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. Optional parameter @code{rl} allows to distinguish different
3311 Clifford algebras, which will commute with each other. The last
3312 optional parameter @code{anticommuting} defines if the anticommuting
3315 $e_i e_j + e_j e_i = 0$)
3318 e~i e~j + e~j e~i = 0)
3320 will be used for contraction of Clifford units. If the @code{metric} is
3321 supplied by a @code{matrix} object, then the value of
3322 @code{anticommuting} is calculated automatically and the supplied one
3323 will be ignored. One can overcome this by giving @code{metric} through
3324 matrix wrapped into an @code{indexed} object.
3326 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3327 something very close to @code{dirac_gamma(mu)}, although
3328 @code{dirac_gamma} have more efficient simplification mechanism.
3329 @cindex @code{clifford::get_metric()}
3330 The method @code{clifford::get_metric()} returns a metric defining this
3332 @cindex @code{clifford::is_anticommuting()}
3333 The method @code{clifford::is_anticommuting()} returns the
3334 @code{anticommuting} property of a unit.
3336 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3337 the Clifford algebra units with a call like that
3340 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3343 since this may yield some further automatic simplifications. Again, for a
3344 metric defined through a @code{matrix} such a symmetry is detected
3347 Individual generators of a Clifford algebra can be accessed in several
3353 varidx nu(symbol("nu"), 4);
3355 ex M = diag_matrix(lst(1, -1, 0, s));
3356 ex e = clifford_unit(nu, M);
3357 ex e0 = e.subs(nu == 0);
3358 ex e1 = e.subs(nu == 1);
3359 ex e2 = e.subs(nu == 2);
3360 ex e3 = e.subs(nu == 3);
3365 will produce four anti-commuting generators of a Clifford algebra with properties
3367 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3370 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3371 @code{pow(e3, 2) = s}.
3374 @cindex @code{lst_to_clifford()}
3375 A similar effect can be achieved from the function
3378 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3379 unsigned char rl = 0, bool anticommuting = false);
3380 ex lst_to_clifford(const ex & v, const ex & e);
3383 which converts a list or vector
3385 $v = (v^0, v^1, ..., v^n)$
3388 @samp{v = (v~0, v~1, ..., v~n)}
3393 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3396 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3399 directly supplied in the second form of the procedure. In the first form
3400 the Clifford unit @samp{e.k} is generated by the call of
3401 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3402 with the help of @code{lst_to_clifford()} as follows
3407 varidx nu(symbol("nu"), 4);
3409 ex M = diag_matrix(lst(1, -1, 0, s));
3410 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3411 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3412 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3413 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3418 @cindex @code{clifford_to_lst()}
3419 There is the inverse function
3422 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3425 which takes an expression @code{e} and tries to find a list
3427 $v = (v^0, v^1, ..., v^n)$
3430 @samp{v = (v~0, v~1, ..., v~n)}
3434 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3437 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3439 with respect to the given Clifford units @code{c} and with none of the
3440 @samp{v~k} containing Clifford units @code{c} (of course, this
3441 may be impossible). This function can use an @code{algebraic} method
3442 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3444 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3447 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3449 is zero or is not @code{numeric} for some @samp{k}
3450 then the method will be automatically changed to symbolic. The same effect
3451 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3453 @cindex @code{clifford_prime()}
3454 @cindex @code{clifford_star()}
3455 @cindex @code{clifford_bar()}
3456 There are several functions for (anti-)automorphisms of Clifford algebras:
3459 ex clifford_prime(const ex & e)
3460 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3461 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3464 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3465 changes signs of all Clifford units in the expression. The reversion
3466 of a Clifford algebra @code{clifford_star()} coincides with the
3467 @code{conjugate()} method and effectively reverses the order of Clifford
3468 units in any product. Finally the main anti-automorphism
3469 of a Clifford algebra @code{clifford_bar()} is the composition of the
3470 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3471 in a product. These functions correspond to the notations
3486 used in Clifford algebra textbooks.
3488 @cindex @code{clifford_norm()}
3492 ex clifford_norm(const ex & e);
3495 @cindex @code{clifford_inverse()}
3496 calculates the norm of a Clifford number from the expression
3498 $||e||^2 = e\overline{e}$.
3501 @code{||e||^2 = e \bar@{e@}}
3503 The inverse of a Clifford expression is returned by the function
3506 ex clifford_inverse(const ex & e);
3509 which calculates it as
3511 $e^{-1} = \overline{e}/||e||^2$.
3514 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3523 then an exception is raised.
3525 @cindex @code{remove_dirac_ONE()}
3526 If a Clifford number happens to be a factor of
3527 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3528 expression by the function
3531 ex remove_dirac_ONE(const ex & e);
3534 @cindex @code{canonicalize_clifford()}
3535 The function @code{canonicalize_clifford()} works for a
3536 generic Clifford algebra in a similar way as for Dirac gammas.
3538 The next provided function is
3540 @cindex @code{clifford_moebius_map()}
3542 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3543 const ex & d, const ex & v, const ex & G,
3544 unsigned char rl = 0, bool anticommuting = false);
3545 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3546 unsigned char rl = 0, bool anticommuting = false);
3549 It takes a list or vector @code{v} and makes the Moebius (conformal or
3550 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3551 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3552 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3553 indexed object, tensormetric, matrix or a Clifford unit, in the later
3554 case the optional parameters @code{rl} and @code{anticommuting} are
3555 ignored even if supplied. Depending from the type of @code{v} the
3556 returned value of this function is either a vector or a list holding vector's
3559 @cindex @code{clifford_max_label()}
3560 Finally the function
3563 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3566 can detect a presence of Clifford objects in the expression @code{e}: if
3567 such objects are found it returns the maximal
3568 @code{representation_label} of them, otherwise @code{-1}. The optional
3569 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3570 be ignored during the search.
3572 LaTeX output for Clifford units looks like
3573 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3574 @code{representation_label} and @code{\nu} is the index of the
3575 corresponding unit. This provides a flexible typesetting with a suitable
3576 defintion of the @code{\clifford} command. For example, the definition
3578 \newcommand@{\clifford@}[1][]@{@}
3580 typesets all Clifford units identically, while the alternative definition
3582 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3584 prints units with @code{representation_label=0} as
3591 with @code{representation_label=1} as
3598 and with @code{representation_label=2} as
3606 @cindex @code{color} (class)
3607 @subsection Color algebra
3609 @cindex @code{color_T()}
3610 For computations in quantum chromodynamics, GiNaC implements the base elements
3611 and structure constants of the su(3) Lie algebra (color algebra). The base
3612 elements @math{T_a} are constructed by the function
3615 ex color_T(const ex & a, unsigned char rl = 0);
3618 which takes two arguments: the index and a @dfn{representation label} in the
3619 range 0 to 255 which is used to distinguish elements of different color
3620 algebras. Objects with different labels commutate with each other. The
3621 dimension of the index must be exactly 8 and it should be of class @code{idx},
3624 @cindex @code{color_ONE()}
3625 The unity element of a color algebra is constructed by
3628 ex color_ONE(unsigned char rl = 0);
3631 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3632 multiples of the unity element, even though it's customary to omit it.
3633 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3634 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3635 GiNaC may produce incorrect results.
3637 @cindex @code{color_d()}
3638 @cindex @code{color_f()}
3642 ex color_d(const ex & a, const ex & b, const ex & c);
3643 ex color_f(const ex & a, const ex & b, const ex & c);
3646 create the symmetric and antisymmetric structure constants @math{d_abc} and
3647 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3648 and @math{[T_a, T_b] = i f_abc T_c}.
3650 These functions evaluate to their numerical values,
3651 if you supply numeric indices to them. The index values should be in
3652 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3653 goes along better with the notations used in physical literature.
3655 @cindex @code{color_h()}
3656 There's an additional function
3659 ex color_h(const ex & a, const ex & b, const ex & c);
3662 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3664 The function @code{simplify_indexed()} performs some simplifications on
3665 expressions containing color objects:
3670 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3671 k(symbol("k"), 8), l(symbol("l"), 8);
3673 e = color_d(a, b, l) * color_f(a, b, k);
3674 cout << e.simplify_indexed() << endl;
3677 e = color_d(a, b, l) * color_d(a, b, k);
3678 cout << e.simplify_indexed() << endl;
3681 e = color_f(l, a, b) * color_f(a, b, k);
3682 cout << e.simplify_indexed() << endl;
3685 e = color_h(a, b, c) * color_h(a, b, c);
3686 cout << e.simplify_indexed() << endl;
3689 e = color_h(a, b, c) * color_T(b) * color_T(c);
3690 cout << e.simplify_indexed() << endl;
3693 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3694 cout << e.simplify_indexed() << endl;
3697 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3698 cout << e.simplify_indexed() << endl;
3699 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3703 @cindex @code{color_trace()}
3704 To calculate the trace of an expression containing color objects you use one
3708 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3709 ex color_trace(const ex & e, const lst & rll);
3710 ex color_trace(const ex & e, unsigned char rl = 0);
3713 These functions take the trace over all color @samp{T} objects in the
3714 specified set @code{rls} or list @code{rll} of representation labels, or the
3715 single label @code{rl}; @samp{T}s with other labels are left standing. For
3720 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3722 // -> -I*f.a.c.b+d.a.c.b
3727 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3728 @c node-name, next, previous, up
3731 @cindex @code{exhashmap} (class)
3733 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3734 that can be used as a drop-in replacement for the STL
3735 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3736 typically constant-time, element look-up than @code{map<>}.
3738 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3739 following differences:
3743 no @code{lower_bound()} and @code{upper_bound()} methods
3745 no reverse iterators, no @code{rbegin()}/@code{rend()}
3747 no @code{operator<(exhashmap, exhashmap)}
3749 the comparison function object @code{key_compare} is hardcoded to
3752 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3753 initial hash table size (the actual table size after construction may be
3754 larger than the specified value)
3756 the method @code{size_t bucket_count()} returns the current size of the hash
3759 @code{insert()} and @code{erase()} operations invalidate all iterators
3763 @node Methods and functions, Information about expressions, Hash maps, Top
3764 @c node-name, next, previous, up
3765 @chapter Methods and functions
3768 In this chapter the most important algorithms provided by GiNaC will be
3769 described. Some of them are implemented as functions on expressions,
3770 others are implemented as methods provided by expression objects. If
3771 they are methods, there exists a wrapper function around it, so you can
3772 alternatively call it in a functional way as shown in the simple
3777 cout << "As method: " << sin(1).evalf() << endl;
3778 cout << "As function: " << evalf(sin(1)) << endl;
3782 @cindex @code{subs()}
3783 The general rule is that wherever methods accept one or more parameters
3784 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3785 wrapper accepts is the same but preceded by the object to act on
3786 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3787 most natural one in an OO model but it may lead to confusion for MapleV
3788 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3789 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3790 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3791 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3792 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3793 here. Also, users of MuPAD will in most cases feel more comfortable
3794 with GiNaC's convention. All function wrappers are implemented
3795 as simple inline functions which just call the corresponding method and
3796 are only provided for users uncomfortable with OO who are dead set to
3797 avoid method invocations. Generally, nested function wrappers are much
3798 harder to read than a sequence of methods and should therefore be
3799 avoided if possible. On the other hand, not everything in GiNaC is a
3800 method on class @code{ex} and sometimes calling a function cannot be
3804 * Information about expressions::
3805 * Numerical evaluation::
3806 * Substituting expressions::
3807 * Pattern matching and advanced substitutions::
3808 * Applying a function on subexpressions::
3809 * Visitors and tree traversal::
3810 * Polynomial arithmetic:: Working with polynomials.
3811 * Rational expressions:: Working with rational functions.
3812 * Symbolic differentiation::
3813 * Series expansion:: Taylor and Laurent expansion.
3815 * Built-in functions:: List of predefined mathematical functions.
3816 * Multiple polylogarithms::
3817 * Complex expressions::
3818 * Solving linear systems of equations::
3819 * Input/output:: Input and output of expressions.
3823 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3824 @c node-name, next, previous, up
3825 @section Getting information about expressions
3827 @subsection Checking expression types
3828 @cindex @code{is_a<@dots{}>()}
3829 @cindex @code{is_exactly_a<@dots{}>()}
3830 @cindex @code{ex_to<@dots{}>()}
3831 @cindex Converting @code{ex} to other classes
3832 @cindex @code{info()}
3833 @cindex @code{return_type()}
3834 @cindex @code{return_type_tinfo()}
3836 Sometimes it's useful to check whether a given expression is a plain number,
3837 a sum, a polynomial with integer coefficients, or of some other specific type.
3838 GiNaC provides a couple of functions for this:
3841 bool is_a<T>(const ex & e);
3842 bool is_exactly_a<T>(const ex & e);
3843 bool ex::info(unsigned flag);
3844 unsigned ex::return_type() const;
3845 unsigned ex::return_type_tinfo() const;
3848 When the test made by @code{is_a<T>()} returns true, it is safe to call
3849 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3850 class names (@xref{The class hierarchy}, for a list of all classes). For
3851 example, assuming @code{e} is an @code{ex}:
3856 if (is_a<numeric>(e))
3857 numeric n = ex_to<numeric>(e);
3862 @code{is_a<T>(e)} allows you to check whether the top-level object of
3863 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3864 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3865 e.g., for checking whether an expression is a number, a sum, or a product:
3872 is_a<numeric>(e1); // true
3873 is_a<numeric>(e2); // false
3874 is_a<add>(e1); // false
3875 is_a<add>(e2); // true
3876 is_a<mul>(e1); // false
3877 is_a<mul>(e2); // false
3881 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3882 top-level object of an expression @samp{e} is an instance of the GiNaC
3883 class @samp{T}, not including parent classes.
3885 The @code{info()} method is used for checking certain attributes of
3886 expressions. The possible values for the @code{flag} argument are defined
3887 in @file{ginac/flags.h}, the most important being explained in the following
3891 @multitable @columnfractions .30 .70
3892 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3893 @item @code{numeric}
3894 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3896 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3897 @item @code{rational}
3898 @tab @dots{}an exact rational number (integers are rational, too)
3899 @item @code{integer}
3900 @tab @dots{}a (non-complex) integer
3901 @item @code{crational}
3902 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3903 @item @code{cinteger}
3904 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3905 @item @code{positive}
3906 @tab @dots{}not complex and greater than 0
3907 @item @code{negative}
3908 @tab @dots{}not complex and less than 0
3909 @item @code{nonnegative}
3910 @tab @dots{}not complex and greater than or equal to 0
3912 @tab @dots{}an integer greater than 0
3914 @tab @dots{}an integer less than 0
3915 @item @code{nonnegint}
3916 @tab @dots{}an integer greater than or equal to 0
3918 @tab @dots{}an even integer
3920 @tab @dots{}an odd integer
3922 @tab @dots{}a prime integer (probabilistic primality test)
3923 @item @code{relation}
3924 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3925 @item @code{relation_equal}
3926 @tab @dots{}a @code{==} relation
3927 @item @code{relation_not_equal}
3928 @tab @dots{}a @code{!=} relation
3929 @item @code{relation_less}
3930 @tab @dots{}a @code{<} relation
3931 @item @code{relation_less_or_equal}
3932 @tab @dots{}a @code{<=} relation
3933 @item @code{relation_greater}
3934 @tab @dots{}a @code{>} relation
3935 @item @code{relation_greater_or_equal}
3936 @tab @dots{}a @code{>=} relation
3938 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3940 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3941 @item @code{polynomial}
3942 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3943 @item @code{integer_polynomial}
3944 @tab @dots{}a polynomial with (non-complex) integer coefficients
3945 @item @code{cinteger_polynomial}
3946 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3947 @item @code{rational_polynomial}
3948 @tab @dots{}a polynomial with (non-complex) rational coefficients
3949 @item @code{crational_polynomial}
3950 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3951 @item @code{rational_function}
3952 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3953 @item @code{algebraic}
3954 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3958 To determine whether an expression is commutative or non-commutative and if
3959 so, with which other expressions it would commutate, you use the methods
3960 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3961 for an explanation of these.
3964 @subsection Accessing subexpressions
3967 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3968 @code{function}, act as containers for subexpressions. For example, the
3969 subexpressions of a sum (an @code{add} object) are the individual terms,
3970 and the subexpressions of a @code{function} are the function's arguments.
3972 @cindex @code{nops()}
3974 GiNaC provides several ways of accessing subexpressions. The first way is to
3979 ex ex::op(size_t i);
3982 @code{nops()} determines the number of subexpressions (operands) contained
3983 in the expression, while @code{op(i)} returns the @code{i}-th
3984 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3985 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3986 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3987 @math{i>0} are the indices.
3990 @cindex @code{const_iterator}
3991 The second way to access subexpressions is via the STL-style random-access
3992 iterator class @code{const_iterator} and the methods
3995 const_iterator ex::begin();
3996 const_iterator ex::end();
3999 @code{begin()} returns an iterator referring to the first subexpression;
4000 @code{end()} returns an iterator which is one-past the last subexpression.
4001 If the expression has no subexpressions, then @code{begin() == end()}. These
4002 iterators can also be used in conjunction with non-modifying STL algorithms.
