1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2001 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistical structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package Tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
216 #include <ginac/ginac.h>
218 using namespace GiNaC;
220 ex HermitePoly(const symbol & x, int n)
222 ex HKer=exp(-pow(x, 2));
223 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
224 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
231 for (int i=0; i<6; ++i)
232 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
238 When run, this will type out
244 H_3(z) == -12*z+8*z^3
245 H_4(z) == -48*z^2+16*z^4+12
246 H_5(z) == 120*z-160*z^3+32*z^5
249 This method of generating the coefficients is of course far from optimal
250 for production purposes.
252 In order to show some more examples of what GiNaC can do we will now use
253 the @command{ginsh}, a simple GiNaC interactive shell that provides a
254 convenient window into GiNaC's capabilities.
257 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
258 @c node-name, next, previous, up
259 @section What it can do for you
261 @cindex @command{ginsh}
262 After invoking @command{ginsh} one can test and experiment with GiNaC's
263 features much like in other Computer Algebra Systems except that it does
264 not provide programming constructs like loops or conditionals. For a
265 concise description of the @command{ginsh} syntax we refer to its
266 accompanied man page. Suffice to say that assignments and comparisons in
267 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
270 It can manipulate arbitrary precision integers in a very fast way.
271 Rational numbers are automatically converted to fractions of coprime
276 369988485035126972924700782451696644186473100389722973815184405301748249
278 123329495011708990974900260817232214728824366796574324605061468433916083
285 Exact numbers are always retained as exact numbers and only evaluated as
286 floating point numbers if requested. For instance, with numeric
287 radicals is dealt pretty much as with symbols. Products of sums of them
291 > expand((1+a^(1/5)-a^(2/5))^3);
292 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
293 > expand((1+3^(1/5)-3^(2/5))^3);
295 > evalf((1+3^(1/5)-3^(2/5))^3);
296 0.33408977534118624228
299 The function @code{evalf} that was used above converts any number in
300 GiNaC's expressions into floating point numbers. This can be done to
301 arbitrary predefined accuracy:
305 0.14285714285714285714
309 0.1428571428571428571428571428571428571428571428571428571428571428571428
310 5714285714285714285714285714285714285
313 Exact numbers other than rationals that can be manipulated in GiNaC
314 include predefined constants like Archimedes' @code{Pi}. They can both
315 be used in symbolic manipulations (as an exact number) as well as in
316 numeric expressions (as an inexact number):
322 9.869604401089358619+x
326 11.869604401089358619
329 Built-in functions evaluate immediately to exact numbers if
330 this is possible. Conversions that can be safely performed are done
331 immediately; conversions that are not generally valid are not done:
342 (Note that converting the last input to @code{x} would allow one to
343 conclude that @code{42*Pi} is equal to @code{0}.)
345 Linear equation systems can be solved along with basic linear
346 algebra manipulations over symbolic expressions. In C++ GiNaC offers
347 a matrix class for this purpose but we can see what it can do using
348 @command{ginsh}'s notation of double brackets to type them in:
351 > lsolve(a+x*y==z,x);
353 > lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
355 > M = [[ [[1, 3]], [[-3, 2]] ]];
356 [[ [[1,3]], [[-3,2]] ]]
359 > charpoly(M,lambda);
363 Multivariate polynomials and rational functions may be expanded,
364 collected and normalized (i.e. converted to a ratio of two coprime
368 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
369 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
370 > b = x^2 + 4*x*y - y^2;
373 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
375 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
377 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
382 You can differentiate functions and expand them as Taylor or Laurent
383 series in a very natural syntax (the second argument of @code{series} is
384 a relation defining the evaluation point, the third specifies the
387 @cindex Zeta function
391 > series(sin(x),x==0,4);
393 > series(1/tan(x),x==0,4);
394 x^(-1)-1/3*x+Order(x^2)
395 > series(tgamma(x),x==0,3);
396 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
397 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
399 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
400 -(0.90747907608088628905)*x^2+Order(x^3)
401 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
402 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
403 -Euler-1/12+Order((x-1/2*Pi)^3)
406 Here we have made use of the @command{ginsh}-command @code{"} to pop the
407 previously evaluated element from @command{ginsh}'s internal stack.
409 If you ever wanted to convert units in C or C++ and found this is
410 cumbersome, here is the solution. Symbolic types can always be used as
411 tags for different types of objects. Converting from wrong units to the
412 metric system is now easy:
420 140613.91592783185568*kg*m^(-2)
424 @node Installation, Prerequisites, What it can do for you, Top
425 @c node-name, next, previous, up
426 @chapter Installation
429 GiNaC's installation follows the spirit of most GNU software. It is
430 easily installed on your system by three steps: configuration, build,
434 * Prerequisites:: Packages upon which GiNaC depends.
435 * Configuration:: How to configure GiNaC.
436 * Building GiNaC:: How to compile GiNaC.
437 * Installing GiNaC:: How to install GiNaC on your system.
441 @node Prerequisites, Configuration, Installation, Installation
442 @c node-name, next, previous, up
443 @section Prerequisites
445 In order to install GiNaC on your system, some prerequisites need to be
446 met. First of all, you need to have a C++-compiler adhering to the
447 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
448 development so if you have a different compiler you are on your own.
449 For the configuration to succeed you need a Posix compliant shell
450 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
451 by the built process as well, since some of the source files are
452 automatically generated by Perl scripts. Last but not least, Bruno
453 Haible's library @acronym{CLN} is extensively used and needs to be
454 installed on your system. Please get it either from
455 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
456 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
457 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
458 site} (it is covered by GPL) and install it prior to trying to install
459 GiNaC. The configure script checks if it can find it and if it cannot
460 it will refuse to continue.
463 @node Configuration, Building GiNaC, Prerequisites, Installation
464 @c node-name, next, previous, up
465 @section Configuration
466 @cindex configuration
469 To configure GiNaC means to prepare the source distribution for
470 building. It is done via a shell script called @command{configure} that
471 is shipped with the sources and was originally generated by GNU
472 Autoconf. Since a configure script generated by GNU Autoconf never
473 prompts, all customization must be done either via command line
474 parameters or environment variables. It accepts a list of parameters,
475 the complete set of which can be listed by calling it with the
476 @option{--help} option. The most important ones will be shortly
477 described in what follows:
482 @option{--disable-shared}: When given, this option switches off the
483 build of a shared library, i.e. a @file{.so} file. This may be convenient
484 when developing because it considerably speeds up compilation.
487 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
488 and headers are installed. It defaults to @file{/usr/local} which means
489 that the library is installed in the directory @file{/usr/local/lib},
490 the header files in @file{/usr/local/include/ginac} and the documentation
491 (like this one) into @file{/usr/local/share/doc/GiNaC}.
494 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
495 the library installed in some other directory than
496 @file{@var{PREFIX}/lib/}.
499 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
500 to have the header files installed in some other directory than
501 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
502 @option{--includedir=/usr/include} you will end up with the header files
503 sitting in the directory @file{/usr/include/ginac/}. Note that the
504 subdirectory @file{ginac} is enforced by this process in order to
505 keep the header files separated from others. This avoids some
506 clashes and allows for an easier deinstallation of GiNaC. This ought
507 to be considered A Good Thing (tm).
510 @option{--datadir=@var{DATADIR}}: This option may be given in case you
511 want to have the documentation installed in some other directory than
512 @file{@var{PREFIX}/share/doc/GiNaC/}.
516 In addition, you may specify some environment variables.
517 @env{CXX} holds the path and the name of the C++ compiler
518 in case you want to override the default in your path. (The
519 @command{configure} script searches your path for @command{c++},
520 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
521 and @command{cc++} in that order.) It may be very useful to
522 define some compiler flags with the @env{CXXFLAGS} environment
523 variable, like optimization, debugging information and warning
524 levels. If omitted, it defaults to @option{-g -O2}.
526 The whole process is illustrated in the following two
527 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
528 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
531 Here is a simple configuration for a site-wide GiNaC library assuming
532 everything is in default paths:
535 $ export CXXFLAGS="-Wall -O2"
539 And here is a configuration for a private static GiNaC library with
540 several components sitting in custom places (site-wide @acronym{GCC} and
541 private @acronym{CLN}). The compiler is pursuaded to be picky and full
542 assertions and debugging information are switched on:
545 $ export CXX=/usr/local/gnu/bin/c++
546 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
547 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
548 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
549 $ ./configure --disable-shared --prefix=$(HOME)
553 @node Building GiNaC, Installing GiNaC, Configuration, Installation
554 @c node-name, next, previous, up
555 @section Building GiNaC
556 @cindex building GiNaC
558 After proper configuration you should just build the whole
563 at the command prompt and go for a cup of coffee. The exact time it
564 takes to compile GiNaC depends not only on the speed of your machines
565 but also on other parameters, for instance what value for @env{CXXFLAGS}
566 you entered. Optimization may be very time-consuming.
568 Just to make sure GiNaC works properly you may run a collection of
569 regression tests by typing
575 This will compile some sample programs, run them and check the output
576 for correctness. The regression tests fall in three categories. First,
577 the so called @emph{exams} are performed, simple tests where some
578 predefined input is evaluated (like a pupils' exam). Second, the
579 @emph{checks} test the coherence of results among each other with
580 possible random input. Third, some @emph{timings} are performed, which
581 benchmark some predefined problems with different sizes and display the
582 CPU time used in seconds. Each individual test should return a message
583 @samp{passed}. This is mostly intended to be a QA-check if something
584 was broken during development, not a sanity check of your system. Some
585 of the tests in sections @emph{checks} and @emph{timings} may require
586 insane amounts of memory and CPU time. Feel free to kill them if your
587 machine catches fire. Another quite important intent is to allow people
588 to fiddle around with optimization.
590 Generally, the top-level Makefile runs recursively to the
591 subdirectories. It is therfore safe to go into any subdirectory
592 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
593 @var{target} there in case something went wrong.
596 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
597 @c node-name, next, previous, up
598 @section Installing GiNaC
601 To install GiNaC on your system, simply type
607 As described in the section about configuration the files will be
608 installed in the following directories (the directories will be created
609 if they don't already exist):
614 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
615 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
616 So will @file{libginac.so} unless the configure script was
617 given the option @option{--disable-shared}. The proper symlinks
618 will be established as well.
621 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
622 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
625 All documentation (HTML and Postscript) will be stuffed into
626 @file{@var{PREFIX}/share/doc/GiNaC/} (or
627 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
631 For the sake of completeness we will list some other useful make
632 targets: @command{make clean} deletes all files generated by
633 @command{make}, i.e. all the object files. In addition @command{make
634 distclean} removes all files generated by the configuration and
635 @command{make maintainer-clean} goes one step further and deletes files
636 that may require special tools to rebuild (like the @command{libtool}
637 for instance). Finally @command{make uninstall} removes the installed
638 library, header files and documentation@footnote{Uninstallation does not
639 work after you have called @command{make distclean} since the
640 @file{Makefile} is itself generated by the configuration from
641 @file{Makefile.in} and hence deleted by @command{make distclean}. There
642 are two obvious ways out of this dilemma. First, you can run the
643 configuration again with the same @var{PREFIX} thus creating a
644 @file{Makefile} with a working @samp{uninstall} target. Second, you can
645 do it by hand since you now know where all the files went during
649 @node Basic Concepts, Expressions, Installing GiNaC, Top
650 @c node-name, next, previous, up
651 @chapter Basic Concepts
653 This chapter will describe the different fundamental objects that can be
654 handled by GiNaC. But before doing so, it is worthwhile introducing you
655 to the more commonly used class of expressions, representing a flexible
656 meta-class for storing all mathematical objects.
659 * Expressions:: The fundamental GiNaC class.
660 * The Class Hierarchy:: Overview of GiNaC's classes.
661 * Symbols:: Symbolic objects.
662 * Numbers:: Numerical objects.
663 * Constants:: Pre-defined constants.
664 * Fundamental containers:: The power, add and mul classes.
665 * Lists:: Lists of expressions.
666 * Mathematical functions:: Mathematical functions.
667 * Relations:: Equality, Inequality and all that.
668 * Indexed objects:: Handling indexed quantities.
669 * Non-commutative objects:: Algebras with non-commutative products.
673 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
674 @c node-name, next, previous, up
676 @cindex expression (class @code{ex})
679 The most common class of objects a user deals with is the expression
680 @code{ex}, representing a mathematical object like a variable, number,
681 function, sum, product, etc... Expressions may be put together to form
682 new expressions, passed as arguments to functions, and so on. Here is a
683 little collection of valid expressions:
686 ex MyEx1 = 5; // simple number
687 ex MyEx2 = x + 2*y; // polynomial in x and y
688 ex MyEx3 = (x + 1)/(x - 1); // rational expression
689 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
690 ex MyEx5 = MyEx4 + 1; // similar to above
693 Expressions are handles to other more fundamental objects, that often
694 contain other expressions thus creating a tree of expressions
695 (@xref{Internal Structures}, for particular examples). Most methods on
696 @code{ex} therefore run top-down through such an expression tree. For
697 example, the method @code{has()} scans recursively for occurrences of
698 something inside an expression. Thus, if you have declared @code{MyEx4}
699 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
700 the argument of @code{sin} and hence return @code{true}.
702 The next sections will outline the general picture of GiNaC's class
703 hierarchy and describe the classes of objects that are handled by
707 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
708 @c node-name, next, previous, up
709 @section The Class Hierarchy
711 GiNaC's class hierarchy consists of several classes representing
712 mathematical objects, all of which (except for @code{ex} and some
713 helpers) are internally derived from one abstract base class called
714 @code{basic}. You do not have to deal with objects of class
715 @code{basic}, instead you'll be dealing with symbols, numbers,
716 containers of expressions and so on.
720 To get an idea about what kinds of symbolic composits may be built we
721 have a look at the most important classes in the class hierarchy and
722 some of the relations among the classes:
724 @image{classhierarchy}
726 The abstract classes shown here (the ones without drop-shadow) are of no
727 interest for the user. They are used internally in order to avoid code
728 duplication if two or more classes derived from them share certain
729 features. An example is @code{expairseq}, a container for a sequence of
730 pairs each consisting of one expression and a number (@code{numeric}).
731 What @emph{is} visible to the user are the derived classes @code{add}
732 and @code{mul}, representing sums and products. @xref{Internal
733 Structures}, where these two classes are described in more detail. The
734 following table shortly summarizes what kinds of mathematical objects
735 are stored in the different classes:
738 @multitable @columnfractions .22 .78
739 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
740 @item @code{constant} @tab Constants like
747 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
748 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
749 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
750 @item @code{ncmul} @tab Products of non-commutative objects
751 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
756 @code{sqrt(}@math{2}@code{)}
759 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
760 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
761 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
762 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
763 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
764 @item @code{indexed} @tab Indexed object like @math{A_ij}
765 @item @code{tensor} @tab Special tensor like the delta and metric tensors
766 @item @code{idx} @tab Index of an indexed object
767 @item @code{varidx} @tab Index with variance
768 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
769 @item @code{wildcard} @tab Wildcard for pattern matching
773 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
774 @c node-name, next, previous, up
776 @cindex @code{symbol} (class)
777 @cindex hierarchy of classes
780 Symbols are for symbolic manipulation what atoms are for chemistry. You
781 can declare objects of class @code{symbol} as any other object simply by
782 saying @code{symbol x,y;}. There is, however, a catch in here having to
783 do with the fact that C++ is a compiled language. The information about
784 the symbol's name is thrown away by the compiler but at a later stage
785 you may want to print expressions holding your symbols. In order to
786 avoid confusion GiNaC's symbols are able to know their own name. This
787 is accomplished by declaring its name for output at construction time in
788 the fashion @code{symbol x("x");}. If you declare a symbol using the
789 default constructor (i.e. without string argument) the system will deal
790 out a unique name. That name may not be suitable for printing but for
791 internal routines when no output is desired it is often enough. We'll
792 come across examples of such symbols later in this tutorial.
794 This implies that the strings passed to symbols at construction time may
795 not be used for comparing two of them. It is perfectly legitimate to
796 write @code{symbol x("x"),y("x");} but it is likely to lead into
797 trouble. Here, @code{x} and @code{y} are different symbols and
798 statements like @code{x-y} will not be simplified to zero although the
799 output @code{x-x} looks funny. Such output may also occur when there
800 are two different symbols in two scopes, for instance when you call a
801 function that declares a symbol with a name already existent in a symbol
802 in the calling function. Again, comparing them (using @code{operator==}
803 for instance) will always reveal their difference. Watch out, please.
