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) 1999-2001 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
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2001 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{} 1999-2001 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * The Class Hierarchy:: Overview of GiNaC's classes.
676 * Error handling:: How the library reports errors.
677 * Symbols:: Symbolic objects.
678 * Numbers:: Numerical objects.
679 * Constants:: Pre-defined constants.
680 * Fundamental containers:: The power, add and mul classes.
681 * Lists:: Lists of expressions.
682 * Mathematical functions:: Mathematical functions.
683 * Relations:: Equality, Inequality and all that.
684 * Matrices:: Matrices.
685 * Indexed objects:: Handling indexed quantities.
686 * Non-commutative objects:: Algebras with non-commutative products.
690 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
691 @c node-name, next, previous, up
693 @cindex expression (class @code{ex})
696 The most common class of objects a user deals with is the expression
697 @code{ex}, representing a mathematical object like a variable, number,
698 function, sum, product, etc@dots{} Expressions may be put together to form
699 new expressions, passed as arguments to functions, and so on. Here is a
700 little collection of valid expressions:
703 ex MyEx1 = 5; // simple number
704 ex MyEx2 = x + 2*y; // polynomial in x and y
705 ex MyEx3 = (x + 1)/(x - 1); // rational expression
706 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
707 ex MyEx5 = MyEx4 + 1; // similar to above
710 Expressions are handles to other more fundamental objects, that often
711 contain other expressions thus creating a tree of expressions
712 (@xref{Internal Structures}, for particular examples). Most methods on
713 @code{ex} therefore run top-down through such an expression tree. For
714 example, the method @code{has()} scans recursively for occurrences of
715 something inside an expression. Thus, if you have declared @code{MyEx4}
716 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
717 the argument of @code{sin} and hence return @code{true}.
719 The next sections will outline the general picture of GiNaC's class
720 hierarchy and describe the classes of objects that are handled by
724 @node The Class Hierarchy, Error handling, Expressions, Basic Concepts
725 @c node-name, next, previous, up
726 @section The Class Hierarchy
728 GiNaC's class hierarchy consists of several classes representing
729 mathematical objects, all of which (except for @code{ex} and some
730 helpers) are internally derived from one abstract base class called
731 @code{basic}. You do not have to deal with objects of class
732 @code{basic}, instead you'll be dealing with symbols, numbers,
733 containers of expressions and so on.
737 To get an idea about what kinds of symbolic composits may be built we
738 have a look at the most important classes in the class hierarchy and
739 some of the relations among the classes:
741 @image{classhierarchy}
743 The abstract classes shown here (the ones without drop-shadow) are of no
744 interest for the user. They are used internally in order to avoid code
745 duplication if two or more classes derived from them share certain
746 features. An example is @code{expairseq}, a container for a sequence of
747 pairs each consisting of one expression and a number (@code{numeric}).
748 What @emph{is} visible to the user are the derived classes @code{add}
749 and @code{mul}, representing sums and products. @xref{Internal
750 Structures}, where these two classes are described in more detail. The
751 following table shortly summarizes what kinds of mathematical objects
752 are stored in the different classes:
755 @multitable @columnfractions .22 .78
756 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
757 @item @code{constant} @tab Constants like
764 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
765 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
766 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
767 @item @code{ncmul} @tab Products of non-commutative objects
768 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
773 @code{sqrt(}@math{2}@code{)}
776 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
777 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
778 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
779 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
780 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
781 @item @code{indexed} @tab Indexed object like @math{A_ij}
782 @item @code{tensor} @tab Special tensor like the delta and metric tensors
783 @item @code{idx} @tab Index of an indexed object
784 @item @code{varidx} @tab Index with variance
785 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
786 @item @code{wildcard} @tab Wildcard for pattern matching
791 @node Error handling, Symbols, The Class Hierarchy, Basic Concepts
792 @c node-name, next, previous, up
793 @section Error handling
795 @cindex @code{pole_error} (class)
797 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
798 generated by GiNaC are subclassed from the standard @code{exception} class
799 defined in the @file{<stdexcept>} header. In addition to the predefined
800 @code{logic_error}, @code{domain_error}, @code{out_of_range},
801 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
802 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
803 exception that gets thrown when trying to evaluate a mathematical function
806 The @code{pole_error} class has a member function
809 int pole_error::degree(void) const;
812 that returns the order of the singularity (or 0 when the pole is
813 logarithmic or the order is undefined).
815 When using GiNaC it is useful to arrange for exceptions to be catched in
816 the main program even if you don't want to do any special error handling.
817 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
818 default exception handler of your C++ compiler's run-time system which
819 usually only aborts the program without giving any information what went
822 Here is an example for a @code{main()} function that catches and prints
823 exceptions generated by GiNaC:
828 #include <ginac/ginac.h>
830 using namespace GiNaC;
838 @} catch (exception &p) @{
839 cerr << p.what() << endl;
847 @node Symbols, Numbers, Error handling, Basic Concepts
848 @c node-name, next, previous, up
850 @cindex @code{symbol} (class)
851 @cindex hierarchy of classes
854 Symbols are for symbolic manipulation what atoms are for chemistry. You
855 can declare objects of class @code{symbol} as any other object simply by
856 saying @code{symbol x,y;}. There is, however, a catch in here having to
857 do with the fact that C++ is a compiled language. The information about
858 the symbol's name is thrown away by the compiler but at a later stage
859 you may want to print expressions holding your symbols. In order to
860 avoid confusion GiNaC's symbols are able to know their own name. This
861 is accomplished by declaring its name for output at construction time in
862 the fashion @code{symbol x("x");}. If you declare a symbol using the
863 default constructor (i.e. without string argument) the system will deal
864 out a unique name. That name may not be suitable for printing but for
865 internal routines when no output is desired it is often enough. We'll
866 come across examples of such symbols later in this tutorial.
868 This implies that the strings passed to symbols at construction time may
869 not be used for comparing two of them. It is perfectly legitimate to
870 write @code{symbol x("x"),y("x");} but it is likely to lead into
871 trouble. Here, @code{x} and @code{y} are different symbols and
872 statements like @code{x-y} will not be simplified to zero although the
873 output @code{x-x} looks funny. Such output may also occur when there
874 are two different symbols in two scopes, for instance when you call a
875 function that declares a symbol with a name already existent in a symbol
876 in the calling function. Again, comparing them (using @code{operator==}
877 for instance) will always reveal their difference. Watch out, please.
879 @cindex @code{subs()}
880 Although symbols can be assigned expressions for internal reasons, you
881 should not do it (and we are not going to tell you how it is done). If
882 you want to replace a symbol with something else in an expression, you
883 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
886 @node Numbers, Constants, Symbols, Basic Concepts
887 @c node-name, next, previous, up
889 @cindex @code{numeric} (class)
895 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
896 The classes therein serve as foundation classes for GiNaC. CLN stands
897 for Class Library for Numbers or alternatively for Common Lisp Numbers.
898 In order to find out more about CLN's internals the reader is refered to
899 the documentation of that library. @inforef{Introduction, , cln}, for
900 more information. Suffice to say that it is by itself build on top of
901 another library, the GNU Multiple Precision library GMP, which is an
902 extremely fast library for arbitrary long integers and rationals as well
903 as arbitrary precision floating point numbers. It is very commonly used
904 by several popular cryptographic applications. CLN extends GMP by
905 several useful things: First, it introduces the complex number field
906 over either reals (i.e. floating point numbers with arbitrary precision)
907 or rationals. Second, it automatically converts rationals to integers
908 if the denominator is unity and complex numbers to real numbers if the
909 imaginary part vanishes and also correctly treats algebraic functions.
910 Third it provides good implementations of state-of-the-art algorithms
911 for all trigonometric and hyperbolic functions as well as for
912 calculation of some useful constants.
914 The user can construct an object of class @code{numeric} in several
915 ways. The following example shows the four most important constructors.
916 It uses construction from C-integer, construction of fractions from two
917 integers, construction from C-float and construction from a string:
921 #include <ginac/ginac.h>
922 using namespace GiNaC;
926 numeric two = 2; // exact integer 2
927 numeric r(2,3); // exact fraction 2/3
928 numeric e(2.71828); // floating point number
929 numeric p = "3.14159265358979323846"; // constructor from string
930 // Trott's constant in scientific notation:
931 numeric trott("1.0841015122311136151E-2");
933 std::cout << two*p << std::endl; // floating point 6.283...
937 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
938 This would, however, call C's built-in operator @code{/} for integers
939 first and result in a numeric holding a plain integer 1. @strong{Never
940 use the operator @code{/} on integers} unless you know exactly what you
941 are doing! Use the constructor from two integers instead, as shown in
942 the example above. Writing @code{numeric(1)/2} may look funny but works
945 @cindex @code{Digits}
947 We have seen now the distinction between exact numbers and floating
948 point numbers. Clearly, the user should never have to worry about
949 dynamically created exact numbers, since their `exactness' always
950 determines how they ought to be handled, i.e. how `long' they are. The
951 situation is different for floating point numbers. Their accuracy is
952 controlled by one @emph{global} variable, called @code{Digits}. (For
953 those readers who know about Maple: it behaves very much like Maple's
954 @code{Digits}). All objects of class numeric that are constructed from
955 then on will be stored with a precision matching that number of decimal
960 #include <ginac/ginac.h>
962 using namespace GiNaC;
966 numeric three(3.0), one(1.0);
967 numeric x = one/three;
969 cout << "in " << Digits << " digits:" << endl;
971 cout << Pi.evalf() << endl;
983 The above example prints the following output to screen:
990 0.333333333333333333333333333333333333333333333333333333333333333333
991 3.14159265358979323846264338327950288419716939937510582097494459231
994 It should be clear that objects of class @code{numeric} should be used
995 for constructing numbers or for doing arithmetic with them. The objects
996 one deals with most of the time are the polymorphic expressions @code{ex}.
998 @subsection Tests on numbers
1000 Once you have declared some numbers, assigned them to expressions and
1001 done some arithmetic with them it is frequently desired to retrieve some
1002 kind of information from them like asking whether that number is
1003 integer, rational, real or complex. For those cases GiNaC provides
1004 several useful methods. (Internally, they fall back to invocations of
1005 certain CLN functions.)
1007 As an example, let's construct some rational number, multiply it with
1008 some multiple of its denominator and test what comes out:
1012 #include <ginac/ginac.h>
1013 using namespace std;
1014 using namespace GiNaC;
1016 // some very important constants:
1017 const numeric twentyone(21);
1018 const numeric ten(10);
1019 const numeric five(5);
1023 numeric answer = twentyone;
1026 cout << answer.is_integer() << endl; // false, it's 21/5
1028 cout << answer.is_integer() << endl; // true, it's 42 now!
1032 Note that the variable @code{answer} is constructed here as an integer
1033 by @code{numeric}'s copy constructor but in an intermediate step it
1034 holds a rational number represented as integer numerator and integer
1035 denominator. When multiplied by 10, the denominator becomes unity and
1036 the result is automatically converted to a pure integer again.
1037 Internally, the underlying CLN is responsible for this behavior and we
1038 refer the reader to CLN's documentation. Suffice to say that
1039 the same behavior applies to complex numbers as well as return values of
1040 certain functions. Complex numbers are automatically converted to real
1041 numbers if the imaginary part becomes zero. The full set of tests that
1042 can be applied is listed in the following table.
1045 @multitable @columnfractions .30 .70
1046 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1047 @item @code{.is_zero()}
1048 @tab @dots{}equal to zero
1049 @item @code{.is_positive()}
1050 @tab @dots{}not complex and greater than 0
1051 @item @code{.is_integer()}
1052 @tab @dots{}a (non-complex) integer
1053 @item @code{.is_pos_integer()}
1054 @tab @dots{}an integer and greater than 0
1055 @item @code{.is_nonneg_integer()}
1056 @tab @dots{}an integer and greater equal 0
1057 @item @code{.is_even()}
1058 @tab @dots{}an even integer
1059 @item @code{.is_odd()}
1060 @tab @dots{}an odd integer
1061 @item @code{.is_prime()}
1062 @tab @dots{}a prime integer (probabilistic primality test)
1063 @item @code{.is_rational()}
1064 @tab @dots{}an exact rational number (integers are rational, too)
1065 @item @code{.is_real()}
1066 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1067 @item @code{.is_cinteger()}
1068 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1069 @item @code{.is_crational()}
1070 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1075 @node Constants, Fundamental containers, Numbers, Basic Concepts
1076 @c node-name, next, previous, up
1078 @cindex @code{constant} (class)
1081 @cindex @code{Catalan}
1082 @cindex @code{Euler}
1083 @cindex @code{evalf()}
1084 Constants behave pretty much like symbols except that they return some
1085 specific number when the method @code{.evalf()} is called.
1087 The predefined known constants are:
1090 @multitable @columnfractions .14 .30 .56
1091 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1093 @tab Archimedes' constant
1094 @tab 3.14159265358979323846264338327950288
1095 @item @code{Catalan}
1096 @tab Catalan's constant
1097 @tab 0.91596559417721901505460351493238411
1099 @tab Euler's (or Euler-Mascheroni) constant
1100 @tab 0.57721566490153286060651209008240243
1105 @node Fundamental containers, Lists, Constants, Basic Concepts
1106 @c node-name, next, previous, up
1107 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1111 @cindex @code{power}
1113 Simple polynomial expressions are written down in GiNaC pretty much like
1114 in other CAS or like expressions involving numerical variables in C.
1115 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1116 been overloaded to achieve this goal. When you run the following
1117 code snippet, the constructor for an object of type @code{mul} is
1118 automatically called to hold the product of @code{a} and @code{b} and
1119 then the constructor for an object of type @code{add} is called to hold
1120 the sum of that @code{mul} object and the number one:
1124 symbol a("a"), b("b");
1129 @cindex @code{pow()}
1130 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1131 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1132 construction is necessary since we cannot safely overload the constructor
1133 @code{^} in C++ to construct a @code{power} object. If we did, it would
1134 have several counterintuitive and undesired effects:
1138 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1140 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1141 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1142 interpret this as @code{x^(a^b)}.
1144 Also, expressions involving integer exponents are very frequently used,
1145 which makes it even more dangerous to overload @code{^} since it is then
1146 hard to distinguish between the semantics as exponentiation and the one
1147 for exclusive or. (It would be embarrassing to return @code{1} where one
1148 has requested @code{2^3}.)
1151 @cindex @command{ginsh}
1152 All effects are contrary to mathematical notation and differ from the
1153 way most other CAS handle exponentiation, therefore overloading @code{^}
1154 is ruled out for GiNaC's C++ part. The situation is different in
1155 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1156 that the other frequently used exponentiation operator @code{**} does
1157 not exist at all in C++).
1159 To be somewhat more precise, objects of the three classes described
1160 here, are all containers for other expressions. An object of class
1161 @code{power} is best viewed as a container with two slots, one for the
1162 basis, one for the exponent. All valid GiNaC expressions can be
1163 inserted. However, basic transformations like simplifying
1164 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1165 when this is mathematically possible. If we replace the outer exponent
1166 three in the example by some symbols @code{a}, the simplification is not
1167 safe and will not be performed, since @code{a} might be @code{1/2} and
1170 Objects of type @code{add} and @code{mul} are containers with an
1171 arbitrary number of slots for expressions to be inserted. Again, simple
1172 and safe simplifications are carried out like transforming
1173 @code{3*x+4-x} to @code{2*x+4}.
1175 The general rule is that when you construct such objects, GiNaC
1176 automatically creates them in canonical form, which might differ from
1177 the form you typed in your program. This allows for rapid comparison of
1178 expressions, since after all @code{a-a} is simply zero. Note, that the
1179 canonical form is not necessarily lexicographical ordering or in any way
1180 easily guessable. It is only guaranteed that constructing the same
1181 expression twice, either implicitly or explicitly, results in the same
1185 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1186 @c node-name, next, previous, up
1187 @section Lists of expressions
1188 @cindex @code{lst} (class)
1190 @cindex @code{nops()}
1192 @cindex @code{append()}
1193 @cindex @code{prepend()}
1194 @cindex @code{remove_first()}
1195 @cindex @code{remove_last()}
1197 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1198 expressions. These are sometimes used to supply a variable number of
1199 arguments of the same type to GiNaC methods such as @code{subs()} and
1200 @code{to_rational()}, so you should have a basic understanding about them.
