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 By default, the only documentation that will be built is this tutorial
606 in @file{.info} format. To build the GiNaC tutorial and reference manual
607 in HTML, DVI, PostScript, or PDF formats, use one of
616 Generally, the top-level Makefile runs recursively to the
617 subdirectories. It is therefore safe to go into any subdirectory
618 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
619 @var{target} there in case something went wrong.
622 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
623 @c node-name, next, previous, up
624 @section Installing GiNaC
627 To install GiNaC on your system, simply type
633 As described in the section about configuration the files will be
634 installed in the following directories (the directories will be created
635 if they don't already exist):
640 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
641 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
642 So will @file{libginac.so} unless the configure script was
643 given the option @option{--disable-shared}. The proper symlinks
644 will be established as well.
647 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
648 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
651 All documentation (info) will be stuffed into
652 @file{@var{PREFIX}/share/doc/GiNaC/} (or
653 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
657 For the sake of completeness we will list some other useful make
658 targets: @command{make clean} deletes all files generated by
659 @command{make}, i.e. all the object files. In addition @command{make
660 distclean} removes all files generated by the configuration and
661 @command{make maintainer-clean} goes one step further and deletes files
662 that may require special tools to rebuild (like the @command{libtool}
663 for instance). Finally @command{make uninstall} removes the installed
664 library, header files and documentation@footnote{Uninstallation does not
665 work after you have called @command{make distclean} since the
666 @file{Makefile} is itself generated by the configuration from
667 @file{Makefile.in} and hence deleted by @command{make distclean}. There
668 are two obvious ways out of this dilemma. First, you can run the
669 configuration again with the same @var{PREFIX} thus creating a
670 @file{Makefile} with a working @samp{uninstall} target. Second, you can
671 do it by hand since you now know where all the files went during
675 @node Basic Concepts, Expressions, Installing GiNaC, Top
676 @c node-name, next, previous, up
677 @chapter Basic Concepts
679 This chapter will describe the different fundamental objects that can be
680 handled by GiNaC. But before doing so, it is worthwhile introducing you
681 to the more commonly used class of expressions, representing a flexible
682 meta-class for storing all mathematical objects.
685 * Expressions:: The fundamental GiNaC class.
686 * Automatic evaluation:: Evaluation and canonicalization.
687 * Error handling:: How the library reports errors.
688 * The Class Hierarchy:: Overview of GiNaC's classes.
689 * Symbols:: Symbolic objects.
690 * Numbers:: Numerical objects.
691 * Constants:: Pre-defined constants.
692 * Fundamental containers:: Sums, products and powers.
693 * Lists:: Lists of expressions.
694 * Mathematical functions:: Mathematical functions.
695 * Relations:: Equality, Inequality and all that.
696 * Integrals:: Symbolic integrals.
697 * Matrices:: Matrices.
698 * Indexed objects:: Handling indexed quantities.
699 * Non-commutative objects:: Algebras with non-commutative products.
700 * Hash Maps:: A faster alternative to std::map<>.
704 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
705 @c node-name, next, previous, up
707 @cindex expression (class @code{ex})
710 The most common class of objects a user deals with is the expression
711 @code{ex}, representing a mathematical object like a variable, number,
712 function, sum, product, etc@dots{} Expressions may be put together to form
713 new expressions, passed as arguments to functions, and so on. Here is a
714 little collection of valid expressions:
717 ex MyEx1 = 5; // simple number
718 ex MyEx2 = x + 2*y; // polynomial in x and y
719 ex MyEx3 = (x + 1)/(x - 1); // rational expression
720 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
721 ex MyEx5 = MyEx4 + 1; // similar to above
724 Expressions are handles to other more fundamental objects, that often
725 contain other expressions thus creating a tree of expressions
726 (@xref{Internal Structures}, for particular examples). Most methods on
727 @code{ex} therefore run top-down through such an expression tree. For
728 example, the method @code{has()} scans recursively for occurrences of
729 something inside an expression. Thus, if you have declared @code{MyEx4}
730 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
731 the argument of @code{sin} and hence return @code{true}.
733 The next sections will outline the general picture of GiNaC's class
734 hierarchy and describe the classes of objects that are handled by
737 @subsection Note: Expressions and STL containers
739 GiNaC expressions (@code{ex} objects) have value semantics (they can be
740 assigned, reassigned and copied like integral types) but the operator
741 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
742 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
744 This implies that in order to use expressions in sorted containers such as
745 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
746 comparison predicate. GiNaC provides such a predicate, called
747 @code{ex_is_less}. For example, a set of expressions should be defined
748 as @code{std::set<ex, ex_is_less>}.
750 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
751 don't pose a problem. A @code{std::vector<ex>} works as expected.
753 @xref{Information About Expressions}, for more about comparing and ordering
757 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
758 @c node-name, next, previous, up
759 @section Automatic evaluation and canonicalization of expressions
762 GiNaC performs some automatic transformations on expressions, to simplify
763 them and put them into a canonical form. Some examples:
766 ex MyEx1 = 2*x - 1 + x; // 3*x-1
767 ex MyEx2 = x - x; // 0
768 ex MyEx3 = cos(2*Pi); // 1
769 ex MyEx4 = x*y/x; // y
772 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
773 evaluation}. GiNaC only performs transformations that are
777 at most of complexity
785 algebraically correct, possibly except for a set of measure zero (e.g.
786 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
789 There are two types of automatic transformations in GiNaC that may not
790 behave in an entirely obvious way at first glance:
794 The terms of sums and products (and some other things like the arguments of
795 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
796 into a canonical form that is deterministic, but not lexicographical or in
797 any other way easy to guess (it almost always depends on the number and
798 order of the symbols you define). However, constructing the same expression
799 twice, either implicitly or explicitly, will always result in the same
802 Expressions of the form 'number times sum' are automatically expanded (this
803 has to do with GiNaC's internal representation of sums and products). For
806 ex MyEx5 = 2*(x + y); // 2*x+2*y
807 ex MyEx6 = z*(x + y); // z*(x+y)
811 The general rule is that when you construct expressions, GiNaC automatically
812 creates them in canonical form, which might differ from the form you typed in
813 your program. This may create some awkward looking output (@samp{-y+x} instead
814 of @samp{x-y}) but allows for more efficient operation and usually yields
815 some immediate simplifications.
817 @cindex @code{eval()}
818 Internally, the anonymous evaluator in GiNaC is implemented by the methods
821 ex ex::eval(int level = 0) const;
822 ex basic::eval(int level = 0) const;
825 but unless you are extending GiNaC with your own classes or functions, there
826 should never be any reason to call them explicitly. All GiNaC methods that
827 transform expressions, like @code{subs()} or @code{normal()}, automatically
828 re-evaluate their results.
831 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
832 @c node-name, next, previous, up
833 @section Error handling
835 @cindex @code{pole_error} (class)
837 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
838 generated by GiNaC are subclassed from the standard @code{exception} class
839 defined in the @file{<stdexcept>} header. In addition to the predefined
840 @code{logic_error}, @code{domain_error}, @code{out_of_range},
841 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
842 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
843 exception that gets thrown when trying to evaluate a mathematical function
846 The @code{pole_error} class has a member function
849 int pole_error::degree() const;
852 that returns the order of the singularity (or 0 when the pole is
853 logarithmic or the order is undefined).
855 When using GiNaC it is useful to arrange for exceptions to be caught in
856 the main program even if you don't want to do any special error handling.
857 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
858 default exception handler of your C++ compiler's run-time system which
859 usually only aborts the program without giving any information what went
862 Here is an example for a @code{main()} function that catches and prints
863 exceptions generated by GiNaC:
868 #include <ginac/ginac.h>
870 using namespace GiNaC;
878 @} catch (exception &p) @{
879 cerr << p.what() << endl;
887 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
888 @c node-name, next, previous, up
889 @section The Class Hierarchy
891 GiNaC's class hierarchy consists of several classes representing
892 mathematical objects, all of which (except for @code{ex} and some
893 helpers) are internally derived from one abstract base class called
894 @code{basic}. You do not have to deal with objects of class
895 @code{basic}, instead you'll be dealing with symbols, numbers,
896 containers of expressions and so on.
900 To get an idea about what kinds of symbolic composites may be built we
901 have a look at the most important classes in the class hierarchy and
902 some of the relations among the classes:
904 @image{classhierarchy}
906 The abstract classes shown here (the ones without drop-shadow) are of no
907 interest for the user. They are used internally in order to avoid code
908 duplication if two or more classes derived from them share certain
909 features. An example is @code{expairseq}, a container for a sequence of
910 pairs each consisting of one expression and a number (@code{numeric}).
911 What @emph{is} visible to the user are the derived classes @code{add}
912 and @code{mul}, representing sums and products. @xref{Internal
913 Structures}, where these two classes are described in more detail. The
914 following table shortly summarizes what kinds of mathematical objects
915 are stored in the different classes:
918 @multitable @columnfractions .22 .78
919 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
920 @item @code{constant} @tab Constants like
927 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
928 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
929 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
930 @item @code{ncmul} @tab Products of non-commutative objects
931 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
936 @code{sqrt(}@math{2}@code{)}
939 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
940 @item @code{function} @tab A symbolic function like
947 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
948 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
949 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
950 @item @code{indexed} @tab Indexed object like @math{A_ij}
951 @item @code{tensor} @tab Special tensor like the delta and metric tensors
952 @item @code{idx} @tab Index of an indexed object
953 @item @code{varidx} @tab Index with variance
954 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
955 @item @code{wildcard} @tab Wildcard for pattern matching
956 @item @code{structure} @tab Template for user-defined classes
961 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
962 @c node-name, next, previous, up
964 @cindex @code{symbol} (class)
965 @cindex hierarchy of classes
968 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
969 manipulation what atoms are for chemistry.
971 A typical symbol definition looks like this:
976 This definition actually contains three very different things:
978 @item a C++ variable named @code{x}
979 @item a @code{symbol} object stored in this C++ variable; this object
980 represents the symbol in a GiNaC expression
981 @item the string @code{"x"} which is the name of the symbol, used (almost)
982 exclusively for printing expressions holding the symbol
985 Symbols have an explicit name, supplied as a string during construction,
986 because in C++, variable names can't be used as values, and the C++ compiler
987 throws them away during compilation.
989 It is possible to omit the symbol name in the definition:
994 In this case, GiNaC will assign the symbol an internal, unique name of the
995 form @code{symbolNNN}. This won't affect the usability of the symbol but
996 the output of your calculations will become more readable if you give your
997 symbols sensible names (for intermediate expressions that are only used
998 internally such anonymous symbols can be quite useful, however).
1000 Now, here is one important property of GiNaC that differentiates it from
1001 other computer algebra programs you may have used: GiNaC does @emph{not} use
1002 the names of symbols to tell them apart, but a (hidden) serial number that
1003 is unique for each newly created @code{symbol} object. In you want to use
1004 one and the same symbol in different places in your program, you must only
1005 create one @code{symbol} object and pass that around. If you create another
1006 symbol, even if it has the same name, GiNaC will treat it as a different
1023 // prints "x^6" which looks right, but...
1025 cout << e.degree(x) << endl;
1026 // ...this doesn't work. The symbol "x" here is different from the one
1027 // in f() and in the expression returned by f(). Consequently, it
1032 One possibility to ensure that @code{f()} and @code{main()} use the same
1033 symbol is to pass the symbol as an argument to @code{f()}:
1035 ex f(int n, const ex & x)
1044 // Now, f() uses the same symbol.
1047 cout << e.degree(x) << endl;
1048 // prints "6", as expected
1052 Another possibility would be to define a global symbol @code{x} that is used
1053 by both @code{f()} and @code{main()}. If you are using global symbols and
1054 multiple compilation units you must take special care, however. Suppose
1055 that you have a header file @file{globals.h} in your program that defines
1056 a @code{symbol x("x");}. In this case, every unit that includes
1057 @file{globals.h} would also get its own definition of @code{x} (because
1058 header files are just inlined into the source code by the C++ preprocessor),
1059 and hence you would again end up with multiple equally-named, but different,
1060 symbols. Instead, the @file{globals.h} header should only contain a
1061 @emph{declaration} like @code{extern symbol x;}, with the definition of
1062 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1064 A different approach to ensuring that symbols used in different parts of
1065 your program are identical is to create them with a @emph{factory} function
1068 const symbol & get_symbol(const string & s)
1070 static map<string, symbol> directory;
1071 map<string, symbol>::iterator i = directory.find(s);
1072 if (i != directory.end())
1075 return directory.insert(make_pair(s, symbol(s))).first->second;
1079 This function returns one newly constructed symbol for each name that is
1080 passed in, and it returns the same symbol when called multiple times with
1081 the same name. Using this symbol factory, we can rewrite our example like
1086 return pow(get_symbol("x"), n);
1093 // Both calls of get_symbol("x") yield the same symbol.
1094 cout << e.degree(get_symbol("x")) << endl;
1099 Instead of creating symbols from strings we could also have
1100 @code{get_symbol()} take, for example, an integer number as its argument.
1101 In this case, we would probably want to give the generated symbols names
1102 that include this number, which can be accomplished with the help of an
1103 @code{ostringstream}.
1105 In general, if you're getting weird results from GiNaC such as an expression
1106 @samp{x-x} that is not simplified to zero, you should check your symbol
1109 As we said, the names of symbols primarily serve for purposes of expression
1110 output. But there are actually two instances where GiNaC uses the names for
1111 identifying symbols: When constructing an expression from a string, and when
1112 recreating an expression from an archive (@pxref{Input/Output}).
1114 In addition to its name, a symbol may contain a special string that is used
1117 symbol x("x", "\\Box");
1120 This creates a symbol that is printed as "@code{x}" in normal output, but
1121 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1122 information about the different output formats of expressions in GiNaC).
1123 GiNaC automatically creates proper LaTeX code for symbols having names of
1124 greek letters (@samp{alpha}, @samp{mu}, etc.).
1126 @cindex @code{subs()}
1127 Symbols in GiNaC can't be assigned values. If you need to store results of
1128 calculations and give them a name, use C++ variables of type @code{ex}.
1129 If you want to replace a symbol in an expression with something else, you
1130 can invoke the expression's @code{.subs()} method
1131 (@pxref{Substituting Expressions}).
1133 @cindex @code{realsymbol()}
1134 By default, symbols are expected to stand in for complex values, i.e. they live
1135 in the complex domain. As a consequence, operations like complex conjugation,
1136 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1137 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1138 because of the unknown imaginary part of @code{x}.
1139 On the other hand, if you are sure that your symbols will hold only real values, you
1140 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1141 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1142 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1145 @node Numbers, Constants, Symbols, Basic Concepts
1146 @c node-name, next, previous, up
1148 @cindex @code{numeric} (class)
1154 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1155 The classes therein serve as foundation classes for GiNaC. CLN stands
1156 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1157 In order to find out more about CLN's internals, the reader is referred to
1158 the documentation of that library. @inforef{Introduction, , cln}, for
1159 more information. Suffice to say that it is by itself build on top of
1160 another library, the GNU Multiple Precision library GMP, which is an
1161 extremely fast library for arbitrary long integers and rationals as well
1162 as arbitrary precision floating point numbers. It is very commonly used
1163 by several popular cryptographic applications. CLN extends GMP by
1164 several useful things: First, it introduces the complex number field
1165 over either reals (i.e. floating point numbers with arbitrary precision)
1166 or rationals. Second, it automatically converts rationals to integers
1167 if the denominator is unity and complex numbers to real numbers if the
1168 imaginary part vanishes and also correctly treats algebraic functions.
1169 Third it provides good implementations of state-of-the-art algorithms
1170 for all trigonometric and hyperbolic functions as well as for
1171 calculation of some useful constants.
1173 The user can construct an object of class @code{numeric} in several
1174 ways. The following example shows the four most important constructors.
1175 It uses construction from C-integer, construction of fractions from two
1176 integers, construction from C-float and construction from a string:
1180 #include <ginac/ginac.h>
1181 using namespace GiNaC;
1185 numeric two = 2; // exact integer 2
1186 numeric r(2,3); // exact fraction 2/3
1187 numeric e(2.71828); // floating point number
1188 numeric p = "3.14159265358979323846"; // constructor from string
1189 // Trott's constant in scientific notation:
1190 numeric trott("1.0841015122311136151E-2");
1192 std::cout << two*p << std::endl; // floating point 6.283...
1197 @cindex complex numbers
1198 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1203 numeric z1 = 2-3*I; // exact complex number 2-3i
1204 numeric z2 = 5.9+1.6*I; // complex floating point number
1208 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1209 This would, however, call C's built-in operator @code{/} for integers
1210 first and result in a numeric holding a plain integer 1. @strong{Never
1211 use the operator @code{/} on integers} unless you know exactly what you
1212 are doing! Use the constructor from two integers instead, as shown in
1213 the example above. Writing @code{numeric(1)/2} may look funny but works
1216 @cindex @code{Digits}
1218 We have seen now the distinction between exact numbers and floating
1219 point numbers. Clearly, the user should never have to worry about
1220 dynamically created exact numbers, since their `exactness' always
1221 determines how they ought to be handled, i.e. how `long' they are. The
1222 situation is different for floating point numbers. Their accuracy is
1223 controlled by one @emph{global} variable, called @code{Digits}. (For
1224 those readers who know about Maple: it behaves very much like Maple's
1225 @code{Digits}). All objects of class numeric that are constructed from
1226 then on will be stored with a precision matching that number of decimal
1231 #include <ginac/ginac.h>
1232 using namespace std;
1233 using namespace GiNaC;
1237 numeric three(3.0), one(1.0);
1238 numeric x = one/three;
1240 cout << "in " << Digits << " digits:" << endl;
1242 cout << Pi.evalf() << endl;
1254 The above example prints the following output to screen:
1258 0.33333333333333333334
1259 3.1415926535897932385
1261 0.33333333333333333333333333333333333333333333333333333333333333333334
1262 3.1415926535897932384626433832795028841971693993751058209749445923078
1266 Note that the last number is not necessarily rounded as you would
1267 naively expect it to be rounded in the decimal system. But note also,
1268 that in both cases you got a couple of extra digits. This is because
1269 numbers are internally stored by CLN as chunks of binary digits in order
1270 to match your machine's word size and to not waste precision. Thus, on
1271 architectures with different word size, the above output might even
1272 differ with regard to actually computed digits.
1274 It should be clear that objects of class @code{numeric} should be used
1275 for constructing numbers or for doing arithmetic with them. The objects
1276 one deals with most of the time are the polymorphic expressions @code{ex}.
1278 @subsection Tests on numbers
1280 Once you have declared some numbers, assigned them to expressions and
1281 done some arithmetic with them it is frequently desired to retrieve some
1282 kind of information from them like asking whether that number is
1283 integer, rational, real or complex. For those cases GiNaC provides
1284 several useful methods. (Internally, they fall back to invocations of
1285 certain CLN functions.)
1287 As an example, let's construct some rational number, multiply it with
1288 some multiple of its denominator and test what comes out:
1292 #include <ginac/ginac.h>
1293 using namespace std;
1294 using namespace GiNaC;
1296 // some very important constants:
1297 const numeric twentyone(21);
1298 const numeric ten(10);
1299 const numeric five(5);
1303 numeric answer = twentyone;
1306 cout << answer.is_integer() << endl; // false, it's 21/5
1308 cout << answer.is_integer() << endl; // true, it's 42 now!
1312 Note that the variable @code{answer} is constructed here as an integer
1313 by @code{numeric}'s copy constructor but in an intermediate step it
1314 holds a rational number represented as integer numerator and integer
1315 denominator. When multiplied by 10, the denominator becomes unity and
1316 the result is automatically converted to a pure integer again.
1317 Internally, the underlying CLN is responsible for this behavior and we
1318 refer the reader to CLN's documentation. Suffice to say that
1319 the same behavior applies to complex numbers as well as return values of
1320 certain functions. Complex numbers are automatically converted to real
1321 numbers if the imaginary part becomes zero. The full set of tests that
1322 can be applied is listed in the following table.
1325 @multitable @columnfractions .30 .70
1326 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1327 @item @code{.is_zero()}
1328 @tab @dots{}equal to zero
1329 @item @code{.is_positive()}
1330 @tab @dots{}not complex and greater than 0
1331 @item @code{.is_integer()}
1332 @tab @dots{}a (non-complex) integer
1333 @item @code{.is_pos_integer()}
1334 @tab @dots{}an integer and greater than 0
1335 @item @code{.is_nonneg_integer()}
1336 @tab @dots{}an integer and greater equal 0
1337 @item @code{.is_even()}
1338 @tab @dots{}an even integer
1339 @item @code{.is_odd()}
1340 @tab @dots{}an odd integer
1341 @item @code{.is_prime()}
1342 @tab @dots{}a prime integer (probabilistic primality test)
1343 @item @code{.is_rational()}
1344 @tab @dots{}an exact rational number (integers are rational, too)
1345 @item @code{.is_real()}
1346 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1347 @item @code{.is_cinteger()}
1348 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1349 @item @code{.is_crational()}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1354 @subsection Numeric functions
1356 The following functions can be applied to @code{numeric} objects and will be
1357 evaluated immediately:
1360 @multitable @columnfractions .30 .70
1361 @item @strong{Name} @tab @strong{Function}
1362 @item @code{inverse(z)}
1363 @tab returns @math{1/z}
1364 @cindex @code{inverse()} (numeric)
1365 @item @code{pow(a, b)}
1366 @tab exponentiation @math{a^b}
1369 @item @code{real(z)}
1371 @cindex @code{real()}
1372 @item @code{imag(z)}
1374 @cindex @code{imag()}
1375 @item @code{csgn(z)}
1376 @tab complex sign (returns an @code{int})
1377 @item @code{numer(z)}
1378 @tab numerator of rational or complex rational number
1379 @item @code{denom(z)}
1380 @tab denominator of rational or complex rational number
1381 @item @code{sqrt(z)}
1383 @item @code{isqrt(n)}
1384 @tab integer square root
1385 @cindex @code{isqrt()}
1392 @item @code{asin(z)}
1394 @item @code{acos(z)}
1396 @item @code{atan(z)}
1397 @tab inverse tangent
1398 @item @code{atan(y, x)}
1399 @tab inverse tangent with two arguments
1400 @item @code{sinh(z)}
1401 @tab hyperbolic sine
1402 @item @code{cosh(z)}
1403 @tab hyperbolic cosine
1404 @item @code{tanh(z)}
1405 @tab hyperbolic tangent
1406 @item @code{asinh(z)}
1407 @tab inverse hyperbolic sine
1408 @item @code{acosh(z)}
1409 @tab inverse hyperbolic cosine
1410 @item @code{atanh(z)}
1411 @tab inverse hyperbolic tangent
1413 @tab exponential function
1415 @tab natural logarithm
1418 @item @code{zeta(z)}
1419 @tab Riemann's zeta function
1420 @item @code{tgamma(z)}
1422 @item @code{lgamma(z)}
1423 @tab logarithm of gamma function
1425 @tab psi (digamma) function
1426 @item @code{psi(n, z)}
1427 @tab derivatives of psi function (polygamma functions)
1428 @item @code{factorial(n)}
1429 @tab factorial function @math{n!}
1430 @item @code{doublefactorial(n)}
1431 @tab double factorial function @math{n!!}
1432 @cindex @code{doublefactorial()}
1433 @item @code{binomial(n, k)}
1434 @tab binomial coefficients
1435 @item @code{bernoulli(n)}
1436 @tab Bernoulli numbers
1437 @cindex @code{bernoulli()}
1438 @item @code{fibonacci(n)}
1439 @tab Fibonacci numbers
1440 @cindex @code{fibonacci()}
1441 @item @code{mod(a, b)}
1442 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1443 @cindex @code{mod()}
1444 @item @code{smod(a, b)}
1445 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1446 @cindex @code{smod()}
1447 @item @code{irem(a, b)}
1448 @tab integer remainder (has the sign of @math{a}, or is zero)
1449 @cindex @code{irem()}
1450 @item @code{irem(a, b, q)}
1451 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1452 @item @code{iquo(a, b)}
1453 @tab integer quotient
1454 @cindex @code{iquo()}
1455 @item @code{iquo(a, b, r)}
1456 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1457 @item @code{gcd(a, b)}
1458 @tab greatest common divisor
1459 @item @code{lcm(a, b)}
1460 @tab least common multiple
1464 Most of these functions are also available as symbolic functions that can be
1465 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1466 as polynomial algorithms.
1468 @subsection Converting numbers
1470 Sometimes it is desirable to convert a @code{numeric} object back to a
1471 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1472 class provides a couple of methods for this purpose:
1474 @cindex @code{to_int()}
1475 @cindex @code{to_long()}
1476 @cindex @code{to_double()}
1477 @cindex @code{to_cl_N()}
1479 int numeric::to_int() const;
1480 long numeric::to_long() const;
1481 double numeric::to_double() const;
1482 cln::cl_N numeric::to_cl_N() const;
1485 @code{to_int()} and @code{to_long()} only work when the number they are
1486 applied on is an exact integer. Otherwise the program will halt with a
1487 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1488 rational number will return a floating-point approximation. Both
1489 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1490 part of complex numbers.
1493 @node Constants, Fundamental containers, Numbers, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{constant} (class)
1499 @cindex @code{Catalan}
1500 @cindex @code{Euler}
1501 @cindex @code{evalf()}
1502 Constants behave pretty much like symbols except that they return some
1503 specific number when the method @code{.evalf()} is called.
