1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2006 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic concepts:: Description of fundamental classes.
85 * Methods and functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A comparison with other CAS:: Compares GiNaC to traditional CAS.
88 * Internal structures:: Description of some internal structures.
89 * Package tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2006 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
483 process as well, since some of the source files are automatically
484 generated by Perl scripts. Last but not least, the CLN library
485 is used extensively and needs to be installed on your system.
486 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
487 (it is covered by GPL) and install it prior to trying to install
488 GiNaC. The configure script checks if it can find it and if it cannot
489 it will refuse to continue.
492 @node Configuration, Building GiNaC, Prerequisites, Installation
493 @c node-name, next, previous, up
494 @section Configuration
495 @cindex configuration
498 To configure GiNaC means to prepare the source distribution for
499 building. It is done via a shell script called @command{configure} that
500 is shipped with the sources and was originally generated by GNU
501 Autoconf. Since a configure script generated by GNU Autoconf never
502 prompts, all customization must be done either via command line
503 parameters or environment variables. It accepts a list of parameters,
504 the complete set of which can be listed by calling it with the
505 @option{--help} option. The most important ones will be shortly
506 described in what follows:
511 @option{--disable-shared}: When given, this option switches off the
512 build of a shared library, i.e. a @file{.so} file. This may be convenient
513 when developing because it considerably speeds up compilation.
516 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
517 and headers are installed. It defaults to @file{/usr/local} which means
518 that the library is installed in the directory @file{/usr/local/lib},
519 the header files in @file{/usr/local/include/ginac} and the documentation
520 (like this one) into @file{/usr/local/share/doc/GiNaC}.
523 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
524 the library installed in some other directory than
525 @file{@var{PREFIX}/lib/}.
528 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
529 to have the header files installed in some other directory than
530 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
531 @option{--includedir=/usr/include} you will end up with the header files
532 sitting in the directory @file{/usr/include/ginac/}. Note that the
533 subdirectory @file{ginac} is enforced by this process in order to
534 keep the header files separated from others. This avoids some
535 clashes and allows for an easier deinstallation of GiNaC. This ought
536 to be considered A Good Thing (tm).
539 @option{--datadir=@var{DATADIR}}: This option may be given in case you
540 want to have the documentation installed in some other directory than
541 @file{@var{PREFIX}/share/doc/GiNaC/}.
545 In addition, you may specify some environment variables. @env{CXX}
546 holds the path and the name of the C++ compiler in case you want to
547 override the default in your path. (The @command{configure} script
548 searches your path for @command{c++}, @command{g++}, @command{gcc},
549 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
550 be very useful to define some compiler flags with the @env{CXXFLAGS}
551 environment variable, like optimization, debugging information and
552 warning levels. If omitted, it defaults to @option{-g
553 -O2}.@footnote{The @command{configure} script is itself generated from
554 the file @file{configure.ac}. It is only distributed in packaged
555 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
556 must generate @command{configure} along with the various
557 @file{Makefile.in} by using the @command{autogen.sh} script. This will
558 require a fair amount of support from your local toolchain, though.}
560 The whole process is illustrated in the following two
561 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
562 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
565 Here is a simple configuration for a site-wide GiNaC library assuming
566 everything is in default paths:
569 $ export CXXFLAGS="-Wall -O2"
573 And here is a configuration for a private static GiNaC library with
574 several components sitting in custom places (site-wide GCC and private
575 CLN). The compiler is persuaded to be picky and full assertions and
576 debugging information are switched on:
579 $ export CXX=/usr/local/gnu/bin/c++
580 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
581 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
582 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
583 $ ./configure --disable-shared --prefix=$(HOME)
587 @node Building GiNaC, Installing GiNaC, Configuration, Installation
588 @c node-name, next, previous, up
589 @section Building GiNaC
590 @cindex building GiNaC
592 After proper configuration you should just build the whole
597 at the command prompt and go for a cup of coffee. The exact time it
598 takes to compile GiNaC depends not only on the speed of your machines
599 but also on other parameters, for instance what value for @env{CXXFLAGS}
600 you entered. Optimization may be very time-consuming.
602 Just to make sure GiNaC works properly you may run a collection of
603 regression tests by typing
609 This will compile some sample programs, run them and check the output
610 for correctness. The regression tests fall in three categories. First,
611 the so called @emph{exams} are performed, simple tests where some
612 predefined input is evaluated (like a pupils' exam). Second, the
613 @emph{checks} test the coherence of results among each other with
614 possible random input. Third, some @emph{timings} are performed, which
615 benchmark some predefined problems with different sizes and display the
616 CPU time used in seconds. Each individual test should return a message
617 @samp{passed}. This is mostly intended to be a QA-check if something
618 was broken during development, not a sanity check of your system. Some
619 of the tests in sections @emph{checks} and @emph{timings} may require
620 insane amounts of memory and CPU time. Feel free to kill them if your
621 machine catches fire. Another quite important intent is to allow people
622 to fiddle around with optimization.
624 By default, the only documentation that will be built is this tutorial
625 in @file{.info} format. To build the GiNaC tutorial and reference manual
626 in HTML, DVI, PostScript, or PDF formats, use one of
635 Generally, the top-level Makefile runs recursively to the
636 subdirectories. It is therefore safe to go into any subdirectory
637 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
638 @var{target} there in case something went wrong.
641 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
642 @c node-name, next, previous, up
643 @section Installing GiNaC
646 To install GiNaC on your system, simply type
652 As described in the section about configuration the files will be
653 installed in the following directories (the directories will be created
654 if they don't already exist):
659 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
660 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
661 So will @file{libginac.so} unless the configure script was
662 given the option @option{--disable-shared}. The proper symlinks
663 will be established as well.
666 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
667 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
670 All documentation (info) will be stuffed into
671 @file{@var{PREFIX}/share/doc/GiNaC/} (or
672 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
676 For the sake of completeness we will list some other useful make
677 targets: @command{make clean} deletes all files generated by
678 @command{make}, i.e. all the object files. In addition @command{make
679 distclean} removes all files generated by the configuration and
680 @command{make maintainer-clean} goes one step further and deletes files
681 that may require special tools to rebuild (like the @command{libtool}
682 for instance). Finally @command{make uninstall} removes the installed
683 library, header files and documentation@footnote{Uninstallation does not
684 work after you have called @command{make distclean} since the
685 @file{Makefile} is itself generated by the configuration from
686 @file{Makefile.in} and hence deleted by @command{make distclean}. There
687 are two obvious ways out of this dilemma. First, you can run the
688 configuration again with the same @var{PREFIX} thus creating a
689 @file{Makefile} with a working @samp{uninstall} target. Second, you can
690 do it by hand since you now know where all the files went during
694 @node Basic concepts, Expressions, Installing GiNaC, Top
695 @c node-name, next, previous, up
696 @chapter Basic concepts
698 This chapter will describe the different fundamental objects that can be
699 handled by GiNaC. But before doing so, it is worthwhile introducing you
700 to the more commonly used class of expressions, representing a flexible
701 meta-class for storing all mathematical objects.
704 * Expressions:: The fundamental GiNaC class.
705 * Automatic evaluation:: Evaluation and canonicalization.
706 * Error handling:: How the library reports errors.
707 * The class hierarchy:: Overview of GiNaC's classes.
708 * Symbols:: Symbolic objects.
709 * Numbers:: Numerical objects.
710 * Constants:: Pre-defined constants.
711 * Fundamental containers:: Sums, products and powers.
712 * Lists:: Lists of expressions.
713 * Mathematical functions:: Mathematical functions.
714 * Relations:: Equality, Inequality and all that.
715 * Integrals:: Symbolic integrals.
716 * Matrices:: Matrices.
717 * Indexed objects:: Handling indexed quantities.
718 * Non-commutative objects:: Algebras with non-commutative products.
719 * Hash maps:: A faster alternative to std::map<>.
723 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
724 @c node-name, next, previous, up
726 @cindex expression (class @code{ex})
729 The most common class of objects a user deals with is the expression
730 @code{ex}, representing a mathematical object like a variable, number,
731 function, sum, product, etc@dots{} Expressions may be put together to form
732 new expressions, passed as arguments to functions, and so on. Here is a
733 little collection of valid expressions:
736 ex MyEx1 = 5; // simple number
737 ex MyEx2 = x + 2*y; // polynomial in x and y
738 ex MyEx3 = (x + 1)/(x - 1); // rational expression
739 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
740 ex MyEx5 = MyEx4 + 1; // similar to above
743 Expressions are handles to other more fundamental objects, that often
744 contain other expressions thus creating a tree of expressions
745 (@xref{Internal structures}, for particular examples). Most methods on
746 @code{ex} therefore run top-down through such an expression tree. For
747 example, the method @code{has()} scans recursively for occurrences of
748 something inside an expression. Thus, if you have declared @code{MyEx4}
749 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
750 the argument of @code{sin} and hence return @code{true}.
752 The next sections will outline the general picture of GiNaC's class
753 hierarchy and describe the classes of objects that are handled by
756 @subsection Note: Expressions and STL containers
758 GiNaC expressions (@code{ex} objects) have value semantics (they can be
759 assigned, reassigned and copied like integral types) but the operator
760 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
761 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
763 This implies that in order to use expressions in sorted containers such as
764 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
765 comparison predicate. GiNaC provides such a predicate, called
766 @code{ex_is_less}. For example, a set of expressions should be defined
767 as @code{std::set<ex, ex_is_less>}.
769 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
770 don't pose a problem. A @code{std::vector<ex>} works as expected.
772 @xref{Information about expressions}, for more about comparing and ordering
776 @node Automatic evaluation, Error handling, Expressions, Basic concepts
777 @c node-name, next, previous, up
778 @section Automatic evaluation and canonicalization of expressions
781 GiNaC performs some automatic transformations on expressions, to simplify
782 them and put them into a canonical form. Some examples:
785 ex MyEx1 = 2*x - 1 + x; // 3*x-1
786 ex MyEx2 = x - x; // 0
787 ex MyEx3 = cos(2*Pi); // 1
788 ex MyEx4 = x*y/x; // y
791 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
792 evaluation}. GiNaC only performs transformations that are
796 at most of complexity
804 algebraically correct, possibly except for a set of measure zero (e.g.
805 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
808 There are two types of automatic transformations in GiNaC that may not
809 behave in an entirely obvious way at first glance:
813 The terms of sums and products (and some other things like the arguments of
814 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
815 into a canonical form that is deterministic, but not lexicographical or in
816 any other way easy to guess (it almost always depends on the number and
817 order of the symbols you define). However, constructing the same expression
818 twice, either implicitly or explicitly, will always result in the same
821 Expressions of the form 'number times sum' are automatically expanded (this
822 has to do with GiNaC's internal representation of sums and products). For
825 ex MyEx5 = 2*(x + y); // 2*x+2*y
826 ex MyEx6 = z*(x + y); // z*(x+y)
830 The general rule is that when you construct expressions, GiNaC automatically
831 creates them in canonical form, which might differ from the form you typed in
832 your program. This may create some awkward looking output (@samp{-y+x} instead
833 of @samp{x-y}) but allows for more efficient operation and usually yields
834 some immediate simplifications.
836 @cindex @code{eval()}
837 Internally, the anonymous evaluator in GiNaC is implemented by the methods
840 ex ex::eval(int level = 0) const;
841 ex basic::eval(int level = 0) const;
844 but unless you are extending GiNaC with your own classes or functions, there
845 should never be any reason to call them explicitly. All GiNaC methods that
846 transform expressions, like @code{subs()} or @code{normal()}, automatically
847 re-evaluate their results.
850 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
851 @c node-name, next, previous, up
852 @section Error handling
854 @cindex @code{pole_error} (class)
856 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
857 generated by GiNaC are subclassed from the standard @code{exception} class
858 defined in the @file{<stdexcept>} header. In addition to the predefined
859 @code{logic_error}, @code{domain_error}, @code{out_of_range},
860 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
861 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
862 exception that gets thrown when trying to evaluate a mathematical function
865 The @code{pole_error} class has a member function
868 int pole_error::degree() const;
871 that returns the order of the singularity (or 0 when the pole is
872 logarithmic or the order is undefined).
874 When using GiNaC it is useful to arrange for exceptions to be caught in
875 the main program even if you don't want to do any special error handling.
876 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
877 default exception handler of your C++ compiler's run-time system which
878 usually only aborts the program without giving any information what went
881 Here is an example for a @code{main()} function that catches and prints
882 exceptions generated by GiNaC:
887 #include <ginac/ginac.h>
889 using namespace GiNaC;
897 @} catch (exception &p) @{
898 cerr << p.what() << endl;
906 @node The class hierarchy, Symbols, Error handling, Basic concepts
907 @c node-name, next, previous, up
908 @section The class hierarchy
910 GiNaC's class hierarchy consists of several classes representing
911 mathematical objects, all of which (except for @code{ex} and some
912 helpers) are internally derived from one abstract base class called
913 @code{basic}. You do not have to deal with objects of class
914 @code{basic}, instead you'll be dealing with symbols, numbers,
915 containers of expressions and so on.
919 To get an idea about what kinds of symbolic composites may be built we
920 have a look at the most important classes in the class hierarchy and
921 some of the relations among the classes:
923 @image{classhierarchy}
925 The abstract classes shown here (the ones without drop-shadow) are of no
926 interest for the user. They are used internally in order to avoid code
927 duplication if two or more classes derived from them share certain
928 features. An example is @code{expairseq}, a container for a sequence of
929 pairs each consisting of one expression and a number (@code{numeric}).
930 What @emph{is} visible to the user are the derived classes @code{add}
931 and @code{mul}, representing sums and products. @xref{Internal
932 structures}, where these two classes are described in more detail. The
933 following table shortly summarizes what kinds of mathematical objects
934 are stored in the different classes:
937 @multitable @columnfractions .22 .78
938 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
939 @item @code{constant} @tab Constants like
946 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
947 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
948 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
949 @item @code{ncmul} @tab Products of non-commutative objects
950 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
955 @code{sqrt(}@math{2}@code{)}
958 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
959 @item @code{function} @tab A symbolic function like
966 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
967 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
968 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
969 @item @code{indexed} @tab Indexed object like @math{A_ij}
970 @item @code{tensor} @tab Special tensor like the delta and metric tensors
971 @item @code{idx} @tab Index of an indexed object
972 @item @code{varidx} @tab Index with variance
973 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
974 @item @code{wildcard} @tab Wildcard for pattern matching
975 @item @code{structure} @tab Template for user-defined classes
980 @node Symbols, Numbers, The class hierarchy, Basic concepts
981 @c node-name, next, previous, up
983 @cindex @code{symbol} (class)
984 @cindex hierarchy of classes
987 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
988 manipulation what atoms are for chemistry.
990 A typical symbol definition looks like this:
995 This definition actually contains three very different things:
997 @item a C++ variable named @code{x}
998 @item a @code{symbol} object stored in this C++ variable; this object
999 represents the symbol in a GiNaC expression
1000 @item the string @code{"x"} which is the name of the symbol, used (almost)
1001 exclusively for printing expressions holding the symbol
1004 Symbols have an explicit name, supplied as a string during construction,
1005 because in C++, variable names can't be used as values, and the C++ compiler
1006 throws them away during compilation.
1008 It is possible to omit the symbol name in the definition:
1013 In this case, GiNaC will assign the symbol an internal, unique name of the
1014 form @code{symbolNNN}. This won't affect the usability of the symbol but
1015 the output of your calculations will become more readable if you give your
1016 symbols sensible names (for intermediate expressions that are only used
1017 internally such anonymous symbols can be quite useful, however).
1019 Now, here is one important property of GiNaC that differentiates it from
1020 other computer algebra programs you may have used: GiNaC does @emph{not} use
1021 the names of symbols to tell them apart, but a (hidden) serial number that
1022 is unique for each newly created @code{symbol} object. In you want to use
1023 one and the same symbol in different places in your program, you must only
1024 create one @code{symbol} object and pass that around. If you create another
1025 symbol, even if it has the same name, GiNaC will treat it as a different
1042 // prints "x^6" which looks right, but...
1044 cout << e.degree(x) << endl;
1045 // ...this doesn't work. The symbol "x" here is different from the one
1046 // in f() and in the expression returned by f(). Consequently, it
1051 One possibility to ensure that @code{f()} and @code{main()} use the same
1052 symbol is to pass the symbol as an argument to @code{f()}:
1054 ex f(int n, const ex & x)
1063 // Now, f() uses the same symbol.
1066 cout << e.degree(x) << endl;
1067 // prints "6", as expected
1071 Another possibility would be to define a global symbol @code{x} that is used
1072 by both @code{f()} and @code{main()}. If you are using global symbols and
1073 multiple compilation units you must take special care, however. Suppose
1074 that you have a header file @file{globals.h} in your program that defines
1075 a @code{symbol x("x");}. In this case, every unit that includes
1076 @file{globals.h} would also get its own definition of @code{x} (because
1077 header files are just inlined into the source code by the C++ preprocessor),
1078 and hence you would again end up with multiple equally-named, but different,
1079 symbols. Instead, the @file{globals.h} header should only contain a
1080 @emph{declaration} like @code{extern symbol x;}, with the definition of
1081 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1083 A different approach to ensuring that symbols used in different parts of
1084 your program are identical is to create them with a @emph{factory} function
1087 const symbol & get_symbol(const string & s)
1089 static map<string, symbol> directory;
1090 map<string, symbol>::iterator i = directory.find(s);
1091 if (i != directory.end())
1094 return directory.insert(make_pair(s, symbol(s))).first->second;
1098 This function returns one newly constructed symbol for each name that is
1099 passed in, and it returns the same symbol when called multiple times with
1100 the same name. Using this symbol factory, we can rewrite our example like
1105 return pow(get_symbol("x"), n);
1112 // Both calls of get_symbol("x") yield the same symbol.
1113 cout << e.degree(get_symbol("x")) << endl;
1118 Instead of creating symbols from strings we could also have
1119 @code{get_symbol()} take, for example, an integer number as its argument.
1120 In this case, we would probably want to give the generated symbols names
1121 that include this number, which can be accomplished with the help of an
1122 @code{ostringstream}.
1124 In general, if you're getting weird results from GiNaC such as an expression
1125 @samp{x-x} that is not simplified to zero, you should check your symbol
1128 As we said, the names of symbols primarily serve for purposes of expression
1129 output. But there are actually two instances where GiNaC uses the names for
1130 identifying symbols: When constructing an expression from a string, and when
1131 recreating an expression from an archive (@pxref{Input/output}).
1133 In addition to its name, a symbol may contain a special string that is used
1136 symbol x("x", "\\Box");
1139 This creates a symbol that is printed as "@code{x}" in normal output, but
1140 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1141 information about the different output formats of expressions in GiNaC).
1142 GiNaC automatically creates proper LaTeX code for symbols having names of
1143 greek letters (@samp{alpha}, @samp{mu}, etc.).
1145 @cindex @code{subs()}
1146 Symbols in GiNaC can't be assigned values. If you need to store results of
1147 calculations and give them a name, use C++ variables of type @code{ex}.
1148 If you want to replace a symbol in an expression with something else, you
1149 can invoke the expression's @code{.subs()} method
1150 (@pxref{Substituting expressions}).
1152 @cindex @code{realsymbol()}
1153 By default, symbols are expected to stand in for complex values, i.e. they live
1154 in the complex domain. As a consequence, operations like complex conjugation,
1155 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1156 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1157 because of the unknown imaginary part of @code{x}.
1158 On the other hand, if you are sure that your symbols will hold only real
1159 values, you would like to have such functions evaluated. Therefore GiNaC
1160 allows you to specify
1161 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1162 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @cindex @code{possymbol()}
1165 Furthermore, it is also possible to declare a symbol as positive. This will,
1166 for instance, enable the automatic simplification of @code{abs(x)} into
1167 @code{x}. This is done by declaying the symbol as @code{possymbol x("x");}.
1170 @node Numbers, Constants, Symbols, Basic concepts
1171 @c node-name, next, previous, up
1173 @cindex @code{numeric} (class)
1179 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1180 The classes therein serve as foundation classes for GiNaC. CLN stands
1181 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1182 In order to find out more about CLN's internals, the reader is referred to
1183 the documentation of that library. @inforef{Introduction, , cln}, for
1184 more information. Suffice to say that it is by itself build on top of
1185 another library, the GNU Multiple Precision library GMP, which is an
1186 extremely fast library for arbitrary long integers and rationals as well
1187 as arbitrary precision floating point numbers. It is very commonly used
1188 by several popular cryptographic applications. CLN extends GMP by
1189 several useful things: First, it introduces the complex number field
1190 over either reals (i.e. floating point numbers with arbitrary precision)
1191 or rationals. Second, it automatically converts rationals to integers
1192 if the denominator is unity and complex numbers to real numbers if the
1193 imaginary part vanishes and also correctly treats algebraic functions.
1194 Third it provides good implementations of state-of-the-art algorithms
1195 for all trigonometric and hyperbolic functions as well as for
1196 calculation of some useful constants.
1198 The user can construct an object of class @code{numeric} in several
1199 ways. The following example shows the four most important constructors.
1200 It uses construction from C-integer, construction of fractions from two
1201 integers, construction from C-float and construction from a string:
1205 #include <ginac/ginac.h>
1206 using namespace GiNaC;
1210 numeric two = 2; // exact integer 2
1211 numeric r(2,3); // exact fraction 2/3
1212 numeric e(2.71828); // floating point number
1213 numeric p = "3.14159265358979323846"; // constructor from string
1214 // Trott's constant in scientific notation:
1215 numeric trott("1.0841015122311136151E-2");
1217 std::cout << two*p << std::endl; // floating point 6.283...
1222 @cindex complex numbers
1223 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1228 numeric z1 = 2-3*I; // exact complex number 2-3i
1229 numeric z2 = 5.9+1.6*I; // complex floating point number
1233 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1234 This would, however, call C's built-in operator @code{/} for integers
1235 first and result in a numeric holding a plain integer 1. @strong{Never
1236 use the operator @code{/} on integers} unless you know exactly what you
1237 are doing! Use the constructor from two integers instead, as shown in
1238 the example above. Writing @code{numeric(1)/2} may look funny but works
1241 @cindex @code{Digits}
1243 We have seen now the distinction between exact numbers and floating
1244 point numbers. Clearly, the user should never have to worry about
1245 dynamically created exact numbers, since their `exactness' always
1246 determines how they ought to be handled, i.e. how `long' they are. The
1247 situation is different for floating point numbers. Their accuracy is
1248 controlled by one @emph{global} variable, called @code{Digits}. (For
1249 those readers who know about Maple: it behaves very much like Maple's
1250 @code{Digits}). All objects of class numeric that are constructed from
1251 then on will be stored with a precision matching that number of decimal
1256 #include <ginac/ginac.h>
1257 using namespace std;
1258 using namespace GiNaC;
1262 numeric three(3.0), one(1.0);
1263 numeric x = one/three;
1265 cout << "in " << Digits << " digits:" << endl;
1267 cout << Pi.evalf() << endl;
1279 The above example prints the following output to screen:
1283 0.33333333333333333334
1284 3.1415926535897932385
1286 0.33333333333333333333333333333333333333333333333333333333333333333334
1287 3.1415926535897932384626433832795028841971693993751058209749445923078
1291 Note that the last number is not necessarily rounded as you would
1292 naively expect it to be rounded in the decimal system. But note also,
1293 that in both cases you got a couple of extra digits. This is because
1294 numbers are internally stored by CLN as chunks of binary digits in order
1295 to match your machine's word size and to not waste precision. Thus, on
1296 architectures with different word size, the above output might even
1297 differ with regard to actually computed digits.
1299 It should be clear that objects of class @code{numeric} should be used
1300 for constructing numbers or for doing arithmetic with them. The objects
1301 one deals with most of the time are the polymorphic expressions @code{ex}.
1303 @subsection Tests on numbers
1305 Once you have declared some numbers, assigned them to expressions and
1306 done some arithmetic with them it is frequently desired to retrieve some
1307 kind of information from them like asking whether that number is
1308 integer, rational, real or complex. For those cases GiNaC provides
1309 several useful methods. (Internally, they fall back to invocations of
1310 certain CLN functions.)
1312 As an example, let's construct some rational number, multiply it with
1313 some multiple of its denominator and test what comes out:
1317 #include <ginac/ginac.h>
1318 using namespace std;
1319 using namespace GiNaC;
1321 // some very important constants:
1322 const numeric twentyone(21);
1323 const numeric ten(10);
1324 const numeric five(5);
1328 numeric answer = twentyone;
1331 cout << answer.is_integer() << endl; // false, it's 21/5
1333 cout << answer.is_integer() << endl; // true, it's 42 now!
1337 Note that the variable @code{answer} is constructed here as an integer
1338 by @code{numeric}'s copy constructor but in an intermediate step it
1339 holds a rational number represented as integer numerator and integer
1340 denominator. When multiplied by 10, the denominator becomes unity and
1341 the result is automatically converted to a pure integer again.
1342 Internally, the underlying CLN is responsible for this behavior and we
1343 refer the reader to CLN's documentation. Suffice to say that
1344 the same behavior applies to complex numbers as well as return values of
1345 certain functions. Complex numbers are automatically converted to real
1346 numbers if the imaginary part becomes zero. The full set of tests that
1347 can be applied is listed in the following table.
