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) 1999-2004 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{} 1999-2004 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{} 1999-2004 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., 59 Temple Place - Suite 330, 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 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
688 * Hash Maps:: A faster alternative to std::map<>.
692 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
693 @c node-name, next, previous, up
695 @cindex expression (class @code{ex})
698 The most common class of objects a user deals with is the expression
699 @code{ex}, representing a mathematical object like a variable, number,
700 function, sum, product, etc@dots{} Expressions may be put together to form
701 new expressions, passed as arguments to functions, and so on. Here is a
702 little collection of valid expressions:
705 ex MyEx1 = 5; // simple number
706 ex MyEx2 = x + 2*y; // polynomial in x and y
707 ex MyEx3 = (x + 1)/(x - 1); // rational expression
708 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
709 ex MyEx5 = MyEx4 + 1; // similar to above
712 Expressions are handles to other more fundamental objects, that often
713 contain other expressions thus creating a tree of expressions
714 (@xref{Internal Structures}, for particular examples). Most methods on
715 @code{ex} therefore run top-down through such an expression tree. For
716 example, the method @code{has()} scans recursively for occurrences of
717 something inside an expression. Thus, if you have declared @code{MyEx4}
718 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
719 the argument of @code{sin} and hence return @code{true}.
721 The next sections will outline the general picture of GiNaC's class
722 hierarchy and describe the classes of objects that are handled by
725 @subsection Note: Expressions and STL containers
727 GiNaC expressions (@code{ex} objects) have value semantics (they can be
728 assigned, reassigned and copied like integral types) but the operator
729 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
730 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
732 This implies that in order to use expressions in sorted containers such as
733 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
734 comparison predicate. GiNaC provides such a predicate, called
735 @code{ex_is_less}. For example, a set of expressions should be defined
736 as @code{std::set<ex, ex_is_less>}.
738 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
739 don't pose a problem. A @code{std::vector<ex>} works as expected.
741 @xref{Information About Expressions}, for more about comparing and ordering
745 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
746 @c node-name, next, previous, up
747 @section Automatic evaluation and canonicalization of expressions
750 GiNaC performs some automatic transformations on expressions, to simplify
751 them and put them into a canonical form. Some examples:
754 ex MyEx1 = 2*x - 1 + x; // 3*x-1
755 ex MyEx2 = x - x; // 0
756 ex MyEx3 = cos(2*Pi); // 1
757 ex MyEx4 = x*y/x; // y
760 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
761 evaluation}. GiNaC only performs transformations that are
765 at most of complexity
773 algebraically correct, possibly except for a set of measure zero (e.g.
774 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
777 There are two types of automatic transformations in GiNaC that may not
778 behave in an entirely obvious way at first glance:
782 The terms of sums and products (and some other things like the arguments of
783 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
784 into a canonical form that is deterministic, but not lexicographical or in
785 any other way easy to guess (it almost always depends on the number and
786 order of the symbols you define). However, constructing the same expression
787 twice, either implicitly or explicitly, will always result in the same
790 Expressions of the form 'number times sum' are automatically expanded (this
791 has to do with GiNaC's internal representation of sums and products). For
794 ex MyEx5 = 2*(x + y); // 2*x+2*y
795 ex MyEx6 = z*(x + y); // z*(x+y)
799 The general rule is that when you construct expressions, GiNaC automatically
800 creates them in canonical form, which might differ from the form you typed in
801 your program. This may create some awkward looking output (@samp{-y+x} instead
802 of @samp{x-y}) but allows for more efficient operation and usually yields
803 some immediate simplifications.
805 @cindex @code{eval()}
806 Internally, the anonymous evaluator in GiNaC is implemented by the methods
809 ex ex::eval(int level = 0) const;
810 ex basic::eval(int level = 0) const;
813 but unless you are extending GiNaC with your own classes or functions, there
814 should never be any reason to call them explicitly. All GiNaC methods that
815 transform expressions, like @code{subs()} or @code{normal()}, automatically
816 re-evaluate their results.
819 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
820 @c node-name, next, previous, up
821 @section Error handling
823 @cindex @code{pole_error} (class)
825 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
826 generated by GiNaC are subclassed from the standard @code{exception} class
827 defined in the @file{<stdexcept>} header. In addition to the predefined
828 @code{logic_error}, @code{domain_error}, @code{out_of_range},
829 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
830 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
831 exception that gets thrown when trying to evaluate a mathematical function
834 The @code{pole_error} class has a member function
837 int pole_error::degree() const;
840 that returns the order of the singularity (or 0 when the pole is
841 logarithmic or the order is undefined).
843 When using GiNaC it is useful to arrange for exceptions to be caught in
844 the main program even if you don't want to do any special error handling.
845 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
846 default exception handler of your C++ compiler's run-time system which
847 usually only aborts the program without giving any information what went
850 Here is an example for a @code{main()} function that catches and prints
851 exceptions generated by GiNaC:
856 #include <ginac/ginac.h>
858 using namespace GiNaC;
866 @} catch (exception &p) @{
867 cerr << p.what() << endl;
875 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
876 @c node-name, next, previous, up
877 @section The Class Hierarchy
879 GiNaC's class hierarchy consists of several classes representing
880 mathematical objects, all of which (except for @code{ex} and some
881 helpers) are internally derived from one abstract base class called
882 @code{basic}. You do not have to deal with objects of class
883 @code{basic}, instead you'll be dealing with symbols, numbers,
884 containers of expressions and so on.
888 To get an idea about what kinds of symbolic composites may be built we
889 have a look at the most important classes in the class hierarchy and
890 some of the relations among the classes:
892 @image{classhierarchy}
894 The abstract classes shown here (the ones without drop-shadow) are of no
895 interest for the user. They are used internally in order to avoid code
896 duplication if two or more classes derived from them share certain
897 features. An example is @code{expairseq}, a container for a sequence of
898 pairs each consisting of one expression and a number (@code{numeric}).
899 What @emph{is} visible to the user are the derived classes @code{add}
900 and @code{mul}, representing sums and products. @xref{Internal
901 Structures}, where these two classes are described in more detail. The
902 following table shortly summarizes what kinds of mathematical objects
903 are stored in the different classes:
906 @multitable @columnfractions .22 .78
907 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
908 @item @code{constant} @tab Constants like
915 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
916 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
917 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
918 @item @code{ncmul} @tab Products of non-commutative objects
919 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
924 @code{sqrt(}@math{2}@code{)}
927 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
928 @item @code{function} @tab A symbolic function like
935 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
936 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
937 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
938 @item @code{indexed} @tab Indexed object like @math{A_ij}
939 @item @code{tensor} @tab Special tensor like the delta and metric tensors
940 @item @code{idx} @tab Index of an indexed object
941 @item @code{varidx} @tab Index with variance
942 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
943 @item @code{wildcard} @tab Wildcard for pattern matching
944 @item @code{structure} @tab Template for user-defined classes
949 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
950 @c node-name, next, previous, up
952 @cindex @code{symbol} (class)
953 @cindex hierarchy of classes
956 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
957 manipulation what atoms are for chemistry.
959 A typical symbol definition looks like this:
964 This definition actually contains three very different things:
966 @item a C++ variable named @code{x}
967 @item a @code{symbol} object stored in this C++ variable; this object
968 represents the symbol in a GiNaC expression
969 @item the string @code{"x"} which is the name of the symbol, used (almost)
970 exclusively for printing expressions holding the symbol
973 Symbols have an explicit name, supplied as a string during construction,
974 because in C++, variable names can't be used as values, and the C++ compiler
975 throws them away during compilation.
977 It is possible to omit the symbol name in the definition:
982 In this case, GiNaC will assign the symbol an internal, unique name of the
983 form @code{symbolNNN}. This won't affect the usability of the symbol but
984 the output of your calculations will become more readable if you give your
985 symbols sensible names (for intermediate expressions that are only used
986 internally such anonymous symbols can be quite useful, however).
988 Now, here is one important property of GiNaC that differentiates it from
989 other computer algebra programs you may have used: GiNaC does @emph{not} use
990 the names of symbols to tell them apart, but a (hidden) serial number that
991 is unique for each newly created @code{symbol} object. In you want to use
992 one and the same symbol in different places in your program, you must only
993 create one @code{symbol} object and pass that around. If you create another
994 symbol, even if it has the same name, GiNaC will treat it as a different
1011 // prints "x^6" which looks right, but...
1013 cout << e.degree(x) << endl;
1014 // ...this doesn't work. The symbol "x" here is different from the one
1015 // in f() and in the expression returned by f(). Consequently, it
1020 One possibility to ensure that @code{f()} and @code{main()} use the same
1021 symbol is to pass the symbol as an argument to @code{f()}:
1023 ex f(int n, const ex & x)
1032 // Now, f() uses the same symbol.
1035 cout << e.degree(x) << endl;
1036 // prints "6", as expected
1040 Another possibility would be to define a global symbol @code{x} that is used
1041 by both @code{f()} and @code{main()}. If you are using global symbols and
1042 multiple compilation units you must take special care, however. Suppose
1043 that you have a header file @file{globals.h} in your program that defines
1044 a @code{symbol x("x");}. In this case, every unit that includes
1045 @file{globals.h} would also get its own definition of @code{x} (because
1046 header files are just inlined into the source code by the C++ preprocessor),
1047 and hence you would again end up with multiple equally-named, but different,
1048 symbols. Instead, the @file{globals.h} header should only contain a
1049 @emph{declaration} like @code{extern symbol x;}, with the definition of
1050 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1052 A different approach to ensuring that symbols used in different parts of
1053 your program are identical is to create them with a @emph{factory} function
1056 const symbol & get_symbol(const string & s)
1058 static map<string, symbol> directory;
1059 map<string, symbol>::iterator i = directory.find(s);
1060 if (i != directory.end())
1063 return directory.insert(make_pair(s, symbol(s))).first->second;
1067 This function returns one newly constructed symbol for each name that is
1068 passed in, and it returns the same symbol when called multiple times with
1069 the same name. Using this symbol factory, we can rewrite our example like
1074 return pow(get_symbol("x"), n);
1081 // Both calls of get_symbol("x") yield the same symbol.
1082 cout << e.degree(get_symbol("x")) << endl;
1087 Instead of creating symbols from strings we could also have
1088 @code{get_symbol()} take, for example, an integer number as its argument.
1089 In this case, we would probably want to give the generated symbols names
1090 that include this number, which can be accomplished with the help of an
1091 @code{ostringstream}.
1093 In general, if you're getting weird results from GiNaC such as an expression
1094 @samp{x-x} that is not simplified to zero, you should check your symbol
1097 As we said, the names of symbols primarily serve for purposes of expression
1098 output. But there are actually two instances where GiNaC uses the names for
1099 identifying symbols: When constructing an expression from a string, and when
1100 recreating an expression from an archive (@pxref{Input/Output}).
1102 In addition to its name, a symbol may contain a special string that is used
1105 symbol x("x", "\\Box");
1108 This creates a symbol that is printed as "@code{x}" in normal output, but
1109 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1110 information about the different output formats of expressions in GiNaC).
1111 GiNaC automatically creates proper LaTeX code for symbols having names of
1112 greek letters (@samp{alpha}, @samp{mu}, etc.).
1114 @cindex @code{subs()}
1115 Symbols in GiNaC can't be assigned values. If you need to store results of
1116 calculations and give them a name, use C++ variables of type @code{ex}.
1117 If you want to replace a symbol in an expression with something else, you
1118 can invoke the expression's @code{.subs()} method
1119 (@pxref{Substituting Expressions}).
1121 @cindex @code{realsymbol()}
1122 By default, symbols are expected to stand in for complex values, i.e. they live
1123 in the complex domain. As a consequence, operations like complex conjugation,
1124 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1125 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1126 because of the unknown imaginary part of @code{x}.
1127 On the other hand, if you are sure that your symbols will hold only real values, you
1128 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1129 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1130 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1133 @node Numbers, Constants, Symbols, Basic Concepts
1134 @c node-name, next, previous, up
1136 @cindex @code{numeric} (class)
1142 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1143 The classes therein serve as foundation classes for GiNaC. CLN stands
1144 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1145 In order to find out more about CLN's internals, the reader is referred to
1146 the documentation of that library. @inforef{Introduction, , cln}, for
1147 more information. Suffice to say that it is by itself build on top of
1148 another library, the GNU Multiple Precision library GMP, which is an
1149 extremely fast library for arbitrary long integers and rationals as well
1150 as arbitrary precision floating point numbers. It is very commonly used
1151 by several popular cryptographic applications. CLN extends GMP by
1152 several useful things: First, it introduces the complex number field
1153 over either reals (i.e. floating point numbers with arbitrary precision)
1154 or rationals. Second, it automatically converts rationals to integers
1155 if the denominator is unity and complex numbers to real numbers if the
1156 imaginary part vanishes and also correctly treats algebraic functions.
1157 Third it provides good implementations of state-of-the-art algorithms
1158 for all trigonometric and hyperbolic functions as well as for
1159 calculation of some useful constants.
1161 The user can construct an object of class @code{numeric} in several
1162 ways. The following example shows the four most important constructors.
1163 It uses construction from C-integer, construction of fractions from two
1164 integers, construction from C-float and construction from a string:
1168 #include <ginac/ginac.h>
1169 using namespace GiNaC;
1173 numeric two = 2; // exact integer 2
1174 numeric r(2,3); // exact fraction 2/3
1175 numeric e(2.71828); // floating point number
1176 numeric p = "3.14159265358979323846"; // constructor from string
1177 // Trott's constant in scientific notation:
1178 numeric trott("1.0841015122311136151E-2");
1180 std::cout << two*p << std::endl; // floating point 6.283...
1185 @cindex complex numbers
1186 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1191 numeric z1 = 2-3*I; // exact complex number 2-3i
1192 numeric z2 = 5.9+1.6*I; // complex floating point number
1196 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1197 This would, however, call C's built-in operator @code{/} for integers
1198 first and result in a numeric holding a plain integer 1. @strong{Never
1199 use the operator @code{/} on integers} unless you know exactly what you
1200 are doing! Use the constructor from two integers instead, as shown in
1201 the example above. Writing @code{numeric(1)/2} may look funny but works
1204 @cindex @code{Digits}
1206 We have seen now the distinction between exact numbers and floating
1207 point numbers. Clearly, the user should never have to worry about
1208 dynamically created exact numbers, since their `exactness' always
1209 determines how they ought to be handled, i.e. how `long' they are. The
1210 situation is different for floating point numbers. Their accuracy is
1211 controlled by one @emph{global} variable, called @code{Digits}. (For
1212 those readers who know about Maple: it behaves very much like Maple's
1213 @code{Digits}). All objects of class numeric that are constructed from
1214 then on will be stored with a precision matching that number of decimal
1219 #include <ginac/ginac.h>
1220 using namespace std;
1221 using namespace GiNaC;
1225 numeric three(3.0), one(1.0);
1226 numeric x = one/three;
1228 cout << "in " << Digits << " digits:" << endl;
1230 cout << Pi.evalf() << endl;
1242 The above example prints the following output to screen:
1246 0.33333333333333333334
1247 3.1415926535897932385
1249 0.33333333333333333333333333333333333333333333333333333333333333333334
1250 3.1415926535897932384626433832795028841971693993751058209749445923078
1254 Note that the last number is not necessarily rounded as you would
1255 naively expect it to be rounded in the decimal system. But note also,
1256 that in both cases you got a couple of extra digits. This is because
1257 numbers are internally stored by CLN as chunks of binary digits in order
1258 to match your machine's word size and to not waste precision. Thus, on
1259 architectures with different word size, the above output might even
1260 differ with regard to actually computed digits.
1262 It should be clear that objects of class @code{numeric} should be used
1263 for constructing numbers or for doing arithmetic with them. The objects
1264 one deals with most of the time are the polymorphic expressions @code{ex}.
1266 @subsection Tests on numbers
1268 Once you have declared some numbers, assigned them to expressions and
1269 done some arithmetic with them it is frequently desired to retrieve some
1270 kind of information from them like asking whether that number is
1271 integer, rational, real or complex. For those cases GiNaC provides
1272 several useful methods. (Internally, they fall back to invocations of
1273 certain CLN functions.)
1275 As an example, let's construct some rational number, multiply it with
1276 some multiple of its denominator and test what comes out:
1280 #include <ginac/ginac.h>
1281 using namespace std;
1282 using namespace GiNaC;
1284 // some very important constants:
1285 const numeric twentyone(21);
1286 const numeric ten(10);
1287 const numeric five(5);
1291 numeric answer = twentyone;
1294 cout << answer.is_integer() << endl; // false, it's 21/5
1296 cout << answer.is_integer() << endl; // true, it's 42 now!
1300 Note that the variable @code{answer} is constructed here as an integer
1301 by @code{numeric}'s copy constructor but in an intermediate step it
1302 holds a rational number represented as integer numerator and integer
1303 denominator. When multiplied by 10, the denominator becomes unity and
1304 the result is automatically converted to a pure integer again.
1305 Internally, the underlying CLN is responsible for this behavior and we
1306 refer the reader to CLN's documentation. Suffice to say that
1307 the same behavior applies to complex numbers as well as return values of
1308 certain functions. Complex numbers are automatically converted to real
1309 numbers if the imaginary part becomes zero. The full set of tests that
1310 can be applied is listed in the following table.
1313 @multitable @columnfractions .30 .70
1314 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1315 @item @code{.is_zero()}
1316 @tab @dots{}equal to zero
1317 @item @code{.is_positive()}
1318 @tab @dots{}not complex and greater than 0
1319 @item @code{.is_integer()}
1320 @tab @dots{}a (non-complex) integer
1321 @item @code{.is_pos_integer()}
1322 @tab @dots{}an integer and greater than 0
1323 @item @code{.is_nonneg_integer()}
1324 @tab @dots{}an integer and greater equal 0
1325 @item @code{.is_even()}
1326 @tab @dots{}an even integer
1327 @item @code{.is_odd()}
1328 @tab @dots{}an odd integer
1329 @item @code{.is_prime()}
1330 @tab @dots{}a prime integer (probabilistic primality test)
1331 @item @code{.is_rational()}
1332 @tab @dots{}an exact rational number (integers are rational, too)
1333 @item @code{.is_real()}
1334 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1335 @item @code{.is_cinteger()}
1336 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1337 @item @code{.is_crational()}
1338 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1342 @subsection Numeric functions
1344 The following functions can be applied to @code{numeric} objects and will be
1345 evaluated immediately:
1348 @multitable @columnfractions .30 .70
1349 @item @strong{Name} @tab @strong{Function}
1350 @item @code{inverse(z)}
1351 @tab returns @math{1/z}
1352 @cindex @code{inverse()} (numeric)
1353 @item @code{pow(a, b)}
1354 @tab exponentiation @math{a^b}
1357 @item @code{real(z)}
1359 @cindex @code{real()}
1360 @item @code{imag(z)}
1362 @cindex @code{imag()}
1363 @item @code{csgn(z)}
1364 @tab complex sign (returns an @code{int})
1365 @item @code{numer(z)}
1366 @tab numerator of rational or complex rational number
1367 @item @code{denom(z)}
1368 @tab denominator of rational or complex rational number
1369 @item @code{sqrt(z)}
1371 @item @code{isqrt(n)}
1372 @tab integer square root
1373 @cindex @code{isqrt()}
1380 @item @code{asin(z)}
1382 @item @code{acos(z)}
1384 @item @code{atan(z)}
1385 @tab inverse tangent
1386 @item @code{atan(y, x)}
1387 @tab inverse tangent with two arguments
1388 @item @code{sinh(z)}
1389 @tab hyperbolic sine
1390 @item @code{cosh(z)}
1391 @tab hyperbolic cosine
1392 @item @code{tanh(z)}
1393 @tab hyperbolic tangent
1394 @item @code{asinh(z)}
1395 @tab inverse hyperbolic sine
1396 @item @code{acosh(z)}
1397 @tab inverse hyperbolic cosine
1398 @item @code{atanh(z)}
1399 @tab inverse hyperbolic tangent
1401 @tab exponential function
1403 @tab natural logarithm
1406 @item @code{zeta(z)}
1407 @tab Riemann's zeta function
1408 @item @code{tgamma(z)}
1410 @item @code{lgamma(z)}
1411 @tab logarithm of gamma function
1413 @tab psi (digamma) function
1414 @item @code{psi(n, z)}
1415 @tab derivatives of psi function (polygamma functions)
1416 @item @code{factorial(n)}
1417 @tab factorial function @math{n!}
1418 @item @code{doublefactorial(n)}
1419 @tab double factorial function @math{n!!}
1420 @cindex @code{doublefactorial()}
1421 @item @code{binomial(n, k)}
1422 @tab binomial coefficients
1423 @item @code{bernoulli(n)}
1424 @tab Bernoulli numbers
1425 @cindex @code{bernoulli()}
1426 @item @code{fibonacci(n)}
1427 @tab Fibonacci numbers
1428 @cindex @code{fibonacci()}
1429 @item @code{mod(a, b)}
1430 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1431 @cindex @code{mod()}
1432 @item @code{smod(a, b)}
1433 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1434 @cindex @code{smod()}
1435 @item @code{irem(a, b)}
1436 @tab integer remainder (has the sign of @math{a}, or is zero)
1437 @cindex @code{irem()}
1438 @item @code{irem(a, b, q)}
1439 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1440 @item @code{iquo(a, b)}
1441 @tab integer quotient
1442 @cindex @code{iquo()}
1443 @item @code{iquo(a, b, r)}
1444 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1445 @item @code{gcd(a, b)}
1446 @tab greatest common divisor
1447 @item @code{lcm(a, b)}
1448 @tab least common multiple
1452 Most of these functions are also available as symbolic functions that can be
1453 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1454 as polynomial algorithms.
1456 @subsection Converting numbers
1458 Sometimes it is desirable to convert a @code{numeric} object back to a
1459 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1460 class provides a couple of methods for this purpose:
1462 @cindex @code{to_int()}
1463 @cindex @code{to_long()}
1464 @cindex @code{to_double()}
1465 @cindex @code{to_cl_N()}
1467 int numeric::to_int() const;
1468 long numeric::to_long() const;
1469 double numeric::to_double() const;
1470 cln::cl_N numeric::to_cl_N() const;
1473 @code{to_int()} and @code{to_long()} only work when the number they are
1474 applied on is an exact integer. Otherwise the program will halt with a
1475 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1476 rational number will return a floating-point approximation. Both
1477 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1478 part of complex numbers.
1481 @node Constants, Fundamental containers, Numbers, Basic Concepts
1482 @c node-name, next, previous, up
1484 @cindex @code{constant} (class)
1487 @cindex @code{Catalan}
1488 @cindex @code{Euler}
1489 @cindex @code{evalf()}
1490 Constants behave pretty much like symbols except that they return some
1491 specific number when the method @code{.evalf()} is called.
1493 The predefined known constants are:
1496 @multitable @columnfractions .14 .30 .56
1497 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1499 @tab Archimedes' constant
1500 @tab 3.14159265358979323846264338327950288
1501 @item @code{Catalan}
1502 @tab Catalan's constant
1503 @tab 0.91596559417721901505460351493238411
1505 @tab Euler's (or Euler-Mascheroni) constant
1506 @tab 0.57721566490153286060651209008240243
1511 @node Fundamental containers, Lists, Constants, Basic Concepts
1512 @c node-name, next, previous, up
1513 @section Sums, products and powers
1517 @cindex @code{power}
1519 Simple rational expressions are written down in GiNaC pretty much like
1520 in other CAS or like expressions involving numerical variables in C.
1521 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1522 been overloaded to achieve this goal. When you run the following
1523 code snippet, the constructor for an object of type @code{mul} is
1524 automatically called to hold the product of @code{a} and @code{b} and
1525 then the constructor for an object of type @code{add} is called to hold
1526 the sum of that @code{mul} object and the number one:
1530 symbol a("a"), b("b");
1535 @cindex @code{pow()}
1536 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1537 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1538 construction is necessary since we cannot safely overload the constructor
1539 @code{^} in C++ to construct a @code{power} object. If we did, it would
1540 have several counterintuitive and undesired effects:
1544 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1546 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1547 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1548 interpret this as @code{x^(a^b)}.