805 @cindex @code{subs()}
806 Although symbols can be assigned expressions for internal reasons, you
807 should not do it (and we are not going to tell you how it is done). If
808 you want to replace a symbol with something else in an expression, you
809 can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
810 for more information).
813 @node Numbers, Constants, Symbols, Basic Concepts
814 @c node-name, next, previous, up
816 @cindex @code{numeric} (class)
822 For storing numerical things, GiNaC uses Bruno Haible's library
823 @acronym{CLN}. The classes therein serve as foundation classes for
824 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
825 alternatively for Common Lisp Numbers. In order to find out more about
826 @acronym{CLN}'s internals the reader is refered to the documentation of
827 that library. @inforef{Introduction, , cln}, for more
828 information. Suffice to say that it is by itself build on top of another
829 library, the GNU Multiple Precision library @acronym{GMP}, which is an
830 extremely fast library for arbitrary long integers and rationals as well
831 as arbitrary precision floating point numbers. It is very commonly used
832 by several popular cryptographic applications. @acronym{CLN} extends
833 @acronym{GMP} by several useful things: First, it introduces the complex
834 number field over either reals (i.e. floating point numbers with
835 arbitrary precision) or rationals. Second, it automatically converts
836 rationals to integers if the denominator is unity and complex numbers to
837 real numbers if the imaginary part vanishes and also correctly treats
838 algebraic functions. Third it provides good implementations of
839 state-of-the-art algorithms for all trigonometric and hyperbolic
840 functions as well as for calculation of some useful constants.
842 The user can construct an object of class @code{numeric} in several
843 ways. The following example shows the four most important constructors.
844 It uses construction from C-integer, construction of fractions from two
845 integers, construction from C-float and construction from a string:
848 #include <ginac/ginac.h>
849 using namespace GiNaC;
853 numeric two(2); // exact integer 2
854 numeric r(2,3); // exact fraction 2/3
855 numeric e(2.71828); // floating point number
856 numeric p("3.1415926535897932385"); // floating point number
857 // Trott's constant in scientific notation:
858 numeric trott("1.0841015122311136151E-2");
860 std::cout << two*p << std::endl; // floating point 6.283...
864 Note that all those constructors are @emph{explicit} which means you are
865 not allowed to write @code{numeric two=2;}. This is because the basic
866 objects to be handled by GiNaC are the expressions @code{ex} and we want
867 to keep things simple and wish objects like @code{pow(x,2)} to be
868 handled the same way as @code{pow(x,a)}, which means that we need to
869 allow a general @code{ex} as base and exponent. Therefore there is an
870 implicit constructor from C-integers directly to expressions handling
871 numerics at work in most of our examples. This design really becomes
872 convenient when one declares own functions having more than one
873 parameter but it forbids using implicit constructors because that would
874 lead to compile-time ambiguities.
876 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
877 This would, however, call C's built-in operator @code{/} for integers
878 first and result in a numeric holding a plain integer 1. @strong{Never
879 use the operator @code{/} on integers} unless you know exactly what you
880 are doing! Use the constructor from two integers instead, as shown in
881 the example above. Writing @code{numeric(1)/2} may look funny but works
884 @cindex @code{Digits}
886 We have seen now the distinction between exact numbers and floating
887 point numbers. Clearly, the user should never have to worry about
888 dynamically created exact numbers, since their `exactness' always
889 determines how they ought to be handled, i.e. how `long' they are. The
890 situation is different for floating point numbers. Their accuracy is
891 controlled by one @emph{global} variable, called @code{Digits}. (For
892 those readers who know about Maple: it behaves very much like Maple's
893 @code{Digits}). All objects of class numeric that are constructed from
894 then on will be stored with a precision matching that number of decimal
898 #include <ginac/ginac.h>
900 using namespace GiNaC;
904 numeric three(3.0), one(1.0);
905 numeric x = one/three;
907 cout << "in " << Digits << " digits:" << endl;
909 cout << Pi.evalf() << endl;
921 The above example prints the following output to screen:
928 0.333333333333333333333333333333333333333333333333333333333333333333
929 3.14159265358979323846264338327950288419716939937510582097494459231
932 It should be clear that objects of class @code{numeric} should be used
933 for constructing numbers or for doing arithmetic with them. The objects
934 one deals with most of the time are the polymorphic expressions @code{ex}.
936 @subsection Tests on numbers
938 Once you have declared some numbers, assigned them to expressions and
939 done some arithmetic with them it is frequently desired to retrieve some
940 kind of information from them like asking whether that number is
941 integer, rational, real or complex. For those cases GiNaC provides
942 several useful methods. (Internally, they fall back to invocations of
943 certain CLN functions.)
945 As an example, let's construct some rational number, multiply it with
946 some multiple of its denominator and test what comes out:
949 #include <ginac/ginac.h>
951 using namespace GiNaC;
953 // some very important constants:
954 const numeric twentyone(21);
955 const numeric ten(10);
956 const numeric five(5);
960 numeric answer = twentyone;
963 cout << answer.is_integer() << endl; // false, it's 21/5
965 cout << answer.is_integer() << endl; // true, it's 42 now!
969 Note that the variable @code{answer} is constructed here as an integer
970 by @code{numeric}'s copy constructor but in an intermediate step it
971 holds a rational number represented as integer numerator and integer
972 denominator. When multiplied by 10, the denominator becomes unity and
973 the result is automatically converted to a pure integer again.
974 Internally, the underlying @acronym{CLN} is responsible for this
975 behaviour and we refer the reader to @acronym{CLN}'s documentation.
976 Suffice to say that the same behaviour applies to complex numbers as
977 well as return values of certain functions. Complex numbers are
978 automatically converted to real numbers if the imaginary part becomes
979 zero. The full set of tests that can be applied is listed in the
983 @multitable @columnfractions .30 .70
984 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
985 @item @code{.is_zero()}
986 @tab @dots{}equal to zero
987 @item @code{.is_positive()}
988 @tab @dots{}not complex and greater than 0
989 @item @code{.is_integer()}
990 @tab @dots{}a (non-complex) integer
991 @item @code{.is_pos_integer()}
992 @tab @dots{}an integer and greater than 0
993 @item @code{.is_nonneg_integer()}
994 @tab @dots{}an integer and greater equal 0
995 @item @code{.is_even()}
996 @tab @dots{}an even integer
997 @item @code{.is_odd()}
998 @tab @dots{}an odd integer
999 @item @code{.is_prime()}
1000 @tab @dots{}a prime integer (probabilistic primality test)
1001 @item @code{.is_rational()}
1002 @tab @dots{}an exact rational number (integers are rational, too)
1003 @item @code{.is_real()}
1004 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1005 @item @code{.is_cinteger()}
1006 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1007 @item @code{.is_crational()}
1008 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1013 @node Constants, Fundamental containers, Numbers, Basic Concepts
1014 @c node-name, next, previous, up
1016 @cindex @code{constant} (class)
1019 @cindex @code{Catalan}
1020 @cindex @code{Euler}
1021 @cindex @code{evalf()}
1022 Constants behave pretty much like symbols except that they return some
1023 specific number when the method @code{.evalf()} is called.
1025 The predefined known constants are:
1028 @multitable @columnfractions .14 .30 .56
1029 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1031 @tab Archimedes' constant
1032 @tab 3.14159265358979323846264338327950288
1033 @item @code{Catalan}
1034 @tab Catalan's constant
1035 @tab 0.91596559417721901505460351493238411
1037 @tab Euler's (or Euler-Mascheroni) constant
1038 @tab 0.57721566490153286060651209008240243
1043 @node Fundamental containers, Lists, Constants, Basic Concepts
1044 @c node-name, next, previous, up
1045 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1049 @cindex @code{power}
1051 Simple polynomial expressions are written down in GiNaC pretty much like
1052 in other CAS or like expressions involving numerical variables in C.
1053 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1054 been overloaded to achieve this goal. When you run the following
1055 code snippet, the constructor for an object of type @code{mul} is
1056 automatically called to hold the product of @code{a} and @code{b} and
1057 then the constructor for an object of type @code{add} is called to hold
1058 the sum of that @code{mul} object and the number one:
1062 symbol a("a"), b("b");
1067 @cindex @code{pow()}
1068 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1069 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1070 construction is necessary since we cannot safely overload the constructor
1071 @code{^} in C++ to construct a @code{power} object. If we did, it would
1072 have several counterintuitive and undesired effects:
1076 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1078 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1079 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1080 interpret this as @code{x^(a^b)}.
1082 Also, expressions involving integer exponents are very frequently used,
1083 which makes it even more dangerous to overload @code{^} since it is then
1084 hard to distinguish between the semantics as exponentiation and the one
1085 for exclusive or. (It would be embarassing to return @code{1} where one
1086 has requested @code{2^3}.)
1089 @cindex @command{ginsh}
1090 All effects are contrary to mathematical notation and differ from the
1091 way most other CAS handle exponentiation, therefore overloading @code{^}
1092 is ruled out for GiNaC's C++ part. The situation is different in
1093 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1094 that the other frequently used exponentiation operator @code{**} does
1095 not exist at all in C++).
1097 To be somewhat more precise, objects of the three classes described
1098 here, are all containers for other expressions. An object of class
1099 @code{power} is best viewed as a container with two slots, one for the
1100 basis, one for the exponent. All valid GiNaC expressions can be
1101 inserted. However, basic transformations like simplifying
1102 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1103 when this is mathematically possible. If we replace the outer exponent
1104 three in the example by some symbols @code{a}, the simplification is not
1105 safe and will not be performed, since @code{a} might be @code{1/2} and
1108 Objects of type @code{add} and @code{mul} are containers with an
1109 arbitrary number of slots for expressions to be inserted. Again, simple
1110 and safe simplifications are carried out like transforming
1111 @code{3*x+4-x} to @code{2*x+4}.
1113 The general rule is that when you construct such objects, GiNaC
1114 automatically creates them in canonical form, which might differ from
1115 the form you typed in your program. This allows for rapid comparison of
1116 expressions, since after all @code{a-a} is simply zero. Note, that the
1117 canonical form is not necessarily lexicographical ordering or in any way
1118 easily guessable. It is only guaranteed that constructing the same
1119 expression twice, either implicitly or explicitly, results in the same
1123 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1124 @c node-name, next, previous, up
1125 @section Lists of expressions
1126 @cindex @code{lst} (class)
1128 @cindex @code{nops()}
1130 @cindex @code{append()}
1131 @cindex @code{prepend()}
1133 The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
1134 These are sometimes used to supply a variable number of arguments of the same
1135 type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
1136 should have a basic understanding about them.
1138 Lists of up to 15 expressions can be directly constructed from single
1143 symbol x("x"), y("y");
1144 lst l(x, 2, y, x+y);
1145 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1149 Use the @code{nops()} method to determine the size (number of expressions) of
1150 a list and the @code{op()} method to access individual elements:
1154 cout << l.nops() << endl; // prints '4'
1155 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1159 Finally you can append or prepend an expression to a list with the
1160 @code{append()} and @code{prepend()} methods:
1164 l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
1165 l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
1170 @node Mathematical functions, Relations, Lists, Basic Concepts
1171 @c node-name, next, previous, up
1172 @section Mathematical functions
1173 @cindex @code{function} (class)
1174 @cindex trigonometric function
1175 @cindex hyperbolic function
1177 There are quite a number of useful functions hard-wired into GiNaC. For
1178 instance, all trigonometric and hyperbolic functions are implemented
1179 (@xref{Built-in Functions}, for a complete list).
1181 These functions are all objects of class @code{function}. They accept
1182 one or more expressions as arguments and return one expression. If the
1183 arguments are not numerical, the evaluation of the function may be
1184 halted, as it does in the next example, showing how a function returns
1185 itself twice and finally an expression that may be really useful:
1187 @cindex Gamma function
1188 @cindex @code{subs()}
1191 symbol x("x"), y("y");
1193 cout << tgamma(foo) << endl;
1194 // -> tgamma(x+(1/2)*y)
1195 ex bar = foo.subs(y==1);
1196 cout << tgamma(bar) << endl;
1198 ex foobar = bar.subs(x==7);
1199 cout << tgamma(foobar) << endl;
1200 // -> (135135/128)*Pi^(1/2)
1204 Besides evaluation most of these functions allow differentiation, series
1205 expansion and so on. Read the next chapter in order to learn more about
1209 @node Relations, Indexed objects, Mathematical functions, Basic Concepts
1210 @c node-name, next, previous, up
1212 @cindex @code{relational} (class)
1214 Sometimes, a relation holding between two expressions must be stored
1215 somehow. The class @code{relational} is a convenient container for such
1216 purposes. A relation is by definition a container for two @code{ex} and
1217 a relation between them that signals equality, inequality and so on.
1218 They are created by simply using the C++ operators @code{==}, @code{!=},
1219 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1221 @xref{Mathematical functions}, for examples where various applications
1222 of the @code{.subs()} method show how objects of class relational are
1223 used as arguments. There they provide an intuitive syntax for
1224 substitutions. They are also used as arguments to the @code{ex::series}
1225 method, where the left hand side of the relation specifies the variable
1226 to expand in and the right hand side the expansion point. They can also
1227 be used for creating systems of equations that are to be solved for
1228 unknown variables. But the most common usage of objects of this class
1229 is rather inconspicuous in statements of the form @code{if
1230 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1231 conversion from @code{relational} to @code{bool} takes place. Note,
1232 however, that @code{==} here does not perform any simplifications, hence
1233 @code{expand()} must be called explicitly.
1236 @node Indexed objects, Non-commutative objects, Relations, Basic Concepts
1237 @c node-name, next, previous, up
1238 @section Indexed objects
1240 GiNaC allows you to handle expressions containing general indexed objects in
1241 arbitrary spaces. It is also able to canonicalize and simplify such
1242 expressions and perform symbolic dummy index summations. There are a number
1243 of predefined indexed objects provided, like delta and metric tensors.
1245 There are few restrictions placed on indexed objects and their indices and
1246 it is easy to construct nonsense expressions, but our intention is to
1247 provide a general framework that allows you to implement algorithms with
1248 indexed quantities, getting in the way as little as possible.
1250 @cindex @code{idx} (class)
1251 @cindex @code{indexed} (class)
1252 @subsection Indexed quantities and their indices
1254 Indexed expressions in GiNaC are constructed of two special types of objects,
1255 @dfn{index objects} and @dfn{indexed objects}.
1259 @cindex contravariant
1262 @item Index objects are of class @code{idx} or a subclass. Every index has
1263 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1264 the index lives in) which can both be arbitrary expressions but are usually
1265 a number or a simple symbol. In addition, indices of class @code{varidx} have
1266 a @dfn{variance} (they can be co- or contravariant), and indices of class
1267 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1269 @item Indexed objects are of class @code{indexed} or a subclass. They
1270 contain a @dfn{base expression} (which is the expression being indexed), and
1271 one or more indices.
1275 @strong{Note:} when printing expressions, covariant indices and indices
1276 without variance are denoted @samp{.i} while contravariant indices are
1277 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1278 value. In the following, we are going to use that notation in the text so
1279 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1280 not visible in the output.
1282 A simple example shall illustrate the concepts:
1285 #include <ginac/ginac.h>
1286 using namespace std;
1287 using namespace GiNaC;
1291 symbol i_sym("i"), j_sym("j");
1292 idx i(i_sym, 3), j(j_sym, 3);
1295 cout << indexed(A, i, j) << endl;
1300 The @code{idx} constructor takes two arguments, the index value and the
1301 index dimension. First we define two index objects, @code{i} and @code{j},
1302 both with the numeric dimension 3. The value of the index @code{i} is the
1303 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1304 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1305 construct an expression containing one indexed object, @samp{A.i.j}. It has
1306 the symbol @code{A} as its base expression and the two indices @code{i} and
1309 Note the difference between the indices @code{i} and @code{j} which are of
1310 class @code{idx}, and the index values which are the sybols @code{i_sym}
1311 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1312 or numbers but must be index objects. For example, the following is not
1313 correct and will raise an exception:
1316 symbol i("i"), j("j");
1317 e = indexed(A, i, j); // ERROR: indices must be of type idx
1320 You can have multiple indexed objects in an expression, index values can
1321 be numeric, and index dimensions symbolic:
1325 symbol B("B"), dim("dim");
1326 cout << 4 * indexed(A, i)
1327 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1332 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1333 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1334 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1335 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1336 @code{simplify_indexed()} for that, see below).