4004 Here is an example that (non-recursively) prints the subexpressions of a
4005 given expression in three different ways:
4012 for (size_t i = 0; i != e.nops(); ++i)
4013 cout << e.op(i) << endl;
4016 for (const_iterator i = e.begin(); i != e.end(); ++i)
4019 // with iterators and STL copy()
4020 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4024 @cindex @code{const_preorder_iterator}
4025 @cindex @code{const_postorder_iterator}
4026 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4027 expression's immediate children. GiNaC provides two additional iterator
4028 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4029 that iterate over all objects in an expression tree, in preorder or postorder,
4030 respectively. They are STL-style forward iterators, and are created with the
4034 const_preorder_iterator ex::preorder_begin();
4035 const_preorder_iterator ex::preorder_end();
4036 const_postorder_iterator ex::postorder_begin();
4037 const_postorder_iterator ex::postorder_end();
4040 The following example illustrates the differences between
4041 @code{const_iterator}, @code{const_preorder_iterator}, and
4042 @code{const_postorder_iterator}:
4046 symbol A("A"), B("B"), C("C");
4047 ex e = lst(lst(A, B), C);
4049 std::copy(e.begin(), e.end(),
4050 std::ostream_iterator<ex>(cout, "\n"));
4054 std::copy(e.preorder_begin(), e.preorder_end(),
4055 std::ostream_iterator<ex>(cout, "\n"));
4062 std::copy(e.postorder_begin(), e.postorder_end(),
4063 std::ostream_iterator<ex>(cout, "\n"));
4072 @cindex @code{relational} (class)
4073 Finally, the left-hand side and right-hand side expressions of objects of
4074 class @code{relational} (and only of these) can also be accessed with the
4083 @subsection Comparing expressions
4084 @cindex @code{is_equal()}
4085 @cindex @code{is_zero()}
4087 Expressions can be compared with the usual C++ relational operators like
4088 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4089 the result is usually not determinable and the result will be @code{false},
4090 except in the case of the @code{!=} operator. You should also be aware that
4091 GiNaC will only do the most trivial test for equality (subtracting both
4092 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4095 Actually, if you construct an expression like @code{a == b}, this will be
4096 represented by an object of the @code{relational} class (@pxref{Relations})
4097 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4099 There are also two methods
4102 bool ex::is_equal(const ex & other);
4106 for checking whether one expression is equal to another, or equal to zero,
4107 respectively. See also the method @code{ex::is_zero_matrix()},
4111 @subsection Ordering expressions
4112 @cindex @code{ex_is_less} (class)
4113 @cindex @code{ex_is_equal} (class)
4114 @cindex @code{compare()}
4116 Sometimes it is necessary to establish a mathematically well-defined ordering
4117 on a set of arbitrary expressions, for example to use expressions as keys
4118 in a @code{std::map<>} container, or to bring a vector of expressions into
4119 a canonical order (which is done internally by GiNaC for sums and products).
4121 The operators @code{<}, @code{>} etc. described in the last section cannot
4122 be used for this, as they don't implement an ordering relation in the
4123 mathematical sense. In particular, they are not guaranteed to be
4124 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4125 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4128 By default, STL classes and algorithms use the @code{<} and @code{==}
4129 operators to compare objects, which are unsuitable for expressions, but GiNaC
4130 provides two functors that can be supplied as proper binary comparison
4131 predicates to the STL:
4134 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4136 bool operator()(const ex &lh, const ex &rh) const;
4139 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4141 bool operator()(const ex &lh, const ex &rh) const;
4145 For example, to define a @code{map} that maps expressions to strings you
4149 std::map<ex, std::string, ex_is_less> myMap;
4152 Omitting the @code{ex_is_less} template parameter will introduce spurious
4153 bugs because the map operates improperly.
4155 Other examples for the use of the functors:
4163 std::sort(v.begin(), v.end(), ex_is_less());
4165 // count the number of expressions equal to '1'
4166 unsigned num_ones = std::count_if(v.begin(), v.end(),
4167 std::bind2nd(ex_is_equal(), 1));
4170 The implementation of @code{ex_is_less} uses the member function
4173 int ex::compare(const ex & other) const;
4176 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4177 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4181 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4182 @c node-name, next, previous, up
4183 @section Numerical evaluation
4184 @cindex @code{evalf()}
4186 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4187 To evaluate them using floating-point arithmetic you need to call
4190 ex ex::evalf(int level = 0) const;
4193 @cindex @code{Digits}
4194 The accuracy of the evaluation is controlled by the global object @code{Digits}
4195 which can be assigned an integer value. The default value of @code{Digits}
4196 is 17. @xref{Numbers}, for more information and examples.
4198 To evaluate an expression to a @code{double} floating-point number you can
4199 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4203 // Approximate sin(x/Pi)
4205 ex e = series(sin(x/Pi), x == 0, 6);
4207 // Evaluate numerically at x=0.1
4208 ex f = evalf(e.subs(x == 0.1));
4210 // ex_to<numeric> is an unsafe cast, so check the type first
4211 if (is_a<numeric>(f)) @{
4212 double d = ex_to<numeric>(f).to_double();
4221 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4222 @c node-name, next, previous, up
4223 @section Substituting expressions
4224 @cindex @code{subs()}
4226 Algebraic objects inside expressions can be replaced with arbitrary
4227 expressions via the @code{.subs()} method:
4230 ex ex::subs(const ex & e, unsigned options = 0);
4231 ex ex::subs(const exmap & m, unsigned options = 0);
4232 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4235 In the first form, @code{subs()} accepts a relational of the form
4236 @samp{object == expression} or a @code{lst} of such relationals:
4240 symbol x("x"), y("y");
4242 ex e1 = 2*x^2-4*x+3;
4243 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4247 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4252 If you specify multiple substitutions, they are performed in parallel, so e.g.
4253 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4255 The second form of @code{subs()} takes an @code{exmap} object which is a
4256 pair associative container that maps expressions to expressions (currently
4257 implemented as a @code{std::map}). This is the most efficient one of the
4258 three @code{subs()} forms and should be used when the number of objects to
4259 be substituted is large or unknown.
4261 Using this form, the second example from above would look like this:
4265 symbol x("x"), y("y");
4271 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4275 The third form of @code{subs()} takes two lists, one for the objects to be
4276 replaced and one for the expressions to be substituted (both lists must
4277 contain the same number of elements). Using this form, you would write
4281 symbol x("x"), y("y");
4284 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4288 The optional last argument to @code{subs()} is a combination of
4289 @code{subs_options} flags. There are three options available:
4290 @code{subs_options::no_pattern} disables pattern matching, which makes
4291 large @code{subs()} operations significantly faster if you are not using
4292 patterns. The second option, @code{subs_options::algebraic} enables
4293 algebraic substitutions in products and powers.
4294 @ref{Pattern matching and advanced substitutions}, for more information
4295 about patterns and algebraic substitutions. The third option,
4296 @code{subs_options::no_index_renaming} disables the feature that dummy
4297 indices are renamed if the subsitution could give a result in which a
4298 dummy index occurs more than two times. This is sometimes necessary if
4299 you want to use @code{subs()} to rename your dummy indices.
4301 @code{subs()} performs syntactic substitution of any complete algebraic
4302 object; it does not try to match sub-expressions as is demonstrated by the
4307 symbol x("x"), y("y"), z("z");
4309 ex e1 = pow(x+y, 2);
4310 cout << e1.subs(x+y == 4) << endl;
4313 ex e2 = sin(x)*sin(y)*cos(x);
4314 cout << e2.subs(sin(x) == cos(x)) << endl;
4315 // -> cos(x)^2*sin(y)
4318 cout << e3.subs(x+y == 4) << endl;
4320 // (and not 4+z as one might expect)
4324 A more powerful form of substitution using wildcards is described in the
4328 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4329 @c node-name, next, previous, up
4330 @section Pattern matching and advanced substitutions
4331 @cindex @code{wildcard} (class)
4332 @cindex Pattern matching
4334 GiNaC allows the use of patterns for checking whether an expression is of a
4335 certain form or contains subexpressions of a certain form, and for
4336 substituting expressions in a more general way.
4338 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4339 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4340 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4341 an unsigned integer number to allow having multiple different wildcards in a
4342 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4343 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4347 ex wild(unsigned label = 0);
4350 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4353 Some examples for patterns:
4355 @multitable @columnfractions .5 .5
4356 @item @strong{Constructed as} @tab @strong{Output as}
4357 @item @code{wild()} @tab @samp{$0}
4358 @item @code{pow(x,wild())} @tab @samp{x^$0}
4359 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4360 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4366 @item Wildcards behave like symbols and are subject to the same algebraic
4367 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4368 @item As shown in the last example, to use wildcards for indices you have to
4369 use them as the value of an @code{idx} object. This is because indices must
4370 always be of class @code{idx} (or a subclass).
4371 @item Wildcards only represent expressions or subexpressions. It is not
4372 possible to use them as placeholders for other properties like index
4373 dimension or variance, representation labels, symmetry of indexed objects
4375 @item Because wildcards are commutative, it is not possible to use wildcards
4376 as part of noncommutative products.
4377 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4378 are also valid patterns.
4381 @subsection Matching expressions
4382 @cindex @code{match()}
4383 The most basic application of patterns is to check whether an expression
4384 matches a given pattern. This is done by the function
4387 bool ex::match(const ex & pattern);
4388 bool ex::match(const ex & pattern, lst & repls);
4391 This function returns @code{true} when the expression matches the pattern
4392 and @code{false} if it doesn't. If used in the second form, the actual
4393 subexpressions matched by the wildcards get returned in the @code{repls}
4394 object as a list of relations of the form @samp{wildcard == expression}.
4395 If @code{match()} returns false, the state of @code{repls} is undefined.
4396 For reproducible results, the list should be empty when passed to
4397 @code{match()}, but it is also possible to find similarities in multiple
4398 expressions by passing in the result of a previous match.
4400 The matching algorithm works as follows:
4403 @item A single wildcard matches any expression. If one wildcard appears
4404 multiple times in a pattern, it must match the same expression in all
4405 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4406 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4407 @item If the expression is not of the same class as the pattern, the match
4408 fails (i.e. a sum only matches a sum, a function only matches a function,
4410 @item If the pattern is a function, it only matches the same function
4411 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4412 @item Except for sums and products, the match fails if the number of
4413 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4415 @item If there are no subexpressions, the expressions and the pattern must
4416 be equal (in the sense of @code{is_equal()}).
4417 @item Except for sums and products, each subexpression (@code{op()}) must
4418 match the corresponding subexpression of the pattern.
4421 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4422 account for their commutativity and associativity:
4425 @item If the pattern contains a term or factor that is a single wildcard,
4426 this one is used as the @dfn{global wildcard}. If there is more than one
4427 such wildcard, one of them is chosen as the global wildcard in a random
4429 @item Every term/factor of the pattern, except the global wildcard, is
4430 matched against every term of the expression in sequence. If no match is
4431 found, the whole match fails. Terms that did match are not considered in
4433 @item If there are no unmatched terms left, the match succeeds. Otherwise
4434 the match fails unless there is a global wildcard in the pattern, in
4435 which case this wildcard matches the remaining terms.
4438 In general, having more than one single wildcard as a term of a sum or a
4439 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4442 Here are some examples in @command{ginsh} to demonstrate how it works (the
4443 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4444 match fails, and the list of wildcard replacements otherwise):
4447 > match((x+y)^a,(x+y)^a);
4449 > match((x+y)^a,(x+y)^b);
4451 > match((x+y)^a,$1^$2);
4453 > match((x+y)^a,$1^$1);
4455 > match((x+y)^(x+y),$1^$1);
4457 > match((x+y)^(x+y),$1^$2);
4459 > match((a+b)*(a+c),($1+b)*($1+c));
4461 > match((a+b)*(a+c),(a+$1)*(a+$2));
4463 (Unpredictable. The result might also be [$1==c,$2==b].)
4464 > match((a+b)*(a+c),($1+$2)*($1+$3));
4465 (The result is undefined. Due to the sequential nature of the algorithm
4466 and the re-ordering of terms in GiNaC, the match for the first factor
4467 may be @{$1==a,$2==b@} in which case the match for the second factor
4468 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4470 > match(a*(x+y)+a*z+b,a*$1+$2);
4471 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4472 @{$1=x+y,$2=a*z+b@}.)
4473 > match(a+b+c+d+e+f,c);
4475 > match(a+b+c+d+e+f,c+$0);
4477 > match(a+b+c+d+e+f,c+e+$0);
4479 > match(a+b,a+b+$0);
4481 > match(a*b^2,a^$1*b^$2);
4483 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4484 even though a==a^1.)
4485 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4487 > match(atan2(y,x^2),atan2(y,$0));
4491 @subsection Matching parts of expressions
4492 @cindex @code{has()}
4493 A more general way to look for patterns in expressions is provided by the
4497 bool ex::has(const ex & pattern);
4500 This function checks whether a pattern is matched by an expression itself or
4501 by any of its subexpressions.
4503 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4504 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4507 > has(x*sin(x+y+2*a),y);
4509 > has(x*sin(x+y+2*a),x+y);
4511 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4512 has the subexpressions "x", "y" and "2*a".)
4513 > has(x*sin(x+y+2*a),x+y+$1);
4515 (But this is possible.)
4516 > has(x*sin(2*(x+y)+2*a),x+y);
4518 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4519 which "x+y" is not a subexpression.)
4522 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4524 > has(4*x^2-x+3,$1*x);
4526 > has(4*x^2+x+3,$1*x);
4528 (Another possible pitfall. The first expression matches because the term
4529 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4530 contains a linear term you should use the coeff() function instead.)
4533 @cindex @code{find()}
4537 bool ex::find(const ex & pattern, lst & found);
4540 works a bit like @code{has()} but it doesn't stop upon finding the first
4541 match. Instead, it appends all found matches to the specified list. If there
4542 are multiple occurrences of the same expression, it is entered only once to
4543 the list. @code{find()} returns false if no matches were found (in
4544 @command{ginsh}, it returns an empty list):
4547 > find(1+x+x^2+x^3,x);
4549 > find(1+x+x^2+x^3,y);
4551 > find(1+x+x^2+x^3,x^$1);
4553 (Note the absence of "x".)
4554 > expand((sin(x)+sin(y))*(a+b));
4555 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4560 @subsection Substituting expressions
4561 @cindex @code{subs()}
4562 Probably the most useful application of patterns is to use them for
4563 substituting expressions with the @code{subs()} method. Wildcards can be
4564 used in the search patterns as well as in the replacement expressions, where
4565 they get replaced by the expressions matched by them. @code{subs()} doesn't
4566 know anything about algebra; it performs purely syntactic substitutions.
4571 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4573 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4575 > subs((a+b+c)^2,a+b==x);
4577 > subs((a+b+c)^2,a+b+$1==x+$1);
4579 > subs(a+2*b,a+b==x);
4581 > subs(4*x^3-2*x^2+5*x-1,x==a);
4583 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4585 > subs(sin(1+sin(x)),sin($1)==cos($1));
4587 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4591 The last example would be written in C++ in this way:
4595 symbol a("a"), b("b"), x("x"), y("y");
4596 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4597 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4598 cout << e.expand() << endl;
4603 @subsection The option algebraic
4604 Both @code{has()} and @code{subs()} take an optional argument to pass them
4605 extra options. This section describes what happens if you give the former
4606 the option @code{has_options::algebraic} or the latter
4607 @code{subs:options::algebraic}. In that case the matching condition for
4608 powers and multiplications is changed in such a way that they become
4609 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4610 If you use these options you will find that
4611 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4612 Besides matching some of the factors of a product also powers match as
4613 often as is possible without getting negative exponents. For example
4614 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4615 @code{x*c^2*z}. This also works with negative powers:
4616 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4617 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4618 and not for locating @code{x+y} within @code{x+y+z}.
4621 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4622 @c node-name, next, previous, up
4623 @section Applying a function on subexpressions
4624 @cindex tree traversal
4625 @cindex @code{map()}
4627 Sometimes you may want to perform an operation on specific parts of an
4628 expression while leaving the general structure of it intact. An example
4629 of this would be a matrix trace operation: the trace of a sum is the sum
4630 of the traces of the individual terms. That is, the trace should @dfn{map}
4631 on the sum, by applying itself to each of the sum's operands. It is possible
4632 to do this manually which usually results in code like this:
4637 if (is_a<matrix>(e))
4638 return ex_to<matrix>(e).trace();
4639 else if (is_a<add>(e)) @{
4641 for (size_t i=0; i<e.nops(); i++)
4642 sum += calc_trace(e.op(i));
4644 @} else if (is_a<mul>)(e)) @{
4652 This is, however, slightly inefficient (if the sum is very large it can take
4653 a long time to add the terms one-by-one), and its applicability is limited to
4654 a rather small class of expressions. If @code{calc_trace()} is called with
4655 a relation or a list as its argument, you will probably want the trace to
4656 be taken on both sides of the relation or of all elements of the list.
4658 GiNaC offers the @code{map()} method to aid in the implementation of such
4662 ex ex::map(map_function & f) const;
4663 ex ex::map(ex (*f)(const ex & e)) const;
4666 In the first (preferred) form, @code{map()} takes a function object that
4667 is subclassed from the @code{map_function} class. In the second form, it
4668 takes a pointer to a function that accepts and returns an expression.