1550 Also, expressions involving integer exponents are very frequently used,
1551 which makes it even more dangerous to overload @code{^} since it is then
1552 hard to distinguish between the semantics as exponentiation and the one
1553 for exclusive or. (It would be embarrassing to return @code{1} where one
1554 has requested @code{2^3}.)
1557 @cindex @command{ginsh}
1558 All effects are contrary to mathematical notation and differ from the
1559 way most other CAS handle exponentiation, therefore overloading @code{^}
1560 is ruled out for GiNaC's C++ part. The situation is different in
1561 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1562 that the other frequently used exponentiation operator @code{**} does
1563 not exist at all in C++).
1565 To be somewhat more precise, objects of the three classes described
1566 here, are all containers for other expressions. An object of class
1567 @code{power} is best viewed as a container with two slots, one for the
1568 basis, one for the exponent. All valid GiNaC expressions can be
1569 inserted. However, basic transformations like simplifying
1570 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1571 when this is mathematically possible. If we replace the outer exponent
1572 three in the example by some symbols @code{a}, the simplification is not
1573 safe and will not be performed, since @code{a} might be @code{1/2} and
1576 Objects of type @code{add} and @code{mul} are containers with an
1577 arbitrary number of slots for expressions to be inserted. Again, simple
1578 and safe simplifications are carried out like transforming
1579 @code{3*x+4-x} to @code{2*x+4}.
1582 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1583 @c node-name, next, previous, up
1584 @section Lists of expressions
1585 @cindex @code{lst} (class)
1587 @cindex @code{nops()}
1589 @cindex @code{append()}
1590 @cindex @code{prepend()}
1591 @cindex @code{remove_first()}
1592 @cindex @code{remove_last()}
1593 @cindex @code{remove_all()}
1595 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1596 expressions. They are not as ubiquitous as in many other computer algebra
1597 packages, but are sometimes used to supply a variable number of arguments of
1598 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1599 constructors, so you should have a basic understanding of them.
1601 Lists can be constructed by assigning a comma-separated sequence of
1606 symbol x("x"), y("y");
1609 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1614 There are also constructors that allow direct creation of lists of up to
1615 16 expressions, which is often more convenient but slightly less efficient:
1619 // This produces the same list 'l' as above:
1620 // lst l(x, 2, y, x+y);
1621 // lst l = lst(x, 2, y, x+y);
1625 Use the @code{nops()} method to determine the size (number of expressions) of
1626 a list and the @code{op()} method or the @code{[]} operator to access
1627 individual elements:
1631 cout << l.nops() << endl; // prints '4'
1632 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1636 As with the standard @code{list<T>} container, accessing random elements of a
1637 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1638 sequential access to the elements of a list is possible with the
1639 iterator types provided by the @code{lst} class:
1642 typedef ... lst::const_iterator;
1643 typedef ... lst::const_reverse_iterator;
1644 lst::const_iterator lst::begin() const;
1645 lst::const_iterator lst::end() const;
1646 lst::const_reverse_iterator lst::rbegin() const;
1647 lst::const_reverse_iterator lst::rend() const;
1650 For example, to print the elements of a list individually you can use:
1655 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1660 which is one order faster than
1665 for (size_t i = 0; i < l.nops(); ++i)
1666 cout << l.op(i) << endl;
1670 These iterators also allow you to use some of the algorithms provided by
1671 the C++ standard library:
1675 // print the elements of the list (requires #include <iterator>)
1676 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1678 // sum up the elements of the list (requires #include <numeric>)
1679 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1680 cout << sum << endl; // prints '2+2*x+2*y'
1684 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1685 (the only other one is @code{matrix}). You can modify single elements:
1689 l[1] = 42; // l is now @{x, 42, y, x+y@}
1690 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1694 You can append or prepend an expression to a list with the @code{append()}
1695 and @code{prepend()} methods:
1699 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1700 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1704 You can remove the first or last element of a list with @code{remove_first()}
1705 and @code{remove_last()}:
1709 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1710 l.remove_last(); // l is now @{x, 7, y, x+y@}
1714 You can remove all the elements of a list with @code{remove_all()}:
1718 l.remove_all(); // l is now empty
1722 You can bring the elements of a list into a canonical order with @code{sort()}:
1731 // l1 and l2 are now equal
1735 Finally, you can remove all but the first element of consecutive groups of
1736 elements with @code{unique()}:
1741 l3 = x, 2, 2, 2, y, x+y, y+x;
1742 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1747 @node Mathematical functions, Relations, Lists, Basic Concepts
1748 @c node-name, next, previous, up
1749 @section Mathematical functions
1750 @cindex @code{function} (class)
1751 @cindex trigonometric function
1752 @cindex hyperbolic function
1754 There are quite a number of useful functions hard-wired into GiNaC. For
1755 instance, all trigonometric and hyperbolic functions are implemented
1756 (@xref{Built-in Functions}, for a complete list).
1758 These functions (better called @emph{pseudofunctions}) are all objects
1759 of class @code{function}. They accept one or more expressions as
1760 arguments and return one expression. If the arguments are not
1761 numerical, the evaluation of the function may be halted, as it does in
1762 the next example, showing how a function returns itself twice and
1763 finally an expression that may be really useful:
1765 @cindex Gamma function
1766 @cindex @code{subs()}
1769 symbol x("x"), y("y");
1771 cout << tgamma(foo) << endl;
1772 // -> tgamma(x+(1/2)*y)
1773 ex bar = foo.subs(y==1);
1774 cout << tgamma(bar) << endl;
1776 ex foobar = bar.subs(x==7);
1777 cout << tgamma(foobar) << endl;
1778 // -> (135135/128)*Pi^(1/2)
1782 Besides evaluation most of these functions allow differentiation, series
1783 expansion and so on. Read the next chapter in order to learn more about
1786 It must be noted that these pseudofunctions are created by inline
1787 functions, where the argument list is templated. This means that
1788 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1789 @code{sin(ex(1))} and will therefore not result in a floating point
1790 number. Unless of course the function prototype is explicitly
1791 overridden -- which is the case for arguments of type @code{numeric}
1792 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1793 point number of class @code{numeric} you should call
1794 @code{sin(numeric(1))}. This is almost the same as calling
1795 @code{sin(1).evalf()} except that the latter will return a numeric
1796 wrapped inside an @code{ex}.
1799 @node Relations, Matrices, Mathematical functions, Basic Concepts
1800 @c node-name, next, previous, up
1802 @cindex @code{relational} (class)
1804 Sometimes, a relation holding between two expressions must be stored
1805 somehow. The class @code{relational} is a convenient container for such
1806 purposes. A relation is by definition a container for two @code{ex} and
1807 a relation between them that signals equality, inequality and so on.
1808 They are created by simply using the C++ operators @code{==}, @code{!=},
1809 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1811 @xref{Mathematical functions}, for examples where various applications
1812 of the @code{.subs()} method show how objects of class relational are
1813 used as arguments. There they provide an intuitive syntax for
1814 substitutions. They are also used as arguments to the @code{ex::series}
1815 method, where the left hand side of the relation specifies the variable
1816 to expand in and the right hand side the expansion point. They can also
1817 be used for creating systems of equations that are to be solved for
1818 unknown variables. But the most common usage of objects of this class
1819 is rather inconspicuous in statements of the form @code{if
1820 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1821 conversion from @code{relational} to @code{bool} takes place. Note,
1822 however, that @code{==} here does not perform any simplifications, hence
1823 @code{expand()} must be called explicitly.
1826 @node Matrices, Indexed objects, Relations, Basic Concepts
1827 @c node-name, next, previous, up
1829 @cindex @code{matrix} (class)
1831 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1832 matrix with @math{m} rows and @math{n} columns are accessed with two
1833 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1834 second one in the range 0@dots{}@math{n-1}.
1836 There are a couple of ways to construct matrices, with or without preset
1837 elements. The constructor
1840 matrix::matrix(unsigned r, unsigned c);
1843 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1846 The fastest way to create a matrix with preinitialized elements is to assign
1847 a list of comma-separated expressions to an empty matrix (see below for an
1848 example). But you can also specify the elements as a (flat) list with
1851 matrix::matrix(unsigned r, unsigned c, const lst & l);
1856 @cindex @code{lst_to_matrix()}
1858 ex lst_to_matrix(const lst & l);
1861 constructs a matrix from a list of lists, each list representing a matrix row.
1863 There is also a set of functions for creating some special types of
1866 @cindex @code{diag_matrix()}
1867 @cindex @code{unit_matrix()}
1868 @cindex @code{symbolic_matrix()}
1870 ex diag_matrix(const lst & l);
1871 ex unit_matrix(unsigned x);
1872 ex unit_matrix(unsigned r, unsigned c);
1873 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1874 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1877 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1878 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1879 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1880 matrix filled with newly generated symbols made of the specified base name
1881 and the position of each element in the matrix.
1883 Matrix elements can be accessed and set using the parenthesis (function call)
1887 const ex & matrix::operator()(unsigned r, unsigned c) const;
1888 ex & matrix::operator()(unsigned r, unsigned c);
1891 It is also possible to access the matrix elements in a linear fashion with
1892 the @code{op()} method. But C++-style subscripting with square brackets
1893 @samp{[]} is not available.
1895 Here are a couple of examples for constructing matrices:
1899 symbol a("a"), b("b");
1913 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1916 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1919 cout << diag_matrix(lst(a, b)) << endl;
1922 cout << unit_matrix(3) << endl;
1923 // -> [[1,0,0],[0,1,0],[0,0,1]]
1925 cout << symbolic_matrix(2, 3, "x") << endl;
1926 // -> [[x00,x01,x02],[x10,x11,x12]]
1930 @cindex @code{transpose()}
1931 There are three ways to do arithmetic with matrices. The first (and most
1932 direct one) is to use the methods provided by the @code{matrix} class:
1935 matrix matrix::add(const matrix & other) const;
1936 matrix matrix::sub(const matrix & other) const;
1937 matrix matrix::mul(const matrix & other) const;
1938 matrix matrix::mul_scalar(const ex & other) const;
1939 matrix matrix::pow(const ex & expn) const;
1940 matrix matrix::transpose() const;
1943 All of these methods return the result as a new matrix object. Here is an
1944 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1949 matrix A(2, 2), B(2, 2), C(2, 2);
1957 matrix result = A.mul(B).sub(C.mul_scalar(2));
1958 cout << result << endl;
1959 // -> [[-13,-6],[1,2]]
1964 @cindex @code{evalm()}
1965 The second (and probably the most natural) way is to construct an expression
1966 containing matrices with the usual arithmetic operators and @code{pow()}.
1967 For efficiency reasons, expressions with sums, products and powers of
1968 matrices are not automatically evaluated in GiNaC. You have to call the
1972 ex ex::evalm() const;
1975 to obtain the result:
1982 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1983 cout << e.evalm() << endl;
1984 // -> [[-13,-6],[1,2]]
1989 The non-commutativity of the product @code{A*B} in this example is
1990 automatically recognized by GiNaC. There is no need to use a special
1991 operator here. @xref{Non-commutative objects}, for more information about
1992 dealing with non-commutative expressions.
1994 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1995 to perform the arithmetic:
2000 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2001 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2003 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2004 cout << e.simplify_indexed() << endl;
2005 // -> [[-13,-6],[1,2]].i.j
2009 Using indices is most useful when working with rectangular matrices and
2010 one-dimensional vectors because you don't have to worry about having to
2011 transpose matrices before multiplying them. @xref{Indexed objects}, for
2012 more information about using matrices with indices, and about indices in
2015 The @code{matrix} class provides a couple of additional methods for
2016 computing determinants, traces, characteristic polynomials and ranks:
2018 @cindex @code{determinant()}
2019 @cindex @code{trace()}
2020 @cindex @code{charpoly()}
2021 @cindex @code{rank()}
2023 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2024 ex matrix::trace() const;
2025 ex matrix::charpoly(const ex & lambda) const;
2026 unsigned matrix::rank() const;
2029 The @samp{algo} argument of @code{determinant()} allows to select
2030 between different algorithms for calculating the determinant. The
2031 asymptotic speed (as parametrized by the matrix size) can greatly differ
2032 between those algorithms, depending on the nature of the matrix'
2033 entries. The possible values are defined in the @file{flags.h} header
2034 file. By default, GiNaC uses a heuristic to automatically select an
2035 algorithm that is likely (but not guaranteed) to give the result most
2038 @cindex @code{inverse()} (matrix)
2039 @cindex @code{solve()}
2040 Matrices may also be inverted using the @code{ex matrix::inverse()}
2041 method and linear systems may be solved with:
2044 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2047 Assuming the matrix object this method is applied on is an @code{m}
2048 times @code{n} matrix, then @code{vars} must be a @code{n} times
2049 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2050 times @code{p} matrix. The returned matrix then has dimension @code{n}
2051 times @code{p} and in the case of an underdetermined system will still
2052 contain some of the indeterminates from @code{vars}. If the system is
2053 overdetermined, an exception is thrown.
2056 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2057 @c node-name, next, previous, up
2058 @section Indexed objects
2060 GiNaC allows you to handle expressions containing general indexed objects in
2061 arbitrary spaces. It is also able to canonicalize and simplify such
2062 expressions and perform symbolic dummy index summations. There are a number
2063 of predefined indexed objects provided, like delta and metric tensors.
2065 There are few restrictions placed on indexed objects and their indices and
2066 it is easy to construct nonsense expressions, but our intention is to
2067 provide a general framework that allows you to implement algorithms with
2068 indexed quantities, getting in the way as little as possible.
2070 @cindex @code{idx} (class)
2071 @cindex @code{indexed} (class)
2072 @subsection Indexed quantities and their indices
2074 Indexed expressions in GiNaC are constructed of two special types of objects,
2075 @dfn{index objects} and @dfn{indexed objects}.
2079 @cindex contravariant
2082 @item Index objects are of class @code{idx} or a subclass. Every index has
2083 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2084 the index lives in) which can both be arbitrary expressions but are usually
2085 a number or a simple symbol. In addition, indices of class @code{varidx} have
2086 a @dfn{variance} (they can be co- or contravariant), and indices of class
2087 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2089 @item Indexed objects are of class @code{indexed} or a subclass. They
2090 contain a @dfn{base expression} (which is the expression being indexed), and
2091 one or more indices.
2095 @strong{Note:} when printing expressions, covariant indices and indices
2096 without variance are denoted @samp{.i} while contravariant indices are
2097 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2098 value. In the following, we are going to use that notation in the text so
2099 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2100 not visible in the output.
2102 A simple example shall illustrate the concepts:
2106 #include <ginac/ginac.h>
2107 using namespace std;
2108 using namespace GiNaC;
2112 symbol i_sym("i"), j_sym("j");
2113 idx i(i_sym, 3), j(j_sym, 3);
2116 cout << indexed(A, i, j) << endl;
2118 cout << index_dimensions << indexed(A, i, j) << endl;
2120 cout << dflt; // reset cout to default output format (dimensions hidden)
2124 The @code{idx} constructor takes two arguments, the index value and the
2125 index dimension. First we define two index objects, @code{i} and @code{j},
2126 both with the numeric dimension 3. The value of the index @code{i} is the
2127 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2128 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2129 construct an expression containing one indexed object, @samp{A.i.j}. It has
2130 the symbol @code{A} as its base expression and the two indices @code{i} and
2133 The dimensions of indices are normally not visible in the output, but one
2134 can request them to be printed with the @code{index_dimensions} manipulator,
2137 Note the difference between the indices @code{i} and @code{j} which are of
2138 class @code{idx}, and the index values which are the symbols @code{i_sym}
2139 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2140 or numbers but must be index objects. For example, the following is not
2141 correct and will raise an exception:
2144 symbol i("i"), j("j");
2145 e = indexed(A, i, j); // ERROR: indices must be of type idx
2148 You can have multiple indexed objects in an expression, index values can
2149 be numeric, and index dimensions symbolic:
2153 symbol B("B"), dim("dim");
2154 cout << 4 * indexed(A, i)
2155 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2160 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2161 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2162 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2163 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2164 @code{simplify_indexed()} for that, see below).
2166 In fact, base expressions, index values and index dimensions can be
2167 arbitrary expressions:
2171 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2176 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2177 get an error message from this but you will probably not be able to do
2178 anything useful with it.
2180 @cindex @code{get_value()}
2181 @cindex @code{get_dimension()}
2185 ex idx::get_value();
2186 ex idx::get_dimension();
2189 return the value and dimension of an @code{idx} object. If you have an index
2190 in an expression, such as returned by calling @code{.op()} on an indexed
2191 object, you can get a reference to the @code{idx} object with the function
2192 @code{ex_to<idx>()} on the expression.
2194 There are also the methods
2197 bool idx::is_numeric();
2198 bool idx::is_symbolic();
2199 bool idx::is_dim_numeric();
2200 bool idx::is_dim_symbolic();
2203 for checking whether the value and dimension are numeric or symbolic
2204 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2205 About Expressions}) returns information about the index value.
2207 @cindex @code{varidx} (class)
2208 If you need co- and contravariant indices, use the @code{varidx} class:
2212 symbol mu_sym("mu"), nu_sym("nu");
2213 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2214 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2216 cout << indexed(A, mu, nu) << endl;
2218 cout << indexed(A, mu_co, nu) << endl;
2220 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2225 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2226 co- or contravariant. The default is a contravariant (upper) index, but
2227 this can be overridden by supplying a third argument to the @code{varidx}
2228 constructor. The two methods
2231 bool varidx::is_covariant();
2232 bool varidx::is_contravariant();
2235 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2236 to get the object reference from an expression). There's also the very useful
2240 ex varidx::toggle_variance();
2243 which makes a new index with the same value and dimension but the opposite
2244 variance. By using it you only have to define the index once.
2246 @cindex @code{spinidx} (class)
2247 The @code{spinidx} class provides dotted and undotted variant indices, as
2248 used in the Weyl-van-der-Waerden spinor formalism:
2252 symbol K("K"), C_sym("C"), D_sym("D");
2253 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2254 // contravariant, undotted
2255 spinidx C_co(C_sym, 2, true); // covariant index
2256 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2257 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2259 cout << indexed(K, C, D) << endl;
2261 cout << indexed(K, C_co, D_dot) << endl;
2263 cout << indexed(K, D_co_dot, D) << endl;
2268 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2269 dotted or undotted. The default is undotted but this can be overridden by
2270 supplying a fourth argument to the @code{spinidx} constructor. The two
2274 bool spinidx::is_dotted();
2275 bool spinidx::is_undotted();
2278 allow you to check whether or not a @code{spinidx} object is dotted (use
2279 @code{ex_to<spinidx>()} to get the object reference from an expression).
2280 Finally, the two methods
2283 ex spinidx::toggle_dot();
2284 ex spinidx::toggle_variance_dot();
2287 create a new index with the same value and dimension but opposite dottedness
2288 and the same or opposite variance.
2290 @subsection Substituting indices
2292 @cindex @code{subs()}
2293 Sometimes you will want to substitute one symbolic index with another
2294 symbolic or numeric index, for example when calculating one specific element
2295 of a tensor expression. This is done with the @code{.subs()} method, as it
2296 is done for symbols (see @ref{Substituting Expressions}).
2298 You have two possibilities here. You can either substitute the whole index
2299 by another index or expression:
2303 ex e = indexed(A, mu_co);
2304 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2305 // -> A.mu becomes A~nu
2306 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2307 // -> A.mu becomes A~0
2308 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2309 // -> A.mu becomes A.0
2313 The third example shows that trying to replace an index with something that
2314 is not an index will substitute the index value instead.
2316 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2321 ex e = indexed(A, mu_co);
2322 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2323 // -> A.mu becomes A.nu
2324 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2325 // -> A.mu becomes A.0
2329 As you see, with the second method only the value of the index will get
2330 substituted. Its other properties, including its dimension, remain unchanged.
2331 If you want to change the dimension of an index you have to substitute the
2332 whole index by another one with the new dimension.
2334 Finally, substituting the base expression of an indexed object works as
2339 ex e = indexed(A, mu_co);
2340 cout << e << " becomes " << e.subs(A == A+B) << endl;
2341 // -> A.mu becomes (B+A).mu
2345 @subsection Symmetries
2346 @cindex @code{symmetry} (class)
2347 @cindex @code{sy_none()}
2348 @cindex @code{sy_symm()}
2349 @cindex @code{sy_anti()}
2350 @cindex @code{sy_cycl()}
2352 Indexed objects can have certain symmetry properties with respect to their
2353 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2354 that is constructed with the helper functions
2357 symmetry sy_none(...);
2358 symmetry sy_symm(...);
2359 symmetry sy_anti(...);
2360 symmetry sy_cycl(...);
2363 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2364 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2365 represents a cyclic symmetry. Each of these functions accepts up to four
2366 arguments which can be either symmetry objects themselves or unsigned integer
2367 numbers that represent an index position (counting from 0). A symmetry
2368 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2369 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2372 Here are some examples of symmetry definitions:
2377 e = indexed(A, i, j);
2378 e = indexed(A, sy_none(), i, j); // equivalent
2379 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2381 // Symmetric in all three indices:
2382 e = indexed(A, sy_symm(), i, j, k);
2383 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2384 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2385 // different canonical order
2387 // Symmetric in the first two indices only:
2388 e = indexed(A, sy_symm(0, 1), i, j, k);
2389 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2391 // Antisymmetric in the first and last index only (index ranges need not
2393 e = indexed(A, sy_anti(0, 2), i, j, k);
2394 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2396 // An example of a mixed symmetry: antisymmetric in the first two and
2397 // last two indices, symmetric when swapping the first and last index
2398 // pairs (like the Riemann curvature tensor):
2399 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2401 // Cyclic symmetry in all three indices:
2402 e = indexed(A, sy_cycl(), i, j, k);
2403 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2405 // The following examples are invalid constructions that will throw
2406 // an exception at run time.
2408 // An index may not appear multiple times:
2409 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2410 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2412 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2413 // same number of indices:
2414 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2416 // And of course, you cannot specify indices which are not there:
2417 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2421 If you need to specify more than four indices, you have to use the
2422 @code{.add()} method of the @code{symmetry} class. For example, to specify
2423 full symmetry in the first six indices you would write
2424 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2426 If an indexed object has a symmetry, GiNaC will automatically bring the
2427 indices into a canonical order which allows for some immediate simplifications:
2431 cout << indexed(A, sy_symm(), i, j)
2432 + indexed(A, sy_symm(), j, i) << endl;
2434 cout << indexed(B, sy_anti(), i, j)
2435 + indexed(B, sy_anti(), j, i) << endl;
2437 cout << indexed(B, sy_anti(), i, j, k)
2438 - indexed(B, sy_anti(), j, k, i) << endl;
2443 @cindex @code{get_free_indices()}
2445 @subsection Dummy indices
2447 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2448 that a summation over the index range is implied. Symbolic indices which are
2449 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2450 dummy nor free indices.
2452 To be recognized as a dummy index pair, the two indices must be of the same
2453 class and their value must be the same single symbol (an index like
2454 @samp{2*n+1} is never a dummy index). If the indices are of class
2455 @code{varidx} they must also be of opposite variance; if they are of class
2456 @code{spinidx} they must be both dotted or both undotted.
