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) 1999-2002 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{} 1999-2002 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{} 1999-2002 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:
987 0.33333333333333333334
988 3.1415926535897932385
990 0.33333333333333333333333333333333333333333333333333333333333333333334
991 3.1415926535897932384626433832795028841971693993751058209749445923078
995 Note that the last number is not necessarily rounded as you would
996 naively expect it to be rounded in the decimal system. But note also,
997 that in both cases you got a couple of extra digits. This is because
998 numbers are internally stored by CLN as chunks of binary digits in order
999 to match your machine's word size and to not waste precision. Thus, on
1000 architectures with differnt word size, the above output might even
1001 differ with regard to actually computed digits.
1003 It should be clear that objects of class @code{numeric} should be used
1004 for constructing numbers or for doing arithmetic with them. The objects
1005 one deals with most of the time are the polymorphic expressions @code{ex}.
1007 @subsection Tests on numbers
1009 Once you have declared some numbers, assigned them to expressions and
1010 done some arithmetic with them it is frequently desired to retrieve some
1011 kind of information from them like asking whether that number is
1012 integer, rational, real or complex. For those cases GiNaC provides
1013 several useful methods. (Internally, they fall back to invocations of
1014 certain CLN functions.)
1016 As an example, let's construct some rational number, multiply it with
1017 some multiple of its denominator and test what comes out:
1021 #include <ginac/ginac.h>
1022 using namespace std;
1023 using namespace GiNaC;
1025 // some very important constants:
1026 const numeric twentyone(21);
1027 const numeric ten(10);
1028 const numeric five(5);
1032 numeric answer = twentyone;
1035 cout << answer.is_integer() << endl; // false, it's 21/5
1037 cout << answer.is_integer() << endl; // true, it's 42 now!
1041 Note that the variable @code{answer} is constructed here as an integer
1042 by @code{numeric}'s copy constructor but in an intermediate step it
1043 holds a rational number represented as integer numerator and integer
1044 denominator. When multiplied by 10, the denominator becomes unity and
1045 the result is automatically converted to a pure integer again.
1046 Internally, the underlying CLN is responsible for this behavior and we
1047 refer the reader to CLN's documentation. Suffice to say that
1048 the same behavior applies to complex numbers as well as return values of
1049 certain functions. Complex numbers are automatically converted to real
1050 numbers if the imaginary part becomes zero. The full set of tests that
1051 can be applied is listed in the following table.
1054 @multitable @columnfractions .30 .70
1055 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1056 @item @code{.is_zero()}
1057 @tab @dots{}equal to zero
1058 @item @code{.is_positive()}
1059 @tab @dots{}not complex and greater than 0
1060 @item @code{.is_integer()}
1061 @tab @dots{}a (non-complex) integer
1062 @item @code{.is_pos_integer()}
1063 @tab @dots{}an integer and greater than 0
1064 @item @code{.is_nonneg_integer()}
1065 @tab @dots{}an integer and greater equal 0
1066 @item @code{.is_even()}
1067 @tab @dots{}an even integer
1068 @item @code{.is_odd()}
1069 @tab @dots{}an odd integer
1070 @item @code{.is_prime()}
1071 @tab @dots{}a prime integer (probabilistic primality test)
1072 @item @code{.is_rational()}
1073 @tab @dots{}an exact rational number (integers are rational, too)
1074 @item @code{.is_real()}
1075 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1076 @item @code{.is_cinteger()}
1077 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1078 @item @code{.is_crational()}
1079 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1084 @node Constants, Fundamental containers, Numbers, Basic Concepts
1085 @c node-name, next, previous, up
1087 @cindex @code{constant} (class)
1090 @cindex @code{Catalan}
1091 @cindex @code{Euler}
1092 @cindex @code{evalf()}
1093 Constants behave pretty much like symbols except that they return some
1094 specific number when the method @code{.evalf()} is called.
1096 The predefined known constants are:
1099 @multitable @columnfractions .14 .30 .56
1100 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1102 @tab Archimedes' constant
1103 @tab 3.14159265358979323846264338327950288
1104 @item @code{Catalan}
1105 @tab Catalan's constant
1106 @tab 0.91596559417721901505460351493238411
1108 @tab Euler's (or Euler-Mascheroni) constant
1109 @tab 0.57721566490153286060651209008240243
1114 @node Fundamental containers, Lists, Constants, Basic Concepts
1115 @c node-name, next, previous, up
1116 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1120 @cindex @code{power}
1122 Simple polynomial expressions are written down in GiNaC pretty much like
1123 in other CAS or like expressions involving numerical variables in C.
1124 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1125 been overloaded to achieve this goal. When you run the following
1126 code snippet, the constructor for an object of type @code{mul} is
1127 automatically called to hold the product of @code{a} and @code{b} and
1128 then the constructor for an object of type @code{add} is called to hold
1129 the sum of that @code{mul} object and the number one:
1133 symbol a("a"), b("b");
1138 @cindex @code{pow()}
1139 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1140 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1141 construction is necessary since we cannot safely overload the constructor
1142 @code{^} in C++ to construct a @code{power} object. If we did, it would
1143 have several counterintuitive and undesired effects:
1147 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1149 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1150 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1151 interpret this as @code{x^(a^b)}.
1153 Also, expressions involving integer exponents are very frequently used,
1154 which makes it even more dangerous to overload @code{^} since it is then
1155 hard to distinguish between the semantics as exponentiation and the one
1156 for exclusive or. (It would be embarrassing to return @code{1} where one
1157 has requested @code{2^3}.)
1160 @cindex @command{ginsh}
1161 All effects are contrary to mathematical notation and differ from the
1162 way most other CAS handle exponentiation, therefore overloading @code{^}
1163 is ruled out for GiNaC's C++ part. The situation is different in
1164 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1165 that the other frequently used exponentiation operator @code{**} does
1166 not exist at all in C++).
1168 To be somewhat more precise, objects of the three classes described
1169 here, are all containers for other expressions. An object of class
1170 @code{power} is best viewed as a container with two slots, one for the
1171 basis, one for the exponent. All valid GiNaC expressions can be
1172 inserted. However, basic transformations like simplifying
1173 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1174 when this is mathematically possible. If we replace the outer exponent
1175 three in the example by some symbols @code{a}, the simplification is not
1176 safe and will not be performed, since @code{a} might be @code{1/2} and
1179 Objects of type @code{add} and @code{mul} are containers with an
1180 arbitrary number of slots for expressions to be inserted. Again, simple
1181 and safe simplifications are carried out like transforming
1182 @code{3*x+4-x} to @code{2*x+4}.
1184 The general rule is that when you construct such objects, GiNaC
1185 automatically creates them in canonical form, which might differ from
1186 the form you typed in your program. This allows for rapid comparison of
1187 expressions, since after all @code{a-a} is simply zero. Note, that the
1188 canonical form is not necessarily lexicographical ordering or in any way
1189 easily guessable. It is only guaranteed that constructing the same
1190 expression twice, either implicitly or explicitly, results in the same
1194 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1195 @c node-name, next, previous, up
1196 @section Lists of expressions
1197 @cindex @code{lst} (class)
1199 @cindex @code{nops()}
1201 @cindex @code{append()}
1202 @cindex @code{prepend()}
1203 @cindex @code{remove_first()}
1204 @cindex @code{remove_last()}
1206 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1207 expressions. These are sometimes used to supply a variable number of
1208 arguments of the same type to GiNaC methods such as @code{subs()} and
1209 @code{to_rational()}, so you should have a basic understanding about them.
1211 Lists of up to 16 expressions can be directly constructed from single
1216 symbol x("x"), y("y");
1217 lst l(x, 2, y, x+y);
1218 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1222 Use the @code{nops()} method to determine the size (number of expressions) of
1223 a list and the @code{op()} method to access individual elements:
1227 cout << l.nops() << endl; // prints '4'
1228 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1232 You can append or prepend an expression to a list with the @code{append()}
1233 and @code{prepend()} methods:
1237 l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
1238 l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
1242 Finally you can remove the first or last element of a list with
1243 @code{remove_first()} and @code{remove_last()}:
1247 l.remove_first(); // l is now @{x, 2, y, x+y, 4*x@}
1248 l.remove_last(); // l is now @{x, 2, y, x+y@}
1253 @node Mathematical functions, Relations, Lists, Basic Concepts
1254 @c node-name, next, previous, up
1255 @section Mathematical functions
1256 @cindex @code{function} (class)
1257 @cindex trigonometric function
1258 @cindex hyperbolic function
1260 There are quite a number of useful functions hard-wired into GiNaC. For
1261 instance, all trigonometric and hyperbolic functions are implemented
1262 (@xref{Built-in Functions}, for a complete list).
1264 These functions (better called @emph{pseudofunctions}) are all objects
1265 of class @code{function}. They accept one or more expressions as
1266 arguments and return one expression. If the arguments are not
1267 numerical, the evaluation of the function may be halted, as it does in
1268 the next example, showing how a function returns itself twice and
1269 finally an expression that may be really useful:
1271 @cindex Gamma function
1272 @cindex @code{subs()}
1275 symbol x("x"), y("y");
1277 cout << tgamma(foo) << endl;
1278 // -> tgamma(x+(1/2)*y)
1279 ex bar = foo.subs(y==1);
1280 cout << tgamma(bar) << endl;
1282 ex foobar = bar.subs(x==7);
1283 cout << tgamma(foobar) << endl;
1284 // -> (135135/128)*Pi^(1/2)
1288 Besides evaluation most of these functions allow differentiation, series
1289 expansion and so on. Read the next chapter in order to learn more about
1292 It must be noted that these pseudofunctions are created by inline
1293 functions, where the argument list is templated. This means that
1294 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1295 @code{sin(ex(1))} and will therefore not result in a floating point
1296 number. Unless of course the function prototype is explicitly
1297 overridden -- which is the case for arguments of type @code{numeric}
1298 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1299 point number of class @code{numeric} you should call
1300 @code{sin(numeric(1))}. This is almost the same as calling
1301 @code{sin(1).evalf()} except that the latter will return a numeric
1302 wrapped inside an @code{ex}.
1305 @node Relations, Matrices, Mathematical functions, Basic Concepts
1306 @c node-name, next, previous, up
1308 @cindex @code{relational} (class)
1310 Sometimes, a relation holding between two expressions must be stored
1311 somehow. The class @code{relational} is a convenient container for such
1312 purposes. A relation is by definition a container for two @code{ex} and
1313 a relation between them that signals equality, inequality and so on.
1314 They are created by simply using the C++ operators @code{==}, @code{!=},
1315 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1317 @xref{Mathematical functions}, for examples where various applications
1318 of the @code{.subs()} method show how objects of class relational are
1319 used as arguments. There they provide an intuitive syntax for
1320 substitutions. They are also used as arguments to the @code{ex::series}
1321 method, where the left hand side of the relation specifies the variable
1322 to expand in and the right hand side the expansion point. They can also
1323 be used for creating systems of equations that are to be solved for
1324 unknown variables. But the most common usage of objects of this class
1325 is rather inconspicuous in statements of the form @code{if
1326 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1327 conversion from @code{relational} to @code{bool} takes place. Note,
1328 however, that @code{==} here does not perform any simplifications, hence
1329 @code{expand()} must be called explicitly.
1332 @node Matrices, Indexed objects, Relations, Basic Concepts
1333 @c node-name, next, previous, up
1335 @cindex @code{matrix} (class)
1337 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1338 matrix with @math{m} rows and @math{n} columns are accessed with two
1339 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1340 second one in the range 0@dots{}@math{n-1}.
1342 There are a couple of ways to construct matrices, with or without preset
1346 matrix::matrix(unsigned r, unsigned c);
1347 matrix::matrix(unsigned r, unsigned c, const lst & l);
1348 ex lst_to_matrix(const lst & l);
1349 ex diag_matrix(const lst & l);
1352 The first two functions are @code{matrix} constructors which create a matrix
1353 with @samp{r} rows and @samp{c} columns. The matrix elements can be
1354 initialized from a (flat) list of expressions @samp{l}. Otherwise they are
1355 all set to zero. The @code{lst_to_matrix()} function constructs a matrix
1356 from a list of lists, each list representing a matrix row. Finally,
1357 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1358 elements. Note that the last two functions return expressions, not matrix
1361 Matrix elements can be accessed and set using the parenthesis (function call)
1365 const ex & matrix::operator()(unsigned r, unsigned c) const;
1366 ex & matrix::operator()(unsigned r, unsigned c);
1369 It is also possible to access the matrix elements in a linear fashion with
1370 the @code{op()} method. But C++-style subscripting with square brackets
1371 @samp{[]} is not available.
1373 Here are a couple of examples that all construct the same 2x2 diagonal
1378 symbol a("a"), b("b");
1386 e = matrix(2, 2, lst(a, 0, 0, b));
1388 e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
1390 e = diag_matrix(lst(a, b));
1397 @cindex @code{transpose()}
1398 @cindex @code{inverse()}
1399 There are three ways to do arithmetic with matrices. The first (and most
1400 efficient one) is to use the methods provided by the @code{matrix} class:
1403 matrix matrix::add(const matrix & other) const;
1404 matrix matrix::sub(const matrix & other) const;
1405 matrix matrix::mul(const matrix & other) const;
1406 matrix matrix::mul_scalar(const ex & other) const;
1407 matrix matrix::pow(const ex & expn) const;
1408 matrix matrix::transpose(void) const;
1409 matrix matrix::inverse(void) const;
1412 All of these methods return the result as a new matrix object. Here is an
1413 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1418 matrix A(2, 2, lst(1, 2, 3, 4));
1419 matrix B(2, 2, lst(-1, 0, 2, 1));
1420 matrix C(2, 2, lst(8, 4, 2, 1));
1422 matrix result = A.mul(B).sub(C.mul_scalar(2));
1423 cout << result << endl;
1424 // -> [[-13,-6],[1,2]]
1429 @cindex @code{evalm()}
1430 The second (and probably the most natural) way is to construct an expression
1431 containing matrices with the usual arithmetic operators and @code{pow()}.
1432 For efficiency reasons, expressions with sums, products and powers of
1433 matrices are not automatically evaluated in GiNaC. You have to call the
1437 ex ex::evalm() const;
1440 to obtain the result:
1447 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1448 cout << e.evalm() << endl;
1449 // -> [[-13,-6],[1,2]]
1454 The non-commutativity of the product @code{A*B} in this example is
1455 automatically recognized by GiNaC. There is no need to use a special
1456 operator here. @xref{Non-commutative objects}, for more information about
1457 dealing with non-commutative expressions.
1459 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1460 to perform the arithmetic:
1465 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1466 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1468 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1469 cout << e.simplify_indexed() << endl;
1470 // -> [[-13,-6],[1,2]].i.j
1474 Using indices is most useful when working with rectangular matrices and
1475 one-dimensional vectors because you don't have to worry about having to
1476 transpose matrices before multiplying them. @xref{Indexed objects}, for
1477 more information about using matrices with indices, and about indices in
1480 The @code{matrix} class provides a couple of additional methods for
1481 computing determinants, traces, and characteristic polynomials:
1484 ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
1485 ex matrix::trace(void) const;
1486 ex matrix::charpoly(const symbol & lambda) const;
1489 The @samp{algo} argument of @code{determinant()} allows to select between
1490 different algorithms for calculating the determinant. The possible values
1491 are defined in the @file{flags.h} header file. By default, GiNaC uses a
1492 heuristic to automatically select an algorithm that is likely to give the
1493 result most quickly.
1496 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1497 @c node-name, next, previous, up
1498 @section Indexed objects
1500 GiNaC allows you to handle expressions containing general indexed objects in
1501 arbitrary spaces. It is also able to canonicalize and simplify such
1502 expressions and perform symbolic dummy index summations. There are a number
1503 of predefined indexed objects provided, like delta and metric tensors.