1338 In fact, base expressions, index values and index dimensions can be
1339 arbitrary expressions:
1343 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1348 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1349 get an error message from this but you will probably not be able to do
1350 anything useful with it.
1352 @cindex @code{get_value()}
1353 @cindex @code{get_dimension()}
1357 ex idx::get_value(void);
1358 ex idx::get_dimension(void);
1361 return the value and dimension of an @code{idx} object. If you have an index
1362 in an expression, such as returned by calling @code{.op()} on an indexed
1363 object, you can get a reference to the @code{idx} object with the function
1364 @code{ex_to_idx()} on the expression.
1366 There are also the methods
1369 bool idx::is_numeric(void);
1370 bool idx::is_symbolic(void);
1371 bool idx::is_dim_numeric(void);
1372 bool idx::is_dim_symbolic(void);
1375 for checking whether the value and dimension are numeric or symbolic
1376 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1377 About Expressions}) returns information about the index value.
1379 @cindex @code{varidx} (class)
1380 If you need co- and contravariant indices, use the @code{varidx} class:
1384 symbol mu_sym("mu"), nu_sym("nu");
1385 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1386 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1388 cout << indexed(A, mu, nu) << endl;
1390 cout << indexed(A, mu_co, nu) << endl;
1392 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1397 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1398 co- or contravariant. The default is a contravariant (upper) index, but
1399 this can be overridden by supplying a third argument to the @code{varidx}
1400 constructor. The two methods
1403 bool varidx::is_covariant(void);
1404 bool varidx::is_contravariant(void);
1407 allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
1408 to get the object reference from an expression). There's also the very useful
1412 ex varidx::toggle_variance(void);
1415 which makes a new index with the same value and dimension but the opposite
1416 variance. By using it you only have to define the index once.
1418 @cindex @code{spinidx} (class)
1419 The @code{spinidx} class provides dotted and undotted variant indices, as
1420 used in the Weyl-van-der-Waerden spinor formalism:
1424 symbol K("K"), C_sym("C"), D_sym("D");
1425 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1426 // contravariant, undotted
1427 spinidx C_co(C_sym, 2, true); // covariant index
1428 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1429 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1431 cout << indexed(K, C, D) << endl;
1433 cout << indexed(K, C_co, D_dot) << endl;
1435 cout << indexed(K, D_co_dot, D) << endl;
1440 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1441 dotted or undotted. The default is undotted but this can be overridden by
1442 supplying a fourth argument to the @code{spinidx} constructor. The two
1446 bool spinidx::is_dotted(void);
1447 bool spinidx::is_undotted(void);
1450 allow you to check whether or not a @code{spinidx} object is dotted (use
1451 @code{ex_to_spinidx()} to get the object reference from an expression).
1452 Finally, the two methods
1455 ex spinidx::toggle_dot(void);
1456 ex spinidx::toggle_variance_dot(void);
1459 create a new index with the same value and dimension but opposite dottedness
1460 and the same or opposite variance.
1462 @subsection Substituting indices
1464 @cindex @code{subs()}
1465 Sometimes you will want to substitute one symbolic index with another
1466 symbolic or numeric index, for example when calculating one specific element
1467 of a tensor expression. This is done with the @code{.subs()} method, as it
1468 is done for symbols (see @ref{Substituting Expressions}).
1470 You have two possibilities here. You can either substitute the whole index
1471 by another index or expression:
1475 ex e = indexed(A, mu_co);
1476 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1477 // -> A.mu becomes A~nu
1478 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1479 // -> A.mu becomes A~0
1480 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1481 // -> A.mu becomes A.0
1485 The third example shows that trying to replace an index with something that
1486 is not an index will substitute the index value instead.
1488 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1493 ex e = indexed(A, mu_co);
1494 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1495 // -> A.mu becomes A.nu
1496 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1497 // -> A.mu becomes A.0
1501 As you see, with the second method only the value of the index will get
1502 substituted. Its other properties, including its dimension, remain unchanged.
1503 If you want to change the dimension of an index you have to substitute the
1504 whole index by another one with the new dimension.
1506 Finally, substituting the base expression of an indexed object works as
1511 ex e = indexed(A, mu_co);
1512 cout << e << " becomes " << e.subs(A == A+B) << endl;
1513 // -> A.mu becomes (B+A).mu
1517 @subsection Symmetries
1519 Indexed objects can be declared as being totally symmetric or antisymmetric
1520 with respect to their indices. In this case, GiNaC will automatically bring
1521 the indices into a canonical order which allows for some immediate
1526 cout << indexed(A, indexed::symmetric, i, j)
1527 + indexed(A, indexed::symmetric, j, i) << endl;
1529 cout << indexed(B, indexed::antisymmetric, i, j)
1530 + indexed(B, indexed::antisymmetric, j, j) << endl;
1532 cout << indexed(B, indexed::antisymmetric, i, j)
1533 + indexed(B, indexed::antisymmetric, j, i) << endl;
1538 @cindex @code{get_free_indices()}
1540 @subsection Dummy indices
1542 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1543 that a summation over the index range is implied. Symbolic indices which are
1544 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1545 dummy nor free indices.
1547 To be recognized as a dummy index pair, the two indices must be of the same
1548 class and dimension and their value must be the same single symbol (an index
1549 like @samp{2*n+1} is never a dummy index). If the indices are of class
1550 @code{varidx} they must also be of opposite variance; if they are of class
1551 @code{spinidx} they must be both dotted or both undotted.
1553 The method @code{.get_free_indices()} returns a vector containing the free
1554 indices of an expression. It also checks that the free indices of the terms
1555 of a sum are consistent:
1559 symbol A("A"), B("B"), C("C");
1561 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1562 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1564 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1565 cout << exprseq(e.get_free_indices()) << endl;
1567 // 'j' and 'l' are dummy indices
1569 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1570 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1572 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1573 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1574 cout << exprseq(e.get_free_indices()) << endl;
1576 // 'nu' is a dummy index, but 'sigma' is not
1578 e = indexed(A, mu, mu);
1579 cout << exprseq(e.get_free_indices()) << endl;
1581 // 'mu' is not a dummy index because it appears twice with the same
1584 e = indexed(A, mu, nu) + 42;
1585 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1586 // this will throw an exception:
1587 // "add::get_free_indices: inconsistent indices in sum"
1591 @cindex @code{simplify_indexed()}
1592 @subsection Simplifying indexed expressions
1594 In addition to the few automatic simplifications that GiNaC performs on
1595 indexed expressions (such as re-ordering the indices of symmetric tensors
1596 and calculating traces and convolutions of matrices and predefined tensors)
1600 ex ex::simplify_indexed(void);
1601 ex ex::simplify_indexed(const scalar_products & sp);
1604 that performs some more expensive operations:
1607 @item it checks the consistency of free indices in sums in the same way
1608 @code{get_free_indices()} does
1609 @item it tries to give dumy indices that appear in different terms of a sum
1610 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1611 @item it (symbolically) calculates all possible dummy index summations/contractions
1612 with the predefined tensors (this will be explained in more detail in the
1614 @item as a special case of dummy index summation, it can replace scalar products
1615 of two tensors with a user-defined value
1618 The last point is done with the help of the @code{scalar_products} class
1619 which is used to store scalar products with known values (this is not an
1620 arithmetic class, you just pass it to @code{simplify_indexed()}):
1624 symbol A("A"), B("B"), C("C"), i_sym("i");
1628 sp.add(A, B, 0); // A and B are orthogonal
1629 sp.add(A, C, 0); // A and C are orthogonal
1630 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1632 e = indexed(A + B, i) * indexed(A + C, i);
1634 // -> (B+A).i*(A+C).i
1636 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1642 The @code{scalar_products} object @code{sp} acts as a storage for the
1643 scalar products added to it with the @code{.add()} method. This method
1644 takes three arguments: the two expressions of which the scalar product is
1645 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1646 @code{simplify_indexed()} will replace all scalar products of indexed
1647 objects that have the symbols @code{A} and @code{B} as base expressions
1648 with the single value 0. The number, type and dimension of the indices
1649 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1651 @cindex @code{expand()}
1652 The example above also illustrates a feature of the @code{expand()} method:
1653 if passed the @code{expand_indexed} option it will distribute indices
1654 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1656 @cindex @code{tensor} (class)
1657 @subsection Predefined tensors
1659 Some frequently used special tensors such as the delta, epsilon and metric
1660 tensors are predefined in GiNaC. They have special properties when
1661 contracted with other tensor expressions and some of them have constant
1662 matrix representations (they will evaluate to a number when numeric
1663 indices are specified).
1665 @cindex @code{delta_tensor()}
1666 @subsubsection Delta tensor
1668 The delta tensor takes two indices, is symmetric and has the matrix
1669 representation @code{diag(1,1,1,...)}. It is constructed by the function
1670 @code{delta_tensor()}:
1674 symbol A("A"), B("B");
1676 idx i(symbol("i"), 3), j(symbol("j"), 3),
1677 k(symbol("k"), 3), l(symbol("l"), 3);
1679 ex e = indexed(A, i, j) * indexed(B, k, l)
1680 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1681 cout << e.simplify_indexed() << endl;
1684 cout << delta_tensor(i, i) << endl;
1689 @cindex @code{metric_tensor()}
1690 @subsubsection General metric tensor
1692 The function @code{metric_tensor()} creates a general symmetric metric
1693 tensor with two indices that can be used to raise/lower tensor indices. The
1694 metric tensor is denoted as @samp{g} in the output and if its indices are of
1695 mixed variance it is automatically replaced by a delta tensor:
1701 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1703 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1704 cout << e.simplify_indexed() << endl;
1707 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1708 cout << e.simplify_indexed() << endl;
1711 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1712 * metric_tensor(nu, rho);
1713 cout << e.simplify_indexed() << endl;
1716 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1717 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1718 + indexed(A, mu.toggle_variance(), rho));
1719 cout << e.simplify_indexed() << endl;
1724 @cindex @code{lorentz_g()}
1725 @subsubsection Minkowski metric tensor
1727 The Minkowski metric tensor is a special metric tensor with a constant
1728 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1729 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1730 It is created with the function @code{lorentz_g()} (although it is output as
1735 varidx mu(symbol("mu"), 4);
1737 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1738 * lorentz_g(mu, varidx(0, 4)); // negative signature
1739 cout << e.simplify_indexed() << endl;
1742 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1743 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1744 cout << e.simplify_indexed() << endl;
1749 @cindex @code{spinor_metric()}
1750 @subsubsection Spinor metric tensor
1752 The function @code{spinor_metric()} creates an antisymmetric tensor with
1753 two indices that is used to raise/lower indices of 2-component spinors.
1754 It is output as @samp{eps}:
1760 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
1761 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
1763 e = spinor_metric(A, B) * indexed(psi, B_co);
1764 cout << e.simplify_indexed() << endl;
1767 e = spinor_metric(A, B) * indexed(psi, A_co);
1768 cout << e.simplify_indexed() << endl;
1771 e = spinor_metric(A_co, B_co) * indexed(psi, B);
1772 cout << e.simplify_indexed() << endl;
1775 e = spinor_metric(A_co, B_co) * indexed(psi, A);
1776 cout << e.simplify_indexed() << endl;
1779 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
1780 cout << e.simplify_indexed() << endl;
1783 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
1784 cout << e.simplify_indexed() << endl;
1789 The matrix representation of the spinor metric is @code{[[ [[ 0, 1 ]], [[ -1, 0 ]] ]]}.
1791 @cindex @code{epsilon_tensor()}
1792 @cindex @code{lorentz_eps()}
1793 @subsubsection Epsilon tensor
1795 The epsilon tensor is totally antisymmetric, its number of indices is equal
1796 to the dimension of the index space (the indices must all be of the same
1797 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1798 defined to be 1. Its behaviour with indices that have a variance also
1799 depends on the signature of the metric. Epsilon tensors are output as
1802 There are three functions defined to create epsilon tensors in 2, 3 and 4
1806 ex epsilon_tensor(const ex & i1, const ex & i2);
1807 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1808 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1811 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1812 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1813 Minkowski space (the last @code{bool} argument specifies whether the metric
1814 has negative or positive signature, as in the case of the Minkowski metric
1817 @subsection Linear algebra
1819 The @code{matrix} class can be used with indices to do some simple linear
1820 algebra (linear combinations and products of vectors and matrices, traces
1821 and scalar products):
1825 idx i(symbol("i"), 2), j(symbol("j"), 2);
1826 symbol x("x"), y("y");
1828 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
1830 cout << indexed(A, i, i) << endl;
1833 ex e = indexed(A, i, j) * indexed(X, j);
1834 cout << e.simplify_indexed() << endl;
1835 // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
1837 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
1838 cout << e.simplify_indexed() << endl;
1839 // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
1843 You can of course obtain the same results with the @code{matrix::add()},
1844 @code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
1845 don't have to worry about transposing matrices.
1847 Matrix indices always start at 0 and their dimension must match the number
1848 of rows/columns of the matrix. Matrices with one row or one column are
1849 vectors and can have one or two indices (it doesn't matter whether it's a
1850 row or a column vector). Other matrices must have two indices.
1852 You should be careful when using indices with variance on matrices. GiNaC
1853 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
1854 @samp{F.mu.nu} are different matrices. In this case you should use only
1855 one form for @samp{F} and explicitly multiply it with a matrix representation
1856 of the metric tensor.
1859 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
1860 @c node-name, next, previous, up
1861 @section Non-commutative objects
1863 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
1864 non-commutative objects are built-in which are mostly of use in high energy
1868 @item Clifford (Dirac) algebra (class @code{clifford})
1869 @item su(3) Lie algebra (class @code{color})
1870 @item Matrices (unindexed) (class @code{matrix})
1873 The @code{clifford} and @code{color} classes are subclasses of
1874 @code{indexed} because the elements of these algebras ususally carry
1877 Unlike most computer algebra systems, GiNaC does not primarily provide an
1878 operator (often denoted @samp{&*}) for representing inert products of
1879 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
1880 classes of objects involved, and non-commutative products are formed with
1881 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
1882 figuring out by itself which objects commute and will group the factors
1883 by their class. Consider this example:
1887 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
1888 idx a(symbol("a"), 8), b(symbol("b"), 8);
1889 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
1891 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
1895 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
1896 groups the non-commutative factors (the gammas and the su(3) generators)
1897 together while preserving the order of factors within each class (because
1898 Clifford objects commute with color objects). The resulting expression is a
1899 @emph{commutative} product with two factors that are themselves non-commutative
1900 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
1901 parentheses are placed around the non-commutative products in the output.
1903 @cindex @code{ncmul} (class)
1904 Non-commutative products are internally represented by objects of the class
1905 @code{ncmul}, as opposed to commutative products which are handled by the
1906 @code{mul} class. You will normally not have to worry about this distinction,
1909 The advantage of this approach is that you never have to worry about using
1910 (or forgetting to use) a special operator when constructing non-commutative
1911 expressions. Also, non-commutative products in GiNaC are more intelligent
1912 than in other computer algebra systems; they can, for example, automatically
1913 canonicalize themselves according to rules specified in the implementation
1914 of the non-commutative classes. The drawback is that to work with other than
1915 the built-in algebras you have to implement new classes yourself. Symbols
1916 always commute and it's not possible to construct non-commutative products
1917 using symbols to represent the algebra elements or generators. User-defined
1918 functions can, however, be specified as being non-commutative.
1920 @cindex @code{return_type()}
1921 @cindex @code{return_type_tinfo()}
1922 Information about the commutativity of an object or expression can be
1923 obtained with the two member functions
1926 unsigned ex::return_type(void) const;
1927 unsigned ex::return_type_tinfo(void) const;
1930 The @code{return_type()} function returns one of three values (defined in
1931 the header file @file{flags.h}), corresponding to three categories of
1932 expressions in GiNaC:
1935 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
1936 classes are of this kind.
1937 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
1938 certain class of non-commutative objects which can be determined with the
1939 @code{return_type_tinfo()} method. Expressions of this category commute
1940 with everything except @code{noncommutative} expressions of the same
1942 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
1943 of non-commutative objects of different classes. Expressions of this
1944 category don't commute with any other @code{noncommutative} or
1945 @code{noncommutative_composite} expressions.
1948 The value returned by the @code{return_type_tinfo()} method is valid only
1949 when the return type of the expression is @code{noncommutative}. It is a
1950 value that is unique to the class of the object and usually one of the
1951 constants in @file{tinfos.h}, or derived therefrom.