4669 @code{map()} constructs a new expression of the same type, applying the
4670 specified function on all subexpressions (in the sense of @code{op()}),
4673 The use of a function object makes it possible to supply more arguments to
4674 the function that is being mapped, or to keep local state information.
4675 The @code{map_function} class declares a virtual function call operator
4676 that you can overload. Here is a sample implementation of @code{calc_trace()}
4677 that uses @code{map()} in a recursive fashion:
4680 struct calc_trace : public map_function @{
4681 ex operator()(const ex &e)
4683 if (is_a<matrix>(e))
4684 return ex_to<matrix>(e).trace();
4685 else if (is_a<mul>(e)) @{
4688 return e.map(*this);
4693 This function object could then be used like this:
4697 ex M = ... // expression with matrices
4698 calc_trace do_trace;
4699 ex tr = do_trace(M);
4703 Here is another example for you to meditate over. It removes quadratic
4704 terms in a variable from an expanded polynomial:
4707 struct map_rem_quad : public map_function @{
4709 map_rem_quad(const ex & var_) : var(var_) @{@}
4711 ex operator()(const ex & e)
4713 if (is_a<add>(e) || is_a<mul>(e))
4714 return e.map(*this);
4715 else if (is_a<power>(e) &&
4716 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4726 symbol x("x"), y("y");
4729 for (int i=0; i<8; i++)
4730 e += pow(x, i) * pow(y, 8-i) * (i+1);
4732 // -> 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
4734 map_rem_quad rem_quad(x);
4735 cout << rem_quad(e) << endl;
4736 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4740 @command{ginsh} offers a slightly different implementation of @code{map()}
4741 that allows applying algebraic functions to operands. The second argument
4742 to @code{map()} is an expression containing the wildcard @samp{$0} which
4743 acts as the placeholder for the operands:
4748 > map(a+2*b,sin($0));
4750 > map(@{a,b,c@},$0^2+$0);
4751 @{a^2+a,b^2+b,c^2+c@}
4754 Note that it is only possible to use algebraic functions in the second
4755 argument. You can not use functions like @samp{diff()}, @samp{op()},
4756 @samp{subs()} etc. because these are evaluated immediately:
4759 > map(@{a,b,c@},diff($0,a));
4761 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4762 to "map(@{a,b,c@},0)".
4766 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4767 @c node-name, next, previous, up
4768 @section Visitors and tree traversal
4769 @cindex tree traversal
4770 @cindex @code{visitor} (class)
4771 @cindex @code{accept()}
4772 @cindex @code{visit()}
4773 @cindex @code{traverse()}
4774 @cindex @code{traverse_preorder()}
4775 @cindex @code{traverse_postorder()}
4777 Suppose that you need a function that returns a list of all indices appearing
4778 in an arbitrary expression. The indices can have any dimension, and for
4779 indices with variance you always want the covariant version returned.
4781 You can't use @code{get_free_indices()} because you also want to include
4782 dummy indices in the list, and you can't use @code{find()} as it needs
4783 specific index dimensions (and it would require two passes: one for indices
4784 with variance, one for plain ones).
4786 The obvious solution to this problem is a tree traversal with a type switch,
4787 such as the following:
4790 void gather_indices_helper(const ex & e, lst & l)
4792 if (is_a<varidx>(e)) @{
4793 const varidx & vi = ex_to<varidx>(e);
4794 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4795 @} else if (is_a<idx>(e)) @{
4798 size_t n = e.nops();
4799 for (size_t i = 0; i < n; ++i)
4800 gather_indices_helper(e.op(i), l);
4804 lst gather_indices(const ex & e)
4807 gather_indices_helper(e, l);
4814 This works fine but fans of object-oriented programming will feel
4815 uncomfortable with the type switch. One reason is that there is a possibility
4816 for subtle bugs regarding derived classes. If we had, for example, written
4819 if (is_a<idx>(e)) @{
4821 @} else if (is_a<varidx>(e)) @{
4825 in @code{gather_indices_helper}, the code wouldn't have worked because the
4826 first line "absorbs" all classes derived from @code{idx}, including
4827 @code{varidx}, so the special case for @code{varidx} would never have been
4830 Also, for a large number of classes, a type switch like the above can get
4831 unwieldy and inefficient (it's a linear search, after all).
4832 @code{gather_indices_helper} only checks for two classes, but if you had to
4833 write a function that required a different implementation for nearly
4834 every GiNaC class, the result would be very hard to maintain and extend.
4836 The cleanest approach to the problem would be to add a new virtual function
4837 to GiNaC's class hierarchy. In our example, there would be specializations
4838 for @code{idx} and @code{varidx} while the default implementation in
4839 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4840 impossible to add virtual member functions to existing classes without
4841 changing their source and recompiling everything. GiNaC comes with source,
4842 so you could actually do this, but for a small algorithm like the one
4843 presented this would be impractical.
4845 One solution to this dilemma is the @dfn{Visitor} design pattern,
4846 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4847 variation, described in detail in
4848 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4849 virtual functions to the class hierarchy to implement operations, GiNaC
4850 provides a single "bouncing" method @code{accept()} that takes an instance
4851 of a special @code{visitor} class and redirects execution to the one
4852 @code{visit()} virtual function of the visitor that matches the type of
4853 object that @code{accept()} was being invoked on.
4855 Visitors in GiNaC must derive from the global @code{visitor} class as well
4856 as from the class @code{T::visitor} of each class @code{T} they want to
4857 visit, and implement the member functions @code{void visit(const T &)} for
4863 void ex::accept(visitor & v) const;
4866 will then dispatch to the correct @code{visit()} member function of the
4867 specified visitor @code{v} for the type of GiNaC object at the root of the
4868 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4870 Here is an example of a visitor:
4874 : public visitor, // this is required
4875 public add::visitor, // visit add objects
4876 public numeric::visitor, // visit numeric objects
4877 public basic::visitor // visit basic objects
4879 void visit(const add & x)
4880 @{ cout << "called with an add object" << endl; @}
4882 void visit(const numeric & x)
4883 @{ cout << "called with a numeric object" << endl; @}
4885 void visit(const basic & x)
4886 @{ cout << "called with a basic object" << endl; @}
4890 which can be used as follows:
4901 // prints "called with a numeric object"
4903 // prints "called with an add object"
4905 // prints "called with a basic object"
4909 The @code{visit(const basic &)} method gets called for all objects that are
4910 not @code{numeric} or @code{add} and acts as an (optional) default.
4912 From a conceptual point of view, the @code{visit()} methods of the visitor
4913 behave like a newly added virtual function of the visited hierarchy.
4914 In addition, visitors can store state in member variables, and they can
4915 be extended by deriving a new visitor from an existing one, thus building
4916 hierarchies of visitors.
4918 We can now rewrite our index example from above with a visitor:
4921 class gather_indices_visitor
4922 : public visitor, public idx::visitor, public varidx::visitor
4926 void visit(const idx & i)
4931 void visit(const varidx & vi)
4933 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4937 const lst & get_result() // utility function
4946 What's missing is the tree traversal. We could implement it in
4947 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4950 void ex::traverse_preorder(visitor & v) const;
4951 void ex::traverse_postorder(visitor & v) const;
4952 void ex::traverse(visitor & v) const;
4955 @code{traverse_preorder()} visits a node @emph{before} visiting its
4956 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4957 visiting its subexpressions. @code{traverse()} is a synonym for
4958 @code{traverse_preorder()}.
4960 Here is a new implementation of @code{gather_indices()} that uses the visitor
4961 and @code{traverse()}:
4964 lst gather_indices(const ex & e)
4966 gather_indices_visitor v;
4968 return v.get_result();
4972 Alternatively, you could use pre- or postorder iterators for the tree
4976 lst gather_indices(const ex & e)
4978 gather_indices_visitor v;
4979 for (const_preorder_iterator i = e.preorder_begin();
4980 i != e.preorder_end(); ++i) @{
4983 return v.get_result();
4988 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4989 @c node-name, next, previous, up
4990 @section Polynomial arithmetic
4992 @subsection Testing whether an expression is a polynomial
4993 @cindex @code{is_polynomial()}
4995 Testing whether an expression is a polynomial in one or more variables
4996 can be done with the method
4998 bool ex::is_polynomial(const ex & vars) const;
5000 In the case of more than
5001 one variable, the variables are given as a list.
5004 (x*y*sin(y)).is_polynomial(x) // Returns true.
5005 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5008 @subsection Expanding and collecting
5009 @cindex @code{expand()}
5010 @cindex @code{collect()}
5011 @cindex @code{collect_common_factors()}
5013 A polynomial in one or more variables has many equivalent
5014 representations. Some useful ones serve a specific purpose. Consider
5015 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5016 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5017 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5018 representations are the recursive ones where one collects for exponents
5019 in one of the three variable. Since the factors are themselves
5020 polynomials in the remaining two variables the procedure can be
5021 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5022 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5025 To bring an expression into expanded form, its method
5028 ex ex::expand(unsigned options = 0);
5031 may be called. In our example above, this corresponds to @math{4*x*y +
5032 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5033 GiNaC is not easy to guess you should be prepared to see different
5034 orderings of terms in such sums!
5036 Another useful representation of multivariate polynomials is as a
5037 univariate polynomial in one of the variables with the coefficients
5038 being polynomials in the remaining variables. The method
5039 @code{collect()} accomplishes this task:
5042 ex ex::collect(const ex & s, bool distributed = false);
5045 The first argument to @code{collect()} can also be a list of objects in which
5046 case the result is either a recursively collected polynomial, or a polynomial
5047 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5048 by the @code{distributed} flag.
5050 Note that the original polynomial needs to be in expanded form (for the
5051 variables concerned) in order for @code{collect()} to be able to find the
5052 coefficients properly.
5054 The following @command{ginsh} transcript shows an application of @code{collect()}
5055 together with @code{find()}:
5058 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5059 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)
5060 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5061 > collect(a,@{p,q@});
5062 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5063 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5064 > collect(a,find(a,sin($1)));
5065 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5066 > collect(a,@{find(a,sin($1)),p,q@});
5067 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5068 > collect(a,@{find(a,sin($1)),d@});
5069 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5072 Polynomials can often be brought into a more compact form by collecting
5073 common factors from the terms of sums. This is accomplished by the function
5076 ex collect_common_factors(const ex & e);
5079 This function doesn't perform a full factorization but only looks for
5080 factors which are already explicitly present:
5083 > collect_common_factors(a*x+a*y);
5085 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5087 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5088 (c+a)*a*(x*y+y^2+x)*b
5091 @subsection Degree and coefficients
5092 @cindex @code{degree()}
5093 @cindex @code{ldegree()}
5094 @cindex @code{coeff()}
5096 The degree and low degree of a polynomial can be obtained using the two
5100 int ex::degree(const ex & s);
5101 int ex::ldegree(const ex & s);
5104 which also work reliably on non-expanded input polynomials (they even work
5105 on rational functions, returning the asymptotic degree). By definition, the
5106 degree of zero is zero. To extract a coefficient with a certain power from
5107 an expanded polynomial you use
5110 ex ex::coeff(const ex & s, int n);
5113 You can also obtain the leading and trailing coefficients with the methods
5116 ex ex::lcoeff(const ex & s);
5117 ex ex::tcoeff(const ex & s);
5120 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5123 An application is illustrated in the next example, where a multivariate
5124 polynomial is analyzed:
5128 symbol x("x"), y("y");
5129 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5130 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5131 ex Poly = PolyInp.expand();
5133 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5134 cout << "The x^" << i << "-coefficient is "
5135 << Poly.coeff(x,i) << endl;
5137 cout << "As polynomial in y: "
5138 << Poly.collect(y) << endl;
5142 When run, it returns an output in the following fashion:
5145 The x^0-coefficient is y^2+11*y
5146 The x^1-coefficient is 5*y^2-2*y
5147 The x^2-coefficient is -1
5148 The x^3-coefficient is 4*y
5149 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5152 As always, the exact output may vary between different versions of GiNaC
5153 or even from run to run since the internal canonical ordering is not
5154 within the user's sphere of influence.
5156 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5157 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5158 with non-polynomial expressions as they not only work with symbols but with
5159 constants, functions and indexed objects as well:
5163 symbol a("a"), b("b"), c("c"), x("x");
5164 idx i(symbol("i"), 3);
5166 ex e = pow(sin(x) - cos(x), 4);
5167 cout << e.degree(cos(x)) << endl;
5169 cout << e.expand().coeff(sin(x), 3) << endl;
5172 e = indexed(a+b, i) * indexed(b+c, i);
5173 e = e.expand(expand_options::expand_indexed);
5174 cout << e.collect(indexed(b, i)) << endl;
5175 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5180 @subsection Polynomial division
5181 @cindex polynomial division
5184 @cindex pseudo-remainder
5185 @cindex @code{quo()}
5186 @cindex @code{rem()}
5187 @cindex @code{prem()}
5188 @cindex @code{divide()}
5193 ex quo(const ex & a, const ex & b, const ex & x);
5194 ex rem(const ex & a, const ex & b, const ex & x);
5197 compute the quotient and remainder of univariate polynomials in the variable
5198 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5200 The additional function
5203 ex prem(const ex & a, const ex & b, const ex & x);
5206 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5207 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5209 Exact division of multivariate polynomials is performed by the function
5212 bool divide(const ex & a, const ex & b, ex & q);
5215 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5216 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5217 in which case the value of @code{q} is undefined.
5220 @subsection Unit, content and primitive part
5221 @cindex @code{unit()}
5222 @cindex @code{content()}
5223 @cindex @code{primpart()}
5224 @cindex @code{unitcontprim()}
5229 ex ex::unit(const ex & x);
5230 ex ex::content(const ex & x);
5231 ex ex::primpart(const ex & x);
5232 ex ex::primpart(const ex & x, const ex & c);
5235 return the unit part, content part, and primitive polynomial of a multivariate
5236 polynomial with respect to the variable @samp{x} (the unit part being the sign
5237 of the leading coefficient, the content part being the GCD of the coefficients,
5238 and the primitive polynomial being the input polynomial divided by the unit and
5239 content parts). The second variant of @code{primpart()} expects the previously
5240 calculated content part of the polynomial in @code{c}, which enables it to
5241 work faster in the case where the content part has already been computed. The
5242 product of unit, content, and primitive part is the original polynomial.
5244 Additionally, the method
5247 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5250 computes the unit, content, and primitive parts in one go, returning them
5251 in @code{u}, @code{c}, and @code{p}, respectively.
5254 @subsection GCD, LCM and resultant
5257 @cindex @code{gcd()}
5258 @cindex @code{lcm()}
5260 The functions for polynomial greatest common divisor and least common
5261 multiple have the synopsis
5264 ex gcd(const ex & a, const ex & b);
5265 ex lcm(const ex & a, const ex & b);
5268 The functions @code{gcd()} and @code{lcm()} accept two expressions
5269 @code{a} and @code{b} as arguments and return a new expression, their
5270 greatest common divisor or least common multiple, respectively. If the
5271 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5272 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5273 the coefficients must be rationals.
5276 #include <ginac/ginac.h>
5277 using namespace GiNaC;
5281 symbol x("x"), y("y"), z("z");
5282 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5283 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5285 ex P_gcd = gcd(P_a, P_b);
5287 ex P_lcm = lcm(P_a, P_b);
5288 // 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
5293 @cindex @code{resultant()}
5295 The resultant of two expressions only makes sense with polynomials.
5296 It is always computed with respect to a specific symbol within the
5297 expressions. The function has the interface
5300 ex resultant(const ex & a, const ex & b, const ex & s);
5303 Resultants are symmetric in @code{a} and @code{b}. The following example
5304 computes the resultant of two expressions with respect to @code{x} and
5305 @code{y}, respectively:
5308 #include <ginac/ginac.h>
5309 using namespace GiNaC;
5313 symbol x("x"), y("y");
5315 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5318 r = resultant(e1, e2, x);
5320 r = resultant(e1, e2, y);
5325 @subsection Square-free decomposition
5326 @cindex square-free decomposition
5327 @cindex factorization
5328 @cindex @code{sqrfree()}
5330 GiNaC still lacks proper factorization support. Some form of
5331 factorization is, however, easily implemented by noting that factors
5332 appearing in a polynomial with power two or more also appear in the
5333 derivative and hence can easily be found by computing the GCD of the
5334 original polynomial and its derivatives. Any decent system has an
5335 interface for this so called square-free factorization. So we provide
5338 ex sqrfree(const ex & a, const lst & l = lst());
5340 Here is an example that by the way illustrates how the exact form of the
5341 result may slightly depend on the order of differentiation, calling for
5342 some care with subsequent processing of the result:
5345 symbol x("x"), y("y");
5346 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5348 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5349 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5351 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5352 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5354 cout << sqrfree(BiVarPol) << endl;
5355 // -> depending on luck, any of the above
5358 Note also, how factors with the same exponents are not fully factorized
5362 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5363 @c node-name, next, previous, up
5364 @section Rational expressions
5366 @subsection The @code{normal} method
5367 @cindex @code{normal()}
5368 @cindex simplification
5369 @cindex temporary replacement
5371 Some basic form of simplification of expressions is called for frequently.
5372 GiNaC provides the method @code{.normal()}, which converts a rational function
5373 into an equivalent rational function of the form @samp{numerator/denominator}
5374 where numerator and denominator are coprime. If the input expression is already
5375 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5376 otherwise it performs fraction addition and multiplication.
