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 values, you
1159 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1160 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1161 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @node Numbers, Constants, Symbols, Basic concepts
1165 @c node-name, next, previous, up
1167 @cindex @code{numeric} (class)
1173 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1174 The classes therein serve as foundation classes for GiNaC. CLN stands
1175 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1176 In order to find out more about CLN's internals, the reader is referred to
1177 the documentation of that library. @inforef{Introduction, , cln}, for
1178 more information. Suffice to say that it is by itself build on top of
1179 another library, the GNU Multiple Precision library GMP, which is an
1180 extremely fast library for arbitrary long integers and rationals as well
1181 as arbitrary precision floating point numbers. It is very commonly used
1182 by several popular cryptographic applications. CLN extends GMP by
1183 several useful things: First, it introduces the complex number field
1184 over either reals (i.e. floating point numbers with arbitrary precision)
1185 or rationals. Second, it automatically converts rationals to integers
1186 if the denominator is unity and complex numbers to real numbers if the
1187 imaginary part vanishes and also correctly treats algebraic functions.
1188 Third it provides good implementations of state-of-the-art algorithms
1189 for all trigonometric and hyperbolic functions as well as for
1190 calculation of some useful constants.
1192 The user can construct an object of class @code{numeric} in several
1193 ways. The following example shows the four most important constructors.
1194 It uses construction from C-integer, construction of fractions from two
1195 integers, construction from C-float and construction from a string:
1199 #include <ginac/ginac.h>
1200 using namespace GiNaC;
1204 numeric two = 2; // exact integer 2
1205 numeric r(2,3); // exact fraction 2/3
1206 numeric e(2.71828); // floating point number
1207 numeric p = "3.14159265358979323846"; // constructor from string
1208 // Trott's constant in scientific notation:
1209 numeric trott("1.0841015122311136151E-2");
1211 std::cout << two*p << std::endl; // floating point 6.283...
1216 @cindex complex numbers
1217 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1222 numeric z1 = 2-3*I; // exact complex number 2-3i
1223 numeric z2 = 5.9+1.6*I; // complex floating point number
1227 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1228 This would, however, call C's built-in operator @code{/} for integers
1229 first and result in a numeric holding a plain integer 1. @strong{Never
1230 use the operator @code{/} on integers} unless you know exactly what you
1231 are doing! Use the constructor from two integers instead, as shown in
1232 the example above. Writing @code{numeric(1)/2} may look funny but works
1235 @cindex @code{Digits}
1237 We have seen now the distinction between exact numbers and floating
1238 point numbers. Clearly, the user should never have to worry about
1239 dynamically created exact numbers, since their `exactness' always
1240 determines how they ought to be handled, i.e. how `long' they are. The
1241 situation is different for floating point numbers. Their accuracy is
1242 controlled by one @emph{global} variable, called @code{Digits}. (For
1243 those readers who know about Maple: it behaves very much like Maple's
1244 @code{Digits}). All objects of class numeric that are constructed from
1245 then on will be stored with a precision matching that number of decimal
1250 #include <ginac/ginac.h>
1251 using namespace std;
1252 using namespace GiNaC;
1256 numeric three(3.0), one(1.0);
1257 numeric x = one/three;
1259 cout << "in " << Digits << " digits:" << endl;
1261 cout << Pi.evalf() << endl;
1273 The above example prints the following output to screen:
1277 0.33333333333333333334
1278 3.1415926535897932385
1280 0.33333333333333333333333333333333333333333333333333333333333333333334
1281 3.1415926535897932384626433832795028841971693993751058209749445923078
1285 Note that the last number is not necessarily rounded as you would
1286 naively expect it to be rounded in the decimal system. But note also,
1287 that in both cases you got a couple of extra digits. This is because
1288 numbers are internally stored by CLN as chunks of binary digits in order
1289 to match your machine's word size and to not waste precision. Thus, on
1290 architectures with different word size, the above output might even
1291 differ with regard to actually computed digits.
1293 It should be clear that objects of class @code{numeric} should be used
1294 for constructing numbers or for doing arithmetic with them. The objects
1295 one deals with most of the time are the polymorphic expressions @code{ex}.
1297 @subsection Tests on numbers
1299 Once you have declared some numbers, assigned them to expressions and
1300 done some arithmetic with them it is frequently desired to retrieve some
1301 kind of information from them like asking whether that number is
1302 integer, rational, real or complex. For those cases GiNaC provides
1303 several useful methods. (Internally, they fall back to invocations of
1304 certain CLN functions.)
1306 As an example, let's construct some rational number, multiply it with
1307 some multiple of its denominator and test what comes out:
1311 #include <ginac/ginac.h>
1312 using namespace std;
1313 using namespace GiNaC;
1315 // some very important constants:
1316 const numeric twentyone(21);
1317 const numeric ten(10);
1318 const numeric five(5);
1322 numeric answer = twentyone;
1325 cout << answer.is_integer() << endl; // false, it's 21/5
1327 cout << answer.is_integer() << endl; // true, it's 42 now!
1331 Note that the variable @code{answer} is constructed here as an integer
1332 by @code{numeric}'s copy constructor but in an intermediate step it
1333 holds a rational number represented as integer numerator and integer
1334 denominator. When multiplied by 10, the denominator becomes unity and
1335 the result is automatically converted to a pure integer again.
1336 Internally, the underlying CLN is responsible for this behavior and we
1337 refer the reader to CLN's documentation. Suffice to say that
1338 the same behavior applies to complex numbers as well as return values of
1339 certain functions. Complex numbers are automatically converted to real
1340 numbers if the imaginary part becomes zero. The full set of tests that
1341 can be applied is listed in the following table.
1344 @multitable @columnfractions .30 .70
1345 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1346 @item @code{.is_zero()}
1347 @tab @dots{}equal to zero
1348 @item @code{.is_positive()}
1349 @tab @dots{}not complex and greater than 0
1350 @item @code{.is_integer()}
1351 @tab @dots{}a (non-complex) integer
1352 @item @code{.is_pos_integer()}
1353 @tab @dots{}an integer and greater than 0
1354 @item @code{.is_nonneg_integer()}
1355 @tab @dots{}an integer and greater equal 0
1356 @item @code{.is_even()}
1357 @tab @dots{}an even integer
1358 @item @code{.is_odd()}
1359 @tab @dots{}an odd integer
1360 @item @code{.is_prime()}
1361 @tab @dots{}a prime integer (probabilistic primality test)
1362 @item @code{.is_rational()}
1363 @tab @dots{}an exact rational number (integers are rational, too)
1364 @item @code{.is_real()}
1365 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1366 @item @code{.is_cinteger()}
1367 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1368 @item @code{.is_crational()}
1369 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1373 @subsection Numeric functions
1375 The following functions can be applied to @code{numeric} objects and will be
1376 evaluated immediately:
1379 @multitable @columnfractions .30 .70
1380 @item @strong{Name} @tab @strong{Function}
1381 @item @code{inverse(z)}
1382 @tab returns @math{1/z}
1383 @cindex @code{inverse()} (numeric)
1384 @item @code{pow(a, b)}
1385 @tab exponentiation @math{a^b}
1388 @item @code{real(z)}
1390 @cindex @code{real()}
1391 @item @code{imag(z)}
1393 @cindex @code{imag()}
1394 @item @code{csgn(z)}
1395 @tab complex sign (returns an @code{int})
1396 @item @code{step(x)}
1397 @tab step function (returns an @code{numeric})
1398 @item @code{numer(z)}
1399 @tab numerator of rational or complex rational number
1400 @item @code{denom(z)}
1401 @tab denominator of rational or complex rational number
1402 @item @code{sqrt(z)}
1404 @item @code{isqrt(n)}
1405 @tab integer square root
1406 @cindex @code{isqrt()}
1413 @item @code{asin(z)}
1415 @item @code{acos(z)}
1417 @item @code{atan(z)}
1418 @tab inverse tangent
1419 @item @code{atan(y, x)}
1420 @tab inverse tangent with two arguments
1421 @item @code{sinh(z)}
1422 @tab hyperbolic sine
1423 @item @code{cosh(z)}
1424 @tab hyperbolic cosine
1425 @item @code{tanh(z)}
1426 @tab hyperbolic tangent
1427 @item @code{asinh(z)}
1428 @tab inverse hyperbolic sine
1429 @item @code{acosh(z)}
1430 @tab inverse hyperbolic cosine
1431 @item @code{atanh(z)}
1432 @tab inverse hyperbolic tangent
1434 @tab exponential function
1436 @tab natural logarithm
1439 @item @code{zeta(z)}
1440 @tab Riemann's zeta function
1441 @item @code{tgamma(z)}
1443 @item @code{lgamma(z)}
1444 @tab logarithm of gamma function
1446 @tab psi (digamma) function
1447 @item @code{psi(n, z)}
1448 @tab derivatives of psi function (polygamma functions)
1449 @item @code{factorial(n)}
1450 @tab factorial function @math{n!}
1451 @item @code{doublefactorial(n)}
1452 @tab double factorial function @math{n!!}
1453 @cindex @code{doublefactorial()}
1454 @item @code{binomial(n, k)}
1455 @tab binomial coefficients
1456 @item @code{bernoulli(n)}
1457 @tab Bernoulli numbers
1458 @cindex @code{bernoulli()}
1459 @item @code{fibonacci(n)}
1460 @tab Fibonacci numbers
1461 @cindex @code{fibonacci()}
1462 @item @code{mod(a, b)}
1463 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1464 @cindex @code{mod()}
1465 @item @code{smod(a, b)}
1466 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1467 @cindex @code{smod()}
1468 @item @code{irem(a, b)}
1469 @tab integer remainder (has the sign of @math{a}, or is zero)
1470 @cindex @code{irem()}
1471 @item @code{irem(a, b, q)}
1472 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1473 @item @code{iquo(a, b)}
1474 @tab integer quotient
1475 @cindex @code{iquo()}
1476 @item @code{iquo(a, b, r)}
1477 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1478 @item @code{gcd(a, b)}
1479 @tab greatest common divisor
1480 @item @code{lcm(a, b)}
1481 @tab least common multiple
1485 Most of these functions are also available as symbolic functions that can be
1486 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1487 as polynomial algorithms.
1489 @subsection Converting numbers
1491 Sometimes it is desirable to convert a @code{numeric} object back to a
1492 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1493 class provides a couple of methods for this purpose:
1495 @cindex @code{to_int()}
1496 @cindex @code{to_long()}
1497 @cindex @code{to_double()}
1498 @cindex @code{to_cl_N()}
1500 int numeric::to_int() const;
1501 long numeric::to_long() const;
1502 double numeric::to_double() const;
1503 cln::cl_N numeric::to_cl_N() const;
1506 @code{to_int()} and @code{to_long()} only work when the number they are
1507 applied on is an exact integer. Otherwise the program will halt with a
1508 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1509 rational number will return a floating-point approximation. Both
1510 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1511 part of complex numbers.
1514 @node Constants, Fundamental containers, Numbers, Basic concepts
1515 @c node-name, next, previous, up
1517 @cindex @code{constant} (class)
1520 @cindex @code{Catalan}
1521 @cindex @code{Euler}
1522 @cindex @code{evalf()}
1523 Constants behave pretty much like symbols except that they return some
1524 specific number when the method @code{.evalf()} is called.
1526 The predefined known constants are:
1529 @multitable @columnfractions .14 .30 .56
1530 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1532 @tab Archimedes' constant
1533 @tab 3.14159265358979323846264338327950288
1534 @item @code{Catalan}
1535 @tab Catalan's constant
1536 @tab 0.91596559417721901505460351493238411
1538 @tab Euler's (or Euler-Mascheroni) constant
1539 @tab 0.57721566490153286060651209008240243
1544 @node Fundamental containers, Lists, Constants, Basic concepts
1545 @c node-name, next, previous, up
1546 @section Sums, products and powers
1550 @cindex @code{power}
1552 Simple rational expressions are written down in GiNaC pretty much like
1553 in other CAS or like expressions involving numerical variables in C.
1554 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1555 been overloaded to achieve this goal. When you run the following
1556 code snippet, the constructor for an object of type @code{mul} is
1557 automatically called to hold the product of @code{a} and @code{b} and
1558 then the constructor for an object of type @code{add} is called to hold
1559 the sum of that @code{mul} object and the number one:
1563 symbol a("a"), b("b");
1568 @cindex @code{pow()}
1569 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1570 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1571 construction is necessary since we cannot safely overload the constructor
1572 @code{^} in C++ to construct a @code{power} object. If we did, it would
1573 have several counterintuitive and undesired effects:
1577 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1579 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1580 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1581 interpret this as @code{x^(a^b)}.
1583 Also, expressions involving integer exponents are very frequently used,
1584 which makes it even more dangerous to overload @code{^} since it is then
1585 hard to distinguish between the semantics as exponentiation and the one
1586 for exclusive or. (It would be embarrassing to return @code{1} where one
1587 has requested @code{2^3}.)
1590 @cindex @command{ginsh}
1591 All effects are contrary to mathematical notation and differ from the
1592 way most other CAS handle exponentiation, therefore overloading @code{^}
1593 is ruled out for GiNaC's C++ part. The situation is different in
1594 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1595 that the other frequently used exponentiation operator @code{**} does
1596 not exist at all in C++).
1598 To be somewhat more precise, objects of the three classes described
1599 here, are all containers for other expressions. An object of class
1600 @code{power} is best viewed as a container with two slots, one for the
1601 basis, one for the exponent. All valid GiNaC expressions can be
1602 inserted. However, basic transformations like simplifying
1603 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1604 when this is mathematically possible. If we replace the outer exponent
1605 three in the example by some symbols @code{a}, the simplification is not
1606 safe and will not be performed, since @code{a} might be @code{1/2} and
1609 Objects of type @code{add} and @code{mul} are containers with an
1610 arbitrary number of slots for expressions to be inserted. Again, simple
1611 and safe simplifications are carried out like transforming
1612 @code{3*x+4-x} to @code{2*x+4}.
1615 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1616 @c node-name, next, previous, up
1617 @section Lists of expressions
1618 @cindex @code{lst} (class)
1620 @cindex @code{nops()}
1622 @cindex @code{append()}
1623 @cindex @code{prepend()}
1624 @cindex @code{remove_first()}
1625 @cindex @code{remove_last()}
1626 @cindex @code{remove_all()}
1628 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1629 expressions. They are not as ubiquitous as in many other computer algebra
1630 packages, but are sometimes used to supply a variable number of arguments of
1631 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1632 constructors, so you should have a basic understanding of them.
1634 Lists can be constructed by assigning a comma-separated sequence of
1639 symbol x("x"), y("y");
1642 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1647 There are also constructors that allow direct creation of lists of up to
1648 16 expressions, which is often more convenient but slightly less efficient:
1652 // This produces the same list 'l' as above:
1653 // lst l(x, 2, y, x+y);
1654 // lst l = lst(x, 2, y, x+y);
1658 Use the @code{nops()} method to determine the size (number of expressions) of
1659 a list and the @code{op()} method or the @code{[]} operator to access
1660 individual elements:
1664 cout << l.nops() << endl; // prints '4'
1665 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1669 As with the standard @code{list<T>} container, accessing random elements of a
1670 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1671 sequential access to the elements of a list is possible with the
1672 iterator types provided by the @code{lst} class:
1675 typedef ... lst::const_iterator;
1676 typedef ... lst::const_reverse_iterator;
1677 lst::const_iterator lst::begin() const;
1678 lst::const_iterator lst::end() const;
1679 lst::const_reverse_iterator lst::rbegin() const;
1680 lst::const_reverse_iterator lst::rend() const;
1683 For example, to print the elements of a list individually you can use:
1688 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1693 which is one order faster than
1698 for (size_t i = 0; i < l.nops(); ++i)
1699 cout << l.op(i) << endl;
1703 These iterators also allow you to use some of the algorithms provided by
1704 the C++ standard library:
1708 // print the elements of the list (requires #include <iterator>)
1709 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1711 // sum up the elements of the list (requires #include <numeric>)
1712 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1713 cout << sum << endl; // prints '2+2*x+2*y'
1717 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1718 (the only other one is @code{matrix}). You can modify single elements:
1722 l[1] = 42; // l is now @{x, 42, y, x+y@}
1723 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1727 You can append or prepend an expression to a list with the @code{append()}
1728 and @code{prepend()} methods:
1732 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1733 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1737 You can remove the first or last element of a list with @code{remove_first()}
1738 and @code{remove_last()}:
1742 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1743 l.remove_last(); // l is now @{x, 7, y, x+y@}
1747 You can remove all the elements of a list with @code{remove_all()}:
1751 l.remove_all(); // l is now empty
1755 You can bring the elements of a list into a canonical order with @code{sort()}:
1764 // l1 and l2 are now equal
1768 Finally, you can remove all but the first element of consecutive groups of
1769 elements with @code{unique()}:
1774 l3 = x, 2, 2, 2, y, x+y, y+x;
1775 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1780 @node Mathematical functions, Relations, Lists, Basic concepts
1781 @c node-name, next, previous, up
1782 @section Mathematical functions
1783 @cindex @code{function} (class)
1784 @cindex trigonometric function
1785 @cindex hyperbolic function
1787 There are quite a number of useful functions hard-wired into GiNaC. For
1788 instance, all trigonometric and hyperbolic functions are implemented
1789 (@xref{Built-in functions}, for a complete list).
1791 These functions (better called @emph{pseudofunctions}) are all objects
1792 of class @code{function}. They accept one or more expressions as
1793 arguments and return one expression. If the arguments are not
1794 numerical, the evaluation of the function may be halted, as it does in
1795 the next example, showing how a function returns itself twice and
1796 finally an expression that may be really useful:
1798 @cindex Gamma function
1799 @cindex @code{subs()}
1802 symbol x("x"), y("y");
1804 cout << tgamma(foo) << endl;
1805 // -> tgamma(x+(1/2)*y)
1806 ex bar = foo.subs(y==1);
1807 cout << tgamma(bar) << endl;
1809 ex foobar = bar.subs(x==7);
1810 cout << tgamma(foobar) << endl;
1811 // -> (135135/128)*Pi^(1/2)
1815 Besides evaluation most of these functions allow differentiation, series
1816 expansion and so on. Read the next chapter in order to learn more about
1819 It must be noted that these pseudofunctions are created by inline
1820 functions, where the argument list is templated. This means that
1821 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1822 @code{sin(ex(1))} and will therefore not result in a floating point
1823 number. Unless of course the function prototype is explicitly
1824 overridden -- which is the case for arguments of type @code{numeric}
1825 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1826 point number of class @code{numeric} you should call
1827 @code{sin(numeric(1))}. This is almost the same as calling
1828 @code{sin(1).evalf()} except that the latter will return a numeric
1829 wrapped inside an @code{ex}.
1832 @node Relations, Integrals, Mathematical functions, Basic concepts
1833 @c node-name, next, previous, up
1835 @cindex @code{relational} (class)
1837 Sometimes, a relation holding between two expressions must be stored
1838 somehow. The class @code{relational} is a convenient container for such
1839 purposes. A relation is by definition a container for two @code{ex} and
1840 a relation between them that signals equality, inequality and so on.
1841 They are created by simply using the C++ operators @code{==}, @code{!=},
1842 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1844 @xref{Mathematical functions}, for examples where various applications
1845 of the @code{.subs()} method show how objects of class relational are
1846 used as arguments. There they provide an intuitive syntax for
1847 substitutions. They are also used as arguments to the @code{ex::series}
1848 method, where the left hand side of the relation specifies the variable
1849 to expand in and the right hand side the expansion point. They can also
1850 be used for creating systems of equations that are to be solved for
1851 unknown variables. But the most common usage of objects of this class
1852 is rather inconspicuous in statements of the form @code{if
1853 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1854 conversion from @code{relational} to @code{bool} takes place. Note,
1855 however, that @code{==} here does not perform any simplifications, hence
1856 @code{expand()} must be called explicitly.
1858 @node Integrals, Matrices, Relations, Basic concepts
1859 @c node-name, next, previous, up
1861 @cindex @code{integral} (class)
1863 An object of class @dfn{integral} can be used to hold a symbolic integral.
1864 If you want to symbolically represent the integral of @code{x*x} from 0 to
1865 1, you would write this as
1867 integral(x, 0, 1, x*x)
1869 The first argument is the integration variable. It should be noted that
1870 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1871 fact, it can only integrate polynomials. An expression containing integrals
1872 can be evaluated symbolically by calling the
1876 method on it. Numerical evaluation is available by calling the
1880 method on an expression containing the integral. This will only evaluate
1881 integrals into a number if @code{subs}ing the integration variable by a
1882 number in the fourth argument of an integral and then @code{evalf}ing the
1883 result always results in a number. Of course, also the boundaries of the
1884 integration domain must @code{evalf} into numbers. It should be noted that
1885 trying to @code{evalf} a function with discontinuities in the integration
1886 domain is not recommended. The accuracy of the numeric evaluation of
1887 integrals is determined by the static member variable
1889 ex integral::relative_integration_error
1891 of the class @code{integral}. The default value of this is 10^-8.
1892 The integration works by halving the interval of integration, until numeric
1893 stability of the answer indicates that the requested accuracy has been
1894 reached. The maximum depth of the halving can be set via the static member
1897 int integral::max_integration_level
1899 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1900 return the integral unevaluated. The function that performs the numerical
1901 evaluation, is also available as
1903 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1906 This function will throw an exception if the maximum depth is exceeded. The
1907 last parameter of the function is optional and defaults to the
1908 @code{relative_integration_error}. To make sure that we do not do too
1909 much work if an expression contains the same integral multiple times,
1910 a lookup table is used.
1912 If you know that an expression holds an integral, you can get the
1913 integration variable, the left boundary, right boundary and integrand by
1914 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1915 @code{.op(3)}. Differentiating integrals with respect to variables works
1916 as expected. Note that it makes no sense to differentiate an integral
1917 with respect to the integration variable.
1919 @node Matrices, Indexed objects, Integrals, Basic concepts
1920 @c node-name, next, previous, up
1922 @cindex @code{matrix} (class)
1924 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1925 matrix with @math{m} rows and @math{n} columns are accessed with two
1926 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1927 second one in the range 0@dots{}@math{n-1}.
1929 There are a couple of ways to construct matrices, with or without preset
1930 elements. The constructor
1933 matrix::matrix(unsigned r, unsigned c);
1936 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1939 The fastest way to create a matrix with preinitialized elements is to assign
1940 a list of comma-separated expressions to an empty matrix (see below for an
1941 example). But you can also specify the elements as a (flat) list with
1944 matrix::matrix(unsigned r, unsigned c, const lst & l);
1949 @cindex @code{lst_to_matrix()}
1951 ex lst_to_matrix(const lst & l);
1954 constructs a matrix from a list of lists, each list representing a matrix row.
1956 There is also a set of functions for creating some special types of
1959 @cindex @code{diag_matrix()}
1960 @cindex @code{unit_matrix()}
1961 @cindex @code{symbolic_matrix()}
1963 ex diag_matrix(const lst & l);
1964 ex unit_matrix(unsigned x);
1965 ex unit_matrix(unsigned r, unsigned c);
1966 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1967 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1968 const string & tex_base_name);
1971 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1972 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1973 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1974 matrix filled with newly generated symbols made of the specified base name
1975 and the position of each element in the matrix.
1977 Matrices often arise by omitting elements of another matrix. For
1978 instance, the submatrix @code{S} of a matrix @code{M} takes a
1979 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1980 by removing one row and one column from a matrix @code{M}. (The
1981 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1982 can be used for computing the inverse using Cramer's rule.)
1984 @cindex @code{sub_matrix()}
1985 @cindex @code{reduced_matrix()}
1987 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1988 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1991 The function @code{sub_matrix()} takes a row offset @code{r} and a
1992 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1993 columns. The function @code{reduced_matrix()} has two integer arguments
1994 that specify which row and column to remove:
2002 cout << reduced_matrix(m, 1, 1) << endl;
2003 // -> [[11,13],[31,33]]
2004 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2005 // -> [[22,23],[32,33]]
2009 Matrix elements can be accessed and set using the parenthesis (function call)
2013 const ex & matrix::operator()(unsigned r, unsigned c) const;
2014 ex & matrix::operator()(unsigned r, unsigned c);
2017 It is also possible to access the matrix elements in a linear fashion with
2018 the @code{op()} method. But C++-style subscripting with square brackets
2019 @samp{[]} is not available.