1953 Here are a couple of examples:
1956 @multitable @columnfractions 0.33 0.33 0.34
1957 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
1958 @item @code{42} @tab @code{commutative} @tab -
1959 @item @code{2*x-y} @tab @code{commutative} @tab -
1960 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1961 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1962 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
1963 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
1967 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
1968 @code{TINFO_clifford} for objects with a representation label of zero.
1969 Other representation labels yield a different @code{return_type_tinfo()},
1970 but it's the same for any two objects with the same label. This is also true
1974 @cindex @code{clifford} (class)
1975 @subsection Clifford algebra
1977 @cindex @code{dirac_gamma()}
1978 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
1979 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
1980 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
1981 is the Minkowski metric tensor. Dirac gammas are constructed by the function
1984 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
1987 which takes two arguments: the index and a @dfn{representation label} in the
1988 range 0 to 255 which is used to distinguish elements of different Clifford
1989 algebras (this is also called a @dfn{spin line index}). Gammas with different
1990 labels commute with each other. The dimension of the index can be 4 or (in
1991 the framework of dimensional regularization) any symbolic value. Spinor
1992 indices on Dirac gammas are not supported in GiNaC.
1994 @cindex @code{dirac_ONE()}
1995 The unity element of a Clifford algebra is constructed by
1998 ex dirac_ONE(unsigned char rl = 0);
2001 @cindex @code{dirac_gamma5()}
2002 and there's a special element @samp{gamma5} that commutes with all other
2003 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2007 ex dirac_gamma5(unsigned char rl = 0);
2010 @cindex @code{dirac_gamma6()}
2011 @cindex @code{dirac_gamma7()}
2012 The two additional functions
2015 ex dirac_gamma6(unsigned char rl = 0);
2016 ex dirac_gamma7(unsigned char rl = 0);
2019 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2022 @cindex @code{dirac_slash()}
2023 Finally, the function
2026 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2029 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2030 whose dimension is given by the @code{dim} argument.
2032 In products of dirac gammas, superfluous unity elements are automatically
2033 removed, squares are replaced by their values and @samp{gamma5} is
2034 anticommuted to the front. The @code{simplify_indexed()} function performs
2035 contractions in gamma strings, for example
2040 symbol a("a"), b("b"), D("D");
2041 varidx mu(symbol("mu"), D);
2042 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2043 * dirac_gamma(mu.toggle_variance());
2045 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2046 e = e.simplify_indexed();
2048 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2049 cout << e.subs(D == 4) << endl;
2050 // -> -2*gamma~symbol10*a.symbol10
2051 // [ == -2 * dirac_slash(a, D) ]
2056 @cindex @code{dirac_trace()}
2057 To calculate the trace of an expression containing strings of Dirac gammas
2058 you use the function
2061 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2064 This function takes the trace of all gammas with the specified representation
2065 label; gammas with other labels are left standing. The last argument to
2066 @code{dirac_trace()} is the value to be returned for the trace of the unity
2067 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2068 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2069 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2070 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2071 This @samp{gamma5} scheme is described in greater detail in
2072 @cite{The Role of gamma5 in Dimensional Regularization}.
2074 The value of the trace itself is also usually different in 4 and in
2075 @math{D != 4} dimensions:
2080 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2081 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2082 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2083 cout << dirac_trace(e).simplify_indexed() << endl;
2090 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2091 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2092 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2093 cout << dirac_trace(e).simplify_indexed() << endl;
2094 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2098 Here is an example for using @code{dirac_trace()} to compute a value that
2099 appears in the calculation of the one-loop vacuum polarization amplitude in
2104 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2105 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2108 sp.add(l, l, pow(l, 2));
2109 sp.add(l, q, ldotq);
2111 ex e = dirac_gamma(mu) *
2112 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2113 dirac_gamma(mu.toggle_variance()) *
2114 (dirac_slash(l, D) + m * dirac_ONE());
2115 e = dirac_trace(e).simplify_indexed(sp);
2116 e = e.collect(lst(l, ldotq, m), true);
2118 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2122 The @code{canonicalize_clifford()} function reorders all gamma products that
2123 appear in an expression to a canonical (but not necessarily simple) form.
2124 You can use this to compare two expressions or for further simplifications:
2128 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2129 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2131 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2133 e = canonicalize_clifford(e);
2140 @cindex @code{color} (class)
2141 @subsection Color algebra
2143 @cindex @code{color_T()}
2144 For computations in quantum chromodynamics, GiNaC implements the base elements
2145 and structure constants of the su(3) Lie algebra (color algebra). The base
2146 elements @math{T_a} are constructed by the function
2149 ex color_T(const ex & a, unsigned char rl = 0);
2152 which takes two arguments: the index and a @dfn{representation label} in the
2153 range 0 to 255 which is used to distinguish elements of different color
2154 algebras. Objects with different labels commute with each other. The
2155 dimension of the index must be exactly 8 and it should be of class @code{idx},
2158 @cindex @code{color_ONE()}
2159 The unity element of a color algebra is constructed by
2162 ex color_ONE(unsigned char rl = 0);
2165 @cindex @code{color_d()}
2166 @cindex @code{color_f()}
2170 ex color_d(const ex & a, const ex & b, const ex & c);
2171 ex color_f(const ex & a, const ex & b, const ex & c);
2174 create the symmetric and antisymmetric structure constants @math{d_abc} and
2175 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2176 and @math{[T_a, T_b] = i f_abc T_c}.
2178 @cindex @code{color_h()}
2179 There's an additional function
2182 ex color_h(const ex & a, const ex & b, const ex & c);
2185 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2187 The function @code{simplify_indexed()} performs some simplifications on
2188 expressions containing color objects:
2193 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2194 k(symbol("k"), 8), l(symbol("l"), 8);
2196 e = color_d(a, b, l) * color_f(a, b, k);
2197 cout << e.simplify_indexed() << endl;
2200 e = color_d(a, b, l) * color_d(a, b, k);
2201 cout << e.simplify_indexed() << endl;
2204 e = color_f(l, a, b) * color_f(a, b, k);
2205 cout << e.simplify_indexed() << endl;
2208 e = color_h(a, b, c) * color_h(a, b, c);
2209 cout << e.simplify_indexed() << endl;
2212 e = color_h(a, b, c) * color_T(b) * color_T(c);
2213 cout << e.simplify_indexed() << endl;
2216 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2217 cout << e.simplify_indexed() << endl;
2220 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2221 cout << e.simplify_indexed() << endl;
2222 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2226 @cindex @code{color_trace()}
2227 To calculate the trace of an expression containing color objects you use the
2231 ex color_trace(const ex & e, unsigned char rl = 0);
2234 This function takes the trace of all color @samp{T} objects with the
2235 specified representation label; @samp{T}s with other labels are left
2236 standing. For example:
2240 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2242 // -> -I*f.a.c.b+d.a.c.b
2247 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2248 @c node-name, next, previous, up
2249 @chapter Methods and Functions
2252 In this chapter the most important algorithms provided by GiNaC will be
2253 described. Some of them are implemented as functions on expressions,
2254 others are implemented as methods provided by expression objects. If
2255 they are methods, there exists a wrapper function around it, so you can
2256 alternatively call it in a functional way as shown in the simple
2261 cout << "As method: " << sin(1).evalf() << endl;
2262 cout << "As function: " << evalf(sin(1)) << endl;
2266 @cindex @code{subs()}
2267 The general rule is that wherever methods accept one or more parameters
2268 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2269 wrapper accepts is the same but preceded by the object to act on
2270 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2271 most natural one in an OO model but it may lead to confusion for MapleV
2272 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2273 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2274 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2275 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2276 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2277 here. Also, users of MuPAD will in most cases feel more comfortable
2278 with GiNaC's convention. All function wrappers are implemented
2279 as simple inline functions which just call the corresponding method and
2280 are only provided for users uncomfortable with OO who are dead set to
2281 avoid method invocations. Generally, nested function wrappers are much
2282 harder to read than a sequence of methods and should therefore be
2283 avoided if possible. On the other hand, not everything in GiNaC is a
2284 method on class @code{ex} and sometimes calling a function cannot be
2288 * Information About Expressions::
2289 * Substituting Expressions::
2290 * Pattern Matching and Advanced Substitutions::
2291 * Polynomial Arithmetic:: Working with polynomials.
2292 * Rational Expressions:: Working with rational functions.
2293 * Symbolic Differentiation::
2294 * Series Expansion:: Taylor and Laurent expansion.
2295 * Built-in Functions:: List of predefined mathematical functions.
2296 * Input/Output:: Input and output of expressions.
2300 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2301 @c node-name, next, previous, up
2302 @section Getting information about expressions
2304 @subsection Checking expression types
2305 @cindex @code{is_ex_of_type()}
2306 @cindex @code{ex_to_numeric()}
2307 @cindex @code{ex_to_@dots{}}
2308 @cindex @code{Converting ex to other classes}
2309 @cindex @code{info()}
2310 @cindex @code{return_type()}
2311 @cindex @code{return_type_tinfo()}
2313 Sometimes it's useful to check whether a given expression is a plain number,
2314 a sum, a polynomial with integer coefficients, or of some other specific type.
2315 GiNaC provides a couple of functions for this (the first one is actually a macro):
2318 bool is_ex_of_type(const ex & e, TYPENAME t);
2319 bool ex::info(unsigned flag);
2320 unsigned ex::return_type(void) const;
2321 unsigned ex::return_type_tinfo(void) const;
2324 When the test made by @code{is_ex_of_type()} returns true, it is safe to
2325 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
2326 one of the class names (@xref{The Class Hierarchy}, for a list of all
2327 classes). For example, assuming @code{e} is an @code{ex}:
2332 if (is_ex_of_type(e, numeric))
2333 numeric n = ex_to_numeric(e);
2338 @code{is_ex_of_type()} allows you to check whether the top-level object of
2339 an expression @samp{e} is an instance of the GiNaC class @samp{t}
2340 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2341 e.g., for checking whether an expression is a number, a sum, or a product:
2348 is_ex_of_type(e1, numeric); // true
2349 is_ex_of_type(e2, numeric); // false
2350 is_ex_of_type(e1, add); // false
2351 is_ex_of_type(e2, add); // true
2352 is_ex_of_type(e1, mul); // false
2353 is_ex_of_type(e2, mul); // false
2357 The @code{info()} method is used for checking certain attributes of
2358 expressions. The possible values for the @code{flag} argument are defined
2359 in @file{ginac/flags.h}, the most important being explained in the following
2363 @multitable @columnfractions .30 .70
2364 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2365 @item @code{numeric}
2366 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
2368 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2369 @item @code{rational}
2370 @tab @dots{}an exact rational number (integers are rational, too)
2371 @item @code{integer}
2372 @tab @dots{}a (non-complex) integer
2373 @item @code{crational}
2374 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2375 @item @code{cinteger}
2376 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2377 @item @code{positive}
2378 @tab @dots{}not complex and greater than 0
2379 @item @code{negative}
2380 @tab @dots{}not complex and less than 0
2381 @item @code{nonnegative}
2382 @tab @dots{}not complex and greater than or equal to 0
2384 @tab @dots{}an integer greater than 0
2386 @tab @dots{}an integer less than 0
2387 @item @code{nonnegint}
2388 @tab @dots{}an integer greater than or equal to 0
2390 @tab @dots{}an even integer
2392 @tab @dots{}an odd integer
2394 @tab @dots{}a prime integer (probabilistic primality test)
2395 @item @code{relation}
2396 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
2397 @item @code{relation_equal}
2398 @tab @dots{}a @code{==} relation
2399 @item @code{relation_not_equal}
2400 @tab @dots{}a @code{!=} relation
2401 @item @code{relation_less}
2402 @tab @dots{}a @code{<} relation
2403 @item @code{relation_less_or_equal}
2404 @tab @dots{}a @code{<=} relation
2405 @item @code{relation_greater}
2406 @tab @dots{}a @code{>} relation
2407 @item @code{relation_greater_or_equal}
2408 @tab @dots{}a @code{>=} relation
2410 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
2412 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
2413 @item @code{polynomial}
2414 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2415 @item @code{integer_polynomial}
2416 @tab @dots{}a polynomial with (non-complex) integer coefficients
2417 @item @code{cinteger_polynomial}
2418 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2419 @item @code{rational_polynomial}
2420 @tab @dots{}a polynomial with (non-complex) rational coefficients
2421 @item @code{crational_polynomial}
2422 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2423 @item @code{rational_function}
2424 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2425 @item @code{algebraic}
2426 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2430 To determine whether an expression is commutative or non-commutative and if
2431 so, with which other expressions it would commute, you use the methods
2432 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2433 for an explanation of these.
2436 @subsection Accessing subexpressions
2437 @cindex @code{nops()}
2440 @cindex @code{relational} (class)
2442 GiNaC provides the two methods
2445 unsigned ex::nops();
2446 ex ex::op(unsigned i);
2449 for accessing the subexpressions in the container-like GiNaC classes like
2450 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2451 determines the number of subexpressions (@samp{operands}) contained, while
2452 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2453 In the case of a @code{power} object, @code{op(0)} will return the basis
2454 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2455 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2457 The left-hand and right-hand side expressions of objects of class
2458 @code{relational} (and only of these) can also be accessed with the methods
2466 @subsection Comparing expressions
2467 @cindex @code{is_equal()}
2468 @cindex @code{is_zero()}
2470 Expressions can be compared with the usual C++ relational operators like
2471 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2472 the result is usually not determinable and the result will be @code{false},
2473 except in the case of the @code{!=} operator. You should also be aware that
2474 GiNaC will only do the most trivial test for equality (subtracting both
2475 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2478 Actually, if you construct an expression like @code{a == b}, this will be
2479 represented by an object of the @code{relational} class (@xref{Relations}.)
2480 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2482 There are also two methods
2485 bool ex::is_equal(const ex & other);
2489 for checking whether one expression is equal to another, or equal to zero,
2492 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2493 GiNaC header files. This method is however only to be used internally by
2494 GiNaC to establish a canonical sort order for terms, and using it to compare
2495 expressions will give very surprising results.
2498 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
2499 @c node-name, next, previous, up
2500 @section Substituting expressions
2501 @cindex @code{subs()}
2503 Algebraic objects inside expressions can be replaced with arbitrary
2504 expressions via the @code{.subs()} method:
2507 ex ex::subs(const ex & e);
2508 ex ex::subs(const lst & syms, const lst & repls);
2511 In the first form, @code{subs()} accepts a relational of the form
2512 @samp{object == expression} or a @code{lst} of such relationals:
2516 symbol x("x"), y("y");
2518 ex e1 = 2*x^2-4*x+3;
2519 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2523 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2528 If you specify multiple substitutions, they are performed in parallel, so e.g.
2529 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2531 The second form of @code{subs()} takes two lists, one for the objects to be
2532 replaced and one for the expressions to be substituted (both lists must
2533 contain the same number of elements). Using this form, you would write
2534 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2536 @code{subs()} performs syntactic substitution of any complete algebraic
2537 object; it does not try to match sub-expressions as is demonstrated by the
2542 symbol x("x"), y("y"), z("z");
2544 ex e1 = pow(x+y, 2);
2545 cout << e1.subs(x+y == 4) << endl;
2548 ex e2 = sin(x)*sin(y)*cos(x);
2549 cout << e2.subs(sin(x) == cos(x)) << endl;
2550 // -> cos(x)^2*sin(y)
2553 cout << e3.subs(x+y == 4) << endl;
2555 // (and not 4+z as one might expect)
2559 A more powerful form of substitution using wildcards is described in the
2563 @node Pattern Matching and Advanced Substitutions, Polynomial Arithmetic, Substituting Expressions, Methods and Functions
2564 @c node-name, next, previous, up
2565 @section Pattern matching and advanced substitutions
2567 GiNaC allows the use of patterns for checking whether an expression is of a
2568 certain form or contains subexpressions of a certain form, and for
2569 substituting expressions in a more general way.
2571 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
2572 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
2573 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
2574 an unsigned integer number to allow having multiple different wildcards in a
2575 pattern. Wildcards are printed as @samp{$label} (this is also the way they
2576 are specified in @command{ginsh}. In C++ code, wildcard objects are created
2580 ex wild(unsigned label = 0);
2583 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
2586 Some examples for patterns:
2588 @multitable @columnfractions .5 .5
2589 @item @strong{Constructed as} @tab @strong{Output as}
2590 @item @code{wild()} @tab @samp{$0}
2591 @item @code{pow(x,wild())} @tab @samp{x^$0}
2592 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
2593 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
2599 @item Wildcards behave like symbols and are subject to the same algebraic
2600 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
2601 @item As shown in the last example, to use wildcards for indices you have to
2602 use them as the value of an @code{idx} object. This is because indices must
2603 always be of class @code{idx} (or a subclass).