1202 Lists of up to 16 expressions can be directly constructed from single
1207 symbol x("x"), y("y");
1208 lst l(x, 2, y, x+y);
1209 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1213 Use the @code{nops()} method to determine the size (number of expressions) of
1214 a list and the @code{op()} method to access individual elements:
1218 cout << l.nops() << endl; // prints '4'
1219 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1223 You can append or prepend an expression to a list with the @code{append()}
1224 and @code{prepend()} methods:
1228 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1229 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1233 Finally you can remove the first or last element of a list with
1234 @code{remove_first()} and @code{remove_last()}:
1238 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1239 l.remove_last(); // l is now @{x, 2, y, x+y@}
1244 @node Mathematical functions, Relations, Lists, Basic Concepts
1245 @c node-name, next, previous, up
1246 @section Mathematical functions
1247 @cindex @code{function} (class)
1248 @cindex trigonometric function
1249 @cindex hyperbolic function
1251 There are quite a number of useful functions hard-wired into GiNaC. For
1252 instance, all trigonometric and hyperbolic functions are implemented
1253 (@xref{Built-in Functions}, for a complete list).
1255 These functions (better called @emph{pseudofunctions}) are all objects
1256 of class @code{function}. They accept one or more expressions as
1257 arguments and return one expression. If the arguments are not
1258 numerical, the evaluation of the function may be halted, as it does in
1259 the next example, showing how a function returns itself twice and
1260 finally an expression that may be really useful:
1262 @cindex Gamma function
1263 @cindex @code{subs()}
1266 symbol x("x"), y("y");
1268 cout << tgamma(foo) << endl;
1269 // -> tgamma(x+(1/2)*y)
1270 ex bar = foo.subs(y==1);
1271 cout << tgamma(bar) << endl;
1273 ex foobar = bar.subs(x==7);
1274 cout << tgamma(foobar) << endl;
1275 // -> (135135/128)*Pi^(1/2)
1279 Besides evaluation most of these functions allow differentiation, series
1280 expansion and so on. Read the next chapter in order to learn more about
1283 It must be noted that these pseudofunctions are created by inline
1284 functions, where the argument list is templated. This means that
1285 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1286 @code{sin(ex(1))} and will therefore not result in a floating point
1287 number. Unless of course the function prototype is explicitly
1288 overridden -- which is the case for arguments of type @code{numeric}
1289 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1290 point number of class @code{numeric} you should call
1291 @code{sin(numeric(1))}. This is almost the same as calling
1292 @code{sin(1).evalf()} except that the latter will return a numeric
1293 wrapped inside an @code{ex}.
1296 @node Relations, Matrices, Mathematical functions, Basic Concepts
1297 @c node-name, next, previous, up
1299 @cindex @code{relational} (class)
1301 Sometimes, a relation holding between two expressions must be stored
1302 somehow. The class @code{relational} is a convenient container for such
1303 purposes. A relation is by definition a container for two @code{ex} and
1304 a relation between them that signals equality, inequality and so on.
1305 They are created by simply using the C++ operators @code{==}, @code{!=},
1306 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1308 @xref{Mathematical functions}, for examples where various applications
1309 of the @code{.subs()} method show how objects of class relational are
1310 used as arguments. There they provide an intuitive syntax for
1311 substitutions. They are also used as arguments to the @code{ex::series}
1312 method, where the left hand side of the relation specifies the variable
1313 to expand in and the right hand side the expansion point. They can also
1314 be used for creating systems of equations that are to be solved for
1315 unknown variables. But the most common usage of objects of this class
1316 is rather inconspicuous in statements of the form @code{if
1317 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1318 conversion from @code{relational} to @code{bool} takes place. Note,
1319 however, that @code{==} here does not perform any simplifications, hence
1320 @code{expand()} must be called explicitly.
1323 @node Matrices, Indexed objects, Relations, Basic Concepts
1324 @c node-name, next, previous, up
1326 @cindex @code{matrix} (class)
1328 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1329 matrix with @math{m} rows and @math{n} columns are accessed with two
1330 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1331 second one in the range 0@dots{}@math{n-1}.
1333 There are a couple of ways to construct matrices, with or without preset
1337 matrix::matrix(unsigned r, unsigned c);
1338 matrix::matrix(unsigned r, unsigned c, const lst & l);
1339 ex lst_to_matrix(const lst & l);
1340 ex diag_matrix(const lst & l);
1343 The first two functions are @code{matrix} constructors which create a matrix
1344 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1345 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1346 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1347 from a list of lists, each list representing a matrix row. Finally,
1348 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1349 elements. Note that the last two functions return expressions, not matrix
1352 Matrix elements can be accessed and set using the parenthesis (function call)
1356 const ex & matrix::operator()(unsigned r, unsigned c) const;
1357 ex & matrix::operator()(unsigned r, unsigned c);
1360 It is also possible to access the matrix elements in a linear fashion with
1361 the @code{op()} method. But C++-style subscripting with square brackets
1362 @samp{[]} is not available.
1364 Here are a couple of examples that all construct the same 2x2 diagonal
1369 symbol a("a"), b("b");
1377 e = matrix(2, 2, lst(a, 0, 0, b));
1379 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1381 e = diag_matrix(lst(a, b));
1388 @cindex @code{transpose()}
1389 @cindex @code{inverse()}
1390 There are three ways to do arithmetic with matrices. The first (and most
1391 efficient one) is to use the methods provided by the @code{matrix} class:
1394 matrix matrix::add(const matrix & other) const;
1395 matrix matrix::sub(const matrix & other) const;
1396 matrix matrix::mul(const matrix & other) const;
1397 matrix matrix::mul_scalar(const ex & other) const;
1398 matrix matrix::pow(const ex & expn) const;
1399 matrix matrix::transpose(void) const;
1400 matrix matrix::inverse(void) const;
1403 All of these methods return the result as a new matrix object. Here is an
1404 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1409 matrix A(2, 2, lst(1, 2, 3, 4));
1410 matrix B(2, 2, lst(-1, 0, 2, 1));
1411 matrix C(2, 2, lst(8, 4, 2, 1));
1413 matrix result = A.mul(B).sub(C.mul_scalar(2));
1414 cout << result << endl;
1415 // -> [[-13,-6],[1,2]]
1420 @cindex @code{evalm()}
1421 The second (and probably the most natural) way is to construct an expression
1422 containing matrices with the usual arithmetic operators and @code{pow()}.
1423 For efficiency reasons, expressions with sums, products and powers of
1424 matrices are not automatically evaluated in GiNaC. You have to call the
1428 ex ex::evalm() const;
1431 to obtain the result:
1438 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1439 cout << e.evalm() << endl;
1440 // -> [[-13,-6],[1,2]]
1445 The non-commutativity of the product @code{A*B} in this example is
1446 automatically recognized by GiNaC. There is no need to use a special
1447 operator here. @xref{Non-commutative objects}, for more information about
1448 dealing with non-commutative expressions.
1450 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1451 to perform the arithmetic:
1456 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1457 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1459 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1460 cout << e.simplify_indexed() << endl;
1461 // -> [[-13,-6],[1,2]].i.j
1465 Using indices is most useful when working with rectangular matrices and
1466 one-dimensional vectors because you don't have to worry about having to
1467 transpose matrices before multiplying them. @xref{Indexed objects}, for
1468 more information about using matrices with indices, and about indices in
1471 The @code{matrix} class provides a couple of additional methods for
1472 computing determinants, traces, and characteristic polynomials:
1475 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1476 ex matrix::trace(void) const;
1477 ex matrix::charpoly(const symbol & lambda) const;
1480 The @samp{algo} argument of @code{determinant()} allows to select between
1481 different algorithms for calculating the determinant. The possible values
1482 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1483 heuristic to automatically select an algorithm that is likely to give the
1484 result most quickly.
1487 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1488 @c node-name, next, previous, up
1489 @section Indexed objects
1491 GiNaC allows you to handle expressions containing general indexed objects in
1492 arbitrary spaces. It is also able to canonicalize and simplify such
1493 expressions and perform symbolic dummy index summations. There are a number
1494 of predefined indexed objects provided, like delta and metric tensors.
1496 There are few restrictions placed on indexed objects and their indices and
1497 it is easy to construct nonsense expressions, but our intention is to
1498 provide a general framework that allows you to implement algorithms with
1499 indexed quantities, getting in the way as little as possible.
1501 @cindex @code{idx} (class)
1502 @cindex @code{indexed} (class)
1503 @subsection Indexed quantities and their indices
1505 Indexed expressions in GiNaC are constructed of two special types of objects,
1506 @dfn{index objects} and @dfn{indexed objects}.
1510 @cindex contravariant
1513 @item Index objects are of class @code{idx} or a subclass. Every index has
1514 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1515 the index lives in) which can both be arbitrary expressions but are usually
1516 a number or a simple symbol. In addition, indices of class @code{varidx} have
1517 a @dfn{variance} (they can be co- or contravariant), and indices of class
1518 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1520 @item Indexed objects are of class @code{indexed} or a subclass. They
1521 contain a @dfn{base expression} (which is the expression being indexed), and
1522 one or more indices.
1526 @strong{Note:} when printing expressions, covariant indices and indices
1527 without variance are denoted @samp{.i} while contravariant indices are
1528 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1529 value. In the following, we are going to use that notation in the text so
1530 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1531 not visible in the output.
1533 A simple example shall illustrate the concepts:
1537 #include <ginac/ginac.h>
1538 using namespace std;
1539 using namespace GiNaC;
1543 symbol i_sym("i"), j_sym("j");
1544 idx i(i_sym, 3), j(j_sym, 3);
1547 cout << indexed(A, i, j) << endl;
1552 The @code{idx} constructor takes two arguments, the index value and the
1553 index dimension. First we define two index objects, @code{i} and @code{j},
1554 both with the numeric dimension 3. The value of the index @code{i} is the
1555 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1556 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1557 construct an expression containing one indexed object, @samp{A.i.j}. It has
1558 the symbol @code{A} as its base expression and the two indices @code{i} and
1561 Note the difference between the indices @code{i} and @code{j} which are of
1562 class @code{idx}, and the index values which are the symbols @code{i_sym}
1563 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1564 or numbers but must be index objects. For example, the following is not
1565 correct and will raise an exception:
1568 symbol i("i"), j("j");
1569 e = indexed(A, i, j); // ERROR: indices must be of type idx
1572 You can have multiple indexed objects in an expression, index values can
1573 be numeric, and index dimensions symbolic:
1577 symbol B("B"), dim("dim");
1578 cout << 4 * indexed(A, i)
1579 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1584 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1585 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1586 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1587 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1588 @code{simplify_indexed()} for that, see below).
1590 In fact, base expressions, index values and index dimensions can be
1591 arbitrary expressions:
1595 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1600 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1601 get an error message from this but you will probably not be able to do
1602 anything useful with it.
1604 @cindex @code{get_value()}
1605 @cindex @code{get_dimension()}
1609 ex idx::get_value(void);
1610 ex idx::get_dimension(void);
1613 return the value and dimension of an @code{idx} object. If you have an index
1614 in an expression, such as returned by calling @code{.op()} on an indexed
1615 object, you can get a reference to the @code{idx} object with the function
1616 @code{ex_to<idx>()} on the expression.
1618 There are also the methods
1621 bool idx::is_numeric(void);
1622 bool idx::is_symbolic(void);
1623 bool idx::is_dim_numeric(void);
1624 bool idx::is_dim_symbolic(void);
1627 for checking whether the value and dimension are numeric or symbolic
1628 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1629 About Expressions}) returns information about the index value.
1631 @cindex @code{varidx} (class)
1632 If you need co- and contravariant indices, use the @code{varidx} class:
1636 symbol mu_sym("mu"), nu_sym("nu");
1637 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1638 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1640 cout << indexed(A, mu, nu) << endl;
1642 cout << indexed(A, mu_co, nu) << endl;
1644 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1649 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1650 co- or contravariant. The default is a contravariant (upper) index, but
1651 this can be overridden by supplying a third argument to the @code{varidx}
1652 constructor. The two methods
1655 bool varidx::is_covariant(void);
1656 bool varidx::is_contravariant(void);
1659 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1660 to get the object reference from an expression). There's also the very useful
1664 ex varidx::toggle_variance(void);
1667 which makes a new index with the same value and dimension but the opposite
1668 variance. By using it you only have to define the index once.
1670 @cindex @code{spinidx} (class)
1671 The @code{spinidx} class provides dotted and undotted variant indices, as
1672 used in the Weyl-van-der-Waerden spinor formalism:
1676 symbol K("K"), C_sym("C"), D_sym("D");
1677 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1678 // contravariant, undotted
1679 spinidx C_co(C_sym, 2, true); // covariant index
1680 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1681 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1683 cout << indexed(K, C, D) << endl;
1685 cout << indexed(K, C_co, D_dot) << endl;
1687 cout << indexed(K, D_co_dot, D) << endl;
1692 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1693 dotted or undotted. The default is undotted but this can be overridden by
1694 supplying a fourth argument to the @code{spinidx} constructor. The two
1698 bool spinidx::is_dotted(void);
1699 bool spinidx::is_undotted(void);
1702 allow you to check whether or not a @code{spinidx} object is dotted (use
1703 @code{ex_to<spinidx>()} to get the object reference from an expression).
1704 Finally, the two methods
1707 ex spinidx::toggle_dot(void);
1708 ex spinidx::toggle_variance_dot(void);
1711 create a new index with the same value and dimension but opposite dottedness
1712 and the same or opposite variance.
1714 @subsection Substituting indices
1716 @cindex @code{subs()}
1717 Sometimes you will want to substitute one symbolic index with another
1718 symbolic or numeric index, for example when calculating one specific element
1719 of a tensor expression. This is done with the @code{.subs()} method, as it
1720 is done for symbols (see @ref{Substituting Expressions}).
1722 You have two possibilities here. You can either substitute the whole index
1723 by another index or expression:
1727 ex e = indexed(A, mu_co);
1728 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1729 // -> A.mu becomes A~nu
1730 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1731 // -> A.mu becomes A~0
1732 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1733 // -> A.mu becomes A.0
1737 The third example shows that trying to replace an index with something that
1738 is not an index will substitute the index value instead.
1740 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1745 ex e = indexed(A, mu_co);
1746 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1747 // -> A.mu becomes A.nu
1748 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1749 // -> A.mu becomes A.0
1753 As you see, with the second method only the value of the index will get
1754 substituted. Its other properties, including its dimension, remain unchanged.
1755 If you want to change the dimension of an index you have to substitute the
1756 whole index by another one with the new dimension.
1758 Finally, substituting the base expression of an indexed object works as
1763 ex e = indexed(A, mu_co);
1764 cout << e << " becomes " << e.subs(A == A+B) << endl;
1765 // -> A.mu becomes (B+A).mu
1769 @subsection Symmetries
1770 @cindex @code{symmetry} (class)
1771 @cindex @code{sy_none()}
1772 @cindex @code{sy_symm()}
1773 @cindex @code{sy_anti()}
1774 @cindex @code{sy_cycl()}
1776 Indexed objects can have certain symmetry properties with respect to their
1777 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1778 that is constructed with the helper functions
1781 symmetry sy_none(...);
1782 symmetry sy_symm(...);
1783 symmetry sy_anti(...);
1784 symmetry sy_cycl(...);
1787 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1788 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1789 represents a cyclic symmetry. Each of these functions accepts up to four
1790 arguments which can be either symmetry objects themselves or unsigned integer
1791 numbers that represent an index position (counting from 0). A symmetry
1792 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1793 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1796 Here are some examples of symmetry definitions:
1801 e = indexed(A, i, j);
1802 e = indexed(A, sy_none(), i, j); // equivalent
1803 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1805 // Symmetric in all three indices:
1806 e = indexed(A, sy_symm(), i, j, k);
1807 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1808 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1809 // different canonical order
1811 // Symmetric in the first two indices only:
1812 e = indexed(A, sy_symm(0, 1), i, j, k);
1813 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1815 // Antisymmetric in the first and last index only (index ranges need not
1817 e = indexed(A, sy_anti(0, 2), i, j, k);
1818 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1820 // An example of a mixed symmetry: antisymmetric in the first two and
1821 // last two indices, symmetric when swapping the first and last index
1822 // pairs (like the Riemann curvature tensor):
1823 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1825 // Cyclic symmetry in all three indices:
1826 e = indexed(A, sy_cycl(), i, j, k);
1827 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1829 // The following examples are invalid constructions that will throw
1830 // an exception at run time.
1832 // An index may not appear multiple times:
1833 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1834 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1836 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1837 // same number of indices:
1838 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1840 // And of course, you cannot specify indices which are not there:
1841 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1845 If you need to specify more than four indices, you have to use the
1846 @code{.add()} method of the @code{symmetry} class. For example, to specify
1847 full symmetry in the first six indices you would write
1848 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1850 If an indexed object has a symmetry, GiNaC will automatically bring the
1851 indices into a canonical order which allows for some immediate simplifications:
1855 cout << indexed(A, sy_symm(), i, j)
1856 + indexed(A, sy_symm(), j, i) << endl;
1858 cout << indexed(B, sy_anti(), i, j)
1859 + indexed(B, sy_anti(), j, i) << endl;
1861 cout << indexed(B, sy_anti(), i, j, k)
1862 + indexed(B, sy_anti(), j, i, k) << endl;
1867 @cindex @code{get_free_indices()}
1869 @subsection Dummy indices
1871 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1872 that a summation over the index range is implied. Symbolic indices which are
1873 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1874 dummy nor free indices.