2021 Here are a couple of examples for constructing matrices:
2025 symbol a("a"), b("b");
2039 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2042 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2045 cout << diag_matrix(lst(a, b)) << endl;
2048 cout << unit_matrix(3) << endl;
2049 // -> [[1,0,0],[0,1,0],[0,0,1]]
2051 cout << symbolic_matrix(2, 3, "x") << endl;
2052 // -> [[x00,x01,x02],[x10,x11,x12]]
2056 @cindex @code{is_zero_matrix()}
2057 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2058 all entries of the matrix are zeros. There is also method
2059 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2060 expression is zero or a zero matrix.
2062 @cindex @code{transpose()}
2063 There are three ways to do arithmetic with matrices. The first (and most
2064 direct one) is to use the methods provided by the @code{matrix} class:
2067 matrix matrix::add(const matrix & other) const;
2068 matrix matrix::sub(const matrix & other) const;
2069 matrix matrix::mul(const matrix & other) const;
2070 matrix matrix::mul_scalar(const ex & other) const;
2071 matrix matrix::pow(const ex & expn) const;
2072 matrix matrix::transpose() const;
2075 All of these methods return the result as a new matrix object. Here is an
2076 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2081 matrix A(2, 2), B(2, 2), C(2, 2);
2089 matrix result = A.mul(B).sub(C.mul_scalar(2));
2090 cout << result << endl;
2091 // -> [[-13,-6],[1,2]]
2096 @cindex @code{evalm()}
2097 The second (and probably the most natural) way is to construct an expression
2098 containing matrices with the usual arithmetic operators and @code{pow()}.
2099 For efficiency reasons, expressions with sums, products and powers of
2100 matrices are not automatically evaluated in GiNaC. You have to call the
2104 ex ex::evalm() const;
2107 to obtain the result:
2114 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2115 cout << e.evalm() << endl;
2116 // -> [[-13,-6],[1,2]]
2121 The non-commutativity of the product @code{A*B} in this example is
2122 automatically recognized by GiNaC. There is no need to use a special
2123 operator here. @xref{Non-commutative objects}, for more information about
2124 dealing with non-commutative expressions.
2126 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2127 to perform the arithmetic:
2132 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2133 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2135 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2136 cout << e.simplify_indexed() << endl;
2137 // -> [[-13,-6],[1,2]].i.j
2141 Using indices is most useful when working with rectangular matrices and
2142 one-dimensional vectors because you don't have to worry about having to
2143 transpose matrices before multiplying them. @xref{Indexed objects}, for
2144 more information about using matrices with indices, and about indices in
2147 The @code{matrix} class provides a couple of additional methods for
2148 computing determinants, traces, characteristic polynomials and ranks:
2150 @cindex @code{determinant()}
2151 @cindex @code{trace()}
2152 @cindex @code{charpoly()}
2153 @cindex @code{rank()}
2155 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2156 ex matrix::trace() const;
2157 ex matrix::charpoly(const ex & lambda) const;
2158 unsigned matrix::rank() const;
2161 The @samp{algo} argument of @code{determinant()} allows to select
2162 between different algorithms for calculating the determinant. The
2163 asymptotic speed (as parametrized by the matrix size) can greatly differ
2164 between those algorithms, depending on the nature of the matrix'
2165 entries. The possible values are defined in the @file{flags.h} header
2166 file. By default, GiNaC uses a heuristic to automatically select an
2167 algorithm that is likely (but not guaranteed) to give the result most
2170 @cindex @code{inverse()} (matrix)
2171 @cindex @code{solve()}
2172 Matrices may also be inverted using the @code{ex matrix::inverse()}
2173 method and linear systems may be solved with:
2176 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2177 unsigned algo=solve_algo::automatic) const;
2180 Assuming the matrix object this method is applied on is an @code{m}
2181 times @code{n} matrix, then @code{vars} must be a @code{n} times
2182 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2183 times @code{p} matrix. The returned matrix then has dimension @code{n}
2184 times @code{p} and in the case of an underdetermined system will still
2185 contain some of the indeterminates from @code{vars}. If the system is
2186 overdetermined, an exception is thrown.
2189 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2190 @c node-name, next, previous, up
2191 @section Indexed objects
2193 GiNaC allows you to handle expressions containing general indexed objects in
2194 arbitrary spaces. It is also able to canonicalize and simplify such
2195 expressions and perform symbolic dummy index summations. There are a number
2196 of predefined indexed objects provided, like delta and metric tensors.
2198 There are few restrictions placed on indexed objects and their indices and
2199 it is easy to construct nonsense expressions, but our intention is to
2200 provide a general framework that allows you to implement algorithms with
2201 indexed quantities, getting in the way as little as possible.
2203 @cindex @code{idx} (class)
2204 @cindex @code{indexed} (class)
2205 @subsection Indexed quantities and their indices
2207 Indexed expressions in GiNaC are constructed of two special types of objects,
2208 @dfn{index objects} and @dfn{indexed objects}.
2212 @cindex contravariant
2215 @item Index objects are of class @code{idx} or a subclass. Every index has
2216 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2217 the index lives in) which can both be arbitrary expressions but are usually
2218 a number or a simple symbol. In addition, indices of class @code{varidx} have
2219 a @dfn{variance} (they can be co- or contravariant), and indices of class
2220 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2222 @item Indexed objects are of class @code{indexed} or a subclass. They
2223 contain a @dfn{base expression} (which is the expression being indexed), and
2224 one or more indices.
2228 @strong{Please notice:} when printing expressions, covariant indices and indices
2229 without variance are denoted @samp{.i} while contravariant indices are
2230 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2231 value. In the following, we are going to use that notation in the text so
2232 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2233 not visible in the output.
2235 A simple example shall illustrate the concepts:
2239 #include <ginac/ginac.h>
2240 using namespace std;
2241 using namespace GiNaC;
2245 symbol i_sym("i"), j_sym("j");
2246 idx i(i_sym, 3), j(j_sym, 3);
2249 cout << indexed(A, i, j) << endl;
2251 cout << index_dimensions << indexed(A, i, j) << endl;
2253 cout << dflt; // reset cout to default output format (dimensions hidden)
2257 The @code{idx} constructor takes two arguments, the index value and the
2258 index dimension. First we define two index objects, @code{i} and @code{j},
2259 both with the numeric dimension 3. The value of the index @code{i} is the
2260 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2261 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2262 construct an expression containing one indexed object, @samp{A.i.j}. It has
2263 the symbol @code{A} as its base expression and the two indices @code{i} and
2266 The dimensions of indices are normally not visible in the output, but one
2267 can request them to be printed with the @code{index_dimensions} manipulator,
2270 Note the difference between the indices @code{i} and @code{j} which are of
2271 class @code{idx}, and the index values which are the symbols @code{i_sym}
2272 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2273 or numbers but must be index objects. For example, the following is not
2274 correct and will raise an exception:
2277 symbol i("i"), j("j");
2278 e = indexed(A, i, j); // ERROR: indices must be of type idx
2281 You can have multiple indexed objects in an expression, index values can
2282 be numeric, and index dimensions symbolic:
2286 symbol B("B"), dim("dim");
2287 cout << 4 * indexed(A, i)
2288 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2293 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2294 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2295 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2296 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2297 @code{simplify_indexed()} for that, see below).
2299 In fact, base expressions, index values and index dimensions can be
2300 arbitrary expressions:
2304 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2309 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2310 get an error message from this but you will probably not be able to do
2311 anything useful with it.
2313 @cindex @code{get_value()}
2314 @cindex @code{get_dimension()}
2318 ex idx::get_value();
2319 ex idx::get_dimension();
2322 return the value and dimension of an @code{idx} object. If you have an index
2323 in an expression, such as returned by calling @code{.op()} on an indexed
2324 object, you can get a reference to the @code{idx} object with the function
2325 @code{ex_to<idx>()} on the expression.
2327 There are also the methods
2330 bool idx::is_numeric();
2331 bool idx::is_symbolic();
2332 bool idx::is_dim_numeric();
2333 bool idx::is_dim_symbolic();
2336 for checking whether the value and dimension are numeric or symbolic
2337 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2338 about expressions}) returns information about the index value.
2340 @cindex @code{varidx} (class)
2341 If you need co- and contravariant indices, use the @code{varidx} class:
2345 symbol mu_sym("mu"), nu_sym("nu");
2346 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2347 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2349 cout << indexed(A, mu, nu) << endl;
2351 cout << indexed(A, mu_co, nu) << endl;
2353 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2358 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2359 co- or contravariant. The default is a contravariant (upper) index, but
2360 this can be overridden by supplying a third argument to the @code{varidx}
2361 constructor. The two methods
2364 bool varidx::is_covariant();
2365 bool varidx::is_contravariant();
2368 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2369 to get the object reference from an expression). There's also the very useful
2373 ex varidx::toggle_variance();
2376 which makes a new index with the same value and dimension but the opposite
2377 variance. By using it you only have to define the index once.
2379 @cindex @code{spinidx} (class)
2380 The @code{spinidx} class provides dotted and undotted variant indices, as
2381 used in the Weyl-van-der-Waerden spinor formalism:
2385 symbol K("K"), C_sym("C"), D_sym("D");
2386 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2387 // contravariant, undotted
2388 spinidx C_co(C_sym, 2, true); // covariant index
2389 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2390 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2392 cout << indexed(K, C, D) << endl;
2394 cout << indexed(K, C_co, D_dot) << endl;
2396 cout << indexed(K, D_co_dot, D) << endl;
2401 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2402 dotted or undotted. The default is undotted but this can be overridden by
2403 supplying a fourth argument to the @code{spinidx} constructor. The two
2407 bool spinidx::is_dotted();
2408 bool spinidx::is_undotted();
2411 allow you to check whether or not a @code{spinidx} object is dotted (use
2412 @code{ex_to<spinidx>()} to get the object reference from an expression).
2413 Finally, the two methods
2416 ex spinidx::toggle_dot();
2417 ex spinidx::toggle_variance_dot();
2420 create a new index with the same value and dimension but opposite dottedness
2421 and the same or opposite variance.
2423 @subsection Substituting indices
2425 @cindex @code{subs()}
2426 Sometimes you will want to substitute one symbolic index with another
2427 symbolic or numeric index, for example when calculating one specific element
2428 of a tensor expression. This is done with the @code{.subs()} method, as it
2429 is done for symbols (see @ref{Substituting expressions}).
2431 You have two possibilities here. You can either substitute the whole index
2432 by another index or expression:
2436 ex e = indexed(A, mu_co);
2437 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2438 // -> A.mu becomes A~nu
2439 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2440 // -> A.mu becomes A~0
2441 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2442 // -> A.mu becomes A.0
2446 The third example shows that trying to replace an index with something that
2447 is not an index will substitute the index value instead.
2449 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2454 ex e = indexed(A, mu_co);
2455 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2456 // -> A.mu becomes A.nu
2457 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2458 // -> A.mu becomes A.0
2462 As you see, with the second method only the value of the index will get
2463 substituted. Its other properties, including its dimension, remain unchanged.
2464 If you want to change the dimension of an index you have to substitute the
2465 whole index by another one with the new dimension.
2467 Finally, substituting the base expression of an indexed object works as
2472 ex e = indexed(A, mu_co);
2473 cout << e << " becomes " << e.subs(A == A+B) << endl;
2474 // -> A.mu becomes (B+A).mu
2478 @subsection Symmetries
2479 @cindex @code{symmetry} (class)
2480 @cindex @code{sy_none()}
2481 @cindex @code{sy_symm()}
2482 @cindex @code{sy_anti()}
2483 @cindex @code{sy_cycl()}
2485 Indexed objects can have certain symmetry properties with respect to their
2486 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2487 that is constructed with the helper functions
2490 symmetry sy_none(...);
2491 symmetry sy_symm(...);
2492 symmetry sy_anti(...);
2493 symmetry sy_cycl(...);
2496 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2497 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2498 represents a cyclic symmetry. Each of these functions accepts up to four
2499 arguments which can be either symmetry objects themselves or unsigned integer
2500 numbers that represent an index position (counting from 0). A symmetry
2501 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2502 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2505 Here are some examples of symmetry definitions:
2510 e = indexed(A, i, j);
2511 e = indexed(A, sy_none(), i, j); // equivalent
2512 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2514 // Symmetric in all three indices:
2515 e = indexed(A, sy_symm(), i, j, k);
2516 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2517 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2518 // different canonical order
2520 // Symmetric in the first two indices only:
2521 e = indexed(A, sy_symm(0, 1), i, j, k);
2522 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2524 // Antisymmetric in the first and last index only (index ranges need not
2526 e = indexed(A, sy_anti(0, 2), i, j, k);
2527 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2529 // An example of a mixed symmetry: antisymmetric in the first two and
2530 // last two indices, symmetric when swapping the first and last index
2531 // pairs (like the Riemann curvature tensor):
2532 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2534 // Cyclic symmetry in all three indices:
2535 e = indexed(A, sy_cycl(), i, j, k);
2536 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2538 // The following examples are invalid constructions that will throw
2539 // an exception at run time.
2541 // An index may not appear multiple times:
2542 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2543 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2545 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2546 // same number of indices:
2547 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2549 // And of course, you cannot specify indices which are not there:
2550 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2554 If you need to specify more than four indices, you have to use the
2555 @code{.add()} method of the @code{symmetry} class. For example, to specify
2556 full symmetry in the first six indices you would write
2557 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2559 If an indexed object has a symmetry, GiNaC will automatically bring the
2560 indices into a canonical order which allows for some immediate simplifications:
2564 cout << indexed(A, sy_symm(), i, j)
2565 + indexed(A, sy_symm(), j, i) << endl;
2567 cout << indexed(B, sy_anti(), i, j)
2568 + indexed(B, sy_anti(), j, i) << endl;
2570 cout << indexed(B, sy_anti(), i, j, k)
2571 - indexed(B, sy_anti(), j, k, i) << endl;
2576 @cindex @code{get_free_indices()}
2578 @subsection Dummy indices
2580 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2581 that a summation over the index range is implied. Symbolic indices which are
2582 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2583 dummy nor free indices.
2585 To be recognized as a dummy index pair, the two indices must be of the same
2586 class and their value must be the same single symbol (an index like
2587 @samp{2*n+1} is never a dummy index). If the indices are of class
2588 @code{varidx} they must also be of opposite variance; if they are of class
2589 @code{spinidx} they must be both dotted or both undotted.
2591 The method @code{.get_free_indices()} returns a vector containing the free
2592 indices of an expression. It also checks that the free indices of the terms
2593 of a sum are consistent:
2597 symbol A("A"), B("B"), C("C");
2599 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2600 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2602 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2603 cout << exprseq(e.get_free_indices()) << endl;
2605 // 'j' and 'l' are dummy indices
2607 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2608 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2610 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2611 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2612 cout << exprseq(e.get_free_indices()) << endl;
2614 // 'nu' is a dummy index, but 'sigma' is not
2616 e = indexed(A, mu, mu);
2617 cout << exprseq(e.get_free_indices()) << endl;
2619 // 'mu' is not a dummy index because it appears twice with the same
2622 e = indexed(A, mu, nu) + 42;
2623 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2624 // this will throw an exception:
2625 // "add::get_free_indices: inconsistent indices in sum"
2629 @cindex @code{expand_dummy_sum()}
2630 A dummy index summation like
2637 can be expanded for indices with numeric
2638 dimensions (e.g. 3) into the explicit sum like
2640 $a_1b^1+a_2b^2+a_3b^3 $.
2643 a.1 b~1 + a.2 b~2 + a.3 b~3.
2645 This is performed by the function
2648 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2651 which takes an expression @code{e} and returns the expanded sum for all
2652 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2653 is set to @code{true} then all substitutions are made by @code{idx} class
2654 indices, i.e. without variance. In this case the above sum
2663 $a_1b_1+a_2b_2+a_3b_3 $.
2666 a.1 b.1 + a.2 b.2 + a.3 b.3.
2670 @cindex @code{simplify_indexed()}
2671 @subsection Simplifying indexed expressions
2673 In addition to the few automatic simplifications that GiNaC performs on
2674 indexed expressions (such as re-ordering the indices of symmetric tensors
2675 and calculating traces and convolutions of matrices and predefined tensors)
2679 ex ex::simplify_indexed();
2680 ex ex::simplify_indexed(const scalar_products & sp);
2683 that performs some more expensive operations:
2686 @item it checks the consistency of free indices in sums in the same way
2687 @code{get_free_indices()} does
2688 @item it tries to give dummy indices that appear in different terms of a sum
2689 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2690 @item it (symbolically) calculates all possible dummy index summations/contractions
2691 with the predefined tensors (this will be explained in more detail in the
2693 @item it detects contractions that vanish for symmetry reasons, for example
2694 the contraction of a symmetric and a totally antisymmetric tensor
2695 @item as a special case of dummy index summation, it can replace scalar products
2696 of two tensors with a user-defined value
2699 The last point is done with the help of the @code{scalar_products} class
2700 which is used to store scalar products with known values (this is not an
2701 arithmetic class, you just pass it to @code{simplify_indexed()}):
2705 symbol A("A"), B("B"), C("C"), i_sym("i");
2709 sp.add(A, B, 0); // A and B are orthogonal
2710 sp.add(A, C, 0); // A and C are orthogonal
2711 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2713 e = indexed(A + B, i) * indexed(A + C, i);
2715 // -> (B+A).i*(A+C).i
2717 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2723 The @code{scalar_products} object @code{sp} acts as a storage for the
2724 scalar products added to it with the @code{.add()} method. This method
2725 takes three arguments: the two expressions of which the scalar product is
2726 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2727 @code{simplify_indexed()} will replace all scalar products of indexed
2728 objects that have the symbols @code{A} and @code{B} as base expressions
2729 with the single value 0. The number, type and dimension of the indices
2730 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2732 @cindex @code{expand()}
2733 The example above also illustrates a feature of the @code{expand()} method:
2734 if passed the @code{expand_indexed} option it will distribute indices
2735 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2737 @cindex @code{tensor} (class)
2738 @subsection Predefined tensors
2740 Some frequently used special tensors such as the delta, epsilon and metric
2741 tensors are predefined in GiNaC. They have special properties when
2742 contracted with other tensor expressions and some of them have constant
2743 matrix representations (they will evaluate to a number when numeric
2744 indices are specified).
2746 @cindex @code{delta_tensor()}
2747 @subsubsection Delta tensor
2749 The delta tensor takes two indices, is symmetric and has the matrix
2750 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2751 @code{delta_tensor()}:
2755 symbol A("A"), B("B");
2757 idx i(symbol("i"), 3), j(symbol("j"), 3),
2758 k(symbol("k"), 3), l(symbol("l"), 3);
2760 ex e = indexed(A, i, j) * indexed(B, k, l)
2761 * delta_tensor(i, k) * delta_tensor(j, l);
2762 cout << e.simplify_indexed() << endl;
2765 cout << delta_tensor(i, i) << endl;
2770 @cindex @code{metric_tensor()}
2771 @subsubsection General metric tensor
2773 The function @code{metric_tensor()} creates a general symmetric metric
2774 tensor with two indices that can be used to raise/lower tensor indices. The
2775 metric tensor is denoted as @samp{g} in the output and if its indices are of
2776 mixed variance it is automatically replaced by a delta tensor:
2782 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2784 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2785 cout << e.simplify_indexed() << endl;
2788 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2789 cout << e.simplify_indexed() << endl;
2792 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2793 * metric_tensor(nu, rho);
2794 cout << e.simplify_indexed() << endl;
2797 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2798 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2799 + indexed(A, mu.toggle_variance(), rho));
2800 cout << e.simplify_indexed() << endl;
2805 @cindex @code{lorentz_g()}
2806 @subsubsection Minkowski metric tensor
2808 The Minkowski metric tensor is a special metric tensor with a constant
2809 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2810 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2811 It is created with the function @code{lorentz_g()} (although it is output as
2816 varidx mu(symbol("mu"), 4);
2818 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2819 * lorentz_g(mu, varidx(0, 4)); // negative signature
2820 cout << e.simplify_indexed() << endl;
2823 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2824 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2825 cout << e.simplify_indexed() << endl;
2830 @cindex @code{spinor_metric()}
2831 @subsubsection Spinor metric tensor
2833 The function @code{spinor_metric()} creates an antisymmetric tensor with
2834 two indices that is used to raise/lower indices of 2-component spinors.
2835 It is output as @samp{eps}:
2841 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2842 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2844 e = spinor_metric(A, B) * indexed(psi, B_co);
2845 cout << e.simplify_indexed() << endl;
2848 e = spinor_metric(A, B) * indexed(psi, A_co);
2849 cout << e.simplify_indexed() << endl;
2852 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2853 cout << e.simplify_indexed() << endl;
2856 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2857 cout << e.simplify_indexed() << endl;
2860 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2861 cout << e.simplify_indexed() << endl;
2864 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2865 cout << e.simplify_indexed() << endl;
2870 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2872 @cindex @code{epsilon_tensor()}
2873 @cindex @code{lorentz_eps()}
2874 @subsubsection Epsilon tensor
2876 The epsilon tensor is totally antisymmetric, its number of indices is equal
2877 to the dimension of the index space (the indices must all be of the same
2878 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2879 defined to be 1. Its behavior with indices that have a variance also
2880 depends on the signature of the metric. Epsilon tensors are output as
2883 There are three functions defined to create epsilon tensors in 2, 3 and 4
2887 ex epsilon_tensor(const ex & i1, const ex & i2);
2888 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2889 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2890 bool pos_sig = false);
2893 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2894 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2895 Minkowski space (the last @code{bool} argument specifies whether the metric
2896 has negative or positive signature, as in the case of the Minkowski metric
2901 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2902 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2903 e = lorentz_eps(mu, nu, rho, sig) *
2904 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2905 cout << simplify_indexed(e) << endl;
2906 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2908 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2909 symbol A("A"), B("B");
2910 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2911 cout << simplify_indexed(e) << endl;
2912 // -> -B.k*A.j*eps.i.k.j
2913 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2914 cout << simplify_indexed(e) << endl;
2919 @subsection Linear algebra
2921 The @code{matrix} class can be used with indices to do some simple linear
2922 algebra (linear combinations and products of vectors and matrices, traces
2923 and scalar products):
2927 idx i(symbol("i"), 2), j(symbol("j"), 2);
2928 symbol x("x"), y("y");
2930 // A is a 2x2 matrix, X is a 2x1 vector
2931 matrix A(2, 2), X(2, 1);
2936 cout << indexed(A, i, i) << endl;
2939 ex e = indexed(A, i, j) * indexed(X, j);
2940 cout << e.simplify_indexed() << endl;
2941 // -> [[2*y+x],[4*y+3*x]].i
2943 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2944 cout << e.simplify_indexed() << endl;
2945 // -> [[3*y+3*x,6*y+2*x]].j
2949 You can of course obtain the same results with the @code{matrix::add()},
2950 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2951 but with indices you don't have to worry about transposing matrices.
2953 Matrix indices always start at 0 and their dimension must match the number
2954 of rows/columns of the matrix. Matrices with one row or one column are
2955 vectors and can have one or two indices (it doesn't matter whether it's a
2956 row or a column vector). Other matrices must have two indices.
2958 You should be careful when using indices with variance on matrices. GiNaC
2959 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2960 @samp{F.mu.nu} are different matrices. In this case you should use only
2961 one form for @samp{F} and explicitly multiply it with a matrix representation
2962 of the metric tensor.
2965 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2966 @c node-name, next, previous, up
2967 @section Non-commutative objects
2969 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2970 non-commutative objects are built-in which are mostly of use in high energy
2974 @item Clifford (Dirac) algebra (class @code{clifford})
2975 @item su(3) Lie algebra (class @code{color})
2976 @item Matrices (unindexed) (class @code{matrix})
2979 The @code{clifford} and @code{color} classes are subclasses of
2980 @code{indexed} because the elements of these algebras usually carry
2981 indices. The @code{matrix} class is described in more detail in
2984 Unlike most computer algebra systems, GiNaC does not primarily provide an
2985 operator (often denoted @samp{&*}) for representing inert products of
2986 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2987 classes of objects involved, and non-commutative products are formed with
2988 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2989 figuring out by itself which objects commutate and will group the factors
2990 by their class. Consider this example:
2994 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2995 idx a(symbol("a"), 8), b(symbol("b"), 8);
2996 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2998 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3002 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3003 groups the non-commutative factors (the gammas and the su(3) generators)
3004 together while preserving the order of factors within each class (because
3005 Clifford objects commutate with color objects). The resulting expression is a
3006 @emph{commutative} product with two factors that are themselves non-commutative
3007 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3008 parentheses are placed around the non-commutative products in the output.