2604 @item Wildcards only represent expressions or subexpressions. It is not
2605 possible to use them as placeholders for other properties like index
2606 dimension or variance, representation labels, symmetry of indexed objects
2608 @item Because wildcards are commutative, it is not possible to use wildcards
2609 as part of noncommutative products.
2610 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
2611 are also valid patterns.
2614 @cindex @code{match()}
2615 The most basic application of patterns is to check whether an expression
2616 matches a given pattern. This is done by the function
2619 bool ex::match(const ex & pattern);
2620 bool ex::match(const ex & pattern, lst & repls);
2623 This function returns @code{true} when the expression matches the pattern
2624 and @code{false} if it doesn't. If used in the second form, the actual
2625 subexpressions matched by the wildcards get returned in the @code{repls}
2626 object as a list of relations of the form @samp{wildcard == expression}.
2627 If @code{match()} returns false, the state of @code{repls} is undefined.
2628 For reproducible results, the list should be empty when passed to
2629 @code{match()}, but it is also possible to find similarities in multiple
2630 expressions by passing in the result of a previous match.
2632 The matching algorithm works as follows:
2635 @item A single wildcard matches any expression. If one wildcard appears
2636 multiple times in a pattern, it must match the same expression in all
2637 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
2638 @samp{x*(x+1)} but not @samp{x*(y+1)}).
2639 @item If the expression is not of the same class as the pattern, the match
2640 fails (i.e. a sum only matches a sum, a function only matches a function,
2642 @item If the pattern is a function, it only matches the same function
2643 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
2644 @item Except for sums and products, the match fails if the number of
2645 subexpressions (@code{nops()}) is not equal to the number of subexpressions
2647 @item If there are no subexpressions, the expressions and the pattern must
2648 be equal (in the sense of @code{is_equal()}).
2649 @item Except for sums and products, each subexpression (@code{op()}) must
2650 match the corresponding subexpression of the pattern.
2653 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
2654 account for their commutativity and associativity:
2657 @item If the pattern contains a term or factor that is a single wildcard,
2658 this one is used as the @dfn{global wildcard}. If there is more than one
2659 such wildcard, one of them is chosen as the global wildcard in a random
2661 @item Every term/factor of the pattern, except the global wildcard, is
2662 matched against every term of the expression in sequence. If no match is
2663 found, the whole match fails. Terms that did match are not considered in
2665 @item If there are no unmatched terms left, the match succeeds. Otherwise
2666 the match fails unless there is a global wildcard in the pattern, in
2667 which case this wildcard matches the remaining terms.
2670 In general, having more than one single wildcard as a term of a sum or a
2671 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
2674 Here are some examples in @command{ginsh} to demonstrate how it works (the
2675 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
2676 match fails, and the list of wildcard replacements otherwise):
2679 > match((x+y)^a,(x+y)^a);
2681 > match((x+y)^a,(x+y)^b);
2683 > match((x+y)^a,$1^$2);
2685 > match((x+y)^a,$1^$1);
2687 > match((x+y)^(x+y),$1^$1);
2689 > match((x+y)^(x+y),$1^$2);
2691 > match((a+b)*(a+c),($1+b)*($1+c));
2693 > match((a+b)*(a+c),(a+$1)*(a+$2));
2695 (Unpredictable. The result might also be [$1==c,$2==b].)
2696 > match((a+b)*(a+c),($1+$2)*($1+$3));
2697 (The result is undefined. Due to the sequential nature of the algorithm
2698 and the re-ordering of terms in GiNaC, the match for the first factor
2699 may be [$1==a,$2==b] in which case the match for the second factor
2700 succeeds, or it may be [$1==b,$2==a] which causes the second match to
2702 > match(a*(x+y)+a*z+b,a*$1+$2);
2703 (This is also ambiguous and may return either [$1==z,$2==a*(x+y)+b] or
2705 > match(a+b+c+d+e+f,c);
2707 > match(a+b+c+d+e+f,c+$0);
2709 > match(a+b+c+d+e+f,c+e+$0);
2711 > match(a+b,a+b+$0);
2713 > match(a*b^2,a^$1*b^$2);
2715 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
2717 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
2719 > match(atan2(y,x^2),atan2(y,$0));
2723 @cindex @code{has()}
2724 A more general way to look for patterns in expressions is provided by the
2728 bool ex::has(const ex & pattern);
2731 This function checks whether a pattern is matched by an expression itself or
2732 by any of its subexpressions.
2734 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
2735 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
2738 > has(x*sin(x+y+2*a),y);
2740 > has(x*sin(x+y+2*a+y),x+y);
2742 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
2743 has the subexpressions "x", "y" and "2*a".)
2744 > has(x*sin(x+y+2*a+y),x+y+$1);
2746 (But this is possible.)
2747 > has(x*sin(2*(x+y)+2*a),x+y);
2749 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
2750 which "x+y" is not a subexpression.)
2753 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
2755 > has(4*x^2-x+3,$1*x);
2757 > has(4*x^2+x+3,$1*x);
2759 (Another possible pitfall. The first expression matches because the term
2760 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
2761 contains a linear term you should use the coeff() function instead.)
2764 @cindex @code{subs()}
2765 Probably the most useful application of patterns is to use them for
2766 substituting expressions with the @code{subs()} method. Wildcards can be
2767 used in the search patterns as well as in the replacement expressions, where
2768 they get replaced by the expressions matched by them. @code{subs()} doesn't
2769 know anything about algebra; it performs purely syntactic substitutions.
2774 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
2776 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
2778 > subs((a+b+c)^2,a+b=x);
2780 > subs((a+b+c)^2,a+b+$1==x+$1);
2782 > subs(a+2*b,a+b=x);
2784 > subs(4*x^3-2*x^2+5*x-1,x==a);
2786 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
2788 > subs(sin(1+sin(x)),sin($1)==cos($1));
2790 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
2794 The last example would be written in C++ in this way:
2798 symbol a("a"), b("b"), x("x"), y("y");
2799 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
2800 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
2801 cout << e.expand() << endl;
2807 @node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
2808 @c node-name, next, previous, up
2809 @section Polynomial arithmetic
2811 @subsection Expanding and collecting
2812 @cindex @code{expand()}
2813 @cindex @code{collect()}
2815 A polynomial in one or more variables has many equivalent
2816 representations. Some useful ones serve a specific purpose. Consider
2817 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2818 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2819 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
2820 representations are the recursive ones where one collects for exponents
2821 in one of the three variable. Since the factors are themselves
2822 polynomials in the remaining two variables the procedure can be
2823 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
2824 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
2827 To bring an expression into expanded form, its method
2833 may be called. In our example above, this corresponds to @math{4*x*y +
2834 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
2835 GiNaC is not easily guessable you should be prepared to see different
2836 orderings of terms in such sums!
2838 Another useful representation of multivariate polynomials is as a
2839 univariate polynomial in one of the variables with the coefficients
2840 being polynomials in the remaining variables. The method
2841 @code{collect()} accomplishes this task:
2844 ex ex::collect(const ex & s, bool distributed = false);
2847 The first argument to @code{collect()} can also be a list of objects in which
2848 case the result is either a recursively collected polynomial, or a polynomial
2849 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
2850 by the @code{distributed} flag.
2852 Note that the original polynomial needs to be in expanded form in order
2853 for @code{collect()} to be able to find the coefficients properly.
2855 @subsection Degree and coefficients
2856 @cindex @code{degree()}
2857 @cindex @code{ldegree()}
2858 @cindex @code{coeff()}
2860 The degree and low degree of a polynomial can be obtained using the two
2864 int ex::degree(const ex & s);
2865 int ex::ldegree(const ex & s);
2868 which also work reliably on non-expanded input polynomials (they even work
2869 on rational functions, returning the asymptotic degree). To extract
2870 a coefficient with a certain power from an expanded polynomial you use
2873 ex ex::coeff(const ex & s, int n);
2876 You can also obtain the leading and trailing coefficients with the methods
2879 ex ex::lcoeff(const ex & s);
2880 ex ex::tcoeff(const ex & s);
2883 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
2886 An application is illustrated in the next example, where a multivariate
2887 polynomial is analyzed:
2890 #include <ginac/ginac.h>
2891 using namespace std;
2892 using namespace GiNaC;
2896 symbol x("x"), y("y");
2897 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
2898 - pow(x+y,2) + 2*pow(y+2,2) - 8;
2899 ex Poly = PolyInp.expand();
2901 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
2902 cout << "The x^" << i << "-coefficient is "
2903 << Poly.coeff(x,i) << endl;
2905 cout << "As polynomial in y: "
2906 << Poly.collect(y) << endl;
2910 When run, it returns an output in the following fashion:
2913 The x^0-coefficient is y^2+11*y
2914 The x^1-coefficient is 5*y^2-2*y
2915 The x^2-coefficient is -1
2916 The x^3-coefficient is 4*y
2917 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
2920 As always, the exact output may vary between different versions of GiNaC
2921 or even from run to run since the internal canonical ordering is not
2922 within the user's sphere of influence.
2924 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
2925 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
2926 with non-polynomial expressions as they not only work with symbols but with
2927 constants, functions and indexed objects as well:
2931 symbol a("a"), b("b"), c("c");
2932 idx i(symbol("i"), 3);
2934 ex e = pow(sin(x) - cos(x), 4);
2935 cout << e.degree(cos(x)) << endl;
2937 cout << e.expand().coeff(sin(x), 3) << endl;
2940 e = indexed(a+b, i) * indexed(b+c, i);
2941 e = e.expand(expand_options::expand_indexed);
2942 cout << e.collect(indexed(b, i)) << endl;
2943 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
2948 @subsection Polynomial division
2949 @cindex polynomial division
2952 @cindex pseudo-remainder
2953 @cindex @code{quo()}
2954 @cindex @code{rem()}
2955 @cindex @code{prem()}
2956 @cindex @code{divide()}
2961 ex quo(const ex & a, const ex & b, const symbol & x);
2962 ex rem(const ex & a, const ex & b, const symbol & x);
2965 compute the quotient and remainder of univariate polynomials in the variable
2966 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
2968 The additional function
2971 ex prem(const ex & a, const ex & b, const symbol & x);
2974 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
2975 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
2977 Exact division of multivariate polynomials is performed by the function
2980 bool divide(const ex & a, const ex & b, ex & q);
2983 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
2984 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
2985 in which case the value of @code{q} is undefined.
2988 @subsection Unit, content and primitive part
2989 @cindex @code{unit()}
2990 @cindex @code{content()}
2991 @cindex @code{primpart()}
2996 ex ex::unit(const symbol & x);
2997 ex ex::content(const symbol & x);
2998 ex ex::primpart(const symbol & x);
3001 return the unit part, content part, and primitive polynomial of a multivariate
3002 polynomial with respect to the variable @samp{x} (the unit part being the sign
3003 of the leading coefficient, the content part being the GCD of the coefficients,
3004 and the primitive polynomial being the input polynomial divided by the unit and
3005 content parts). The product of unit, content, and primitive part is the
3006 original polynomial.
3009 @subsection GCD and LCM
3012 @cindex @code{gcd()}
3013 @cindex @code{lcm()}
3015 The functions for polynomial greatest common divisor and least common
3016 multiple have the synopsis
3019 ex gcd(const ex & a, const ex & b);
3020 ex lcm(const ex & a, const ex & b);
3023 The functions @code{gcd()} and @code{lcm()} accept two expressions
3024 @code{a} and @code{b} as arguments and return a new expression, their
3025 greatest common divisor or least common multiple, respectively. If the
3026 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
3027 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
3030 #include <ginac/ginac.h>
3031 using namespace GiNaC;
3035 symbol x("x"), y("y"), z("z");
3036 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
3037 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
3039 ex P_gcd = gcd(P_a, P_b);
3041 ex P_lcm = lcm(P_a, P_b);
3042 // 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
3047 @subsection Square-free decomposition
3048 @cindex square-free decomposition
3049 @cindex factorization
3050 @cindex @code{sqrfree()}
3052 GiNaC still lacks proper factorization support. Some form of
3053 factorization is, however, easily implemented by noting that factors
3054 appearing in a polynomial with power two or more also appear in the
3055 derivative and hence can easily be found by computing the GCD of the
3056 original polynomial and its derivatives. Any system has an interface
3057 for this so called square-free factorization. So we provide one, too:
3059 ex sqrfree(const ex & a, const lst & l = lst());
3061 Here is an example that by the way illustrates how the result may depend
3062 on the order of differentiation:
3065 symbol x("x"), y("y");
3066 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
3068 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
3069 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
3071 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
3072 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
3074 cout << sqrfree(BiVarPol) << endl;
3075 // -> depending on luck, any of the above
3080 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
3081 @c node-name, next, previous, up
3082 @section Rational expressions
3084 @subsection The @code{normal} method
3085 @cindex @code{normal()}
3086 @cindex simplification
3087 @cindex temporary replacement
3089 Some basic form of simplification of expressions is called for frequently.
3090 GiNaC provides the method @code{.normal()}, which converts a rational function
3091 into an equivalent rational function of the form @samp{numerator/denominator}
3092 where numerator and denominator are coprime. If the input expression is already
3093 a fraction, it just finds the GCD of numerator and denominator and cancels it,
3094 otherwise it performs fraction addition and multiplication.
3096 @code{.normal()} can also be used on expressions which are not rational functions
3097 as it will replace all non-rational objects (like functions or non-integer
3098 powers) by temporary symbols to bring the expression to the domain of rational
3099 functions before performing the normalization, and re-substituting these
3100 symbols afterwards. This algorithm is also available as a separate method
3101 @code{.to_rational()}, described below.
3103 This means that both expressions @code{t1} and @code{t2} are indeed
3104 simplified in this little program:
3107 #include <ginac/ginac.h>
3108 using namespace GiNaC;
3113 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
3114 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
3115 std::cout << "t1 is " << t1.normal() << std::endl;
3116 std::cout << "t2 is " << t2.normal() << std::endl;
3120 Of course this works for multivariate polynomials too, so the ratio of
3121 the sample-polynomials from the section about GCD and LCM above would be
3122 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
3125 @subsection Numerator and denominator
3128 @cindex @code{numer()}
3129 @cindex @code{denom()}
3131 The numerator and denominator of an expression can be obtained with
3138 These functions will first normalize the expression as described above and
3139 then return the numerator or denominator, respectively.
3142 @subsection Converting to a rational expression
3143 @cindex @code{to_rational()}
3145 Some of the methods described so far only work on polynomials or rational
3146 functions. GiNaC provides a way to extend the domain of these functions to
3147 general expressions by using the temporary replacement algorithm described
3148 above. You do this by calling
3151 ex ex::to_rational(lst &l);
3154 on the expression to be converted. The supplied @code{lst} will be filled
3155 with the generated temporary symbols and their replacement expressions in
3156 a format that can be used directly for the @code{subs()} method. It can also
3157 already contain a list of replacements from an earlier application of
3158 @code{.to_rational()}, so it's possible to use it on multiple expressions
3159 and get consistent results.
3166 ex a = pow(sin(x), 2) - pow(cos(x), 2);
3167 ex b = sin(x) + cos(x);
3170 divide(a.to_rational(l), b.to_rational(l), q);
3171 cout << q.subs(l) << endl;
3175 will print @samp{sin(x)-cos(x)}.
3178 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
3179 @c node-name, next, previous, up
3180 @section Symbolic differentiation
3181 @cindex differentiation
3182 @cindex @code{diff()}
3184 @cindex product rule
3186 GiNaC's objects know how to differentiate themselves. Thus, a
3187 polynomial (class @code{add}) knows that its derivative is the sum of
3188 the derivatives of all the monomials:
3191 #include <ginac/ginac.h>
3192 using namespace GiNaC;
3196 symbol x("x"), y("y"), z("z");
3197 ex P = pow(x, 5) + pow(x, 2) + y;
3199 cout << P.diff(x,2) << endl; // 20*x^3 + 2
3200 cout << P.diff(y) << endl; // 1
3201 cout << P.diff(z) << endl; // 0
3205 If a second integer parameter @var{n} is given, the @code{diff} method
3206 returns the @var{n}th derivative.