1350 @multitable @columnfractions .30 .70
1351 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1352 @item @code{.is_zero()}
1353 @tab @dots{}equal to zero
1354 @item @code{.is_positive()}
1355 @tab @dots{}not complex and greater than 0
1356 @item @code{.is_integer()}
1357 @tab @dots{}a (non-complex) integer
1358 @item @code{.is_pos_integer()}
1359 @tab @dots{}an integer and greater than 0
1360 @item @code{.is_nonneg_integer()}
1361 @tab @dots{}an integer and greater equal 0
1362 @item @code{.is_even()}
1363 @tab @dots{}an even integer
1364 @item @code{.is_odd()}
1365 @tab @dots{}an odd integer
1366 @item @code{.is_prime()}
1367 @tab @dots{}a prime integer (probabilistic primality test)
1368 @item @code{.is_rational()}
1369 @tab @dots{}an exact rational number (integers are rational, too)
1370 @item @code{.is_real()}
1371 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1372 @item @code{.is_cinteger()}
1373 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1374 @item @code{.is_crational()}
1375 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1379 @subsection Numeric functions
1381 The following functions can be applied to @code{numeric} objects and will be
1382 evaluated immediately:
1385 @multitable @columnfractions .30 .70
1386 @item @strong{Name} @tab @strong{Function}
1387 @item @code{inverse(z)}
1388 @tab returns @math{1/z}
1389 @cindex @code{inverse()} (numeric)
1390 @item @code{pow(a, b)}
1391 @tab exponentiation @math{a^b}
1394 @item @code{real(z)}
1396 @cindex @code{real()}
1397 @item @code{imag(z)}
1399 @cindex @code{imag()}
1400 @item @code{csgn(z)}
1401 @tab complex sign (returns an @code{int})
1402 @item @code{step(x)}
1403 @tab step function (returns an @code{numeric})
1404 @item @code{numer(z)}
1405 @tab numerator of rational or complex rational number
1406 @item @code{denom(z)}
1407 @tab denominator of rational or complex rational number
1408 @item @code{sqrt(z)}
1410 @item @code{isqrt(n)}
1411 @tab integer square root
1412 @cindex @code{isqrt()}
1419 @item @code{asin(z)}
1421 @item @code{acos(z)}
1423 @item @code{atan(z)}
1424 @tab inverse tangent
1425 @item @code{atan(y, x)}
1426 @tab inverse tangent with two arguments
1427 @item @code{sinh(z)}
1428 @tab hyperbolic sine
1429 @item @code{cosh(z)}
1430 @tab hyperbolic cosine
1431 @item @code{tanh(z)}
1432 @tab hyperbolic tangent
1433 @item @code{asinh(z)}
1434 @tab inverse hyperbolic sine
1435 @item @code{acosh(z)}
1436 @tab inverse hyperbolic cosine
1437 @item @code{atanh(z)}
1438 @tab inverse hyperbolic tangent
1440 @tab exponential function
1442 @tab natural logarithm
1445 @item @code{zeta(z)}
1446 @tab Riemann's zeta function
1447 @item @code{tgamma(z)}
1449 @item @code{lgamma(z)}
1450 @tab logarithm of gamma function
1452 @tab psi (digamma) function
1453 @item @code{psi(n, z)}
1454 @tab derivatives of psi function (polygamma functions)
1455 @item @code{factorial(n)}
1456 @tab factorial function @math{n!}
1457 @item @code{doublefactorial(n)}
1458 @tab double factorial function @math{n!!}
1459 @cindex @code{doublefactorial()}
1460 @item @code{binomial(n, k)}
1461 @tab binomial coefficients
1462 @item @code{bernoulli(n)}
1463 @tab Bernoulli numbers
1464 @cindex @code{bernoulli()}
1465 @item @code{fibonacci(n)}
1466 @tab Fibonacci numbers
1467 @cindex @code{fibonacci()}
1468 @item @code{mod(a, b)}
1469 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1470 @cindex @code{mod()}
1471 @item @code{smod(a, b)}
1472 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1473 @cindex @code{smod()}
1474 @item @code{irem(a, b)}
1475 @tab integer remainder (has the sign of @math{a}, or is zero)
1476 @cindex @code{irem()}
1477 @item @code{irem(a, b, q)}
1478 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1479 @item @code{iquo(a, b)}
1480 @tab integer quotient
1481 @cindex @code{iquo()}
1482 @item @code{iquo(a, b, r)}
1483 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1484 @item @code{gcd(a, b)}
1485 @tab greatest common divisor
1486 @item @code{lcm(a, b)}
1487 @tab least common multiple
1491 Most of these functions are also available as symbolic functions that can be
1492 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1493 as polynomial algorithms.
1495 @subsection Converting numbers
1497 Sometimes it is desirable to convert a @code{numeric} object back to a
1498 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1499 class provides a couple of methods for this purpose:
1501 @cindex @code{to_int()}
1502 @cindex @code{to_long()}
1503 @cindex @code{to_double()}
1504 @cindex @code{to_cl_N()}
1506 int numeric::to_int() const;
1507 long numeric::to_long() const;
1508 double numeric::to_double() const;
1509 cln::cl_N numeric::to_cl_N() const;
1512 @code{to_int()} and @code{to_long()} only work when the number they are
1513 applied on is an exact integer. Otherwise the program will halt with a
1514 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1515 rational number will return a floating-point approximation. Both
1516 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1517 part of complex numbers.
1520 @node Constants, Fundamental containers, Numbers, Basic concepts
1521 @c node-name, next, previous, up
1523 @cindex @code{constant} (class)
1526 @cindex @code{Catalan}
1527 @cindex @code{Euler}
1528 @cindex @code{evalf()}
1529 Constants behave pretty much like symbols except that they return some
1530 specific number when the method @code{.evalf()} is called.
1532 The predefined known constants are:
1535 @multitable @columnfractions .14 .30 .56
1536 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1538 @tab Archimedes' constant
1539 @tab 3.14159265358979323846264338327950288
1540 @item @code{Catalan}
1541 @tab Catalan's constant
1542 @tab 0.91596559417721901505460351493238411
1544 @tab Euler's (or Euler-Mascheroni) constant
1545 @tab 0.57721566490153286060651209008240243
1550 @node Fundamental containers, Lists, Constants, Basic concepts
1551 @c node-name, next, previous, up
1552 @section Sums, products and powers
1556 @cindex @code{power}
1558 Simple rational expressions are written down in GiNaC pretty much like
1559 in other CAS or like expressions involving numerical variables in C.
1560 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1561 been overloaded to achieve this goal. When you run the following
1562 code snippet, the constructor for an object of type @code{mul} is
1563 automatically called to hold the product of @code{a} and @code{b} and
1564 then the constructor for an object of type @code{add} is called to hold
1565 the sum of that @code{mul} object and the number one:
1569 symbol a("a"), b("b");
1574 @cindex @code{pow()}
1575 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1576 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1577 construction is necessary since we cannot safely overload the constructor
1578 @code{^} in C++ to construct a @code{power} object. If we did, it would
1579 have several counterintuitive and undesired effects:
1583 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1585 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1586 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1587 interpret this as @code{x^(a^b)}.
1589 Also, expressions involving integer exponents are very frequently used,
1590 which makes it even more dangerous to overload @code{^} since it is then
1591 hard to distinguish between the semantics as exponentiation and the one
1592 for exclusive or. (It would be embarrassing to return @code{1} where one
1593 has requested @code{2^3}.)
1596 @cindex @command{ginsh}
1597 All effects are contrary to mathematical notation and differ from the
1598 way most other CAS handle exponentiation, therefore overloading @code{^}
1599 is ruled out for GiNaC's C++ part. The situation is different in
1600 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1601 that the other frequently used exponentiation operator @code{**} does
1602 not exist at all in C++).
1604 To be somewhat more precise, objects of the three classes described
1605 here, are all containers for other expressions. An object of class
1606 @code{power} is best viewed as a container with two slots, one for the
1607 basis, one for the exponent. All valid GiNaC expressions can be
1608 inserted. However, basic transformations like simplifying
1609 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1610 when this is mathematically possible. If we replace the outer exponent
1611 three in the example by some symbols @code{a}, the simplification is not
1612 safe and will not be performed, since @code{a} might be @code{1/2} and
1615 Objects of type @code{add} and @code{mul} are containers with an
1616 arbitrary number of slots for expressions to be inserted. Again, simple
1617 and safe simplifications are carried out like transforming
1618 @code{3*x+4-x} to @code{2*x+4}.
1621 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1622 @c node-name, next, previous, up
1623 @section Lists of expressions
1624 @cindex @code{lst} (class)
1626 @cindex @code{nops()}
1628 @cindex @code{append()}
1629 @cindex @code{prepend()}
1630 @cindex @code{remove_first()}
1631 @cindex @code{remove_last()}
1632 @cindex @code{remove_all()}
1634 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1635 expressions. They are not as ubiquitous as in many other computer algebra
1636 packages, but are sometimes used to supply a variable number of arguments of
1637 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1638 constructors, so you should have a basic understanding of them.
1640 Lists can be constructed by assigning a comma-separated sequence of
1645 symbol x("x"), y("y");
1648 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1653 There are also constructors that allow direct creation of lists of up to
1654 16 expressions, which is often more convenient but slightly less efficient:
1658 // This produces the same list 'l' as above:
1659 // lst l(x, 2, y, x+y);
1660 // lst l = lst(x, 2, y, x+y);
1664 Use the @code{nops()} method to determine the size (number of expressions) of
1665 a list and the @code{op()} method or the @code{[]} operator to access
1666 individual elements:
1670 cout << l.nops() << endl; // prints '4'
1671 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1675 As with the standard @code{list<T>} container, accessing random elements of a
1676 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1677 sequential access to the elements of a list is possible with the
1678 iterator types provided by the @code{lst} class:
1681 typedef ... lst::const_iterator;
1682 typedef ... lst::const_reverse_iterator;
1683 lst::const_iterator lst::begin() const;
1684 lst::const_iterator lst::end() const;
1685 lst::const_reverse_iterator lst::rbegin() const;
1686 lst::const_reverse_iterator lst::rend() const;
1689 For example, to print the elements of a list individually you can use:
1694 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1699 which is one order faster than
1704 for (size_t i = 0; i < l.nops(); ++i)
1705 cout << l.op(i) << endl;
1709 These iterators also allow you to use some of the algorithms provided by
1710 the C++ standard library:
1714 // print the elements of the list (requires #include <iterator>)
1715 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1717 // sum up the elements of the list (requires #include <numeric>)
1718 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1719 cout << sum << endl; // prints '2+2*x+2*y'
1723 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1724 (the only other one is @code{matrix}). You can modify single elements:
1728 l[1] = 42; // l is now @{x, 42, y, x+y@}
1729 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1733 You can append or prepend an expression to a list with the @code{append()}
1734 and @code{prepend()} methods:
1738 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1739 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1743 You can remove the first or last element of a list with @code{remove_first()}
1744 and @code{remove_last()}:
1748 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1749 l.remove_last(); // l is now @{x, 7, y, x+y@}
1753 You can remove all the elements of a list with @code{remove_all()}:
1757 l.remove_all(); // l is now empty
1761 You can bring the elements of a list into a canonical order with @code{sort()}:
1770 // l1 and l2 are now equal
1774 Finally, you can remove all but the first element of consecutive groups of
1775 elements with @code{unique()}:
1780 l3 = x, 2, 2, 2, y, x+y, y+x;
1781 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1786 @node Mathematical functions, Relations, Lists, Basic concepts
1787 @c node-name, next, previous, up
1788 @section Mathematical functions
1789 @cindex @code{function} (class)
1790 @cindex trigonometric function
1791 @cindex hyperbolic function
1793 There are quite a number of useful functions hard-wired into GiNaC. For
1794 instance, all trigonometric and hyperbolic functions are implemented
1795 (@xref{Built-in functions}, for a complete list).
1797 These functions (better called @emph{pseudofunctions}) are all objects
1798 of class @code{function}. They accept one or more expressions as
1799 arguments and return one expression. If the arguments are not
1800 numerical, the evaluation of the function may be halted, as it does in
1801 the next example, showing how a function returns itself twice and
1802 finally an expression that may be really useful:
1804 @cindex Gamma function
1805 @cindex @code{subs()}
1808 symbol x("x"), y("y");
1810 cout << tgamma(foo) << endl;
1811 // -> tgamma(x+(1/2)*y)
1812 ex bar = foo.subs(y==1);
1813 cout << tgamma(bar) << endl;
1815 ex foobar = bar.subs(x==7);
1816 cout << tgamma(foobar) << endl;
1817 // -> (135135/128)*Pi^(1/2)
1821 Besides evaluation most of these functions allow differentiation, series
1822 expansion and so on. Read the next chapter in order to learn more about
1825 It must be noted that these pseudofunctions are created by inline
1826 functions, where the argument list is templated. This means that
1827 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1828 @code{sin(ex(1))} and will therefore not result in a floating point
1829 number. Unless of course the function prototype is explicitly
1830 overridden -- which is the case for arguments of type @code{numeric}
1831 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1832 point number of class @code{numeric} you should call
1833 @code{sin(numeric(1))}. This is almost the same as calling
1834 @code{sin(1).evalf()} except that the latter will return a numeric
1835 wrapped inside an @code{ex}.
1838 @node Relations, Integrals, Mathematical functions, Basic concepts
1839 @c node-name, next, previous, up
1841 @cindex @code{relational} (class)
1843 Sometimes, a relation holding between two expressions must be stored
1844 somehow. The class @code{relational} is a convenient container for such
1845 purposes. A relation is by definition a container for two @code{ex} and
1846 a relation between them that signals equality, inequality and so on.
1847 They are created by simply using the C++ operators @code{==}, @code{!=},
1848 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1850 @xref{Mathematical functions}, for examples where various applications
1851 of the @code{.subs()} method show how objects of class relational are
1852 used as arguments. There they provide an intuitive syntax for
1853 substitutions. They are also used as arguments to the @code{ex::series}
1854 method, where the left hand side of the relation specifies the variable
1855 to expand in and the right hand side the expansion point. They can also
1856 be used for creating systems of equations that are to be solved for
1857 unknown variables. But the most common usage of objects of this class
1858 is rather inconspicuous in statements of the form @code{if
1859 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1860 conversion from @code{relational} to @code{bool} takes place. Note,
1861 however, that @code{==} here does not perform any simplifications, hence
1862 @code{expand()} must be called explicitly.
1864 @node Integrals, Matrices, Relations, Basic concepts
1865 @c node-name, next, previous, up
1867 @cindex @code{integral} (class)
1869 An object of class @dfn{integral} can be used to hold a symbolic integral.
1870 If you want to symbolically represent the integral of @code{x*x} from 0 to
1871 1, you would write this as
1873 integral(x, 0, 1, x*x)
1875 The first argument is the integration variable. It should be noted that
1876 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1877 fact, it can only integrate polynomials. An expression containing integrals
1878 can be evaluated symbolically by calling the
1882 method on it. Numerical evaluation is available by calling the
1886 method on an expression containing the integral. This will only evaluate
1887 integrals into a number if @code{subs}ing the integration variable by a
1888 number in the fourth argument of an integral and then @code{evalf}ing the
1889 result always results in a number. Of course, also the boundaries of the
1890 integration domain must @code{evalf} into numbers. It should be noted that
1891 trying to @code{evalf} a function with discontinuities in the integration
1892 domain is not recommended. The accuracy of the numeric evaluation of
1893 integrals is determined by the static member variable
1895 ex integral::relative_integration_error
1897 of the class @code{integral}. The default value of this is 10^-8.
1898 The integration works by halving the interval of integration, until numeric
1899 stability of the answer indicates that the requested accuracy has been
1900 reached. The maximum depth of the halving can be set via the static member
1903 int integral::max_integration_level
1905 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1906 return the integral unevaluated. The function that performs the numerical
1907 evaluation, is also available as
1909 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1912 This function will throw an exception if the maximum depth is exceeded. The
1913 last parameter of the function is optional and defaults to the
1914 @code{relative_integration_error}. To make sure that we do not do too
1915 much work if an expression contains the same integral multiple times,
1916 a lookup table is used.
1918 If you know that an expression holds an integral, you can get the
1919 integration variable, the left boundary, right boundary and integrand by
1920 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1921 @code{.op(3)}. Differentiating integrals with respect to variables works
1922 as expected. Note that it makes no sense to differentiate an integral
1923 with respect to the integration variable.
1925 @node Matrices, Indexed objects, Integrals, Basic concepts
1926 @c node-name, next, previous, up
1928 @cindex @code{matrix} (class)
1930 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1931 matrix with @math{m} rows and @math{n} columns are accessed with two
1932 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1933 second one in the range 0@dots{}@math{n-1}.
1935 There are a couple of ways to construct matrices, with or without preset
1936 elements. The constructor
1939 matrix::matrix(unsigned r, unsigned c);
1942 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1945 The fastest way to create a matrix with preinitialized elements is to assign
1946 a list of comma-separated expressions to an empty matrix (see below for an
1947 example). But you can also specify the elements as a (flat) list with
1950 matrix::matrix(unsigned r, unsigned c, const lst & l);
1955 @cindex @code{lst_to_matrix()}
1957 ex lst_to_matrix(const lst & l);
1960 constructs a matrix from a list of lists, each list representing a matrix row.
1962 There is also a set of functions for creating some special types of
1965 @cindex @code{diag_matrix()}
1966 @cindex @code{unit_matrix()}
1967 @cindex @code{symbolic_matrix()}
1969 ex diag_matrix(const lst & l);
1970 ex unit_matrix(unsigned x);
1971 ex unit_matrix(unsigned r, unsigned c);
1972 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1973 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1974 const string & tex_base_name);
1977 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1978 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1979 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1980 matrix filled with newly generated symbols made of the specified base name
1981 and the position of each element in the matrix.
1983 Matrices often arise by omitting elements of another matrix. For
1984 instance, the submatrix @code{S} of a matrix @code{M} takes a
1985 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1986 by removing one row and one column from a matrix @code{M}. (The
1987 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1988 can be used for computing the inverse using Cramer's rule.)
1990 @cindex @code{sub_matrix()}
1991 @cindex @code{reduced_matrix()}
1993 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1994 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1997 The function @code{sub_matrix()} takes a row offset @code{r} and a
1998 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1999 columns. The function @code{reduced_matrix()} has two integer arguments
2000 that specify which row and column to remove:
2008 cout << reduced_matrix(m, 1, 1) << endl;
2009 // -> [[11,13],[31,33]]
2010 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2011 // -> [[22,23],[32,33]]
2015 Matrix elements can be accessed and set using the parenthesis (function call)
2019 const ex & matrix::operator()(unsigned r, unsigned c) const;
2020 ex & matrix::operator()(unsigned r, unsigned c);
2023 It is also possible to access the matrix elements in a linear fashion with
2024 the @code{op()} method. But C++-style subscripting with square brackets
2025 @samp{[]} is not available.
2027 Here are a couple of examples for constructing matrices:
2031 symbol a("a"), b("b");
2045 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2048 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2051 cout << diag_matrix(lst(a, b)) << endl;
2054 cout << unit_matrix(3) << endl;
2055 // -> [[1,0,0],[0,1,0],[0,0,1]]
2057 cout << symbolic_matrix(2, 3, "x") << endl;
2058 // -> [[x00,x01,x02],[x10,x11,x12]]
2062 @cindex @code{is_zero_matrix()}
2063 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2064 all entries of the matrix are zeros. There is also method
2065 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2066 expression is zero or a zero matrix.
2068 @cindex @code{transpose()}
2069 There are three ways to do arithmetic with matrices. The first (and most
2070 direct one) is to use the methods provided by the @code{matrix} class:
2073 matrix matrix::add(const matrix & other) const;
2074 matrix matrix::sub(const matrix & other) const;
2075 matrix matrix::mul(const matrix & other) const;
2076 matrix matrix::mul_scalar(const ex & other) const;
2077 matrix matrix::pow(const ex & expn) const;
2078 matrix matrix::transpose() const;
2081 All of these methods return the result as a new matrix object. Here is an
2082 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2087 matrix A(2, 2), B(2, 2), C(2, 2);
2095 matrix result = A.mul(B).sub(C.mul_scalar(2));
2096 cout << result << endl;
2097 // -> [[-13,-6],[1,2]]
2102 @cindex @code{evalm()}
2103 The second (and probably the most natural) way is to construct an expression
2104 containing matrices with the usual arithmetic operators and @code{pow()}.
2105 For efficiency reasons, expressions with sums, products and powers of
2106 matrices are not automatically evaluated in GiNaC. You have to call the
2110 ex ex::evalm() const;
2113 to obtain the result:
2120 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2121 cout << e.evalm() << endl;
2122 // -> [[-13,-6],[1,2]]
2127 The non-commutativity of the product @code{A*B} in this example is
2128 automatically recognized by GiNaC. There is no need to use a special
2129 operator here. @xref{Non-commutative objects}, for more information about
2130 dealing with non-commutative expressions.
2132 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2133 to perform the arithmetic:
2138 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2139 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2141 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2142 cout << e.simplify_indexed() << endl;
2143 // -> [[-13,-6],[1,2]].i.j
2147 Using indices is most useful when working with rectangular matrices and
2148 one-dimensional vectors because you don't have to worry about having to
2149 transpose matrices before multiplying them. @xref{Indexed objects}, for
2150 more information about using matrices with indices, and about indices in
2153 The @code{matrix} class provides a couple of additional methods for
2154 computing determinants, traces, characteristic polynomials and ranks:
2156 @cindex @code{determinant()}
2157 @cindex @code{trace()}
2158 @cindex @code{charpoly()}
2159 @cindex @code{rank()}
2161 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2162 ex matrix::trace() const;
2163 ex matrix::charpoly(const ex & lambda) const;
2164 unsigned matrix::rank() const;
2167 The @samp{algo} argument of @code{determinant()} allows to select
2168 between different algorithms for calculating the determinant. The
2169 asymptotic speed (as parametrized by the matrix size) can greatly differ
2170 between those algorithms, depending on the nature of the matrix'
2171 entries. The possible values are defined in the @file{flags.h} header
2172 file. By default, GiNaC uses a heuristic to automatically select an
2173 algorithm that is likely (but not guaranteed) to give the result most
2176 @cindex @code{inverse()} (matrix)
2177 @cindex @code{solve()}
2178 Matrices may also be inverted using the @code{ex matrix::inverse()}
2179 method and linear systems may be solved with:
2182 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2183 unsigned algo=solve_algo::automatic) const;
2186 Assuming the matrix object this method is applied on is an @code{m}
2187 times @code{n} matrix, then @code{vars} must be a @code{n} times
2188 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2189 times @code{p} matrix. The returned matrix then has dimension @code{n}
2190 times @code{p} and in the case of an underdetermined system will still
2191 contain some of the indeterminates from @code{vars}. If the system is
2192 overdetermined, an exception is thrown.
2195 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2196 @c node-name, next, previous, up
2197 @section Indexed objects
2199 GiNaC allows you to handle expressions containing general indexed objects in
2200 arbitrary spaces. It is also able to canonicalize and simplify such
2201 expressions and perform symbolic dummy index summations. There are a number
2202 of predefined indexed objects provided, like delta and metric tensors.
2204 There are few restrictions placed on indexed objects and their indices and
2205 it is easy to construct nonsense expressions, but our intention is to
2206 provide a general framework that allows you to implement algorithms with
2207 indexed quantities, getting in the way as little as possible.
2209 @cindex @code{idx} (class)
2210 @cindex @code{indexed} (class)
2211 @subsection Indexed quantities and their indices
2213 Indexed expressions in GiNaC are constructed of two special types of objects,
2214 @dfn{index objects} and @dfn{indexed objects}.
2218 @cindex contravariant
2221 @item Index objects are of class @code{idx} or a subclass. Every index has
2222 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2223 the index lives in) which can both be arbitrary expressions but are usually
2224 a number or a simple symbol. In addition, indices of class @code{varidx} have
2225 a @dfn{variance} (they can be co- or contravariant), and indices of class
2226 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2228 @item Indexed objects are of class @code{indexed} or a subclass. They
2229 contain a @dfn{base expression} (which is the expression being indexed), and
2230 one or more indices.
2234 @strong{Please notice:} when printing expressions, covariant indices and indices
2235 without variance are denoted @samp{.i} while contravariant indices are
2236 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2237 value. In the following, we are going to use that notation in the text so
2238 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2239 not visible in the output.
2241 A simple example shall illustrate the concepts:
2245 #include <ginac/ginac.h>
2246 using namespace std;
2247 using namespace GiNaC;
2251 symbol i_sym("i"), j_sym("j");
2252 idx i(i_sym, 3), j(j_sym, 3);
2255 cout << indexed(A, i, j) << endl;
2257 cout << index_dimensions << indexed(A, i, j) << endl;
2259 cout << dflt; // reset cout to default output format (dimensions hidden)
2263 The @code{idx} constructor takes two arguments, the index value and the
2264 index dimension. First we define two index objects, @code{i} and @code{j},
2265 both with the numeric dimension 3. The value of the index @code{i} is the
2266 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2267 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2268 construct an expression containing one indexed object, @samp{A.i.j}. It has
2269 the symbol @code{A} as its base expression and the two indices @code{i} and
2272 The dimensions of indices are normally not visible in the output, but one
2273 can request them to be printed with the @code{index_dimensions} manipulator,
2276 Note the difference between the indices @code{i} and @code{j} which are of
2277 class @code{idx}, and the index values which are the symbols @code{i_sym}
2278 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2279 or numbers but must be index objects. For example, the following is not
2280 correct and will raise an exception:
2283 symbol i("i"), j("j");
2284 e = indexed(A, i, j); // ERROR: indices must be of type idx
2287 You can have multiple indexed objects in an expression, index values can
2288 be numeric, and index dimensions symbolic:
2292 symbol B("B"), dim("dim");
2293 cout << 4 * indexed(A, i)
2294 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2299 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2300 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2301 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2302 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2303 @code{simplify_indexed()} for that, see below).
2305 In fact, base expressions, index values and index dimensions can be
2306 arbitrary expressions:
2310 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2315 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2316 get an error message from this but you will probably not be able to do
2317 anything useful with it.
2319 @cindex @code{get_value()}
2320 @cindex @code{get_dimension()}
2324 ex idx::get_value();
2325 ex idx::get_dimension();
2328 return the value and dimension of an @code{idx} object. If you have an index
2329 in an expression, such as returned by calling @code{.op()} on an indexed
2330 object, you can get a reference to the @code{idx} object with the function
2331 @code{ex_to<idx>()} on the expression.
2333 There are also the methods
2336 bool idx::is_numeric();
2337 bool idx::is_symbolic();
2338 bool idx::is_dim_numeric();
2339 bool idx::is_dim_symbolic();
2342 for checking whether the value and dimension are numeric or symbolic
2343 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2344 about expressions}) returns information about the index value.
2346 @cindex @code{varidx} (class)
2347 If you need co- and contravariant indices, use the @code{varidx} class:
2351 symbol mu_sym("mu"), nu_sym("nu");
2352 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2353 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2355 cout << indexed(A, mu, nu) << endl;
2357 cout << indexed(A, mu_co, nu) << endl;
2359 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2364 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2365 co- or contravariant. The default is a contravariant (upper) index, but
2366 this can be overridden by supplying a third argument to the @code{varidx}
2367 constructor. The two methods
2370 bool varidx::is_covariant();
2371 bool varidx::is_contravariant();
2374 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2375 to get the object reference from an expression). There's also the very useful
2379 ex varidx::toggle_variance();
2382 which makes a new index with the same value and dimension but the opposite
2383 variance. By using it you only have to define the index once.