1505 The predefined known constants are:
1508 @multitable @columnfractions .14 .30 .56
1509 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1511 @tab Archimedes' constant
1512 @tab 3.14159265358979323846264338327950288
1513 @item @code{Catalan}
1514 @tab Catalan's constant
1515 @tab 0.91596559417721901505460351493238411
1517 @tab Euler's (or Euler-Mascheroni) constant
1518 @tab 0.57721566490153286060651209008240243
1523 @node Fundamental containers, Lists, Constants, Basic Concepts
1524 @c node-name, next, previous, up
1525 @section Sums, products and powers
1529 @cindex @code{power}
1531 Simple rational expressions are written down in GiNaC pretty much like
1532 in other CAS or like expressions involving numerical variables in C.
1533 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1534 been overloaded to achieve this goal. When you run the following
1535 code snippet, the constructor for an object of type @code{mul} is
1536 automatically called to hold the product of @code{a} and @code{b} and
1537 then the constructor for an object of type @code{add} is called to hold
1538 the sum of that @code{mul} object and the number one:
1542 symbol a("a"), b("b");
1547 @cindex @code{pow()}
1548 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1549 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1550 construction is necessary since we cannot safely overload the constructor
1551 @code{^} in C++ to construct a @code{power} object. If we did, it would
1552 have several counterintuitive and undesired effects:
1556 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1558 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1559 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1560 interpret this as @code{x^(a^b)}.
1562 Also, expressions involving integer exponents are very frequently used,
1563 which makes it even more dangerous to overload @code{^} since it is then
1564 hard to distinguish between the semantics as exponentiation and the one
1565 for exclusive or. (It would be embarrassing to return @code{1} where one
1566 has requested @code{2^3}.)
1569 @cindex @command{ginsh}
1570 All effects are contrary to mathematical notation and differ from the
1571 way most other CAS handle exponentiation, therefore overloading @code{^}
1572 is ruled out for GiNaC's C++ part. The situation is different in
1573 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1574 that the other frequently used exponentiation operator @code{**} does
1575 not exist at all in C++).
1577 To be somewhat more precise, objects of the three classes described
1578 here, are all containers for other expressions. An object of class
1579 @code{power} is best viewed as a container with two slots, one for the
1580 basis, one for the exponent. All valid GiNaC expressions can be
1581 inserted. However, basic transformations like simplifying
1582 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1583 when this is mathematically possible. If we replace the outer exponent
1584 three in the example by some symbols @code{a}, the simplification is not
1585 safe and will not be performed, since @code{a} might be @code{1/2} and
1588 Objects of type @code{add} and @code{mul} are containers with an
1589 arbitrary number of slots for expressions to be inserted. Again, simple
1590 and safe simplifications are carried out like transforming
1591 @code{3*x+4-x} to @code{2*x+4}.
1594 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1595 @c node-name, next, previous, up
1596 @section Lists of expressions
1597 @cindex @code{lst} (class)
1599 @cindex @code{nops()}
1601 @cindex @code{append()}
1602 @cindex @code{prepend()}
1603 @cindex @code{remove_first()}
1604 @cindex @code{remove_last()}
1605 @cindex @code{remove_all()}
1607 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1608 expressions. They are not as ubiquitous as in many other computer algebra
1609 packages, but are sometimes used to supply a variable number of arguments of
1610 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1611 constructors, so you should have a basic understanding of them.
1613 Lists can be constructed by assigning a comma-separated sequence of
1618 symbol x("x"), y("y");
1621 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1626 There are also constructors that allow direct creation of lists of up to
1627 16 expressions, which is often more convenient but slightly less efficient:
1631 // This produces the same list 'l' as above:
1632 // lst l(x, 2, y, x+y);
1633 // lst l = lst(x, 2, y, x+y);
1637 Use the @code{nops()} method to determine the size (number of expressions) of
1638 a list and the @code{op()} method or the @code{[]} operator to access
1639 individual elements:
1643 cout << l.nops() << endl; // prints '4'
1644 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1648 As with the standard @code{list<T>} container, accessing random elements of a
1649 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1650 sequential access to the elements of a list is possible with the
1651 iterator types provided by the @code{lst} class:
1654 typedef ... lst::const_iterator;
1655 typedef ... lst::const_reverse_iterator;
1656 lst::const_iterator lst::begin() const;
1657 lst::const_iterator lst::end() const;
1658 lst::const_reverse_iterator lst::rbegin() const;
1659 lst::const_reverse_iterator lst::rend() const;
1662 For example, to print the elements of a list individually you can use:
1667 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1672 which is one order faster than
1677 for (size_t i = 0; i < l.nops(); ++i)
1678 cout << l.op(i) << endl;
1682 These iterators also allow you to use some of the algorithms provided by
1683 the C++ standard library:
1687 // print the elements of the list (requires #include <iterator>)
1688 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1690 // sum up the elements of the list (requires #include <numeric>)
1691 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1692 cout << sum << endl; // prints '2+2*x+2*y'
1696 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1697 (the only other one is @code{matrix}). You can modify single elements:
1701 l[1] = 42; // l is now @{x, 42, y, x+y@}
1702 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1706 You can append or prepend an expression to a list with the @code{append()}
1707 and @code{prepend()} methods:
1711 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1712 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1716 You can remove the first or last element of a list with @code{remove_first()}
1717 and @code{remove_last()}:
1721 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1722 l.remove_last(); // l is now @{x, 7, y, x+y@}
1726 You can remove all the elements of a list with @code{remove_all()}:
1730 l.remove_all(); // l is now empty
1734 You can bring the elements of a list into a canonical order with @code{sort()}:
1743 // l1 and l2 are now equal
1747 Finally, you can remove all but the first element of consecutive groups of
1748 elements with @code{unique()}:
1753 l3 = x, 2, 2, 2, y, x+y, y+x;
1754 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1759 @node Mathematical functions, Relations, Lists, Basic Concepts
1760 @c node-name, next, previous, up
1761 @section Mathematical functions
1762 @cindex @code{function} (class)
1763 @cindex trigonometric function
1764 @cindex hyperbolic function
1766 There are quite a number of useful functions hard-wired into GiNaC. For
1767 instance, all trigonometric and hyperbolic functions are implemented
1768 (@xref{Built-in Functions}, for a complete list).
1770 These functions (better called @emph{pseudofunctions}) are all objects
1771 of class @code{function}. They accept one or more expressions as
1772 arguments and return one expression. If the arguments are not
1773 numerical, the evaluation of the function may be halted, as it does in
1774 the next example, showing how a function returns itself twice and
1775 finally an expression that may be really useful:
1777 @cindex Gamma function
1778 @cindex @code{subs()}
1781 symbol x("x"), y("y");
1783 cout << tgamma(foo) << endl;
1784 // -> tgamma(x+(1/2)*y)
1785 ex bar = foo.subs(y==1);
1786 cout << tgamma(bar) << endl;
1788 ex foobar = bar.subs(x==7);
1789 cout << tgamma(foobar) << endl;
1790 // -> (135135/128)*Pi^(1/2)
1794 Besides evaluation most of these functions allow differentiation, series
1795 expansion and so on. Read the next chapter in order to learn more about
1798 It must be noted that these pseudofunctions are created by inline
1799 functions, where the argument list is templated. This means that
1800 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1801 @code{sin(ex(1))} and will therefore not result in a floating point
1802 number. Unless of course the function prototype is explicitly
1803 overridden -- which is the case for arguments of type @code{numeric}
1804 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1805 point number of class @code{numeric} you should call
1806 @code{sin(numeric(1))}. This is almost the same as calling
1807 @code{sin(1).evalf()} except that the latter will return a numeric
1808 wrapped inside an @code{ex}.
1811 @node Relations, Integrals, Mathematical functions, Basic Concepts
1812 @c node-name, next, previous, up
1814 @cindex @code{relational} (class)
1816 Sometimes, a relation holding between two expressions must be stored
1817 somehow. The class @code{relational} is a convenient container for such
1818 purposes. A relation is by definition a container for two @code{ex} and
1819 a relation between them that signals equality, inequality and so on.
1820 They are created by simply using the C++ operators @code{==}, @code{!=},
1821 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1823 @xref{Mathematical functions}, for examples where various applications
1824 of the @code{.subs()} method show how objects of class relational are
1825 used as arguments. There they provide an intuitive syntax for
1826 substitutions. They are also used as arguments to the @code{ex::series}
1827 method, where the left hand side of the relation specifies the variable
1828 to expand in and the right hand side the expansion point. They can also
1829 be used for creating systems of equations that are to be solved for
1830 unknown variables. But the most common usage of objects of this class
1831 is rather inconspicuous in statements of the form @code{if
1832 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1833 conversion from @code{relational} to @code{bool} takes place. Note,
1834 however, that @code{==} here does not perform any simplifications, hence
1835 @code{expand()} must be called explicitly.
1837 @node Integrals, Matrices, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{integral} (class)
1842 An object of class @dfn{integral} can be used to hold a symbolic integral.
1843 If you want to symbolically represent the integral of @code{x*x} from 0 to
1844 1, you would write this as
1846 integral(x, 0, 1, x*x)
1848 The first argument is the integration variable. It should be noted that
1849 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1850 fact, it can only integrate polynomials. An expression containing integrals
1851 can be evaluated symbolically by calling the
1855 method on it. Numerical evaluation is available by calling the
1859 method on an expression containing the integral. This will only evaluate
1860 integrals into a number if @code{subs}ing the integration variable by a
1861 number in the fourth argument of an integral and then @code{evalf}ing the
1862 result always results in a number. Of course, also the boundaries of the
1863 integration domain must @code{evalf} into numbers. It should be noted that
1864 trying to @code{evalf} a function with discontinuities in the integration
1865 domain is not recommended. The accuracy of the numeric evaluation of
1866 integrals is determined by the static member variable
1868 ex integral::relative_integration_error
1870 of the class @code{integral}. The default value of this is 10^-8.
1871 The integration works by halving the interval of integration, until numeric
1872 stability of the answer indicates that the requested accuracy has been
1873 reached. The maximum depth of the halving can be set via the static member
1876 int integral::max_integration_level
1878 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1879 return the integral unevaluated. The function that performs the numerical
1880 evaluation, is also available as
1882 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1885 This function will throw an exception if the maximum depth is exceeded. The
1886 last parameter of the function is optional and defaults to the
1887 @code{relative_integration_error}. To make sure that we do not do too
1888 much work if an expression contains the same integral multiple times,
1889 a lookup table is used.
1891 If you know that an expression holds an integral, you can get the
1892 integration variable, the left boundary, right boundary and integrant by
1893 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1894 @code{.op(3)}. Differentiating integrals with respect to variables works
1895 as expected. Note that it makes no sense to differentiate an integral
1896 with respect to the integration variable.
1898 @node Matrices, Indexed objects, Integrals, Basic Concepts
1899 @c node-name, next, previous, up
1901 @cindex @code{matrix} (class)
1903 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1904 matrix with @math{m} rows and @math{n} columns are accessed with two
1905 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1906 second one in the range 0@dots{}@math{n-1}.
1908 There are a couple of ways to construct matrices, with or without preset
1909 elements. The constructor
1912 matrix::matrix(unsigned r, unsigned c);
1915 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1918 The fastest way to create a matrix with preinitialized elements is to assign
1919 a list of comma-separated expressions to an empty matrix (see below for an
1920 example). But you can also specify the elements as a (flat) list with
1923 matrix::matrix(unsigned r, unsigned c, const lst & l);
1928 @cindex @code{lst_to_matrix()}
1930 ex lst_to_matrix(const lst & l);
1933 constructs a matrix from a list of lists, each list representing a matrix row.
1935 There is also a set of functions for creating some special types of
1938 @cindex @code{diag_matrix()}
1939 @cindex @code{unit_matrix()}
1940 @cindex @code{symbolic_matrix()}
1942 ex diag_matrix(const lst & l);
1943 ex unit_matrix(unsigned x);
1944 ex unit_matrix(unsigned r, unsigned c);
1945 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1946 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1949 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1950 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1951 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1952 matrix filled with newly generated symbols made of the specified base name
1953 and the position of each element in the matrix.
1955 Matrix elements can be accessed and set using the parenthesis (function call)
1959 const ex & matrix::operator()(unsigned r, unsigned c) const;
1960 ex & matrix::operator()(unsigned r, unsigned c);
1963 It is also possible to access the matrix elements in a linear fashion with
1964 the @code{op()} method. But C++-style subscripting with square brackets
1965 @samp{[]} is not available.
1967 Here are a couple of examples for constructing matrices:
1971 symbol a("a"), b("b");
1985 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1988 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1991 cout << diag_matrix(lst(a, b)) << endl;
1994 cout << unit_matrix(3) << endl;
1995 // -> [[1,0,0],[0,1,0],[0,0,1]]
1997 cout << symbolic_matrix(2, 3, "x") << endl;
1998 // -> [[x00,x01,x02],[x10,x11,x12]]
2002 @cindex @code{transpose()}
2003 There are three ways to do arithmetic with matrices. The first (and most
2004 direct one) is to use the methods provided by the @code{matrix} class:
2007 matrix matrix::add(const matrix & other) const;
2008 matrix matrix::sub(const matrix & other) const;
2009 matrix matrix::mul(const matrix & other) const;
2010 matrix matrix::mul_scalar(const ex & other) const;
2011 matrix matrix::pow(const ex & expn) const;
2012 matrix matrix::transpose() const;
2015 All of these methods return the result as a new matrix object. Here is an
2016 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2021 matrix A(2, 2), B(2, 2), C(2, 2);
2029 matrix result = A.mul(B).sub(C.mul_scalar(2));
2030 cout << result << endl;
2031 // -> [[-13,-6],[1,2]]
2036 @cindex @code{evalm()}
2037 The second (and probably the most natural) way is to construct an expression
2038 containing matrices with the usual arithmetic operators and @code{pow()}.
2039 For efficiency reasons, expressions with sums, products and powers of
2040 matrices are not automatically evaluated in GiNaC. You have to call the
2044 ex ex::evalm() const;
2047 to obtain the result:
2054 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2055 cout << e.evalm() << endl;
2056 // -> [[-13,-6],[1,2]]
2061 The non-commutativity of the product @code{A*B} in this example is
2062 automatically recognized by GiNaC. There is no need to use a special
2063 operator here. @xref{Non-commutative objects}, for more information about
2064 dealing with non-commutative expressions.
2066 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2067 to perform the arithmetic:
2072 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2073 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2075 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2076 cout << e.simplify_indexed() << endl;
2077 // -> [[-13,-6],[1,2]].i.j
2081 Using indices is most useful when working with rectangular matrices and
2082 one-dimensional vectors because you don't have to worry about having to
2083 transpose matrices before multiplying them. @xref{Indexed objects}, for
2084 more information about using matrices with indices, and about indices in
2087 The @code{matrix} class provides a couple of additional methods for
2088 computing determinants, traces, characteristic polynomials and ranks:
2090 @cindex @code{determinant()}
2091 @cindex @code{trace()}
2092 @cindex @code{charpoly()}
2093 @cindex @code{rank()}
2095 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2096 ex matrix::trace() const;
2097 ex matrix::charpoly(const ex & lambda) const;
2098 unsigned matrix::rank() const;
2101 The @samp{algo} argument of @code{determinant()} allows to select
2102 between different algorithms for calculating the determinant. The
2103 asymptotic speed (as parametrized by the matrix size) can greatly differ
2104 between those algorithms, depending on the nature of the matrix'
2105 entries. The possible values are defined in the @file{flags.h} header
2106 file. By default, GiNaC uses a heuristic to automatically select an
2107 algorithm that is likely (but not guaranteed) to give the result most
2110 @cindex @code{inverse()} (matrix)
2111 @cindex @code{solve()}
2112 Matrices may also be inverted using the @code{ex matrix::inverse()}
2113 method and linear systems may be solved with:
2116 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2119 Assuming the matrix object this method is applied on is an @code{m}
2120 times @code{n} matrix, then @code{vars} must be a @code{n} times
2121 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2122 times @code{p} matrix. The returned matrix then has dimension @code{n}
2123 times @code{p} and in the case of an underdetermined system will still
2124 contain some of the indeterminates from @code{vars}. If the system is
2125 overdetermined, an exception is thrown.
2128 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2129 @c node-name, next, previous, up
2130 @section Indexed objects
2132 GiNaC allows you to handle expressions containing general indexed objects in
2133 arbitrary spaces. It is also able to canonicalize and simplify such
2134 expressions and perform symbolic dummy index summations. There are a number
2135 of predefined indexed objects provided, like delta and metric tensors.
2137 There are few restrictions placed on indexed objects and their indices and
2138 it is easy to construct nonsense expressions, but our intention is to
2139 provide a general framework that allows you to implement algorithms with
2140 indexed quantities, getting in the way as little as possible.
2142 @cindex @code{idx} (class)
2143 @cindex @code{indexed} (class)
2144 @subsection Indexed quantities and their indices
2146 Indexed expressions in GiNaC are constructed of two special types of objects,
2147 @dfn{index objects} and @dfn{indexed objects}.
2151 @cindex contravariant
2154 @item Index objects are of class @code{idx} or a subclass. Every index has
2155 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2156 the index lives in) which can both be arbitrary expressions but are usually
2157 a number or a simple symbol. In addition, indices of class @code{varidx} have
2158 a @dfn{variance} (they can be co- or contravariant), and indices of class
2159 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2161 @item Indexed objects are of class @code{indexed} or a subclass. They
2162 contain a @dfn{base expression} (which is the expression being indexed), and
2163 one or more indices.
2167 @strong{Note:} when printing expressions, covariant indices and indices
2168 without variance are denoted @samp{.i} while contravariant indices are
2169 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2170 value. In the following, we are going to use that notation in the text so
2171 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2172 not visible in the output.
2174 A simple example shall illustrate the concepts:
2178 #include <ginac/ginac.h>
2179 using namespace std;
2180 using namespace GiNaC;
2184 symbol i_sym("i"), j_sym("j");
2185 idx i(i_sym, 3), j(j_sym, 3);
2188 cout << indexed(A, i, j) << endl;
2190 cout << index_dimensions << indexed(A, i, j) << endl;
2192 cout << dflt; // reset cout to default output format (dimensions hidden)
2196 The @code{idx} constructor takes two arguments, the index value and the
2197 index dimension. First we define two index objects, @code{i} and @code{j},
2198 both with the numeric dimension 3. The value of the index @code{i} is the
2199 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2200 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2201 construct an expression containing one indexed object, @samp{A.i.j}. It has
2202 the symbol @code{A} as its base expression and the two indices @code{i} and
2205 The dimensions of indices are normally not visible in the output, but one
2206 can request them to be printed with the @code{index_dimensions} manipulator,
2209 Note the difference between the indices @code{i} and @code{j} which are of
2210 class @code{idx}, and the index values which are the symbols @code{i_sym}
2211 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2212 or numbers but must be index objects. For example, the following is not
2213 correct and will raise an exception:
2216 symbol i("i"), j("j");
2217 e = indexed(A, i, j); // ERROR: indices must be of type idx
2220 You can have multiple indexed objects in an expression, index values can
2221 be numeric, and index dimensions symbolic:
2225 symbol B("B"), dim("dim");
2226 cout << 4 * indexed(A, i)
2227 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2232 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2233 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2234 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2235 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2236 @code{simplify_indexed()} for that, see below).
2238 In fact, base expressions, index values and index dimensions can be
2239 arbitrary expressions:
2243 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2248 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2249 get an error message from this but you will probably not be able to do
2250 anything useful with it.
2252 @cindex @code{get_value()}
2253 @cindex @code{get_dimension()}
2257 ex idx::get_value();
2258 ex idx::get_dimension();
2261 return the value and dimension of an @code{idx} object. If you have an index
2262 in an expression, such as returned by calling @code{.op()} on an indexed
2263 object, you can get a reference to the @code{idx} object with the function
2264 @code{ex_to<idx>()} on the expression.
2266 There are also the methods
2269 bool idx::is_numeric();
2270 bool idx::is_symbolic();
2271 bool idx::is_dim_numeric();
2272 bool idx::is_dim_symbolic();
2275 for checking whether the value and dimension are numeric or symbolic
2276 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2277 About Expressions}) returns information about the index value.
2279 @cindex @code{varidx} (class)
2280 If you need co- and contravariant indices, use the @code{varidx} class:
2284 symbol mu_sym("mu"), nu_sym("nu");
2285 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2286 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2288 cout << indexed(A, mu, nu) << endl;
2290 cout << indexed(A, mu_co, nu) << endl;
2292 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2297 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2298 co- or contravariant. The default is a contravariant (upper) index, but
2299 this can be overridden by supplying a third argument to the @code{varidx}
2300 constructor. The two methods
2303 bool varidx::is_covariant();
2304 bool varidx::is_contravariant();
2307 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2308 to get the object reference from an expression). There's also the very useful
2312 ex varidx::toggle_variance();
2315 which makes a new index with the same value and dimension but the opposite
2316 variance. By using it you only have to define the index once.
2318 @cindex @code{spinidx} (class)
2319 The @code{spinidx} class provides dotted and undotted variant indices, as
2320 used in the Weyl-van-der-Waerden spinor formalism:
2324 symbol K("K"), C_sym("C"), D_sym("D");
2325 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2326 // contravariant, undotted
2327 spinidx C_co(C_sym, 2, true); // covariant index
2328 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2329 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2331 cout << indexed(K, C, D) << endl;
2333 cout << indexed(K, C_co, D_dot) << endl;
2335 cout << indexed(K, D_co_dot, D) << endl;
2340 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2341 dotted or undotted. The default is undotted but this can be overridden by
2342 supplying a fourth argument to the @code{spinidx} constructor. The two
2346 bool spinidx::is_dotted();
2347 bool spinidx::is_undotted();
2350 allow you to check whether or not a @code{spinidx} object is dotted (use
2351 @code{ex_to<spinidx>()} to get the object reference from an expression).
2352 Finally, the two methods
2355 ex spinidx::toggle_dot();
2356 ex spinidx::toggle_variance_dot();
2359 create a new index with the same value and dimension but opposite dottedness
2360 and the same or opposite variance.
2362 @subsection Substituting indices
2364 @cindex @code{subs()}
2365 Sometimes you will want to substitute one symbolic index with another
2366 symbolic or numeric index, for example when calculating one specific element
2367 of a tensor expression. This is done with the @code{.subs()} method, as it
2368 is done for symbols (see @ref{Substituting Expressions}).
2370 You have two possibilities here. You can either substitute the whole index
2371 by another index or expression:
2375 ex e = indexed(A, mu_co);
2376 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2377 // -> A.mu becomes A~nu
2378 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2379 // -> A.mu becomes A~0
2380 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2381 // -> A.mu becomes A.0
2385 The third example shows that trying to replace an index with something that
2386 is not an index will substitute the index value instead.
2388 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2393 ex e = indexed(A, mu_co);
2394 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2395 // -> A.mu becomes A.nu
2396 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2397 // -> A.mu becomes A.0
2401 As you see, with the second method only the value of the index will get
2402 substituted. Its other properties, including its dimension, remain unchanged.
2403 If you want to change the dimension of an index you have to substitute the
2404 whole index by another one with the new dimension.
2406 Finally, substituting the base expression of an indexed object works as
2411 ex e = indexed(A, mu_co);
2412 cout << e << " becomes " << e.subs(A == A+B) << endl;
2413 // -> A.mu becomes (B+A).mu
2417 @subsection Symmetries
2418 @cindex @code{symmetry} (class)
2419 @cindex @code{sy_none()}
2420 @cindex @code{sy_symm()}
2421 @cindex @code{sy_anti()}
2422 @cindex @code{sy_cycl()}
2424 Indexed objects can have certain symmetry properties with respect to their
2425 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2426 that is constructed with the helper functions
2429 symmetry sy_none(...);
2430 symmetry sy_symm(...);
2431 symmetry sy_anti(...);
2432 symmetry sy_cycl(...);
2435 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2436 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2437 represents a cyclic symmetry. Each of these functions accepts up to four
2438 arguments which can be either symmetry objects themselves or unsigned integer
2439 numbers that represent an index position (counting from 0). A symmetry
2440 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2441 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2444 Here are some examples of symmetry definitions:
2449 e = indexed(A, i, j);
2450 e = indexed(A, sy_none(), i, j); // equivalent
2451 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2453 // Symmetric in all three indices:
2454 e = indexed(A, sy_symm(), i, j, k);
2455 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2456 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2457 // different canonical order
2459 // Symmetric in the first two indices only:
2460 e = indexed(A, sy_symm(0, 1), i, j, k);
2461 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2463 // Antisymmetric in the first and last index only (index ranges need not
2465 e = indexed(A, sy_anti(0, 2), i, j, k);
2466 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2468 // An example of a mixed symmetry: antisymmetric in the first two and
2469 // last two indices, symmetric when swapping the first and last index
2470 // pairs (like the Riemann curvature tensor):
2471 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2473 // Cyclic symmetry in all three indices:
2474 e = indexed(A, sy_cycl(), i, j, k);
2475 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2477 // The following examples are invalid constructions that will throw
2478 // an exception at run time.