3010 @cindex @code{ncmul} (class)
3011 Non-commutative products are internally represented by objects of the class
3012 @code{ncmul}, as opposed to commutative products which are handled by the
3013 @code{mul} class. You will normally not have to worry about this distinction,
3016 The advantage of this approach is that you never have to worry about using
3017 (or forgetting to use) a special operator when constructing non-commutative
3018 expressions. Also, non-commutative products in GiNaC are more intelligent
3019 than in other computer algebra systems; they can, for example, automatically
3020 canonicalize themselves according to rules specified in the implementation
3021 of the non-commutative classes. The drawback is that to work with other than
3022 the built-in algebras you have to implement new classes yourself. Both
3023 symbols and user-defined functions can be specified as being non-commutative.
3025 @cindex @code{return_type()}
3026 @cindex @code{return_type_tinfo()}
3027 Information about the commutativity of an object or expression can be
3028 obtained with the two member functions
3031 unsigned ex::return_type() const;
3032 unsigned ex::return_type_tinfo() const;
3035 The @code{return_type()} function returns one of three values (defined in
3036 the header file @file{flags.h}), corresponding to three categories of
3037 expressions in GiNaC:
3040 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3041 classes are of this kind.
3042 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3043 certain class of non-commutative objects which can be determined with the
3044 @code{return_type_tinfo()} method. Expressions of this category commutate
3045 with everything except @code{noncommutative} expressions of the same
3047 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3048 of non-commutative objects of different classes. Expressions of this
3049 category don't commutate with any other @code{noncommutative} or
3050 @code{noncommutative_composite} expressions.
3053 The value returned by the @code{return_type_tinfo()} method is valid only
3054 when the return type of the expression is @code{noncommutative}. It is a
3055 value that is unique to the class of the object and usually one of the
3056 constants in @file{tinfos.h}, or derived therefrom.
3058 Here are a couple of examples:
3061 @multitable @columnfractions 0.33 0.33 0.34
3062 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3063 @item @code{42} @tab @code{commutative} @tab -
3064 @item @code{2*x-y} @tab @code{commutative} @tab -
3065 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3066 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3067 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3068 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3072 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3073 @code{TINFO_clifford} for objects with a representation label of zero.
3074 Other representation labels yield a different @code{return_type_tinfo()},
3075 but it's the same for any two objects with the same label. This is also true
3078 A last note: With the exception of matrices, positive integer powers of
3079 non-commutative objects are automatically expanded in GiNaC. For example,
3080 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3081 non-commutative expressions).
3084 @cindex @code{clifford} (class)
3085 @subsection Clifford algebra
3088 Clifford algebras are supported in two flavours: Dirac gamma
3089 matrices (more physical) and generic Clifford algebras (more
3092 @cindex @code{dirac_gamma()}
3093 @subsubsection Dirac gamma matrices
3094 Dirac gamma matrices (note that GiNaC doesn't treat them
3095 as matrices) are designated as @samp{gamma~mu} and satisfy
3096 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3097 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3098 constructed by the function
3101 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3104 which takes two arguments: the index and a @dfn{representation label} in the
3105 range 0 to 255 which is used to distinguish elements of different Clifford
3106 algebras (this is also called a @dfn{spin line index}). Gammas with different
3107 labels commutate with each other. The dimension of the index can be 4 or (in
3108 the framework of dimensional regularization) any symbolic value. Spinor
3109 indices on Dirac gammas are not supported in GiNaC.
3111 @cindex @code{dirac_ONE()}
3112 The unity element of a Clifford algebra is constructed by
3115 ex dirac_ONE(unsigned char rl = 0);
3118 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3119 multiples of the unity element, even though it's customary to omit it.
3120 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3121 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3122 GiNaC will complain and/or produce incorrect results.
3124 @cindex @code{dirac_gamma5()}
3125 There is a special element @samp{gamma5} that commutates with all other
3126 gammas, has a unit square, and in 4 dimensions equals
3127 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3130 ex dirac_gamma5(unsigned char rl = 0);
3133 @cindex @code{dirac_gammaL()}
3134 @cindex @code{dirac_gammaR()}
3135 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3136 objects, constructed by
3139 ex dirac_gammaL(unsigned char rl = 0);
3140 ex dirac_gammaR(unsigned char rl = 0);
3143 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3144 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3146 @cindex @code{dirac_slash()}
3147 Finally, the function
3150 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3153 creates a term that represents a contraction of @samp{e} with the Dirac
3154 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3155 with a unique index whose dimension is given by the @code{dim} argument).
3156 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3158 In products of dirac gammas, superfluous unity elements are automatically
3159 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3160 and @samp{gammaR} are moved to the front.
3162 The @code{simplify_indexed()} function performs contractions in gamma strings,
3168 symbol a("a"), b("b"), D("D");
3169 varidx mu(symbol("mu"), D);
3170 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3171 * dirac_gamma(mu.toggle_variance());
3173 // -> gamma~mu*a\*gamma.mu
3174 e = e.simplify_indexed();
3177 cout << e.subs(D == 4) << endl;
3183 @cindex @code{dirac_trace()}
3184 To calculate the trace of an expression containing strings of Dirac gammas
3185 you use one of the functions
3188 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3189 const ex & trONE = 4);
3190 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3191 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3194 These functions take the trace over all gammas in the specified set @code{rls}
3195 or list @code{rll} of representation labels, or the single label @code{rl};
3196 gammas with other labels are left standing. The last argument to
3197 @code{dirac_trace()} is the value to be returned for the trace of the unity
3198 element, which defaults to 4.
3200 The @code{dirac_trace()} function is a linear functional that is equal to the
3201 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3202 functional is not cyclic in
3205 dimensions when acting on
3206 expressions containing @samp{gamma5}, so it's not a proper trace. This
3207 @samp{gamma5} scheme is described in greater detail in
3208 @cite{The Role of gamma5 in Dimensional Regularization}.
3210 The value of the trace itself is also usually different in 4 and in
3218 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3219 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3220 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3221 cout << dirac_trace(e).simplify_indexed() << endl;
3228 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3229 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3230 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3231 cout << dirac_trace(e).simplify_indexed() << endl;
3232 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3236 Here is an example for using @code{dirac_trace()} to compute a value that
3237 appears in the calculation of the one-loop vacuum polarization amplitude in
3242 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3243 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3246 sp.add(l, l, pow(l, 2));
3247 sp.add(l, q, ldotq);
3249 ex e = dirac_gamma(mu) *
3250 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3251 dirac_gamma(mu.toggle_variance()) *
3252 (dirac_slash(l, D) + m * dirac_ONE());
3253 e = dirac_trace(e).simplify_indexed(sp);
3254 e = e.collect(lst(l, ldotq, m));
3256 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3260 The @code{canonicalize_clifford()} function reorders all gamma products that
3261 appear in an expression to a canonical (but not necessarily simple) form.
3262 You can use this to compare two expressions or for further simplifications:
3266 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3267 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3269 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3271 e = canonicalize_clifford(e);
3273 // -> 2*ONE*eta~mu~nu
3277 @cindex @code{clifford_unit()}
3278 @subsubsection A generic Clifford algebra
3280 A generic Clifford algebra, i.e. a
3284 dimensional algebra with
3288 satisfying the identities
3290 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3293 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3295 for some bilinear form (@code{metric})
3296 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3297 and contain symbolic entries. Such generators are created by the
3301 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3302 bool anticommuting = false);
3305 where @code{mu} should be a @code{varidx} class object indexing the
3306 generators, an index @code{mu} with a numeric value may be of type
3308 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3309 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3310 object. In fact, any expression either with two free indices or without
3311 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3312 object with two newly created indices with @code{metr} as its
3313 @code{op(0)} will be used.
3314 Optional parameter @code{rl} allows to distinguish different
3315 Clifford algebras, which will commute with each other. The last
3316 optional parameter @code{anticommuting} defines if the anticommuting
3319 $e_i e_j + e_j e_i = 0$)
3322 e~i e~j + e~j e~i = 0)
3324 will be used for contraction of Clifford units. If the @code{metric} is
3325 supplied by a @code{matrix} object, then the value of
3326 @code{anticommuting} is calculated automatically and the supplied one
3327 will be ignored. One can overcome this by giving @code{metric} through
3328 matrix wrapped into an @code{indexed} object.
3330 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3331 something very close to @code{dirac_gamma(mu)}, although
3332 @code{dirac_gamma} have more efficient simplification mechanism.
3333 @cindex @code{clifford::get_metric()}
3334 The method @code{clifford::get_metric()} returns a metric defining this
3336 @cindex @code{clifford::is_anticommuting()}
3337 The method @code{clifford::is_anticommuting()} returns the
3338 @code{anticommuting} property of a unit.
3340 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3341 the Clifford algebra units with a call like that
3344 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3347 since this may yield some further automatic simplifications. Again, for a
3348 metric defined through a @code{matrix} such a symmetry is detected
3351 Individual generators of a Clifford algebra can be accessed in several
3357 varidx nu(symbol("nu"), 4);
3359 ex M = diag_matrix(lst(1, -1, 0, s));
3360 ex e = clifford_unit(nu, M);
3361 ex e0 = e.subs(nu == 0);
3362 ex e1 = e.subs(nu == 1);
3363 ex e2 = e.subs(nu == 2);
3364 ex e3 = e.subs(nu == 3);
3369 will produce four anti-commuting generators of a Clifford algebra with properties
3371 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3374 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3375 @code{pow(e3, 2) = s}.
3378 @cindex @code{lst_to_clifford()}
3379 A similar effect can be achieved from the function
3382 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3383 unsigned char rl = 0, bool anticommuting = false);
3384 ex lst_to_clifford(const ex & v, const ex & e);
3387 which converts a list or vector
3389 $v = (v^0, v^1, ..., v^n)$
3392 @samp{v = (v~0, v~1, ..., v~n)}
3397 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3400 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3403 directly supplied in the second form of the procedure. In the first form
3404 the Clifford unit @samp{e.k} is generated by the call of
3405 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3406 with the help of @code{lst_to_clifford()} as follows
3411 varidx nu(symbol("nu"), 4);
3413 ex M = diag_matrix(lst(1, -1, 0, s));
3414 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3415 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3416 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3417 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3422 @cindex @code{clifford_to_lst()}
3423 There is the inverse function
3426 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3429 which takes an expression @code{e} and tries to find a list
3431 $v = (v^0, v^1, ..., v^n)$
3434 @samp{v = (v~0, v~1, ..., v~n)}
3438 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3441 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3443 with respect to the given Clifford units @code{c} and with none of the
3444 @samp{v~k} containing Clifford units @code{c} (of course, this
3445 may be impossible). This function can use an @code{algebraic} method
3446 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3448 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3451 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3453 is zero or is not @code{numeric} for some @samp{k}
3454 then the method will be automatically changed to symbolic. The same effect
3455 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3457 @cindex @code{clifford_prime()}
3458 @cindex @code{clifford_star()}
3459 @cindex @code{clifford_bar()}
3460 There are several functions for (anti-)automorphisms of Clifford algebras:
3463 ex clifford_prime(const ex & e)
3464 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3465 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3468 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3469 changes signs of all Clifford units in the expression. The reversion
3470 of a Clifford algebra @code{clifford_star()} coincides with the
3471 @code{conjugate()} method and effectively reverses the order of Clifford
3472 units in any product. Finally the main anti-automorphism
3473 of a Clifford algebra @code{clifford_bar()} is the composition of the
3474 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3475 in a product. These functions correspond to the notations
3490 used in Clifford algebra textbooks.
3492 @cindex @code{clifford_norm()}
3496 ex clifford_norm(const ex & e);
3499 @cindex @code{clifford_inverse()}
3500 calculates the norm of a Clifford number from the expression
3502 $||e||^2 = e\overline{e}$.
3505 @code{||e||^2 = e \bar@{e@}}
3507 The inverse of a Clifford expression is returned by the function
3510 ex clifford_inverse(const ex & e);
3513 which calculates it as
3515 $e^{-1} = \overline{e}/||e||^2$.
3518 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3527 then an exception is raised.
3529 @cindex @code{remove_dirac_ONE()}
3530 If a Clifford number happens to be a factor of
3531 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3532 expression by the function
3535 ex remove_dirac_ONE(const ex & e);
3538 @cindex @code{canonicalize_clifford()}
3539 The function @code{canonicalize_clifford()} works for a
3540 generic Clifford algebra in a similar way as for Dirac gammas.
3542 The next provided function is
3544 @cindex @code{clifford_moebius_map()}
3546 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3547 const ex & d, const ex & v, const ex & G,
3548 unsigned char rl = 0, bool anticommuting = false);
3549 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3550 unsigned char rl = 0, bool anticommuting = false);
3553 It takes a list or vector @code{v} and makes the Moebius (conformal or
3554 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3555 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3556 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3557 indexed object, tensormetric, matrix or a Clifford unit, in the later
3558 case the optional parameters @code{rl} and @code{anticommuting} are
3559 ignored even if supplied. Depending from the type of @code{v} the
3560 returned value of this function is either a vector or a list holding vector's
3563 @cindex @code{clifford_max_label()}
3564 Finally the function
3567 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3570 can detect a presence of Clifford objects in the expression @code{e}: if
3571 such objects are found it returns the maximal
3572 @code{representation_label} of them, otherwise @code{-1}. The optional
3573 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3574 be ignored during the search.
3576 LaTeX output for Clifford units looks like
3577 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3578 @code{representation_label} and @code{\nu} is the index of the
3579 corresponding unit. This provides a flexible typesetting with a suitable
3580 defintion of the @code{\clifford} command. For example, the definition
3582 \newcommand@{\clifford@}[1][]@{@}
3584 typesets all Clifford units identically, while the alternative definition
3586 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3588 prints units with @code{representation_label=0} as
3595 with @code{representation_label=1} as
3602 and with @code{representation_label=2} as
3610 @cindex @code{color} (class)
3611 @subsection Color algebra
3613 @cindex @code{color_T()}
3614 For computations in quantum chromodynamics, GiNaC implements the base elements
3615 and structure constants of the su(3) Lie algebra (color algebra). The base
3616 elements @math{T_a} are constructed by the function
3619 ex color_T(const ex & a, unsigned char rl = 0);
3622 which takes two arguments: the index and a @dfn{representation label} in the
3623 range 0 to 255 which is used to distinguish elements of different color
3624 algebras. Objects with different labels commutate with each other. The
3625 dimension of the index must be exactly 8 and it should be of class @code{idx},
3628 @cindex @code{color_ONE()}
3629 The unity element of a color algebra is constructed by
3632 ex color_ONE(unsigned char rl = 0);
3635 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3636 multiples of the unity element, even though it's customary to omit it.
3637 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3638 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3639 GiNaC may produce incorrect results.
3641 @cindex @code{color_d()}
3642 @cindex @code{color_f()}
3646 ex color_d(const ex & a, const ex & b, const ex & c);
3647 ex color_f(const ex & a, const ex & b, const ex & c);
3650 create the symmetric and antisymmetric structure constants @math{d_abc} and
3651 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3652 and @math{[T_a, T_b] = i f_abc T_c}.
3654 These functions evaluate to their numerical values,
3655 if you supply numeric indices to them. The index values should be in
3656 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3657 goes along better with the notations used in physical literature.
3659 @cindex @code{color_h()}
3660 There's an additional function
3663 ex color_h(const ex & a, const ex & b, const ex & c);
3666 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3668 The function @code{simplify_indexed()} performs some simplifications on
3669 expressions containing color objects:
3674 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3675 k(symbol("k"), 8), l(symbol("l"), 8);
3677 e = color_d(a, b, l) * color_f(a, b, k);
3678 cout << e.simplify_indexed() << endl;
3681 e = color_d(a, b, l) * color_d(a, b, k);
3682 cout << e.simplify_indexed() << endl;
3685 e = color_f(l, a, b) * color_f(a, b, k);
3686 cout << e.simplify_indexed() << endl;
3689 e = color_h(a, b, c) * color_h(a, b, c);
3690 cout << e.simplify_indexed() << endl;
3693 e = color_h(a, b, c) * color_T(b) * color_T(c);
3694 cout << e.simplify_indexed() << endl;
3697 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3698 cout << e.simplify_indexed() << endl;
3701 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3702 cout << e.simplify_indexed() << endl;
3703 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3707 @cindex @code{color_trace()}
3708 To calculate the trace of an expression containing color objects you use one
3712 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3713 ex color_trace(const ex & e, const lst & rll);
3714 ex color_trace(const ex & e, unsigned char rl = 0);
3717 These functions take the trace over all color @samp{T} objects in the
3718 specified set @code{rls} or list @code{rll} of representation labels, or the
3719 single label @code{rl}; @samp{T}s with other labels are left standing. For
3724 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3726 // -> -I*f.a.c.b+d.a.c.b
3731 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3732 @c node-name, next, previous, up
3735 @cindex @code{exhashmap} (class)
3737 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3738 that can be used as a drop-in replacement for the STL
3739 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3740 typically constant-time, element look-up than @code{map<>}.
3742 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3743 following differences:
3747 no @code{lower_bound()} and @code{upper_bound()} methods
3749 no reverse iterators, no @code{rbegin()}/@code{rend()}
3751 no @code{operator<(exhashmap, exhashmap)}
3753 the comparison function object @code{key_compare} is hardcoded to
3756 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3757 initial hash table size (the actual table size after construction may be
3758 larger than the specified value)
3760 the method @code{size_t bucket_count()} returns the current size of the hash
3763 @code{insert()} and @code{erase()} operations invalidate all iterators
3767 @node Methods and functions, Information about expressions, Hash maps, Top
3768 @c node-name, next, previous, up
3769 @chapter Methods and functions
3772 In this chapter the most important algorithms provided by GiNaC will be
3773 described. Some of them are implemented as functions on expressions,
3774 others are implemented as methods provided by expression objects. If
3775 they are methods, there exists a wrapper function around it, so you can
3776 alternatively call it in a functional way as shown in the simple
3781 cout << "As method: " << sin(1).evalf() << endl;
3782 cout << "As function: " << evalf(sin(1)) << endl;
3786 @cindex @code{subs()}
3787 The general rule is that wherever methods accept one or more parameters
3788 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3789 wrapper accepts is the same but preceded by the object to act on
3790 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3791 most natural one in an OO model but it may lead to confusion for MapleV
3792 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3793 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3794 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3795 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3796 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3797 here. Also, users of MuPAD will in most cases feel more comfortable
3798 with GiNaC's convention. All function wrappers are implemented
3799 as simple inline functions which just call the corresponding method and
3800 are only provided for users uncomfortable with OO who are dead set to
3801 avoid method invocations. Generally, nested function wrappers are much
3802 harder to read than a sequence of methods and should therefore be
3803 avoided if possible. On the other hand, not everything in GiNaC is a
3804 method on class @code{ex} and sometimes calling a function cannot be
3808 * Information about expressions::
3809 * Numerical evaluation::
3810 * Substituting expressions::
3811 * Pattern matching and advanced substitutions::
3812 * Applying a function on subexpressions::
3813 * Visitors and tree traversal::
3814 * Polynomial arithmetic:: Working with polynomials.
3815 * Rational expressions:: Working with rational functions.
3816 * Symbolic differentiation::
3817 * Series expansion:: Taylor and Laurent expansion.
3819 * Built-in functions:: List of predefined mathematical functions.
3820 * Multiple polylogarithms::
3821 * Complex expressions::
3822 * Solving linear systems of equations::
3823 * Input/output:: Input and output of expressions.
3827 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3828 @c node-name, next, previous, up
3829 @section Getting information about expressions
3831 @subsection Checking expression types
3832 @cindex @code{is_a<@dots{}>()}
3833 @cindex @code{is_exactly_a<@dots{}>()}
3834 @cindex @code{ex_to<@dots{}>()}
3835 @cindex Converting @code{ex} to other classes
3836 @cindex @code{info()}
3837 @cindex @code{return_type()}
3838 @cindex @code{return_type_tinfo()}
3840 Sometimes it's useful to check whether a given expression is a plain number,
3841 a sum, a polynomial with integer coefficients, or of some other specific type.
3842 GiNaC provides a couple of functions for this:
3845 bool is_a<T>(const ex & e);
3846 bool is_exactly_a<T>(const ex & e);
3847 bool ex::info(unsigned flag);
3848 unsigned ex::return_type() const;
3849 unsigned ex::return_type_tinfo() const;
3852 When the test made by @code{is_a<T>()} returns true, it is safe to call
3853 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3854 class names (@xref{The class hierarchy}, for a list of all classes). For
3855 example, assuming @code{e} is an @code{ex}:
3860 if (is_a<numeric>(e))
3861 numeric n = ex_to<numeric>(e);
3866 @code{is_a<T>(e)} allows you to check whether the top-level object of
3867 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3868 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3869 e.g., for checking whether an expression is a number, a sum, or a product:
3876 is_a<numeric>(e1); // true
3877 is_a<numeric>(e2); // false
3878 is_a<add>(e1); // false
3879 is_a<add>(e2); // true
3880 is_a<mul>(e1); // false
3881 is_a<mul>(e2); // false
3885 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3886 top-level object of an expression @samp{e} is an instance of the GiNaC
3887 class @samp{T}, not including parent classes.
3889 The @code{info()} method is used for checking certain attributes of
3890 expressions. The possible values for the @code{flag} argument are defined
3891 in @file{ginac/flags.h}, the most important being explained in the following
3895 @multitable @columnfractions .30 .70
3896 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3897 @item @code{numeric}
3898 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3900 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3901 @item @code{rational}
3902 @tab @dots{}an exact rational number (integers are rational, too)
3903 @item @code{integer}
3904 @tab @dots{}a (non-complex) integer
3905 @item @code{crational}
3906 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3907 @item @code{cinteger}
3908 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3909 @item @code{positive}
3910 @tab @dots{}not complex and greater than 0
3911 @item @code{negative}
3912 @tab @dots{}not complex and less than 0
3913 @item @code{nonnegative}
3914 @tab @dots{}not complex and greater than or equal to 0
3916 @tab @dots{}an integer greater than 0
3918 @tab @dots{}an integer less than 0
3919 @item @code{nonnegint}
3920 @tab @dots{}an integer greater than or equal to 0
3922 @tab @dots{}an even integer
3924 @tab @dots{}an odd integer
3926 @tab @dots{}a prime integer (probabilistic primality test)
3927 @item @code{relation}
3928 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3929 @item @code{relation_equal}
3930 @tab @dots{}a @code{==} relation
3931 @item @code{relation_not_equal}
3932 @tab @dots{}a @code{!=} relation
3933 @item @code{relation_less}
3934 @tab @dots{}a @code{<} relation
3935 @item @code{relation_less_or_equal}
3936 @tab @dots{}a @code{<=} relation
3937 @item @code{relation_greater}
3938 @tab @dots{}a @code{>} relation
3939 @item @code{relation_greater_or_equal}
3940 @tab @dots{}a @code{>=} relation
3942 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3944 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3945 @item @code{polynomial}
3946 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3947 @item @code{integer_polynomial}
3948 @tab @dots{}a polynomial with (non-complex) integer coefficients
3949 @item @code{cinteger_polynomial}
3950 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3951 @item @code{rational_polynomial}
3952 @tab @dots{}a polynomial with (non-complex) rational coefficients
3953 @item @code{crational_polynomial}
3954 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3955 @item @code{rational_function}
3956 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3957 @item @code{algebraic}
3958 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3962 To determine whether an expression is commutative or non-commutative and if
3963 so, with which other expressions it would commutate, you use the methods
3964 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3965 for an explanation of these.
3968 @subsection Accessing subexpressions
3971 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3972 @code{function}, act as containers for subexpressions. For example, the
3973 subexpressions of a sum (an @code{add} object) are the individual terms,
3974 and the subexpressions of a @code{function} are the function's arguments.