2385 @cindex @code{spinidx} (class)
2386 The @code{spinidx} class provides dotted and undotted variant indices, as
2387 used in the Weyl-van-der-Waerden spinor formalism:
2391 symbol K("K"), C_sym("C"), D_sym("D");
2392 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2393 // contravariant, undotted
2394 spinidx C_co(C_sym, 2, true); // covariant index
2395 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2396 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2398 cout << indexed(K, C, D) << endl;
2400 cout << indexed(K, C_co, D_dot) << endl;
2402 cout << indexed(K, D_co_dot, D) << endl;
2407 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2408 dotted or undotted. The default is undotted but this can be overridden by
2409 supplying a fourth argument to the @code{spinidx} constructor. The two
2413 bool spinidx::is_dotted();
2414 bool spinidx::is_undotted();
2417 allow you to check whether or not a @code{spinidx} object is dotted (use
2418 @code{ex_to<spinidx>()} to get the object reference from an expression).
2419 Finally, the two methods
2422 ex spinidx::toggle_dot();
2423 ex spinidx::toggle_variance_dot();
2426 create a new index with the same value and dimension but opposite dottedness
2427 and the same or opposite variance.
2429 @subsection Substituting indices
2431 @cindex @code{subs()}
2432 Sometimes you will want to substitute one symbolic index with another
2433 symbolic or numeric index, for example when calculating one specific element
2434 of a tensor expression. This is done with the @code{.subs()} method, as it
2435 is done for symbols (see @ref{Substituting expressions}).
2437 You have two possibilities here. You can either substitute the whole index
2438 by another index or expression:
2442 ex e = indexed(A, mu_co);
2443 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2444 // -> A.mu becomes A~nu
2445 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2446 // -> A.mu becomes A~0
2447 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2448 // -> A.mu becomes A.0
2452 The third example shows that trying to replace an index with something that
2453 is not an index will substitute the index value instead.
2455 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2460 ex e = indexed(A, mu_co);
2461 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2462 // -> A.mu becomes A.nu
2463 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2464 // -> A.mu becomes A.0
2468 As you see, with the second method only the value of the index will get
2469 substituted. Its other properties, including its dimension, remain unchanged.
2470 If you want to change the dimension of an index you have to substitute the
2471 whole index by another one with the new dimension.
2473 Finally, substituting the base expression of an indexed object works as
2478 ex e = indexed(A, mu_co);
2479 cout << e << " becomes " << e.subs(A == A+B) << endl;
2480 // -> A.mu becomes (B+A).mu
2484 @subsection Symmetries
2485 @cindex @code{symmetry} (class)
2486 @cindex @code{sy_none()}
2487 @cindex @code{sy_symm()}
2488 @cindex @code{sy_anti()}
2489 @cindex @code{sy_cycl()}
2491 Indexed objects can have certain symmetry properties with respect to their
2492 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2493 that is constructed with the helper functions
2496 symmetry sy_none(...);
2497 symmetry sy_symm(...);
2498 symmetry sy_anti(...);
2499 symmetry sy_cycl(...);
2502 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2503 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2504 represents a cyclic symmetry. Each of these functions accepts up to four
2505 arguments which can be either symmetry objects themselves or unsigned integer
2506 numbers that represent an index position (counting from 0). A symmetry
2507 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2508 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2511 Here are some examples of symmetry definitions:
2516 e = indexed(A, i, j);
2517 e = indexed(A, sy_none(), i, j); // equivalent
2518 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2520 // Symmetric in all three indices:
2521 e = indexed(A, sy_symm(), i, j, k);
2522 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2523 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2524 // different canonical order
2526 // Symmetric in the first two indices only:
2527 e = indexed(A, sy_symm(0, 1), i, j, k);
2528 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2530 // Antisymmetric in the first and last index only (index ranges need not
2532 e = indexed(A, sy_anti(0, 2), i, j, k);
2533 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2535 // An example of a mixed symmetry: antisymmetric in the first two and
2536 // last two indices, symmetric when swapping the first and last index
2537 // pairs (like the Riemann curvature tensor):
2538 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2540 // Cyclic symmetry in all three indices:
2541 e = indexed(A, sy_cycl(), i, j, k);
2542 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2544 // The following examples are invalid constructions that will throw
2545 // an exception at run time.
2547 // An index may not appear multiple times:
2548 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2549 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2551 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2552 // same number of indices:
2553 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2555 // And of course, you cannot specify indices which are not there:
2556 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2560 If you need to specify more than four indices, you have to use the
2561 @code{.add()} method of the @code{symmetry} class. For example, to specify
2562 full symmetry in the first six indices you would write
2563 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2565 If an indexed object has a symmetry, GiNaC will automatically bring the
2566 indices into a canonical order which allows for some immediate simplifications:
2570 cout << indexed(A, sy_symm(), i, j)
2571 + indexed(A, sy_symm(), j, i) << endl;
2573 cout << indexed(B, sy_anti(), i, j)
2574 + indexed(B, sy_anti(), j, i) << endl;
2576 cout << indexed(B, sy_anti(), i, j, k)
2577 - indexed(B, sy_anti(), j, k, i) << endl;
2582 @cindex @code{get_free_indices()}
2584 @subsection Dummy indices
2586 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2587 that a summation over the index range is implied. Symbolic indices which are
2588 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2589 dummy nor free indices.
2591 To be recognized as a dummy index pair, the two indices must be of the same
2592 class and their value must be the same single symbol (an index like
2593 @samp{2*n+1} is never a dummy index). If the indices are of class
2594 @code{varidx} they must also be of opposite variance; if they are of class
2595 @code{spinidx} they must be both dotted or both undotted.
2597 The method @code{.get_free_indices()} returns a vector containing the free
2598 indices of an expression. It also checks that the free indices of the terms
2599 of a sum are consistent:
2603 symbol A("A"), B("B"), C("C");
2605 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2606 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2608 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2609 cout << exprseq(e.get_free_indices()) << endl;
2611 // 'j' and 'l' are dummy indices
2613 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2614 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2616 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2617 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2618 cout << exprseq(e.get_free_indices()) << endl;
2620 // 'nu' is a dummy index, but 'sigma' is not
2622 e = indexed(A, mu, mu);
2623 cout << exprseq(e.get_free_indices()) << endl;
2625 // 'mu' is not a dummy index because it appears twice with the same
2628 e = indexed(A, mu, nu) + 42;
2629 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2630 // this will throw an exception:
2631 // "add::get_free_indices: inconsistent indices in sum"
2635 @cindex @code{expand_dummy_sum()}
2636 A dummy index summation like
2643 can be expanded for indices with numeric
2644 dimensions (e.g. 3) into the explicit sum like
2646 $a_1b^1+a_2b^2+a_3b^3 $.
2649 a.1 b~1 + a.2 b~2 + a.3 b~3.
2651 This is performed by the function
2654 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2657 which takes an expression @code{e} and returns the expanded sum for all
2658 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2659 is set to @code{true} then all substitutions are made by @code{idx} class
2660 indices, i.e. without variance. In this case the above sum
2669 $a_1b_1+a_2b_2+a_3b_3 $.
2672 a.1 b.1 + a.2 b.2 + a.3 b.3.
2676 @cindex @code{simplify_indexed()}
2677 @subsection Simplifying indexed expressions
2679 In addition to the few automatic simplifications that GiNaC performs on
2680 indexed expressions (such as re-ordering the indices of symmetric tensors
2681 and calculating traces and convolutions of matrices and predefined tensors)
2685 ex ex::simplify_indexed();
2686 ex ex::simplify_indexed(const scalar_products & sp);
2689 that performs some more expensive operations:
2692 @item it checks the consistency of free indices in sums in the same way
2693 @code{get_free_indices()} does
2694 @item it tries to give dummy indices that appear in different terms of a sum
2695 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2696 @item it (symbolically) calculates all possible dummy index summations/contractions
2697 with the predefined tensors (this will be explained in more detail in the
2699 @item it detects contractions that vanish for symmetry reasons, for example
2700 the contraction of a symmetric and a totally antisymmetric tensor
2701 @item as a special case of dummy index summation, it can replace scalar products
2702 of two tensors with a user-defined value
2705 The last point is done with the help of the @code{scalar_products} class
2706 which is used to store scalar products with known values (this is not an
2707 arithmetic class, you just pass it to @code{simplify_indexed()}):
2711 symbol A("A"), B("B"), C("C"), i_sym("i");
2715 sp.add(A, B, 0); // A and B are orthogonal
2716 sp.add(A, C, 0); // A and C are orthogonal
2717 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2719 e = indexed(A + B, i) * indexed(A + C, i);
2721 // -> (B+A).i*(A+C).i
2723 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2729 The @code{scalar_products} object @code{sp} acts as a storage for the
2730 scalar products added to it with the @code{.add()} method. This method
2731 takes three arguments: the two expressions of which the scalar product is
2732 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2733 @code{simplify_indexed()} will replace all scalar products of indexed
2734 objects that have the symbols @code{A} and @code{B} as base expressions
2735 with the single value 0. The number, type and dimension of the indices
2736 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2738 @cindex @code{expand()}
2739 The example above also illustrates a feature of the @code{expand()} method:
2740 if passed the @code{expand_indexed} option it will distribute indices
2741 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2743 @cindex @code{tensor} (class)
2744 @subsection Predefined tensors
2746 Some frequently used special tensors such as the delta, epsilon and metric
2747 tensors are predefined in GiNaC. They have special properties when
2748 contracted with other tensor expressions and some of them have constant
2749 matrix representations (they will evaluate to a number when numeric
2750 indices are specified).
2752 @cindex @code{delta_tensor()}
2753 @subsubsection Delta tensor
2755 The delta tensor takes two indices, is symmetric and has the matrix
2756 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2757 @code{delta_tensor()}:
2761 symbol A("A"), B("B");
2763 idx i(symbol("i"), 3), j(symbol("j"), 3),
2764 k(symbol("k"), 3), l(symbol("l"), 3);
2766 ex e = indexed(A, i, j) * indexed(B, k, l)
2767 * delta_tensor(i, k) * delta_tensor(j, l);
2768 cout << e.simplify_indexed() << endl;
2771 cout << delta_tensor(i, i) << endl;
2776 @cindex @code{metric_tensor()}
2777 @subsubsection General metric tensor
2779 The function @code{metric_tensor()} creates a general symmetric metric
2780 tensor with two indices that can be used to raise/lower tensor indices. The
2781 metric tensor is denoted as @samp{g} in the output and if its indices are of
2782 mixed variance it is automatically replaced by a delta tensor:
2788 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2790 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2791 cout << e.simplify_indexed() << endl;
2794 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2795 cout << e.simplify_indexed() << endl;
2798 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2799 * metric_tensor(nu, rho);
2800 cout << e.simplify_indexed() << endl;
2803 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2804 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2805 + indexed(A, mu.toggle_variance(), rho));
2806 cout << e.simplify_indexed() << endl;
2811 @cindex @code{lorentz_g()}
2812 @subsubsection Minkowski metric tensor
2814 The Minkowski metric tensor is a special metric tensor with a constant
2815 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2816 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2817 It is created with the function @code{lorentz_g()} (although it is output as
2822 varidx mu(symbol("mu"), 4);
2824 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2825 * lorentz_g(mu, varidx(0, 4)); // negative signature
2826 cout << e.simplify_indexed() << endl;
2829 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2830 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2831 cout << e.simplify_indexed() << endl;
2836 @cindex @code{spinor_metric()}
2837 @subsubsection Spinor metric tensor
2839 The function @code{spinor_metric()} creates an antisymmetric tensor with
2840 two indices that is used to raise/lower indices of 2-component spinors.
2841 It is output as @samp{eps}:
2847 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2848 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2850 e = spinor_metric(A, B) * indexed(psi, B_co);
2851 cout << e.simplify_indexed() << endl;
2854 e = spinor_metric(A, B) * indexed(psi, A_co);
2855 cout << e.simplify_indexed() << endl;
2858 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2859 cout << e.simplify_indexed() << endl;
2862 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2863 cout << e.simplify_indexed() << endl;
2866 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2867 cout << e.simplify_indexed() << endl;
2870 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2871 cout << e.simplify_indexed() << endl;
2876 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2878 @cindex @code{epsilon_tensor()}
2879 @cindex @code{lorentz_eps()}
2880 @subsubsection Epsilon tensor
2882 The epsilon tensor is totally antisymmetric, its number of indices is equal
2883 to the dimension of the index space (the indices must all be of the same
2884 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2885 defined to be 1. Its behavior with indices that have a variance also
2886 depends on the signature of the metric. Epsilon tensors are output as
2889 There are three functions defined to create epsilon tensors in 2, 3 and 4
2893 ex epsilon_tensor(const ex & i1, const ex & i2);
2894 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2895 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2896 bool pos_sig = false);
2899 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2900 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2901 Minkowski space (the last @code{bool} argument specifies whether the metric
2902 has negative or positive signature, as in the case of the Minkowski metric
2907 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2908 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2909 e = lorentz_eps(mu, nu, rho, sig) *
2910 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2911 cout << simplify_indexed(e) << endl;
2912 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2914 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2915 symbol A("A"), B("B");
2916 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2917 cout << simplify_indexed(e) << endl;
2918 // -> -B.k*A.j*eps.i.k.j
2919 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2920 cout << simplify_indexed(e) << endl;
2925 @subsection Linear algebra
2927 The @code{matrix} class can be used with indices to do some simple linear
2928 algebra (linear combinations and products of vectors and matrices, traces
2929 and scalar products):
2933 idx i(symbol("i"), 2), j(symbol("j"), 2);
2934 symbol x("x"), y("y");
2936 // A is a 2x2 matrix, X is a 2x1 vector
2937 matrix A(2, 2), X(2, 1);
2942 cout << indexed(A, i, i) << endl;
2945 ex e = indexed(A, i, j) * indexed(X, j);
2946 cout << e.simplify_indexed() << endl;
2947 // -> [[2*y+x],[4*y+3*x]].i
2949 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2950 cout << e.simplify_indexed() << endl;
2951 // -> [[3*y+3*x,6*y+2*x]].j
2955 You can of course obtain the same results with the @code{matrix::add()},
2956 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2957 but with indices you don't have to worry about transposing matrices.
2959 Matrix indices always start at 0 and their dimension must match the number
2960 of rows/columns of the matrix. Matrices with one row or one column are
2961 vectors and can have one or two indices (it doesn't matter whether it's a
2962 row or a column vector). Other matrices must have two indices.
2964 You should be careful when using indices with variance on matrices. GiNaC
2965 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2966 @samp{F.mu.nu} are different matrices. In this case you should use only
2967 one form for @samp{F} and explicitly multiply it with a matrix representation
2968 of the metric tensor.
2971 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2972 @c node-name, next, previous, up
2973 @section Non-commutative objects
2975 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2976 non-commutative objects are built-in which are mostly of use in high energy
2980 @item Clifford (Dirac) algebra (class @code{clifford})
2981 @item su(3) Lie algebra (class @code{color})
2982 @item Matrices (unindexed) (class @code{matrix})
2985 The @code{clifford} and @code{color} classes are subclasses of
2986 @code{indexed} because the elements of these algebras usually carry
2987 indices. The @code{matrix} class is described in more detail in
2990 Unlike most computer algebra systems, GiNaC does not primarily provide an
2991 operator (often denoted @samp{&*}) for representing inert products of
2992 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2993 classes of objects involved, and non-commutative products are formed with
2994 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2995 figuring out by itself which objects commutate and will group the factors
2996 by their class. Consider this example:
3000 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3001 idx a(symbol("a"), 8), b(symbol("b"), 8);
3002 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3004 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3008 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3009 groups the non-commutative factors (the gammas and the su(3) generators)
3010 together while preserving the order of factors within each class (because
3011 Clifford objects commutate with color objects). The resulting expression is a
3012 @emph{commutative} product with two factors that are themselves non-commutative
3013 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3014 parentheses are placed around the non-commutative products in the output.
3016 @cindex @code{ncmul} (class)
3017 Non-commutative products are internally represented by objects of the class
3018 @code{ncmul}, as opposed to commutative products which are handled by the
3019 @code{mul} class. You will normally not have to worry about this distinction,
3022 The advantage of this approach is that you never have to worry about using
3023 (or forgetting to use) a special operator when constructing non-commutative
3024 expressions. Also, non-commutative products in GiNaC are more intelligent
3025 than in other computer algebra systems; they can, for example, automatically
3026 canonicalize themselves according to rules specified in the implementation
3027 of the non-commutative classes. The drawback is that to work with other than
3028 the built-in algebras you have to implement new classes yourself. Both
3029 symbols and user-defined functions can be specified as being non-commutative.
3031 @cindex @code{return_type()}
3032 @cindex @code{return_type_tinfo()}
3033 Information about the commutativity of an object or expression can be
3034 obtained with the two member functions
3037 unsigned ex::return_type() const;
3038 unsigned ex::return_type_tinfo() const;
3041 The @code{return_type()} function returns one of three values (defined in
3042 the header file @file{flags.h}), corresponding to three categories of
3043 expressions in GiNaC:
3046 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3047 classes are of this kind.
3048 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3049 certain class of non-commutative objects which can be determined with the
3050 @code{return_type_tinfo()} method. Expressions of this category commutate
3051 with everything except @code{noncommutative} expressions of the same
3053 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3054 of non-commutative objects of different classes. Expressions of this
3055 category don't commutate with any other @code{noncommutative} or
3056 @code{noncommutative_composite} expressions.
3059 The value returned by the @code{return_type_tinfo()} method is valid only
3060 when the return type of the expression is @code{noncommutative}. It is a
3061 value that is unique to the class of the object and usually one of the
3062 constants in @file{tinfos.h}, or derived therefrom.
3064 Here are a couple of examples:
3067 @multitable @columnfractions 0.33 0.33 0.34
3068 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3069 @item @code{42} @tab @code{commutative} @tab -
3070 @item @code{2*x-y} @tab @code{commutative} @tab -
3071 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3072 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3073 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3074 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3078 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3079 @code{TINFO_clifford} for objects with a representation label of zero.
3080 Other representation labels yield a different @code{return_type_tinfo()},
3081 but it's the same for any two objects with the same label. This is also true
3084 A last note: With the exception of matrices, positive integer powers of
3085 non-commutative objects are automatically expanded in GiNaC. For example,
3086 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3087 non-commutative expressions).
3090 @cindex @code{clifford} (class)
3091 @subsection Clifford algebra
3094 Clifford algebras are supported in two flavours: Dirac gamma
3095 matrices (more physical) and generic Clifford algebras (more
3098 @cindex @code{dirac_gamma()}
3099 @subsubsection Dirac gamma matrices
3100 Dirac gamma matrices (note that GiNaC doesn't treat them
3101 as matrices) are designated as @samp{gamma~mu} and satisfy
3102 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3103 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3104 constructed by the function
3107 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3110 which takes two arguments: the index and a @dfn{representation label} in the
3111 range 0 to 255 which is used to distinguish elements of different Clifford
3112 algebras (this is also called a @dfn{spin line index}). Gammas with different
3113 labels commutate with each other. The dimension of the index can be 4 or (in
3114 the framework of dimensional regularization) any symbolic value. Spinor
3115 indices on Dirac gammas are not supported in GiNaC.
3117 @cindex @code{dirac_ONE()}
3118 The unity element of a Clifford algebra is constructed by
3121 ex dirac_ONE(unsigned char rl = 0);
3124 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3125 multiples of the unity element, even though it's customary to omit it.
3126 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3127 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3128 GiNaC will complain and/or produce incorrect results.
3130 @cindex @code{dirac_gamma5()}
3131 There is a special element @samp{gamma5} that commutates with all other
3132 gammas, has a unit square, and in 4 dimensions equals
3133 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3136 ex dirac_gamma5(unsigned char rl = 0);
3139 @cindex @code{dirac_gammaL()}
3140 @cindex @code{dirac_gammaR()}
3141 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3142 objects, constructed by
3145 ex dirac_gammaL(unsigned char rl = 0);
3146 ex dirac_gammaR(unsigned char rl = 0);
3149 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3150 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3152 @cindex @code{dirac_slash()}
3153 Finally, the function
3156 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3159 creates a term that represents a contraction of @samp{e} with the Dirac
3160 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3161 with a unique index whose dimension is given by the @code{dim} argument).
3162 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3164 In products of dirac gammas, superfluous unity elements are automatically
3165 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3166 and @samp{gammaR} are moved to the front.
3168 The @code{simplify_indexed()} function performs contractions in gamma strings,
3174 symbol a("a"), b("b"), D("D");
3175 varidx mu(symbol("mu"), D);
3176 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3177 * dirac_gamma(mu.toggle_variance());
3179 // -> gamma~mu*a\*gamma.mu
3180 e = e.simplify_indexed();
3183 cout << e.subs(D == 4) << endl;
3189 @cindex @code{dirac_trace()}
3190 To calculate the trace of an expression containing strings of Dirac gammas
3191 you use one of the functions
3194 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3195 const ex & trONE = 4);
3196 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3197 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3200 These functions take the trace over all gammas in the specified set @code{rls}
3201 or list @code{rll} of representation labels, or the single label @code{rl};
3202 gammas with other labels are left standing. The last argument to
3203 @code{dirac_trace()} is the value to be returned for the trace of the unity
3204 element, which defaults to 4.
3206 The @code{dirac_trace()} function is a linear functional that is equal to the
3207 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3208 functional is not cyclic in
3211 dimensions when acting on
3212 expressions containing @samp{gamma5}, so it's not a proper trace. This
3213 @samp{gamma5} scheme is described in greater detail in
3214 @cite{The Role of gamma5 in Dimensional Regularization}.
3216 The value of the trace itself is also usually different in 4 and in
3224 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3225 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3226 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3227 cout << dirac_trace(e).simplify_indexed() << endl;
3234 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3235 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3236 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3237 cout << dirac_trace(e).simplify_indexed() << endl;
3238 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3242 Here is an example for using @code{dirac_trace()} to compute a value that
3243 appears in the calculation of the one-loop vacuum polarization amplitude in
3248 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3249 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3252 sp.add(l, l, pow(l, 2));
3253 sp.add(l, q, ldotq);
3255 ex e = dirac_gamma(mu) *
3256 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3257 dirac_gamma(mu.toggle_variance()) *
3258 (dirac_slash(l, D) + m * dirac_ONE());
3259 e = dirac_trace(e).simplify_indexed(sp);
3260 e = e.collect(lst(l, ldotq, m));
3262 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3266 The @code{canonicalize_clifford()} function reorders all gamma products that
3267 appear in an expression to a canonical (but not necessarily simple) form.
3268 You can use this to compare two expressions or for further simplifications:
3272 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3273 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3275 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3277 e = canonicalize_clifford(e);
3279 // -> 2*ONE*eta~mu~nu
3283 @cindex @code{clifford_unit()}
3284 @subsubsection A generic Clifford algebra
3286 A generic Clifford algebra, i.e. a
3290 dimensional algebra with
3294 satisfying the identities
3296 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3299 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3301 for some bilinear form (@code{metric})
3302 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3303 and contain symbolic entries. Such generators are created by the
3307 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3308 bool anticommuting = false);
3311 where @code{mu} should be a @code{varidx} class object indexing the
3312 generators, an index @code{mu} with a numeric value may be of type
3314 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3315 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3316 object. In fact, any expression either with two free indices or without
3317 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3318 object with two newly created indices with @code{metr} as its
3319 @code{op(0)} will be used.
3320 Optional parameter @code{rl} allows to distinguish different
3321 Clifford algebras, which will commute with each other. The last
3322 optional parameter @code{anticommuting} defines if the anticommuting
3325 $e_i e_j + e_j e_i = 0$)
3328 e~i e~j + e~j e~i = 0)
3330 will be used for contraction of Clifford units. If the @code{metric} is
3331 supplied by a @code{matrix} object, then the value of
3332 @code{anticommuting} is calculated automatically and the supplied one
3333 will be ignored. One can overcome this by giving @code{metric} through
3334 matrix wrapped into an @code{indexed} object.
3336 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3337 something very close to @code{dirac_gamma(mu)}, although
3338 @code{dirac_gamma} have more efficient simplification mechanism.
3339 @cindex @code{clifford::get_metric()}
3340 The method @code{clifford::get_metric()} returns a metric defining this
3342 @cindex @code{clifford::is_anticommuting()}
3343 The method @code{clifford::is_anticommuting()} returns the
3344 @code{anticommuting} property of a unit.
3346 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3347 the Clifford algebra units with a call like that
3350 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3353 since this may yield some further automatic simplifications. Again, for a
3354 metric defined through a @code{matrix} such a symmetry is detected
3357 Individual generators of a Clifford algebra can be accessed in several
3363 varidx nu(symbol("nu"), 4);
3365 ex M = diag_matrix(lst(1, -1, 0, s));
3366 ex e = clifford_unit(nu, M);
3367 ex e0 = e.subs(nu == 0);
3368 ex e1 = e.subs(nu == 1);
3369 ex e2 = e.subs(nu == 2);
3370 ex e3 = e.subs(nu == 3);
3375 will produce four anti-commuting generators of a Clifford algebra with properties
3377 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3380 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3381 @code{pow(e3, 2) = s}.
3384 @cindex @code{lst_to_clifford()}
3385 A similar effect can be achieved from the function
3388 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3389 unsigned char rl = 0, bool anticommuting = false);
3390 ex lst_to_clifford(const ex & v, const ex & e);
3393 which converts a list or vector
3395 $v = (v^0, v^1, ..., v^n)$
3398 @samp{v = (v~0, v~1, ..., v~n)}
3403 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3406 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3409 directly supplied in the second form of the procedure. In the first form
3410 the Clifford unit @samp{e.k} is generated by the call of
3411 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3412 with the help of @code{lst_to_clifford()} as follows
3417 varidx nu(symbol("nu"), 4);
3419 ex M = diag_matrix(lst(1, -1, 0, s));
3420 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3421 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3422 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3423 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3428 @cindex @code{clifford_to_lst()}
3429 There is the inverse function
3432 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3435 which takes an expression @code{e} and tries to find a list
3437 $v = (v^0, v^1, ..., v^n)$
3440 @samp{v = (v~0, v~1, ..., v~n)}
3444 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3447 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3449 with respect to the given Clifford units @code{c} and with none of the
3450 @samp{v~k} containing Clifford units @code{c} (of course, this
3451 may be impossible). This function can use an @code{algebraic} method
3452 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3454 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3457 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3459 is zero or is not @code{numeric} for some @samp{k}
3460 then the method will be automatically changed to symbolic. The same effect
3461 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3463 @cindex @code{clifford_prime()}
3464 @cindex @code{clifford_star()}
3465 @cindex @code{clifford_bar()}
3466 There are several functions for (anti-)automorphisms of Clifford algebras:
3469 ex clifford_prime(const ex & e)
3470 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3471 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3474 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3475 changes signs of all Clifford units in the expression. The reversion
3476 of a Clifford algebra @code{clifford_star()} coincides with the
3477 @code{conjugate()} method and effectively reverses the order of Clifford
3478 units in any product. Finally the main anti-automorphism
3479 of a Clifford algebra @code{clifford_bar()} is the composition of the
3480 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3481 in a product. These functions correspond to the notations
3496 used in Clifford algebra textbooks.