2480 // An index may not appear multiple times:
2481 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2482 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2484 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2485 // same number of indices:
2486 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2488 // And of course, you cannot specify indices which are not there:
2489 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2493 If you need to specify more than four indices, you have to use the
2494 @code{.add()} method of the @code{symmetry} class. For example, to specify
2495 full symmetry in the first six indices you would write
2496 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2498 If an indexed object has a symmetry, GiNaC will automatically bring the
2499 indices into a canonical order which allows for some immediate simplifications:
2503 cout << indexed(A, sy_symm(), i, j)
2504 + indexed(A, sy_symm(), j, i) << endl;
2506 cout << indexed(B, sy_anti(), i, j)
2507 + indexed(B, sy_anti(), j, i) << endl;
2509 cout << indexed(B, sy_anti(), i, j, k)
2510 - indexed(B, sy_anti(), j, k, i) << endl;
2515 @cindex @code{get_free_indices()}
2517 @subsection Dummy indices
2519 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2520 that a summation over the index range is implied. Symbolic indices which are
2521 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2522 dummy nor free indices.
2524 To be recognized as a dummy index pair, the two indices must be of the same
2525 class and their value must be the same single symbol (an index like
2526 @samp{2*n+1} is never a dummy index). If the indices are of class
2527 @code{varidx} they must also be of opposite variance; if they are of class
2528 @code{spinidx} they must be both dotted or both undotted.
2530 The method @code{.get_free_indices()} returns a vector containing the free
2531 indices of an expression. It also checks that the free indices of the terms
2532 of a sum are consistent:
2536 symbol A("A"), B("B"), C("C");
2538 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2539 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2541 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2542 cout << exprseq(e.get_free_indices()) << endl;
2544 // 'j' and 'l' are dummy indices
2546 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2547 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2549 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2550 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2551 cout << exprseq(e.get_free_indices()) << endl;
2553 // 'nu' is a dummy index, but 'sigma' is not
2555 e = indexed(A, mu, mu);
2556 cout << exprseq(e.get_free_indices()) << endl;
2558 // 'mu' is not a dummy index because it appears twice with the same
2561 e = indexed(A, mu, nu) + 42;
2562 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2563 // this will throw an exception:
2564 // "add::get_free_indices: inconsistent indices in sum"
2568 @cindex @code{simplify_indexed()}
2569 @subsection Simplifying indexed expressions
2571 In addition to the few automatic simplifications that GiNaC performs on
2572 indexed expressions (such as re-ordering the indices of symmetric tensors
2573 and calculating traces and convolutions of matrices and predefined tensors)
2577 ex ex::simplify_indexed();
2578 ex ex::simplify_indexed(const scalar_products & sp);
2581 that performs some more expensive operations:
2584 @item it checks the consistency of free indices in sums in the same way
2585 @code{get_free_indices()} does
2586 @item it tries to give dummy indices that appear in different terms of a sum
2587 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2588 @item it (symbolically) calculates all possible dummy index summations/contractions
2589 with the predefined tensors (this will be explained in more detail in the
2591 @item it detects contractions that vanish for symmetry reasons, for example
2592 the contraction of a symmetric and a totally antisymmetric tensor
2593 @item as a special case of dummy index summation, it can replace scalar products
2594 of two tensors with a user-defined value
2597 The last point is done with the help of the @code{scalar_products} class
2598 which is used to store scalar products with known values (this is not an
2599 arithmetic class, you just pass it to @code{simplify_indexed()}):
2603 symbol A("A"), B("B"), C("C"), i_sym("i");
2607 sp.add(A, B, 0); // A and B are orthogonal
2608 sp.add(A, C, 0); // A and C are orthogonal
2609 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2611 e = indexed(A + B, i) * indexed(A + C, i);
2613 // -> (B+A).i*(A+C).i
2615 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2621 The @code{scalar_products} object @code{sp} acts as a storage for the
2622 scalar products added to it with the @code{.add()} method. This method
2623 takes three arguments: the two expressions of which the scalar product is
2624 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2625 @code{simplify_indexed()} will replace all scalar products of indexed
2626 objects that have the symbols @code{A} and @code{B} as base expressions
2627 with the single value 0. The number, type and dimension of the indices
2628 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2630 @cindex @code{expand()}
2631 The example above also illustrates a feature of the @code{expand()} method:
2632 if passed the @code{expand_indexed} option it will distribute indices
2633 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2635 @cindex @code{tensor} (class)
2636 @subsection Predefined tensors
2638 Some frequently used special tensors such as the delta, epsilon and metric
2639 tensors are predefined in GiNaC. They have special properties when
2640 contracted with other tensor expressions and some of them have constant
2641 matrix representations (they will evaluate to a number when numeric
2642 indices are specified).
2644 @cindex @code{delta_tensor()}
2645 @subsubsection Delta tensor
2647 The delta tensor takes two indices, is symmetric and has the matrix
2648 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2649 @code{delta_tensor()}:
2653 symbol A("A"), B("B");
2655 idx i(symbol("i"), 3), j(symbol("j"), 3),
2656 k(symbol("k"), 3), l(symbol("l"), 3);
2658 ex e = indexed(A, i, j) * indexed(B, k, l)
2659 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2660 cout << e.simplify_indexed() << endl;
2663 cout << delta_tensor(i, i) << endl;
2668 @cindex @code{metric_tensor()}
2669 @subsubsection General metric tensor
2671 The function @code{metric_tensor()} creates a general symmetric metric
2672 tensor with two indices that can be used to raise/lower tensor indices. The
2673 metric tensor is denoted as @samp{g} in the output and if its indices are of
2674 mixed variance it is automatically replaced by a delta tensor:
2680 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2682 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2683 cout << e.simplify_indexed() << endl;
2686 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2687 cout << e.simplify_indexed() << endl;
2690 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2691 * metric_tensor(nu, rho);
2692 cout << e.simplify_indexed() << endl;
2695 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2696 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2697 + indexed(A, mu.toggle_variance(), rho));
2698 cout << e.simplify_indexed() << endl;
2703 @cindex @code{lorentz_g()}
2704 @subsubsection Minkowski metric tensor
2706 The Minkowski metric tensor is a special metric tensor with a constant
2707 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2708 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2709 It is created with the function @code{lorentz_g()} (although it is output as
2714 varidx mu(symbol("mu"), 4);
2716 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2717 * lorentz_g(mu, varidx(0, 4)); // negative signature
2718 cout << e.simplify_indexed() << endl;
2721 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2722 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2723 cout << e.simplify_indexed() << endl;
2728 @cindex @code{spinor_metric()}
2729 @subsubsection Spinor metric tensor
2731 The function @code{spinor_metric()} creates an antisymmetric tensor with
2732 two indices that is used to raise/lower indices of 2-component spinors.
2733 It is output as @samp{eps}:
2739 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2740 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2742 e = spinor_metric(A, B) * indexed(psi, B_co);
2743 cout << e.simplify_indexed() << endl;
2746 e = spinor_metric(A, B) * indexed(psi, A_co);
2747 cout << e.simplify_indexed() << endl;
2750 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2751 cout << e.simplify_indexed() << endl;
2754 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2755 cout << e.simplify_indexed() << endl;
2758 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2759 cout << e.simplify_indexed() << endl;
2762 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2763 cout << e.simplify_indexed() << endl;
2768 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2770 @cindex @code{epsilon_tensor()}
2771 @cindex @code{lorentz_eps()}
2772 @subsubsection Epsilon tensor
2774 The epsilon tensor is totally antisymmetric, its number of indices is equal
2775 to the dimension of the index space (the indices must all be of the same
2776 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2777 defined to be 1. Its behavior with indices that have a variance also
2778 depends on the signature of the metric. Epsilon tensors are output as
2781 There are three functions defined to create epsilon tensors in 2, 3 and 4
2785 ex epsilon_tensor(const ex & i1, const ex & i2);
2786 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2787 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2790 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2791 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2792 Minkowski space (the last @code{bool} argument specifies whether the metric
2793 has negative or positive signature, as in the case of the Minkowski metric
2798 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2799 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2800 e = lorentz_eps(mu, nu, rho, sig) *
2801 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2802 cout << simplify_indexed(e) << endl;
2803 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2805 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2806 symbol A("A"), B("B");
2807 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2808 cout << simplify_indexed(e) << endl;
2809 // -> -B.k*A.j*eps.i.k.j
2810 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2811 cout << simplify_indexed(e) << endl;
2816 @subsection Linear algebra
2818 The @code{matrix} class can be used with indices to do some simple linear
2819 algebra (linear combinations and products of vectors and matrices, traces
2820 and scalar products):
2824 idx i(symbol("i"), 2), j(symbol("j"), 2);
2825 symbol x("x"), y("y");
2827 // A is a 2x2 matrix, X is a 2x1 vector
2828 matrix A(2, 2), X(2, 1);
2833 cout << indexed(A, i, i) << endl;
2836 ex e = indexed(A, i, j) * indexed(X, j);
2837 cout << e.simplify_indexed() << endl;
2838 // -> [[2*y+x],[4*y+3*x]].i
2840 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2841 cout << e.simplify_indexed() << endl;
2842 // -> [[3*y+3*x,6*y+2*x]].j
2846 You can of course obtain the same results with the @code{matrix::add()},
2847 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2848 but with indices you don't have to worry about transposing matrices.
2850 Matrix indices always start at 0 and their dimension must match the number
2851 of rows/columns of the matrix. Matrices with one row or one column are
2852 vectors and can have one or two indices (it doesn't matter whether it's a
2853 row or a column vector). Other matrices must have two indices.
2855 You should be careful when using indices with variance on matrices. GiNaC
2856 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2857 @samp{F.mu.nu} are different matrices. In this case you should use only
2858 one form for @samp{F} and explicitly multiply it with a matrix representation
2859 of the metric tensor.
2862 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2863 @c node-name, next, previous, up
2864 @section Non-commutative objects
2866 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2867 non-commutative objects are built-in which are mostly of use in high energy
2871 @item Clifford (Dirac) algebra (class @code{clifford})
2872 @item su(3) Lie algebra (class @code{color})
2873 @item Matrices (unindexed) (class @code{matrix})
2876 The @code{clifford} and @code{color} classes are subclasses of
2877 @code{indexed} because the elements of these algebras usually carry
2878 indices. The @code{matrix} class is described in more detail in
2881 Unlike most computer algebra systems, GiNaC does not primarily provide an
2882 operator (often denoted @samp{&*}) for representing inert products of
2883 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2884 classes of objects involved, and non-commutative products are formed with
2885 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2886 figuring out by itself which objects commutate and will group the factors
2887 by their class. Consider this example:
2891 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2892 idx a(symbol("a"), 8), b(symbol("b"), 8);
2893 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2895 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2899 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2900 groups the non-commutative factors (the gammas and the su(3) generators)
2901 together while preserving the order of factors within each class (because
2902 Clifford objects commutate with color objects). The resulting expression is a
2903 @emph{commutative} product with two factors that are themselves non-commutative
2904 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2905 parentheses are placed around the non-commutative products in the output.
2907 @cindex @code{ncmul} (class)
2908 Non-commutative products are internally represented by objects of the class
2909 @code{ncmul}, as opposed to commutative products which are handled by the
2910 @code{mul} class. You will normally not have to worry about this distinction,
2913 The advantage of this approach is that you never have to worry about using
2914 (or forgetting to use) a special operator when constructing non-commutative
2915 expressions. Also, non-commutative products in GiNaC are more intelligent
2916 than in other computer algebra systems; they can, for example, automatically
2917 canonicalize themselves according to rules specified in the implementation
2918 of the non-commutative classes. The drawback is that to work with other than
2919 the built-in algebras you have to implement new classes yourself. Symbols
2920 always commutate and it's not possible to construct non-commutative products
2921 using symbols to represent the algebra elements or generators. User-defined
2922 functions can, however, be specified as being non-commutative.
2924 @cindex @code{return_type()}
2925 @cindex @code{return_type_tinfo()}
2926 Information about the commutativity of an object or expression can be
2927 obtained with the two member functions
2930 unsigned ex::return_type() const;
2931 unsigned ex::return_type_tinfo() const;
2934 The @code{return_type()} function returns one of three values (defined in
2935 the header file @file{flags.h}), corresponding to three categories of
2936 expressions in GiNaC:
2939 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2940 classes are of this kind.
2941 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2942 certain class of non-commutative objects which can be determined with the
2943 @code{return_type_tinfo()} method. Expressions of this category commutate
2944 with everything except @code{noncommutative} expressions of the same
2946 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2947 of non-commutative objects of different classes. Expressions of this
2948 category don't commutate with any other @code{noncommutative} or
2949 @code{noncommutative_composite} expressions.
2952 The value returned by the @code{return_type_tinfo()} method is valid only
2953 when the return type of the expression is @code{noncommutative}. It is a
2954 value that is unique to the class of the object and usually one of the
2955 constants in @file{tinfos.h}, or derived therefrom.
2957 Here are a couple of examples:
2960 @multitable @columnfractions 0.33 0.33 0.34
2961 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2962 @item @code{42} @tab @code{commutative} @tab -
2963 @item @code{2*x-y} @tab @code{commutative} @tab -
2964 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2965 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2966 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2967 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2971 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2972 @code{TINFO_clifford} for objects with a representation label of zero.
2973 Other representation labels yield a different @code{return_type_tinfo()},
2974 but it's the same for any two objects with the same label. This is also true
2977 A last note: With the exception of matrices, positive integer powers of
2978 non-commutative objects are automatically expanded in GiNaC. For example,
2979 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2980 non-commutative expressions).
2983 @cindex @code{clifford} (class)
2984 @subsection Clifford algebra
2987 Clifford algebras are supported in two flavours: Dirac gamma
2988 matrices (more physical) and generic Clifford algebras (more
2991 @cindex @code{dirac_gamma()}
2992 @subsubsection Dirac gamma matrices
2993 Dirac gamma matrices (note that GiNaC doesn't treat them
2994 as matrices) are designated as @samp{gamma~mu} and satisfy
2995 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
2996 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
2997 constructed by the function
3000 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3003 which takes two arguments: the index and a @dfn{representation label} in the
3004 range 0 to 255 which is used to distinguish elements of different Clifford
3005 algebras (this is also called a @dfn{spin line index}). Gammas with different
3006 labels commutate with each other. The dimension of the index can be 4 or (in
3007 the framework of dimensional regularization) any symbolic value. Spinor
3008 indices on Dirac gammas are not supported in GiNaC.
3010 @cindex @code{dirac_ONE()}
3011 The unity element of a Clifford algebra is constructed by
3014 ex dirac_ONE(unsigned char rl = 0);
3017 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
3018 multiples of the unity element, even though it's customary to omit it.
3019 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3020 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3021 GiNaC will complain and/or produce incorrect results.
3023 @cindex @code{dirac_gamma5()}
3024 There is a special element @samp{gamma5} that commutates with all other
3025 gammas, has a unit square, and in 4 dimensions equals
3026 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3029 ex dirac_gamma5(unsigned char rl = 0);
3032 @cindex @code{dirac_gammaL()}
3033 @cindex @code{dirac_gammaR()}
3034 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3035 objects, constructed by
3038 ex dirac_gammaL(unsigned char rl = 0);
3039 ex dirac_gammaR(unsigned char rl = 0);
3042 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3043 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3045 @cindex @code{dirac_slash()}
3046 Finally, the function
3049 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3052 creates a term that represents a contraction of @samp{e} with the Dirac
3053 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3054 with a unique index whose dimension is given by the @code{dim} argument).
3055 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3057 In products of dirac gammas, superfluous unity elements are automatically
3058 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3059 and @samp{gammaR} are moved to the front.
3061 The @code{simplify_indexed()} function performs contractions in gamma strings,
3067 symbol a("a"), b("b"), D("D");
3068 varidx mu(symbol("mu"), D);
3069 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3070 * dirac_gamma(mu.toggle_variance());
3072 // -> gamma~mu*a\*gamma.mu
3073 e = e.simplify_indexed();
3076 cout << e.subs(D == 4) << endl;
3082 @cindex @code{dirac_trace()}
3083 To calculate the trace of an expression containing strings of Dirac gammas
3084 you use one of the functions
3087 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
3088 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3089 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3092 These functions take the trace over all gammas in the specified set @code{rls}
3093 or list @code{rll} of representation labels, or the single label @code{rl};
3094 gammas with other labels are left standing. The last argument to
3095 @code{dirac_trace()} is the value to be returned for the trace of the unity
3096 element, which defaults to 4.
3098 The @code{dirac_trace()} function is a linear functional that is equal to the
3099 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3100 functional is not cyclic in @math{D != 4} dimensions when acting on
3101 expressions containing @samp{gamma5}, so it's not a proper trace. This
3102 @samp{gamma5} scheme is described in greater detail in
3103 @cite{The Role of gamma5 in Dimensional Regularization}.
3105 The value of the trace itself is also usually different in 4 and in
3106 @math{D != 4} dimensions:
3111 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3112 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3113 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3114 cout << dirac_trace(e).simplify_indexed() << endl;
3121 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3122 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3123 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3124 cout << dirac_trace(e).simplify_indexed() << endl;
3125 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3129 Here is an example for using @code{dirac_trace()} to compute a value that
3130 appears in the calculation of the one-loop vacuum polarization amplitude in
3135 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3136 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3139 sp.add(l, l, pow(l, 2));
3140 sp.add(l, q, ldotq);
3142 ex e = dirac_gamma(mu) *
3143 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3144 dirac_gamma(mu.toggle_variance()) *
3145 (dirac_slash(l, D) + m * dirac_ONE());
3146 e = dirac_trace(e).simplify_indexed(sp);
3147 e = e.collect(lst(l, ldotq, m));
3149 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3153 The @code{canonicalize_clifford()} function reorders all gamma products that
3154 appear in an expression to a canonical (but not necessarily simple) form.
3155 You can use this to compare two expressions or for further simplifications:
3159 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3160 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3162 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3164 e = canonicalize_clifford(e);
3166 // -> 2*ONE*eta~mu~nu
3170 @cindex @code{clifford_unit()}
3171 @subsubsection A generic Clifford algebra
3173 A generic Clifford algebra, i.e. a
3177 dimensional algebra with
3178 generators @samp{e~k} satisfying the identities
3179 @samp{e~i e~j + e~j e~i = B(i, j)} for some symmetric matrix (@code{metric})
3180 @math{B(i, j)}. Such generators are created by the function
3183 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3186 where @code{mu} should be a @code{varidx} class object indexing the
3187 generators, @code{metr} defines the metric @math{B(i, j)} and can be
3188 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3189 object, optional parameter @code{rl} allows to distinguish different
3190 Clifford algebras (which will commute with each other). Note that the call
3191 @code{clifford_unit(mu, minkmetric())} creates something very close to
3192 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3193 metric defining this Clifford number.
3195 Individual generators of a Clifford algebra can be accessed in several
3201 varidx nu(symbol("nu"), 3);
3202 matrix M(3, 3) = 1, 0, 0,
3205 ex e = clifford_unit(nu, M);
3206 ex e0 = e.subs(nu == 0);
3207 ex e1 = e.subs(nu == 1);
3208 ex e2 = e.subs(nu == 2);
3213 will produce three generators of a Clifford algebra with properties
3214 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1} and @code{pow(e2, 2) = 0}.
3216 @cindex @code{lst_to_clifford()}
3217 A similar effect can be achieved from the function
3220 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3221 unsigned char rl = 0);
3224 which converts a list or vector @samp{v = (v~0, v~1, ..., v~n)} into
3225 the Clifford number @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n} with @samp{e.k}
3226 being created by @code{clifford_unit(mu, metr, rl)}. The previous code
3227 may be rewritten with the help of @code{lst_to_clifford()} as follows
3232 varidx nu(symbol("nu"), 3);
3233 matrix M(3, 3) = 1, 0, 0,
3236 ex e0 = lst_to_clifford(lst(1, 0, 0), nu, M);
3237 ex e1 = lst_to_clifford(lst(0, 1, 0), nu, M);
3238 ex e2 = lst_to_clifford(lst(0, 0, 1), nu, M);
3243 @cindex @code{clifford_to_lst()}
3244 There is the inverse function
3247 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3250 which takes an expression @code{e} and tries to find a list
3251 @samp{v = (v~0, v~1, ..., v~n)} such that @samp{e = v~0 c.0 + v~1 c.1 + ...
3252 + v~n c.n} with respect to the given Clifford units @code{c} and none of
3253 @samp{v~k} contains the Clifford units @code{c} (of course, this
3254 may be impossible). This function can use an @code{algebraic} method
3255 (default) or a symbolic one. With the @code{algebraic} method @samp{v~k} are calculated as
3256 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2) = 0} for some @samp{k}
3257 then the method will be automatically changed to symbolic. The same effect
3258 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3260 @cindex @code{clifford_prime()}
3261 @cindex @code{clifford_star()}
3262 @cindex @code{clifford_bar()}
3263 There are several functions for (anti-)automorphisms of Clifford algebras:
3266 ex clifford_prime(const ex & e)
3267 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3268 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3271 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3272 changes signs of all Clifford units in the expression. The reversion
3273 of a Clifford algebra @code{clifford_star()} coincides with the
3274 @code{conjugate()} method and effectively reverses the order of Clifford
3275 units in any product. Finally the main anti-automorphism
3276 of a Clifford algebra @code{clifford_bar()} is the composition of the
3277 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3278 in a product. These functions correspond to the notations
3287 used in Clifford algebra textbooks.
3289 @cindex @code{clifford_norm()}
3293 ex clifford_norm(const ex & e);
3296 @cindex @code{clifford_inverse()}
3297 calculates the norm of a Clifford number from the expression
3299 $||e||^2 = e\overline{e}$
3301 . The inverse of a Clifford expression is returned
3305 ex clifford_inverse(const ex & e);
3308 which calculates it as
3310 $e^{-1} = e/||e||^2$
3316 then an exception is raised.
3318 @cindex @code{remove_dirac_ONE()}
3319 If a Clifford number happens to be a factor of
3320 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3321 expression by the function
3324 ex remove_dirac_ONE(const ex & e);
3327 @cindex @code{canonicalize_clifford()}
3328 The function @code{canonicalize_clifford()} works for a
3329 generic Clifford algebra in a similar way as for Dirac gammas.
3331 The last provided function is
3333 @cindex @code{clifford_moebius_map()}
3335 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3336 const ex & d, const ex & v, const ex & G);
3339 It takes a list or vector @code{v} and makes the Moebius
3340 (conformal or linear-fractional) transformation @samp{v ->
3341 (av+b)/(cv+d)} defined by the matrix @samp{[[a, b], [c, d]]}. The last
3342 parameter @code{G} defines the metric of the surrounding
3343 (pseudo-)Euclidean space. The returned value of this function is a list
3344 of components of the resulting vector.
3347 @cindex @code{color} (class)
3348 @subsection Color algebra
3350 @cindex @code{color_T()}
3351 For computations in quantum chromodynamics, GiNaC implements the base elements
3352 and structure constants of the su(3) Lie algebra (color algebra). The base
3353 elements @math{T_a} are constructed by the function
3356 ex color_T(const ex & a, unsigned char rl = 0);
3359 which takes two arguments: the index and a @dfn{representation label} in the
3360 range 0 to 255 which is used to distinguish elements of different color
3361 algebras. Objects with different labels commutate with each other. The
3362 dimension of the index must be exactly 8 and it should be of class @code{idx},
3365 @cindex @code{color_ONE()}
3366 The unity element of a color algebra is constructed by
3369 ex color_ONE(unsigned char rl = 0);
3372 @strong{Note:} You must always use @code{color_ONE()} when referring to
3373 multiples of the unity element, even though it's customary to omit it.
3374 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3375 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3376 GiNaC may produce incorrect results.
3378 @cindex @code{color_d()}
3379 @cindex @code{color_f()}
3383 ex color_d(const ex & a, const ex & b, const ex & c);
3384 ex color_f(const ex & a, const ex & b, const ex & c);
3387 create the symmetric and antisymmetric structure constants @math{d_abc} and
3388 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3389 and @math{[T_a, T_b] = i f_abc T_c}.
3391 @cindex @code{color_h()}
3392 There's an additional function
3395 ex color_h(const ex & a, const ex & b, const ex & c);
3398 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3400 The function @code{simplify_indexed()} performs some simplifications on
3401 expressions containing color objects:
3406 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3407 k(symbol("k"), 8), l(symbol("l"), 8);
3409 e = color_d(a, b, l) * color_f(a, b, k);
3410 cout << e.simplify_indexed() << endl;
3413 e = color_d(a, b, l) * color_d(a, b, k);
3414 cout << e.simplify_indexed() << endl;
3417 e = color_f(l, a, b) * color_f(a, b, k);
3418 cout << e.simplify_indexed() << endl;
3421 e = color_h(a, b, c) * color_h(a, b, c);
3422 cout << e.simplify_indexed() << endl;
3425 e = color_h(a, b, c) * color_T(b) * color_T(c);
3426 cout << e.simplify_indexed() << endl;
3429 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3430 cout << e.simplify_indexed() << endl;
3433 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3434 cout << e.simplify_indexed() << endl;
3435 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3439 @cindex @code{color_trace()}
3440 To calculate the trace of an expression containing color objects you use one
3444 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3445 ex color_trace(const ex & e, const lst & rll);
3446 ex color_trace(const ex & e, unsigned char rl = 0);
3449 These functions take the trace over all color @samp{T} objects in the
3450 specified set @code{rls} or list @code{rll} of representation labels, or the
3451 single label @code{rl}; @samp{T}s with other labels are left standing. For
3456 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3458 // -> -I*f.a.c.b+d.a.c.b
3463 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3464 @c node-name, next, previous, up
3467 @cindex @code{exhashmap} (class)
3469 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3470 that can be used as a drop-in replacement for the STL
3471 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3472 typically constant-time, element look-up than @code{map<>}.