3208 If @emph{every} object and every function is told what its derivative
3209 is, all derivatives of composed objects can be calculated using the
3210 chain rule and the product rule. Consider, for instance the expression
3211 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
3212 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
3213 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
3214 out that the composition is the generating function for Euler Numbers,
3215 i.e. the so called @var{n}th Euler number is the coefficient of
3216 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
3217 identity to code a function that generates Euler numbers in just three
3220 @cindex Euler numbers
3222 #include <ginac/ginac.h>
3223 using namespace GiNaC;
3225 ex EulerNumber(unsigned n)
3228 const ex generator = pow(cosh(x),-1);
3229 return generator.diff(x,n).subs(x==0);
3234 for (unsigned i=0; i<11; i+=2)
3235 std::cout << EulerNumber(i) << std::endl;
3240 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3241 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3242 @code{i} by two since all odd Euler numbers vanish anyways.
3245 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
3246 @c node-name, next, previous, up
3247 @section Series expansion
3248 @cindex @code{series()}
3249 @cindex Taylor expansion
3250 @cindex Laurent expansion
3251 @cindex @code{pseries} (class)
3253 Expressions know how to expand themselves as a Taylor series or (more
3254 generally) a Laurent series. As in most conventional Computer Algebra
3255 Systems, no distinction is made between those two. There is a class of
3256 its own for storing such series (@code{class pseries}) and a built-in
3257 function (called @code{Order}) for storing the order term of the series.
3258 As a consequence, if you want to work with series, i.e. multiply two
3259 series, you need to call the method @code{ex::series} again to convert
3260 it to a series object with the usual structure (expansion plus order
3261 term). A sample application from special relativity could read:
3264 #include <ginac/ginac.h>
3265 using namespace std;
3266 using namespace GiNaC;
3270 symbol v("v"), c("c");
3272 ex gamma = 1/sqrt(1 - pow(v/c,2));
3273 ex mass_nonrel = gamma.series(v==0, 10);
3275 cout << "the relativistic mass increase with v is " << endl
3276 << mass_nonrel << endl;
3278 cout << "the inverse square of this series is " << endl
3279 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3283 Only calling the series method makes the last output simplify to
3284 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3285 series raised to the power @math{-2}.
3287 @cindex M@'echain's formula
3288 As another instructive application, let us calculate the numerical
3289 value of Archimedes' constant
3293 (for which there already exists the built-in constant @code{Pi})
3294 using M@'echain's amazing formula
3296 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3299 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3301 We may expand the arcus tangent around @code{0} and insert the fractions
3302 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3303 carries an order term with it and the question arises what the system is
3304 supposed to do when the fractions are plugged into that order term. The
3305 solution is to use the function @code{series_to_poly()} to simply strip
3309 #include <ginac/ginac.h>
3310 using namespace GiNaC;
3312 ex mechain_pi(int degr)
3315 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3316 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3317 -4*pi_expansion.subs(x==numeric(1,239));
3323 using std::cout; // just for fun, another way of...
3324 using std::endl; // ...dealing with this namespace std.
3326 for (int i=2; i<12; i+=2) @{
3327 pi_frac = mechain_pi(i);
3328 cout << i << ":\t" << pi_frac << endl
3329 << "\t" << pi_frac.evalf() << endl;
3335 Note how we just called @code{.series(x,degr)} instead of
3336 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3337 method @code{series()}: if the first argument is a symbol the expression
3338 is expanded in that symbol around point @code{0}. When you run this
3339 program, it will type out:
3343 3.1832635983263598326
3344 4: 5359397032/1706489875
3345 3.1405970293260603143
3346 6: 38279241713339684/12184551018734375
3347 3.141621029325034425
3348 8: 76528487109180192540976/24359780855939418203125
3349 3.141591772182177295
3350 10: 327853873402258685803048818236/104359128170408663038552734375
3351 3.1415926824043995174
3355 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
3356 @c node-name, next, previous, up
3357 @section Predefined mathematical functions
3359 GiNaC contains the following predefined mathematical functions:
3362 @multitable @columnfractions .30 .70
3363 @item @strong{Name} @tab @strong{Function}
3366 @item @code{csgn(x)}
3368 @item @code{sqrt(x)}
3369 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3376 @item @code{asin(x)}
3378 @item @code{acos(x)}
3380 @item @code{atan(x)}
3381 @tab inverse tangent
3382 @item @code{atan2(y, x)}
3383 @tab inverse tangent with two arguments
3384 @item @code{sinh(x)}
3385 @tab hyperbolic sine
3386 @item @code{cosh(x)}
3387 @tab hyperbolic cosine
3388 @item @code{tanh(x)}
3389 @tab hyperbolic tangent
3390 @item @code{asinh(x)}
3391 @tab inverse hyperbolic sine
3392 @item @code{acosh(x)}
3393 @tab inverse hyperbolic cosine
3394 @item @code{atanh(x)}
3395 @tab inverse hyperbolic tangent
3397 @tab exponential function
3399 @tab natural logarithm
3402 @item @code{zeta(x)}
3403 @tab Riemann's zeta function
3404 @item @code{zeta(n, x)}
3405 @tab derivatives of Riemann's zeta function
3406 @item @code{tgamma(x)}
3408 @item @code{lgamma(x)}
3409 @tab logarithm of Gamma function
3410 @item @code{beta(x, y)}
3411 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3413 @tab psi (digamma) function
3414 @item @code{psi(n, x)}
3415 @tab derivatives of psi function (polygamma functions)
3416 @item @code{factorial(n)}
3417 @tab factorial function
3418 @item @code{binomial(n, m)}
3419 @tab binomial coefficients
3420 @item @code{Order(x)}
3421 @tab order term function in truncated power series
3422 @item @code{Derivative(x, l)}
3423 @tab inert partial differentiation operator (used internally)
3428 For functions that have a branch cut in the complex plane GiNaC follows
3429 the conventions for C++ as defined in the ANSI standard as far as
3430 possible. In particular: the natural logarithm (@code{log}) and the
3431 square root (@code{sqrt}) both have their branch cuts running along the
3432 negative real axis where the points on the axis itself belong to the
3433 upper part (i.e. continuous with quadrant II). The inverse
3434 trigonometric and hyperbolic functions are not defined for complex
3435 arguments by the C++ standard, however. In GiNaC we follow the
3436 conventions used by CLN, which in turn follow the carefully designed
3437 definitions in the Common Lisp standard. It should be noted that this
3438 convention is identical to the one used by the C99 standard and by most
3439 serious CAS. It is to be expected that future revisions of the C++
3440 standard incorporate these functions in the complex domain in a manner
3441 compatible with C99.
3444 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3445 @c node-name, next, previous, up
3446 @section Input and output of expressions
3449 @subsection Expression output
3451 @cindex output of expressions
3453 The easiest way to print an expression is to write it to a stream:
3458 ex e = 4.5+pow(x,2)*3/2;
3459 cout << e << endl; // prints '(4.5)+3/2*x^2'
3463 The output format is identical to the @command{ginsh} input syntax and
3464 to that used by most computer algebra systems, but not directly pastable
3465 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3466 is printed as @samp{x^2}).
3468 It is possible to print expressions in a number of different formats with
3472 void ex::print(const print_context & c, unsigned level = 0);
3475 @cindex @code{print_context} (class)
3476 The type of @code{print_context} object passed in determines the format
3477 of the output. The possible types are defined in @file{ginac/print.h}.
3478 All constructors of @code{print_context} and derived classes take an
3479 @code{ostream &} as their first argument.
3481 To print an expression in a way that can be directly used in a C or C++
3482 program, you pass a @code{print_csrc} object like this:
3486 cout << "float f = ";
3487 e.print(print_csrc_float(cout));
3490 cout << "double d = ";
3491 e.print(print_csrc_double(cout));
3494 cout << "cl_N n = ";
3495 e.print(print_csrc_cl_N(cout));
3500 The three possible types mostly affect the way in which floating point
3501 numbers are written.
3503 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3506 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3507 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3508 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3511 The @code{print_context} type @code{print_tree} provides a dump of the
3512 internal structure of an expression for debugging purposes:
3516 e.print(print_tree(cout));
3523 add, hash=0x0, flags=0x3, nops=2
3524 power, hash=0x9, flags=0x3, nops=2
3525 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
3526 2 (numeric), hash=0x80000042, flags=0xf
3527 3/2 (numeric), hash=0x80000061, flags=0xf
3530 4.5L0 (numeric), hash=0x8000004b, flags=0xf
3534 This kind of output is also available in @command{ginsh} as the @code{print()}
3537 Another useful output format is for LaTeX parsing in mathematical mode.
3538 It is rather similar to the default @code{print_context} but provides
3539 some braces needed by LaTeX for delimiting boxes and also converts some
3540 common objects to conventional LaTeX names. It is possible to give symbols
3541 a special name for LaTeX output by supplying it as a second argument to
3542 the @code{symbol} constructor.
3544 For example, the code snippet
3549 ex foo = lgamma(x).series(x==0,3);
3550 foo.print(print_latex(std::cout));
3556 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
3559 If you need any fancy special output format, e.g. for interfacing GiNaC
3560 with other algebra systems or for producing code for different
3561 programming languages, you can always traverse the expression tree yourself:
3564 static void my_print(const ex & e)
3566 if (is_ex_of_type(e, function))
3567 cout << ex_to_function(e).get_name();
3569 cout << e.bp->class_name();
3571 unsigned n = e.nops();
3573 for (unsigned i=0; i<n; i++) @{
3585 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
3593 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
3594 symbol(y))),numeric(-2)))
3597 If you need an output format that makes it possible to accurately
3598 reconstruct an expression by feeding the output to a suitable parser or
3599 object factory, you should consider storing the expression in an
3600 @code{archive} object and reading the object properties from there.
3601 See the section on archiving for more information.
3604 @subsection Expression input
3605 @cindex input of expressions
3607 GiNaC provides no way to directly read an expression from a stream because
3608 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
3609 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
3610 @code{y} you defined in your program and there is no way to specify the
3611 desired symbols to the @code{>>} stream input operator.
3613 Instead, GiNaC lets you construct an expression from a string, specifying the
3614 list of symbols to be used:
3618 symbol x("x"), y("y");
3619 ex e("2*x+sin(y)", lst(x, y));
3623 The input syntax is the same as that used by @command{ginsh} and the stream
3624 output operator @code{<<}. The symbols in the string are matched by name to
3625 the symbols in the list and if GiNaC encounters a symbol not specified in
3626 the list it will throw an exception.
3628 With this constructor, it's also easy to implement interactive GiNaC programs:
3633 #include <stdexcept>
3634 #include <ginac/ginac.h>
3635 using namespace std;
3636 using namespace GiNaC;
3643 cout << "Enter an expression containing 'x': ";
3648 cout << "The derivative of " << e << " with respect to x is ";
3649 cout << e.diff(x) << ".\n";
3650 @} catch (exception &p) @{
3651 cerr << p.what() << endl;
3657 @subsection Archiving
3658 @cindex @code{archive} (class)
3661 GiNaC allows creating @dfn{archives} of expressions which can be stored
3662 to or retrieved from files. To create an archive, you declare an object
3663 of class @code{archive} and archive expressions in it, giving each
3664 expression a unique name:
3668 using namespace std;
3669 #include <ginac/ginac.h>
3670 using namespace GiNaC;
3674 symbol x("x"), y("y"), z("z");
3676 ex foo = sin(x + 2*y) + 3*z + 41;
3680 a.archive_ex(foo, "foo");
3681 a.archive_ex(bar, "the second one");
3685 The archive can then be written to a file:
3689 ofstream out("foobar.gar");
3695 The file @file{foobar.gar} contains all information that is needed to
3696 reconstruct the expressions @code{foo} and @code{bar}.
3698 @cindex @command{viewgar}
3699 The tool @command{viewgar} that comes with GiNaC can be used to view
3700 the contents of GiNaC archive files:
3703 $ viewgar foobar.gar
3704 foo = 41+sin(x+2*y)+3*z
3705 the second one = 42+sin(x+2*y)+3*z
3708 The point of writing archive files is of course that they can later be
3714 ifstream in("foobar.gar");
3719 And the stored expressions can be retrieved by their name:
3725 ex ex1 = a2.unarchive_ex(syms, "foo");
3726 ex ex2 = a2.unarchive_ex(syms, "the second one");
3728 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
3729 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
3730 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
3734 Note that you have to supply a list of the symbols which are to be inserted
3735 in the expressions. Symbols in archives are stored by their name only and
3736 if you don't specify which symbols you have, unarchiving the expression will
3737 create new symbols with that name. E.g. if you hadn't included @code{x} in
3738 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
3739 have had no effect because the @code{x} in @code{ex1} would have been a
3740 different symbol than the @code{x} which was defined at the beginning of
3741 the program, altough both would appear as @samp{x} when printed.
3743 You can also use the information stored in an @code{archive} object to
3744 output expressions in a format suitable for exact reconstruction. The
3745 @code{archive} and @code{archive_node} classes have a couple of member
3746 functions that let you access the stored properties:
3749 static void my_print2(const archive_node & n)
3752 n.find_string("class", class_name);
3753 cout << class_name << "(";
3755 archive_node::propinfovector p;
3756 n.get_properties(p);
3758 unsigned num = p.size();
3759 for (unsigned i=0; i<num; i++) @{
3760 const string &name = p[i].name;
3761 if (name == "class")
3763 cout << name << "=";
3765 unsigned count = p[i].count;
3769 for (unsigned j=0; j<count; j++) @{
3770 switch (p[i].type) @{
3771 case archive_node::PTYPE_BOOL: @{
3773 n.find_bool(name, x);
3774 cout << (x ? "true" : "false");
3777 case archive_node::PTYPE_UNSIGNED: @{
3779 n.find_unsigned(name, x);
3783 case archive_node::PTYPE_STRING: @{
3785 n.find_string(name, x);
3786 cout << '\"' << x << '\"';
3789 case archive_node::PTYPE_NODE: @{
3790 const archive_node &x = n.find_ex_node(name, j);
3812 ex e = pow(2, x) - y;
3814 my_print2(ar.get_top_node(0)); cout << endl;
3822 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
3823 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
3824 overall_coeff=numeric(number="0"))
3827 Be warned, however, that the set of properties and their meaning for each
3828 class may change between GiNaC versions.
3831 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
3832 @c node-name, next, previous, up
3833 @chapter Extending GiNaC
3835 By reading so far you should have gotten a fairly good understanding of
3836 GiNaC's design-patterns. From here on you should start reading the
3837 sources. All we can do now is issue some recommendations how to tackle
3838 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
3839 develop some useful extension please don't hesitate to contact the GiNaC
3840 authors---they will happily incorporate them into future versions.
3843 * What does not belong into GiNaC:: What to avoid.
3844 * Symbolic functions:: Implementing symbolic functions.
3845 * Adding classes:: Defining new algebraic classes.
3849 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
3850 @c node-name, next, previous, up
3851 @section What doesn't belong into GiNaC
3853 @cindex @command{ginsh}
3854 First of all, GiNaC's name must be read literally. It is designed to be
3855 a library for use within C++. The tiny @command{ginsh} accompanying
3856 GiNaC makes this even more clear: it doesn't even attempt to provide a
3857 language. There are no loops or conditional expressions in
3858 @command{ginsh}, it is merely a window into the library for the
3859 programmer to test stuff (or to show off). Still, the design of a
3860 complete CAS with a language of its own, graphical capabilites and all
3861 this on top of GiNaC is possible and is without doubt a nice project for
3864 There are many built-in functions in GiNaC that do not know how to
3865 evaluate themselves numerically to a precision declared at runtime
3866 (using @code{Digits}). Some may be evaluated at certain points, but not
3867 generally. This ought to be fixed. However, doing numerical
3868 computations with GiNaC's quite abstract classes is doomed to be
3869 inefficient. For this purpose, the underlying foundation classes
3870 provided by @acronym{CLN} are much better suited.
3873 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
3874 @c node-name, next, previous, up
3875 @section Symbolic functions
3877 The easiest and most instructive way to start with is probably to
3878 implement your own function. GiNaC's functions are objects of class
3879 @code{function}. The preprocessor is then used to convert the function
3880 names to objects with a corresponding serial number that is used
3881 internally to identify them. You usually need not worry about this
3882 number. New functions may be inserted into the system via a kind of
3883 `registry'. It is your responsibility to care for some functions that
3884 are called when the user invokes certain methods. These are usual
3885 C++-functions accepting a number of @code{ex} as arguments and returning
3886 one @code{ex}. As an example, if we have a look at a simplified
3887 implementation of the cosine trigonometric function, we first need a
3888 function that is called when one wishes to @code{eval} it. It could
3889 look something like this:
3892 static ex cos_eval_method(const ex & x)
3894 // if (!x%(2*Pi)) return 1
3895 // if (!x%Pi) return -1
3896 // if (!x%Pi/2) return 0
3897 // care for other cases...
