiation, series
1822 expansion and so on. Read the next chapter in order to learn more about
1825 It must be noted that these pseudofunctions are created by inline
1826 functions, where the argument list is templated. This means that
1827 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1828 @code{sin(ex(1))} and will therefore not result in a floating point
1829 number. Unless of course the function prototype is explicitly
1830 overridden -- which is the case for arguments of type @code{numeric}
1831 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1832 point number of class @code{numeric} you should call
1833 @code{sin(numeric(1))}. This is almost the same as calling
1834 @code{sin(1).evalf()} except that the latter will return a numeric
1835 wrapped inside an @code{ex}.
1838 @node Relations, Integrals, Mathematical functions, Basic concepts
1839 @c node-name, next, previous, up
1841 @cindex @code{relational} (class)
1843 Sometimes, a relation holding between two expressions must be stored
1844 somehow. The class @code{relational} is a convenient container for such
1845 purposes. A relation is by definition a container for two @code{ex} and
1846 a relation between them that signals equality, inequality and so on.
1847 They are created by simply using the C++ operators @code{==}, @code{!=},
1848 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1850 @xref{Mathematical functions}, for examples where various applications
1851 of the @code{.subs()} method show how objects of class relational are
1852 used as arguments. There they provide an intuitive syntax for
1853 substitutions. They are also used as arguments to the @code{ex::series}
1854 method, where the left hand side of the relation specifies the variable
1855 to expand in and the right hand side the expansion point. They can also
1856 be used for creating systems of equations that are to be solved for
1857 unknown variables. But the most common usage of objects of this class
1858 is rather inconspicuous in statements of the form @code{if
1859 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1860 conversion from @code{relational} to @code{bool} takes place. Note,
1861 however, that @code{==} here does not perform any simplifications, hence
1862 @code{expand()} must be called explicitly.
1864 @node Integrals, Matrices, Relations, Basic concepts
1865 @c node-name, next, previous, up
1867 @cindex @code{integral} (class)
1869 An object of class @dfn{integral} can be used to hold a symbolic integral.
1870 If you want to symbolically represent the integral of @code{x*x} from 0 to
1871 1, you would write this as
1873 integral(x, 0, 1, x*x)
1875 The first argument is the integration variable. It should be noted that
1876 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1877 fact, it can only integrate polynomials. An expression containing integrals
1878 can be evaluated symbolically by calling the
1882 method on it. Numerical evaluation is available by calling the
1886 method on an expression containing the integral. This will only evaluate
1887 integrals into a number if @code{subs}ing the integration variable by a
1888 number in the fourth argument of an integral and then @code{evalf}ing the
1889 result always results in a number. Of course, also the boundaries of the
1890 integration domain must @code{evalf} into numbers. It should be noted that
1891 trying to @code{evalf} a function with discontinuities in the integration
1892 domain is not recommended. The accuracy of the numeric evaluation of
1893 integrals is determined by the static member variable
1895 ex integral::relative_integration_error
1897 of the class @code{integral}. The default value of this is 10^-8.
1898 The integration works by halving the interval of integration, until numeric
1899 stability of the answer indicates that the requested accuracy has been
1900 reached. The maximum depth of the halving can be set via the static member
1903 int integral::max_integration_level
1905 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1906 return the integral unevaluated. The function that performs the numerical
1907 evaluation, is also available as
1909 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1912 This function will throw an exception if the maximum depth is exceeded. The
1913 last parameter of the function is optional and defaults to the
1914 @code{relative_integration_error}. To make sure that we do not do too
1915 much work if an expression contains the same integral multiple times,
1916 a lookup table is used.
1918 If you know that an expression holds an integral, you can get the
1919 integration variable, the left boundary, right boundary and integrand by
1920 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1921 @code{.op(3)}. Differentiating integrals with respect to variables works
1922 as expected. Note that it makes no sense to differentiate an integral
1923 with respect to the integration variable.
1925 @node Matrices, Indexed objects, Integrals, Basic concepts
1926 @c node-name, next, previous, up
1928 @cindex @code{matrix} (class)
1930 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1931 matrix with @math{m} rows and @math{n} columns are accessed with two
1932 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1933 second one in the range 0@dots{}@math{n-1}.
1935 There are a couple of ways to construct matrices, with or without preset
1936 elements. The constructor
1939 matrix::matrix(unsigned r, unsigned c);
1942 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1945 The fastest way to create a matrix with preinitialized elements is to assign
1946 a list of comma-separated expressions to an empty matrix (see below for an
1947 example). But you can also specify the elements as a (flat) list with
1950 matrix::matrix(unsigned r, unsigned c, const lst & l);
1955 @cindex @code{lst_to_matrix()}
1957 ex lst_to_matrix(const lst & l);
1960 constructs a matrix from a list of lists, each list representing a matrix row.
1962 There is also a set of functions for creating some special types of
1965 @cindex @code{diag_matrix()}
1966 @cindex @code{unit_matrix()}
1967 @cindex @code{symbolic_matrix()}
1969 ex diag_matrix(const lst & l);
1970 ex unit_matrix(unsigned x);
1971 ex unit_matrix(unsigned r, unsigned c);
1972 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1973 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1974 const string & tex_base_name);
1977 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1978 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1979 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1980 matrix filled with newly generated symbols made of the specified base name
1981 and the position of each element in the matrix.
1983 Matrices often arise by omitting elements of another matrix. For
1984 instance, the submatrix @code{S} of a matrix @code{M} takes a
1985 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1986 by removing one row and one column from a matrix @code{M}. (The
1987 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1988 can be used for computing the inverse using Cramer's rule.)
1990 @cindex @code{sub_matrix()}
1991 @cindex @code{reduced_matrix()}
1993 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1994 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1997 The function @code{sub_matrix()} takes a row offset @code{r} and a
1998 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1999 columns. The function @code{reduced_matrix()} has two integer arguments
2000 that specify which row and column to remove:
2008 cout << reduced_matrix(m, 1, 1) << endl;
2009 // -> [[11,13],[31,33]]
2010 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2011 // -> [[22,23],[32,33]]
2015 Matrix elements can be accessed and set using the parenthesis (function call)
2019 const ex & matrix::operator()(unsigned r, unsigned c) const;
2020 ex & matrix::operator()(unsigned r, unsigned c);
2023 It is also possible to access the matrix elements in a linear fashion with
2024 the @code{op()} method. But C++-style subscripting with square brackets
2025 @samp{[]} is not available.
2027 Here are a couple of examples for constructing matrices:
2031 symbol a("a"), b("b");
2045 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2048 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2051 cout << diag_matrix(lst(a, b)) << endl;
2054 cout << unit_matrix(3) << endl;
2055 // -> [[1,0,0],[0,1,0],[0,0,1]]
2057 cout << symbolic_matrix(2, 3, "x") << endl;
2058 // -> [[x00,x01,x02],[x10,x11,x12]]
2062 @cindex @code{is_zero_matrix()}
2063 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2064 all entries of the matrix are zeros. There is also method
2065 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2066 expression is zero or a zero matrix.
2068 @cindex @code{transpose()}
2069 There are three ways to do arithmetic with matrices. The first (and most
2070 direct one) is to use the methods provided by the @code{matrix} class:
2073 matrix matrix::add(const matrix & other) const;
2074 matrix matrix::sub(const matrix & other) const;
2075 matrix matrix::mul(const matrix & other) const;
2076 matrix matrix::mul_scalar(const ex & other) const;
2077 matrix matrix::pow(const ex & expn) const;
2078 matrix matrix::transpose() const;
2081 All of these methods return the result as a new matrix object. Here is an
2082 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2087 matrix A(2, 2), B(2, 2), C(2, 2);
2095 matrix result = A.mul(B).sub(C.mul_scalar(2));
2096 cout << result << endl;
2097 // -> [[-13,-6],[1,2]]
2102 @cindex @code{evalm()}
2103 The second (and probably the most natural) way is to construct an expression
2104 containing matrices with the usual arithmetic operators and @code{pow()}.
2105 For efficiency reasons, expressions with sums, products and powers of
2106 matrices are not automatically evaluated in GiNaC. You have to call the
2110 ex ex::evalm() const;
2113 to obtain the result:
2120 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2121 cout << e.evalm() << endl;
2122 // -> [[-13,-6],[1,2]]
2127 The non-commutativity of the product @code{A*B} in this example is
2128 automatically recognized by GiNaC. There is no need to use a special
2129 operator here. @xref{Non-commutative objects}, for more information about
2130 dealing with non-commutative expressions.
2132 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2133 to perform the arithmetic:
2138 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2139 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2141 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2142 cout << e.simplify_indexed() << endl;
2143 // -> [[-13,-6],[1,2]].i.j
2147 Using indices is most useful when working with rectangular matrices and
2148 one-dimensional vectors because you don't have to worry about having to
2149 transpose matrices before multiplying them. @xref{Indexed objects}, for
2150 more information about using matrices with indices, and about indices in
2153 The @code{matrix} class provides a couple of additional methods for
2154 computing determinants, traces, characteristic polynomials and ranks:
2156 @cindex @code{determinant()}
2157 @cindex @code{trace()}
2158 @cindex @code{charpoly()}
2159 @cindex @code{rank()}
2161 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2162 ex matrix::trace() const;
2163 ex matrix::charpoly(const ex & lambda) const;
2164 unsigned matrix::rank() const;
2167 The @samp{algo} argument of @code{determinant()} allows to select
2168 between different algorithms for calculating the determinant. The
2169 asymptotic speed (as parametrized by the matrix size) can greatly differ
2170 between those algorithms, depending on the nature of the matrix'
2171 entries. The possible values are defined in the @file{flags.h} header
2172 file. By default, GiNaC uses a heuristic to automatically select an
2173 algorithm that is likely (but not guaranteed) to give the result most
2176 @cindex @code{inverse()} (matrix)
2177 @cindex @code{solve()}
2178 Matrices may also be inverted using the @code{ex matrix::inverse()}
2179 method and linear systems may be solved with:
2182 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2183 unsigned algo=solve_algo::automatic) const;
2186 Assuming the matrix object this method is applied on is an @code{m}
2187 times @code{n} matrix, then @code{vars} must be a @code{n} times
2188 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2189 times @code{p} matrix. The returned matrix then has dimension @code{n}
2190 times @code{p} and in the case of an underdetermined system will still
2191 contain some of the indeterminates from @code{vars}. If the system is
2192 overdetermined, an exception is thrown.
2195 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2196 @c node-name, next, previous, up
2197 @section Indexed objects
2199 GiNaC allows you to handle expressions containing general indexed objects in
2200 arbitrary spaces. It is also able to canonicalize and simplify such
2201 expressions and perform symbolic dummy index summations. There are a number
2202 of predefined indexed objects provided, like delta and metric tensors.
2204 There are few restrictions placed on indexed objects and their indices and
2205 it is easy to construct nonsense expressions, but our intention is to
2206 provide a general framework that allows you to implement algorithms with
2207 indexed quantities, getting in the way as little as possible.
2209 @cindex @code{idx} (class)
2210 @cindex @code{indexed} (class)
2211 @subsection Indexed quantities and their indices
2213 Indexed expressions in GiNaC are constructed of two special types of objects,
2214 @dfn{index objects} and @dfn{indexed objects}.
2218 @cindex contravariant
2221 @item Index objects are of class @code{idx} or a subclass. Every index has
2222 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2223 the index lives in) which can both be arbitrary expressions but are usually
2224 a number or a simple symbol. In addition, indices of class @code{varidx} have
2225 a @dfn{variance} (they can be co- or contravariant), and indices of class
2226 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2228 @item Indexed objects are of class @code{indexed} or a subclass. They
2229 contain a @dfn{base expression} (which is the expression being indexed), and
2230 one or more indices.
2234 @strong{Please notice:} when printing expressions, covariant indices and indices
2235 without variance are denoted @samp{.i} while contravariant indices are
2236 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2237 value. In the following, we are going to use that notation in the text so
2238 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2239 not visible in the output.
2241 A simple example shall illustrate the concepts:
2245 #include <ginac/ginac.h>
2246 using namespace std;
2247 using namespace GiNaC;
2251 symbol i_sym("i"), j_sym("j");
2252 idx i(i_sym, 3), j(j_sym, 3);
2255 cout << indexed(A, i, j) << endl;
2257 cout << index_dimensions << indexed(A, i, j) << endl;
2259 cout << dflt; // reset cout to default output format (dimensions hidden)
2263 The @code{idx} constructor takes two arguments, the index value and the
2264 index dimension. First we define two index objects, @code{i} and @code{j},
2265 both with the numeric dimension 3. The value of the index @code{i} is the
2266 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2267 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2268 construct an expression containing one indexed object, @samp{A.i.j}. It has
2269 the symbol @code{A} as its base expression and the two indices @code{i} and
2272 The dimensions of indices are normally not visible in the output, but one
2273 can request them to be printed with the @code{index_dimensions} manipulator,
2276 Note the difference between the indices @code{i} and @code{j} which are of
2277 class @code{idx}, and the index values which are the symbols @code{i_sym}
2278 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2279 or numbers but must be index objects. For example, the following is not
2280 correct and will raise an exception:
2283 symbol i("i"), j("j");
2284 e = indexed(A, i, j); // ERROR: indices must be of type idx
2287 You can have multiple indexed objects in an expression, index values can
2288 be numeric, and index dimensions symbolic:
2292 symbol B("B"), dim("dim");
2293 cout << 4 * indexed(A, i)
2294 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2299 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2300 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2301 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2302 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2303 @code{simplify_indexed()} for that, see below).
2305 In fact, base expressions, index values and index dimensions can be
2306 arbitrary expressions:
2310 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2315 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2316 get an error message from this but you will probably not be able to do
2317 anything useful with it.
2319 @cindex @code{get_value()}
2320 @cindex @code{get_dimension()}
2324 ex idx::get_value();
2325 ex idx::get_dimension();
2328 return the value and dimension of an @code{idx} object. If you have an index
2329 in an expression, such as returned by calling @code{.op()} on an indexed
2330 object, you can get a reference to the @code{idx} object with the function
2331 @code{ex_to<idx>()} on the expression.
2333 There are also the methods
2336 bool idx::is_numeric();
2337 bool idx::is_symbolic();
2338 bool idx::is_dim_numeric();
2339 bool idx::is_dim_symbolic();
2342 for checking whether the value and dimension are numeric or symbolic
2343 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2344 about expressions}) returns information about the index value.
2346 @cindex @code{varidx} (class)
2347 If you need co- and contravariant indices, use the @code{varidx} class:
2351 symbol mu_sym("mu"), nu_sym("nu");
2352 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2353 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2355 cout << indexed(A, mu, nu) << endl;
2357 cout << indexed(A, mu_co, nu) << endl;
2359 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2364 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2365 co- or contravariant. The default is a contravariant (upper) index, but
2366 this can be overridden by supplying a third argument to the @code{varidx}
2367 constructor. The two methods
2370 bool varidx::is_covariant();
2371 bool varidx::is_contravariant();
2374 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2375 to get the object reference from an expression). There's also the very useful
2379 ex varidx::toggle_variance();
2382 which makes a new index with the same value and dimension but the opposite
2383 variance. By using it you only have to define the index once.
2385 @cindex @code{spinidx} (class)
2386 The @code{spinidx} class provides dotted and undotted variant indices, as
2387 used in the Weyl-van-der-Waerden spinor formalism:
2391 symbol K("K"), C_sym("C"), D_sym("D");
2392 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2393 // contravariant, undotted
2394 spinidx C_co(C_sym, 2, true); // covariant index
2395 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2396 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2398 cout << indexed(K, C, D) << endl;
2400 cout << indexed(K, C_co, D_dot) << endl;
2402 cout << indexed(K, D_co_dot, D) << endl;
2407 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2408 dotted or undotted. The default is undotted but this can be overridden by
2409 supplying a fourth argument to the @code{spinidx} constructor. The two
2413 bool spinidx::is_dotted();
2414 bool spinidx::is_undotted();
2417 allow you to check whether or not a @code{spinidx} object is dotted (use
2418 @code{ex_to<spinidx>()} to get the object reference from an expression).
2419 Finally, the two methods
2422 ex spinidx::toggle_dot();
2423 ex spinidx::toggle_variance_dot();
2426 create a new index with the same value and dimension but opposite dottedness
2427 and the same or opposite variance.
2429 @subsection Substituting indices
2431 @cindex @code{subs()}
2432 Sometimes you will want to substitute one symbolic index with another
2433 symbolic or numeric index, for example when calculating one specific element
2434 of a tensor expression. This is done with the @code{.subs()} method, as it
2435 is done for symbols (see @ref{Substituting expressions}).
2437 You have two possibilities here. You can either substitute the whole index
2438 by another index or expression:
2442 ex e = indexed(A, mu_co);
2443 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2444 // -> A.mu becomes A~nu
2445 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2446 // -> A.mu becomes A~0
2447 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2448 // -> A.mu becomes A.0
2452 The third example shows that trying to replace an index with something that
2453 is not an index will substitute the index value instead.
2455 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2460 ex e = indexed(A, mu_co);
2461 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2462 // -> A.mu becomes A.nu
2463 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2464 // -> A.mu becomes A.0
2468 As you see, with the second method only the value of the index will get
2469 substituted. Its other properties, including its dimension, remain unchanged.
2470 If you want to change the dimension of an index you have to substitute the
2471 whole index by another one with the new dimension.
2473 Finally, substituting the base expression of an indexed object works as
2478 ex e = indexed(A, mu_co);
2479 cout << e << " becomes " << e.subs(A == A+B) << endl;
2480 // -> A.mu becomes (B+A).mu
2484 @subsection Symmetries
2485 @cindex @code{symmetry} (class)
2486 @cindex @code{sy_none()}
2487 @cindex @code{sy_symm()}
2488 @cindex @code{sy_anti()}
2489 @cindex @code{sy_cycl()}
2491 Indexed objects can have certain symmetry properties with respect to their
2492 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2493 that is constructed with the helper functions
2496 symmetry sy_none(...);
2497 symmetry sy_symm(...);
2498 symmetry sy_anti(...);
2499 symmetry sy_cycl(...);
2502 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2503 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2504 represents a cyclic symmetry. Each of these functions accepts up to four
2505 arguments which can be either symmetry objects themselves or unsigned integer
2506 numbers that represent an index position (counting from 0). A symmetry
2507 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2508 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2511 Here are some examples of symmetry definitions:
2516 e = indexed(A, i, j);
2517 e = indexed(A, sy_none(), i, j); // equivalent
2518 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2520 // Symmetric in all three indices:
2521 e = indexed(A, sy_symm(), i, j, k);
2522 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2523 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2524 // different canonical order
2526 // Symmetric in the first two indices only:
2527 e = indexed(A, sy_symm(0, 1), i, j, k);
2528 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2530 // Antisymmetric in the first and last index only (index ranges need not
2532 e = indexed(A, sy_anti(0, 2), i, j, k);
2533 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2535 // An example of a mixed symmetry: antisymmetric in the first two and
2536 // last two indices, symmetric when swapping the first and last index
2537 // pairs (like the Riemann curvature tensor):
2538 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2540 // Cyclic symmetry in all three indices:
2541 e = indexed(A, sy_cycl(), i, j, k);
2542 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2544 // The following examples are invalid constructions that will throw
2545 // an exception at run time.