3474 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3475 following differences:
3479 no @code{lower_bound()} and @code{upper_bound()} methods
3481 no reverse iterators, no @code{rbegin()}/@code{rend()}
3483 no @code{operator<(exhashmap, exhashmap)}
3485 the comparison function object @code{key_compare} is hardcoded to
3488 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3489 initial hash table size (the actual table size after construction may be
3490 larger than the specified value)
3492 the method @code{size_t bucket_count()} returns the current size of the hash
3495 @code{insert()} and @code{erase()} operations invalidate all iterators
3499 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3500 @c node-name, next, previous, up
3501 @chapter Methods and Functions
3504 In this chapter the most important algorithms provided by GiNaC will be
3505 described. Some of them are implemented as functions on expressions,
3506 others are implemented as methods provided by expression objects. If
3507 they are methods, there exists a wrapper function around it, so you can
3508 alternatively call it in a functional way as shown in the simple
3513 cout << "As method: " << sin(1).evalf() << endl;
3514 cout << "As function: " << evalf(sin(1)) << endl;
3518 @cindex @code{subs()}
3519 The general rule is that wherever methods accept one or more parameters
3520 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3521 wrapper accepts is the same but preceded by the object to act on
3522 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3523 most natural one in an OO model but it may lead to confusion for MapleV
3524 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3525 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3526 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3527 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3528 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3529 here. Also, users of MuPAD will in most cases feel more comfortable
3530 with GiNaC's convention. All function wrappers are implemented
3531 as simple inline functions which just call the corresponding method and
3532 are only provided for users uncomfortable with OO who are dead set to
3533 avoid method invocations. Generally, nested function wrappers are much
3534 harder to read than a sequence of methods and should therefore be
3535 avoided if possible. On the other hand, not everything in GiNaC is a
3536 method on class @code{ex} and sometimes calling a function cannot be
3540 * Information About Expressions::
3541 * Numerical Evaluation::
3542 * Substituting Expressions::
3543 * Pattern Matching and Advanced Substitutions::
3544 * Applying a Function on Subexpressions::
3545 * Visitors and Tree Traversal::
3546 * Polynomial Arithmetic:: Working with polynomials.
3547 * Rational Expressions:: Working with rational functions.
3548 * Symbolic Differentiation::
3549 * Series Expansion:: Taylor and Laurent expansion.
3551 * Built-in Functions:: List of predefined mathematical functions.
3552 * Multiple polylogarithms::
3553 * Complex Conjugation::
3554 * Built-in Functions:: List of predefined mathematical functions.
3555 * Solving Linear Systems of Equations::
3556 * Input/Output:: Input and output of expressions.
3560 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3561 @c node-name, next, previous, up
3562 @section Getting information about expressions
3564 @subsection Checking expression types
3565 @cindex @code{is_a<@dots{}>()}
3566 @cindex @code{is_exactly_a<@dots{}>()}
3567 @cindex @code{ex_to<@dots{}>()}
3568 @cindex Converting @code{ex} to other classes
3569 @cindex @code{info()}
3570 @cindex @code{return_type()}
3571 @cindex @code{return_type_tinfo()}
3573 Sometimes it's useful to check whether a given expression is a plain number,
3574 a sum, a polynomial with integer coefficients, or of some other specific type.
3575 GiNaC provides a couple of functions for this:
3578 bool is_a<T>(const ex & e);
3579 bool is_exactly_a<T>(const ex & e);
3580 bool ex::info(unsigned flag);
3581 unsigned ex::return_type() const;
3582 unsigned ex::return_type_tinfo() const;
3585 When the test made by @code{is_a<T>()} returns true, it is safe to call
3586 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3587 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3588 example, assuming @code{e} is an @code{ex}:
3593 if (is_a<numeric>(e))
3594 numeric n = ex_to<numeric>(e);
3599 @code{is_a<T>(e)} allows you to check whether the top-level object of
3600 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3601 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3602 e.g., for checking whether an expression is a number, a sum, or a product:
3609 is_a<numeric>(e1); // true
3610 is_a<numeric>(e2); // false
3611 is_a<add>(e1); // false
3612 is_a<add>(e2); // true
3613 is_a<mul>(e1); // false
3614 is_a<mul>(e2); // false
3618 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3619 top-level object of an expression @samp{e} is an instance of the GiNaC
3620 class @samp{T}, not including parent classes.
3622 The @code{info()} method is used for checking certain attributes of
3623 expressions. The possible values for the @code{flag} argument are defined
3624 in @file{ginac/flags.h}, the most important being explained in the following
3628 @multitable @columnfractions .30 .70
3629 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3630 @item @code{numeric}
3631 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3633 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3634 @item @code{rational}
3635 @tab @dots{}an exact rational number (integers are rational, too)
3636 @item @code{integer}
3637 @tab @dots{}a (non-complex) integer
3638 @item @code{crational}
3639 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3640 @item @code{cinteger}
3641 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3642 @item @code{positive}
3643 @tab @dots{}not complex and greater than 0
3644 @item @code{negative}
3645 @tab @dots{}not complex and less than 0
3646 @item @code{nonnegative}
3647 @tab @dots{}not complex and greater than or equal to 0
3649 @tab @dots{}an integer greater than 0
3651 @tab @dots{}an integer less than 0
3652 @item @code{nonnegint}
3653 @tab @dots{}an integer greater than or equal to 0
3655 @tab @dots{}an even integer
3657 @tab @dots{}an odd integer
3659 @tab @dots{}a prime integer (probabilistic primality test)
3660 @item @code{relation}
3661 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3662 @item @code{relation_equal}
3663 @tab @dots{}a @code{==} relation
3664 @item @code{relation_not_equal}
3665 @tab @dots{}a @code{!=} relation
3666 @item @code{relation_less}
3667 @tab @dots{}a @code{<} relation
3668 @item @code{relation_less_or_equal}
3669 @tab @dots{}a @code{<=} relation
3670 @item @code{relation_greater}
3671 @tab @dots{}a @code{>} relation
3672 @item @code{relation_greater_or_equal}
3673 @tab @dots{}a @code{>=} relation
3675 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3677 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3678 @item @code{polynomial}
3679 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3680 @item @code{integer_polynomial}
3681 @tab @dots{}a polynomial with (non-complex) integer coefficients
3682 @item @code{cinteger_polynomial}
3683 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3684 @item @code{rational_polynomial}
3685 @tab @dots{}a polynomial with (non-complex) rational coefficients
3686 @item @code{crational_polynomial}
3687 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3688 @item @code{rational_function}
3689 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3690 @item @code{algebraic}
3691 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3695 To determine whether an expression is commutative or non-commutative and if
3696 so, with which other expressions it would commutate, you use the methods
3697 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3698 for an explanation of these.
3701 @subsection Accessing subexpressions
3704 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3705 @code{function}, act as containers for subexpressions. For example, the
3706 subexpressions of a sum (an @code{add} object) are the individual terms,
3707 and the subexpressions of a @code{function} are the function's arguments.
3709 @cindex @code{nops()}
3711 GiNaC provides several ways of accessing subexpressions. The first way is to
3716 ex ex::op(size_t i);
3719 @code{nops()} determines the number of subexpressions (operands) contained
3720 in the expression, while @code{op(i)} returns the @code{i}-th
3721 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3722 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3723 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3724 @math{i>0} are the indices.
3727 @cindex @code{const_iterator}
3728 The second way to access subexpressions is via the STL-style random-access
3729 iterator class @code{const_iterator} and the methods
3732 const_iterator ex::begin();
3733 const_iterator ex::end();
3736 @code{begin()} returns an iterator referring to the first subexpression;
3737 @code{end()} returns an iterator which is one-past the last subexpression.
3738 If the expression has no subexpressions, then @code{begin() == end()}. These
3739 iterators can also be used in conjunction with non-modifying STL algorithms.
3741 Here is an example that (non-recursively) prints the subexpressions of a
3742 given expression in three different ways:
3749 for (size_t i = 0; i != e.nops(); ++i)
3750 cout << e.op(i) << endl;
3753 for (const_iterator i = e.begin(); i != e.end(); ++i)
3756 // with iterators and STL copy()
3757 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3761 @cindex @code{const_preorder_iterator}
3762 @cindex @code{const_postorder_iterator}
3763 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3764 expression's immediate children. GiNaC provides two additional iterator
3765 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3766 that iterate over all objects in an expression tree, in preorder or postorder,
3767 respectively. They are STL-style forward iterators, and are created with the
3771 const_preorder_iterator ex::preorder_begin();
3772 const_preorder_iterator ex::preorder_end();
3773 const_postorder_iterator ex::postorder_begin();
3774 const_postorder_iterator ex::postorder_end();
3777 The following example illustrates the differences between
3778 @code{const_iterator}, @code{const_preorder_iterator}, and
3779 @code{const_postorder_iterator}:
3783 symbol A("A"), B("B"), C("C");
3784 ex e = lst(lst(A, B), C);
3786 std::copy(e.begin(), e.end(),
3787 std::ostream_iterator<ex>(cout, "\n"));
3791 std::copy(e.preorder_begin(), e.preorder_end(),
3792 std::ostream_iterator<ex>(cout, "\n"));
3799 std::copy(e.postorder_begin(), e.postorder_end(),
3800 std::ostream_iterator<ex>(cout, "\n"));
3809 @cindex @code{relational} (class)
3810 Finally, the left-hand side and right-hand side expressions of objects of
3811 class @code{relational} (and only of these) can also be accessed with the
3820 @subsection Comparing expressions
3821 @cindex @code{is_equal()}
3822 @cindex @code{is_zero()}
3824 Expressions can be compared with the usual C++ relational operators like
3825 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3826 the result is usually not determinable and the result will be @code{false},
3827 except in the case of the @code{!=} operator. You should also be aware that
3828 GiNaC will only do the most trivial test for equality (subtracting both
3829 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3832 Actually, if you construct an expression like @code{a == b}, this will be
3833 represented by an object of the @code{relational} class (@pxref{Relations})
3834 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3836 There are also two methods
3839 bool ex::is_equal(const ex & other);
3843 for checking whether one expression is equal to another, or equal to zero,
3847 @subsection Ordering expressions
3848 @cindex @code{ex_is_less} (class)
3849 @cindex @code{ex_is_equal} (class)
3850 @cindex @code{compare()}
3852 Sometimes it is necessary to establish a mathematically well-defined ordering
3853 on a set of arbitrary expressions, for example to use expressions as keys
3854 in a @code{std::map<>} container, or to bring a vector of expressions into
3855 a canonical order (which is done internally by GiNaC for sums and products).
3857 The operators @code{<}, @code{>} etc. described in the last section cannot
3858 be used for this, as they don't implement an ordering relation in the
3859 mathematical sense. In particular, they are not guaranteed to be
3860 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3861 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3864 By default, STL classes and algorithms use the @code{<} and @code{==}
3865 operators to compare objects, which are unsuitable for expressions, but GiNaC
3866 provides two functors that can be supplied as proper binary comparison
3867 predicates to the STL:
3870 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3872 bool operator()(const ex &lh, const ex &rh) const;
3875 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3877 bool operator()(const ex &lh, const ex &rh) const;
3881 For example, to define a @code{map} that maps expressions to strings you
3885 std::map<ex, std::string, ex_is_less> myMap;
3888 Omitting the @code{ex_is_less} template parameter will introduce spurious
3889 bugs because the map operates improperly.
3891 Other examples for the use of the functors:
3899 std::sort(v.begin(), v.end(), ex_is_less());
3901 // count the number of expressions equal to '1'
3902 unsigned num_ones = std::count_if(v.begin(), v.end(),
3903 std::bind2nd(ex_is_equal(), 1));
3906 The implementation of @code{ex_is_less} uses the member function
3909 int ex::compare(const ex & other) const;
3912 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3913 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3917 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3918 @c node-name, next, previous, up
3919 @section Numerical Evaluation
3920 @cindex @code{evalf()}
3922 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3923 To evaluate them using floating-point arithmetic you need to call
3926 ex ex::evalf(int level = 0) const;
3929 @cindex @code{Digits}
3930 The accuracy of the evaluation is controlled by the global object @code{Digits}
3931 which can be assigned an integer value. The default value of @code{Digits}
3932 is 17. @xref{Numbers}, for more information and examples.
3934 To evaluate an expression to a @code{double} floating-point number you can
3935 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3939 // Approximate sin(x/Pi)
3941 ex e = series(sin(x/Pi), x == 0, 6);
3943 // Evaluate numerically at x=0.1
3944 ex f = evalf(e.subs(x == 0.1));
3946 // ex_to<numeric> is an unsafe cast, so check the type first
3947 if (is_a<numeric>(f)) @{
3948 double d = ex_to<numeric>(f).to_double();
3957 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3958 @c node-name, next, previous, up
3959 @section Substituting expressions
3960 @cindex @code{subs()}
3962 Algebraic objects inside expressions can be replaced with arbitrary
3963 expressions via the @code{.subs()} method:
3966 ex ex::subs(const ex & e, unsigned options = 0);
3967 ex ex::subs(const exmap & m, unsigned options = 0);
3968 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3971 In the first form, @code{subs()} accepts a relational of the form
3972 @samp{object == expression} or a @code{lst} of such relationals:
3976 symbol x("x"), y("y");
3978 ex e1 = 2*x^2-4*x+3;
3979 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3983 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3988 If you specify multiple substitutions, they are performed in parallel, so e.g.
3989 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3991 The second form of @code{subs()} takes an @code{exmap} object which is a
3992 pair associative container that maps expressions to expressions (currently
3993 implemented as a @code{std::map}). This is the most efficient one of the
3994 three @code{subs()} forms and should be used when the number of objects to
3995 be substituted is large or unknown.
3997 Using this form, the second example from above would look like this:
4001 symbol x("x"), y("y");
4007 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4011 The third form of @code{subs()} takes two lists, one for the objects to be
4012 replaced and one for the expressions to be substituted (both lists must
4013 contain the same number of elements). Using this form, you would write
4017 symbol x("x"), y("y");
4020 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4024 The optional last argument to @code{subs()} is a combination of
4025 @code{subs_options} flags. There are two options available:
4026 @code{subs_options::no_pattern} disables pattern matching, which makes
4027 large @code{subs()} operations significantly faster if you are not using
4028 patterns. The second option, @code{subs_options::algebraic} enables
4029 algebraic substitutions in products and powers.
4030 @ref{Pattern Matching and Advanced Substitutions}, for more information
4031 about patterns and algebraic substitutions.
4033 @code{subs()} performs syntactic substitution of any complete algebraic
4034 object; it does not try to match sub-expressions as is demonstrated by the
4039 symbol x("x"), y("y"), z("z");
4041 ex e1 = pow(x+y, 2);
4042 cout << e1.subs(x+y == 4) << endl;
4045 ex e2 = sin(x)*sin(y)*cos(x);
4046 cout << e2.subs(sin(x) == cos(x)) << endl;
4047 // -> cos(x)^2*sin(y)
4050 cout << e3.subs(x+y == 4) << endl;
4052 // (and not 4+z as one might expect)
4056 A more powerful form of substitution using wildcards is described in the
4060 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4061 @c node-name, next, previous, up
4062 @section Pattern matching and advanced substitutions
4063 @cindex @code{wildcard} (class)
4064 @cindex Pattern matching
4066 GiNaC allows the use of patterns for checking whether an expression is of a
4067 certain form or contains subexpressions of a certain form, and for
4068 substituting expressions in a more general way.
4070 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4071 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4072 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4073 an unsigned integer number to allow having multiple different wildcards in a
4074 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4075 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4079 ex wild(unsigned label = 0);
4082 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4085 Some examples for patterns:
4087 @multitable @columnfractions .5 .5
4088 @item @strong{Constructed as} @tab @strong{Output as}
4089 @item @code{wild()} @tab @samp{$0}
4090 @item @code{pow(x,wild())} @tab @samp{x^$0}
4091 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4092 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4098 @item Wildcards behave like symbols and are subject to the same algebraic
4099 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4100 @item As shown in the last example, to use wildcards for indices you have to
4101 use them as the value of an @code{idx} object. This is because indices must
4102 always be of class @code{idx} (or a subclass).
4103 @item Wildcards only represent expressions or subexpressions. It is not
4104 possible to use them as placeholders for other properties like index
4105 dimension or variance, representation labels, symmetry of indexed objects
4107 @item Because wildcards are commutative, it is not possible to use wildcards
4108 as part of noncommutative products.
4109 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4110 are also valid patterns.
4113 @subsection Matching expressions
4114 @cindex @code{match()}
4115 The most basic application of patterns is to check whether an expression
4116 matches a given pattern. This is done by the function
4119 bool ex::match(const ex & pattern);
4120 bool ex::match(const ex & pattern, lst & repls);
4123 This function returns @code{true} when the expression matches the pattern
4124 and @code{false} if it doesn't. If used in the second form, the actual
4125 subexpressions matched by the wildcards get returned in the @code{repls}
4126 object as a list of relations of the form @samp{wildcard == expression}.
4127 If @code{match()} returns false, the state of @code{repls} is undefined.
4128 For reproducible results, the list should be empty when passed to
4129 @code{match()}, but it is also possible to find similarities in multiple
4130 expressions by passing in the result of a previous match.
4132 The matching algorithm works as follows:
4135 @item A single wildcard matches any expression. If one wildcard appears
4136 multiple times in a pattern, it must match the same expression in all
4137 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4138 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4139 @item If the expression is not of the same class as the pattern, the match
4140 fails (i.e. a sum only matches a sum, a function only matches a function,
4142 @item If the pattern is a function, it only matches the same function
4143 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4144 @item Except for sums and products, the match fails if the number of
4145 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4147 @item If there are no subexpressions, the expressions and the pattern must
4148 be equal (in the sense of @code{is_equal()}).
4149 @item Except for sums and products, each subexpression (@code{op()}) must
4150 match the corresponding subexpression of the pattern.
4153 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4154 account for their commutativity and associativity:
4157 @item If the pattern contains a term or factor that is a single wildcard,
4158 this one is used as the @dfn{global wildcard}. If there is more than one
4159 such wildcard, one of them is chosen as the global wildcard in a random
4161 @item Every term/factor of the pattern, except the global wildcard, is
4162 matched against every term of the expression in sequence. If no match is
4163 found, the whole match fails. Terms that did match are not considered in
4165 @item If there are no unmatched terms left, the match succeeds. Otherwise
4166 the match fails unless there is a global wildcard in the pattern, in
4167 which case this wildcard matches the remaining terms.
4170 In general, having more than one single wildcard as a term of a sum or a
4171 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4174 Here are some examples in @command{ginsh} to demonstrate how it works (the
4175 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4176 match fails, and the list of wildcard replacements otherwise):
4179 > match((x+y)^a,(x+y)^a);
4181 > match((x+y)^a,(x+y)^b);
4183 > match((x+y)^a,$1^$2);
4185 > match((x+y)^a,$1^$1);
4187 > match((x+y)^(x+y),$1^$1);
4189 > match((x+y)^(x+y),$1^$2);
4191 > match((a+b)*(a+c),($1+b)*($1+c));
4193 > match((a+b)*(a+c),(a+$1)*(a+$2));
4195 (Unpredictable. The result might also be [$1==c,$2==b].)
4196 > match((a+b)*(a+c),($1+$2)*($1+$3));
4197 (The result is undefined. Due to the sequential nature of the algorithm
4198 and the re-ordering of terms in GiNaC, the match for the first factor
4199 may be @{$1==a,$2==b@} in which case the match for the second factor
4200 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4202 > match(a*(x+y)+a*z+b,a*$1+$2);
4203 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4204 @{$1=x+y,$2=a*z+b@}.)
4205 > match(a+b+c+d+e+f,c);
4207 > match(a+b+c+d+e+f,c+$0);
4209 > match(a+b+c+d+e+f,c+e+$0);
4211 > match(a+b,a+b+$0);
4213 > match(a*b^2,a^$1*b^$2);
4215 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4216 even though a==a^1.)
4217 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4219 > match(atan2(y,x^2),atan2(y,$0));
4223 @subsection Matching parts of expressions
4224 @cindex @code{has()}
4225 A more general way to look for patterns in expressions is provided by the
4229 bool ex::has(const ex & pattern);
4232 This function checks whether a pattern is matched by an expression itself or
4233 by any of its subexpressions.
4235 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4236 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4239 > has(x*sin(x+y+2*a),y);
4241 > has(x*sin(x+y+2*a),x+y);
4243 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4244 has the subexpressions "x", "y" and "2*a".)
4245 > has(x*sin(x+y+2*a),x+y+$1);
4247 (But this is possible.)
4248 > has(x*sin(2*(x+y)+2*a),x+y);
4250 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4251 which "x+y" is not a subexpression.)
4254 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4256 > has(4*x^2-x+3,$1*x);
4258 > has(4*x^2+x+3,$1*x);
4260 (Another possible pitfall. The first expression matches because the term
4261 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4262 contains a linear term you should use the coeff() function instead.)
4265 @cindex @code{find()}
4269 bool ex::find(const ex & pattern, lst & found);
4272 works a bit like @code{has()} but it doesn't stop upon finding the first
4273 match. Instead, it appends all found matches to the specified list. If there
4274 are multiple occurrences of the same expression, it is entered only once to
4275 the list. @code{find()} returns false if no matches were found (in
4276 @command{ginsh}, it returns an empty list):
4279 > find(1+x+x^2+x^3,x);
4281 > find(1+x+x^2+x^3,y);
4283 > find(1+x+x^2+x^3,x^$1);
4285 (Note the absence of "x".)
4286 > expand((sin(x)+sin(y))*(a+b));
4287 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4292 @subsection Substituting expressions
4293 @cindex @code{subs()}
4294 Probably the most useful application of patterns is to use them for
4295 substituting expressions with the @code{subs()} method. Wildcards can be
4296 used in the search patterns as well as in the replacement expressions, where
4297 they get replaced by the expressions matched by them. @code{subs()} doesn't
4298 know anything about algebra; it performs purely syntactic substitutions.
4303 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4305 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4307 > subs((a+b+c)^2,a+b==x);
4309 > subs((a+b+c)^2,a+b+$1==x+$1);
4311 > subs(a+2*b,a+b==x);
4313 > subs(4*x^3-2*x^2+5*x-1,x==a);
4315 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4317 > subs(sin(1+sin(x)),sin($1)==cos($1));
4319 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4323 The last example would be written in C++ in this way:
4327 symbol a("a"), b("b"), x("x"), y("y");
4328 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4329 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4330 cout << e.expand() << endl;
4335 @subsection Algebraic substitutions
4336 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4337 enables smarter, algebraic substitutions in products and powers. If you want
4338 to substitute some factors of a product, you only need to list these factors
4339 in your pattern. Furthermore, if an (integer) power of some expression occurs
4340 in your pattern and in the expression that you want the substitution to occur
4341 in, it can be substituted as many times as possible, without getting negative
4344 An example clarifies it all (hopefully):
4347 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4348 subs_options::algebraic) << endl;
4349 // --> (y+x)^6+b^6+a^6
4351 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4353 // Powers and products are smart, but addition is just the same.
4355 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4358 // As I said: addition is just the same.
4360 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4361 // --> x^3*b*a^2+2*b
4363 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4365 // --> 2*b+x^3*b^(-1)*a^(-2)
4367 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4368 // --> -1-2*a^2+4*a^3+5*a
4370 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4371 subs_options::algebraic) << endl;
4372 // --> -1+5*x+4*x^3-2*x^2
4373 // You should not really need this kind of patterns very often now.
4374 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4376 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4377 subs_options::algebraic) << endl;
4378 // --> cos(1+cos(x))
4380 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4381 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4382 subs_options::algebraic)) << endl;
4387 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4388 @c node-name, next, previous, up
4389 @section Applying a Function on Subexpressions
4390 @cindex tree traversal
4391 @cindex @code{map()}
4393 Sometimes you may want to perform an operation on specific parts of an
4394 expression while leaving the general structure of it intact. An example
4395 of this would be a matrix trace operation: the trace of a sum is the sum
4396 of the traces of the individual terms. That is, the trace should @dfn{map}
4397 on the sum, by applying itself to each of the sum's operands. It is possible
4398 to do this manually which usually results in code like this:
4403 if (is_a<matrix>(e))
4404 return ex_to<matrix>(e).trace();
4405 else if (is_a<add>(e)) @{
4407 for (size_t i=0; i<e.nops(); i++)
4408 sum += calc_trace(e.op(i));
4410 @} else if (is_a<mul>)(e)) @{
4418 This is, however, slightly inefficient (if the sum is very large it can take
4419 a long time to add the terms one-by-one), and its applicability is limited to
4420 a rather small class of expressions. If @code{calc_trace()} is called with
4421 a relation or a list as its argument, you will probably want the trace to
4422 be taken on both sides of the relation or of all elements of the list.