3498 @cindex @code{clifford_norm()}
3502 ex clifford_norm(const ex & e);
3505 @cindex @code{clifford_inverse()}
3506 calculates the norm of a Clifford number from the expression
3508 $||e||^2 = e\overline{e}$.
3511 @code{||e||^2 = e \bar@{e@}}
3513 The inverse of a Clifford expression is returned by the function
3516 ex clifford_inverse(const ex & e);
3519 which calculates it as
3521 $e^{-1} = \overline{e}/||e||^2$.
3524 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3533 then an exception is raised.
3535 @cindex @code{remove_dirac_ONE()}
3536 If a Clifford number happens to be a factor of
3537 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3538 expression by the function
3541 ex remove_dirac_ONE(const ex & e);
3544 @cindex @code{canonicalize_clifford()}
3545 The function @code{canonicalize_clifford()} works for a
3546 generic Clifford algebra in a similar way as for Dirac gammas.
3548 The next provided function is
3550 @cindex @code{clifford_moebius_map()}
3552 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3553 const ex & d, const ex & v, const ex & G,
3554 unsigned char rl = 0, bool anticommuting = false);
3555 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3556 unsigned char rl = 0, bool anticommuting = false);
3559 It takes a list or vector @code{v} and makes the Moebius (conformal or
3560 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3561 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3562 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3563 indexed object, tensormetric, matrix or a Clifford unit, in the later
3564 case the optional parameters @code{rl} and @code{anticommuting} are
3565 ignored even if supplied. Depending from the type of @code{v} the
3566 returned value of this function is either a vector or a list holding vector's
3569 @cindex @code{clifford_max_label()}
3570 Finally the function
3573 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3576 can detect a presence of Clifford objects in the expression @code{e}: if
3577 such objects are found it returns the maximal
3578 @code{representation_label} of them, otherwise @code{-1}. The optional
3579 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3580 be ignored during the search.
3582 LaTeX output for Clifford units looks like
3583 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3584 @code{representation_label} and @code{\nu} is the index of the
3585 corresponding unit. This provides a flexible typesetting with a suitable
3586 defintion of the @code{\clifford} command. For example, the definition
3588 \newcommand@{\clifford@}[1][]@{@}
3590 typesets all Clifford units identically, while the alternative definition
3592 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3594 prints units with @code{representation_label=0} as
3601 with @code{representation_label=1} as
3608 and with @code{representation_label=2} as
3616 @cindex @code{color} (class)
3617 @subsection Color algebra
3619 @cindex @code{color_T()}
3620 For computations in quantum chromodynamics, GiNaC implements the base elements
3621 and structure constants of the su(3) Lie algebra (color algebra). The base
3622 elements @math{T_a} are constructed by the function
3625 ex color_T(const ex & a, unsigned char rl = 0);
3628 which takes two arguments: the index and a @dfn{representation label} in the
3629 range 0 to 255 which is used to distinguish elements of different color
3630 algebras. Objects with different labels commutate with each other. The
3631 dimension of the index must be exactly 8 and it should be of class @code{idx},
3634 @cindex @code{color_ONE()}
3635 The unity element of a color algebra is constructed by
3638 ex color_ONE(unsigned char rl = 0);
3641 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3642 multiples of the unity element, even though it's customary to omit it.
3643 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3644 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3645 GiNaC may produce incorrect results.
3647 @cindex @code{color_d()}
3648 @cindex @code{color_f()}
3652 ex color_d(const ex & a, const ex & b, const ex & c);
3653 ex color_f(const ex & a, const ex & b, const ex & c);
3656 create the symmetric and antisymmetric structure constants @math{d_abc} and
3657 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3658 and @math{[T_a, T_b] = i f_abc T_c}.
3660 These functions evaluate to their numerical values,
3661 if you supply numeric indices to them. The index values should be in
3662 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3663 goes along better with the notations used in physical literature.
3665 @cindex @code{color_h()}
3666 There's an additional function
3669 ex color_h(const ex & a, const ex & b, const ex & c);
3672 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3674 The function @code{simplify_indexed()} performs some simplifications on
3675 expressions containing color objects:
3680 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3681 k(symbol("k"), 8), l(symbol("l"), 8);
3683 e = color_d(a, b, l) * color_f(a, b, k);
3684 cout << e.simplify_indexed() << endl;
3687 e = color_d(a, b, l) * color_d(a, b, k);
3688 cout << e.simplify_indexed() << endl;
3691 e = color_f(l, a, b) * color_f(a, b, k);
3692 cout << e.simplify_indexed() << endl;
3695 e = color_h(a, b, c) * color_h(a, b, c);
3696 cout << e.simplify_indexed() << endl;
3699 e = color_h(a, b, c) * color_T(b) * color_T(c);
3700 cout << e.simplify_indexed() << endl;
3703 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3704 cout << e.simplify_indexed() << endl;
3707 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3708 cout << e.simplify_indexed() << endl;
3709 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3713 @cindex @code{color_trace()}
3714 To calculate the trace of an expression containing color objects you use one
3718 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3719 ex color_trace(const ex & e, const lst & rll);
3720 ex color_trace(const ex & e, unsigned char rl = 0);
3723 These functions take the trace over all color @samp{T} objects in the
3724 specified set @code{rls} or list @code{rll} of representation labels, or the
3725 single label @code{rl}; @samp{T}s with other labels are left standing. For
3730 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3732 // -> -I*f.a.c.b+d.a.c.b
3737 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3738 @c node-name, next, previous, up
3741 @cindex @code{exhashmap} (class)
3743 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3744 that can be used as a drop-in replacement for the STL
3745 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3746 typically constant-time, element look-up than @code{map<>}.
3748 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3749 following differences:
3753 no @code{lower_bound()} and @code{upper_bound()} methods
3755 no reverse iterators, no @code{rbegin()}/@code{rend()}
3757 no @code{operator<(exhashmap, exhashmap)}
3759 the comparison function object @code{key_compare} is hardcoded to
3762 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3763 initial hash table size (the actual table size after construction may be
3764 larger than the specified value)
3766 the method @code{size_t bucket_count()} returns the current size of the hash
3769 @code{insert()} and @code{erase()} operations invalidate all iterators
3773 @node Methods and functions, Information about expressions, Hash maps, Top
3774 @c node-name, next, previous, up
3775 @chapter Methods and functions
3778 In this chapter the most important algorithms provided by GiNaC will be
3779 described. Some of them are implemented as functions on expressions,
3780 others are implemented as methods provided by expression objects. If
3781 they are methods, there exists a wrapper function around it, so you can
3782 alternatively call it in a functional way as shown in the simple
3787 cout << "As method: " << sin(1).evalf() << endl;
3788 cout << "As function: " << evalf(sin(1)) << endl;
3792 @cindex @code{subs()}
3793 The general rule is that wherever methods accept one or more parameters
3794 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3795 wrapper accepts is the same but preceded by the object to act on
3796 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3797 most natural one in an OO model but it may lead to confusion for MapleV
3798 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3799 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3800 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3801 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3802 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3803 here. Also, users of MuPAD will in most cases feel more comfortable
3804 with GiNaC's convention. All function wrappers are implemented
3805 as simple inline functions which just call the corresponding method and
3806 are only provided for users uncomfortable with OO who are dead set to
3807 avoid method invocations. Generally, nested function wrappers are much
3808 harder to read than a sequence of methods and should therefore be
3809 avoided if possible. On the other hand, not everything in GiNaC is a
3810 method on class @code{ex} and sometimes calling a function cannot be
3814 * Information about expressions::
3815 * Numerical evaluation::
3816 * Substituting expressions::
3817 * Pattern matching and advanced substitutions::
3818 * Applying a function on subexpressions::
3819 * Visitors and tree traversal::
3820 * Polynomial arithmetic:: Working with polynomials.
3821 * Rational expressions:: Working with rational functions.
3822 * Symbolic differentiation::
3823 * Series expansion:: Taylor and Laurent expansion.
3825 * Built-in functions:: List of predefined mathematical functions.
3826 * Multiple polylogarithms::
3827 * Complex expressions::
3828 * Solving linear systems of equations::
3829 * Input/output:: Input and output of expressions.
3833 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3834 @c node-name, next, previous, up
3835 @section Getting information about expressions
3837 @subsection Checking expression types
3838 @cindex @code{is_a<@dots{}>()}
3839 @cindex @code{is_exactly_a<@dots{}>()}
3840 @cindex @code{ex_to<@dots{}>()}
3841 @cindex Converting @code{ex} to other classes
3842 @cindex @code{info()}
3843 @cindex @code{return_type()}
3844 @cindex @code{return_type_tinfo()}
3846 Sometimes it's useful to check whether a given expression is a plain number,
3847 a sum, a polynomial with integer coefficients, or of some other specific type.
3848 GiNaC provides a couple of functions for this:
3851 bool is_a<T>(const ex & e);
3852 bool is_exactly_a<T>(const ex & e);
3853 bool ex::info(unsigned flag);
3854 unsigned ex::return_type() const;
3855 unsigned ex::return_type_tinfo() const;
3858 When the test made by @code{is_a<T>()} returns true, it is safe to call
3859 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3860 class names (@xref{The class hierarchy}, for a list of all classes). For
3861 example, assuming @code{e} is an @code{ex}:
3866 if (is_a<numeric>(e))
3867 numeric n = ex_to<numeric>(e);
3872 @code{is_a<T>(e)} allows you to check whether the top-level object of
3873 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3874 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3875 e.g., for checking whether an expression is a number, a sum, or a product:
3882 is_a<numeric>(e1); // true
3883 is_a<numeric>(e2); // false
3884 is_a<add>(e1); // false
3885 is_a<add>(e2); // true
3886 is_a<mul>(e1); // false
3887 is_a<mul>(e2); // false
3891 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3892 top-level object of an expression @samp{e} is an instance of the GiNaC
3893 class @samp{T}, not including parent classes.
3895 The @code{info()} method is used for checking certain attributes of
3896 expressions. The possible values for the @code{flag} argument are defined
3897 in @file{ginac/flags.h}, the most important being explained in the following
3901 @multitable @columnfractions .30 .70
3902 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3903 @item @code{numeric}
3904 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3906 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3907 @item @code{rational}
3908 @tab @dots{}an exact rational number (integers are rational, too)
3909 @item @code{integer}
3910 @tab @dots{}a (non-complex) integer
3911 @item @code{crational}
3912 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3913 @item @code{cinteger}
3914 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3915 @item @code{positive}
3916 @tab @dots{}not complex and greater than 0
3917 @item @code{negative}
3918 @tab @dots{}not complex and less than 0
3919 @item @code{nonnegative}
3920 @tab @dots{}not complex and greater than or equal to 0
3922 @tab @dots{}an integer greater than 0
3924 @tab @dots{}an integer less than 0
3925 @item @code{nonnegint}
3926 @tab @dots{}an integer greater than or equal to 0
3928 @tab @dots{}an even integer
3930 @tab @dots{}an odd integer
3932 @tab @dots{}a prime integer (probabilistic primality test)
3933 @item @code{relation}
3934 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3935 @item @code{relation_equal}
3936 @tab @dots{}a @code{==} relation
3937 @item @code{relation_not_equal}
3938 @tab @dots{}a @code{!=} relation
3939 @item @code{relation_less}
3940 @tab @dots{}a @code{<} relation
3941 @item @code{relation_less_or_equal}
3942 @tab @dots{}a @code{<=} relation
3943 @item @code{relation_greater}
3944 @tab @dots{}a @code{>} relation
3945 @item @code{relation_greater_or_equal}
3946 @tab @dots{}a @code{>=} relation
3948 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3950 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3951 @item @code{polynomial}
3952 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3953 @item @code{integer_polynomial}
3954 @tab @dots{}a polynomial with (non-complex) integer coefficients
3955 @item @code{cinteger_polynomial}
3956 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3957 @item @code{rational_polynomial}
3958 @tab @dots{}a polynomial with (non-complex) rational coefficients
3959 @item @code{crational_polynomial}
3960 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3961 @item @code{rational_function}
3962 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3963 @item @code{algebraic}
3964 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3968 To determine whether an expression is commutative or non-commutative and if
3969 so, with which other expressions it would commutate, you use the methods
3970 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3971 for an explanation of these.
3974 @subsection Accessing subexpressions
3977 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3978 @code{function}, act as containers for subexpressions. For example, the
3979 subexpressions of a sum (an @code{add} object) are the individual terms,
3980 and the subexpressions of a @code{function} are the function's arguments.
3982 @cindex @code{nops()}
3984 GiNaC provides several ways of accessing subexpressions. The first way is to
3989 ex ex::op(size_t i);
3992 @code{nops()} determines the number of subexpressions (operands) contained
3993 in the expression, while @code{op(i)} returns the @code{i}-th
3994 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3995 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3996 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3997 @math{i>0} are the indices.
4000 @cindex @code{const_iterator}
4001 The second way to access subexpressions is via the STL-style random-access
4002 iterator class @code{const_iterator} and the methods
4005 const_iterator ex::begin();
4006 const_iterator ex::end();
4009 @code{begin()} returns an iterator referring to the first subexpression;
4010 @code{end()} returns an iterator which is one-past the last subexpression.
4011 If the expression has no subexpressions, then @code{begin() == end()}. These
4012 iterators can also be used in conjunction with non-modifying STL algorithms.
4014 Here is an example that (non-recursively) prints the subexpressions of a
4015 given expression in three different ways:
4022 for (size_t i = 0; i != e.nops(); ++i)
4023 cout << e.op(i) << endl;
4026 for (const_iterator i = e.begin(); i != e.end(); ++i)
4029 // with iterators and STL copy()
4030 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4034 @cindex @code{const_preorder_iterator}
4035 @cindex @code{const_postorder_iterator}
4036 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4037 expression's immediate children. GiNaC provides two additional iterator
4038 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4039 that iterate over all objects in an expression tree, in preorder or postorder,
4040 respectively. They are STL-style forward iterators, and are created with the
4044 const_preorder_iterator ex::preorder_begin();
4045 const_preorder_iterator ex::preorder_end();
4046 const_postorder_iterator ex::postorder_begin();
4047 const_postorder_iterator ex::postorder_end();
4050 The following example illustrates the differences between
4051 @code{const_iterator}, @code{const_preorder_iterator}, and
4052 @code{const_postorder_iterator}:
4056 symbol A("A"), B("B"), C("C");
4057 ex e = lst(lst(A, B), C);
4059 std::copy(e.begin(), e.end(),
4060 std::ostream_iterator<ex>(cout, "\n"));
4064 std::copy(e.preorder_begin(), e.preorder_end(),
4065 std::ostream_iterator<ex>(cout, "\n"));
4072 std::copy(e.postorder_begin(), e.postorder_end(),
4073 std::ostream_iterator<ex>(cout, "\n"));
4082 @cindex @code{relational} (class)
4083 Finally, the left-hand side and right-hand side expressions of objects of
4084 class @code{relational} (and only of these) can also be accessed with the
4093 @subsection Comparing expressions
4094 @cindex @code{is_equal()}
4095 @cindex @code{is_zero()}
4097 Expressions can be compared with the usual C++ relational operators like
4098 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4099 the result is usually not determinable and the result will be @code{false},
4100 except in the case of the @code{!=} operator. You should also be aware that
4101 GiNaC will only do the most trivial test for equality (subtracting both
4102 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4105 Actually, if you construct an expression like @code{a == b}, this will be
4106 represented by an object of the @code{relational} class (@pxref{Relations})
4107 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4109 There are also two methods
4112 bool ex::is_equal(const ex & other);
4116 for checking whether one expression is equal to another, or equal to zero,
4117 respectively. See also the method @code{ex::is_zero_matrix()},
4121 @subsection Ordering expressions
4122 @cindex @code{ex_is_less} (class)
4123 @cindex @code{ex_is_equal} (class)
4124 @cindex @code{compare()}
4126 Sometimes it is necessary to establish a mathematically well-defined ordering
4127 on a set of arbitrary expressions, for example to use expressions as keys
4128 in a @code{std::map<>} container, or to bring a vector of expressions into
4129 a canonical order (which is done internally by GiNaC for sums and products).
4131 The operators @code{<}, @code{>} etc. described in the last section cannot
4132 be used for this, as they don't implement an ordering relation in the
4133 mathematical sense. In particular, they are not guaranteed to be
4134 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4135 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4138 By default, STL classes and algorithms use the @code{<} and @code{==}
4139 operators to compare objects, which are unsuitable for expressions, but GiNaC
4140 provides two functors that can be supplied as proper binary comparison
4141 predicates to the STL:
4144 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4146 bool operator()(const ex &lh, const ex &rh) const;
4149 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4151 bool operator()(const ex &lh, const ex &rh) const;
4155 For example, to define a @code{map} that maps expressions to strings you
4159 std::map<ex, std::string, ex_is_less> myMap;
4162 Omitting the @code{ex_is_less} template parameter will introduce spurious
4163 bugs because the map operates improperly.
4165 Other examples for the use of the functors:
4173 std::sort(v.begin(), v.end(), ex_is_less());
4175 // count the number of expressions equal to '1'
4176 unsigned num_ones = std::count_if(v.begin(), v.end(),
4177 std::bind2nd(ex_is_equal(), 1));
4180 The implementation of @code{ex_is_less} uses the member function
4183 int ex::compare(const ex & other) const;
4186 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4187 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4191 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4192 @c node-name, next, previous, up
4193 @section Numerical evaluation
4194 @cindex @code{evalf()}
4196 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4197 To evaluate them using floating-point arithmetic you need to call
4200 ex ex::evalf(int level = 0) const;
4203 @cindex @code{Digits}
4204 The accuracy of the evaluation is controlled by the global object @code{Digits}
4205 which can be assigned an integer value. The default value of @code{Digits}
4206 is 17. @xref{Numbers}, for more information and examples.
4208 To evaluate an expression to a @code{double} floating-point number you can
4209 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4213 // Approximate sin(x/Pi)
4215 ex e = series(sin(x/Pi), x == 0, 6);
4217 // Evaluate numerically at x=0.1
4218 ex f = evalf(e.subs(x == 0.1));
4220 // ex_to<numeric> is an unsafe cast, so check the type first
4221 if (is_a<numeric>(f)) @{
4222 double d = ex_to<numeric>(f).to_double();
4231 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4232 @c node-name, next, previous, up
4233 @section Substituting expressions
4234 @cindex @code{subs()}
4236 Algebraic objects inside expressions can be replaced with arbitrary
4237 expressions via the @code{.subs()} method:
4240 ex ex::subs(const ex & e, unsigned options = 0);
4241 ex ex::subs(const exmap & m, unsigned options = 0);
4242 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4245 In the first form, @code{subs()} accepts a relational of the form
4246 @samp{object == expression} or a @code{lst} of such relationals:
4250 symbol x("x"), y("y");
4252 ex e1 = 2*x^2-4*x+3;
4253 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4257 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4262 If you specify multiple substitutions, they are performed in parallel, so e.g.
4263 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4265 The second form of @code{subs()} takes an @code{exmap} object which is a
4266 pair associative container that maps expressions to expressions (currently
4267 implemented as a @code{std::map}). This is the most efficient one of the
4268 three @code{subs()} forms and should be used when the number of objects to
4269 be substituted is large or unknown.
4271 Using this form, the second example from above would look like this:
4275 symbol x("x"), y("y");
4281 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4285 The third form of @code{subs()} takes two lists, one for the objects to be
4286 replaced and one for the expressions to be substituted (both lists must
4287 contain the same number of elements). Using this form, you would write
4291 symbol x("x"), y("y");
4294 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4298 The optional last argument to @code{subs()} is a combination of
4299 @code{subs_options} flags. There are three options available:
4300 @code{subs_options::no_pattern} disables pattern matching, which makes
4301 large @code{subs()} operations significantly faster if you are not using
4302 patterns. The second option, @code{subs_options::algebraic} enables
4303 algebraic substitutions in products and powers.
4304 @ref{Pattern matching and advanced substitutions}, for more information
4305 about patterns and algebraic substitutions. The third option,
4306 @code{subs_options::no_index_renaming} disables the feature that dummy
4307 indices are renamed if the subsitution could give a result in which a
4308 dummy index occurs more than two times. This is sometimes necessary if
4309 you want to use @code{subs()} to rename your dummy indices.
4311 @code{subs()} performs syntactic substitution of any complete algebraic
4312 object; it does not try to match sub-expressions as is demonstrated by the
4317 symbol x("x"), y("y"), z("z");
4319 ex e1 = pow(x+y, 2);
4320 cout << e1.subs(x+y == 4) << endl;
4323 ex e2 = sin(x)*sin(y)*cos(x);
4324 cout << e2.subs(sin(x) == cos(x)) << endl;
4325 // -> cos(x)^2*sin(y)
4328 cout << e3.subs(x+y == 4) << endl;
4330 // (and not 4+z as one might expect)
4334 A more powerful form of substitution using wildcards is described in the
4338 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4339 @c node-name, next, previous, up
4340 @section Pattern matching and advanced substitutions
4341 @cindex @code{wildcard} (class)
4342 @cindex Pattern matching
4344 GiNaC allows the use of patterns for checking whether an expression is of a
4345 certain form or contains subexpressions of a certain form, and for
4346 substituting expressions in a more general way.
4348 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4349 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4350 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4351 an unsigned integer number to allow having multiple different wildcards in a
4352 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4353 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4357 ex wild(unsigned label = 0);
4360 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4363 Some examples for patterns:
4365 @multitable @columnfractions .5 .5
4366 @item @strong{Constructed as} @tab @strong{Output as}
4367 @item @code{wild()} @tab @samp{$0}
4368 @item @code{pow(x,wild())} @tab @samp{x^$0}
4369 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4370 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4376 @item Wildcards behave like symbols and are subject to the same algebraic
4377 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4378 @item As shown in the last example, to use wildcards for indices you have to
4379 use them as the value of an @code{idx} object. This is because indices must
4380 always be of class @code{idx} (or a subclass).
4381 @item Wildcards only represent expressions or subexpressions. It is not
4382 possible to use them as placeholders for other properties like index
4383 dimension or variance, representation labels, symmetry of indexed objects
4385 @item Because wildcards are commutative, it is not possible to use wildcards
4386 as part of noncommutative products.
4387 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4388 are also valid patterns.
4391 @subsection Matching expressions
4392 @cindex @code{match()}
4393 The most basic application of patterns is to check whether an expression
4394 matches a given pattern. This is done by the function
4397 bool ex::match(const ex & pattern);
4398 bool ex::match(const ex & pattern, lst & repls);
4401 This function returns @code{true} when the expression matches the pattern
4402 and @code{false} if it doesn't. If used in the second form, the actual
4403 subexpressions matched by the wildcards get returned in the @code{repls}
4404 object as a list of relations of the form @samp{wildcard == expression}.
4405 If @code{match()} returns false, the state of @code{repls} is undefined.
4406 For reproducible results, the list should be empty when passed to
4407 @code{match()}, but it is also possible to find similarities in multiple
4408 expressions by passing in the result of a previous match.
4410 The matching algorithm works as follows:
4413 @item A single wildcard matches any expression. If one wildcard appears
4414 multiple times in a pattern, it must match the same expression in all
4415 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4416 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4417 @item If the expression is not of the same class as the pattern, the match
4418 fails (i.e. a sum only matches a sum, a function only matches a function,
4420 @item If the pattern is a function, it only matches the same function
4421 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4422 @item Except for sums and products, the match fails if the number of
4423 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4425 @item If there are no subexpressions, the expressions and the pattern must
4426 be equal (in the sense of @code{is_equal()}).
4427 @item Except for sums and products, each subexpression (@code{op()}) must
4428 match the corresponding subexpression of the pattern.
4431 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4432 account for their commutativity and associativity:
4435 @item If the pattern contains a term or factor that is a single wildcard,
4436 this one is used as the @dfn{global wildcard}. If there is more than one
4437 such wildcard, one of them is chosen as the global wildcard in a random
4439 @item Every term/factor of the pattern, except the global wildcard, is
4440 matched against every term of the expression in sequence. If no match is
4441 found, the whole match fails. Terms that did match are not considered in
4443 @item If there are no unmatched terms left, the match succeeds. Otherwise
4444 the match fails unless there is a global wildcard in the pattern, in
4445 which case this wildcard matches the remaining terms.
4448 In general, having more than one single wildcard as a term of a sum or a
4449 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4452 Here are some examples in @command{ginsh} to demonstrate how it works (the
4453 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4454 match fails, and the list of wildcard replacements otherwise):
4457 > match((x+y)^a,(x+y)^a);
4459 > match((x+y)^a,(x+y)^b);
4461 > match((x+y)^a,$1^$2);
4463 > match((x+y)^a,$1^$1);
4465 > match((x+y)^(x+y),$1^$1);
4467 > match((x+y)^(x+y),$1^$2);
4469 > match((a+b)*(a+c),($1+b)*($1+c));
4471 > match((a+b)*(a+c),(a+$1)*(a+$2));
4473 (Unpredictable. The result might also be [$1==c,$2==b].)
4474 > match((a+b)*(a+c),($1+$2)*($1+$3));
4475 (The result is undefined. Due to the sequential nature of the algorithm
4476 and the re-ordering of terms in GiNaC, the match for the first factor
4477 may be @{$1==a,$2==b@} in which case the match for the second factor
4478 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4480 > match(a*(x+y)+a*z+b,a*$1+$2);
4481 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4482 @{$1=x+y,$2=a*z+b@}.)
4483 > match(a+b+c+d+e+f,c);
4485 > match(a+b+c+d+e+f,c+$0);
4487 > match(a+b+c+d+e+f,c+e+$0);
4489 > match(a+b,a+b+$0);
4491 > match(a*b^2,a^$1*b^$2);
4493 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4494 even though a==a^1.)
4495 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4497 > match(atan2(y,x^2),atan2(y,$0));
4501 @subsection Matching parts of expressions
4502 @cindex @code{has()}
4503 A more general way to look for patterns in expressions is provided by the
4507 bool ex::has(const ex & pattern);
4510 This function checks whether a pattern is matched by an expression itself or
4511 by any of its subexpressions.
4513 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4514 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4517 > has(x*sin(x+y+2*a),y);
4519 > has(x*sin(x+y+2*a),x+y);
4521 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4522 has the subexpressions "x", "y" and "2*a".)
4523 > has(x*sin(x+y+2*a),x+y+$1);
4525 (But this is possible.)
4526 > has(x*sin(2*(x+y)+2*a),x+y);
4528 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4529 which "x+y" is not a subexpression.)
4532 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4534 > has(4*x^2-x+3,$1*x);
4536 > has(4*x^2+x+3,$1*x);
4538 (Another possible pitfall. The first expression matches because the term
4539 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4540 contains a linear term you should use the coeff() function instead.)