1505 There are few restrictions placed on indexed objects and their indices and
1506 it is easy to construct nonsense expressions, but our intention is to
1507 provide a general framework that allows you to implement algorithms with
1508 indexed quantities, getting in the way as little as possible.
1510 @cindex @code{idx} (class)
1511 @cindex @code{indexed} (class)
1512 @subsection Indexed quantities and their indices
1514 Indexed expressions in GiNaC are constructed of two special types of objects,
1515 @dfn{index objects} and @dfn{indexed objects}.
1519 @cindex contravariant
1522 @item Index objects are of class @code{idx} or a subclass. Every index has
1523 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1524 the index lives in) which can both be arbitrary expressions but are usually
1525 a number or a simple symbol. In addition, indices of class @code{varidx} have
1526 a @dfn{variance} (they can be co- or contravariant), and indices of class
1527 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1529 @item Indexed objects are of class @code{indexed} or a subclass. They
1530 contain a @dfn{base expression} (which is the expression being indexed), and
1531 one or more indices.
1535 @strong{Note:} when printing expressions, covariant indices and indices
1536 without variance are denoted @samp{.i} while contravariant indices are
1537 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1538 value. In the following, we are going to use that notation in the text so
1539 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1540 not visible in the output.
1542 A simple example shall illustrate the concepts:
1546 #include <ginac/ginac.h>
1547 using namespace std;
1548 using namespace GiNaC;
1552 symbol i_sym("i"), j_sym("j");
1553 idx i(i_sym, 3), j(j_sym, 3);
1556 cout << indexed(A, i, j) << endl;
1561 The @code{idx} constructor takes two arguments, the index value and the
1562 index dimension. First we define two index objects, @code{i} and @code{j},
1563 both with the numeric dimension 3. The value of the index @code{i} is the
1564 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1565 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1566 construct an expression containing one indexed object, @samp{A.i.j}. It has
1567 the symbol @code{A} as its base expression and the two indices @code{i} and
1570 Note the difference between the indices @code{i} and @code{j} which are of
1571 class @code{idx}, and the index values which are the symbols @code{i_sym}
1572 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1573 or numbers but must be index objects. For example, the following is not
1574 correct and will raise an exception:
1577 symbol i("i"), j("j");
1578 e = indexed(A, i, j); // ERROR: indices must be of type idx
1581 You can have multiple indexed objects in an expression, index values can
1582 be numeric, and index dimensions symbolic:
1586 symbol B("B"), dim("dim");
1587 cout << 4 * indexed(A, i)
1588 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1593 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1594 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1595 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1596 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1597 @code{simplify_indexed()} for that, see below).
1599 In fact, base expressions, index values and index dimensions can be
1600 arbitrary expressions:
1604 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1609 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1610 get an error message from this but you will probably not be able to do
1611 anything useful with it.
1613 @cindex @code{get_value()}
1614 @cindex @code{get_dimension()}
1618 ex idx::get_value(void);
1619 ex idx::get_dimension(void);
1622 return the value and dimension of an @code{idx} object. If you have an index
1623 in an expression, such as returned by calling @code{.op()} on an indexed
1624 object, you can get a reference to the @code{idx} object with the function
1625 @code{ex_to<idx>()} on the expression.
1627 There are also the methods
1630 bool idx::is_numeric(void);
1631 bool idx::is_symbolic(void);
1632 bool idx::is_dim_numeric(void);
1633 bool idx::is_dim_symbolic(void);
1636 for checking whether the value and dimension are numeric or symbolic
1637 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1638 About Expressions}) returns information about the index value.
1640 @cindex @code{varidx} (class)
1641 If you need co- and contravariant indices, use the @code{varidx} class:
1645 symbol mu_sym("mu"), nu_sym("nu");
1646 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1647 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1649 cout << indexed(A, mu, nu) << endl;
1651 cout << indexed(A, mu_co, nu) << endl;
1653 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1658 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1659 co- or contravariant. The default is a contravariant (upper) index, but
1660 this can be overridden by supplying a third argument to the @code{varidx}
1661 constructor. The two methods
1664 bool varidx::is_covariant(void);
1665 bool varidx::is_contravariant(void);
1668 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1669 to get the object reference from an expression). There's also the very useful
1673 ex varidx::toggle_variance(void);
1676 which makes a new index with the same value and dimension but the opposite
1677 variance. By using it you only have to define the index once.
1679 @cindex @code{spinidx} (class)
1680 The @code{spinidx} class provides dotted and undotted variant indices, as
1681 used in the Weyl-van-der-Waerden spinor formalism:
1685 symbol K("K"), C_sym("C"), D_sym("D");
1686 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1687 // contravariant, undotted
1688 spinidx C_co(C_sym, 2, true); // covariant index
1689 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1690 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1692 cout << indexed(K, C, D) << endl;
1694 cout << indexed(K, C_co, D_dot) << endl;
1696 cout << indexed(K, D_co_dot, D) << endl;
1701 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1702 dotted or undotted. The default is undotted but this can be overridden by
1703 supplying a fourth argument to the @code{spinidx} constructor. The two
1707 bool spinidx::is_dotted(void);
1708 bool spinidx::is_undotted(void);
1711 allow you to check whether or not a @code{spinidx} object is dotted (use
1712 @code{ex_to<spinidx>()} to get the object reference from an expression).
1713 Finally, the two methods
1716 ex spinidx::toggle_dot(void);
1717 ex spinidx::toggle_variance_dot(void);
1720 create a new index with the same value and dimension but opposite dottedness
1721 and the same or opposite variance.
1723 @subsection Substituting indices
1725 @cindex @code{subs()}
1726 Sometimes you will want to substitute one symbolic index with another
1727 symbolic or numeric index, for example when calculating one specific element
1728 of a tensor expression. This is done with the @code{.subs()} method, as it
1729 is done for symbols (see @ref{Substituting Expressions}).
1731 You have two possibilities here. You can either substitute the whole index
1732 by another index or expression:
1736 ex e = indexed(A, mu_co);
1737 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1738 // -> A.mu becomes A~nu
1739 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1740 // -> A.mu becomes A~0
1741 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1742 // -> A.mu becomes A.0
1746 The third example shows that trying to replace an index with something that
1747 is not an index will substitute the index value instead.
1749 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1754 ex e = indexed(A, mu_co);
1755 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1756 // -> A.mu becomes A.nu
1757 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1758 // -> A.mu becomes A.0
1762 As you see, with the second method only the value of the index will get
1763 substituted. Its other properties, including its dimension, remain unchanged.
1764 If you want to change the dimension of an index you have to substitute the
1765 whole index by another one with the new dimension.
1767 Finally, substituting the base expression of an indexed object works as
1772 ex e = indexed(A, mu_co);
1773 cout << e << " becomes " << e.subs(A == A+B) << endl;
1774 // -> A.mu becomes (B+A).mu
1778 @subsection Symmetries
1779 @cindex @code{symmetry} (class)
1780 @cindex @code{sy_none()}
1781 @cindex @code{sy_symm()}
1782 @cindex @code{sy_anti()}
1783 @cindex @code{sy_cycl()}
1785 Indexed objects can have certain symmetry properties with respect to their
1786 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
1787 that is constructed with the helper functions
1790 symmetry sy_none(...);
1791 symmetry sy_symm(...);
1792 symmetry sy_anti(...);
1793 symmetry sy_cycl(...);
1796 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
1797 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
1798 represents a cyclic symmetry. Each of these functions accepts up to four
1799 arguments which can be either symmetry objects themselves or unsigned integer
1800 numbers that represent an index position (counting from 0). A symmetry
1801 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
1802 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
1805 Here are some examples of symmetry definitions:
1810 e = indexed(A, i, j);
1811 e = indexed(A, sy_none(), i, j); // equivalent
1812 e = indexed(A, sy_none(0, 1), i, j); // equivalent
1814 // Symmetric in all three indices:
1815 e = indexed(A, sy_symm(), i, j, k);
1816 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
1817 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
1818 // different canonical order
1820 // Symmetric in the first two indices only:
1821 e = indexed(A, sy_symm(0, 1), i, j, k);
1822 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
1824 // Antisymmetric in the first and last index only (index ranges need not
1826 e = indexed(A, sy_anti(0, 2), i, j, k);
1827 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
1829 // An example of a mixed symmetry: antisymmetric in the first two and
1830 // last two indices, symmetric when swapping the first and last index
1831 // pairs (like the Riemann curvature tensor):
1832 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
1834 // Cyclic symmetry in all three indices:
1835 e = indexed(A, sy_cycl(), i, j, k);
1836 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
1838 // The following examples are invalid constructions that will throw
1839 // an exception at run time.
1841 // An index may not appear multiple times:
1842 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
1843 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
1845 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
1846 // same number of indices:
1847 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
1849 // And of course, you cannot specify indices which are not there:
1850 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
1854 If you need to specify more than four indices, you have to use the
1855 @code{.add()} method of the @code{symmetry} class. For example, to specify
1856 full symmetry in the first six indices you would write
1857 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
1859 If an indexed object has a symmetry, GiNaC will automatically bring the
1860 indices into a canonical order which allows for some immediate simplifications:
1864 cout << indexed(A, sy_symm(), i, j)
1865 + indexed(A, sy_symm(), j, i) << endl;
1867 cout << indexed(B, sy_anti(), i, j)
1868 + indexed(B, sy_anti(), j, i) << endl;
1870 cout << indexed(B, sy_anti(), i, j, k)
1871 + indexed(B, sy_anti(), j, i, k) << endl;
1876 @cindex @code{get_free_indices()}
1878 @subsection Dummy indices
1880 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1881 that a summation over the index range is implied. Symbolic indices which are
1882 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1883 dummy nor free indices.
1885 To be recognized as a dummy index pair, the two indices must be of the same
1886 class and dimension and their value must be the same single symbol (an index
1887 like @samp{2*n+1} is never a dummy index). If the indices are of class
1888 @code{varidx} they must also be of opposite variance; if they are of class
1889 @code{spinidx} they must be both dotted or both undotted.
1891 The method @code{.get_free_indices()} returns a vector containing the free
1892 indices of an expression. It also checks that the free indices of the terms
1893 of a sum are consistent:
1897 symbol A("A"), B("B"), C("C");
1899 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1900 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1902 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1903 cout << exprseq(e.get_free_indices()) << endl;
1905 // 'j' and 'l' are dummy indices
1907 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1908 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1910 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1911 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1912 cout << exprseq(e.get_free_indices()) << endl;
1914 // 'nu' is a dummy index, but 'sigma' is not
1916 e = indexed(A, mu, mu);
1917 cout << exprseq(e.get_free_indices()) << endl;
1919 // 'mu' is not a dummy index because it appears twice with the same
1922 e = indexed(A, mu, nu) + 42;
1923 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1924 // this will throw an exception:
1925 // "add::get_free_indices: inconsistent indices in sum"
1929 @cindex @code{simplify_indexed()}
1930 @subsection Simplifying indexed expressions
1932 In addition to the few automatic simplifications that GiNaC performs on
1933 indexed expressions (such as re-ordering the indices of symmetric tensors
1934 and calculating traces and convolutions of matrices and predefined tensors)
1938 ex ex::simplify_indexed(void);
1939 ex ex::simplify_indexed(const scalar_products & sp);
1942 that performs some more expensive operations:
1945 @item it checks the consistency of free indices in sums in the same way
1946 @code{get_free_indices()} does
1947 @item it tries to give dummy indices that appear in different terms of a sum
1948 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1949 @item it (symbolically) calculates all possible dummy index summations/contractions
1950 with the predefined tensors (this will be explained in more detail in the
1952 @item it detects contractions that vanish for symmetry reasons, for example
1953 the contraction of a symmetric and a totally antisymmetric tensor
1954 @item as a special case of dummy index summation, it can replace scalar products
1955 of two tensors with a user-defined value
1958 The last point is done with the help of the @code{scalar_products} class
1959 which is used to store scalar products with known values (this is not an
1960 arithmetic class, you just pass it to @code{simplify_indexed()}):
1964 symbol A("A"), B("B"), C("C"), i_sym("i");
1968 sp.add(A, B, 0); // A and B are orthogonal
1969 sp.add(A, C, 0); // A and C are orthogonal
1970 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1972 e = indexed(A + B, i) * indexed(A + C, i);
1974 // -> (B+A).i*(A+C).i
1976 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1982 The @code{scalar_products} object @code{sp} acts as a storage for the
1983 scalar products added to it with the @code{.add()} method. This method
1984 takes three arguments: the two expressions of which the scalar product is
1985 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1986 @code{simplify_indexed()} will replace all scalar products of indexed
1987 objects that have the symbols @code{A} and @code{B} as base expressions
1988 with the single value 0. The number, type and dimension of the indices
1989 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1991 @cindex @code{expand()}
1992 The example above also illustrates a feature of the @code{expand()} method:
1993 if passed the @code{expand_indexed} option it will distribute indices
1994 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1996 @cindex @code{tensor} (class)
1997 @subsection Predefined tensors
1999 Some frequently used special tensors such as the delta, epsilon and metric
2000 tensors are predefined in GiNaC. They have special properties when
2001 contracted with other tensor expressions and some of them have constant
2002 matrix representations (they will evaluate to a number when numeric
2003 indices are specified).
2005 @cindex @code{delta_tensor()}
2006 @subsubsection Delta tensor
2008 The delta tensor takes two indices, is symmetric and has the matrix
2009 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2010 @code{delta_tensor()}:
2014 symbol A("A"), B("B");
2016 idx i(symbol("i"), 3), j(symbol("j"), 3),
2017 k(symbol("k"), 3), l(symbol("l"), 3);
2019 ex e = indexed(A, i, j) * indexed(B, k, l)
2020 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2021 cout << e.simplify_indexed() << endl;
2024 cout << delta_tensor(i, i) << endl;
2029 @cindex @code{metric_tensor()}
2030 @subsubsection General metric tensor
2032 The function @code{metric_tensor()} creates a general symmetric metric
2033 tensor with two indices that can be used to raise/lower tensor indices. The
2034 metric tensor is denoted as @samp{g} in the output and if its indices are of
2035 mixed variance it is automatically replaced by a delta tensor:
2041 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2043 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2044 cout << e.simplify_indexed() << endl;
2047 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2048 cout << e.simplify_indexed() << endl;
2051 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2052 * metric_tensor(nu, rho);
2053 cout << e.simplify_indexed() << endl;
2056 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2057 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2058 + indexed(A, mu.toggle_variance(), rho));
2059 cout << e.simplify_indexed() << endl;
2064 @cindex @code{lorentz_g()}
2065 @subsubsection Minkowski metric tensor
2067 The Minkowski metric tensor is a special metric tensor with a constant
2068 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2069 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2070 It is created with the function @code{lorentz_g()} (although it is output as
2075 varidx mu(symbol("mu"), 4);
2077 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2078 * lorentz_g(mu, varidx(0, 4)); // negative signature
2079 cout << e.simplify_indexed() << endl;
2082 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2083 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2084 cout << e.simplify_indexed() << endl;
2089 @cindex @code{spinor_metric()}
2090 @subsubsection Spinor metric tensor
2092 The function @code{spinor_metric()} creates an antisymmetric tensor with
2093 two indices that is used to raise/lower indices of 2-component spinors.