1876 To be recognized as a dummy index pair, the two indices must be of the same
1877 class and dimension and their value must be the same single symbol (an index
1878 like @samp{2*n+1} is never a dummy index). If the indices are of class
1879 @code{varidx} they must also be of opposite variance; if they are of class
1880 @code{spinidx} they must be both dotted or both undotted.
1882 The method @code{.get_free_indices()} returns a vector containing the free
1883 indices of an expression. It also checks that the free indices of the terms
1884 of a sum are consistent:
1888 symbol A("A"), B("B"), C("C");
1890 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1891 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1893 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1894 cout << exprseq(e.get_free_indices()) << endl;
1896 // 'j' and 'l' are dummy indices
1898 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1899 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1901 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1902 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1903 cout << exprseq(e.get_free_indices()) << endl;
1905 // 'nu' is a dummy index, but 'sigma' is not
1907 e = indexed(A, mu, mu);
1908 cout << exprseq(e.get_free_indices()) << endl;
1910 // 'mu' is not a dummy index because it appears twice with the same
1913 e = indexed(A, mu, nu) + 42;
1914 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1915 // this will throw an exception:
1916 // "add::get_free_indices: inconsistent indices in sum"
1920 @cindex @code{simplify_indexed()}
1921 @subsection Simplifying indexed expressions
1923 In addition to the few automatic simplifications that GiNaC performs on
1924 indexed expressions (such as re-ordering the indices of symmetric tensors
1925 and calculating traces and convolutions of matrices and predefined tensors)
1929 ex ex::simplify_indexed(void);
1930 ex ex::simplify_indexed(const scalar_products & sp);
1933 that performs some more expensive operations:
1936 @item it checks the consistency of free indices in sums in the same way
1937 @code{get_free_indices()} does
1938 @item it tries to give dummy indices that appear in different terms of a sum
1939 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1940 @item it (symbolically) calculates all possible dummy index summations/contractions
1941 with the predefined tensors (this will be explained in more detail in the
1943 @item it detects contractions that vanish for symmetry reasons, for example
1944 the contraction of a symmetric and a totally antisymmetric tensor
1945 @item as a special case of dummy index summation, it can replace scalar products
1946 of two tensors with a user-defined value
1949 The last point is done with the help of the @code{scalar_products} class
1950 which is used to store scalar products with known values (this is not an
1951 arithmetic class, you just pass it to @code{simplify_indexed()}):
1955 symbol A("A"), B("B"), C("C"), i_sym("i");
1959 sp.add(A, B, 0); // A and B are orthogonal
1960 sp.add(A, C, 0); // A and C are orthogonal
1961 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1963 e = indexed(A + B, i) * indexed(A + C, i);
1965 // -> (B+A).i*(A+C).i
1967 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1973 The @code{scalar_products} object @code{sp} acts as a storage for the
1974 scalar products added to it with the @code{.add()} method. This method
1975 takes three arguments: the two expressions of which the scalar product is
1976 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1977 @code{simplify_indexed()} will replace all scalar products of indexed
1978 objects that have the symbols @code{A} and @code{B} as base expressions
1979 with the single value 0. The number, type and dimension of the indices
1980 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1982 @cindex @code{expand()}
1983 The example above also illustrates a feature of the @code{expand()} method:
1984 if passed the @code{expand_indexed} option it will distribute indices
1985 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1987 @cindex @code{tensor} (class)
1988 @subsection Predefined tensors
1990 Some frequently used special tensors such as the delta, epsilon and metric
1991 tensors are predefined in GiNaC. They have special properties when
1992 contracted with other tensor expressions and some of them have constant
1993 matrix representations (they will evaluate to a number when numeric
1994 indices are specified).
1996 @cindex @code{delta_tensor()}
1997 @subsubsection Delta tensor
1999 The delta tensor takes two indices, is symmetric and has the matrix
2000 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2001 @code{delta_tensor()}:
2005 symbol A("A"), B("B");
2007 idx i(symbol("i"), 3), j(symbol("j"), 3),
2008 k(symbol("k"), 3), l(symbol("l"), 3);
2010 ex e = indexed(A, i, j) * indexed(B, k, l)
2011 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2012 cout << e.simplify_indexed() << endl;
2015 cout << delta_tensor(i, i) << endl;
2020 @cindex @code{metric_tensor()}
2021 @subsubsection General metric tensor
2023 The function @code{metric_tensor()} creates a general symmetric metric
2024 tensor with two indices that can be used to raise/lower tensor indices. The
2025 metric tensor is denoted as @samp{g} in the output and if its indices are of
2026 mixed variance it is automatically replaced by a delta tensor:
2032 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2034 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2035 cout << e.simplify_indexed() << endl;
2038 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2039 cout << e.simplify_indexed() << endl;
2042 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2043 * metric_tensor(nu, rho);
2044 cout << e.simplify_indexed() << endl;
2047 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2048 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2049 + indexed(A, mu.toggle_variance(), rho));
2050 cout << e.simplify_indexed() << endl;
2055 @cindex @code{lorentz_g()}
2056 @subsubsection Minkowski metric tensor
2058 The Minkowski metric tensor is a special metric tensor with a constant
2059 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2060 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2061 It is created with the function @code{lorentz_g()} (although it is output as
2066 varidx mu(symbol("mu"), 4);
2068 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2069 * lorentz_g(mu, varidx(0, 4)); // negative signature
2070 cout << e.simplify_indexed() << endl;
2073 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2074 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2075 cout << e.simplify_indexed() << endl;
2080 @cindex @code{spinor_metric()}
2081 @subsubsection Spinor metric tensor
2083 The function @code{spinor_metric()} creates an antisymmetric tensor with
2084 two indices that is used to raise/lower indices of 2-component spinors.
2085 It is output as @samp{eps}:
2091 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2092 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2094 e = spinor_metric(A, B) * indexed(psi, B_co);
2095 cout << e.simplify_indexed() << endl;
2098 e = spinor_metric(A, B) * indexed(psi, A_co);
2099 cout << e.simplify_indexed() << endl;
2102 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2103 cout << e.simplify_indexed() << endl;
2106 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2107 cout << e.simplify_indexed() << endl;
2110 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2111 cout << e.simplify_indexed() << endl;
2114 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2115 cout << e.simplify_indexed() << endl;
2120 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2122 @cindex @code{epsilon_tensor()}
2123 @cindex @code{lorentz_eps()}
2124 @subsubsection Epsilon tensor
2126 The epsilon tensor is totally antisymmetric, its number of indices is equal
2127 to the dimension of the index space (the indices must all be of the same
2128 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2129 defined to be 1. Its behavior with indices that have a variance also
2130 depends on the signature of the metric. Epsilon tensors are output as
2133 There are three functions defined to create epsilon tensors in 2, 3 and 4
2137 ex epsilon_tensor(const ex & i1, const ex & i2);
2138 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2139 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2142 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2143 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2144 Minkowski space (the last @code{bool} argument specifies whether the metric
2145 has negative or positive signature, as in the case of the Minkowski metric
2150 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2151 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2152 e = lorentz_eps(mu, nu, rho, sig) *
2153 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2154 cout << simplify_indexed(e) << endl;
2155 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2157 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2158 symbol A("A"), B("B");
2159 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2160 cout << simplify_indexed(e) << endl;
2161 // -> -B.k*A.j*eps.i.k.j
2162 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2163 cout << simplify_indexed(e) << endl;
2168 @subsection Linear algebra
2170 The @code{matrix} class can be used with indices to do some simple linear
2171 algebra (linear combinations and products of vectors and matrices, traces
2172 and scalar products):
2176 idx i(symbol("i"), 2), j(symbol("j"), 2);
2177 symbol x("x"), y("y");
2179 // A is a 2x2 matrix, X is a 2x1 vector
2180 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2182 cout << indexed(A, i, i) << endl;
2185 ex e = indexed(A, i, j) * indexed(X, j);
2186 cout << e.simplify_indexed() << endl;
2187 // -> [[2*y+x],[4*y+3*x]].i
2189 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2190 cout << e.simplify_indexed() << endl;
2191 // -> [[3*y+3*x,6*y+2*x]].j
2195 You can of course obtain the same results with the @code{matrix::add()},
2196 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2197 but with indices you don't have to worry about transposing matrices.
2199 Matrix indices always start at 0 and their dimension must match the number
2200 of rows/columns of the matrix. Matrices with one row or one column are
2201 vectors and can have one or two indices (it doesn't matter whether it's a
2202 row or a column vector). Other matrices must have two indices.
2204 You should be careful when using indices with variance on matrices. GiNaC
2205 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2206 @samp{F.mu.nu} are different matrices. In this case you should use only
2207 one form for @samp{F} and explicitly multiply it with a matrix representation
2208 of the metric tensor.
2211 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2212 @c node-name, next, previous, up
2213 @section Non-commutative objects
2215 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2216 non-commutative objects are built-in which are mostly of use in high energy
2220 @item Clifford (Dirac) algebra (class @code{clifford})
2221 @item su(3) Lie algebra (class @code{color})
2222 @item Matrices (unindexed) (class @code{matrix})
2225 The @code{clifford} and @code{color} classes are subclasses of
2226 @code{indexed} because the elements of these algebras usually carry
2227 indices. The @code{matrix} class is described in more detail in
2230 Unlike most computer algebra systems, GiNaC does not primarily provide an
2231 operator (often denoted @samp{&*}) for representing inert products of
2232 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2233 classes of objects involved, and non-commutative products are formed with
2234 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2235 figuring out by itself which objects commute and will group the factors
2236 by their class. Consider this example:
2240 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2241 idx a(symbol("a"), 8), b(symbol("b"), 8);
2242 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2244 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2248 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2249 groups the non-commutative factors (the gammas and the su(3) generators)
2250 together while preserving the order of factors within each class (because
2251 Clifford objects commute with color objects). The resulting expression is a
2252 @emph{commutative} product with two factors that are themselves non-commutative
2253 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2254 parentheses are placed around the non-commutative products in the output.
2256 @cindex @code{ncmul} (class)
2257 Non-commutative products are internally represented by objects of the class
2258 @code{ncmul}, as opposed to commutative products which are handled by the
2259 @code{mul} class. You will normally not have to worry about this distinction,
2262 The advantage of this approach is that you never have to worry about using
2263 (or forgetting to use) a special operator when constructing non-commutative
2264 expressions. Also, non-commutative products in GiNaC are more intelligent
2265 than in other computer algebra systems; they can, for example, automatically
2266 canonicalize themselves according to rules specified in the implementation
2267 of the non-commutative classes. The drawback is that to work with other than
2268 the built-in algebras you have to implement new classes yourself. Symbols
2269 always commute and it's not possible to construct non-commutative products
2270 using symbols to represent the algebra elements or generators. User-defined
2271 functions can, however, be specified as being non-commutative.
2273 @cindex @code{return_type()}
2274 @cindex @code{return_type_tinfo()}
2275 Information about the commutativity of an object or expression can be
2276 obtained with the two member functions
2279 unsigned ex::return_type(void) const;
2280 unsigned ex::return_type_tinfo(void) const;
2283 The @code{return_type()} function returns one of three values (defined in
2284 the header file @file{flags.h}), corresponding to three categories of
2285 expressions in GiNaC:
2288 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2289 classes are of this kind.
2290 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2291 certain class of non-commutative objects which can be determined with the
2292 @code{return_type_tinfo()} method. Expressions of this category commute
2293 with everything except @code{noncommutative} expressions of the same
2295 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2296 of non-commutative objects of different classes. Expressions of this
2297 category don't commute with any other @code{noncommutative} or
2298 @code{noncommutative_composite} expressions.
2301 The value returned by the @code{return_type_tinfo()} method is valid only
2302 when the return type of the expression is @code{noncommutative}. It is a
2303 value that is unique to the class of the object and usually one of the
2304 constants in @file{tinfos.h}, or derived therefrom.
2306 Here are a couple of examples:
2309 @multitable @columnfractions 0.33 0.33 0.34
2310 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2311 @item @code{42} @tab @code{commutative} @tab -
2312 @item @code{2*x-y} @tab @code{commutative} @tab -
2313 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2314 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2315 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2316 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2320 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2321 @code{TINFO_clifford} for objects with a representation label of zero.
2322 Other representation labels yield a different @code{return_type_tinfo()},
2323 but it's the same for any two objects with the same label. This is also true
2326 A last note: With the exception of matrices, positive integer powers of
2327 non-commutative objects are automatically expanded in GiNaC. For example,
2328 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2329 non-commutative expressions).
2332 @cindex @code{clifford} (class)
2333 @subsection Clifford algebra
2335 @cindex @code{dirac_gamma()}
2336 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2337 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2338 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2339 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2342 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2345 which takes two arguments: the index and a @dfn{representation label} in the
2346 range 0 to 255 which is used to distinguish elements of different Clifford
2347 algebras (this is also called a @dfn{spin line index}). Gammas with different
2348 labels commute with each other. The dimension of the index can be 4 or (in
2349 the framework of dimensional regularization) any symbolic value. Spinor
2350 indices on Dirac gammas are not supported in GiNaC.
2352 @cindex @code{dirac_ONE()}
2353 The unity element of a Clifford algebra is constructed by
2356 ex dirac_ONE(unsigned char rl = 0);
2359 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2360 multiples of the unity element, even though it's customary to omit it.
2361 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2362 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2363 GiNaC may produce incorrect results.
2365 @cindex @code{dirac_gamma5()}
2366 There's a special element @samp{gamma5} that commutes with all other
2367 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2371 ex dirac_gamma5(unsigned char rl = 0);
2374 @cindex @code{dirac_gamma6()}
2375 @cindex @code{dirac_gamma7()}
2376 The two additional functions
2379 ex dirac_gamma6(unsigned char rl = 0);
2380 ex dirac_gamma7(unsigned char rl = 0);
2383 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2386 @cindex @code{dirac_slash()}
2387 Finally, the function
2390 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2393 creates a term that represents a contraction of @samp{e} with the Dirac
2394 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2395 with a unique index whose dimension is given by the @code{dim} argument).
2396 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2398 In products of dirac gammas, superfluous unity elements are automatically
2399 removed, squares are replaced by their values and @samp{gamma5} is
2400 anticommuted to the front. The @code{simplify_indexed()} function performs
2401 contractions in gamma strings, for example
2406 symbol a("a"), b("b"), D("D");
2407 varidx mu(symbol("mu"), D);
2408 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2409 * dirac_gamma(mu.toggle_variance());
2411 // -> gamma~mu*a\*gamma.mu
2412 e = e.simplify_indexed();
2415 cout << e.subs(D == 4) << endl;
2421 @cindex @code{dirac_trace()}
2422 To calculate the trace of an expression containing strings of Dirac gammas
2423 you use the function
2426 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2429 This function takes the trace of all gammas with the specified representation
2430 label; gammas with other labels are left standing. The last argument to
2431 @code{dirac_trace()} is the value to be returned for the trace of the unity
2432 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2433 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2434 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2435 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2436 This @samp{gamma5} scheme is described in greater detail in
2437 @cite{The Role of gamma5 in Dimensional Regularization}.
2439 The value of the trace itself is also usually different in 4 and in
2440 @math{D != 4} dimensions:
2445 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2446 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2447 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2448 cout << dirac_trace(e).simplify_indexed() << endl;
2455 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2456 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2457 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2458 cout << dirac_trace(e).simplify_indexed() << endl;
2459 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2463 Here is an example for using @code{dirac_trace()} to compute a value that
2464 appears in the calculation of the one-loop vacuum polarization amplitude in
2469 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2470 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2473 sp.add(l, l, pow(l, 2));
2474 sp.add(l, q, ldotq);
2476 ex e = dirac_gamma(mu) *
2477 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2478 dirac_gamma(mu.toggle_variance()) *
2479 (dirac_slash(l, D) + m * dirac_ONE());
2480 e = dirac_trace(e).simplify_indexed(sp);
2481 e = e.collect(lst(l, ldotq, m));
2483 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2487 The @code{canonicalize_clifford()} function reorders all gamma products that
2488 appear in an expression to a canonical (but not necessarily simple) form.