4424 GiNaC offers the @code{map()} method to aid in the implementation of such
4428 ex ex::map(map_function & f) const;
4429 ex ex::map(ex (*f)(const ex & e)) const;
4432 In the first (preferred) form, @code{map()} takes a function object that
4433 is subclassed from the @code{map_function} class. In the second form, it
4434 takes a pointer to a function that accepts and returns an expression.
4435 @code{map()} constructs a new expression of the same type, applying the
4436 specified function on all subexpressions (in the sense of @code{op()}),
4439 The use of a function object makes it possible to supply more arguments to
4440 the function that is being mapped, or to keep local state information.
4441 The @code{map_function} class declares a virtual function call operator
4442 that you can overload. Here is a sample implementation of @code{calc_trace()}
4443 that uses @code{map()} in a recursive fashion:
4446 struct calc_trace : public map_function @{
4447 ex operator()(const ex &e)
4449 if (is_a<matrix>(e))
4450 return ex_to<matrix>(e).trace();
4451 else if (is_a<mul>(e)) @{
4454 return e.map(*this);
4459 This function object could then be used like this:
4463 ex M = ... // expression with matrices
4464 calc_trace do_trace;
4465 ex tr = do_trace(M);
4469 Here is another example for you to meditate over. It removes quadratic
4470 terms in a variable from an expanded polynomial:
4473 struct map_rem_quad : public map_function @{
4475 map_rem_quad(const ex & var_) : var(var_) @{@}
4477 ex operator()(const ex & e)
4479 if (is_a<add>(e) || is_a<mul>(e))
4480 return e.map(*this);
4481 else if (is_a<power>(e) &&
4482 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4492 symbol x("x"), y("y");
4495 for (int i=0; i<8; i++)
4496 e += pow(x, i) * pow(y, 8-i) * (i+1);
4498 // -> 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
4500 map_rem_quad rem_quad(x);
4501 cout << rem_quad(e) << endl;
4502 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4506 @command{ginsh} offers a slightly different implementation of @code{map()}
4507 that allows applying algebraic functions to operands. The second argument
4508 to @code{map()} is an expression containing the wildcard @samp{$0} which
4509 acts as the placeholder for the operands:
4514 > map(a+2*b,sin($0));
4516 > map(@{a,b,c@},$0^2+$0);
4517 @{a^2+a,b^2+b,c^2+c@}
4520 Note that it is only possible to use algebraic functions in the second
4521 argument. You can not use functions like @samp{diff()}, @samp{op()},
4522 @samp{subs()} etc. because these are evaluated immediately:
4525 > map(@{a,b,c@},diff($0,a));
4527 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4528 to "map(@{a,b,c@},0)".
4532 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4533 @c node-name, next, previous, up
4534 @section Visitors and Tree Traversal
4535 @cindex tree traversal
4536 @cindex @code{visitor} (class)
4537 @cindex @code{accept()}
4538 @cindex @code{visit()}
4539 @cindex @code{traverse()}
4540 @cindex @code{traverse_preorder()}
4541 @cindex @code{traverse_postorder()}
4543 Suppose that you need a function that returns a list of all indices appearing
4544 in an arbitrary expression. The indices can have any dimension, and for
4545 indices with variance you always want the covariant version returned.
4547 You can't use @code{get_free_indices()} because you also want to include
4548 dummy indices in the list, and you can't use @code{find()} as it needs
4549 specific index dimensions (and it would require two passes: one for indices
4550 with variance, one for plain ones).
4552 The obvious solution to this problem is a tree traversal with a type switch,
4553 such as the following:
4556 void gather_indices_helper(const ex & e, lst & l)
4558 if (is_a<varidx>(e)) @{
4559 const varidx & vi = ex_to<varidx>(e);
4560 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4561 @} else if (is_a<idx>(e)) @{
4564 size_t n = e.nops();
4565 for (size_t i = 0; i < n; ++i)
4566 gather_indices_helper(e.op(i), l);
4570 lst gather_indices(const ex & e)
4573 gather_indices_helper(e, l);
4580 This works fine but fans of object-oriented programming will feel
4581 uncomfortable with the type switch. One reason is that there is a possibility
4582 for subtle bugs regarding derived classes. If we had, for example, written
4585 if (is_a<idx>(e)) @{
4587 @} else if (is_a<varidx>(e)) @{
4591 in @code{gather_indices_helper}, the code wouldn't have worked because the
4592 first line "absorbs" all classes derived from @code{idx}, including
4593 @code{varidx}, so the special case for @code{varidx} would never have been
4596 Also, for a large number of classes, a type switch like the above can get
4597 unwieldy and inefficient (it's a linear search, after all).
4598 @code{gather_indices_helper} only checks for two classes, but if you had to
4599 write a function that required a different implementation for nearly
4600 every GiNaC class, the result would be very hard to maintain and extend.
4602 The cleanest approach to the problem would be to add a new virtual function
4603 to GiNaC's class hierarchy. In our example, there would be specializations
4604 for @code{idx} and @code{varidx} while the default implementation in
4605 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4606 impossible to add virtual member functions to existing classes without
4607 changing their source and recompiling everything. GiNaC comes with source,
4608 so you could actually do this, but for a small algorithm like the one
4609 presented this would be impractical.
4611 One solution to this dilemma is the @dfn{Visitor} design pattern,
4612 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4613 variation, described in detail in
4614 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4615 virtual functions to the class hierarchy to implement operations, GiNaC
4616 provides a single "bouncing" method @code{accept()} that takes an instance
4617 of a special @code{visitor} class and redirects execution to the one
4618 @code{visit()} virtual function of the visitor that matches the type of
4619 object that @code{accept()} was being invoked on.
4621 Visitors in GiNaC must derive from the global @code{visitor} class as well
4622 as from the class @code{T::visitor} of each class @code{T} they want to
4623 visit, and implement the member functions @code{void visit(const T &)} for
4629 void ex::accept(visitor & v) const;
4632 will then dispatch to the correct @code{visit()} member function of the
4633 specified visitor @code{v} for the type of GiNaC object at the root of the
4634 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4636 Here is an example of a visitor:
4640 : public visitor, // this is required
4641 public add::visitor, // visit add objects
4642 public numeric::visitor, // visit numeric objects
4643 public basic::visitor // visit basic objects
4645 void visit(const add & x)
4646 @{ cout << "called with an add object" << endl; @}
4648 void visit(const numeric & x)
4649 @{ cout << "called with a numeric object" << endl; @}
4651 void visit(const basic & x)
4652 @{ cout << "called with a basic object" << endl; @}
4656 which can be used as follows:
4667 // prints "called with a numeric object"
4669 // prints "called with an add object"
4671 // prints "called with a basic object"
4675 The @code{visit(const basic &)} method gets called for all objects that are
4676 not @code{numeric} or @code{add} and acts as an (optional) default.
4678 From a conceptual point of view, the @code{visit()} methods of the visitor
4679 behave like a newly added virtual function of the visited hierarchy.
4680 In addition, visitors can store state in member variables, and they can
4681 be extended by deriving a new visitor from an existing one, thus building
4682 hierarchies of visitors.
4684 We can now rewrite our index example from above with a visitor:
4687 class gather_indices_visitor
4688 : public visitor, public idx::visitor, public varidx::visitor
4692 void visit(const idx & i)
4697 void visit(const varidx & vi)
4699 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4703 const lst & get_result() // utility function
4712 What's missing is the tree traversal. We could implement it in
4713 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4716 void ex::traverse_preorder(visitor & v) const;
4717 void ex::traverse_postorder(visitor & v) const;
4718 void ex::traverse(visitor & v) const;
4721 @code{traverse_preorder()} visits a node @emph{before} visiting its
4722 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4723 visiting its subexpressions. @code{traverse()} is a synonym for
4724 @code{traverse_preorder()}.
4726 Here is a new implementation of @code{gather_indices()} that uses the visitor
4727 and @code{traverse()}:
4730 lst gather_indices(const ex & e)
4732 gather_indices_visitor v;
4734 return v.get_result();
4738 Alternatively, you could use pre- or postorder iterators for the tree
4742 lst gather_indices(const ex & e)
4744 gather_indices_visitor v;
4745 for (const_preorder_iterator i = e.preorder_begin();
4746 i != e.preorder_end(); ++i) @{
4749 return v.get_result();
4754 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4755 @c node-name, next, previous, up
4756 @section Polynomial arithmetic
4758 @subsection Expanding and collecting
4759 @cindex @code{expand()}
4760 @cindex @code{collect()}
4761 @cindex @code{collect_common_factors()}
4763 A polynomial in one or more variables has many equivalent
4764 representations. Some useful ones serve a specific purpose. Consider
4765 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4766 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4767 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4768 representations are the recursive ones where one collects for exponents
4769 in one of the three variable. Since the factors are themselves
4770 polynomials in the remaining two variables the procedure can be
4771 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4772 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4775 To bring an expression into expanded form, its method
4778 ex ex::expand(unsigned options = 0);
4781 may be called. In our example above, this corresponds to @math{4*x*y +
4782 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4783 GiNaC is not easy to guess you should be prepared to see different
4784 orderings of terms in such sums!
4786 Another useful representation of multivariate polynomials is as a
4787 univariate polynomial in one of the variables with the coefficients
4788 being polynomials in the remaining variables. The method
4789 @code{collect()} accomplishes this task:
4792 ex ex::collect(const ex & s, bool distributed = false);
4795 The first argument to @code{collect()} can also be a list of objects in which
4796 case the result is either a recursively collected polynomial, or a polynomial
4797 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4798 by the @code{distributed} flag.
4800 Note that the original polynomial needs to be in expanded form (for the
4801 variables concerned) in order for @code{collect()} to be able to find the
4802 coefficients properly.
4804 The following @command{ginsh} transcript shows an application of @code{collect()}
4805 together with @code{find()}:
4808 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4809 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
4810 > collect(a,@{p,q@});
4811 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)
4812 > collect(a,find(a,sin($1)));
4813 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4814 > collect(a,@{find(a,sin($1)),p,q@});
4815 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4816 > collect(a,@{find(a,sin($1)),d@});
4817 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4820 Polynomials can often be brought into a more compact form by collecting
4821 common factors from the terms of sums. This is accomplished by the function
4824 ex collect_common_factors(const ex & e);
4827 This function doesn't perform a full factorization but only looks for
4828 factors which are already explicitly present:
4831 > collect_common_factors(a*x+a*y);
4833 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4835 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4836 (c+a)*a*(x*y+y^2+x)*b
4839 @subsection Degree and coefficients
4840 @cindex @code{degree()}
4841 @cindex @code{ldegree()}
4842 @cindex @code{coeff()}
4844 The degree and low degree of a polynomial can be obtained using the two
4848 int ex::degree(const ex & s);
4849 int ex::ldegree(const ex & s);
4852 which also work reliably on non-expanded input polynomials (they even work
4853 on rational functions, returning the asymptotic degree). By definition, the
4854 degree of zero is zero. To extract a coefficient with a certain power from
4855 an expanded polynomial you use
4858 ex ex::coeff(const ex & s, int n);
4861 You can also obtain the leading and trailing coefficients with the methods
4864 ex ex::lcoeff(const ex & s);
4865 ex ex::tcoeff(const ex & s);
4868 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4871 An application is illustrated in the next example, where a multivariate
4872 polynomial is analyzed:
4876 symbol x("x"), y("y");
4877 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4878 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4879 ex Poly = PolyInp.expand();
4881 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4882 cout << "The x^" << i << "-coefficient is "
4883 << Poly.coeff(x,i) << endl;
4885 cout << "As polynomial in y: "
4886 << Poly.collect(y) << endl;
4890 When run, it returns an output in the following fashion:
4893 The x^0-coefficient is y^2+11*y
4894 The x^1-coefficient is 5*y^2-2*y
4895 The x^2-coefficient is -1
4896 The x^3-coefficient is 4*y
4897 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4900 As always, the exact output may vary between different versions of GiNaC
4901 or even from run to run since the internal canonical ordering is not
4902 within the user's sphere of influence.
4904 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4905 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4906 with non-polynomial expressions as they not only work with symbols but with
4907 constants, functions and indexed objects as well:
4911 symbol a("a"), b("b"), c("c"), x("x");
4912 idx i(symbol("i"), 3);
4914 ex e = pow(sin(x) - cos(x), 4);
4915 cout << e.degree(cos(x)) << endl;
4917 cout << e.expand().coeff(sin(x), 3) << endl;
4920 e = indexed(a+b, i) * indexed(b+c, i);
4921 e = e.expand(expand_options::expand_indexed);
4922 cout << e.collect(indexed(b, i)) << endl;
4923 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4928 @subsection Polynomial division
4929 @cindex polynomial division
4932 @cindex pseudo-remainder
4933 @cindex @code{quo()}
4934 @cindex @code{rem()}
4935 @cindex @code{prem()}
4936 @cindex @code{divide()}
4941 ex quo(const ex & a, const ex & b, const ex & x);
4942 ex rem(const ex & a, const ex & b, const ex & x);
4945 compute the quotient and remainder of univariate polynomials in the variable
4946 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4948 The additional function
4951 ex prem(const ex & a, const ex & b, const ex & x);
4954 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4955 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4957 Exact division of multivariate polynomials is performed by the function
4960 bool divide(const ex & a, const ex & b, ex & q);
4963 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4964 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4965 in which case the value of @code{q} is undefined.
4968 @subsection Unit, content and primitive part
4969 @cindex @code{unit()}
4970 @cindex @code{content()}
4971 @cindex @code{primpart()}
4972 @cindex @code{unitcontprim()}
4977 ex ex::unit(const ex & x);
4978 ex ex::content(const ex & x);
4979 ex ex::primpart(const ex & x);
4980 ex ex::primpart(const ex & x, const ex & c);
4983 return the unit part, content part, and primitive polynomial of a multivariate
4984 polynomial with respect to the variable @samp{x} (the unit part being the sign
4985 of the leading coefficient, the content part being the GCD of the coefficients,
4986 and the primitive polynomial being the input polynomial divided by the unit and
4987 content parts). The second variant of @code{primpart()} expects the previously
4988 calculated content part of the polynomial in @code{c}, which enables it to
4989 work faster in the case where the content part has already been computed. The
4990 product of unit, content, and primitive part is the original polynomial.
4992 Additionally, the method
4995 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
4998 computes the unit, content, and primitive parts in one go, returning them
4999 in @code{u}, @code{c}, and @code{p}, respectively.
5002 @subsection GCD, LCM and resultant
5005 @cindex @code{gcd()}
5006 @cindex @code{lcm()}
5008 The functions for polynomial greatest common divisor and least common
5009 multiple have the synopsis
5012 ex gcd(const ex & a, const ex & b);
5013 ex lcm(const ex & a, const ex & b);
5016 The functions @code{gcd()} and @code{lcm()} accept two expressions
5017 @code{a} and @code{b} as arguments and return a new expression, their
5018 greatest common divisor or least common multiple, respectively. If the
5019 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5020 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
5023 #include <ginac/ginac.h>
5024 using namespace GiNaC;
5028 symbol x("x"), y("y"), z("z");
5029 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5030 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5032 ex P_gcd = gcd(P_a, P_b);
5034 ex P_lcm = lcm(P_a, P_b);
5035 // 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
5040 @cindex @code{resultant()}
5042 The resultant of two expressions only makes sense with polynomials.
5043 It is always computed with respect to a specific symbol within the
5044 expressions. The function has the interface
5047 ex resultant(const ex & a, const ex & b, const ex & s);
5050 Resultants are symmetric in @code{a} and @code{b}. The following example
5051 computes the resultant of two expressions with respect to @code{x} and
5052 @code{y}, respectively:
5055 #include <ginac/ginac.h>
5056 using namespace GiNaC;
5060 symbol x("x"), y("y");
5062 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5065 r = resultant(e1, e2, x);
5067 r = resultant(e1, e2, y);
5072 @subsection Square-free decomposition
5073 @cindex square-free decomposition
5074 @cindex factorization
5075 @cindex @code{sqrfree()}
5077 GiNaC still lacks proper factorization support. Some form of
5078 factorization is, however, easily implemented by noting that factors
5079 appearing in a polynomial with power two or more also appear in the
5080 derivative and hence can easily be found by computing the GCD of the
5081 original polynomial and its derivatives. Any decent system has an
5082 interface for this so called square-free factorization. So we provide
5085 ex sqrfree(const ex & a, const lst & l = lst());
5087 Here is an example that by the way illustrates how the exact form of the
5088 result may slightly depend on the order of differentiation, calling for
5089 some care with subsequent processing of the result:
5092 symbol x("x"), y("y");
5093 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5095 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5096 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5098 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5099 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5101 cout << sqrfree(BiVarPol) << endl;
5102 // -> depending on luck, any of the above
5105 Note also, how factors with the same exponents are not fully factorized
5109 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5110 @c node-name, next, previous, up
5111 @section Rational expressions
5113 @subsection The @code{normal} method
5114 @cindex @code{normal()}
5115 @cindex simplification
5116 @cindex temporary replacement
5118 Some basic form of simplification of expressions is called for frequently.
5119 GiNaC provides the method @code{.normal()}, which converts a rational function
5120 into an equivalent rational function of the form @samp{numerator/denominator}
5121 where numerator and denominator are coprime. If the input expression is already
5122 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5123 otherwise it performs fraction addition and multiplication.
5125 @code{.normal()} can also be used on expressions which are not rational functions
5126 as it will replace all non-rational objects (like functions or non-integer
5127 powers) by temporary symbols to bring the expression to the domain of rational
5128 functions before performing the normalization, and re-substituting these
5129 symbols afterwards. This algorithm is also available as a separate method
5130 @code{.to_rational()}, described below.
5132 This means that both expressions @code{t1} and @code{t2} are indeed
5133 simplified in this little code snippet:
5138 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5139 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5140 std::cout << "t1 is " << t1.normal() << std::endl;
5141 std::cout << "t2 is " << t2.normal() << std::endl;
5145 Of course this works for multivariate polynomials too, so the ratio of
5146 the sample-polynomials from the section about GCD and LCM above would be
5147 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5150 @subsection Numerator and denominator
5153 @cindex @code{numer()}
5154 @cindex @code{denom()}
5155 @cindex @code{numer_denom()}
5157 The numerator and denominator of an expression can be obtained with
5162 ex ex::numer_denom();
5165 These functions will first normalize the expression as described above and
5166 then return the numerator, denominator, or both as a list, respectively.
5167 If you need both numerator and denominator, calling @code{numer_denom()} is
5168 faster than using @code{numer()} and @code{denom()} separately.
5171 @subsection Converting to a polynomial or rational expression
5172 @cindex @code{to_polynomial()}
5173 @cindex @code{to_rational()}
5175 Some of the methods described so far only work on polynomials or rational
5176 functions. GiNaC provides a way to extend the domain of these functions to
5177 general expressions by using the temporary replacement algorithm described
5178 above. You do this by calling
5181 ex ex::to_polynomial(exmap & m);
5182 ex ex::to_polynomial(lst & l);
5186 ex ex::to_rational(exmap & m);
5187 ex ex::to_rational(lst & l);
5190 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5191 will be filled with the generated temporary symbols and their replacement
5192 expressions in a format that can be used directly for the @code{subs()}
5193 method. It can also already contain a list of replacements from an earlier
5194 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5195 possible to use it on multiple expressions and get consistent results.
5197 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5198 is probably best illustrated with an example:
5202 symbol x("x"), y("y");
5203 ex a = 2*x/sin(x) - y/(3*sin(x));
5207 ex p = a.to_polynomial(lp);
5208 cout << " = " << p << "\n with " << lp << endl;
5209 // = symbol3*symbol2*y+2*symbol2*x
5210 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5213 ex r = a.to_rational(lr);
5214 cout << " = " << r << "\n with " << lr << endl;
5215 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5216 // with @{symbol4==sin(x)@}
5220 The following more useful example will print @samp{sin(x)-cos(x)}:
5225 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5226 ex b = sin(x) + cos(x);
5229 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5230 cout << q.subs(m) << endl;
5235 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5236 @c node-name, next, previous, up
5237 @section Symbolic differentiation
5238 @cindex differentiation
5239 @cindex @code{diff()}
5241 @cindex product rule
5243 GiNaC's objects know how to differentiate themselves. Thus, a
5244 polynomial (class @code{add}) knows that its derivative is the sum of
5245 the derivatives of all the monomials:
5249 symbol x("x"), y("y"), z("z");
5250 ex P = pow(x, 5) + pow(x, 2) + y;
5252 cout << P.diff(x,2) << endl;
5254 cout << P.diff(y) << endl; // 1
5256 cout << P.diff(z) << endl; // 0
5261 If a second integer parameter @var{n} is given, the @code{diff} method
5262 returns the @var{n}th derivative.
5264 If @emph{every} object and every function is told what its derivative
5265 is, all derivatives of composed objects can be calculated using the
5266 chain rule and the product rule. Consider, for instance the expression
5267 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5268 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5269 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5270 out that the composition is the generating function for Euler Numbers,
5271 i.e. the so called @var{n}th Euler number is the coefficient of
5272 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5273 identity to code a function that generates Euler numbers in just three
5276 @cindex Euler numbers
5278 #include <ginac/ginac.h>
5279 using namespace GiNaC;
5281 ex EulerNumber(unsigned n)
5284 const ex generator = pow(cosh(x),-1);
5285 return generator.diff(x,n).subs(x==0);
5290 for (unsigned i=0; i<11; i+=2)
5291 std::cout << EulerNumber(i) << std::endl;
5296 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5297 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5298 @code{i} by two since all odd Euler numbers vanish anyways.
5301 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5302 @c node-name, next, previous, up
5303 @section Series expansion
5304 @cindex @code{series()}
5305 @cindex Taylor expansion
5306 @cindex Laurent expansion
5307 @cindex @code{pseries} (class)
5308 @cindex @code{Order()}
5310 Expressions know how to expand themselves as a Taylor series or (more
5311 generally) a Laurent series. As in most conventional Computer Algebra
5312 Systems, no distinction is made between those two. There is a class of
5313 its own for storing such series (@code{class pseries}) and a built-in
5314 function (called @code{Order}) for storing the order term of the series.
5315 As a consequence, if you want to work with series, i.e. multiply two
5316 series, you need to call the method @code{ex::series} again to convert
5317 it to a series object with the usual structure (expansion plus order
5318 term). A sample application from special relativity could read:
5321 #include <ginac/ginac.h>
5322 using namespace std;
5323 using namespace GiNaC;
5327 symbol v("v"), c("c");
5329 ex gamma = 1/sqrt(1 - pow(v/c,2));
5330 ex mass_nonrel = gamma.series(v==0, 10);
5332 cout << "the relativistic mass increase with v is " << endl
5333 << mass_nonrel << endl;
5335 cout << "the inverse square of this series is " << endl
5336 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5340 Only calling the series method makes the last output simplify to
5341 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5342 series raised to the power @math{-2}.
5344 @cindex Machin's formula
5345 As another instructive application, let us calculate the numerical
5346 value of Archimedes' constant
5350 (for which there already exists the built-in constant @code{Pi})
5351 using John Machin's amazing formula
5353 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5356 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5358 This equation (and similar ones) were used for over 200 years for
5359 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5360 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5361 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5362 order term with it and the question arises what the system is supposed
5363 to do when the fractions are plugged into that order term. The solution
5364 is to use the function @code{series_to_poly()} to simply strip the order
5368 #include <ginac/ginac.h>
5369 using namespace GiNaC;
5371 ex machin_pi(int degr)
5374 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5375 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5376 -4*pi_expansion.subs(x==numeric(1,239));
5382 using std::cout; // just for fun, another way of...
5383 using std::endl; // ...dealing with this namespace std.
5385 for (int i=2; i<12; i+=2) @{
5386 pi_frac = machin_pi(i);
5387 cout << i << ":\t" << pi_frac << endl
5388 << "\t" << pi_frac.evalf() << endl;
5394 Note how we just called @code{.series(x,degr)} instead of
5395 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5396 method @code{series()}: if the first argument is a symbol the expression
5397 is expanded in that symbol around point @code{0}. When you run this
5398 program, it will type out:
5402 3.1832635983263598326
5403 4: 5359397032/1706489875
5404 3.1405970293260603143
5405 6: 38279241713339684/12184551018734375
5406 3.141621029325034425
5407 8: 76528487109180192540976/24359780855939418203125
5408 3.141591772182177295
5409 10: 327853873402258685803048818236/104359128170408663038552734375
5410 3.1415926824043995174
5414 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5415 @c node-name, next, previous, up
5416 @section Symmetrization
5417 @cindex @code{symmetrize()}
5418 @cindex @code{antisymmetrize()}
5419 @cindex @code{symmetrize_cyclic()}
5424 ex ex::symmetrize(const lst & l);
5425 ex ex::antisymmetrize(const lst & l);
5426 ex ex::symmetrize_cyclic(const lst & l);
5429 symmetrize an expression by returning the sum over all symmetric,
5430 antisymmetric or cyclic permutations of the specified list of objects,
5431 weighted by the number of permutations.
5433 The three additional methods
5436 ex ex::symmetrize();
5437 ex ex::antisymmetrize();
5438 ex ex::symmetrize_cyclic();
5441 symmetrize or antisymmetrize an expression over its free indices.