2094 It is output as @samp{eps}:
2100 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2101 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2103 e = spinor_metric(A, B) * indexed(psi, B_co);
2104 cout << e.simplify_indexed() << endl;
2107 e = spinor_metric(A, B) * indexed(psi, A_co);
2108 cout << e.simplify_indexed() << endl;
2111 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2112 cout << e.simplify_indexed() << endl;
2115 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2116 cout << e.simplify_indexed() << endl;
2119 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2120 cout << e.simplify_indexed() << endl;
2123 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2124 cout << e.simplify_indexed() << endl;
2129 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2131 @cindex @code{epsilon_tensor()}
2132 @cindex @code{lorentz_eps()}
2133 @subsubsection Epsilon tensor
2135 The epsilon tensor is totally antisymmetric, its number of indices is equal
2136 to the dimension of the index space (the indices must all be of the same
2137 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2138 defined to be 1. Its behavior with indices that have a variance also
2139 depends on the signature of the metric. Epsilon tensors are output as
2142 There are three functions defined to create epsilon tensors in 2, 3 and 4
2146 ex epsilon_tensor(const ex & i1, const ex & i2);
2147 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2148 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2151 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2152 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2153 Minkowski space (the last @code{bool} argument specifies whether the metric
2154 has negative or positive signature, as in the case of the Minkowski metric
2159 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2160 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2161 e = lorentz_eps(mu, nu, rho, sig) *
2162 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2163 cout << simplify_indexed(e) << endl;
2164 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2166 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2167 symbol A("A"), B("B");
2168 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2169 cout << simplify_indexed(e) << endl;
2170 // -> -B.k*A.j*eps.i.k.j
2171 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2172 cout << simplify_indexed(e) << endl;
2177 @subsection Linear algebra
2179 The @code{matrix} class can be used with indices to do some simple linear
2180 algebra (linear combinations and products of vectors and matrices, traces
2181 and scalar products):
2185 idx i(symbol("i"), 2), j(symbol("j"), 2);
2186 symbol x("x"), y("y");
2188 // A is a 2x2 matrix, X is a 2x1 vector
2189 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
2191 cout << indexed(A, i, i) << endl;
2194 ex e = indexed(A, i, j) * indexed(X, j);
2195 cout << e.simplify_indexed() << endl;
2196 // -> [[2*y+x],[4*y+3*x]].i
2198 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2199 cout << e.simplify_indexed() << endl;
2200 // -> [[3*y+3*x,6*y+2*x]].j
2204 You can of course obtain the same results with the @code{matrix::add()},
2205 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2206 but with indices you don't have to worry about transposing matrices.
2208 Matrix indices always start at 0 and their dimension must match the number
2209 of rows/columns of the matrix. Matrices with one row or one column are
2210 vectors and can have one or two indices (it doesn't matter whether it's a
2211 row or a column vector). Other matrices must have two indices.
2213 You should be careful when using indices with variance on matrices. GiNaC
2214 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2215 @samp{F.mu.nu} are different matrices. In this case you should use only
2216 one form for @samp{F} and explicitly multiply it with a matrix representation
2217 of the metric tensor.
2220 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2221 @c node-name, next, previous, up
2222 @section Non-commutative objects
2224 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2225 non-commutative objects are built-in which are mostly of use in high energy
2229 @item Clifford (Dirac) algebra (class @code{clifford})
2230 @item su(3) Lie algebra (class @code{color})
2231 @item Matrices (unindexed) (class @code{matrix})
2234 The @code{clifford} and @code{color} classes are subclasses of
2235 @code{indexed} because the elements of these algebras usually carry
2236 indices. The @code{matrix} class is described in more detail in
2239 Unlike most computer algebra systems, GiNaC does not primarily provide an
2240 operator (often denoted @samp{&*}) for representing inert products of
2241 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2242 classes of objects involved, and non-commutative products are formed with
2243 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2244 figuring out by itself which objects commute and will group the factors
2245 by their class. Consider this example:
2249 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2250 idx a(symbol("a"), 8), b(symbol("b"), 8);
2251 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2253 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2257 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2258 groups the non-commutative factors (the gammas and the su(3) generators)
2259 together while preserving the order of factors within each class (because
2260 Clifford objects commute with color objects). The resulting expression is a
2261 @emph{commutative} product with two factors that are themselves non-commutative
2262 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2263 parentheses are placed around the non-commutative products in the output.
2265 @cindex @code{ncmul} (class)
2266 Non-commutative products are internally represented by objects of the class
2267 @code{ncmul}, as opposed to commutative products which are handled by the
2268 @code{mul} class. You will normally not have to worry about this distinction,
2271 The advantage of this approach is that you never have to worry about using
2272 (or forgetting to use) a special operator when constructing non-commutative
2273 expressions. Also, non-commutative products in GiNaC are more intelligent
2274 than in other computer algebra systems; they can, for example, automatically
2275 canonicalize themselves according to rules specified in the implementation
2276 of the non-commutative classes. The drawback is that to work with other than
2277 the built-in algebras you have to implement new classes yourself. Symbols
2278 always commute and it's not possible to construct non-commutative products
2279 using symbols to represent the algebra elements or generators. User-defined
2280 functions can, however, be specified as being non-commutative.
2282 @cindex @code{return_type()}
2283 @cindex @code{return_type_tinfo()}
2284 Information about the commutativity of an object or expression can be
2285 obtained with the two member functions
2288 unsigned ex::return_type(void) const;
2289 unsigned ex::return_type_tinfo(void) const;
2292 The @code{return_type()} function returns one of three values (defined in
2293 the header file @file{flags.h}), corresponding to three categories of
2294 expressions in GiNaC:
2297 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2298 classes are of this kind.
2299 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2300 certain class of non-commutative objects which can be determined with the
2301 @code{return_type_tinfo()} method. Expressions of this category commute
2302 with everything except @code{noncommutative} expressions of the same
2304 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2305 of non-commutative objects of different classes. Expressions of this
2306 category don't commute with any other @code{noncommutative} or
2307 @code{noncommutative_composite} expressions.
2310 The value returned by the @code{return_type_tinfo()} method is valid only
2311 when the return type of the expression is @code{noncommutative}. It is a
2312 value that is unique to the class of the object and usually one of the
2313 constants in @file{tinfos.h}, or derived therefrom.
2315 Here are a couple of examples:
2318 @multitable @columnfractions 0.33 0.33 0.34
2319 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2320 @item @code{42} @tab @code{commutative} @tab -
2321 @item @code{2*x-y} @tab @code{commutative} @tab -
2322 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2323 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2324 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2325 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2329 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2330 @code{TINFO_clifford} for objects with a representation label of zero.
2331 Other representation labels yield a different @code{return_type_tinfo()},
2332 but it's the same for any two objects with the same label. This is also true
2335 A last note: With the exception of matrices, positive integer powers of
2336 non-commutative objects are automatically expanded in GiNaC. For example,
2337 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2338 non-commutative expressions).
2341 @cindex @code{clifford} (class)
2342 @subsection Clifford algebra
2344 @cindex @code{dirac_gamma()}
2345 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2346 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2347 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2348 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2351 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2354 which takes two arguments: the index and a @dfn{representation label} in the
2355 range 0 to 255 which is used to distinguish elements of different Clifford
2356 algebras (this is also called a @dfn{spin line index}). Gammas with different
2357 labels commute with each other. The dimension of the index can be 4 or (in
2358 the framework of dimensional regularization) any symbolic value. Spinor
2359 indices on Dirac gammas are not supported in GiNaC.
2361 @cindex @code{dirac_ONE()}
2362 The unity element of a Clifford algebra is constructed by
2365 ex dirac_ONE(unsigned char rl = 0);
2368 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2369 multiples of the unity element, even though it's customary to omit it.
2370 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2371 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2372 GiNaC may produce incorrect results.
2374 @cindex @code{dirac_gamma5()}
2375 There's a special element @samp{gamma5} that commutes with all other
2376 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2380 ex dirac_gamma5(unsigned char rl = 0);
2383 @cindex @code{dirac_gamma6()}
2384 @cindex @code{dirac_gamma7()}
2385 The two additional functions
2388 ex dirac_gamma6(unsigned char rl = 0);
2389 ex dirac_gamma7(unsigned char rl = 0);
2392 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2395 @cindex @code{dirac_slash()}
2396 Finally, the function
2399 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2402 creates a term that represents a contraction of @samp{e} with the Dirac
2403 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2404 with a unique index whose dimension is given by the @code{dim} argument).
2405 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2407 In products of dirac gammas, superfluous unity elements are automatically
2408 removed, squares are replaced by their values and @samp{gamma5} is
2409 anticommuted to the front. The @code{simplify_indexed()} function performs
2410 contractions in gamma strings, for example
2415 symbol a("a"), b("b"), D("D");
2416 varidx mu(symbol("mu"), D);
2417 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2418 * dirac_gamma(mu.toggle_variance());
2420 // -> gamma~mu*a\*gamma.mu
2421 e = e.simplify_indexed();
2424 cout << e.subs(D == 4) << endl;
2430 @cindex @code{dirac_trace()}
2431 To calculate the trace of an expression containing strings of Dirac gammas
2432 you use the function
2435 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2438 This function takes the trace of all gammas with the specified representation
2439 label; gammas with other labels are left standing. The last argument to
2440 @code{dirac_trace()} is the value to be returned for the trace of the unity
2441 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2442 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2443 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2444 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2445 This @samp{gamma5} scheme is described in greater detail in
2446 @cite{The Role of gamma5 in Dimensional Regularization}.
2448 The value of the trace itself is also usually different in 4 and in
2449 @math{D != 4} dimensions:
2454 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2455 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2456 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2457 cout << dirac_trace(e).simplify_indexed() << endl;
2464 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2465 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2466 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2467 cout << dirac_trace(e).simplify_indexed() << endl;
2468 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2472 Here is an example for using @code{dirac_trace()} to compute a value that
2473 appears in the calculation of the one-loop vacuum polarization amplitude in
2478 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2479 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2482 sp.add(l, l, pow(l, 2));
2483 sp.add(l, q, ldotq);
2485 ex e = dirac_gamma(mu) *
2486 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2487 dirac_gamma(mu.toggle_variance()) *
2488 (dirac_slash(l, D) + m * dirac_ONE());
2489 e = dirac_trace(e).simplify_indexed(sp);
2490 e = e.collect(lst(l, ldotq, m));
2492 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2496 The @code{canonicalize_clifford()} function reorders all gamma products that
2497 appear in an expression to a canonical (but not necessarily simple) form.
2498 You can use this to compare two expressions or for further simplifications:
2502 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2503 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2505 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2507 e = canonicalize_clifford(e);
2514 @cindex @code{color} (class)
2515 @subsection Color algebra
2517 @cindex @code{color_T()}
2518 For computations in quantum chromodynamics, GiNaC implements the base elements
2519 and structure constants of the su(3) Lie algebra (color algebra). The base
2520 elements @math{T_a} are constructed by the function
2523 ex color_T(const ex & a, unsigned char rl = 0);
2526 which takes two arguments: the index and a @dfn{representation label} in the
2527 range 0 to 255 which is used to distinguish elements of different color
2528 algebras. Objects with different labels commute with each other. The
2529 dimension of the index must be exactly 8 and it should be of class @code{idx},
2532 @cindex @code{color_ONE()}
2533 The unity element of a color algebra is constructed by
2536 ex color_ONE(unsigned char rl = 0);
2539 @strong{Note:} You must always use @code{color_ONE()} when referring to
2540 multiples of the unity element, even though it's customary to omit it.
2541 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2542 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2543 GiNaC may produce incorrect results.
2545 @cindex @code{color_d()}
2546 @cindex @code{color_f()}
2550 ex color_d(const ex & a, const ex & b, const ex & c);
2551 ex color_f(const ex & a, const ex & b, const ex & c);
2554 create the symmetric and antisymmetric structure constants @math{d_abc} and
2555 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2556 and @math{[T_a, T_b] = i f_abc T_c}.
2558 @cindex @code{color_h()}
2559 There's an additional function
2562 ex color_h(const ex & a, const ex & b, const ex & c);
2565 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2567 The function @code{simplify_indexed()} performs some simplifications on
2568 expressions containing color objects:
2573 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2574 k(symbol("k"), 8), l(symbol("l"), 8);
2576 e = color_d(a, b, l) * color_f(a, b, k);
2577 cout << e.simplify_indexed() << endl;
2580 e = color_d(a, b, l) * color_d(a, b, k);
2581 cout << e.simplify_indexed() << endl;
2584 e = color_f(l, a, b) * color_f(a, b, k);
2585 cout << e.simplify_indexed() << endl;
2588 e = color_h(a, b, c) * color_h(a, b, c);
2589 cout << e.simplify_indexed() << endl;
2592 e = color_h(a, b, c) * color_T(b) * color_T(c);
2593 cout << e.simplify_indexed() << endl;
2596 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2597 cout << e.simplify_indexed() << endl;
2600 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2601 cout << e.simplify_indexed() << endl;
2602 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2606 @cindex @code{color_trace()}
2607 To calculate the trace of an expression containing color objects you use the
2611 ex color_trace(const ex & e, unsigned char rl = 0);
2614 This function takes the trace of all color @samp{T} objects with the
2615 specified representation label; @samp{T}s with other labels are left
2616 standing. For example:
2620 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2622 // -> -I*f.a.c.b+d.a.c.b
2627 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2628 @c node-name, next, previous, up
2629 @chapter Methods and Functions
2632 In this chapter the most important algorithms provided by GiNaC will be
2633 described. Some of them are implemented as functions on expressions,
2634 others are implemented as methods provided by expression objects. If
2635 they are methods, there exists a wrapper function around it, so you can
2636 alternatively call it in a functional way as shown in the simple
2641 cout << "As method: " << sin(1).evalf() << endl;
2642 cout << "As function: " << evalf(sin(1)) << endl;
2646 @cindex @code{subs()}
2647 The general rule is that wherever methods accept one or more parameters
2648 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2649 wrapper accepts is the same but preceded by the object to act on
2650 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2651 most natural one in an OO model but it may lead to confusion for MapleV
2652 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2653 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2654 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2655 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2656 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2657 here. Also, users of MuPAD will in most cases feel more comfortable
2658 with GiNaC's convention. All function wrappers are implemented
2659 as simple inline functions which just call the corresponding method and
2660 are only provided for users uncomfortable with OO who are dead set to
2661 avoid method invocations. Generally, nested function wrappers are much
2662 harder to read than a sequence of methods and should therefore be
2663 avoided if possible. On the other hand, not everything in GiNaC is a
2664 method on class @code{ex} and sometimes calling a function cannot be
2668 * Information About Expressions::
2669 * Substituting Expressions::
2670 * Pattern Matching and Advanced Substitutions::
2671 * Applying a Function on Subexpressions::
2672 * Polynomial Arithmetic:: Working with polynomials.
2673 * Rational Expressions:: Working with rational functions.
2674 * Symbolic Differentiation::
2675 * Series Expansion:: Taylor and Laurent expansion.
2677 * Built-in Functions:: List of predefined mathematical functions.
2678 * Input/Output:: Input and output of expressions.
2682 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2683 @c node-name, next, previous, up
2684 @section Getting information about expressions
2686 @subsection Checking expression types
2687 @cindex @code{is_a<@dots{}>()}
2688 @cindex @code{is_exactly_a<@dots{}>()}
2689 @cindex @code{ex_to<@dots{}>()}
2690 @cindex Converting @code{ex} to other classes
2691 @cindex @code{info()}
2692 @cindex @code{return_type()}
2693 @cindex @code{return_type_tinfo()}
2695 Sometimes it's useful to check whether a given expression is a plain number,
2696 a sum, a polynomial with integer coefficients, or of some other specific type.
2697 GiNaC provides a couple of functions for this:
2700 bool is_a<T>(const ex & e);
2701 bool is_exactly_a<T>(const ex & e);
2702 bool ex::info(unsigned flag);
2703 unsigned ex::return_type(void) const;
2704 unsigned ex::return_type_tinfo(void) const;
2707 When the test made by @code{is_a<T>()} returns true, it is safe to call
2708 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
2709 class names (@xref{The Class Hierarchy}, for a list of all classes). For
2710 example, assuming @code{e} is an @code{ex}:
2715 if (is_a<numeric>(e))
2716 numeric n = ex_to<numeric>(e);
2721 @code{is_a<T>(e)} allows you to check whether the top-level object of
2722 an expression @samp{e} is an instance of the GiNaC class @samp{T}
2723 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2724 e.g., for checking whether an expression is a number, a sum, or a product:
2731 is_a<numeric>(e1); // true
2732 is_a<numeric>(e2); // false
2733 is_a<add>(e1); // false
2734 is_a<add>(e2); // true
2735 is_a<mul>(e1); // false
2736 is_a<mul>(e2); // false
2740 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
2741 top-level object of an expression @samp{e} is an instance of the GiNaC
2742 class @samp{T}, not including parent classes.