2489 You can use this to compare two expressions or for further simplifications:
2493 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2494 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2496 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2498 e = canonicalize_clifford(e);
2505 @cindex @code{color} (class)
2506 @subsection Color algebra
2508 @cindex @code{color_T()}
2509 For computations in quantum chromodynamics, GiNaC implements the base elements
2510 and structure constants of the su(3) Lie algebra (color algebra). The base
2511 elements @math{T_a} are constructed by the function
2514 ex color_T(const ex & a, unsigned char rl = 0);
2517 which takes two arguments: the index and a @dfn{representation label} in the
2518 range 0 to 255 which is used to distinguish elements of different color
2519 algebras. Objects with different labels commute with each other. The
2520 dimension of the index must be exactly 8 and it should be of class @code{idx},
2523 @cindex @code{color_ONE()}
2524 The unity element of a color algebra is constructed by
2527 ex color_ONE(unsigned char rl = 0);
2530 @strong{Note:} You must always use @code{color_ONE()} when referring to
2531 multiples of the unity element, even though it's customary to omit it.
2532 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2533 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2534 GiNaC may produce incorrect results.
2536 @cindex @code{color_d()}
2537 @cindex @code{color_f()}
2541 ex color_d(const ex & a, const ex & b, const ex & c);
2542 ex color_f(const ex & a, const ex & b, const ex & c);
2545 create the symmetric and antisymmetric structure constants @math{d_abc} and
2546 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2547 and @math{[T_a, T_b] = i f_abc T_c}.
2549 @cindex @code{color_h()}
2550 There's an additional function
2553 ex color_h(const ex & a, const ex & b, const ex & c);
2556 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2558 The function @code{simplify_indexed()} performs some simplifications on
2559 expressions containing color objects:
2564 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2565 k(symbol("k"), 8), l(symbol("l"), 8);
2567 e = color_d(a, b, l) * color_f(a, b, k);
2568 cout << e.simplify_indexed() << endl;
2571 e = color_d(a, b, l) * color_d(a, b, k);
2572 cout << e.simplify_indexed() << endl;
2575 e = color_f(l, a, b) * color_f(a, b, k);
2576 cout << e.simplify_indexed() << endl;
2579 e = color_h(a, b, c) * color_h(a, b, c);
2580 cout << e.simplify_indexed() << endl;
2583 e = color_h(a, b, c) * color_T(b) * color_T(c);
2584 cout << e.simplify_indexed() << endl;
2587 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2588 cout << e.simplify_indexed() << endl;
2591 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2592 cout << e.simplify_indexed() << endl;
2593 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2597 @cindex @code{color_trace()}
2598 To calculate the trace of an expression containing color objects you use the
2602 ex color_trace(const ex & e, unsigned char rl = 0);
2605 This function takes the trace of all color @samp{T} objects with the
2606 specified representation label; @samp{T}s with other labels are left
2607 standing. For example:
2611 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2613 // -> -I*f.a.c.b+d.a.c.b
2618 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2619 @c node-name, next, previous, up
2620 @chapter Methods and Functions
2623 In this chapter the most important algorithms provided by GiNaC will be
2624 described. Some of them are implemented as functions on expressions,
2625 others are implemented as methods provided by expression objects. If
2626 they are methods, there exists a wrapper function around it, so you can
2627 alternatively call it in a functional way as shown in the simple
2632 cout << "As method: " << sin(1).evalf() << endl;
2633 cout << "As function: " << evalf(sin(1)) << endl;
2637 @cindex @code{subs()}
2638 The general rule is that wherever methods accept one or more parameters
2639 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2640 wrapper accepts is the same but preceded by the object to act on
2641 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2642 most natural one in an OO model but it may lead to confusion for MapleV
2643 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2644 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2645 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2646 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2647 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2648 here. Also, users of MuPAD will in most cases feel more comfortable
2649 with GiNaC's convention. All function wrappers are implemented
2650 as simple inline functions which just call the corresponding method and
2651 are only provided for users uncomfortable with OO who are dead set to
2652 avoid method invocations. Generally, nested function wrappers are much
2653 harder to read than a sequence of methods and should therefore be
2654 avoided if possible. On the other hand, not everything in GiNaC is a
2655 method on class @code{ex} and sometimes calling a function cannot be
2659 * Information About Expressions::
2660 * Substituting Expressions::
2661 * Pattern Matching and Advanced Substitutions::
2662 * Applying a Function on Subexpressions::
2663 * Polynomial Arithmetic:: Working with polynomials.
2664 * Rational Expressions:: Working with rational functions.
2665 * Symbolic Differentiation::
2666 * Series Expansion:: Taylor and Laurent expansion.
2668 * Built-in Functions:: List of predefined mathematical functions.
2669 * Input/Output:: Input and output of expressions.
2673 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2674 @c node-name, next, previous, up
2675 @section Getting information about expressions
2677 @subsection Checking expression types
2678 @cindex @code{is_a<@dots{}>()}
2679 @cindex @code{is_exactly_a<@dots{}>()}
2680 @cindex @code{ex_to<@dots{}>()}
2681 @cindex Converting @code{ex} to other classes
2682 @cindex @code{info()}
2683 @cindex @code{return_type()}
2684 @cindex @code{return_type_tinfo()}
2686 Sometimes it's useful to check whether a given expression is a plain number,
2687 a sum, a polynomial with integer coefficients, or of some other specific type.
2688 GiNaC provides a couple of functions for this:
2691 bool is_a<T>(const ex & e);
2692 bool is_exactly_a<T>(const ex & e);
2693 bool ex::info(unsigned flag);
2694 unsigned ex::return_type(void) const;
2695 unsigned ex::return_type_tinfo(void) const;
2698 When the test made by @code{is_a<T>()} returns true, it is safe to call
2699 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2700 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2701 example, assuming @code{e} is an @code{ex}:
2706 if (is_a<numeric>(e))
2707 numeric n = ex_to<numeric>(e);
2712 @code{is_a<T>(e)} allows you to check whether the top-level object of
2713 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2714 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2715 e.g., for checking whether an expression is a number, a sum, or a product:
2722 is_a<numeric>(e1); // true
2723 is_a<numeric>(e2); // false
2724 is_a<add>(e1); // false
2725 is_a<add>(e2); // true
2726 is_a<mul>(e1); // false
2727 is_a<mul>(e2); // false
2731 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2732 top-level object of an expression @samp{e} is an instance of the GiNaC
2733 class @samp{T}, not including parent classes.
2735 The @code{info()} method is used for checking certain attributes of
2736 expressions. The possible values for the @code{flag} argument are defined
2737 in @file{ginac/flags.h}, the most important being explained in the following
2741 @multitable @columnfractions .30 .70
2742 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2743 @item @code{numeric}
2744 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2746 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2747 @item @code{rational}
2748 @tab @dots{}an exact rational number (integers are rational, too)
2749 @item @code{integer}
2750 @tab @dots{}a (non-complex) integer
2751 @item @code{crational}
2752 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2753 @item @code{cinteger}
2754 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2755 @item @code{positive}
2756 @tab @dots{}not complex and greater than 0
2757 @item @code{negative}
2758 @tab @dots{}not complex and less than 0
2759 @item @code{nonnegative}
2760 @tab @dots{}not complex and greater than or equal to 0
2762 @tab @dots{}an integer greater than 0
2764 @tab @dots{}an integer less than 0
2765 @item @code{nonnegint}
2766 @tab @dots{}an integer greater than or equal to 0
2768 @tab @dots{}an even integer
2770 @tab @dots{}an odd integer
2772 @tab @dots{}a prime integer (probabilistic primality test)
2773 @item @code{relation}
2774 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2775 @item @code{relation_equal}
2776 @tab @dots{}a @code{==} relation
2777 @item @code{relation_not_equal}
2778 @tab @dots{}a @code{!=} relation
2779 @item @code{relation_less}
2780 @tab @dots{}a @code{<} relation
2781 @item @code{relation_less_or_equal}
2782 @tab @dots{}a @code{<=} relation
2783 @item @code{relation_greater}
2784 @tab @dots{}a @code{>} relation
2785 @item @code{relation_greater_or_equal}
2786 @tab @dots{}a @code{>=} relation
2788 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2790 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2791 @item @code{polynomial}
2792 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2793 @item @code{integer_polynomial}
2794 @tab @dots{}a polynomial with (non-complex) integer coefficients
2795 @item @code{cinteger_polynomial}
2796 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2797 @item @code{rational_polynomial}
2798 @tab @dots{}a polynomial with (non-complex) rational coefficients
2799 @item @code{crational_polynomial}
2800 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2801 @item @code{rational_function}
2802 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2803 @item @code{algebraic}
2804 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2808 To determine whether an expression is commutative or non-commutative and if
2809 so, with which other expressions it would commute, you use the methods
2810 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2811 for an explanation of these.
2814 @subsection Accessing subexpressions
2815 @cindex @code{nops()}
2818 @cindex @code{relational} (class)
2820 GiNaC provides the two methods
2823 unsigned ex::nops();
2824 ex ex::op(unsigned i);
2827 for accessing the subexpressions in the container-like GiNaC classes like
2828 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2829 determines the number of subexpressions (@samp{operands}) contained, while
2830 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2831 In the case of a @code{power} object, @code{op(0)} will return the basis
2832 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2833 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2835 The left-hand and right-hand side expressions of objects of class
2836 @code{relational} (and only of these) can also be accessed with the methods
2844 @subsection Comparing expressions
2845 @cindex @code{is_equal()}
2846 @cindex @code{is_zero()}
2848 Expressions can be compared with the usual C++ relational operators like
2849 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2850 the result is usually not determinable and the result will be @code{false},
2851 except in the case of the @code{!=} operator. You should also be aware that
2852 GiNaC will only do the most trivial test for equality (subtracting both
2853 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2856 Actually, if you construct an expression like @code{a == b}, this will be
2857 represented by an object of the @code{relational} class (@pxref{Relations})
2858 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
2860 There are also two methods
2863 bool ex::is_equal(const ex & other);
2867 for checking whether one expression is equal to another, or equal to zero,
2870 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2871 GiNaC header files. This method is however only to be used internally by
2872 GiNaC to establish a canonical sort order for terms, and using it to compare
2873 expressions will give very surprising results.
2876 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2877 @c node-name, next, previous, up
2878 @section Substituting expressions
2879 @cindex @code{subs()}
2881 Algebraic objects inside expressions can be replaced with arbitrary
2882 expressions via the @code{.subs()} method:
2885 ex ex::subs(const ex & e);
2886 ex ex::subs(const lst & syms, const lst & repls);
2889 In the first form, @code{subs()} accepts a relational of the form
2890 @samp{object == expression} or a @code{lst} of such relationals:
2894 symbol x("x"), y("y");
2896 ex e1 = 2*x^2-4*x+3;
2897 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2901 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2906 If you specify multiple substitutions, they are performed in parallel, so e.g.
2907 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2909 The second form of @code{subs()} takes two lists, one for the objects to be
2910 replaced and one for the expressions to be substituted (both lists must
2911 contain the same number of elements). Using this form, you would write
2912 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2914 @code{subs()} performs syntactic substitution of any complete algebraic
2915 object; it does not try to match sub-expressions as is demonstrated by the
2920 symbol x("x"), y("y"), z("z");
2922 ex e1 = pow(x+y, 2);
2923 cout << e1.subs(x+y == 4) << endl;
2926 ex e2 = sin(x)*sin(y)*cos(x);
2927 cout << e2.subs(sin(x) == cos(x)) << endl;
2928 // -> cos(x)^2*sin(y)
2931 cout << e3.subs(x+y == 4) << endl;
2933 // (and not 4+z as one might expect)
2937 A more powerful form of substitution using wildcards is described in the
2941 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2942 @c node-name, next, previous, up
2943 @section Pattern matching and advanced substitutions
2944 @cindex @code{wildcard} (class)
2945 @cindex Pattern matching
2947 GiNaC allows the use of patterns for checking whether an expression is of a
2948 certain form or contains subexpressions of a certain form, and for
2949 substituting expressions in a more general way.
2951 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2952 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2953 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2954 an unsigned integer number to allow having multiple different wildcards in a
2955 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2956 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2960 ex wild(unsigned label = 0);
2963 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2966 Some examples for patterns:
2968 @multitable @columnfractions .5 .5
2969 @item @strong{Constructed as} @tab @strong{Output as}
2970 @item @code{wild()} @tab @samp{$0}
2971 @item @code{pow(x,wild())} @tab @samp{x^$0}
2972 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2973 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2979 @item Wildcards behave like symbols and are subject to the same algebraic
2980 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2981 @item As shown in the last example, to use wildcards for indices you have to
2982 use them as the value of an @code{idx} object. This is because indices must
2983 always be of class @code{idx} (or a subclass).
2984 @item Wildcards only represent expressions or subexpressions. It is not
2985 possible to use them as placeholders for other properties like index
2986 dimension or variance, representation labels, symmetry of indexed objects
2988 @item Because wildcards are commutative, it is not possible to use wildcards
2989 as part of noncommutative products.
2990 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2991 are also valid patterns.
2994 @cindex @code{match()}
2995 The most basic application of patterns is to check whether an expression
2996 matches a given pattern. This is done by the function
2999 bool ex::match(const ex & pattern);
3000 bool ex::match(const ex & pattern, lst & repls);
3003 This function returns @code{true} when the expression matches the pattern
3004 and @code{false} if it doesn't. If used in the second form, the actual
3005 subexpressions matched by the wildcards get returned in the @code{repls}
3006 object as a list of relations of the form @samp{wildcard == expression}.
3007 If @code{match()} returns false, the state of @code{repls} is undefined.
3008 For reproducible results, the list should be empty when passed to
3009 @code{match()}, but it is also possible to find similarities in multiple
3010 expressions by passing in the result of a previous match.
3012 The matching algorithm works as follows:
3015 @item A single wildcard matches any expression. If one wildcard appears
3016 multiple times in a pattern, it must match the same expression in all
3017 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3018 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3019 @item If the expression is not of the same class as the pattern, the match
3020 fails (i.e. a sum only matches a sum, a function only matches a function,
3022 @item If the pattern is a function, it only matches the same function
3023 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3024 @item Except for sums and products, the match fails if the number of
3025 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3027 @item If there are no subexpressions, the expressions and the pattern must
3028 be equal (in the sense of @code{is_equal()}).
3029 @item Except for sums and products, each subexpression (@code{op()}) must
3030 match the corresponding subexpression of the pattern.
3033 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3034 account for their commutativity and associativity:
3037 @item If the pattern contains a term or factor that is a single wildcard,
3038 this one is used as the @dfn{global wildcard}. If there is more than one
3039 such wildcard, one of them is chosen as the global wildcard in a random
3041 @item Every term/factor of the pattern, except the global wildcard, is
3042 matched against every term of the expression in sequence. If no match is
3043 found, the whole match fails. Terms that did match are not considered in
3045 @item If there are no unmatched terms left, the match succeeds. Otherwise
3046 the match fails unless there is a global wildcard in the pattern, in
3047 which case this wildcard matches the remaining terms.
3050 In general, having more than one single wildcard as a term of a sum or a
3051 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3054 Here are some examples in @command{ginsh} to demonstrate how it works (the
3055 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3056 match fails, and the list of wildcard replacements otherwise):
3059 > match((x+y)^a,(x+y)^a);
3061 > match((x+y)^a,(x+y)^b);
3063 > match((x+y)^a,$1^$2);
3065 > match((x+y)^a,$1^$1);
3067 > match((x+y)^(x+y),$1^$1);
3069 > match((x+y)^(x+y),$1^$2);
3071 > match((a+b)*(a+c),($1+b)*($1+c));
3073 > match((a+b)*(a+c),(a+$1)*(a+$2));
3075 (Unpredictable. The result might also be [$1==c,$2==b].)
3076 > match((a+b)*(a+c),($1+$2)*($1+$3));
3077 (The result is undefined. Due to the sequential nature of the algorithm
3078 and the re-ordering of terms in GiNaC, the match for the first factor
3079 may be @{$1==a,$2==b@} in which case the match for the second factor
3080 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3082 > match(a*(x+y)+a*z+b,a*$1+$2);
3083 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3084 @{$1=x+y,$2=a*z+b@}.)
3085 > match(a+b+c+d+e+f,c);
3087 > match(a+b+c+d+e+f,c+$0);
3089 > match(a+b+c+d+e+f,c+e+$0);
3091 > match(a+b,a+b+$0);
3093 > match(a*b^2,a^$1*b^$2);
3095 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3096 even though a==a^1.)
3097 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3099 > match(atan2(y,x^2),atan2(y,$0));
3103 @cindex @code{has()}
3104 A more general way to look for patterns in expressions is provided by the
3108 bool ex::has(const ex & pattern);
3111 This function checks whether a pattern is matched by an expression itself or
3112 by any of its subexpressions.
3114 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3115 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3118 > has(x*sin(x+y+2*a),y);
3120 > has(x*sin(x+y+2*a),x+y);
3122 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3123 has the subexpressions "x", "y" and "2*a".)