3976 @cindex @code{nops()}
3978 GiNaC provides several ways of accessing subexpressions. The first way is to
3983 ex ex::op(size_t i);
3986 @code{nops()} determines the number of subexpressions (operands) contained
3987 in the expression, while @code{op(i)} returns the @code{i}-th
3988 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3989 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3990 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3991 @math{i>0} are the indices.
3994 @cindex @code{const_iterator}
3995 The second way to access subexpressions is via the STL-style random-access
3996 iterator class @code{const_iterator} and the methods
3999 const_iterator ex::begin();
4000 const_iterator ex::end();
4003 @code{begin()} returns an iterator referring to the first subexpression;
4004 @code{end()} returns an iterator which is one-past the last subexpression.
4005 If the expression has no subexpressions, then @code{begin() == end()}. These
4006 iterators can also be used in conjunction with non-modifying STL algorithms.
4008 Here is an example that (non-recursively) prints the subexpressions of a
4009 given expression in three different ways:
4016 for (size_t i = 0; i != e.nops(); ++i)
4017 cout << e.op(i) << endl;
4020 for (const_iterator i = e.begin(); i != e.end(); ++i)
4023 // with iterators and STL copy()
4024 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4028 @cindex @code{const_preorder_iterator}
4029 @cindex @code{const_postorder_iterator}
4030 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4031 expression's immediate children. GiNaC provides two additional iterator
4032 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4033 that iterate over all objects in an expression tree, in preorder or postorder,
4034 respectively. They are STL-style forward iterators, and are created with the
4038 const_preorder_iterator ex::preorder_begin();
4039 const_preorder_iterator ex::preorder_end();
4040 const_postorder_iterator ex::postorder_begin();
4041 const_postorder_iterator ex::postorder_end();
4044 The following example illustrates the differences between
4045 @code{const_iterator}, @code{const_preorder_iterator}, and
4046 @code{const_postorder_iterator}:
4050 symbol A("A"), B("B"), C("C");
4051 ex e = lst(lst(A, B), C);
4053 std::copy(e.begin(), e.end(),
4054 std::ostream_iterator<ex>(cout, "\n"));
4058 std::copy(e.preorder_begin(), e.preorder_end(),
4059 std::ostream_iterator<ex>(cout, "\n"));
4066 std::copy(e.postorder_begin(), e.postorder_end(),
4067 std::ostream_iterator<ex>(cout, "\n"));
4076 @cindex @code{relational} (class)
4077 Finally, the left-hand side and right-hand side expressions of objects of
4078 class @code{relational} (and only of these) can also be accessed with the
4087 @subsection Comparing expressions
4088 @cindex @code{is_equal()}
4089 @cindex @code{is_zero()}
4091 Expressions can be compared with the usual C++ relational operators like
4092 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4093 the result is usually not determinable and the result will be @code{false},
4094 except in the case of the @code{!=} operator. You should also be aware that
4095 GiNaC will only do the most trivial test for equality (subtracting both
4096 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4099 Actually, if you construct an expression like @code{a == b}, this will be
4100 represented by an object of the @code{relational} class (@pxref{Relations})
4101 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4103 There are also two methods
4106 bool ex::is_equal(const ex & other);
4110 for checking whether one expression is equal to another, or equal to zero,
4111 respectively. See also the method @code{ex::is_zero_matrix()},
4115 @subsection Ordering expressions
4116 @cindex @code{ex_is_less} (class)
4117 @cindex @code{ex_is_equal} (class)
4118 @cindex @code{compare()}
4120 Sometimes it is necessary to establish a mathematically well-defined ordering
4121 on a set of arbitrary expressions, for example to use expressions as keys
4122 in a @code{std::map<>} container, or to bring a vector of expressions into
4123 a canonical order (which is done internally by GiNaC for sums and products).
4125 The operators @code{<}, @code{>} etc. described in the last section cannot
4126 be used for this, as they don't implement an ordering relation in the
4127 mathematical sense. In particular, they are not guaranteed to be
4128 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4129 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4132 By default, STL classes and algorithms use the @code{<} and @code{==}
4133 operators to compare objects, which are unsuitable for expressions, but GiNaC
4134 provides two functors that can be supplied as proper binary comparison
4135 predicates to the STL:
4138 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4140 bool operator()(const ex &lh, const ex &rh) const;
4143 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4145 bool operator()(const ex &lh, const ex &rh) const;
4149 For example, to define a @code{map} that maps expressions to strings you
4153 std::map<ex, std::string, ex_is_less> myMap;
4156 Omitting the @code{ex_is_less} template parameter will introduce spurious
4157 bugs because the map operates improperly.
4159 Other examples for the use of the functors:
4167 std::sort(v.begin(), v.end(), ex_is_less());
4169 // count the number of expressions equal to '1'
4170 unsigned num_ones = std::count_if(v.begin(), v.end(),
4171 std::bind2nd(ex_is_equal(), 1));
4174 The implementation of @code{ex_is_less} uses the member function
4177 int ex::compare(const ex & other) const;
4180 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4181 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4185 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4186 @c node-name, next, previous, up
4187 @section Numerical evaluation
4188 @cindex @code{evalf()}
4190 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4191 To evaluate them using floating-point arithmetic you need to call
4194 ex ex::evalf(int level = 0) const;
4197 @cindex @code{Digits}
4198 The accuracy of the evaluation is controlled by the global object @code{Digits}
4199 which can be assigned an integer value. The default value of @code{Digits}
4200 is 17. @xref{Numbers}, for more information and examples.
4202 To evaluate an expression to a @code{double} floating-point number you can
4203 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4207 // Approximate sin(x/Pi)
4209 ex e = series(sin(x/Pi), x == 0, 6);
4211 // Evaluate numerically at x=0.1
4212 ex f = evalf(e.subs(x == 0.1));
4214 // ex_to<numeric> is an unsafe cast, so check the type first
4215 if (is_a<numeric>(f)) @{
4216 double d = ex_to<numeric>(f).to_double();
4225 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4226 @c node-name, next, previous, up
4227 @section Substituting expressions
4228 @cindex @code{subs()}
4230 Algebraic objects inside expressions can be replaced with arbitrary
4231 expressions via the @code{.subs()} method:
4234 ex ex::subs(const ex & e, unsigned options = 0);
4235 ex ex::subs(const exmap & m, unsigned options = 0);
4236 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4239 In the first form, @code{subs()} accepts a relational of the form
4240 @samp{object == expression} or a @code{lst} of such relationals:
4244 symbol x("x"), y("y");
4246 ex e1 = 2*x^2-4*x+3;
4247 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4251 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4256 If you specify multiple substitutions, they are performed in parallel, so e.g.
4257 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4259 The second form of @code{subs()} takes an @code{exmap} object which is a
4260 pair associative container that maps expressions to expressions (currently
4261 implemented as a @code{std::map}). This is the most efficient one of the
4262 three @code{subs()} forms and should be used when the number of objects to
4263 be substituted is large or unknown.
4265 Using this form, the second example from above would look like this:
4269 symbol x("x"), y("y");
4275 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4279 The third form of @code{subs()} takes two lists, one for the objects to be
4280 replaced and one for the expressions to be substituted (both lists must
4281 contain the same number of elements). Using this form, you would write
4285 symbol x("x"), y("y");
4288 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4292 The optional last argument to @code{subs()} is a combination of
4293 @code{subs_options} flags. There are three options available:
4294 @code{subs_options::no_pattern} disables pattern matching, which makes
4295 large @code{subs()} operations significantly faster if you are not using
4296 patterns. The second option, @code{subs_options::algebraic} enables
4297 algebraic substitutions in products and powers.
4298 @ref{Pattern matching and advanced substitutions}, for more information
4299 about patterns and algebraic substitutions. The third option,
4300 @code{subs_options::no_index_renaming} disables the feature that dummy
4301 indices are renamed if the subsitution could give a result in which a
4302 dummy index occurs more than two times. This is sometimes necessary if
4303 you want to use @code{subs()} to rename your dummy indices.
4305 @code{subs()} performs syntactic substitution of any complete algebraic
4306 object; it does not try to match sub-expressions as is demonstrated by the
4311 symbol x("x"), y("y"), z("z");
4313 ex e1 = pow(x+y, 2);
4314 cout << e1.subs(x+y == 4) << endl;
4317 ex e2 = sin(x)*sin(y)*cos(x);
4318 cout << e2.subs(sin(x) == cos(x)) << endl;
4319 // -> cos(x)^2*sin(y)
4322 cout << e3.subs(x+y == 4) << endl;
4324 // (and not 4+z as one might expect)
4328 A more powerful form of substitution using wildcards is described in the
4332 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4333 @c node-name, next, previous, up
4334 @section Pattern matching and advanced substitutions
4335 @cindex @code{wildcard} (class)
4336 @cindex Pattern matching
4338 GiNaC allows the use of patterns for checking whether an expression is of a
4339 certain form or contains subexpressions of a certain form, and for
4340 substituting expressions in a more general way.
4342 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4343 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4344 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4345 an unsigned integer number to allow having multiple different wildcards in a
4346 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4347 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4351 ex wild(unsigned label = 0);
4354 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4357 Some examples for patterns:
4359 @multitable @columnfractions .5 .5
4360 @item @strong{Constructed as} @tab @strong{Output as}
4361 @item @code{wild()} @tab @samp{$0}
4362 @item @code{pow(x,wild())} @tab @samp{x^$0}
4363 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4364 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4370 @item Wildcards behave like symbols and are subject to the same algebraic
4371 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4372 @item As shown in the last example, to use wildcards for indices you have to
4373 use them as the value of an @code{idx} object. This is because indices must
4374 always be of class @code{idx} (or a subclass).
4375 @item Wildcards only represent expressions or subexpressions. It is not
4376 possible to use them as placeholders for other properties like index
4377 dimension or variance, representation labels, symmetry of indexed objects
4379 @item Because wildcards are commutative, it is not possible to use wildcards
4380 as part of noncommutative products.
4381 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4382 are also valid patterns.
4385 @subsection Matching expressions
4386 @cindex @code{match()}
4387 The most basic application of patterns is to check whether an expression
4388 matches a given pattern. This is done by the function
4391 bool ex::match(const ex & pattern);
4392 bool ex::match(const ex & pattern, lst & repls);
4395 This function returns @code{true} when the expression matches the pattern
4396 and @code{false} if it doesn't. If used in the second form, the actual
4397 subexpressions matched by the wildcards get returned in the @code{repls}
4398 object as a list of relations of the form @samp{wildcard == expression}.
4399 If @code{match()} returns false, the state of @code{repls} is undefined.
4400 For reproducible results, the list should be empty when passed to
4401 @code{match()}, but it is also possible to find similarities in multiple
4402 expressions by passing in the result of a previous match.
4404 The matching algorithm works as follows:
4407 @item A single wildcard matches any expression. If one wildcard appears
4408 multiple times in a pattern, it must match the same expression in all
4409 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4410 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4411 @item If the expression is not of the same class as the pattern, the match
4412 fails (i.e. a sum only matches a sum, a function only matches a function,
4414 @item If the pattern is a function, it only matches the same function
4415 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4416 @item Except for sums and products, the match fails if the number of
4417 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4419 @item If there are no subexpressions, the expressions and the pattern must
4420 be equal (in the sense of @code{is_equal()}).
4421 @item Except for sums and products, each subexpression (@code{op()}) must
4422 match the corresponding subexpression of the pattern.
4425 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4426 account for their commutativity and associativity:
4429 @item If the pattern contains a term or factor that is a single wildcard,
4430 this one is used as the @dfn{global wildcard}. If there is more than one
4431 such wildcard, one of them is chosen as the global wildcard in a random
4433 @item Every term/factor of the pattern, except the global wildcard, is
4434 matched against every term of the expression in sequence. If no match is
4435 found, the whole match fails. Terms that did match are not considered in
4437 @item If there are no unmatched terms left, the match succeeds. Otherwise
4438 the match fails unless there is a global wildcard in the pattern, in
4439 which case this wildcard matches the remaining terms.
4442 In general, having more than one single wildcard as a term of a sum or a
4443 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4446 Here are some examples in @command{ginsh} to demonstrate how it works (the
4447 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4448 match fails, and the list of wildcard replacements otherwise):
4451 > match((x+y)^a,(x+y)^a);
4453 > match((x+y)^a,(x+y)^b);
4455 > match((x+y)^a,$1^$2);
4457 > match((x+y)^a,$1^$1);
4459 > match((x+y)^(x+y),$1^$1);
4461 > match((x+y)^(x+y),$1^$2);
4463 > match((a+b)*(a+c),($1+b)*($1+c));
4465 > match((a+b)*(a+c),(a+$1)*(a+$2));
4467 (Unpredictable. The result might also be [$1==c,$2==b].)
4468 > match((a+b)*(a+c),($1+$2)*($1+$3));
4469 (The result is undefined. Due to the sequential nature of the algorithm
4470 and the re-ordering of terms in GiNaC, the match for the first factor
4471 may be @{$1==a,$2==b@} in which case the match for the second factor
4472 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4474 > match(a*(x+y)+a*z+b,a*$1+$2);
4475 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4476 @{$1=x+y,$2=a*z+b@}.)
4477 > match(a+b+c+d+e+f,c);
4479 > match(a+b+c+d+e+f,c+$0);
4481 > match(a+b+c+d+e+f,c+e+$0);
4483 > match(a+b,a+b+$0);
4485 > match(a*b^2,a^$1*b^$2);
4487 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4488 even though a==a^1.)
4489 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4491 > match(atan2(y,x^2),atan2(y,$0));
4495 @subsection Matching parts of expressions
4496 @cindex @code{has()}
4497 A more general way to look for patterns in expressions is provided by the
4501 bool ex::has(const ex & pattern);
4504 This function checks whether a pattern is matched by an expression itself or
4505 by any of its subexpressions.
4507 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4508 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4511 > has(x*sin(x+y+2*a),y);
4513 > has(x*sin(x+y+2*a),x+y);
4515 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4516 has the subexpressions "x", "y" and "2*a".)
4517 > has(x*sin(x+y+2*a),x+y+$1);
4519 (But this is possible.)
4520 > has(x*sin(2*(x+y)+2*a),x+y);
4522 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4523 which "x+y" is not a subexpression.)
4526 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4528 > has(4*x^2-x+3,$1*x);
4530 > has(4*x^2+x+3,$1*x);
4532 (Another possible pitfall. The first expression matches because the term
4533 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4534 contains a linear term you should use the coeff() function instead.)
4537 @cindex @code{find()}
4541 bool ex::find(const ex & pattern, lst & found);
4544 works a bit like @code{has()} but it doesn't stop upon finding the first
4545 match. Instead, it appends all found matches to the specified list. If there
4546 are multiple occurrences of the same expression, it is entered only once to
4547 the list. @code{find()} returns false if no matches were found (in
4548 @command{ginsh}, it returns an empty list):
4551 > find(1+x+x^2+x^3,x);
4553 > find(1+x+x^2+x^3,y);
4555 > find(1+x+x^2+x^3,x^$1);
4557 (Note the absence of "x".)
4558 > expand((sin(x)+sin(y))*(a+b));
4559 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4564 @subsection Substituting expressions
4565 @cindex @code{subs()}
4566 Probably the most useful application of patterns is to use them for
4567 substituting expressions with the @code{subs()} method. Wildcards can be
4568 used in the search patterns as well as in the replacement expressions, where
4569 they get replaced by the expressions matched by them. @code{subs()} doesn't
4570 know anything about algebra; it performs purely syntactic substitutions.
4575 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4577 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4579 > subs((a+b+c)^2,a+b==x);
4581 > subs((a+b+c)^2,a+b+$1==x+$1);
4583 > subs(a+2*b,a+b==x);
4585 > subs(4*x^3-2*x^2+5*x-1,x==a);
4587 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4589 > subs(sin(1+sin(x)),sin($1)==cos($1));
4591 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4595 The last example would be written in C++ in this way:
4599 symbol a("a"), b("b"), x("x"), y("y");
4600 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4601 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4602 cout << e.expand() << endl;
4607 @subsection The option algebraic
4608 Both @code{has()} and @code{subs()} take an optional argument to pass them
4609 extra options. This section describes what happens if you give the former
4610 the option @code{has_options::algebraic} or the latter
4611 @code{subs:options::algebraic}. In that case the matching condition for
4612 powers and multiplications is changed in such a way that they become
4613 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4614 If you use these options you will find that
4615 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4616 Besides matching some of the factors of a product also powers match as
4617 often as is possible without getting negative exponents. For example
4618 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4619 @code{x*c^2*z}. This also works with negative powers:
4620 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4621 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4622 and not for locating @code{x+y} within @code{x+y+z}.
4625 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4626 @c node-name, next, previous, up
4627 @section Applying a function on subexpressions
4628 @cindex tree traversal
4629 @cindex @code{map()}
4631 Sometimes you may want to perform an operation on specific parts of an
4632 expression while leaving the general structure of it intact. An example
4633 of this would be a matrix trace operation: the trace of a sum is the sum
4634 of the traces of the individual terms. That is, the trace should @dfn{map}
4635 on the sum, by applying itself to each of the sum's operands. It is possible
4636 to do this manually which usually results in code like this:
4641 if (is_a<matrix>(e))
4642 return ex_to<matrix>(e).trace();
4643 else if (is_a<add>(e)) @{
4645 for (size_t i=0; i<e.nops(); i++)
4646 sum += calc_trace(e.op(i));
4648 @} else if (is_a<mul>)(e)) @{
4656 This is, however, slightly inefficient (if the sum is very large it can take
4657 a long time to add the terms one-by-one), and its applicability is limited to
4658 a rather small class of expressions. If @code{calc_trace()} is called with
4659 a relation or a list as its argument, you will probably want the trace to
4660 be taken on both sides of the relation or of all elements of the list.
4662 GiNaC offers the @code{map()} method to aid in the implementation of such
4666 ex ex::map(map_function & f) const;
4667 ex ex::map(ex (*f)(const ex & e)) const;
4670 In the first (preferred) form, @code{map()} takes a function object that
4671 is subclassed from the @code{map_function} class. In the second form, it
4672 takes a pointer to a function that accepts and returns an expression.
4673 @code{map()} constructs a new expression of the same type, applying the
4674 specified function on all subexpressions (in the sense of @code{op()}),
4677 The use of a function object makes it possible to supply more arguments to
4678 the function that is being mapped, or to keep local state information.
4679 The @code{map_function} class declares a virtual function call operator
4680 that you can overload. Here is a sample implementation of @code{calc_trace()}
4681 that uses @code{map()} in a recursive fashion:
4684 struct calc_trace : public map_function @{
4685 ex operator()(const ex &e)
4687 if (is_a<matrix>(e))
4688 return ex_to<matrix>(e).trace();
4689 else if (is_a<mul>(e)) @{
4692 return e.map(*this);
4697 This function object could then be used like this:
4701 ex M = ... // expression with matrices
4702 calc_trace do_trace;
4703 ex tr = do_trace(M);
4707 Here is another example for you to meditate over. It removes quadratic
4708 terms in a variable from an expanded polynomial:
4711 struct map_rem_quad : public map_function @{
4713 map_rem_quad(const ex & var_) : var(var_) @{@}
4715 ex operator()(const ex & e)
4717 if (is_a<add>(e) || is_a<mul>(e))
4718 return e.map(*this);
4719 else if (is_a<power>(e) &&
4720 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4730 symbol x("x"), y("y");
4733 for (int i=0; i<8; i++)
4734 e += pow(x, i) * pow(y, 8-i) * (i+1);
4736 // -> 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
4738 map_rem_quad rem_quad(x);
4739 cout << rem_quad(e) << endl;
4740 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4744 @command{ginsh} offers a slightly different implementation of @code{map()}
4745 that allows applying algebraic functions to operands. The second argument
4746 to @code{map()} is an expression containing the wildcard @samp{$0} which
4747 acts as the placeholder for the operands:
4752 > map(a+2*b,sin($0));
4754 > map(@{a,b,c@},$0^2+$0);
4755 @{a^2+a,b^2+b,c^2+c@}
4758 Note that it is only possible to use algebraic functions in the second
4759 argument. You can not use functions like @samp{diff()}, @samp{op()},
4760 @samp{subs()} etc. because these are evaluated immediately:
4763 > map(@{a,b,c@},diff($0,a));
4765 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4766 to "map(@{a,b,c@},0)".
4770 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4771 @c node-name, next, previous, up
4772 @section Visitors and tree traversal
4773 @cindex tree traversal
4774 @cindex @code{visitor} (class)
4775 @cindex @code{accept()}
4776 @cindex @code{visit()}
4777 @cindex @code{traverse()}
4778 @cindex @code{traverse_preorder()}
4779 @cindex @code{traverse_postorder()}
4781 Suppose that you need a function that returns a list of all indices appearing
4782 in an arbitrary expression. The indices can have any dimension, and for
4783 indices with variance you always want the covariant version returned.
4785 You can't use @code{get_free_indices()} because you also want to include
4786 dummy indices in the list, and you can't use @code{find()} as it needs
4787 specific index dimensions (and it would require two passes: one for indices
4788 with variance, one for plain ones).
4790 The obvious solution to this problem is a tree traversal with a type switch,
4791 such as the following:
4794 void gather_indices_helper(const ex & e, lst & l)
4796 if (is_a<varidx>(e)) @{
4797 const varidx & vi = ex_to<varidx>(e);
4798 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4799 @} else if (is_a<idx>(e)) @{
4802 size_t n = e.nops();
4803 for (size_t i = 0; i < n; ++i)
4804 gather_indices_helper(e.op(i), l);
4808 lst gather_indices(const ex & e)
4811 gather_indices_helper(e, l);
4818 This works fine but fans of object-oriented programming will feel
4819 uncomfortable with the type switch. One reason is that there is a possibility
4820 for subtle bugs regarding derived classes. If we had, for example, written
4823 if (is_a<idx>(e)) @{
4825 @} else if (is_a<varidx>(e)) @{
4829 in @code{gather_indices_helper}, the code wouldn't have worked because the
4830 first line "absorbs" all classes derived from @code{idx}, including
4831 @code{varidx}, so the special case for @code{varidx} would never have been
4834 Also, for a large number of classes, a type switch like the above can get
4835 unwieldy and inefficient (it's a linear search, after all).
4836 @code{gather_indices_helper} only checks for two classes, but if you had to
4837 write a function that required a different implementation for nearly
4838 every GiNaC class, the result would be very hard to maintain and extend.
4840 The cleanest approach to the problem would be to add a new virtual function
4841 to GiNaC's class hierarchy. In our example, there would be specializations
4842 for @code{idx} and @code{varidx} while the default implementation in
4843 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4844 impossible to add virtual member functions to existing classes without
4845 changing their source and recompiling everything. GiNaC comes with source,
4846 so you could actually do this, but for a small algorithm like the one
4847 presented this would be impractical.
4849 One solution to this dilemma is the @dfn{Visitor} design pattern,
4850 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4851 variation, described in detail in
4852 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4853 virtual functions to the class hierarchy to implement operations, GiNaC
4854 provides a single "bouncing" method @code{accept()} that takes an instance
4855 of a special @code{visitor} class and redirects execution to the one
4856 @code{visit()} virtual function of the visitor that matches the type of
4857 object that @code{accept()} was being invoked on.
4859 Visitors in GiNaC must derive from the global @code{visitor} class as well
4860 as from the class @code{T::visitor} of each class @code{T} they want to
4861 visit, and implement the member functions @code{void visit(const T &)} for
4867 void ex::accept(visitor & v) const;
4870 will then dispatch to the correct @code{visit()} member function of the
4871 specified visitor @code{v} for the type of GiNaC object at the root of the
4872 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4874 Here is an example of a visitor:
4878 : public visitor, // this is required
4879 public add::visitor, // visit add objects
4880 public numeric::visitor, // visit numeric objects
4881 public basic::visitor // visit basic objects
4883 void visit(const add & x)
4884 @{ cout << "called with an add object" << endl; @}
4886 void visit(const numeric & x)
4887 @{ cout << "called with a numeric object" << endl; @}
4889 void visit(const basic & x)
4890 @{ cout << "called with a basic object" << endl; @}
4894 which can be used as follows:
4905 // prints "called with a numeric object"
4907 // prints "called with an add object"
4909 // prints "called with a basic object"
4913 The @code{visit(const basic &)} method gets called for all objects that are
4914 not @code{numeric} or @code{add} and acts as an (optional) default.