2458 The method @code{.get_free_indices()} returns a vector containing the free
2459 indices of an expression. It also checks that the free indices of the terms
2460 of a sum are consistent:
2464 symbol A("A"), B("B"), C("C");
2466 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2467 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2469 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2470 cout << exprseq(e.get_free_indices()) << endl;
2472 // 'j' and 'l' are dummy indices
2474 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2475 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2477 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2478 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2479 cout << exprseq(e.get_free_indices()) << endl;
2481 // 'nu' is a dummy index, but 'sigma' is not
2483 e = indexed(A, mu, mu);
2484 cout << exprseq(e.get_free_indices()) << endl;
2486 // 'mu' is not a dummy index because it appears twice with the same
2489 e = indexed(A, mu, nu) + 42;
2490 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2491 // this will throw an exception:
2492 // "add::get_free_indices: inconsistent indices in sum"
2496 @cindex @code{simplify_indexed()}
2497 @subsection Simplifying indexed expressions
2499 In addition to the few automatic simplifications that GiNaC performs on
2500 indexed expressions (such as re-ordering the indices of symmetric tensors
2501 and calculating traces and convolutions of matrices and predefined tensors)
2505 ex ex::simplify_indexed();
2506 ex ex::simplify_indexed(const scalar_products & sp);
2509 that performs some more expensive operations:
2512 @item it checks the consistency of free indices in sums in the same way
2513 @code{get_free_indices()} does
2514 @item it tries to give dummy indices that appear in different terms of a sum
2515 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2516 @item it (symbolically) calculates all possible dummy index summations/contractions
2517 with the predefined tensors (this will be explained in more detail in the
2519 @item it detects contractions that vanish for symmetry reasons, for example
2520 the contraction of a symmetric and a totally antisymmetric tensor
2521 @item as a special case of dummy index summation, it can replace scalar products
2522 of two tensors with a user-defined value
2525 The last point is done with the help of the @code{scalar_products} class
2526 which is used to store scalar products with known values (this is not an
2527 arithmetic class, you just pass it to @code{simplify_indexed()}):
2531 symbol A("A"), B("B"), C("C"), i_sym("i");
2535 sp.add(A, B, 0); // A and B are orthogonal
2536 sp.add(A, C, 0); // A and C are orthogonal
2537 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2539 e = indexed(A + B, i) * indexed(A + C, i);
2541 // -> (B+A).i*(A+C).i
2543 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2549 The @code{scalar_products} object @code{sp} acts as a storage for the
2550 scalar products added to it with the @code{.add()} method. This method
2551 takes three arguments: the two expressions of which the scalar product is
2552 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2553 @code{simplify_indexed()} will replace all scalar products of indexed
2554 objects that have the symbols @code{A} and @code{B} as base expressions
2555 with the single value 0. The number, type and dimension of the indices
2556 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2558 @cindex @code{expand()}
2559 The example above also illustrates a feature of the @code{expand()} method:
2560 if passed the @code{expand_indexed} option it will distribute indices
2561 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2563 @cindex @code{tensor} (class)
2564 @subsection Predefined tensors
2566 Some frequently used special tensors such as the delta, epsilon and metric
2567 tensors are predefined in GiNaC. They have special properties when
2568 contracted with other tensor expressions and some of them have constant
2569 matrix representations (they will evaluate to a number when numeric
2570 indices are specified).
2572 @cindex @code{delta_tensor()}
2573 @subsubsection Delta tensor
2575 The delta tensor takes two indices, is symmetric and has the matrix
2576 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2577 @code{delta_tensor()}:
2581 symbol A("A"), B("B");
2583 idx i(symbol("i"), 3), j(symbol("j"), 3),
2584 k(symbol("k"), 3), l(symbol("l"), 3);
2586 ex e = indexed(A, i, j) * indexed(B, k, l)
2587 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2588 cout << e.simplify_indexed() << endl;
2591 cout << delta_tensor(i, i) << endl;
2596 @cindex @code{metric_tensor()}
2597 @subsubsection General metric tensor
2599 The function @code{metric_tensor()} creates a general symmetric metric
2600 tensor with two indices that can be used to raise/lower tensor indices. The
2601 metric tensor is denoted as @samp{g} in the output and if its indices are of
2602 mixed variance it is automatically replaced by a delta tensor:
2608 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2610 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2611 cout << e.simplify_indexed() << endl;
2614 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2615 cout << e.simplify_indexed() << endl;
2618 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2619 * metric_tensor(nu, rho);
2620 cout << e.simplify_indexed() << endl;
2623 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2624 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2625 + indexed(A, mu.toggle_variance(), rho));
2626 cout << e.simplify_indexed() << endl;
2631 @cindex @code{lorentz_g()}
2632 @subsubsection Minkowski metric tensor
2634 The Minkowski metric tensor is a special metric tensor with a constant
2635 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2636 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2637 It is created with the function @code{lorentz_g()} (although it is output as
2642 varidx mu(symbol("mu"), 4);
2644 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2645 * lorentz_g(mu, varidx(0, 4)); // negative signature
2646 cout << e.simplify_indexed() << endl;
2649 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2650 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2651 cout << e.simplify_indexed() << endl;
2656 @cindex @code{spinor_metric()}
2657 @subsubsection Spinor metric tensor
2659 The function @code{spinor_metric()} creates an antisymmetric tensor with
2660 two indices that is used to raise/lower indices of 2-component spinors.
2661 It is output as @samp{eps}:
2667 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2668 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2670 e = spinor_metric(A, B) * indexed(psi, B_co);
2671 cout << e.simplify_indexed() << endl;
2674 e = spinor_metric(A, B) * indexed(psi, A_co);
2675 cout << e.simplify_indexed() << endl;
2678 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2679 cout << e.simplify_indexed() << endl;
2682 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2683 cout << e.simplify_indexed() << endl;
2686 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2687 cout << e.simplify_indexed() << endl;
2690 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2691 cout << e.simplify_indexed() << endl;
2696 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2698 @cindex @code{epsilon_tensor()}
2699 @cindex @code{lorentz_eps()}
2700 @subsubsection Epsilon tensor
2702 The epsilon tensor is totally antisymmetric, its number of indices is equal
2703 to the dimension of the index space (the indices must all be of the same
2704 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2705 defined to be 1. Its behavior with indices that have a variance also
2706 depends on the signature of the metric. Epsilon tensors are output as
2709 There are three functions defined to create epsilon tensors in 2, 3 and 4
2713 ex epsilon_tensor(const ex & i1, const ex & i2);
2714 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2715 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2718 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2719 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2720 Minkowski space (the last @code{bool} argument specifies whether the metric
2721 has negative or positive signature, as in the case of the Minkowski metric
2726 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2727 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2728 e = lorentz_eps(mu, nu, rho, sig) *
2729 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2730 cout << simplify_indexed(e) << endl;
2731 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2733 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2734 symbol A("A"), B("B");
2735 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2736 cout << simplify_indexed(e) << endl;
2737 // -> -B.k*A.j*eps.i.k.j
2738 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2739 cout << simplify_indexed(e) << endl;
2744 @subsection Linear algebra
2746 The @code{matrix} class can be used with indices to do some simple linear
2747 algebra (linear combinations and products of vectors and matrices, traces
2748 and scalar products):
2752 idx i(symbol("i"), 2), j(symbol("j"), 2);
2753 symbol x("x"), y("y");
2755 // A is a 2x2 matrix, X is a 2x1 vector
2756 matrix A(2, 2), X(2, 1);
2761 cout << indexed(A, i, i) << endl;
2764 ex e = indexed(A, i, j) * indexed(X, j);
2765 cout << e.simplify_indexed() << endl;
2766 // -> [[2*y+x],[4*y+3*x]].i
2768 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2769 cout << e.simplify_indexed() << endl;
2770 // -> [[3*y+3*x,6*y+2*x]].j
2774 You can of course obtain the same results with the @code{matrix::add()},
2775 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2776 but with indices you don't have to worry about transposing matrices.
2778 Matrix indices always start at 0 and their dimension must match the number
2779 of rows/columns of the matrix. Matrices with one row or one column are
2780 vectors and can have one or two indices (it doesn't matter whether it's a
2781 row or a column vector). Other matrices must have two indices.
2783 You should be careful when using indices with variance on matrices. GiNaC
2784 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2785 @samp{F.mu.nu} are different matrices. In this case you should use only
2786 one form for @samp{F} and explicitly multiply it with a matrix representation
2787 of the metric tensor.
2790 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2791 @c node-name, next, previous, up
2792 @section Non-commutative objects
2794 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2795 non-commutative objects are built-in which are mostly of use in high energy
2799 @item Clifford (Dirac) algebra (class @code{clifford})
2800 @item su(3) Lie algebra (class @code{color})
2801 @item Matrices (unindexed) (class @code{matrix})
2804 The @code{clifford} and @code{color} classes are subclasses of
2805 @code{indexed} because the elements of these algebras usually carry
2806 indices. The @code{matrix} class is described in more detail in
2809 Unlike most computer algebra systems, GiNaC does not primarily provide an
2810 operator (often denoted @samp{&*}) for representing inert products of
2811 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2812 classes of objects involved, and non-commutative products are formed with
2813 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2814 figuring out by itself which objects commutate and will group the factors
2815 by their class. Consider this example:
2819 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2820 idx a(symbol("a"), 8), b(symbol("b"), 8);
2821 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2823 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2827 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2828 groups the non-commutative factors (the gammas and the su(3) generators)
2829 together while preserving the order of factors within each class (because
2830 Clifford objects commutate with color objects). The resulting expression is a
2831 @emph{commutative} product with two factors that are themselves non-commutative
2832 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2833 parentheses are placed around the non-commutative products in the output.
2835 @cindex @code{ncmul} (class)
2836 Non-commutative products are internally represented by objects of the class
2837 @code{ncmul}, as opposed to commutative products which are handled by the
2838 @code{mul} class. You will normally not have to worry about this distinction,
2841 The advantage of this approach is that you never have to worry about using
2842 (or forgetting to use) a special operator when constructing non-commutative
2843 expressions. Also, non-commutative products in GiNaC are more intelligent
2844 than in other computer algebra systems; they can, for example, automatically
2845 canonicalize themselves according to rules specified in the implementation
2846 of the non-commutative classes. The drawback is that to work with other than
2847 the built-in algebras you have to implement new classes yourself. Symbols
2848 always commutate and it's not possible to construct non-commutative products
2849 using symbols to represent the algebra elements or generators. User-defined
2850 functions can, however, be specified as being non-commutative.
2852 @cindex @code{return_type()}
2853 @cindex @code{return_type_tinfo()}
2854 Information about the commutativity of an object or expression can be
2855 obtained with the two member functions
2858 unsigned ex::return_type() const;
2859 unsigned ex::return_type_tinfo() const;
2862 The @code{return_type()} function returns one of three values (defined in
2863 the header file @file{flags.h}), corresponding to three categories of
2864 expressions in GiNaC:
2867 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2868 classes are of this kind.
2869 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2870 certain class of non-commutative objects which can be determined with the
2871 @code{return_type_tinfo()} method. Expressions of this category commutate
2872 with everything except @code{noncommutative} expressions of the same
2874 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2875 of non-commutative objects of different classes. Expressions of this
2876 category don't commutate with any other @code{noncommutative} or
2877 @code{noncommutative_composite} expressions.
2880 The value returned by the @code{return_type_tinfo()} method is valid only
2881 when the return type of the expression is @code{noncommutative}. It is a
2882 value that is unique to the class of the object and usually one of the
2883 constants in @file{tinfos.h}, or derived therefrom.
2885 Here are a couple of examples:
2888 @multitable @columnfractions 0.33 0.33 0.34
2889 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2890 @item @code{42} @tab @code{commutative} @tab -
2891 @item @code{2*x-y} @tab @code{commutative} @tab -
2892 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2893 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2894 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2895 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2899 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2900 @code{TINFO_clifford} for objects with a representation label of zero.
2901 Other representation labels yield a different @code{return_type_tinfo()},
2902 but it's the same for any two objects with the same label. This is also true
2905 A last note: With the exception of matrices, positive integer powers of
2906 non-commutative objects are automatically expanded in GiNaC. For example,
2907 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2908 non-commutative expressions).
2911 @cindex @code{clifford} (class)
2912 @subsection Clifford algebra
2914 @cindex @code{dirac_gamma()}
2915 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2916 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2917 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2918 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2921 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2924 which takes two arguments: the index and a @dfn{representation label} in the
2925 range 0 to 255 which is used to distinguish elements of different Clifford
2926 algebras (this is also called a @dfn{spin line index}). Gammas with different
2927 labels commutate with each other. The dimension of the index can be 4 or (in
2928 the framework of dimensional regularization) any symbolic value. Spinor
2929 indices on Dirac gammas are not supported in GiNaC.
2931 @cindex @code{dirac_ONE()}
2932 The unity element of a Clifford algebra is constructed by
2935 ex dirac_ONE(unsigned char rl = 0);
2938 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2939 multiples of the unity element, even though it's customary to omit it.
2940 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2941 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2942 GiNaC will complain and/or produce incorrect results.
2944 @cindex @code{dirac_gamma5()}
2945 There is a special element @samp{gamma5} that commutates with all other
2946 gammas, has a unit square, and in 4 dimensions equals
2947 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2950 ex dirac_gamma5(unsigned char rl = 0);
2953 @cindex @code{dirac_gammaL()}
2954 @cindex @code{dirac_gammaR()}
2955 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2956 objects, constructed by
2959 ex dirac_gammaL(unsigned char rl = 0);
2960 ex dirac_gammaR(unsigned char rl = 0);
2963 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2964 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2966 @cindex @code{dirac_slash()}
2967 Finally, the function
2970 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2973 creates a term that represents a contraction of @samp{e} with the Dirac
2974 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2975 with a unique index whose dimension is given by the @code{dim} argument).
2976 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2978 In products of dirac gammas, superfluous unity elements are automatically
2979 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2980 and @samp{gammaR} are moved to the front.
2982 The @code{simplify_indexed()} function performs contractions in gamma strings,
2988 symbol a("a"), b("b"), D("D");
2989 varidx mu(symbol("mu"), D);
2990 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2991 * dirac_gamma(mu.toggle_variance());
2993 // -> gamma~mu*a\*gamma.mu
2994 e = e.simplify_indexed();
2997 cout << e.subs(D == 4) << endl;
3003 @cindex @code{dirac_trace()}
3004 To calculate the trace of an expression containing strings of Dirac gammas
3005 you use one of the functions
3008 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
3009 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3010 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3013 These functions take the trace over all gammas in the specified set @code{rls}
3014 or list @code{rll} of representation labels, or the single label @code{rl};
3015 gammas with other labels are left standing. The last argument to
3016 @code{dirac_trace()} is the value to be returned for the trace of the unity
3017 element, which defaults to 4.
3019 The @code{dirac_trace()} function is a linear functional that is equal to the
3020 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3021 functional is not cyclic in @math{D != 4} dimensions when acting on
3022 expressions containing @samp{gamma5}, so it's not a proper trace. This
3023 @samp{gamma5} scheme is described in greater detail in
3024 @cite{The Role of gamma5 in Dimensional Regularization}.
3026 The value of the trace itself is also usually different in 4 and in
3027 @math{D != 4} dimensions:
3032 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3033 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3034 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3035 cout << dirac_trace(e).simplify_indexed() << endl;
3042 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3043 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3044 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3045 cout << dirac_trace(e).simplify_indexed() << endl;
3046 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3050 Here is an example for using @code{dirac_trace()} to compute a value that
3051 appears in the calculation of the one-loop vacuum polarization amplitude in
3056 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3057 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3060 sp.add(l, l, pow(l, 2));
3061 sp.add(l, q, ldotq);
3063 ex e = dirac_gamma(mu) *
3064 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3065 dirac_gamma(mu.toggle_variance()) *
3066 (dirac_slash(l, D) + m * dirac_ONE());
3067 e = dirac_trace(e).simplify_indexed(sp);
3068 e = e.collect(lst(l, ldotq, m));
3070 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3074 The @code{canonicalize_clifford()} function reorders all gamma products that
3075 appear in an expression to a canonical (but not necessarily simple) form.
3076 You can use this to compare two expressions or for further simplifications:
3080 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3081 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3083 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3085 e = canonicalize_clifford(e);
3087 // -> 2*ONE*eta~mu~nu
3092 @cindex @code{color} (class)
3093 @subsection Color algebra
3095 @cindex @code{color_T()}
3096 For computations in quantum chromodynamics, GiNaC implements the base elements
3097 and structure constants of the su(3) Lie algebra (color algebra). The base
3098 elements @math{T_a} are constructed by the function
3101 ex color_T(const ex & a, unsigned char rl = 0);
3104 which takes two arguments: the index and a @dfn{representation label} in the
3105 range 0 to 255 which is used to distinguish elements of different color
3106 algebras. Objects with different labels commutate with each other. The
3107 dimension of the index must be exactly 8 and it should be of class @code{idx},
3110 @cindex @code{color_ONE()}
3111 The unity element of a color algebra is constructed by
3114 ex color_ONE(unsigned char rl = 0);
3117 @strong{Note:} You must always use @code{color_ONE()} when referring to
3118 multiples of the unity element, even though it's customary to omit it.
3119 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3120 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3121 GiNaC may produce incorrect results.
3123 @cindex @code{color_d()}
3124 @cindex @code{color_f()}
3128 ex color_d(const ex & a, const ex & b, const ex & c);
3129 ex color_f(const ex & a, const ex & b, const ex & c);
3132 create the symmetric and antisymmetric structure constants @math{d_abc} and
3133 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3134 and @math{[T_a, T_b] = i f_abc T_c}.
3136 @cindex @code{color_h()}
3137 There's an additional function
3140 ex color_h(const ex & a, const ex & b, const ex & c);
3143 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3145 The function @code{simplify_indexed()} performs some simplifications on
3146 expressions containing color objects:
3151 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3152 k(symbol("k"), 8), l(symbol("l"), 8);
3154 e = color_d(a, b, l) * color_f(a, b, k);
3155 cout << e.simplify_indexed() << endl;
3158 e = color_d(a, b, l) * color_d(a, b, k);
3159 cout << e.simplify_indexed() << endl;
3162 e = color_f(l, a, b) * color_f(a, b, k);
3163 cout << e.simplify_indexed() << endl;
3166 e = color_h(a, b, c) * color_h(a, b, c);
3167 cout << e.simplify_indexed() << endl;
3170 e = color_h(a, b, c) * color_T(b) * color_T(c);
3171 cout << e.simplify_indexed() << endl;
3174 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3175 cout << e.simplify_indexed() << endl;
3178 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3179 cout << e.simplify_indexed() << endl;
3180 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3184 @cindex @code{color_trace()}
3185 To calculate the trace of an expression containing color objects you use one
3189 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3190 ex color_trace(const ex & e, const lst & rll);
3191 ex color_trace(const ex & e, unsigned char rl = 0);
3194 These functions take the trace over all color @samp{T} objects in the
3195 specified set @code{rls} or list @code{rll} of representation labels, or the
3196 single label @code{rl}; @samp{T}s with other labels are left standing. For
3201 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3203 // -> -I*f.a.c.b+d.a.c.b
3208 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3209 @c node-name, next, previous, up
3212 @cindex @code{exhashmap} (class)
3214 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3215 that can be used as a drop-in replacement for the STL
3216 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3217 typically constant-time, element look-up than @code{map<>}.
3219 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3220 following differences:
3224 no @code{lower_bound()} and @code{upper_bound()} methods
3226 no reverse iterators, no @code{rbegin()}/@code{rend()}
3228 no @code{operator<(exhashmap, exhashmap)}
3230 the comparison function object @code{key_compare} is hardcoded to
3233 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3234 initial hash table size (the actual table size after construction may be
3235 larger than the specified value)
3237 the method @code{size_t bucket_count()} returns the current size of the hash
3240 @code{insert()} and @code{erase()} operations invalidate all iterators
3244 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3245 @c node-name, next, previous, up
3246 @chapter Methods and Functions
3249 In this chapter the most important algorithms provided by GiNaC will be
3250 described. Some of them are implemented as functions on expressions,
3251 others are implemented as methods provided by expression objects. If
3252 they are methods, there exists a wrapper function around it, so you can
3253 alternatively call it in a functional way as shown in the simple
3258 cout << "As method: " << sin(1).evalf() << endl;
3259 cout << "As function: " << evalf(sin(1)) << endl;
3263 @cindex @code{subs()}
3264 The general rule is that wherever methods accept one or more parameters
3265 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3266 wrapper accepts is the same but preceded by the object to act on
3267 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3268 most natural one in an OO model but it may lead to confusion for MapleV
3269 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3270 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3271 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3272 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3273 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3274 here. Also, users of MuPAD will in most cases feel more comfortable
3275 with GiNaC's convention. All function wrappers are implemented
3276 as simple inline functions which just call the corresponding method and
3277 are only provided for users uncomfortable with OO who are dead set to
3278 avoid method invocations. Generally, nested function wrappers are much
3279 harder to read than a sequence of methods and should therefore be
3280 avoided if possible. On the other hand, not everything in GiNaC is a
3281 method on class @code{ex} and sometimes calling a function cannot be
3285 * Information About Expressions::
3286 * Numerical Evaluation::
3287 * Substituting Expressions::
3288 * Pattern Matching and Advanced Substitutions::
3289 * Applying a Function on Subexpressions::
3290 * Visitors and Tree Traversal::
3291 * Polynomial Arithmetic:: Working with polynomials.
3292 * Rational Expressions:: Working with rational functions.
3293 * Symbolic Differentiation::
3294 * Series Expansion:: Taylor and Laurent expansion.
3296 * Built-in Functions:: List of predefined mathematical functions.
3297 * Multiple polylogarithms::
3298 * Complex Conjugation::
3299 * Built-in Functions:: List of predefined mathematical functions.
3300 * Solving Linear Systems of Equations::
3301 * Input/Output:: Input and output of expressions.
3305 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3306 @c node-name, next, previous, up
3307 @section Getting information about expressions
3309 @subsection Checking expression types
3310 @cindex @code{is_a<@dots{}>()}
3311 @cindex @code{is_exactly_a<@dots{}>()}
3312 @cindex @code{ex_to<@dots{}>()}
3313 @cindex Converting @code{ex} to other classes
3314 @cindex @code{info()}
3315 @cindex @code{return_type()}
3316 @cindex @code{return_type_tinfo()}
3318 Sometimes it's useful to check whether a given expression is a plain number,
3319 a sum, a polynomial with integer coefficients, or of some other specific type.
3320 GiNaC provides a couple of functions for this:
3323 bool is_a<T>(const ex & e);
3324 bool is_exactly_a<T>(const ex & e);
3325 bool ex::info(unsigned flag);
3326 unsigned ex::return_type() const;
3327 unsigned ex::return_type_tinfo() const;
3330 When the test made by @code{is_a<T>()} returns true, it is safe to call
3331 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3332 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3333 example, assuming @code{e} is an @code{ex}:
3338 if (is_a<numeric>(e))
3339 numeric n = ex_to<numeric>(e);
3344 @code{is_a<T>(e)} allows you to check whether the top-level object of
3345 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3346 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3347 e.g., for checking whether an expression is a number, a sum, or a product:
3354 is_a<numeric>(e1); // true
3355 is_a<numeric>(e2); // false
3356 is_a<add>(e1); // false
3357 is_a<add>(e2); // true
3358 is_a<mul>(e1); // false
3359 is_a<mul>(e2); // false
3363 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3364 top-level object of an expression @samp{e} is an instance of the GiNaC
3365 class @samp{T}, not including parent classes.