2744 The @code{info()} method is used for checking certain attributes of
2745 expressions. The possible values for the @code{flag} argument are defined
2746 in @file{ginac/flags.h}, the most important being explained in the following
2750 @multitable @columnfractions .30 .70
2751 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2752 @item @code{numeric}
2753 @tab @dots{}a number (same as @code{is_<numeric>(...)})
2755 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2756 @item @code{rational}
2757 @tab @dots{}an exact rational number (integers are rational, too)
2758 @item @code{integer}
2759 @tab @dots{}a (non-complex) integer
2760 @item @code{crational}
2761 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2762 @item @code{cinteger}
2763 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2764 @item @code{positive}
2765 @tab @dots{}not complex and greater than 0
2766 @item @code{negative}
2767 @tab @dots{}not complex and less than 0
2768 @item @code{nonnegative}
2769 @tab @dots{}not complex and greater than or equal to 0
2771 @tab @dots{}an integer greater than 0
2773 @tab @dots{}an integer less than 0
2774 @item @code{nonnegint}
2775 @tab @dots{}an integer greater than or equal to 0
2777 @tab @dots{}an even integer
2779 @tab @dots{}an odd integer
2781 @tab @dots{}a prime integer (probabilistic primality test)
2782 @item @code{relation}
2783 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
2784 @item @code{relation_equal}
2785 @tab @dots{}a @code{==} relation
2786 @item @code{relation_not_equal}
2787 @tab @dots{}a @code{!=} relation
2788 @item @code{relation_less}
2789 @tab @dots{}a @code{<} relation
2790 @item @code{relation_less_or_equal}
2791 @tab @dots{}a @code{<=} relation
2792 @item @code{relation_greater}
2793 @tab @dots{}a @code{>} relation
2794 @item @code{relation_greater_or_equal}
2795 @tab @dots{}a @code{>=} relation
2797 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
2799 @tab @dots{}a list (same as @code{is_a<lst>(...)})
2800 @item @code{polynomial}
2801 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2802 @item @code{integer_polynomial}
2803 @tab @dots{}a polynomial with (non-complex) integer coefficients
2804 @item @code{cinteger_polynomial}
2805 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2806 @item @code{rational_polynomial}
2807 @tab @dots{}a polynomial with (non-complex) rational coefficients
2808 @item @code{crational_polynomial}
2809 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2810 @item @code{rational_function}
2811 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2812 @item @code{algebraic}
2813 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2817 To determine whether an expression is commutative or non-commutative and if
2818 so, with which other expressions it would commute, you use the methods
2819 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2820 for an explanation of these.
2823 @subsection Accessing subexpressions
2824 @cindex @code{nops()}
2827 @cindex @code{relational} (class)
2829 GiNaC provides the two methods
2832 unsigned ex::nops();
2833 ex ex::op(unsigned i);
2836 for accessing the subexpressions in the container-like GiNaC classes like
2837 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2838 determines the number of subexpressions (@samp{operands}) contained, while
2839 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2840 In the case of a @code{power} object, @code{op(0)} will return the basis
2841 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2842 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2844 The left-hand and right-hand side expressions of objects of class
2845 @code{relational} (and only of these) can also be accessed with the methods
2853 @subsection Comparing expressions
2854 @cindex @code{is_equal()}
2855 @cindex @code{is_zero()}
2857 Expressions can be compared with the usual C++ relational operators like
2858 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2859 the result is usually not determinable and the result will be @code{false},
2860 except in the case of the @code{!=} operator. You should also be aware that
2861 GiNaC will only do the most trivial test for equality (subtracting both
2862 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2865 Actually, if you construct an expression like @code{a == b}, this will be
2866 represented by an object of the @code{relational} class (@pxref{Relations})
2867 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
2869 There are also two methods
2872 bool ex::is_equal(const ex & other);
2876 for checking whether one expression is equal to another, or equal to zero,
2879 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2880 GiNaC header files. This method is however only to be used internally by
2881 GiNaC to establish a canonical sort order for terms, and using it to compare
2882 expressions will give very surprising results.
2885 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2886 @c node-name, next, previous, up
2887 @section Substituting expressions
2888 @cindex @code{subs()}
2890 Algebraic objects inside expressions can be replaced with arbitrary
2891 expressions via the @code{.subs()} method:
2894 ex ex::subs(const ex & e);
2895 ex ex::subs(const lst & syms, const lst & repls);
2898 In the first form, @code{subs()} accepts a relational of the form
2899 @samp{object == expression} or a @code{lst} of such relationals:
2903 symbol x("x"), y("y");
2905 ex e1 = 2*x^2-4*x+3;
2906 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2910 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2915 If you specify multiple substitutions, they are performed in parallel, so e.g.
2916 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2918 The second form of @code{subs()} takes two lists, one for the objects to be
2919 replaced and one for the expressions to be substituted (both lists must
2920 contain the same number of elements). Using this form, you would write
2921 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2923 @code{subs()} performs syntactic substitution of any complete algebraic
2924 object; it does not try to match sub-expressions as is demonstrated by the
2929 symbol x("x"), y("y"), z("z");
2931 ex e1 = pow(x+y, 2);
2932 cout << e1.subs(x+y == 4) << endl;
2935 ex e2 = sin(x)*sin(y)*cos(x);
2936 cout << e2.subs(sin(x) == cos(x)) << endl;
2937 // -> cos(x)^2*sin(y)
2940 cout << e3.subs(x+y == 4) << endl;
2942 // (and not 4+z as one might expect)
2946 A more powerful form of substitution using wildcards is described in the
2950 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
2951 @c node-name, next, previous, up
2952 @section Pattern matching and advanced substitutions
2953 @cindex @code{wildcard} (class)
2954 @cindex Pattern matching
2956 GiNaC allows the use of patterns for checking whether an expression is of a
2957 certain form or contains subexpressions of a certain form, and for
2958 substituting expressions in a more general way.
2960 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2961 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2962 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2963 an unsigned integer number to allow having multiple different wildcards in a
2964 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2965 are specified in @command{ginsh}). In C++ code, wildcard objects are created
2969 ex wild(unsigned label = 0);
2972 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2975 Some examples for patterns:
2977 @multitable @columnfractions .5 .5
2978 @item @strong{Constructed as} @tab @strong{Output as}
2979 @item @code{wild()} @tab @samp{$0}
2980 @item @code{pow(x,wild())} @tab @samp{x^$0}
2981 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2982 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2988 @item Wildcards behave like symbols and are subject to the same algebraic
2989 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2990 @item As shown in the last example, to use wildcards for indices you have to
2991 use them as the value of an @code{idx} object. This is because indices must
2992 always be of class @code{idx} (or a subclass).
2993 @item Wildcards only represent expressions or subexpressions. It is not
2994 possible to use them as placeholders for other properties like index
2995 dimension or variance, representation labels, symmetry of indexed objects
2997 @item Because wildcards are commutative, it is not possible to use wildcards
2998 as part of noncommutative products.
2999 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3000 are also valid patterns.
3003 @cindex @code{match()}
3004 The most basic application of patterns is to check whether an expression
3005 matches a given pattern. This is done by the function
3008 bool ex::match(const ex & pattern);
3009 bool ex::match(const ex & pattern, lst & repls);
3012 This function returns @code{true} when the expression matches the pattern
3013 and @code{false} if it doesn't. If used in the second form, the actual
3014 subexpressions matched by the wildcards get returned in the @code{repls}
3015 object as a list of relations of the form @samp{wildcard == expression}.
3016 If @code{match()} returns false, the state of @code{repls} is undefined.
3017 For reproducible results, the list should be empty when passed to
3018 @code{match()}, but it is also possible to find similarities in multiple
3019 expressions by passing in the result of a previous match.
3021 The matching algorithm works as follows:
3024 @item A single wildcard matches any expression. If one wildcard appears
3025 multiple times in a pattern, it must match the same expression in all
3026 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3027 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3028 @item If the expression is not of the same class as the pattern, the match
3029 fails (i.e. a sum only matches a sum, a function only matches a function,
3031 @item If the pattern is a function, it only matches the same function
3032 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3033 @item Except for sums and products, the match fails if the number of
3034 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3036 @item If there are no subexpressions, the expressions and the pattern must
3037 be equal (in the sense of @code{is_equal()}).
3038 @item Except for sums and products, each subexpression (@code{op()}) must
3039 match the corresponding subexpression of the pattern.
3042 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3043 account for their commutativity and associativity:
3046 @item If the pattern contains a term or factor that is a single wildcard,
3047 this one is used as the @dfn{global wildcard}. If there is more than one
3048 such wildcard, one of them is chosen as the global wildcard in a random
3050 @item Every term/factor of the pattern, except the global wildcard, is
3051 matched against every term of the expression in sequence. If no match is
3052 found, the whole match fails. Terms that did match are not considered in
3054 @item If there are no unmatched terms left, the match succeeds. Otherwise
3055 the match fails unless there is a global wildcard in the pattern, in
3056 which case this wildcard matches the remaining terms.
3059 In general, having more than one single wildcard as a term of a sum or a
3060 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3063 Here are some examples in @command{ginsh} to demonstrate how it works (the
3064 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3065 match fails, and the list of wildcard replacements otherwise):
3068 > match((x+y)^a,(x+y)^a);
3070 > match((x+y)^a,(x+y)^b);
3072 > match((x+y)^a,$1^$2);
3074 > match((x+y)^a,$1^$1);
3076 > match((x+y)^(x+y),$1^$1);
3078 > match((x+y)^(x+y),$1^$2);
3080 > match((a+b)*(a+c),($1+b)*($1+c));
3082 > match((a+b)*(a+c),(a+$1)*(a+$2));
3084 (Unpredictable. The result might also be [$1==c,$2==b].)
3085 > match((a+b)*(a+c),($1+$2)*($1+$3));
3086 (The result is undefined. Due to the sequential nature of the algorithm
3087 and the re-ordering of terms in GiNaC, the match for the first factor
3088 may be @{$1==a,$2==b@} in which case the match for the second factor
3089 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3091 > match(a*(x+y)+a*z+b,a*$1+$2);
3092 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3093 @{$1=x+y,$2=a*z+b@}.)
3094 > match(a+b+c+d+e+f,c);
3096 > match(a+b+c+d+e+f,c+$0);
3098 > match(a+b+c+d+e+f,c+e+$0);
3100 > match(a+b,a+b+$0);
3102 > match(a*b^2,a^$1*b^$2);
3104 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3105 even though a==a^1.)
3106 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3108 > match(atan2(y,x^2),atan2(y,$0));
3112 @cindex @code{has()}
3113 A more general way to look for patterns in expressions is provided by the
3117 bool ex::has(const ex & pattern);
3120 This function checks whether a pattern is matched by an expression itself or
3121 by any of its subexpressions.
3123 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3124 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3127 > has(x*sin(x+y+2*a),y);
3129 > has(x*sin(x+y+2*a),x+y);
3131 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3132 has the subexpressions "x", "y" and "2*a".)
3133 > has(x*sin(x+y+2*a),x+y+$1);
3135 (But this is possible.)
3136 > has(x*sin(2*(x+y)+2*a),x+y);
3138 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3139 which "x+y" is not a subexpression.)
3142 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3144 > has(4*x^2-x+3,$1*x);
3146 > has(4*x^2+x+3,$1*x);
3148 (Another possible pitfall. The first expression matches because the term
3149 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3150 contains a linear term you should use the coeff() function instead.)
3153 @cindex @code{find()}
3157 bool ex::find(const ex & pattern, lst & found);
3160 works a bit like @code{has()} but it doesn't stop upon finding the first
3161 match. Instead, it appends all found matches to the specified list. If there
3162 are multiple occurrences of the same expression, it is entered only once to
3163 the list. @code{find()} returns false if no matches were found (in
3164 @command{ginsh}, it returns an empty list):
3167 > find(1+x+x^2+x^3,x);
3169 > find(1+x+x^2+x^3,y);
3171 > find(1+x+x^2+x^3,x^$1);
3173 (Note the absence of "x".)
3174 > expand((sin(x)+sin(y))*(a+b));
3175 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3180 @cindex @code{subs()}
3181 Probably the most useful application of patterns is to use them for
3182 substituting expressions with the @code{subs()} method. Wildcards can be
3183 used in the search patterns as well as in the replacement expressions, where
3184 they get replaced by the expressions matched by them. @code{subs()} doesn't
3185 know anything about algebra; it performs purely syntactic substitutions.
3190 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3192 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3194 > subs((a+b+c)^2,a+b==x);
3196 > subs((a+b+c)^2,a+b+$1==x+$1);
3198 > subs(a+2*b,a+b==x);
3200 > subs(4*x^3-2*x^2+5*x-1,x==a);
3202 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3204 > subs(sin(1+sin(x)),sin($1)==cos($1));
3206 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3210 The last example would be written in C++ in this way:
3214 symbol a("a"), b("b"), x("x"), y("y");
3215 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3216 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3217 cout << e.expand() << endl;
3223 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3224 @c node-name, next, previous, up
3225 @section Applying a Function on Subexpressions
3226 @cindex Tree traversal
3227 @cindex @code{map()}
3229 Sometimes you may want to perform an operation on specific parts of an
3230 expression while leaving the general structure of it intact. An example
3231 of this would be a matrix trace operation: the trace of a sum is the sum
3232 of the traces of the individual terms. That is, the trace should @dfn{map}
3233 on the sum, by applying itself to each of the sum's operands. It is possible
3234 to do this manually which usually results in code like this:
3239 if (is_a<matrix>(e))
3240 return ex_to<matrix>(e).trace();
3241 else if (is_a<add>(e)) @{
3243 for (unsigned i=0; i<e.nops(); i++)
3244 sum += calc_trace(e.op(i));
3246 @} else if (is_a<mul>)(e)) @{
3254 This is, however, slightly inefficient (if the sum is very large it can take
3255 a long time to add the terms one-by-one), and its applicability is limited to
3256 a rather small class of expressions. If @code{calc_trace()} is called with
3257 a relation or a list as its argument, you will probably want the trace to
3258 be taken on both sides of the relation or of all elements of the list.
3260 GiNaC offers the @code{map()} method to aid in the implementation of such
3264 ex ex::map(map_function & f) const;
3265 ex ex::map(ex (*f)(const ex & e)) const;
3268 In the first (preferred) form, @code{map()} takes a function object that
3269 is subclassed from the @code{map_function} class. In the second form, it
3270 takes a pointer to a function that accepts and returns an expression.
3271 @code{map()} constructs a new expression of the same type, applying the
3272 specified function on all subexpressions (in the sense of @code{op()}),
3275 The use of a function object makes it possible to supply more arguments to
3276 the function that is being mapped, or to keep local state information.