4916 From a conceptual point of view, the @code{visit()} methods of the visitor
4917 behave like a newly added virtual function of the visited hierarchy.
4918 In addition, visitors can store state in member variables, and they can
4919 be extended by deriving a new visitor from an existing one, thus building
4920 hierarchies of visitors.
4922 We can now rewrite our index example from above with a visitor:
4925 class gather_indices_visitor
4926 : public visitor, public idx::visitor, public varidx::visitor
4930 void visit(const idx & i)
4935 void visit(const varidx & vi)
4937 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4941 const lst & get_result() // utility function
4950 What's missing is the tree traversal. We could implement it in
4951 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4954 void ex::traverse_preorder(visitor & v) const;
4955 void ex::traverse_postorder(visitor & v) const;
4956 void ex::traverse(visitor & v) const;
4959 @code{traverse_preorder()} visits a node @emph{before} visiting its
4960 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4961 visiting its subexpressions. @code{traverse()} is a synonym for
4962 @code{traverse_preorder()}.
4964 Here is a new implementation of @code{gather_indices()} that uses the visitor
4965 and @code{traverse()}:
4968 lst gather_indices(const ex & e)
4970 gather_indices_visitor v;
4972 return v.get_result();
4976 Alternatively, you could use pre- or postorder iterators for the tree
4980 lst gather_indices(const ex & e)
4982 gather_indices_visitor v;
4983 for (const_preorder_iterator i = e.preorder_begin();
4984 i != e.preorder_end(); ++i) @{
4987 return v.get_result();
4992 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4993 @c node-name, next, previous, up
4994 @section Polynomial arithmetic
4996 @subsection Testing whether an expression is a polynomial
4997 @cindex @code{is_polynomial()}
4999 Testing whether an expression is a polynomial in one or more variables
5000 can be done with the method
5002 bool ex::is_polynomial(const ex & vars) const;
5004 In the case of more than
5005 one variable, the variables are given as a list.
5008 (x*y*sin(y)).is_polynomial(x) // Returns true.
5009 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5012 @subsection Expanding and collecting
5013 @cindex @code{expand()}
5014 @cindex @code{collect()}
5015 @cindex @code{collect_common_factors()}
5017 A polynomial in one or more variables has many equivalent
5018 representations. Some useful ones serve a specific purpose. Consider
5019 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5020 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5021 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5022 representations are the recursive ones where one collects for exponents
5023 in one of the three variable. Since the factors are themselves
5024 polynomials in the remaining two variables the procedure can be
5025 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5026 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5029 To bring an expression into expanded form, its method
5032 ex ex::expand(unsigned options = 0);
5035 may be called. In our example above, this corresponds to @math{4*x*y +
5036 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5037 GiNaC is not easy to guess you should be prepared to see different
5038 orderings of terms in such sums!
5040 Another useful representation of multivariate polynomials is as a
5041 univariate polynomial in one of the variables with the coefficients
5042 being polynomials in the remaining variables. The method
5043 @code{collect()} accomplishes this task:
5046 ex ex::collect(const ex & s, bool distributed = false);
5049 The first argument to @code{collect()} can also be a list of objects in which
5050 case the result is either a recursively collected polynomial, or a polynomial
5051 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5052 by the @code{distributed} flag.
5054 Note that the original polynomial needs to be in expanded form (for the
5055 variables concerned) in order for @code{collect()} to be able to find the
5056 coefficients properly.
5058 The following @command{ginsh} transcript shows an application of @code{collect()}
5059 together with @code{find()}:
5062 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5063 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)
5064 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5065 > collect(a,@{p,q@});
5066 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5067 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5068 > collect(a,find(a,sin($1)));
5069 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5070 > collect(a,@{find(a,sin($1)),p,q@});
5071 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5072 > collect(a,@{find(a,sin($1)),d@});
5073 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5076 Polynomials can often be brought into a more compact form by collecting
5077 common factors from the terms of sums. This is accomplished by the function
5080 ex collect_common_factors(const ex & e);
5083 This function doesn't perform a full factorization but only looks for
5084 factors which are already explicitly present:
5087 > collect_common_factors(a*x+a*y);
5089 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5091 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5092 (c+a)*a*(x*y+y^2+x)*b
5095 @subsection Degree and coefficients
5096 @cindex @code{degree()}
5097 @cindex @code{ldegree()}
5098 @cindex @code{coeff()}
5100 The degree and low degree of a polynomial can be obtained using the two
5104 int ex::degree(const ex & s);
5105 int ex::ldegree(const ex & s);
5108 which also work reliably on non-expanded input polynomials (they even work
5109 on rational functions, returning the asymptotic degree). By definition, the
5110 degree of zero is zero. To extract a coefficient with a certain power from
5111 an expanded polynomial you use
5114 ex ex::coeff(const ex & s, int n);
5117 You can also obtain the leading and trailing coefficients with the methods
5120 ex ex::lcoeff(const ex & s);
5121 ex ex::tcoeff(const ex & s);
5124 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5127 An application is illustrated in the next example, where a multivariate
5128 polynomial is analyzed:
5132 symbol x("x"), y("y");
5133 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5134 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5135 ex Poly = PolyInp.expand();
5137 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5138 cout << "The x^" << i << "-coefficient is "
5139 << Poly.coeff(x,i) << endl;
5141 cout << "As polynomial in y: "
5142 << Poly.collect(y) << endl;
5146 When run, it returns an output in the following fashion:
5149 The x^0-coefficient is y^2+11*y
5150 The x^1-coefficient is 5*y^2-2*y
5151 The x^2-coefficient is -1
5152 The x^3-coefficient is 4*y
5153 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5156 As always, the exact output may vary between different versions of GiNaC
5157 or even from run to run since the internal canonical ordering is not
5158 within the user's sphere of influence.
5160 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5161 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5162 with non-polynomial expressions as they not only work with symbols but with
5163 constants, functions and indexed objects as well:
5167 symbol a("a"), b("b"), c("c"), x("x");
5168 idx i(symbol("i"), 3);
5170 ex e = pow(sin(x) - cos(x), 4);
5171 cout << e.degree(cos(x)) << endl;
5173 cout << e.expand().coeff(sin(x), 3) << endl;
5176 e = indexed(a+b, i) * indexed(b+c, i);
5177 e = e.expand(expand_options::expand_indexed);
5178 cout << e.collect(indexed(b, i)) << endl;
5179 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5184 @subsection Polynomial division
5185 @cindex polynomial division
5188 @cindex pseudo-remainder
5189 @cindex @code{quo()}
5190 @cindex @code{rem()}
5191 @cindex @code{prem()}
5192 @cindex @code{divide()}
5197 ex quo(const ex & a, const ex & b, const ex & x);
5198 ex rem(const ex & a, const ex & b, const ex & x);
5201 compute the quotient and remainder of univariate polynomials in the variable
5202 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5204 The additional function
5207 ex prem(const ex & a, const ex & b, const ex & x);
5210 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5211 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5213 Exact division of multivariate polynomials is performed by the function
5216 bool divide(const ex & a, const ex & b, ex & q);
5219 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5220 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5221 in which case the value of @code{q} is undefined.
5224 @subsection Unit, content and primitive part
5225 @cindex @code{unit()}
5226 @cindex @code{content()}
5227 @cindex @code{primpart()}
5228 @cindex @code{unitcontprim()}
5233 ex ex::unit(const ex & x);
5234 ex ex::content(const ex & x);
5235 ex ex::primpart(const ex & x);
5236 ex ex::primpart(const ex & x, const ex & c);
5239 return the unit part, content part, and primitive polynomial of a multivariate
5240 polynomial with respect to the variable @samp{x} (the unit part being the sign
5241 of the leading coefficient, the content part being the GCD of the coefficients,
5242 and the primitive polynomial being the input polynomial divided by the unit and
5243 content parts). The second variant of @code{primpart()} expects the previously
5244 calculated content part of the polynomial in @code{c}, which enables it to
5245 work faster in the case where the content part has already been computed. The
5246 product of unit, content, and primitive part is the original polynomial.
5248 Additionally, the method
5251 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5254 computes the unit, content, and primitive parts in one go, returning them
5255 in @code{u}, @code{c}, and @code{p}, respectively.
5258 @subsection GCD, LCM and resultant
5261 @cindex @code{gcd()}
5262 @cindex @code{lcm()}
5264 The functions for polynomial greatest common divisor and least common
5265 multiple have the synopsis
5268 ex gcd(const ex & a, const ex & b);
5269 ex lcm(const ex & a, const ex & b);
5272 The functions @code{gcd()} and @code{lcm()} accept two expressions
5273 @code{a} and @code{b} as arguments and return a new expression, their
5274 greatest common divisor or least common multiple, respectively. If the
5275 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5276 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5277 the coefficients must be rationals.
5280 #include <ginac/ginac.h>
5281 using namespace GiNaC;
5285 symbol x("x"), y("y"), z("z");
5286 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5287 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5289 ex P_gcd = gcd(P_a, P_b);
5291 ex P_lcm = lcm(P_a, P_b);
5292 // 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
5297 @cindex @code{resultant()}
5299 The resultant of two expressions only makes sense with polynomials.
5300 It is always computed with respect to a specific symbol within the
5301 expressions. The function has the interface
5304 ex resultant(const ex & a, const ex & b, const ex & s);
5307 Resultants are symmetric in @code{a} and @code{b}. The following example
5308 computes the resultant of two expressions with respect to @code{x} and
5309 @code{y}, respectively:
5312 #include <ginac/ginac.h>
5313 using namespace GiNaC;
5317 symbol x("x"), y("y");
5319 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5322 r = resultant(e1, e2, x);
5324 r = resultant(e1, e2, y);
5329 @subsection Square-free decomposition
5330 @cindex square-free decomposition
5331 @cindex factorization
5332 @cindex @code{sqrfree()}
5334 GiNaC still lacks proper factorization support. Some form of
5335 factorization is, however, easily implemented by noting that factors
5336 appearing in a polynomial with power two or more also appear in the
5337 derivative and hence can easily be found by computing the GCD of the
5338 original polynomial and its derivatives. Any decent system has an
5339 interface for this so called square-free factorization. So we provide
5342 ex sqrfree(const ex & a, const lst & l = lst());
5344 Here is an example that by the way illustrates how the exact form of the
5345 result may slightly depend on the order of differentiation, calling for
5346 some care with subsequent processing of the result:
5349 symbol x("x"), y("y");
5350 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5352 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5353 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5355 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5356 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5358 cout << sqrfree(BiVarPol) << endl;
5359 // -> depending on luck, any of the above
5362 Note also, how factors with the same exponents are not fully factorized
5366 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5367 @c node-name, next, previous, up
5368 @section Rational expressions
5370 @subsection The @code{normal} method
5371 @cindex @code{normal()}
5372 @cindex simplification
5373 @cindex temporary replacement
5375 Some basic form of simplification of expressions is called for frequently.
5376 GiNaC provides the method @code{.normal()}, which converts a rational function
5377 into an equivalent rational function of the form @samp{numerator/denominator}
5378 where numerator and denominator are coprime. If the input expression is already
5379 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5380 otherwise it performs fraction addition and multiplication.
5382 @code{.normal()} can also be used on expressions which are not rational functions
5383 as it will replace all non-rational objects (like functions or non-integer
5384 powers) by temporary symbols to bring the expression to the domain of rational
5385 functions before performing the normalization, and re-substituting these
5386 symbols afterwards. This algorithm is also available as a separate method
5387 @code{.to_rational()}, described below.
5389 This means that both expressions @code{t1} and @code{t2} are indeed
5390 simplified in this little code snippet:
5395 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5396 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5397 std::cout << "t1 is " << t1.normal() << std::endl;
5398 std::cout << "t2 is " << t2.normal() << std::endl;
5402 Of course this works for multivariate polynomials too, so the ratio of
5403 the sample-polynomials from the section about GCD and LCM above would be
5404 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5407 @subsection Numerator and denominator
5410 @cindex @code{numer()}
5411 @cindex @code{denom()}
5412 @cindex @code{numer_denom()}
5414 The numerator and denominator of an expression can be obtained with
5419 ex ex::numer_denom();
5422 These functions will first normalize the expression as described above and
5423 then return the numerator, denominator, or both as a list, respectively.
5424 If you need both numerator and denominator, calling @code{numer_denom()} is
5425 faster than using @code{numer()} and @code{denom()} separately.
5428 @subsection Converting to a polynomial or rational expression
5429 @cindex @code{to_polynomial()}
5430 @cindex @code{to_rational()}
5432 Some of the methods described so far only work on polynomials or rational
5433 functions. GiNaC provides a way to extend the domain of these functions to
5434 general expressions by using the temporary replacement algorithm described
5435 above. You do this by calling
5438 ex ex::to_polynomial(exmap & m);
5439 ex ex::to_polynomial(lst & l);
5443 ex ex::to_rational(exmap & m);
5444 ex ex::to_rational(lst & l);
5447 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5448 will be filled with the generated temporary symbols and their replacement
5449 expressions in a format that can be used directly for the @code{subs()}
5450 method. It can also already contain a list of replacements from an earlier
5451 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5452 possible to use it on multiple expressions and get consistent results.
5454 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5455 is probably best illustrated with an example:
5459 symbol x("x"), y("y");
5460 ex a = 2*x/sin(x) - y/(3*sin(x));
5464 ex p = a.to_polynomial(lp);
5465 cout << " = " << p << "\n with " << lp << endl;
5466 // = symbol3*symbol2*y+2*symbol2*x
5467 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5470 ex r = a.to_rational(lr);
5471 cout << " = " << r << "\n with " << lr << endl;
5472 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5473 // with @{symbol4==sin(x)@}
5477 The following more useful example will print @samp{sin(x)-cos(x)}:
5482 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5483 ex b = sin(x) + cos(x);
5486 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5487 cout << q.subs(m) << endl;
5492 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5493 @c node-name, next, previous, up
5494 @section Symbolic differentiation
5495 @cindex differentiation
5496 @cindex @code{diff()}
5498 @cindex product rule
5500 GiNaC's objects know how to differentiate themselves. Thus, a
5501 polynomial (class @code{add}) knows that its derivative is the sum of
5502 the derivatives of all the monomials:
5506 symbol x("x"), y("y"), z("z");
5507 ex P = pow(x, 5) + pow(x, 2) + y;
5509 cout << P.diff(x,2) << endl;
5511 cout << P.diff(y) << endl; // 1
5513 cout << P.diff(z) << endl; // 0
5518 If a second integer parameter @var{n} is given, the @code{diff} method
5519 returns the @var{n}th derivative.
5521 If @emph{every} object and every function is told what its derivative
5522 is, all derivatives of composed objects can be calculated using the
5523 chain rule and the product rule. Consider, for instance the expression
5524 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5525 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5526 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5527 out that the composition is the generating function for Euler Numbers,
5528 i.e. the so called @var{n}th Euler number is the coefficient of
5529 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5530 identity to code a function that generates Euler numbers in just three
5533 @cindex Euler numbers
5535 #include <ginac/ginac.h>
5536 using namespace GiNaC;
5538 ex EulerNumber(unsigned n)
5541 const ex generator = pow(cosh(x),-1);
5542 return generator.diff(x,n).subs(x==0);
5547 for (unsigned i=0; i<11; i+=2)
5548 std::cout << EulerNumber(i) << std::endl;
5553 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5554 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5555 @code{i} by two since all odd Euler numbers vanish anyways.
5558 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5559 @c node-name, next, previous, up
5560 @section Series expansion
5561 @cindex @code{series()}
5562 @cindex Taylor expansion
5563 @cindex Laurent expansion
5564 @cindex @code{pseries} (class)
5565 @cindex @code{Order()}
5567 Expressions know how to expand themselves as a Taylor series or (more
5568 generally) a Laurent series. As in most conventional Computer Algebra
5569 Systems, no distinction is made between those two. There is a class of
5570 its own for storing such series (@code{class pseries}) and a built-in
5571 function (called @code{Order}) for storing the order term of the series.
5572 As a consequence, if you want to work with series, i.e. multiply two
5573 series, you need to call the method @code{ex::series} again to convert
5574 it to a series object with the usual structure (expansion plus order
5575 term). A sample application from special relativity could read:
5578 #include <ginac/ginac.h>
5579 using namespace std;
5580 using namespace GiNaC;
5584 symbol v("v"), c("c");
5586 ex gamma = 1/sqrt(1 - pow(v/c,2));
5587 ex mass_nonrel = gamma.series(v==0, 10);
5589 cout << "the relativistic mass increase with v is " << endl
5590 << mass_nonrel << endl;
5592 cout << "the inverse square of this series is " << endl
5593 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5597 Only calling the series method makes the last output simplify to
5598 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5599 series raised to the power @math{-2}.
5601 @cindex Machin's formula
5602 As another instructive application, let us calculate the numerical
5603 value of Archimedes' constant
5607 (for which there already exists the built-in constant @code{Pi})
5608 using John Machin's amazing formula
5610 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5613 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5615 This equation (and similar ones) were used for over 200 years for
5616 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5617 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5618 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5619 order term with it and the question arises what the system is supposed
5620 to do when the fractions are plugged into that order term. The solution
5621 is to use the function @code{series_to_poly()} to simply strip the order
5625 #include <ginac/ginac.h>
5626 using namespace GiNaC;
5628 ex machin_pi(int degr)
5631 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5632 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5633 -4*pi_expansion.subs(x==numeric(1,239));
5639 using std::cout; // just for fun, another way of...
5640 using std::endl; // ...dealing with this namespace std.
5642 for (int i=2; i<12; i+=2) @{
5643 pi_frac = machin_pi(i);
5644 cout << i << ":\t" << pi_frac << endl
5645 << "\t" << pi_frac.evalf() << endl;
5651 Note how we just called @code{.series(x,degr)} instead of
5652 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5653 method @code{series()}: if the first argument is a symbol the expression
5654 is expanded in that symbol around point @code{0}. When you run this
5655 program, it will type out:
5659 3.1832635983263598326
5660 4: 5359397032/1706489875
5661 3.1405970293260603143
5662 6: 38279241713339684/12184551018734375
5663 3.141621029325034425
5664 8: 76528487109180192540976/24359780855939418203125
5665 3.141591772182177295
5666 10: 327853873402258685803048818236/104359128170408663038552734375
5667 3.1415926824043995174
5671 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5672 @c node-name, next, previous, up
5673 @section Symmetrization
5674 @cindex @code{symmetrize()}
5675 @cindex @code{antisymmetrize()}
5676 @cindex @code{symmetrize_cyclic()}
5681 ex ex::symmetrize(const lst & l);
5682 ex ex::antisymmetrize(const lst & l);
5683 ex ex::symmetrize_cyclic(const lst & l);
5686 symmetrize an expression by returning the sum over all symmetric,
5687 antisymmetric or cyclic permutations of the specified list of objects,
5688 weighted by the number of permutations.
5690 The three additional methods
5693 ex ex::symmetrize();
5694 ex ex::antisymmetrize();
5695 ex ex::symmetrize_cyclic();
5698 symmetrize or antisymmetrize an expression over its free indices.
5700 Symmetrization is most useful with indexed expressions but can be used with
5701 almost any kind of object (anything that is @code{subs()}able):
5705 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5706 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5708 cout << indexed(A, i, j).symmetrize() << endl;
5709 // -> 1/2*A.j.i+1/2*A.i.j
5710 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5711 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5712 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5713 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5717 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5718 @c node-name, next, previous, up
5719 @section Predefined mathematical functions
5721 @subsection Overview
5723 GiNaC contains the following predefined mathematical functions:
5726 @multitable @columnfractions .30 .70
5727 @item @strong{Name} @tab @strong{Function}
5730 @cindex @code{abs()}
5731 @item @code{step(x)}
5733 @cindex @code{step()}
5734 @item @code{csgn(x)}
5736 @cindex @code{conjugate()}
5737 @item @code{conjugate(x)}
5738 @tab complex conjugation
5739 @cindex @code{real_part()}
5740 @item @code{real_part(x)}
5742 @cindex @code{imag_part()}
5743 @item @code{imag_part(x)}
5745 @item @code{sqrt(x)}
5746 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5747 @cindex @code{sqrt()}
5750 @cindex @code{sin()}
5753 @cindex @code{cos()}
5756 @cindex @code{tan()}
5757 @item @code{asin(x)}
5759 @cindex @code{asin()}
5760 @item @code{acos(x)}
5762 @cindex @code{acos()}
5763 @item @code{atan(x)}
5764 @tab inverse tangent
5765 @cindex @code{atan()}
5766 @item @code{atan2(y, x)}
5767 @tab inverse tangent with two arguments
5768 @item @code{sinh(x)}
5769 @tab hyperbolic sine
5770 @cindex @code{sinh()}
5771 @item @code{cosh(x)}
5772 @tab hyperbolic cosine
5773 @cindex @code{cosh()}
5774 @item @code{tanh(x)}
5775 @tab hyperbolic tangent
5776 @cindex @code{tanh()}
5777 @item @code{asinh(x)}
5778 @tab inverse hyperbolic sine
5779 @cindex @code{asinh()}
5780 @item @code{acosh(x)}
5781 @tab inverse hyperbolic cosine
5782 @cindex @code{acosh()}
5783 @item @code{atanh(x)}
5784 @tab inverse hyperbolic tangent
5785 @cindex @code{atanh()}
5787 @tab exponential function
5788 @cindex @code{exp()}
5790 @tab natural logarithm
5791 @cindex @code{log()}
5794 @cindex @code{Li2()}
5795 @item @code{Li(m, x)}
5796 @tab classical polylogarithm as well as multiple polylogarithm
5798 @item @code{G(a, y)}
5799 @tab multiple polylogarithm
5801 @item @code{G(a, s, y)}
5802 @tab multiple polylogarithm with explicit signs for the imaginary parts
5804 @item @code{S(n, p, x)}
5805 @tab Nielsen's generalized polylogarithm
5807 @item @code{H(m, x)}
5808 @tab harmonic polylogarithm
5810 @item @code{zeta(m)}
5811 @tab Riemann's zeta function as well as multiple zeta value
5812 @cindex @code{zeta()}
5813 @item @code{zeta(m, s)}
5814 @tab alternating Euler sum
5815 @cindex @code{zeta()}
5816 @item @code{zetaderiv(n, x)}
5817 @tab derivatives of Riemann's zeta function
5818 @item @code{tgamma(x)}
5820 @cindex @code{tgamma()}
5821 @cindex gamma function
5822 @item @code{lgamma(x)}
5823 @tab logarithm of gamma function
5824 @cindex @code{lgamma()}
5825 @item @code{beta(x, y)}
5826 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5827 @cindex @code{beta()}
5829 @tab psi (digamma) function
5830 @cindex @code{psi()}
5831 @item @code{psi(n, x)}
5832 @tab derivatives of psi function (polygamma functions)
5833 @item @code{factorial(n)}
5834 @tab factorial function @math{n!}
5835 @cindex @code{factorial()}
5836 @item @code{binomial(n, k)}
5837 @tab binomial coefficients
5838 @cindex @code{binomial()}
5839 @item @code{Order(x)}
5840 @tab order term function in truncated power series
5841 @cindex @code{Order()}
5846 For functions that have a branch cut in the complex plane GiNaC follows
5847 the conventions for C++ as defined in the ANSI standard as far as
5848 possible. In particular: the natural logarithm (@code{log}) and the
5849 square root (@code{sqrt}) both have their branch cuts running along the
5850 negative real axis where the points on the axis itself belong to the
5851 upper part (i.e. continuous with quadrant II). The inverse
5852 trigonometric and hyperbolic functions are not defined for complex
5853 arguments by the C++ standard, however. In GiNaC we follow the
5854 conventions used by CLN, which in turn follow the carefully designed
5855 definitions in the Common Lisp standard. It should be noted that this
5856 convention is identical to the one used by the C99 standard and by most
5857 serious CAS. It is to be expected that future revisions of the C++
5858 standard incorporate these functions in the complex domain in a manner
5859 compatible with C99.