2547 // An index may not appear multiple times:
2548 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2549 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2551 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2552 // same number of indices:
2553 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2555 // And of course, you cannot specify indices which are not there:
2556 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2560 If you need to specify more than four indices, you have to use the
2561 @code{.add()} method of the @code{symmetry} class. For example, to specify
2562 full symmetry in the first six indices you would write
2563 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2565 If an indexed object has a symmetry, GiNaC will automatically bring the
2566 indices into a canonical order which allows for some immediate simplifications:
2570 cout << indexed(A, sy_symm(), i, j)
2571 + indexed(A, sy_symm(), j, i) << endl;
2573 cout << indexed(B, sy_anti(), i, j)
2574 + indexed(B, sy_anti(), j, i) << endl;
2576 cout << indexed(B, sy_anti(), i, j, k)
2577 - indexed(B, sy_anti(), j, k, i) << endl;
2582 @cindex @code{get_free_indices()}
2584 @subsection Dummy indices
2586 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2587 that a summation over the index range is implied. Symbolic indices which are
2588 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2589 dummy nor free indices.
2591 To be recognized as a dummy index pair, the two indices must be of the same
2592 class and their value must be the same single symbol (an index like
2593 @samp{2*n+1} is never a dummy index). If the indices are of class
2594 @code{varidx} they must also be of opposite variance; if they are of class
2595 @code{spinidx} they must be both dotted or both undotted.
2597 The method @code{.get_free_indices()} returns a vector containing the free
2598 indices of an expression. It also checks that the free indices of the terms
2599 of a sum are consistent:
2603 symbol A("A"), B("B"), C("C");
2605 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2606 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2608 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2609 cout << exprseq(e.get_free_indices()) << endl;
2611 // 'j' and 'l' are dummy indices
2613 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2614 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2616 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2617 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2618 cout << exprseq(e.get_free_indices()) << endl;
2620 // 'nu' is a dummy index, but 'sigma' is not
2622 e = indexed(A, mu, mu);
2623 cout << exprseq(e.get_free_indices()) << endl;
2625 // 'mu' is not a dummy index because it appears twice with the same
2628 e = indexed(A, mu, nu) + 42;
2629 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2630 // this will throw an exception:
2631 // "add::get_free_indices: inconsistent indices in sum"
2635 @cindex @code{expand_dummy_sum()}
2636 A dummy index summation like
2643 can be expanded for indices with numeric
2644 dimensions (e.g. 3) into the explicit sum like
2646 $a_1b^1+a_2b^2+a_3b^3 $.
2649 a.1 b~1 + a.2 b~2 + a.3 b~3.
2651 This is performed by the function
2654 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2657 which takes an expression @code{e} and returns the expanded sum for all
2658 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2659 is set to @code{true} then all substitutions are made by @code{idx} class
2660 indices, i.e. without variance. In this case the above sum
2669 $a_1b_1+a_2b_2+a_3b_3 $.
2672 a.1 b.1 + a.2 b.2 + a.3 b.3.
2676 @cindex @code{simplify_indexed()}
2677 @subsection Simplifying indexed expressions
2679 In addition to the few automatic simplifications that GiNaC performs on
2680 indexed expressions (such as re-ordering the indices of symmetric tensors
2681 and calculating traces and convolutions of matrices and predefined tensors)
2685 ex ex::simplify_indexed();
2686 ex ex::simplify_indexed(const scalar_products & sp);
2689 that performs some more expensive operations:
2692 @item it checks the consistency of free indices in sums in the same way
2693 @code{get_free_indices()} does
2694 @item it tries to give dummy indices that appear in different terms of a sum
2695 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2696 @item it (symbolically) calculates all possible dummy index summations/contractions
2697 with the predefined tensors (this will be explained in more detail in the
2699 @item it detects contractions that vanish for symmetry reasons, for example
2700 the contraction of a symmetric and a totally antisymmetric tensor
2701 @item as a special case of dummy index summation, it can replace scalar products
2702 of two tensors with a user-defined value
2705 The last point is done with the help of the @code{scalar_products} class
2706 which is used to store scalar products with known values (this is not an
2707 arithmetic class, you just pass it to @code{simplify_indexed()}):
2711 symbol A("A"), B("B"), C("C"), i_sym("i");
2715 sp.add(A, B, 0); // A and B are orthogonal
2716 sp.add(A, C, 0); // A and C are orthogonal
2717 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2719 e = indexed(A + B, i) * indexed(A + C, i);
2721 // -> (B+A).i*(A+C).i
2723 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2729 The @code{scalar_products} object @code{sp} acts as a storage for the
2730 scalar products added to it with the @code{.add()} method. This method
2731 takes three arguments: the two expressions of which the scalar product is
2732 taken, and the expression to replace it with.
2734 @cindex @code{expand()}
2735 The example above also illustrates a feature of the @code{expand()} method:
2736 if passed the @code{expand_indexed} option it will distribute indices
2737 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2739 @cindex @code{tensor} (class)
2740 @subsection Predefined tensors
2742 Some frequently used special tensors such as the delta, epsilon and metric
2743 tensors are predefined in GiNaC. They have special properties when
2744 contracted with other tensor expressions and some of them have constant
2745 matrix representations (they will evaluate to a number when numeric
2746 indices are specified).
2748 @cindex @code{delta_tensor()}
2749 @subsubsection Delta tensor
2751 The delta tensor takes two indices, is symmetric and has the matrix
2752 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2753 @code{delta_tensor()}:
2757 symbol A("A"), B("B");
2759 idx i(symbol("i"), 3), j(symbol("j"), 3),
2760 k(symbol("k"), 3), l(symbol("l"), 3);
2762 ex e = indexed(A, i, j) * indexed(B, k, l)
2763 * delta_tensor(i, k) * delta_tensor(j, l);
2764 cout << e.simplify_indexed() << endl;
2767 cout << delta_tensor(i, i) << endl;
2772 @cindex @code{metric_tensor()}
2773 @subsubsection General metric tensor
2775 The function @code{metric_tensor()} creates a general symmetric metric
2776 tensor with two indices that can be used to raise/lower tensor indices. The
2777 metric tensor is denoted as @samp{g} in the output and if its indices are of
2778 mixed variance it is automatically replaced by a delta tensor:
2784 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2786 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2787 cout << e.simplify_indexed() << endl;
2790 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2791 cout << e.simplify_indexed() << endl;
2794 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2795 * metric_tensor(nu, rho);
2796 cout << e.simplify_indexed() << endl;
2799 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2800 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2801 + indexed(A, mu.toggle_variance(), rho));
2802 cout << e.simplify_indexed() << endl;
2807 @cindex @code{lorentz_g()}
2808 @subsubsection Minkowski metric tensor
2810 The Minkowski metric tensor is a special metric tensor with a constant
2811 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2812 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2813 It is created with the function @code{lorentz_g()} (although it is output as
2818 varidx mu(symbol("mu"), 4);
2820 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2821 * lorentz_g(mu, varidx(0, 4)); // negative signature
2822 cout << e.simplify_indexed() << endl;
2825 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2826 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2827 cout << e.simplify_indexed() << endl;
2832 @cindex @code{spinor_metric()}
2833 @subsubsection Spinor metric tensor
2835 The function @code{spinor_metric()} creates an antisymmetric tensor with
2836 two indices that is used to raise/lower indices of 2-component spinors.
2837 It is output as @samp{eps}:
2843 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2844 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2846 e = spinor_metric(A, B) * indexed(psi, B_co);
2847 cout << e.simplify_indexed() << endl;
2850 e = spinor_metric(A, B) * indexed(psi, A_co);
2851 cout << e.simplify_indexed() << endl;
2854 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2855 cout << e.simplify_indexed() << endl;
2858 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2859 cout << e.simplify_indexed() << endl;
2862 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2863 cout << e.simplify_indexed() << endl;
2866 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2867 cout << e.simplify_indexed() << endl;
2872 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2874 @cindex @code{epsilon_tensor()}
2875 @cindex @code{lorentz_eps()}
2876 @subsubsection Epsilon tensor
2878 The epsilon tensor is totally antisymmetric, its number of indices is equal
2879 to the dimension of the index space (the indices must all be of the same
2880 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2881 defined to be 1. Its behavior with indices that have a variance also
2882 depends on the signature of the metric. Epsilon tensors are output as
2885 There are three functions defined to create epsilon tensors in 2, 3 and 4
2889 ex epsilon_tensor(const ex & i1, const ex & i2);
2890 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2891 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2892 bool pos_sig = false);
2895 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2896 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2897 Minkowski space (the last @code{bool} argument specifies whether the metric
2898 has negative or positive signature, as in the case of the Minkowski metric
2903 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2904 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2905 e = lorentz_eps(mu, nu, rho, sig) *
2906 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2907 cout << simplify_indexed(e) << endl;
2908 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2910 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2911 symbol A("A"), B("B");
2912 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2913 cout << simplify_indexed(e) << endl;
2914 // -> -B.k*A.j*eps.i.k.j
2915 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2916 cout << simplify_indexed(e) << endl;
2921 @subsection Linear algebra
2923 The @code{matrix} class can be used with indices to do some simple linear
2924 algebra (linear combinations and products of vectors and matrices, traces
2925 and scalar products):
2929 idx i(symbol("i"), 2), j(symbol("j"), 2);
2930 symbol x("x"), y("y");
2932 // A is a 2x2 matrix, X is a 2x1 vector
2933 matrix A(2, 2), X(2, 1);
2938 cout << indexed(A, i, i) << endl;
2941 ex e = indexed(A, i, j) * indexed(X, j);
2942 cout << e.simplify_indexed() << endl;
2943 // -> [[2*y+x],[4*y+3*x]].i
2945 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2946 cout << e.simplify_indexed() << endl;
2947 // -> [[3*y+3*x,6*y+2*x]].j
2951 You can of course obtain the same results with the @code{matrix::add()},
2952 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2953 but with indices you don't have to worry about transposing matrices.
2955 Matrix indices always start at 0 and their dimension must match the number
2956 of rows/columns of the matrix. Matrices with one row or one column are
2957 vectors and can have one or two indices (it doesn't matter whether it's a
2958 row or a column vector). Other matrices must have two indices.
2960 You should be careful when using indices with variance on matrices. GiNaC
2961 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2962 @samp{F.mu.nu} are different matrices. In this case you should use only
2963 one form for @samp{F} and explicitly multiply it with a matrix representation
2964 of the metric tensor.
2967 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2968 @c node-name, next, previous, up
2969 @section Non-commutative objects
2971 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2972 non-commutative objects are built-in which are mostly of use in high energy
2976 @item Clifford (Dirac) algebra (class @code{clifford})
2977 @item su(3) Lie algebra (class @code{color})
2978 @item Matrices (unindexed) (class @code{matrix})
2981 The @code{clifford} and @code{color} classes are subclasses of
2982 @code{indexed} because the elements of these algebras usually carry
2983 indices. The @code{matrix} class is described in more detail in
2986 Unlike most computer algebra systems, GiNaC does not primarily provide an
2987 operator (often denoted @samp{&*}) for representing inert products of
2988 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2989 classes of objects involved, and non-commutative products are formed with
2990 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2991 figuring out by itself which objects commutate and will group the factors
2992 by their class. Consider this example:
2996 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2997 idx a(symbol("a"), 8), b(symbol("b"), 8);
2998 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3000 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3004 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3005 groups the non-commutative factors (the gammas and the su(3) generators)
3006 together while preserving the order of factors within each class (because
3007 Clifford objects commutate with color objects). The resulting expression is a
3008 @emph{commutative} product with two factors that are themselves non-commutative
3009 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3010 parentheses are placed around the non-commutative products in the output.
3012 @cindex @code{ncmul} (class)
3013 Non-commutative products are internally represented by objects of the class
3014 @code{ncmul}, as opposed to commutative products which are handled by the
3015 @code{mul} class. You will normally not have to worry about this distinction,
3018 The advantage of this approach is that you never have to worry about using
3019 (or forgetting to use) a special operator when constructing non-commutative
3020 expressions. Also, non-commutative products in GiNaC are more intelligent
3021 than in other computer algebra systems; they can, for example, automatically
3022 canonicalize themselves according to rules specified in the implementation
3023 of the non-commutative classes. The drawback is that to work with other than
3024 the built-in algebras you have to implement new classes yourself. Both
3025 symbols and user-defined functions can be specified as being non-commutative.
3027 @cindex @code{return_type()}
3028 @cindex @code{return_type_tinfo()}
3029 Information about the commutativity of an object or expression can be
3030 obtained with the two member functions
3033 unsigned ex::return_type() const;
3034 unsigned ex::return_type_tinfo() const;
3037 The @code{return_type()} function returns one of three values (defined in
3038 the header file @file{flags.h}), corresponding to three categories of
3039 expressions in GiNaC:
3042 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3043 classes are of this kind.
3044 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3045 certain class of non-commutative objects which can be determined with the
3046 @code{return_type_tinfo()} method. Expressions of this category commutate
3047 with everything except @code{noncommutative} expressions of the same
3049 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3050 of non-commutative objects of different classes. Expressions of this
3051 category don't commutate with any other @code{noncommutative} or
3052 @code{noncommutative_composite} expressions.
3055 The value returned by the @code{return_type_tinfo()} method is valid only
3056 when the return type of the expression is @code{noncommutative}. It is a
3057 value that is unique to the class of the object and usually one of the
3058 constants in @file{tinfos.h}, or derived therefrom.
3060 Here are a couple of examples:
3063 @multitable @columnfractions 0.33 0.33 0.34
3064 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3065 @item @code{42} @tab @code{commutative} @tab -
3066 @item @code{2*x-y} @tab @code{commutative} @tab -
3067 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3068 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3069 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3070 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3074 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3075 @code{TINFO_clifford} for objects with a representation label of zero.
3076 Other representation labels yield a different @code{return_type_tinfo()},
3077 but it's the same for any two objects with the same label. This is also true
3080 A last note: With the exception of matrices, positive integer powers of
3081 non-commutative objects are automatically expanded in GiNaC. For example,
3082 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3083 non-commutative expressions).
3086 @cindex @code{clifford} (class)
3087 @subsection Clifford algebra
3090 Clifford algebras are supported in two flavours: Dirac gamma
3091 matrices (more physical) and generic Clifford algebras (more
3094 @cindex @code{dirac_gamma()}
3095 @subsubsection Dirac gamma matrices
3096 Dirac gamma matrices (note that GiNaC doesn't treat them
3097 as matrices) are designated as @samp{gamma~mu} and satisfy
3098 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3099 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3100 constructed by the function
3103 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3106 which takes two arguments: the index and a @dfn{representation label} in the
3107 range 0 to 255 which is used to distinguish elements of different Clifford
3108 algebras (this is also called a @dfn{spin line index}). Gammas with different
3109 labels commutate with each other. The dimension of the index can be 4 or (in
3110 the framework of dimensional regularization) any symbolic value. Spinor
3111 indices on Dirac gammas are not supported in GiNaC.
3113 @cindex @code{dirac_ONE()}
3114 The unity element of a Clifford algebra is constructed by
3117 ex dirac_ONE(unsigned char rl = 0);
3120 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3121 multiples of the unity element, even though it's customary to omit it.
3122 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3123 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3124 GiNaC will complain and/or produce incorrect results.
3126 @cindex @code{dirac_gamma5()}
3127 There is a special element @samp{gamma5} that commutates with all other
3128 gammas, has a unit square, and in 4 dimensions equals
3129 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3132 ex dirac_gamma5(unsigned char rl = 0);
3135 @cindex @code{dirac_gammaL()}
3136 @cindex @code{dirac_gammaR()}
3137 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3138 objects, constructed by
3141 ex dirac_gammaL(unsigned char rl = 0);
3142 ex dirac_gammaR(unsigned char rl = 0);
3145 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3146 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3148 @cindex @code{dirac_slash()}
3149 Finally, the function
3152 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3155 creates a term that represents a contraction of @samp{e} with the Dirac
3156 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3157 with a unique index whose dimension is given by the @code{dim} argument).
3158 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3160 In products of dirac gammas, superfluous unity elements are automatically
3161 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3162 and @samp{gammaR} are moved to the front.
3164 The @code{simplify_indexed()} function performs contractions in gamma strings,
3170 symbol a("a"), b("b"), D("D");
3171 varidx mu(symbol("mu"), D);
3172 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3173 * dirac_gamma(mu.toggle_variance());
3175 // -> gamma~mu*a\*gamma.mu
3176 e = e.simplify_indexed();
3179 cout << e.subs(D == 4) << endl;
3185 @cindex @code{dirac_trace()}
3186 To calculate the trace of an expression containing strings of Dirac gammas
3187 you use one of the functions
3190 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3191 const ex & trONE = 4);
3192 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3193 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3196 These functions take the trace over all gammas in the specified set @code{rls}
3197 or list @code{rll} of representation labels, or the single label @code{rl};
3198 gammas with other labels are left standing. The last argument to
3199 @code{dirac_trace()} is the value to be returned for the trace of the unity
3200 element, which defaults to 4.
3202 The @code{dirac_trace()} function is a linear functional that is equal to the
3203 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3204 functional is not cyclic in
3207 dimensions when acting on
3208 expressions containing @samp{gamma5}, so it's not a proper trace. This
3209 @samp{gamma5} scheme is described in greater detail in
3210 @cite{The Role of gamma5 in Dimensional Regularization}.
3212 The value of the trace itself is also usually different in 4 and in
3220 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3221 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3222 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3223 cout << dirac_trace(e).simplify_indexed() << endl;
3230 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3231 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3232 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3233 cout << dirac_trace(e).simplify_indexed() << endl;
3234 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3238 Here is an example for using @code{dirac_trace()} to compute a value that
3239 appears in the calculation of the one-loop vacuum polarization amplitude in
3244 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3245 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3248 sp.add(l, l, pow(l, 2));
3249 sp.add(l, q, ldotq);
3251 ex e = dirac_gamma(mu) *
3252 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3253 dirac_gamma(mu.toggle_variance()) *
3254 (dirac_slash(l, D) + m * dirac_ONE());
3255 e = dirac_trace(e).simplify_indexed(sp);
3256 e = e.collect(lst(l, ldotq, m));
3258 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3262 The @code{canonicalize_clifford()} function reorders all gamma products that
3263 appear in an expression to a canonical (but not necessarily simple) form.
3264 You can use this to compare two expressions or for further simplifications:
3268 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3269 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3271 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3273 e = canonicalize_clifford(e);
3275 // -> 2*ONE*eta~mu~nu
3279 @cindex @code{clifford_unit()}
3280 @subsubsection A generic Clifford algebra
3282 A generic Clifford algebra, i.e. a
3286 dimensional algebra with
3290 satisfying the identities
3292 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3295 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3297 for some bilinear form (@code{metric})
3298 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3299 and contain symbolic entries. Such generators are created by the
3303 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3304 bool anticommuting = false);
3307 where @code{mu} should be a @code{varidx} class object indexing the
3308 generators, an index @code{mu} with a numeric value may be of type
3310 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3311 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3312 object. In fact, any expression either with two free indices or without
3313 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3314 object with two newly created indices with @code{metr} as its
3315 @code{op(0)} will be used.
3316 Optional parameter @code{rl} allows to distinguish different
3317 Clifford algebras, which will commute with each other. The last
3318 optional parameter @code{anticommuting} defines if the anticommuting
3321 $e_i e_j + e_j e_i = 0$)
3324 e~i e~j + e~j e~i = 0)
3326 will be used for contraction of Clifford units. If the @code{metric} is
3327 supplied by a @code{matrix} object, then the value of
3328 @code{anticommuting} is calculated automatically and the supplied one
3329 will be ignored. One can overcome this by giving @code{metric} through
3330 matrix wrapped into an @code{indexed} object.
3332 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3333 something very close to @code{dirac_gamma(mu)}, although
3334 @code{dirac_gamma} have more efficient simplification mechanism.
3335 @cindex @code{clifford::get_metric()}
3336 The method @code{clifford::get_metric()} returns a metric defining this
3338 @cindex @code{clifford::is_anticommuting()}
3339 The method @code{clifford::is_anticommuting()} returns the
3340 @code{anticommuting} property of a unit.