3277 The @code{map_function} class declares a virtual function call operator
3278 that you can overload. Here is a sample implementation of @code{calc_trace()}
3279 that uses @code{map()} in a recursive fashion:
3282 struct calc_trace : public map_function @{
3283 ex operator()(const ex &e)
3285 if (is_a<matrix>(e))
3286 return ex_to<matrix>(e).trace();
3287 else if (is_a<mul>(e)) @{
3290 return e.map(*this);
3295 This function object could then be used like this:
3299 ex M = ... // expression with matrices
3300 calc_trace do_trace;
3301 ex tr = do_trace(M);
3305 Here is another example for you to meditate over. It removes quadratic
3306 terms in a variable from an expanded polynomial:
3309 struct map_rem_quad : public map_function @{
3311 map_rem_quad(const ex & var_) : var(var_) @{@}
3313 ex operator()(const ex & e)
3315 if (is_a<add>(e) || is_a<mul>(e))
3316 return e.map(*this);
3317 else if (is_a<power>(e) &&
3318 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3328 symbol x("x"), y("y");
3331 for (int i=0; i<8; i++)
3332 e += pow(x, i) * pow(y, 8-i) * (i+1);
3334 // -> 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
3336 map_rem_quad rem_quad(x);
3337 cout << rem_quad(e) << endl;
3338 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3342 @command{ginsh} offers a slightly different implementation of @code{map()}
3343 that allows applying algebraic functions to operands. The second argument
3344 to @code{map()} is an expression containing the wildcard @samp{$0} which
3345 acts as the placeholder for the operands:
3350 > map(a+2*b,sin($0));
3352 > map(@{a,b,c@},$0^2+$0);
3353 @{a^2+a,b^2+b,c^2+c@}
3356 Note that it is only possible to use algebraic functions in the second
3357 argument. You can not use functions like @samp{diff()}, @samp{op()},
3358 @samp{subs()} etc. because these are evaluated immediately:
3361 > map(@{a,b,c@},diff($0,a));
3363 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3364 to "map(@{a,b,c@},0)".
3368 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3369 @c node-name, next, previous, up
3370 @section Polynomial arithmetic
3372 @subsection Expanding and collecting
3373 @cindex @code{expand()}
3374 @cindex @code{collect()}
3376 A polynomial in one or more variables has many equivalent
3377 representations. Some useful ones serve a specific purpose. Consider
3378 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3379 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3380 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3381 representations are the recursive ones where one collects for exponents
3382 in one of the three variable. Since the factors are themselves
3383 polynomials in the remaining two variables the procedure can be
3384 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3385 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3388 To bring an expression into expanded form, its method
3394 may be called. In our example above, this corresponds to @math{4*x*y +
3395 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3396 GiNaC is not easily guessable you should be prepared to see different
3397 orderings of terms in such sums!
3399 Another useful representation of multivariate polynomials is as a
3400 univariate polynomial in one of the variables with the coefficients
3401 being polynomials in the remaining variables. The method
3402 @code{collect()} accomplishes this task:
3405 ex ex::collect(const ex & s, bool distributed = false);
3408 The first argument to @code{collect()} can also be a list of objects in which
3409 case the result is either a recursively collected polynomial, or a polynomial
3410 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3411 by the @code{distributed} flag.
3413 Note that the original polynomial needs to be in expanded form (for the
3414 variables concerned) in order for @code{collect()} to be able to find the
3415 coefficients properly.
3417 The following @command{ginsh} transcript shows an application of @code{collect()}
3418 together with @code{find()}:
3421 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3422 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
3423 > collect(a,@{p,q@});
3424 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)
3425 > collect(a,find(a,sin($1)));
3426 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3427 > collect(a,@{find(a,sin($1)),p,q@});
3428 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3429 > collect(a,@{find(a,sin($1)),d@});
3430 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3433 @subsection Degree and coefficients
3434 @cindex @code{degree()}
3435 @cindex @code{ldegree()}
3436 @cindex @code{coeff()}
3438 The degree and low degree of a polynomial can be obtained using the two
3442 int ex::degree(const ex & s);
3443 int ex::ldegree(const ex & s);
3446 These functions only work reliably if the input polynomial is collected in
3447 terms of the object @samp{s}. Otherwise, they are only guaranteed to return
3448 the upper/lower bounds of the exponents. If you need accurate results, you
3449 have to call @code{expand()} and/or @code{collect()} on the input polynomial.
3457 > degree(expand(a),x);
3461 @code{degree()} also works on rational functions, returning the asymptotic
3465 > degree((x+1)/(x^3+1),x);
3469 If the input is not a polynomial or rational function in the variable @samp{s},
3470 the behavior of @code{degree()} and @code{ldegree()} is undefined.
3472 To extract a coefficient with a certain power from an expanded
3476 ex ex::coeff(const ex & s, int n);
3479 You can also obtain the leading and trailing coefficients with the methods
3482 ex ex::lcoeff(const ex & s);
3483 ex ex::tcoeff(const ex & s);
3486 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3489 An application is illustrated in the next example, where a multivariate
3490 polynomial is analyzed:
3494 symbol x("x"), y("y");
3495 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3496 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3497 ex Poly = PolyInp.expand();
3499 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3500 cout << "The x^" << i << "-coefficient is "
3501 << Poly.coeff(x,i) << endl;
3503 cout << "As polynomial in y: "
3504 << Poly.collect(y) << endl;
3508 When run, it returns an output in the following fashion:
3511 The x^0-coefficient is y^2+11*y
3512 The x^1-coefficient is 5*y^2-2*y
3513 The x^2-coefficient is -1
3514 The x^3-coefficient is 4*y
3515 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3518 As always, the exact output may vary between different versions of GiNaC
3519 or even from run to run since the internal canonical ordering is not
3520 within the user's sphere of influence.
3522 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3523 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3524 with non-polynomial expressions as they not only work with symbols but with
3525 constants, functions and indexed objects as well:
3529 symbol a("a"), b("b"), c("c");
3530 idx i(symbol("i"), 3);
3532 ex e = pow(sin(x) - cos(x), 4);
3533 cout << e.degree(cos(x)) << endl;
3535 cout << e.expand().coeff(sin(x), 3) << endl;
3538 e = indexed(a+b, i) * indexed(b+c, i);
3539 e = e.expand(expand_options::expand_indexed);
3540 cout << e.collect(indexed(b, i)) << endl;
3541 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
3546 @subsection Polynomial division
3547 @cindex polynomial division
3550 @cindex pseudo-remainder
3551 @cindex @code{quo()}
3552 @cindex @code{rem()}
3553 @cindex @code{prem()}
3554 @cindex @code{divide()}
3559 ex quo(const ex & a, const ex & b, const symbol & x);
3560 ex rem(const ex & a, const ex & b, const symbol & x);
3563 compute the quotient and remainder of univariate polynomials in the variable
3564 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
3566 The additional function
3569 ex prem(const ex & a, const ex & b, const symbol & x);
3572 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
3573 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
3575 Exact division of multivariate polynomials is performed by the function
3578 bool divide(const ex & a, const ex & b, ex & q);
3581 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
3582 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
3583 in which case the value of @code{q} is undefined.
3586 @subsection Unit, content and primitive part
3587 @cindex @code{unit()}
3588 @cindex @code{content()}
3589 @cindex @code{primpart()}
3594 ex ex::unit(const symbol & x);
3595 ex ex::content(const symbol & x);
3596 ex ex::primpart(const symbol & x);
3599 return the unit part, content part, and primitive polynomial of a multivariate
3600 polynomial with respect to the variable @samp{x} (the unit part being the sign
3601 of the leading coefficient, the content part being the GCD of the coefficients,
3602 and the primitive polynomial being the input polynomial divided by the unit and
3603 content parts). The product of unit, content, and primitive part is the
3604 original polynomial.
3607 @subsection GCD and LCM
3610 @cindex @code{gcd()}
3611 @cindex @code{lcm()}
3613 The functions for polynomial greatest common divisor and least common
3614 multiple have the synopsis
3617 ex gcd(const ex & a, const ex & b);
3618 ex lcm(const ex & a, const ex & b);
3621 The functions @code{gcd()} and @code{lcm()} accept two expressions
3622 @code{a} and @code{b} as arguments and return a new expression, their
3623 greatest common divisor or least common multiple, respectively. If the
3624 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3625 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3628 #include <ginac/ginac.h>
3629 using namespace GiNaC;
3633 symbol x("x"), y("y"), z("z");
3634 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3635 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3637 ex P_gcd = gcd(P_a, P_b);
3639 ex P_lcm = lcm(P_a, P_b);
3640 // 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
3645 @subsection Square-free decomposition
3646 @cindex square-free decomposition
3647 @cindex factorization
3648 @cindex @code{sqrfree()}
3650 GiNaC still lacks proper factorization support. Some form of
3651 factorization is, however, easily implemented by noting that factors
3652 appearing in a polynomial with power two or more also appear in the
3653 derivative and hence can easily be found by computing the GCD of the
3654 original polynomial and its derivatives. Any decent system has an
3655 interface for this so called square-free factorization. So we provide
3658 ex sqrfree(const ex & a, const lst & l = lst());
3660 Here is an example that by the way illustrates how the exact form of the
3661 result may slightly depend on the order of differentiation, calling for
3662 some care with subsequent processing of the result:
3665 symbol x("x"), y("y");
3666 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
3668 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3669 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
3671 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3672 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
3674 cout << sqrfree(BiVarPol) << endl;
3675 // -> depending on luck, any of the above
3678 Note also, how factors with the same exponents are not fully factorized
3682 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3683 @c node-name, next, previous, up
3684 @section Rational expressions
3686 @subsection The @code{normal} method
3687 @cindex @code{normal()}
3688 @cindex simplification
3689 @cindex temporary replacement
3691 Some basic form of simplification of expressions is called for frequently.
3692 GiNaC provides the method @code{.normal()}, which converts a rational function
3693 into an equivalent rational function of the form @samp{numerator/denominator}
3694 where numerator and denominator are coprime. If the input expression is already
3695 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3696 otherwise it performs fraction addition and multiplication.
3698 @code{.normal()} can also be used on expressions which are not rational functions
3699 as it will replace all non-rational objects (like functions or non-integer
3700 powers) by temporary symbols to bring the expression to the domain of rational
3701 functions before performing the normalization, and re-substituting these
3702 symbols afterwards. This algorithm is also available as a separate method
3703 @code{.to_rational()}, described below.
3705 This means that both expressions @code{t1} and @code{t2} are indeed
3706 simplified in this little code snippet:
3711 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3712 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3713 std::cout << "t1 is " << t1.normal() << std::endl;
3714 std::cout << "t2 is " << t2.normal() << std::endl;
3718 Of course this works for multivariate polynomials too, so the ratio of
3719 the sample-polynomials from the section about GCD and LCM above would be
3720 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3723 @subsection Numerator and denominator
3726 @cindex @code{numer()}
3727 @cindex @code{denom()}
3728 @cindex @code{numer_denom()}
3730 The numerator and denominator of an expression can be obtained with
3735 ex ex::numer_denom();
3738 These functions will first normalize the expression as described above and
3739 then return the numerator, denominator, or both as a list, respectively.
3740 If you need both numerator and denominator, calling @code{numer_denom()} is
3741 faster than using @code{numer()} and @code{denom()} separately.
3744 @subsection Converting to a rational expression
3745 @cindex @code{to_rational()}
3747 Some of the methods described so far only work on polynomials or rational
3748 functions. GiNaC provides a way to extend the domain of these functions to
3749 general expressions by using the temporary replacement algorithm described
3750 above. You do this by calling
3753 ex ex::to_rational(lst &l);
3756 on the expression to be converted. The supplied @code{lst} will be filled
3757 with the generated temporary symbols and their replacement expressions in
3758 a format that can be used directly for the @code{subs()} method. It can also
3759 already contain a list of replacements from an earlier application of
3760 @code{.to_rational()}, so it's possible to use it on multiple expressions
3761 and get consistent results.
3768 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3769 ex b = sin(x) + cos(x);
3772 divide(a.to_rational(l), b.to_rational(l), q);
3773 cout << q.subs(l) << endl;
3777 will print @samp{sin(x)-cos(x)}.
3780 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3781 @c node-name, next, previous, up
3782 @section Symbolic differentiation
3783 @cindex differentiation
3784 @cindex @code{diff()}
3786 @cindex product rule
3788 GiNaC's objects know how to differentiate themselves. Thus, a
3789 polynomial (class @code{add}) knows that its derivative is the sum of
3790 the derivatives of all the monomials:
3794 symbol x("x"), y("y"), z("z");
3795 ex P = pow(x, 5) + pow(x, 2) + y;
3797 cout << P.diff(x,2) << endl;
3799 cout << P.diff(y) << endl; // 1
3801 cout << P.diff(z) << endl; // 0
3806 If a second integer parameter @var{n} is given, the @code{diff} method
3807 returns the @var{n}th derivative.
3809 If @emph{every} object and every function is told what its derivative
3810 is, all derivatives of composed objects can be calculated using the
3811 chain rule and the product rule. Consider, for instance the expression
3812 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3813 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3814 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3815 out that the composition is the generating function for Euler Numbers,
3816 i.e. the so called @var{n}th Euler number is the coefficient of
3817 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3818 identity to code a function that generates Euler numbers in just three
3821 @cindex Euler numbers
3823 #include <ginac/ginac.h>
3824 using namespace GiNaC;
3826 ex EulerNumber(unsigned n)
3829 const ex generator = pow(cosh(x),-1);
3830 return generator.diff(x,n).subs(x==0);
3835 for (unsigned i=0; i<11; i+=2)
3836 std::cout << EulerNumber(i) << std::endl;
3841 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3842 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3843 @code{i} by two since all odd Euler numbers vanish anyways.
3846 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
3847 @c node-name, next, previous, up
3848 @section Series expansion
3849 @cindex @code{series()}
3850 @cindex Taylor expansion
3851 @cindex Laurent expansion
3852 @cindex @code{pseries} (class)
3853 @cindex @code{Order()}
3855 Expressions know how to expand themselves as a Taylor series or (more
3856 generally) a Laurent series. As in most conventional Computer Algebra
3857 Systems, no distinction is made between those two. There is a class of
3858 its own for storing such series (@code{class pseries}) and a built-in
3859 function (called @code{Order}) for storing the order term of the series.
3860 As a consequence, if you want to work with series, i.e. multiply two
3861 series, you need to call the method @code{ex::series} again to convert
3862 it to a series object with the usual structure (expansion plus order
3863 term). A sample application from special relativity could read:
3866 #include <ginac/ginac.h>
3867 using namespace std;
3868 using namespace GiNaC;
3872 symbol v("v"), c("c");
3874 ex gamma = 1/sqrt(1 - pow(v/c,2));
3875 ex mass_nonrel = gamma.series(v==0, 10);
3877 cout << "the relativistic mass increase with v is " << endl
3878 << mass_nonrel << endl;
3880 cout << "the inverse square of this series is " << endl
3881 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3885 Only calling the series method makes the last output simplify to
3886 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3887 series raised to the power @math{-2}.
3889 @cindex M@'echain's formula
3890 As another instructive application, let us calculate the numerical
3891 value of Archimedes' constant
3895 (for which there already exists the built-in constant @code{Pi})
3896 using M@'echain's amazing formula
3898 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3901 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3903 We may expand the arcus tangent around @code{0} and insert the fractions
3904 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3905 carries an order term with it and the question arises what the system is
3906 supposed to do when the fractions are plugged into that order term. The
3907 solution is to use the function @code{series_to_poly()} to simply strip
3911 #include <ginac/ginac.h>
3912 using namespace GiNaC;
3914 ex mechain_pi(int degr)
3917 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3918 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3919 -4*pi_expansion.subs(x==numeric(1,239));
3925 using std::cout; // just for fun, another way of...
3926 using std::endl; // ...dealing with this namespace std.
3928 for (int i=2; i<12; i+=2) @{
3929 pi_frac = mechain_pi(i);
3930 cout << i << ":\t" << pi_frac << endl
3931 << "\t" << pi_frac.evalf() << endl;
3937 Note how we just called @code{.series(x,degr)} instead of
3938 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3939 method @code{series()}: if the first argument is a symbol the expression
3940 is expanded in that symbol around point @code{0}. When you run this
3941 program, it will type out:
3945 3.1832635983263598326
3946 4: 5359397032/1706489875
3947 3.1405970293260603143
3948 6: 38279241713339684/12184551018734375
3949 3.141621029325034425
3950 8: 76528487109180192540976/24359780855939418203125
3951 3.141591772182177295
3952 10: 327853873402258685803048818236/104359128170408663038552734375
3953 3.1415926824043995174
3957 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
3958 @c node-name, next, previous, up
3959 @section Symmetrization
3960 @cindex @code{symmetrize()}
3961 @cindex @code{antisymmetrize()}
3962 @cindex @code{symmetrize_cyclic()}
3967 ex ex::symmetrize(const lst & l);
3968 ex ex::antisymmetrize(const lst & l);
3969 ex ex::symmetrize_cyclic(const lst & l);
3972 symmetrize an expression by returning the sum over all symmetric,
3973 antisymmetric or cyclic permutations of the specified list of objects,
3974 weighted by the number of permutations.