5378 @code{.normal()} can also be used on expressions which are not rational functions
5379 as it will replace all non-rational objects (like functions or non-integer
5380 powers) by temporary symbols to bring the expression to the domain of rational
5381 functions before performing the normalization, and re-substituting these
5382 symbols afterwards. This algorithm is also available as a separate method
5383 @code{.to_rational()}, described below.
5385 This means that both expressions @code{t1} and @code{t2} are indeed
5386 simplified in this little code snippet:
5391 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5392 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5393 std::cout << "t1 is " << t1.normal() << std::endl;
5394 std::cout << "t2 is " << t2.normal() << std::endl;
5398 Of course this works for multivariate polynomials too, so the ratio of
5399 the sample-polynomials from the section about GCD and LCM above would be
5400 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5403 @subsection Numerator and denominator
5406 @cindex @code{numer()}
5407 @cindex @code{denom()}
5408 @cindex @code{numer_denom()}
5410 The numerator and denominator of an expression can be obtained with
5415 ex ex::numer_denom();
5418 These functions will first normalize the expression as described above and
5419 then return the numerator, denominator, or both as a list, respectively.
5420 If you need both numerator and denominator, calling @code{numer_denom()} is
5421 faster than using @code{numer()} and @code{denom()} separately.
5424 @subsection Converting to a polynomial or rational expression
5425 @cindex @code{to_polynomial()}
5426 @cindex @code{to_rational()}
5428 Some of the methods described so far only work on polynomials or rational
5429 functions. GiNaC provides a way to extend the domain of these functions to
5430 general expressions by using the temporary replacement algorithm described
5431 above. You do this by calling
5434 ex ex::to_polynomial(exmap & m);
5435 ex ex::to_polynomial(lst & l);
5439 ex ex::to_rational(exmap & m);
5440 ex ex::to_rational(lst & l);
5443 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5444 will be filled with the generated temporary symbols and their replacement
5445 expressions in a format that can be used directly for the @code{subs()}
5446 method. It can also already contain a list of replacements from an earlier
5447 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5448 possible to use it on multiple expressions and get consistent results.
5450 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5451 is probably best illustrated with an example:
5455 symbol x("x"), y("y");
5456 ex a = 2*x/sin(x) - y/(3*sin(x));
5460 ex p = a.to_polynomial(lp);
5461 cout << " = " << p << "\n with " << lp << endl;
5462 // = symbol3*symbol2*y+2*symbol2*x
5463 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5466 ex r = a.to_rational(lr);
5467 cout << " = " << r << "\n with " << lr << endl;
5468 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5469 // with @{symbol4==sin(x)@}
5473 The following more useful example will print @samp{sin(x)-cos(x)}:
5478 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5479 ex b = sin(x) + cos(x);
5482 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5483 cout << q.subs(m) << endl;
5488 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5489 @c node-name, next, previous, up
5490 @section Symbolic differentiation
5491 @cindex differentiation
5492 @cindex @code{diff()}
5494 @cindex product rule
5496 GiNaC's objects know how to differentiate themselves. Thus, a
5497 polynomial (class @code{add}) knows that its derivative is the sum of
5498 the derivatives of all the monomials:
5502 symbol x("x"), y("y"), z("z");
5503 ex P = pow(x, 5) + pow(x, 2) + y;
5505 cout << P.diff(x,2) << endl;
5507 cout << P.diff(y) << endl; // 1
5509 cout << P.diff(z) << endl; // 0
5514 If a second integer parameter @var{n} is given, the @code{diff} method
5515 returns the @var{n}th derivative.
5517 If @emph{every} object and every function is told what its derivative
5518 is, all derivatives of composed objects can be calculated using the
5519 chain rule and the product rule. Consider, for instance the expression
5520 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5521 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5522 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5523 out that the composition is the generating function for Euler Numbers,
5524 i.e. the so called @var{n}th Euler number is the coefficient of
5525 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5526 identity to code a function that generates Euler numbers in just three
5529 @cindex Euler numbers
5531 #include <ginac/ginac.h>
5532 using namespace GiNaC;
5534 ex EulerNumber(unsigned n)
5537 const ex generator = pow(cosh(x),-1);
5538 return generator.diff(x,n).subs(x==0);
5543 for (unsigned i=0; i<11; i+=2)
5544 std::cout << EulerNumber(i) << std::endl;
5549 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5550 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5551 @code{i} by two since all odd Euler numbers vanish anyways.
5554 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5555 @c node-name, next, previous, up
5556 @section Series expansion
5557 @cindex @code{series()}
5558 @cindex Taylor expansion
5559 @cindex Laurent expansion
5560 @cindex @code{pseries} (class)
5561 @cindex @code{Order()}
5563 Expressions know how to expand themselves as a Taylor series or (more
5564 generally) a Laurent series. As in most conventional Computer Algebra
5565 Systems, no distinction is made between those two. There is a class of
5566 its own for storing such series (@code{class pseries}) and a built-in
5567 function (called @code{Order}) for storing the order term of the series.
5568 As a consequence, if you want to work with series, i.e. multiply two
5569 series, you need to call the method @code{ex::series} again to convert
5570 it to a series object with the usual structure (expansion plus order
5571 term). A sample application from special relativity could read:
5574 #include <ginac/ginac.h>
5575 using namespace std;
5576 using namespace GiNaC;
5580 symbol v("v"), c("c");
5582 ex gamma = 1/sqrt(1 - pow(v/c,2));
5583 ex mass_nonrel = gamma.series(v==0, 10);
5585 cout << "the relativistic mass increase with v is " << endl
5586 << mass_nonrel << endl;
5588 cout << "the inverse square of this series is " << endl
5589 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5593 Only calling the series method makes the last output simplify to
5594 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5595 series raised to the power @math{-2}.
5597 @cindex Machin's formula
5598 As another instructive application, let us calculate the numerical
5599 value of Archimedes' constant
5603 (for which there already exists the built-in constant @code{Pi})
5604 using John Machin's amazing formula
5606 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5609 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5611 This equation (and similar ones) were used for over 200 years for
5612 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5613 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5614 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5615 order term with it and the question arises what the system is supposed
5616 to do when the fractions are plugged into that order term. The solution
5617 is to use the function @code{series_to_poly()} to simply strip the order
5621 #include <ginac/ginac.h>
5622 using namespace GiNaC;
5624 ex machin_pi(int degr)
5627 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5628 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5629 -4*pi_expansion.subs(x==numeric(1,239));
5635 using std::cout; // just for fun, another way of...
5636 using std::endl; // ...dealing with this namespace std.
5638 for (int i=2; i<12; i+=2) @{
5639 pi_frac = machin_pi(i);
5640 cout << i << ":\t" << pi_frac << endl
5641 << "\t" << pi_frac.evalf() << endl;
5647 Note how we just called @code{.series(x,degr)} instead of
5648 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5649 method @code{series()}: if the first argument is a symbol the expression
5650 is expanded in that symbol around point @code{0}. When you run this
5651 program, it will type out:
5655 3.1832635983263598326
5656 4: 5359397032/1706489875
5657 3.1405970293260603143
5658 6: 38279241713339684/12184551018734375
5659 3.141621029325034425
5660 8: 76528487109180192540976/24359780855939418203125
5661 3.141591772182177295
5662 10: 327853873402258685803048818236/104359128170408663038552734375
5663 3.1415926824043995174
5667 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5668 @c node-name, next, previous, up
5669 @section Symmetrization
5670 @cindex @code{symmetrize()}
5671 @cindex @code{antisymmetrize()}
5672 @cindex @code{symmetrize_cyclic()}
5677 ex ex::symmetrize(const lst & l);
5678 ex ex::antisymmetrize(const lst & l);
5679 ex ex::symmetrize_cyclic(const lst & l);
5682 symmetrize an expression by returning the sum over all symmetric,
5683 antisymmetric or cyclic permutations of the specified list of objects,
5684 weighted by the number of permutations.
5686 The three additional methods
5689 ex ex::symmetrize();
5690 ex ex::antisymmetrize();
5691 ex ex::symmetrize_cyclic();
5694 symmetrize or antisymmetrize an expression over its free indices.
5696 Symmetrization is most useful with indexed expressions but can be used with
5697 almost any kind of object (anything that is @code{subs()}able):
5701 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5702 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5704 cout << indexed(A, i, j).symmetrize() << endl;
5705 // -> 1/2*A.j.i+1/2*A.i.j
5706 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5707 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5708 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5709 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5713 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5714 @c node-name, next, previous, up
5715 @section Predefined mathematical functions
5717 @subsection Overview
5719 GiNaC contains the following predefined mathematical functions:
5722 @multitable @columnfractions .30 .70
5723 @item @strong{Name} @tab @strong{Function}
5726 @cindex @code{abs()}
5727 @item @code{step(x)}
5729 @cindex @code{step()}
5730 @item @code{csgn(x)}
5732 @cindex @code{conjugate()}
5733 @item @code{conjugate(x)}
5734 @tab complex conjugation
5735 @cindex @code{real_part()}
5736 @item @code{real_part(x)}
5738 @cindex @code{imag_part()}
5739 @item @code{imag_part(x)}
5741 @item @code{sqrt(x)}
5742 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5743 @cindex @code{sqrt()}
5746 @cindex @code{sin()}
5749 @cindex @code{cos()}
5752 @cindex @code{tan()}
5753 @item @code{asin(x)}
5755 @cindex @code{asin()}
5756 @item @code{acos(x)}
5758 @cindex @code{acos()}
5759 @item @code{atan(x)}
5760 @tab inverse tangent
5761 @cindex @code{atan()}
5762 @item @code{atan2(y, x)}
5763 @tab inverse tangent with two arguments
5764 @item @code{sinh(x)}
5765 @tab hyperbolic sine
5766 @cindex @code{sinh()}
5767 @item @code{cosh(x)}
5768 @tab hyperbolic cosine
5769 @cindex @code{cosh()}
5770 @item @code{tanh(x)}
5771 @tab hyperbolic tangent
5772 @cindex @code{tanh()}
5773 @item @code{asinh(x)}
5774 @tab inverse hyperbolic sine
5775 @cindex @code{asinh()}
5776 @item @code{acosh(x)}
5777 @tab inverse hyperbolic cosine
5778 @cindex @code{acosh()}
5779 @item @code{atanh(x)}
5780 @tab inverse hyperbolic tangent
5781 @cindex @code{atanh()}
5783 @tab exponential function
5784 @cindex @code{exp()}
5786 @tab natural logarithm
5787 @cindex @code{log()}
5790 @cindex @code{Li2()}
5791 @item @code{Li(m, x)}
5792 @tab classical polylogarithm as well as multiple polylogarithm
5794 @item @code{G(a, y)}
5795 @tab multiple polylogarithm
5797 @item @code{G(a, s, y)}
5798 @tab multiple polylogarithm with explicit signs for the imaginary parts
5800 @item @code{S(n, p, x)}
5801 @tab Nielsen's generalized polylogarithm
5803 @item @code{H(m, x)}
5804 @tab harmonic polylogarithm
5806 @item @code{zeta(m)}
5807 @tab Riemann's zeta function as well as multiple zeta value
5808 @cindex @code{zeta()}
5809 @item @code{zeta(m, s)}
5810 @tab alternating Euler sum
5811 @cindex @code{zeta()}
5812 @item @code{zetaderiv(n, x)}
5813 @tab derivatives of Riemann's zeta function
5814 @item @code{tgamma(x)}
5816 @cindex @code{tgamma()}
5817 @cindex gamma function
5818 @item @code{lgamma(x)}
5819 @tab logarithm of gamma function
5820 @cindex @code{lgamma()}
5821 @item @code{beta(x, y)}
5822 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5823 @cindex @code{beta()}
5825 @tab psi (digamma) function
5826 @cindex @code{psi()}
5827 @item @code{psi(n, x)}
5828 @tab derivatives of psi function (polygamma functions)
5829 @item @code{factorial(n)}
5830 @tab factorial function @math{n!}
5831 @cindex @code{factorial()}
5832 @item @code{binomial(n, k)}
5833 @tab binomial coefficients
5834 @cindex @code{binomial()}
5835 @item @code{Order(x)}
5836 @tab order term function in truncated power series
5837 @cindex @code{Order()}
5842 For functions that have a branch cut in the complex plane GiNaC follows
5843 the conventions for C++ as defined in the ANSI standard as far as
5844 possible. In particular: the natural logarithm (@code{log}) and the
5845 square root (@code{sqrt}) both have their branch cuts running along the
5846 negative real axis where the points on the axis itself belong to the
5847 upper part (i.e. continuous with quadrant II). The inverse
5848 trigonometric and hyperbolic functions are not defined for complex
5849 arguments by the C++ standard, however. In GiNaC we follow the
5850 conventions used by CLN, which in turn follow the carefully designed
5851 definitions in the Common Lisp standard. It should be noted that this
5852 convention is identical to the one used by the C99 standard and by most
5853 serious CAS. It is to be expected that future revisions of the C++
5854 standard incorporate these functions in the complex domain in a manner
5855 compatible with C99.
5857 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5858 @c node-name, next, previous, up
5859 @subsection Multiple polylogarithms
5861 @cindex polylogarithm
5862 @cindex Nielsen's generalized polylogarithm
5863 @cindex harmonic polylogarithm
5864 @cindex multiple zeta value
5865 @cindex alternating Euler sum
5866 @cindex multiple polylogarithm
5868 The multiple polylogarithm is the most generic member of a family of functions,
5869 to which others like the harmonic polylogarithm, Nielsen's generalized
5870 polylogarithm and the multiple zeta value belong.
5871 Everyone of these functions can also be written as a multiple polylogarithm with specific
5872 parameters. This whole family of functions is therefore often referred to simply as
5873 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5874 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5875 @code{Li} and @code{G} in principle represent the same function, the different
5876 notations are more natural to the series representation or the integral
5877 representation, respectively.
5879 To facilitate the discussion of these functions we distinguish between indices and
5880 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5881 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5883 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5884 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5885 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5886 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5887 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5888 @code{s} is not given, the signs default to +1.
5889 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5890 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5891 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5892 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5893 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5895 The functions print in LaTeX format as
5897 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5903 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5906 $\zeta(m_1,m_2,\ldots,m_k)$.
5908 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5909 are printed with a line above, e.g.
5911 $\zeta(5,\overline{2})$.
5913 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5915 Definitions and analytical as well as numerical properties of multiple polylogarithms
5916 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5917 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5918 except for a few differences which will be explicitly stated in the following.
5920 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5921 that the indices and arguments are understood to be in the same order as in which they appear in
5922 the series representation. This means
5924 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5927 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5930 $\zeta(1,2)$ evaluates to infinity.
5932 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5935 The functions only evaluate if the indices are integers greater than zero, except for the indices
5936 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5937 will be interpreted as the sequence of signs for the corresponding indices
5938 @code{m} or the sign of the imaginary part for the
5939 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5940 @code{zeta(lst(3,4), lst(-1,1))} means
5942 $\zeta(\overline{3},4)$
5945 @code{G(lst(a,b), lst(-1,1), c)} means
5947 $G(a-0\epsilon,b+0\epsilon;c)$.
5949 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5950 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5951 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
5952 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5953 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5954 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5955 evaluates also for negative integers and positive even integers. For example:
5958 > Li(@{3,1@},@{x,1@});
5961 -zeta(@{3,2@},@{-1,-1@})
5966 It is easy to tell for a given function into which other function it can be rewritten, may
5967 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5968 with negative indices or trailing zeros (the example above gives a hint). Signs can
5969 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5970 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5971 @code{Li} (@code{eval()} already cares for the possible downgrade):
5974 > convert_H_to_Li(@{0,-2,-1,3@},x);
5975 Li(@{3,1,3@},@{-x,1,-1@})
5976 > convert_H_to_Li(@{2,-1,0@},x);
5977 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5980 Every function can be numerically evaluated for
5981 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5982 global variable @code{Digits}:
5987 > evalf(zeta(@{3,1,3,1@}));
5988 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5991 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5992 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5994 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5999 In long expressions this helps a lot with debugging, because you can easily spot
6000 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6001 cancellations of divergencies happen.
6003 Useful publications:
6005 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6006 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6008 @cite{Harmonic Polylogarithms},
6009 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6011 @cite{Special Values of Multiple Polylogarithms},
6012 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6014 @cite{Numerical Evaluation of Multiple Polylogarithms},
6015 J.Vollinga, S.Weinzierl, hep-ph/0410259
6017 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6018 @c node-name, next, previous, up
6019 @section Complex expressions
6021 @cindex @code{conjugate()}
6023 For dealing with complex expressions there are the methods
6031 that return respectively the complex conjugate, the real part and the
6032 imaginary part of an expression. Complex conjugation works as expected
6033 for all built-in functinos and objects. Taking real and imaginary
6034 parts has not yet been implemented for all built-in functions. In cases where
6035 it is not known how to conjugate or take a real/imaginary part one
6036 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6037 is returned. For instance, in case of a complex symbol @code{x}
6038 (symbols are complex by default), one could not simplify
6039 @code{conjugate(x)}. In the case of strings of gamma matrices,
6040 the @code{conjugate} method takes the Dirac conjugate.