3342 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3343 the Clifford algebra units with a call like that
3346 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3349 since this may yield some further automatic simplifications. Again, for a
3350 metric defined through a @code{matrix} such a symmetry is detected
3353 Individual generators of a Clifford algebra can be accessed in several
3359 varidx nu(symbol("nu"), 4);
3361 ex M = diag_matrix(lst(1, -1, 0, s));
3362 ex e = clifford_unit(nu, M);
3363 ex e0 = e.subs(nu == 0);
3364 ex e1 = e.subs(nu == 1);
3365 ex e2 = e.subs(nu == 2);
3366 ex e3 = e.subs(nu == 3);
3371 will produce four anti-commuting generators of a Clifford algebra with properties
3373 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3376 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3377 @code{pow(e3, 2) = s}.
3380 @cindex @code{lst_to_clifford()}
3381 A similar effect can be achieved from the function
3384 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3385 unsigned char rl = 0, bool anticommuting = false);
3386 ex lst_to_clifford(const ex & v, const ex & e);
3389 which converts a list or vector
3391 $v = (v^0, v^1, ..., v^n)$
3394 @samp{v = (v~0, v~1, ..., v~n)}
3399 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3402 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3405 directly supplied in the second form of the procedure. In the first form
3406 the Clifford unit @samp{e.k} is generated by the call of
3407 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3408 with the help of @code{lst_to_clifford()} as follows
3413 varidx nu(symbol("nu"), 4);
3415 ex M = diag_matrix(lst(1, -1, 0, s));
3416 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3417 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3418 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3419 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3424 @cindex @code{clifford_to_lst()}
3425 There is the inverse function
3428 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3431 which takes an expression @code{e} and tries to find a list
3433 $v = (v^0, v^1, ..., v^n)$
3436 @samp{v = (v~0, v~1, ..., v~n)}
3440 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3443 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3445 with respect to the given Clifford units @code{c} and with none of the
3446 @samp{v~k} containing Clifford units @code{c} (of course, this
3447 may be impossible). This function can use an @code{algebraic} method
3448 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3450 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3453 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3455 is zero or is not @code{numeric} for some @samp{k}
3456 then the method will be automatically changed to symbolic. The same effect
3457 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3459 @cindex @code{clifford_prime()}
3460 @cindex @code{clifford_star()}
3461 @cindex @code{clifford_bar()}
3462 There are several functions for (anti-)automorphisms of Clifford algebras:
3465 ex clifford_prime(const ex & e)
3466 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3467 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3470 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3471 changes signs of all Clifford units in the expression. The reversion
3472 of a Clifford algebra @code{clifford_star()} coincides with the
3473 @code{conjugate()} method and effectively reverses the order of Clifford
3474 units in any product. Finally the main anti-automorphism
3475 of a Clifford algebra @code{clifford_bar()} is the composition of the
3476 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3477 in a product. These functions correspond to the notations
3492 used in Clifford algebra textbooks.
3494 @cindex @code{clifford_norm()}
3498 ex clifford_norm(const ex & e);
3501 @cindex @code{clifford_inverse()}
3502 calculates the norm of a Clifford number from the expression
3504 $||e||^2 = e\overline{e}$.
3507 @code{||e||^2 = e \bar@{e@}}
3509 The inverse of a Clifford expression is returned by the function
3512 ex clifford_inverse(const ex & e);
3515 which calculates it as
3517 $e^{-1} = \overline{e}/||e||^2$.
3520 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3529 then an exception is raised.
3531 @cindex @code{remove_dirac_ONE()}
3532 If a Clifford number happens to be a factor of
3533 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3534 expression by the function
3537 ex remove_dirac_ONE(const ex & e);
3540 @cindex @code{canonicalize_clifford()}
3541 The function @code{canonicalize_clifford()} works for a
3542 generic Clifford algebra in a similar way as for Dirac gammas.
3544 The next provided function is
3546 @cindex @code{clifford_moebius_map()}
3548 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3549 const ex & d, const ex & v, const ex & G,
3550 unsigned char rl = 0, bool anticommuting = false);
3551 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3552 unsigned char rl = 0, bool anticommuting = false);
3555 It takes a list or vector @code{v} and makes the Moebius (conformal or
3556 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3557 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3558 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3559 indexed object, tensormetric, matrix or a Clifford unit, in the later
3560 case the optional parameters @code{rl} and @code{anticommuting} are
3561 ignored even if supplied. Depending from the type of @code{v} the
3562 returned value of this function is either a vector or a list holding vector's
3565 @cindex @code{clifford_max_label()}
3566 Finally the function
3569 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3572 can detect a presence of Clifford objects in the expression @code{e}: if
3573 such objects are found it returns the maximal
3574 @code{representation_label} of them, otherwise @code{-1}. The optional
3575 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3576 be ignored during the search.
3578 LaTeX output for Clifford units looks like
3579 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3580 @code{representation_label} and @code{\nu} is the index of the
3581 corresponding unit. This provides a flexible typesetting with a suitable
3582 defintion of the @code{\clifford} command. For example, the definition
3584 \newcommand@{\clifford@}[1][]@{@}
3586 typesets all Clifford units identically, while the alternative definition
3588 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3590 prints units with @code{representation_label=0} as
3597 with @code{representation_label=1} as
3604 and with @code{representation_label=2} as
3612 @cindex @code{color} (class)
3613 @subsection Color algebra
3615 @cindex @code{color_T()}
3616 For computations in quantum chromodynamics, GiNaC implements the base elements
3617 and structure constants of the su(3) Lie algebra (color algebra). The base
3618 elements @math{T_a} are constructed by the function
3621 ex color_T(const ex & a, unsigned char rl = 0);
3624 which takes two arguments: the index and a @dfn{representation label} in the
3625 range 0 to 255 which is used to distinguish elements of different color
3626 algebras. Objects with different labels commutate with each other. The
3627 dimension of the index must be exactly 8 and it should be of class @code{idx},
3630 @cindex @code{color_ONE()}
3631 The unity element of a color algebra is constructed by
3634 ex color_ONE(unsigned char rl = 0);
3637 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3638 multiples of the unity element, even though it's customary to omit it.
3639 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3640 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3641 GiNaC may produce incorrect results.
3643 @cindex @code{color_d()}
3644 @cindex @code{color_f()}
3648 ex color_d(const ex & a, const ex & b, const ex & c);
3649 ex color_f(const ex & a, const ex & b, const ex & c);
3652 create the symmetric and antisymmetric structure constants @math{d_abc} and
3653 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3654 and @math{[T_a, T_b] = i f_abc T_c}.
3656 These functions evaluate to their numerical values,
3657 if you supply numeric indices to them. The index values should be in
3658 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3659 goes along better with the notations used in physical literature.
3661 @cindex @code{color_h()}
3662 There's an additional function
3665 ex color_h(const ex & a, const ex & b, const ex & c);
3668 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3670 The function @code{simplify_indexed()} performs some simplifications on
3671 expressions containing color objects:
3676 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3677 k(symbol("k"), 8), l(symbol("l"), 8);
3679 e = color_d(a, b, l) * color_f(a, b, k);
3680 cout << e.simplify_indexed() << endl;
3683 e = color_d(a, b, l) * color_d(a, b, k);
3684 cout << e.simplify_indexed() << endl;
3687 e = color_f(l, a, b) * color_f(a, b, k);
3688 cout << e.simplify_indexed() << endl;
3691 e = color_h(a, b, c) * color_h(a, b, c);
3692 cout << e.simplify_indexed() << endl;
3695 e = color_h(a, b, c) * color_T(b) * color_T(c);
3696 cout << e.simplify_indexed() << endl;
3699 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3700 cout << e.simplify_indexed() << endl;
3703 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3704 cout << e.simplify_indexed() << endl;
3705 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3709 @cindex @code{color_trace()}
3710 To calculate the trace of an expression containing color objects you use one
3714 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3715 ex color_trace(const ex & e, const lst & rll);
3716 ex color_trace(const ex & e, unsigned char rl = 0);
3719 These functions take the trace over all color @samp{T} objects in the
3720 specified set @code{rls} or list @code{rll} of representation labels, or the
3721 single label @code{rl}; @samp{T}s with other labels are left standing. For
3726 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3728 // -> -I*f.a.c.b+d.a.c.b
3733 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3734 @c node-name, next, previous, up
3737 @cindex @code{exhashmap} (class)
3739 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3740 that can be used as a drop-in replacement for the STL
3741 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3742 typically constant-time, element look-up than @code{map<>}.
3744 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3745 following differences:
3749 no @code{lower_bound()} and @code{upper_bound()} methods
3751 no reverse iterators, no @code{rbegin()}/@code{rend()}
3753 no @code{operator<(exhashmap, exhashmap)}
3755 the comparison function object @code{key_compare} is hardcoded to
3758 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3759 initial hash table size (the actual table size after construction may be
3760 larger than the specified value)
3762 the method @code{size_t bucket_count()} returns the current size of the hash
3765 @code{insert()} and @code{erase()} operations invalidate all iterators
3769 @node Methods and functions, Information about expressions, Hash maps, Top
3770 @c node-name, next, previous, up
3771 @chapter Methods and functions
3774 In this chapter the most important algorithms provided by GiNaC will be
3775 described. Some of them are implemented as functions on expressions,
3776 others are implemented as methods provided by expression objects. If
3777 they are methods, there exists a wrapper function around it, so you can
3778 alternatively call it in a functional way as shown in the simple
3783 cout << "As method: " << sin(1).evalf() << endl;
3784 cout << "As function: " << evalf(sin(1)) << endl;
3788 @cindex @code{subs()}
3789 The general rule is that wherever methods accept one or more parameters
3790 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3791 wrapper accepts is the same but preceded by the object to act on
3792 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3793 most natural one in an OO model but it may lead to confusion for MapleV
3794 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3795 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3796 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3797 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3798 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3799 here. Also, users of MuPAD will in most cases feel more comfortable
3800 with GiNaC's convention. All function wrappers are implemented
3801 as simple inline functions which just call the corresponding method and
3802 are only provided for users uncomfortable with OO who are dead set to
3803 avoid method invocations. Generally, nested function wrappers are much
3804 harder to read than a sequence of methods and should therefore be
3805 avoided if possible. On the other hand, not everything in GiNaC is a
3806 method on class @code{ex} and sometimes calling a function cannot be
3810 * Information about expressions::
3811 * Numerical evaluation::
3812 * Substituting expressions::
3813 * Pattern matching and advanced substitutions::
3814 * Applying a function on subexpressions::
3815 * Visitors and tree traversal::
3816 * Polynomial arithmetic:: Working with polynomials.
3817 * Rational expressions:: Working with rational functions.
3818 * Symbolic differentiation::
3819 * Series expansion:: Taylor and Laurent expansion.
3821 * Built-in functions:: List of predefined mathematical functions.
3822 * Multiple polylogarithms::
3823 * Complex expressions::
3824 * Solving linear systems of equations::
3825 * Input/output:: Input and output of expressions.
3829 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3830 @c node-name, next, previous, up
3831 @section Getting information about expressions
3833 @subsection Checking expression types
3834 @cindex @code{is_a<@dots{}>()}
3835 @cindex @code{is_exactly_a<@dots{}>()}
3836 @cindex @code{ex_to<@dots{}>()}
3837 @cindex Converting @code{ex} to other classes
3838 @cindex @code{info()}
3839 @cindex @code{return_type()}
3840 @cindex @code{return_type_tinfo()}
3842 Sometimes it's useful to check whether a given expression is a plain number,
3843 a sum, a polynomial with integer coefficients, or of some other specific type.
3844 GiNaC provides a couple of functions for this:
3847 bool is_a<T>(const ex & e);
3848 bool is_exactly_a<T>(const ex & e);
3849 bool ex::info(unsigned flag);
3850 unsigned ex::return_type() const;
3851 unsigned ex::return_type_tinfo() const;
3854 When the test made by @code{is_a<T>()} returns true, it is safe to call
3855 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3856 class names (@xref{The class hierarchy}, for a list of all classes). For
3857 example, assuming @code{e} is an @code{ex}:
3862 if (is_a<numeric>(e))
3863 numeric n = ex_to<numeric>(e);
3868 @code{is_a<T>(e)} allows you to check whether the top-level object of
3869 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3870 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3871 e.g., for checking whether an expression is a number, a sum, or a product:
3878 is_a<numeric>(e1); // true
3879 is_a<numeric>(e2); // false
3880 is_a<add>(e1); // false
3881 is_a<add>(e2); // true
3882 is_a<mul>(e1); // false
3883 is_a<mul>(e2); // false
3887 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3888 top-level object of an expression @samp{e} is an instance of the GiNaC
3889 class @samp{T}, not including parent classes.
3891 The @code{info()} method is used for checking certain attributes of
3892 expressions. The possible values for the @code{flag} argument are defined
3893 in @file{ginac/flags.h}, the most important being explained in the following
3897 @multitable @columnfractions .30 .70
3898 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3899 @item @code{numeric}
3900 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3902 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3903 @item @code{rational}
3904 @tab @dots{}an exact rational number (integers are rational, too)
3905 @item @code{integer}
3906 @tab @dots{}a (non-complex) integer
3907 @item @code{crational}
3908 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3909 @item @code{cinteger}
3910 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3911 @item @code{positive}
3912 @tab @dots{}not complex and greater than 0
3913 @item @code{negative}
3914 @tab @dots{}not complex and less than 0
3915 @item @code{nonnegative}
3916 @tab @dots{}not complex and greater than or equal to 0
3918 @tab @dots{}an integer greater than 0
3920 @tab @dots{}an integer less than 0
3921 @item @code{nonnegint}
3922 @tab @dots{}an integer greater than or equal to 0
3924 @tab @dots{}an even integer
3926 @tab @dots{}an odd integer
3928 @tab @dots{}a prime integer (probabilistic primality test)
3929 @item @code{relation}
3930 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3931 @item @code{relation_equal}
3932 @tab @dots{}a @code{==} relation
3933 @item @code{relation_not_equal}
3934 @tab @dots{}a @code{!=} relation
3935 @item @code{relation_less}
3936 @tab @dots{}a @code{<} relation
3937 @item @code{relation_less_or_equal}
3938 @tab @dots{}a @code{<=} relation
3939 @item @code{relation_greater}
3940 @tab @dots{}a @code{>} relation
3941 @item @code{relation_greater_or_equal}
3942 @tab @dots{}a @code{>=} relation
3944 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3946 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3947 @item @code{polynomial}
3948 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3949 @item @code{integer_polynomial}
3950 @tab @dots{}a polynomial with (non-complex) integer coefficients
3951 @item @code{cinteger_polynomial}
3952 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3953 @item @code{rational_polynomial}
3954 @tab @dots{}a polynomial with (non-complex) rational coefficients
3955 @item @code{crational_polynomial}
3956 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3957 @item @code{rational_function}
3958 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3959 @item @code{algebraic}
3960 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3964 To determine whether an expression is commutative or non-commutative and if
3965 so, with which other expressions it would commutate, you use the methods
3966 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3967 for an explanation of these.
3970 @subsection Accessing subexpressions
3973 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3974 @code{function}, act as containers for subexpressions. For example, the
3975 subexpressions of a sum (an @code{add} object) are the individual terms,
3976 and the subexpressions of a @code{function} are the function's arguments.
3978 @cindex @code{nops()}
3980 GiNaC provides several ways of accessing subexpressions. The first way is to
3985 ex ex::op(size_t i);
3988 @code{nops()} determines the number of subexpressions (operands) contained
3989 in the expression, while @code{op(i)} returns the @code{i}-th
3990 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3991 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3992 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3993 @math{i>0} are the indices.
3996 @cindex @code{const_iterator}
3997 The second way to access subexpressions is via the STL-style random-access
3998 iterator class @code{const_iterator} and the methods
4001 const_iterator ex::begin();
4002 const_iterator ex::end();
4005 @code{begin()} returns an iterator referring to the first subexpression;
4006 @code{end()} returns an iterator which is one-past the last subexpression.
4007 If the expression has no subexpressions, then @code{begin() == end()}. These
4008 iterators can also be used in conjunction with non-modifying STL algorithms.
4010 Here is an example that (non-recursively) prints the subexpressions of a
4011 given expression in three different ways:
4018 for (size_t i = 0; i != e.nops(); ++i)
4019 cout << e.op(i) << endl;
4022 for (const_iterator i = e.begin(); i != e.end(); ++i)
4025 // with iterators and STL copy()
4026 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4030 @cindex @code{const_preorder_iterator}
4031 @cindex @code{const_postorder_iterator}
4032 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4033 expression's immediate children. GiNaC provides two additional iterator
4034 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4035 that iterate over all objects in an expression tree, in preorder or postorder,
4036 respectively. They are STL-style forward iterators, and are created with the
4040 const_preorder_iterator ex::preorder_begin();
4041 const_preorder_iterator ex::preorder_end();
4042 const_postorder_iterator ex::postorder_begin();
4043 const_postorder_iterator ex::postorder_end();
4046 The following example illustrates the differences between
4047 @code{const_iterator}, @code{const_preorder_iterator}, and
4048 @code{const_postorder_iterator}:
4052 symbol A("A"), B("B"), C("C");
4053 ex e = lst(lst(A, B), C);
4055 std::copy(e.begin(), e.end(),
4056 std::ostream_iterator<ex>(cout, "\n"));
4060 std::copy(e.preorder_begin(), e.preorder_end(),
4061 std::ostream_iterator<ex>(cout, "\n"));
4068 std::copy(e.postorder_begin(), e.postorder_end(),
4069 std::ostream_iterator<ex>(cout, "\n"));
4078 @cindex @code{relational} (class)
4079 Finally, the left-hand side and right-hand side expressions of objects of
4080 class @code{relational} (and only of these) can also be accessed with the
4089 @subsection Comparing expressions
4090 @cindex @code{is_equal()}
4091 @cindex @code{is_zero()}
4093 Expressions can be compared with the usual C++ relational operators like
4094 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4095 the result is usually not determinable and the result will be @code{false},
4096 except in the case of the @code{!=} operator. You should also be aware that
4097 GiNaC will only do the most trivial test for equality (subtracting both
4098 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4101 Actually, if you construct an expression like @code{a == b}, this will be
4102 represented by an object of the @code{relational} class (@pxref{Relations})
4103 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4105 There are also two methods
4108 bool ex::is_equal(const ex & other);
4112 for checking whether one expression is equal to another, or equal to zero,
4113 respectively. See also the method @code{ex::is_zero_matrix()},
4117 @subsection Ordering expressions
4118 @cindex @code{ex_is_less} (class)
4119 @cindex @code{ex_is_equal} (class)
4120 @cindex @code{compare()}
4122 Sometimes it is necessary to establish a mathematically well-defined ordering
4123 on a set of arbitrary expressions, for example to use expressions as keys
4124 in a @code{std::map<>} container, or to bring a vector of expressions into
4125 a canonical order (which is done internally by GiNaC for sums and products).
4127 The operators @code{<}, @code{>} etc. described in the last section cannot
4128 be used for this, as they don't implement an ordering relation in the
4129 mathematical sense. In particular, they are not guaranteed to be
4130 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4131 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4134 By default, STL classes and algorithms use the @code{<} and @code{==}
4135 operators to compare objects, which are unsuitable for expressions, but GiNaC
4136 provides two functors that can be supplied as proper binary comparison
4137 predicates to the STL:
4140 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4142 bool operator()(const ex &lh, const ex &rh) const;
4145 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4147 bool operator()(const ex &lh, const ex &rh) const;
4151 For example, to define a @code{map} that maps expressions to strings you
4155 std::map<ex, std::string, ex_is_less> myMap;
4158 Omitting the @code{ex_is_less} template parameter will introduce spurious
4159 bugs because the map operates improperly.