3976 The three additional methods
3979 ex ex::symmetrize();
3980 ex ex::antisymmetrize();
3981 ex ex::symmetrize_cyclic();
3984 symmetrize or antisymmetrize an expression over its free indices.
3986 Symmetrization is most useful with indexed expressions but can be used with
3987 almost any kind of object (anything that is @code{subs()}able):
3991 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
3992 symbol A("A"), B("B"), a("a"), b("b"), c("c");
3994 cout << indexed(A, i, j).symmetrize() << endl;
3995 // -> 1/2*A.j.i+1/2*A.i.j
3996 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
3997 // -> -1/2*A.j.i.k+1/2*A.i.j.k
3998 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
3999 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4004 @node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
4005 @c node-name, next, previous, up
4006 @section Predefined mathematical functions
4008 GiNaC contains the following predefined mathematical functions:
4011 @multitable @columnfractions .30 .70
4012 @item @strong{Name} @tab @strong{Function}
4015 @cindex @code{abs()}
4016 @item @code{csgn(x)}
4018 @cindex @code{csgn()}
4019 @item @code{sqrt(x)}
4020 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4021 @cindex @code{sqrt()}
4024 @cindex @code{sin()}
4027 @cindex @code{cos()}
4030 @cindex @code{tan()}
4031 @item @code{asin(x)}
4033 @cindex @code{asin()}
4034 @item @code{acos(x)}
4036 @cindex @code{acos()}
4037 @item @code{atan(x)}
4038 @tab inverse tangent
4039 @cindex @code{atan()}
4040 @item @code{atan2(y, x)}
4041 @tab inverse tangent with two arguments
4042 @item @code{sinh(x)}
4043 @tab hyperbolic sine
4044 @cindex @code{sinh()}
4045 @item @code{cosh(x)}
4046 @tab hyperbolic cosine
4047 @cindex @code{cosh()}
4048 @item @code{tanh(x)}
4049 @tab hyperbolic tangent
4050 @cindex @code{tanh()}
4051 @item @code{asinh(x)}
4052 @tab inverse hyperbolic sine
4053 @cindex @code{asinh()}
4054 @item @code{acosh(x)}
4055 @tab inverse hyperbolic cosine
4056 @cindex @code{acosh()}
4057 @item @code{atanh(x)}
4058 @tab inverse hyperbolic tangent
4059 @cindex @code{atanh()}
4061 @tab exponential function
4062 @cindex @code{exp()}
4064 @tab natural logarithm
4065 @cindex @code{log()}
4068 @cindex @code{Li2()}
4069 @item @code{zeta(x)}
4070 @tab Riemann's zeta function
4071 @cindex @code{zeta()}
4072 @item @code{zeta(n, x)}
4073 @tab derivatives of Riemann's zeta function
4074 @item @code{tgamma(x)}
4076 @cindex @code{tgamma()}
4077 @cindex Gamma function
4078 @item @code{lgamma(x)}
4079 @tab logarithm of Gamma function
4080 @cindex @code{lgamma()}
4081 @item @code{beta(x, y)}
4082 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4083 @cindex @code{beta()}
4085 @tab psi (digamma) function
4086 @cindex @code{psi()}
4087 @item @code{psi(n, x)}
4088 @tab derivatives of psi function (polygamma functions)
4089 @item @code{factorial(n)}
4090 @tab factorial function
4091 @cindex @code{factorial()}
4092 @item @code{binomial(n, m)}
4093 @tab binomial coefficients
4094 @cindex @code{binomial()}
4095 @item @code{Order(x)}
4096 @tab order term function in truncated power series
4097 @cindex @code{Order()}
4102 For functions that have a branch cut in the complex plane GiNaC follows
4103 the conventions for C++ as defined in the ANSI standard as far as
4104 possible. In particular: the natural logarithm (@code{log}) and the
4105 square root (@code{sqrt}) both have their branch cuts running along the
4106 negative real axis where the points on the axis itself belong to the
4107 upper part (i.e. continuous with quadrant II). The inverse
4108 trigonometric and hyperbolic functions are not defined for complex
4109 arguments by the C++ standard, however. In GiNaC we follow the
4110 conventions used by CLN, which in turn follow the carefully designed
4111 definitions in the Common Lisp standard. It should be noted that this
4112 convention is identical to the one used by the C99 standard and by most
4113 serious CAS. It is to be expected that future revisions of the C++
4114 standard incorporate these functions in the complex domain in a manner
4115 compatible with C99.
4118 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
4119 @c node-name, next, previous, up
4120 @section Input and output of expressions
4123 @subsection Expression output
4125 @cindex output of expressions
4127 The easiest way to print an expression is to write it to a stream:
4132 ex e = 4.5+pow(x,2)*3/2;
4133 cout << e << endl; // prints '(4.5)+3/2*x^2'
4137 The output format is identical to the @command{ginsh} input syntax and
4138 to that used by most computer algebra systems, but not directly pastable
4139 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4140 is printed as @samp{x^2}).
4142 It is possible to print expressions in a number of different formats with
4146 void ex::print(const print_context & c, unsigned level = 0);
4149 @cindex @code{print_context} (class)
4150 The type of @code{print_context} object passed in determines the format
4151 of the output. The possible types are defined in @file{ginac/print.h}.
4152 All constructors of @code{print_context} and derived classes take an
4153 @code{ostream &} as their first argument.
4155 To print an expression in a way that can be directly used in a C or C++
4156 program, you pass a @code{print_csrc} object like this:
4160 cout << "float f = ";
4161 e.print(print_csrc_float(cout));
4164 cout << "double d = ";
4165 e.print(print_csrc_double(cout));
4168 cout << "cl_N n = ";
4169 e.print(print_csrc_cl_N(cout));
4174 The three possible types mostly affect the way in which floating point
4175 numbers are written.
4177 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
4180 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4181 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
4182 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
4185 The @code{print_context} type @code{print_tree} provides a dump of the
4186 internal structure of an expression for debugging purposes:
4190 e.print(print_tree(cout));
4197 add, hash=0x0, flags=0x3, nops=2
4198 power, hash=0x9, flags=0x3, nops=2
4199 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
4200 2 (numeric), hash=0x80000042, flags=0xf
4201 3/2 (numeric), hash=0x80000061, flags=0xf
4204 4.5L0 (numeric), hash=0x8000004b, flags=0xf
4208 This kind of output is also available in @command{ginsh} as the @code{print()}
4211 Another useful output format is for LaTeX parsing in mathematical mode.
4212 It is rather similar to the default @code{print_context} but provides
4213 some braces needed by LaTeX for delimiting boxes and also converts some
4214 common objects to conventional LaTeX names. It is possible to give symbols
4215 a special name for LaTeX output by supplying it as a second argument to
4216 the @code{symbol} constructor.
4218 For example, the code snippet
4223 ex foo = lgamma(x).series(x==0,3);
4224 foo.print(print_latex(std::cout));
4230 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
4233 @cindex Tree traversal
4234 If you need any fancy special output format, e.g. for interfacing GiNaC
4235 with other algebra systems or for producing code for different
4236 programming languages, you can always traverse the expression tree yourself:
4239 static void my_print(const ex & e)
4241 if (is_a<function>(e))
4242 cout << ex_to<function>(e).get_name();
4244 cout << e.bp->class_name();
4246 unsigned n = e.nops();
4248 for (unsigned i=0; i<n; i++) @{
4260 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4268 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4269 symbol(y))),numeric(-2)))
4272 If you need an output format that makes it possible to accurately
4273 reconstruct an expression by feeding the output to a suitable parser or
4274 object factory, you should consider storing the expression in an
4275 @code{archive} object and reading the object properties from there.
4276 See the section on archiving for more information.
4279 @subsection Expression input
4280 @cindex input of expressions
4282 GiNaC provides no way to directly read an expression from a stream because
4283 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4284 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4285 @code{y} you defined in your program and there is no way to specify the
4286 desired symbols to the @code{>>} stream input operator.
4288 Instead, GiNaC lets you construct an expression from a string, specifying the
4289 list of symbols to be used:
4293 symbol x("x"), y("y");
4294 ex e("2*x+sin(y)", lst(x, y));
4298 The input syntax is the same as that used by @command{ginsh} and the stream
4299 output operator @code{<<}. The symbols in the string are matched by name to
4300 the symbols in the list and if GiNaC encounters a symbol not specified in
4301 the list it will throw an exception.
4303 With this constructor, it's also easy to implement interactive GiNaC programs:
4308 #include <stdexcept>
4309 #include <ginac/ginac.h>
4310 using namespace std;
4311 using namespace GiNaC;
4318 cout << "Enter an expression containing 'x': ";
4323 cout << "The derivative of " << e << " with respect to x is ";
4324 cout << e.diff(x) << ".\n";
4325 @} catch (exception &p) @{
4326 cerr << p.what() << endl;
4332 @subsection Archiving
4333 @cindex @code{archive} (class)
4336 GiNaC allows creating @dfn{archives} of expressions which can be stored
4337 to or retrieved from files. To create an archive, you declare an object
4338 of class @code{archive} and archive expressions in it, giving each
4339 expression a unique name:
4343 using namespace std;
4344 #include <ginac/ginac.h>
4345 using namespace GiNaC;
4349 symbol x("x"), y("y"), z("z");
4351 ex foo = sin(x + 2*y) + 3*z + 41;
4355 a.archive_ex(foo, "foo");
4356 a.archive_ex(bar, "the second one");
4360 The archive can then be written to a file:
4364 ofstream out("foobar.gar");
4370 The file @file{foobar.gar} contains all information that is needed to
4371 reconstruct the expressions @code{foo} and @code{bar}.
4373 @cindex @command{viewgar}
4374 The tool @command{viewgar} that comes with GiNaC can be used to view
4375 the contents of GiNaC archive files:
4378 $ viewgar foobar.gar
4379 foo = 41+sin(x+2*y)+3*z
4380 the second one = 42+sin(x+2*y)+3*z
4383 The point of writing archive files is of course that they can later be
4389 ifstream in("foobar.gar");
4394 And the stored expressions can be retrieved by their name:
4400 ex ex1 = a2.unarchive_ex(syms, "foo");
4401 ex ex2 = a2.unarchive_ex(syms, "the second one");
4403 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
4404 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
4405 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
4409 Note that you have to supply a list of the symbols which are to be inserted
4410 in the expressions. Symbols in archives are stored by their name only and
4411 if you don't specify which symbols you have, unarchiving the expression will
4412 create new symbols with that name. E.g. if you hadn't included @code{x} in
4413 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
4414 have had no effect because the @code{x} in @code{ex1} would have been a
4415 different symbol than the @code{x} which was defined at the beginning of
4416 the program, although both would appear as @samp{x} when printed.
4418 You can also use the information stored in an @code{archive} object to
4419 output expressions in a format suitable for exact reconstruction. The
4420 @code{archive} and @code{archive_node} classes have a couple of member
4421 functions that let you access the stored properties:
4424 static void my_print2(const archive_node & n)
4427 n.find_string("class", class_name);
4428 cout << class_name << "(";
4430 archive_node::propinfovector p;
4431 n.get_properties(p);
4433 unsigned num = p.size();
4434 for (unsigned i=0; i<num; i++) @{
4435 const string &name = p[i].name;
4436 if (name == "class")
4438 cout << name << "=";
4440 unsigned count = p[i].count;
4444 for (unsigned j=0; j<count; j++) @{
4445 switch (p[i].type) @{
4446 case archive_node::PTYPE_BOOL: @{
4448 n.find_bool(name, x, j);
4449 cout << (x ? "true" : "false");
4452 case archive_node::PTYPE_UNSIGNED: @{
4454 n.find_unsigned(name, x, j);
4458 case archive_node::PTYPE_STRING: @{
4460 n.find_string(name, x, j);
4461 cout << '\"' << x << '\"';
4464 case archive_node::PTYPE_NODE: @{
4465 const archive_node &x = n.find_ex_node(name, j);
4487 ex e = pow(2, x) - y;
4489 my_print2(ar.get_top_node(0)); cout << endl;
4497 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
4498 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
4499 overall_coeff=numeric(number="0"))
4502 Be warned, however, that the set of properties and their meaning for each
4503 class may change between GiNaC versions.
4506 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
4507 @c node-name, next, previous, up
4508 @chapter Extending GiNaC
4510 By reading so far you should have gotten a fairly good understanding of
4511 GiNaC's design-patterns. From here on you should start reading the
4512 sources. All we can do now is issue some recommendations how to tackle
4513 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
4514 develop some useful extension please don't hesitate to contact the GiNaC
4515 authors---they will happily incorporate them into future versions.
4518 * What does not belong into GiNaC:: What to avoid.
4519 * Symbolic functions:: Implementing symbolic functions.
4520 * Adding classes:: Defining new algebraic classes.
4524 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
4525 @c node-name, next, previous, up
4526 @section What doesn't belong into GiNaC
4528 @cindex @command{ginsh}
4529 First of all, GiNaC's name must be read literally. It is designed to be
4530 a library for use within C++. The tiny @command{ginsh} accompanying
4531 GiNaC makes this even more clear: it doesn't even attempt to provide a
4532 language. There are no loops or conditional expressions in
4533 @command{ginsh}, it is merely a window into the library for the
4534 programmer to test stuff (or to show off). Still, the design of a
4535 complete CAS with a language of its own, graphical capabilities and all
4536 this on top of GiNaC is possible and is without doubt a nice project for
4539 There are many built-in functions in GiNaC that do not know how to
4540 evaluate themselves numerically to a precision declared at runtime
4541 (using @code{Digits}). Some may be evaluated at certain points, but not
4542 generally. This ought to be fixed. However, doing numerical
4543 computations with GiNaC's quite abstract classes is doomed to be
4544 inefficient. For this purpose, the underlying foundation classes
4545 provided by CLN are much better suited.
4548 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
4549 @c node-name, next, previous, up
4550 @section Symbolic functions
4552 The easiest and most instructive way to start with is probably to
4553 implement your own function. GiNaC's functions are objects of class
4554 @code{function}. The preprocessor is then used to convert the function
4555 names to objects with a corresponding serial number that is used
4556 internally to identify them. You usually need not worry about this
4557 number. New functions may be inserted into the system via a kind of
4558 `registry'. It is your responsibility to care for some functions that
4559 are called when the user invokes certain methods. These are usual
4560 C++-functions accepting a number of @code{ex} as arguments and returning
4561 one @code{ex}. As an example, if we have a look at a simplified
4562 implementation of the cosine trigonometric function, we first need a
4563 function that is called when one wishes to @code{eval} it. It could
4564 look something like this:
4567 static ex cos_eval_method(const ex & x)
4569 // if (!x%(2*Pi)) return 1
4570 // if (!x%Pi) return -1
4571 // if (!x%Pi/2) return 0
4572 // care for other cases...
4573 return cos(x).hold();
4577 @cindex @code{hold()}
4579 The last line returns @code{cos(x)} if we don't know what else to do and
4580 stops a potential recursive evaluation by saying @code{.hold()}, which
4581 sets a flag to the expression signaling that it has been evaluated. We
4582 should also implement a method for numerical evaluation and since we are
4583 lazy we sweep the problem under the rug by calling someone else's
4584 function that does so, in this case the one in class @code{numeric}:
4587 static ex cos_evalf(const ex & x)
4589 if (is_a<numeric>(x))
4590 return cos(ex_to<numeric>(x));
4592 return cos(x).hold();
4596 Differentiation will surely turn up and so we need to tell @code{cos}
4597 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
4598 instance are then handled automatically by @code{basic::diff} and
4602 static ex cos_deriv(const ex & x, unsigned diff_param)
4608 @cindex product rule
4609 The second parameter is obligatory but uninteresting at this point. It
4610 specifies which parameter to differentiate in a partial derivative in
4611 case the function has more than one parameter and its main application
4612 is for correct handling of the chain rule. For Taylor expansion, it is
4613 enough to know how to differentiate. But if the function you want to
4614 implement does have a pole somewhere in the complex plane, you need to
4615 write another method for Laurent expansion around that point.