6045 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6049 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6050 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6051 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6052 // -> -gamma5*gamma~b*gamma~a
6056 If you declare your own GiNaC functions, then they will conjugate themselves
6057 by conjugating their arguments. This is the default strategy. If you want to
6058 change this behavior, you have to supply a specialized conjugation method
6059 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6060 for @code{abs} as an example). Also, specialized methods can be provided
6061 to take real and imaginary parts of user-defined functions.
6063 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6064 @c node-name, next, previous, up
6065 @section Solving linear systems of equations
6066 @cindex @code{lsolve()}
6068 The function @code{lsolve()} provides a convenient wrapper around some
6069 matrix operations that comes in handy when a system of linear equations
6073 ex lsolve(const ex & eqns, const ex & symbols,
6074 unsigned options = solve_algo::automatic);
6077 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6078 @code{relational}) while @code{symbols} is a @code{lst} of
6079 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6082 It returns the @code{lst} of solutions as an expression. As an example,
6083 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6087 symbol a("a"), b("b"), x("x"), y("y");
6089 eqns = a*x+b*y==3, x-y==b;
6091 cout << lsolve(eqns, vars) << endl;
6092 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6095 When the linear equations @code{eqns} are underdetermined, the solution
6096 will contain one or more tautological entries like @code{x==x},
6097 depending on the rank of the system. When they are overdetermined, the
6098 solution will be an empty @code{lst}. Note the third optional parameter
6099 to @code{lsolve()}: it accepts the same parameters as
6100 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6104 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6105 @c node-name, next, previous, up
6106 @section Input and output of expressions
6109 @subsection Expression output
6111 @cindex output of expressions
6113 Expressions can simply be written to any stream:
6118 ex e = 4.5*I+pow(x,2)*3/2;
6119 cout << e << endl; // prints '4.5*I+3/2*x^2'
6123 The default output format is identical to the @command{ginsh} input syntax and
6124 to that used by most computer algebra systems, but not directly pastable
6125 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6126 is printed as @samp{x^2}).
6128 It is possible to print expressions in a number of different formats with
6129 a set of stream manipulators;
6132 std::ostream & dflt(std::ostream & os);
6133 std::ostream & latex(std::ostream & os);
6134 std::ostream & tree(std::ostream & os);
6135 std::ostream & csrc(std::ostream & os);
6136 std::ostream & csrc_float(std::ostream & os);
6137 std::ostream & csrc_double(std::ostream & os);
6138 std::ostream & csrc_cl_N(std::ostream & os);
6139 std::ostream & index_dimensions(std::ostream & os);
6140 std::ostream & no_index_dimensions(std::ostream & os);
6143 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6144 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6145 @code{print_csrc()} functions, respectively.
6148 All manipulators affect the stream state permanently. To reset the output
6149 format to the default, use the @code{dflt} manipulator:
6153 cout << latex; // all output to cout will be in LaTeX format from
6155 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6156 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6157 cout << dflt; // revert to default output format
6158 cout << e << endl; // prints '4.5*I+3/2*x^2'
6162 If you don't want to affect the format of the stream you're working with,
6163 you can output to a temporary @code{ostringstream} like this:
6168 s << latex << e; // format of cout remains unchanged
6169 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6174 @cindex @code{csrc_float}
6175 @cindex @code{csrc_double}
6176 @cindex @code{csrc_cl_N}
6177 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6178 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6179 format that can be directly used in a C or C++ program. The three possible
6180 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6181 classes provided by the CLN library):
6185 cout << "f = " << csrc_float << e << ";\n";
6186 cout << "d = " << csrc_double << e << ";\n";
6187 cout << "n = " << csrc_cl_N << e << ";\n";
6191 The above example will produce (note the @code{x^2} being converted to
6195 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6196 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6197 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6201 The @code{tree} manipulator allows dumping the internal structure of an
6202 expression for debugging purposes:
6213 add, hash=0x0, flags=0x3, nops=2
6214 power, hash=0x0, flags=0x3, nops=2
6215 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6216 2 (numeric), hash=0x6526b0fa, flags=0xf
6217 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6220 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6224 @cindex @code{latex}
6225 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6226 It is rather similar to the default format but provides some braces needed
6227 by LaTeX for delimiting boxes and also converts some common objects to
6228 conventional LaTeX names. It is possible to give symbols a special name for
6229 LaTeX output by supplying it as a second argument to the @code{symbol}
6232 For example, the code snippet
6236 symbol x("x", "\\circ");
6237 ex e = lgamma(x).series(x==0,3);
6238 cout << latex << e << endl;
6245 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6246 +\mathcal@{O@}(\circ^@{3@})
6249 @cindex @code{index_dimensions}
6250 @cindex @code{no_index_dimensions}
6251 Index dimensions are normally hidden in the output. To make them visible, use
6252 the @code{index_dimensions} manipulator. The dimensions will be written in
6253 square brackets behind each index value in the default and LaTeX output
6258 symbol x("x"), y("y");
6259 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6260 ex e = indexed(x, mu) * indexed(y, nu);
6263 // prints 'x~mu*y~nu'
6264 cout << index_dimensions << e << endl;
6265 // prints 'x~mu[4]*y~nu[4]'
6266 cout << no_index_dimensions << e << endl;
6267 // prints 'x~mu*y~nu'
6272 @cindex Tree traversal
6273 If you need any fancy special output format, e.g. for interfacing GiNaC
6274 with other algebra systems or for producing code for different
6275 programming languages, you can always traverse the expression tree yourself:
6278 static void my_print(const ex & e)
6280 if (is_a<function>(e))
6281 cout << ex_to<function>(e).get_name();
6283 cout << ex_to<basic>(e).class_name();
6285 size_t n = e.nops();
6287 for (size_t i=0; i<n; i++) @{
6299 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6307 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6308 symbol(y))),numeric(-2)))
6311 If you need an output format that makes it possible to accurately
6312 reconstruct an expression by feeding the output to a suitable parser or
6313 object factory, you should consider storing the expression in an
6314 @code{archive} object and reading the object properties from there.
6315 See the section on archiving for more information.
6318 @subsection Expression input
6319 @cindex input of expressions
6321 GiNaC provides no way to directly read an expression from a stream because
6322 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6323 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6324 @code{y} you defined in your program and there is no way to specify the
6325 desired symbols to the @code{>>} stream input operator.
6327 Instead, GiNaC lets you construct an expression from a string, specifying the
6328 list of symbols to be used:
6332 symbol x("x"), y("y");
6333 ex e("2*x+sin(y)", lst(x, y));
6337 The input syntax is the same as that used by @command{ginsh} and the stream
6338 output operator @code{<<}. The symbols in the string are matched by name to
6339 the symbols in the list and if GiNaC encounters a symbol not specified in
6340 the list it will throw an exception.
6342 With this constructor, it's also easy to implement interactive GiNaC programs:
6347 #include <stdexcept>
6348 #include <ginac/ginac.h>
6349 using namespace std;
6350 using namespace GiNaC;
6357 cout << "Enter an expression containing 'x': ";
6362 cout << "The derivative of " << e << " with respect to x is ";
6363 cout << e.diff(x) << ".\n";
6364 @} catch (exception &p) @{
6365 cerr << p.what() << endl;
6371 @subsection Archiving
6372 @cindex @code{archive} (class)
6375 GiNaC allows creating @dfn{archives} of expressions which can be stored
6376 to or retrieved from files. To create an archive, you declare an object
6377 of class @code{archive} and archive expressions in it, giving each
6378 expression a unique name:
6382 using namespace std;
6383 #include <ginac/ginac.h>
6384 using namespace GiNaC;
6388 symbol x("x"), y("y"), z("z");
6390 ex foo = sin(x + 2*y) + 3*z + 41;
6394 a.archive_ex(foo, "foo");
6395 a.archive_ex(bar, "the second one");
6399 The archive can then be written to a file:
6403 ofstream out("foobar.gar");
6409 The file @file{foobar.gar} contains all information that is needed to
6410 reconstruct the expressions @code{foo} and @code{bar}.
6412 @cindex @command{viewgar}
6413 The tool @command{viewgar} that comes with GiNaC can be used to view
6414 the contents of GiNaC archive files:
6417 $ viewgar foobar.gar
6418 foo = 41+sin(x+2*y)+3*z
6419 the second one = 42+sin(x+2*y)+3*z
6422 The point of writing archive files is of course that they can later be
6428 ifstream in("foobar.gar");
6433 And the stored expressions can be retrieved by their name:
6440 ex ex1 = a2.unarchive_ex(syms, "foo");
6441 ex ex2 = a2.unarchive_ex(syms, "the second one");
6443 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6444 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6445 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6449 Note that you have to supply a list of the symbols which are to be inserted
6450 in the expressions. Symbols in archives are stored by their name only and
6451 if you don't specify which symbols you have, unarchiving the expression will
6452 create new symbols with that name. E.g. if you hadn't included @code{x} in
6453 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6454 have had no effect because the @code{x} in @code{ex1} would have been a
6455 different symbol than the @code{x} which was defined at the beginning of
6456 the program, although both would appear as @samp{x} when printed.
6458 You can also use the information stored in an @code{archive} object to
6459 output expressions in a format suitable for exact reconstruction. The
6460 @code{archive} and @code{archive_node} classes have a couple of member
6461 functions that let you access the stored properties:
6464 static void my_print2(const archive_node & n)
6467 n.find_string("class", class_name);
6468 cout << class_name << "(";
6470 archive_node::propinfovector p;
6471 n.get_properties(p);
6473 size_t num = p.size();
6474 for (size_t i=0; i<num; i++) @{
6475 const string &name = p[i].name;
6476 if (name == "class")
6478 cout << name << "=";
6480 unsigned count = p[i].count;
6484 for (unsigned j=0; j<count; j++) @{
6485 switch (p[i].type) @{
6486 case archive_node::PTYPE_BOOL: @{
6488 n.find_bool(name, x, j);
6489 cout << (x ? "true" : "false");
6492 case archive_node::PTYPE_UNSIGNED: @{
6494 n.find_unsigned(name, x, j);
6498 case archive_node::PTYPE_STRING: @{
6500 n.find_string(name, x, j);
6501 cout << '\"' << x << '\"';
6504 case archive_node::PTYPE_NODE: @{
6505 const archive_node &x = n.find_ex_node(name, j);
6527 ex e = pow(2, x) - y;
6529 my_print2(ar.get_top_node(0)); cout << endl;
6537 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6538 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6539 overall_coeff=numeric(number="0"))
6542 Be warned, however, that the set of properties and their meaning for each
6543 class may change between GiNaC versions.
6546 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6547 @c node-name, next, previous, up
6548 @chapter Extending GiNaC
6550 By reading so far you should have gotten a fairly good understanding of
6551 GiNaC's design patterns. From here on you should start reading the
6552 sources. All we can do now is issue some recommendations how to tackle
6553 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6554 develop some useful extension please don't hesitate to contact the GiNaC
6555 authors---they will happily incorporate them into future versions.
6558 * What does not belong into GiNaC:: What to avoid.
6559 * Symbolic functions:: Implementing symbolic functions.
6560 * Printing:: Adding new output formats.
6561 * Structures:: Defining new algebraic classes (the easy way).
6562 * Adding classes:: Defining new algebraic classes (the hard way).
6566 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6567 @c node-name, next, previous, up
6568 @section What doesn't belong into GiNaC
6570 @cindex @command{ginsh}
6571 First of all, GiNaC's name must be read literally. It is designed to be
6572 a library for use within C++. The tiny @command{ginsh} accompanying
6573 GiNaC makes this even more clear: it doesn't even attempt to provide a
6574 language. There are no loops or conditional expressions in
6575 @command{ginsh}, it is merely a window into the library for the
6576 programmer to test stuff (or to show off). Still, the design of a
6577 complete CAS with a language of its own, graphical capabilities and all
6578 this on top of GiNaC is possible and is without doubt a nice project for
6581 There are many built-in functions in GiNaC that do not know how to
6582 evaluate themselves numerically to a precision declared at runtime
6583 (using @code{Digits}). Some may be evaluated at certain points, but not
6584 generally. This ought to be fixed. However, doing numerical
6585 computations with GiNaC's quite abstract classes is doomed to be
6586 inefficient. For this purpose, the underlying foundation classes
6587 provided by CLN are much better suited.
6590 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6591 @c node-name, next, previous, up
6592 @section Symbolic functions
6594 The easiest and most instructive way to start extending GiNaC is probably to
6595 create your own symbolic functions. These are implemented with the help of
6596 two preprocessor macros:
6598 @cindex @code{DECLARE_FUNCTION}
6599 @cindex @code{REGISTER_FUNCTION}
6601 DECLARE_FUNCTION_<n>P(<name>)
6602 REGISTER_FUNCTION(<name>, <options>)
6605 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6606 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6607 parameters of type @code{ex} and returns a newly constructed GiNaC
6608 @code{function} object that represents your function.
6610 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6611 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6612 set of options that associate the symbolic function with C++ functions you
6613 provide to implement the various methods such as evaluation, derivative,
6614 series expansion etc. They also describe additional attributes the function
6615 might have, such as symmetry and commutation properties, and a name for
6616 LaTeX output. Multiple options are separated by the member access operator
6617 @samp{.} and can be given in an arbitrary order.
6619 (By the way: in case you are worrying about all the macros above we can
6620 assure you that functions are GiNaC's most macro-intense classes. We have
6621 done our best to avoid macros where we can.)
6623 @subsection A minimal example
6625 Here is an example for the implementation of a function with two arguments
6626 that is not further evaluated:
6629 DECLARE_FUNCTION_2P(myfcn)
6631 REGISTER_FUNCTION(myfcn, dummy())
6634 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6635 in algebraic expressions:
6641 ex e = 2*myfcn(42, 1+3*x) - x;
6643 // prints '2*myfcn(42,1+3*x)-x'
6648 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6649 "no options". A function with no options specified merely acts as a kind of
6650 container for its arguments. It is a pure "dummy" function with no associated
6651 logic (which is, however, sometimes perfectly sufficient).
6653 Let's now have a look at the implementation of GiNaC's cosine function for an
6654 example of how to make an "intelligent" function.
6656 @subsection The cosine function
6658 The GiNaC header file @file{inifcns.h} contains the line
6661 DECLARE_FUNCTION_1P(cos)
6664 which declares to all programs using GiNaC that there is a function @samp{cos}
6665 that takes one @code{ex} as an argument. This is all they need to know to use
6666 this function in expressions.
6668 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6669 is its @code{REGISTER_FUNCTION} line:
6672 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6673 evalf_func(cos_evalf).
6674 derivative_func(cos_deriv).
6675 latex_name("\\cos"));
6678 There are four options defined for the cosine function. One of them
6679 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6680 other three indicate the C++ functions in which the "brains" of the cosine
6681 function are defined.
6683 @cindex @code{hold()}
6685 The @code{eval_func()} option specifies the C++ function that implements
6686 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6687 the same number of arguments as the associated symbolic function (one in this
6688 case) and returns the (possibly transformed or in some way simplified)
6689 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6690 of the automatic evaluation process). If no (further) evaluation is to take
6691 place, the @code{eval_func()} function must return the original function
6692 with @code{.hold()}, to avoid a potential infinite recursion. If your
6693 symbolic functions produce a segmentation fault or stack overflow when
6694 using them in expressions, you are probably missing a @code{.hold()}
6697 The @code{eval_func()} function for the cosine looks something like this
6698 (actually, it doesn't look like this at all, but it should give you an idea
6702 static ex cos_eval(const ex & x)
6704 if ("x is a multiple of 2*Pi")
6706 else if ("x is a multiple of Pi")
6708 else if ("x is a multiple of Pi/2")
6712 else if ("x has the form 'acos(y)'")
6714 else if ("x has the form 'asin(y)'")
6719 return cos(x).hold();
6723 This function is called every time the cosine is used in a symbolic expression:
6729 // this calls cos_eval(Pi), and inserts its return value into
6730 // the actual expression
6737 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6738 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6739 symbolic transformation can be done, the unmodified function is returned
6740 with @code{.hold()}.
6742 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6743 The user has to call @code{evalf()} for that. This is implemented in a
6747 static ex cos_evalf(const ex & x)
6749 if (is_a<numeric>(x))
6750 return cos(ex_to<numeric>(x));
6752 return cos(x).hold();
6756 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6757 in this case the @code{cos()} function for @code{numeric} objects, which in
6758 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6759 isn't really needed here, but reminds us that the corresponding @code{eval()}
6760 function would require it in this place.
6762 Differentiation will surely turn up and so we need to tell @code{cos}
6763 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6764 instance, are then handled automatically by @code{basic::diff} and
6768 static ex cos_deriv(const ex & x, unsigned diff_param)
6774 @cindex product rule
6775 The second parameter is obligatory but uninteresting at this point. It
6776 specifies which parameter to differentiate in a partial derivative in
6777 case the function has more than one parameter, and its main application
6778 is for correct handling of the chain rule.