4161 Other examples for the use of the functors:
4169 std::sort(v.begin(), v.end(), ex_is_less());
4171 // count the number of expressions equal to '1'
4172 unsigned num_ones = std::count_if(v.begin(), v.end(),
4173 std::bind2nd(ex_is_equal(), 1));
4176 The implementation of @code{ex_is_less} uses the member function
4179 int ex::compare(const ex & other) const;
4182 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4183 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4187 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4188 @c node-name, next, previous, up
4189 @section Numerical evaluation
4190 @cindex @code{evalf()}
4192 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4193 To evaluate them using floating-point arithmetic you need to call
4196 ex ex::evalf(int level = 0) const;
4199 @cindex @code{Digits}
4200 The accuracy of the evaluation is controlled by the global object @code{Digits}
4201 which can be assigned an integer value. The default value of @code{Digits}
4202 is 17. @xref{Numbers}, for more information and examples.
4204 To evaluate an expression to a @code{double} floating-point number you can
4205 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4209 // Approximate sin(x/Pi)
4211 ex e = series(sin(x/Pi), x == 0, 6);
4213 // Evaluate numerically at x=0.1
4214 ex f = evalf(e.subs(x == 0.1));
4216 // ex_to<numeric> is an unsafe cast, so check the type first
4217 if (is_a<numeric>(f)) @{
4218 double d = ex_to<numeric>(f).to_double();
4227 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4228 @c node-name, next, previous, up
4229 @section Substituting expressions
4230 @cindex @code{subs()}
4232 Algebraic objects inside expressions can be replaced with arbitrary
4233 expressions via the @code{.subs()} method:
4236 ex ex::subs(const ex & e, unsigned options = 0);
4237 ex ex::subs(const exmap & m, unsigned options = 0);
4238 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4241 In the first form, @code{subs()} accepts a relational of the form
4242 @samp{object == expression} or a @code{lst} of such relationals:
4246 symbol x("x"), y("y");
4248 ex e1 = 2*x^2-4*x+3;
4249 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4253 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4258 If you specify multiple substitutions, they are performed in parallel, so e.g.
4259 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4261 The second form of @code{subs()} takes an @code{exmap} object which is a
4262 pair associative container that maps expressions to expressions (currently
4263 implemented as a @code{std::map}). This is the most efficient one of the
4264 three @code{subs()} forms and should be used when the number of objects to
4265 be substituted is large or unknown.
4267 Using this form, the second example from above would look like this:
4271 symbol x("x"), y("y");
4277 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4281 The third form of @code{subs()} takes two lists, one for the objects to be
4282 replaced and one for the expressions to be substituted (both lists must
4283 contain the same number of elements). Using this form, you would write
4287 symbol x("x"), y("y");
4290 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4294 The optional last argument to @code{subs()} is a combination of
4295 @code{subs_options} flags. There are three options available:
4296 @code{subs_options::no_pattern} disables pattern matching, which makes
4297 large @code{subs()} operations significantly faster if you are not using
4298 patterns. The second option, @code{subs_options::algebraic} enables
4299 algebraic substitutions in products and powers.
4300 @ref{Pattern matching and advanced substitutions}, for more information
4301 about patterns and algebraic substitutions. The third option,
4302 @code{subs_options::no_index_renaming} disables the feature that dummy
4303 indices are renamed if the subsitution could give a result in which a
4304 dummy index occurs more than two times. This is sometimes necessary if
4305 you want to use @code{subs()} to rename your dummy indices.
4307 @code{subs()} performs syntactic substitution of any complete algebraic
4308 object; it does not try to match sub-expressions as is demonstrated by the
4313 symbol x("x"), y("y"), z("z");
4315 ex e1 = pow(x+y, 2);
4316 cout << e1.subs(x+y == 4) << endl;
4319 ex e2 = sin(x)*sin(y)*cos(x);
4320 cout << e2.subs(sin(x) == cos(x)) << endl;
4321 // -> cos(x)^2*sin(y)
4324 cout << e3.subs(x+y == 4) << endl;
4326 // (and not 4+z as one might expect)
4330 A more powerful form of substitution using wildcards is described in the
4334 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4335 @c node-name, next, previous, up
4336 @section Pattern matching and advanced substitutions
4337 @cindex @code{wildcard} (class)
4338 @cindex Pattern matching
4340 GiNaC allows the use of patterns for checking whether an expression is of a
4341 certain form or contains subexpressions of a certain form, and for
4342 substituting expressions in a more general way.
4344 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4345 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4346 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4347 an unsigned integer number to allow having multiple different wildcards in a
4348 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4349 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4353 ex wild(unsigned label = 0);
4356 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4359 Some examples for patterns:
4361 @multitable @columnfractions .5 .5
4362 @item @strong{Constructed as} @tab @strong{Output as}
4363 @item @code{wild()} @tab @samp{$0}
4364 @item @code{pow(x,wild())} @tab @samp{x^$0}
4365 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4366 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4372 @item Wildcards behave like symbols and are subject to the same algebraic
4373 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4374 @item As shown in the last example, to use wildcards for indices you have to
4375 use them as the value of an @code{idx} object. This is because indices must
4376 always be of class @code{idx} (or a subclass).
4377 @item Wildcards only represent expressions or subexpressions. It is not
4378 possible to use them as placeholders for other properties like index
4379 dimension or variance, representation labels, symmetry of indexed objects
4381 @item Because wildcards are commutative, it is not possible to use wildcards
4382 as part of noncommutative products.
4383 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4384 are also valid patterns.
4387 @subsection Matching expressions
4388 @cindex @code{match()}
4389 The most basic application of patterns is to check whether an expression
4390 matches a given pattern. This is done by the function
4393 bool ex::match(const ex & pattern);
4394 bool ex::match(const ex & pattern, lst & repls);
4397 This function returns @code{true} when the expression matches the pattern
4398 and @code{false} if it doesn't. If used in the second form, the actual
4399 subexpressions matched by the wildcards get returned in the @code{repls}
4400 object as a list of relations of the form @samp{wildcard == expression}.
4401 If @code{match()} returns false, the state of @code{repls} is undefined.
4402 For reproducible results, the list should be empty when passed to
4403 @code{match()}, but it is also possible to find similarities in multiple
4404 expressions by passing in the result of a previous match.
4406 The matching algorithm works as follows:
4409 @item A single wildcard matches any expression. If one wildcard appears
4410 multiple times in a pattern, it must match the same expression in all
4411 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4412 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4413 @item If the expression is not of the same class as the pattern, the match
4414 fails (i.e. a sum only matches a sum, a function only matches a function,
4416 @item If the pattern is a function, it only matches the same function
4417 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4418 @item Except for sums and products, the match fails if the number of
4419 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4421 @item If there are no subexpressions, the expressions and the pattern must
4422 be equal (in the sense of @code{is_equal()}).
4423 @item Except for sums and products, each subexpression (@code{op()}) must
4424 match the corresponding subexpression of the pattern.
4427 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4428 account for their commutativity and associativity:
4431 @item If the pattern contains a term or factor that is a single wildcard,
4432 this one is used as the @dfn{global wildcard}. If there is more than one
4433 such wildcard, one of them is chosen as the global wildcard in a random
4435 @item Every term/factor of the pattern, except the global wildcard, is
4436 matched against every term of the expression in sequence. If no match is
4437 found, the whole match fails. Terms that did match are not considered in
4439 @item If there are no unmatched terms left, the match succeeds. Otherwise
4440 the match fails unless there is a global wildcard in the pattern, in
4441 which case this wildcard matches the remaining terms.
4444 In general, having more than one single wildcard as a term of a sum or a
4445 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4448 Here are some examples in @command{ginsh} to demonstrate how it works (the
4449 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4450 match fails, and the list of wildcard replacements otherwise):
4453 > match((x+y)^a,(x+y)^a);
4455 > match((x+y)^a,(x+y)^b);
4457 > match((x+y)^a,$1^$2);
4459 > match((x+y)^a,$1^$1);
4461 > match((x+y)^(x+y),$1^$1);
4463 > match((x+y)^(x+y),$1^$2);
4465 > match((a+b)*(a+c),($1+b)*($1+c));
4467 > match((a+b)*(a+c),(a+$1)*(a+$2));
4469 (Unpredictable. The result might also be [$1==c,$2==b].)
4470 > match((a+b)*(a+c),($1+$2)*($1+$3));
4471 (The result is undefined. Due to the sequential nature of the algorithm
4472 and the re-ordering of terms in GiNaC, the match for the first factor
4473 may be @{$1==a,$2==b@} in which case the match for the second factor
4474 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4476 > match(a*(x+y)+a*z+b,a*$1+$2);
4477 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4478 @{$1=x+y,$2=a*z+b@}.)
4479 > match(a+b+c+d+e+f,c);
4481 > match(a+b+c+d+e+f,c+$0);
4483 > match(a+b+c+d+e+f,c+e+$0);
4485 > match(a+b,a+b+$0);
4487 > match(a*b^2,a^$1*b^$2);
4489 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4490 even though a==a^1.)
4491 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4493 > match(atan2(y,x^2),atan2(y,$0));
4497 @subsection Matching parts of expressions
4498 @cindex @code{has()}
4499 A more general way to look for patterns in expressions is provided by the
4503 bool ex::has(const ex & pattern);
4506 This function checks whether a pattern is matched by an expression itself or
4507 by any of its subexpressions.
4509 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4510 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4513 > has(x*sin(x+y+2*a),y);
4515 > has(x*sin(x+y+2*a),x+y);
4517 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4518 has the subexpressions "x", "y" and "2*a".)
4519 > has(x*sin(x+y+2*a),x+y+$1);
4521 (But this is possible.)
4522 > has(x*sin(2*(x+y)+2*a),x+y);
4524 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4525 which "x+y" is not a subexpression.)
4528 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4530 > has(4*x^2-x+3,$1*x);
4532 > has(4*x^2+x+3,$1*x);
4534 (Another possible pitfall. The first expression matches because the term
4535 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4536 contains a linear term you should use the coeff() function instead.)
4539 @cindex @code{find()}
4543 bool ex::find(const ex & pattern, lst & found);
4546 works a bit like @code{has()} but it doesn't stop upon finding the first
4547 match. Instead, it appends all found matches to the specified list. If there
4548 are multiple occurrences of the same expression, it is entered only once to
4549 the list. @code{find()} returns false if no matches were found (in
4550 @command{ginsh}, it returns an empty list):
4553 > find(1+x+x^2+x^3,x);
4555 > find(1+x+x^2+x^3,y);
4557 > find(1+x+x^2+x^3,x^$1);
4559 (Note the absence of "x".)
4560 > expand((sin(x)+sin(y))*(a+b));
4561 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4566 @subsection Substituting expressions
4567 @cindex @code{subs()}
4568 Probably the most useful application of patterns is to use them for
4569 substituting expressions with the @code{subs()} method. Wildcards can be
4570 used in the search patterns as well as in the replacement expressions, where
4571 they get replaced by the expressions matched by them. @code{subs()} doesn't
4572 know anything about algebra; it performs purely syntactic substitutions.
4577 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4579 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4581 > subs((a+b+c)^2,a+b==x);
4583 > subs((a+b+c)^2,a+b+$1==x+$1);
4585 > subs(a+2*b,a+b==x);
4587 > subs(4*x^3-2*x^2+5*x-1,x==a);
4589 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4591 > subs(sin(1+sin(x)),sin($1)==cos($1));
4593 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4597 The last example would be written in C++ in this way:
4601 symbol a("a"), b("b"), x("x"), y("y");
4602 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4603 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4604 cout << e.expand() << endl;
4609 @subsection The option algebraic
4610 Both @code{has()} and @code{subs()} take an optional argument to pass them
4611 extra options. This section describes what happens if you give the former
4612 the option @code{has_options::algebraic} or the latter
4613 @code{subs:options::algebraic}. In that case the matching condition for
4614 powers and multiplications is changed in such a way that they become
4615 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4616 If you use these options you will find that
4617 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4618 Besides matching some of the factors of a product also powers match as
4619 often as is possible without getting negative exponents. For example
4620 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4621 @code{x*c^2*z}. This also works with negative powers:
4622 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4623 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4624 and not for locating @code{x+y} within @code{x+y+z}.
4627 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4628 @c node-name, next, previous, up
4629 @section Applying a function on subexpressions
4630 @cindex tree traversal
4631 @cindex @code{map()}
4633 Sometimes you may want to perform an operation on specific parts of an
4634 expression while leaving the general structure of it intact. An example
4635 of this would be a matrix trace operation: the trace of a sum is the sum
4636 of the traces of the individual terms. That is, the trace should @dfn{map}
4637 on the sum, by applying itself to each of the sum's operands. It is possible
4638 to do this manually which usually results in code like this:
4643 if (is_a<matrix>(e))
4644 return ex_to<matrix>(e).trace();
4645 else if (is_a<add>(e)) @{
4647 for (size_t i=0; i<e.nops(); i++)
4648 sum += calc_trace(e.op(i));
4650 @} else if (is_a<mul>)(e)) @{
4658 This is, however, slightly inefficient (if the sum is very large it can take
4659 a long time to add the terms one-by-one), and its applicability is limited to
4660 a rather small class of expressions. If @code{calc_trace()} is called with
4661 a relation or a list as its argument, you will probably want the trace to
4662 be taken on both sides of the relation or of all elements of the list.
4664 GiNaC offers the @code{map()} method to aid in the implementation of such
4668 ex ex::map(map_function & f) const;
4669 ex ex::map(ex (*f)(const ex & e)) const;
4672 In the first (preferred) form, @code{map()} takes a function object that
4673 is subclassed from the @code{map_function} class. In the second form, it
4674 takes a pointer to a function that accepts and returns an expression.
4675 @code{map()} constructs a new expression of the same type, applying the
4676 specified function on all subexpressions (in the sense of @code{op()}),
4679 The use of a function object makes it possible to supply more arguments to
4680 the function that is being mapped, or to keep local state information.
4681 The @code{map_function} class declares a virtual function call operator
4682 that you can overload. Here is a sample implementation of @code{calc_trace()}
4683 that uses @code{map()} in a recursive fashion:
4686 struct calc_trace : public map_function @{
4687 ex operator()(const ex &e)
4689 if (is_a<matrix>(e))
4690 return ex_to<matrix>(e).trace();
4691 else if (is_a<mul>(e)) @{
4694 return e.map(*this);
4699 This function object could then be used like this:
4703 ex M = ... // expression with matrices
4704 calc_trace do_trace;
4705 ex tr = do_trace(M);
4709 Here is another example for you to meditate over. It removes quadratic
4710 terms in a variable from an expanded polynomial:
4713 struct map_rem_quad : public map_function @{
4715 map_rem_quad(const ex & var_) : var(var_) @{@}
4717 ex operator()(const ex & e)
4719 if (is_a<add>(e) || is_a<mul>(e))
4720 return e.map(*this);
4721 else if (is_a<power>(e) &&
4722 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4732 symbol x("x"), y("y");
4735 for (int i=0; i<8; i++)
4736 e += pow(x, i) * pow(y, 8-i) * (i+1);
4738 // -> 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
4740 map_rem_quad rem_quad(x);
4741 cout << rem_quad(e) << endl;
4742 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4746 @command{ginsh} offers a slightly different implementation of @code{map()}
4747 that allows applying algebraic functions to operands. The second argument
4748 to @code{map()} is an expression containing the wildcard @samp{$0} which
4749 acts as the placeholder for the operands:
4754 > map(a+2*b,sin($0));
4756 > map(@{a,b,c@},$0^2+$0);
4757 @{a^2+a,b^2+b,c^2+c@}
4760 Note that it is only possible to use algebraic functions in the second
4761 argument. You can not use functions like @samp{diff()}, @samp{op()},
4762 @samp{subs()} etc. because these are evaluated immediately:
4765 > map(@{a,b,c@},diff($0,a));
4767 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4768 to "map(@{a,b,c@},0)".
4772 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4773 @c node-name, next, previous, up
4774 @section Visitors and tree traversal
4775 @cindex tree traversal
4776 @cindex @code{visitor} (class)
4777 @cindex @code{accept()}
4778 @cindex @code{visit()}
4779 @cindex @code{traverse()}
4780 @cindex @code{traverse_preorder()}
4781 @cindex @code{traverse_postorder()}
4783 Suppose that you need a function that returns a list of all indices appearing
4784 in an arbitrary expression. The indices can have any dimension, and for
4785 indices with variance you always want the covariant version returned.
4787 You can't use @code{get_free_indices()} because you also want to include
4788 dummy indices in the list, and you can't use @code{find()} as it needs
4789 specific index dimensions (and it would require two passes: one for indices
4790 with variance, one for plain ones).
4792 The obvious solution to this problem is a tree traversal with a type switch,
4793 such as the following:
4796 void gather_indices_helper(const ex & e, lst & l)
4798 if (is_a<varidx>(e)) @{
4799 const varidx & vi = ex_to<varidx>(e);
4800 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4801 @} else if (is_a<idx>(e)) @{
4804 size_t n = e.nops();
4805 for (size_t i = 0; i < n; ++i)
4806 gather_indices_helper(e.op(i), l);
4810 lst gather_indices(const ex & e)
4813 gather_indices_helper(e, l);
4820 This works fine but fans of object-oriented programming will feel
4821 uncomfortable with the type switch. One reason is that there is a possibility
4822 for subtle bugs regarding derived classes. If we had, for example, written
4825 if (is_a<idx>(e)) @{
4827 @} else if (is_a<varidx>(e)) @{
4831 in @code{gather_indices_helper}, the code wouldn't have worked because the
4832 first line "absorbs" all classes derived from @code{idx}, including
4833 @code{varidx}, so the special case for @code{varidx} would never have been
4836 Also, for a large number of classes, a type switch like the above can get
4837 unwieldy and inefficient (it's a linear search, after all).
4838 @code{gather_indices_helper} only checks for two classes, but if you had to
4839 write a function that required a different implementation for nearly
4840 every GiNaC class, the result would be very hard to maintain and extend.
4842 The cleanest approach to the problem would be to add a new virtual function
4843 to GiNaC's class hierarchy. In our example, there would be specializations
4844 for @code{idx} and @code{varidx} while the default implementation in
4845 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4846 impossible to add virtual member functions to existing classes without
4847 changing their source and recompiling everything. GiNaC comes with source,
4848 so you could actually do this, but for a small algorithm like the one
4849 presented this would be impractical.
4851 One solution to this dilemma is the @dfn{Visitor} design pattern,
4852 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4853 variation, described in detail in
4854 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4855 virtual functions to the class hierarchy to implement operations, GiNaC
4856 provides a single "bouncing" method @code{accept()} that takes an instance
4857 of a special @code{visitor} class and redirects execution to the one
4858 @code{visit()} virtual function of the visitor that matches the type of
4859 object that @code{accept()} was being invoked on.
4861 Visitors in GiNaC must derive from the global @code{visitor} class as well
4862 as from the class @code{T::visitor} of each class @code{T} they want to
4863 visit, and implement the member functions @code{void visit(const T &)} for
4869 void ex::accept(visitor & v) const;
4872 will then dispatch to the correct @code{visit()} member function of the
4873 specified visitor @code{v} for the type of GiNaC object at the root of the
4874 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4876 Here is an example of a visitor:
4880 : public visitor, // this is required
4881 public add::visitor, // visit add objects
4882 public numeric::visitor, // visit numeric objects
4883 public basic::visitor // visit basic objects
4885 void visit(const add & x)
4886 @{ cout << "called with an add object" << endl; @}
4888 void visit(const numeric & x)
4889 @{ cout << "called with a numeric object" << endl; @}
4891 void visit(const basic & x)
4892 @{ cout << "called with a basic object" << endl; @}
4896 which can be used as follows:
4907 // prints "called with a numeric object"
4909 // prints "called with an add object"
4911 // prints "called with a basic object"
4915 The @code{visit(const basic &)} method gets called for all objects that are
4916 not @code{numeric} or @code{add} and acts as an (optional) default.