4543 @cindex @code{find()}
4547 bool ex::find(const ex & pattern, lst & found);
4550 works a bit like @code{has()} but it doesn't stop upon finding the first
4551 match. Instead, it appends all found matches to the specified list. If there
4552 are multiple occurrences of the same expression, it is entered only once to
4553 the list. @code{find()} returns false if no matches were found (in
4554 @command{ginsh}, it returns an empty list):
4557 > find(1+x+x^2+x^3,x);
4559 > find(1+x+x^2+x^3,y);
4561 > find(1+x+x^2+x^3,x^$1);
4563 (Note the absence of "x".)
4564 > expand((sin(x)+sin(y))*(a+b));
4565 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4570 @subsection Substituting expressions
4571 @cindex @code{subs()}
4572 Probably the most useful application of patterns is to use them for
4573 substituting expressions with the @code{subs()} method. Wildcards can be
4574 used in the search patterns as well as in the replacement expressions, where
4575 they get replaced by the expressions matched by them. @code{subs()} doesn't
4576 know anything about algebra; it performs purely syntactic substitutions.
4581 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4583 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4585 > subs((a+b+c)^2,a+b==x);
4587 > subs((a+b+c)^2,a+b+$1==x+$1);
4589 > subs(a+2*b,a+b==x);
4591 > subs(4*x^3-2*x^2+5*x-1,x==a);
4593 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4595 > subs(sin(1+sin(x)),sin($1)==cos($1));
4597 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4601 The last example would be written in C++ in this way:
4605 symbol a("a"), b("b"), x("x"), y("y");
4606 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4607 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4608 cout << e.expand() << endl;
4613 @subsection The option algebraic
4614 Both @code{has()} and @code{subs()} take an optional argument to pass them
4615 extra options. This section describes what happens if you give the former
4616 the option @code{has_options::algebraic} or the latter
4617 @code{subs:options::algebraic}. In that case the matching condition for
4618 powers and multiplications is changed in such a way that they become
4619 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4620 If you use these options you will find that
4621 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4622 Besides matching some of the factors of a product also powers match as
4623 often as is possible without getting negative exponents. For example
4624 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4625 @code{x*c^2*z}. This also works with negative powers:
4626 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4627 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4628 and not for locating @code{x+y} within @code{x+y+z}.
4631 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4632 @c node-name, next, previous, up
4633 @section Applying a function on subexpressions
4634 @cindex tree traversal
4635 @cindex @code{map()}
4637 Sometimes you may want to perform an operation on specific parts of an
4638 expression while leaving the general structure of it intact. An example
4639 of this would be a matrix trace operation: the trace of a sum is the sum
4640 of the traces of the individual terms. That is, the trace should @dfn{map}
4641 on the sum, by applying itself to each of the sum's operands. It is possible
4642 to do this manually which usually results in code like this:
4647 if (is_a<matrix>(e))
4648 return ex_to<matrix>(e).trace();
4649 else if (is_a<add>(e)) @{
4651 for (size_t i=0; i<e.nops(); i++)
4652 sum += calc_trace(e.op(i));
4654 @} else if (is_a<mul>)(e)) @{
4662 This is, however, slightly inefficient (if the sum is very large it can take
4663 a long time to add the terms one-by-one), and its applicability is limited to
4664 a rather small class of expressions. If @code{calc_trace()} is called with
4665 a relation or a list as its argument, you will probably want the trace to
4666 be taken on both sides of the relation or of all elements of the list.
4668 GiNaC offers the @code{map()} method to aid in the implementation of such
4672 ex ex::map(map_function & f) const;
4673 ex ex::map(ex (*f)(const ex & e)) const;
4676 In the first (preferred) form, @code{map()} takes a function object that
4677 is subclassed from the @code{map_function} class. In the second form, it
4678 takes a pointer to a function that accepts and returns an expression.
4679 @code{map()} constructs a new expression of the same type, applying the
4680 specified function on all subexpressions (in the sense of @code{op()}),
4683 The use of a function object makes it possible to supply more arguments to
4684 the function that is being mapped, or to keep local state information.
4685 The @code{map_function} class declares a virtual function call operator
4686 that you can overload. Here is a sample implementation of @code{calc_trace()}
4687 that uses @code{map()} in a recursive fashion:
4690 struct calc_trace : public map_function @{
4691 ex operator()(const ex &e)
4693 if (is_a<matrix>(e))
4694 return ex_to<matrix>(e).trace();
4695 else if (is_a<mul>(e)) @{
4698 return e.map(*this);
4703 This function object could then be used like this:
4707 ex M = ... // expression with matrices
4708 calc_trace do_trace;
4709 ex tr = do_trace(M);
4713 Here is another example for you to meditate over. It removes quadratic
4714 terms in a variable from an expanded polynomial:
4717 struct map_rem_quad : public map_function @{
4719 map_rem_quad(const ex & var_) : var(var_) @{@}
4721 ex operator()(const ex & e)
4723 if (is_a<add>(e) || is_a<mul>(e))
4724 return e.map(*this);
4725 else if (is_a<power>(e) &&
4726 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4736 symbol x("x"), y("y");
4739 for (int i=0; i<8; i++)
4740 e += pow(x, i) * pow(y, 8-i) * (i+1);
4742 // -> 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
4744 map_rem_quad rem_quad(x);
4745 cout << rem_quad(e) << endl;
4746 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4750 @command{ginsh} offers a slightly different implementation of @code{map()}
4751 that allows applying algebraic functions to operands. The second argument
4752 to @code{map()} is an expression containing the wildcard @samp{$0} which
4753 acts as the placeholder for the operands:
4758 > map(a+2*b,sin($0));
4760 > map(@{a,b,c@},$0^2+$0);
4761 @{a^2+a,b^2+b,c^2+c@}
4764 Note that it is only possible to use algebraic functions in the second
4765 argument. You can not use functions like @samp{diff()}, @samp{op()},
4766 @samp{subs()} etc. because these are evaluated immediately:
4769 > map(@{a,b,c@},diff($0,a));
4771 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4772 to "map(@{a,b,c@},0)".
4776 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4777 @c node-name, next, previous, up
4778 @section Visitors and tree traversal
4779 @cindex tree traversal
4780 @cindex @code{visitor} (class)
4781 @cindex @code{accept()}
4782 @cindex @code{visit()}
4783 @cindex @code{traverse()}
4784 @cindex @code{traverse_preorder()}
4785 @cindex @code{traverse_postorder()}
4787 Suppose that you need a function that returns a list of all indices appearing
4788 in an arbitrary expression. The indices can have any dimension, and for
4789 indices with variance you always want the covariant version returned.
4791 You can't use @code{get_free_indices()} because you also want to include
4792 dummy indices in the list, and you can't use @code{find()} as it needs
4793 specific index dimensions (and it would require two passes: one for indices
4794 with variance, one for plain ones).
4796 The obvious solution to this problem is a tree traversal with a type switch,
4797 such as the following:
4800 void gather_indices_helper(const ex & e, lst & l)
4802 if (is_a<varidx>(e)) @{
4803 const varidx & vi = ex_to<varidx>(e);
4804 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4805 @} else if (is_a<idx>(e)) @{
4808 size_t n = e.nops();
4809 for (size_t i = 0; i < n; ++i)
4810 gather_indices_helper(e.op(i), l);
4814 lst gather_indices(const ex & e)
4817 gather_indices_helper(e, l);
4824 This works fine but fans of object-oriented programming will feel
4825 uncomfortable with the type switch. One reason is that there is a possibility
4826 for subtle bugs regarding derived classes. If we had, for example, written
4829 if (is_a<idx>(e)) @{
4831 @} else if (is_a<varidx>(e)) @{
4835 in @code{gather_indices_helper}, the code wouldn't have worked because the
4836 first line "absorbs" all classes derived from @code{idx}, including
4837 @code{varidx}, so the special case for @code{varidx} would never have been
4840 Also, for a large number of classes, a type switch like the above can get
4841 unwieldy and inefficient (it's a linear search, after all).
4842 @code{gather_indices_helper} only checks for two classes, but if you had to
4843 write a function that required a different implementation for nearly
4844 every GiNaC class, the result would be very hard to maintain and extend.
4846 The cleanest approach to the problem would be to add a new virtual function
4847 to GiNaC's class hierarchy. In our example, there would be specializations
4848 for @code{idx} and @code{varidx} while the default implementation in
4849 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4850 impossible to add virtual member functions to existing classes without
4851 changing their source and recompiling everything. GiNaC comes with source,
4852 so you could actually do this, but for a small algorithm like the one
4853 presented this would be impractical.
4855 One solution to this dilemma is the @dfn{Visitor} design pattern,
4856 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4857 variation, described in detail in
4858 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4859 virtual functions to the class hierarchy to implement operations, GiNaC
4860 provides a single "bouncing" method @code{accept()} that takes an instance
4861 of a special @code{visitor} class and redirects execution to the one
4862 @code{visit()} virtual function of the visitor that matches the type of
4863 object that @code{accept()} was being invoked on.
4865 Visitors in GiNaC must derive from the global @code{visitor} class as well
4866 as from the class @code{T::visitor} of each class @code{T} they want to
4867 visit, and implement the member functions @code{void visit(const T &)} for
4873 void ex::accept(visitor & v) const;
4876 will then dispatch to the correct @code{visit()} member function of the
4877 specified visitor @code{v} for the type of GiNaC object at the root of the
4878 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4880 Here is an example of a visitor:
4884 : public visitor, // this is required
4885 public add::visitor, // visit add objects
4886 public numeric::visitor, // visit numeric objects
4887 public basic::visitor // visit basic objects
4889 void visit(const add & x)
4890 @{ cout << "called with an add object" << endl; @}
4892 void visit(const numeric & x)
4893 @{ cout << "called with a numeric object" << endl; @}
4895 void visit(const basic & x)
4896 @{ cout << "called with a basic object" << endl; @}
4900 which can be used as follows:
4911 // prints "called with a numeric object"
4913 // prints "called with an add object"
4915 // prints "called with a basic object"
4919 The @code{visit(const basic &)} method gets called for all objects that are
4920 not @code{numeric} or @code{add} and acts as an (optional) default.
4922 From a conceptual point of view, the @code{visit()} methods of the visitor
4923 behave like a newly added virtual function of the visited hierarchy.
4924 In addition, visitors can store state in member variables, and they can
4925 be extended by deriving a new visitor from an existing one, thus building
4926 hierarchies of visitors.
4928 We can now rewrite our index example from above with a visitor:
4931 class gather_indices_visitor
4932 : public visitor, public idx::visitor, public varidx::visitor
4936 void visit(const idx & i)
4941 void visit(const varidx & vi)
4943 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4947 const lst & get_result() // utility function
4956 What's missing is the tree traversal. We could implement it in
4957 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4960 void ex::traverse_preorder(visitor & v) const;
4961 void ex::traverse_postorder(visitor & v) const;
4962 void ex::traverse(visitor & v) const;
4965 @code{traverse_preorder()} visits a node @emph{before} visiting its
4966 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4967 visiting its subexpressions. @code{traverse()} is a synonym for
4968 @code{traverse_preorder()}.
4970 Here is a new implementation of @code{gather_indices()} that uses the visitor
4971 and @code{traverse()}:
4974 lst gather_indices(const ex & e)
4976 gather_indices_visitor v;
4978 return v.get_result();
4982 Alternatively, you could use pre- or postorder iterators for the tree
4986 lst gather_indices(const ex & e)
4988 gather_indices_visitor v;
4989 for (const_preorder_iterator i = e.preorder_begin();
4990 i != e.preorder_end(); ++i) @{
4993 return v.get_result();
4998 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4999 @c node-name, next, previous, up
5000 @section Polynomial arithmetic
5002 @subsection Testing whether an expression is a polynomial
5003 @cindex @code{is_polynomial()}
5005 Testing whether an expression is a polynomial in one or more variables
5006 can be done with the method
5008 bool ex::is_polynomial(const ex & vars) const;
5010 In the case of more than
5011 one variable, the variables are given as a list.
5014 (x*y*sin(y)).is_polynomial(x) // Returns true.
5015 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5018 @subsection Expanding and collecting
5019 @cindex @code{expand()}
5020 @cindex @code{collect()}
5021 @cindex @code{collect_common_factors()}
5023 A polynomial in one or more variables has many equivalent
5024 representations. Some useful ones serve a specific purpose. Consider
5025 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5026 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5027 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5028 representations are the recursive ones where one collects for exponents
5029 in one of the three variable. Since the factors are themselves
5030 polynomials in the remaining two variables the procedure can be
5031 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5032 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5035 To bring an expression into expanded form, its method
5038 ex ex::expand(unsigned options = 0);
5041 may be called. In our example above, this corresponds to @math{4*x*y +
5042 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5043 GiNaC is not easy to guess you should be prepared to see different
5044 orderings of terms in such sums!
5046 Another useful representation of multivariate polynomials is as a
5047 univariate polynomial in one of the variables with the coefficients
5048 being polynomials in the remaining variables. The method
5049 @code{collect()} accomplishes this task:
5052 ex ex::collect(const ex & s, bool distributed = false);
5055 The first argument to @code{collect()} can also be a list of objects in which
5056 case the result is either a recursively collected polynomial, or a polynomial
5057 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5058 by the @code{distributed} flag.
5060 Note that the original polynomial needs to be in expanded form (for the
5061 variables concerned) in order for @code{collect()} to be able to find the
5062 coefficients properly.
5064 The following @command{ginsh} transcript shows an application of @code{collect()}
5065 together with @code{find()}:
5068 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5069 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)
5070 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5071 > collect(a,@{p,q@});
5072 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5073 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5074 > collect(a,find(a,sin($1)));
5075 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5076 > collect(a,@{find(a,sin($1)),p,q@});
5077 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5078 > collect(a,@{find(a,sin($1)),d@});
5079 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5082 Polynomials can often be brought into a more compact form by collecting
5083 common factors from the terms of sums. This is accomplished by the function
5086 ex collect_common_factors(const ex & e);
5089 This function doesn't perform a full factorization but only looks for
5090 factors which are already explicitly present:
5093 > collect_common_factors(a*x+a*y);
5095 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5097 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5098 (c+a)*a*(x*y+y^2+x)*b
5101 @subsection Degree and coefficients
5102 @cindex @code{degree()}
5103 @cindex @code{ldegree()}
5104 @cindex @code{coeff()}
5106 The degree and low degree of a polynomial can be obtained using the two
5110 int ex::degree(const ex & s);
5111 int ex::ldegree(const ex & s);
5114 which also work reliably on non-expanded input polynomials (they even work
5115 on rational functions, returning the asymptotic degree). By definition, the
5116 degree of zero is zero. To extract a coefficient with a certain power from
5117 an expanded polynomial you use
5120 ex ex::coeff(const ex & s, int n);
5123 You can also obtain the leading and trailing coefficients with the methods
5126 ex ex::lcoeff(const ex & s);
5127 ex ex::tcoeff(const ex & s);
5130 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5133 An application is illustrated in the next example, where a multivariate
5134 polynomial is analyzed:
5138 symbol x("x"), y("y");
5139 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5140 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5141 ex Poly = PolyInp.expand();
5143 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5144 cout << "The x^" << i << "-coefficient is "
5145 << Poly.coeff(x,i) << endl;
5147 cout << "As polynomial in y: "
5148 << Poly.collect(y) << endl;
5152 When run, it returns an output in the following fashion:
5155 The x^0-coefficient is y^2+11*y
5156 The x^1-coefficient is 5*y^2-2*y
5157 The x^2-coefficient is -1
5158 The x^3-coefficient is 4*y
5159 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5162 As always, the exact output may vary between different versions of GiNaC
5163 or even from run to run since the internal canonical ordering is not
5164 within the user's sphere of influence.
5166 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5167 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5168 with non-polynomial expressions as they not only work with symbols but with
5169 constants, functions and indexed objects as well:
5173 symbol a("a"), b("b"), c("c"), x("x");
5174 idx i(symbol("i"), 3);
5176 ex e = pow(sin(x) - cos(x), 4);
5177 cout << e.degree(cos(x)) << endl;
5179 cout << e.expand().coeff(sin(x), 3) << endl;
5182 e = indexed(a+b, i) * indexed(b+c, i);
5183 e = e.expand(expand_options::expand_indexed);
5184 cout << e.collect(indexed(b, i)) << endl;
5185 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5190 @subsection Polynomial division
5191 @cindex polynomial division
5194 @cindex pseudo-remainder
5195 @cindex @code{quo()}
5196 @cindex @code{rem()}
5197 @cindex @code{prem()}
5198 @cindex @code{divide()}
5203 ex quo(const ex & a, const ex & b, const ex & x);
5204 ex rem(const ex & a, const ex & b, const ex & x);
5207 compute the quotient and remainder of univariate polynomials in the variable
5208 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5210 The additional function
5213 ex prem(const ex & a, const ex & b, const ex & x);
5216 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5217 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5219 Exact division of multivariate polynomials is performed by the function
5222 bool divide(const ex & a, const ex & b, ex & q);
5225 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5226 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5227 in which case the value of @code{q} is undefined.
5230 @subsection Unit, content and primitive part
5231 @cindex @code{unit()}
5232 @cindex @code{content()}
5233 @cindex @code{primpart()}
5234 @cindex @code{unitcontprim()}
5239 ex ex::unit(const ex & x);
5240 ex ex::content(const ex & x);
5241 ex ex::primpart(const ex & x);
5242 ex ex::primpart(const ex & x, const ex & c);
5245 return the unit part, content part, and primitive polynomial of a multivariate
5246 polynomial with respect to the variable @samp{x} (the unit part being the sign
5247 of the leading coefficient, the content part being the GCD of the coefficients,
5248 and the primitive polynomial being the input polynomial divided by the unit and
5249 content parts). The second variant of @code{primpart()} expects the previously
5250 calculated content part of the polynomial in @code{c}, which enables it to
5251 work faster in the case where the content part has already been computed. The
5252 product of unit, content, and primitive part is the original polynomial.
5254 Additionally, the method
5257 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5260 computes the unit, content, and primitive parts in one go, returning them
5261 in @code{u}, @code{c}, and @code{p}, respectively.
5264 @subsection GCD, LCM and resultant
5267 @cindex @code{gcd()}
5268 @cindex @code{lcm()}
5270 The functions for polynomial greatest common divisor and least common
5271 multiple have the synopsis
5274 ex gcd(const ex & a, const ex & b);
5275 ex lcm(const ex & a, const ex & b);
5278 The functions @code{gcd()} and @code{lcm()} accept two expressions
5279 @code{a} and @code{b} as arguments and return a new expression, their
5280 greatest common divisor or least common multiple, respectively. If the
5281 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5282 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5283 the coefficients must be rationals.
5286 #include <ginac/ginac.h>
5287 using namespace GiNaC;
5291 symbol x("x"), y("y"), z("z");
5292 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5293 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5295 ex P_gcd = gcd(P_a, P_b);
5297 ex P_lcm = lcm(P_a, P_b);
5298 // 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
5303 @cindex @code{resultant()}
5305 The resultant of two expressions only makes sense with polynomials.
5306 It is always computed with respect to a specific symbol within the
5307 expressions. The function has the interface
5310 ex resultant(const ex & a, const ex & b, const ex & s);
5313 Resultants are symmetric in @code{a} and @code{b}. The following example
5314 computes the resultant of two expressions with respect to @code{x} and
5315 @code{y}, respectively:
5318 #include <ginac/ginac.h>
5319 using namespace GiNaC;
5323 symbol x("x"), y("y");
5325 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5328 r = resultant(e1, e2, x);
5330 r = resultant(e1, e2, y);
5335 @subsection Square-free decomposition
5336 @cindex square-free decomposition
5337 @cindex factorization
5338 @cindex @code{sqrfree()}
5340 GiNaC still lacks proper factorization support. Some form of
5341 factorization is, however, easily implemented by noting that factors
5342 appearing in a polynomial with power two or more also appear in the
5343 derivative and hence can easily be found by computing the GCD of the
5344 original polynomial and its derivatives. Any decent system has an
5345 interface for this so called square-free factorization. So we provide
5348 ex sqrfree(const ex & a, const lst & l = lst());
5350 Here is an example that by the way illustrates how the exact form of the
5351 result may slightly depend on the order of differentiation, calling for
5352 some care with subsequent processing of the result:
5355 symbol x("x"), y("y");
5356 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5358 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5359 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5361 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5362 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5364 cout << sqrfree(BiVarPol) << endl;
5365 // -> depending on luck, any of the above
5368 Note also, how factors with the same exponents are not fully factorized
5372 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5373 @c node-name, next, previous, up
5374 @section Rational expressions
5376 @subsection The @code{normal} method
5377 @cindex @code{normal()}
5378 @cindex simplification
5379 @cindex temporary replacement
5381 Some basic form of simplification of expressions is called for frequently.
5382 GiNaC provides the method @code{.normal()}, which converts a rational function
5383 into an equivalent rational function of the form @samp{numerator/denominator}
5384 where numerator and denominator are coprime. If the input expression is already
5385 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5386 otherwise it performs fraction addition and multiplication.
5388 @code{.normal()} can also be used on expressions which are not rational functions
5389 as it will replace all non-rational objects (like functions or non-integer
5390 powers) by temporary symbols to bring the expression to the domain of rational
5391 functions before performing the normalization, and re-substituting these
5392 symbols afterwards. This algorithm is also available as a separate method
5393 @code{.to_rational()}, described below.
5395 This means that both expressions @code{t1} and @code{t2} are indeed
5396 simplified in this little code snippet:
5401 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5402 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5403 std::cout << "t1 is " << t1.normal() << std::endl;
5404 std::cout << "t2 is " << t2.normal() << std::endl;
5408 Of course this works for multivariate polynomials too, so the ratio of
5409 the sample-polynomials from the section about GCD and LCM above would be
5410 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5413 @subsection Numerator and denominator
5416 @cindex @code{numer()}
5417 @cindex @code{denom()}
5418 @cindex @code{numer_denom()}
5420 The numerator and denominator of an expression can be obtained with
5425 ex ex::numer_denom();
5428 These functions will first normalize the expression as described above and
5429 then return the numerator, denominator, or both as a list, respectively.
5430 If you need both numerator and denominator, calling @code{numer_denom()} is
5431 faster than using @code{numer()} and @code{denom()} separately.
5434 @subsection Converting to a polynomial or rational expression
5435 @cindex @code{to_polynomial()}
5436 @cindex @code{to_rational()}
5438 Some of the methods described so far only work on polynomials or rational
5439 functions. GiNaC provides a way to extend the domain of these functions to
5440 general expressions by using the temporary replacement algorithm described
5441 above. You do this by calling
5444 ex ex::to_polynomial(exmap & m);
5445 ex ex::to_polynomial(lst & l);
5449 ex ex::to_rational(exmap & m);
5450 ex ex::to_rational(lst & l);
5453 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5454 will be filled with the generated temporary symbols and their replacement
5455 expressions in a format that can be used directly for the @code{subs()}
5456 method. It can also already contain a list of replacements from an earlier
5457 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5458 possible to use it on multiple expressions and get consistent results.
5460 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5461 is probably best illustrated with an example:
5465 symbol x("x"), y("y");
5466 ex a = 2*x/sin(x) - y/(3*sin(x));
5470 ex p = a.to_polynomial(lp);
5471 cout << " = " << p << "\n with " << lp << endl;
5472 // = symbol3*symbol2*y+2*symbol2*x
5473 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5476 ex r = a.to_rational(lr);
5477 cout << " = " << r << "\n with " << lr << endl;
5478 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5479 // with @{symbol4==sin(x)@}
5483 The following more useful example will print @samp{sin(x)-cos(x)}:
5488 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5489 ex b = sin(x) + cos(x);
5492 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5493 cout << q.subs(m) << endl;
5498 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5499 @c node-name, next, previous, up
5500 @section Symbolic differentiation
5501 @cindex differentiation
5502 @cindex @code{diff()}
5504 @cindex product rule
5506 GiNaC's objects know how to differentiate themselves. Thus, a
5507 polynomial (class @code{add}) knows that its derivative is the sum of
5508 the derivatives of all the monomials:
5512 symbol x("x"), y("y"), z("z");
5513 ex P = pow(x, 5) + pow(x, 2) + y;
5515 cout << P.diff(x,2) << endl;
5517 cout << P.diff(y) << endl; // 1
5519 cout << P.diff(z) << endl; // 0
5524 If a second integer parameter @var{n} is given, the @code{diff} method
5525 returns the @var{n}th derivative.
5527 If @emph{every} object and every function is told what its derivative
5528 is, all derivatives of composed objects can be calculated using the
5529 chain rule and the product rule. Consider, for instance the expression
5530 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5531 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5532 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5533 out that the composition is the generating function for Euler Numbers,
5534 i.e. the so called @var{n}th Euler number is the coefficient of
5535 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5536 identity to code a function that generates Euler numbers in just three
5539 @cindex Euler numbers
5541 #include <ginac/ginac.h>
5542 using namespace GiNaC;
5544 ex EulerNumber(unsigned n)
5547 const ex generator = pow(cosh(x),-1);
5548 return generator.diff(x,n).subs(x==0);
5553 for (unsigned i=0; i<11; i+=2)
5554 std::cout << EulerNumber(i) << std::endl;
5559 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5560 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5561 @code{i} by two since all odd Euler numbers vanish anyways.
5564 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5565 @c node-name, next, previous, up
5566 @section Series expansion
5567 @cindex @code{series()}
5568 @cindex Taylor expansion
5569 @cindex Laurent expansion
5570 @cindex @code{pseries} (class)
5571 @cindex @code{Order()}
5573 Expressions know how to expand themselves as a Taylor series or (more
5574 generally) a Laurent series. As in most conventional Computer Algebra
5575 Systems, no distinction is made between those two. There is a class of
5576 its own for storing such series (@code{class pseries}) and a built-in
5577 function (called @code{Order}) for storing the order term of the series.
5578 As a consequence, if you want to work with series, i.e. multiply two
5579 series, you need to call the method @code{ex::series} again to convert
5580 it to a series object with the usual structure (expansion plus order
5581 term). A sample application from special relativity could read:
5584 #include <ginac/ginac.h>
5585 using namespace std;
5586 using namespace GiNaC;
5590 symbol v("v"), c("c");
5592 ex gamma = 1/sqrt(1 - pow(v/c,2));
5593 ex mass_nonrel = gamma.series(v==0, 10);
5595 cout << "the relativistic mass increase with v is " << endl
5596 << mass_nonrel << endl;
5598 cout << "the inverse square of this series is " << endl
5599 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5603 Only calling the series method makes the last output simplify to
5604 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5605 series raised to the power @math{-2}.