3367 The @code{info()} method is used for checking certain attributes of
3368 expressions. The possible values for the @code{flag} argument are defined
3369 in @file{ginac/flags.h}, the most important being explained in the following
3373 @multitable @columnfractions .30 .70
3374 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3375 @item @code{numeric}
3376 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3378 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3379 @item @code{rational}
3380 @tab @dots{}an exact rational number (integers are rational, too)
3381 @item @code{integer}
3382 @tab @dots{}a (non-complex) integer
3383 @item @code{crational}
3384 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3385 @item @code{cinteger}
3386 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3387 @item @code{positive}
3388 @tab @dots{}not complex and greater than 0
3389 @item @code{negative}
3390 @tab @dots{}not complex and less than 0
3391 @item @code{nonnegative}
3392 @tab @dots{}not complex and greater than or equal to 0
3394 @tab @dots{}an integer greater than 0
3396 @tab @dots{}an integer less than 0
3397 @item @code{nonnegint}
3398 @tab @dots{}an integer greater than or equal to 0
3400 @tab @dots{}an even integer
3402 @tab @dots{}an odd integer
3404 @tab @dots{}a prime integer (probabilistic primality test)
3405 @item @code{relation}
3406 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3407 @item @code{relation_equal}
3408 @tab @dots{}a @code{==} relation
3409 @item @code{relation_not_equal}
3410 @tab @dots{}a @code{!=} relation
3411 @item @code{relation_less}
3412 @tab @dots{}a @code{<} relation
3413 @item @code{relation_less_or_equal}
3414 @tab @dots{}a @code{<=} relation
3415 @item @code{relation_greater}
3416 @tab @dots{}a @code{>} relation
3417 @item @code{relation_greater_or_equal}
3418 @tab @dots{}a @code{>=} relation
3420 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3422 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3423 @item @code{polynomial}
3424 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3425 @item @code{integer_polynomial}
3426 @tab @dots{}a polynomial with (non-complex) integer coefficients
3427 @item @code{cinteger_polynomial}
3428 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3429 @item @code{rational_polynomial}
3430 @tab @dots{}a polynomial with (non-complex) rational coefficients
3431 @item @code{crational_polynomial}
3432 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3433 @item @code{rational_function}
3434 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3435 @item @code{algebraic}
3436 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3440 To determine whether an expression is commutative or non-commutative and if
3441 so, with which other expressions it would commutate, you use the methods
3442 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3443 for an explanation of these.
3446 @subsection Accessing subexpressions
3449 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3450 @code{function}, act as containers for subexpressions. For example, the
3451 subexpressions of a sum (an @code{add} object) are the individual terms,
3452 and the subexpressions of a @code{function} are the function's arguments.
3454 @cindex @code{nops()}
3456 GiNaC provides several ways of accessing subexpressions. The first way is to
3461 ex ex::op(size_t i);
3464 @code{nops()} determines the number of subexpressions (operands) contained
3465 in the expression, while @code{op(i)} returns the @code{i}-th
3466 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3467 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3468 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3469 @math{i>0} are the indices.
3472 @cindex @code{const_iterator}
3473 The second way to access subexpressions is via the STL-style random-access
3474 iterator class @code{const_iterator} and the methods
3477 const_iterator ex::begin();
3478 const_iterator ex::end();
3481 @code{begin()} returns an iterator referring to the first subexpression;
3482 @code{end()} returns an iterator which is one-past the last subexpression.
3483 If the expression has no subexpressions, then @code{begin() == end()}. These
3484 iterators can also be used in conjunction with non-modifying STL algorithms.
3486 Here is an example that (non-recursively) prints the subexpressions of a
3487 given expression in three different ways:
3494 for (size_t i = 0; i != e.nops(); ++i)
3495 cout << e.op(i) << endl;
3498 for (const_iterator i = e.begin(); i != e.end(); ++i)
3501 // with iterators and STL copy()
3502 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3506 @cindex @code{const_preorder_iterator}
3507 @cindex @code{const_postorder_iterator}
3508 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3509 expression's immediate children. GiNaC provides two additional iterator
3510 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3511 that iterate over all objects in an expression tree, in preorder or postorder,
3512 respectively. They are STL-style forward iterators, and are created with the
3516 const_preorder_iterator ex::preorder_begin();
3517 const_preorder_iterator ex::preorder_end();
3518 const_postorder_iterator ex::postorder_begin();
3519 const_postorder_iterator ex::postorder_end();
3522 The following example illustrates the differences between
3523 @code{const_iterator}, @code{const_preorder_iterator}, and
3524 @code{const_postorder_iterator}:
3528 symbol A("A"), B("B"), C("C");
3529 ex e = lst(lst(A, B), C);
3531 std::copy(e.begin(), e.end(),
3532 std::ostream_iterator<ex>(cout, "\n"));
3536 std::copy(e.preorder_begin(), e.preorder_end(),
3537 std::ostream_iterator<ex>(cout, "\n"));
3544 std::copy(e.postorder_begin(), e.postorder_end(),
3545 std::ostream_iterator<ex>(cout, "\n"));
3554 @cindex @code{relational} (class)
3555 Finally, the left-hand side and right-hand side expressions of objects of
3556 class @code{relational} (and only of these) can also be accessed with the
3565 @subsection Comparing expressions
3566 @cindex @code{is_equal()}
3567 @cindex @code{is_zero()}
3569 Expressions can be compared with the usual C++ relational operators like
3570 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3571 the result is usually not determinable and the result will be @code{false},
3572 except in the case of the @code{!=} operator. You should also be aware that
3573 GiNaC will only do the most trivial test for equality (subtracting both
3574 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3577 Actually, if you construct an expression like @code{a == b}, this will be
3578 represented by an object of the @code{relational} class (@pxref{Relations})
3579 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3581 There are also two methods
3584 bool ex::is_equal(const ex & other);
3588 for checking whether one expression is equal to another, or equal to zero,
3592 @subsection Ordering expressions
3593 @cindex @code{ex_is_less} (class)
3594 @cindex @code{ex_is_equal} (class)
3595 @cindex @code{compare()}
3597 Sometimes it is necessary to establish a mathematically well-defined ordering
3598 on a set of arbitrary expressions, for example to use expressions as keys
3599 in a @code{std::map<>} container, or to bring a vector of expressions into
3600 a canonical order (which is done internally by GiNaC for sums and products).
3602 The operators @code{<}, @code{>} etc. described in the last section cannot
3603 be used for this, as they don't implement an ordering relation in the
3604 mathematical sense. In particular, they are not guaranteed to be
3605 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3606 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3609 By default, STL classes and algorithms use the @code{<} and @code{==}
3610 operators to compare objects, which are unsuitable for expressions, but GiNaC
3611 provides two functors that can be supplied as proper binary comparison
3612 predicates to the STL:
3615 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3617 bool operator()(const ex &lh, const ex &rh) const;
3620 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3622 bool operator()(const ex &lh, const ex &rh) const;
3626 For example, to define a @code{map} that maps expressions to strings you
3630 std::map<ex, std::string, ex_is_less> myMap;
3633 Omitting the @code{ex_is_less} template parameter will introduce spurious
3634 bugs because the map operates improperly.
3636 Other examples for the use of the functors:
3644 std::sort(v.begin(), v.end(), ex_is_less());
3646 // count the number of expressions equal to '1'
3647 unsigned num_ones = std::count_if(v.begin(), v.end(),
3648 std::bind2nd(ex_is_equal(), 1));
3651 The implementation of @code{ex_is_less} uses the member function
3654 int ex::compare(const ex & other) const;
3657 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3658 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3662 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3663 @c node-name, next, previous, up
3664 @section Numerical Evaluation
3665 @cindex @code{evalf()}
3667 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3668 To evaluate them using floating-point arithmetic you need to call
3671 ex ex::evalf(int level = 0) const;
3674 @cindex @code{Digits}
3675 The accuracy of the evaluation is controlled by the global object @code{Digits}
3676 which can be assigned an integer value. The default value of @code{Digits}
3677 is 17. @xref{Numbers}, for more information and examples.
3679 To evaluate an expression to a @code{double} floating-point number you can
3680 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3684 // Approximate sin(x/Pi)
3686 ex e = series(sin(x/Pi), x == 0, 6);
3688 // Evaluate numerically at x=0.1
3689 ex f = evalf(e.subs(x == 0.1));
3691 // ex_to<numeric> is an unsafe cast, so check the type first
3692 if (is_a<numeric>(f)) @{
3693 double d = ex_to<numeric>(f).to_double();
3702 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3703 @c node-name, next, previous, up
3704 @section Substituting expressions
3705 @cindex @code{subs()}
3707 Algebraic objects inside expressions can be replaced with arbitrary
3708 expressions via the @code{.subs()} method:
3711 ex ex::subs(const ex & e, unsigned options = 0);
3712 ex ex::subs(const exmap & m, unsigned options = 0);
3713 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3716 In the first form, @code{subs()} accepts a relational of the form
3717 @samp{object == expression} or a @code{lst} of such relationals:
3721 symbol x("x"), y("y");
3723 ex e1 = 2*x^2-4*x+3;
3724 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3728 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3733 If you specify multiple substitutions, they are performed in parallel, so e.g.
3734 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3736 The second form of @code{subs()} takes an @code{exmap} object which is a
3737 pair associative container that maps expressions to expressions (currently
3738 implemented as a @code{std::map}). This is the most efficient one of the
3739 three @code{subs()} forms and should be used when the number of objects to
3740 be substituted is large or unknown.
3742 Using this form, the second example from above would look like this:
3746 symbol x("x"), y("y");
3752 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3756 The third form of @code{subs()} takes two lists, one for the objects to be
3757 replaced and one for the expressions to be substituted (both lists must
3758 contain the same number of elements). Using this form, you would write
3762 symbol x("x"), y("y");
3765 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3769 The optional last argument to @code{subs()} is a combination of
3770 @code{subs_options} flags. There are two options available:
3771 @code{subs_options::no_pattern} disables pattern matching, which makes
3772 large @code{subs()} operations significantly faster if you are not using
3773 patterns. The second option, @code{subs_options::algebraic} enables
3774 algebraic substitutions in products and powers.
3775 @ref{Pattern Matching and Advanced Substitutions}, for more information
3776 about patterns and algebraic substitutions.
3778 @code{subs()} performs syntactic substitution of any complete algebraic
3779 object; it does not try to match sub-expressions as is demonstrated by the
3784 symbol x("x"), y("y"), z("z");
3786 ex e1 = pow(x+y, 2);
3787 cout << e1.subs(x+y == 4) << endl;
3790 ex e2 = sin(x)*sin(y)*cos(x);
3791 cout << e2.subs(sin(x) == cos(x)) << endl;
3792 // -> cos(x)^2*sin(y)
3795 cout << e3.subs(x+y == 4) << endl;
3797 // (and not 4+z as one might expect)
3801 A more powerful form of substitution using wildcards is described in the
3805 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3806 @c node-name, next, previous, up
3807 @section Pattern matching and advanced substitutions
3808 @cindex @code{wildcard} (class)
3809 @cindex Pattern matching
3811 GiNaC allows the use of patterns for checking whether an expression is of a
3812 certain form or contains subexpressions of a certain form, and for
3813 substituting expressions in a more general way.
3815 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3816 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3817 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3818 an unsigned integer number to allow having multiple different wildcards in a
3819 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3820 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3824 ex wild(unsigned label = 0);
3827 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3830 Some examples for patterns:
3832 @multitable @columnfractions .5 .5
3833 @item @strong{Constructed as} @tab @strong{Output as}
3834 @item @code{wild()} @tab @samp{$0}
3835 @item @code{pow(x,wild())} @tab @samp{x^$0}
3836 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3837 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3843 @item Wildcards behave like symbols and are subject to the same algebraic
3844 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3845 @item As shown in the last example, to use wildcards for indices you have to
3846 use them as the value of an @code{idx} object. This is because indices must
3847 always be of class @code{idx} (or a subclass).
3848 @item Wildcards only represent expressions or subexpressions. It is not
3849 possible to use them as placeholders for other properties like index
3850 dimension or variance, representation labels, symmetry of indexed objects
3852 @item Because wildcards are commutative, it is not possible to use wildcards
3853 as part of noncommutative products.
3854 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3855 are also valid patterns.
3858 @subsection Matching expressions
3859 @cindex @code{match()}
3860 The most basic application of patterns is to check whether an expression
3861 matches a given pattern. This is done by the function
3864 bool ex::match(const ex & pattern);
3865 bool ex::match(const ex & pattern, lst & repls);
3868 This function returns @code{true} when the expression matches the pattern
3869 and @code{false} if it doesn't. If used in the second form, the actual
3870 subexpressions matched by the wildcards get returned in the @code{repls}
3871 object as a list of relations of the form @samp{wildcard == expression}.
3872 If @code{match()} returns false, the state of @code{repls} is undefined.
3873 For reproducible results, the list should be empty when passed to
3874 @code{match()}, but it is also possible to find similarities in multiple
3875 expressions by passing in the result of a previous match.
3877 The matching algorithm works as follows:
3880 @item A single wildcard matches any expression. If one wildcard appears
3881 multiple times in a pattern, it must match the same expression in all
3882 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3883 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3884 @item If the expression is not of the same class as the pattern, the match
3885 fails (i.e. a sum only matches a sum, a function only matches a function,
3887 @item If the pattern is a function, it only matches the same function
3888 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3889 @item Except for sums and products, the match fails if the number of
3890 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3892 @item If there are no subexpressions, the expressions and the pattern must
3893 be equal (in the sense of @code{is_equal()}).
3894 @item Except for sums and products, each subexpression (@code{op()}) must
3895 match the corresponding subexpression of the pattern.
3898 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3899 account for their commutativity and associativity:
3902 @item If the pattern contains a term or factor that is a single wildcard,
3903 this one is used as the @dfn{global wildcard}. If there is more than one
3904 such wildcard, one of them is chosen as the global wildcard in a random
3906 @item Every term/factor of the pattern, except the global wildcard, is
3907 matched against every term of the expression in sequence. If no match is
3908 found, the whole match fails. Terms that did match are not considered in
3910 @item If there are no unmatched terms left, the match succeeds. Otherwise
3911 the match fails unless there is a global wildcard in the pattern, in
3912 which case this wildcard matches the remaining terms.
3915 In general, having more than one single wildcard as a term of a sum or a
3916 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3919 Here are some examples in @command{ginsh} to demonstrate how it works (the
3920 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3921 match fails, and the list of wildcard replacements otherwise):
3924 > match((x+y)^a,(x+y)^a);
3926 > match((x+y)^a,(x+y)^b);
3928 > match((x+y)^a,$1^$2);
3930 > match((x+y)^a,$1^$1);
3932 > match((x+y)^(x+y),$1^$1);
3934 > match((x+y)^(x+y),$1^$2);
3936 > match((a+b)*(a+c),($1+b)*($1+c));
3938 > match((a+b)*(a+c),(a+$1)*(a+$2));
3940 (Unpredictable. The result might also be [$1==c,$2==b].)
3941 > match((a+b)*(a+c),($1+$2)*($1+$3));
3942 (The result is undefined. Due to the sequential nature of the algorithm
3943 and the re-ordering of terms in GiNaC, the match for the first factor
3944 may be @{$1==a,$2==b@} in which case the match for the second factor
3945 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3947 > match(a*(x+y)+a*z+b,a*$1+$2);
3948 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3949 @{$1=x+y,$2=a*z+b@}.)
3950 > match(a+b+c+d+e+f,c);
3952 > match(a+b+c+d+e+f,c+$0);
3954 > match(a+b+c+d+e+f,c+e+$0);
3956 > match(a+b,a+b+$0);
3958 > match(a*b^2,a^$1*b^$2);
3960 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3961 even though a==a^1.)
3962 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3964 > match(atan2(y,x^2),atan2(y,$0));
3968 @subsection Matching parts of expressions
3969 @cindex @code{has()}
3970 A more general way to look for patterns in expressions is provided by the
3974 bool ex::has(const ex & pattern);
3977 This function checks whether a pattern is matched by an expression itself or
3978 by any of its subexpressions.
3980 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3981 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3984 > has(x*sin(x+y+2*a),y);
3986 > has(x*sin(x+y+2*a),x+y);
3988 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3989 has the subexpressions "x", "y" and "2*a".)
3990 > has(x*sin(x+y+2*a),x+y+$1);
3992 (But this is possible.)
3993 > has(x*sin(2*(x+y)+2*a),x+y);
3995 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3996 which "x+y" is not a subexpression.)
3999 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4001 > has(4*x^2-x+3,$1*x);
4003 > has(4*x^2+x+3,$1*x);
4005 (Another possible pitfall. The first expression matches because the term
4006 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4007 contains a linear term you should use the coeff() function instead.)
4010 @cindex @code{find()}
4014 bool ex::find(const ex & pattern, lst & found);
4017 works a bit like @code{has()} but it doesn't stop upon finding the first
4018 match. Instead, it appends all found matches to the specified list. If there
4019 are multiple occurrences of the same expression, it is entered only once to
4020 the list. @code{find()} returns false if no matches were found (in
4021 @command{ginsh}, it returns an empty list):
4024 > find(1+x+x^2+x^3,x);
4026 > find(1+x+x^2+x^3,y);
4028 > find(1+x+x^2+x^3,x^$1);
4030 (Note the absence of "x".)
4031 > expand((sin(x)+sin(y))*(a+b));
4032 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4037 @subsection Substituting expressions
4038 @cindex @code{subs()}
4039 Probably the most useful application of patterns is to use them for
4040 substituting expressions with the @code{subs()} method. Wildcards can be
4041 used in the search patterns as well as in the replacement expressions, where
4042 they get replaced by the expressions matched by them. @code{subs()} doesn't
4043 know anything about algebra; it performs purely syntactic substitutions.
4048 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4050 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4052 > subs((a+b+c)^2,a+b==x);
4054 > subs((a+b+c)^2,a+b+$1==x+$1);
4056 > subs(a+2*b,a+b==x);
4058 > subs(4*x^3-2*x^2+5*x-1,x==a);
4060 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4062 > subs(sin(1+sin(x)),sin($1)==cos($1));
4064 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4068 The last example would be written in C++ in this way:
4072 symbol a("a"), b("b"), x("x"), y("y");
4073 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4074 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4075 cout << e.expand() << endl;
4080 @subsection Algebraic substitutions
4081 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4082 enables smarter, algebraic substitutions in products and powers. If you want
4083 to substitute some factors of a product, you only need to list these factors
4084 in your pattern. Furthermore, if an (integer) power of some expression occurs
4085 in your pattern and in the expression that you want the substitution to occur
4086 in, it can be substituted as many times as possible, without getting negative
4089 An example clarifies it all (hopefully):
4092 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4093 subs_options::algebraic) << endl;
4094 // --> (y+x)^6+b^6+a^6
4096 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4098 // Powers and products are smart, but addition is just the same.
4100 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4103 // As I said: addition is just the same.
4105 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4106 // --> x^3*b*a^2+2*b
4108 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4110 // --> 2*b+x^3*b^(-1)*a^(-2)
4112 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4113 // --> -1-2*a^2+4*a^3+5*a
4115 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4116 subs_options::algebraic) << endl;
4117 // --> -1+5*x+4*x^3-2*x^2
4118 // You should not really need this kind of patterns very often now.
4119 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4121 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4122 subs_options::algebraic) << endl;
4123 // --> cos(1+cos(x))
4125 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4126 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4127 subs_options::algebraic)) << endl;
4132 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4133 @c node-name, next, previous, up
4134 @section Applying a Function on Subexpressions
4135 @cindex tree traversal
4136 @cindex @code{map()}
4138 Sometimes you may want to perform an operation on specific parts of an
4139 expression while leaving the general structure of it intact. An example
4140 of this would be a matrix trace operation: the trace of a sum is the sum
4141 of the traces of the individual terms. That is, the trace should @dfn{map}
4142 on the sum, by applying itself to each of the sum's operands. It is possible
4143 to do this manually which usually results in code like this:
4148 if (is_a<matrix>(e))
4149 return ex_to<matrix>(e).trace();
4150 else if (is_a<add>(e)) @{
4152 for (size_t i=0; i<e.nops(); i++)
4153 sum += calc_trace(e.op(i));
4155 @} else if (is_a<mul>)(e)) @{
4163 This is, however, slightly inefficient (if the sum is very large it can take
4164 a long time to add the terms one-by-one), and its applicability is limited to
4165 a rather small class of expressions. If @code{calc_trace()} is called with
4166 a relation or a list as its argument, you will probably want the trace to
4167 be taken on both sides of the relation or of all elements of the list.
4169 GiNaC offers the @code{map()} method to aid in the implementation of such
4173 ex ex::map(map_function & f) const;
4174 ex ex::map(ex (*f)(const ex & e)) const;
4177 In the first (preferred) form, @code{map()} takes a function object that
4178 is subclassed from the @code{map_function} class. In the second form, it
4179 takes a pointer to a function that accepts and returns an expression.
4180 @code{map()} constructs a new expression of the same type, applying the
4181 specified function on all subexpressions (in the sense of @code{op()}),
4184 The use of a function object makes it possible to supply more arguments to
4185 the function that is being mapped, or to keep local state information.
4186 The @code{map_function} class declares a virtual function call operator
4187 that you can overload. Here is a sample implementation of @code{calc_trace()}
4188 that uses @code{map()} in a recursive fashion:
4191 struct calc_trace : public map_function @{
4192 ex operator()(const ex &e)
4194 if (is_a<matrix>(e))
4195 return ex_to<matrix>(e).trace();
4196 else if (is_a<mul>(e)) @{
4199 return e.map(*this);
4204 This function object could then be used like this:
4208 ex M = ... // expression with matrices
4209 calc_trace do_trace;
4210 ex tr = do_trace(M);
4214 Here is another example for you to meditate over. It removes quadratic
4215 terms in a variable from an expanded polynomial:
4218 struct map_rem_quad : public map_function @{
4220 map_rem_quad(const ex & var_) : var(var_) @{@}
4222 ex operator()(const ex & e)
4224 if (is_a<add>(e) || is_a<mul>(e))
4225 return e.map(*this);
4226 else if (is_a<power>(e) &&
4227 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4237 symbol x("x"), y("y");
4240 for (int i=0; i<8; i++)
4241 e += pow(x, i) * pow(y, 8-i) * (i+1);
4243 // -> 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
4245 map_rem_quad rem_quad(x);
4246 cout << rem_quad(e) << endl;
4247 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4251 @command{ginsh} offers a slightly different implementation of @code{map()}
4252 that allows applying algebraic functions to operands. The second argument
4253 to @code{map()} is an expression containing the wildcard @samp{$0} which
4254 acts as the placeholder for the operands:
4259 > map(a+2*b,sin($0));
4261 > map(@{a,b,c@},$0^2+$0);
4262 @{a^2+a,b^2+b,c^2+c@}
4265 Note that it is only possible to use algebraic functions in the second
4266 argument. You can not use functions like @samp{diff()}, @samp{op()},
4267 @samp{subs()} etc. because these are evaluated immediately:
4270 > map(@{a,b,c@},diff($0,a));
4272 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4273 to "map(@{a,b,c@},0)".
4277 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4278 @c node-name, next, previous, up
4279 @section Visitors and Tree Traversal
4280 @cindex tree traversal
4281 @cindex @code{visitor} (class)
4282 @cindex @code{accept()}
4283 @cindex @code{visit()}
4284 @cindex @code{traverse()}
4285 @cindex @code{traverse_preorder()}
4286 @cindex @code{traverse_postorder()}
4288 Suppose that you need a function that returns a list of all indices appearing
4289 in an arbitrary expression. The indices can have any dimension, and for
4290 indices with variance you always want the covariant version returned.
4292 You can't use @code{get_free_indices()} because you also want to include
4293 dummy indices in the list, and you can't use @code{find()} as it needs
4294 specific index dimensions (and it would require two passes: one for indices
4295 with variance, one for plain ones).
4297 The obvious solution to this problem is a tree traversal with a type switch,
4298 such as the following:
4301 void gather_indices_helper(const ex & e, lst & l)
4303 if (is_a<varidx>(e)) @{
4304 const varidx & vi = ex_to<varidx>(e);
4305 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4306 @} else if (is_a<idx>(e)) @{
4309 size_t n = e.nops();
4310 for (size_t i = 0; i < n; ++i)
4311 gather_indices_helper(e.op(i), l);
4315 lst gather_indices(const ex & e)
4318 gather_indices_helper(e, l);
4325 This works fine but fans of object-oriented programming will feel
4326 uncomfortable with the type switch. One reason is that there is a possibility
4327 for subtle bugs regarding derived classes. If we had, for example, written
4330 if (is_a<idx>(e)) @{
4332 @} else if (is_a<varidx>(e)) @{
4336 in @code{gather_indices_helper}, the code wouldn't have worked because the
4337 first line "absorbs" all classes derived from @code{idx}, including
4338 @code{varidx}, so the special case for @code{varidx} would never have been
4341 Also, for a large number of classes, a type switch like the above can get
4342 unwieldy and inefficient (it's a linear search, after all).