5861 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5862 @c node-name, next, previous, up
5863 @subsection Multiple polylogarithms
5865 @cindex polylogarithm
5866 @cindex Nielsen's generalized polylogarithm
5867 @cindex harmonic polylogarithm
5868 @cindex multiple zeta value
5869 @cindex alternating Euler sum
5870 @cindex multiple polylogarithm
5872 The multiple polylogarithm is the most generic member of a family of functions,
5873 to which others like the harmonic polylogarithm, Nielsen's generalized
5874 polylogarithm and the multiple zeta value belong.
5875 Everyone of these functions can also be written as a multiple polylogarithm with specific
5876 parameters. This whole family of functions is therefore often referred to simply as
5877 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5878 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5879 @code{Li} and @code{G} in principle represent the same function, the different
5880 notations are more natural to the series representation or the integral
5881 representation, respectively.
5883 To facilitate the discussion of these functions we distinguish between indices and
5884 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5885 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5887 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5888 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5889 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5890 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5891 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5892 @code{s} is not given, the signs default to +1.
5893 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5894 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5895 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5896 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5897 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5899 The functions print in LaTeX format as
5901 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5907 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5910 $\zeta(m_1,m_2,\ldots,m_k)$.
5912 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5913 are printed with a line above, e.g.
5915 $\zeta(5,\overline{2})$.
5917 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5919 Definitions and analytical as well as numerical properties of multiple polylogarithms
5920 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5921 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5922 except for a few differences which will be explicitly stated in the following.
5924 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5925 that the indices and arguments are understood to be in the same order as in which they appear in
5926 the series representation. This means
5928 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5931 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5934 $\zeta(1,2)$ evaluates to infinity.
5936 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5939 The functions only evaluate if the indices are integers greater than zero, except for the indices
5940 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5941 will be interpreted as the sequence of signs for the corresponding indices
5942 @code{m} or the sign of the imaginary part for the
5943 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5944 @code{zeta(lst(3,4), lst(-1,1))} means
5946 $\zeta(\overline{3},4)$
5949 @code{G(lst(a,b), lst(-1,1), c)} means
5951 $G(a-0\epsilon,b+0\epsilon;c)$.
5953 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5954 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5955 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
5956 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5957 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5958 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5959 evaluates also for negative integers and positive even integers. For example:
5962 > Li(@{3,1@},@{x,1@});
5965 -zeta(@{3,2@},@{-1,-1@})
5970 It is easy to tell for a given function into which other function it can be rewritten, may
5971 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5972 with negative indices or trailing zeros (the example above gives a hint). Signs can
5973 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5974 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5975 @code{Li} (@code{eval()} already cares for the possible downgrade):
5978 > convert_H_to_Li(@{0,-2,-1,3@},x);
5979 Li(@{3,1,3@},@{-x,1,-1@})
5980 > convert_H_to_Li(@{2,-1,0@},x);
5981 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5984 Every function can be numerically evaluated for
5985 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5986 global variable @code{Digits}:
5991 > evalf(zeta(@{3,1,3,1@}));
5992 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5995 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5996 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5998 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6003 In long expressions this helps a lot with debugging, because you can easily spot
6004 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6005 cancellations of divergencies happen.
6007 Useful publications:
6009 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6010 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6012 @cite{Harmonic Polylogarithms},
6013 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6015 @cite{Special Values of Multiple Polylogarithms},
6016 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6018 @cite{Numerical Evaluation of Multiple Polylogarithms},
6019 J.Vollinga, S.Weinzierl, hep-ph/0410259
6021 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6022 @c node-name, next, previous, up
6023 @section Complex expressions
6025 @cindex @code{conjugate()}
6027 For dealing with complex expressions there are the methods
6035 that return respectively the complex conjugate, the real part and the
6036 imaginary part of an expression. Complex conjugation works as expected
6037 for all built-in functinos and objects. Taking real and imaginary
6038 parts has not yet been implemented for all built-in functions. In cases where
6039 it is not known how to conjugate or take a real/imaginary part one
6040 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6041 is returned. For instance, in case of a complex symbol @code{x}
6042 (symbols are complex by default), one could not simplify
6043 @code{conjugate(x)}. In the case of strings of gamma matrices,
6044 the @code{conjugate} method takes the Dirac conjugate.
6049 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6053 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6054 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6055 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6056 // -> -gamma5*gamma~b*gamma~a
6060 If you declare your own GiNaC functions, then they will conjugate themselves
6061 by conjugating their arguments. This is the default strategy. If you want to
6062 change this behavior, you have to supply a specialized conjugation method
6063 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6064 for @code{abs} as an example). Also, specialized methods can be provided
6065 to take real and imaginary parts of user-defined functions.
6067 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6068 @c node-name, next, previous, up
6069 @section Solving linear systems of equations
6070 @cindex @code{lsolve()}
6072 The function @code{lsolve()} provides a convenient wrapper around some
6073 matrix operations that comes in handy when a system of linear equations
6077 ex lsolve(const ex & eqns, const ex & symbols,
6078 unsigned options = solve_algo::automatic);
6081 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6082 @code{relational}) while @code{symbols} is a @code{lst} of
6083 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6086 It returns the @code{lst} of solutions as an expression. As an example,
6087 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6091 symbol a("a"), b("b"), x("x"), y("y");
6093 eqns = a*x+b*y==3, x-y==b;
6095 cout << lsolve(eqns, vars) << endl;
6096 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6099 When the linear equations @code{eqns} are underdetermined, the solution
6100 will contain one or more tautological entries like @code{x==x},
6101 depending on the rank of the system. When they are overdetermined, the
6102 solution will be an empty @code{lst}. Note the third optional parameter
6103 to @code{lsolve()}: it accepts the same parameters as
6104 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6108 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6109 @c node-name, next, previous, up
6110 @section Input and output of expressions
6113 @subsection Expression output
6115 @cindex output of expressions
6117 Expressions can simply be written to any stream:
6122 ex e = 4.5*I+pow(x,2)*3/2;
6123 cout << e << endl; // prints '4.5*I+3/2*x^2'
6127 The default output format is identical to the @command{ginsh} input syntax and
6128 to that used by most computer algebra systems, but not directly pastable
6129 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6130 is printed as @samp{x^2}).
6132 It is possible to print expressions in a number of different formats with
6133 a set of stream manipulators;
6136 std::ostream & dflt(std::ostream & os);
6137 std::ostream & latex(std::ostream & os);
6138 std::ostream & tree(std::ostream & os);
6139 std::ostream & csrc(std::ostream & os);
6140 std::ostream & csrc_float(std::ostream & os);
6141 std::ostream & csrc_double(std::ostream & os);
6142 std::ostream & csrc_cl_N(std::ostream & os);
6143 std::ostream & index_dimensions(std::ostream & os);
6144 std::ostream & no_index_dimensions(std::ostream & os);
6147 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6148 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6149 @code{print_csrc()} functions, respectively.
6152 All manipulators affect the stream state permanently. To reset the output
6153 format to the default, use the @code{dflt} manipulator:
6157 cout << latex; // all output to cout will be in LaTeX format from
6159 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6160 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6161 cout << dflt; // revert to default output format
6162 cout << e << endl; // prints '4.5*I+3/2*x^2'
6166 If you don't want to affect the format of the stream you're working with,
6167 you can output to a temporary @code{ostringstream} like this:
6172 s << latex << e; // format of cout remains unchanged
6173 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6178 @cindex @code{csrc_float}
6179 @cindex @code{csrc_double}
6180 @cindex @code{csrc_cl_N}
6181 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6182 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6183 format that can be directly used in a C or C++ program. The three possible
6184 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6185 classes provided by the CLN library):
6189 cout << "f = " << csrc_float << e << ";\n";
6190 cout << "d = " << csrc_double << e << ";\n";
6191 cout << "n = " << csrc_cl_N << e << ";\n";
6195 The above example will produce (note the @code{x^2} being converted to
6199 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6200 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6201 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6205 The @code{tree} manipulator allows dumping the internal structure of an
6206 expression for debugging purposes:
6217 add, hash=0x0, flags=0x3, nops=2
6218 power, hash=0x0, flags=0x3, nops=2
6219 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6220 2 (numeric), hash=0x6526b0fa, flags=0xf
6221 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6224 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6228 @cindex @code{latex}
6229 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6230 It is rather similar to the default format but provides some braces needed
6231 by LaTeX for delimiting boxes and also converts some common objects to
6232 conventional LaTeX names. It is possible to give symbols a special name for
6233 LaTeX output by supplying it as a second argument to the @code{symbol}
6236 For example, the code snippet
6240 symbol x("x", "\\circ");
6241 ex e = lgamma(x).series(x==0,3);
6242 cout << latex << e << endl;
6249 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6250 +\mathcal@{O@}(\circ^@{3@})
6253 @cindex @code{index_dimensions}
6254 @cindex @code{no_index_dimensions}
6255 Index dimensions are normally hidden in the output. To make them visible, use
6256 the @code{index_dimensions} manipulator. The dimensions will be written in
6257 square brackets behind each index value in the default and LaTeX output
6262 symbol x("x"), y("y");
6263 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6264 ex e = indexed(x, mu) * indexed(y, nu);
6267 // prints 'x~mu*y~nu'
6268 cout << index_dimensions << e << endl;
6269 // prints 'x~mu[4]*y~nu[4]'
6270 cout << no_index_dimensions << e << endl;
6271 // prints 'x~mu*y~nu'
6276 @cindex Tree traversal
6277 If you need any fancy special output format, e.g. for interfacing GiNaC
6278 with other algebra systems or for producing code for different
6279 programming languages, you can always traverse the expression tree yourself:
6282 static void my_print(const ex & e)
6284 if (is_a<function>(e))
6285 cout << ex_to<function>(e).get_name();
6287 cout << ex_to<basic>(e).class_name();
6289 size_t n = e.nops();
6291 for (size_t i=0; i<n; i++) @{
6303 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6311 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6312 symbol(y))),numeric(-2)))
6315 If you need an output format that makes it possible to accurately
6316 reconstruct an expression by feeding the output to a suitable parser or
6317 object factory, you should consider storing the expression in an
6318 @code{archive} object and reading the object properties from there.
6319 See the section on archiving for more information.
6322 @subsection Expression input
6323 @cindex input of expressions
6325 GiNaC provides no way to directly read an expression from a stream because
6326 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6327 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6328 @code{y} you defined in your program and there is no way to specify the
6329 desired symbols to the @code{>>} stream input operator.
6331 Instead, GiNaC lets you construct an expression from a string, specifying the
6332 list of symbols to be used:
6336 symbol x("x"), y("y");
6337 ex e("2*x+sin(y)", lst(x, y));
6341 The input syntax is the same as that used by @command{ginsh} and the stream
6342 output operator @code{<<}. The symbols in the string are matched by name to
6343 the symbols in the list and if GiNaC encounters a symbol not specified in
6344 the list it will throw an exception.
6346 With this constructor, it's also easy to implement interactive GiNaC programs:
6351 #include <stdexcept>
6352 #include <ginac/ginac.h>
6353 using namespace std;
6354 using namespace GiNaC;
6361 cout << "Enter an expression containing 'x': ";
6366 cout << "The derivative of " << e << " with respect to x is ";
6367 cout << e.diff(x) << ".\n";
6368 @} catch (exception &p) @{
6369 cerr << p.what() << endl;
6375 @subsection Archiving
6376 @cindex @code{archive} (class)
6379 GiNaC allows creating @dfn{archives} of expressions which can be stored
6380 to or retrieved from files. To create an archive, you declare an object
6381 of class @code{archive} and archive expressions in it, giving each
6382 expression a unique name:
6386 using namespace std;
6387 #include <ginac/ginac.h>
6388 using namespace GiNaC;
6392 symbol x("x"), y("y"), z("z");
6394 ex foo = sin(x + 2*y) + 3*z + 41;
6398 a.archive_ex(foo, "foo");
6399 a.archive_ex(bar, "the second one");
6403 The archive can then be written to a file:
6407 ofstream out("foobar.gar");
6413 The file @file{foobar.gar} contains all information that is needed to
6414 reconstruct the expressions @code{foo} and @code{bar}.
6416 @cindex @command{viewgar}
6417 The tool @command{viewgar} that comes with GiNaC can be used to view
6418 the contents of GiNaC archive files:
6421 $ viewgar foobar.gar
6422 foo = 41+sin(x+2*y)+3*z
6423 the second one = 42+sin(x+2*y)+3*z
6426 The point of writing archive files is of course that they can later be
6432 ifstream in("foobar.gar");
6437 And the stored expressions can be retrieved by their name:
6444 ex ex1 = a2.unarchive_ex(syms, "foo");
6445 ex ex2 = a2.unarchive_ex(syms, "the second one");
6447 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6448 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6449 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6453 Note that you have to supply a list of the symbols which are to be inserted
6454 in the expressions. Symbols in archives are stored by their name only and
6455 if you don't specify which symbols you have, unarchiving the expression will
6456 create new symbols with that name. E.g. if you hadn't included @code{x} in
6457 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6458 have had no effect because the @code{x} in @code{ex1} would have been a
6459 different symbol than the @code{x} which was defined at the beginning of
6460 the program, although both would appear as @samp{x} when printed.
6462 You can also use the information stored in an @code{archive} object to
6463 output expressions in a format suitable for exact reconstruction. The
6464 @code{archive} and @code{archive_node} classes have a couple of member
6465 functions that let you access the stored properties:
6468 static void my_print2(const archive_node & n)
6471 n.find_string("class", class_name);
6472 cout << class_name << "(";
6474 archive_node::propinfovector p;
6475 n.get_properties(p);
6477 size_t num = p.size();
6478 for (size_t i=0; i<num; i++) @{
6479 const string &name = p[i].name;
6480 if (name == "class")
6482 cout << name << "=";
6484 unsigned count = p[i].count;
6488 for (unsigned j=0; j<count; j++) @{
6489 switch (p[i].type) @{
6490 case archive_node::PTYPE_BOOL: @{
6492 n.find_bool(name, x, j);
6493 cout << (x ? "true" : "false");
6496 case archive_node::PTYPE_UNSIGNED: @{
6498 n.find_unsigned(name, x, j);
6502 case archive_node::PTYPE_STRING: @{
6504 n.find_string(name, x, j);
6505 cout << '\"' << x << '\"';
6508 case archive_node::PTYPE_NODE: @{
6509 const archive_node &x = n.find_ex_node(name, j);
6531 ex e = pow(2, x) - y;
6533 my_print2(ar.get_top_node(0)); cout << endl;
6541 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6542 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6543 overall_coeff=numeric(number="0"))
6546 Be warned, however, that the set of properties and their meaning for each
6547 class may change between GiNaC versions.
6550 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6551 @c node-name, next, previous, up
6552 @chapter Extending GiNaC
6554 By reading so far you should have gotten a fairly good understanding of
6555 GiNaC's design patterns. From here on you should start reading the
6556 sources. All we can do now is issue some recommendations how to tackle
6557 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6558 develop some useful extension please don't hesitate to contact the GiNaC
6559 authors---they will happily incorporate them into future versions.
6562 * What does not belong into GiNaC:: What to avoid.
6563 * Symbolic functions:: Implementing symbolic functions.
6564 * Printing:: Adding new output formats.
6565 * Structures:: Defining new algebraic classes (the easy way).
6566 * Adding classes:: Defining new algebraic classes (the hard way).
6570 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6571 @c node-name, next, previous, up
6572 @section What doesn't belong into GiNaC
6574 @cindex @command{ginsh}
6575 First of all, GiNaC's name must be read literally. It is designed to be
6576 a library for use within C++. The tiny @command{ginsh} accompanying
6577 GiNaC makes this even more clear: it doesn't even attempt to provide a
6578 language. There are no loops or conditional expressions in
6579 @command{ginsh}, it is merely a window into the library for the
6580 programmer to test stuff (or to show off). Still, the design of a
6581 complete CAS with a language of its own, graphical capabilities and all
6582 this on top of GiNaC is possible and is without doubt a nice project for
6585 There are many built-in functions in GiNaC that do not know how to
6586 evaluate themselves numerically to a precision declared at runtime
6587 (using @code{Digits}). Some may be evaluated at certain points, but not
6588 generally. This ought to be fixed. However, doing numerical
6589 computations with GiNaC's quite abstract classes is doomed to be
6590 inefficient. For this purpose, the underlying foundation classes
6591 provided by CLN are much better suited.
6594 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6595 @c node-name, next, previous, up
6596 @section Symbolic functions
6598 The easiest and most instructive way to start extending GiNaC is probably to
6599 create your own symbolic functions. These are implemented with the help of
6600 two preprocessor macros:
6602 @cindex @code{DECLARE_FUNCTION}
6603 @cindex @code{REGISTER_FUNCTION}
6605 DECLARE_FUNCTION_<n>P(<name>)
6606 REGISTER_FUNCTION(<name>, <options>)
6609 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6610 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6611 parameters of type @code{ex} and returns a newly constructed GiNaC
6612 @code{function} object that represents your function.
6614 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6615 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6616 set of options that associate the symbolic function with C++ functions you
6617 provide to implement the various methods such as evaluation, derivative,
6618 series expansion etc. They also describe additional attributes the function
6619 might have, such as symmetry and commutation properties, and a name for
6620 LaTeX output. Multiple options are separated by the member access operator
6621 @samp{.} and can be given in an arbitrary order.
6623 (By the way: in case you are worrying about all the macros above we can
6624 assure you that functions are GiNaC's most macro-intense classes. We have
6625 done our best to avoid macros where we can.)
6627 @subsection A minimal example
6629 Here is an example for the implementation of a function with two arguments
6630 that is not further evaluated:
6633 DECLARE_FUNCTION_2P(myfcn)
6635 REGISTER_FUNCTION(myfcn, dummy())
6638 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6639 in algebraic expressions:
6645 ex e = 2*myfcn(42, 1+3*x) - x;
6647 // prints '2*myfcn(42,1+3*x)-x'
6652 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6653 "no options". A function with no options specified merely acts as a kind of
6654 container for its arguments. It is a pure "dummy" function with no associated
6655 logic (which is, however, sometimes perfectly sufficient).
6657 Let's now have a look at the implementation of GiNaC's cosine function for an
6658 example of how to make an "intelligent" function.
6660 @subsection The cosine function
6662 The GiNaC header file @file{inifcns.h} contains the line
6665 DECLARE_FUNCTION_1P(cos)
6668 which declares to all programs using GiNaC that there is a function @samp{cos}
6669 that takes one @code{ex} as an argument. This is all they need to know to use
6670 this function in expressions.
6672 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6673 is its @code{REGISTER_FUNCTION} line:
6676 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6677 evalf_func(cos_evalf).
6678 derivative_func(cos_deriv).
6679 latex_name("\\cos"));
6682 There are four options defined for the cosine function. One of them
6683 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6684 other three indicate the C++ functions in which the "brains" of the cosine
6685 function are defined.
6687 @cindex @code{hold()}
6689 The @code{eval_func()} option specifies the C++ function that implements
6690 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6691 the same number of arguments as the associated symbolic function (one in this
6692 case) and returns the (possibly transformed or in some way simplified)
6693 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6694 of the automatic evaluation process). If no (further) evaluation is to take
6695 place, the @code{eval_func()} function must return the original function
6696 with @code{.hold()}, to avoid a potential infinite recursion. If your
6697 symbolic functions produce a segmentation fault or stack overflow when
6698 using them in expressions, you are probably missing a @code{.hold()}
6701 The @code{eval_func()} function for the cosine looks something like this
6702 (actually, it doesn't look like this at all, but it should give you an idea
6706 static ex cos_eval(const ex & x)
6708 if ("x is a multiple of 2*Pi")
6710 else if ("x is a multiple of Pi")
6712 else if ("x is a multiple of Pi/2")
6716 else if ("x has the form 'acos(y)'")
6718 else if ("x has the form 'asin(y)'")
6723 return cos(x).hold();
6727 This function is called every time the cosine is used in a symbolic expression:
6733 // this calls cos_eval(Pi), and inserts its return value into
6734 // the actual expression
6741 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6742 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6743 symbolic transformation can be done, the unmodified function is returned
6744 with @code{.hold()}.
6746 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6747 The user has to call @code{evalf()} for that. This is implemented in a
6751 static ex cos_evalf(const ex & x)
6753 if (is_a<numeric>(x))
6754 return cos(ex_to<numeric>(x));
6756 return cos(x).hold();
6760 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6761 in this case the @code{cos()} function for @code{numeric} objects, which in
6762 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6763 isn't really needed here, but reminds us that the corresponding @code{eval()}
6764 function would require it in this place.
6766 Differentiation will surely turn up and so we need to tell @code{cos}
6767 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6768 instance, are then handled automatically by @code{basic::diff} and
6772 static ex cos_deriv(const ex & x, unsigned diff_param)
6778 @cindex product rule
6779 The second parameter is obligatory but uninteresting at this point. It
6780 specifies which parameter to differentiate in a partial derivative in
6781 case the function has more than one parameter, and its main application
6782 is for correct handling of the chain rule.
6784 An implementation of the series expansion is not needed for @code{cos()} as
6785 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6786 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6787 the other hand, does have poles and may need to do Laurent expansion:
6790 static ex tan_series(const ex & x, const relational & rel,
6791 int order, unsigned options)
6793 // Find the actual expansion point
6794 const ex x_pt = x.subs(rel);
6796 if ("x_pt is not an odd multiple of Pi/2")
6797 throw do_taylor(); // tell function::series() to do Taylor expansion
6799 // On a pole, expand sin()/cos()
6800 return (sin(x)/cos(x)).series(rel, order+2, options);
6804 The @code{series()} implementation of a function @emph{must} return a
6805 @code{pseries} object, otherwise your code will crash.
6807 @subsection Function options
6809 GiNaC functions understand several more options which are always
6810 specified as @code{.option(params)}. None of them are required, but you
6811 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6812 is a do-nothing option called @code{dummy()} which you can use to define
6813 functions without any special options.
6816 eval_func(<C++ function>)
6817 evalf_func(<C++ function>)
6818 derivative_func(<C++ function>)
6819 series_func(<C++ function>)
6820 conjugate_func(<C++ function>)
6823 These specify the C++ functions that implement symbolic evaluation,
6824 numeric evaluation, partial derivatives, and series expansion, respectively.
6825 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6826 @code{diff()} and @code{series()}.
6828 The @code{eval_func()} function needs to use @code{.hold()} if no further
6829 automatic evaluation is desired or possible.
6831 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6832 expansion, which is correct if there are no poles involved. If the function
6833 has poles in the complex plane, the @code{series_func()} needs to check
6834 whether the expansion point is on a pole and fall back to Taylor expansion
6835 if it isn't. Otherwise, the pole usually needs to be regularized by some
6836 suitable transformation.
6839 latex_name(const string & n)
6842 specifies the LaTeX code that represents the name of the function in LaTeX
6843 output. The default is to put the function name in an @code{\mbox@{@}}.
6846 do_not_evalf_params()
6849 This tells @code{evalf()} to not recursively evaluate the parameters of the
6850 function before calling the @code{evalf_func()}.
6853 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6856 This allows you to explicitly specify the commutation properties of the
6857 function (@xref{Non-commutative objects}, for an explanation of
6858 (non)commutativity in GiNaC). For example, you can use
6859 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6860 GiNaC treat your function like a matrix. By default, functions inherit the
6861 commutation properties of their first argument.
6864 set_symmetry(const symmetry & s)
6867 specifies the symmetry properties of the function with respect to its
6868 arguments. @xref{Indexed objects}, for an explanation of symmetry
6869 specifications. GiNaC will automatically rearrange the arguments of
6870 symmetric functions into a canonical order.
6872 Sometimes you may want to have finer control over how functions are
6873 displayed in the output. For example, the @code{abs()} function prints
6874 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6875 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6879 print_func<C>(<C++ function>)
6882 option which is explained in the next section.
6884 @subsection Functions with a variable number of arguments
6886 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6887 functions with a fixed number of arguments. Sometimes, though, you may need
6888 to have a function that accepts a variable number of expressions. One way to
6889 accomplish this is to pass variable-length lists as arguments. The
6890 @code{Li()} function uses this method for multiple polylogarithms.