6780 An implementation of the series expansion is not needed for @code{cos()} as
6781 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6782 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6783 the other hand, does have poles and may need to do Laurent expansion:
6786 static ex tan_series(const ex & x, const relational & rel,
6787 int order, unsigned options)
6789 // Find the actual expansion point
6790 const ex x_pt = x.subs(rel);
6792 if ("x_pt is not an odd multiple of Pi/2")
6793 throw do_taylor(); // tell function::series() to do Taylor expansion
6795 // On a pole, expand sin()/cos()
6796 return (sin(x)/cos(x)).series(rel, order+2, options);
6800 The @code{series()} implementation of a function @emph{must} return a
6801 @code{pseries} object, otherwise your code will crash.
6803 @subsection Function options
6805 GiNaC functions understand several more options which are always
6806 specified as @code{.option(params)}. None of them are required, but you
6807 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6808 is a do-nothing option called @code{dummy()} which you can use to define
6809 functions without any special options.
6812 eval_func(<C++ function>)
6813 evalf_func(<C++ function>)
6814 derivative_func(<C++ function>)
6815 series_func(<C++ function>)
6816 conjugate_func(<C++ function>)
6819 These specify the C++ functions that implement symbolic evaluation,
6820 numeric evaluation, partial derivatives, and series expansion, respectively.
6821 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6822 @code{diff()} and @code{series()}.
6824 The @code{eval_func()} function needs to use @code{.hold()} if no further
6825 automatic evaluation is desired or possible.
6827 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6828 expansion, which is correct if there are no poles involved. If the function
6829 has poles in the complex plane, the @code{series_func()} needs to check
6830 whether the expansion point is on a pole and fall back to Taylor expansion
6831 if it isn't. Otherwise, the pole usually needs to be regularized by some
6832 suitable transformation.
6835 latex_name(const string & n)
6838 specifies the LaTeX code that represents the name of the function in LaTeX
6839 output. The default is to put the function name in an @code{\mbox@{@}}.
6842 do_not_evalf_params()
6845 This tells @code{evalf()} to not recursively evaluate the parameters of the
6846 function before calling the @code{evalf_func()}.
6849 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6852 This allows you to explicitly specify the commutation properties of the
6853 function (@xref{Non-commutative objects}, for an explanation of
6854 (non)commutativity in GiNaC). For example, you can use
6855 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6856 GiNaC treat your function like a matrix. By default, functions inherit the
6857 commutation properties of their first argument.
6860 set_symmetry(const symmetry & s)
6863 specifies the symmetry properties of the function with respect to its
6864 arguments. @xref{Indexed objects}, for an explanation of symmetry
6865 specifications. GiNaC will automatically rearrange the arguments of
6866 symmetric functions into a canonical order.
6868 Sometimes you may want to have finer control over how functions are
6869 displayed in the output. For example, the @code{abs()} function prints
6870 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6871 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6875 print_func<C>(<C++ function>)
6878 option which is explained in the next section.
6880 @subsection Functions with a variable number of arguments
6882 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6883 functions with a fixed number of arguments. Sometimes, though, you may need
6884 to have a function that accepts a variable number of expressions. One way to
6885 accomplish this is to pass variable-length lists as arguments. The
6886 @code{Li()} function uses this method for multiple polylogarithms.
6888 It is also possible to define functions that accept a different number of
6889 parameters under the same function name, such as the @code{psi()} function
6890 which can be called either as @code{psi(z)} (the digamma function) or as
6891 @code{psi(n, z)} (polygamma functions). These are actually two different
6892 functions in GiNaC that, however, have the same name. Defining such
6893 functions is not possible with the macros but requires manually fiddling
6894 with GiNaC internals. If you are interested, please consult the GiNaC source
6895 code for the @code{psi()} function (@file{inifcns.h} and
6896 @file{inifcns_gamma.cpp}).
6899 @node Printing, Structures, Symbolic functions, Extending GiNaC
6900 @c node-name, next, previous, up
6901 @section GiNaC's expression output system
6903 GiNaC allows the output of expressions in a variety of different formats
6904 (@pxref{Input/output}). This section will explain how expression output
6905 is implemented internally, and how to define your own output formats or
6906 change the output format of built-in algebraic objects. You will also want
6907 to read this section if you plan to write your own algebraic classes or
6910 @cindex @code{print_context} (class)
6911 @cindex @code{print_dflt} (class)
6912 @cindex @code{print_latex} (class)
6913 @cindex @code{print_tree} (class)
6914 @cindex @code{print_csrc} (class)
6915 All the different output formats are represented by a hierarchy of classes
6916 rooted in the @code{print_context} class, defined in the @file{print.h}
6921 the default output format
6923 output in LaTeX mathematical mode
6925 a dump of the internal expression structure (for debugging)
6927 the base class for C source output
6928 @item print_csrc_float
6929 C source output using the @code{float} type
6930 @item print_csrc_double
6931 C source output using the @code{double} type
6932 @item print_csrc_cl_N
6933 C source output using CLN types
6936 The @code{print_context} base class provides two public data members:
6948 @code{s} is a reference to the stream to output to, while @code{options}
6949 holds flags and modifiers. Currently, there is only one flag defined:
6950 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6951 to print the index dimension which is normally hidden.
6953 When you write something like @code{std::cout << e}, where @code{e} is
6954 an object of class @code{ex}, GiNaC will construct an appropriate
6955 @code{print_context} object (of a class depending on the selected output
6956 format), fill in the @code{s} and @code{options} members, and call
6958 @cindex @code{print()}
6960 void ex::print(const print_context & c, unsigned level = 0) const;
6963 which in turn forwards the call to the @code{print()} method of the
6964 top-level algebraic object contained in the expression.
6966 Unlike other methods, GiNaC classes don't usually override their
6967 @code{print()} method to implement expression output. Instead, the default
6968 implementation @code{basic::print(c, level)} performs a run-time double
6969 dispatch to a function selected by the dynamic type of the object and the
6970 passed @code{print_context}. To this end, GiNaC maintains a separate method
6971 table for each class, similar to the virtual function table used for ordinary
6972 (single) virtual function dispatch.
6974 The method table contains one slot for each possible @code{print_context}
6975 type, indexed by the (internally assigned) serial number of the type. Slots
6976 may be empty, in which case GiNaC will retry the method lookup with the
6977 @code{print_context} object's parent class, possibly repeating the process
6978 until it reaches the @code{print_context} base class. If there's still no
6979 method defined, the method table of the algebraic object's parent class
6980 is consulted, and so on, until a matching method is found (eventually it
6981 will reach the combination @code{basic/print_context}, which prints the
6982 object's class name enclosed in square brackets).
6984 You can think of the print methods of all the different classes and output
6985 formats as being arranged in a two-dimensional matrix with one axis listing
6986 the algebraic classes and the other axis listing the @code{print_context}
6989 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6990 to implement printing, but then they won't get any of the benefits of the
6991 double dispatch mechanism (such as the ability for derived classes to
6992 inherit only certain print methods from its parent, or the replacement of
6993 methods at run-time).
6995 @subsection Print methods for classes
6997 The method table for a class is set up either in the definition of the class,
6998 by passing the appropriate @code{print_func<C>()} option to
6999 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7000 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7001 can also be used to override existing methods dynamically.
7003 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7004 be a member function of the class (or one of its parent classes), a static
7005 member function, or an ordinary (global) C++ function. The @code{C} template
7006 parameter specifies the appropriate @code{print_context} type for which the
7007 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7008 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7009 the class is the one being implemented by
7010 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7012 For print methods that are member functions, their first argument must be of
7013 a type convertible to a @code{const C &}, and the second argument must be an
7016 For static members and global functions, the first argument must be of a type
7017 convertible to a @code{const T &}, the second argument must be of a type
7018 convertible to a @code{const C &}, and the third argument must be an
7019 @code{unsigned}. A global function will, of course, not have access to
7020 private and protected members of @code{T}.
7022 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7023 and @code{basic::print()}) is used for proper parenthesizing of the output
7024 (and by @code{print_tree} for proper indentation). It can be used for similar
7025 purposes if you write your own output formats.
7027 The explanations given above may seem complicated, but in practice it's
7028 really simple, as shown in the following example. Suppose that we want to
7029 display exponents in LaTeX output not as superscripts but with little
7030 upwards-pointing arrows. This can be achieved in the following way:
7033 void my_print_power_as_latex(const power & p,
7034 const print_latex & c,
7037 // get the precedence of the 'power' class
7038 unsigned power_prec = p.precedence();
7040 // if the parent operator has the same or a higher precedence
7041 // we need parentheses around the power
7042 if (level >= power_prec)
7045 // print the basis and exponent, each enclosed in braces, and
7046 // separated by an uparrow
7048 p.op(0).print(c, power_prec);
7049 c.s << "@}\\uparrow@{";
7050 p.op(1).print(c, power_prec);
7053 // don't forget the closing parenthesis
7054 if (level >= power_prec)
7060 // a sample expression
7061 symbol x("x"), y("y");
7062 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7064 // switch to LaTeX mode
7067 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7070 // now we replace the method for the LaTeX output of powers with
7072 set_print_func<power, print_latex>(my_print_power_as_latex);
7074 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7085 The first argument of @code{my_print_power_as_latex} could also have been
7086 a @code{const basic &}, the second one a @code{const print_context &}.
7089 The above code depends on @code{mul} objects converting their operands to
7090 @code{power} objects for the purpose of printing.
7093 The output of products including negative powers as fractions is also
7094 controlled by the @code{mul} class.
7097 The @code{power/print_latex} method provided by GiNaC prints square roots
7098 using @code{\sqrt}, but the above code doesn't.
7102 It's not possible to restore a method table entry to its previous or default
7103 value. Once you have called @code{set_print_func()}, you can only override
7104 it with another call to @code{set_print_func()}, but you can't easily go back
7105 to the default behavior again (you can, of course, dig around in the GiNaC
7106 sources, find the method that is installed at startup
7107 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7108 one; that is, after you circumvent the C++ member access control@dots{}).
7110 @subsection Print methods for functions
7112 Symbolic functions employ a print method dispatch mechanism similar to the
7113 one used for classes. The methods are specified with @code{print_func<C>()}
7114 function options. If you don't specify any special print methods, the function
7115 will be printed with its name (or LaTeX name, if supplied), followed by a
7116 comma-separated list of arguments enclosed in parentheses.
7118 For example, this is what GiNaC's @samp{abs()} function is defined like:
7121 static ex abs_eval(const ex & arg) @{ ... @}
7122 static ex abs_evalf(const ex & arg) @{ ... @}
7124 static void abs_print_latex(const ex & arg, const print_context & c)
7126 c.s << "@{|"; arg.print(c); c.s << "|@}";
7129 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7131 c.s << "fabs("; arg.print(c); c.s << ")";
7134 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7135 evalf_func(abs_evalf).
7136 print_func<print_latex>(abs_print_latex).
7137 print_func<print_csrc_float>(abs_print_csrc_float).
7138 print_func<print_csrc_double>(abs_print_csrc_float));
7141 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7142 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7144 There is currently no equivalent of @code{set_print_func()} for functions.
7146 @subsection Adding new output formats
7148 Creating a new output format involves subclassing @code{print_context},
7149 which is somewhat similar to adding a new algebraic class
7150 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7151 that needs to go into the class definition, and a corresponding macro
7152 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7153 Every @code{print_context} class needs to provide a default constructor
7154 and a constructor from an @code{std::ostream} and an @code{unsigned}
7157 Here is an example for a user-defined @code{print_context} class:
7160 class print_myformat : public print_dflt
7162 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7164 print_myformat(std::ostream & os, unsigned opt = 0)
7165 : print_dflt(os, opt) @{@}
7168 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7170 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7173 That's all there is to it. None of the actual expression output logic is
7174 implemented in this class. It merely serves as a selector for choosing
7175 a particular format. The algorithms for printing expressions in the new
7176 format are implemented as print methods, as described above.
7178 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7179 exactly like GiNaC's default output format:
7184 ex e = pow(x, 2) + 1;
7186 // this prints "1+x^2"
7189 // this also prints "1+x^2"
7190 e.print(print_myformat()); cout << endl;
7196 To fill @code{print_myformat} with life, we need to supply appropriate
7197 print methods with @code{set_print_func()}, like this:
7200 // This prints powers with '**' instead of '^'. See the LaTeX output
7201 // example above for explanations.
7202 void print_power_as_myformat(const power & p,
7203 const print_myformat & c,
7206 unsigned power_prec = p.precedence();
7207 if (level >= power_prec)
7209 p.op(0).print(c, power_prec);
7211 p.op(1).print(c, power_prec);
7212 if (level >= power_prec)
7218 // install a new print method for power objects
7219 set_print_func<power, print_myformat>(print_power_as_myformat);
7221 // now this prints "1+x**2"
7222 e.print(print_myformat()); cout << endl;
7224 // but the default format is still "1+x^2"
7230 @node Structures, Adding classes, Printing, Extending GiNaC
7231 @c node-name, next, previous, up
7234 If you are doing some very specialized things with GiNaC, or if you just
7235 need some more organized way to store data in your expressions instead of
7236 anonymous lists, you may want to implement your own algebraic classes.
7237 ('algebraic class' means any class directly or indirectly derived from
7238 @code{basic} that can be used in GiNaC expressions).
7240 GiNaC offers two ways of accomplishing this: either by using the
7241 @code{structure<T>} template class, or by rolling your own class from
7242 scratch. This section will discuss the @code{structure<T>} template which
7243 is easier to use but more limited, while the implementation of custom
7244 GiNaC classes is the topic of the next section. However, you may want to
7245 read both sections because many common concepts and member functions are
7246 shared by both concepts, and it will also allow you to decide which approach
7247 is most suited to your needs.
7249 The @code{structure<T>} template, defined in the GiNaC header file
7250 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7251 or @code{class}) into a GiNaC object that can be used in expressions.
7253 @subsection Example: scalar products
7255 Let's suppose that we need a way to handle some kind of abstract scalar
7256 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7257 product class have to store their left and right operands, which can in turn
7258 be arbitrary expressions. Here is a possible way to represent such a
7259 product in a C++ @code{struct}:
7263 using namespace std;
7265 #include <ginac/ginac.h>
7266 using namespace GiNaC;
7272 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7276 The default constructor is required. Now, to make a GiNaC class out of this
7277 data structure, we need only one line:
7280 typedef structure<sprod_s> sprod;
7283 That's it. This line constructs an algebraic class @code{sprod} which
7284 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7285 expressions like any other GiNaC class:
7289 symbol a("a"), b("b");
7290 ex e = sprod(sprod_s(a, b));
7294 Note the difference between @code{sprod} which is the algebraic class, and
7295 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7296 and @code{right} data members. As shown above, an @code{sprod} can be
7297 constructed from an @code{sprod_s} object.
7299 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7300 you could define a little wrapper function like this:
7303 inline ex make_sprod(ex left, ex right)
7305 return sprod(sprod_s(left, right));
7309 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7310 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7311 @code{get_struct()}:
7315 cout << ex_to<sprod>(e)->left << endl;
7317 cout << ex_to<sprod>(e).get_struct().right << endl;
7322 You only have read access to the members of @code{sprod_s}.
7324 The type definition of @code{sprod} is enough to write your own algorithms
7325 that deal with scalar products, for example:
7330 if (is_a<sprod>(p)) @{
7331 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7332 return make_sprod(sp.right, sp.left);
7343 @subsection Structure output
7345 While the @code{sprod} type is useable it still leaves something to be
7346 desired, most notably proper output:
7351 // -> [structure object]
7355 By default, any structure types you define will be printed as
7356 @samp{[structure object]}. To override this you can either specialize the
7357 template's @code{print()} member function, or specify print methods with
7358 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7359 it's not possible to supply class options like @code{print_func<>()} to
7360 structures, so for a self-contained structure type you need to resort to
7361 overriding the @code{print()} function, which is also what we will do here.
7363 The member functions of GiNaC classes are described in more detail in the
7364 next section, but it shouldn't be hard to figure out what's going on here:
7367 void sprod::print(const print_context & c, unsigned level) const
7369 // tree debug output handled by superclass
7370 if (is_a<print_tree>(c))
7371 inherited::print(c, level);
7373 // get the contained sprod_s object
7374 const sprod_s & sp = get_struct();
7376 // print_context::s is a reference to an ostream
7377 c.s << "<" << sp.left << "|" << sp.right << ">";
7381 Now we can print expressions containing scalar products:
7387 cout << swap_sprod(e) << endl;
7392 @subsection Comparing structures
7394 The @code{sprod} class defined so far still has one important drawback: all
7395 scalar products are treated as being equal because GiNaC doesn't know how to
7396 compare objects of type @code{sprod_s}. This can lead to some confusing
7397 and undesired behavior:
7401 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7403 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7404 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7408 To remedy this, we first need to define the operators @code{==} and @code{<}
7409 for objects of type @code{sprod_s}:
7412 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7414 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7417 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7419 return lhs.left.compare(rhs.left) < 0
7420 ? true : lhs.right.compare(rhs.right) < 0;
7424 The ordering established by the @code{<} operator doesn't have to make any
7425 algebraic sense, but it needs to be well defined. Note that we can't use
7426 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7427 in the implementation of these operators because they would construct
7428 GiNaC @code{relational} objects which in the case of @code{<} do not
7429 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7430 decide which one is algebraically 'less').