4343 @code{gather_indices_helper} only checks for two classes, but if you had to
4344 write a function that required a different implementation for nearly
4345 every GiNaC class, the result would be very hard to maintain and extend.
4347 The cleanest approach to the problem would be to add a new virtual function
4348 to GiNaC's class hierarchy. In our example, there would be specializations
4349 for @code{idx} and @code{varidx} while the default implementation in
4350 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4351 impossible to add virtual member functions to existing classes without
4352 changing their source and recompiling everything. GiNaC comes with source,
4353 so you could actually do this, but for a small algorithm like the one
4354 presented this would be impractical.
4356 One solution to this dilemma is the @dfn{Visitor} design pattern,
4357 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4358 variation, described in detail in
4359 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4360 virtual functions to the class hierarchy to implement operations, GiNaC
4361 provides a single "bouncing" method @code{accept()} that takes an instance
4362 of a special @code{visitor} class and redirects execution to the one
4363 @code{visit()} virtual function of the visitor that matches the type of
4364 object that @code{accept()} was being invoked on.
4366 Visitors in GiNaC must derive from the global @code{visitor} class as well
4367 as from the class @code{T::visitor} of each class @code{T} they want to
4368 visit, and implement the member functions @code{void visit(const T &)} for
4374 void ex::accept(visitor & v) const;
4377 will then dispatch to the correct @code{visit()} member function of the
4378 specified visitor @code{v} for the type of GiNaC object at the root of the
4379 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4381 Here is an example of a visitor:
4385 : public visitor, // this is required
4386 public add::visitor, // visit add objects
4387 public numeric::visitor, // visit numeric objects
4388 public basic::visitor // visit basic objects
4390 void visit(const add & x)
4391 @{ cout << "called with an add object" << endl; @}
4393 void visit(const numeric & x)
4394 @{ cout << "called with a numeric object" << endl; @}
4396 void visit(const basic & x)
4397 @{ cout << "called with a basic object" << endl; @}
4401 which can be used as follows:
4412 // prints "called with a numeric object"
4414 // prints "called with an add object"
4416 // prints "called with a basic object"
4420 The @code{visit(const basic &)} method gets called for all objects that are
4421 not @code{numeric} or @code{add} and acts as an (optional) default.
4423 From a conceptual point of view, the @code{visit()} methods of the visitor
4424 behave like a newly added virtual function of the visited hierarchy.
4425 In addition, visitors can store state in member variables, and they can
4426 be extended by deriving a new visitor from an existing one, thus building
4427 hierarchies of visitors.
4429 We can now rewrite our index example from above with a visitor:
4432 class gather_indices_visitor
4433 : public visitor, public idx::visitor, public varidx::visitor
4437 void visit(const idx & i)
4442 void visit(const varidx & vi)
4444 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4448 const lst & get_result() // utility function
4457 What's missing is the tree traversal. We could implement it in
4458 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4461 void ex::traverse_preorder(visitor & v) const;
4462 void ex::traverse_postorder(visitor & v) const;
4463 void ex::traverse(visitor & v) const;
4466 @code{traverse_preorder()} visits a node @emph{before} visiting its
4467 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4468 visiting its subexpressions. @code{traverse()} is a synonym for
4469 @code{traverse_preorder()}.
4471 Here is a new implementation of @code{gather_indices()} that uses the visitor
4472 and @code{traverse()}:
4475 lst gather_indices(const ex & e)
4477 gather_indices_visitor v;
4479 return v.get_result();
4483 Alternatively, you could use pre- or postorder iterators for the tree
4487 lst gather_indices(const ex & e)
4489 gather_indices_visitor v;
4490 for (const_preorder_iterator i = e.preorder_begin();
4491 i != e.preorder_end(); ++i) @{
4494 return v.get_result();
4499 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4500 @c node-name, next, previous, up
4501 @section Polynomial arithmetic
4503 @subsection Expanding and collecting
4504 @cindex @code{expand()}
4505 @cindex @code{collect()}
4506 @cindex @code{collect_common_factors()}
4508 A polynomial in one or more variables has many equivalent
4509 representations. Some useful ones serve a specific purpose. Consider
4510 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4511 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4512 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4513 representations are the recursive ones where one collects for exponents
4514 in one of the three variable. Since the factors are themselves
4515 polynomials in the remaining two variables the procedure can be
4516 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4517 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4520 To bring an expression into expanded form, its method
4523 ex ex::expand(unsigned options = 0);
4526 may be called. In our example above, this corresponds to @math{4*x*y +
4527 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4528 GiNaC is not easy to guess you should be prepared to see different
4529 orderings of terms in such sums!
4531 Another useful representation of multivariate polynomials is as a
4532 univariate polynomial in one of the variables with the coefficients
4533 being polynomials in the remaining variables. The method
4534 @code{collect()} accomplishes this task:
4537 ex ex::collect(const ex & s, bool distributed = false);
4540 The first argument to @code{collect()} can also be a list of objects in which
4541 case the result is either a recursively collected polynomial, or a polynomial
4542 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4543 by the @code{distributed} flag.
4545 Note that the original polynomial needs to be in expanded form (for the
4546 variables concerned) in order for @code{collect()} to be able to find the
4547 coefficients properly.
4549 The following @command{ginsh} transcript shows an application of @code{collect()}
4550 together with @code{find()}:
4553 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4554 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)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4555 > collect(a,@{p,q@});
4556 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4557 > collect(a,find(a,sin($1)));
4558 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4559 > collect(a,@{find(a,sin($1)),p,q@});
4560 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4561 > collect(a,@{find(a,sin($1)),d@});
4562 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4565 Polynomials can often be brought into a more compact form by collecting
4566 common factors from the terms of sums. This is accomplished by the function
4569 ex collect_common_factors(const ex & e);
4572 This function doesn't perform a full factorization but only looks for
4573 factors which are already explicitly present:
4576 > collect_common_factors(a*x+a*y);
4578 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4580 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4581 (c+a)*a*(x*y+y^2+x)*b
4584 @subsection Degree and coefficients
4585 @cindex @code{degree()}
4586 @cindex @code{ldegree()}
4587 @cindex @code{coeff()}
4589 The degree and low degree of a polynomial can be obtained using the two
4593 int ex::degree(const ex & s);
4594 int ex::ldegree(const ex & s);
4597 which also work reliably on non-expanded input polynomials (they even work
4598 on rational functions, returning the asymptotic degree). By definition, the
4599 degree of zero is zero. To extract a coefficient with a certain power from
4600 an expanded polynomial you use
4603 ex ex::coeff(const ex & s, int n);
4606 You can also obtain the leading and trailing coefficients with the methods
4609 ex ex::lcoeff(const ex & s);
4610 ex ex::tcoeff(const ex & s);
4613 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4616 An application is illustrated in the next example, where a multivariate
4617 polynomial is analyzed:
4621 symbol x("x"), y("y");
4622 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4623 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4624 ex Poly = PolyInp.expand();
4626 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4627 cout << "The x^" << i << "-coefficient is "
4628 << Poly.coeff(x,i) << endl;
4630 cout << "As polynomial in y: "
4631 << Poly.collect(y) << endl;
4635 When run, it returns an output in the following fashion:
4638 The x^0-coefficient is y^2+11*y
4639 The x^1-coefficient is 5*y^2-2*y
4640 The x^2-coefficient is -1
4641 The x^3-coefficient is 4*y
4642 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4645 As always, the exact output may vary between different versions of GiNaC
4646 or even from run to run since the internal canonical ordering is not
4647 within the user's sphere of influence.
4649 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4650 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4651 with non-polynomial expressions as they not only work with symbols but with
4652 constants, functions and indexed objects as well:
4656 symbol a("a"), b("b"), c("c"), x("x");
4657 idx i(symbol("i"), 3);
4659 ex e = pow(sin(x) - cos(x), 4);
4660 cout << e.degree(cos(x)) << endl;
4662 cout << e.expand().coeff(sin(x), 3) << endl;
4665 e = indexed(a+b, i) * indexed(b+c, i);
4666 e = e.expand(expand_options::expand_indexed);
4667 cout << e.collect(indexed(b, i)) << endl;
4668 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4673 @subsection Polynomial division
4674 @cindex polynomial division
4677 @cindex pseudo-remainder
4678 @cindex @code{quo()}
4679 @cindex @code{rem()}
4680 @cindex @code{prem()}
4681 @cindex @code{divide()}
4686 ex quo(const ex & a, const ex & b, const ex & x);
4687 ex rem(const ex & a, const ex & b, const ex & x);
4690 compute the quotient and remainder of univariate polynomials in the variable
4691 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4693 The additional function
4696 ex prem(const ex & a, const ex & b, const ex & x);
4699 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4700 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4702 Exact division of multivariate polynomials is performed by the function
4705 bool divide(const ex & a, const ex & b, ex & q);
4708 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4709 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4710 in which case the value of @code{q} is undefined.
4713 @subsection Unit, content and primitive part
4714 @cindex @code{unit()}
4715 @cindex @code{content()}
4716 @cindex @code{primpart()}
4721 ex ex::unit(const ex & x);
4722 ex ex::content(const ex & x);
4723 ex ex::primpart(const ex & x);
4726 return the unit part, content part, and primitive polynomial of a multivariate
4727 polynomial with respect to the variable @samp{x} (the unit part being the sign
4728 of the leading coefficient, the content part being the GCD of the coefficients,
4729 and the primitive polynomial being the input polynomial divided by the unit and
4730 content parts). The product of unit, content, and primitive part is the
4731 original polynomial.
4734 @subsection GCD, LCM and resultant
4737 @cindex @code{gcd()}
4738 @cindex @code{lcm()}
4740 The functions for polynomial greatest common divisor and least common
4741 multiple have the synopsis
4744 ex gcd(const ex & a, const ex & b);
4745 ex lcm(const ex & a, const ex & b);
4748 The functions @code{gcd()} and @code{lcm()} accept two expressions
4749 @code{a} and @code{b} as arguments and return a new expression, their
4750 greatest common divisor or least common multiple, respectively. If the
4751 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4752 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4755 #include <ginac/ginac.h>
4756 using namespace GiNaC;
4760 symbol x("x"), y("y"), z("z");
4761 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4762 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4764 ex P_gcd = gcd(P_a, P_b);
4766 ex P_lcm = lcm(P_a, P_b);
4767 // 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
4772 @cindex @code{resultant()}
4774 The resultant of two expressions only makes sense with polynomials.
4775 It is always computed with respect to a specific symbol within the
4776 expressions. The function has the interface
4779 ex resultant(const ex & a, const ex & b, const ex & s);
4782 Resultants are symmetric in @code{a} and @code{b}. The following example
4783 computes the resultant of two expressions with respect to @code{x} and
4784 @code{y}, respectively:
4787 #include <ginac/ginac.h>
4788 using namespace GiNaC;
4792 symbol x("x"), y("y");
4794 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
4797 r = resultant(e1, e2, x);
4799 r = resultant(e1, e2, y);
4804 @subsection Square-free decomposition
4805 @cindex square-free decomposition
4806 @cindex factorization
4807 @cindex @code{sqrfree()}
4809 GiNaC still lacks proper factorization support. Some form of
4810 factorization is, however, easily implemented by noting that factors
4811 appearing in a polynomial with power two or more also appear in the
4812 derivative and hence can easily be found by computing the GCD of the
4813 original polynomial and its derivatives. Any decent system has an
4814 interface for this so called square-free factorization. So we provide
4817 ex sqrfree(const ex & a, const lst & l = lst());
4819 Here is an example that by the way illustrates how the exact form of the
4820 result may slightly depend on the order of differentiation, calling for
4821 some care with subsequent processing of the result:
4824 symbol x("x"), y("y");
4825 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4827 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4828 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4830 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4831 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4833 cout << sqrfree(BiVarPol) << endl;
4834 // -> depending on luck, any of the above
4837 Note also, how factors with the same exponents are not fully factorized
4841 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4842 @c node-name, next, previous, up
4843 @section Rational expressions
4845 @subsection The @code{normal} method
4846 @cindex @code{normal()}
4847 @cindex simplification
4848 @cindex temporary replacement
4850 Some basic form of simplification of expressions is called for frequently.
4851 GiNaC provides the method @code{.normal()}, which converts a rational function
4852 into an equivalent rational function of the form @samp{numerator/denominator}
4853 where numerator and denominator are coprime. If the input expression is already
4854 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4855 otherwise it performs fraction addition and multiplication.
4857 @code{.normal()} can also be used on expressions which are not rational functions
4858 as it will replace all non-rational objects (like functions or non-integer
4859 powers) by temporary symbols to bring the expression to the domain of rational
4860 functions before performing the normalization, and re-substituting these
4861 symbols afterwards. This algorithm is also available as a separate method
4862 @code{.to_rational()}, described below.
4864 This means that both expressions @code{t1} and @code{t2} are indeed
4865 simplified in this little code snippet:
4870 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4871 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4872 std::cout << "t1 is " << t1.normal() << std::endl;
4873 std::cout << "t2 is " << t2.normal() << std::endl;
4877 Of course this works for multivariate polynomials too, so the ratio of
4878 the sample-polynomials from the section about GCD and LCM above would be
4879 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4882 @subsection Numerator and denominator
4885 @cindex @code{numer()}
4886 @cindex @code{denom()}
4887 @cindex @code{numer_denom()}
4889 The numerator and denominator of an expression can be obtained with
4894 ex ex::numer_denom();
4897 These functions will first normalize the expression as described above and
4898 then return the numerator, denominator, or both as a list, respectively.
4899 If you need both numerator and denominator, calling @code{numer_denom()} is
4900 faster than using @code{numer()} and @code{denom()} separately.
4903 @subsection Converting to a polynomial or rational expression
4904 @cindex @code{to_polynomial()}
4905 @cindex @code{to_rational()}
4907 Some of the methods described so far only work on polynomials or rational
4908 functions. GiNaC provides a way to extend the domain of these functions to
4909 general expressions by using the temporary replacement algorithm described
4910 above. You do this by calling
4913 ex ex::to_polynomial(exmap & m);
4914 ex ex::to_polynomial(lst & l);
4918 ex ex::to_rational(exmap & m);
4919 ex ex::to_rational(lst & l);
4922 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4923 will be filled with the generated temporary symbols and their replacement
4924 expressions in a format that can be used directly for the @code{subs()}
4925 method. It can also already contain a list of replacements from an earlier
4926 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4927 possible to use it on multiple expressions and get consistent results.
4929 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4930 is probably best illustrated with an example:
4934 symbol x("x"), y("y");
4935 ex a = 2*x/sin(x) - y/(3*sin(x));
4939 ex p = a.to_polynomial(lp);
4940 cout << " = " << p << "\n with " << lp << endl;
4941 // = symbol3*symbol2*y+2*symbol2*x
4942 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4945 ex r = a.to_rational(lr);
4946 cout << " = " << r << "\n with " << lr << endl;
4947 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4948 // with @{symbol4==sin(x)@}
4952 The following more useful example will print @samp{sin(x)-cos(x)}:
4957 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4958 ex b = sin(x) + cos(x);
4961 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4962 cout << q.subs(m) << endl;
4967 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4968 @c node-name, next, previous, up
4969 @section Symbolic differentiation
4970 @cindex differentiation
4971 @cindex @code{diff()}
4973 @cindex product rule
4975 GiNaC's objects know how to differentiate themselves. Thus, a
4976 polynomial (class @code{add}) knows that its derivative is the sum of
4977 the derivatives of all the monomials:
4981 symbol x("x"), y("y"), z("z");
4982 ex P = pow(x, 5) + pow(x, 2) + y;
4984 cout << P.diff(x,2) << endl;
4986 cout << P.diff(y) << endl; // 1
4988 cout << P.diff(z) << endl; // 0
4993 If a second integer parameter @var{n} is given, the @code{diff} method
4994 returns the @var{n}th derivative.
4996 If @emph{every} object and every function is told what its derivative
4997 is, all derivatives of composed objects can be calculated using the
4998 chain rule and the product rule. Consider, for instance the expression
4999 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5000 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5001 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5002 out that the composition is the generating function for Euler Numbers,
5003 i.e. the so called @var{n}th Euler number is the coefficient of
5004 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5005 identity to code a function that generates Euler numbers in just three
5008 @cindex Euler numbers
5010 #include <ginac/ginac.h>
5011 using namespace GiNaC;
5013 ex EulerNumber(unsigned n)
5016 const ex generator = pow(cosh(x),-1);
5017 return generator.diff(x,n).subs(x==0);
5022 for (unsigned i=0; i<11; i+=2)
5023 std::cout << EulerNumber(i) << std::endl;
5028 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5029 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5030 @code{i} by two since all odd Euler numbers vanish anyways.
5033 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5034 @c node-name, next, previous, up
5035 @section Series expansion
5036 @cindex @code{series()}
5037 @cindex Taylor expansion
5038 @cindex Laurent expansion
5039 @cindex @code{pseries} (class)
5040 @cindex @code{Order()}
5042 Expressions know how to expand themselves as a Taylor series or (more
5043 generally) a Laurent series. As in most conventional Computer Algebra
5044 Systems, no distinction is made between those two. There is a class of
5045 its own for storing such series (@code{class pseries}) and a built-in
5046 function (called @code{Order}) for storing the order term of the series.
5047 As a consequence, if you want to work with series, i.e. multiply two
5048 series, you need to call the method @code{ex::series} again to convert
5049 it to a series object with the usual structure (expansion plus order
5050 term). A sample application from special relativity could read:
5053 #include <ginac/ginac.h>
5054 using namespace std;
5055 using namespace GiNaC;
5059 symbol v("v"), c("c");
5061 ex gamma = 1/sqrt(1 - pow(v/c,2));
5062 ex mass_nonrel = gamma.series(v==0, 10);
5064 cout << "the relativistic mass increase with v is " << endl
5065 << mass_nonrel << endl;
5067 cout << "the inverse square of this series is " << endl
5068 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5072 Only calling the series method makes the last output simplify to
5073 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5074 series raised to the power @math{-2}.
5076 @cindex Machin's formula
5077 As another instructive application, let us calculate the numerical
5078 value of Archimedes' constant
5082 (for which there already exists the built-in constant @code{Pi})
5083 using John Machin's amazing formula
5085 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5088 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5090 This equation (and similar ones) were used for over 200 years for
5091 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5092 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5093 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5094 order term with it and the question arises what the system is supposed
5095 to do when the fractions are plugged into that order term. The solution
5096 is to use the function @code{series_to_poly()} to simply strip the order
5100 #include <ginac/ginac.h>
5101 using namespace GiNaC;
5103 ex machin_pi(int degr)
5106 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5107 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5108 -4*pi_expansion.subs(x==numeric(1,239));
5114 using std::cout; // just for fun, another way of...
5115 using std::endl; // ...dealing with this namespace std.
5117 for (int i=2; i<12; i+=2) @{
5118 pi_frac = machin_pi(i);
5119 cout << i << ":\t" << pi_frac << endl
5120 << "\t" << pi_frac.evalf() << endl;
5126 Note how we just called @code{.series(x,degr)} instead of
5127 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5128 method @code{series()}: if the first argument is a symbol the expression
5129 is expanded in that symbol around point @code{0}. When you run this
5130 program, it will type out:
5134 3.1832635983263598326
5135 4: 5359397032/1706489875
5136 3.1405970293260603143
5137 6: 38279241713339684/12184551018734375
5138 3.141621029325034425
5139 8: 76528487109180192540976/24359780855939418203125
5140 3.141591772182177295
5141 10: 327853873402258685803048818236/104359128170408663038552734375
5142 3.1415926824043995174
5146 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5147 @c node-name, next, previous, up
5148 @section Symmetrization
5149 @cindex @code{symmetrize()}
5150 @cindex @code{antisymmetrize()}
5151 @cindex @code{symmetrize_cyclic()}
5156 ex ex::symmetrize(const lst & l);
5157 ex ex::antisymmetrize(const lst & l);
5158 ex ex::symmetrize_cyclic(const lst & l);
5161 symmetrize an expression by returning the sum over all symmetric,
5162 antisymmetric or cyclic permutations of the specified list of objects,
5163 weighted by the number of permutations.
5165 The three additional methods
5168 ex ex::symmetrize();
5169 ex ex::antisymmetrize();
5170 ex ex::symmetrize_cyclic();
5173 symmetrize or antisymmetrize an expression over its free indices.
5175 Symmetrization is most useful with indexed expressions but can be used with
5176 almost any kind of object (anything that is @code{subs()}able):
5180 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5181 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5183 cout << indexed(A, i, j).symmetrize() << endl;
5184 // -> 1/2*A.j.i+1/2*A.i.j
5185 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5186 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5187 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5188 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5192 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5193 @c node-name, next, previous, up
5194 @section Predefined mathematical functions
5196 @subsection Overview
5198 GiNaC contains the following predefined mathematical functions:
5201 @multitable @columnfractions .30 .70
5202 @item @strong{Name} @tab @strong{Function}
5205 @cindex @code{abs()}
5206 @item @code{csgn(x)}
5208 @cindex @code{conjugate()}
5209 @item @code{conjugate(x)}
5210 @tab complex conjugation
5211 @cindex @code{csgn()}
5212 @item @code{sqrt(x)}
5213 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5214 @cindex @code{sqrt()}
5217 @cindex @code{sin()}
5220 @cindex @code{cos()}
5223 @cindex @code{tan()}
5224 @item @code{asin(x)}
5226 @cindex @code{asin()}
5227 @item @code{acos(x)}
5229 @cindex @code{acos()}
5230 @item @code{atan(x)}
5231 @tab inverse tangent
5232 @cindex @code{atan()}
5233 @item @code{atan2(y, x)}
5234 @tab inverse tangent with two arguments
5235 @item @code{sinh(x)}
5236 @tab hyperbolic sine
5237 @cindex @code{sinh()}
5238 @item @code{cosh(x)}
5239 @tab hyperbolic cosine
5240 @cindex @code{cosh()}
5241 @item @code{tanh(x)}
5242 @tab hyperbolic tangent
5243 @cindex @code{tanh()}
5244 @item @code{asinh(x)}
5245 @tab inverse hyperbolic sine
5246 @cindex @code{asinh()}
5247 @item @code{acosh(x)}
5248 @tab inverse hyperbolic cosine
5249 @cindex @code{acosh()}
5250 @item @code{atanh(x)}
5251 @tab inverse hyperbolic tangent
5252 @cindex @code{atanh()}
5254 @tab exponential function
5255 @cindex @code{exp()}
5257 @tab natural logarithm
5258 @cindex @code{log()}
5261 @cindex @code{Li2()}
5262 @item @code{Li(m, x)}
5263 @tab classical polylogarithm as well as multiple polylogarithm
5265 @item @code{S(n, p, x)}
5266 @tab Nielsen's generalized polylogarithm
5268 @item @code{H(m, x)}
5269 @tab harmonic polylogarithm
5271 @item @code{zeta(m)}
5272 @tab Riemann's zeta function as well as multiple zeta value
5273 @cindex @code{zeta()}
5274 @item @code{zeta(m, s)}
5275 @tab alternating Euler sum
5276 @cindex @code{zeta()}
5277 @item @code{zetaderiv(n, x)}
5278 @tab derivatives of Riemann's zeta function
5279 @item @code{tgamma(x)}
5281 @cindex @code{tgamma()}
5282 @cindex gamma function
5283 @item @code{lgamma(x)}
5284 @tab logarithm of gamma function
5285 @cindex @code{lgamma()}
5286 @item @code{beta(x, y)}
5287 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5288 @cindex @code{beta()}
5290 @tab psi (digamma) function
5291 @cindex @code{psi()}
5292 @item @code{psi(n, x)}
5293 @tab derivatives of psi function (polygamma functions)
5294 @item @code{factorial(n)}
5295 @tab factorial function @math{n!}
5296 @cindex @code{factorial()}
5297 @item @code{binomial(n, k)}
5298 @tab binomial coefficients
5299 @cindex @code{binomial()}
5300 @item @code{Order(x)}
5301 @tab order term function in truncated power series
5302 @cindex @code{Order()}
5307 For functions that have a branch cut in the complex plane GiNaC follows
5308 the conventions for C++ as defined in the ANSI standard as far as
5309 possible. In particular: the natural logarithm (@code{log}) and the
5310 square root (@code{sqrt}) both have their branch cuts running along the
5311 negative real axis where the points on the axis itself belong to the
5312 upper part (i.e. continuous with quadrant II). The inverse
5313 trigonometric and hyperbolic functions are not defined for complex
5314 arguments by the C++ standard, however. In GiNaC we follow the
5315 conventions used by CLN, which in turn follow the carefully designed
5316 definitions in the Common Lisp standard. It should be noted that this
5317 convention is identical to the one used by the C99 standard and by most
5318 serious CAS. It is to be expected that future revisions of the C++
5319 standard incorporate these functions in the complex domain in a manner
5320 compatible with C99.