3124 > has(x*sin(x+y+2*a),x+y+$1);
3126 (But this is possible.)
3127 > has(x*sin(2*(x+y)+2*a),x+y);
3129 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3130 which "x+y" is not a subexpression.)
3133 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3135 > has(4*x^2-x+3,$1*x);
3137 > has(4*x^2+x+3,$1*x);
3139 (Another possible pitfall. The first expression matches because the term
3140 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3141 contains a linear term you should use the coeff() function instead.)
3144 @cindex @code{find()}
3148 bool ex::find(const ex & pattern, lst & found);
3151 works a bit like @code{has()} but it doesn't stop upon finding the first
3152 match. Instead, it appends all found matches to the specified list. If there
3153 are multiple occurrences of the same expression, it is entered only once to
3154 the list. @code{find()} returns false if no matches were found (in
3155 @command{ginsh}, it returns an empty list):
3158 > find(1+x+x^2+x^3,x);
3160 > find(1+x+x^2+x^3,y);
3162 > find(1+x+x^2+x^3,x^$1);
3164 (Note the absence of "x".)
3165 > expand((sin(x)+sin(y))*(a+b));
3166 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3171 @cindex @code{subs()}
3172 Probably the most useful application of patterns is to use them for
3173 substituting expressions with the @code{subs()} method. Wildcards can be
3174 used in the search patterns as well as in the replacement expressions, where
3175 they get replaced by the expressions matched by them. @code{subs()} doesn't
3176 know anything about algebra; it performs purely syntactic substitutions.
3181 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3183 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3185 > subs((a+b+c)^2,a+b=x);
3187 > subs((a+b+c)^2,a+b+$1==x+$1);
3189 > subs(a+2*b,a+b=x);
3191 > subs(4*x^3-2*x^2+5*x-1,x==a);
3193 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3195 > subs(sin(1+sin(x)),sin($1)==cos($1));
3197 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3201 The last example would be written in C++ in this way:
3205 symbol a("a"), b("b"), x("x"), y("y");
3206 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3207 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3208 cout << e.expand() << endl;
3214 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3215 @c node-name, next, previous, up
3216 @section Applying a Function on Subexpressions
3217 @cindex Tree traversal
3218 @cindex @code{map()}
3220 Sometimes you may want to perform an operation on specific parts of an
3221 expression while leaving the general structure of it intact. An example
3222 of this would be a matrix trace operation: the trace of a sum is the sum
3223 of the traces of the individual terms. That is, the trace should @dfn{map}
3224 on the sum, by applying itself to each of the sum's operands. It is possible
3225 to do this manually which usually results in code like this:
3230 if (is_a<matrix>(e))
3231 return ex_to<matrix>(e).trace();
3232 else if (is_a<add>(e)) @{
3234 for (unsigned i=0; i<e.nops(); i++)
3235 sum += calc_trace(e.op(i));
3237 @} else if (is_a<mul>)(e)) @{
3245 This is, however, slightly inefficient (if the sum is very large it can take
3246 a long time to add the terms one-by-one), and its applicability is limited to
3247 a rather small class of expressions. If @code{calc_trace()} is called with
3248 a relation or a list as its argument, you will probably want the trace to
3249 be taken on both sides of the relation or of all elements of the list.
3251 GiNaC offers the @code{map()} method to aid in the implementation of such
3255 ex ex::map(map_function & f) const;
3256 ex ex::map(ex (*f)(const ex & e)) const;
3259 In the first (preferred) form, @code{map()} takes a function object that
3260 is subclassed from the @code{map_function} class. In the second form, it
3261 takes a pointer to a function that accepts and returns an expression.
3262 @code{map()} constructs a new expression of the same type, applying the
3263 specified function on all subexpressions (in the sense of @code{op()}),
3266 The use of a function object makes it possible to supply more arguments to
3267 the function that is being mapped, or to keep local state information.
3268 The @code{map_function} class declares a virtual function call operator
3269 that you can overload. Here is a sample implementation of @code{calc_trace()}
3270 that uses @code{map()} in a recursive fashion:
3273 struct calc_trace : public map_function @{
3274 ex operator()(const ex &e)
3276 if (is_a<matrix>(e))
3277 return ex_to<matrix>(e).trace();
3278 else if (is_a<mul>(e)) @{
3281 return e.map(*this);
3286 This function object could then be used like this:
3290 ex M = ... // expression with matrices
3291 calc_trace do_trace;
3292 ex tr = do_trace(M);
3296 Here is another example for you to meditate over. It removes quadratic
3297 terms in a variable from an expanded polynomial:
3300 struct map_rem_quad : public map_function @{
3302 map_rem_quad(const ex & var_) : var(var_) @{@}
3304 ex operator()(const ex & e)
3306 if (is_a<add>(e) || is_a<mul>(e))
3307 return e.map(*this);
3308 else if (is_a<power>(e) &&
3309 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3319 symbol x("x"), y("y");
3322 for (int i=0; i<8; i++)
3323 e += pow(x, i) * pow(y, 8-i) * (i+1);
3325 // -> 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
3327 map_rem_quad rem_quad(x);
3328 cout << rem_quad(e) << endl;
3329 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3333 @command{ginsh} offers a slightly different implementation of @code{map()}
3334 that allows applying algebraic functions to operands. The second argument
3335 to @code{map()} is an expression containing the wildcard @samp{$0} which
3336 acts as the placeholder for the operands:
3341 > map(a+2*b,sin($0));
3343 > map(@{a,b,c@},$0^2+$0);
3344 @{a^2+a,b^2+b,c^2+c@}
3347 Note that it is only possible to use algebraic functions in the second
3348 argument. You can not use functions like @samp{diff()}, @samp{op()},
3349 @samp{subs()} etc. because these are evaluated immediately:
3352 > map(@{a,b,c@},diff($0,a));
3354 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3355 to "map(@{a,b,c@},0)".
3359 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3360 @c node-name, next, previous, up
3361 @section Polynomial arithmetic
3363 @subsection Expanding and collecting
3364 @cindex @code{expand()}
3365 @cindex @code{collect()}
3367 A polynomial in one or more variables has many equivalent
3368 representations. Some useful ones serve a specific purpose. Consider
3369 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3370 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3371 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3372 representations are the recursive ones where one collects for exponents
3373 in one of the three variable. Since the factors are themselves
3374 polynomials in the remaining two variables the procedure can be
3375 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3376 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3379 To bring an expression into expanded form, its method
3385 may be called. In our example above, this corresponds to @math{4*x*y +
3386 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3387 GiNaC is not easily guessable you should be prepared to see different
3388 orderings of terms in such sums!
3390 Another useful representation of multivariate polynomials is as a
3391 univariate polynomial in one of the variables with the coefficients
3392 being polynomials in the remaining variables. The method
3393 @code{collect()} accomplishes this task:
3396 ex ex::collect(const ex & s, bool distributed = false);
3399 The first argument to @code{collect()} can also be a list of objects in which
3400 case the result is either a recursively collected polynomial, or a polynomial
3401 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3402 by the @code{distributed} flag.
3404 Note that the original polynomial needs to be in expanded form (for the
3405 variables concerned) in order for @code{collect()} to be able to find the
3406 coefficients properly.
3408 The following @command{ginsh} transcript shows an application of @code{collect()}
3409 together with @code{find()}:
3412 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3413 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
3414 > collect(a,@{p,q@});
3415 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)
3416 > collect(a,find(a,sin($1)));
3417 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3418 > collect(a,@{find(a,sin($1)),p,q@});
3419 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3420 > collect(a,@{find(a,sin($1)),d@});
3421 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3424 @subsection Degree and coefficients
3425 @cindex @code{degree()}
3426 @cindex @code{ldegree()}
3427 @cindex @code{coeff()}
3429 The degree and low degree of a polynomial can be obtained using the two
3433 int ex::degree(const ex & s);
3434 int ex::ldegree(const ex & s);
3437 which also work reliably on non-expanded input polynomials (they even work
3438 on rational functions, returning the asymptotic degree). To extract
3439 a coefficient with a certain power from an expanded polynomial you use
3442 ex ex::coeff(const ex & s, int n);
3445 You can also obtain the leading and trailing coefficients with the methods
3448 ex ex::lcoeff(const ex & s);
3449 ex ex::tcoeff(const ex & s);
3452 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3455 An application is illustrated in the next example, where a multivariate
3456 polynomial is analyzed:
3460 symbol x("x"), y("y");
3461 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3462 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3463 ex Poly = PolyInp.expand();
3465 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3466 cout << "The x^" << i << "-coefficient is "
3467 << Poly.coeff(x,i) << endl;
3469 cout << "As polynomial in y: "
3470 << Poly.collect(y) << endl;
3474 When run, it returns an output in the following fashion:
3477 The x^0-coefficient is y^2+11*y
3478 The x^1-coefficient is 5*y^2-2*y
3479 The x^2-coefficient is -1
3480 The x^3-coefficient is 4*y
3481 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3484 As always, the exact output may vary between different versions of GiNaC
3485 or even from run to run since the internal canonical ordering is not
3486 within the user's sphere of influence.
3488 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3489 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3490 with non-polynomial expressions as they not only work with symbols but with
3491 constants, functions and indexed objects as well:
3495 symbol a("a"), b("b"), c("c");
3496 idx i(symbol("i"), 3);
3498 ex e = pow(sin(x) - cos(x), 4);
3499 cout << e.degree(cos(x)) << endl;
3501 cout << e.expand().coeff(sin(x), 3) << endl;
3504 e = indexed(a+b, i) * indexed(b+c, i);
3505 e = e.expand(expand_options::expand_indexed);
3506 cout << e.collect(indexed(b, i)) << endl;
3507 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3512 @subsection Polynomial division
3513 @cindex polynomial division
3516 @cindex pseudo-remainder
3517 @cindex @code{quo()}
3518 @cindex @code{rem()}
3519 @cindex @code{prem()}
3520 @cindex @code{divide()}
3525 ex quo(const ex & a, const ex & b, const symbol & x);
3526 ex rem(const ex & a, const ex & b, const symbol & x);
3529 compute the quotient and remainder of univariate polynomials in the variable
3530 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3532 The additional function
3535 ex prem(const ex & a, const ex & b, const symbol & x);
3538 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3539 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3541 Exact division of multivariate polynomials is performed by the function
3544 bool divide(const ex & a, const ex & b, ex & q);
3547 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3548 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3549 in which case the value of @code{q} is undefined.
3552 @subsection Unit, content and primitive part
3553 @cindex @code{unit()}
3554 @cindex @code{content()}
3555 @cindex @code{primpart()}
3560 ex ex::unit(const symbol & x);
3561 ex ex::content(const symbol & x);
3562 ex ex::primpart(const symbol & x);
3565 return the unit part, content part, and primitive polynomial of a multivariate
3566 polynomial with respect to the variable @samp{x} (the unit part being the sign
3567 of the leading coefficient, the content part being the GCD of the coefficients,
3568 and the primitive polynomial being the input polynomial divided by the unit and
3569 content parts). The product of unit, content, and primitive part is the
3570 original polynomial.
3573 @subsection GCD and LCM
3576 @cindex @code{gcd()}
3577 @cindex @code{lcm()}
3579 The functions for polynomial greatest common divisor and least common
3580 multiple have the synopsis
3583 ex gcd(const ex & a, const ex & b);
3584 ex lcm(const ex & a, const ex & b);
3587 The functions @code{gcd()} and @code{lcm()} accept two expressions
3588 @code{a} and @code{b} as arguments and return a new expression, their
3589 greatest common divisor or least common multiple, respectively. If the
3590 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3591 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3594 #include <ginac/ginac.h>
3595 using namespace GiNaC;
3599 symbol x("x"), y("y"), z("z");
3600 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3601 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3603 ex P_gcd = gcd(P_a, P_b);
3605 ex P_lcm = lcm(P_a, P_b);
3606 // 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
3611 @subsection Square-free decomposition
3612 @cindex square-free decomposition
3613 @cindex factorization
3614 @cindex @code{sqrfree()}
3616 GiNaC still lacks proper factorization support. Some form of
3617 factorization is, however, easily implemented by noting that factors
3618 appearing in a polynomial with power two or more also appear in the
3619 derivative and hence can easily be found by computing the GCD of the
3620 original polynomial and its derivatives. Any decent system has an
3621 interface for this so called square-free factorization. So we provide
3624 ex sqrfree(const ex & a, const lst & l = lst());
3626 Here is an example that by the way illustrates how the exact form of the
3627 result may slightly depend on the order of differentiation, calling for
3628 some care with subsequent processing of the result:
3631 symbol x("x"), y("y");
3632 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
3634 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3635 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
3637 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3638 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
3640 cout << sqrfree(BiVarPol) << endl;
3641 // -> depending on luck, any of the above
3644 Note also, how factors with the same exponents are not fully factorized
3648 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3649 @c node-name, next, previous, up
3650 @section Rational expressions
3652 @subsection The @code{normal} method
3653 @cindex @code{normal()}
3654 @cindex simplification
3655 @cindex temporary replacement
3657 Some basic form of simplification of expressions is called for frequently.
3658 GiNaC provides the method @code{.normal()}, which converts a rational function
3659 into an equivalent rational function of the form @samp{numerator/denominator}
3660 where numerator and denominator are coprime. If the input expression is already
3661 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3662 otherwise it performs fraction addition and multiplication.
3664 @code{.normal()} can also be used on expressions which are not rational functions
3665 as it will replace all non-rational objects (like functions or non-integer
3666 powers) by temporary symbols to bring the expression to the domain of rational
3667 functions before performing the normalization, and re-substituting these
3668 symbols afterwards. This algorithm is also available as a separate method
3669 @code{.to_rational()}, described below.
3671 This means that both expressions @code{t1} and @code{t2} are indeed
3672 simplified in this little code snippet:
3677 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3678 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3679 std::cout << "t1 is " << t1.normal() << std::endl;
3680 std::cout << "t2 is " << t2.normal() << std::endl;
3684 Of course this works for multivariate polynomials too, so the ratio of
3685 the sample-polynomials from the section about GCD and LCM above would be
3686 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3689 @subsection Numerator and denominator
3692 @cindex @code{numer()}
3693 @cindex @code{denom()}
3694 @cindex @code{numer_denom()}
3696 The numerator and denominator of an expression can be obtained with
3701 ex ex::numer_denom();
3704 These functions will first normalize the expression as described above and
3705 then return the numerator, denominator, or both as a list, respectively.
3706 If you need both numerator and denominator, calling @code{numer_denom()} is
3707 faster than using @code{numer()} and @code{denom()} separately.
3710 @subsection Converting to a rational expression
3711 @cindex @code{to_rational()}
3713 Some of the methods described so far only work on polynomials or rational
3714 functions. GiNaC provides a way to extend the domain of these functions to
3715 general expressions by using the temporary replacement algorithm described
3716 above. You do this by calling
3719 ex ex::to_rational(lst &l);
3722 on the expression to be converted. The supplied @code{lst} will be filled
3723 with the generated temporary symbols and their replacement expressions in
3724 a format that can be used directly for the @code{subs()} method. It can also
3725 already contain a list of replacements from an earlier application of
3726 @code{.to_rational()}, so it's possible to use it on multiple expressions
3727 and get consistent results.
3734 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3735 ex b = sin(x) + cos(x);
3738 divide(a.to_rational(l), b.to_rational(l), q);
3739 cout << q.subs(l) << endl;
3743 will print @samp{sin(x)-cos(x)}.
3746 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3747 @c node-name, next, previous, up
3748 @section Symbolic differentiation
3749 @cindex differentiation
3750 @cindex @code{diff()}
3752 @cindex product rule
3754 GiNaC's objects know how to differentiate themselves. Thus, a
3755 polynomial (class @code{add}) knows that its derivative is the sum of
3756 the derivatives of all the monomials:
3760 symbol x("x"), y("y"), z("z");
3761 ex P = pow(x, 5) + pow(x, 2) + y;
3763 cout << P.diff(x,2) << endl;
3765 cout << P.diff(y) << endl; // 1
3767 cout << P.diff(z) << endl; // 0
3772 If a second integer parameter @var{n} is given, the @code{diff} method
3773 returns the @var{n}th derivative.
3775 If @emph{every} object and every function is told what its derivative
3776 is, all derivatives of composed objects can be calculated using the
3777 chain rule and the product rule. Consider, for instance the expression
3778 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3779 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3780 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3781 out that the composition is the generating function for Euler Numbers,
3782 i.e. the so called @var{n}th Euler number is the coefficient of
3783 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3784 identity to code a function that generates Euler numbers in just three
3787 @cindex Euler numbers
3789 #include <ginac/ginac.h>
3790 using namespace GiNaC;
3792 ex EulerNumber(unsigned n)
3795 const ex generator = pow(cosh(x),-1);
3796 return generator.diff(x,n).subs(x==0);
3801 for (unsigned i=0; i<11; i+=2)
3802 std::cout << EulerNumber(i) << std::endl;
3807 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3808 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3809 @code{i} by two since all odd Euler numbers vanish anyways.