4918 From a conceptual point of view, the @code{visit()} methods of the visitor
4919 behave like a newly added virtual function of the visited hierarchy.
4920 In addition, visitors can store state in member variables, and they can
4921 be extended by deriving a new visitor from an existing one, thus building
4922 hierarchies of visitors.
4924 We can now rewrite our index example from above with a visitor:
4927 class gather_indices_visitor
4928 : public visitor, public idx::visitor, public varidx::visitor
4932 void visit(const idx & i)
4937 void visit(const varidx & vi)
4939 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4943 const lst & get_result() // utility function
4952 What's missing is the tree traversal. We could implement it in
4953 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4956 void ex::traverse_preorder(visitor & v) const;
4957 void ex::traverse_postorder(visitor & v) const;
4958 void ex::traverse(visitor & v) const;
4961 @code{traverse_preorder()} visits a node @emph{before} visiting its
4962 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4963 visiting its subexpressions. @code{traverse()} is a synonym for
4964 @code{traverse_preorder()}.
4966 Here is a new implementation of @code{gather_indices()} that uses the visitor
4967 and @code{traverse()}:
4970 lst gather_indices(const ex & e)
4972 gather_indices_visitor v;
4974 return v.get_result();
4978 Alternatively, you could use pre- or postorder iterators for the tree
4982 lst gather_indices(const ex & e)
4984 gather_indices_visitor v;
4985 for (const_preorder_iterator i = e.preorder_begin();
4986 i != e.preorder_end(); ++i) @{
4989 return v.get_result();
4994 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4995 @c node-name, next, previous, up
4996 @section Polynomial arithmetic
4998 @subsection Testing whether an expression is a polynomial
4999 @cindex @code{is_polynomial()}
5001 Testing whether an expression is a polynomial in one or more variables
5002 can be done with the method
5004 bool ex::is_polynomial(const ex & vars) const;
5006 In the case of more than
5007 one variable, the variables are given as a list.
5010 (x*y*sin(y)).is_polynomial(x) // Returns true.
5011 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
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{step(x)}
5735 @cindex @code{step()}
5736 @item @code{csgn(x)}
5738 @cindex @code{conjugate()}
5739 @item @code{conjugate(x)}
5740 @tab complex conjugation
5741 @cindex @code{real_part()}
5742 @item @code{real_part(x)}
5744 @cindex @code{imag_part()}
5745 @item @code{imag_part(x)}
5747 @item @code{sqrt(x)}
5748 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5749 @cindex @code{sqrt()}
5752 @cindex @code{sin()}
5755 @cindex @code{cos()}
5758 @cindex @code{tan()}
5759 @item @code{asin(x)}
5761 @cindex @code{asin()}
5762 @item @code{acos(x)}
5764 @cindex @code{acos()}
5765 @item @code{atan(x)}
5766 @tab inverse tangent
5767 @cindex @code{atan()}
5768 @item @code{atan2(y, x)}
5769 @tab inverse tangent with two arguments
5770 @item @code{sinh(x)}
5771 @tab hyperbolic sine
5772 @cindex @code{sinh()}
5773 @item @code{cosh(x)}
5774 @tab hyperbolic cosine
5775 @cindex @code{cosh()}
5776 @item @code{tanh(x)}
5777 @tab hyperbolic tangent
5778 @cindex @code{tanh()}
5779 @item @code{asinh(x)}
5780 @tab inverse hyperbolic sine
5781 @cindex @code{asinh()}
5782 @item @code{acosh(x)}
5783 @tab inverse hyperbolic cosine
5784 @cindex @code{acosh()}
5785 @item @code{atanh(x)}
5786 @tab inverse hyperbolic tangent
5787 @cindex @code{atanh()}
5789 @tab exponential function
5790 @cindex @code{exp()}
5792 @tab natural logarithm
5793 @cindex @code{log()}
5796 @cindex @code{Li2()}
5797 @item @code{Li(m, x)}
5798 @tab classical polylogarithm as well as multiple polylogarithm
5800 @item @code{G(a, y)}
5801 @tab multiple polylogarithm
5803 @item @code{G(a, s, y)}
5804 @tab multiple polylogarithm with explicit signs for the imaginary parts
5806 @item @code{S(n, p, x)}
5807 @tab Nielsen's generalized polylogarithm
5809 @item @code{H(m, x)}
5810 @tab harmonic polylogarithm
5812 @item @code{zeta(m)}
5813 @tab Riemann's zeta function as well as multiple zeta value
5814 @cindex @code{zeta()}
5815 @item @code{zeta(m, s)}
5816 @tab alternating Euler sum
5817 @cindex @code{zeta()}
5818 @item @code{zetaderiv(n, x)}
5819 @tab derivatives of Riemann's zeta function
5820 @item @code{tgamma(x)}
5822 @cindex @code{tgamma()}
5823 @cindex gamma function
5824 @item @code{lgamma(x)}
5825 @tab logarithm of gamma function
5826 @cindex @code{lgamma()}
5827 @item @code{beta(x, y)}
5828 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5829 @cindex @code{beta()}
5831 @tab psi (digamma) function
5832 @cindex @code{psi()}
5833 @item @code{psi(n, x)}
5834 @tab derivatives of psi function (polygamma functions)
5835 @item @code{factorial(n)}
5836 @tab factorial function @math{n!}
5837 @cindex @code{factorial()}
5838 @item @code{binomial(n, k)}
5839 @tab binomial coefficients
5840 @cindex @code{binomial()}
5841 @item @code{Order(x)}
5842 @tab order term function in truncated power series
5843 @cindex @code{Order()}
5848 For functions that have a branch cut in the complex plane GiNaC follows
5849 the conventions for C++ as defined in the ANSI standard as far as
5850 possible. In particular: the natural logarithm (@code{log}) and the
5851 square root (@code{sqrt}) both have their branch cuts running along the
5852 negative real axis where the points on the axis itself belong to the
5853 upper part (i.e. continuous with quadrant II). The inverse
5854 trigonometric and hyperbolic functions are not defined for complex
5855 arguments by the C++ standard, however. In GiNaC we follow the
5856 conventions used by CLN, which in turn follow the carefully designed
5857 definitions in the Common Lisp standard. It should be noted that this
5858 convention is identical to the one used by the C99 standard and by most
5859 serious CAS. It is to be expected that future revisions of the C++
5860 standard incorporate these functions in the complex domain in a manner
5861 compatible with C99.
5863 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5864 @c node-name, next, previous, up
5865 @subsection Multiple polylogarithms
5867 @cindex polylogarithm
5868 @cindex Nielsen's generalized polylogarithm
5869 @cindex harmonic polylogarithm
5870 @cindex multiple zeta value
5871 @cindex alternating Euler sum
5872 @cindex multiple polylogarithm
5874 The multiple polylogarithm is the most generic member of a family of functions,
5875 to which others like the harmonic polylogarithm, Nielsen's generalized
5876 polylogarithm and the multiple zeta value belong.
5877 Everyone of these functions can also be written as a multiple polylogarithm with specific
5878 parameters. This whole family of functions is therefore often referred to simply as
5879 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5880 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5881 @code{Li} and @code{G} in principle represent the same function, the different
5882 notations are more natural to the series representation or the integral
5883 representation, respectively.
5885 To facilitate the discussion of these functions we distinguish between indices and
5886 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5887 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5889 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5890 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5891 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5892 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5893 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5894 @code{s} is not given, the signs default to +1.
5895 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5896 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5897 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5898 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5899 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5901 The functions print in LaTeX format as
5903 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5909 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5912 $\zeta(m_1,m_2,\ldots,m_k)$.
5914 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5915 are printed with a line above, e.g.
5917 $\zeta(5,\overline{2})$.
5919 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5921 Definitions and analytical as well as numerical properties of multiple polylogarithms
5922 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5923 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5924 except for a few differences which will be explicitly stated in the following.
5926 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5927 that the indices and arguments are understood to be in the same order as in which they appear in
5928 the series representation. This means
5930 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5933 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5936 $\zeta(1,2)$ evaluates to infinity.
5938 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5941 The functions only evaluate if the indices are integers greater than zero, except for the indices
5942 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5943 will be interpreted as the sequence of signs for the corresponding indices
5944 @code{m} or the sign of the imaginary part for the
5945 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5946 @code{zeta(lst(3,4), lst(-1,1))} means
5948 $\zeta(\overline{3},4)$
5951 @code{G(lst(a,b), lst(-1,1), c)} means
5953 $G(a-0\epsilon,b+0\epsilon;c)$.
5955 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5956 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5957 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
5958 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5959 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5960 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5961 evaluates also for negative integers and positive even integers. For example:
5964 > Li(@{3,1@},@{x,1@});
5967 -zeta(@{3,2@},@{-1,-1@})
5972 It is easy to tell for a given function into which other function it can be rewritten, may
5973 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5974 with negative indices or trailing zeros (the example above gives a hint). Signs can
5975 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5976 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5977 @code{Li} (@code{eval()} already cares for the possible downgrade):
5980 > convert_H_to_Li(@{0,-2,-1,3@},x);
5981 Li(@{3,1,3@},@{-x,1,-1@})
5982 > convert_H_to_Li(@{2,-1,0@},x);
5983 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5986 Every function can be numerically evaluated for
5987 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5988 global variable @code{Digits}:
5993 > evalf(zeta(@{3,1,3,1@}));
5994 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5997 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5998 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6000 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6005 In long expressions this helps a lot with debugging, because you can easily spot
6006 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6007 cancellations of divergencies happen.
6009 Useful publications:
6011 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6012 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6014 @cite{Harmonic Polylogarithms},
6015 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6017 @cite{Special Values of Multiple Polylogarithms},
6018 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6020 @cite{Numerical Evaluation of Multiple Polylogarithms},
6021 J.Vollinga, S.Weinzierl, hep-ph/0410259
6023 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6024 @c node-name, next, previous, up
6025 @section Complex expressions
6027 @cindex @code{conjugate()}
6029 For dealing with complex expressions there are the methods
6037 that return respectively the complex conjugate, the real part and the
6038 imaginary part of an expression. Complex conjugation works as expected
6039 for all built-in functinos and objects. Taking real and imaginary
6040 parts has not yet been implemented for all built-in functions. In cases where
6041 it is not known how to conjugate or take a real/imaginary part one
6042 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6043 is returned. For instance, in case of a complex symbol @code{x}
6044 (symbols are complex by default), one could not simplify
6045 @code{conjugate(x)}. In the case of strings of gamma matrices,
6046 the @code{conjugate} method takes the Dirac conjugate.
6051 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6055 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6056 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6057 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6058 // -> -gamma5*gamma~b*gamma~a
6062 If you declare your own GiNaC functions, then they will conjugate themselves
6063 by conjugating their arguments. This is the default strategy. If you want to
6064 change this behavior, you have to supply a specialized conjugation method
6065 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6066 for @code{abs} as an example). Also, specialized methods can be provided
6067 to take real and imaginary parts of user-defined functions.
6069 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6070 @c node-name, next, previous, up
6071 @section Solving linear systems of equations
6072 @cindex @code{lsolve()}
6074 The function @code{lsolve()} provides a convenient wrapper around some
6075 matrix operations that comes in handy when a system of linear equations
6079 ex lsolve(const ex & eqns, const ex & symbols,
6080 unsigned options = solve_algo::automatic);
6083 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6084 @code{relational}) while @code{symbols} is a @code{lst} of
6085 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6088 It returns the @code{lst} of solutions as an expression. As an example,
6089 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6093 symbol a("a"), b("b"), x("x"), y("y");
6095 eqns = a*x+b*y==3, x-y==b;
6097 cout << lsolve(eqns, vars) << endl;
6098 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6101 When the linear equations @code{eqns} are underdetermined, the solution
6102 will contain one or more tautological entries like @code{x==x},
6103 depending on the rank of the system. When they are overdetermined, the
6104 solution will be an empty @code{lst}. Note the third optional parameter
6105 to @code{lsolve()}: it accepts the same parameters as
6106 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6110 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6111 @c node-name, next, previous, up
6112 @section Input and output of expressions
6115 @subsection Expression output
6117 @cindex output of expressions
6119 Expressions can simply be written to any stream:
6124 ex e = 4.5*I+pow(x,2)*3/2;
6125 cout << e << endl; // prints '4.5*I+3/2*x^2'
6129 The default output format is identical to the @command{ginsh} input syntax and
6130 to that used by most computer algebra systems, but not directly pastable
6131 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6132 is printed as @samp{x^2}).
6134 It is possible to print expressions in a number of different formats with
6135 a set of stream manipulators;
6138 std::ostream & dflt(std::ostream & os);
6139 std::ostream & latex(std::ostream & os);
6140 std::ostream & tree(std::ostream & os);
6141 std::ostream & csrc(std::ostream & os);
6142 std::ostream & csrc_float(std::ostream & os);
6143 std::ostream & csrc_double(std::ostream & os);
6144 std::ostream & csrc_cl_N(std::ostream & os);
6145 std::ostream & index_dimensions(std::ostream & os);
6146 std::ostream & no_index_dimensions(std::ostream & os);
6149 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6150 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6151 @code{print_csrc()} functions, respectively.
6154 All manipulators affect the stream state permanently. To reset the output
6155 format to the default, use the @code{dflt} manipulator:
6159 cout << latex; // all output to cout will be in LaTeX format from
6161 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6162 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6163 cout << dflt; // revert to default output format
6164 cout << e << endl; // prints '4.5*I+3/2*x^2'
6168 If you don't want to affect the format of the stream you're working with,
6169 you can output to a temporary @code{ostringstream} like this:
6174 s << latex << e; // format of cout remains unchanged
6175 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6180 @cindex @code{csrc_float}
6181 @cindex @code{csrc_double}
6182 @cindex @code{csrc_cl_N}
6183 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6184 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6185 format that can be directly used in a C or C++ program. The three possible
6186 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6187 classes provided by the CLN library):
6191 cout << "f = " << csrc_float << e << ";\n";
6192 cout << "d = " << csrc_double << e << ";\n";
6193 cout << "n = " << csrc_cl_N << e << ";\n";
6197 The above example will produce (note the @code{x^2} being converted to
6201 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6202 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6203 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6207 The @code{tree} manipulator allows dumping the internal structure of an
6208 expression for debugging purposes:
6219 add, hash=0x0, flags=0x3, nops=2
6220 power, hash=0x0, flags=0x3, nops=2
6221 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6222 2 (numeric), hash=0x6526b0fa, flags=0xf
6223 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6226 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6230 @cindex @code{latex}
6231 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6232 It is rather similar to the default format but provides some braces needed
6233 by LaTeX for delimiting boxes and also converts some common objects to
6234 conventional LaTeX names. It is possible to give symbols a special name for
6235 LaTeX output by supplying it as a second argument to the @code{symbol}
6238 For example, the code snippet
6242 symbol x("x", "\\circ");
6243 ex e = lgamma(x).series(x==0,3);
6244 cout << latex << e << endl;
6251 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6252 +\mathcal@{O@}(\circ^@{3@})
6255 @cindex @code{index_dimensions}
6256 @cindex @code{no_index_dimensions}
6257 Index dimensions are normally hidden in the output. To make them visible, use
6258 the @code{index_dimensions} manipulator. The dimensions will be written in
6259 square brackets behind each index value in the default and LaTeX output
6264 symbol x("x"), y("y");
6265 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6266 ex e = indexed(x, mu) * indexed(y, nu);
6269 // prints 'x~mu*y~nu'
6270 cout << index_dimensions << e << endl;
6271 // prints 'x~mu[4]*y~nu[4]'
6272 cout << no_index_dimensions << e << endl;
6273 // prints 'x~mu*y~nu'
6278 @cindex Tree traversal
6279 If you need any fancy special output format, e.g. for interfacing GiNaC
6280 with other algebra systems or for producing code for different
6281 programming languages, you can always traverse the expression tree yourself:
6284 static void my_print(const ex & e)
6286 if (is_a<function>(e))
6287 cout << ex_to<function>(e).get_name();
6289 cout << ex_to<basic>(e).class_name();
6291 size_t n = e.nops();
6293 for (size_t i=0; i<n; i++) @{
6305 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6313 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6314 symbol(y))),numeric(-2)))
6317 If you need an output format that makes it possible to accurately
6318 reconstruct an expression by feeding the output to a suitable parser or
6319 object factory, you should consider storing the expression in an
6320 @code{archive} object and reading the object properties from there.
6321 See the section on archiving for more information.
6324 @subsection Expression input
6325 @cindex input of expressions
6327 GiNaC provides no way to directly read an expression from a stream because
6328 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6329 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6330 @code{y} you defined in your program and there is no way to specify the
6331 desired symbols to the @code{>>} stream input operator.
6333 Instead, GiNaC lets you construct an expression from a string, specifying the
6334 list of symbols to be used:
6338 symbol x("x"), y("y");
6339 ex e("2*x+sin(y)", lst(x, y));
6343 The input syntax is the same as that used by @command{ginsh} and the stream
6344 output operator @code{<<}. The symbols in the string are matched by name to
6345 the symbols in the list and if GiNaC encounters a symbol not specified in
6346 the list it will throw an exception.
6348 With this constructor, it's also easy to implement interactive GiNaC programs:
6353 #include <stdexcept>
6354 #include <ginac/ginac.h>
6355 using namespace std;
6356 using namespace GiNaC;
6363 cout << "Enter an expression containing 'x': ";
6368 cout << "The derivative of " << e << " with respect to x is ";
6369 cout << e.diff(x) << ".\n";
6370 @} catch (exception &p) @{
6371 cerr << p.what() << endl;
6377 @subsection Archiving
6378 @cindex @code{archive} (class)
6381 GiNaC allows creating @dfn{archives} of expressions which can be stored
6382 to or retrieved from files. To create an archive, you declare an object
6383 of class @code{archive} and archive expressions in it, giving each
6384 expression a unique name:
6388 using namespace std;
6389 #include <ginac/ginac.h>
6390 using namespace GiNaC;
6394 symbol x("x"), y("y"), z("z");
6396 ex foo = sin(x + 2*y) + 3*z + 41;
6400 a.archive_ex(foo, "foo");
6401 a.archive_ex(bar, "the second one");
6405 The archive can then be written to a file:
6409 ofstream out("foobar.gar");
6415 The file @file{foobar.gar} contains all information that is needed to
6416 reconstruct the expressions @code{foo} and @code{bar}.
6418 @cindex @command{viewgar}
6419 The tool @command{viewgar} that comes with GiNaC can be used to view
6420 the contents of GiNaC archive files:
6423 $ viewgar foobar.gar
6424 foo = 41+sin(x+2*y)+3*z
6425 the second one = 42+sin(x+2*y)+3*z
6428 The point of writing archive files is of course that they can later be
6434 ifstream in("foobar.gar");
6439 And the stored expressions can be retrieved by their name:
6446 ex ex1 = a2.unarchive_ex(syms, "foo");
6447 ex ex2 = a2.unarchive_ex(syms, "the second one");
6449 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6450 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6451 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6455 Note that you have to supply a list of the symbols which are to be inserted
6456 in the expressions. Symbols in archives are stored by their name only and
6457 if you don't specify which symbols you have, unarchiving the expression will
6458 create new symbols with that name. E.g. if you hadn't included @code{x} in
6459 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6460 have had no effect because the @code{x} in @code{ex1} would have been a
6461 different symbol than the @code{x} which was defined at the beginning of
6462 the program, although both would appear as @samp{x} when printed.