4617 Now that all the ingredients for @code{cos} have been set up, we need
4618 to tell the system about it. This is done by a macro and we are not
4619 going to describe how it expands, please consult your preprocessor if you
4623 REGISTER_FUNCTION(cos, eval_func(cos_eval).
4624 evalf_func(cos_evalf).
4625 derivative_func(cos_deriv));
4628 The first argument is the function's name used for calling it and for
4629 output. The second binds the corresponding methods as options to this
4630 object. Options are separated by a dot and can be given in an arbitrary
4631 order. GiNaC functions understand several more options which are always
4632 specified as @code{.option(params)}, for example a method for series
4633 expansion @code{.series_func(cos_series)}. Again, if no series
4634 expansion method is given, GiNaC defaults to simple Taylor expansion,
4635 which is correct if there are no poles involved as is the case for the
4636 @code{cos} function. The way GiNaC handles poles in case there are any
4637 is best understood by studying one of the examples, like the Gamma
4638 (@code{tgamma}) function for instance. (In essence the function first
4639 checks if there is a pole at the evaluation point and falls back to
4640 Taylor expansion if there isn't. Then, the pole is regularized by some
4641 suitable transformation.) Also, the new function needs to be declared
4642 somewhere. This may also be done by a convenient preprocessor macro:
4645 DECLARE_FUNCTION_1P(cos)
4648 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
4649 implementation of @code{cos} is very incomplete and lacks several safety
4650 mechanisms. Please, have a look at the real implementation in GiNaC.
4651 (By the way: in case you are worrying about all the macros above we can
4652 assure you that functions are GiNaC's most macro-intense classes. We
4653 have done our best to avoid macros where we can.)
4656 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
4657 @c node-name, next, previous, up
4658 @section Adding classes
4660 If you are doing some very specialized things with GiNaC you may find that
4661 you have to implement your own algebraic classes to fit your needs. This
4662 section will explain how to do this by giving the example of a simple
4663 'string' class. After reading this section you will know how to properly
4664 declare a GiNaC class and what the minimum required member functions are
4665 that you have to implement. We only cover the implementation of a 'leaf'
4666 class here (i.e. one that doesn't contain subexpressions). Creating a
4667 container class like, for example, a class representing tensor products is
4668 more involved but this section should give you enough information so you can
4669 consult the source to GiNaC's predefined classes if you want to implement
4670 something more complicated.
4672 @subsection GiNaC's run-time type information system
4674 @cindex hierarchy of classes
4676 All algebraic classes (that is, all classes that can appear in expressions)
4677 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4678 @code{basic *} (which is essentially what an @code{ex} is) represents a
4679 generic pointer to an algebraic class. Occasionally it is necessary to find
4680 out what the class of an object pointed to by a @code{basic *} really is.
4681 Also, for the unarchiving of expressions it must be possible to find the
4682 @code{unarchive()} function of a class given the class name (as a string). A
4683 system that provides this kind of information is called a run-time type
4684 information (RTTI) system. The C++ language provides such a thing (see the
4685 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4686 implements its own, simpler RTTI.
4688 The RTTI in GiNaC is based on two mechanisms:
4693 The @code{basic} class declares a member variable @code{tinfo_key} which
4694 holds an unsigned integer that identifies the object's class. These numbers
4695 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4696 classes. They all start with @code{TINFO_}.
4699 By means of some clever tricks with static members, GiNaC maintains a list
4700 of information for all classes derived from @code{basic}. The information
4701 available includes the class names, the @code{tinfo_key}s, and pointers
4702 to the unarchiving functions. This class registry is defined in the
4703 @file{registrar.h} header file.
4707 The disadvantage of this proprietary RTTI implementation is that there's
4708 a little more to do when implementing new classes (C++'s RTTI works more
4709 or less automatic) but don't worry, most of the work is simplified by
4712 @subsection A minimalistic example
4714 Now we will start implementing a new class @code{mystring} that allows
4715 placing character strings in algebraic expressions (this is not very useful,
4716 but it's just an example). This class will be a direct subclass of
4717 @code{basic}. You can use this sample implementation as a starting point
4718 for your own classes.
4720 The code snippets given here assume that you have included some header files
4726 #include <stdexcept>
4727 using namespace std;
4729 #include <ginac/ginac.h>
4730 using namespace GiNaC;
4733 The first thing we have to do is to define a @code{tinfo_key} for our new
4734 class. This can be any arbitrary unsigned number that is not already taken
4735 by one of the existing classes but it's better to come up with something
4736 that is unlikely to clash with keys that might be added in the future. The
4737 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4738 which is not a requirement but we are going to stick with this scheme:
4741 const unsigned TINFO_mystring = 0x42420001U;
4744 Now we can write down the class declaration. The class stores a C++
4745 @code{string} and the user shall be able to construct a @code{mystring}
4746 object from a C or C++ string:
4749 class mystring : public basic
4751 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4754 mystring(const string &s);
4755 mystring(const char *s);
4761 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4764 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4765 macros are defined in @file{registrar.h}. They take the name of the class
4766 and its direct superclass as arguments and insert all required declarations
4767 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4768 the first line after the opening brace of the class definition. The
4769 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4770 source (at global scope, of course, not inside a function).
4772 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4773 declarations of the default and copy constructor, the destructor, the
4774 assignment operator and a couple of other functions that are required. It
4775 also defines a type @code{inherited} which refers to the superclass so you
4776 don't have to modify your code every time you shuffle around the class
4777 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4778 constructor, the destructor and the assignment operator.
4780 Now there are nine member functions we have to implement to get a working
4786 @code{mystring()}, the default constructor.
4789 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4790 assignment operator to free dynamically allocated members. The @code{call_parent}
4791 specifies whether the @code{destroy()} function of the superclass is to be
4795 @code{void copy(const mystring &other)}, which is used in the copy constructor
4796 and assignment operator to copy the member variables over from another
4797 object of the same class.
4800 @code{void archive(archive_node &n)}, the archiving function. This stores all
4801 information needed to reconstruct an object of this class inside an
4802 @code{archive_node}.
4805 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4806 constructor. This constructs an instance of the class from the information
4807 found in an @code{archive_node}.
4810 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4811 unarchiving function. It constructs a new instance by calling the unarchiving
4815 @code{int compare_same_type(const basic &other)}, which is used internally
4816 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4817 -1, depending on the relative order of this object and the @code{other}
4818 object. If it returns 0, the objects are considered equal.
4819 @strong{Note:} This has nothing to do with the (numeric) ordering
4820 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4821 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4822 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4823 must provide a @code{compare_same_type()} function, even those representing
4824 objects for which no reasonable algebraic ordering relationship can be
4828 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4829 which are the two constructors we declared.
4833 Let's proceed step-by-step. The default constructor looks like this:
4836 mystring::mystring() : inherited(TINFO_mystring)
4838 // dynamically allocate resources here if required
4842 The golden rule is that in all constructors you have to set the
4843 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4844 it will be set by the constructor of the superclass and all hell will break
4845 loose in the RTTI. For your convenience, the @code{basic} class provides
4846 a constructor that takes a @code{tinfo_key} value, which we are using here
4847 (remember that in our case @code{inherited = basic}). If the superclass
4848 didn't have such a constructor, we would have to set the @code{tinfo_key}
4849 to the right value manually.
4851 In the default constructor you should set all other member variables to
4852 reasonable default values (we don't need that here since our @code{str}
4853 member gets set to an empty string automatically). The constructor(s) are of
4854 course also the right place to allocate any dynamic resources you require.
4856 Next, the @code{destroy()} function:
4859 void mystring::destroy(bool call_parent)
4861 // free dynamically allocated resources here if required
4863 inherited::destroy(call_parent);
4867 This function is where we free all dynamically allocated resources. We
4868 don't have any so we're not doing anything here, but if we had, for
4869 example, used a C-style @code{char *} to store our string, this would be
4870 the place to @code{delete[]} the string storage. If @code{call_parent}
4871 is true, we have to call the @code{destroy()} function of the superclass
4872 after we're done (to mimic C++'s automatic invocation of superclass
4873 destructors where @code{destroy()} is called from outside a destructor).
4875 The @code{copy()} function just copies over the member variables from
4879 void mystring::copy(const mystring &other)
4881 inherited::copy(other);
4886 We can simply overwrite the member variables here. There's no need to worry
4887 about dynamically allocated storage. The assignment operator (which is
4888 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4889 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4890 explicitly call the @code{copy()} function of the superclass here so
4891 all the member variables will get copied.
4893 Next are the three functions for archiving. You have to implement them even
4894 if you don't plan to use archives, but the minimum required implementation
4895 is really simple. First, the archiving function:
4898 void mystring::archive(archive_node &n) const
4900 inherited::archive(n);
4901 n.add_string("string", str);
4905 The only thing that is really required is calling the @code{archive()}
4906 function of the superclass. Optionally, you can store all information you
4907 deem necessary for representing the object into the passed
4908 @code{archive_node}. We are just storing our string here. For more
4909 information on how the archiving works, consult the @file{archive.h} header
4912 The unarchiving constructor is basically the inverse of the archiving
4916 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4918 n.find_string("string", str);
4922 If you don't need archiving, just leave this function empty (but you must
4923 invoke the unarchiving constructor of the superclass). Note that we don't
4924 have to set the @code{tinfo_key} here because it is done automatically
4925 by the unarchiving constructor of the @code{basic} class.
4927 Finally, the unarchiving function:
4930 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4932 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4936 You don't have to understand how exactly this works. Just copy these
4937 four lines into your code literally (replacing the class name, of
4938 course). It calls the unarchiving constructor of the class and unless
4939 you are doing something very special (like matching @code{archive_node}s
4940 to global objects) you don't need a different implementation. For those
4941 who are interested: setting the @code{dynallocated} flag puts the object
4942 under the control of GiNaC's garbage collection. It will get deleted
4943 automatically once it is no longer referenced.
4945 Our @code{compare_same_type()} function uses a provided function to compare
4949 int mystring::compare_same_type(const basic &other) const
4951 const mystring &o = static_cast<const mystring &>(other);
4952 int cmpval = str.compare(o.str);
4955 else if (cmpval < 0)
4962 Although this function takes a @code{basic &}, it will always be a reference
4963 to an object of exactly the same class (objects of different classes are not
4964 comparable), so the cast is safe. If this function returns 0, the two objects
4965 are considered equal (in the sense that @math{A-B=0}), so you should compare
4966 all relevant member variables.
4968 Now the only thing missing is our two new constructors:
4971 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4973 // dynamically allocate resources here if required
4976 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4978 // dynamically allocate resources here if required
4982 No surprises here. We set the @code{str} member from the argument and
4983 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4985 That's it! We now have a minimal working GiNaC class that can store
4986 strings in algebraic expressions. Let's confirm that the RTTI works:
4989 ex e = mystring("Hello, world!");
4990 cout << is_a<mystring>(e) << endl;
4993 cout << e.bp->class_name() << endl;
4997 Obviously it does. Let's see what the expression @code{e} looks like:
5001 // -> [mystring object]
5004 Hm, not exactly what we expect, but of course the @code{mystring} class
5005 doesn't yet know how to print itself. This is done in the @code{print()}
5006 member function. Let's say that we wanted to print the string surrounded
5010 class mystring : public basic
5014 void print(const print_context &c, unsigned level = 0) const;
5018 void mystring::print(const print_context &c, unsigned level) const
5020 // print_context::s is a reference to an ostream
5021 c.s << '\"' << str << '\"';
5025 The @code{level} argument is only required for container classes to
5026 correctly parenthesize the output. Let's try again to print the expression:
5030 // -> "Hello, world!"
5033 Much better. The @code{mystring} class can be used in arbitrary expressions:
5036 e += mystring("GiNaC rulez");
5038 // -> "GiNaC rulez"+"Hello, world!"
5041 (GiNaC's automatic term reordering is in effect here), or even
5044 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
5046 // -> "One string"^(2*sin(-"Another string"+Pi))
5049 Whether this makes sense is debatable but remember that this is only an
5050 example. At least it allows you to implement your own symbolic algorithms
5053 Note that GiNaC's algebraic rules remain unchanged:
5056 e = mystring("Wow") * mystring("Wow");
5060 e = pow(mystring("First")-mystring("Second"), 2);
5061 cout << e.expand() << endl;
5062 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5065 There's no way to, for example, make GiNaC's @code{add} class perform string
5066 concatenation. You would have to implement this yourself.
5068 @subsection Automatic evaluation
5070 @cindex @code{hold()}
5071 @cindex @code{eval()}
5073 When dealing with objects that are just a little more complicated than the
5074 simple string objects we have implemented, chances are that you will want to
5075 have some automatic simplifications or canonicalizations performed on them.
5076 This is done in the evaluation member function @code{eval()}. Let's say that
5077 we wanted all strings automatically converted to lowercase with
5078 non-alphabetic characters stripped, and empty strings removed:
5081 class mystring : public basic
5085 ex eval(int level = 0) const;
5089 ex mystring::eval(int level) const
5092 for (int i=0; i<str.length(); i++) @{
5094 if (c >= 'A' && c <= 'Z')
5095 new_str += tolower(c);
5096 else if (c >= 'a' && c <= 'z')
5100 if (new_str.length() == 0)
5103 return mystring(new_str).hold();
5107 The @code{level} argument is used to limit the recursion depth of the
5108 evaluation. We don't have any subexpressions in the @code{mystring}
5109 class so we are not concerned with this. If we had, we would call the
5110 @code{eval()} functions of the subexpressions with @code{level - 1} as
5111 the argument if @code{level != 1}. The @code{hold()} member function
5112 sets a flag in the object that prevents further evaluation. Otherwise
5113 we might end up in an endless loop. When you want to return the object
5114 unmodified, use @code{return this->hold();}.
5116 Let's confirm that it works:
5119 ex e = mystring("Hello, world!") + mystring("!?#");
5123 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5128 @subsection Other member functions
5130 We have implemented only a small set of member functions to make the class
5131 work in the GiNaC framework. For a real algebraic class, there are probably
5132 some more functions that you will want to re-implement, such as
5133 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
5134 or the header file of the class you want to make a subclass of to see
5135 what's there. One member function that you will most likely want to
5136 implement for terminal classes like the described string class is
5137 @code{calcchash()} that returns an @code{unsigned} hash value for the object
5138 which will allow GiNaC to compare and canonicalize expressions much more
5141 You can, of course, also add your own new member functions. Remember,
5142 that the RTTI may be used to get information about what kinds of objects
5143 you are dealing with (the position in the class hierarchy) and that you
5144 can always extract the bare object from an @code{ex} by stripping the
5145 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5146 should become a need.
5148 That's it. May the source be with you!
5151 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5152 @c node-name, next, previous, up
5153 @chapter A Comparison With Other CAS
5156 This chapter will give you some information on how GiNaC compares to
5157 other, traditional Computer Algebra Systems, like @emph{Maple},
5158 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5159 disadvantages over these systems.
5162 * Advantages:: Strengths of the GiNaC approach.
5163 * Disadvantages:: Weaknesses of the GiNaC approach.
5164 * Why C++?:: Attractiveness of C++.
5167 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5168 @c node-name, next, previous, up
5171 GiNaC has several advantages over traditional Computer
5172 Algebra Systems, like
5177 familiar language: all common CAS implement their own proprietary
5178 grammar which you have to learn first (and maybe learn again when your
5179 vendor decides to `enhance' it). With GiNaC you can write your program
5180 in common C++, which is standardized.