3812 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3813 @c node-name, next, previous, up
3814 @section Series expansion
3815 @cindex @code{series()}
3816 @cindex Taylor expansion
3817 @cindex Laurent expansion
3818 @cindex @code{pseries} (class)
3819 @cindex @code{Order()}
3821 Expressions know how to expand themselves as a Taylor series or (more
3822 generally) a Laurent series. As in most conventional Computer Algebra
3823 Systems, no distinction is made between those two. There is a class of
3824 its own for storing such series (@code{class pseries}) and a built-in
3825 function (called @code{Order}) for storing the order term of the series.
3826 As a consequence, if you want to work with series, i.e. multiply two
3827 series, you need to call the method @code{ex::series} again to convert
3828 it to a series object with the usual structure (expansion plus order
3829 term). A sample application from special relativity could read:
3832 #include <ginac/ginac.h>
3833 using namespace std;
3834 using namespace GiNaC;
3838 symbol v("v"), c("c");
3840 ex gamma = 1/sqrt(1 - pow(v/c,2));
3841 ex mass_nonrel = gamma.series(v==0, 10);
3843 cout << "the relativistic mass increase with v is " << endl
3844 << mass_nonrel << endl;
3846 cout << "the inverse square of this series is " << endl
3847 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3851 Only calling the series method makes the last output simplify to
3852 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3853 series raised to the power @math{-2}.
3855 @cindex M@'echain's formula
3856 As another instructive application, let us calculate the numerical
3857 value of Archimedes' constant
3861 (for which there already exists the built-in constant @code{Pi})
3862 using M@'echain's amazing formula
3864 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3867 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3869 We may expand the arcus tangent around @code{0} and insert the fractions
3870 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3871 carries an order term with it and the question arises what the system is
3872 supposed to do when the fractions are plugged into that order term. The
3873 solution is to use the function @code{series_to_poly()} to simply strip
3877 #include <ginac/ginac.h>
3878 using namespace GiNaC;
3880 ex mechain_pi(int degr)
3883 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3884 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3885 -4*pi_expansion.subs(x==numeric(1,239));
3891 using std::cout; // just for fun, another way of...
3892 using std::endl; // ...dealing with this namespace std.
3894 for (int i=2; i<12; i+=2) @{
3895 pi_frac = mechain_pi(i);
3896 cout << i << ":\t" << pi_frac << endl
3897 << "\t" << pi_frac.evalf() << endl;
3903 Note how we just called @code{.series(x,degr)} instead of
3904 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3905 method @code{series()}: if the first argument is a symbol the expression
3906 is expanded in that symbol around point @code{0}. When you run this
3907 program, it will type out:
3911 3.1832635983263598326
3912 4: 5359397032/1706489875
3913 3.1405970293260603143
3914 6: 38279241713339684/12184551018734375
3915 3.141621029325034425
3916 8: 76528487109180192540976/24359780855939418203125
3917 3.141591772182177295
3918 10: 327853873402258685803048818236/104359128170408663038552734375
3919 3.1415926824043995174
3923 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3924 @c node-name, next, previous, up
3925 @section Symmetrization
3926 @cindex @code{symmetrize()}
3927 @cindex @code{antisymmetrize()}
3928 @cindex @code{symmetrize_cyclic()}
3933 ex ex::symmetrize(const lst & l);
3934 ex ex::antisymmetrize(const lst & l);
3935 ex ex::symmetrize_cyclic(const lst & l);
3938 symmetrize an expression by returning the sum over all symmetric,
3939 antisymmetric or cyclic permutations of the specified list of objects,
3940 weighted by the number of permutations.
3942 The three additional methods
3945 ex ex::symmetrize();
3946 ex ex::antisymmetrize();
3947 ex ex::symmetrize_cyclic();
3950 symmetrize or antisymmetrize an expression over its free indices.
3952 Symmetrization is most useful with indexed expressions but can be used with
3953 almost any kind of object (anything that is @code{subs()}able):
3957 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3958 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3960 cout << indexed(A, i, j).symmetrize() << endl;
3961 // -> 1/2*A.j.i+1/2*A.i.j
3962 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3963 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3964 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3965 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
3970 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
3971 @c node-name, next, previous, up
3972 @section Predefined mathematical functions
3974 GiNaC contains the following predefined mathematical functions:
3977 @multitable @columnfractions .30 .70
3978 @item @strong{Name} @tab @strong{Function}
3981 @cindex @code{abs()}
3982 @item @code{csgn(x)}
3984 @cindex @code{csgn()}
3985 @item @code{sqrt(x)}
3986 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
3987 @cindex @code{sqrt()}
3990 @cindex @code{sin()}
3993 @cindex @code{cos()}
3996 @cindex @code{tan()}
3997 @item @code{asin(x)}
3999 @cindex @code{asin()}
4000 @item @code{acos(x)}
4002 @cindex @code{acos()}
4003 @item @code{atan(x)}
4004 @tab inverse tangent
4005 @cindex @code{atan()}
4006 @item @code{atan2(y, x)}
4007 @tab inverse tangent with two arguments
4008 @item @code{sinh(x)}
4009 @tab hyperbolic sine
4010 @cindex @code{sinh()}
4011 @item @code{cosh(x)}
4012 @tab hyperbolic cosine
4013 @cindex @code{cosh()}
4014 @item @code{tanh(x)}
4015 @tab hyperbolic tangent
4016 @cindex @code{tanh()}
4017 @item @code{asinh(x)}
4018 @tab inverse hyperbolic sine
4019 @cindex @code{asinh()}
4020 @item @code{acosh(x)}
4021 @tab inverse hyperbolic cosine
4022 @cindex @code{acosh()}
4023 @item @code{atanh(x)}
4024 @tab inverse hyperbolic tangent
4025 @cindex @code{atanh()}
4027 @tab exponential function
4028 @cindex @code{exp()}
4030 @tab natural logarithm
4031 @cindex @code{log()}
4034 @cindex @code{Li2()}
4035 @item @code{zeta(x)}
4036 @tab Riemann's zeta function
4037 @cindex @code{zeta()}
4038 @item @code{zeta(n, x)}
4039 @tab derivatives of Riemann's zeta function
4040 @item @code{tgamma(x)}
4042 @cindex @code{tgamma()}
4043 @cindex Gamma function
4044 @item @code{lgamma(x)}
4045 @tab logarithm of Gamma function
4046 @cindex @code{lgamma()}
4047 @item @code{beta(x, y)}
4048 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4049 @cindex @code{beta()}
4051 @tab psi (digamma) function
4052 @cindex @code{psi()}
4053 @item @code{psi(n, x)}
4054 @tab derivatives of psi function (polygamma functions)
4055 @item @code{factorial(n)}
4056 @tab factorial function
4057 @cindex @code{factorial()}
4058 @item @code{binomial(n, m)}
4059 @tab binomial coefficients
4060 @cindex @code{binomial()}
4061 @item @code{Order(x)}
4062 @tab order term function in truncated power series
4063 @cindex @code{Order()}
4068 For functions that have a branch cut in the complex plane GiNaC follows
4069 the conventions for C++ as defined in the ANSI standard as far as
4070 possible. In particular: the natural logarithm (@code{log}) and the
4071 square root (@code{sqrt}) both have their branch cuts running along the
4072 negative real axis where the points on the axis itself belong to the
4073 upper part (i.e. continuous with quadrant II). The inverse
4074 trigonometric and hyperbolic functions are not defined for complex
4075 arguments by the C++ standard, however. In GiNaC we follow the
4076 conventions used by CLN, which in turn follow the carefully designed
4077 definitions in the Common Lisp standard. It should be noted that this
4078 convention is identical to the one used by the C99 standard and by most
4079 serious CAS. It is to be expected that future revisions of the C++
4080 standard incorporate these functions in the complex domain in a manner
4081 compatible with C99.
4084 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
4085 @c node-name, next, previous, up
4086 @section Input and output of expressions
4089 @subsection Expression output
4091 @cindex output of expressions
4093 The easiest way to print an expression is to write it to a stream:
4098 ex e = 4.5+pow(x,2)*3/2;
4099 cout << e << endl; // prints '(4.5)+3/2*x^2'
4103 The output format is identical to the @command{ginsh} input syntax and
4104 to that used by most computer algebra systems, but not directly pastable
4105 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4106 is printed as @samp{x^2}).
4108 It is possible to print expressions in a number of different formats with
4112 void ex::print(const print_context & c, unsigned level = 0);
4115 @cindex @code{print_context} (class)
4116 The type of @code{print_context} object passed in determines the format
4117 of the output. The possible types are defined in @file{ginac/print.h}.
4118 All constructors of @code{print_context} and derived classes take an
4119 @code{ostream &} as their first argument.
4121 To print an expression in a way that can be directly used in a C or C++
4122 program, you pass a @code{print_csrc} object like this:
4126 cout << "float f = ";
4127 e.print(print_csrc_float(cout));
4130 cout << "double d = ";
4131 e.print(print_csrc_double(cout));
4134 cout << "cl_N n = ";
4135 e.print(print_csrc_cl_N(cout));
4140 The three possible types mostly affect the way in which floating point
4141 numbers are written.
4143 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4146 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4147 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4148 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4151 The @code{print_context} type @code{print_tree} provides a dump of the
4152 internal structure of an expression for debugging purposes:
4156 e.print(print_tree(cout));
4163 add, hash=0x0, flags=0x3, nops=2
4164 power, hash=0x9, flags=0x3, nops=2
4165 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4166 2 (numeric), hash=0x80000042, flags=0xf
4167 3/2 (numeric), hash=0x80000061, flags=0xf
4170 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4174 This kind of output is also available in @command{ginsh} as the @code{print()}
4177 Another useful output format is for LaTeX parsing in mathematical mode.
4178 It is rather similar to the default @code{print_context} but provides
4179 some braces needed by LaTeX for delimiting boxes and also converts some
4180 common objects to conventional LaTeX names. It is possible to give symbols
4181 a special name for LaTeX output by supplying it as a second argument to
4182 the @code{symbol} constructor.
4184 For example, the code snippet
4189 ex foo = lgamma(x).series(x==0,3);
4190 foo.print(print_latex(std::cout));
4196 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4199 @cindex Tree traversal
4200 If you need any fancy special output format, e.g. for interfacing GiNaC
4201 with other algebra systems or for producing code for different
4202 programming languages, you can always traverse the expression tree yourself:
4205 static void my_print(const ex & e)
4207 if (is_a<function>(e))
4208 cout << ex_to<function>(e).get_name();
4210 cout << e.bp->class_name();
4212 unsigned n = e.nops();
4214 for (unsigned i=0; i<n; i++) @{
4226 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4234 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4235 symbol(y))),numeric(-2)))
4238 If you need an output format that makes it possible to accurately
4239 reconstruct an expression by feeding the output to a suitable parser or
4240 object factory, you should consider storing the expression in an
4241 @code{archive} object and reading the object properties from there.
4242 See the section on archiving for more information.
4245 @subsection Expression input
4246 @cindex input of expressions
4248 GiNaC provides no way to directly read an expression from a stream because
4249 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4250 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4251 @code{y} you defined in your program and there is no way to specify the
4252 desired symbols to the @code{>>} stream input operator.
4254 Instead, GiNaC lets you construct an expression from a string, specifying the
4255 list of symbols to be used:
4259 symbol x("x"), y("y");
4260 ex e("2*x+sin(y)", lst(x, y));
4264 The input syntax is the same as that used by @command{ginsh} and the stream
4265 output operator @code{<<}. The symbols in the string are matched by name to
4266 the symbols in the list and if GiNaC encounters a symbol not specified in
4267 the list it will throw an exception.
4269 With this constructor, it's also easy to implement interactive GiNaC programs:
4274 #include <stdexcept>
4275 #include <ginac/ginac.h>
4276 using namespace std;
4277 using namespace GiNaC;
4284 cout << "Enter an expression containing 'x': ";
4289 cout << "The derivative of " << e << " with respect to x is ";
4290 cout << e.diff(x) << ".\n";
4291 @} catch (exception &p) @{
4292 cerr << p.what() << endl;
4298 @subsection Archiving
4299 @cindex @code{archive} (class)
4302 GiNaC allows creating @dfn{archives} of expressions which can be stored
4303 to or retrieved from files. To create an archive, you declare an object
4304 of class @code{archive} and archive expressions in it, giving each
4305 expression a unique name:
4309 using namespace std;
4310 #include <ginac/ginac.h>
4311 using namespace GiNaC;
4315 symbol x("x"), y("y"), z("z");
4317 ex foo = sin(x + 2*y) + 3*z + 41;
4321 a.archive_ex(foo, "foo");
4322 a.archive_ex(bar, "the second one");
4326 The archive can then be written to a file:
4330 ofstream out("foobar.gar");
4336 The file @file{foobar.gar} contains all information that is needed to
4337 reconstruct the expressions @code{foo} and @code{bar}.
4339 @cindex @command{viewgar}
4340 The tool @command{viewgar} that comes with GiNaC can be used to view
4341 the contents of GiNaC archive files:
4344 $ viewgar foobar.gar
4345 foo = 41+sin(x+2*y)+3*z
4346 the second one = 42+sin(x+2*y)+3*z
4349 The point of writing archive files is of course that they can later be
4355 ifstream in("foobar.gar");
4360 And the stored expressions can be retrieved by their name:
4366 ex ex1 = a2.unarchive_ex(syms, "foo");
4367 ex ex2 = a2.unarchive_ex(syms, "the second one");
4369 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4370 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4371 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4375 Note that you have to supply a list of the symbols which are to be inserted
4376 in the expressions. Symbols in archives are stored by their name only and
4377 if you don't specify which symbols you have, unarchiving the expression will
4378 create new symbols with that name. E.g. if you hadn't included @code{x} in
4379 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4380 have had no effect because the @code{x} in @code{ex1} would have been a
4381 different symbol than the @code{x} which was defined at the beginning of
4382 the program, although both would appear as @samp{x} when printed.
4384 You can also use the information stored in an @code{archive} object to
4385 output expressions in a format suitable for exact reconstruction. The
4386 @code{archive} and @code{archive_node} classes have a couple of member
4387 functions that let you access the stored properties:
4390 static void my_print2(const archive_node & n)
4393 n.find_string("class", class_name);
4394 cout << class_name << "(";
4396 archive_node::propinfovector p;
4397 n.get_properties(p);
4399 unsigned num = p.size();
4400 for (unsigned i=0; i<num; i++) @{
4401 const string &name = p[i].name;
4402 if (name == "class")
4404 cout << name << "=";
4406 unsigned count = p[i].count;
4410 for (unsigned j=0; j<count; j++) @{
4411 switch (p[i].type) @{
4412 case archive_node::PTYPE_BOOL: @{
4414 n.find_bool(name, x, j);
4415 cout << (x ? "true" : "false");
4418 case archive_node::PTYPE_UNSIGNED: @{
4420 n.find_unsigned(name, x, j);
4424 case archive_node::PTYPE_STRING: @{
4426 n.find_string(name, x, j);
4427 cout << '\"' << x << '\"';
4430 case archive_node::PTYPE_NODE: @{
4431 const archive_node &x = n.find_ex_node(name, j);
4453 ex e = pow(2, x) - y;
4455 my_print2(ar.get_top_node(0)); cout << endl;
4463 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4464 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4465 overall_coeff=numeric(number="0"))
4468 Be warned, however, that the set of properties and their meaning for each
4469 class may change between GiNaC versions.
4472 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4473 @c node-name, next, previous, up
4474 @chapter Extending GiNaC
4476 By reading so far you should have gotten a fairly good understanding of
4477 GiNaC's design-patterns. From here on you should start reading the
4478 sources. All we can do now is issue some recommendations how to tackle
4479 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4480 develop some useful extension please don't hesitate to contact the GiNaC
4481 authors---they will happily incorporate them into future versions.
4484 * What does not belong into GiNaC:: What to avoid.
4485 * Symbolic functions:: Implementing symbolic functions.
4486 * Adding classes:: Defining new algebraic classes.