7432 Next, we need to change our definition of the @code{sprod} type to let
7433 GiNaC know that an ordering relation exists for the embedded objects:
7436 typedef structure<sprod_s, compare_std_less> sprod;
7439 @code{sprod} objects then behave as expected:
7443 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7444 // -> <a|b>-<a^2|b^2>
7445 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7446 // -> <a|b>+<a^2|b^2>
7447 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7449 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7454 The @code{compare_std_less} policy parameter tells GiNaC to use the
7455 @code{std::less} and @code{std::equal_to} functors to compare objects of
7456 type @code{sprod_s}. By default, these functors forward their work to the
7457 standard @code{<} and @code{==} operators, which we have overloaded.
7458 Alternatively, we could have specialized @code{std::less} and
7459 @code{std::equal_to} for class @code{sprod_s}.
7461 GiNaC provides two other comparison policies for @code{structure<T>}
7462 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7463 which does a bit-wise comparison of the contained @code{T} objects.
7464 This should be used with extreme care because it only works reliably with
7465 built-in integral types, and it also compares any padding (filler bytes of
7466 undefined value) that the @code{T} class might have.
7468 @subsection Subexpressions
7470 Our scalar product class has two subexpressions: the left and right
7471 operands. It might be a good idea to make them accessible via the standard
7472 @code{nops()} and @code{op()} methods:
7475 size_t sprod::nops() const
7480 ex sprod::op(size_t i) const
7484 return get_struct().left;
7486 return get_struct().right;
7488 throw std::range_error("sprod::op(): no such operand");
7493 Implementing @code{nops()} and @code{op()} for container types such as
7494 @code{sprod} has two other nice side effects:
7498 @code{has()} works as expected
7500 GiNaC generates better hash keys for the objects (the default implementation
7501 of @code{calchash()} takes subexpressions into account)
7504 @cindex @code{let_op()}
7505 There is a non-const variant of @code{op()} called @code{let_op()} that
7506 allows replacing subexpressions:
7509 ex & sprod::let_op(size_t i)
7511 // every non-const member function must call this
7512 ensure_if_modifiable();
7516 return get_struct().left;
7518 return get_struct().right;
7520 throw std::range_error("sprod::let_op(): no such operand");
7525 Once we have provided @code{let_op()} we also get @code{subs()} and
7526 @code{map()} for free. In fact, every container class that returns a non-null
7527 @code{nops()} value must either implement @code{let_op()} or provide custom
7528 implementations of @code{subs()} and @code{map()}.
7530 In turn, the availability of @code{map()} enables the recursive behavior of a
7531 couple of other default method implementations, in particular @code{evalf()},
7532 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7533 we probably want to provide our own version of @code{expand()} for scalar
7534 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7535 This is left as an exercise for the reader.
7537 The @code{structure<T>} template defines many more member functions that
7538 you can override by specialization to customize the behavior of your
7539 structures. You are referred to the next section for a description of
7540 some of these (especially @code{eval()}). There is, however, one topic
7541 that shall be addressed here, as it demonstrates one peculiarity of the
7542 @code{structure<T>} template: archiving.
7544 @subsection Archiving structures
7546 If you don't know how the archiving of GiNaC objects is implemented, you
7547 should first read the next section and then come back here. You're back?
7550 To implement archiving for structures it is not enough to provide
7551 specializations for the @code{archive()} member function and the
7552 unarchiving constructor (the @code{unarchive()} function has a default
7553 implementation). You also need to provide a unique name (as a string literal)
7554 for each structure type you define. This is because in GiNaC archives,
7555 the class of an object is stored as a string, the class name.
7557 By default, this class name (as returned by the @code{class_name()} member
7558 function) is @samp{structure} for all structure classes. This works as long
7559 as you have only defined one structure type, but if you use two or more you
7560 need to provide a different name for each by specializing the
7561 @code{get_class_name()} member function. Here is a sample implementation
7562 for enabling archiving of the scalar product type defined above:
7565 const char *sprod::get_class_name() @{ return "sprod"; @}
7567 void sprod::archive(archive_node & n) const
7569 inherited::archive(n);
7570 n.add_ex("left", get_struct().left);
7571 n.add_ex("right", get_struct().right);
7574 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7576 n.find_ex("left", get_struct().left, sym_lst);
7577 n.find_ex("right", get_struct().right, sym_lst);
7581 Note that the unarchiving constructor is @code{sprod::structure} and not
7582 @code{sprod::sprod}, and that we don't need to supply an
7583 @code{sprod::unarchive()} function.
7586 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7587 @c node-name, next, previous, up
7588 @section Adding classes
7590 The @code{structure<T>} template provides an way to extend GiNaC with custom
7591 algebraic classes that is easy to use but has its limitations, the most
7592 severe of which being that you can't add any new member functions to
7593 structures. To be able to do this, you need to write a new class definition
7596 This section will explain how to implement new algebraic classes in GiNaC by
7597 giving the example of a simple 'string' class. After reading this section
7598 you will know how to properly declare a GiNaC class and what the minimum
7599 required member functions are that you have to implement. We only cover the
7600 implementation of a 'leaf' class here (i.e. one that doesn't contain
7601 subexpressions). Creating a container class like, for example, a class
7602 representing tensor products is more involved but this section should give
7603 you enough information so you can consult the source to GiNaC's predefined
7604 classes if you want to implement something more complicated.
7606 @subsection GiNaC's run-time type information system
7608 @cindex hierarchy of classes
7610 All algebraic classes (that is, all classes that can appear in expressions)
7611 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7612 @code{basic *} (which is essentially what an @code{ex} is) represents a
7613 generic pointer to an algebraic class. Occasionally it is necessary to find
7614 out what the class of an object pointed to by a @code{basic *} really is.
7615 Also, for the unarchiving of expressions it must be possible to find the
7616 @code{unarchive()} function of a class given the class name (as a string). A
7617 system that provides this kind of information is called a run-time type
7618 information (RTTI) system. The C++ language provides such a thing (see the
7619 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7620 implements its own, simpler RTTI.
7622 The RTTI in GiNaC is based on two mechanisms:
7627 The @code{basic} class declares a member variable @code{tinfo_key} which
7628 holds an unsigned integer that identifies the object's class. These numbers
7629 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7630 classes. They all start with @code{TINFO_}.
7633 By means of some clever tricks with static members, GiNaC maintains a list
7634 of information for all classes derived from @code{basic}. The information
7635 available includes the class names, the @code{tinfo_key}s, and pointers
7636 to the unarchiving functions. This class registry is defined in the
7637 @file{registrar.h} header file.
7641 The disadvantage of this proprietary RTTI implementation is that there's
7642 a little more to do when implementing new classes (C++'s RTTI works more
7643 or less automatically) but don't worry, most of the work is simplified by
7646 @subsection A minimalistic example
7648 Now we will start implementing a new class @code{mystring} that allows
7649 placing character strings in algebraic expressions (this is not very useful,
7650 but it's just an example). This class will be a direct subclass of
7651 @code{basic}. You can use this sample implementation as a starting point
7652 for your own classes.
7654 The code snippets given here assume that you have included some header files
7660 #include <stdexcept>
7661 using namespace std;
7663 #include <ginac/ginac.h>
7664 using namespace GiNaC;
7667 The first thing we have to do is to define a @code{tinfo_key} for our new
7668 class. This can be any arbitrary unsigned number that is not already taken
7669 by one of the existing classes but it's better to come up with something
7670 that is unlikely to clash with keys that might be added in the future. The
7671 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7672 which is not a requirement but we are going to stick with this scheme:
7675 const unsigned TINFO_mystring = 0x42420001U;
7678 Now we can write down the class declaration. The class stores a C++
7679 @code{string} and the user shall be able to construct a @code{mystring}
7680 object from a C or C++ string:
7683 class mystring : public basic
7685 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7688 mystring(const string &s);
7689 mystring(const char *s);
7695 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7698 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7699 macros are defined in @file{registrar.h}. They take the name of the class
7700 and its direct superclass as arguments and insert all required declarations
7701 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7702 the first line after the opening brace of the class definition. The
7703 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7704 source (at global scope, of course, not inside a function).
7706 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7707 declarations of the default constructor and a couple of other functions that
7708 are required. It also defines a type @code{inherited} which refers to the
7709 superclass so you don't have to modify your code every time you shuffle around
7710 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7711 class with the GiNaC RTTI (there is also a
7712 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7713 options for the class, and which we will be using instead in a few minutes).
7715 Now there are seven member functions we have to implement to get a working
7721 @code{mystring()}, the default constructor.
7724 @code{void archive(archive_node &n)}, the archiving function. This stores all
7725 information needed to reconstruct an object of this class inside an
7726 @code{archive_node}.
7729 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7730 constructor. This constructs an instance of the class from the information
7731 found in an @code{archive_node}.
7734 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7735 unarchiving function. It constructs a new instance by calling the unarchiving
7739 @cindex @code{compare_same_type()}
7740 @code{int compare_same_type(const basic &other)}, which is used internally
7741 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7742 -1, depending on the relative order of this object and the @code{other}
7743 object. If it returns 0, the objects are considered equal.
7744 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7745 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7746 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7747 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7748 must provide a @code{compare_same_type()} function, even those representing
7749 objects for which no reasonable algebraic ordering relationship can be
7753 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7754 which are the two constructors we declared.
7758 Let's proceed step-by-step. The default constructor looks like this:
7761 mystring::mystring() : inherited(TINFO_mystring) @{@}
7764 The golden rule is that in all constructors you have to set the
7765 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7766 it will be set by the constructor of the superclass and all hell will break
7767 loose in the RTTI. For your convenience, the @code{basic} class provides
7768 a constructor that takes a @code{tinfo_key} value, which we are using here
7769 (remember that in our case @code{inherited == basic}). If the superclass
7770 didn't have such a constructor, we would have to set the @code{tinfo_key}
7771 to the right value manually.
7773 In the default constructor you should set all other member variables to
7774 reasonable default values (we don't need that here since our @code{str}
7775 member gets set to an empty string automatically).
7777 Next are the three functions for archiving. You have to implement them even
7778 if you don't plan to use archives, but the minimum required implementation
7779 is really simple. First, the archiving function:
7782 void mystring::archive(archive_node &n) const
7784 inherited::archive(n);
7785 n.add_string("string", str);
7789 The only thing that is really required is calling the @code{archive()}
7790 function of the superclass. Optionally, you can store all information you
7791 deem necessary for representing the object into the passed
7792 @code{archive_node}. We are just storing our string here. For more
7793 information on how the archiving works, consult the @file{archive.h} header
7796 The unarchiving constructor is basically the inverse of the archiving
7800 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7802 n.find_string("string", str);
7806 If you don't need archiving, just leave this function empty (but you must
7807 invoke the unarchiving constructor of the superclass). Note that we don't
7808 have to set the @code{tinfo_key} here because it is done automatically
7809 by the unarchiving constructor of the @code{basic} class.
7811 Finally, the unarchiving function:
7814 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7816 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7820 You don't have to understand how exactly this works. Just copy these
7821 four lines into your code literally (replacing the class name, of
7822 course). It calls the unarchiving constructor of the class and unless
7823 you are doing something very special (like matching @code{archive_node}s
7824 to global objects) you don't need a different implementation. For those
7825 who are interested: setting the @code{dynallocated} flag puts the object
7826 under the control of GiNaC's garbage collection. It will get deleted
7827 automatically once it is no longer referenced.
7829 Our @code{compare_same_type()} function uses a provided function to compare
7833 int mystring::compare_same_type(const basic &other) const
7835 const mystring &o = static_cast<const mystring &>(other);
7836 int cmpval = str.compare(o.str);
7839 else if (cmpval < 0)
7846 Although this function takes a @code{basic &}, it will always be a reference
7847 to an object of exactly the same class (objects of different classes are not
7848 comparable), so the cast is safe. If this function returns 0, the two objects
7849 are considered equal (in the sense that @math{A-B=0}), so you should compare
7850 all relevant member variables.
7852 Now the only thing missing is our two new constructors:
7855 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7856 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7859 No surprises here. We set the @code{str} member from the argument and
7860 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7862 That's it! We now have a minimal working GiNaC class that can store
7863 strings in algebraic expressions. Let's confirm that the RTTI works:
7866 ex e = mystring("Hello, world!");
7867 cout << is_a<mystring>(e) << endl;
7870 cout << ex_to<basic>(e).class_name() << endl;
7874 Obviously it does. Let's see what the expression @code{e} looks like:
7878 // -> [mystring object]
7881 Hm, not exactly what we expect, but of course the @code{mystring} class
7882 doesn't yet know how to print itself. This can be done either by implementing
7883 the @code{print()} member function, or, preferably, by specifying a
7884 @code{print_func<>()} class option. Let's say that we want to print the string
7885 surrounded by double quotes:
7888 class mystring : public basic
7892 void do_print(const print_context &c, unsigned level = 0) const;
7896 void mystring::do_print(const print_context &c, unsigned level) const
7898 // print_context::s is a reference to an ostream
7899 c.s << '\"' << str << '\"';
7903 The @code{level} argument is only required for container classes to
7904 correctly parenthesize the output.
7906 Now we need to tell GiNaC that @code{mystring} objects should use the
7907 @code{do_print()} member function for printing themselves. For this, we
7911 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7917 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7918 print_func<print_context>(&mystring::do_print))
7921 Let's try again to print the expression:
7925 // -> "Hello, world!"
7928 Much better. If we wanted to have @code{mystring} objects displayed in a
7929 different way depending on the output format (default, LaTeX, etc.), we
7930 would have supplied multiple @code{print_func<>()} options with different
7931 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7932 separated by dots. This is similar to the way options are specified for
7933 symbolic functions. @xref{Printing}, for a more in-depth description of the
7934 way expression output is implemented in GiNaC.
7936 The @code{mystring} class can be used in arbitrary expressions:
7939 e += mystring("GiNaC rulez");
7941 // -> "GiNaC rulez"+"Hello, world!"
7944 (GiNaC's automatic term reordering is in effect here), or even
7947 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7949 // -> "One string"^(2*sin(-"Another string"+Pi))
7952 Whether this makes sense is debatable but remember that this is only an
7953 example. At least it allows you to implement your own symbolic algorithms
7956 Note that GiNaC's algebraic rules remain unchanged:
7959 e = mystring("Wow") * mystring("Wow");
7963 e = pow(mystring("First")-mystring("Second"), 2);
7964 cout << e.expand() << endl;
7965 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7968 There's no way to, for example, make GiNaC's @code{add} class perform string
7969 concatenation. You would have to implement this yourself.
7971 @subsection Automatic evaluation
7974 @cindex @code{eval()}
7975 @cindex @code{hold()}
7976 When dealing with objects that are just a little more complicated than the
7977 simple string objects we have implemented, chances are that you will want to
7978 have some automatic simplifications or canonicalizations performed on them.
7979 This is done in the evaluation member function @code{eval()}. Let's say that
7980 we wanted all strings automatically converted to lowercase with
7981 non-alphabetic characters stripped, and empty strings removed:
7984 class mystring : public basic
7988 ex eval(int level = 0) const;
7992 ex mystring::eval(int level) const
7995 for (int i=0; i<str.length(); i++) @{
7997 if (c >= 'A' && c <= 'Z')
7998 new_str += tolower(c);
7999 else if (c >= 'a' && c <= 'z')
8003 if (new_str.length() == 0)
8006 return mystring(new_str).hold();
8010 The @code{level} argument is used to limit the recursion depth of the
8011 evaluation. We don't have any subexpressions in the @code{mystring}
8012 class so we are not concerned with this. If we had, we would call the
8013 @code{eval()} functions of the subexpressions with @code{level - 1} as
8014 the argument if @code{level != 1}. The @code{hold()} member function
8015 sets a flag in the object that prevents further evaluation. Otherwise
8016 we might end up in an endless loop. When you want to return the object
8017 unmodified, use @code{return this->hold();}.
8019 Let's confirm that it works:
8022 ex e = mystring("Hello, world!") + mystring("!?#");
8026 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8031 @subsection Optional member functions
8033 We have implemented only a small set of member functions to make the class
8034 work in the GiNaC framework. There are two functions that are not strictly
8035 required but will make operations with objects of the class more efficient:
8037 @cindex @code{calchash()}
8038 @cindex @code{is_equal_same_type()}
8040 unsigned calchash() const;
8041 bool is_equal_same_type(const basic &other) const;
8044 The @code{calchash()} method returns an @code{unsigned} hash value for the
8045 object which will allow GiNaC to compare and canonicalize expressions much
8046 more efficiently. You should consult the implementation of some of the built-in
8047 GiNaC classes for examples of hash functions. The default implementation of
8048 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8049 class and all subexpressions that are accessible via @code{op()}.
8051 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8052 tests for equality without establishing an ordering relation, which is often
8053 faster. The default implementation of @code{is_equal_same_type()} just calls
8054 @code{compare_same_type()} and tests its result for zero.
8056 @subsection Other member functions
8058 For a real algebraic class, there are probably some more functions that you
8059 might want to provide:
8062 bool info(unsigned inf) const;
8063 ex evalf(int level = 0) const;
8064 ex series(const relational & r, int order, unsigned options = 0) const;
8065 ex derivative(const symbol & s) const;
8068 If your class stores sub-expressions (see the scalar product example in the
8069 previous section) you will probably want to override
8071 @cindex @code{let_op()}
8074 ex op(size_t i) const;
8075 ex & let_op(size_t i);
8076 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8077 ex map(map_function & f) const;
8080 @code{let_op()} is a variant of @code{op()} that allows write access. The
8081 default implementations of @code{subs()} and @code{map()} use it, so you have
8082 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8084 You can, of course, also add your own new member functions. Remember
8085 that the RTTI may be used to get information about what kinds of objects
8086 you are dealing with (the position in the class hierarchy) and that you
8087 can always extract the bare object from an @code{ex} by stripping the
8088 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8089 should become a need.