5443 Symmetrization is most useful with indexed expressions but can be used with
5444 almost any kind of object (anything that is @code{subs()}able):
5448 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5449 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5451 cout << indexed(A, i, j).symmetrize() << endl;
5452 // -> 1/2*A.j.i+1/2*A.i.j
5453 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5454 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5455 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5456 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5460 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5461 @c node-name, next, previous, up
5462 @section Predefined mathematical functions
5464 @subsection Overview
5466 GiNaC contains the following predefined mathematical functions:
5469 @multitable @columnfractions .30 .70
5470 @item @strong{Name} @tab @strong{Function}
5473 @cindex @code{abs()}
5474 @item @code{csgn(x)}
5476 @cindex @code{conjugate()}
5477 @item @code{conjugate(x)}
5478 @tab complex conjugation
5479 @cindex @code{csgn()}
5480 @item @code{sqrt(x)}
5481 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5482 @cindex @code{sqrt()}
5485 @cindex @code{sin()}
5488 @cindex @code{cos()}
5491 @cindex @code{tan()}
5492 @item @code{asin(x)}
5494 @cindex @code{asin()}
5495 @item @code{acos(x)}
5497 @cindex @code{acos()}
5498 @item @code{atan(x)}
5499 @tab inverse tangent
5500 @cindex @code{atan()}
5501 @item @code{atan2(y, x)}
5502 @tab inverse tangent with two arguments
5503 @item @code{sinh(x)}
5504 @tab hyperbolic sine
5505 @cindex @code{sinh()}
5506 @item @code{cosh(x)}
5507 @tab hyperbolic cosine
5508 @cindex @code{cosh()}
5509 @item @code{tanh(x)}
5510 @tab hyperbolic tangent
5511 @cindex @code{tanh()}
5512 @item @code{asinh(x)}
5513 @tab inverse hyperbolic sine
5514 @cindex @code{asinh()}
5515 @item @code{acosh(x)}
5516 @tab inverse hyperbolic cosine
5517 @cindex @code{acosh()}
5518 @item @code{atanh(x)}
5519 @tab inverse hyperbolic tangent
5520 @cindex @code{atanh()}
5522 @tab exponential function
5523 @cindex @code{exp()}
5525 @tab natural logarithm
5526 @cindex @code{log()}
5529 @cindex @code{Li2()}
5530 @item @code{Li(m, x)}
5531 @tab classical polylogarithm as well as multiple polylogarithm
5533 @item @code{G(a, y)}
5534 @tab multiple polylogarithm
5536 @item @code{G(a, s, y)}
5537 @tab multiple polylogarithm with explicit signs for the imaginary parts
5539 @item @code{S(n, p, x)}
5540 @tab Nielsen's generalized polylogarithm
5542 @item @code{H(m, x)}
5543 @tab harmonic polylogarithm
5545 @item @code{zeta(m)}
5546 @tab Riemann's zeta function as well as multiple zeta value
5547 @cindex @code{zeta()}
5548 @item @code{zeta(m, s)}
5549 @tab alternating Euler sum
5550 @cindex @code{zeta()}
5551 @item @code{zetaderiv(n, x)}
5552 @tab derivatives of Riemann's zeta function
5553 @item @code{tgamma(x)}
5555 @cindex @code{tgamma()}
5556 @cindex gamma function
5557 @item @code{lgamma(x)}
5558 @tab logarithm of gamma function
5559 @cindex @code{lgamma()}
5560 @item @code{beta(x, y)}
5561 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5562 @cindex @code{beta()}
5564 @tab psi (digamma) function
5565 @cindex @code{psi()}
5566 @item @code{psi(n, x)}
5567 @tab derivatives of psi function (polygamma functions)
5568 @item @code{factorial(n)}
5569 @tab factorial function @math{n!}
5570 @cindex @code{factorial()}
5571 @item @code{binomial(n, k)}
5572 @tab binomial coefficients
5573 @cindex @code{binomial()}
5574 @item @code{Order(x)}
5575 @tab order term function in truncated power series
5576 @cindex @code{Order()}
5581 For functions that have a branch cut in the complex plane GiNaC follows
5582 the conventions for C++ as defined in the ANSI standard as far as
5583 possible. In particular: the natural logarithm (@code{log}) and the
5584 square root (@code{sqrt}) both have their branch cuts running along the
5585 negative real axis where the points on the axis itself belong to the
5586 upper part (i.e. continuous with quadrant II). The inverse
5587 trigonometric and hyperbolic functions are not defined for complex
5588 arguments by the C++ standard, however. In GiNaC we follow the
5589 conventions used by CLN, which in turn follow the carefully designed
5590 definitions in the Common Lisp standard. It should be noted that this
5591 convention is identical to the one used by the C99 standard and by most
5592 serious CAS. It is to be expected that future revisions of the C++
5593 standard incorporate these functions in the complex domain in a manner
5594 compatible with C99.
5596 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5597 @c node-name, next, previous, up
5598 @subsection Multiple polylogarithms
5600 @cindex polylogarithm
5601 @cindex Nielsen's generalized polylogarithm
5602 @cindex harmonic polylogarithm
5603 @cindex multiple zeta value
5604 @cindex alternating Euler sum
5605 @cindex multiple polylogarithm
5607 The multiple polylogarithm is the most generic member of a family of functions,
5608 to which others like the harmonic polylogarithm, Nielsen's generalized
5609 polylogarithm and the multiple zeta value belong.
5610 Everyone of these functions can also be written as a multiple polylogarithm with specific
5611 parameters. This whole family of functions is therefore often referred to simply as
5612 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5613 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5614 @code{Li} and @code{G} in principle represent the same function, the different
5615 notations are more natural to the series representation or the integral
5616 representation, respectively.
5618 To facilitate the discussion of these functions we distinguish between indices and
5619 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5620 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5622 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5623 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5624 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5625 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5626 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5627 @code{s} is not given, the signs default to +1.
5628 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5629 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5630 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5631 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5632 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5634 The functions print in LaTeX format as
5636 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5642 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5645 $\zeta(m_1,m_2,\ldots,m_k)$.
5647 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5648 are printed with a line above, e.g.
5650 $\zeta(5,\overline{2})$.
5652 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5654 Definitions and analytical as well as numerical properties of multiple polylogarithms
5655 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5656 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5657 except for a few differences which will be explicitly stated in the following.
5659 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5660 that the indices and arguments are understood to be in the same order as in which they appear in
5661 the series representation. This means
5663 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5666 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5669 $\zeta(1,2)$ evaluates to infinity.
5671 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5674 The functions only evaluate if the indices are integers greater than zero, except for the indices
5675 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5676 will be interpreted as the sequence of signs for the corresponding indices
5677 @code{m} or the sign of the imaginary part for the
5678 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5679 @code{zeta(lst(3,4), lst(-1,1))} means
5681 $\zeta(\overline{3},4)$
5684 @code{G(lst(a,b), lst(-1,1), c)} means
5686 $G(a-0\epsilon,b+0\epsilon;c)$.
5688 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5689 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5690 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
5691 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5692 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5693 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5694 evaluates also for negative integers and positive even integers. For example:
5697 > Li(@{3,1@},@{x,1@});
5700 -zeta(@{3,2@},@{-1,-1@})
5705 It is easy to tell for a given function into which other function it can be rewritten, may
5706 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5707 with negative indices or trailing zeros (the example above gives a hint). Signs can
5708 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5709 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5710 @code{Li} (@code{eval()} already cares for the possible downgrade):
5713 > convert_H_to_Li(@{0,-2,-1,3@},x);
5714 Li(@{3,1,3@},@{-x,1,-1@})
5715 > convert_H_to_Li(@{2,-1,0@},x);
5716 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5719 Every function can be numerically evaluated for
5720 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5721 global variable @code{Digits}:
5726 > evalf(zeta(@{3,1,3,1@}));
5727 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5730 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5731 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5733 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5738 In long expressions this helps a lot with debugging, because you can easily spot
5739 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5740 cancellations of divergencies happen.
5742 Useful publications:
5744 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5745 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5747 @cite{Harmonic Polylogarithms},
5748 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5750 @cite{Special Values of Multiple Polylogarithms},
5751 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5753 @cite{Numerical Evaluation of Multiple Polylogarithms},
5754 J.Vollinga, S.Weinzierl, hep-ph/0410259
5756 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5757 @c node-name, next, previous, up
5758 @section Complex Conjugation
5760 @cindex @code{conjugate()}
5768 returns the complex conjugate of the expression. For all built-in functions and objects the
5769 conjugation gives the expected results:
5773 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5777 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5778 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5779 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5780 // -> -gamma5*gamma~b*gamma~a
5784 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5785 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5786 arguments. This is the default strategy. If you want to define your own functions and want to
5787 change this behavior, you have to supply a specialized conjugation method for your function
5788 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5790 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5791 @c node-name, next, previous, up
5792 @section Solving Linear Systems of Equations
5793 @cindex @code{lsolve()}
5795 The function @code{lsolve()} provides a convenient wrapper around some
5796 matrix operations that comes in handy when a system of linear equations
5800 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5803 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5804 @code{relational}) while @code{symbols} is a @code{lst} of
5805 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5808 It returns the @code{lst} of solutions as an expression. As an example,
5809 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5813 symbol a("a"), b("b"), x("x"), y("y");
5815 eqns = a*x+b*y==3, x-y==b;
5817 cout << lsolve(eqns, vars) << endl;
5818 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5821 When the linear equations @code{eqns} are underdetermined, the solution
5822 will contain one or more tautological entries like @code{x==x},
5823 depending on the rank of the system. When they are overdetermined, the
5824 solution will be an empty @code{lst}. Note the third optional parameter
5825 to @code{lsolve()}: it accepts the same parameters as
5826 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5830 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5831 @c node-name, next, previous, up
5832 @section Input and output of expressions
5835 @subsection Expression output
5837 @cindex output of expressions
5839 Expressions can simply be written to any stream:
5844 ex e = 4.5*I+pow(x,2)*3/2;
5845 cout << e << endl; // prints '4.5*I+3/2*x^2'
5849 The default output format is identical to the @command{ginsh} input syntax and
5850 to that used by most computer algebra systems, but not directly pastable
5851 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5852 is printed as @samp{x^2}).
5854 It is possible to print expressions in a number of different formats with
5855 a set of stream manipulators;
5858 std::ostream & dflt(std::ostream & os);
5859 std::ostream & latex(std::ostream & os);
5860 std::ostream & tree(std::ostream & os);
5861 std::ostream & csrc(std::ostream & os);
5862 std::ostream & csrc_float(std::ostream & os);
5863 std::ostream & csrc_double(std::ostream & os);
5864 std::ostream & csrc_cl_N(std::ostream & os);
5865 std::ostream & index_dimensions(std::ostream & os);
5866 std::ostream & no_index_dimensions(std::ostream & os);
5869 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5870 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5871 @code{print_csrc()} functions, respectively.
5874 All manipulators affect the stream state permanently. To reset the output
5875 format to the default, use the @code{dflt} manipulator:
5879 cout << latex; // all output to cout will be in LaTeX format from now on
5880 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5881 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5882 cout << dflt; // revert to default output format
5883 cout << e << endl; // prints '4.5*I+3/2*x^2'
5887 If you don't want to affect the format of the stream you're working with,
5888 you can output to a temporary @code{ostringstream} like this:
5893 s << latex << e; // format of cout remains unchanged
5894 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5899 @cindex @code{csrc_float}
5900 @cindex @code{csrc_double}
5901 @cindex @code{csrc_cl_N}
5902 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5903 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5904 format that can be directly used in a C or C++ program. The three possible
5905 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5906 classes provided by the CLN library):
5910 cout << "f = " << csrc_float << e << ";\n";
5911 cout << "d = " << csrc_double << e << ";\n";
5912 cout << "n = " << csrc_cl_N << e << ";\n";
5916 The above example will produce (note the @code{x^2} being converted to
5920 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5921 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5922 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5926 The @code{tree} manipulator allows dumping the internal structure of an
5927 expression for debugging purposes:
5938 add, hash=0x0, flags=0x3, nops=2
5939 power, hash=0x0, flags=0x3, nops=2
5940 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5941 2 (numeric), hash=0x6526b0fa, flags=0xf
5942 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5945 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5949 @cindex @code{latex}
5950 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5951 It is rather similar to the default format but provides some braces needed
5952 by LaTeX for delimiting boxes and also converts some common objects to
5953 conventional LaTeX names. It is possible to give symbols a special name for
5954 LaTeX output by supplying it as a second argument to the @code{symbol}
5957 For example, the code snippet
5961 symbol x("x", "\\circ");
5962 ex e = lgamma(x).series(x==0,3);
5963 cout << latex << e << endl;
5970 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5973 @cindex @code{index_dimensions}
5974 @cindex @code{no_index_dimensions}
5975 Index dimensions are normally hidden in the output. To make them visible, use
5976 the @code{index_dimensions} manipulator. The dimensions will be written in
5977 square brackets behind each index value in the default and LaTeX output
5982 symbol x("x"), y("y");
5983 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5984 ex e = indexed(x, mu) * indexed(y, nu);
5987 // prints 'x~mu*y~nu'
5988 cout << index_dimensions << e << endl;
5989 // prints 'x~mu[4]*y~nu[4]'
5990 cout << no_index_dimensions << e << endl;
5991 // prints 'x~mu*y~nu'
5996 @cindex Tree traversal
5997 If you need any fancy special output format, e.g. for interfacing GiNaC
5998 with other algebra systems or for producing code for different
5999 programming languages, you can always traverse the expression tree yourself:
6002 static void my_print(const ex & e)
6004 if (is_a<function>(e))
6005 cout << ex_to<function>(e).get_name();
6007 cout << ex_to<basic>(e).class_name();
6009 size_t n = e.nops();
6011 for (size_t i=0; i<n; i++) @{
6023 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6031 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6032 symbol(y))),numeric(-2)))
6035 If you need an output format that makes it possible to accurately
6036 reconstruct an expression by feeding the output to a suitable parser or
6037 object factory, you should consider storing the expression in an
6038 @code{archive} object and reading the object properties from there.
6039 See the section on archiving for more information.
6042 @subsection Expression input
6043 @cindex input of expressions
6045 GiNaC provides no way to directly read an expression from a stream because
6046 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6047 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6048 @code{y} you defined in your program and there is no way to specify the
6049 desired symbols to the @code{>>} stream input operator.
6051 Instead, GiNaC lets you construct an expression from a string, specifying the
6052 list of symbols to be used:
6056 symbol x("x"), y("y");
6057 ex e("2*x+sin(y)", lst(x, y));
6061 The input syntax is the same as that used by @command{ginsh} and the stream
6062 output operator @code{<<}. The symbols in the string are matched by name to
6063 the symbols in the list and if GiNaC encounters a symbol not specified in
6064 the list it will throw an exception.
6066 With this constructor, it's also easy to implement interactive GiNaC programs:
6071 #include <stdexcept>
6072 #include <ginac/ginac.h>
6073 using namespace std;
6074 using namespace GiNaC;
6081 cout << "Enter an expression containing 'x': ";
6086 cout << "The derivative of " << e << " with respect to x is ";
6087 cout << e.diff(x) << ".\n";
6088 @} catch (exception &p) @{
6089 cerr << p.what() << endl;
6095 @subsection Archiving
6096 @cindex @code{archive} (class)
6099 GiNaC allows creating @dfn{archives} of expressions which can be stored
6100 to or retrieved from files. To create an archive, you declare an object
6101 of class @code{archive} and archive expressions in it, giving each
6102 expression a unique name:
6106 using namespace std;
6107 #include <ginac/ginac.h>
6108 using namespace GiNaC;
6112 symbol x("x"), y("y"), z("z");
6114 ex foo = sin(x + 2*y) + 3*z + 41;
6118 a.archive_ex(foo, "foo");
6119 a.archive_ex(bar, "the second one");
6123 The archive can then be written to a file:
6127 ofstream out("foobar.gar");
6133 The file @file{foobar.gar} contains all information that is needed to
6134 reconstruct the expressions @code{foo} and @code{bar}.
6136 @cindex @command{viewgar}
6137 The tool @command{viewgar} that comes with GiNaC can be used to view
6138 the contents of GiNaC archive files:
6141 $ viewgar foobar.gar
6142 foo = 41+sin(x+2*y)+3*z
6143 the second one = 42+sin(x+2*y)+3*z
6146 The point of writing archive files is of course that they can later be
6152 ifstream in("foobar.gar");
6157 And the stored expressions can be retrieved by their name:
6164 ex ex1 = a2.unarchive_ex(syms, "foo");
6165 ex ex2 = a2.unarchive_ex(syms, "the second one");
6167 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6168 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6169 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6173 Note that you have to supply a list of the symbols which are to be inserted
6174 in the expressions. Symbols in archives are stored by their name only and
6175 if you don't specify which symbols you have, unarchiving the expression will
6176 create new symbols with that name. E.g. if you hadn't included @code{x} in
6177 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6178 have had no effect because the @code{x} in @code{ex1} would have been a
6179 different symbol than the @code{x} which was defined at the beginning of
6180 the program, although both would appear as @samp{x} when printed.
6182 You can also use the information stored in an @code{archive} object to
6183 output expressions in a format suitable for exact reconstruction. The
6184 @code{archive} and @code{archive_node} classes have a couple of member
6185 functions that let you access the stored properties:
6188 static void my_print2(const archive_node & n)
6191 n.find_string("class", class_name);
6192 cout << class_name << "(";
6194 archive_node::propinfovector p;
6195 n.get_properties(p);
6197 size_t num = p.size();
6198 for (size_t i=0; i<num; i++) @{
6199 const string &name = p[i].name;
6200 if (name == "class")
6202 cout << name << "=";
6204 unsigned count = p[i].count;
6208 for (unsigned j=0; j<count; j++) @{
6209 switch (p[i].type) @{
6210 case archive_node::PTYPE_BOOL: @{
6212 n.find_bool(name, x, j);
6213 cout << (x ? "true" : "false");
6216 case archive_node::PTYPE_UNSIGNED: @{
6218 n.find_unsigned(name, x, j);
6222 case archive_node::PTYPE_STRING: @{
6224 n.find_string(name, x, j);
6225 cout << '\"' << x << '\"';
6228 case archive_node::PTYPE_NODE: @{
6229 const archive_node &x = n.find_ex_node(name, j);
6251 ex e = pow(2, x) - y;
6253 my_print2(ar.get_top_node(0)); cout << endl;
6261 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6262 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6263 overall_coeff=numeric(number="0"))
6266 Be warned, however, that the set of properties and their meaning for each
6267 class may change between GiNaC versions.
6270 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6271 @c node-name, next, previous, up
6272 @chapter Extending GiNaC
6274 By reading so far you should have gotten a fairly good understanding of
6275 GiNaC's design patterns. From here on you should start reading the
6276 sources. All we can do now is issue some recommendations how to tackle
6277 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6278 develop some useful extension please don't hesitate to contact the GiNaC
6279 authors---they will happily incorporate them into future versions.
6282 * What does not belong into GiNaC:: What to avoid.
6283 * Symbolic functions:: Implementing symbolic functions.
6284 * Printing:: Adding new output formats.
6285 * Structures:: Defining new algebraic classes (the easy way).
6286 * Adding classes:: Defining new algebraic classes (the hard way).
6290 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6291 @c node-name, next, previous, up
6292 @section What doesn't belong into GiNaC
6294 @cindex @command{ginsh}
6295 First of all, GiNaC's name must be read literally. It is designed to be
6296 a library for use within C++. The tiny @command{ginsh} accompanying
6297 GiNaC makes this even more clear: it doesn't even attempt to provide a
6298 language. There are no loops or conditional expressions in
6299 @command{ginsh}, it is merely a window into the library for the
6300 programmer to test stuff (or to show off). Still, the design of a
6301 complete CAS with a language of its own, graphical capabilities and all
6302 this on top of GiNaC is possible and is without doubt a nice project for
6305 There are many built-in functions in GiNaC that do not know how to
6306 evaluate themselves numerically to a precision declared at runtime
6307 (using @code{Digits}). Some may be evaluated at certain points, but not
6308 generally. This ought to be fixed. However, doing numerical
6309 computations with GiNaC's quite abstract classes is doomed to be
6310 inefficient. For this purpose, the underlying foundation classes
6311 provided by CLN are much better suited.
6314 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6315 @c node-name, next, previous, up
6316 @section Symbolic functions
6318 The easiest and most instructive way to start extending GiNaC is probably to
6319 create your own symbolic functions. These are implemented with the help of
6320 two preprocessor macros:
6322 @cindex @code{DECLARE_FUNCTION}
6323 @cindex @code{REGISTER_FUNCTION}
6325 DECLARE_FUNCTION_<n>P(<name>)
6326 REGISTER_FUNCTION(<name>, <options>)
6329 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6330 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6331 parameters of type @code{ex} and returns a newly constructed GiNaC
6332 @code{function} object that represents your function.
6334 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6335 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6336 set of options that associate the symbolic function with C++ functions you
6337 provide to implement the various methods such as evaluation, derivative,
6338 series expansion etc. They also describe additional attributes the function
6339 might have, such as symmetry and commutation properties, and a name for
6340 LaTeX output. Multiple options are separated by the member access operator
6341 @samp{.} and can be given in an arbitrary order.
6343 (By the way: in case you are worrying about all the macros above we can
6344 assure you that functions are GiNaC's most macro-intense classes. We have
6345 done our best to avoid macros where we can.)
6347 @subsection A minimal example
6349 Here is an example for the implementation of a function with two arguments
6350 that is not further evaluated:
6353 DECLARE_FUNCTION_2P(myfcn)
6355 REGISTER_FUNCTION(myfcn, dummy())
6358 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6359 in algebraic expressions:
6365 ex e = 2*myfcn(42, 1+3*x) - x;
6367 // prints '2*myfcn(42,1+3*x)-x'
6372 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6373 "no options". A function with no options specified merely acts as a kind of
6374 container for its arguments. It is a pure "dummy" function with no associated
6375 logic (which is, however, sometimes perfectly sufficient).
6377 Let's now have a look at the implementation of GiNaC's cosine function for an
6378 example of how to make an "intelligent" function.
6380 @subsection The cosine function
6382 The GiNaC header file @file{inifcns.h} contains the line
6385 DECLARE_FUNCTION_1P(cos)
6388 which declares to all programs using GiNaC that there is a function @samp{cos}
6389 that takes one @code{ex} as an argument. This is all they need to know to use
6390 this function in expressions.
6392 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6393 is its @code{REGISTER_FUNCTION} line:
6396 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6397 evalf_func(cos_evalf).
6398 derivative_func(cos_deriv).
6399 latex_name("\\cos"));
6402 There are four options defined for the cosine function. One of them
6403 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6404 other three indicate the C++ functions in which the "brains" of the cosine
6405 function are defined.
6407 @cindex @code{hold()}
6409 The @code{eval_func()} option specifies the C++ function that implements
6410 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6411 the same number of arguments as the associated symbolic function (one in this
6412 case) and returns the (possibly transformed or in some way simplified)
6413 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6414 of the automatic evaluation process). If no (further) evaluation is to take
6415 place, the @code{eval_func()} function must return the original function
6416 with @code{.hold()}, to avoid a potential infinite recursion. If your
6417 symbolic functions produce a segmentation fault or stack overflow when
6418 using them in expressions, you are probably missing a @code{.hold()}
6421 The @code{eval_func()} function for the cosine looks something like this
6422 (actually, it doesn't look like this at all, but it should give you an idea
6426 static ex cos_eval(const ex & x)
6428 if ("x is a multiple of 2*Pi")
6430 else if ("x is a multiple of Pi")
6432 else if ("x is a multiple of Pi/2")
6436 else if ("x has the form 'acos(y)'")
6438 else if ("x has the form 'asin(y)'")
6443 return cos(x).hold();
6447 This function is called every time the cosine is used in a symbolic expression:
6453 // this calls cos_eval(Pi), and inserts its return value into
6454 // the actual expression
6461 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6462 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6463 symbolic transformation can be done, the unmodified function is returned
6464 with @code{.hold()}.
6466 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6467 The user has to call @code{evalf()} for that. This is implemented in a
6471 static ex cos_evalf(const ex & x)
6473 if (is_a<numeric>(x))
6474 return cos(ex_to<numeric>(x));
6476 return cos(x).hold();
6480 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6481 in this case the @code{cos()} function for @code{numeric} objects, which in
6482 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6483 isn't really needed here, but reminds us that the corresponding @code{eval()}
6484 function would require it in this place.
6486 Differentiation will surely turn up and so we need to tell @code{cos}
6487 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6488 instance, are then handled automatically by @code{basic::diff} and
6492 static ex cos_deriv(const ex & x, unsigned diff_param)
6498 @cindex product rule
6499 The second parameter is obligatory but uninteresting at this point. It
6500 specifies which parameter to differentiate in a partial derivative in
6501 case the function has more than one parameter, and its main application
6502 is for correct handling of the chain rule.
6504 An implementation of the series expansion is not needed for @code{cos()} as
6505 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6506 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6507 the other hand, does have poles and may need to do Laurent expansion:
6510 static ex tan_series(const ex & x, const relational & rel,
6511 int order, unsigned options)
6513 // Find the actual expansion point
6514 const ex x_pt = x.subs(rel);
6516 if ("x_pt is not an odd multiple of Pi/2")
6517 throw do_taylor(); // tell function::series() to do Taylor expansion
6519 // On a pole, expand sin()/cos()
6520 return (sin(x)/cos(x)).series(rel, order+2, options);
6524 The @code{series()} implementation of a function @emph{must} return a
6525 @code{pseries} object, otherwise your code will crash.