4490 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4491 @c node-name, next, previous, up
4492 @section What doesn't belong into GiNaC
4494 @cindex @command{ginsh}
4495 First of all, GiNaC's name must be read literally. It is designed to be
4496 a library for use within C++. The tiny @command{ginsh} accompanying
4497 GiNaC makes this even more clear: it doesn't even attempt to provide a
4498 language. There are no loops or conditional expressions in
4499 @command{ginsh}, it is merely a window into the library for the
4500 programmer to test stuff (or to show off). Still, the design of a
4501 complete CAS with a language of its own, graphical capabilities and all
4502 this on top of GiNaC is possible and is without doubt a nice project for
4505 There are many built-in functions in GiNaC that do not know how to
4506 evaluate themselves numerically to a precision declared at runtime
4507 (using @code{Digits}). Some may be evaluated at certain points, but not
4508 generally. This ought to be fixed. However, doing numerical
4509 computations with GiNaC's quite abstract classes is doomed to be
4510 inefficient. For this purpose, the underlying foundation classes
4511 provided by CLN are much better suited.
4514 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4515 @c node-name, next, previous, up
4516 @section Symbolic functions
4518 The easiest and most instructive way to start with is probably to
4519 implement your own function. GiNaC's functions are objects of class
4520 @code{function}. The preprocessor is then used to convert the function
4521 names to objects with a corresponding serial number that is used
4522 internally to identify them. You usually need not worry about this
4523 number. New functions may be inserted into the system via a kind of
4524 `registry'. It is your responsibility to care for some functions that
4525 are called when the user invokes certain methods. These are usual
4526 C++-functions accepting a number of @code{ex} as arguments and returning
4527 one @code{ex}. As an example, if we have a look at a simplified
4528 implementation of the cosine trigonometric function, we first need a
4529 function that is called when one wishes to @code{eval} it. It could
4530 look something like this:
4533 static ex cos_eval_method(const ex & x)
4535 // if (!x%(2*Pi)) return 1
4536 // if (!x%Pi) return -1
4537 // if (!x%Pi/2) return 0
4538 // care for other cases...
4539 return cos(x).hold();
4543 @cindex @code{hold()}
4545 The last line returns @code{cos(x)} if we don't know what else to do and
4546 stops a potential recursive evaluation by saying @code{.hold()}, which
4547 sets a flag to the expression signaling that it has been evaluated. We
4548 should also implement a method for numerical evaluation and since we are
4549 lazy we sweep the problem under the rug by calling someone else's
4550 function that does so, in this case the one in class @code{numeric}:
4553 static ex cos_evalf(const ex & x)
4555 if (is_a<numeric>(x))
4556 return cos(ex_to<numeric>(x));
4558 return cos(x).hold();
4562 Differentiation will surely turn up and so we need to tell @code{cos}
4563 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4564 instance are then handled automatically by @code{basic::diff} and
4568 static ex cos_deriv(const ex & x, unsigned diff_param)
4574 @cindex product rule
4575 The second parameter is obligatory but uninteresting at this point. It
4576 specifies which parameter to differentiate in a partial derivative in
4577 case the function has more than one parameter and its main application
4578 is for correct handling of the chain rule. For Taylor expansion, it is
4579 enough to know how to differentiate. But if the function you want to
4580 implement does have a pole somewhere in the complex plane, you need to
4581 write another method for Laurent expansion around that point.
4583 Now that all the ingredients for @code{cos} have been set up, we need
4584 to tell the system about it. This is done by a macro and we are not
4585 going to describe how it expands, please consult your preprocessor if you
4589 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4590 evalf_func(cos_evalf).
4591 derivative_func(cos_deriv));
4594 The first argument is the function's name used for calling it and for
4595 output. The second binds the corresponding methods as options to this
4596 object. Options are separated by a dot and can be given in an arbitrary
4597 order. GiNaC functions understand several more options which are always
4598 specified as @code{.option(params)}, for example a method for series
4599 expansion @code{.series_func(cos_series)}. Again, if no series
4600 expansion method is given, GiNaC defaults to simple Taylor expansion,
4601 which is correct if there are no poles involved as is the case for the
4602 @code{cos} function. The way GiNaC handles poles in case there are any
4603 is best understood by studying one of the examples, like the Gamma
4604 (@code{tgamma}) function for instance. (In essence the function first
4605 checks if there is a pole at the evaluation point and falls back to
4606 Taylor expansion if there isn't. Then, the pole is regularized by some
4607 suitable transformation.) Also, the new function needs to be declared
4608 somewhere. This may also be done by a convenient preprocessor macro:
4611 DECLARE_FUNCTION_1P(cos)
4614 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4615 implementation of @code{cos} is very incomplete and lacks several safety
4616 mechanisms. Please, have a look at the real implementation in GiNaC.
4617 (By the way: in case you are worrying about all the macros above we can
4618 assure you that functions are GiNaC's most macro-intense classes. We
4619 have done our best to avoid macros where we can.)
4622 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4623 @c node-name, next, previous, up
4624 @section Adding classes
4626 If you are doing some very specialized things with GiNaC you may find that
4627 you have to implement your own algebraic classes to fit your needs. This
4628 section will explain how to do this by giving the example of a simple
4629 'string' class. After reading this section you will know how to properly
4630 declare a GiNaC class and what the minimum required member functions are
4631 that you have to implement. We only cover the implementation of a 'leaf'
4632 class here (i.e. one that doesn't contain subexpressions). Creating a
4633 container class like, for example, a class representing tensor products is
4634 more involved but this section should give you enough information so you can
4635 consult the source to GiNaC's predefined classes if you want to implement
4636 something more complicated.
4638 @subsection GiNaC's run-time type information system
4640 @cindex hierarchy of classes
4642 All algebraic classes (that is, all classes that can appear in expressions)
4643 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4644 @code{basic *} (which is essentially what an @code{ex} is) represents a
4645 generic pointer to an algebraic class. Occasionally it is necessary to find
4646 out what the class of an object pointed to by a @code{basic *} really is.
4647 Also, for the unarchiving of expressions it must be possible to find the
4648 @code{unarchive()} function of a class given the class name (as a string). A
4649 system that provides this kind of information is called a run-time type
4650 information (RTTI) system. The C++ language provides such a thing (see the
4651 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4652 implements its own, simpler RTTI.
4654 The RTTI in GiNaC is based on two mechanisms:
4659 The @code{basic} class declares a member variable @code{tinfo_key} which
4660 holds an unsigned integer that identifies the object's class. These numbers
4661 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4662 classes. They all start with @code{TINFO_}.
4665 By means of some clever tricks with static members, GiNaC maintains a list
4666 of information for all classes derived from @code{basic}. The information
4667 available includes the class names, the @code{tinfo_key}s, and pointers
4668 to the unarchiving functions. This class registry is defined in the
4669 @file{registrar.h} header file.
4673 The disadvantage of this proprietary RTTI implementation is that there's
4674 a little more to do when implementing new classes (C++'s RTTI works more
4675 or less automatic) but don't worry, most of the work is simplified by
4678 @subsection A minimalistic example
4680 Now we will start implementing a new class @code{mystring} that allows
4681 placing character strings in algebraic expressions (this is not very useful,
4682 but it's just an example). This class will be a direct subclass of
4683 @code{basic}. You can use this sample implementation as a starting point
4684 for your own classes.
4686 The code snippets given here assume that you have included some header files
4692 #include <stdexcept>
4693 using namespace std;
4695 #include <ginac/ginac.h>
4696 using namespace GiNaC;
4699 The first thing we have to do is to define a @code{tinfo_key} for our new
4700 class. This can be any arbitrary unsigned number that is not already taken
4701 by one of the existing classes but it's better to come up with something
4702 that is unlikely to clash with keys that might be added in the future. The
4703 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4704 which is not a requirement but we are going to stick with this scheme:
4707 const unsigned TINFO_mystring = 0x42420001U;
4710 Now we can write down the class declaration. The class stores a C++
4711 @code{string} and the user shall be able to construct a @code{mystring}
4712 object from a C or C++ string:
4715 class mystring : public basic
4717 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4720 mystring(const string &s);
4721 mystring(const char *s);
4727 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4730 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4731 macros are defined in @file{registrar.h}. They take the name of the class
4732 and its direct superclass as arguments and insert all required declarations
4733 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4734 the first line after the opening brace of the class definition. The
4735 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4736 source (at global scope, of course, not inside a function).
4738 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4739 declarations of the default and copy constructor, the destructor, the
4740 assignment operator and a couple of other functions that are required. It
4741 also defines a type @code{inherited} which refers to the superclass so you
4742 don't have to modify your code every time you shuffle around the class
4743 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4744 constructor, the destructor and the assignment operator.
4746 Now there are nine member functions we have to implement to get a working
4752 @code{mystring()}, the default constructor.
4755 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4756 assignment operator to free dynamically allocated members. The @code{call_parent}
4757 specifies whether the @code{destroy()} function of the superclass is to be
4761 @code{void copy(const mystring &other)}, which is used in the copy constructor
4762 and assignment operator to copy the member variables over from another
4763 object of the same class.
4766 @code{void archive(archive_node &n)}, the archiving function. This stores all
4767 information needed to reconstruct an object of this class inside an
4768 @code{archive_node}.
4771 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4772 constructor. This constructs an instance of the class from the information
4773 found in an @code{archive_node}.
4776 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4777 unarchiving function. It constructs a new instance by calling the unarchiving
4781 @code{int compare_same_type(const basic &other)}, which is used internally
4782 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4783 -1, depending on the relative order of this object and the @code{other}
4784 object. If it returns 0, the objects are considered equal.
4785 @strong{Note:} This has nothing to do with the (numeric) ordering
4786 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4787 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4788 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4789 must provide a @code{compare_same_type()} function, even those representing
4790 objects for which no reasonable algebraic ordering relationship can be
4794 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4795 which are the two constructors we declared.
4799 Let's proceed step-by-step. The default constructor looks like this:
4802 mystring::mystring() : inherited(TINFO_mystring)
4804 // dynamically allocate resources here if required
4808 The golden rule is that in all constructors you have to set the
4809 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4810 it will be set by the constructor of the superclass and all hell will break
4811 loose in the RTTI. For your convenience, the @code{basic} class provides
4812 a constructor that takes a @code{tinfo_key} value, which we are using here
4813 (remember that in our case @code{inherited = basic}). If the superclass
4814 didn't have such a constructor, we would have to set the @code{tinfo_key}
4815 to the right value manually.
4817 In the default constructor you should set all other member variables to
4818 reasonable default values (we don't need that here since our @code{str}
4819 member gets set to an empty string automatically). The constructor(s) are of
4820 course also the right place to allocate any dynamic resources you require.
4822 Next, the @code{destroy()} function:
4825 void mystring::destroy(bool call_parent)
4827 // free dynamically allocated resources here if required
4829 inherited::destroy(call_parent);
4833 This function is where we free all dynamically allocated resources. We
4834 don't have any so we're not doing anything here, but if we had, for
4835 example, used a C-style @code{char *} to store our string, this would be
4836 the place to @code{delete[]} the string storage. If @code{call_parent}
4837 is true, we have to call the @code{destroy()} function of the superclass
4838 after we're done (to mimic C++'s automatic invocation of superclass
4839 destructors where @code{destroy()} is called from outside a destructor).
4841 The @code{copy()} function just copies over the member variables from
4845 void mystring::copy(const mystring &other)
4847 inherited::copy(other);
4852 We can simply overwrite the member variables here. There's no need to worry
4853 about dynamically allocated storage. The assignment operator (which is
4854 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4855 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4856 explicitly call the @code{copy()} function of the superclass here so
4857 all the member variables will get copied.
4859 Next are the three functions for archiving. You have to implement them even
4860 if you don't plan to use archives, but the minimum required implementation
4861 is really simple. First, the archiving function:
4864 void mystring::archive(archive_node &n) const
4866 inherited::archive(n);
4867 n.add_string("string", str);
4871 The only thing that is really required is calling the @code{archive()}
4872 function of the superclass. Optionally, you can store all information you
4873 deem necessary for representing the object into the passed
4874 @code{archive_node}. We are just storing our string here. For more
4875 information on how the archiving works, consult the @file{archive.h} header
4878 The unarchiving constructor is basically the inverse of the archiving
4882 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4884 n.find_string("string", str);
4888 If you don't need archiving, just leave this function empty (but you must
4889 invoke the unarchiving constructor of the superclass). Note that we don't
4890 have to set the @code{tinfo_key} here because it is done automatically
4891 by the unarchiving constructor of the @code{basic} class.
4893 Finally, the unarchiving function:
4896 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4898 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4902 You don't have to understand how exactly this works. Just copy these
4903 four lines into your code literally (replacing the class name, of
4904 course). It calls the unarchiving constructor of the class and unless
4905 you are doing something very special (like matching @code{archive_node}s
4906 to global objects) you don't need a different implementation. For those
4907 who are interested: setting the @code{dynallocated} flag puts the object
4908 under the control of GiNaC's garbage collection. It will get deleted
4909 automatically once it is no longer referenced.
4911 Our @code{compare_same_type()} function uses a provided function to compare
4915 int mystring::compare_same_type(const basic &other) const
4917 const mystring &o = static_cast<const mystring &>(other);
4918 int cmpval = str.compare(o.str);
4921 else if (cmpval < 0)
4928 Although this function takes a @code{basic &}, it will always be a reference
4929 to an object of exactly the same class (objects of different classes are not
4930 comparable), so the cast is safe. If this function returns 0, the two objects
4931 are considered equal (in the sense that @math{A-B=0}), so you should compare
4932 all relevant member variables.
4934 Now the only thing missing is our two new constructors:
4937 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4939 // dynamically allocate resources here if required
4942 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4944 // dynamically allocate resources here if required
4948 No surprises here. We set the @code{str} member from the argument and
4949 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4951 That's it! We now have a minimal working GiNaC class that can store
4952 strings in algebraic expressions. Let's confirm that the RTTI works:
4955 ex e = mystring("Hello, world!");
4956 cout << is_a<mystring>(e) << endl;
4959 cout << e.bp->class_name() << endl;
4963 Obviously it does. Let's see what the expression @code{e} looks like:
4967 // -> [mystring object]
4970 Hm, not exactly what we expect, but of course the @code{mystring} class
4971 doesn't yet know how to print itself. This is done in the @code{print()}
4972 member function. Let's say that we wanted to print the string surrounded
4976 class mystring : public basic
4980 void print(const print_context &c, unsigned level = 0) const;
4984 void mystring::print(const print_context &c, unsigned level) const
4986 // print_context::s is a reference to an ostream
4987 c.s << '\"' << str << '\"';
4991 The @code{level} argument is only required for container classes to
4992 correctly parenthesize the output. Let's try again to print the expression:
4996 // -> "Hello, world!"
4999 Much better. The @code{mystring} class can be used in arbitrary expressions:
5002 e += mystring("GiNaC rulez");
5004 // -> "GiNaC rulez"+"Hello, world!"
5007 (GiNaC's automatic term reordering is in effect here), or even
5010 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
5012 // -> "One string"^(2*sin(-"Another string"+Pi))
5015 Whether this makes sense is debatable but remember that this is only an
5016 example. At least it allows you to implement your own symbolic algorithms
5019 Note that GiNaC's algebraic rules remain unchanged:
5022 e = mystring("Wow") * mystring("Wow");
5026 e = pow(mystring("First")-mystring("Second"), 2);
5027 cout << e.expand() << endl;
5028 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5031 There's no way to, for example, make GiNaC's @code{add} class perform string
5032 concatenation. You would have to implement this yourself.
5034 @subsection Automatic evaluation
5036 @cindex @code{hold()}
5037 @cindex @code{eval()}
5039 When dealing with objects that are just a little more complicated than the
5040 simple string objects we have implemented, chances are that you will want to
5041 have some automatic simplifications or canonicalizations performed on them.
5042 This is done in the evaluation member function @code{eval()}. Let's say that
5043 we wanted all strings automatically converted to lowercase with
5044 non-alphabetic characters stripped, and empty strings removed:
5047 class mystring : public basic
5051 ex eval(int level = 0) const;
5055 ex mystring::eval(int level) const
5058 for (int i=0; i<str.length(); i++) @{
5060 if (c >= 'A' && c <= 'Z')
5061 new_str += tolower(c);
5062 else if (c >= 'a' && c <= 'z')
5066 if (new_str.length() == 0)
5069 return mystring(new_str).hold();
5073 The @code{level} argument is used to limit the recursion depth of the
5074 evaluation. We don't have any subexpressions in the @code{mystring}
5075 class so we are not concerned with this. If we had, we would call the
5076 @code{eval()} functions of the subexpressions with @code{level - 1} as
5077 the argument if @code{level != 1}. The @code{hold()} member function
5078 sets a flag in the object that prevents further evaluation. Otherwise
5079 we might end up in an endless loop. When you want to return the object
5080 unmodified, use @code{return this->hold();}.