5322 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5323 @c node-name, next, previous, up
5324 @subsection Multiple polylogarithms
5326 @cindex polylogarithm
5327 @cindex Nielsen's generalized polylogarithm
5328 @cindex harmonic polylogarithm
5329 @cindex multiple zeta value
5330 @cindex alternating Euler sum
5331 @cindex multiple polylogarithm
5333 The multiple polylogarithm is the most generic member of a family of functions,
5334 to which others like the harmonic polylogarithm, Nielsen's generalized
5335 polylogarithm and the multiple zeta value belong.
5336 Everyone of these functions can also be written as a multiple polylogarithm with specific
5337 parameters. This whole family of functions is therefore often referred to simply as
5338 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5340 To facilitate the discussion of these functions we distinguish between indices and
5341 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5342 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5344 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5345 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5346 for the argument @code{x} as well.
5347 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5348 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5349 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5350 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5351 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5353 The functions print in LaTeX format as
5355 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5361 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5364 $\zeta(m_1,m_2,\ldots,m_k)$.
5366 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5367 are printed with a line above, e.g.
5369 $\zeta(5,\overline{2})$.
5371 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5373 Definitions and analytical as well as numerical properties of multiple polylogarithms
5374 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5375 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5376 except for a few differences which will be explicitly stated in the following.
5378 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5379 that the indices and arguments are understood to be in the same order as in which they appear in
5380 the series representation. This means
5382 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5385 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5388 $\zeta(1,2)$ evaluates to infinity.
5390 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5393 The functions only evaluate if the indices are integers greater than zero, except for the indices
5394 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5395 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5396 @code{zeta(lst(3,4), lst(-1,1))} means
5398 $\zeta(\overline{3},4)$.
5400 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5401 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5402 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
5403 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5404 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5405 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5406 evaluates also for negative integers and positive even integers. For example:
5409 > Li(@{3,1@},@{x,1@});
5412 -zeta(@{3,2@},@{-1,-1@})
5417 It is easy to tell for a given function into which other function it can be rewritten, may
5418 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5419 with negative indices or trailing zeros (the example above gives a hint). Signs can
5420 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5421 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5422 @code{Li} (@code{eval()} already cares for the possible downgrade):
5425 > convert_H_to_Li(@{0,-2,-1,3@},x);
5426 Li(@{3,1,3@},@{-x,1,-1@})
5427 > convert_H_to_Li(@{2,-1,0@},x);
5428 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5431 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5432 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5437 $x_1x_2\cdots x_i < 1$ holds.
5443 > evalf(zeta(@{3,1,3,1@}));
5444 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5447 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5448 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5450 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5455 In long expressions this helps a lot with debugging, because you can easily spot
5456 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5457 cancellations of divergencies happen.
5459 Useful publications:
5461 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5462 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5464 @cite{Harmonic Polylogarithms},
5465 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5467 @cite{Special Values of Multiple Polylogarithms},
5468 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5470 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5471 @c node-name, next, previous, up
5472 @section Complex Conjugation
5474 @cindex @code{conjugate()}
5482 returns the complex conjugate of the expression. For all built-in functions and objects the
5483 conjugation gives the expected results:
5487 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5491 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5492 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5493 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5494 // -> -gamma5*gamma~b*gamma~a
5498 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5499 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5500 arguments. This is the default strategy. If you want to define your own functions and want to
5501 change this behavior, you have to supply a specialized conjugation method for your function
5502 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5504 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5505 @c node-name, next, previous, up
5506 @section Solving Linear Systems of Equations
5507 @cindex @code{lsolve()}
5509 The function @code{lsolve()} provides a convenient wrapper around some
5510 matrix operations that comes in handy when a system of linear equations
5514 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5517 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5518 @code{relational}) while @code{symbols} is a @code{lst} of
5519 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5522 It returns the @code{lst} of solutions as an expression. As an example,
5523 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5527 symbol a("a"), b("b"), x("x"), y("y");
5529 eqns = a*x+b*y==3, x-y==b;
5531 cout << lsolve(eqns, vars) << endl;
5532 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5535 When the linear equations @code{eqns} are underdetermined, the solution
5536 will contain one or more tautological entries like @code{x==x},
5537 depending on the rank of the system. When they are overdetermined, the
5538 solution will be an empty @code{lst}. Note the third optional parameter
5539 to @code{lsolve()}: it accepts the same parameters as
5540 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5544 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5545 @c node-name, next, previous, up
5546 @section Input and output of expressions
5549 @subsection Expression output
5551 @cindex output of expressions
5553 Expressions can simply be written to any stream:
5558 ex e = 4.5*I+pow(x,2)*3/2;
5559 cout << e << endl; // prints '4.5*I+3/2*x^2'
5563 The default output format is identical to the @command{ginsh} input syntax and
5564 to that used by most computer algebra systems, but not directly pastable
5565 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5566 is printed as @samp{x^2}).
5568 It is possible to print expressions in a number of different formats with
5569 a set of stream manipulators;
5572 std::ostream & dflt(std::ostream & os);
5573 std::ostream & latex(std::ostream & os);
5574 std::ostream & tree(std::ostream & os);
5575 std::ostream & csrc(std::ostream & os);
5576 std::ostream & csrc_float(std::ostream & os);
5577 std::ostream & csrc_double(std::ostream & os);
5578 std::ostream & csrc_cl_N(std::ostream & os);
5579 std::ostream & index_dimensions(std::ostream & os);
5580 std::ostream & no_index_dimensions(std::ostream & os);
5583 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5584 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5585 @code{print_csrc()} functions, respectively.
5588 All manipulators affect the stream state permanently. To reset the output
5589 format to the default, use the @code{dflt} manipulator:
5593 cout << latex; // all output to cout will be in LaTeX format from now on
5594 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5595 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5596 cout << dflt; // revert to default output format
5597 cout << e << endl; // prints '4.5*I+3/2*x^2'
5601 If you don't want to affect the format of the stream you're working with,
5602 you can output to a temporary @code{ostringstream} like this:
5607 s << latex << e; // format of cout remains unchanged
5608 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5613 @cindex @code{csrc_float}
5614 @cindex @code{csrc_double}
5615 @cindex @code{csrc_cl_N}
5616 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5617 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5618 format that can be directly used in a C or C++ program. The three possible
5619 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5620 classes provided by the CLN library):
5624 cout << "f = " << csrc_float << e << ";\n";
5625 cout << "d = " << csrc_double << e << ";\n";
5626 cout << "n = " << csrc_cl_N << e << ";\n";
5630 The above example will produce (note the @code{x^2} being converted to
5634 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5635 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5636 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5640 The @code{tree} manipulator allows dumping the internal structure of an
5641 expression for debugging purposes:
5652 add, hash=0x0, flags=0x3, nops=2
5653 power, hash=0x0, flags=0x3, nops=2
5654 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5655 2 (numeric), hash=0x6526b0fa, flags=0xf
5656 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5659 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5663 @cindex @code{latex}
5664 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5665 It is rather similar to the default format but provides some braces needed
5666 by LaTeX for delimiting boxes and also converts some common objects to
5667 conventional LaTeX names. It is possible to give symbols a special name for
5668 LaTeX output by supplying it as a second argument to the @code{symbol}
5671 For example, the code snippet
5675 symbol x("x", "\\circ");
5676 ex e = lgamma(x).series(x==0,3);
5677 cout << latex << e << endl;
5684 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5687 @cindex @code{index_dimensions}
5688 @cindex @code{no_index_dimensions}
5689 Index dimensions are normally hidden in the output. To make them visible, use
5690 the @code{index_dimensions} manipulator. The dimensions will be written in
5691 square brackets behind each index value in the default and LaTeX output
5696 symbol x("x"), y("y");
5697 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5698 ex e = indexed(x, mu) * indexed(y, nu);
5701 // prints 'x~mu*y~nu'
5702 cout << index_dimensions << e << endl;
5703 // prints 'x~mu[4]*y~nu[4]'
5704 cout << no_index_dimensions << e << endl;
5705 // prints 'x~mu*y~nu'
5710 @cindex Tree traversal
5711 If you need any fancy special output format, e.g. for interfacing GiNaC
5712 with other algebra systems or for producing code for different
5713 programming languages, you can always traverse the expression tree yourself:
5716 static void my_print(const ex & e)
5718 if (is_a<function>(e))
5719 cout << ex_to<function>(e).get_name();
5721 cout << ex_to<basic>(e).class_name();
5723 size_t n = e.nops();
5725 for (size_t i=0; i<n; i++) @{
5737 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5745 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5746 symbol(y))),numeric(-2)))
5749 If you need an output format that makes it possible to accurately
5750 reconstruct an expression by feeding the output to a suitable parser or
5751 object factory, you should consider storing the expression in an
5752 @code{archive} object and reading the object properties from there.
5753 See the section on archiving for more information.
5756 @subsection Expression input
5757 @cindex input of expressions
5759 GiNaC provides no way to directly read an expression from a stream because
5760 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5761 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5762 @code{y} you defined in your program and there is no way to specify the
5763 desired symbols to the @code{>>} stream input operator.
5765 Instead, GiNaC lets you construct an expression from a string, specifying the
5766 list of symbols to be used:
5770 symbol x("x"), y("y");
5771 ex e("2*x+sin(y)", lst(x, y));
5775 The input syntax is the same as that used by @command{ginsh} and the stream
5776 output operator @code{<<}. The symbols in the string are matched by name to
5777 the symbols in the list and if GiNaC encounters a symbol not specified in
5778 the list it will throw an exception.
5780 With this constructor, it's also easy to implement interactive GiNaC programs:
5785 #include <stdexcept>
5786 #include <ginac/ginac.h>
5787 using namespace std;
5788 using namespace GiNaC;
5795 cout << "Enter an expression containing 'x': ";
5800 cout << "The derivative of " << e << " with respect to x is ";
5801 cout << e.diff(x) << ".\n";
5802 @} catch (exception &p) @{
5803 cerr << p.what() << endl;
5809 @subsection Archiving
5810 @cindex @code{archive} (class)
5813 GiNaC allows creating @dfn{archives} of expressions which can be stored
5814 to or retrieved from files. To create an archive, you declare an object
5815 of class @code{archive} and archive expressions in it, giving each
5816 expression a unique name:
5820 using namespace std;
5821 #include <ginac/ginac.h>
5822 using namespace GiNaC;
5826 symbol x("x"), y("y"), z("z");
5828 ex foo = sin(x + 2*y) + 3*z + 41;
5832 a.archive_ex(foo, "foo");
5833 a.archive_ex(bar, "the second one");
5837 The archive can then be written to a file:
5841 ofstream out("foobar.gar");
5847 The file @file{foobar.gar} contains all information that is needed to
5848 reconstruct the expressions @code{foo} and @code{bar}.
5850 @cindex @command{viewgar}
5851 The tool @command{viewgar} that comes with GiNaC can be used to view
5852 the contents of GiNaC archive files:
5855 $ viewgar foobar.gar
5856 foo = 41+sin(x+2*y)+3*z
5857 the second one = 42+sin(x+2*y)+3*z
5860 The point of writing archive files is of course that they can later be
5866 ifstream in("foobar.gar");
5871 And the stored expressions can be retrieved by their name:
5878 ex ex1 = a2.unarchive_ex(syms, "foo");
5879 ex ex2 = a2.unarchive_ex(syms, "the second one");
5881 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5882 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5883 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5887 Note that you have to supply a list of the symbols which are to be inserted
5888 in the expressions. Symbols in archives are stored by their name only and
5889 if you don't specify which symbols you have, unarchiving the expression will
5890 create new symbols with that name. E.g. if you hadn't included @code{x} in
5891 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5892 have had no effect because the @code{x} in @code{ex1} would have been a
5893 different symbol than the @code{x} which was defined at the beginning of
5894 the program, although both would appear as @samp{x} when printed.
5896 You can also use the information stored in an @code{archive} object to
5897 output expressions in a format suitable for exact reconstruction. The
5898 @code{archive} and @code{archive_node} classes have a couple of member
5899 functions that let you access the stored properties:
5902 static void my_print2(const archive_node & n)
5905 n.find_string("class", class_name);
5906 cout << class_name << "(";
5908 archive_node::propinfovector p;
5909 n.get_properties(p);
5911 size_t num = p.size();
5912 for (size_t i=0; i<num; i++) @{
5913 const string &name = p[i].name;
5914 if (name == "class")
5916 cout << name << "=";
5918 unsigned count = p[i].count;
5922 for (unsigned j=0; j<count; j++) @{
5923 switch (p[i].type) @{
5924 case archive_node::PTYPE_BOOL: @{
5926 n.find_bool(name, x, j);
5927 cout << (x ? "true" : "false");
5930 case archive_node::PTYPE_UNSIGNED: @{
5932 n.find_unsigned(name, x, j);
5936 case archive_node::PTYPE_STRING: @{
5938 n.find_string(name, x, j);
5939 cout << '\"' << x << '\"';
5942 case archive_node::PTYPE_NODE: @{
5943 const archive_node &x = n.find_ex_node(name, j);
5965 ex e = pow(2, x) - y;
5967 my_print2(ar.get_top_node(0)); cout << endl;
5975 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5976 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5977 overall_coeff=numeric(number="0"))
5980 Be warned, however, that the set of properties and their meaning for each
5981 class may change between GiNaC versions.
5984 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5985 @c node-name, next, previous, up
5986 @chapter Extending GiNaC
5988 By reading so far you should have gotten a fairly good understanding of
5989 GiNaC's design patterns. From here on you should start reading the
5990 sources. All we can do now is issue some recommendations how to tackle
5991 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5992 develop some useful extension please don't hesitate to contact the GiNaC
5993 authors---they will happily incorporate them into future versions.
5996 * What does not belong into GiNaC:: What to avoid.
5997 * Symbolic functions:: Implementing symbolic functions.
5998 * Printing:: Adding new output formats.
5999 * Structures:: Defining new algebraic classes (the easy way).
6000 * Adding classes:: Defining new algebraic classes (the hard way).
6004 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6005 @c node-name, next, previous, up
6006 @section What doesn't belong into GiNaC
6008 @cindex @command{ginsh}
6009 First of all, GiNaC's name must be read literally. It is designed to be
6010 a library for use within C++. The tiny @command{ginsh} accompanying
6011 GiNaC makes this even more clear: it doesn't even attempt to provide a
6012 language. There are no loops or conditional expressions in
6013 @command{ginsh}, it is merely a window into the library for the
6014 programmer to test stuff (or to show off). Still, the design of a
6015 complete CAS with a language of its own, graphical capabilities and all
6016 this on top of GiNaC is possible and is without doubt a nice project for
6019 There are many built-in functions in GiNaC that do not know how to
6020 evaluate themselves numerically to a precision declared at runtime
6021 (using @code{Digits}). Some may be evaluated at certain points, but not
6022 generally. This ought to be fixed. However, doing numerical
6023 computations with GiNaC's quite abstract classes is doomed to be
6024 inefficient. For this purpose, the underlying foundation classes
6025 provided by CLN are much better suited.
6028 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6029 @c node-name, next, previous, up
6030 @section Symbolic functions
6032 The easiest and most instructive way to start extending GiNaC is probably to
6033 create your own symbolic functions. These are implemented with the help of
6034 two preprocessor macros:
6036 @cindex @code{DECLARE_FUNCTION}
6037 @cindex @code{REGISTER_FUNCTION}
6039 DECLARE_FUNCTION_<n>P(<name>)
6040 REGISTER_FUNCTION(<name>, <options>)
6043 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6044 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6045 parameters of type @code{ex} and returns a newly constructed GiNaC
6046 @code{function} object that represents your function.
6048 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6049 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6050 set of options that associate the symbolic function with C++ functions you
6051 provide to implement the various methods such as evaluation, derivative,
6052 series expansion etc. They also describe additional attributes the function
6053 might have, such as symmetry and commutation properties, and a name for
6054 LaTeX output. Multiple options are separated by the member access operator
6055 @samp{.} and can be given in an arbitrary order.
6057 (By the way: in case you are worrying about all the macros above we can
6058 assure you that functions are GiNaC's most macro-intense classes. We have
6059 done our best to avoid macros where we can.)
6061 @subsection A minimal example
6063 Here is an example for the implementation of a function with two arguments
6064 that is not further evaluated:
6067 DECLARE_FUNCTION_2P(myfcn)
6069 REGISTER_FUNCTION(myfcn, dummy())
6072 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6073 in algebraic expressions:
6079 ex e = 2*myfcn(42, 1+3*x) - x;
6081 // prints '2*myfcn(42,1+3*x)-x'
6086 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6087 "no options". A function with no options specified merely acts as a kind of
6088 container for its arguments. It is a pure "dummy" function with no associated
6089 logic (which is, however, sometimes perfectly sufficient).
6091 Let's now have a look at the implementation of GiNaC's cosine function for an
6092 example of how to make an "intelligent" function.
6094 @subsection The cosine function
6096 The GiNaC header file @file{inifcns.h} contains the line
6099 DECLARE_FUNCTION_1P(cos)
6102 which declares to all programs using GiNaC that there is a function @samp{cos}
6103 that takes one @code{ex} as an argument. This is all they need to know to use
6104 this function in expressions.
6106 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6107 is its @code{REGISTER_FUNCTION} line:
6110 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6111 evalf_func(cos_evalf).
6112 derivative_func(cos_deriv).
6113 latex_name("\\cos"));
6116 There are four options defined for the cosine function. One of them
6117 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6118 other three indicate the C++ functions in which the "brains" of the cosine
6119 function are defined.
6121 @cindex @code{hold()}
6123 The @code{eval_func()} option specifies the C++ function that implements
6124 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6125 the same number of arguments as the associated symbolic function (one in this
6126 case) and returns the (possibly transformed or in some way simplified)
6127 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6128 of the automatic evaluation process). If no (further) evaluation is to take
6129 place, the @code{eval_func()} function must return the original function
6130 with @code{.hold()}, to avoid a potential infinite recursion. If your
6131 symbolic functions produce a segmentation fault or stack overflow when
6132 using them in expressions, you are probably missing a @code{.hold()}
6135 The @code{eval_func()} function for the cosine looks something like this
6136 (actually, it doesn't look like this at all, but it should give you an idea
6140 static ex cos_eval(const ex & x)
6142 if ("x is a multiple of 2*Pi")
6144 else if ("x is a multiple of Pi")
6146 else if ("x is a multiple of Pi/2")
6150 else if ("x has the form 'acos(y)'")
6152 else if ("x has the form 'asin(y)'")
6157 return cos(x).hold();
6161 This function is called every time the cosine is used in a symbolic expression:
6167 // this calls cos_eval(Pi), and inserts its return value into
6168 // the actual expression
6175 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6176 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6177 symbolic transformation can be done, the unmodified function is returned
6178 with @code{.hold()}.
6180 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6181 The user has to call @code{evalf()} for that. This is implemented in a
6185 static ex cos_evalf(const ex & x)
6187 if (is_a<numeric>(x))
6188 return cos(ex_to<numeric>(x));
6190 return cos(x).hold();
6194 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6195 in this case the @code{cos()} function for @code{numeric} objects, which in
6196 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6197 isn't really needed here, but reminds us that the corresponding @code{eval()}
6198 function would require it in this place.
6200 Differentiation will surely turn up and so we need to tell @code{cos}
6201 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6202 instance, are then handled automatically by @code{basic::diff} and
6206 static ex cos_deriv(const ex & x, unsigned diff_param)
6212 @cindex product rule
6213 The second parameter is obligatory but uninteresting at this point. It
6214 specifies which parameter to differentiate in a partial derivative in
6215 case the function has more than one parameter, and its main application
6216 is for correct handling of the chain rule.
6218 An implementation of the series expansion is not needed for @code{cos()} as
6219 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6220 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6221 the other hand, does have poles and may need to do Laurent expansion:
6224 static ex tan_series(const ex & x, const relational & rel,
6225 int order, unsigned options)
6227 // Find the actual expansion point
6228 const ex x_pt = x.subs(rel);
6230 if ("x_pt is not an odd multiple of Pi/2")
6231 throw do_taylor(); // tell function::series() to do Taylor expansion
6233 // On a pole, expand sin()/cos()
6234 return (sin(x)/cos(x)).series(rel, order+2, options);
6238 The @code{series()} implementation of a function @emph{must} return a
6239 @code{pseries} object, otherwise your code will crash.
6241 @subsection Function options
6243 GiNaC functions understand several more options which are always
6244 specified as @code{.option(params)}. None of them are required, but you
6245 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6246 is a do-nothing option called @code{dummy()} which you can use to define
6247 functions without any special options.
6250 eval_func(<C++ function>)
6251 evalf_func(<C++ function>)
6252 derivative_func(<C++ function>)
6253 series_func(<C++ function>)
6254 conjugate_func(<C++ function>)
6257 These specify the C++ functions that implement symbolic evaluation,
6258 numeric evaluation, partial derivatives, and series expansion, respectively.
6259 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6260 @code{diff()} and @code{series()}.
6262 The @code{eval_func()} function needs to use @code{.hold()} if no further
6263 automatic evaluation is desired or possible.
6265 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6266 expansion, which is correct if there are no poles involved. If the function
6267 has poles in the complex plane, the @code{series_func()} needs to check
6268 whether the expansion point is on a pole and fall back to Taylor expansion
6269 if it isn't. Otherwise, the pole usually needs to be regularized by some
6270 suitable transformation.
6273 latex_name(const string & n)
6276 specifies the LaTeX code that represents the name of the function in LaTeX
6277 output. The default is to put the function name in an @code{\mbox@{@}}.
6280 do_not_evalf_params()
6283 This tells @code{evalf()} to not recursively evaluate the parameters of the
6284 function before calling the @code{evalf_func()}.
6287 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6290 This allows you to explicitly specify the commutation properties of the
6291 function (@xref{Non-commutative objects}, for an explanation of
6292 (non)commutativity in GiNaC). For example, you can use
6293 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6294 GiNaC treat your function like a matrix. By default, functions inherit the
6295 commutation properties of their first argument.
6298 set_symmetry(const symmetry & s)
6301 specifies the symmetry properties of the function with respect to its
6302 arguments. @xref{Indexed objects}, for an explanation of symmetry
6303 specifications. GiNaC will automatically rearrange the arguments of
6304 symmetric functions into a canonical order.
6306 Sometimes you may want to have finer control over how functions are
6307 displayed in the output. For example, the @code{abs()} function prints
6308 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6309 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6313 print_func<C>(<C++ function>)
6316 option which is explained in the next section.
6318 @subsection Functions with a variable number of arguments
6320 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6321 functions with a fixed number of arguments. Sometimes, though, you may need
6322 to have a function that accepts a variable number of expressions. One way to
6323 accomplish this is to pass variable-length lists as arguments. The
6324 @code{Li()} function uses this method for multiple polylogarithms.