3898 return cos(x).hold();
3902 @cindex @code{hold()}
3904 The last line returns @code{cos(x)} if we don't know what else to do and
3905 stops a potential recursive evaluation by saying @code{.hold()}, which
3906 sets a flag to the expression signaling that it has been evaluated. We
3907 should also implement a method for numerical evaluation and since we are
3908 lazy we sweep the problem under the rug by calling someone else's
3909 function that does so, in this case the one in class @code{numeric}:
3912 static ex cos_evalf(const ex & x)
3914 return cos(ex_to_numeric(x));
3918 Differentiation will surely turn up and so we need to tell @code{cos}
3919 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
3920 instance are then handled automatically by @code{basic::diff} and
3924 static ex cos_deriv(const ex & x, unsigned diff_param)
3930 @cindex product rule
3931 The second parameter is obligatory but uninteresting at this point. It
3932 specifies which parameter to differentiate in a partial derivative in
3933 case the function has more than one parameter and its main application
3934 is for correct handling of the chain rule. For Taylor expansion, it is
3935 enough to know how to differentiate. But if the function you want to
3936 implement does have a pole somewhere in the complex plane, you need to
3937 write another method for Laurent expansion around that point.
3939 Now that all the ingredients for @code{cos} have been set up, we need
3940 to tell the system about it. This is done by a macro and we are not
3941 going to descibe how it expands, please consult your preprocessor if you
3945 REGISTER_FUNCTION(cos, eval_func(cos_eval).
3946 evalf_func(cos_evalf).
3947 derivative_func(cos_deriv));
3950 The first argument is the function's name used for calling it and for
3951 output. The second binds the corresponding methods as options to this
3952 object. Options are separated by a dot and can be given in an arbitrary
3953 order. GiNaC functions understand several more options which are always
3954 specified as @code{.option(params)}, for example a method for series
3955 expansion @code{.series_func(cos_series)}. Again, if no series
3956 expansion method is given, GiNaC defaults to simple Taylor expansion,
3957 which is correct if there are no poles involved as is the case for the
3958 @code{cos} function. The way GiNaC handles poles in case there are any
3959 is best understood by studying one of the examples, like the Gamma
3960 (@code{tgamma}) function for instance. (In essence the function first
3961 checks if there is a pole at the evaluation point and falls back to
3962 Taylor expansion if there isn't. Then, the pole is regularized by some
3963 suitable transformation.) Also, the new function needs to be declared
3964 somewhere. This may also be done by a convenient preprocessor macro:
3967 DECLARE_FUNCTION_1P(cos)
3970 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
3971 implementation of @code{cos} is very incomplete and lacks several safety
3972 mechanisms. Please, have a look at the real implementation in GiNaC.
3973 (By the way: in case you are worrying about all the macros above we can
3974 assure you that functions are GiNaC's most macro-intense classes. We
3975 have done our best to avoid macros where we can.)
3978 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
3979 @c node-name, next, previous, up
3980 @section Adding classes
3982 If you are doing some very specialized things with GiNaC you may find that
3983 you have to implement your own algebraic classes to fit your needs. This
3984 section will explain how to do this by giving the example of a simple
3985 'string' class. After reading this section you will know how to properly
3986 declare a GiNaC class and what the minimum required member functions are
3987 that you have to implement. We only cover the implementation of a 'leaf'
3988 class here (i.e. one that doesn't contain subexpressions). Creating a
3989 container class like, for example, a class representing tensor products is
3990 more involved but this section should give you enough information so you can
3991 consult the source to GiNaC's predefined classes if you want to implement
3992 something more complicated.
3994 @subsection GiNaC's run-time type information system
3996 @cindex hierarchy of classes
3998 All algebraic classes (that is, all classes that can appear in expressions)
3999 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
4000 @code{basic *} (which is essentially what an @code{ex} is) represents a
4001 generic pointer to an algebraic class. Occasionally it is necessary to find
4002 out what the class of an object pointed to by a @code{basic *} really is.
4003 Also, for the unarchiving of expressions it must be possible to find the
4004 @code{unarchive()} function of a class given the class name (as a string). A
4005 system that provides this kind of information is called a run-time type
4006 information (RTTI) system. The C++ language provides such a thing (see the
4007 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
4008 implements its own, simpler RTTI.
4010 The RTTI in GiNaC is based on two mechanisms:
4015 The @code{basic} class declares a member variable @code{tinfo_key} which
4016 holds an unsigned integer that identifies the object's class. These numbers
4017 are defined in the @file{tinfos.h} header file for the built-in GiNaC
4018 classes. They all start with @code{TINFO_}.
4021 By means of some clever tricks with static members, GiNaC maintains a list
4022 of information for all classes derived from @code{basic}. The information
4023 available includes the class names, the @code{tinfo_key}s, and pointers
4024 to the unarchiving functions. This class registry is defined in the
4025 @file{registrar.h} header file.
4029 The disadvantage of this proprietary RTTI implementation is that there's
4030 a little more to do when implementing new classes (C++'s RTTI works more
4031 or less automatic) but don't worry, most of the work is simplified by
4034 @subsection A minimalistic example
4036 Now we will start implementing a new class @code{mystring} that allows
4037 placing character strings in algebraic expressions (this is not very useful,
4038 but it's just an example). This class will be a direct subclass of
4039 @code{basic}. You can use this sample implementation as a starting point
4040 for your own classes.
4042 The code snippets given here assume that you have included some header files
4048 #include <stdexcept>
4049 using namespace std;
4051 #include <ginac/ginac.h>
4052 using namespace GiNaC;
4055 The first thing we have to do is to define a @code{tinfo_key} for our new
4056 class. This can be any arbitrary unsigned number that is not already taken
4057 by one of the existing classes but it's better to come up with something
4058 that is unlikely to clash with keys that might be added in the future. The
4059 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
4060 which is not a requirement but we are going to stick with this scheme:
4063 const unsigned TINFO_mystring = 0x42420001U;
4066 Now we can write down the class declaration. The class stores a C++
4067 @code{string} and the user shall be able to construct a @code{mystring}
4068 object from a C or C++ string:
4071 class mystring : public basic
4073 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
4076 mystring(const string &s);
4077 mystring(const char *s);
4083 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
4086 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
4087 macros are defined in @file{registrar.h}. They take the name of the class
4088 and its direct superclass as arguments and insert all required declarations
4089 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
4090 the first line after the opening brace of the class definition. The
4091 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
4092 source (at global scope, of course, not inside a function).
4094 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
4095 declarations of the default and copy constructor, the destructor, the
4096 assignment operator and a couple of other functions that are required. It
4097 also defines a type @code{inherited} which refers to the superclass so you
4098 don't have to modify your code every time you shuffle around the class
4099 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
4100 constructor, the destructor and the assignment operator.
4102 Now there are nine member functions we have to implement to get a working
4108 @code{mystring()}, the default constructor.
4111 @code{void destroy(bool call_parent)}, which is used in the destructor and the
4112 assignment operator to free dynamically allocated members. The @code{call_parent}
4113 specifies whether the @code{destroy()} function of the superclass is to be
4117 @code{void copy(const mystring &other)}, which is used in the copy constructor
4118 and assignment operator to copy the member variables over from another
4119 object of the same class.
4122 @code{void archive(archive_node &n)}, the archiving function. This stores all
4123 information needed to reconstruct an object of this class inside an
4124 @code{archive_node}.
4127 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
4128 constructor. This constructs an instance of the class from the information
4129 found in an @code{archive_node}.
4132 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
4133 unarchiving function. It constructs a new instance by calling the unarchiving
4137 @code{int compare_same_type(const basic &other)}, which is used internally
4138 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
4139 -1, depending on the relative order of this object and the @code{other}
4140 object. If it returns 0, the objects are considered equal.
4141 @strong{Note:} This has nothing to do with the (numeric) ordering
4142 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
4143 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
4144 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
4145 must provide a @code{compare_same_type()} function, even those representing
4146 objects for which no reasonable algebraic ordering relationship can be
4150 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
4151 which are the two constructors we declared.
4155 Let's proceed step-by-step. The default constructor looks like this:
4158 mystring::mystring() : inherited(TINFO_mystring)
4160 // dynamically allocate resources here if required
4164 The golden rule is that in all constructors you have to set the
4165 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
4166 it will be set by the constructor of the superclass and all hell will break
4167 loose in the RTTI. For your convenience, the @code{basic} class provides
4168 a constructor that takes a @code{tinfo_key} value, which we are using here
4169 (remember that in our case @code{inherited = basic}). If the superclass
4170 didn't have such a constructor, we would have to set the @code{tinfo_key}
4171 to the right value manually.
4173 In the default constructor you should set all other member variables to
4174 reasonable default values (we don't need that here since our @code{str}
4175 member gets set to an empty string automatically). The constructor(s) are of
4176 course also the right place to allocate any dynamic resources you require.
4178 Next, the @code{destroy()} function:
4181 void mystring::destroy(bool call_parent)
4183 // free dynamically allocated resources here if required
4185 inherited::destroy(call_parent);
4189 This function is where we free all dynamically allocated resources. We don't
4190 have any so we're not doing anything here, but if we had, for example, used
4191 a C-style @code{char *} to store our string, this would be the place to
4192 @code{delete[]} the string storage. If @code{call_parent} is true, we have
4193 to call the @code{destroy()} function of the superclass after we're done
4194 (to mimic C++'s automatic invocation of superclass destructors where
4195 @code{destroy()} is called from outside a destructor).
4197 The @code{copy()} function just copies over the member variables from
4201 void mystring::copy(const mystring &other)
4203 inherited::copy(other);
4208 We can simply overwrite the member variables here. There's no need to worry
4209 about dynamically allocated storage. The assignment operator (which is
4210 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
4211 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
4212 explicitly call the @code{copy()} function of the superclass here so
4213 all the member variables will get copied.
4215 Next are the three functions for archiving. You have to implement them even
4216 if you don't plan to use archives, but the minimum required implementation
4217 is really simple. First, the archiving function:
4220 void mystring::archive(archive_node &n) const
4222 inherited::archive(n);
4223 n.add_string("string", str);
4227 The only thing that is really required is calling the @code{archive()}
4228 function of the superclass. Optionally, you can store all information you
4229 deem necessary for representing the object into the passed
4230 @code{archive_node}. We are just storing our string here. For more
4231 information on how the archiving works, consult the @file{archive.h} header
4234 The unarchiving constructor is basically the inverse of the archiving
4238 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4240 n.find_string("string", str);
4244 If you don't need archiving, just leave this function empty (but you must
4245 invoke the unarchiving constructor of the superclass). Note that we don't
4246 have to set the @code{tinfo_key} here because it is done automatically
4247 by the unarchiving constructor of the @code{basic} class.
4249 Finally, the unarchiving function:
4252 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4254 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4258 You don't have to understand how exactly this works. Just copy these four
4259 lines into your code literally (replacing the class name, of course). It
4260 calls the unarchiving constructor of the class and unless you are doing
4261 something very special (like matching @code{archive_node}s to global
4262 objects) you don't need a different implementation. For those who are
4263 interested: setting the @code{dynallocated} flag puts the object under
4264 the control of GiNaC's garbage collection. It will get deleted automatically
4265 once it is no longer referenced.
4267 Our @code{compare_same_type()} function uses a provided function to compare
4271 int mystring::compare_same_type(const basic &other) const
4273 const mystring &o = static_cast<const mystring &>(other);
4274 int cmpval = str.compare(o.str);
4277 else if (cmpval < 0)
4284 Although this function takes a @code{basic &}, it will always be a reference
4285 to an object of exactly the same class (objects of different classes are not
4286 comparable), so the cast is safe. If this function returns 0, the two objects
4287 are considered equal (in the sense that @math{A-B=0}), so you should compare
4288 all relevant member variables.
4290 Now the only thing missing is our two new constructors:
4293 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4295 // dynamically allocate resources here if required
4298 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4300 // dynamically allocate resources here if required
4304 No surprises here. We set the @code{str} member from the argument and
4305 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4307 That's it! We now have a minimal working GiNaC class that can store
4308 strings in algebraic expressions. Let's confirm that the RTTI works:
4311 ex e = mystring("Hello, world!");
4312 cout << is_ex_of_type(e, mystring) << endl;
4315 cout << e.bp->class_name() << endl;
4319 Obviously it does. Let's see what the expression @code{e} looks like:
4323 // -> [mystring object]
4326 Hm, not exactly what we expect, but of course the @code{mystring} class
4327 doesn't yet know how to print itself. This is done in the @code{print()}
4328 member function. Let's say that we wanted to print the string surrounded
4332 class mystring : public basic
4336 void print(const print_context &c, unsigned level = 0) const;
4340 void mystring::print(const print_context &c, unsigned level) const
4342 // print_context::s is a reference to an ostream
4343 c.s << '\"' << str << '\"';
4347 The @code{level} argument is only required for container classes to
4348 correctly parenthesize the output. Let's try again to print the expression:
4352 // -> "Hello, world!"
4355 Much better. The @code{mystring} class can be used in arbitrary expressions:
4358 e += mystring("GiNaC rulez");
4360 // -> "GiNaC rulez"+"Hello, world!"
4363 (note that GiNaC's automatic term reordering is in effect here), or even
4366 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4368 // -> "One string"^(2*sin(-"Another string"+Pi))
4371 Whether this makes sense is debatable but remember that this is only an
4372 example. At least it allows you to implement your own symbolic algorithms
4375 Note that GiNaC's algebraic rules remain unchanged:
4378 e = mystring("Wow") * mystring("Wow");
4382 e = pow(mystring("First")-mystring("Second"), 2);
4383 cout << e.expand() << endl;
4384 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4387 There's no way to, for example, make GiNaC's @code{add} class perform string
4388 concatenation. You would have to implement this yourself.
4390 @subsection Automatic evaluation
4392 @cindex @code{hold()}
4394 When dealing with objects that are just a little more complicated than the
4395 simple string objects we have implemented, chances are that you will want to
4396 have some automatic simplifications or canonicalizations performed on them.
4397 This is done in the evaluation member function @code{eval()}. Let's say that
4398 we wanted all strings automatically converted to lowercase with
4399 non-alphabetic characters stripped, and empty strings removed:
4402 class mystring : public basic
4406 ex eval(int level = 0) const;
4410 ex mystring::eval(int level) const
4413 for (int i=0; i<str.length(); i++) @{
4415 if (c >= 'A' && c <= 'Z')
4416 new_str += tolower(c);
4417 else if (c >= 'a' && c <= 'z')
4421 if (new_str.length() == 0)
4424 return mystring(new_str).hold();
4428 The @code{level} argument is used to limit the recursion depth of the
4429 evaluation. We don't have any subexpressions in the @code{mystring} class
4430 so we are not concerned with this. If we had, we would call the @code{eval()}
4431 functions of the subexpressions with @code{level - 1} as the argument if
4432 @code{level != 1}. The @code{hold()} member function sets a flag in the
4433 object that prevents further evaluation. Otherwise we might end up in an
4434 endless loop. When you want to return the object unmodified, use
4435 @code{return this->hold();}.
4437 Let's confirm that it works:
4440 ex e = mystring("Hello, world!") + mystring("!?#");
4444 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4449 @subsection Other member functions
4451 We have implemented only a small set of member functions to make the class
4452 work in the GiNaC framework. For a real algebraic class, there are probably
4453 some more functions that you will want to re-implement, such as
4454 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4455 or the header file of the class you want to make a subclass of to see
4456 what's there. One member function that you will most likely want to
4457 implement for terminal classes like the described string class is
4458 @code{calcchash()} that returns an @code{unsigned} hash value for the object
4459 which will allow GiNaC to compare and canonicalize expressions much more
4462 You can, of course, also add your own new member functions. In this case you
4463 will probably want to define a little helper function like
4466 inline const mystring &ex_to_mystring(const ex &e)
4468 return static_cast<const mystring &>(*e.bp);
4472 that let's you get at the object inside an expression (after you have
4473 verified that the type is correct) so you can call member functions that are
4474 specific to the class.