6892 It is also possible to define functions that accept a different number of
6893 parameters under the same function name, such as the @code{psi()} function
6894 which can be called either as @code{psi(z)} (the digamma function) or as
6895 @code{psi(n, z)} (polygamma functions). These are actually two different
6896 functions in GiNaC that, however, have the same name. Defining such
6897 functions is not possible with the macros but requires manually fiddling
6898 with GiNaC internals. If you are interested, please consult the GiNaC source
6899 code for the @code{psi()} function (@file{inifcns.h} and
6900 @file{inifcns_gamma.cpp}).
6903 @node Printing, Structures, Symbolic functions, Extending GiNaC
6904 @c node-name, next, previous, up
6905 @section GiNaC's expression output system
6907 GiNaC allows the output of expressions in a variety of different formats
6908 (@pxref{Input/output}). This section will explain how expression output
6909 is implemented internally, and how to define your own output formats or
6910 change the output format of built-in algebraic objects. You will also want
6911 to read this section if you plan to write your own algebraic classes or
6914 @cindex @code{print_context} (class)
6915 @cindex @code{print_dflt} (class)
6916 @cindex @code{print_latex} (class)
6917 @cindex @code{print_tree} (class)
6918 @cindex @code{print_csrc} (class)
6919 All the different output formats are represented by a hierarchy of classes
6920 rooted in the @code{print_context} class, defined in the @file{print.h}
6925 the default output format
6927 output in LaTeX mathematical mode
6929 a dump of the internal expression structure (for debugging)
6931 the base class for C source output
6932 @item print_csrc_float
6933 C source output using the @code{float} type
6934 @item print_csrc_double
6935 C source output using the @code{double} type
6936 @item print_csrc_cl_N
6937 C source output using CLN types
6940 The @code{print_context} base class provides two public data members:
6952 @code{s} is a reference to the stream to output to, while @code{options}
6953 holds flags and modifiers. Currently, there is only one flag defined:
6954 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6955 to print the index dimension which is normally hidden.
6957 When you write something like @code{std::cout << e}, where @code{e} is
6958 an object of class @code{ex}, GiNaC will construct an appropriate
6959 @code{print_context} object (of a class depending on the selected output
6960 format), fill in the @code{s} and @code{options} members, and call
6962 @cindex @code{print()}
6964 void ex::print(const print_context & c, unsigned level = 0) const;
6967 which in turn forwards the call to the @code{print()} method of the
6968 top-level algebraic object contained in the expression.
6970 Unlike other methods, GiNaC classes don't usually override their
6971 @code{print()} method to implement expression output. Instead, the default
6972 implementation @code{basic::print(c, level)} performs a run-time double
6973 dispatch to a function selected by the dynamic type of the object and the
6974 passed @code{print_context}. To this end, GiNaC maintains a separate method
6975 table for each class, similar to the virtual function table used for ordinary
6976 (single) virtual function dispatch.
6978 The method table contains one slot for each possible @code{print_context}
6979 type, indexed by the (internally assigned) serial number of the type. Slots
6980 may be empty, in which case GiNaC will retry the method lookup with the
6981 @code{print_context} object's parent class, possibly repeating the process
6982 until it reaches the @code{print_context} base class. If there's still no
6983 method defined, the method table of the algebraic object's parent class
6984 is consulted, and so on, until a matching method is found (eventually it
6985 will reach the combination @code{basic/print_context}, which prints the
6986 object's class name enclosed in square brackets).
6988 You can think of the print methods of all the different classes and output
6989 formats as being arranged in a two-dimensional matrix with one axis listing
6990 the algebraic classes and the other axis listing the @code{print_context}
6993 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6994 to implement printing, but then they won't get any of the benefits of the
6995 double dispatch mechanism (such as the ability for derived classes to
6996 inherit only certain print methods from its parent, or the replacement of
6997 methods at run-time).
6999 @subsection Print methods for classes
7001 The method table for a class is set up either in the definition of the class,
7002 by passing the appropriate @code{print_func<C>()} option to
7003 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7004 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7005 can also be used to override existing methods dynamically.
7007 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7008 be a member function of the class (or one of its parent classes), a static
7009 member function, or an ordinary (global) C++ function. The @code{C} template
7010 parameter specifies the appropriate @code{print_context} type for which the
7011 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7012 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7013 the class is the one being implemented by
7014 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7016 For print methods that are member functions, their first argument must be of
7017 a type convertible to a @code{const C &}, and the second argument must be an
7020 For static members and global functions, the first argument must be of a type
7021 convertible to a @code{const T &}, the second argument must be of a type
7022 convertible to a @code{const C &}, and the third argument must be an
7023 @code{unsigned}. A global function will, of course, not have access to
7024 private and protected members of @code{T}.
7026 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7027 and @code{basic::print()}) is used for proper parenthesizing of the output
7028 (and by @code{print_tree} for proper indentation). It can be used for similar
7029 purposes if you write your own output formats.
7031 The explanations given above may seem complicated, but in practice it's
7032 really simple, as shown in the following example. Suppose that we want to
7033 display exponents in LaTeX output not as superscripts but with little
7034 upwards-pointing arrows. This can be achieved in the following way:
7037 void my_print_power_as_latex(const power & p,
7038 const print_latex & c,
7041 // get the precedence of the 'power' class
7042 unsigned power_prec = p.precedence();
7044 // if the parent operator has the same or a higher precedence
7045 // we need parentheses around the power
7046 if (level >= power_prec)
7049 // print the basis and exponent, each enclosed in braces, and
7050 // separated by an uparrow
7052 p.op(0).print(c, power_prec);
7053 c.s << "@}\\uparrow@{";
7054 p.op(1).print(c, power_prec);
7057 // don't forget the closing parenthesis
7058 if (level >= power_prec)
7064 // a sample expression
7065 symbol x("x"), y("y");
7066 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7068 // switch to LaTeX mode
7071 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7074 // now we replace the method for the LaTeX output of powers with
7076 set_print_func<power, print_latex>(my_print_power_as_latex);
7078 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7089 The first argument of @code{my_print_power_as_latex} could also have been
7090 a @code{const basic &}, the second one a @code{const print_context &}.
7093 The above code depends on @code{mul} objects converting their operands to
7094 @code{power} objects for the purpose of printing.
7097 The output of products including negative powers as fractions is also
7098 controlled by the @code{mul} class.
7101 The @code{power/print_latex} method provided by GiNaC prints square roots
7102 using @code{\sqrt}, but the above code doesn't.
7106 It's not possible to restore a method table entry to its previous or default
7107 value. Once you have called @code{set_print_func()}, you can only override
7108 it with another call to @code{set_print_func()}, but you can't easily go back
7109 to the default behavior again (you can, of course, dig around in the GiNaC
7110 sources, find the method that is installed at startup
7111 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7112 one; that is, after you circumvent the C++ member access control@dots{}).
7114 @subsection Print methods for functions
7116 Symbolic functions employ a print method dispatch mechanism similar to the
7117 one used for classes. The methods are specified with @code{print_func<C>()}
7118 function options. If you don't specify any special print methods, the function
7119 will be printed with its name (or LaTeX name, if supplied), followed by a
7120 comma-separated list of arguments enclosed in parentheses.
7122 For example, this is what GiNaC's @samp{abs()} function is defined like:
7125 static ex abs_eval(const ex & arg) @{ ... @}
7126 static ex abs_evalf(const ex & arg) @{ ... @}
7128 static void abs_print_latex(const ex & arg, const print_context & c)
7130 c.s << "@{|"; arg.print(c); c.s << "|@}";
7133 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7135 c.s << "fabs("; arg.print(c); c.s << ")";
7138 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7139 evalf_func(abs_evalf).
7140 print_func<print_latex>(abs_print_latex).
7141 print_func<print_csrc_float>(abs_print_csrc_float).
7142 print_func<print_csrc_double>(abs_print_csrc_float));
7145 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7146 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7148 There is currently no equivalent of @code{set_print_func()} for functions.
7150 @subsection Adding new output formats
7152 Creating a new output format involves subclassing @code{print_context},
7153 which is somewhat similar to adding a new algebraic class
7154 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7155 that needs to go into the class definition, and a corresponding macro
7156 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7157 Every @code{print_context} class needs to provide a default constructor
7158 and a constructor from an @code{std::ostream} and an @code{unsigned}
7161 Here is an example for a user-defined @code{print_context} class:
7164 class print_myformat : public print_dflt
7166 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7168 print_myformat(std::ostream & os, unsigned opt = 0)
7169 : print_dflt(os, opt) @{@}
7172 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7174 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7177 That's all there is to it. None of the actual expression output logic is
7178 implemented in this class. It merely serves as a selector for choosing
7179 a particular format. The algorithms for printing expressions in the new
7180 format are implemented as print methods, as described above.
7182 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7183 exactly like GiNaC's default output format:
7188 ex e = pow(x, 2) + 1;
7190 // this prints "1+x^2"
7193 // this also prints "1+x^2"
7194 e.print(print_myformat()); cout << endl;
7200 To fill @code{print_myformat} with life, we need to supply appropriate
7201 print methods with @code{set_print_func()}, like this:
7204 // This prints powers with '**' instead of '^'. See the LaTeX output
7205 // example above for explanations.
7206 void print_power_as_myformat(const power & p,
7207 const print_myformat & c,
7210 unsigned power_prec = p.precedence();
7211 if (level >= power_prec)
7213 p.op(0).print(c, power_prec);
7215 p.op(1).print(c, power_prec);
7216 if (level >= power_prec)
7222 // install a new print method for power objects
7223 set_print_func<power, print_myformat>(print_power_as_myformat);
7225 // now this prints "1+x**2"
7226 e.print(print_myformat()); cout << endl;
7228 // but the default format is still "1+x^2"
7234 @node Structures, Adding classes, Printing, Extending GiNaC
7235 @c node-name, next, previous, up
7238 If you are doing some very specialized things with GiNaC, or if you just
7239 need some more organized way to store data in your expressions instead of
7240 anonymous lists, you may want to implement your own algebraic classes.
7241 ('algebraic class' means any class directly or indirectly derived from
7242 @code{basic} that can be used in GiNaC expressions).
7244 GiNaC offers two ways of accomplishing this: either by using the
7245 @code{structure<T>} template class, or by rolling your own class from
7246 scratch. This section will discuss the @code{structure<T>} template which
7247 is easier to use but more limited, while the implementation of custom
7248 GiNaC classes is the topic of the next section. However, you may want to
7249 read both sections because many common concepts and member functions are
7250 shared by both concepts, and it will also allow you to decide which approach
7251 is most suited to your needs.
7253 The @code{structure<T>} template, defined in the GiNaC header file
7254 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7255 or @code{class}) into a GiNaC object that can be used in expressions.
7257 @subsection Example: scalar products
7259 Let's suppose that we need a way to handle some kind of abstract scalar
7260 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7261 product class have to store their left and right operands, which can in turn
7262 be arbitrary expressions. Here is a possible way to represent such a
7263 product in a C++ @code{struct}:
7267 using namespace std;
7269 #include <ginac/ginac.h>
7270 using namespace GiNaC;
7276 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7280 The default constructor is required. Now, to make a GiNaC class out of this
7281 data structure, we need only one line:
7284 typedef structure<sprod_s> sprod;
7287 That's it. This line constructs an algebraic class @code{sprod} which
7288 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7289 expressions like any other GiNaC class:
7293 symbol a("a"), b("b");
7294 ex e = sprod(sprod_s(a, b));
7298 Note the difference between @code{sprod} which is the algebraic class, and
7299 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7300 and @code{right} data members. As shown above, an @code{sprod} can be
7301 constructed from an @code{sprod_s} object.
7303 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7304 you could define a little wrapper function like this:
7307 inline ex make_sprod(ex left, ex right)
7309 return sprod(sprod_s(left, right));
7313 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7314 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7315 @code{get_struct()}:
7319 cout << ex_to<sprod>(e)->left << endl;
7321 cout << ex_to<sprod>(e).get_struct().right << endl;
7326 You only have read access to the members of @code{sprod_s}.
7328 The type definition of @code{sprod} is enough to write your own algorithms
7329 that deal with scalar products, for example:
7334 if (is_a<sprod>(p)) @{
7335 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7336 return make_sprod(sp.right, sp.left);
7347 @subsection Structure output
7349 While the @code{sprod} type is useable it still leaves something to be
7350 desired, most notably proper output:
7355 // -> [structure object]
7359 By default, any structure types you define will be printed as
7360 @samp{[structure object]}. To override this you can either specialize the
7361 template's @code{print()} member function, or specify print methods with
7362 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7363 it's not possible to supply class options like @code{print_func<>()} to
7364 structures, so for a self-contained structure type you need to resort to
7365 overriding the @code{print()} function, which is also what we will do here.
7367 The member functions of GiNaC classes are described in more detail in the
7368 next section, but it shouldn't be hard to figure out what's going on here:
7371 void sprod::print(const print_context & c, unsigned level) const
7373 // tree debug output handled by superclass
7374 if (is_a<print_tree>(c))
7375 inherited::print(c, level);
7377 // get the contained sprod_s object
7378 const sprod_s & sp = get_struct();
7380 // print_context::s is a reference to an ostream
7381 c.s << "<" << sp.left << "|" << sp.right << ">";
7385 Now we can print expressions containing scalar products:
7391 cout << swap_sprod(e) << endl;
7396 @subsection Comparing structures
7398 The @code{sprod} class defined so far still has one important drawback: all
7399 scalar products are treated as being equal because GiNaC doesn't know how to
7400 compare objects of type @code{sprod_s}. This can lead to some confusing
7401 and undesired behavior:
7405 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7407 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7408 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7412 To remedy this, we first need to define the operators @code{==} and @code{<}
7413 for objects of type @code{sprod_s}:
7416 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7418 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7421 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7423 return lhs.left.compare(rhs.left) < 0
7424 ? true : lhs.right.compare(rhs.right) < 0;
7428 The ordering established by the @code{<} operator doesn't have to make any
7429 algebraic sense, but it needs to be well defined. Note that we can't use
7430 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7431 in the implementation of these operators because they would construct
7432 GiNaC @code{relational} objects which in the case of @code{<} do not
7433 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7434 decide which one is algebraically 'less').
7436 Next, we need to change our definition of the @code{sprod} type to let
7437 GiNaC know that an ordering relation exists for the embedded objects:
7440 typedef structure<sprod_s, compare_std_less> sprod;
7443 @code{sprod} objects then behave as expected:
7447 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7448 // -> <a|b>-<a^2|b^2>
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, b) << endl;
7453 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7458 The @code{compare_std_less} policy parameter tells GiNaC to use the
7459 @code{std::less} and @code{std::equal_to} functors to compare objects of
7460 type @code{sprod_s}. By default, these functors forward their work to the
7461 standard @code{<} and @code{==} operators, which we have overloaded.
7462 Alternatively, we could have specialized @code{std::less} and
7463 @code{std::equal_to} for class @code{sprod_s}.
7465 GiNaC provides two other comparison policies for @code{structure<T>}
7466 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7467 which does a bit-wise comparison of the contained @code{T} objects.
7468 This should be used with extreme care because it only works reliably with
7469 built-in integral types, and it also compares any padding (filler bytes of
7470 undefined value) that the @code{T} class might have.
7472 @subsection Subexpressions
7474 Our scalar product class has two subexpressions: the left and right
7475 operands. It might be a good idea to make them accessible via the standard
7476 @code{nops()} and @code{op()} methods:
7479 size_t sprod::nops() const
7484 ex sprod::op(size_t i) const
7488 return get_struct().left;
7490 return get_struct().right;
7492 throw std::range_error("sprod::op(): no such operand");
7497 Implementing @code{nops()} and @code{op()} for container types such as
7498 @code{sprod} has two other nice side effects:
7502 @code{has()} works as expected
7504 GiNaC generates better hash keys for the objects (the default implementation
7505 of @code{calchash()} takes subexpressions into account)
7508 @cindex @code{let_op()}
7509 There is a non-const variant of @code{op()} called @code{let_op()} that
7510 allows replacing subexpressions:
7513 ex & sprod::let_op(size_t i)
7515 // every non-const member function must call this
7516 ensure_if_modifiable();
7520 return get_struct().left;
7522 return get_struct().right;
7524 throw std::range_error("sprod::let_op(): no such operand");
7529 Once we have provided @code{let_op()} we also get @code{subs()} and
7530 @code{map()} for free. In fact, every container class that returns a non-null
7531 @code{nops()} value must either implement @code{let_op()} or provide custom
7532 implementations of @code{subs()} and @code{map()}.
7534 In turn, the availability of @code{map()} enables the recursive behavior of a
7535 couple of other default method implementations, in particular @code{evalf()},
7536 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7537 we probably want to provide our own version of @code{expand()} for scalar
7538 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7539 This is left as an exercise for the reader.
7541 The @code{structure<T>} template defines many more member functions that
7542 you can override by specialization to customize the behavior of your
7543 structures. You are referred to the next section for a description of
7544 some of these (especially @code{eval()}). There is, however, one topic
7545 that shall be addressed here, as it demonstrates one peculiarity of the
7546 @code{structure<T>} template: archiving.
7548 @subsection Archiving structures
7550 If you don't know how the archiving of GiNaC objects is implemented, you
7551 should first read the next section and then come back here. You're back?
7554 To implement archiving for structures it is not enough to provide
7555 specializations for the @code{archive()} member function and the
7556 unarchiving constructor (the @code{unarchive()} function has a default
7557 implementation). You also need to provide a unique name (as a string literal)
7558 for each structure type you define. This is because in GiNaC archives,
7559 the class of an object is stored as a string, the class name.
7561 By default, this class name (as returned by the @code{class_name()} member
7562 function) is @samp{structure} for all structure classes. This works as long
7563 as you have only defined one structure type, but if you use two or more you
7564 need to provide a different name for each by specializing the
7565 @code{get_class_name()} member function. Here is a sample implementation
7566 for enabling archiving of the scalar product type defined above:
7569 const char *sprod::get_class_name() @{ return "sprod"; @}
7571 void sprod::archive(archive_node & n) const
7573 inherited::archive(n);
7574 n.add_ex("left", get_struct().left);
7575 n.add_ex("right", get_struct().right);
7578 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7580 n.find_ex("left", get_struct().left, sym_lst);
7581 n.find_ex("right", get_struct().right, sym_lst);
7585 Note that the unarchiving constructor is @code{sprod::structure} and not
7586 @code{sprod::sprod}, and that we don't need to supply an
7587 @code{sprod::unarchive()} function.
7590 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7591 @c node-name, next, previous, up
7592 @section Adding classes
7594 The @code{structure<T>} template provides an way to extend GiNaC with custom
7595 algebraic classes that is easy to use but has its limitations, the most
7596 severe of which being that you can't add any new member functions to
7597 structures. To be able to do this, you need to write a new class definition
7600 This section will explain how to implement new algebraic classes in GiNaC by
7601 giving the example of a simple 'string' class. After reading this section
7602 you will know how to properly declare a GiNaC class and what the minimum
7603 required member functions are that you have to implement. We only cover the
7604 implementation of a 'leaf' class here (i.e. one that doesn't contain
7605 subexpressions). Creating a container class like, for example, a class
7606 representing tensor products is more involved but this section should give
7607 you enough information so you can consult the source to GiNaC's predefined
7608 classes if you want to implement something more complicated.
7610 @subsection GiNaC's run-time type information system
7612 @cindex hierarchy of classes
7614 All algebraic classes (that is, all classes that can appear in expressions)
7615 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7616 @code{basic *} (which is essentially what an @code{ex} is) represents a
7617 generic pointer to an algebraic class. Occasionally it is necessary to find
7618 out what the class of an object pointed to by a @code{basic *} really is.
7619 Also, for the unarchiving of expressions it must be possible to find the
7620 @code{unarchive()} function of a class given the class name (as a string). A
7621 system that provides this kind of information is called a run-time type
7622 information (RTTI) system. The C++ language provides such a thing (see the
7623 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7624 implements its own, simpler RTTI.
7626 The RTTI in GiNaC is based on two mechanisms:
7631 The @code{basic} class declares a member variable @code{tinfo_key} which
7632 holds an unsigned integer that identifies the object's class. These numbers
7633 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7634 classes. They all start with @code{TINFO_}.
7637 By means of some clever tricks with static members, GiNaC maintains a list
7638 of information for all classes derived from @code{basic}. The information
7639 available includes the class names, the @code{tinfo_key}s, and pointers
7640 to the unarchiving functions. This class registry is defined in the
7641 @file{registrar.h} header file.
7645 The disadvantage of this proprietary RTTI implementation is that there's
7646 a little more to do when implementing new classes (C++'s RTTI works more
7647 or less automatically) but don't worry, most of the work is simplified by
7650 @subsection A minimalistic example
7652 Now we will start implementing a new class @code{mystring} that allows
7653 placing character strings in algebraic expressions (this is not very useful,
7654 but it's just an example). This class will be a direct subclass of
7655 @code{basic}. You can use this sample implementation as a starting point
7656 for your own classes.
7658 The code snippets given here assume that you have included some header files
7664 #include <stdexcept>
7665 using namespace std;
7667 #include <ginac/ginac.h>
7668 using namespace GiNaC;
7671 The first thing we have to do is to define a @code{tinfo_key} for our new
7672 class. This can be any arbitrary unsigned number that is not already taken
7673 by one of the existing classes but it's better to come up with something
7674 that is unlikely to clash with keys that might be added in the future. The
7675 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7676 which is not a requirement but we are going to stick with this scheme:
7679 const unsigned TINFO_mystring = 0x42420001U;
7682 Now we can write down the class declaration. The class stores a C++
7683 @code{string} and the user shall be able to construct a @code{mystring}
7684 object from a C or C++ string:
7687 class mystring : public basic
7689 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7692 mystring(const string &s);
7693 mystring(const char *s);
7699 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7702 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7703 macros are defined in @file{registrar.h}. They take the name of the class
7704 and its direct superclass as arguments and insert all required declarations
7705 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7706 the first line after the opening brace of the class definition. The
7707 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7708 source (at global scope, of course, not inside a function).
7710 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7711 declarations of the default constructor and a couple of other functions that
7712 are required. It also defines a type @code{inherited} which refers to the
7713 superclass so you don't have to modify your code every time you shuffle around
7714 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7715 class with the GiNaC RTTI (there is also a
7716 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7717 options for the class, and which we will be using instead in a few minutes).
7719 Now there are seven member functions we have to implement to get a working
7725 @code{mystring()}, the default constructor.
7728 @code{void archive(archive_node &n)}, the archiving function. This stores all
7729 information needed to reconstruct an object of this class inside an
7730 @code{archive_node}.
7733 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7734 constructor. This constructs an instance of the class from the information
7735 found in an @code{archive_node}.
7738 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7739 unarchiving function. It constructs a new instance by calling the unarchiving
7743 @cindex @code{compare_same_type()}
7744 @code{int compare_same_type(const basic &other)}, which is used internally
7745 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7746 -1, depending on the relative order of this object and the @code{other}
7747 object. If it returns 0, the objects are considered equal.
7748 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7749 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7750 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7751 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7752 must provide a @code{compare_same_type()} function, even those representing
7753 objects for which no reasonable algebraic ordering relationship can be
7757 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7758 which are the two constructors we declared.
7762 Let's proceed step-by-step. The default constructor looks like this:
7765 mystring::mystring() : inherited(TINFO_mystring) @{@}
7768 The golden rule is that in all constructors you have to set the
7769 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7770 it will be set by the constructor of the superclass and all hell will break
7771 loose in the RTTI. For your convenience, the @code{basic} class provides
7772 a constructor that takes a @code{tinfo_key} value, which we are using here
7773 (remember that in our case @code{inherited == basic}). If the superclass
7774 didn't have such a constructor, we would have to set the @code{tinfo_key}
7775 to the right value manually.
7777 In the default constructor you should set all other member variables to
7778 reasonable default values (we don't need that here since our @code{str}
7779 member gets set to an empty string automatically).