6464 You can also use the information stored in an @code{archive} object to
6465 output expressions in a format suitable for exact reconstruction. The
6466 @code{archive} and @code{archive_node} classes have a couple of member
6467 functions that let you access the stored properties:
6470 static void my_print2(const archive_node & n)
6473 n.find_string("class", class_name);
6474 cout << class_name << "(";
6476 archive_node::propinfovector p;
6477 n.get_properties(p);
6479 size_t num = p.size();
6480 for (size_t i=0; i<num; i++) @{
6481 const string &name = p[i].name;
6482 if (name == "class")
6484 cout << name << "=";
6486 unsigned count = p[i].count;
6490 for (unsigned j=0; j<count; j++) @{
6491 switch (p[i].type) @{
6492 case archive_node::PTYPE_BOOL: @{
6494 n.find_bool(name, x, j);
6495 cout << (x ? "true" : "false");
6498 case archive_node::PTYPE_UNSIGNED: @{
6500 n.find_unsigned(name, x, j);
6504 case archive_node::PTYPE_STRING: @{
6506 n.find_string(name, x, j);
6507 cout << '\"' << x << '\"';
6510 case archive_node::PTYPE_NODE: @{
6511 const archive_node &x = n.find_ex_node(name, j);
6533 ex e = pow(2, x) - y;
6535 my_print2(ar.get_top_node(0)); cout << endl;
6543 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6544 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6545 overall_coeff=numeric(number="0"))
6548 Be warned, however, that the set of properties and their meaning for each
6549 class may change between GiNaC versions.
6552 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6553 @c node-name, next, previous, up
6554 @chapter Extending GiNaC
6556 By reading so far you should have gotten a fairly good understanding of
6557 GiNaC's design patterns. From here on you should start reading the
6558 sources. All we can do now is issue some recommendations how to tackle
6559 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6560 develop some useful extension please don't hesitate to contact the GiNaC
6561 authors---they will happily incorporate them into future versions.
6564 * What does not belong into GiNaC:: What to avoid.
6565 * Symbolic functions:: Implementing symbolic functions.
6566 * Printing:: Adding new output formats.
6567 * Structures:: Defining new algebraic classes (the easy way).
6568 * Adding classes:: Defining new algebraic classes (the hard way).
6572 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6573 @c node-name, next, previous, up
6574 @section What doesn't belong into GiNaC
6576 @cindex @command{ginsh}
6577 First of all, GiNaC's name must be read literally. It is designed to be
6578 a library for use within C++. The tiny @command{ginsh} accompanying
6579 GiNaC makes this even more clear: it doesn't even attempt to provide a
6580 language. There are no loops or conditional expressions in
6581 @command{ginsh}, it is merely a window into the library for the
6582 programmer to test stuff (or to show off). Still, the design of a
6583 complete CAS with a language of its own, graphical capabilities and all
6584 this on top of GiNaC is possible and is without doubt a nice project for
6587 There are many built-in functions in GiNaC that do not know how to
6588 evaluate themselves numerically to a precision declared at runtime
6589 (using @code{Digits}). Some may be evaluated at certain points, but not
6590 generally. This ought to be fixed. However, doing numerical
6591 computations with GiNaC's quite abstract classes is doomed to be
6592 inefficient. For this purpose, the underlying foundation classes
6593 provided by CLN are much better suited.
6596 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6597 @c node-name, next, previous, up
6598 @section Symbolic functions
6600 The easiest and most instructive way to start extending GiNaC is probably to
6601 create your own symbolic functions. These are implemented with the help of
6602 two preprocessor macros:
6604 @cindex @code{DECLARE_FUNCTION}
6605 @cindex @code{REGISTER_FUNCTION}
6607 DECLARE_FUNCTION_<n>P(<name>)
6608 REGISTER_FUNCTION(<name>, <options>)
6611 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6612 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6613 parameters of type @code{ex} and returns a newly constructed GiNaC
6614 @code{function} object that represents your function.
6616 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6617 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6618 set of options that associate the symbolic function with C++ functions you
6619 provide to implement the various methods such as evaluation, derivative,
6620 series expansion etc. They also describe additional attributes the function
6621 might have, such as symmetry and commutation properties, and a name for
6622 LaTeX output. Multiple options are separated by the member access operator
6623 @samp{.} and can be given in an arbitrary order.
6625 (By the way: in case you are worrying about all the macros above we can
6626 assure you that functions are GiNaC's most macro-intense classes. We have
6627 done our best to avoid macros where we can.)
6629 @subsection A minimal example
6631 Here is an example for the implementation of a function with two arguments
6632 that is not further evaluated:
6635 DECLARE_FUNCTION_2P(myfcn)
6637 REGISTER_FUNCTION(myfcn, dummy())
6640 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6641 in algebraic expressions:
6647 ex e = 2*myfcn(42, 1+3*x) - x;
6649 // prints '2*myfcn(42,1+3*x)-x'
6654 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6655 "no options". A function with no options specified merely acts as a kind of
6656 container for its arguments. It is a pure "dummy" function with no associated
6657 logic (which is, however, sometimes perfectly sufficient).
6659 Let's now have a look at the implementation of GiNaC's cosine function for an
6660 example of how to make an "intelligent" function.
6662 @subsection The cosine function
6664 The GiNaC header file @file{inifcns.h} contains the line
6667 DECLARE_FUNCTION_1P(cos)
6670 which declares to all programs using GiNaC that there is a function @samp{cos}
6671 that takes one @code{ex} as an argument. This is all they need to know to use
6672 this function in expressions.
6674 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6675 is its @code{REGISTER_FUNCTION} line:
6678 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6679 evalf_func(cos_evalf).
6680 derivative_func(cos_deriv).
6681 latex_name("\\cos"));
6684 There are four options defined for the cosine function. One of them
6685 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6686 other three indicate the C++ functions in which the "brains" of the cosine
6687 function are defined.
6689 @cindex @code{hold()}
6691 The @code{eval_func()} option specifies the C++ function that implements
6692 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6693 the same number of arguments as the associated symbolic function (one in this
6694 case) and returns the (possibly transformed or in some way simplified)
6695 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6696 of the automatic evaluation process). If no (further) evaluation is to take
6697 place, the @code{eval_func()} function must return the original function
6698 with @code{.hold()}, to avoid a potential infinite recursion. If your
6699 symbolic functions produce a segmentation fault or stack overflow when
6700 using them in expressions, you are probably missing a @code{.hold()}
6703 The @code{eval_func()} function for the cosine looks something like this
6704 (actually, it doesn't look like this at all, but it should give you an idea
6708 static ex cos_eval(const ex & x)
6710 if ("x is a multiple of 2*Pi")
6712 else if ("x is a multiple of Pi")
6714 else if ("x is a multiple of Pi/2")
6718 else if ("x has the form 'acos(y)'")
6720 else if ("x has the form 'asin(y)'")
6725 return cos(x).hold();
6729 This function is called every time the cosine is used in a symbolic expression:
6735 // this calls cos_eval(Pi), and inserts its return value into
6736 // the actual expression
6743 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6744 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6745 symbolic transformation can be done, the unmodified function is returned
6746 with @code{.hold()}.
6748 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6749 The user has to call @code{evalf()} for that. This is implemented in a
6753 static ex cos_evalf(const ex & x)
6755 if (is_a<numeric>(x))
6756 return cos(ex_to<numeric>(x));
6758 return cos(x).hold();
6762 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6763 in this case the @code{cos()} function for @code{numeric} objects, which in
6764 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6765 isn't really needed here, but reminds us that the corresponding @code{eval()}
6766 function would require it in this place.
6768 Differentiation will surely turn up and so we need to tell @code{cos}
6769 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6770 instance, are then handled automatically by @code{basic::diff} and
6774 static ex cos_deriv(const ex & x, unsigned diff_param)
6780 @cindex product rule
6781 The second parameter is obligatory but uninteresting at this point. It
6782 specifies which parameter to differentiate in a partial derivative in
6783 case the function has more than one parameter, and its main application
6784 is for correct handling of the chain rule.
6786 An implementation of the series expansion is not needed for @code{cos()} as
6787 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6788 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6789 the other hand, does have poles and may need to do Laurent expansion:
6792 static ex tan_series(const ex & x, const relational & rel,
6793 int order, unsigned options)
6795 // Find the actual expansion point
6796 const ex x_pt = x.subs(rel);
6798 if ("x_pt is not an odd multiple of Pi/2")
6799 throw do_taylor(); // tell function::series() to do Taylor expansion
6801 // On a pole, expand sin()/cos()
6802 return (sin(x)/cos(x)).series(rel, order+2, options);
6806 The @code{series()} implementation of a function @emph{must} return a
6807 @code{pseries} object, otherwise your code will crash.
6809 @subsection Function options
6811 GiNaC functions understand several more options which are always
6812 specified as @code{.option(params)}. None of them are required, but you
6813 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6814 is a do-nothing option called @code{dummy()} which you can use to define
6815 functions without any special options.
6818 eval_func(<C++ function>)
6819 evalf_func(<C++ function>)
6820 derivative_func(<C++ function>)
6821 series_func(<C++ function>)
6822 conjugate_func(<C++ function>)
6825 These specify the C++ functions that implement symbolic evaluation,
6826 numeric evaluation, partial derivatives, and series expansion, respectively.
6827 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6828 @code{diff()} and @code{series()}.
6830 The @code{eval_func()} function needs to use @code{.hold()} if no further
6831 automatic evaluation is desired or possible.
6833 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6834 expansion, which is correct if there are no poles involved. If the function
6835 has poles in the complex plane, the @code{series_func()} needs to check
6836 whether the expansion point is on a pole and fall back to Taylor expansion
6837 if it isn't. Otherwise, the pole usually needs to be regularized by some
6838 suitable transformation.
6841 latex_name(const string & n)
6844 specifies the LaTeX code that represents the name of the function in LaTeX
6845 output. The default is to put the function name in an @code{\mbox@{@}}.
6848 do_not_evalf_params()
6851 This tells @code{evalf()} to not recursively evaluate the parameters of the
6852 function before calling the @code{evalf_func()}.
6855 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6858 This allows you to explicitly specify the commutation properties of the
6859 function (@xref{Non-commutative objects}, for an explanation of
6860 (non)commutativity in GiNaC). For example, you can use
6861 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6862 GiNaC treat your function like a matrix. By default, functions inherit the
6863 commutation properties of their first argument.
6866 set_symmetry(const symmetry & s)
6869 specifies the symmetry properties of the function with respect to its
6870 arguments. @xref{Indexed objects}, for an explanation of symmetry
6871 specifications. GiNaC will automatically rearrange the arguments of
6872 symmetric functions into a canonical order.
6874 Sometimes you may want to have finer control over how functions are
6875 displayed in the output. For example, the @code{abs()} function prints
6876 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6877 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6881 print_func<C>(<C++ function>)
6884 option which is explained in the next section.
6886 @subsection Functions with a variable number of arguments
6888 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6889 functions with a fixed number of arguments. Sometimes, though, you may need
6890 to have a function that accepts a variable number of expressions. One way to
6891 accomplish this is to pass variable-length lists as arguments. The
6892 @code{Li()} function uses this method for multiple polylogarithms.
6894 It is also possible to define functions that accept a different number of
6895 parameters under the same function name, such as the @code{psi()} function
6896 which can be called either as @code{psi(z)} (the digamma function) or as
6897 @code{psi(n, z)} (polygamma functions). These are actually two different
6898 functions in GiNaC that, however, have the same name. Defining such
6899 functions is not possible with the macros but requires manually fiddling
6900 with GiNaC internals. If you are interested, please consult the GiNaC source
6901 code for the @code{psi()} function (@file{inifcns.h} and
6902 @file{inifcns_gamma.cpp}).
6905 @node Printing, Structures, Symbolic functions, Extending GiNaC
6906 @c node-name, next, previous, up
6907 @section GiNaC's expression output system
6909 GiNaC allows the output of expressions in a variety of different formats
6910 (@pxref{Input/output}). This section will explain how expression output
6911 is implemented internally, and how to define your own output formats or
6912 change the output format of built-in algebraic objects. You will also want
6913 to read this section if you plan to write your own algebraic classes or
6916 @cindex @code{print_context} (class)
6917 @cindex @code{print_dflt} (class)
6918 @cindex @code{print_latex} (class)
6919 @cindex @code{print_tree} (class)
6920 @cindex @code{print_csrc} (class)
6921 All the different output formats are represented by a hierarchy of classes
6922 rooted in the @code{print_context} class, defined in the @file{print.h}
6927 the default output format
6929 output in LaTeX mathematical mode
6931 a dump of the internal expression structure (for debugging)
6933 the base class for C source output
6934 @item print_csrc_float
6935 C source output using the @code{float} type
6936 @item print_csrc_double
6937 C source output using the @code{double} type
6938 @item print_csrc_cl_N
6939 C source output using CLN types
6942 The @code{print_context} base class provides two public data members:
6954 @code{s} is a reference to the stream to output to, while @code{options}
6955 holds flags and modifiers. Currently, there is only one flag defined:
6956 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6957 to print the index dimension which is normally hidden.
6959 When you write something like @code{std::cout << e}, where @code{e} is
6960 an object of class @code{ex}, GiNaC will construct an appropriate
6961 @code{print_context} object (of a class depending on the selected output
6962 format), fill in the @code{s} and @code{options} members, and call
6964 @cindex @code{print()}
6966 void ex::print(const print_context & c, unsigned level = 0) const;
6969 which in turn forwards the call to the @code{print()} method of the
6970 top-level algebraic object contained in the expression.
6972 Unlike other methods, GiNaC classes don't usually override their
6973 @code{print()} method to implement expression output. Instead, the default
6974 implementation @code{basic::print(c, level)} performs a run-time double
6975 dispatch to a function selected by the dynamic type of the object and the
6976 passed @code{print_context}. To this end, GiNaC maintains a separate method
6977 table for each class, similar to the virtual function table used for ordinary
6978 (single) virtual function dispatch.
6980 The method table contains one slot for each possible @code{print_context}
6981 type, indexed by the (internally assigned) serial number of the type. Slots
6982 may be empty, in which case GiNaC will retry the method lookup with the
6983 @code{print_context} object's parent class, possibly repeating the process
6984 until it reaches the @code{print_context} base class. If there's still no
6985 method defined, the method table of the algebraic object's parent class
6986 is consulted, and so on, until a matching method is found (eventually it
6987 will reach the combination @code{basic/print_context}, which prints the
6988 object's class name enclosed in square brackets).
6990 You can think of the print methods of all the different classes and output
6991 formats as being arranged in a two-dimensional matrix with one axis listing
6992 the algebraic classes and the other axis listing the @code{print_context}
6995 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6996 to implement printing, but then they won't get any of the benefits of the
6997 double dispatch mechanism (such as the ability for derived classes to
6998 inherit only certain print methods from its parent, or the replacement of
6999 methods at run-time).
7001 @subsection Print methods for classes
7003 The method table for a class is set up either in the definition of the class,
7004 by passing the appropriate @code{print_func<C>()} option to
7005 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7006 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7007 can also be used to override existing methods dynamically.
7009 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7010 be a member function of the class (or one of its parent classes), a static
7011 member function, or an ordinary (global) C++ function. The @code{C} template
7012 parameter specifies the appropriate @code{print_context} type for which the
7013 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7014 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7015 the class is the one being implemented by
7016 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7018 For print methods that are member functions, their first argument must be of
7019 a type convertible to a @code{const C &}, and the second argument must be an
7022 For static members and global functions, the first argument must be of a type
7023 convertible to a @code{const T &}, the second argument must be of a type
7024 convertible to a @code{const C &}, and the third argument must be an
7025 @code{unsigned}. A global function will, of course, not have access to
7026 private and protected members of @code{T}.
7028 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7029 and @code{basic::print()}) is used for proper parenthesizing of the output
7030 (and by @code{print_tree} for proper indentation). It can be used for similar
7031 purposes if you write your own output formats.
7033 The explanations given above may seem complicated, but in practice it's
7034 really simple, as shown in the following example. Suppose that we want to
7035 display exponents in LaTeX output not as superscripts but with little
7036 upwards-pointing arrows. This can be achieved in the following way:
7039 void my_print_power_as_latex(const power & p,
7040 const print_latex & c,
7043 // get the precedence of the 'power' class
7044 unsigned power_prec = p.precedence();
7046 // if the parent operator has the same or a higher precedence
7047 // we need parentheses around the power
7048 if (level >= power_prec)
7051 // print the basis and exponent, each enclosed in braces, and
7052 // separated by an uparrow
7054 p.op(0).print(c, power_prec);
7055 c.s << "@}\\uparrow@{";
7056 p.op(1).print(c, power_prec);
7059 // don't forget the closing parenthesis
7060 if (level >= power_prec)
7066 // a sample expression
7067 symbol x("x"), y("y");
7068 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7070 // switch to LaTeX mode
7073 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7076 // now we replace the method for the LaTeX output of powers with
7078 set_print_func<power, print_latex>(my_print_power_as_latex);
7080 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7091 The first argument of @code{my_print_power_as_latex} could also have been
7092 a @code{const basic &}, the second one a @code{const print_context &}.
7095 The above code depends on @code{mul} objects converting their operands to
7096 @code{power} objects for the purpose of printing.
7099 The output of products including negative powers as fractions is also
7100 controlled by the @code{mul} class.
7103 The @code{power/print_latex} method provided by GiNaC prints square roots
7104 using @code{\sqrt}, but the above code doesn't.
7108 It's not possible to restore a method table entry to its previous or default
7109 value. Once you have called @code{set_print_func()}, you can only override
7110 it with another call to @code{set_print_func()}, but you can't easily go back
7111 to the default behavior again (you can, of course, dig around in the GiNaC
7112 sources, find the method that is installed at startup
7113 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7114 one; that is, after you circumvent the C++ member access control@dots{}).
7116 @subsection Print methods for functions
7118 Symbolic functions employ a print method dispatch mechanism similar to the
7119 one used for classes. The methods are specified with @code{print_func<C>()}
7120 function options. If you don't specify any special print methods, the function
7121 will be printed with its name (or LaTeX name, if supplied), followed by a
7122 comma-separated list of arguments enclosed in parentheses.
7124 For example, this is what GiNaC's @samp{abs()} function is defined like:
7127 static ex abs_eval(const ex & arg) @{ ... @}
7128 static ex abs_evalf(const ex & arg) @{ ... @}
7130 static void abs_print_latex(const ex & arg, const print_context & c)
7132 c.s << "@{|"; arg.print(c); c.s << "|@}";
7135 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7137 c.s << "fabs("; arg.print(c); c.s << ")";
7140 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7141 evalf_func(abs_evalf).
7142 print_func<print_latex>(abs_print_latex).
7143 print_func<print_csrc_float>(abs_print_csrc_float).
7144 print_func<print_csrc_double>(abs_print_csrc_float));
7147 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7148 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7150 There is currently no equivalent of @code{set_print_func()} for functions.
7152 @subsection Adding new output formats
7154 Creating a new output format involves subclassing @code{print_context},
7155 which is somewhat similar to adding a new algebraic class
7156 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7157 that needs to go into the class definition, and a corresponding macro
7158 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7159 Every @code{print_context} class needs to provide a default constructor
7160 and a constructor from an @code{std::ostream} and an @code{unsigned}
7163 Here is an example for a user-defined @code{print_context} class:
7166 class print_myformat : public print_dflt
7168 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7170 print_myformat(std::ostream & os, unsigned opt = 0)
7171 : print_dflt(os, opt) @{@}
7174 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7176 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7179 That's all there is to it. None of the actual expression output logic is
7180 implemented in this class. It merely serves as a selector for choosing
7181 a particular format. The algorithms for printing expressions in the new
7182 format are implemented as print methods, as described above.
7184 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7185 exactly like GiNaC's default output format:
7190 ex e = pow(x, 2) + 1;
7192 // this prints "1+x^2"
7195 // this also prints "1+x^2"
7196 e.print(print_myformat()); cout << endl;
7202 To fill @code{print_myformat} with life, we need to supply appropriate
7203 print methods with @code{set_print_func()}, like this:
7206 // This prints powers with '**' instead of '^'. See the LaTeX output
7207 // example above for explanations.