6326 It is also possible to define functions that accept a different number of
6327 parameters under the same function name, such as the @code{psi()} function
6328 which can be called either as @code{psi(z)} (the digamma function) or as
6329 @code{psi(n, z)} (polygamma functions). These are actually two different
6330 functions in GiNaC that, however, have the same name. Defining such
6331 functions is not possible with the macros but requires manually fiddling
6332 with GiNaC internals. If you are interested, please consult the GiNaC source
6333 code for the @code{psi()} function (@file{inifcns.h} and
6334 @file{inifcns_gamma.cpp}).
6337 @node Printing, Structures, Symbolic functions, Extending GiNaC
6338 @c node-name, next, previous, up
6339 @section GiNaC's expression output system
6341 GiNaC allows the output of expressions in a variety of different formats
6342 (@pxref{Input/Output}). This section will explain how expression output
6343 is implemented internally, and how to define your own output formats or
6344 change the output format of built-in algebraic objects. You will also want
6345 to read this section if you plan to write your own algebraic classes or
6348 @cindex @code{print_context} (class)
6349 @cindex @code{print_dflt} (class)
6350 @cindex @code{print_latex} (class)
6351 @cindex @code{print_tree} (class)
6352 @cindex @code{print_csrc} (class)
6353 All the different output formats are represented by a hierarchy of classes
6354 rooted in the @code{print_context} class, defined in the @file{print.h}
6359 the default output format
6361 output in LaTeX mathematical mode
6363 a dump of the internal expression structure (for debugging)
6365 the base class for C source output
6366 @item print_csrc_float
6367 C source output using the @code{float} type
6368 @item print_csrc_double
6369 C source output using the @code{double} type
6370 @item print_csrc_cl_N
6371 C source output using CLN types
6374 The @code{print_context} base class provides two public data members:
6386 @code{s} is a reference to the stream to output to, while @code{options}
6387 holds flags and modifiers. Currently, there is only one flag defined:
6388 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6389 to print the index dimension which is normally hidden.
6391 When you write something like @code{std::cout << e}, where @code{e} is
6392 an object of class @code{ex}, GiNaC will construct an appropriate
6393 @code{print_context} object (of a class depending on the selected output
6394 format), fill in the @code{s} and @code{options} members, and call
6396 @cindex @code{print()}
6398 void ex::print(const print_context & c, unsigned level = 0) const;
6401 which in turn forwards the call to the @code{print()} method of the
6402 top-level algebraic object contained in the expression.
6404 Unlike other methods, GiNaC classes don't usually override their
6405 @code{print()} method to implement expression output. Instead, the default
6406 implementation @code{basic::print(c, level)} performs a run-time double
6407 dispatch to a function selected by the dynamic type of the object and the
6408 passed @code{print_context}. To this end, GiNaC maintains a separate method
6409 table for each class, similar to the virtual function table used for ordinary
6410 (single) virtual function dispatch.
6412 The method table contains one slot for each possible @code{print_context}
6413 type, indexed by the (internally assigned) serial number of the type. Slots
6414 may be empty, in which case GiNaC will retry the method lookup with the
6415 @code{print_context} object's parent class, possibly repeating the process
6416 until it reaches the @code{print_context} base class. If there's still no
6417 method defined, the method table of the algebraic object's parent class
6418 is consulted, and so on, until a matching method is found (eventually it
6419 will reach the combination @code{basic/print_context}, which prints the
6420 object's class name enclosed in square brackets).
6422 You can think of the print methods of all the different classes and output
6423 formats as being arranged in a two-dimensional matrix with one axis listing
6424 the algebraic classes and the other axis listing the @code{print_context}
6427 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6428 to implement printing, but then they won't get any of the benefits of the
6429 double dispatch mechanism (such as the ability for derived classes to
6430 inherit only certain print methods from its parent, or the replacement of
6431 methods at run-time).
6433 @subsection Print methods for classes
6435 The method table for a class is set up either in the definition of the class,
6436 by passing the appropriate @code{print_func<C>()} option to
6437 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6438 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6439 can also be used to override existing methods dynamically.
6441 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6442 be a member function of the class (or one of its parent classes), a static
6443 member function, or an ordinary (global) C++ function. The @code{C} template
6444 parameter specifies the appropriate @code{print_context} type for which the
6445 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6446 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6447 the class is the one being implemented by
6448 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6450 For print methods that are member functions, their first argument must be of
6451 a type convertible to a @code{const C &}, and the second argument must be an
6454 For static members and global functions, the first argument must be of a type
6455 convertible to a @code{const T &}, the second argument must be of a type
6456 convertible to a @code{const C &}, and the third argument must be an
6457 @code{unsigned}. A global function will, of course, not have access to
6458 private and protected members of @code{T}.
6460 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6461 and @code{basic::print()}) is used for proper parenthesizing of the output
6462 (and by @code{print_tree} for proper indentation). It can be used for similar
6463 purposes if you write your own output formats.
6465 The explanations given above may seem complicated, but in practice it's
6466 really simple, as shown in the following example. Suppose that we want to
6467 display exponents in LaTeX output not as superscripts but with little
6468 upwards-pointing arrows. This can be achieved in the following way:
6471 void my_print_power_as_latex(const power & p,
6472 const print_latex & c,
6475 // get the precedence of the 'power' class
6476 unsigned power_prec = p.precedence();
6478 // if the parent operator has the same or a higher precedence
6479 // we need parentheses around the power
6480 if (level >= power_prec)
6483 // print the basis and exponent, each enclosed in braces, and
6484 // separated by an uparrow
6486 p.op(0).print(c, power_prec);
6487 c.s << "@}\\uparrow@{";
6488 p.op(1).print(c, power_prec);
6491 // don't forget the closing parenthesis
6492 if (level >= power_prec)
6498 // a sample expression
6499 symbol x("x"), y("y");
6500 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6502 // switch to LaTeX mode
6505 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6508 // now we replace the method for the LaTeX output of powers with
6510 set_print_func<power, print_latex>(my_print_power_as_latex);
6512 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6522 The first argument of @code{my_print_power_as_latex} could also have been
6523 a @code{const basic &}, the second one a @code{const print_context &}.
6526 The above code depends on @code{mul} objects converting their operands to
6527 @code{power} objects for the purpose of printing.
6530 The output of products including negative powers as fractions is also
6531 controlled by the @code{mul} class.
6534 The @code{power/print_latex} method provided by GiNaC prints square roots
6535 using @code{\sqrt}, but the above code doesn't.
6539 It's not possible to restore a method table entry to its previous or default
6540 value. Once you have called @code{set_print_func()}, you can only override
6541 it with another call to @code{set_print_func()}, but you can't easily go back
6542 to the default behavior again (you can, of course, dig around in the GiNaC
6543 sources, find the method that is installed at startup
6544 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6545 one; that is, after you circumvent the C++ member access control@dots{}).
6547 @subsection Print methods for functions
6549 Symbolic functions employ a print method dispatch mechanism similar to the
6550 one used for classes. The methods are specified with @code{print_func<C>()}
6551 function options. If you don't specify any special print methods, the function
6552 will be printed with its name (or LaTeX name, if supplied), followed by a
6553 comma-separated list of arguments enclosed in parentheses.
6555 For example, this is what GiNaC's @samp{abs()} function is defined like:
6558 static ex abs_eval(const ex & arg) @{ ... @}
6559 static ex abs_evalf(const ex & arg) @{ ... @}
6561 static void abs_print_latex(const ex & arg, const print_context & c)
6563 c.s << "@{|"; arg.print(c); c.s << "|@}";
6566 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6568 c.s << "fabs("; arg.print(c); c.s << ")";
6571 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6572 evalf_func(abs_evalf).
6573 print_func<print_latex>(abs_print_latex).
6574 print_func<print_csrc_float>(abs_print_csrc_float).
6575 print_func<print_csrc_double>(abs_print_csrc_float));
6578 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6579 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6581 There is currently no equivalent of @code{set_print_func()} for functions.
6583 @subsection Adding new output formats
6585 Creating a new output format involves subclassing @code{print_context},
6586 which is somewhat similar to adding a new algebraic class
6587 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6588 that needs to go into the class definition, and a corresponding macro
6589 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6590 Every @code{print_context} class needs to provide a default constructor
6591 and a constructor from an @code{std::ostream} and an @code{unsigned}
6594 Here is an example for a user-defined @code{print_context} class:
6597 class print_myformat : public print_dflt
6599 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6601 print_myformat(std::ostream & os, unsigned opt = 0)
6602 : print_dflt(os, opt) @{@}
6605 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6607 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6610 That's all there is to it. None of the actual expression output logic is
6611 implemented in this class. It merely serves as a selector for choosing
6612 a particular format. The algorithms for printing expressions in the new
6613 format are implemented as print methods, as described above.
6615 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6616 exactly like GiNaC's default output format:
6621 ex e = pow(x, 2) + 1;
6623 // this prints "1+x^2"
6626 // this also prints "1+x^2"
6627 e.print(print_myformat()); cout << endl;
6633 To fill @code{print_myformat} with life, we need to supply appropriate
6634 print methods with @code{set_print_func()}, like this:
6637 // This prints powers with '**' instead of '^'. See the LaTeX output
6638 // example above for explanations.
6639 void print_power_as_myformat(const power & p,
6640 const print_myformat & c,
6643 unsigned power_prec = p.precedence();
6644 if (level >= power_prec)
6646 p.op(0).print(c, power_prec);
6648 p.op(1).print(c, power_prec);
6649 if (level >= power_prec)
6655 // install a new print method for power objects
6656 set_print_func<power, print_myformat>(print_power_as_myformat);
6658 // now this prints "1+x**2"
6659 e.print(print_myformat()); cout << endl;
6661 // but the default format is still "1+x^2"
6667 @node Structures, Adding classes, Printing, Extending GiNaC
6668 @c node-name, next, previous, up
6671 If you are doing some very specialized things with GiNaC, or if you just
6672 need some more organized way to store data in your expressions instead of
6673 anonymous lists, you may want to implement your own algebraic classes.
6674 ('algebraic class' means any class directly or indirectly derived from
6675 @code{basic} that can be used in GiNaC expressions).
6677 GiNaC offers two ways of accomplishing this: either by using the
6678 @code{structure<T>} template class, or by rolling your own class from
6679 scratch. This section will discuss the @code{structure<T>} template which
6680 is easier to use but more limited, while the implementation of custom
6681 GiNaC classes is the topic of the next section. However, you may want to
6682 read both sections because many common concepts and member functions are
6683 shared by both concepts, and it will also allow you to decide which approach
6684 is most suited to your needs.
6686 The @code{structure<T>} template, defined in the GiNaC header file
6687 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6688 or @code{class}) into a GiNaC object that can be used in expressions.
6690 @subsection Example: scalar products
6692 Let's suppose that we need a way to handle some kind of abstract scalar
6693 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6694 product class have to store their left and right operands, which can in turn
6695 be arbitrary expressions. Here is a possible way to represent such a
6696 product in a C++ @code{struct}:
6700 using namespace std;
6702 #include <ginac/ginac.h>
6703 using namespace GiNaC;
6709 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6713 The default constructor is required. Now, to make a GiNaC class out of this
6714 data structure, we need only one line:
6717 typedef structure<sprod_s> sprod;
6720 That's it. This line constructs an algebraic class @code{sprod} which
6721 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6722 expressions like any other GiNaC class:
6726 symbol a("a"), b("b");
6727 ex e = sprod(sprod_s(a, b));
6731 Note the difference between @code{sprod} which is the algebraic class, and
6732 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6733 and @code{right} data members. As shown above, an @code{sprod} can be
6734 constructed from an @code{sprod_s} object.
6736 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6737 you could define a little wrapper function like this:
6740 inline ex make_sprod(ex left, ex right)
6742 return sprod(sprod_s(left, right));
6746 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6747 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6748 @code{get_struct()}:
6752 cout << ex_to<sprod>(e)->left << endl;
6754 cout << ex_to<sprod>(e).get_struct().right << endl;
6759 You only have read access to the members of @code{sprod_s}.
6761 The type definition of @code{sprod} is enough to write your own algorithms
6762 that deal with scalar products, for example:
6767 if (is_a<sprod>(p)) @{
6768 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6769 return make_sprod(sp.right, sp.left);
6780 @subsection Structure output
6782 While the @code{sprod} type is useable it still leaves something to be
6783 desired, most notably proper output:
6788 // -> [structure object]
6792 By default, any structure types you define will be printed as
6793 @samp{[structure object]}. To override this you can either specialize the
6794 template's @code{print()} member function, or specify print methods with
6795 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6796 it's not possible to supply class options like @code{print_func<>()} to
6797 structures, so for a self-contained structure type you need to resort to
6798 overriding the @code{print()} function, which is also what we will do here.
6800 The member functions of GiNaC classes are described in more detail in the
6801 next section, but it shouldn't be hard to figure out what's going on here:
6804 void sprod::print(const print_context & c, unsigned level) const
6806 // tree debug output handled by superclass
6807 if (is_a<print_tree>(c))
6808 inherited::print(c, level);
6810 // get the contained sprod_s object
6811 const sprod_s & sp = get_struct();
6813 // print_context::s is a reference to an ostream
6814 c.s << "<" << sp.left << "|" << sp.right << ">";
6818 Now we can print expressions containing scalar products:
6824 cout << swap_sprod(e) << endl;
6829 @subsection Comparing structures
6831 The @code{sprod} class defined so far still has one important drawback: all
6832 scalar products are treated as being equal because GiNaC doesn't know how to
6833 compare objects of type @code{sprod_s}. This can lead to some confusing
6834 and undesired behavior:
6838 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6840 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6841 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6845 To remedy this, we first need to define the operators @code{==} and @code{<}
6846 for objects of type @code{sprod_s}:
6849 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6851 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6854 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6856 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6860 The ordering established by the @code{<} operator doesn't have to make any
6861 algebraic sense, but it needs to be well defined. Note that we can't use
6862 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6863 in the implementation of these operators because they would construct
6864 GiNaC @code{relational} objects which in the case of @code{<} do not
6865 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6866 decide which one is algebraically 'less').
6868 Next, we need to change our definition of the @code{sprod} type to let
6869 GiNaC know that an ordering relation exists for the embedded objects:
6872 typedef structure<sprod_s, compare_std_less> sprod;
6875 @code{sprod} objects then behave as expected:
6879 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6880 // -> <a|b>-<a^2|b^2>
6881 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6882 // -> <a|b>+<a^2|b^2>
6883 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6885 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6890 The @code{compare_std_less} policy parameter tells GiNaC to use the
6891 @code{std::less} and @code{std::equal_to} functors to compare objects of
6892 type @code{sprod_s}. By default, these functors forward their work to the
6893 standard @code{<} and @code{==} operators, which we have overloaded.
6894 Alternatively, we could have specialized @code{std::less} and
6895 @code{std::equal_to} for class @code{sprod_s}.
6897 GiNaC provides two other comparison policies for @code{structure<T>}
6898 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6899 which does a bit-wise comparison of the contained @code{T} objects.
6900 This should be used with extreme care because it only works reliably with
6901 built-in integral types, and it also compares any padding (filler bytes of
6902 undefined value) that the @code{T} class might have.
6904 @subsection Subexpressions
6906 Our scalar product class has two subexpressions: the left and right
6907 operands. It might be a good idea to make them accessible via the standard
6908 @code{nops()} and @code{op()} methods:
6911 size_t sprod::nops() const
6916 ex sprod::op(size_t i) const
6920 return get_struct().left;
6922 return get_struct().right;
6924 throw std::range_error("sprod::op(): no such operand");
6929 Implementing @code{nops()} and @code{op()} for container types such as
6930 @code{sprod} has two other nice side effects:
6934 @code{has()} works as expected
6936 GiNaC generates better hash keys for the objects (the default implementation
6937 of @code{calchash()} takes subexpressions into account)
6940 @cindex @code{let_op()}
6941 There is a non-const variant of @code{op()} called @code{let_op()} that
6942 allows replacing subexpressions:
6945 ex & sprod::let_op(size_t i)
6947 // every non-const member function must call this
6948 ensure_if_modifiable();
6952 return get_struct().left;
6954 return get_struct().right;
6956 throw std::range_error("sprod::let_op(): no such operand");
6961 Once we have provided @code{let_op()} we also get @code{subs()} and
6962 @code{map()} for free. In fact, every container class that returns a non-null
6963 @code{nops()} value must either implement @code{let_op()} or provide custom
6964 implementations of @code{subs()} and @code{map()}.
6966 In turn, the availability of @code{map()} enables the recursive behavior of a
6967 couple of other default method implementations, in particular @code{evalf()},
6968 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6969 we probably want to provide our own version of @code{expand()} for scalar
6970 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6971 This is left as an exercise for the reader.
6973 The @code{structure<T>} template defines many more member functions that
6974 you can override by specialization to customize the behavior of your
6975 structures. You are referred to the next section for a description of
6976 some of these (especially @code{eval()}). There is, however, one topic
6977 that shall be addressed here, as it demonstrates one peculiarity of the
6978 @code{structure<T>} template: archiving.
6980 @subsection Archiving structures
6982 If you don't know how the archiving of GiNaC objects is implemented, you
6983 should first read the next section and then come back here. You're back?
6986 To implement archiving for structures it is not enough to provide
6987 specializations for the @code{archive()} member function and the
6988 unarchiving constructor (the @code{unarchive()} function has a default
6989 implementation). You also need to provide a unique name (as a string literal)
6990 for each structure type you define. This is because in GiNaC archives,
6991 the class of an object is stored as a string, the class name.
6993 By default, this class name (as returned by the @code{class_name()} member
6994 function) is @samp{structure} for all structure classes. This works as long
6995 as you have only defined one structure type, but if you use two or more you
6996 need to provide a different name for each by specializing the
6997 @code{get_class_name()} member function. Here is a sample implementation
6998 for enabling archiving of the scalar product type defined above:
7001 const char *sprod::get_class_name() @{ return "sprod"; @}
7003 void sprod::archive(archive_node & n) const
7005 inherited::archive(n);
7006 n.add_ex("left", get_struct().left);
7007 n.add_ex("right", get_struct().right);
7010 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7012 n.find_ex("left", get_struct().left, sym_lst);
7013 n.find_ex("right", get_struct().right, sym_lst);
7017 Note that the unarchiving constructor is @code{sprod::structure} and not
7018 @code{sprod::sprod}, and that we don't need to supply an
7019 @code{sprod::unarchive()} function.
7022 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7023 @c node-name, next, previous, up
7024 @section Adding classes
7026 The @code{structure<T>} template provides an way to extend GiNaC with custom
7027 algebraic classes that is easy to use but has its limitations, the most
7028 severe of which being that you can't add any new member functions to
7029 structures. To be able to do this, you need to write a new class definition
7032 This section will explain how to implement new algebraic classes in GiNaC by
7033 giving the example of a simple 'string' class. After reading this section
7034 you will know how to properly declare a GiNaC class and what the minimum
7035 required member functions are that you have to implement. We only cover the
7036 implementation of a 'leaf' class here (i.e. one that doesn't contain
7037 subexpressions). Creating a container class like, for example, a class
7038 representing tensor products is more involved but this section should give
7039 you enough information so you can consult the source to GiNaC's predefined
7040 classes if you want to implement something more complicated.
7042 @subsection GiNaC's run-time type information system
7044 @cindex hierarchy of classes
7046 All algebraic classes (that is, all classes that can appear in expressions)
7047 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7048 @code{basic *} (which is essentially what an @code{ex} is) represents a
7049 generic pointer to an algebraic class. Occasionally it is necessary to find
7050 out what the class of an object pointed to by a @code{basic *} really is.
7051 Also, for the unarchiving of expressions it must be possible to find the
7052 @code{unarchive()} function of a class given the class name (as a string). A
7053 system that provides this kind of information is called a run-time type
7054 information (RTTI) system. The C++ language provides such a thing (see the
7055 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7056 implements its own, simpler RTTI.
7058 The RTTI in GiNaC is based on two mechanisms:
7063 The @code{basic} class declares a member variable @code{tinfo_key} which
7064 holds an unsigned integer that identifies the object's class. These numbers
7065 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7066 classes. They all start with @code{TINFO_}.
7069 By means of some clever tricks with static members, GiNaC maintains a list
7070 of information for all classes derived from @code{basic}. The information
7071 available includes the class names, the @code{tinfo_key}s, and pointers
7072 to the unarchiving functions. This class registry is defined in the
7073 @file{registrar.h} header file.
7077 The disadvantage of this proprietary RTTI implementation is that there's
7078 a little more to do when implementing new classes (C++'s RTTI works more
7079 or less automatically) but don't worry, most of the work is simplified by
7082 @subsection A minimalistic example
7084 Now we will start implementing a new class @code{mystring} that allows
7085 placing character strings in algebraic expressions (this is not very useful,
7086 but it's just an example). This class will be a direct subclass of
7087 @code{basic}. You can use this sample implementation as a starting point
7088 for your own classes.
7090 The code snippets given here assume that you have included some header files
7096 #include <stdexcept>
7097 using namespace std;
7099 #include <ginac/ginac.h>
7100 using namespace GiNaC;
7103 The first thing we have to do is to define a @code{tinfo_key} for our new
7104 class. This can be any arbitrary unsigned number that is not already taken
7105 by one of the existing classes but it's better to come up with something
7106 that is unlikely to clash with keys that might be added in the future. The
7107 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7108 which is not a requirement but we are going to stick with this scheme:
7111 const unsigned TINFO_mystring = 0x42420001U;
7114 Now we can write down the class declaration. The class stores a C++
7115 @code{string} and the user shall be able to construct a @code{mystring}
7116 object from a C or C++ string:
7119 class mystring : public basic
7121 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7124 mystring(const string &s);
7125 mystring(const char *s);
7131 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7134 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7135 macros are defined in @file{registrar.h}. They take the name of the class
7136 and its direct superclass as arguments and insert all required declarations
7137 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7138 the first line after the opening brace of the class definition. The
7139 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7140 source (at global scope, of course, not inside a function).
7142 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7143 declarations of the default constructor and a couple of other functions that
7144 are required. It also defines a type @code{inherited} which refers to the
7145 superclass so you don't have to modify your code every time you shuffle around
7146 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7147 class with the GiNaC RTTI (there is also a
7148 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7149 options for the class, and which we will be using instead in a few minutes).
7151 Now there are seven member functions we have to implement to get a working
7157 @code{mystring()}, the default constructor.
7160 @code{void archive(archive_node &n)}, the archiving function. This stores all
7161 information needed to reconstruct an object of this class inside an
7162 @code{archive_node}.
7165 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7166 constructor. This constructs an instance of the class from the information
7167 found in an @code{archive_node}.
7170 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7171 unarchiving function. It constructs a new instance by calling the unarchiving
7175 @cindex @code{compare_same_type()}
7176 @code{int compare_same_type(const basic &other)}, which is used internally
7177 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7178 -1, depending on the relative order of this object and the @code{other}
7179 object. If it returns 0, the objects are considered equal.
7180 @strong{Note:} This has nothing to do with the (numeric) ordering
7181 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7182 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7183 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7184 must provide a @code{compare_same_type()} function, even those representing
7185 objects for which no reasonable algebraic ordering relationship can be
7189 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7190 which are the two constructors we declared.
7194 Let's proceed step-by-step. The default constructor looks like this:
7197 mystring::mystring() : inherited(TINFO_mystring) @{@}
7200 The golden rule is that in all constructors you have to set the
7201 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7202 it will be set by the constructor of the superclass and all hell will break
7203 loose in the RTTI. For your convenience, the @code{basic} class provides
7204 a constructor that takes a @code{tinfo_key} value, which we are using here
7205 (remember that in our case @code{inherited == basic}). If the superclass
7206 didn't have such a constructor, we would have to set the @code{tinfo_key}
7207 to the right value manually.
7209 In the default constructor you should set all other member variables to
7210 reasonable default values (we don't need that here since our @code{str}
7211 member gets set to an empty string automatically).