5082 Let's confirm that it works:
5085 ex e = mystring("Hello, world!") + mystring("!?#");
5089 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5094 @subsection Other member functions
5096 We have implemented only a small set of member functions to make the class
5097 work in the GiNaC framework. For a real algebraic class, there are probably
5098 some more functions that you will want to re-implement, such as
5099 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5100 or the header file of the class you want to make a subclass of to see
5101 what's there. One member function that you will most likely want to
5102 implement for terminal classes like the described string class is
5103 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5104 which will allow GiNaC to compare and canonicalize expressions much more
5107 You can, of course, also add your own new member functions. Remember,
5108 that the RTTI may be used to get information about what kinds of objects
5109 you are dealing with (the position in the class hierarchy) and that you
5110 can always extract the bare object from an @code{ex} by stripping the
5111 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5112 should become a need.
5114 That's it. May the source be with you!
5117 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5118 @c node-name, next, previous, up
5119 @chapter A Comparison With Other CAS
5122 This chapter will give you some information on how GiNaC compares to
5123 other, traditional Computer Algebra Systems, like @emph{Maple},
5124 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5125 disadvantages over these systems.
5128 * Advantages:: Strengths of the GiNaC approach.
5129 * Disadvantages:: Weaknesses of the GiNaC approach.
5130 * Why C++?:: Attractiveness of C++.
5133 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5134 @c node-name, next, previous, up
5137 GiNaC has several advantages over traditional Computer
5138 Algebra Systems, like
5143 familiar language: all common CAS implement their own proprietary
5144 grammar which you have to learn first (and maybe learn again when your
5145 vendor decides to `enhance' it). With GiNaC you can write your program
5146 in common C++, which is standardized.
5150 structured data types: you can build up structured data types using
5151 @code{struct}s or @code{class}es together with STL features instead of
5152 using unnamed lists of lists of lists.
5155 strongly typed: in CAS, you usually have only one kind of variables
5156 which can hold contents of an arbitrary type. This 4GL like feature is
5157 nice for novice programmers, but dangerous.
5160 development tools: powerful development tools exist for C++, like fancy
5161 editors (e.g. with automatic indentation and syntax highlighting),
5162 debuggers, visualization tools, documentation generators@dots{}
5165 modularization: C++ programs can easily be split into modules by
5166 separating interface and implementation.
5169 price: GiNaC is distributed under the GNU Public License which means
5170 that it is free and available with source code. And there are excellent
5171 C++-compilers for free, too.
5174 extendable: you can add your own classes to GiNaC, thus extending it on
5175 a very low level. Compare this to a traditional CAS that you can
5176 usually only extend on a high level by writing in the language defined
5177 by the parser. In particular, it turns out to be almost impossible to
5178 fix bugs in a traditional system.
5181 multiple interfaces: Though real GiNaC programs have to be written in
5182 some editor, then be compiled, linked and executed, there are more ways
5183 to work with the GiNaC engine. Many people want to play with
5184 expressions interactively, as in traditional CASs. Currently, two such
5185 windows into GiNaC have been implemented and many more are possible: the
5186 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5187 types to a command line and second, as a more consistent approach, an
5188 interactive interface to the Cint C++ interpreter has been put together
5189 (called GiNaC-cint) that allows an interactive scripting interface
5190 consistent with the C++ language. It is available from the usual GiNaC
5194 seamless integration: it is somewhere between difficult and impossible
5195 to call CAS functions from within a program written in C++ or any other
5196 programming language and vice versa. With GiNaC, your symbolic routines
5197 are part of your program. You can easily call third party libraries,
5198 e.g. for numerical evaluation or graphical interaction. All other
5199 approaches are much more cumbersome: they range from simply ignoring the
5200 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5201 system (i.e. @emph{Yacas}).
5204 efficiency: often large parts of a program do not need symbolic
5205 calculations at all. Why use large integers for loop variables or
5206 arbitrary precision arithmetics where @code{int} and @code{double} are
5207 sufficient? For pure symbolic applications, GiNaC is comparable in
5208 speed with other CAS.
5213 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5214 @c node-name, next, previous, up
5215 @section Disadvantages
5217 Of course it also has some disadvantages:
5222 advanced features: GiNaC cannot compete with a program like
5223 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5224 which grows since 1981 by the work of dozens of programmers, with
5225 respect to mathematical features. Integration, factorization,
5226 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5227 not planned for the near future).
5230 portability: While the GiNaC library itself is designed to avoid any
5231 platform dependent features (it should compile on any ANSI compliant C++
5232 compiler), the currently used version of the CLN library (fast large
5233 integer and arbitrary precision arithmetics) can only by compiled
5234 without hassle on systems with the C++ compiler from the GNU Compiler
5235 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
5236 macros to let the compiler gather all static initializations, which
5237 works for GNU C++ only. Feel free to contact the authors in case you
5238 really believe that you need to use a different compiler. We have
5239 occasionally used other compilers and may be able to give you advice.}
5240 GiNaC uses recent language features like explicit constructors, mutable
5241 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
5242 literally. Recent GCC versions starting at 2.95.3, although itself not
5243 yet ANSI compliant, support all needed features.
5248 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5249 @c node-name, next, previous, up
5252 Why did we choose to implement GiNaC in C++ instead of Java or any other
5253 language? C++ is not perfect: type checking is not strict (casting is
5254 possible), separation between interface and implementation is not
5255 complete, object oriented design is not enforced. The main reason is
5256 the often scolded feature of operator overloading in C++. While it may
5257 be true that operating on classes with a @code{+} operator is rarely
5258 meaningful, it is perfectly suited for algebraic expressions. Writing
5259 @math{3x+5y} as @code{3*x+5*y} instead of
5260 @code{x.times(3).plus(y.times(5))} looks much more natural.
5261 Furthermore, the main developers are more familiar with C++ than with
5262 any other programming language.
5265 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5266 @c node-name, next, previous, up
5267 @appendix Internal Structures
5270 * Expressions are reference counted::
5271 * Internal representation of products and sums::
5274 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5275 @c node-name, next, previous, up
5276 @appendixsection Expressions are reference counted
5278 @cindex reference counting
5279 @cindex copy-on-write
5280 @cindex garbage collection
5281 An expression is extremely light-weight since internally it works like a
5282 handle to the actual representation and really holds nothing more than a
5283 pointer to some other object. What this means in practice is that
5284 whenever you create two @code{ex} and set the second equal to the first
5285 no copying process is involved. Instead, the copying takes place as soon
5286 as you try to change the second. Consider the simple sequence of code:
5290 #include <ginac/ginac.h>
5291 using namespace std;
5292 using namespace GiNaC;
5296 symbol x("x"), y("y"), z("z");
5299 e1 = sin(x + 2*y) + 3*z + 41;
5300 e2 = e1; // e2 points to same object as e1
5301 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5302 e2 += 1; // e2 is copied into a new object
5303 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5307 The line @code{e2 = e1;} creates a second expression pointing to the
5308 object held already by @code{e1}. The time involved for this operation
5309 is therefore constant, no matter how large @code{e1} was. Actual
5310 copying, however, must take place in the line @code{e2 += 1;} because
5311 @code{e1} and @code{e2} are not handles for the same object any more.
5312 This concept is called @dfn{copy-on-write semantics}. It increases
5313 performance considerably whenever one object occurs multiple times and
5314 represents a simple garbage collection scheme because when an @code{ex}
5315 runs out of scope its destructor checks whether other expressions handle
5316 the object it points to too and deletes the object from memory if that
5317 turns out not to be the case. A slightly less trivial example of
5318 differentiation using the chain-rule should make clear how powerful this
5323 symbol x("x"), y("y");
5327 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5328 cout << e1 << endl // prints x+3*y
5329 << e2 << endl // prints (x+3*y)^3
5330 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5334 Here, @code{e1} will actually be referenced three times while @code{e2}
5335 will be referenced two times. When the power of an expression is built,
5336 that expression needs not be copied. Likewise, since the derivative of
5337 a power of an expression can be easily expressed in terms of that
5338 expression, no copying of @code{e1} is involved when @code{e3} is
5339 constructed. So, when @code{e3} is constructed it will print as
5340 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5341 holds a reference to @code{e2} and the factor in front is just
5344 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5345 semantics. When you insert an expression into a second expression, the
5346 result behaves exactly as if the contents of the first expression were
5347 inserted. But it may be useful to remember that this is not what
5348 happens. Knowing this will enable you to write much more efficient
5349 code. If you still have an uncertain feeling with copy-on-write
5350 semantics, we recommend you have a look at the
5351 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5352 Marshall Cline. Chapter 16 covers this issue and presents an
5353 implementation which is pretty close to the one in GiNaC.
5356 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5357 @c node-name, next, previous, up
5358 @appendixsection Internal representation of products and sums
5360 @cindex representation
5363 @cindex @code{power}
5364 Although it should be completely transparent for the user of
5365 GiNaC a short discussion of this topic helps to understand the sources
5366 and also explain performance to a large degree. Consider the
5367 unexpanded symbolic expression
5369 $2d^3 \left( 4a + 5b - 3 \right)$
5372 @math{2*d^3*(4*a+5*b-3)}
5374 which could naively be represented by a tree of linear containers for
5375 addition and multiplication, one container for exponentiation with base
5376 and exponent and some atomic leaves of symbols and numbers in this
5381 @cindex pair-wise representation
5382 However, doing so results in a rather deeply nested tree which will
5383 quickly become inefficient to manipulate. We can improve on this by
5384 representing the sum as a sequence of terms, each one being a pair of a
5385 purely numeric multiplicative coefficient and its rest. In the same
5386 spirit we can store the multiplication as a sequence of terms, each
5387 having a numeric exponent and a possibly complicated base, the tree
5388 becomes much more flat:
5392 The number @code{3} above the symbol @code{d} shows that @code{mul}
5393 objects are treated similarly where the coefficients are interpreted as
5394 @emph{exponents} now. Addition of sums of terms or multiplication of
5395 products with numerical exponents can be coded to be very efficient with
5396 such a pair-wise representation. Internally, this handling is performed
5397 by most CAS in this way. It typically speeds up manipulations by an
5398 order of magnitude. The overall multiplicative factor @code{2} and the
5399 additive term @code{-3} look somewhat out of place in this
5400 representation, however, since they are still carrying a trivial
5401 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5402 this is avoided by adding a field that carries an overall numeric
5403 coefficient. This results in the realistic picture of internal
5406 $2d^3 \left( 4a + 5b - 3 \right)$:
5409 @math{2*d^3*(4*a+5*b-3)}:
5415 This also allows for a better handling of numeric radicals, since
5416 @code{sqrt(2)} can now be carried along calculations. Now it should be
5417 clear, why both classes @code{add} and @code{mul} are derived from the
5418 same abstract class: the data representation is the same, only the
5419 semantics differs. In the class hierarchy, methods for polynomial
5420 expansion and the like are reimplemented for @code{add} and @code{mul},
5421 but the data structure is inherited from @code{expairseq}.
5424 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5425 @c node-name, next, previous, up
5426 @appendix Package Tools
5428 If you are creating a software package that uses the GiNaC library,
5429 setting the correct command line options for the compiler and linker
5430 can be difficult. GiNaC includes two tools to make this process easier.
5433 * ginac-config:: A shell script to detect compiler and linker flags.
5434 * AM_PATH_GINAC:: Macro for GNU automake.
5438 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5439 @c node-name, next, previous, up
5440 @section @command{ginac-config}
5441 @cindex ginac-config
5443 @command{ginac-config} is a shell script that you can use to determine
5444 the compiler and linker command line options required to compile and
5445 link a program with the GiNaC library.
5447 @command{ginac-config} takes the following flags:
5451 Prints out the version of GiNaC installed.
5453 Prints '-I' flags pointing to the installed header files.
5455 Prints out the linker flags necessary to link a program against GiNaC.
5456 @item --prefix[=@var{PREFIX}]
5457 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5458 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5459 Otherwise, prints out the configured value of @env{$prefix}.
5460 @item --exec-prefix[=@var{PREFIX}]
5461 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5462 Otherwise, prints out the configured value of @env{$exec_prefix}.
5465 Typically, @command{ginac-config} will be used within a configure
5466 script, as described below. It, however, can also be used directly from
5467 the command line using backquotes to compile a simple program. For
5471 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5474 This command line might expand to (for example):
5477 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5478 -lginac -lcln -lstdc++
5481 Not only is the form using @command{ginac-config} easier to type, it will
5482 work on any system, no matter how GiNaC was configured.
5485 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5486 @c node-name, next, previous, up
5487 @section @samp{AM_PATH_GINAC}
5488 @cindex AM_PATH_GINAC
5490 For packages configured using GNU automake, GiNaC also provides
5491 a macro to automate the process of checking for GiNaC.
5494 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5502 Determines the location of GiNaC using @command{ginac-config}, which is
5503 either found in the user's path, or from the environment variable
5504 @env{GINACLIB_CONFIG}.
5507 Tests the installed libraries to make sure that their version
5508 is later than @var{MINIMUM-VERSION}. (A default version will be used
5512 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5513 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5514 variable to the output of @command{ginac-config --libs}, and calls
5515 @samp{AC_SUBST()} for these variables so they can be used in generated
5516 makefiles, and then executes @var{ACTION-IF-FOUND}.
5519 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5520 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5524 This macro is in file @file{ginac.m4} which is installed in
5525 @file{$datadir/aclocal}. Note that if automake was installed with a
5526 different @samp{--prefix} than GiNaC, you will either have to manually
5527 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5528 aclocal the @samp{-I} option when running it.
5531 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5532 * Example package:: Example of a package using AM_PATH_GINAC.
5536 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5537 @c node-name, next, previous, up
5538 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5540 Simply make sure that @command{ginac-config} is in your path, and run
5541 the configure script.
5548 The directory where the GiNaC libraries are installed needs
5549 to be found by your system's dynamic linker.
5551 This is generally done by
5554 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5560 setting the environment variable @env{LD_LIBRARY_PATH},
5563 or, as a last resort,
5566 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5567 running configure, for instance:
5570 LDFLAGS=-R/home/cbauer/lib ./configure
5575 You can also specify a @command{ginac-config} not in your path by
5576 setting the @env{GINACLIB_CONFIG} environment variable to the
5577 name of the executable
5580 If you move the GiNaC package from its installed location,
5581 you will either need to modify @command{ginac-config} script
5582 manually to point to the new location or rebuild GiNaC.
5593 --with-ginac-prefix=@var{PREFIX}
5594 --with-ginac-exec-prefix=@var{PREFIX}
5597 are provided to override the prefix and exec-prefix that were stored
5598 in the @command{ginac-config} shell script by GiNaC's configure. You are
5599 generally better off configuring GiNaC with the right path to begin with.
5603 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5604 @c node-name, next, previous, up
5605 @subsection Example of a package using @samp{AM_PATH_GINAC}
5607 The following shows how to build a simple package using automake
5608 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5611 #include <ginac/ginac.h>
5615 GiNaC::symbol x("x");
5616 GiNaC::ex a = GiNaC::sin(x);
5617 std::cout << "Derivative of " << a
5618 << " is " << a.diff(x) << std::endl;
5623 You should first read the introductory portions of the automake
5624 Manual, if you are not already familiar with it.
5626 Two files are needed, @file{configure.in}, which is used to build the
5630 dnl Process this file with autoconf to produce a configure script.
5632 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5638 AM_PATH_GINAC(0.9.0, [
5639 LIBS="$LIBS $GINACLIB_LIBS"
5640 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5641 ], AC_MSG_ERROR([need to have GiNaC installed]))
5646 The only command in this which is not standard for automake
5647 is the @samp{AM_PATH_GINAC} macro.
5649 That command does the following: If a GiNaC version greater or equal
5650 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5651 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5652 the error message `need to have GiNaC installed'
5654 And the @file{Makefile.am}, which will be used to build the Makefile.
5657 ## Process this file with automake to produce Makefile.in
5658 bin_PROGRAMS = simple
5659 simple_SOURCES = simple.cpp
5662 This @file{Makefile.am}, says that we are building a single executable,
5663 from a single sourcefile @file{simple.cpp}. Since every program
5664 we are building uses GiNaC we simply added the GiNaC options
5665 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5666 want to specify them on a per-program basis: for instance by
5670 simple_LDADD = $(GINACLIB_LIBS)
5671 INCLUDES = $(GINACLIB_CPPFLAGS)
5674 to the @file{Makefile.am}.
5676 To try this example out, create a new directory and add the three
5679 Now execute the following commands:
5682 $ automake --add-missing
5687 You now have a package that can be built in the normal fashion
5696 @node Bibliography, Concept Index, Example package, Top
5697 @c node-name, next, previous, up
5698 @appendix Bibliography
5703 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5706 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5709 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5712 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5715 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5716 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5719 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5720 James H. Davenport, Yvon Siret, and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
5721 Academic Press, London
5724 @cite{Computer Algebra Systems - A Practical Guide},
5725 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
5728 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5729 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5732 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
5737 @node Concept Index, , Bibliography, Top
5738 @c node-name, next, previous, up
5739 @unnumbered Concept Index