5607 @cindex Machin's formula
5608 As another instructive application, let us calculate the numerical
5609 value of Archimedes' constant
5613 (for which there already exists the built-in constant @code{Pi})
5614 using John Machin's amazing formula
5616 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5619 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5621 This equation (and similar ones) were used for over 200 years for
5622 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5623 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5624 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5625 order term with it and the question arises what the system is supposed
5626 to do when the fractions are plugged into that order term. The solution
5627 is to use the function @code{series_to_poly()} to simply strip the order
5631 #include <ginac/ginac.h>
5632 using namespace GiNaC;
5634 ex machin_pi(int degr)
5637 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5638 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5639 -4*pi_expansion.subs(x==numeric(1,239));
5645 using std::cout; // just for fun, another way of...
5646 using std::endl; // ...dealing with this namespace std.
5648 for (int i=2; i<12; i+=2) @{
5649 pi_frac = machin_pi(i);
5650 cout << i << ":\t" << pi_frac << endl
5651 << "\t" << pi_frac.evalf() << endl;
5657 Note how we just called @code{.series(x,degr)} instead of
5658 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5659 method @code{series()}: if the first argument is a symbol the expression
5660 is expanded in that symbol around point @code{0}. When you run this
5661 program, it will type out:
5665 3.1832635983263598326
5666 4: 5359397032/1706489875
5667 3.1405970293260603143
5668 6: 38279241713339684/12184551018734375
5669 3.141621029325034425
5670 8: 76528487109180192540976/24359780855939418203125
5671 3.141591772182177295
5672 10: 327853873402258685803048818236/104359128170408663038552734375
5673 3.1415926824043995174
5677 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5678 @c node-name, next, previous, up
5679 @section Symmetrization
5680 @cindex @code{symmetrize()}
5681 @cindex @code{antisymmetrize()}
5682 @cindex @code{symmetrize_cyclic()}
5687 ex ex::symmetrize(const lst & l);
5688 ex ex::antisymmetrize(const lst & l);
5689 ex ex::symmetrize_cyclic(const lst & l);
5692 symmetrize an expression by returning the sum over all symmetric,
5693 antisymmetric or cyclic permutations of the specified list of objects,
5694 weighted by the number of permutations.
5696 The three additional methods
5699 ex ex::symmetrize();
5700 ex ex::antisymmetrize();
5701 ex ex::symmetrize_cyclic();
5704 symmetrize or antisymmetrize an expression over its free indices.
5706 Symmetrization is most useful with indexed expressions but can be used with
5707 almost any kind of object (anything that is @code{subs()}able):
5711 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5712 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5714 cout << indexed(A, i, j).symmetrize() << endl;
5715 // -> 1/2*A.j.i+1/2*A.i.j
5716 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5717 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5718 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5719 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5723 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5724 @c node-name, next, previous, up
5725 @section Predefined mathematical functions
5727 @subsection Overview
5729 GiNaC contains the following predefined mathematical functions:
5732 @multitable @columnfractions .30 .70
5733 @item @strong{Name} @tab @strong{Function}
5736 @cindex @code{abs()}
5737 @item @code{step(x)}
5739 @cindex @code{step()}
5740 @item @code{csgn(x)}
5742 @cindex @code{conjugate()}
5743 @item @code{conjugate(x)}
5744 @tab complex conjugation
5745 @cindex @code{real_part()}
5746 @item @code{real_part(x)}
5748 @cindex @code{imag_part()}
5749 @item @code{imag_part(x)}
5751 @item @code{sqrt(x)}
5752 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5753 @cindex @code{sqrt()}
5756 @cindex @code{sin()}
5759 @cindex @code{cos()}
5762 @cindex @code{tan()}
5763 @item @code{asin(x)}
5765 @cindex @code{asin()}
5766 @item @code{acos(x)}
5768 @cindex @code{acos()}
5769 @item @code{atan(x)}
5770 @tab inverse tangent
5771 @cindex @code{atan()}
5772 @item @code{atan2(y, x)}
5773 @tab inverse tangent with two arguments
5774 @item @code{sinh(x)}
5775 @tab hyperbolic sine
5776 @cindex @code{sinh()}
5777 @item @code{cosh(x)}
5778 @tab hyperbolic cosine
5779 @cindex @code{cosh()}
5780 @item @code{tanh(x)}
5781 @tab hyperbolic tangent
5782 @cindex @code{tanh()}
5783 @item @code{asinh(x)}
5784 @tab inverse hyperbolic sine
5785 @cindex @code{asinh()}
5786 @item @code{acosh(x)}
5787 @tab inverse hyperbolic cosine
5788 @cindex @code{acosh()}
5789 @item @code{atanh(x)}
5790 @tab inverse hyperbolic tangent
5791 @cindex @code{atanh()}
5793 @tab exponential function
5794 @cindex @code{exp()}
5796 @tab natural logarithm
5797 @cindex @code{log()}
5800 @cindex @code{Li2()}
5801 @item @code{Li(m, x)}
5802 @tab classical polylogarithm as well as multiple polylogarithm
5804 @item @code{G(a, y)}
5805 @tab multiple polylogarithm
5807 @item @code{G(a, s, y)}
5808 @tab multiple polylogarithm with explicit signs for the imaginary parts
5810 @item @code{S(n, p, x)}
5811 @tab Nielsen's generalized polylogarithm
5813 @item @code{H(m, x)}
5814 @tab harmonic polylogarithm
5816 @item @code{zeta(m)}
5817 @tab Riemann's zeta function as well as multiple zeta value
5818 @cindex @code{zeta()}
5819 @item @code{zeta(m, s)}
5820 @tab alternating Euler sum
5821 @cindex @code{zeta()}
5822 @item @code{zetaderiv(n, x)}
5823 @tab derivatives of Riemann's zeta function
5824 @item @code{tgamma(x)}
5826 @cindex @code{tgamma()}
5827 @cindex gamma function
5828 @item @code{lgamma(x)}
5829 @tab logarithm of gamma function
5830 @cindex @code{lgamma()}
5831 @item @code{beta(x, y)}
5832 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5833 @cindex @code{beta()}
5835 @tab psi (digamma) function
5836 @cindex @code{psi()}
5837 @item @code{psi(n, x)}
5838 @tab derivatives of psi function (polygamma functions)
5839 @item @code{factorial(n)}
5840 @tab factorial function @math{n!}
5841 @cindex @code{factorial()}
5842 @item @code{binomial(n, k)}
5843 @tab binomial coefficients
5844 @cindex @code{binomial()}
5845 @item @code{Order(x)}
5846 @tab order term function in truncated power series
5847 @cindex @code{Order()}
5852 For functions that have a branch cut in the complex plane GiNaC follows
5853 the conventions for C++ as defined in the ANSI standard as far as
5854 possible. In particular: the natural logarithm (@code{log}) and the
5855 square root (@code{sqrt}) both have their branch cuts running along the
5856 negative real axis where the points on the axis itself belong to the
5857 upper part (i.e. continuous with quadrant II). The inverse
5858 trigonometric and hyperbolic functions are not defined for complex
5859 arguments by the C++ standard, however. In GiNaC we follow the
5860 conventions used by CLN, which in turn follow the carefully designed
5861 definitions in the Common Lisp standard. It should be noted that this
5862 convention is identical to the one used by the C99 standard and by most
5863 serious CAS. It is to be expected that future revisions of the C++
5864 standard incorporate these functions in the complex domain in a manner
5865 compatible with C99.
5867 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5868 @c node-name, next, previous, up
5869 @subsection Multiple polylogarithms
5871 @cindex polylogarithm
5872 @cindex Nielsen's generalized polylogarithm
5873 @cindex harmonic polylogarithm
5874 @cindex multiple zeta value
5875 @cindex alternating Euler sum
5876 @cindex multiple polylogarithm
5878 The multiple polylogarithm is the most generic member of a family of functions,
5879 to which others like the harmonic polylogarithm, Nielsen's generalized
5880 polylogarithm and the multiple zeta value belong.
5881 Everyone of these functions can also be written as a multiple polylogarithm with specific
5882 parameters. This whole family of functions is therefore often referred to simply as
5883 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5884 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5885 @code{Li} and @code{G} in principle represent the same function, the different
5886 notations are more natural to the series representation or the integral
5887 representation, respectively.
5889 To facilitate the discussion of these functions we distinguish between indices and
5890 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5891 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5893 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5894 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5895 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5896 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5897 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5898 @code{s} is not given, the signs default to +1.
5899 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5900 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5901 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5902 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5903 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5905 The functions print in LaTeX format as
5907 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5913 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5916 $\zeta(m_1,m_2,\ldots,m_k)$.
5918 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5919 are printed with a line above, e.g.
5921 $\zeta(5,\overline{2})$.
5923 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5925 Definitions and analytical as well as numerical properties of multiple polylogarithms
5926 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5927 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5928 except for a few differences which will be explicitly stated in the following.
5930 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5931 that the indices and arguments are understood to be in the same order as in which they appear in
5932 the series representation. This means
5934 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5937 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5940 $\zeta(1,2)$ evaluates to infinity.
5942 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5945 The functions only evaluate if the indices are integers greater than zero, except for the indices
5946 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5947 will be interpreted as the sequence of signs for the corresponding indices
5948 @code{m} or the sign of the imaginary part for the
5949 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5950 @code{zeta(lst(3,4), lst(-1,1))} means
5952 $\zeta(\overline{3},4)$
5955 @code{G(lst(a,b), lst(-1,1), c)} means
5957 $G(a-0\epsilon,b+0\epsilon;c)$.
5959 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5960 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5961 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
5962 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5963 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5964 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5965 evaluates also for negative integers and positive even integers. For example:
5968 > Li(@{3,1@},@{x,1@});
5971 -zeta(@{3,2@},@{-1,-1@})
5976 It is easy to tell for a given function into which other function it can be rewritten, may
5977 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5978 with negative indices or trailing zeros (the example above gives a hint). Signs can
5979 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5980 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5981 @code{Li} (@code{eval()} already cares for the possible downgrade):
5984 > convert_H_to_Li(@{0,-2,-1,3@},x);
5985 Li(@{3,1,3@},@{-x,1,-1@})
5986 > convert_H_to_Li(@{2,-1,0@},x);
5987 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5990 Every function can be numerically evaluated for
5991 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5992 global variable @code{Digits}:
5997 > evalf(zeta(@{3,1,3,1@}));
5998 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6001 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6002 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6004 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6009 In long expressions this helps a lot with debugging, because you can easily spot
6010 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6011 cancellations of divergencies happen.
6013 Useful publications:
6015 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6016 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6018 @cite{Harmonic Polylogarithms},
6019 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6021 @cite{Special Values of Multiple Polylogarithms},
6022 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6024 @cite{Numerical Evaluation of Multiple Polylogarithms},
6025 J.Vollinga, S.Weinzierl, hep-ph/0410259
6027 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6028 @c node-name, next, previous, up
6029 @section Complex expressions
6031 @cindex @code{conjugate()}
6033 For dealing with complex expressions there are the methods
6041 that return respectively the complex conjugate, the real part and the
6042 imaginary part of an expression. Complex conjugation works as expected
6043 for all built-in functinos and objects. Taking real and imaginary
6044 parts has not yet been implemented for all built-in functions. In cases where
6045 it is not known how to conjugate or take a real/imaginary part one
6046 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6047 is returned. For instance, in case of a complex symbol @code{x}
6048 (symbols are complex by default), one could not simplify
6049 @code{conjugate(x)}. In the case of strings of gamma matrices,
6050 the @code{conjugate} method takes the Dirac conjugate.
6055 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6059 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6060 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6061 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6062 // -> -gamma5*gamma~b*gamma~a
6066 If you declare your own GiNaC functions, then they will conjugate themselves
6067 by conjugating their arguments. This is the default strategy. If you want to
6068 change this behavior, you have to supply a specialized conjugation method
6069 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6070 for @code{abs} as an example). Also, specialized methods can be provided
6071 to take real and imaginary parts of user-defined functions.
6073 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6074 @c node-name, next, previous, up
6075 @section Solving linear systems of equations
6076 @cindex @code{lsolve()}
6078 The function @code{lsolve()} provides a convenient wrapper around some
6079 matrix operations that comes in handy when a system of linear equations
6083 ex lsolve(const ex & eqns, const ex & symbols,
6084 unsigned options = solve_algo::automatic);
6087 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6088 @code{relational}) while @code{symbols} is a @code{lst} of
6089 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6092 It returns the @code{lst} of solutions as an expression. As an example,
6093 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6097 symbol a("a"), b("b"), x("x"), y("y");
6099 eqns = a*x+b*y==3, x-y==b;
6101 cout << lsolve(eqns, vars) << endl;
6102 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6105 When the linear equations @code{eqns} are underdetermined, the solution
6106 will contain one or more tautological entries like @code{x==x},
6107 depending on the rank of the system. When they are overdetermined, the
6108 solution will be an empty @code{lst}. Note the third optional parameter
6109 to @code{lsolve()}: it accepts the same parameters as
6110 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6114 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6115 @c node-name, next, previous, up
6116 @section Input and output of expressions
6119 @subsection Expression output
6121 @cindex output of expressions
6123 Expressions can simply be written to any stream:
6128 ex e = 4.5*I+pow(x,2)*3/2;
6129 cout << e << endl; // prints '4.5*I+3/2*x^2'
6133 The default output format is identical to the @command{ginsh} input syntax and
6134 to that used by most computer algebra systems, but not directly pastable
6135 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6136 is printed as @samp{x^2}).
6138 It is possible to print expressions in a number of different formats with
6139 a set of stream manipulators;
6142 std::ostream & dflt(std::ostream & os);
6143 std::ostream & latex(std::ostream & os);
6144 std::ostream & tree(std::ostream & os);
6145 std::ostream & csrc(std::ostream & os);
6146 std::ostream & csrc_float(std::ostream & os);
6147 std::ostream & csrc_double(std::ostream & os);
6148 std::ostream & csrc_cl_N(std::ostream & os);
6149 std::ostream & index_dimensions(std::ostream & os);
6150 std::ostream & no_index_dimensions(std::ostream & os);
6153 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6154 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6155 @code{print_csrc()} functions, respectively.
6158 All manipulators affect the stream state permanently. To reset the output
6159 format to the default, use the @code{dflt} manipulator:
6163 cout << latex; // all output to cout will be in LaTeX format from
6165 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6166 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6167 cout << dflt; // revert to default output format
6168 cout << e << endl; // prints '4.5*I+3/2*x^2'
6172 If you don't want to affect the format of the stream you're working with,
6173 you can output to a temporary @code{ostringstream} like this:
6178 s << latex << e; // format of cout remains unchanged
6179 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6184 @cindex @code{csrc_float}
6185 @cindex @code{csrc_double}
6186 @cindex @code{csrc_cl_N}
6187 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6188 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6189 format that can be directly used in a C or C++ program. The three possible
6190 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6191 classes provided by the CLN library):
6195 cout << "f = " << csrc_float << e << ";\n";
6196 cout << "d = " << csrc_double << e << ";\n";
6197 cout << "n = " << csrc_cl_N << e << ";\n";
6201 The above example will produce (note the @code{x^2} being converted to
6205 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6206 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6207 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6211 The @code{tree} manipulator allows dumping the internal structure of an
6212 expression for debugging purposes:
6223 add, hash=0x0, flags=0x3, nops=2
6224 power, hash=0x0, flags=0x3, nops=2
6225 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6226 2 (numeric), hash=0x6526b0fa, flags=0xf
6227 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6230 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6234 @cindex @code{latex}
6235 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6236 It is rather similar to the default format but provides some braces needed
6237 by LaTeX for delimiting boxes and also converts some common objects to
6238 conventional LaTeX names. It is possible to give symbols a special name for
6239 LaTeX output by supplying it as a second argument to the @code{symbol}
6242 For example, the code snippet
6246 symbol x("x", "\\circ");
6247 ex e = lgamma(x).series(x==0,3);
6248 cout << latex << e << endl;
6255 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6256 +\mathcal@{O@}(\circ^@{3@})
6259 @cindex @code{index_dimensions}
6260 @cindex @code{no_index_dimensions}
6261 Index dimensions are normally hidden in the output. To make them visible, use
6262 the @code{index_dimensions} manipulator. The dimensions will be written in
6263 square brackets behind each index value in the default and LaTeX output
6268 symbol x("x"), y("y");
6269 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6270 ex e = indexed(x, mu) * indexed(y, nu);
6273 // prints 'x~mu*y~nu'
6274 cout << index_dimensions << e << endl;
6275 // prints 'x~mu[4]*y~nu[4]'
6276 cout << no_index_dimensions << e << endl;
6277 // prints 'x~mu*y~nu'
6282 @cindex Tree traversal
6283 If you need any fancy special output format, e.g. for interfacing GiNaC
6284 with other algebra systems or for producing code for different
6285 programming languages, you can always traverse the expression tree yourself:
6288 static void my_print(const ex & e)
6290 if (is_a<function>(e))
6291 cout << ex_to<function>(e).get_name();
6293 cout << ex_to<basic>(e).class_name();
6295 size_t n = e.nops();
6297 for (size_t i=0; i<n; i++) @{
6309 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6317 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6318 symbol(y))),numeric(-2)))
6321 If you need an output format that makes it possible to accurately
6322 reconstruct an expression by feeding the output to a suitable parser or
6323 object factory, you should consider storing the expression in an
6324 @code{archive} object and reading the object properties from there.
6325 See the section on archiving for more information.
6328 @subsection Expression input
6329 @cindex input of expressions
6331 GiNaC provides no way to directly read an expression from a stream because
6332 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6333 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6334 @code{y} you defined in your program and there is no way to specify the
6335 desired symbols to the @code{>>} stream input operator.
6337 Instead, GiNaC lets you construct an expression from a string, specifying the
6338 list of symbols to be used:
6342 symbol x("x"), y("y");
6343 ex e("2*x+sin(y)", lst(x, y));
6347 The input syntax is the same as that used by @command{ginsh} and the stream
6348 output operator @code{<<}. The symbols in the string are matched by name to
6349 the symbols in the list and if GiNaC encounters a symbol not specified in
6350 the list it will throw an exception.
6352 With this constructor, it's also easy to implement interactive GiNaC programs:
6357 #include <stdexcept>
6358 #include <ginac/ginac.h>
6359 using namespace std;
6360 using namespace GiNaC;
6367 cout << "Enter an expression containing 'x': ";
6372 cout << "The derivative of " << e << " with respect to x is ";
6373 cout << e.diff(x) << ".\n";
6374 @} catch (exception &p) @{
6375 cerr << p.what() << endl;
6381 @subsection Archiving
6382 @cindex @code{archive} (class)
6385 GiNaC allows creating @dfn{archives} of expressions which can be stored
6386 to or retrieved from files. To create an archive, you declare an object
6387 of class @code{archive} and archive expressions in it, giving each
6388 expression a unique name:
6392 using namespace std;
6393 #include <ginac/ginac.h>
6394 using namespace GiNaC;
6398 symbol x("x"), y("y"), z("z");
6400 ex foo = sin(x + 2*y) + 3*z + 41;
6404 a.archive_ex(foo, "foo");
6405 a.archive_ex(bar, "the second one");
6409 The archive can then be written to a file:
6413 ofstream out("foobar.gar");
6419 The file @file{foobar.gar} contains all information that is needed to
6420 reconstruct the expressions @code{foo} and @code{bar}.
6422 @cindex @command{viewgar}
6423 The tool @command{viewgar} that comes with GiNaC can be used to view
6424 the contents of GiNaC archive files:
6427 $ viewgar foobar.gar
6428 foo = 41+sin(x+2*y)+3*z
6429 the second one = 42+sin(x+2*y)+3*z
6432 The point of writing archive files is of course that they can later be
6438 ifstream in("foobar.gar");
6443 And the stored expressions can be retrieved by their name:
6450 ex ex1 = a2.unarchive_ex(syms, "foo");
6451 ex ex2 = a2.unarchive_ex(syms, "the second one");
6453 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6454 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6455 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6459 Note that you have to supply a list of the symbols which are to be inserted
6460 in the expressions. Symbols in archives are stored by their name only and
6461 if you don't specify which symbols you have, unarchiving the expression will
6462 create new symbols with that name. E.g. if you hadn't included @code{x} in
6463 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6464 have had no effect because the @code{x} in @code{ex1} would have been a
6465 different symbol than the @code{x} which was defined at the beginning of
6466 the program, although both would appear as @samp{x} when printed.
6468 You can also use the information stored in an @code{archive} object to
6469 output expressions in a format suitable for exact reconstruction. The
6470 @code{archive} and @code{archive_node} classes have a couple of member
6471 functions that let you access the stored properties:
6474 static void my_print2(const archive_node & n)
6477 n.find_string("class", class_name);
6478 cout << class_name << "(";
6480 archive_node::propinfovector p;
6481 n.get_properties(p);
6483 size_t num = p.size();
6484 for (size_t i=0; i<num; i++) @{
6485 const string &name = p[i].name;
6486 if (name == "class")
6488 cout << name << "=";
6490 unsigned count = p[i].count;
6494 for (unsigned j=0; j<count; j++) @{
6495 switch (p[i].type) @{
6496 case archive_node::PTYPE_BOOL: @{
6498 n.find_bool(name, x, j);
6499 cout << (x ? "true" : "false");
6502 case archive_node::PTYPE_UNSIGNED: @{
6504 n.find_unsigned(name, x, j);
6508 case archive_node::PTYPE_STRING: @{
6510 n.find_string(name, x, j);
6511 cout << '\"' << x << '\"';
6514 case archive_node::PTYPE_NODE: @{
6515 const archive_node &x = n.find_ex_node(name, j);
6537 ex e = pow(2, x) - y;
6539 my_print2(ar.get_top_node(0)); cout << endl;
6547 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6548 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6549 overall_coeff=numeric(number="0"))
6552 Be warned, however, that the set of properties and their meaning for each
6553 class may change between GiNaC versions.
6556 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6557 @c node-name, next, previous, up
6558 @chapter Extending GiNaC
6560 By reading so far you should have gotten a fairly good understanding of
6561 GiNaC's design patterns. From here on you should start reading the
6562 sources. All we can do now is issue some recommendations how to tackle
6563 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6564 develop some useful extension please don't hesitate to contact the GiNaC
6565 authors---they will happily incorporate them into future versions.
6568 * What does not belong into GiNaC:: What to avoid.
6569 * Symbolic functions:: Implementing symbolic functions.
6570 * Printing:: Adding new output formats.
6571 * Structures:: Defining new algebraic classes (the easy way).
6572 * Adding classes:: Defining new algebraic classes (the hard way).
6576 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6577 @c node-name, next, previous, up
6578 @section What doesn't belong into GiNaC
6580 @cindex @command{ginsh}
6581 First of all, GiNaC's name must be read literally. It is designed to be
6582 a library for use within C++. The tiny @command{ginsh} accompanying
6583 GiNaC makes this even more clear: it doesn't even attempt to provide a
6584 language. There are no loops or conditional expressions in
6585 @command{ginsh}, it is merely a window into the library for the
6586 programmer to test stuff (or to show off). Still, the design of a
6587 complete CAS with a language of its own, graphical capabilities and all
6588 this on top of GiNaC is possible and is without doubt a nice project for
6591 There are many built-in functions in GiNaC that do not know how to
6592 evaluate themselves numerically to a precision declared at runtime
6593 (using @code{Digits}). Some may be evaluated at certain points, but not
6594 generally. This ought to be fixed. However, doing numerical
6595 computations with GiNaC's quite abstract classes is doomed to be
6596 inefficient. For this purpose, the underlying foundation classes
6597 provided by CLN are much better suited.
6600 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6601 @c node-name, next, previous, up
6602 @section Symbolic functions
6604 The easiest and most instructive way to start extending GiNaC is probably to
6605 create your own symbolic functions. These are implemented with the help of
6606 two preprocessor macros:
6608 @cindex @code{DECLARE_FUNCTION}
6609 @cindex @code{REGISTER_FUNCTION}
6611 DECLARE_FUNCTION_<n>P(<name>)
6612 REGISTER_FUNCTION(<name>, <options>)
6615 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6616 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6617 parameters of type @code{ex} and returns a newly constructed GiNaC
6618 @code{function} object that represents your function.
6620 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6621 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6622 set of options that associate the symbolic function with C++ functions you
6623 provide to implement the various methods such as evaluation, derivative,
6624 series expansion etc. They also describe additional attributes the function
6625 might have, such as symmetry and commutation properties, and a name for
6626 LaTeX output. Multiple options are separated by the member access operator
6627 @samp{.} and can be given in an arbitrary order.
6629 (By the way: in case you are worrying about all the macros above we can
6630 assure you that functions are GiNaC's most macro-intense classes. We have
6631 done our best to avoid macros where we can.)
6633 @subsection A minimal example
6635 Here is an example for the implementation of a function with two arguments
6636 that is not further evaluated:
6639 DECLARE_FUNCTION_2P(myfcn)
6641 REGISTER_FUNCTION(myfcn, dummy())
6644 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6645 in algebraic expressions:
6651 ex e = 2*myfcn(42, 1+3*x) - x;
6653 // prints '2*myfcn(42,1+3*x)-x'
6658 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6659 "no options". A function with no options specified merely acts as a kind of
6660 container for its arguments. It is a pure "dummy" function with no associated
6661 logic (which is, however, sometimes perfectly sufficient).
6663 Let's now have a look at the implementation of GiNaC's cosine function for an
6664 example of how to make an "intelligent" function.
6666 @subsection The cosine function
6668 The GiNaC header file @file{inifcns.h} contains the line
6671 DECLARE_FUNCTION_1P(cos)
6674 which declares to all programs using GiNaC that there is a function @samp{cos}
6675 that takes one @code{ex} as an argument. This is all they need to know to use
6676 this function in expressions.
6678 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6679 is its @code{REGISTER_FUNCTION} line:
6682 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6683 evalf_func(cos_evalf).
6684 derivative_func(cos_deriv).
6685 latex_name("\\cos"));
6688 There are four options defined for the cosine function. One of them
6689 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6690 other three indicate the C++ functions in which the "brains" of the cosine
6691 function are defined.