7781 Next are the three functions for archiving. You have to implement them even
7782 if you don't plan to use archives, but the minimum required implementation
7783 is really simple. First, the archiving function:
7786 void mystring::archive(archive_node &n) const
7788 inherited::archive(n);
7789 n.add_string("string", str);
7793 The only thing that is really required is calling the @code{archive()}
7794 function of the superclass. Optionally, you can store all information you
7795 deem necessary for representing the object into the passed
7796 @code{archive_node}. We are just storing our string here. For more
7797 information on how the archiving works, consult the @file{archive.h} header
7800 The unarchiving constructor is basically the inverse of the archiving
7804 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7806 n.find_string("string", str);
7810 If you don't need archiving, just leave this function empty (but you must
7811 invoke the unarchiving constructor of the superclass). Note that we don't
7812 have to set the @code{tinfo_key} here because it is done automatically
7813 by the unarchiving constructor of the @code{basic} class.
7815 Finally, the unarchiving function:
7818 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7820 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7824 You don't have to understand how exactly this works. Just copy these
7825 four lines into your code literally (replacing the class name, of
7826 course). It calls the unarchiving constructor of the class and unless
7827 you are doing something very special (like matching @code{archive_node}s
7828 to global objects) you don't need a different implementation. For those
7829 who are interested: setting the @code{dynallocated} flag puts the object
7830 under the control of GiNaC's garbage collection. It will get deleted
7831 automatically once it is no longer referenced.
7833 Our @code{compare_same_type()} function uses a provided function to compare
7837 int mystring::compare_same_type(const basic &other) const
7839 const mystring &o = static_cast<const mystring &>(other);
7840 int cmpval = str.compare(o.str);
7843 else if (cmpval < 0)
7850 Although this function takes a @code{basic &}, it will always be a reference
7851 to an object of exactly the same class (objects of different classes are not
7852 comparable), so the cast is safe. If this function returns 0, the two objects
7853 are considered equal (in the sense that @math{A-B=0}), so you should compare
7854 all relevant member variables.
7856 Now the only thing missing is our two new constructors:
7859 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7860 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7863 No surprises here. We set the @code{str} member from the argument and
7864 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7866 That's it! We now have a minimal working GiNaC class that can store
7867 strings in algebraic expressions. Let's confirm that the RTTI works:
7870 ex e = mystring("Hello, world!");
7871 cout << is_a<mystring>(e) << endl;
7874 cout << ex_to<basic>(e).class_name() << endl;
7878 Obviously it does. Let's see what the expression @code{e} looks like:
7882 // -> [mystring object]
7885 Hm, not exactly what we expect, but of course the @code{mystring} class
7886 doesn't yet know how to print itself. This can be done either by implementing
7887 the @code{print()} member function, or, preferably, by specifying a
7888 @code{print_func<>()} class option. Let's say that we want to print the string
7889 surrounded by double quotes:
7892 class mystring : public basic
7896 void do_print(const print_context &c, unsigned level = 0) const;
7900 void mystring::do_print(const print_context &c, unsigned level) const
7902 // print_context::s is a reference to an ostream
7903 c.s << '\"' << str << '\"';
7907 The @code{level} argument is only required for container classes to
7908 correctly parenthesize the output.
7910 Now we need to tell GiNaC that @code{mystring} objects should use the
7911 @code{do_print()} member function for printing themselves. For this, we
7915 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7921 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7922 print_func<print_context>(&mystring::do_print))
7925 Let's try again to print the expression:
7929 // -> "Hello, world!"
7932 Much better. If we wanted to have @code{mystring} objects displayed in a
7933 different way depending on the output format (default, LaTeX, etc.), we
7934 would have supplied multiple @code{print_func<>()} options with different
7935 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7936 separated by dots. This is similar to the way options are specified for
7937 symbolic functions. @xref{Printing}, for a more in-depth description of the
7938 way expression output is implemented in GiNaC.
7940 The @code{mystring} class can be used in arbitrary expressions:
7943 e += mystring("GiNaC rulez");
7945 // -> "GiNaC rulez"+"Hello, world!"
7948 (GiNaC's automatic term reordering is in effect here), or even
7951 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7953 // -> "One string"^(2*sin(-"Another string"+Pi))
7956 Whether this makes sense is debatable but remember that this is only an
7957 example. At least it allows you to implement your own symbolic algorithms
7960 Note that GiNaC's algebraic rules remain unchanged:
7963 e = mystring("Wow") * mystring("Wow");
7967 e = pow(mystring("First")-mystring("Second"), 2);
7968 cout << e.expand() << endl;
7969 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7972 There's no way to, for example, make GiNaC's @code{add} class perform string
7973 concatenation. You would have to implement this yourself.
7975 @subsection Automatic evaluation
7978 @cindex @code{eval()}
7979 @cindex @code{hold()}
7980 When dealing with objects that are just a little more complicated than the
7981 simple string objects we have implemented, chances are that you will want to
7982 have some automatic simplifications or canonicalizations performed on them.
7983 This is done in the evaluation member function @code{eval()}. Let's say that
7984 we wanted all strings automatically converted to lowercase with
7985 non-alphabetic characters stripped, and empty strings removed:
7988 class mystring : public basic
7992 ex eval(int level = 0) const;
7996 ex mystring::eval(int level) const
7999 for (int i=0; i<str.length(); i++) @{
8001 if (c >= 'A' && c <= 'Z')
8002 new_str += tolower(c);
8003 else if (c >= 'a' && c <= 'z')
8007 if (new_str.length() == 0)
8010 return mystring(new_str).hold();
8014 The @code{level} argument is used to limit the recursion depth of the
8015 evaluation. We don't have any subexpressions in the @code{mystring}
8016 class so we are not concerned with this. If we had, we would call the
8017 @code{eval()} functions of the subexpressions with @code{level - 1} as
8018 the argument if @code{level != 1}. The @code{hold()} member function
8019 sets a flag in the object that prevents further evaluation. Otherwise
8020 we might end up in an endless loop. When you want to return the object
8021 unmodified, use @code{return this->hold();}.
8023 Let's confirm that it works:
8026 ex e = mystring("Hello, world!") + mystring("!?#");
8030 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8035 @subsection Optional member functions
8037 We have implemented only a small set of member functions to make the class
8038 work in the GiNaC framework. There are two functions that are not strictly
8039 required but will make operations with objects of the class more efficient:
8041 @cindex @code{calchash()}
8042 @cindex @code{is_equal_same_type()}
8044 unsigned calchash() const;
8045 bool is_equal_same_type(const basic &other) const;
8048 The @code{calchash()} method returns an @code{unsigned} hash value for the
8049 object which will allow GiNaC to compare and canonicalize expressions much
8050 more efficiently. You should consult the implementation of some of the built-in
8051 GiNaC classes for examples of hash functions. The default implementation of
8052 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8053 class and all subexpressions that are accessible via @code{op()}.
8055 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8056 tests for equality without establishing an ordering relation, which is often
8057 faster. The default implementation of @code{is_equal_same_type()} just calls
8058 @code{compare_same_type()} and tests its result for zero.
8060 @subsection Other member functions
8062 For a real algebraic class, there are probably some more functions that you
8063 might want to provide:
8066 bool info(unsigned inf) const;
8067 ex evalf(int level = 0) const;
8068 ex series(const relational & r, int order, unsigned options = 0) const;
8069 ex derivative(const symbol & s) const;
8072 If your class stores sub-expressions (see the scalar product example in the
8073 previous section) you will probably want to override
8075 @cindex @code{let_op()}
8078 ex op(size_t i) const;
8079 ex & let_op(size_t i);
8080 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8081 ex map(map_function & f) const;
8084 @code{let_op()} is a variant of @code{op()} that allows write access. The
8085 default implementations of @code{subs()} and @code{map()} use it, so you have
8086 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8088 You can, of course, also add your own new member functions. Remember
8089 that the RTTI may be used to get information about what kinds of objects
8090 you are dealing with (the position in the class hierarchy) and that you
8091 can always extract the bare object from an @code{ex} by stripping the
8092 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8093 should become a need.
8095 That's it. May the source be with you!
8098 @node A comparison with other CAS, Advantages, Adding classes, Top
8099 @c node-name, next, previous, up
8100 @chapter A Comparison With Other CAS
8103 This chapter will give you some information on how GiNaC compares to
8104 other, traditional Computer Algebra Systems, like @emph{Maple},
8105 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8106 disadvantages over these systems.
8109 * Advantages:: Strengths of the GiNaC approach.
8110 * Disadvantages:: Weaknesses of the GiNaC approach.
8111 * Why C++?:: Attractiveness of C++.
8114 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8115 @c node-name, next, previous, up
8118 GiNaC has several advantages over traditional Computer
8119 Algebra Systems, like
8124 familiar language: all common CAS implement their own proprietary
8125 grammar which you have to learn first (and maybe learn again when your
8126 vendor decides to `enhance' it). With GiNaC you can write your program
8127 in common C++, which is standardized.
8131 structured data types: you can build up structured data types using
8132 @code{struct}s or @code{class}es together with STL features instead of
8133 using unnamed lists of lists of lists.
8136 strongly typed: in CAS, you usually have only one kind of variables
8137 which can hold contents of an arbitrary type. This 4GL like feature is
8138 nice for novice programmers, but dangerous.
8141 development tools: powerful development tools exist for C++, like fancy
8142 editors (e.g. with automatic indentation and syntax highlighting),
8143 debuggers, visualization tools, documentation generators@dots{}
8146 modularization: C++ programs can easily be split into modules by
8147 separating interface and implementation.
8150 price: GiNaC is distributed under the GNU Public License which means
8151 that it is free and available with source code. And there are excellent
8152 C++-compilers for free, too.
8155 extendable: you can add your own classes to GiNaC, thus extending it on
8156 a very low level. Compare this to a traditional CAS that you can
8157 usually only extend on a high level by writing in the language defined
8158 by the parser. In particular, it turns out to be almost impossible to
8159 fix bugs in a traditional system.
8162 multiple interfaces: Though real GiNaC programs have to be written in
8163 some editor, then be compiled, linked and executed, there are more ways
8164 to work with the GiNaC engine. Many people want to play with
8165 expressions interactively, as in traditional CASs. Currently, two such
8166 windows into GiNaC have been implemented and many more are possible: the
8167 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8168 types to a command line and second, as a more consistent approach, an
8169 interactive interface to the Cint C++ interpreter has been put together
8170 (called GiNaC-cint) that allows an interactive scripting interface
8171 consistent with the C++ language. It is available from the usual GiNaC
8175 seamless integration: it is somewhere between difficult and impossible
8176 to call CAS functions from within a program written in C++ or any other
8177 programming language and vice versa. With GiNaC, your symbolic routines
8178 are part of your program. You can easily call third party libraries,
8179 e.g. for numerical evaluation or graphical interaction. All other
8180 approaches are much more cumbersome: they range from simply ignoring the
8181 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8182 system (i.e. @emph{Yacas}).
8185 efficiency: often large parts of a program do not need symbolic
8186 calculations at all. Why use large integers for loop variables or
8187 arbitrary precision arithmetics where @code{int} and @code{double} are
8188 sufficient? For pure symbolic applications, GiNaC is comparable in
8189 speed with other CAS.
8194 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8195 @c node-name, next, previous, up
8196 @section Disadvantages
8198 Of course it also has some disadvantages:
8203 advanced features: GiNaC cannot compete with a program like
8204 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8205 which grows since 1981 by the work of dozens of programmers, with
8206 respect to mathematical features. Integration, factorization,
8207 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8208 not planned for the near future).
8211 portability: While the GiNaC library itself is designed to avoid any
8212 platform dependent features (it should compile on any ANSI compliant C++
8213 compiler), the currently used version of the CLN library (fast large
8214 integer and arbitrary precision arithmetics) can only by compiled
8215 without hassle on systems with the C++ compiler from the GNU Compiler
8216 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8217 macros to let the compiler gather all static initializations, which
8218 works for GNU C++ only. Feel free to contact the authors in case you
8219 really believe that you need to use a different compiler. We have
8220 occasionally used other compilers and may be able to give you advice.}
8221 GiNaC uses recent language features like explicit constructors, mutable
8222 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8223 literally. Recent GCC versions starting at 2.95.3, although itself not
8224 yet ANSI compliant, support all needed features.
8229 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8230 @c node-name, next, previous, up
8233 Why did we choose to implement GiNaC in C++ instead of Java or any other
8234 language? C++ is not perfect: type checking is not strict (casting is
8235 possible), separation between interface and implementation is not
8236 complete, object oriented design is not enforced. The main reason is
8237 the often scolded feature of operator overloading in C++. While it may
8238 be true that operating on classes with a @code{+} operator is rarely
8239 meaningful, it is perfectly suited for algebraic expressions. Writing
8240 @math{3x+5y} as @code{3*x+5*y} instead of
8241 @code{x.times(3).plus(y.times(5))} looks much more natural.
8242 Furthermore, the main developers are more familiar with C++ than with
8243 any other programming language.
8246 @node Internal structures, Expressions are reference counted, Why C++? , Top
8247 @c node-name, next, previous, up
8248 @appendix Internal structures
8251 * Expressions are reference counted::
8252 * Internal representation of products and sums::
8255 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8256 @c node-name, next, previous, up
8257 @appendixsection Expressions are reference counted
8259 @cindex reference counting
8260 @cindex copy-on-write
8261 @cindex garbage collection
8262 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8263 where the counter belongs to the algebraic objects derived from class
8264 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8265 which @code{ex} contains an instance. If you understood that, you can safely
8266 skip the rest of this passage.
8268 Expressions are extremely light-weight since internally they work like
8269 handles to the actual representation. They really hold nothing more
8270 than a pointer to some other object. What this means in practice is
8271 that whenever you create two @code{ex} and set the second equal to the
8272 first no copying process is involved. Instead, the copying takes place
8273 as soon as you try to change the second. Consider the simple sequence
8278 #include <ginac/ginac.h>
8279 using namespace std;
8280 using namespace GiNaC;
8284 symbol x("x"), y("y"), z("z");
8287 e1 = sin(x + 2*y) + 3*z + 41;
8288 e2 = e1; // e2 points to same object as e1
8289 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8290 e2 += 1; // e2 is copied into a new object
8291 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8295 The line @code{e2 = e1;} creates a second expression pointing to the
8296 object held already by @code{e1}. The time involved for this operation
8297 is therefore constant, no matter how large @code{e1} was. Actual
8298 copying, however, must take place in the line @code{e2 += 1;} because
8299 @code{e1} and @code{e2} are not handles for the same object any more.
8300 This concept is called @dfn{copy-on-write semantics}. It increases
8301 performance considerably whenever one object occurs multiple times and
8302 represents a simple garbage collection scheme because when an @code{ex}
8303 runs out of scope its destructor checks whether other expressions handle
8304 the object it points to too and deletes the object from memory if that
8305 turns out not to be the case. A slightly less trivial example of
8306 differentiation using the chain-rule should make clear how powerful this
8311 symbol x("x"), y("y");
8315 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8316 cout << e1 << endl // prints x+3*y
8317 << e2 << endl // prints (x+3*y)^3
8318 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8322 Here, @code{e1} will actually be referenced three times while @code{e2}
8323 will be referenced two times. When the power of an expression is built,
8324 that expression needs not be copied. Likewise, since the derivative of
8325 a power of an expression can be easily expressed in terms of that
8326 expression, no copying of @code{e1} is involved when @code{e3} is
8327 constructed. So, when @code{e3} is constructed it will print as
8328 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8329 holds a reference to @code{e2} and the factor in front is just
8332 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8333 semantics. When you insert an expression into a second expression, the
8334 result behaves exactly as if the contents of the first expression were
8335 inserted. But it may be useful to remember that this is not what
8336 happens. Knowing this will enable you to write much more efficient
8337 code. If you still have an uncertain feeling with copy-on-write
8338 semantics, we recommend you have a look at the
8339 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8340 Marshall Cline. Chapter 16 covers this issue and presents an
8341 implementation which is pretty close to the one in GiNaC.
8344 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8345 @c node-name, next, previous, up
8346 @appendixsection Internal representation of products and sums
8348 @cindex representation
8351 @cindex @code{power}
8352 Although it should be completely transparent for the user of
8353 GiNaC a short discussion of this topic helps to understand the sources
8354 and also explain performance to a large degree. Consider the
8355 unexpanded symbolic expression
8357 $2d^3 \left( 4a + 5b - 3 \right)$
8360 @math{2*d^3*(4*a+5*b-3)}
8362 which could naively be represented by a tree of linear containers for
8363 addition and multiplication, one container for exponentiation with base
8364 and exponent and some atomic leaves of symbols and numbers in this
8369 @cindex pair-wise representation
8370 However, doing so results in a rather deeply nested tree which will
8371 quickly become inefficient to manipulate. We can improve on this by
8372 representing the sum as a sequence of terms, each one being a pair of a
8373 purely numeric multiplicative coefficient and its rest. In the same
8374 spirit we can store the multiplication as a sequence of terms, each
8375 having a numeric exponent and a possibly complicated base, the tree
8376 becomes much more flat:
8380 The number @code{3} above the symbol @code{d} shows that @code{mul}
8381 objects are treated similarly where the coefficients are interpreted as
8382 @emph{exponents} now. Addition of sums of terms or multiplication of
8383 products with numerical exponents can be coded to be very efficient with
8384 such a pair-wise representation. Internally, this handling is performed
8385 by most CAS in this way. It typically speeds up manipulations by an
8386 order of magnitude. The overall multiplicative factor @code{2} and the
8387 additive term @code{-3} look somewhat out of place in this
8388 representation, however, since they are still carrying a trivial
8389 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8390 this is avoided by adding a field that carries an overall numeric
8391 coefficient. This results in the realistic picture of internal
8394 $2d^3 \left( 4a + 5b - 3 \right)$:
8397 @math{2*d^3*(4*a+5*b-3)}:
8403 This also allows for a better handling of numeric radicals, since
8404 @code{sqrt(2)} can now be carried along calculations. Now it should be
8405 clear, why both classes @code{add} and @code{mul} are derived from the
8406 same abstract class: the data representation is the same, only the
8407 semantics differs. In the class hierarchy, methods for polynomial
8408 expansion and the like are reimplemented for @code{add} and @code{mul},
8409 but the data structure is inherited from @code{expairseq}.
8412 @node Package tools, ginac-config, Internal representation of products and sums, Top
8413 @c node-name, next, previous, up
8414 @appendix Package tools
8416 If you are creating a software package that uses the GiNaC library,
8417 setting the correct command line options for the compiler and linker
8418 can be difficult. GiNaC includes two tools to make this process easier.
8421 * ginac-config:: A shell script to detect compiler and linker flags.
8422 * AM_PATH_GINAC:: Macro for GNU automake.
8426 @node ginac-config, AM_PATH_GINAC, Package tools, Package tools
8427 @c node-name, next, previous, up
8428 @section @command{ginac-config}
8429 @cindex ginac-config
8431 @command{ginac-config} is a shell script that you can use to determine
8432 the compiler and linker command line options required to compile and
8433 link a program with the GiNaC library.
8435 @command{ginac-config} takes the following flags:
8439 Prints out the version of GiNaC installed.
8441 Prints '-I' flags pointing to the installed header files.
8443 Prints out the linker flags necessary to link a program against GiNaC.
8444 @item --prefix[=@var{PREFIX}]
8445 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8446 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8447 Otherwise, prints out the configured value of @env{$prefix}.
8448 @item --exec-prefix[=@var{PREFIX}]
8449 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8450 Otherwise, prints out the configured value of @env{$exec_prefix}.
8453 Typically, @command{ginac-config} will be used within a configure
8454 script, as described below. It, however, can also be used directly from
8455 the command line using backquotes to compile a simple program. For
8459 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8462 This command line might expand to (for example):
8465 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8466 -lginac -lcln -lstdc++
8469 Not only is the form using @command{ginac-config} easier to type, it will
8470 work on any system, no matter how GiNaC was configured.
8473 @node AM_PATH_GINAC, Configure script options, ginac-config, Package tools
8474 @c node-name, next, previous, up
8475 @section @samp{AM_PATH_GINAC}
8476 @cindex AM_PATH_GINAC
8478 For packages configured using GNU automake, GiNaC also provides
8479 a macro to automate the process of checking for GiNaC.
8482 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8483 [, @var{ACTION-IF-NOT-FOUND}]]])
8491 Determines the location of GiNaC using @command{ginac-config}, which is
8492 either found in the user's path, or from the environment variable
8493 @env{GINACLIB_CONFIG}.
8496 Tests the installed libraries to make sure that their version
8497 is later than @var{MINIMUM-VERSION}. (A default version will be used
8501 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8502 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8503 variable to the output of @command{ginac-config --libs}, and calls
8504 @samp{AC_SUBST()} for these variables so they can be used in generated
8505 makefiles, and then executes @var{ACTION-IF-FOUND}.
8508 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8509 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8513 This macro is in file @file{ginac.m4} which is installed in
8514 @file{$datadir/aclocal}. Note that if automake was installed with a
8515 different @samp{--prefix} than GiNaC, you will either have to manually
8516 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8517 aclocal the @samp{-I} option when running it.
8520 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8521 * Example package:: Example of a package using AM_PATH_GINAC.
8525 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8526 @c node-name, next, previous, up
8527 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8529 Simply make sure that @command{ginac-config} is in your path, and run
8530 the configure script.
8537 The directory where the GiNaC libraries are installed needs
8538 to be found by your system's dynamic linker.
8540 This is generally done by
8543 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8549 setting the environment variable @env{LD_LIBRARY_PATH},
8552 or, as a last resort,
8555 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8556 running configure, for instance:
8559 LDFLAGS=-R/home/cbauer/lib ./configure
8564 You can also specify a @command{ginac-config} not in your path by
8565 setting the @env{GINACLIB_CONFIG} environment variable to the
8566 name of the executable
8569 If you move the GiNaC package from its installed location,
8570 you will either need to modify @command{ginac-config} script
8571 manually to point to the new location or rebuild GiNaC.
8582 --with-ginac-prefix=@var{PREFIX}
8583 --with-ginac-exec-prefix=@var{PREFIX}
8586 are provided to override the prefix and exec-prefix that were stored
8587 in the @command{ginac-config} shell script by GiNaC's configure. You are
8588 generally better off configuring GiNaC with the right path to begin with.
8592 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8593 @c node-name, next, previous, up
8594 @subsection Example of a package using @samp{AM_PATH_GINAC}
8596 The following shows how to build a simple package using automake
8597 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8601 #include <ginac/ginac.h>
8605 GiNaC::symbol x("x");
8606 GiNaC::ex a = GiNaC::sin(x);
8607 std::cout << "Derivative of " << a
8608 << " is " << a.diff(x) << std::endl;
8613 You should first read the introductory portions of the automake
8614 Manual, if you are not already familiar with it.
8616 Two files are needed, @file{configure.in}, which is used to build the
8620 dnl Process this file with autoconf to produce a configure script.
8622 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8628 AM_PATH_GINAC(0.9.0, [
8629 LIBS="$LIBS $GINACLIB_LIBS"
8630 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8631 ], AC_MSG_ERROR([need to have GiNaC installed]))
8636 The only command in this which is not standard for automake
8637 is the @samp{AM_PATH_GINAC} macro.
8639 That command does the following: If a GiNaC version greater or equal
8640 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8641 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8642 the error message `need to have GiNaC installed'
8644 And the @file{Makefile.am}, which will be used to build the Makefile.
8647 ## Process this file with automake to produce Makefile.in
8648 bin_PROGRAMS = simple
8649 simple_SOURCES = simple.cpp
8652 This @file{Makefile.am}, says that we are building a single executable,
8653 from a single source file @file{simple.cpp}. Since every program
8654 we are building uses GiNaC we simply added the GiNaC options
8655 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8656 want to specify them on a per-program basis: for instance by
8660 simple_LDADD = $(GINACLIB_LIBS)
8661 INCLUDES = $(GINACLIB_CPPFLAGS)
8664 to the @file{Makefile.am}.
8666 To try this example out, create a new directory and add the three
8669 Now execute the following commands:
8672 $ automake --add-missing
8677 You now have a package that can be built in the normal fashion
8686 @node Bibliography, Concept index, Example package, Top
8687 @c node-name, next, previous, up
8688 @appendix Bibliography
8693 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8696 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8699 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8702 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8705 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8706 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8709 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8710 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8711 Academic Press, London
8714 @cite{Computer Algebra Systems - A Practical Guide},
8715 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8718 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8719 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8722 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8723 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8726 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8731 @node Concept index, , Bibliography, Top
8732 @c node-name, next, previous, up
8733 @unnumbered Concept index