5184 structured data types: you can build up structured data types using
5185 @code{struct}s or @code{class}es together with STL features instead of
5186 using unnamed lists of lists of lists.
5189 strongly typed: in CAS, you usually have only one kind of variables
5190 which can hold contents of an arbitrary type. This 4GL like feature is
5191 nice for novice programmers, but dangerous.
5194 development tools: powerful development tools exist for C++, like fancy
5195 editors (e.g. with automatic indentation and syntax highlighting),
5196 debuggers, visualization tools, documentation generators@dots{}
5199 modularization: C++ programs can easily be split into modules by
5200 separating interface and implementation.
5203 price: GiNaC is distributed under the GNU Public License which means
5204 that it is free and available with source code. And there are excellent
5205 C++-compilers for free, too.
5208 extendable: you can add your own classes to GiNaC, thus extending it on
5209 a very low level. Compare this to a traditional CAS that you can
5210 usually only extend on a high level by writing in the language defined
5211 by the parser. In particular, it turns out to be almost impossible to
5212 fix bugs in a traditional system.
5215 multiple interfaces: Though real GiNaC programs have to be written in
5216 some editor, then be compiled, linked and executed, there are more ways
5217 to work with the GiNaC engine. Many people want to play with
5218 expressions interactively, as in traditional CASs. Currently, two such
5219 windows into GiNaC have been implemented and many more are possible: the
5220 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
5221 types to a command line and second, as a more consistent approach, an
5222 interactive interface to the Cint C++ interpreter has been put together
5223 (called GiNaC-cint) that allows an interactive scripting interface
5224 consistent with the C++ language. It is available from the usual GiNaC
5228 seamless integration: it is somewhere between difficult and impossible
5229 to call CAS functions from within a program written in C++ or any other
5230 programming language and vice versa. With GiNaC, your symbolic routines
5231 are part of your program. You can easily call third party libraries,
5232 e.g. for numerical evaluation or graphical interaction. All other
5233 approaches are much more cumbersome: they range from simply ignoring the
5234 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
5235 system (i.e. @emph{Yacas}).
5238 efficiency: often large parts of a program do not need symbolic
5239 calculations at all. Why use large integers for loop variables or
5240 arbitrary precision arithmetics where @code{int} and @code{double} are
5241 sufficient? For pure symbolic applications, GiNaC is comparable in
5242 speed with other CAS.
5247 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
5248 @c node-name, next, previous, up
5249 @section Disadvantages
5251 Of course it also has some disadvantages:
5256 advanced features: GiNaC cannot compete with a program like
5257 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
5258 which grows since 1981 by the work of dozens of programmers, with
5259 respect to mathematical features. Integration, factorization,
5260 non-trivial simplifications, limits etc. are missing in GiNaC (and are
5261 not planned for the near future).
5264 portability: While the GiNaC library itself is designed to avoid any
5265 platform dependent features (it should compile on any ANSI compliant C++
5266 compiler), the currently used version of the CLN library (fast large
5267 integer and arbitrary precision arithmetics) can only by compiled
5268 without hassle on systems with the C++ compiler from the GNU Compiler
5269 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
5270 macros to let the compiler gather all static initializations, which
5271 works for GNU C++ only. Feel free to contact the authors in case you
5272 really believe that you need to use a different compiler. We have
5273 occasionally used other compilers and may be able to give you advice.}
5274 GiNaC uses recent language features like explicit constructors, mutable
5275 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
5276 literally. Recent GCC versions starting at 2.95.3, although itself not
5277 yet ANSI compliant, support all needed features.
5282 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
5283 @c node-name, next, previous, up
5286 Why did we choose to implement GiNaC in C++ instead of Java or any other
5287 language? C++ is not perfect: type checking is not strict (casting is
5288 possible), separation between interface and implementation is not
5289 complete, object oriented design is not enforced. The main reason is
5290 the often scolded feature of operator overloading in C++. While it may
5291 be true that operating on classes with a @code{+} operator is rarely
5292 meaningful, it is perfectly suited for algebraic expressions. Writing
5293 @math{3x+5y} as @code{3*x+5*y} instead of
5294 @code{x.times(3).plus(y.times(5))} looks much more natural.
5295 Furthermore, the main developers are more familiar with C++ than with
5296 any other programming language.
5299 @node Internal Structures, Expressions are reference counted, Why C++? , Top
5300 @c node-name, next, previous, up
5301 @appendix Internal Structures
5304 * Expressions are reference counted::
5305 * Internal representation of products and sums::
5308 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
5309 @c node-name, next, previous, up
5310 @appendixsection Expressions are reference counted
5312 @cindex reference counting
5313 @cindex copy-on-write
5314 @cindex garbage collection
5315 An expression is extremely light-weight since internally it works like a
5316 handle to the actual representation and really holds nothing more than a
5317 pointer to some other object. What this means in practice is that
5318 whenever you create two @code{ex} and set the second equal to the first
5319 no copying process is involved. Instead, the copying takes place as soon
5320 as you try to change the second. Consider the simple sequence of code:
5324 #include <ginac/ginac.h>
5325 using namespace std;
5326 using namespace GiNaC;
5330 symbol x("x"), y("y"), z("z");
5333 e1 = sin(x + 2*y) + 3*z + 41;
5334 e2 = e1; // e2 points to same object as e1
5335 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
5336 e2 += 1; // e2 is copied into a new object
5337 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
5341 The line @code{e2 = e1;} creates a second expression pointing to the
5342 object held already by @code{e1}. The time involved for this operation
5343 is therefore constant, no matter how large @code{e1} was. Actual
5344 copying, however, must take place in the line @code{e2 += 1;} because
5345 @code{e1} and @code{e2} are not handles for the same object any more.
5346 This concept is called @dfn{copy-on-write semantics}. It increases
5347 performance considerably whenever one object occurs multiple times and
5348 represents a simple garbage collection scheme because when an @code{ex}
5349 runs out of scope its destructor checks whether other expressions handle
5350 the object it points to too and deletes the object from memory if that
5351 turns out not to be the case. A slightly less trivial example of
5352 differentiation using the chain-rule should make clear how powerful this
5357 symbol x("x"), y("y");
5361 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
5362 cout << e1 << endl // prints x+3*y
5363 << e2 << endl // prints (x+3*y)^3
5364 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
5368 Here, @code{e1} will actually be referenced three times while @code{e2}
5369 will be referenced two times. When the power of an expression is built,
5370 that expression needs not be copied. Likewise, since the derivative of
5371 a power of an expression can be easily expressed in terms of that
5372 expression, no copying of @code{e1} is involved when @code{e3} is
5373 constructed. So, when @code{e3} is constructed it will print as
5374 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
5375 holds a reference to @code{e2} and the factor in front is just
5378 As a user of GiNaC, you cannot see this mechanism of copy-on-write
5379 semantics. When you insert an expression into a second expression, the
5380 result behaves exactly as if the contents of the first expression were
5381 inserted. But it may be useful to remember that this is not what
5382 happens. Knowing this will enable you to write much more efficient
5383 code. If you still have an uncertain feeling with copy-on-write
5384 semantics, we recommend you have a look at the
5385 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
5386 Marshall Cline. Chapter 16 covers this issue and presents an
5387 implementation which is pretty close to the one in GiNaC.
5390 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
5391 @c node-name, next, previous, up
5392 @appendixsection Internal representation of products and sums
5394 @cindex representation
5397 @cindex @code{power}
5398 Although it should be completely transparent for the user of
5399 GiNaC a short discussion of this topic helps to understand the sources
5400 and also explain performance to a large degree. Consider the
5401 unexpanded symbolic expression
5403 $2d^3 \left( 4a + 5b - 3 \right)$
5406 @math{2*d^3*(4*a+5*b-3)}
5408 which could naively be represented by a tree of linear containers for
5409 addition and multiplication, one container for exponentiation with base
5410 and exponent and some atomic leaves of symbols and numbers in this
5415 @cindex pair-wise representation
5416 However, doing so results in a rather deeply nested tree which will
5417 quickly become inefficient to manipulate. We can improve on this by
5418 representing the sum as a sequence of terms, each one being a pair of a
5419 purely numeric multiplicative coefficient and its rest. In the same
5420 spirit we can store the multiplication as a sequence of terms, each
5421 having a numeric exponent and a possibly complicated base, the tree
5422 becomes much more flat:
5426 The number @code{3} above the symbol @code{d} shows that @code{mul}
5427 objects are treated similarly where the coefficients are interpreted as
5428 @emph{exponents} now. Addition of sums of terms or multiplication of
5429 products with numerical exponents can be coded to be very efficient with
5430 such a pair-wise representation. Internally, this handling is performed
5431 by most CAS in this way. It typically speeds up manipulations by an
5432 order of magnitude. The overall multiplicative factor @code{2} and the
5433 additive term @code{-3} look somewhat out of place in this
5434 representation, however, since they are still carrying a trivial
5435 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
5436 this is avoided by adding a field that carries an overall numeric
5437 coefficient. This results in the realistic picture of internal
5440 $2d^3 \left( 4a + 5b - 3 \right)$:
5443 @math{2*d^3*(4*a+5*b-3)}:
5449 This also allows for a better handling of numeric radicals, since
5450 @code{sqrt(2)} can now be carried along calculations. Now it should be
5451 clear, why both classes @code{add} and @code{mul} are derived from the
5452 same abstract class: the data representation is the same, only the
5453 semantics differs. In the class hierarchy, methods for polynomial
5454 expansion and the like are reimplemented for @code{add} and @code{mul},
5455 but the data structure is inherited from @code{expairseq}.
5458 @node Package Tools, ginac-config, Internal representation of products and sums, Top
5459 @c node-name, next, previous, up
5460 @appendix Package Tools
5462 If you are creating a software package that uses the GiNaC library,
5463 setting the correct command line options for the compiler and linker
5464 can be difficult. GiNaC includes two tools to make this process easier.
5467 * ginac-config:: A shell script to detect compiler and linker flags.
5468 * AM_PATH_GINAC:: Macro for GNU automake.
5472 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
5473 @c node-name, next, previous, up
5474 @section @command{ginac-config}
5475 @cindex ginac-config
5477 @command{ginac-config} is a shell script that you can use to determine
5478 the compiler and linker command line options required to compile and
5479 link a program with the GiNaC library.
5481 @command{ginac-config} takes the following flags:
5485 Prints out the version of GiNaC installed.
5487 Prints '-I' flags pointing to the installed header files.
5489 Prints out the linker flags necessary to link a program against GiNaC.
5490 @item --prefix[=@var{PREFIX}]
5491 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
5492 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
5493 Otherwise, prints out the configured value of @env{$prefix}.
5494 @item --exec-prefix[=@var{PREFIX}]
5495 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
5496 Otherwise, prints out the configured value of @env{$exec_prefix}.
5499 Typically, @command{ginac-config} will be used within a configure
5500 script, as described below. It, however, can also be used directly from
5501 the command line using backquotes to compile a simple program. For
5505 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
5508 This command line might expand to (for example):
5511 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
5512 -lginac -lcln -lstdc++
5515 Not only is the form using @command{ginac-config} easier to type, it will
5516 work on any system, no matter how GiNaC was configured.
5519 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
5520 @c node-name, next, previous, up
5521 @section @samp{AM_PATH_GINAC}
5522 @cindex AM_PATH_GINAC
5524 For packages configured using GNU automake, GiNaC also provides
5525 a macro to automate the process of checking for GiNaC.
5528 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
5536 Determines the location of GiNaC using @command{ginac-config}, which is
5537 either found in the user's path, or from the environment variable
5538 @env{GINACLIB_CONFIG}.
5541 Tests the installed libraries to make sure that their version
5542 is later than @var{MINIMUM-VERSION}. (A default version will be used
5546 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
5547 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
5548 variable to the output of @command{ginac-config --libs}, and calls
5549 @samp{AC_SUBST()} for these variables so they can be used in generated
5550 makefiles, and then executes @var{ACTION-IF-FOUND}.
5553 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
5554 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
5558 This macro is in file @file{ginac.m4} which is installed in
5559 @file{$datadir/aclocal}. Note that if automake was installed with a
5560 different @samp{--prefix} than GiNaC, you will either have to manually
5561 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
5562 aclocal the @samp{-I} option when running it.
5565 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
5566 * Example package:: Example of a package using AM_PATH_GINAC.
5570 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
5571 @c node-name, next, previous, up
5572 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
5574 Simply make sure that @command{ginac-config} is in your path, and run
5575 the configure script.
5582 The directory where the GiNaC libraries are installed needs
5583 to be found by your system's dynamic linker.
5585 This is generally done by
5588 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
5594 setting the environment variable @env{LD_LIBRARY_PATH},
5597 or, as a last resort,
5600 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
5601 running configure, for instance:
5604 LDFLAGS=-R/home/cbauer/lib ./configure
5609 You can also specify a @command{ginac-config} not in your path by
5610 setting the @env{GINACLIB_CONFIG} environment variable to the
5611 name of the executable
5614 If you move the GiNaC package from its installed location,
5615 you will either need to modify @command{ginac-config} script
5616 manually to point to the new location or rebuild GiNaC.
5627 --with-ginac-prefix=@var{PREFIX}
5628 --with-ginac-exec-prefix=@var{PREFIX}
5631 are provided to override the prefix and exec-prefix that were stored
5632 in the @command{ginac-config} shell script by GiNaC's configure. You are
5633 generally better off configuring GiNaC with the right path to begin with.
5637 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
5638 @c node-name, next, previous, up
5639 @subsection Example of a package using @samp{AM_PATH_GINAC}
5641 The following shows how to build a simple package using automake
5642 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
5645 #include <ginac/ginac.h>
5649 GiNaC::symbol x("x");
5650 GiNaC::ex a = GiNaC::sin(x);
5651 std::cout << "Derivative of " << a
5652 << " is " << a.diff(x) << std::endl;
5657 You should first read the introductory portions of the automake
5658 Manual, if you are not already familiar with it.
5660 Two files are needed, @file{configure.in}, which is used to build the
5664 dnl Process this file with autoconf to produce a configure script.
5666 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5672 AM_PATH_GINAC(0.9.0, [
5673 LIBS="$LIBS $GINACLIB_LIBS"
5674 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5675 ], AC_MSG_ERROR([need to have GiNaC installed]))
5680 The only command in this which is not standard for automake
5681 is the @samp{AM_PATH_GINAC} macro.
5683 That command does the following: If a GiNaC version greater or equal
5684 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5685 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5686 the error message `need to have GiNaC installed'
5688 And the @file{Makefile.am}, which will be used to build the Makefile.
5691 ## Process this file with automake to produce Makefile.in
5692 bin_PROGRAMS = simple
5693 simple_SOURCES = simple.cpp
5696 This @file{Makefile.am}, says that we are building a single executable,
5697 from a single sourcefile @file{simple.cpp}. Since every program
5698 we are building uses GiNaC we simply added the GiNaC options
5699 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5700 want to specify them on a per-program basis: for instance by
5704 simple_LDADD = $(GINACLIB_LIBS)
5705 INCLUDES = $(GINACLIB_CPPFLAGS)
5708 to the @file{Makefile.am}.
5710 To try this example out, create a new directory and add the three
5713 Now execute the following commands:
5716 $ automake --add-missing
5721 You now have a package that can be built in the normal fashion
5730 @node Bibliography, Concept Index, Example package, Top
5731 @c node-name, next, previous, up
5732 @appendix Bibliography
5737 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5740 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5743 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5746 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5749 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5750 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5753 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5754 James H. Davenport, Yvon Siret, and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
5755 Academic Press, London
5758 @cite{Computer Algebra Systems - A Practical Guide},
5759 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
5762 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
5763 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
5766 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
5771 @node Concept Index, , Bibliography, Top
5772 @c node-name, next, previous, up
5773 @unnumbered Concept Index