7208 void print_power_as_myformat(const power & p,
7209 const print_myformat & c,
7212 unsigned power_prec = p.precedence();
7213 if (level >= power_prec)
7215 p.op(0).print(c, power_prec);
7217 p.op(1).print(c, power_prec);
7218 if (level >= power_prec)
7224 // install a new print method for power objects
7225 set_print_func<power, print_myformat>(print_power_as_myformat);
7227 // now this prints "1+x**2"
7228 e.print(print_myformat()); cout << endl;
7230 // but the default format is still "1+x^2"
7236 @node Structures, Adding classes, Printing, Extending GiNaC
7237 @c node-name, next, previous, up
7240 If you are doing some very specialized things with GiNaC, or if you just
7241 need some more organized way to store data in your expressions instead of
7242 anonymous lists, you may want to implement your own algebraic classes.
7243 ('algebraic class' means any class directly or indirectly derived from
7244 @code{basic} that can be used in GiNaC expressions).
7246 GiNaC offers two ways of accomplishing this: either by using the
7247 @code{structure<T>} template class, or by rolling your own class from
7248 scratch. This section will discuss the @code{structure<T>} template which
7249 is easier to use but more limited, while the implementation of custom
7250 GiNaC classes is the topic of the next section. However, you may want to
7251 read both sections because many common concepts and member functions are
7252 shared by both concepts, and it will also allow you to decide which approach
7253 is most suited to your needs.
7255 The @code{structure<T>} template, defined in the GiNaC header file
7256 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7257 or @code{class}) into a GiNaC object that can be used in expressions.
7259 @subsection Example: scalar products
7261 Let's suppose that we need a way to handle some kind of abstract scalar
7262 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7263 product class have to store their left and right operands, which can in turn
7264 be arbitrary expressions. Here is a possible way to represent such a
7265 product in a C++ @code{struct}:
7269 using namespace std;
7271 #include <ginac/ginac.h>
7272 using namespace GiNaC;
7278 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7282 The default constructor is required. Now, to make a GiNaC class out of this
7283 data structure, we need only one line:
7286 typedef structure<sprod_s> sprod;
7289 That's it. This line constructs an algebraic class @code{sprod} which
7290 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7291 expressions like any other GiNaC class:
7295 symbol a("a"), b("b");
7296 ex e = sprod(sprod_s(a, b));
7300 Note the difference between @code{sprod} which is the algebraic class, and
7301 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7302 and @code{right} data members. As shown above, an @code{sprod} can be
7303 constructed from an @code{sprod_s} object.
7305 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7306 you could define a little wrapper function like this:
7309 inline ex make_sprod(ex left, ex right)
7311 return sprod(sprod_s(left, right));
7315 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7316 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7317 @code{get_struct()}:
7321 cout << ex_to<sprod>(e)->left << endl;
7323 cout << ex_to<sprod>(e).get_struct().right << endl;
7328 You only have read access to the members of @code{sprod_s}.
7330 The type definition of @code{sprod} is enough to write your own algorithms
7331 that deal with scalar products, for example:
7336 if (is_a<sprod>(p)) @{
7337 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7338 return make_sprod(sp.right, sp.left);
7349 @subsection Structure output
7351 While the @code{sprod} type is useable it still leaves something to be
7352 desired, most notably proper output:
7357 // -> [structure object]
7361 By default, any structure types you define will be printed as
7362 @samp{[structure object]}. To override this you can either specialize the
7363 template's @code{print()} member function, or specify print methods with
7364 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7365 it's not possible to supply class options like @code{print_func<>()} to
7366 structures, so for a self-contained structure type you need to resort to
7367 overriding the @code{print()} function, which is also what we will do here.
7369 The member functions of GiNaC classes are described in more detail in the
7370 next section, but it shouldn't be hard to figure out what's going on here:
7373 void sprod::print(const print_context & c, unsigned level) const
7375 // tree debug output handled by superclass
7376 if (is_a<print_tree>(c))
7377 inherited::print(c, level);
7379 // get the contained sprod_s object
7380 const sprod_s & sp = get_struct();
7382 // print_context::s is a reference to an ostream
7383 c.s << "<" << sp.left << "|" << sp.right << ">";
7387 Now we can print expressions containing scalar products:
7393 cout << swap_sprod(e) << endl;
7398 @subsection Comparing structures
7400 The @code{sprod} class defined so far still has one important drawback: all
7401 scalar products are treated as being equal because GiNaC doesn't know how to
7402 compare objects of type @code{sprod_s}. This can lead to some confusing
7403 and undesired behavior:
7407 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7409 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7410 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7414 To remedy this, we first need to define the operators @code{==} and @code{<}
7415 for objects of type @code{sprod_s}:
7418 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7420 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7423 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7425 return lhs.left.compare(rhs.left) < 0
7426 ? true : lhs.right.compare(rhs.right) < 0;
7430 The ordering established by the @code{<} operator doesn't have to make any
7431 algebraic sense, but it needs to be well defined. Note that we can't use
7432 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7433 in the implementation of these operators because they would construct
7434 GiNaC @code{relational} objects which in the case of @code{<} do not
7435 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7436 decide which one is algebraically 'less').
7438 Next, we need to change our definition of the @code{sprod} type to let
7439 GiNaC know that an ordering relation exists for the embedded objects:
7442 typedef structure<sprod_s, compare_std_less> sprod;
7445 @code{sprod} objects then behave as expected:
7449 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7450 // -> <a|b>-<a^2|b^2>
7451 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7452 // -> <a|b>+<a^2|b^2>
7453 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7455 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7460 The @code{compare_std_less} policy parameter tells GiNaC to use the
7461 @code{std::less} and @code{std::equal_to} functors to compare objects of
7462 type @code{sprod_s}. By default, these functors forward their work to the
7463 standard @code{<} and @code{==} operators, which we have overloaded.
7464 Alternatively, we could have specialized @code{std::less} and
7465 @code{std::equal_to} for class @code{sprod_s}.
7467 GiNaC provides two other comparison policies for @code{structure<T>}
7468 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7469 which does a bit-wise comparison of the contained @code{T} objects.
7470 This should be used with extreme care because it only works reliably with
7471 built-in integral types, and it also compares any padding (filler bytes of
7472 undefined value) that the @code{T} class might have.
7474 @subsection Subexpressions
7476 Our scalar product class has two subexpressions: the left and right
7477 operands. It might be a good idea to make them accessible via the standard
7478 @code{nops()} and @code{op()} methods:
7481 size_t sprod::nops() const
7486 ex sprod::op(size_t i) const
7490 return get_struct().left;
7492 return get_struct().right;
7494 throw std::range_error("sprod::op(): no such operand");
7499 Implementing @code{nops()} and @code{op()} for container types such as
7500 @code{sprod} has two other nice side effects:
7504 @code{has()} works as expected
7506 GiNaC generates better hash keys for the objects (the default implementation
7507 of @code{calchash()} takes subexpressions into account)
7510 @cindex @code{let_op()}
7511 There is a non-const variant of @code{op()} called @code{let_op()} that
7512 allows replacing subexpressions:
7515 ex & sprod::let_op(size_t i)
7517 // every non-const member function must call this
7518 ensure_if_modifiable();
7522 return get_struct().left;
7524 return get_struct().right;
7526 throw std::range_error("sprod::let_op(): no such operand");
7531 Once we have provided @code{let_op()} we also get @code{subs()} and
7532 @code{map()} for free. In fact, every container class that returns a non-null
7533 @code{nops()} value must either implement @code{let_op()} or provide custom
7534 implementations of @code{subs()} and @code{map()}.
7536 In turn, the availability of @code{map()} enables the recursive behavior of a
7537 couple of other default method implementations, in particular @code{evalf()},
7538 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7539 we probably want to provide our own version of @code{expand()} for scalar
7540 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7541 This is left as an exercise for the reader.
7543 The @code{structure<T>} template defines many more member functions that
7544 you can override by specialization to customize the behavior of your
7545 structures. You are referred to the next section for a description of
7546 some of these (especially @code{eval()}). There is, however, one topic
7547 that shall be addressed here, as it demonstrates one peculiarity of the
7548 @code{structure<T>} template: archiving.
7550 @subsection Archiving structures
7552 If you don't know how the archiving of GiNaC objects is implemented, you
7553 should first read the next section and then come back here. You're back?
7556 To implement archiving for structures it is not enough to provide
7557 specializations for the @code{archive()} member function and the
7558 unarchiving constructor (the @code{unarchive()} function has a default
7559 implementation). You also need to provide a unique name (as a string literal)
7560 for each structure type you define. This is because in GiNaC archives,
7561 the class of an object is stored as a string, the class name.
7563 By default, this class name (as returned by the @code{class_name()} member
7564 function) is @samp{structure} for all structure classes. This works as long
7565 as you have only defined one structure type, but if you use two or more you
7566 need to provide a different name for each by specializing the
7567 @code{get_class_name()} member function. Here is a sample implementation
7568 for enabling archiving of the scalar product type defined above:
7571 const char *sprod::get_class_name() @{ return "sprod"; @}
7573 void sprod::archive(archive_node & n) const
7575 inherited::archive(n);
7576 n.add_ex("left", get_struct().left);
7577 n.add_ex("right", get_struct().right);
7580 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7582 n.find_ex("left", get_struct().left, sym_lst);
7583 n.find_ex("right", get_struct().right, sym_lst);
7587 Note that the unarchiving constructor is @code{sprod::structure} and not
7588 @code{sprod::sprod}, and that we don't need to supply an
7589 @code{sprod::unarchive()} function.
7592 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7593 @c node-name, next, previous, up
7594 @section Adding classes
7596 The @code{structure<T>} template provides an way to extend GiNaC with custom
7597 algebraic classes that is easy to use but has its limitations, the most
7598 severe of which being that you can't add any new member functions to
7599 structures. To be able to do this, you need to write a new class definition
7602 This section will explain how to implement new algebraic classes in GiNaC by
7603 giving the example of a simple 'string' class. After reading this section
7604 you will know how to properly declare a GiNaC class and what the minimum
7605 required member functions are that you have to implement. We only cover the
7606 implementation of a 'leaf' class here (i.e. one that doesn't contain
7607 subexpressions). Creating a container class like, for example, a class
7608 representing tensor products is more involved but this section should give
7609 you enough information so you can consult the source to GiNaC's predefined
7610 classes if you want to implement something more complicated.
7612 @subsection GiNaC's run-time type information system
7614 @cindex hierarchy of classes
7616 All algebraic classes (that is, all classes that can appear in expressions)
7617 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7618 @code{basic *} (which is essentially what an @code{ex} is) represents a
7619 generic pointer to an algebraic class. Occasionally it is necessary to find
7620 out what the class of an object pointed to by a @code{basic *} really is.
7621 Also, for the unarchiving of expressions it must be possible to find the
7622 @code{unarchive()} function of a class given the class name (as a string). A
7623 system that provides this kind of information is called a run-time type
7624 information (RTTI) system. The C++ language provides such a thing (see the
7625 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7626 implements its own, simpler RTTI.
7628 The RTTI in GiNaC is based on two mechanisms:
7633 The @code{basic} class declares a member variable @code{tinfo_key} which
7634 holds an unsigned integer that identifies the object's class. These numbers
7635 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7636 classes. They all start with @code{TINFO_}.
7639 By means of some clever tricks with static members, GiNaC maintains a list
7640 of information for all classes derived from @code{basic}. The information
7641 available includes the class names, the @code{tinfo_key}s, and pointers
7642 to the unarchiving functions. This class registry is defined in the
7643 @file{registrar.h} header file.
7647 The disadvantage of this proprietary RTTI implementation is that there's
7648 a little more to do when implementing new classes (C++'s RTTI works more
7649 or less automatically) but don't worry, most of the work is simplified by
7652 @subsection A minimalistic example
7654 Now we will start implementing a new class @code{mystring} that allows
7655 placing character strings in algebraic expressions (this is not very useful,
7656 but it's just an example). This class will be a direct subclass of
7657 @code{basic}. You can use this sample implementation as a starting point
7658 for your own classes.
7660 The code snippets given here assume that you have included some header files
7666 #include <stdexcept>
7667 using namespace std;
7669 #include <ginac/ginac.h>
7670 using namespace GiNaC;
7673 The first thing we have to do is to define a @code{tinfo_key} for our new
7674 class. This can be any arbitrary unsigned number that is not already taken
7675 by one of the existing classes but it's better to come up with something
7676 that is unlikely to clash with keys that might be added in the future. The
7677 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7678 which is not a requirement but we are going to stick with this scheme:
7681 const unsigned TINFO_mystring = 0x42420001U;
7684 Now we can write down the class declaration. The class stores a C++
7685 @code{string} and the user shall be able to construct a @code{mystring}
7686 object from a C or C++ string:
7689 class mystring : public basic
7691 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7694 mystring(const string &s);
7695 mystring(const char *s);
7701 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7704 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7705 macros are defined in @file{registrar.h}. They take the name of the class
7706 and its direct superclass as arguments and insert all required declarations
7707 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7708 the first line after the opening brace of the class definition. The
7709 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7710 source (at global scope, of course, not inside a function).
7712 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7713 declarations of the default constructor and a couple of other functions that
7714 are required. It also defines a type @code{inherited} which refers to the
7715 superclass so you don't have to modify your code every time you shuffle around
7716 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7717 class with the GiNaC RTTI (there is also a
7718 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7719 options for the class, and which we will be using instead in a few minutes).
7721 Now there are seven member functions we have to implement to get a working
7727 @code{mystring()}, the default constructor.
7730 @code{void archive(archive_node &n)}, the archiving function. This stores all
7731 information needed to reconstruct an object of this class inside an
7732 @code{archive_node}.
7735 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7736 constructor. This constructs an instance of the class from the information
7737 found in an @code{archive_node}.
7740 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7741 unarchiving function. It constructs a new instance by calling the unarchiving
7745 @cindex @code{compare_same_type()}
7746 @code{int compare_same_type(const basic &other)}, which is used internally
7747 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7748 -1, depending on the relative order of this object and the @code{other}
7749 object. If it returns 0, the objects are considered equal.
7750 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7751 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7752 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7753 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7754 must provide a @code{compare_same_type()} function, even those representing
7755 objects for which no reasonable algebraic ordering relationship can be
7759 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7760 which are the two constructors we declared.
7764 Let's proceed step-by-step. The default constructor looks like this:
7767 mystring::mystring() : inherited(TINFO_mystring) @{@}
7770 The golden rule is that in all constructors you have to set the
7771 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7772 it will be set by the constructor of the superclass and all hell will break
7773 loose in the RTTI. For your convenience, the @code{basic} class provides
7774 a constructor that takes a @code{tinfo_key} value, which we are using here
7775 (remember that in our case @code{inherited == basic}). If the superclass
7776 didn't have such a constructor, we would have to set the @code{tinfo_key}
7777 to the right value manually.
7779 In the default constructor you should set all other member variables to
7780 reasonable default values (we don't need that here since our @code{str}
7781 member gets set to an empty string automatically).
7783 Next are the three functions for archiving. You have to implement them even
7784 if you don't plan to use archives, but the minimum required implementation
7785 is really simple. First, the archiving function:
7788 void mystring::archive(archive_node &n) const
7790 inherited::archive(n);
7791 n.add_string("string", str);
7795 The only thing that is really required is calling the @code{archive()}
7796 function of the superclass. Optionally, you can store all information you
7797 deem necessary for representing the object into the passed
7798 @code{archive_node}. We are just storing our string here. For more
7799 information on how the archiving works, consult the @file{archive.h} header
7802 The unarchiving constructor is basically the inverse of the archiving
7806 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7808 n.find_string("string", str);
7812 If you don't need archiving, just leave this function empty (but you must
7813 invoke the unarchiving constructor of the superclass). Note that we don't
7814 have to set the @code{tinfo_key} here because it is done automatically
7815 by the unarchiving constructor of the @code{basic} class.
7817 Finally, the unarchiving function:
7820 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7822 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7826 You don't have to understand how exactly this works. Just copy these
7827 four lines into your code literally (replacing the class name, of
7828 course). It calls the unarchiving constructor of the class and unless
7829 you are doing something very special (like matching @code{archive_node}s
7830 to global objects) you don't need a different implementation. For those
7831 who are interested: setting the @code{dynallocated} flag puts the object
7832 under the control of GiNaC's garbage collection. It will get deleted
7833 automatically once it is no longer referenced.
7835 Our @code{compare_same_type()} function uses a provided function to compare
7839 int mystring::compare_same_type(const basic &other) const
7841 const mystring &o = static_cast<const mystring &>(other);
7842 int cmpval = str.compare(o.str);
7845 else if (cmpval < 0)
7852 Although this function takes a @code{basic &}, it will always be a reference
7853 to an object of exactly the same class (objects of different classes are not
7854 comparable), so the cast is safe. If this function returns 0, the two objects
7855 are considered equal (in the sense that @math{A-B=0}), so you should compare
7856 all relevant member variables.
7858 Now the only thing missing is our two new constructors:
7861 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7862 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7865 No surprises here. We set the @code{str} member from the argument and
7866 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7868 That's it! We now have a minimal working GiNaC class that can store
7869 strings in algebraic expressions. Let's confirm that the RTTI works:
7872 ex e = mystring("Hello, world!");
7873 cout << is_a<mystring>(e) << endl;
7876 cout << ex_to<basic>(e).class_name() << endl;
7880 Obviously it does. Let's see what the expression @code{e} looks like:
7884 // -> [mystring object]
7887 Hm, not exactly what we expect, but of course the @code{mystring} class
7888 doesn't yet know how to print itself. This can be done either by implementing
7889 the @code{print()} member function, or, preferably, by specifying a
7890 @code{print_func<>()} class option. Let's say that we want to print the string
7891 surrounded by double quotes:
7894 class mystring : public basic
7898 void do_print(const print_context &c, unsigned level = 0) const;
7902 void mystring::do_print(const print_context &c, unsigned level) const
7904 // print_context::s is a reference to an ostream
7905 c.s << '\"' << str << '\"';
7909 The @code{level} argument is only required for container classes to
7910 correctly parenthesize the output.
7912 Now we need to tell GiNaC that @code{mystring} objects should use the
7913 @code{do_print()} member function for printing themselves. For this, we
7917 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7923 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7924 print_func<print_context>(&mystring::do_print))
7927 Let's try again to print the expression:
7931 // -> "Hello, world!"
7934 Much better. If we wanted to have @code{mystring} objects displayed in a
7935 different way depending on the output format (default, LaTeX, etc.), we
7936 would have supplied multiple @code{print_func<>()} options with different
7937 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7938 separated by dots. This is similar to the way options are specified for
7939 symbolic functions. @xref{Printing}, for a more in-depth description of the
7940 way expression output is implemented in GiNaC.
7942 The @code{mystring} class can be used in arbitrary expressions:
7945 e += mystring("GiNaC rulez");
7947 // -> "GiNaC rulez"+"Hello, world!"
7950 (GiNaC's automatic term reordering is in effect here), or even
7953 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7955 // -> "One string"^(2*sin(-"Another string"+Pi))
7958 Whether this makes sense is debatable but remember that this is only an
7959 example. At least it allows you to implement your own symbolic algorithms
7962 Note that GiNaC's algebraic rules remain unchanged:
7965 e = mystring("Wow") * mystring("Wow");
7969 e = pow(mystring("First")-mystring("Second"), 2);
7970 cout << e.expand() << endl;
7971 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7974 There's no way to, for example, make GiNaC's @code{add} class perform string
7975 concatenation. You would have to implement this yourself.
7977 @subsection Automatic evaluation
7980 @cindex @code{eval()}
7981 @cindex @code{hold()}
7982 When dealing with objects that are just a little more complicated than the
7983 simple string objects we have implemented, chances are that you will want to
7984 have some automatic simplifications or canonicalizations performed on them.