4476 That's it. May the source be with you!
4479 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4480 @c node-name, next, previous, up
4481 @chapter A Comparison With Other CAS
4484 This chapter will give you some information on how GiNaC compares to
4485 other, traditional Computer Algebra Systems, like @emph{Maple},
4486 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4487 disadvantages over these systems.
4490 * Advantages:: Stengths of the GiNaC approach.
4491 * Disadvantages:: Weaknesses of the GiNaC approach.
4492 * Why C++?:: Attractiveness of C++.
4495 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4496 @c node-name, next, previous, up
4499 GiNaC has several advantages over traditional Computer
4500 Algebra Systems, like
4505 familiar language: all common CAS implement their own proprietary
4506 grammar which you have to learn first (and maybe learn again when your
4507 vendor decides to `enhance' it). With GiNaC you can write your program
4508 in common C++, which is standardized.
4512 structured data types: you can build up structured data types using
4513 @code{struct}s or @code{class}es together with STL features instead of
4514 using unnamed lists of lists of lists.
4517 strongly typed: in CAS, you usually have only one kind of variables
4518 which can hold contents of an arbitrary type. This 4GL like feature is
4519 nice for novice programmers, but dangerous.
4522 development tools: powerful development tools exist for C++, like fancy
4523 editors (e.g. with automatic indentation and syntax highlighting),
4524 debuggers, visualization tools, documentation generators...
4527 modularization: C++ programs can easily be split into modules by
4528 separating interface and implementation.
4531 price: GiNaC is distributed under the GNU Public License which means
4532 that it is free and available with source code. And there are excellent
4533 C++-compilers for free, too.
4536 extendable: you can add your own classes to GiNaC, thus extending it on
4537 a very low level. Compare this to a traditional CAS that you can
4538 usually only extend on a high level by writing in the language defined
4539 by the parser. In particular, it turns out to be almost impossible to
4540 fix bugs in a traditional system.
4543 multiple interfaces: Though real GiNaC programs have to be written in
4544 some editor, then be compiled, linked and executed, there are more ways
4545 to work with the GiNaC engine. Many people want to play with
4546 expressions interactively, as in traditional CASs. Currently, two such
4547 windows into GiNaC have been implemented and many more are possible: the
4548 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
4549 types to a command line and second, as a more consistent approach, an
4550 interactive interface to the @acronym{Cint} C++ interpreter has been put
4551 together (called @acronym{GiNaC-cint}) that allows an interactive
4552 scripting interface consistent with the C++ language.
4555 seemless integration: it is somewhere between difficult and impossible
4556 to call CAS functions from within a program written in C++ or any other
4557 programming language and vice versa. With GiNaC, your symbolic routines
4558 are part of your program. You can easily call third party libraries,
4559 e.g. for numerical evaluation or graphical interaction. All other
4560 approaches are much more cumbersome: they range from simply ignoring the
4561 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
4562 system (i.e. @emph{Yacas}).
4565 efficiency: often large parts of a program do not need symbolic
4566 calculations at all. Why use large integers for loop variables or
4567 arbitrary precision arithmetics where @code{int} and @code{double} are
4568 sufficient? For pure symbolic applications, GiNaC is comparable in
4569 speed with other CAS.
4574 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
4575 @c node-name, next, previous, up
4576 @section Disadvantages
4578 Of course it also has some disadvantages:
4583 advanced features: GiNaC cannot compete with a program like
4584 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
4585 which grows since 1981 by the work of dozens of programmers, with
4586 respect to mathematical features. Integration, factorization,
4587 non-trivial simplifications, limits etc. are missing in GiNaC (and are
4588 not planned for the near future).
4591 portability: While the GiNaC library itself is designed to avoid any
4592 platform dependent features (it should compile on any ANSI compliant C++
4593 compiler), the currently used version of the CLN library (fast large
4594 integer and arbitrary precision arithmetics) can be compiled only on
4595 systems with a recently new C++ compiler from the GNU Compiler
4596 Collection (@acronym{GCC}).@footnote{This is because CLN uses
4597 PROVIDE/REQUIRE like macros to let the compiler gather all static
4598 initializations, which works for GNU C++ only.} GiNaC uses recent
4599 language features like explicit constructors, mutable members, RTTI,
4600 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
4601 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
4602 ANSI compliant, support all needed features.
4607 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
4608 @c node-name, next, previous, up
4611 Why did we choose to implement GiNaC in C++ instead of Java or any other
4612 language? C++ is not perfect: type checking is not strict (casting is
4613 possible), separation between interface and implementation is not
4614 complete, object oriented design is not enforced. The main reason is
4615 the often scolded feature of operator overloading in C++. While it may
4616 be true that operating on classes with a @code{+} operator is rarely
4617 meaningful, it is perfectly suited for algebraic expressions. Writing
4618 @math{3x+5y} as @code{3*x+5*y} instead of
4619 @code{x.times(3).plus(y.times(5))} looks much more natural.
4620 Furthermore, the main developers are more familiar with C++ than with
4621 any other programming language.
4624 @node Internal Structures, Expressions are reference counted, Why C++? , Top
4625 @c node-name, next, previous, up
4626 @appendix Internal Structures
4629 * Expressions are reference counted::
4630 * Internal representation of products and sums::
4633 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
4634 @c node-name, next, previous, up
4635 @appendixsection Expressions are reference counted
4637 @cindex reference counting
4638 @cindex copy-on-write
4639 @cindex garbage collection
4640 An expression is extremely light-weight since internally it works like a
4641 handle to the actual representation and really holds nothing more than a
4642 pointer to some other object. What this means in practice is that
4643 whenever you create two @code{ex} and set the second equal to the first
4644 no copying process is involved. Instead, the copying takes place as soon
4645 as you try to change the second. Consider the simple sequence of code:
4648 #include <ginac/ginac.h>
4649 using namespace std;
4650 using namespace GiNaC;
4654 symbol x("x"), y("y"), z("z");
4657 e1 = sin(x + 2*y) + 3*z + 41;
4658 e2 = e1; // e2 points to same object as e1
4659 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
4660 e2 += 1; // e2 is copied into a new object
4661 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
4665 The line @code{e2 = e1;} creates a second expression pointing to the
4666 object held already by @code{e1}. The time involved for this operation
4667 is therefore constant, no matter how large @code{e1} was. Actual
4668 copying, however, must take place in the line @code{e2 += 1;} because
4669 @code{e1} and @code{e2} are not handles for the same object any more.
4670 This concept is called @dfn{copy-on-write semantics}. It increases
4671 performance considerably whenever one object occurs multiple times and
4672 represents a simple garbage collection scheme because when an @code{ex}
4673 runs out of scope its destructor checks whether other expressions handle
4674 the object it points to too and deletes the object from memory if that
4675 turns out not to be the case. A slightly less trivial example of
4676 differentiation using the chain-rule should make clear how powerful this
4680 #include <ginac/ginac.h>
4681 using namespace std;
4682 using namespace GiNaC;
4686 symbol x("x"), y("y");
4690 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
4691 cout << e1 << endl // prints x+3*y
4692 << e2 << endl // prints (x+3*y)^3
4693 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
4697 Here, @code{e1} will actually be referenced three times while @code{e2}
4698 will be referenced two times. When the power of an expression is built,
4699 that expression needs not be copied. Likewise, since the derivative of
4700 a power of an expression can be easily expressed in terms of that
4701 expression, no copying of @code{e1} is involved when @code{e3} is
4702 constructed. So, when @code{e3} is constructed it will print as
4703 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
4704 holds a reference to @code{e2} and the factor in front is just
4707 As a user of GiNaC, you cannot see this mechanism of copy-on-write
4708 semantics. When you insert an expression into a second expression, the
4709 result behaves exactly as if the contents of the first expression were
4710 inserted. But it may be useful to remember that this is not what
4711 happens. Knowing this will enable you to write much more efficient
4712 code. If you still have an uncertain feeling with copy-on-write
4713 semantics, we recommend you have a look at the
4714 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
4715 Marshall Cline. Chapter 16 covers this issue and presents an
4716 implementation which is pretty close to the one in GiNaC.
4719 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
4720 @c node-name, next, previous, up
4721 @appendixsection Internal representation of products and sums
4723 @cindex representation
4726 @cindex @code{power}
4727 Although it should be completely transparent for the user of
4728 GiNaC a short discussion of this topic helps to understand the sources
4729 and also explain performance to a large degree. Consider the
4730 unexpanded symbolic expression
4732 $2d^3 \left( 4a + 5b - 3 \right)$
4735 @math{2*d^3*(4*a+5*b-3)}
4737 which could naively be represented by a tree of linear containers for
4738 addition and multiplication, one container for exponentiation with base
4739 and exponent and some atomic leaves of symbols and numbers in this
4744 @cindex pair-wise representation
4745 However, doing so results in a rather deeply nested tree which will
4746 quickly become inefficient to manipulate. We can improve on this by
4747 representing the sum as a sequence of terms, each one being a pair of a
4748 purely numeric multiplicative coefficient and its rest. In the same
4749 spirit we can store the multiplication as a sequence of terms, each
4750 having a numeric exponent and a possibly complicated base, the tree
4751 becomes much more flat:
4755 The number @code{3} above the symbol @code{d} shows that @code{mul}
4756 objects are treated similarly where the coefficients are interpreted as
4757 @emph{exponents} now. Addition of sums of terms or multiplication of
4758 products with numerical exponents can be coded to be very efficient with
4759 such a pair-wise representation. Internally, this handling is performed
4760 by most CAS in this way. It typically speeds up manipulations by an
4761 order of magnitude. The overall multiplicative factor @code{2} and the
4762 additive term @code{-3} look somewhat out of place in this
4763 representation, however, since they are still carrying a trivial
4764 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
4765 this is avoided by adding a field that carries an overall numeric
4766 coefficient. This results in the realistic picture of internal
4769 $2d^3 \left( 4a + 5b - 3 \right)$:
4772 @math{2*d^3*(4*a+5*b-3)}:
4778 This also allows for a better handling of numeric radicals, since
4779 @code{sqrt(2)} can now be carried along calculations. Now it should be
4780 clear, why both classes @code{add} and @code{mul} are derived from the
4781 same abstract class: the data representation is the same, only the
4782 semantics differs. In the class hierarchy, methods for polynomial
4783 expansion and the like are reimplemented for @code{add} and @code{mul},
4784 but the data structure is inherited from @code{expairseq}.
4787 @node Package Tools, ginac-config, Internal representation of products and sums, Top
4788 @c node-name, next, previous, up
4789 @appendix Package Tools
4791 If you are creating a software package that uses the GiNaC library,
4792 setting the correct command line options for the compiler and linker
4793 can be difficult. GiNaC includes two tools to make this process easier.
4796 * ginac-config:: A shell script to detect compiler and linker flags.
4797 * AM_PATH_GINAC:: Macro for GNU automake.
4801 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
4802 @c node-name, next, previous, up
4803 @section @command{ginac-config}
4804 @cindex ginac-config
4806 @command{ginac-config} is a shell script that you can use to determine
4807 the compiler and linker command line options required to compile and
4808 link a program with the GiNaC library.
4810 @command{ginac-config} takes the following flags:
4814 Prints out the version of GiNaC installed.
4816 Prints '-I' flags pointing to the installed header files.
4818 Prints out the linker flags necessary to link a program against GiNaC.
4819 @item --prefix[=@var{PREFIX}]
4820 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
4821 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
4822 Otherwise, prints out the configured value of @env{$prefix}.
4823 @item --exec-prefix[=@var{PREFIX}]
4824 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
4825 Otherwise, prints out the configured value of @env{$exec_prefix}.
4828 Typically, @command{ginac-config} will be used within a configure
4829 script, as described below. It, however, can also be used directly from
4830 the command line using backquotes to compile a simple program. For
4834 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
4837 This command line might expand to (for example):
4840 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
4841 -lginac -lcln -lstdc++
4844 Not only is the form using @command{ginac-config} easier to type, it will
4845 work on any system, no matter how GiNaC was configured.
4848 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
4849 @c node-name, next, previous, up
4850 @section @samp{AM_PATH_GINAC}
4851 @cindex AM_PATH_GINAC
4853 For packages configured using GNU automake, GiNaC also provides
4854 a macro to automate the process of checking for GiNaC.
4857 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
4865 Determines the location of GiNaC using @command{ginac-config}, which is
4866 either found in the user's path, or from the environment variable
4867 @env{GINACLIB_CONFIG}.
4870 Tests the installed libraries to make sure that their version
4871 is later than @var{MINIMUM-VERSION}. (A default version will be used
4875 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
4876 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
4877 variable to the output of @command{ginac-config --libs}, and calls
4878 @samp{AC_SUBST()} for these variables so they can be used in generated
4879 makefiles, and then executes @var{ACTION-IF-FOUND}.
4882 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
4883 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
4887 This macro is in file @file{ginac.m4} which is installed in
4888 @file{$datadir/aclocal}. Note that if automake was installed with a
4889 different @samp{--prefix} than GiNaC, you will either have to manually
4890 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
4891 aclocal the @samp{-I} option when running it.
4894 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
4895 * Example package:: Example of a package using AM_PATH_GINAC.
4899 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
4900 @c node-name, next, previous, up
4901 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
4903 Simply make sure that @command{ginac-config} is in your path, and run
4904 the configure script.
4911 The directory where the GiNaC libraries are installed needs
4912 to be found by your system's dynamic linker.
4914 This is generally done by
4917 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
4923 setting the environment variable @env{LD_LIBRARY_PATH},
4926 or, as a last resort,
4929 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
4930 running configure, for instance:
4933 LDFLAGS=-R/home/cbauer/lib ./configure
4938 You can also specify a @command{ginac-config} not in your path by
4939 setting the @env{GINACLIB_CONFIG} environment variable to the
4940 name of the executable
4943 If you move the GiNaC package from its installed location,
4944 you will either need to modify @command{ginac-config} script
4945 manually to point to the new location or rebuild GiNaC.
4956 --with-ginac-prefix=@var{PREFIX}
4957 --with-ginac-exec-prefix=@var{PREFIX}
4960 are provided to override the prefix and exec-prefix that were stored
4961 in the @command{ginac-config} shell script by GiNaC's configure. You are
4962 generally better off configuring GiNaC with the right path to begin with.
4966 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
4967 @c node-name, next, previous, up
4968 @subsection Example of a package using @samp{AM_PATH_GINAC}
4970 The following shows how to build a simple package using automake
4971 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
4974 #include <ginac/ginac.h>
4978 GiNaC::symbol x("x");
4979 GiNaC::ex a = GiNaC::sin(x);
4980 std::cout << "Derivative of " << a
4981 << " is " << a.diff(x) << std::endl;
4986 You should first read the introductory portions of the automake
4987 Manual, if you are not already familiar with it.
4989 Two files are needed, @file{configure.in}, which is used to build the
4993 dnl Process this file with autoconf to produce a configure script.
4995 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
5001 AM_PATH_GINAC(0.7.0, [
5002 LIBS="$LIBS $GINACLIB_LIBS"
5003 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
5004 ], AC_MSG_ERROR([need to have GiNaC installed]))
5009 The only command in this which is not standard for automake
5010 is the @samp{AM_PATH_GINAC} macro.
5012 That command does the following: If a GiNaC version greater or equal
5013 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
5014 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
5015 the error message `need to have GiNaC installed'
5017 And the @file{Makefile.am}, which will be used to build the Makefile.
5020 ## Process this file with automake to produce Makefile.in
5021 bin_PROGRAMS = simple
5022 simple_SOURCES = simple.cpp
5025 This @file{Makefile.am}, says that we are building a single executable,
5026 from a single sourcefile @file{simple.cpp}. Since every program
5027 we are building uses GiNaC we simply added the GiNaC options
5028 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
5029 want to specify them on a per-program basis: for instance by
5033 simple_LDADD = $(GINACLIB_LIBS)
5034 INCLUDES = $(GINACLIB_CPPFLAGS)
5037 to the @file{Makefile.am}.
5039 To try this example out, create a new directory and add the three
5042 Now execute the following commands:
5045 $ automake --add-missing
5050 You now have a package that can be built in the normal fashion
5059 @node Bibliography, Concept Index, Example package, Top
5060 @c node-name, next, previous, up
5061 @appendix Bibliography
5066 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
5069 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
5072 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
5075 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
5078 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
5079 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
5082 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
5083 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
5084 Academic Press, London
5087 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
5092 @node Concept Index, , Bibliography, Top
5093 @c node-name, next, previous, up
5094 @unnumbered Concept Index