8091 That's it. May the source be with you!
8094 @node A comparison with other CAS, Advantages, Adding classes, Top
8095 @c node-name, next, previous, up
8096 @chapter A Comparison With Other CAS
8099 This chapter will give you some information on how GiNaC compares to
8100 other, traditional Computer Algebra Systems, like @emph{Maple},
8101 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8102 disadvantages over these systems.
8105 * Advantages:: Strengths of the GiNaC approach.
8106 * Disadvantages:: Weaknesses of the GiNaC approach.
8107 * Why C++?:: Attractiveness of C++.
8110 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8111 @c node-name, next, previous, up
8114 GiNaC has several advantages over traditional Computer
8115 Algebra Systems, like
8120 familiar language: all common CAS implement their own proprietary
8121 grammar which you have to learn first (and maybe learn again when your
8122 vendor decides to `enhance' it). With GiNaC you can write your program
8123 in common C++, which is standardized.
8127 structured data types: you can build up structured data types using
8128 @code{struct}s or @code{class}es together with STL features instead of
8129 using unnamed lists of lists of lists.
8132 strongly typed: in CAS, you usually have only one kind of variables
8133 which can hold contents of an arbitrary type. This 4GL like feature is
8134 nice for novice programmers, but dangerous.
8137 development tools: powerful development tools exist for C++, like fancy
8138 editors (e.g. with automatic indentation and syntax highlighting),
8139 debuggers, visualization tools, documentation generators@dots{}
8142 modularization: C++ programs can easily be split into modules by
8143 separating interface and implementation.
8146 price: GiNaC is distributed under the GNU Public License which means
8147 that it is free and available with source code. And there are excellent
8148 C++-compilers for free, too.
8151 extendable: you can add your own classes to GiNaC, thus extending it on
8152 a very low level. Compare this to a traditional CAS that you can
8153 usually only extend on a high level by writing in the language defined
8154 by the parser. In particular, it turns out to be almost impossible to
8155 fix bugs in a traditional system.
8158 multiple interfaces: Though real GiNaC programs have to be written in
8159 some editor, then be compiled, linked and executed, there are more ways
8160 to work with the GiNaC engine. Many people want to play with
8161 expressions interactively, as in traditional CASs. Currently, two such
8162 windows into GiNaC have been implemented and many more are possible: the
8163 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8164 types to a command line and second, as a more consistent approach, an
8165 interactive interface to the Cint C++ interpreter has been put together
8166 (called GiNaC-cint) that allows an interactive scripting interface
8167 consistent with the C++ language. It is available from the usual GiNaC
8171 seamless integration: it is somewhere between difficult and impossible
8172 to call CAS functions from within a program written in C++ or any other
8173 programming language and vice versa. With GiNaC, your symbolic routines
8174 are part of your program. You can easily call third party libraries,
8175 e.g. for numerical evaluation or graphical interaction. All other
8176 approaches are much more cumbersome: they range from simply ignoring the
8177 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8178 system (i.e. @emph{Yacas}).
8181 efficiency: often large parts of a program do not need symbolic
8182 calculations at all. Why use large integers for loop variables or
8183 arbitrary precision arithmetics where @code{int} and @code{double} are
8184 sufficient? For pure symbolic applications, GiNaC is comparable in
8185 speed with other CAS.
8190 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8191 @c node-name, next, previous, up
8192 @section Disadvantages
8194 Of course it also has some disadvantages:
8199 advanced features: GiNaC cannot compete with a program like
8200 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8201 which grows since 1981 by the work of dozens of programmers, with
8202 respect to mathematical features. Integration, factorization,
8203 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8204 not planned for the near future).
8207 portability: While the GiNaC library itself is designed to avoid any
8208 platform dependent features (it should compile on any ANSI compliant C++
8209 compiler), the currently used version of the CLN library (fast large
8210 integer and arbitrary precision arithmetics) can only by compiled
8211 without hassle on systems with the C++ compiler from the GNU Compiler
8212 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8213 macros to let the compiler gather all static initializations, which
8214 works for GNU C++ only. Feel free to contact the authors in case you
8215 really believe that you need to use a different compiler. We have
8216 occasionally used other compilers and may be able to give you advice.}
8217 GiNaC uses recent language features like explicit constructors, mutable
8218 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8219 literally. Recent GCC versions starting at 2.95.3, although itself not
8220 yet ANSI compliant, support all needed features.
8225 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8226 @c node-name, next, previous, up
8229 Why did we choose to implement GiNaC in C++ instead of Java or any other
8230 language? C++ is not perfect: type checking is not strict (casting is
8231 possible), separation between interface and implementation is not
8232 complete, object oriented design is not enforced. The main reason is
8233 the often scolded feature of operator overloading in C++. While it may
8234 be true that operating on classes with a @code{+} operator is rarely
8235 meaningful, it is perfectly suited for algebraic expressions. Writing
8236 @math{3x+5y} as @code{3*x+5*y} instead of
8237 @code{x.times(3).plus(y.times(5))} looks much more natural.
8238 Furthermore, the main developers are more familiar with C++ than with
8239 any other programming language.
8242 @node Internal structures, Expressions are reference counted, Why C++? , Top
8243 @c node-name, next, previous, up
8244 @appendix Internal structures
8247 * Expressions are reference counted::
8248 * Internal representation of products and sums::
8251 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8252 @c node-name, next, previous, up
8253 @appendixsection Expressions are reference counted
8255 @cindex reference counting
8256 @cindex copy-on-write
8257 @cindex garbage collection
8258 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8259 where the counter belongs to the algebraic objects derived from class
8260 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8261 which @code{ex} contains an instance. If you understood that, you can safely
8262 skip the rest of this passage.
8264 Expressions are extremely light-weight since internally they work like
8265 handles to the actual representation. They really hold nothing more
8266 than a pointer to some other object. What this means in practice is
8267 that whenever you create two @code{ex} and set the second equal to the
8268 first no copying process is involved. Instead, the copying takes place
8269 as soon as you try to change the second. Consider the simple sequence
8274 #include <ginac/ginac.h>
8275 using namespace std;
8276 using namespace GiNaC;
8280 symbol x("x"), y("y"), z("z");
8283 e1 = sin(x + 2*y) + 3*z + 41;
8284 e2 = e1; // e2 points to same object as e1
8285 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8286 e2 += 1; // e2 is copied into a new object
8287 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8291 The line @code{e2 = e1;} creates a second expression pointing to the
8292 object held already by @code{e1}. The time involved for this operation
8293 is therefore constant, no matter how large @code{e1} was. Actual
8294 copying, however, must take place in the line @code{e2 += 1;} because
8295 @code{e1} and @code{e2} are not handles for the same object any more.
8296 This concept is called @dfn{copy-on-write semantics}. It increases
8297 performance considerably whenever one object occurs multiple times and
8298 represents a simple garbage collection scheme because when an @code{ex}
8299 runs out of scope its destructor checks whether other expressions handle
8300 the object it points to too and deletes the object from memory if that
8301 turns out not to be the case. A slightly less trivial example of
8302 differentiation using the chain-rule should make clear how powerful this
8307 symbol x("x"), y("y");
8311 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8312 cout << e1 << endl // prints x+3*y
8313 << e2 << endl // prints (x+3*y)^3
8314 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8318 Here, @code{e1} will actually be referenced three times while @code{e2}
8319 will be referenced two times. When the power of an expression is built,
8320 that expression needs not be copied. Likewise, since the derivative of
8321 a power of an expression can be easily expressed in terms of that
8322 expression, no copying of @code{e1} is involved when @code{e3} is
8323 constructed. So, when @code{e3} is constructed it will print as
8324 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8325 holds a reference to @code{e2} and the factor in front is just
8328 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8329 semantics. When you insert an expression into a second expression, the
8330 result behaves exactly as if the contents of the first expression were
8331 inserted. But it may be useful to remember that this is not what
8332 happens. Knowing this will enable you to write much more efficient
8333 code. If you still have an uncertain feeling with copy-on-write
8334 semantics, we recommend you have a look at the
8335 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8336 Marshall Cline. Chapter 16 covers this issue and presents an
8337 implementation which is pretty close to the one in GiNaC.
8340 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8341 @c node-name, next, previous, up
8342 @appendixsection Internal representation of products and sums
8344 @cindex representation
8347 @cindex @code{power}
8348 Although it should be completely transparent for the user of
8349 GiNaC a short discussion of this topic helps to understand the sources
8350 and also explain performance to a large degree. Consider the
8351 unexpanded symbolic expression
8353 $2d^3 \left( 4a + 5b - 3 \right)$
8356 @math{2*d^3*(4*a+5*b-3)}
8358 which could naively be represented by a tree of linear containers for
8359 addition and multiplication, one container for exponentiation with base
8360 and exponent and some atomic leaves of symbols and numbers in this
8365 @cindex pair-wise representation
8366 However, doing so results in a rather deeply nested tree which will
8367 quickly become inefficient to manipulate. We can improve on this by
8368 representing the sum as a sequence of terms, each one being a pair of a
8369 purely numeric multiplicative coefficient and its rest. In the same
8370 spirit we can store the multiplication as a sequence of terms, each
8371 having a numeric exponent and a possibly complicated base, the tree
8372 becomes much more flat:
8376 The number @code{3} above the symbol @code{d} shows that @code{mul}
8377 objects are treated similarly where the coefficients are interpreted as
8378 @emph{exponents} now. Addition of sums of terms or multiplication of
8379 products with numerical exponents can be coded to be very efficient with
8380 such a pair-wise representation. Internally, this handling is performed
8381 by most CAS in this way. It typically speeds up manipulations by an
8382 order of magnitude. The overall multiplicative factor @code{2} and the
8383 additive term @code{-3} look somewhat out of place in this
8384 representation, however, since they are still carrying a trivial
8385 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8386 this is avoided by adding a field that carries an overall numeric
8387 coefficient. This results in the realistic picture of internal
8390 $2d^3 \left( 4a + 5b - 3 \right)$:
8393 @math{2*d^3*(4*a+5*b-3)}:
8399 This also allows for a better handling of numeric radicals, since
8400 @code{sqrt(2)} can now be carried along calculations. Now it should be
8401 clear, why both classes @code{add} and @code{mul} are derived from the
8402 same abstract class: the data representation is the same, only the
8403 semantics differs. In the class hierarchy, methods for polynomial
8404 expansion and the like are reimplemented for @code{add} and @code{mul},
8405 but the data structure is inherited from @code{expairseq}.
8408 @node Package tools, ginac-config, Internal representation of products and sums, Top
8409 @c node-name, next, previous, up
8410 @appendix Package tools
8412 If you are creating a software package that uses the GiNaC library,
8413 setting the correct command line options for the compiler and linker
8414 can be difficult. GiNaC includes two tools to make this process easier.
8417 * ginac-config:: A shell script to detect compiler and linker flags.
8418 * AM_PATH_GINAC:: Macro for GNU automake.
8422 @node ginac-config, AM_PATH_GINAC, Package tools, Package tools
8423 @c node-name, next, previous, up
8424 @section @command{ginac-config}
8425 @cindex ginac-config
8427 @command{ginac-config} is a shell script that you can use to determine
8428 the compiler and linker command line options required to compile and
8429 link a program with the GiNaC library.
8431 @command{ginac-config} takes the following flags:
8435 Prints out the version of GiNaC installed.
8437 Prints '-I' flags pointing to the installed header files.
8439 Prints out the linker flags necessary to link a program against GiNaC.
8440 @item --prefix[=@var{PREFIX}]
8441 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8442 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8443 Otherwise, prints out the configured value of @env{$prefix}.
8444 @item --exec-prefix[=@var{PREFIX}]
8445 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8446 Otherwise, prints out the configured value of @env{$exec_prefix}.
8449 Typically, @command{ginac-config} will be used within a configure
8450 script, as described below. It, however, can also be used directly from
8451 the command line using backquotes to compile a simple program. For
8455 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8458 This command line might expand to (for example):
8461 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8462 -lginac -lcln -lstdc++
8465 Not only is the form using @command{ginac-config} easier to type, it will
8466 work on any system, no matter how GiNaC was configured.
8469 @node AM_PATH_GINAC, Configure script options, ginac-config, Package tools
8470 @c node-name, next, previous, up
8471 @section @samp{AM_PATH_GINAC}
8472 @cindex AM_PATH_GINAC
8474 For packages configured using GNU automake, GiNaC also provides
8475 a macro to automate the process of checking for GiNaC.
8478 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8479 [, @var{ACTION-IF-NOT-FOUND}]]])
8487 Determines the location of GiNaC using @command{ginac-config}, which is
8488 either found in the user's path, or from the environment variable
8489 @env{GINACLIB_CONFIG}.
8492 Tests the installed libraries to make sure that their version
8493 is later than @var{MINIMUM-VERSION}. (A default version will be used
8497 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8498 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8499 variable to the output of @command{ginac-config --libs}, and calls
8500 @samp{AC_SUBST()} for these variables so they can be used in generated
8501 makefiles, and then executes @var{ACTION-IF-FOUND}.
8504 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8505 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8509 This macro is in file @file{ginac.m4} which is installed in
8510 @file{$datadir/aclocal}. Note that if automake was installed with a
8511 different @samp{--prefix} than GiNaC, you will either have to manually
8512 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8513 aclocal the @samp{-I} option when running it.
8516 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8517 * Example package:: Example of a package using AM_PATH_GINAC.
8521 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8522 @c node-name, next, previous, up
8523 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8525 Simply make sure that @command{ginac-config} is in your path, and run
8526 the configure script.
8533 The directory where the GiNaC libraries are installed needs
8534 to be found by your system's dynamic linker.
8536 This is generally done by
8539 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8545 setting the environment variable @env{LD_LIBRARY_PATH},
8548 or, as a last resort,
8551 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8552 running configure, for instance:
8555 LDFLAGS=-R/home/cbauer/lib ./configure
8560 You can also specify a @command{ginac-config} not in your path by
8561 setting the @env{GINACLIB_CONFIG} environment variable to the
8562 name of the executable
8565 If you move the GiNaC package from its installed location,
8566 you will either need to modify @command{ginac-config} script
8567 manually to point to the new location or rebuild GiNaC.
8578 --with-ginac-prefix=@var{PREFIX}
8579 --with-ginac-exec-prefix=@var{PREFIX}
8582 are provided to override the prefix and exec-prefix that were stored
8583 in the @command{ginac-config} shell script by GiNaC's configure. You are
8584 generally better off configuring GiNaC with the right path to begin with.
8588 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8589 @c node-name, next, previous, up
8590 @subsection Example of a package using @samp{AM_PATH_GINAC}
8592 The following shows how to build a simple package using automake
8593 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8597 #include <ginac/ginac.h>
8601 GiNaC::symbol x("x");
8602 GiNaC::ex a = GiNaC::sin(x);
8603 std::cout << "Derivative of " << a
8604 << " is " << a.diff(x) << std::endl;
8609 You should first read the introductory portions of the automake
8610 Manual, if you are not already familiar with it.
8612 Two files are needed, @file{configure.in}, which is used to build the
8616 dnl Process this file with autoconf to produce a configure script.
8618 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8624 AM_PATH_GINAC(0.9.0, [
8625 LIBS="$LIBS $GINACLIB_LIBS"
8626 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8627 ], AC_MSG_ERROR([need to have GiNaC installed]))
8632 The only command in this which is not standard for automake
8633 is the @samp{AM_PATH_GINAC} macro.
8635 That command does the following: If a GiNaC version greater or equal
8636 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8637 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8638 the error message `need to have GiNaC installed'
8640 And the @file{Makefile.am}, which will be used to build the Makefile.
8643 ## Process this file with automake to produce Makefile.in
8644 bin_PROGRAMS = simple
8645 simple_SOURCES = simple.cpp
8648 This @file{Makefile.am}, says that we are building a single executable,
8649 from a single source file @file{simple.cpp}. Since every program
8650 we are building uses GiNaC we simply added the GiNaC options
8651 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8652 want to specify them on a per-program basis: for instance by
8656 simple_LDADD = $(GINACLIB_LIBS)
8657 INCLUDES = $(GINACLIB_CPPFLAGS)
8660 to the @file{Makefile.am}.
8662 To try this example out, create a new directory and add the three
8665 Now execute the following commands:
8668 $ automake --add-missing
8673 You now have a package that can be built in the normal fashion
8682 @node Bibliography, Concept index, Example package, Top
8683 @c node-name, next, previous, up
8684 @appendix Bibliography
8689 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8692 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8695 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8698 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8701 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8702 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8705 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8706 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8707 Academic Press, London
8710 @cite{Computer Algebra Systems - A Practical Guide},
8711 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8714 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8715 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8718 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8719 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8722 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8727 @node Concept index, , Bibliography, Top
8728 @c node-name, next, previous, up
8729 @unnumbered Concept index