7985 This is done in the evaluation member function @code{eval()}. Let's say that
7986 we wanted all strings automatically converted to lowercase with
7987 non-alphabetic characters stripped, and empty strings removed:
7990 class mystring : public basic
7994 ex eval(int level = 0) const;
7998 ex mystring::eval(int level) const
8001 for (int i=0; i<str.length(); i++) @{
8003 if (c >= 'A' && c <= 'Z')
8004 new_str += tolower(c);
8005 else if (c >= 'a' && c <= 'z')
8009 if (new_str.length() == 0)
8012 return mystring(new_str).hold();
8016 The @code{level} argument is used to limit the recursion depth of the
8017 evaluation. We don't have any subexpressions in the @code{mystring}
8018 class so we are not concerned with this. If we had, we would call the
8019 @code{eval()} functions of the subexpressions with @code{level - 1} as
8020 the argument if @code{level != 1}. The @code{hold()} member function
8021 sets a flag in the object that prevents further evaluation. Otherwise
8022 we might end up in an endless loop. When you want to return the object
8023 unmodified, use @code{return this->hold();}.
8025 Let's confirm that it works:
8028 ex e = mystring("Hello, world!") + mystring("!?#");
8032 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8037 @subsection Optional member functions
8039 We have implemented only a small set of member functions to make the class
8040 work in the GiNaC framework. There are two functions that are not strictly
8041 required but will make operations with objects of the class more efficient:
8043 @cindex @code{calchash()}
8044 @cindex @code{is_equal_same_type()}
8046 unsigned calchash() const;
8047 bool is_equal_same_type(const basic &other) const;
8050 The @code{calchash()} method returns an @code{unsigned} hash value for the
8051 object which will allow GiNaC to compare and canonicalize expressions much
8052 more efficiently. You should consult the implementation of some of the built-in
8053 GiNaC classes for examples of hash functions. The default implementation of
8054 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8055 class and all subexpressions that are accessible via @code{op()}.
8057 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8058 tests for equality without establishing an ordering relation, which is often
8059 faster. The default implementation of @code{is_equal_same_type()} just calls
8060 @code{compare_same_type()} and tests its result for zero.
8062 @subsection Other member functions
8064 For a real algebraic class, there are probably some more functions that you
8065 might want to provide:
8068 bool info(unsigned inf) const;
8069 ex evalf(int level = 0) const;
8070 ex series(const relational & r, int order, unsigned options = 0) const;
8071 ex derivative(const symbol & s) const;
8074 If your class stores sub-expressions (see the scalar product example in the
8075 previous section) you will probably want to override
8077 @cindex @code{let_op()}
8080 ex op(size_t i) const;
8081 ex & let_op(size_t i);
8082 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8083 ex map(map_function & f) const;
8086 @code{let_op()} is a variant of @code{op()} that allows write access. The
8087 default implementations of @code{subs()} and @code{map()} use it, so you have
8088 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8090 You can, of course, also add your own new member functions. Remember
8091 that the RTTI may be used to get information about what kinds of objects
8092 you are dealing with (the position in the class hierarchy) and that you
8093 can always extract the bare object from an @code{ex} by stripping the
8094 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8095 should become a need.
8097 That's it. May the source be with you!
8100 @node A comparison with other CAS, Advantages, Adding classes, Top
8101 @c node-name, next, previous, up
8102 @chapter A Comparison With Other CAS
8105 This chapter will give you some information on how GiNaC compares to
8106 other, traditional Computer Algebra Systems, like @emph{Maple},
8107 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8108 disadvantages over these systems.
8111 * Advantages:: Strengths of the GiNaC approach.
8112 * Disadvantages:: Weaknesses of the GiNaC approach.
8113 * Why C++?:: Attractiveness of C++.
8116 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8117 @c node-name, next, previous, up
8120 GiNaC has several advantages over traditional Computer
8121 Algebra Systems, like
8126 familiar language: all common CAS implement their own proprietary
8127 grammar which you have to learn first (and maybe learn again when your
8128 vendor decides to `enhance' it). With GiNaC you can write your program
8129 in common C++, which is standardized.
8133 structured data types: you can build up structured data types using
8134 @code{struct}s or @code{class}es together with STL features instead of
8135 using unnamed lists of lists of lists.
8138 strongly typed: in CAS, you usually have only one kind of variables
8139 which can hold contents of an arbitrary type. This 4GL like feature is
8140 nice for novice programmers, but dangerous.
8143 development tools: powerful development tools exist for C++, like fancy
8144 editors (e.g. with automatic indentation and syntax highlighting),
8145 debuggers, visualization tools, documentation generators@dots{}
8148 modularization: C++ programs can easily be split into modules by
8149 separating interface and implementation.
8152 price: GiNaC is distributed under the GNU Public License which means
8153 that it is free and available with source code. And there are excellent
8154 C++-compilers for free, too.
8157 extendable: you can add your own classes to GiNaC, thus extending it on
8158 a very low level. Compare this to a traditional CAS that you can
8159 usually only extend on a high level by writing in the language defined
8160 by the parser. In particular, it turns out to be almost impossible to
8161 fix bugs in a traditional system.
8164 multiple interfaces: Though real GiNaC programs have to be written in
8165 some editor, then be compiled, linked and executed, there are more ways
8166 to work with the GiNaC engine. Many people want to play with
8167 expressions interactively, as in traditional CASs. Currently, two such
8168 windows into GiNaC have been implemented and many more are possible: the
8169 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8170 types to a command line and second, as a more consistent approach, an
8171 interactive interface to the Cint C++ interpreter has been put together
8172 (called GiNaC-cint) that allows an interactive scripting interface
8173 consistent with the C++ language. It is available from the usual GiNaC
8177 seamless integration: it is somewhere between difficult and impossible
8178 to call CAS functions from within a program written in C++ or any other
8179 programming language and vice versa. With GiNaC, your symbolic routines
8180 are part of your program. You can easily call third party libraries,
8181 e.g. for numerical evaluation or graphical interaction. All other
8182 approaches are much more cumbersome: they range from simply ignoring the
8183 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8184 system (i.e. @emph{Yacas}).
8187 efficiency: often large parts of a program do not need symbolic
8188 calculations at all. Why use large integers for loop variables or
8189 arbitrary precision arithmetics where @code{int} and @code{double} are
8190 sufficient? For pure symbolic applications, GiNaC is comparable in
8191 speed with other CAS.
8196 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8197 @c node-name, next, previous, up
8198 @section Disadvantages
8200 Of course it also has some disadvantages:
8205 advanced features: GiNaC cannot compete with a program like
8206 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8207 which grows since 1981 by the work of dozens of programmers, with
8208 respect to mathematical features. Integration, factorization,
8209 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8210 not planned for the near future).
8213 portability: While the GiNaC library itself is designed to avoid any
8214 platform dependent features (it should compile on any ANSI compliant C++
8215 compiler), the currently used version of the CLN library (fast large
8216 integer and arbitrary precision arithmetics) can only by compiled
8217 without hassle on systems with the C++ compiler from the GNU Compiler
8218 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8219 macros to let the compiler gather all static initializations, which
8220 works for GNU C++ only. Feel free to contact the authors in case you
8221 really believe that you need to use a different compiler. We have
8222 occasionally used other compilers and may be able to give you advice.}
8223 GiNaC uses recent language features like explicit constructors, mutable
8224 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8225 literally. Recent GCC versions starting at 2.95.3, although itself not
8226 yet ANSI compliant, support all needed features.
8231 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8232 @c node-name, next, previous, up
8235 Why did we choose to implement GiNaC in C++ instead of Java or any other
8236 language? C++ is not perfect: type checking is not strict (casting is
8237 possible), separation between interface and implementation is not
8238 complete, object oriented design is not enforced. The main reason is
8239 the often scolded feature of operator overloading in C++. While it may
8240 be true that operating on classes with a @code{+} operator is rarely
8241 meaningful, it is perfectly suited for algebraic expressions. Writing
8242 @math{3x+5y} as @code{3*x+5*y} instead of
8243 @code{x.times(3).plus(y.times(5))} looks much more natural.
8244 Furthermore, the main developers are more familiar with C++ than with
8245 any other programming language.
8248 @node Internal structures, Expressions are reference counted, Why C++? , Top
8249 @c node-name, next, previous, up
8250 @appendix Internal structures
8253 * Expressions are reference counted::
8254 * Internal representation of products and sums::
8257 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8258 @c node-name, next, previous, up
8259 @appendixsection Expressions are reference counted
8261 @cindex reference counting
8262 @cindex copy-on-write
8263 @cindex garbage collection
8264 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8265 where the counter belongs to the algebraic objects derived from class
8266 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8267 which @code{ex} contains an instance. If you understood that, you can safely
8268 skip the rest of this passage.
8270 Expressions are extremely light-weight since internally they work like
8271 handles to the actual representation. They really hold nothing more
8272 than a pointer to some other object. What this means in practice is
8273 that whenever you create two @code{ex} and set the second equal to the
8274 first no copying process is involved. Instead, the copying takes place
8275 as soon as you try to change the second. Consider the simple sequence
8280 #include <ginac/ginac.h>
8281 using namespace std;
8282 using namespace GiNaC;
8286 symbol x("x"), y("y"), z("z");
8289 e1 = sin(x + 2*y) + 3*z + 41;
8290 e2 = e1; // e2 points to same object as e1
8291 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8292 e2 += 1; // e2 is copied into a new object
8293 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8297 The line @code{e2 = e1;} creates a second expression pointing to the
8298 object held already by @code{e1}. The time involved for this operation
8299 is therefore constant, no matter how large @code{e1} was. Actual
8300 copying, however, must take place in the line @code{e2 += 1;} because
8301 @code{e1} and @code{e2} are not handles for the same object any more.
8302 This concept is called @dfn{copy-on-write semantics}. It increases
8303 performance considerably whenever one object occurs multiple times and
8304 represents a simple garbage collection scheme because when an @code{ex}
8305 runs out of scope its destructor checks whether other expressions handle
8306 the object it points to too and deletes the object from memory if that
8307 turns out not to be the case. A slightly less trivial example of
8308 differentiation using the chain-rule should make clear how powerful this
8313 symbol x("x"), y("y");
8317 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8318 cout << e1 << endl // prints x+3*y
8319 << e2 << endl // prints (x+3*y)^3
8320 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8324 Here, @code{e1} will actually be referenced three times while @code{e2}
8325 will be referenced two times. When the power of an expression is built,
8326 that expression needs not be copied. Likewise, since the derivative of
8327 a power of an expression can be easily expressed in terms of that
8328 expression, no copying of @code{e1} is involved when @code{e3} is
8329 constructed. So, when @code{e3} is constructed it will print as
8330 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8331 holds a reference to @code{e2} and the factor in front is just
8334 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8335 semantics. When you insert an expression into a second expression, the
8336 result behaves exactly as if the contents of the first expression were
8337 inserted. But it may be useful to remember that this is not what
8338 happens. Knowing this will enable you to write much more efficient
8339 code. If you still have an uncertain feeling with copy-on-write
8340 semantics, we recommend you have a look at the
8341 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8342 Marshall Cline. Chapter 16 covers this issue and presents an
8343 implementation which is pretty close to the one in GiNaC.
8346 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8347 @c node-name, next, previous, up
8348 @appendixsection Internal representation of products and sums
8350 @cindex representation
8353 @cindex @code{power}
8354 Although it should be completely transparent for the user of
8355 GiNaC a short discussion of this topic helps to understand the sources
8356 and also explain performance to a large degree. Consider the
8357 unexpanded symbolic expression
8359 $2d^3 \left( 4a + 5b - 3 \right)$
8362 @math{2*d^3*(4*a+5*b-3)}
8364 which could naively be represented by a tree of linear containers for
8365 addition and multiplication, one container for exponentiation with base
8366 and exponent and some atomic leaves of symbols and numbers in this
8371 @cindex pair-wise representation
8372 However, doing so results in a rather deeply nested tree which will
8373 quickly become inefficient to manipulate. We can improve on this by
8374 representing the sum as a sequence of terms, each one being a pair of a
8375 purely numeric multiplicative coefficient and its rest. In the same
8376 spirit we can store the multiplication as a sequence of terms, each
8377 having a numeric exponent and a possibly complicated base, the tree
8378 becomes much more flat:
8382 The number @code{3} above the symbol @code{d} shows that @code{mul}
8383 objects are treated similarly where the coefficients are interpreted as
8384 @emph{exponents} now. Addition of sums of terms or multiplication of
8385 products with numerical exponents can be coded to be very efficient with
8386 such a pair-wise representation. Internally, this handling is performed
8387 by most CAS in this way. It typically speeds up manipulations by an
8388 order of magnitude. The overall multiplicative factor @code{2} and the
8389 additive term @code{-3} look somewhat out of place in this
8390 representation, however, since they are still carrying a trivial
8391 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8392 this is avoided by adding a field that carries an overall numeric
8393 coefficient. This results in the realistic picture of internal
8396 $2d^3 \left( 4a + 5b - 3 \right)$:
8399 @math{2*d^3*(4*a+5*b-3)}:
8405 This also allows for a better handling of numeric radicals, since
8406 @code{sqrt(2)} can now be carried along calculations. Now it should be
8407 clear, why both classes @code{add} and @code{mul} are derived from the
8408 same abstract class: the data representation is the same, only the
8409 semantics differs. In the class hierarchy, methods for polynomial
8410 expansion and the like are reimplemented for @code{add} and @code{mul},
8411 but the data structure is inherited from @code{expairseq}.
8414 @node Package tools, ginac-config, Internal representation of products and sums, Top
8415 @c node-name, next, previous, up
8416 @appendix Package tools
8418 If you are creating a software package that uses the GiNaC library,
8419 setting the correct command line options for the compiler and linker
8420 can be difficult. GiNaC includes two tools to make this process easier.
8423 * ginac-config:: A shell script to detect compiler and linker flags.
8424 * AM_PATH_GINAC:: Macro for GNU automake.
8428 @node ginac-config, AM_PATH_GINAC, Package tools, Package tools
8429 @c node-name, next, previous, up
8430 @section @command{ginac-config}
8431 @cindex ginac-config
8433 @command{ginac-config} is a shell script that you can use to determine
8434 the compiler and linker command line options required to compile and
8435 link a program with the GiNaC library.
8437 @command{ginac-config} takes the following flags:
8441 Prints out the version of GiNaC installed.
8443 Prints '-I' flags pointing to the installed header files.
8445 Prints out the linker flags necessary to link a program against GiNaC.
8446 @item --prefix[=@var{PREFIX}]
8447 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8448 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8449 Otherwise, prints out the configured value of @env{$prefix}.
8450 @item --exec-prefix[=@var{PREFIX}]
8451 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8452 Otherwise, prints out the configured value of @env{$exec_prefix}.
8455 Typically, @command{ginac-config} will be used within a configure
8456 script, as described below. It, however, can also be used directly from
8457 the command line using backquotes to compile a simple program. For
8461 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8464 This command line might expand to (for example):
8467 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8468 -lginac -lcln -lstdc++
8471 Not only is the form using @command{ginac-config} easier to type, it will
8472 work on any system, no matter how GiNaC was configured.
8475 @node AM_PATH_GINAC, Configure script options, ginac-config, Package tools
8476 @c node-name, next, previous, up
8477 @section @samp{AM_PATH_GINAC}
8478 @cindex AM_PATH_GINAC
8480 For packages configured using GNU automake, GiNaC also provides
8481 a macro to automate the process of checking for GiNaC.
8484 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8485 [, @var{ACTION-IF-NOT-FOUND}]]])
8493 Determines the location of GiNaC using @command{ginac-config}, which is
8494 either found in the user's path, or from the environment variable
8495 @env{GINACLIB_CONFIG}.
8498 Tests the installed libraries to make sure that their version
8499 is later than @var{MINIMUM-VERSION}. (A default version will be used
8503 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8504 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8505 variable to the output of @command{ginac-config --libs}, and calls
8506 @samp{AC_SUBST()} for these variables so they can be used in generated
8507 makefiles, and then executes @var{ACTION-IF-FOUND}.
8510 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8511 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8515 This macro is in file @file{ginac.m4} which is installed in
8516 @file{$datadir/aclocal}. Note that if automake was installed with a
8517 different @samp{--prefix} than GiNaC, you will either have to manually
8518 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8519 aclocal the @samp{-I} option when running it.
8522 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8523 * Example package:: Example of a package using AM_PATH_GINAC.
8527 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8528 @c node-name, next, previous, up
8529 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8531 Simply make sure that @command{ginac-config} is in your path, and run
8532 the configure script.
8539 The directory where the GiNaC libraries are installed needs
8540 to be found by your system's dynamic linker.
8542 This is generally done by
8545 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8551 setting the environment variable @env{LD_LIBRARY_PATH},
8554 or, as a last resort,
8557 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8558 running configure, for instance:
8561 LDFLAGS=-R/home/cbauer/lib ./configure
8566 You can also specify a @command{ginac-config} not in your path by
8567 setting the @env{GINACLIB_CONFIG} environment variable to the
8568 name of the executable
8571 If you move the GiNaC package from its installed location,
8572 you will either need to modify @command{ginac-config} script
8573 manually to point to the new location or rebuild GiNaC.
8584 --with-ginac-prefix=@var{PREFIX}
8585 --with-ginac-exec-prefix=@var{PREFIX}
8588 are provided to override the prefix and exec-prefix that were stored
8589 in the @command{ginac-config} shell script by GiNaC's configure. You are
8590 generally better off configuring GiNaC with the right path to begin with.
8594 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8595 @c node-name, next, previous, up
8596 @subsection Example of a package using @samp{AM_PATH_GINAC}
8598 The following shows how to build a simple package using automake
8599 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8603 #include <ginac/ginac.h>
8607 GiNaC::symbol x("x");
8608 GiNaC::ex a = GiNaC::sin(x);
8609 std::cout << "Derivative of " << a
8610 << " is " << a.diff(x) << std::endl;
8615 You should first read the introductory portions of the automake
8616 Manual, if you are not already familiar with it.
8618 Two files are needed, @file{configure.in}, which is used to build the
8622 dnl Process this file with autoconf to produce a configure script.
8624 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8630 AM_PATH_GINAC(0.9.0, [
8631 LIBS="$LIBS $GINACLIB_LIBS"
8632 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8633 ], AC_MSG_ERROR([need to have GiNaC installed]))
8638 The only command in this which is not standard for automake
8639 is the @samp{AM_PATH_GINAC} macro.
8641 That command does the following: If a GiNaC version greater or equal
8642 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8643 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8644 the error message `need to have GiNaC installed'
8646 And the @file{Makefile.am}, which will be used to build the Makefile.
8649 ## Process this file with automake to produce Makefile.in
8650 bin_PROGRAMS = simple
8651 simple_SOURCES = simple.cpp
8654 This @file{Makefile.am}, says that we are building a single executable,
8655 from a single source file @file{simple.cpp}. Since every program
8656 we are building uses GiNaC we simply added the GiNaC options
8657 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8658 want to specify them on a per-program basis: for instance by
8662 simple_LDADD = $(GINACLIB_LIBS)
8663 INCLUDES = $(GINACLIB_CPPFLAGS)
8666 to the @file{Makefile.am}.
8668 To try this example out, create a new directory and add the three
8671 Now execute the following commands:
8674 $ automake --add-missing
8679 You now have a package that can be built in the normal fashion
8688 @node Bibliography, Concept index, Example package, Top
8689 @c node-name, next, previous, up
8690 @appendix Bibliography
8695 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8698 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8701 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8704 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8707 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8708 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8711 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8712 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8713 Academic Press, London
8716 @cite{Computer Algebra Systems - A Practical Guide},
8717 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8720 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8721 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8724 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8725 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8728 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8733 @node Concept index, , Bibliography, Top
8734 @c node-name, next, previous, up
8735 @unnumbered Concept index