6693 @cindex @code{hold()}
6695 The @code{eval_func()} option specifies the C++ function that implements
6696 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6697 the same number of arguments as the associated symbolic function (one in this
6698 case) and returns the (possibly transformed or in some way simplified)
6699 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6700 of the automatic evaluation process). If no (further) evaluation is to take
6701 place, the @code{eval_func()} function must return the original function
6702 with @code{.hold()}, to avoid a potential infinite recursion. If your
6703 symbolic functions produce a segmentation fault or stack overflow when
6704 using them in expressions, you are probably missing a @code{.hold()}
6707 The @code{eval_func()} function for the cosine looks something like this
6708 (actually, it doesn't look like this at all, but it should give you an idea
6712 static ex cos_eval(const ex & x)
6714 if ("x is a multiple of 2*Pi")
6716 else if ("x is a multiple of Pi")
6718 else if ("x is a multiple of Pi/2")
6722 else if ("x has the form 'acos(y)'")
6724 else if ("x has the form 'asin(y)'")
6729 return cos(x).hold();
6733 This function is called every time the cosine is used in a symbolic expression:
6739 // this calls cos_eval(Pi), and inserts its return value into
6740 // the actual expression
6747 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6748 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6749 symbolic transformation can be done, the unmodified function is returned
6750 with @code{.hold()}.
6752 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6753 The user has to call @code{evalf()} for that. This is implemented in a
6757 static ex cos_evalf(const ex & x)
6759 if (is_a<numeric>(x))
6760 return cos(ex_to<numeric>(x));
6762 return cos(x).hold();
6766 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6767 in this case the @code{cos()} function for @code{numeric} objects, which in
6768 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6769 isn't really needed here, but reminds us that the corresponding @code{eval()}
6770 function would require it in this place.
6772 Differentiation will surely turn up and so we need to tell @code{cos}
6773 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6774 instance, are then handled automatically by @code{basic::diff} and
6778 static ex cos_deriv(const ex & x, unsigned diff_param)
6784 @cindex product rule
6785 The second parameter is obligatory but uninteresting at this point. It
6786 specifies which parameter to differentiate in a partial derivative in
6787 case the function has more than one parameter, and its main application
6788 is for correct handling of the chain rule.
6790 An implementation of the series expansion is not needed for @code{cos()} as
6791 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6792 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6793 the other hand, does have poles and may need to do Laurent expansion:
6796 static ex tan_series(const ex & x, const relational & rel,
6797 int order, unsigned options)
6799 // Find the actual expansion point
6800 const ex x_pt = x.subs(rel);
6802 if ("x_pt is not an odd multiple of Pi/2")
6803 throw do_taylor(); // tell function::series() to do Taylor expansion
6805 // On a pole, expand sin()/cos()
6806 return (sin(x)/cos(x)).series(rel, order+2, options);
6810 The @code{series()} implementation of a function @emph{must} return a
6811 @code{pseries} object, otherwise your code will crash.
6813 @subsection Function options
6815 GiNaC functions understand several more options which are always
6816 specified as @code{.option(params)}. None of them are required, but you
6817 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6818 is a do-nothing option called @code{dummy()} which you can use to define
6819 functions without any special options.
6822 eval_func(<C++ function>)
6823 evalf_func(<C++ function>)
6824 derivative_func(<C++ function>)
6825 series_func(<C++ function>)
6826 conjugate_func(<C++ function>)
6829 These specify the C++ functions that implement symbolic evaluation,
6830 numeric evaluation, partial derivatives, and series expansion, respectively.
6831 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6832 @code{diff()} and @code{series()}.
6834 The @code{eval_func()} function needs to use @code{.hold()} if no further
6835 automatic evaluation is desired or possible.
6837 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6838 expansion, which is correct if there are no poles involved. If the function
6839 has poles in the complex plane, the @code{series_func()} needs to check
6840 whether the expansion point is on a pole and fall back to Taylor expansion
6841 if it isn't. Otherwise, the pole usually needs to be regularized by some
6842 suitable transformation.
6845 latex_name(const string & n)
6848 specifies the LaTeX code that represents the name of the function in LaTeX
6849 output. The default is to put the function name in an @code{\mbox@{@}}.
6852 do_not_evalf_params()
6855 This tells @code{evalf()} to not recursively evaluate the parameters of the
6856 function before calling the @code{evalf_func()}.
6859 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6862 This allows you to explicitly specify the commutation properties of the
6863 function (@xref{Non-commutative objects}, for an explanation of
6864 (non)commutativity in GiNaC). For example, you can use
6865 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6866 GiNaC treat your function like a matrix. By default, functions inherit the
6867 commutation properties of their first argument.
6870 set_symmetry(const symmetry & s)
6873 specifies the symmetry properties of the function with respect to its
6874 arguments. @xref{Indexed objects}, for an explanation of symmetry
6875 specifications. GiNaC will automatically rearrange the arguments of
6876 symmetric functions into a canonical order.
6878 Sometimes you may want to have finer control over how functions are
6879 displayed in the output. For example, the @code{abs()} function prints
6880 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6881 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6885 print_func<C>(<C++ function>)
6888 option which is explained in the next section.
6890 @subsection Functions with a variable number of arguments
6892 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6893 functions with a fixed number of arguments. Sometimes, though, you may need
6894 to have a function that accepts a variable number of expressions. One way to
6895 accomplish this is to pass variable-length lists as arguments. The
6896 @code{Li()} function uses this method for multiple polylogarithms.
6898 It is also possible to define functions that accept a different number of
6899 parameters under the same function name, such as the @code{psi()} function
6900 which can be called either as @code{psi(z)} (the digamma function) or as
6901 @code{psi(n, z)} (polygamma functions). These are actually two different
6902 functions in GiNaC that, however, have the same name. Defining such
6903 functions is not possible with the macros but requires manually fiddling
6904 with GiNaC internals. If you are interested, please consult the GiNaC source
6905 code for the @code{psi()} function (@file{inifcns.h} and
6906 @file{inifcns_gamma.cpp}).
6909 @node Printing, Structures, Symbolic functions, Extending GiNaC
6910 @c node-name, next, previous, up
6911 @section GiNaC's expression output system
6913 GiNaC allows the output of expressions in a variety of different formats
6914 (@pxref{Input/output}). This section will explain how expression output
6915 is implemented internally, and how to define your own output formats or
6916 change the output format of built-in algebraic objects. You will also want
6917 to read this section if you plan to write your own algebraic classes or
6920 @cindex @code{print_context} (class)
6921 @cindex @code{print_dflt} (class)
6922 @cindex @code{print_latex} (class)
6923 @cindex @code{print_tree} (class)
6924 @cindex @code{print_csrc} (class)
6925 All the different output formats are represented by a hierarchy of classes
6926 rooted in the @code{print_context} class, defined in the @file{print.h}
6931 the default output format
6933 output in LaTeX mathematical mode
6935 a dump of the internal expression structure (for debugging)
6937 the base class for C source output
6938 @item print_csrc_float
6939 C source output using the @code{float} type
6940 @item print_csrc_double
6941 C source output using the @code{double} type
6942 @item print_csrc_cl_N
6943 C source output using CLN types
6946 The @code{print_context} base class provides two public data members:
6958 @code{s} is a reference to the stream to output to, while @code{options}
6959 holds flags and modifiers. Currently, there is only one flag defined:
6960 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6961 to print the index dimension which is normally hidden.
6963 When you write something like @code{std::cout << e}, where @code{e} is
6964 an object of class @code{ex}, GiNaC will construct an appropriate
6965 @code{print_context} object (of a class depending on the selected output
6966 format), fill in the @code{s} and @code{options} members, and call
6968 @cindex @code{print()}
6970 void ex::print(const print_context & c, unsigned level = 0) const;
6973 which in turn forwards the call to the @code{print()} method of the
6974 top-level algebraic object contained in the expression.
6976 Unlike other methods, GiNaC classes don't usually override their
6977 @code{print()} method to implement expression output. Instead, the default
6978 implementation @code{basic::print(c, level)} performs a run-time double
6979 dispatch to a function selected by the dynamic type of the object and the
6980 passed @code{print_context}. To this end, GiNaC maintains a separate method
6981 table for each class, similar to the virtual function table used for ordinary
6982 (single) virtual function dispatch.
6984 The method table contains one slot for each possible @code{print_context}
6985 type, indexed by the (internally assigned) serial number of the type. Slots
6986 may be empty, in which case GiNaC will retry the method lookup with the
6987 @code{print_context} object's parent class, possibly repeating the process
6988 until it reaches the @code{print_context} base class. If there's still no
6989 method defined, the method table of the algebraic object's parent class
6990 is consulted, and so on, until a matching method is found (eventually it
6991 will reach the combination @code{basic/print_context}, which prints the
6992 object's class name enclosed in square brackets).
6994 You can think of the print methods of all the different classes and output
6995 formats as being arranged in a two-dimensional matrix with one axis listing
6996 the algebraic classes and the other axis listing the @code{print_context}
6999 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7000 to implement printing, but then they won't get any of the benefits of the
7001 double dispatch mechanism (such as the ability for derived classes to
7002 inherit only certain print methods from its parent, or the replacement of
7003 methods at run-time).
7005 @subsection Print methods for classes
7007 The method table for a class is set up either in the definition of the class,
7008 by passing the appropriate @code{print_func<C>()} option to
7009 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7010 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7011 can also be used to override existing methods dynamically.
7013 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7014 be a member function of the class (or one of its parent classes), a static
7015 member function, or an ordinary (global) C++ function. The @code{C} template
7016 parameter specifies the appropriate @code{print_context} type for which the
7017 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7018 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7019 the class is the one being implemented by
7020 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7022 For print methods that are member functions, their first argument must be of
7023 a type convertible to a @code{const C &}, and the second argument must be an
7026 For static members and global functions, the first argument must be of a type
7027 convertible to a @code{const T &}, the second argument must be of a type
7028 convertible to a @code{const C &}, and the third argument must be an
7029 @code{unsigned}. A global function will, of course, not have access to
7030 private and protected members of @code{T}.
7032 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7033 and @code{basic::print()}) is used for proper parenthesizing of the output
7034 (and by @code{print_tree} for proper indentation). It can be used for similar
7035 purposes if you write your own output formats.
7037 The explanations given above may seem complicated, but in practice it's
7038 really simple, as shown in the following example. Suppose that we want to
7039 display exponents in LaTeX output not as superscripts but with little
7040 upwards-pointing arrows. This can be achieved in the following way:
7043 void my_print_power_as_latex(const power & p,
7044 const print_latex & c,
7047 // get the precedence of the 'power' class
7048 unsigned power_prec = p.precedence();
7050 // if the parent operator has the same or a higher precedence
7051 // we need parentheses around the power
7052 if (level >= power_prec)
7055 // print the basis and exponent, each enclosed in braces, and
7056 // separated by an uparrow
7058 p.op(0).print(c, power_prec);
7059 c.s << "@}\\uparrow@{";
7060 p.op(1).print(c, power_prec);
7063 // don't forget the closing parenthesis
7064 if (level >= power_prec)
7070 // a sample expression
7071 symbol x("x"), y("y");
7072 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7074 // switch to LaTeX mode
7077 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7080 // now we replace the method for the LaTeX output of powers with
7082 set_print_func<power, print_latex>(my_print_power_as_latex);
7084 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7095 The first argument of @code{my_print_power_as_latex} could also have been
7096 a @code{const basic &}, the second one a @code{const print_context &}.
7099 The above code depends on @code{mul} objects converting their operands to
7100 @code{power} objects for the purpose of printing.
7103 The output of products including negative powers as fractions is also
7104 controlled by the @code{mul} class.
7107 The @code{power/print_latex} method provided by GiNaC prints square roots
7108 using @code{\sqrt}, but the above code doesn't.
7112 It's not possible to restore a method table entry to its previous or default
7113 value. Once you have called @code{set_print_func()}, you can only override
7114 it with another call to @code{set_print_func()}, but you can't easily go back
7115 to the default behavior again (you can, of course, dig around in the GiNaC
7116 sources, find the method that is installed at startup
7117 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7118 one; that is, after you circumvent the C++ member access control@dots{}).
7120 @subsection Print methods for functions
7122 Symbolic functions employ a print method dispatch mechanism similar to the
7123 one used for classes. The methods are specified with @code{print_func<C>()}
7124 function options. If you don't specify any special print methods, the function
7125 will be printed with its name (or LaTeX name, if supplied), followed by a
7126 comma-separated list of arguments enclosed in parentheses.
7128 For example, this is what GiNaC's @samp{abs()} function is defined like:
7131 static ex abs_eval(const ex & arg) @{ ... @}
7132 static ex abs_evalf(const ex & arg) @{ ... @}
7134 static void abs_print_latex(const ex & arg, const print_context & c)
7136 c.s << "@{|"; arg.print(c); c.s << "|@}";
7139 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7141 c.s << "fabs("; arg.print(c); c.s << ")";
7144 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7145 evalf_func(abs_evalf).
7146 print_func<print_latex>(abs_print_latex).
7147 print_func<print_csrc_float>(abs_print_csrc_float).
7148 print_func<print_csrc_double>(abs_print_csrc_float));
7151 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7152 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7154 There is currently no equivalent of @code{set_print_func()} for functions.
7156 @subsection Adding new output formats
7158 Creating a new output format involves subclassing @code{print_context},
7159 which is somewhat similar to adding a new algebraic class
7160 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7161 that needs to go into the class definition, and a corresponding macro
7162 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7163 Every @code{print_context} class needs to provide a default constructor
7164 and a constructor from an @code{std::ostream} and an @code{unsigned}
7167 Here is an example for a user-defined @code{print_context} class:
7170 class print_myformat : public print_dflt
7172 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7174 print_myformat(std::ostream & os, unsigned opt = 0)
7175 : print_dflt(os, opt) @{@}
7178 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7180 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7183 That's all there is to it. None of the actual expression output logic is
7184 implemented in this class. It merely serves as a selector for choosing
7185 a particular format. The algorithms for printing expressions in the new
7186 format are implemented as print methods, as described above.
7188 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7189 exactly like GiNaC's default output format:
7194 ex e = pow(x, 2) + 1;
7196 // this prints "1+x^2"
7199 // this also prints "1+x^2"
7200 e.print(print_myformat()); cout << endl;
7206 To fill @code{print_myformat} with life, we need to supply appropriate
7207 print methods with @code{set_print_func()}, like this:
7210 // This prints powers with '**' instead of '^'. See the LaTeX output
7211 // example above for explanations.
7212 void print_power_as_myformat(const power & p,
7213 const print_myformat & c,
7216 unsigned power_prec = p.precedence();
7217 if (level >= power_prec)
7219 p.op(0).print(c, power_prec);
7221 p.op(1).print(c, power_prec);
7222 if (level >= power_prec)
7228 // install a new print method for power objects
7229 set_print_func<power, print_myformat>(print_power_as_myformat);
7231 // now this prints "1+x**2"
7232 e.print(print_myformat()); cout << endl;
7234 // but the default format is still "1+x^2"
7240 @node Structures, Adding classes, Printing, Extending GiNaC
7241 @c node-name, next, previous, up
7244 If you are doing some very specialized things with GiNaC, or if you just
7245 need some more organized way to store data in your expressions instead of
7246 anonymous lists, you may want to implement your own algebraic classes.
7247 ('algebraic class' means any class directly or indirectly derived from
7248 @code{basic} that can be used in GiNaC expressions).
7250 GiNaC offers two ways of accomplishing this: either by using the
7251 @code{structure<T>} template class, or by rolling your own class from
7252 scratch. This section will discuss the @code{structure<T>} template which
7253 is easier to use but more limited, while the implementation of custom
7254 GiNaC classes is the topic of the next section. However, you may want to
7255 read both sections because many common concepts and member functions are
7256 shared by both concepts, and it will also allow you to decide which approach
7257 is most suited to your needs.
7259 The @code{structure<T>} template, defined in the GiNaC header file
7260 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7261 or @code{class}) into a GiNaC object that can be used in expressions.
7263 @subsection Example: scalar products
7265 Let's suppose that we need a way to handle some kind of abstract scalar
7266 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7267 product class have to store their left and right operands, which can in turn
7268 be arbitrary expressions. Here is a possible way to represent such a
7269 product in a C++ @code{struct}:
7273 using namespace std;
7275 #include <ginac/ginac.h>
7276 using namespace GiNaC;
7282 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7286 The default constructor is required. Now, to make a GiNaC class out of this
7287 data structure, we need only one line:
7290 typedef structure<sprod_s> sprod;
7293 That's it. This line constructs an algebraic class @code{sprod} which
7294 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7295 expressions like any other GiNaC class:
7299 symbol a("a"), b("b");
7300 ex e = sprod(sprod_s(a, b));
7304 Note the difference between @code{sprod} which is the algebraic class, and
7305 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7306 and @code{right} data members. As shown above, an @code{sprod} can be
7307 constructed from an @code{sprod_s} object.
7309 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7310 you could define a little wrapper function like this:
7313 inline ex make_sprod(ex left, ex right)
7315 return sprod(sprod_s(left, right));
7319 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7320 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7321 @code{get_struct()}:
7325 cout << ex_to<sprod>(e)->left << endl;
7327 cout << ex_to<sprod>(e).get_struct().right << endl;
7332 You only have read access to the members of @code{sprod_s}.
7334 The type definition of @code{sprod} is enough to write your own algorithms
7335 that deal with scalar products, for example:
7340 if (is_a<sprod>(p)) @{
7341 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7342 return make_sprod(sp.right, sp.left);
7353 @subsection Structure output
7355 While the @code{sprod} type is useable it still leaves something to be
7356 desired, most notably proper output:
7361 // -> [structure object]
7365 By default, any structure types you define will be printed as
7366 @samp{[structure object]}. To override this you can either specialize the
7367 template's @code{print()} member function, or specify print methods with
7368 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7369 it's not possible to supply class options like @code{print_func<>()} to
7370 structures, so for a self-contained structure type you need to resort to
7371 overriding the @code{print()} function, which is also what we will do here.
7373 The member functions of GiNaC classes are described in more detail in the
7374 next section, but it shouldn't be hard to figure out what's going on here:
7377 void sprod::print(const print_context & c, unsigned level) const
7379 // tree debug output handled by superclass
7380 if (is_a<print_tree>(c))
7381 inherited::print(c, level);
7383 // get the contained sprod_s object
7384 const sprod_s & sp = get_struct();
7386 // print_context::s is a reference to an ostream
7387 c.s << "<" << sp.left << "|" << sp.right << ">";
7391 Now we can print expressions containing scalar products:
7397 cout << swap_sprod(e) << endl;
7402 @subsection Comparing structures
7404 The @code{sprod} class defined so far still has one important drawback: all
7405 scalar products are treated as being equal because GiNaC doesn't know how to
7406 compare objects of type @code{sprod_s}. This can lead to some confusing
7407 and undesired behavior:
7411 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7413 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7414 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7418 To remedy this, we first need to define the operators @code{==} and @code{<}
7419 for objects of type @code{sprod_s}:
7422 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7424 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7427 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7429 return lhs.left.compare(rhs.left) < 0
7430 ? true : lhs.right.compare(rhs.right) < 0;
7434 The ordering established by the @code{<} operator doesn't have to make any
7435 algebraic sense, but it needs to be well defined. Note that we can't use
7436 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7437 in the implementation of these operators because they would construct
7438 GiNaC @code{relational} objects which in the case of @code{<} do not
7439 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7440 decide which one is algebraically 'less').
7442 Next, we need to change our definition of the @code{sprod} type to let
7443 GiNaC know that an ordering relation exists for the embedded objects:
7446 typedef structure<sprod_s, compare_std_less> sprod;
7449 @code{sprod} objects then behave as expected:
7453 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7454 // -> <a|b>-<a^2|b^2>
7455 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7456 // -> <a|b>+<a^2|b^2>
7457 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7459 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7464 The @code{compare_std_less} policy parameter tells GiNaC to use the
7465 @code{std::less} and @code{std::equal_to} functors to compare objects of
7466 type @code{sprod_s}. By default, these functors forward their work to the
7467 standard @code{<} and @code{==} operators, which we have overloaded.
7468 Alternatively, we could have specialized @code{std::less} and
7469 @code{std::equal_to} for class @code{sprod_s}.
7471 GiNaC provides two other comparison policies for @code{structure<T>}
7472 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7473 which does a bit-wise comparison of the contained @code{T} objects.
7474 This should be used with extreme care because it only works reliably with
7475 built-in integral types, and it also compares any padding (filler bytes of
7476 undefined value) that the @code{T} class might have.
7478 @subsection Subexpressions
7480 Our scalar product class has two subexpressions: the left and right
7481 operands. It might be a good idea to make them accessible via the standard
7482 @code{nops()} and @code{op()} methods:
7485 size_t sprod::nops() const
7490 ex sprod::op(size_t i) const
7494 return get_struct().left;
7496 return get_struct().right;
7498 throw std::range_error("sprod::op(): no such operand");
7503 Implementing @code{nops()} and @code{op()} for container types such as
7504 @code{sprod} has two other nice side effects:
7508 @code{has()} works as expected
7510 GiNaC generates better hash keys for the objects (the default implementation
7511 of @code{calchash()} takes subexpressions into account)
7514 @cindex @code{let_op()}
7515 There is a non-const variant of @code{op()} called @code{let_op()} that
7516 allows replacing subexpressions:
7519 ex & sprod::let_op(size_t i)
7521 // every non-const member function must call this
7522 ensure_if_modifiable();
7526 return get_struct().left;
7528 return get_struct().right;
7530 throw std::range_error("sprod::let_op(): no such operand");
7535 Once we have provided @code{let_op()} we also get @code{subs()} and
7536 @code{map()} for free. In fact, every container class that returns a non-null
7537 @code{nops()} value must either implement @code{let_op()} or provide custom
7538 implementations of @code{subs()} and @code{map()}.
7540 In turn, the availability of @code{map()} enables the recursive behavior of a
7541 couple of other default method implementations, in particular @code{evalf()},
7542 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7543 we probably want to provide our own version of @code{expand()} for scalar
7544 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7545 This is left as an exercise for the reader.
7547 The @code{structure<T>} template defines many more member functions that
7548 you can override by specialization to customize the behavior of your
7549 structures. You are referred to the next section for a description of
7550 some of these (especially @code{eval()}). There is, however, one topic
7551 that shall be addressed here, as it demonstrates one peculiarity of the
7552 @code{structure<T>} template: archiving.
7554 @subsection Archiving structures
7556 If you don't know how the archiving of GiNaC objects is implemented, you
7557 should first read the next section and then come back here. You're back?
7560 To implement archiving for structures it is not enough to provide
7561 specializations for the @code{archive()} member function and the
7562 unarchiving constructor (the @code{unarchive()} function has a default
7563 implementation). You also need to provide a unique name (as a string literal)
7564 for each structure type you define. This is because in GiNaC archives,
7565 the class of an object is stored as a string, the class name.
7567 By default, this class name (as returned by the @code{class_name()} member
7568 function) is @samp{structure} for all structure classes. This works as long
7569 as you have only defined one structure type, but if you use two or more you
7570 need to provide a different name for each by specializing the
7571 @code{get_class_name()} member function. Here is a sample implementation
7572 for enabling archiving of the scalar product type defined above:
7575 const char *sprod::get_class_name() @{ return "sprod"; @}
7577 void sprod::archive(archive_node & n) const
7579 inherited::archive(n);
7580 n.add_ex("left", get_struct().left);
7581 n.add_ex("right", get_struct().right);
7584 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7586 n.find_ex("left", get_struct().left, sym_lst);
7587 n.find_ex("right", get_struct().right, sym_lst);
7591 Note that the unarchiving constructor is @code{sprod::structure} and not
7592 @code{sprod::sprod}, and that we don't need to supply an
7593 @code{sprod::unarchive()} function.
7596 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7597 @c node-name, next, previous, up
7598 @section Adding classes
7600 The @code{structure<T>} template provides an way to extend GiNaC with custom
7601 algebraic classes that is easy to use but has its limitations, the most
7602 severe of which being that you can't add any new member functions to
7603 structures. To be able to do this, you need to write a new class definition
7606 This section will explain how to implement new algebraic classes in GiNaC by
7607 giving the example of a simple 'string' class. After reading this section
7608 you will know how to properly declare a GiNaC class and what the minimum
7609 required member functions are that you have to implement. We only cover the
7610 implementation of a 'leaf' class here (i.e. one that doesn't contain
7611 subexpressions). Creating a container class like, for example, a class
7612 representing tensor products is more involved but this section should give
7613 you enough information so you can consult the source to GiNaC's predefined
7614 classes if you want to implement something more complicated.
7616 @subsection GiNaC's run-time type information system
7618 @cindex hierarchy of classes
7620 All algebraic classes (that is, all classes that can appear in expressions)
7621 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7622 @code{basic *} (which is essentially what an @code{ex} is) represents a
7623 generic pointer to an algebraic class. Occasionally it is necessary to find
7624 out what the class of an object pointed to by a @code{basic *} really is.
7625 Also, for the unarchiving of expressions it must be possible to find the
7626 @code{unarchive()} function of a class given the class name (as a string). A
7627 system that provides this kind of information is called a run-time type
7628 information (RTTI) system. The C++ language provides such a thing (see the
7629 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7630 implements its own, simpler RTTI.
7632 The RTTI in GiNaC is based on two mechanisms:
7637 The @code{basic} class declares a member variable @code{tinfo_key} which
7638 holds an unsigned integer that identifies the object's class. These numbers
7639 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7640 classes. They all start with @code{TINFO_}.
7643 By means of some clever tricks with static members, GiNaC maintains a list
7644 of information for all classes derived from @code{basic}. The information
7645 available includes the class names, the @code{tinfo_key}s, and pointers
7646 to the unarchiving functions. This class registry is defined in the
7647 @file{registrar.h} header file.
7651 The disadvantage of this proprietary RTTI implementation is that there's
7652 a little more to do when implementing new classes (C++'s RTTI works more
7653 or less automatically) but don't worry, most of the work is simplified by
7656 @subsection A minimalistic example
7658 Now we will start implementing a new class @code{mystring} that allows
7659 placing character strings in algebraic expressions (this is not very useful,
7660 but it's just an example). This class will be a direct subclass of
7661 @code{basic}. You can use this sample implementation as a starting point
7662 for your own classes.
7664 The code snippets given here assume that you have included some header files
7670 #include <stdexcept>
7671 using namespace std;
7673 #include <ginac/ginac.h>
7674 using namespace GiNaC;
7677 The first thing we have to do is to define a @code{tinfo_key} for our new
7678 class. This can be any arbitrary unsigned number that is not already taken
7679 by one of the existing classes but it's better to come up with something
7680 that is unlikely to clash with keys that might be added in the future. The
7681 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7682 which is not a requirement but we are going to stick with this scheme:
7685 const unsigned TINFO_mystring = 0x42420001U;
7688 Now we can write down the class declaration. The class stores a C++
7689 @code{string} and the user shall be able to construct a @code{mystring}
7690 object from a C or C++ string:
7693 class mystring : public basic
7695 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7698 mystring(const string &s);
7699 mystring(const char *s);
7705 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7708 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7709 macros are defined in @file{registrar.h}. They take the name of the class
7710 and its direct superclass as arguments and insert all required declarations
7711 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7712 the first line after the opening brace of the class definition. The
7713 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7714 source (at global scope, of course, not inside a function).