6527 @subsection Function options
6529 GiNaC functions understand several more options which are always
6530 specified as @code{.option(params)}. None of them are required, but you
6531 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6532 is a do-nothing option called @code{dummy()} which you can use to define
6533 functions without any special options.
6536 eval_func(<C++ function>)
6537 evalf_func(<C++ function>)
6538 derivative_func(<C++ function>)
6539 series_func(<C++ function>)
6540 conjugate_func(<C++ function>)
6543 These specify the C++ functions that implement symbolic evaluation,
6544 numeric evaluation, partial derivatives, and series expansion, respectively.
6545 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6546 @code{diff()} and @code{series()}.
6548 The @code{eval_func()} function needs to use @code{.hold()} if no further
6549 automatic evaluation is desired or possible.
6551 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6552 expansion, which is correct if there are no poles involved. If the function
6553 has poles in the complex plane, the @code{series_func()} needs to check
6554 whether the expansion point is on a pole and fall back to Taylor expansion
6555 if it isn't. Otherwise, the pole usually needs to be regularized by some
6556 suitable transformation.
6559 latex_name(const string & n)
6562 specifies the LaTeX code that represents the name of the function in LaTeX
6563 output. The default is to put the function name in an @code{\mbox@{@}}.
6566 do_not_evalf_params()
6569 This tells @code{evalf()} to not recursively evaluate the parameters of the
6570 function before calling the @code{evalf_func()}.
6573 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6576 This allows you to explicitly specify the commutation properties of the
6577 function (@xref{Non-commutative objects}, for an explanation of
6578 (non)commutativity in GiNaC). For example, you can use
6579 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6580 GiNaC treat your function like a matrix. By default, functions inherit the
6581 commutation properties of their first argument.
6584 set_symmetry(const symmetry & s)
6587 specifies the symmetry properties of the function with respect to its
6588 arguments. @xref{Indexed objects}, for an explanation of symmetry
6589 specifications. GiNaC will automatically rearrange the arguments of
6590 symmetric functions into a canonical order.
6592 Sometimes you may want to have finer control over how functions are
6593 displayed in the output. For example, the @code{abs()} function prints
6594 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6595 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6599 print_func<C>(<C++ function>)
6602 option which is explained in the next section.
6604 @subsection Functions with a variable number of arguments
6606 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6607 functions with a fixed number of arguments. Sometimes, though, you may need
6608 to have a function that accepts a variable number of expressions. One way to
6609 accomplish this is to pass variable-length lists as arguments. The
6610 @code{Li()} function uses this method for multiple polylogarithms.
6612 It is also possible to define functions that accept a different number of
6613 parameters under the same function name, such as the @code{psi()} function
6614 which can be called either as @code{psi(z)} (the digamma function) or as
6615 @code{psi(n, z)} (polygamma functions). These are actually two different
6616 functions in GiNaC that, however, have the same name. Defining such
6617 functions is not possible with the macros but requires manually fiddling
6618 with GiNaC internals. If you are interested, please consult the GiNaC source
6619 code for the @code{psi()} function (@file{inifcns.h} and
6620 @file{inifcns_gamma.cpp}).
6623 @node Printing, Structures, Symbolic functions, Extending GiNaC
6624 @c node-name, next, previous, up
6625 @section GiNaC's expression output system
6627 GiNaC allows the output of expressions in a variety of different formats
6628 (@pxref{Input/Output}). This section will explain how expression output
6629 is implemented internally, and how to define your own output formats or
6630 change the output format of built-in algebraic objects. You will also want
6631 to read this section if you plan to write your own algebraic classes or
6634 @cindex @code{print_context} (class)
6635 @cindex @code{print_dflt} (class)
6636 @cindex @code{print_latex} (class)
6637 @cindex @code{print_tree} (class)
6638 @cindex @code{print_csrc} (class)
6639 All the different output formats are represented by a hierarchy of classes
6640 rooted in the @code{print_context} class, defined in the @file{print.h}
6645 the default output format
6647 output in LaTeX mathematical mode
6649 a dump of the internal expression structure (for debugging)
6651 the base class for C source output
6652 @item print_csrc_float
6653 C source output using the @code{float} type
6654 @item print_csrc_double
6655 C source output using the @code{double} type
6656 @item print_csrc_cl_N
6657 C source output using CLN types
6660 The @code{print_context} base class provides two public data members:
6672 @code{s} is a reference to the stream to output to, while @code{options}
6673 holds flags and modifiers. Currently, there is only one flag defined:
6674 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6675 to print the index dimension which is normally hidden.
6677 When you write something like @code{std::cout << e}, where @code{e} is
6678 an object of class @code{ex}, GiNaC will construct an appropriate
6679 @code{print_context} object (of a class depending on the selected output
6680 format), fill in the @code{s} and @code{options} members, and call
6682 @cindex @code{print()}
6684 void ex::print(const print_context & c, unsigned level = 0) const;
6687 which in turn forwards the call to the @code{print()} method of the
6688 top-level algebraic object contained in the expression.
6690 Unlike other methods, GiNaC classes don't usually override their
6691 @code{print()} method to implement expression output. Instead, the default
6692 implementation @code{basic::print(c, level)} performs a run-time double
6693 dispatch to a function selected by the dynamic type of the object and the
6694 passed @code{print_context}. To this end, GiNaC maintains a separate method
6695 table for each class, similar to the virtual function table used for ordinary
6696 (single) virtual function dispatch.
6698 The method table contains one slot for each possible @code{print_context}
6699 type, indexed by the (internally assigned) serial number of the type. Slots
6700 may be empty, in which case GiNaC will retry the method lookup with the
6701 @code{print_context} object's parent class, possibly repeating the process
6702 until it reaches the @code{print_context} base class. If there's still no
6703 method defined, the method table of the algebraic object's parent class
6704 is consulted, and so on, until a matching method is found (eventually it
6705 will reach the combination @code{basic/print_context}, which prints the
6706 object's class name enclosed in square brackets).
6708 You can think of the print methods of all the different classes and output
6709 formats as being arranged in a two-dimensional matrix with one axis listing
6710 the algebraic classes and the other axis listing the @code{print_context}
6713 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6714 to implement printing, but then they won't get any of the benefits of the
6715 double dispatch mechanism (such as the ability for derived classes to
6716 inherit only certain print methods from its parent, or the replacement of
6717 methods at run-time).
6719 @subsection Print methods for classes
6721 The method table for a class is set up either in the definition of the class,
6722 by passing the appropriate @code{print_func<C>()} option to
6723 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6724 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6725 can also be used to override existing methods dynamically.
6727 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6728 be a member function of the class (or one of its parent classes), a static
6729 member function, or an ordinary (global) C++ function. The @code{C} template
6730 parameter specifies the appropriate @code{print_context} type for which the
6731 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6732 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6733 the class is the one being implemented by
6734 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6736 For print methods that are member functions, their first argument must be of
6737 a type convertible to a @code{const C &}, and the second argument must be an
6740 For static members and global functions, the first argument must be of a type
6741 convertible to a @code{const T &}, the second argument must be of a type
6742 convertible to a @code{const C &}, and the third argument must be an
6743 @code{unsigned}. A global function will, of course, not have access to
6744 private and protected members of @code{T}.
6746 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6747 and @code{basic::print()}) is used for proper parenthesizing of the output
6748 (and by @code{print_tree} for proper indentation). It can be used for similar
6749 purposes if you write your own output formats.
6751 The explanations given above may seem complicated, but in practice it's
6752 really simple, as shown in the following example. Suppose that we want to
6753 display exponents in LaTeX output not as superscripts but with little
6754 upwards-pointing arrows. This can be achieved in the following way:
6757 void my_print_power_as_latex(const power & p,
6758 const print_latex & c,
6761 // get the precedence of the 'power' class
6762 unsigned power_prec = p.precedence();
6764 // if the parent operator has the same or a higher precedence
6765 // we need parentheses around the power
6766 if (level >= power_prec)
6769 // print the basis and exponent, each enclosed in braces, and
6770 // separated by an uparrow
6772 p.op(0).print(c, power_prec);
6773 c.s << "@}\\uparrow@{";
6774 p.op(1).print(c, power_prec);
6777 // don't forget the closing parenthesis
6778 if (level >= power_prec)
6784 // a sample expression
6785 symbol x("x"), y("y");
6786 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6788 // switch to LaTeX mode
6791 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6794 // now we replace the method for the LaTeX output of powers with
6796 set_print_func<power, print_latex>(my_print_power_as_latex);
6798 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6808 The first argument of @code{my_print_power_as_latex} could also have been
6809 a @code{const basic &}, the second one a @code{const print_context &}.
6812 The above code depends on @code{mul} objects converting their operands to
6813 @code{power} objects for the purpose of printing.
6816 The output of products including negative powers as fractions is also
6817 controlled by the @code{mul} class.
6820 The @code{power/print_latex} method provided by GiNaC prints square roots
6821 using @code{\sqrt}, but the above code doesn't.
6825 It's not possible to restore a method table entry to its previous or default
6826 value. Once you have called @code{set_print_func()}, you can only override
6827 it with another call to @code{set_print_func()}, but you can't easily go back
6828 to the default behavior again (you can, of course, dig around in the GiNaC
6829 sources, find the method that is installed at startup
6830 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6831 one; that is, after you circumvent the C++ member access control@dots{}).
6833 @subsection Print methods for functions
6835 Symbolic functions employ a print method dispatch mechanism similar to the
6836 one used for classes. The methods are specified with @code{print_func<C>()}
6837 function options. If you don't specify any special print methods, the function
6838 will be printed with its name (or LaTeX name, if supplied), followed by a
6839 comma-separated list of arguments enclosed in parentheses.
6841 For example, this is what GiNaC's @samp{abs()} function is defined like:
6844 static ex abs_eval(const ex & arg) @{ ... @}
6845 static ex abs_evalf(const ex & arg) @{ ... @}
6847 static void abs_print_latex(const ex & arg, const print_context & c)
6849 c.s << "@{|"; arg.print(c); c.s << "|@}";
6852 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6854 c.s << "fabs("; arg.print(c); c.s << ")";
6857 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6858 evalf_func(abs_evalf).
6859 print_func<print_latex>(abs_print_latex).
6860 print_func<print_csrc_float>(abs_print_csrc_float).
6861 print_func<print_csrc_double>(abs_print_csrc_float));
6864 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6865 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6867 There is currently no equivalent of @code{set_print_func()} for functions.
6869 @subsection Adding new output formats
6871 Creating a new output format involves subclassing @code{print_context},
6872 which is somewhat similar to adding a new algebraic class
6873 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6874 that needs to go into the class definition, and a corresponding macro
6875 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6876 Every @code{print_context} class needs to provide a default constructor
6877 and a constructor from an @code{std::ostream} and an @code{unsigned}
6880 Here is an example for a user-defined @code{print_context} class:
6883 class print_myformat : public print_dflt
6885 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6887 print_myformat(std::ostream & os, unsigned opt = 0)
6888 : print_dflt(os, opt) @{@}
6891 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6893 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6896 That's all there is to it. None of the actual expression output logic is
6897 implemented in this class. It merely serves as a selector for choosing
6898 a particular format. The algorithms for printing expressions in the new
6899 format are implemented as print methods, as described above.
6901 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6902 exactly like GiNaC's default output format:
6907 ex e = pow(x, 2) + 1;
6909 // this prints "1+x^2"
6912 // this also prints "1+x^2"
6913 e.print(print_myformat()); cout << endl;
6919 To fill @code{print_myformat} with life, we need to supply appropriate
6920 print methods with @code{set_print_func()}, like this:
6923 // This prints powers with '**' instead of '^'. See the LaTeX output
6924 // example above for explanations.
6925 void print_power_as_myformat(const power & p,
6926 const print_myformat & c,
6929 unsigned power_prec = p.precedence();
6930 if (level >= power_prec)
6932 p.op(0).print(c, power_prec);
6934 p.op(1).print(c, power_prec);
6935 if (level >= power_prec)
6941 // install a new print method for power objects
6942 set_print_func<power, print_myformat>(print_power_as_myformat);
6944 // now this prints "1+x**2"
6945 e.print(print_myformat()); cout << endl;
6947 // but the default format is still "1+x^2"
6953 @node Structures, Adding classes, Printing, Extending GiNaC
6954 @c node-name, next, previous, up
6957 If you are doing some very specialized things with GiNaC, or if you just
6958 need some more organized way to store data in your expressions instead of
6959 anonymous lists, you may want to implement your own algebraic classes.
6960 ('algebraic class' means any class directly or indirectly derived from
6961 @code{basic} that can be used in GiNaC expressions).
6963 GiNaC offers two ways of accomplishing this: either by using the
6964 @code{structure<T>} template class, or by rolling your own class from
6965 scratch. This section will discuss the @code{structure<T>} template which
6966 is easier to use but more limited, while the implementation of custom
6967 GiNaC classes is the topic of the next section. However, you may want to
6968 read both sections because many common concepts and member functions are
6969 shared by both concepts, and it will also allow you to decide which approach
6970 is most suited to your needs.
6972 The @code{structure<T>} template, defined in the GiNaC header file
6973 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6974 or @code{class}) into a GiNaC object that can be used in expressions.
6976 @subsection Example: scalar products
6978 Let's suppose that we need a way to handle some kind of abstract scalar
6979 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6980 product class have to store their left and right operands, which can in turn
6981 be arbitrary expressions. Here is a possible way to represent such a
6982 product in a C++ @code{struct}:
6986 using namespace std;
6988 #include <ginac/ginac.h>
6989 using namespace GiNaC;
6995 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6999 The default constructor is required. Now, to make a GiNaC class out of this
7000 data structure, we need only one line:
7003 typedef structure<sprod_s> sprod;
7006 That's it. This line constructs an algebraic class @code{sprod} which
7007 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7008 expressions like any other GiNaC class:
7012 symbol a("a"), b("b");
7013 ex e = sprod(sprod_s(a, b));
7017 Note the difference between @code{sprod} which is the algebraic class, and
7018 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7019 and @code{right} data members. As shown above, an @code{sprod} can be
7020 constructed from an @code{sprod_s} object.
7022 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7023 you could define a little wrapper function like this:
7026 inline ex make_sprod(ex left, ex right)
7028 return sprod(sprod_s(left, right));
7032 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7033 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7034 @code{get_struct()}:
7038 cout << ex_to<sprod>(e)->left << endl;
7040 cout << ex_to<sprod>(e).get_struct().right << endl;
7045 You only have read access to the members of @code{sprod_s}.
7047 The type definition of @code{sprod} is enough to write your own algorithms
7048 that deal with scalar products, for example:
7053 if (is_a<sprod>(p)) @{
7054 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7055 return make_sprod(sp.right, sp.left);
7066 @subsection Structure output
7068 While the @code{sprod} type is useable it still leaves something to be
7069 desired, most notably proper output:
7074 // -> [structure object]
7078 By default, any structure types you define will be printed as
7079 @samp{[structure object]}. To override this you can either specialize the
7080 template's @code{print()} member function, or specify print methods with
7081 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7082 it's not possible to supply class options like @code{print_func<>()} to
7083 structures, so for a self-contained structure type you need to resort to
7084 overriding the @code{print()} function, which is also what we will do here.
7086 The member functions of GiNaC classes are described in more detail in the
7087 next section, but it shouldn't be hard to figure out what's going on here:
7090 void sprod::print(const print_context & c, unsigned level) const
7092 // tree debug output handled by superclass
7093 if (is_a<print_tree>(c))
7094 inherited::print(c, level);
7096 // get the contained sprod_s object
7097 const sprod_s & sp = get_struct();
7099 // print_context::s is a reference to an ostream
7100 c.s << "<" << sp.left << "|" << sp.right << ">";
7104 Now we can print expressions containing scalar products:
7110 cout << swap_sprod(e) << endl;
7115 @subsection Comparing structures
7117 The @code{sprod} class defined so far still has one important drawback: all
7118 scalar products are treated as being equal because GiNaC doesn't know how to
7119 compare objects of type @code{sprod_s}. This can lead to some confusing
7120 and undesired behavior:
7124 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7126 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7127 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7131 To remedy this, we first need to define the operators @code{==} and @code{<}
7132 for objects of type @code{sprod_s}:
7135 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7137 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7140 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7142 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
7146 The ordering established by the @code{<} operator doesn't have to make any
7147 algebraic sense, but it needs to be well defined. Note that we can't use
7148 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7149 in the implementation of these operators because they would construct
7150 GiNaC @code{relational} objects which in the case of @code{<} do not
7151 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7152 decide which one is algebraically 'less').
7154 Next, we need to change our definition of the @code{sprod} type to let
7155 GiNaC know that an ordering relation exists for the embedded objects:
7158 typedef structure<sprod_s, compare_std_less> sprod;
7161 @code{sprod} objects then behave as expected:
7165 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7166 // -> <a|b>-<a^2|b^2>
7167 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7168 // -> <a|b>+<a^2|b^2>
7169 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7171 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7176 The @code{compare_std_less} policy parameter tells GiNaC to use the
7177 @code{std::less} and @code{std::equal_to} functors to compare objects of
7178 type @code{sprod_s}. By default, these functors forward their work to the
7179 standard @code{<} and @code{==} operators, which we have overloaded.
7180 Alternatively, we could have specialized @code{std::less} and
7181 @code{std::equal_to} for class @code{sprod_s}.
7183 GiNaC provides two other comparison policies for @code{structure<T>}
7184 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7185 which does a bit-wise comparison of the contained @code{T} objects.
7186 This should be used with extreme care because it only works reliably with
7187 built-in integral types, and it also compares any padding (filler bytes of
7188 undefined value) that the @code{T} class might have.
7190 @subsection Subexpressions
7192 Our scalar product class has two subexpressions: the left and right
7193 operands. It might be a good idea to make them accessible via the standard
7194 @code{nops()} and @code{op()} methods:
7197 size_t sprod::nops() const
7202 ex sprod::op(size_t i) const
7206 return get_struct().left;
7208 return get_struct().right;
7210 throw std::range_error("sprod::op(): no such operand");
7215 Implementing @code{nops()} and @code{op()} for container types such as
7216 @code{sprod} has two other nice side effects:
7220 @code{has()} works as expected
7222 GiNaC generates better hash keys for the objects (the default implementation
7223 of @code{calchash()} takes subexpressions into account)
7226 @cindex @code{let_op()}
7227 There is a non-const variant of @code{op()} called @code{let_op()} that
7228 allows replacing subexpressions:
7231 ex & sprod::let_op(size_t i)
7233 // every non-const member function must call this
7234 ensure_if_modifiable();
7238 return get_struct().left;
7240 return get_struct().right;
7242 throw std::range_error("sprod::let_op(): no such operand");
7247 Once we have provided @code{let_op()} we also get @code{subs()} and
7248 @code{map()} for free. In fact, every container class that returns a non-null
7249 @code{nops()} value must either implement @code{let_op()} or provide custom
7250 implementations of @code{subs()} and @code{map()}.
7252 In turn, the availability of @code{map()} enables the recursive behavior of a
7253 couple of other default method implementations, in particular @code{evalf()},
7254 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7255 we probably want to provide our own version of @code{expand()} for scalar
7256 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7257 This is left as an exercise for the reader.
7259 The @code{structure<T>} template defines many more member functions that
7260 you can override by specialization to customize the behavior of your
7261 structures. You are referred to the next section for a description of
7262 some of these (especially @code{eval()}). There is, however, one topic
7263 that shall be addressed here, as it demonstrates one peculiarity of the
7264 @code{structure<T>} template: archiving.
7266 @subsection Archiving structures
7268 If you don't know how the archiving of GiNaC objects is implemented, you
7269 should first read the next section and then come back here. You're back?
7272 To implement archiving for structures it is not enough to provide
7273 specializations for the @code{archive()} member function and the
7274 unarchiving constructor (the @code{unarchive()} function has a default
7275 implementation). You also need to provide a unique name (as a string literal)
7276 for each structure type you define. This is because in GiNaC archives,
7277 the class of an object is stored as a string, the class name.
7279 By default, this class name (as returned by the @code{class_name()} member
7280 function) is @samp{structure} for all structure classes. This works as long
7281 as you have only defined one structure type, but if you use two or more you
7282 need to provide a different name for each by specializing the
7283 @code{get_class_name()} member function. Here is a sample implementation
7284 for enabling archiving of the scalar product type defined above:
7287 const char *sprod::get_class_name() @{ return "sprod"; @}
7289 void sprod::archive(archive_node & n) const
7291 inherited::archive(n);
7292 n.add_ex("left", get_struct().left);
7293 n.add_ex("right", get_struct().right);
7296 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7298 n.find_ex("left", get_struct().left, sym_lst);
7299 n.find_ex("right", get_struct().right, sym_lst);
7303 Note that the unarchiving constructor is @code{sprod::structure} and not
7304 @code{sprod::sprod}, and that we don't need to supply an
7305 @code{sprod::unarchive()} function.
7308 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7309 @c node-name, next, previous, up
7310 @section Adding classes
7312 The @code{structure<T>} template provides an way to extend GiNaC with custom
7313 algebraic classes that is easy to use but has its limitations, the most
7314 severe of which being that you can't add any new member functions to
7315 structures. To be able to do this, you need to write a new class definition
7318 This section will explain how to implement new algebraic classes in GiNaC by
7319 giving the example of a simple 'string' class. After reading this section
7320 you will know how to properly declare a GiNaC class and what the minimum
7321 required member functions are that you have to implement. We only cover the
7322 implementation of a 'leaf' class here (i.e. one that doesn't contain
7323 subexpressions). Creating a container class like, for example, a class
7324 representing tensor products is more involved but this section should give
7325 you enough information so you can consult the source to GiNaC's predefined
7326 classes if you want to implement something more complicated.
7328 @subsection GiNaC's run-time type information system
7330 @cindex hierarchy of classes
7332 All algebraic classes (that is, all classes that can appear in expressions)
7333 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7334 @code{basic *} (which is essentially what an @code{ex} is) represents a
7335 generic pointer to an algebraic class. Occasionally it is necessary to find
7336 out what the class of an object pointed to by a @code{basic *} really is.
7337 Also, for the unarchiving of expressions it must be possible to find the
7338 @code{unarchive()} function of a class given the class name (as a string). A
7339 system that provides this kind of information is called a run-time type
7340 information (RTTI) system. The C++ language provides such a thing (see the
7341 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7342 implements its own, simpler RTTI.
7344 The RTTI in GiNaC is based on two mechanisms:
7349 The @code{basic} class declares a member variable @code{tinfo_key} which
7350 holds an unsigned integer that identifies the object's class. These numbers
7351 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7352 classes. They all start with @code{TINFO_}.
7355 By means of some clever tricks with static members, GiNaC maintains a list
7356 of information for all classes derived from @code{basic}. The information
7357 available includes the class names, the @code{tinfo_key}s, and pointers
7358 to the unarchiving functions. This class registry is defined in the
7359 @file{registrar.h} header file.
7363 The disadvantage of this proprietary RTTI implementation is that there's
7364 a little more to do when implementing new classes (C++'s RTTI works more
7365 or less automatically) but don't worry, most of the work is simplified by
7368 @subsection A minimalistic example
7370 Now we will start implementing a new class @code{mystring} that allows
7371 placing character strings in algebraic expressions (this is not very useful,
7372 but it's just an example). This class will be a direct subclass of
7373 @code{basic}. You can use this sample implementation as a starting point
7374 for your own classes.
7376 The code snippets given here assume that you have included some header files
7382 #include <stdexcept>
7383 using namespace std;
7385 #include <ginac/ginac.h>
7386 using namespace GiNaC;
7389 The first thing we have to do is to define a @code{tinfo_key} for our new
7390 class. This can be any arbitrary unsigned number that is not already taken
7391 by one of the existing classes but it's better to come up with something
7392 that is unlikely to clash with keys that might be added in the future. The
7393 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7394 which is not a requirement but we are going to stick with this scheme:
7397 const unsigned TINFO_mystring = 0x42420001U;
7400 Now we can write down the class declaration. The class stores a C++
7401 @code{string} and the user shall be able to construct a @code{mystring}
7402 object from a C or C++ string:
7405 class mystring : public basic
7407 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7410 mystring(const string &s);
7411 mystring(const char *s);
7417 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7420 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7421 macros are defined in @file{registrar.h}. They take the name of the class
7422 and its direct superclass as arguments and insert all required declarations
7423 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7424 the first line after the opening brace of the class definition. The
7425 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7426 source (at global scope, of course, not inside a function).
7428 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7429 declarations of the default constructor and a couple of other functions that
7430 are required. It also defines a type @code{inherited} which refers to the
7431 superclass so you don't have to modify your code every time you shuffle around
7432 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7433 class with the GiNaC RTTI (there is also a
7434 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7435 options for the class, and which we will be using instead in a few minutes).
7437 Now there are seven member functions we have to implement to get a working
7443 @code{mystring()}, the default constructor.
7446 @code{void archive(archive_node &n)}, the archiving function. This stores all
7447 information needed to reconstruct an object of this class inside an
7448 @code{archive_node}.
7451 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7452 constructor. This constructs an instance of the class from the information
7453 found in an @code{archive_node}.
7456 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7457 unarchiving function. It constructs a new instance by calling the unarchiving
7461 @cindex @code{compare_same_type()}
7462 @code{int compare_same_type(const basic &other)}, which is used internally
7463 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7464 -1, depending on the relative order of this object and the @code{other}
7465 object. If it returns 0, the objects are considered equal.