7213 Next are the three functions for archiving. You have to implement them even
7214 if you don't plan to use archives, but the minimum required implementation
7215 is really simple. First, the archiving function:
7218 void mystring::archive(archive_node &n) const
7220 inherited::archive(n);
7221 n.add_string("string", str);
7225 The only thing that is really required is calling the @code{archive()}
7226 function of the superclass. Optionally, you can store all information you
7227 deem necessary for representing the object into the passed
7228 @code{archive_node}. We are just storing our string here. For more
7229 information on how the archiving works, consult the @file{archive.h} header
7232 The unarchiving constructor is basically the inverse of the archiving
7236 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7238 n.find_string("string", str);
7242 If you don't need archiving, just leave this function empty (but you must
7243 invoke the unarchiving constructor of the superclass). Note that we don't
7244 have to set the @code{tinfo_key} here because it is done automatically
7245 by the unarchiving constructor of the @code{basic} class.
7247 Finally, the unarchiving function:
7250 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7252 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7256 You don't have to understand how exactly this works. Just copy these
7257 four lines into your code literally (replacing the class name, of
7258 course). It calls the unarchiving constructor of the class and unless
7259 you are doing something very special (like matching @code{archive_node}s
7260 to global objects) you don't need a different implementation. For those
7261 who are interested: setting the @code{dynallocated} flag puts the object
7262 under the control of GiNaC's garbage collection. It will get deleted
7263 automatically once it is no longer referenced.
7265 Our @code{compare_same_type()} function uses a provided function to compare
7269 int mystring::compare_same_type(const basic &other) const
7271 const mystring &o = static_cast<const mystring &>(other);
7272 int cmpval = str.compare(o.str);
7275 else if (cmpval < 0)
7282 Although this function takes a @code{basic &}, it will always be a reference
7283 to an object of exactly the same class (objects of different classes are not
7284 comparable), so the cast is safe. If this function returns 0, the two objects
7285 are considered equal (in the sense that @math{A-B=0}), so you should compare
7286 all relevant member variables.
7288 Now the only thing missing is our two new constructors:
7291 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7292 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7295 No surprises here. We set the @code{str} member from the argument and
7296 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7298 That's it! We now have a minimal working GiNaC class that can store
7299 strings in algebraic expressions. Let's confirm that the RTTI works:
7302 ex e = mystring("Hello, world!");
7303 cout << is_a<mystring>(e) << endl;
7306 cout << e.bp->class_name() << endl;
7310 Obviously it does. Let's see what the expression @code{e} looks like:
7314 // -> [mystring object]
7317 Hm, not exactly what we expect, but of course the @code{mystring} class
7318 doesn't yet know how to print itself. This can be done either by implementing
7319 the @code{print()} member function, or, preferably, by specifying a
7320 @code{print_func<>()} class option. Let's say that we want to print the string
7321 surrounded by double quotes:
7324 class mystring : public basic
7328 void do_print(const print_context &c, unsigned level = 0) const;
7332 void mystring::do_print(const print_context &c, unsigned level) const
7334 // print_context::s is a reference to an ostream
7335 c.s << '\"' << str << '\"';
7339 The @code{level} argument is only required for container classes to
7340 correctly parenthesize the output.
7342 Now we need to tell GiNaC that @code{mystring} objects should use the
7343 @code{do_print()} member function for printing themselves. For this, we
7347 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7353 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7354 print_func<print_context>(&mystring::do_print))
7357 Let's try again to print the expression:
7361 // -> "Hello, world!"
7364 Much better. If we wanted to have @code{mystring} objects displayed in a
7365 different way depending on the output format (default, LaTeX, etc.), we
7366 would have supplied multiple @code{print_func<>()} options with different
7367 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7368 separated by dots. This is similar to the way options are specified for
7369 symbolic functions. @xref{Printing}, for a more in-depth description of the
7370 way expression output is implemented in GiNaC.
7372 The @code{mystring} class can be used in arbitrary expressions:
7375 e += mystring("GiNaC rulez");
7377 // -> "GiNaC rulez"+"Hello, world!"
7380 (GiNaC's automatic term reordering is in effect here), or even
7383 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7385 // -> "One string"^(2*sin(-"Another string"+Pi))
7388 Whether this makes sense is debatable but remember that this is only an
7389 example. At least it allows you to implement your own symbolic algorithms
7392 Note that GiNaC's algebraic rules remain unchanged:
7395 e = mystring("Wow") * mystring("Wow");
7399 e = pow(mystring("First")-mystring("Second"), 2);
7400 cout << e.expand() << endl;
7401 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7404 There's no way to, for example, make GiNaC's @code{add} class perform string
7405 concatenation. You would have to implement this yourself.
7407 @subsection Automatic evaluation
7410 @cindex @code{eval()}
7411 @cindex @code{hold()}
7412 When dealing with objects that are just a little more complicated than the
7413 simple string objects we have implemented, chances are that you will want to
7414 have some automatic simplifications or canonicalizations performed on them.
7415 This is done in the evaluation member function @code{eval()}. Let's say that
7416 we wanted all strings automatically converted to lowercase with
7417 non-alphabetic characters stripped, and empty strings removed:
7420 class mystring : public basic
7424 ex eval(int level = 0) const;
7428 ex mystring::eval(int level) const
7431 for (int i=0; i<str.length(); i++) @{
7433 if (c >= 'A' && c <= 'Z')
7434 new_str += tolower(c);
7435 else if (c >= 'a' && c <= 'z')
7439 if (new_str.length() == 0)
7442 return mystring(new_str).hold();
7446 The @code{level} argument is used to limit the recursion depth of the
7447 evaluation. We don't have any subexpressions in the @code{mystring}
7448 class so we are not concerned with this. If we had, we would call the
7449 @code{eval()} functions of the subexpressions with @code{level - 1} as
7450 the argument if @code{level != 1}. The @code{hold()} member function
7451 sets a flag in the object that prevents further evaluation. Otherwise
7452 we might end up in an endless loop. When you want to return the object
7453 unmodified, use @code{return this->hold();}.
7455 Let's confirm that it works:
7458 ex e = mystring("Hello, world!") + mystring("!?#");
7462 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7467 @subsection Optional member functions
7469 We have implemented only a small set of member functions to make the class
7470 work in the GiNaC framework. There are two functions that are not strictly
7471 required but will make operations with objects of the class more efficient:
7473 @cindex @code{calchash()}
7474 @cindex @code{is_equal_same_type()}
7476 unsigned calchash() const;
7477 bool is_equal_same_type(const basic &other) const;
7480 The @code{calchash()} method returns an @code{unsigned} hash value for the
7481 object which will allow GiNaC to compare and canonicalize expressions much
7482 more efficiently. You should consult the implementation of some of the built-in
7483 GiNaC classes for examples of hash functions. The default implementation of
7484 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7485 class and all subexpressions that are accessible via @code{op()}.
7487 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7488 tests for equality without establishing an ordering relation, which is often
7489 faster. The default implementation of @code{is_equal_same_type()} just calls
7490 @code{compare_same_type()} and tests its result for zero.
7492 @subsection Other member functions
7494 For a real algebraic class, there are probably some more functions that you
7495 might want to provide:
7498 bool info(unsigned inf) const;
7499 ex evalf(int level = 0) const;
7500 ex series(const relational & r, int order, unsigned options = 0) const;
7501 ex derivative(const symbol & s) const;
7504 If your class stores sub-expressions (see the scalar product example in the
7505 previous section) you will probably want to override
7507 @cindex @code{let_op()}
7510 ex op(size_t i) const;
7511 ex & let_op(size_t i);
7512 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7513 ex map(map_function & f) const;
7516 @code{let_op()} is a variant of @code{op()} that allows write access. The
7517 default implementations of @code{subs()} and @code{map()} use it, so you have
7518 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7520 You can, of course, also add your own new member functions. Remember
7521 that the RTTI may be used to get information about what kinds of objects
7522 you are dealing with (the position in the class hierarchy) and that you
7523 can always extract the bare object from an @code{ex} by stripping the
7524 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7525 should become a need.
7527 That's it. May the source be with you!
7530 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7531 @c node-name, next, previous, up
7532 @chapter A Comparison With Other CAS
7535 This chapter will give you some information on how GiNaC compares to
7536 other, traditional Computer Algebra Systems, like @emph{Maple},
7537 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7538 disadvantages over these systems.
7541 * Advantages:: Strengths of the GiNaC approach.
7542 * Disadvantages:: Weaknesses of the GiNaC approach.
7543 * Why C++?:: Attractiveness of C++.
7546 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7547 @c node-name, next, previous, up
7550 GiNaC has several advantages over traditional Computer
7551 Algebra Systems, like
7556 familiar language: all common CAS implement their own proprietary
7557 grammar which you have to learn first (and maybe learn again when your
7558 vendor decides to `enhance' it). With GiNaC you can write your program
7559 in common C++, which is standardized.
7563 structured data types: you can build up structured data types using
7564 @code{struct}s or @code{class}es together with STL features instead of
7565 using unnamed lists of lists of lists.
7568 strongly typed: in CAS, you usually have only one kind of variables
7569 which can hold contents of an arbitrary type. This 4GL like feature is
7570 nice for novice programmers, but dangerous.
7573 development tools: powerful development tools exist for C++, like fancy
7574 editors (e.g. with automatic indentation and syntax highlighting),
7575 debuggers, visualization tools, documentation generators@dots{}
7578 modularization: C++ programs can easily be split into modules by
7579 separating interface and implementation.
7582 price: GiNaC is distributed under the GNU Public License which means
7583 that it is free and available with source code. And there are excellent
7584 C++-compilers for free, too.
7587 extendable: you can add your own classes to GiNaC, thus extending it on
7588 a very low level. Compare this to a traditional CAS that you can
7589 usually only extend on a high level by writing in the language defined
7590 by the parser. In particular, it turns out to be almost impossible to
7591 fix bugs in a traditional system.
7594 multiple interfaces: Though real GiNaC programs have to be written in
7595 some editor, then be compiled, linked and executed, there are more ways
7596 to work with the GiNaC engine. Many people want to play with
7597 expressions interactively, as in traditional CASs. Currently, two such
7598 windows into GiNaC have been implemented and many more are possible: the
7599 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7600 types to a command line and second, as a more consistent approach, an
7601 interactive interface to the Cint C++ interpreter has been put together
7602 (called GiNaC-cint) that allows an interactive scripting interface
7603 consistent with the C++ language. It is available from the usual GiNaC
7607 seamless integration: it is somewhere between difficult and impossible
7608 to call CAS functions from within a program written in C++ or any other
7609 programming language and vice versa. With GiNaC, your symbolic routines
7610 are part of your program. You can easily call third party libraries,
7611 e.g. for numerical evaluation or graphical interaction. All other
7612 approaches are much more cumbersome: they range from simply ignoring the
7613 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7614 system (i.e. @emph{Yacas}).
7617 efficiency: often large parts of a program do not need symbolic
7618 calculations at all. Why use large integers for loop variables or
7619 arbitrary precision arithmetics where @code{int} and @code{double} are
7620 sufficient? For pure symbolic applications, GiNaC is comparable in
7621 speed with other CAS.
7626 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7627 @c node-name, next, previous, up
7628 @section Disadvantages
7630 Of course it also has some disadvantages:
7635 advanced features: GiNaC cannot compete with a program like
7636 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7637 which grows since 1981 by the work of dozens of programmers, with
7638 respect to mathematical features. Integration, factorization,
7639 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7640 not planned for the near future).
7643 portability: While the GiNaC library itself is designed to avoid any
7644 platform dependent features (it should compile on any ANSI compliant C++
7645 compiler), the currently used version of the CLN library (fast large
7646 integer and arbitrary precision arithmetics) can only by compiled
7647 without hassle on systems with the C++ compiler from the GNU Compiler
7648 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7649 macros to let the compiler gather all static initializations, which
7650 works for GNU C++ only. Feel free to contact the authors in case you
7651 really believe that you need to use a different compiler. We have
7652 occasionally used other compilers and may be able to give you advice.}
7653 GiNaC uses recent language features like explicit constructors, mutable
7654 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7655 literally. Recent GCC versions starting at 2.95.3, although itself not
7656 yet ANSI compliant, support all needed features.
7661 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7662 @c node-name, next, previous, up
7665 Why did we choose to implement GiNaC in C++ instead of Java or any other
7666 language? C++ is not perfect: type checking is not strict (casting is
7667 possible), separation between interface and implementation is not
7668 complete, object oriented design is not enforced. The main reason is
7669 the often scolded feature of operator overloading in C++. While it may
7670 be true that operating on classes with a @code{+} operator is rarely
7671 meaningful, it is perfectly suited for algebraic expressions. Writing
7672 @math{3x+5y} as @code{3*x+5*y} instead of
7673 @code{x.times(3).plus(y.times(5))} looks much more natural.
7674 Furthermore, the main developers are more familiar with C++ than with
7675 any other programming language.
7678 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7679 @c node-name, next, previous, up
7680 @appendix Internal Structures
7683 * Expressions are reference counted::
7684 * Internal representation of products and sums::
7687 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7688 @c node-name, next, previous, up
7689 @appendixsection Expressions are reference counted
7691 @cindex reference counting
7692 @cindex copy-on-write
7693 @cindex garbage collection
7694 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7695 where the counter belongs to the algebraic objects derived from class
7696 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7697 which @code{ex} contains an instance. If you understood that, you can safely
7698 skip the rest of this passage.
7700 Expressions are extremely light-weight since internally they work like
7701 handles to the actual representation. They really hold nothing more
7702 than a pointer to some other object. What this means in practice is
7703 that whenever you create two @code{ex} and set the second equal to the
7704 first no copying process is involved. Instead, the copying takes place
7705 as soon as you try to change the second. Consider the simple sequence
7710 #include <ginac/ginac.h>
7711 using namespace std;
7712 using namespace GiNaC;
7716 symbol x("x"), y("y"), z("z");
7719 e1 = sin(x + 2*y) + 3*z + 41;
7720 e2 = e1; // e2 points to same object as e1
7721 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7722 e2 += 1; // e2 is copied into a new object
7723 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7727 The line @code{e2 = e1;} creates a second expression pointing to the
7728 object held already by @code{e1}. The time involved for this operation
7729 is therefore constant, no matter how large @code{e1} was. Actual
7730 copying, however, must take place in the line @code{e2 += 1;} because
7731 @code{e1} and @code{e2} are not handles for the same object any more.
7732 This concept is called @dfn{copy-on-write semantics}. It increases
7733 performance considerably whenever one object occurs multiple times and
7734 represents a simple garbage collection scheme because when an @code{ex}
7735 runs out of scope its destructor checks whether other expressions handle
7736 the object it points to too and deletes the object from memory if that
7737 turns out not to be the case. A slightly less trivial example of
7738 differentiation using the chain-rule should make clear how powerful this
7743 symbol x("x"), y("y");
7747 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7748 cout << e1 << endl // prints x+3*y
7749 << e2 << endl // prints (x+3*y)^3
7750 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7754 Here, @code{e1} will actually be referenced three times while @code{e2}
7755 will be referenced two times. When the power of an expression is built,
7756 that expression needs not be copied. Likewise, since the derivative of
7757 a power of an expression can be easily expressed in terms of that
7758 expression, no copying of @code{e1} is involved when @code{e3} is
7759 constructed. So, when @code{e3} is constructed it will print as
7760 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7761 holds a reference to @code{e2} and the factor in front is just
7764 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7765 semantics. When you insert an expression into a second expression, the
7766 result behaves exactly as if the contents of the first expression were
7767 inserted. But it may be useful to remember that this is not what
7768 happens. Knowing this will enable you to write much more efficient
7769 code. If you still have an uncertain feeling with copy-on-write
7770 semantics, we recommend you have a look at the
7771 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7772 Marshall Cline. Chapter 16 covers this issue and presents an
7773 implementation which is pretty close to the one in GiNaC.
7776 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7777 @c node-name, next, previous, up
7778 @appendixsection Internal representation of products and sums
7780 @cindex representation
7783 @cindex @code{power}
7784 Although it should be completely transparent for the user of
7785 GiNaC a short discussion of this topic helps to understand the sources
7786 and also explain performance to a large degree. Consider the
7787 unexpanded symbolic expression
7789 $2d^3 \left( 4a + 5b - 3 \right)$
7792 @math{2*d^3*(4*a+5*b-3)}
7794 which could naively be represented by a tree of linear containers for
7795 addition and multiplication, one container for exponentiation with base
7796 and exponent and some atomic leaves of symbols and numbers in this
7801 @cindex pair-wise representation
7802 However, doing so results in a rather deeply nested tree which will
7803 quickly become inefficient to manipulate. We can improve on this by
7804 representing the sum as a sequence of terms, each one being a pair of a
7805 purely numeric multiplicative coefficient and its rest. In the same
7806 spirit we can store the multiplication as a sequence of terms, each
7807 having a numeric exponent and a possibly complicated base, the tree
7808 becomes much more flat:
7812 The number @code{3} above the symbol @code{d} shows that @code{mul}
7813 objects are treated similarly where the coefficients are interpreted as
7814 @emph{exponents} now. Addition of sums of terms or multiplication of
7815 products with numerical exponents can be coded to be very efficient with
7816 such a pair-wise representation. Internally, this handling is performed
7817 by most CAS in this way. It typically speeds up manipulations by an
7818 order of magnitude. The overall multiplicative factor @code{2} and the
7819 additive term @code{-3} look somewhat out of place in this
7820 representation, however, since they are still carrying a trivial
7821 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7822 this is avoided by adding a field that carries an overall numeric
7823 coefficient. This results in the realistic picture of internal
7826 $2d^3 \left( 4a + 5b - 3 \right)$:
7829 @math{2*d^3*(4*a+5*b-3)}:
7835 This also allows for a better handling of numeric radicals, since
7836 @code{sqrt(2)} can now be carried along calculations. Now it should be
7837 clear, why both classes @code{add} and @code{mul} are derived from the
7838 same abstract class: the data representation is the same, only the
7839 semantics differs. In the class hierarchy, methods for polynomial
7840 expansion and the like are reimplemented for @code{add} and @code{mul},
7841 but the data structure is inherited from @code{expairseq}.
7844 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7845 @c node-name, next, previous, up
7846 @appendix Package Tools
7848 If you are creating a software package that uses the GiNaC library,
7849 setting the correct command line options for the compiler and linker
7850 can be difficult. GiNaC includes two tools to make this process easier.
7853 * ginac-config:: A shell script to detect compiler and linker flags.
7854 * AM_PATH_GINAC:: Macro for GNU automake.
7858 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7859 @c node-name, next, previous, up
7860 @section @command{ginac-config}
7861 @cindex ginac-config
7863 @command{ginac-config} is a shell script that you can use to determine
7864 the compiler and linker command line options required to compile and
7865 link a program with the GiNaC library.
7867 @command{ginac-config} takes the following flags:
7871 Prints out the version of GiNaC installed.
7873 Prints '-I' flags pointing to the installed header files.
7875 Prints out the linker flags necessary to link a program against GiNaC.
7876 @item --prefix[=@var{PREFIX}]
7877 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7878 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7879 Otherwise, prints out the configured value of @env{$prefix}.
7880 @item --exec-prefix[=@var{PREFIX}]
7881 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7882 Otherwise, prints out the configured value of @env{$exec_prefix}.
7885 Typically, @command{ginac-config} will be used within a configure
7886 script, as described below. It, however, can also be used directly from
7887 the command line using backquotes to compile a simple program. For
7891 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7894 This command line might expand to (for example):
7897 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7898 -lginac -lcln -lstdc++
7901 Not only is the form using @command{ginac-config} easier to type, it will
7902 work on any system, no matter how GiNaC was configured.
7905 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7906 @c node-name, next, previous, up
7907 @section @samp{AM_PATH_GINAC}
7908 @cindex AM_PATH_GINAC
7910 For packages configured using GNU automake, GiNaC also provides
7911 a macro to automate the process of checking for GiNaC.
7914 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7922 Determines the location of GiNaC using @command{ginac-config}, which is
7923 either found in the user's path, or from the environment variable
7924 @env{GINACLIB_CONFIG}.
7927 Tests the installed libraries to make sure that their version
7928 is later than @var{MINIMUM-VERSION}. (A default version will be used
7932 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7933 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7934 variable to the output of @command{ginac-config --libs}, and calls
7935 @samp{AC_SUBST()} for these variables so they can be used in generated
7936 makefiles, and then executes @var{ACTION-IF-FOUND}.
7939 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7940 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7944 This macro is in file @file{ginac.m4} which is installed in
7945 @file{$datadir/aclocal}. Note that if automake was installed with a
7946 different @samp{--prefix} than GiNaC, you will either have to manually
7947 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7948 aclocal the @samp{-I} option when running it.
7951 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7952 * Example package:: Example of a package using AM_PATH_GINAC.
7956 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7957 @c node-name, next, previous, up
7958 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7960 Simply make sure that @command{ginac-config} is in your path, and run
7961 the configure script.
7968 The directory where the GiNaC libraries are installed needs
7969 to be found by your system's dynamic linker.
7971 This is generally done by
7974 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7980 setting the environment variable @env{LD_LIBRARY_PATH},
7983 or, as a last resort,
7986 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7987 running configure, for instance:
7990 LDFLAGS=-R/home/cbauer/lib ./configure
7995 You can also specify a @command{ginac-config} not in your path by
7996 setting the @env{GINACLIB_CONFIG} environment variable to the
7997 name of the executable
8000 If you move the GiNaC package from its installed location,
8001 you will either need to modify @command{ginac-config} script
8002 manually to point to the new location or rebuild GiNaC.
8013 --with-ginac-prefix=@var{PREFIX}
8014 --with-ginac-exec-prefix=@var{PREFIX}
8017 are provided to override the prefix and exec-prefix that were stored
8018 in the @command{ginac-config} shell script by GiNaC's configure. You are
8019 generally better off configuring GiNaC with the right path to begin with.
8023 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8024 @c node-name, next, previous, up
8025 @subsection Example of a package using @samp{AM_PATH_GINAC}
8027 The following shows how to build a simple package using automake
8028 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8032 #include <ginac/ginac.h>
8036 GiNaC::symbol x("x");
8037 GiNaC::ex a = GiNaC::sin(x);
8038 std::cout << "Derivative of " << a
8039 << " is " << a.diff(x) << std::endl;
8044 You should first read the introductory portions of the automake
8045 Manual, if you are not already familiar with it.
8047 Two files are needed, @file{configure.in}, which is used to build the
8051 dnl Process this file with autoconf to produce a configure script.
8053 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8059 AM_PATH_GINAC(0.9.0, [
8060 LIBS="$LIBS $GINACLIB_LIBS"
8061 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8062 ], AC_MSG_ERROR([need to have GiNaC installed]))
8067 The only command in this which is not standard for automake
8068 is the @samp{AM_PATH_GINAC} macro.
8070 That command does the following: If a GiNaC version greater or equal
8071 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8072 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8073 the error message `need to have GiNaC installed'
8075 And the @file{Makefile.am}, which will be used to build the Makefile.
8078 ## Process this file with automake to produce Makefile.in
8079 bin_PROGRAMS = simple
8080 simple_SOURCES = simple.cpp
8083 This @file{Makefile.am}, says that we are building a single executable,
8084 from a single source file @file{simple.cpp}. Since every program
8085 we are building uses GiNaC we simply added the GiNaC options
8086 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8087 want to specify them on a per-program basis: for instance by
8091 simple_LDADD = $(GINACLIB_LIBS)
8092 INCLUDES = $(GINACLIB_CPPFLAGS)
8095 to the @file{Makefile.am}.
8097 To try this example out, create a new directory and add the three
8100 Now execute the following commands:
8103 $ automake --add-missing
8108 You now have a package that can be built in the normal fashion
8117 @node Bibliography, Concept Index, Example package, Top
8118 @c node-name, next, previous, up
8119 @appendix Bibliography
8124 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8127 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8130 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8133 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8136 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8137 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8140 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8141 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8142 Academic Press, London
8145 @cite{Computer Algebra Systems - A Practical Guide},
8146 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8149 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8150 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8153 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8154 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8157 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8162 @node Concept Index, , Bibliography, Top
8163 @c node-name, next, previous, up
8164 @unnumbered Concept Index