7716 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7717 declarations of the default constructor and a couple of other functions that
7718 are required. It also defines a type @code{inherited} which refers to the
7719 superclass so you don't have to modify your code every time you shuffle around
7720 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7721 class with the GiNaC RTTI (there is also a
7722 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7723 options for the class, and which we will be using instead in a few minutes).
7725 Now there are seven member functions we have to implement to get a working
7731 @code{mystring()}, the default constructor.
7734 @code{void archive(archive_node &n)}, the archiving function. This stores all
7735 information needed to reconstruct an object of this class inside an
7736 @code{archive_node}.
7739 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7740 constructor. This constructs an instance of the class from the information
7741 found in an @code{archive_node}.
7744 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7745 unarchiving function. It constructs a new instance by calling the unarchiving
7749 @cindex @code{compare_same_type()}
7750 @code{int compare_same_type(const basic &other)}, which is used internally
7751 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7752 -1, depending on the relative order of this object and the @code{other}
7753 object. If it returns 0, the objects are considered equal.
7754 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7755 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7756 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7757 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7758 must provide a @code{compare_same_type()} function, even those representing
7759 objects for which no reasonable algebraic ordering relationship can be
7763 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7764 which are the two constructors we declared.
7768 Let's proceed step-by-step. The default constructor looks like this:
7771 mystring::mystring() : inherited(TINFO_mystring) @{@}
7774 The golden rule is that in all constructors you have to set the
7775 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7776 it will be set by the constructor of the superclass and all hell will break
7777 loose in the RTTI. For your convenience, the @code{basic} class provides
7778 a constructor that takes a @code{tinfo_key} value, which we are using here
7779 (remember that in our case @code{inherited == basic}). If the superclass
7780 didn't have such a constructor, we would have to set the @code{tinfo_key}
7781 to the right value manually.
7783 In the default constructor you should set all other member variables to
7784 reasonable default values (we don't need that here since our @code{str}
7785 member gets set to an empty string automatically).
7787 Next are the three functions for archiving. You have to implement them even
7788 if you don't plan to use archives, but the minimum required implementation
7789 is really simple. First, the archiving function:
7792 void mystring::archive(archive_node &n) const
7794 inherited::archive(n);
7795 n.add_string("string", str);
7799 The only thing that is really required is calling the @code{archive()}
7800 function of the superclass. Optionally, you can store all information you
7801 deem necessary for representing the object into the passed
7802 @code{archive_node}. We are just storing our string here. For more
7803 information on how the archiving works, consult the @file{archive.h} header
7806 The unarchiving constructor is basically the inverse of the archiving
7810 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7812 n.find_string("string", str);
7816 If you don't need archiving, just leave this function empty (but you must
7817 invoke the unarchiving constructor of the superclass). Note that we don't
7818 have to set the @code{tinfo_key} here because it is done automatically
7819 by the unarchiving constructor of the @code{basic} class.
7821 Finally, the unarchiving function:
7824 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7826 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7830 You don't have to understand how exactly this works. Just copy these
7831 four lines into your code literally (replacing the class name, of
7832 course). It calls the unarchiving constructor of the class and unless
7833 you are doing something very special (like matching @code{archive_node}s
7834 to global objects) you don't need a different implementation. For those
7835 who are interested: setting the @code{dynallocated} flag puts the object
7836 under the control of GiNaC's garbage collection. It will get deleted
7837 automatically once it is no longer referenced.
7839 Our @code{compare_same_type()} function uses a provided function to compare
7843 int mystring::compare_same_type(const basic &other) const
7845 const mystring &o = static_cast<const mystring &>(other);
7846 int cmpval = str.compare(o.str);
7849 else if (cmpval < 0)
7856 Although this function takes a @code{basic &}, it will always be a reference
7857 to an object of exactly the same class (objects of different classes are not
7858 comparable), so the cast is safe. If this function returns 0, the two objects
7859 are considered equal (in the sense that @math{A-B=0}), so you should compare
7860 all relevant member variables.
7862 Now the only thing missing is our two new constructors:
7865 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7866 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7869 No surprises here. We set the @code{str} member from the argument and
7870 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7872 That's it! We now have a minimal working GiNaC class that can store
7873 strings in algebraic expressions. Let's confirm that the RTTI works:
7876 ex e = mystring("Hello, world!");
7877 cout << is_a<mystring>(e) << endl;
7880 cout << ex_to<basic>(e).class_name() << endl;
7884 Obviously it does. Let's see what the expression @code{e} looks like:
7888 // -> [mystring object]
7891 Hm, not exactly what we expect, but of course the @code{mystring} class
7892 doesn't yet know how to print itself. This can be done either by implementing
7893 the @code{print()} member function, or, preferably, by specifying a
7894 @code{print_func<>()} class option. Let's say that we want to print the string
7895 surrounded by double quotes:
7898 class mystring : public basic
7902 void do_print(const print_context &c, unsigned level = 0) const;
7906 void mystring::do_print(const print_context &c, unsigned level) const
7908 // print_context::s is a reference to an ostream
7909 c.s << '\"' << str << '\"';
7913 The @code{level} argument is only required for container classes to
7914 correctly parenthesize the output.
7916 Now we need to tell GiNaC that @code{mystring} objects should use the
7917 @code{do_print()} member function for printing themselves. For this, we
7921 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7927 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7928 print_func<print_context>(&mystring::do_print))
7931 Let's try again to print the expression:
7935 // -> "Hello, world!"
7938 Much better. If we wanted to have @code{mystring} objects displayed in a
7939 different way depending on the output format (default, LaTeX, etc.), we
7940 would have supplied multiple @code{print_func<>()} options with different
7941 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7942 separated by dots. This is similar to the way options are specified for
7943 symbolic functions. @xref{Printing}, for a more in-depth description of the
7944 way expression output is implemented in GiNaC.
7946 The @code{mystring} class can be used in arbitrary expressions:
7949 e += mystring("GiNaC rulez");
7951 // -> "GiNaC rulez"+"Hello, world!"
7954 (GiNaC's automatic term reordering is in effect here), or even
7957 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7959 // -> "One string"^(2*sin(-"Another string"+Pi))
7962 Whether this makes sense is debatable but remember that this is only an
7963 example. At least it allows you to implement your own symbolic algorithms
7966 Note that GiNaC's algebraic rules remain unchanged:
7969 e = mystring("Wow") * mystring("Wow");
7973 e = pow(mystring("First")-mystring("Second"), 2);
7974 cout << e.expand() << endl;
7975 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7978 There's no way to, for example, make GiNaC's @code{add} class perform string
7979 concatenation. You would have to implement this yourself.
7981 @subsection Automatic evaluation
7984 @cindex @code{eval()}
7985 @cindex @code{hold()}
7986 When dealing with objects that are just a little more complicated than the
7987 simple string objects we have implemented, chances are that you will want to
7988 have some automatic simplifications or canonicalizations performed on them.
7989 This is done in the evaluation member function @code{eval()}. Let's say that
7990 we wanted all strings automatically converted to lowercase with
7991 non-alphabetic characters stripped, and empty strings removed:
7994 class mystring : public basic
7998 ex eval(int level = 0) const;
8002 ex mystring::eval(int level) const
8005 for (int i=0; i<str.length(); i++) @{
8007 if (c >= 'A' && c <= 'Z')
8008 new_str += tolower(c);
8009 else if (c >= 'a' && c <= 'z')
8013 if (new_str.length() == 0)
8016 return mystring(new_str).hold();
8020 The @code{level} argument is used to limit the recursion depth of the
8021 evaluation. We don't have any subexpressions in the @code{mystring}
8022 class so we are not concerned with this. If we had, we would call the
8023 @code{eval()} functions of the subexpressions with @code{level - 1} as
8024 the argument if @code{level != 1}. The @code{hold()} member function
8025 sets a flag in the object that prevents further evaluation. Otherwise
8026 we might end up in an endless loop. When you want to return the object
8027 unmodified, use @code{return this->hold();}.
8029 Let's confirm that it works:
8032 ex e = mystring("Hello, world!") + mystring("!?#");
8036 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8041 @subsection Optional member functions
8043 We have implemented only a small set of member functions to make the class
8044 work in the GiNaC framework. There are two functions that are not strictly
8045 required but will make operations with objects of the class more efficient:
8047 @cindex @code{calchash()}
8048 @cindex @code{is_equal_same_type()}
8050 unsigned calchash() const;
8051 bool is_equal_same_type(const basic &other) const;
8054 The @code{calchash()} method returns an @code{unsigned} hash value for the
8055 object which will allow GiNaC to compare and canonicalize expressions much
8056 more efficiently. You should consult the implementation of some of the built-in
8057 GiNaC classes for examples of hash functions. The default implementation of
8058 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8059 class and all subexpressions that are accessible via @code{op()}.
8061 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8062 tests for equality without establishing an ordering relation, which is often
8063 faster. The default implementation of @code{is_equal_same_type()} just calls
8064 @code{compare_same_type()} and tests its result for zero.
8066 @subsection Other member functions
8068 For a real algebraic class, there are probably some more functions that you
8069 might want to provide:
8072 bool info(unsigned inf) const;
8073 ex evalf(int level = 0) const;
8074 ex series(const relational & r, int order, unsigned options = 0) const;
8075 ex derivative(const symbol & s) const;
8078 If your class stores sub-expressions (see the scalar product example in the
8079 previous section) you will probably want to override
8081 @cindex @code{let_op()}
8084 ex op(size_t i) const;
8085 ex & let_op(size_t i);
8086 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8087 ex map(map_function & f) const;
8090 @code{let_op()} is a variant of @code{op()} that allows write access. The
8091 default implementations of @code{subs()} and @code{map()} use it, so you have
8092 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8094 You can, of course, also add your own new member functions. Remember
8095 that the RTTI may be used to get information about what kinds of objects
8096 you are dealing with (the position in the class hierarchy) and that you
8097 can always extract the bare object from an @code{ex} by stripping the
8098 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8099 should become a need.
8101 That's it. May the source be with you!
8104 @node A comparison with other CAS, Advantages, Adding classes, Top
8105 @c node-name, next, previous, up
8106 @chapter A Comparison With Other CAS
8109 This chapter will give you some information on how GiNaC compares to
8110 other, traditional Computer Algebra Systems, like @emph{Maple},
8111 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8112 disadvantages over these systems.
8115 * Advantages:: Strengths of the GiNaC approach.
8116 * Disadvantages:: Weaknesses of the GiNaC approach.
8117 * Why C++?:: Attractiveness of C++.
8120 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8121 @c node-name, next, previous, up
8124 GiNaC has several advantages over traditional Computer
8125 Algebra Systems, like
8130 familiar language: all common CAS implement their own proprietary
8131 grammar which you have to learn first (and maybe learn again when your
8132 vendor decides to `enhance' it). With GiNaC you can write your program
8133 in common C++, which is standardized.
8137 structured data types: you can build up structured data types using
8138 @code{struct}s or @code{class}es together with STL features instead of
8139 using unnamed lists of lists of lists.
8142 strongly typed: in CAS, you usually have only one kind of variables
8143 which can hold contents of an arbitrary type. This 4GL like feature is
8144 nice for novice programmers, but dangerous.
8147 development tools: powerful development tools exist for C++, like fancy
8148 editors (e.g. with automatic indentation and syntax highlighting),
8149 debuggers, visualization tools, documentation generators@dots{}
8152 modularization: C++ programs can easily be split into modules by
8153 separating interface and implementation.
8156 price: GiNaC is distributed under the GNU Public License which means
8157 that it is free and available with source code. And there are excellent
8158 C++-compilers for free, too.
8161 extendable: you can add your own classes to GiNaC, thus extending it on
8162 a very low level. Compare this to a traditional CAS that you can
8163 usually only extend on a high level by writing in the language defined
8164 by the parser. In particular, it turns out to be almost impossible to
8165 fix bugs in a traditional system.
8168 multiple interfaces: Though real GiNaC programs have to be written in
8169 some editor, then be compiled, linked and executed, there are more ways
8170 to work with the GiNaC engine. Many people want to play with
8171 expressions interactively, as in traditional CASs. Currently, two such
8172 windows into GiNaC have been implemented and many more are possible: the
8173 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8174 types to a command line and second, as a more consistent approach, an
8175 interactive interface to the Cint C++ interpreter has been put together
8176 (called GiNaC-cint) that allows an interactive scripting interface
8177 consistent with the C++ language. It is available from the usual GiNaC
8181 seamless integration: it is somewhere between difficult and impossible
8182 to call CAS functions from within a program written in C++ or any other
8183 programming language and vice versa. With GiNaC, your symbolic routines
8184 are part of your program. You can easily call third party libraries,
8185 e.g. for numerical evaluation or graphical interaction. All other
8186 approaches are much more cumbersome: they range from simply ignoring the
8187 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8188 system (i.e. @emph{Yacas}).
8191 efficiency: often large parts of a program do not need symbolic
8192 calculations at all. Why use large integers for loop variables or
8193 arbitrary precision arithmetics where @code{int} and @code{double} are
8194 sufficient? For pure symbolic applications, GiNaC is comparable in
8195 speed with other CAS.
8200 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8201 @c node-name, next, previous, up
8202 @section Disadvantages
8204 Of course it also has some disadvantages:
8209 advanced features: GiNaC cannot compete with a program like
8210 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8211 which grows since 1981 by the work of dozens of programmers, with
8212 respect to mathematical features. Integration, factorization,
8213 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8214 not planned for the near future).
8217 portability: While the GiNaC library itself is designed to avoid any
8218 platform dependent features (it should compile on any ANSI compliant C++
8219 compiler), the currently used version of the CLN library (fast large
8220 integer and arbitrary precision arithmetics) can only by compiled
8221 without hassle on systems with the C++ compiler from the GNU Compiler
8222 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8223 macros to let the compiler gather all static initializations, which
8224 works for GNU C++ only. Feel free to contact the authors in case you
8225 really believe that you need to use a different compiler. We have
8226 occasionally used other compilers and may be able to give you advice.}
8227 GiNaC uses recent language features like explicit constructors, mutable
8228 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8229 literally. Recent GCC versions starting at 2.95.3, although itself not
8230 yet ANSI compliant, support all needed features.
8235 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8236 @c node-name, next, previous, up
8239 Why did we choose to implement GiNaC in C++ instead of Java or any other
8240 language? C++ is not perfect: type checking is not strict (casting is
8241 possible), separation between interface and implementation is not
8242 complete, object oriented design is not enforced. The main reason is
8243 the often scolded feature of operator overloading in C++. While it may
8244 be true that operating on classes with a @code{+} operator is rarely
8245 meaningful, it is perfectly suited for algebraic expressions. Writing
8246 @math{3x+5y} as @code{3*x+5*y} instead of
8247 @code{x.times(3).plus(y.times(5))} looks much more natural.
8248 Furthermore, the main developers are more familiar with C++ than with
8249 any other programming language.
8252 @node Internal structures, Expressions are reference counted, Why C++? , Top
8253 @c node-name, next, previous, up
8254 @appendix Internal structures
8257 * Expressions are reference counted::
8258 * Internal representation of products and sums::
8261 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8262 @c node-name, next, previous, up
8263 @appendixsection Expressions are reference counted
8265 @cindex reference counting
8266 @cindex copy-on-write
8267 @cindex garbage collection
8268 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8269 where the counter belongs to the algebraic objects derived from class
8270 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8271 which @code{ex} contains an instance. If you understood that, you can safely
8272 skip the rest of this passage.
8274 Expressions are extremely light-weight since internally they work like
8275 handles to the actual representation. They really hold nothing more
8276 than a pointer to some other object. What this means in practice is
8277 that whenever you create two @code{ex} and set the second equal to the
8278 first no copying process is involved. Instead, the copying takes place
8279 as soon as you try to change the second. Consider the simple sequence
8284 #include <ginac/ginac.h>
8285 using namespace std;
8286 using namespace GiNaC;
8290 symbol x("x"), y("y"), z("z");
8293 e1 = sin(x + 2*y) + 3*z + 41;
8294 e2 = e1; // e2 points to same object as e1
8295 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8296 e2 += 1; // e2 is copied into a new object
8297 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8301 The line @code{e2 = e1;} creates a second expression pointing to the
8302 object held already by @code{e1}. The time involved for this operation
8303 is therefore constant, no matter how large @code{e1} was. Actual
8304 copying, however, must take place in the line @code{e2 += 1;} because
8305 @code{e1} and @code{e2} are not handles for the same object any more.
8306 This concept is called @dfn{copy-on-write semantics}. It increases
8307 performance considerably whenever one object occurs multiple times and
8308 represents a simple garbage collection scheme because when an @code{ex}
8309 runs out of scope its destructor checks whether other expressions handle
8310 the object it points to too and deletes the object from memory if that
8311 turns out not to be the case. A slightly less trivial example of
8312 differentiation using the chain-rule should make clear how powerful this
8317 symbol x("x"), y("y");
8321 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8322 cout << e1 << endl // prints x+3*y
8323 << e2 << endl // prints (x+3*y)^3
8324 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8328 Here, @code{e1} will actually be referenced three times while @code{e2}
8329 will be referenced two times. When the power of an expression is built,
8330 that expression needs not be copied. Likewise, since the derivative of
8331 a power of an expression can be easily expressed in terms of that
8332 expression, no copying of @code{e1} is involved when @code{e3} is
8333 constructed. So, when @code{e3} is constructed it will print as
8334 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8335 holds a reference to @code{e2} and the factor in front is just
8338 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8339 semantics. When you insert an expression into a second expression, the
8340 result behaves exactly as if the contents of the first expression were
8341 inserted. But it may be useful to remember that this is not what
8342 happens. Knowing this will enable you to write much more efficient
8343 code. If you still have an uncertain feeling with copy-on-write
8344 semantics, we recommend you have a look at the
8345 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8346 Marshall Cline. Chapter 16 covers this issue and presents an
8347 implementation which is pretty close to the one in GiNaC.
8350 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8351 @c node-name, next, previous, up
8352 @appendixsection Internal representation of products and sums
8354 @cindex representation
8357 @cindex @code{power}
8358 Although it should be completely transparent for the user of
8359 GiNaC a short discussion of this topic helps to understand the sources
8360 and also explain performance to a large degree. Consider the
8361 unexpanded symbolic expression
8363 $2d^3 \left( 4a + 5b - 3 \right)$
8366 @math{2*d^3*(4*a+5*b-3)}
8368 which could naively be represented by a tree of linear containers for
8369 addition and multiplication, one container for exponentiation with base
8370 and exponent and some atomic leaves of symbols and numbers in this
8375 @cindex pair-wise representation
8376 However, doing so results in a rather deeply nested tree which will
8377 quickly become inefficient to manipulate. We can improve on this by
8378 representing the sum as a sequence of terms, each one being a pair of a
8379 purely numeric multiplicative coefficient and its rest. In the same
8380 spirit we can store the multiplication as a sequence of terms, each
8381 having a numeric exponent and a possibly complicated base, the tree
8382 becomes much more flat:
8386 The number @code{3} above the symbol @code{d} shows that @code{mul}
8387 objects are treated similarly where the coefficients are interpreted as
8388 @emph{exponents} now. Addition of sums of terms or multiplication of
8389 products with numerical exponents can be coded to be very efficient with
8390 such a pair-wise representation. Internally, this handling is performed
8391 by most CAS in this way. It typically speeds up manipulations by an
8392 order of magnitude. The overall multiplicative factor @code{2} and the
8393 additive term @code{-3} look somewhat out of place in this
8394 representation, however, since they are still carrying a trivial
8395 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8396 this is avoided by adding a field that carries an overall numeric
8397 coefficient. This results in the realistic picture of internal
8400 $2d^3 \left( 4a + 5b - 3 \right)$:
8403 @math{2*d^3*(4*a+5*b-3)}:
8409 This also allows for a better handling of numeric radicals, since
8410 @code{sqrt(2)} can now be carried along calculations. Now it should be
8411 clear, why both classes @code{add} and @code{mul} are derived from the
8412 same abstract class: the data representation is the same, only the
8413 semantics differs. In the class hierarchy, methods for polynomial
8414 expansion and the like are reimplemented for @code{add} and @code{mul},
8415 but the data structure is inherited from @code{expairseq}.
8418 @node Package tools, ginac-config, Internal representation of products and sums, Top
8419 @c node-name, next, previous, up
8420 @appendix Package tools
8422 If you are creating a software package that uses the GiNaC library,
8423 setting the correct command line options for the compiler and linker
8424 can be difficult. GiNaC includes two tools to make this process easier.
8427 * ginac-config:: A shell script to detect compiler and linker flags.
8428 * AM_PATH_GINAC:: Macro for GNU automake.
8432 @node ginac-config, AM_PATH_GINAC, Package tools, Package tools
8433 @c node-name, next, previous, up
8434 @section @command{ginac-config}
8435 @cindex ginac-config
8437 @command{ginac-config} is a shell script that you can use to determine
8438 the compiler and linker command line options required to compile and
8439 link a program with the GiNaC library.
8441 @command{ginac-config} takes the following flags:
8445 Prints out the version of GiNaC installed.
8447 Prints '-I' flags pointing to the installed header files.
8449 Prints out the linker flags necessary to link a program against GiNaC.
8450 @item --prefix[=@var{PREFIX}]
8451 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8452 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8453 Otherwise, prints out the configured value of @env{$prefix}.
8454 @item --exec-prefix[=@var{PREFIX}]
8455 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8456 Otherwise, prints out the configured value of @env{$exec_prefix}.
8459 Typically, @command{ginac-config} will be used within a configure
8460 script, as described below. It, however, can also be used directly from
8461 the command line using backquotes to compile a simple program. For
8465 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8468 This command line might expand to (for example):
8471 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8472 -lginac -lcln -lstdc++
8475 Not only is the form using @command{ginac-config} easier to type, it will
8476 work on any system, no matter how GiNaC was configured.
8479 @node AM_PATH_GINAC, Configure script options, ginac-config, Package tools
8480 @c node-name, next, previous, up
8481 @section @samp{AM_PATH_GINAC}
8482 @cindex AM_PATH_GINAC
8484 For packages configured using GNU automake, GiNaC also provides
8485 a macro to automate the process of checking for GiNaC.
8488 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8489 [, @var{ACTION-IF-NOT-FOUND}]]])
8497 Determines the location of GiNaC using @command{ginac-config}, which is
8498 either found in the user's path, or from the environment variable
8499 @env{GINACLIB_CONFIG}.
8502 Tests the installed libraries to make sure that their version
8503 is later than @var{MINIMUM-VERSION}. (A default version will be used
8507 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8508 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8509 variable to the output of @command{ginac-config --libs}, and calls
8510 @samp{AC_SUBST()} for these variables so they can be used in generated
8511 makefiles, and then executes @var{ACTION-IF-FOUND}.
8514 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8515 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8519 This macro is in file @file{ginac.m4} which is installed in
8520 @file{$datadir/aclocal}. Note that if automake was installed with a
8521 different @samp{--prefix} than GiNaC, you will either have to manually
8522 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8523 aclocal the @samp{-I} option when running it.
8526 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8527 * Example package:: Example of a package using AM_PATH_GINAC.
8531 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8532 @c node-name, next, previous, up
8533 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8535 Simply make sure that @command{ginac-config} is in your path, and run
8536 the configure script.
8543 The directory where the GiNaC libraries are installed needs
8544 to be found by your system's dynamic linker.
8546 This is generally done by
8549 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8555 setting the environment variable @env{LD_LIBRARY_PATH},
8558 or, as a last resort,
8561 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8562 running configure, for instance:
8565 LDFLAGS=-R/home/cbauer/lib ./configure
8570 You can also specify a @command{ginac-config} not in your path by
8571 setting the @env{GINACLIB_CONFIG} environment variable to the
8572 name of the executable
8575 If you move the GiNaC package from its installed location,
8576 you will either need to modify @command{ginac-config} script
8577 manually to point to the new location or rebuild GiNaC.
8588 --with-ginac-prefix=@var{PREFIX}
8589 --with-ginac-exec-prefix=@var{PREFIX}
8592 are provided to override the prefix and exec-prefix that were stored
8593 in the @command{ginac-config} shell script by GiNaC's configure. You are
8594 generally better off configuring GiNaC with the right path to begin with.
8598 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8599 @c node-name, next, previous, up
8600 @subsection Example of a package using @samp{AM_PATH_GINAC}
8602 The following shows how to build a simple package using automake
8603 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8607 #include <ginac/ginac.h>
8611 GiNaC::symbol x("x");
8612 GiNaC::ex a = GiNaC::sin(x);
8613 std::cout << "Derivative of " << a
8614 << " is " << a.diff(x) << std::endl;
8619 You should first read the introductory portions of the automake
8620 Manual, if you are not already familiar with it.
8622 Two files are needed, @file{configure.in}, which is used to build the
8626 dnl Process this file with autoconf to produce a configure script.
8628 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8634 AM_PATH_GINAC(0.9.0, [
8635 LIBS="$LIBS $GINACLIB_LIBS"
8636 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8637 ], AC_MSG_ERROR([need to have GiNaC installed]))
8642 The only command in this which is not standard for automake
8643 is the @samp{AM_PATH_GINAC} macro.
8645 That command does the following: If a GiNaC version greater or equal
8646 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8647 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8648 the error message `need to have GiNaC installed'
8650 And the @file{Makefile.am}, which will be used to build the Makefile.
8653 ## Process this file with automake to produce Makefile.in
8654 bin_PROGRAMS = simple
8655 simple_SOURCES = simple.cpp
8658 This @file{Makefile.am}, says that we are building a single executable,
8659 from a single source file @file{simple.cpp}. Since every program
8660 we are building uses GiNaC we simply added the GiNaC options
8661 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8662 want to specify them on a per-program basis: for instance by
8666 simple_LDADD = $(GINACLIB_LIBS)
8667 INCLUDES = $(GINACLIB_CPPFLAGS)
8670 to the @file{Makefile.am}.
8672 To try this example out, create a new directory and add the three
8675 Now execute the following commands:
8678 $ automake --add-missing
8683 You now have a package that can be built in the normal fashion
8692 @node Bibliography, Concept index, Example package, Top
8693 @c node-name, next, previous, up
8694 @appendix Bibliography
8699 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8702 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8705 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8708 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8711 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8712 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8715 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8716 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8717 Academic Press, London
8720 @cite{Computer Algebra Systems - A Practical Guide},
8721 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8724 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8725 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8728 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8729 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8732 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8737 @node Concept index, , Bibliography, Top
8738 @c node-name, next, previous, up
8739 @unnumbered Concept index