7466 @strong{Note:} This has nothing to do with the (numeric) ordering
7467 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7468 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7469 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7470 must provide a @code{compare_same_type()} function, even those representing
7471 objects for which no reasonable algebraic ordering relationship can be
7475 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7476 which are the two constructors we declared.
7480 Let's proceed step-by-step. The default constructor looks like this:
7483 mystring::mystring() : inherited(TINFO_mystring) @{@}
7486 The golden rule is that in all constructors you have to set the
7487 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7488 it will be set by the constructor of the superclass and all hell will break
7489 loose in the RTTI. For your convenience, the @code{basic} class provides
7490 a constructor that takes a @code{tinfo_key} value, which we are using here
7491 (remember that in our case @code{inherited == basic}). If the superclass
7492 didn't have such a constructor, we would have to set the @code{tinfo_key}
7493 to the right value manually.
7495 In the default constructor you should set all other member variables to
7496 reasonable default values (we don't need that here since our @code{str}
7497 member gets set to an empty string automatically).
7499 Next are the three functions for archiving. You have to implement them even
7500 if you don't plan to use archives, but the minimum required implementation
7501 is really simple. First, the archiving function:
7504 void mystring::archive(archive_node &n) const
7506 inherited::archive(n);
7507 n.add_string("string", str);
7511 The only thing that is really required is calling the @code{archive()}
7512 function of the superclass. Optionally, you can store all information you
7513 deem necessary for representing the object into the passed
7514 @code{archive_node}. We are just storing our string here. For more
7515 information on how the archiving works, consult the @file{archive.h} header
7518 The unarchiving constructor is basically the inverse of the archiving
7522 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7524 n.find_string("string", str);
7528 If you don't need archiving, just leave this function empty (but you must
7529 invoke the unarchiving constructor of the superclass). Note that we don't
7530 have to set the @code{tinfo_key} here because it is done automatically
7531 by the unarchiving constructor of the @code{basic} class.
7533 Finally, the unarchiving function:
7536 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7538 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7542 You don't have to understand how exactly this works. Just copy these
7543 four lines into your code literally (replacing the class name, of
7544 course). It calls the unarchiving constructor of the class and unless
7545 you are doing something very special (like matching @code{archive_node}s
7546 to global objects) you don't need a different implementation. For those
7547 who are interested: setting the @code{dynallocated} flag puts the object
7548 under the control of GiNaC's garbage collection. It will get deleted
7549 automatically once it is no longer referenced.
7551 Our @code{compare_same_type()} function uses a provided function to compare
7555 int mystring::compare_same_type(const basic &other) const
7557 const mystring &o = static_cast<const mystring &>(other);
7558 int cmpval = str.compare(o.str);
7561 else if (cmpval < 0)
7568 Although this function takes a @code{basic &}, it will always be a reference
7569 to an object of exactly the same class (objects of different classes are not
7570 comparable), so the cast is safe. If this function returns 0, the two objects
7571 are considered equal (in the sense that @math{A-B=0}), so you should compare
7572 all relevant member variables.
7574 Now the only thing missing is our two new constructors:
7577 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7578 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7581 No surprises here. We set the @code{str} member from the argument and
7582 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7584 That's it! We now have a minimal working GiNaC class that can store
7585 strings in algebraic expressions. Let's confirm that the RTTI works:
7588 ex e = mystring("Hello, world!");
7589 cout << is_a<mystring>(e) << endl;
7592 cout << e.bp->class_name() << endl;
7596 Obviously it does. Let's see what the expression @code{e} looks like:
7600 // -> [mystring object]
7603 Hm, not exactly what we expect, but of course the @code{mystring} class
7604 doesn't yet know how to print itself. This can be done either by implementing
7605 the @code{print()} member function, or, preferably, by specifying a
7606 @code{print_func<>()} class option. Let's say that we want to print the string
7607 surrounded by double quotes:
7610 class mystring : public basic
7614 void do_print(const print_context &c, unsigned level = 0) const;
7618 void mystring::do_print(const print_context &c, unsigned level) const
7620 // print_context::s is a reference to an ostream
7621 c.s << '\"' << str << '\"';
7625 The @code{level} argument is only required for container classes to
7626 correctly parenthesize the output.
7628 Now we need to tell GiNaC that @code{mystring} objects should use the
7629 @code{do_print()} member function for printing themselves. For this, we
7633 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7639 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7640 print_func<print_context>(&mystring::do_print))
7643 Let's try again to print the expression:
7647 // -> "Hello, world!"
7650 Much better. If we wanted to have @code{mystring} objects displayed in a
7651 different way depending on the output format (default, LaTeX, etc.), we
7652 would have supplied multiple @code{print_func<>()} options with different
7653 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7654 separated by dots. This is similar to the way options are specified for
7655 symbolic functions. @xref{Printing}, for a more in-depth description of the
7656 way expression output is implemented in GiNaC.
7658 The @code{mystring} class can be used in arbitrary expressions:
7661 e += mystring("GiNaC rulez");
7663 // -> "GiNaC rulez"+"Hello, world!"
7666 (GiNaC's automatic term reordering is in effect here), or even
7669 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7671 // -> "One string"^(2*sin(-"Another string"+Pi))
7674 Whether this makes sense is debatable but remember that this is only an
7675 example. At least it allows you to implement your own symbolic algorithms
7678 Note that GiNaC's algebraic rules remain unchanged:
7681 e = mystring("Wow") * mystring("Wow");
7685 e = pow(mystring("First")-mystring("Second"), 2);
7686 cout << e.expand() << endl;
7687 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7690 There's no way to, for example, make GiNaC's @code{add} class perform string
7691 concatenation. You would have to implement this yourself.
7693 @subsection Automatic evaluation
7696 @cindex @code{eval()}
7697 @cindex @code{hold()}
7698 When dealing with objects that are just a little more complicated than the
7699 simple string objects we have implemented, chances are that you will want to
7700 have some automatic simplifications or canonicalizations performed on them.
7701 This is done in the evaluation member function @code{eval()}. Let's say that
7702 we wanted all strings automatically converted to lowercase with
7703 non-alphabetic characters stripped, and empty strings removed:
7706 class mystring : public basic
7710 ex eval(int level = 0) const;
7714 ex mystring::eval(int level) const
7717 for (int i=0; i<str.length(); i++) @{
7719 if (c >= 'A' && c <= 'Z')
7720 new_str += tolower(c);
7721 else if (c >= 'a' && c <= 'z')
7725 if (new_str.length() == 0)
7728 return mystring(new_str).hold();
7732 The @code{level} argument is used to limit the recursion depth of the
7733 evaluation. We don't have any subexpressions in the @code{mystring}
7734 class so we are not concerned with this. If we had, we would call the
7735 @code{eval()} functions of the subexpressions with @code{level - 1} as
7736 the argument if @code{level != 1}. The @code{hold()} member function
7737 sets a flag in the object that prevents further evaluation. Otherwise
7738 we might end up in an endless loop. When you want to return the object
7739 unmodified, use @code{return this->hold();}.
7741 Let's confirm that it works:
7744 ex e = mystring("Hello, world!") + mystring("!?#");
7748 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7753 @subsection Optional member functions
7755 We have implemented only a small set of member functions to make the class
7756 work in the GiNaC framework. There are two functions that are not strictly
7757 required but will make operations with objects of the class more efficient:
7759 @cindex @code{calchash()}
7760 @cindex @code{is_equal_same_type()}
7762 unsigned calchash() const;
7763 bool is_equal_same_type(const basic &other) const;
7766 The @code{calchash()} method returns an @code{unsigned} hash value for the
7767 object which will allow GiNaC to compare and canonicalize expressions much
7768 more efficiently. You should consult the implementation of some of the built-in
7769 GiNaC classes for examples of hash functions. The default implementation of
7770 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7771 class and all subexpressions that are accessible via @code{op()}.
7773 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7774 tests for equality without establishing an ordering relation, which is often
7775 faster. The default implementation of @code{is_equal_same_type()} just calls
7776 @code{compare_same_type()} and tests its result for zero.
7778 @subsection Other member functions
7780 For a real algebraic class, there are probably some more functions that you
7781 might want to provide:
7784 bool info(unsigned inf) const;
7785 ex evalf(int level = 0) const;
7786 ex series(const relational & r, int order, unsigned options = 0) const;
7787 ex derivative(const symbol & s) const;
7790 If your class stores sub-expressions (see the scalar product example in the
7791 previous section) you will probably want to override
7793 @cindex @code{let_op()}
7796 ex op(size_t i) const;
7797 ex & let_op(size_t i);
7798 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7799 ex map(map_function & f) const;
7802 @code{let_op()} is a variant of @code{op()} that allows write access. The
7803 default implementations of @code{subs()} and @code{map()} use it, so you have
7804 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7806 You can, of course, also add your own new member functions. Remember
7807 that the RTTI may be used to get information about what kinds of objects
7808 you are dealing with (the position in the class hierarchy) and that you
7809 can always extract the bare object from an @code{ex} by stripping the
7810 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7811 should become a need.
7813 That's it. May the source be with you!
7816 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7817 @c node-name, next, previous, up
7818 @chapter A Comparison With Other CAS
7821 This chapter will give you some information on how GiNaC compares to
7822 other, traditional Computer Algebra Systems, like @emph{Maple},
7823 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7824 disadvantages over these systems.
7827 * Advantages:: Strengths of the GiNaC approach.
7828 * Disadvantages:: Weaknesses of the GiNaC approach.
7829 * Why C++?:: Attractiveness of C++.
7832 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7833 @c node-name, next, previous, up
7836 GiNaC has several advantages over traditional Computer
7837 Algebra Systems, like
7842 familiar language: all common CAS implement their own proprietary
7843 grammar which you have to learn first (and maybe learn again when your
7844 vendor decides to `enhance' it). With GiNaC you can write your program
7845 in common C++, which is standardized.
7849 structured data types: you can build up structured data types using
7850 @code{struct}s or @code{class}es together with STL features instead of
7851 using unnamed lists of lists of lists.
7854 strongly typed: in CAS, you usually have only one kind of variables
7855 which can hold contents of an arbitrary type. This 4GL like feature is
7856 nice for novice programmers, but dangerous.
7859 development tools: powerful development tools exist for C++, like fancy
7860 editors (e.g. with automatic indentation and syntax highlighting),
7861 debuggers, visualization tools, documentation generators@dots{}
7864 modularization: C++ programs can easily be split into modules by
7865 separating interface and implementation.
7868 price: GiNaC is distributed under the GNU Public License which means
7869 that it is free and available with source code. And there are excellent
7870 C++-compilers for free, too.
7873 extendable: you can add your own classes to GiNaC, thus extending it on
7874 a very low level. Compare this to a traditional CAS that you can
7875 usually only extend on a high level by writing in the language defined
7876 by the parser. In particular, it turns out to be almost impossible to
7877 fix bugs in a traditional system.
7880 multiple interfaces: Though real GiNaC programs have to be written in
7881 some editor, then be compiled, linked and executed, there are more ways
7882 to work with the GiNaC engine. Many people want to play with
7883 expressions interactively, as in traditional CASs. Currently, two such
7884 windows into GiNaC have been implemented and many more are possible: the
7885 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7886 types to a command line and second, as a more consistent approach, an
7887 interactive interface to the Cint C++ interpreter has been put together
7888 (called GiNaC-cint) that allows an interactive scripting interface
7889 consistent with the C++ language. It is available from the usual GiNaC
7893 seamless integration: it is somewhere between difficult and impossible
7894 to call CAS functions from within a program written in C++ or any other
7895 programming language and vice versa. With GiNaC, your symbolic routines
7896 are part of your program. You can easily call third party libraries,
7897 e.g. for numerical evaluation or graphical interaction. All other
7898 approaches are much more cumbersome: they range from simply ignoring the
7899 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7900 system (i.e. @emph{Yacas}).
7903 efficiency: often large parts of a program do not need symbolic
7904 calculations at all. Why use large integers for loop variables or
7905 arbitrary precision arithmetics where @code{int} and @code{double} are
7906 sufficient? For pure symbolic applications, GiNaC is comparable in
7907 speed with other CAS.
7912 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7913 @c node-name, next, previous, up
7914 @section Disadvantages
7916 Of course it also has some disadvantages:
7921 advanced features: GiNaC cannot compete with a program like
7922 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7923 which grows since 1981 by the work of dozens of programmers, with
7924 respect to mathematical features. Integration, factorization,
7925 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7926 not planned for the near future).
7929 portability: While the GiNaC library itself is designed to avoid any
7930 platform dependent features (it should compile on any ANSI compliant C++
7931 compiler), the currently used version of the CLN library (fast large
7932 integer and arbitrary precision arithmetics) can only by compiled
7933 without hassle on systems with the C++ compiler from the GNU Compiler
7934 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7935 macros to let the compiler gather all static initializations, which
7936 works for GNU C++ only. Feel free to contact the authors in case you
7937 really believe that you need to use a different compiler. We have
7938 occasionally used other compilers and may be able to give you advice.}
7939 GiNaC uses recent language features like explicit constructors, mutable
7940 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7941 literally. Recent GCC versions starting at 2.95.3, although itself not
7942 yet ANSI compliant, support all needed features.
7947 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7948 @c node-name, next, previous, up
7951 Why did we choose to implement GiNaC in C++ instead of Java or any other
7952 language? C++ is not perfect: type checking is not strict (casting is
7953 possible), separation between interface and implementation is not
7954 complete, object oriented design is not enforced. The main reason is
7955 the often scolded feature of operator overloading in C++. While it may
7956 be true that operating on classes with a @code{+} operator is rarely
7957 meaningful, it is perfectly suited for algebraic expressions. Writing
7958 @math{3x+5y} as @code{3*x+5*y} instead of
7959 @code{x.times(3).plus(y.times(5))} looks much more natural.
7960 Furthermore, the main developers are more familiar with C++ than with
7961 any other programming language.
7964 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7965 @c node-name, next, previous, up
7966 @appendix Internal Structures
7969 * Expressions are reference counted::
7970 * Internal representation of products and sums::
7973 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7974 @c node-name, next, previous, up
7975 @appendixsection Expressions are reference counted
7977 @cindex reference counting
7978 @cindex copy-on-write
7979 @cindex garbage collection
7980 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7981 where the counter belongs to the algebraic objects derived from class
7982 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7983 which @code{ex} contains an instance. If you understood that, you can safely
7984 skip the rest of this passage.
7986 Expressions are extremely light-weight since internally they work like
7987 handles to the actual representation. They really hold nothing more
7988 than a pointer to some other object. What this means in practice is
7989 that whenever you create two @code{ex} and set the second equal to the
7990 first no copying process is involved. Instead, the copying takes place
7991 as soon as you try to change the second. Consider the simple sequence
7996 #include <ginac/ginac.h>
7997 using namespace std;
7998 using namespace GiNaC;
8002 symbol x("x"), y("y"), z("z");
8005 e1 = sin(x + 2*y) + 3*z + 41;
8006 e2 = e1; // e2 points to same object as e1
8007 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8008 e2 += 1; // e2 is copied into a new object
8009 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8013 The line @code{e2 = e1;} creates a second expression pointing to the
8014 object held already by @code{e1}. The time involved for this operation
8015 is therefore constant, no matter how large @code{e1} was. Actual
8016 copying, however, must take place in the line @code{e2 += 1;} because
8017 @code{e1} and @code{e2} are not handles for the same object any more.
8018 This concept is called @dfn{copy-on-write semantics}. It increases
8019 performance considerably whenever one object occurs multiple times and
8020 represents a simple garbage collection scheme because when an @code{ex}
8021 runs out of scope its destructor checks whether other expressions handle
8022 the object it points to too and deletes the object from memory if that
8023 turns out not to be the case. A slightly less trivial example of
8024 differentiation using the chain-rule should make clear how powerful this
8029 symbol x("x"), y("y");
8033 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8034 cout << e1 << endl // prints x+3*y
8035 << e2 << endl // prints (x+3*y)^3
8036 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8040 Here, @code{e1} will actually be referenced three times while @code{e2}
8041 will be referenced two times. When the power of an expression is built,
8042 that expression needs not be copied. Likewise, since the derivative of
8043 a power of an expression can be easily expressed in terms of that
8044 expression, no copying of @code{e1} is involved when @code{e3} is
8045 constructed. So, when @code{e3} is constructed it will print as
8046 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8047 holds a reference to @code{e2} and the factor in front is just
8050 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8051 semantics. When you insert an expression into a second expression, the
8052 result behaves exactly as if the contents of the first expression were
8053 inserted. But it may be useful to remember that this is not what
8054 happens. Knowing this will enable you to write much more efficient
8055 code. If you still have an uncertain feeling with copy-on-write
8056 semantics, we recommend you have a look at the
8057 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8058 Marshall Cline. Chapter 16 covers this issue and presents an
8059 implementation which is pretty close to the one in GiNaC.
8062 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8063 @c node-name, next, previous, up
8064 @appendixsection Internal representation of products and sums
8066 @cindex representation
8069 @cindex @code{power}
8070 Although it should be completely transparent for the user of
8071 GiNaC a short discussion of this topic helps to understand the sources
8072 and also explain performance to a large degree. Consider the
8073 unexpanded symbolic expression
8075 $2d^3 \left( 4a + 5b - 3 \right)$
8078 @math{2*d^3*(4*a+5*b-3)}
8080 which could naively be represented by a tree of linear containers for
8081 addition and multiplication, one container for exponentiation with base
8082 and exponent and some atomic leaves of symbols and numbers in this
8087 @cindex pair-wise representation
8088 However, doing so results in a rather deeply nested tree which will
8089 quickly become inefficient to manipulate. We can improve on this by
8090 representing the sum as a sequence of terms, each one being a pair of a
8091 purely numeric multiplicative coefficient and its rest. In the same
8092 spirit we can store the multiplication as a sequence of terms, each
8093 having a numeric exponent and a possibly complicated base, the tree
8094 becomes much more flat:
8098 The number @code{3} above the symbol @code{d} shows that @code{mul}
8099 objects are treated similarly where the coefficients are interpreted as
8100 @emph{exponents} now. Addition of sums of terms or multiplication of
8101 products with numerical exponents can be coded to be very efficient with
8102 such a pair-wise representation. Internally, this handling is performed
8103 by most CAS in this way. It typically speeds up manipulations by an
8104 order of magnitude. The overall multiplicative factor @code{2} and the
8105 additive term @code{-3} look somewhat out of place in this
8106 representation, however, since they are still carrying a trivial
8107 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8108 this is avoided by adding a field that carries an overall numeric
8109 coefficient. This results in the realistic picture of internal
8112 $2d^3 \left( 4a + 5b - 3 \right)$:
8115 @math{2*d^3*(4*a+5*b-3)}:
8121 This also allows for a better handling of numeric radicals, since
8122 @code{sqrt(2)} can now be carried along calculations. Now it should be
8123 clear, why both classes @code{add} and @code{mul} are derived from the
8124 same abstract class: the data representation is the same, only the
8125 semantics differs. In the class hierarchy, methods for polynomial
8126 expansion and the like are reimplemented for @code{add} and @code{mul},
8127 but the data structure is inherited from @code{expairseq}.
8130 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8131 @c node-name, next, previous, up
8132 @appendix Package Tools
8134 If you are creating a software package that uses the GiNaC library,
8135 setting the correct command line options for the compiler and linker
8136 can be difficult. GiNaC includes two tools to make this process easier.
8139 * ginac-config:: A shell script to detect compiler and linker flags.
8140 * AM_PATH_GINAC:: Macro for GNU automake.
8144 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8145 @c node-name, next, previous, up
8146 @section @command{ginac-config}
8147 @cindex ginac-config
8149 @command{ginac-config} is a shell script that you can use to determine
8150 the compiler and linker command line options required to compile and
8151 link a program with the GiNaC library.
8153 @command{ginac-config} takes the following flags:
8157 Prints out the version of GiNaC installed.
8159 Prints '-I' flags pointing to the installed header files.
8161 Prints out the linker flags necessary to link a program against GiNaC.
8162 @item --prefix[=@var{PREFIX}]
8163 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8164 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8165 Otherwise, prints out the configured value of @env{$prefix}.
8166 @item --exec-prefix[=@var{PREFIX}]
8167 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8168 Otherwise, prints out the configured value of @env{$exec_prefix}.
8171 Typically, @command{ginac-config} will be used within a configure
8172 script, as described below. It, however, can also be used directly from
8173 the command line using backquotes to compile a simple program. For
8177 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8180 This command line might expand to (for example):
8183 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8184 -lginac -lcln -lstdc++
8187 Not only is the form using @command{ginac-config} easier to type, it will
8188 work on any system, no matter how GiNaC was configured.
8191 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8192 @c node-name, next, previous, up
8193 @section @samp{AM_PATH_GINAC}
8194 @cindex AM_PATH_GINAC
8196 For packages configured using GNU automake, GiNaC also provides
8197 a macro to automate the process of checking for GiNaC.
8200 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
8208 Determines the location of GiNaC using @command{ginac-config}, which is
8209 either found in the user's path, or from the environment variable
8210 @env{GINACLIB_CONFIG}.
8213 Tests the installed libraries to make sure that their version
8214 is later than @var{MINIMUM-VERSION}. (A default version will be used
8218 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8219 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8220 variable to the output of @command{ginac-config --libs}, and calls
8221 @samp{AC_SUBST()} for these variables so they can be used in generated
8222 makefiles, and then executes @var{ACTION-IF-FOUND}.
8225 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8226 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8230 This macro is in file @file{ginac.m4} which is installed in
8231 @file{$datadir/aclocal}. Note that if automake was installed with a
8232 different @samp{--prefix} than GiNaC, you will either have to manually
8233 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8234 aclocal the @samp{-I} option when running it.
8237 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8238 * Example package:: Example of a package using AM_PATH_GINAC.
8242 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8243 @c node-name, next, previous, up
8244 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8246 Simply make sure that @command{ginac-config} is in your path, and run
8247 the configure script.
8254 The directory where the GiNaC libraries are installed needs
8255 to be found by your system's dynamic linker.
8257 This is generally done by
8260 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8266 setting the environment variable @env{LD_LIBRARY_PATH},
8269 or, as a last resort,
8272 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8273 running configure, for instance:
8276 LDFLAGS=-R/home/cbauer/lib ./configure
8281 You can also specify a @command{ginac-config} not in your path by
8282 setting the @env{GINACLIB_CONFIG} environment variable to the
8283 name of the executable
8286 If you move the GiNaC package from its installed location,
8287 you will either need to modify @command{ginac-config} script
8288 manually to point to the new location or rebuild GiNaC.
8299 --with-ginac-prefix=@var{PREFIX}
8300 --with-ginac-exec-prefix=@var{PREFIX}
8303 are provided to override the prefix and exec-prefix that were stored
8304 in the @command{ginac-config} shell script by GiNaC's configure. You are
8305 generally better off configuring GiNaC with the right path to begin with.
8309 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8310 @c node-name, next, previous, up
8311 @subsection Example of a package using @samp{AM_PATH_GINAC}
8313 The following shows how to build a simple package using automake
8314 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8318 #include <ginac/ginac.h>
8322 GiNaC::symbol x("x");
8323 GiNaC::ex a = GiNaC::sin(x);
8324 std::cout << "Derivative of " << a
8325 << " is " << a.diff(x) << std::endl;
8330 You should first read the introductory portions of the automake
8331 Manual, if you are not already familiar with it.
8333 Two files are needed, @file{configure.in}, which is used to build the
8337 dnl Process this file with autoconf to produce a configure script.
8339 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8345 AM_PATH_GINAC(0.9.0, [
8346 LIBS="$LIBS $GINACLIB_LIBS"
8347 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8348 ], AC_MSG_ERROR([need to have GiNaC installed]))
8353 The only command in this which is not standard for automake
8354 is the @samp{AM_PATH_GINAC} macro.
8356 That command does the following: If a GiNaC version greater or equal
8357 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8358 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8359 the error message `need to have GiNaC installed'
8361 And the @file{Makefile.am}, which will be used to build the Makefile.
8364 ## Process this file with automake to produce Makefile.in
8365 bin_PROGRAMS = simple
8366 simple_SOURCES = simple.cpp
8369 This @file{Makefile.am}, says that we are building a single executable,
8370 from a single source file @file{simple.cpp}. Since every program
8371 we are building uses GiNaC we simply added the GiNaC options
8372 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8373 want to specify them on a per-program basis: for instance by
8377 simple_LDADD = $(GINACLIB_LIBS)
8378 INCLUDES = $(GINACLIB_CPPFLAGS)
8381 to the @file{Makefile.am}.
8383 To try this example out, create a new directory and add the three
8386 Now execute the following commands:
8389 $ automake --add-missing
8394 You now have a package that can be built in the normal fashion
8403 @node Bibliography, Concept Index, Example package, Top
8404 @c node-name, next, previous, up
8405 @appendix Bibliography
8410 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8413 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8416 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8419 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8422 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8423 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8426 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8427 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8428 Academic Press, London
8431 @cite{Computer Algebra Systems - A Practical Guide},
8432 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8435 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8436 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8439 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8440 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8443 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8448 @node Concept Index, , Bibliography, Top
8449 @c node-name, next, previous, up
8450 @unnumbered Concept Index