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)
772 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
773 @c node-name, next, previous, up
775 @cindex @code{symbol} (class)
776 @cindex hierarchy of classes
779 Symbols are for symbolic manipulation what atoms are for chemistry. You
780 can declare objects of class @code{symbol} as any other object simply by
781 saying @code{symbol x,y;}. There is, however, a catch in here having to
782 do with the fact that C++ is a compiled language. The information about
783 the symbol's name is thrown away by the compiler but at a later stage
784 you may want to print expressions holding your symbols. In order to
785 avoid confusion GiNaC's symbols are able to know their own name. This
786 is accomplished by declaring its name for output at construction time in
787 the fashion @code{symbol x("x");}. If you declare a symbol using the
788 default constructor (i.e. without string argument) the system will deal
789 out a unique name. That name may not be suitable for printing but for
790 internal routines when no output is desired it is often enough. We'll
791 come across examples of such symbols later in this tutorial.
793 This implies that the strings passed to symbols at construction time may
794 not be used for comparing two of them. It is perfectly legitimate to
795 write @code{symbol x("x"),y("x");} but it is likely to lead into
796 trouble. Here, @code{x} and @code{y} are different symbols and
797 statements like @code{x-y} will not be simplified to zero although the
798 output @code{x-x} looks funny. Such output may also occur when there
799 are two different symbols in two scopes, for instance when you call a
800 function that declares a symbol with a name already existent in a symbol
801 in the calling function. Again, comparing them (using @code{operator==}
802 for instance) will always reveal their difference. Watch out, please.
804 @cindex @code{subs()}
805 Although symbols can be assigned expressions for internal reasons, you
806 should not do it (and we are not going to tell you how it is done). If
807 you want to replace a symbol with something else in an expression, you
808 can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
809 for more information).
812 @node Numbers, Constants, Symbols, Basic Concepts
813 @c node-name, next, previous, up
815 @cindex @code{numeric} (class)
821 For storing numerical things, GiNaC uses Bruno Haible's library
822 @acronym{CLN}. The classes therein serve as foundation classes for
823 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
824 alternatively for Common Lisp Numbers. In order to find out more about
825 @acronym{CLN}'s internals the reader is refered to the documentation of
826 that library. @inforef{Introduction, , cln}, for more
827 information. Suffice to say that it is by itself build on top of another
828 library, the GNU Multiple Precision library @acronym{GMP}, which is an
829 extremely fast library for arbitrary long integers and rationals as well
830 as arbitrary precision floating point numbers. It is very commonly used
831 by several popular cryptographic applications. @acronym{CLN} extends
832 @acronym{GMP} by several useful things: First, it introduces the complex
833 number field over either reals (i.e. floating point numbers with
834 arbitrary precision) or rationals. Second, it automatically converts
835 rationals to integers if the denominator is unity and complex numbers to
836 real numbers if the imaginary part vanishes and also correctly treats
837 algebraic functions. Third it provides good implementations of
838 state-of-the-art algorithms for all trigonometric and hyperbolic
839 functions as well as for calculation of some useful constants.
841 The user can construct an object of class @code{numeric} in several
842 ways. The following example shows the four most important constructors.
843 It uses construction from C-integer, construction of fractions from two
844 integers, construction from C-float and construction from a string:
847 #include <ginac/ginac.h>
848 using namespace GiNaC;
852 numeric two(2); // exact integer 2
853 numeric r(2,3); // exact fraction 2/3
854 numeric e(2.71828); // floating point number
855 numeric p("3.1415926535897932385"); // floating point number
856 // Trott's constant in scientific notation:
857 numeric trott("1.0841015122311136151E-2");
859 std::cout << two*p << std::endl; // floating point 6.283...
863 Note that all those constructors are @emph{explicit} which means you are
864 not allowed to write @code{numeric two=2;}. This is because the basic
865 objects to be handled by GiNaC are the expressions @code{ex} and we want
866 to keep things simple and wish objects like @code{pow(x,2)} to be
867 handled the same way as @code{pow(x,a)}, which means that we need to
868 allow a general @code{ex} as base and exponent. Therefore there is an
869 implicit constructor from C-integers directly to expressions handling
870 numerics at work in most of our examples. This design really becomes
871 convenient when one declares own functions having more than one
872 parameter but it forbids using implicit constructors because that would
873 lead to compile-time ambiguities.
875 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
876 This would, however, call C's built-in operator @code{/} for integers
877 first and result in a numeric holding a plain integer 1. @strong{Never
878 use the operator @code{/} on integers} unless you know exactly what you
879 are doing! Use the constructor from two integers instead, as shown in
880 the example above. Writing @code{numeric(1)/2} may look funny but works
883 @cindex @code{Digits}
885 We have seen now the distinction between exact numbers and floating
886 point numbers. Clearly, the user should never have to worry about
887 dynamically created exact numbers, since their `exactness' always
888 determines how they ought to be handled, i.e. how `long' they are. The
889 situation is different for floating point numbers. Their accuracy is
890 controlled by one @emph{global} variable, called @code{Digits}. (For
891 those readers who know about Maple: it behaves very much like Maple's
892 @code{Digits}). All objects of class numeric that are constructed from
893 then on will be stored with a precision matching that number of decimal
897 #include <ginac/ginac.h>
899 using namespace GiNaC;
903 numeric three(3.0), one(1.0);
904 numeric x = one/three;
906 cout << "in " << Digits << " digits:" << endl;
908 cout << Pi.evalf() << endl;
920 The above example prints the following output to screen:
927 0.333333333333333333333333333333333333333333333333333333333333333333
928 3.14159265358979323846264338327950288419716939937510582097494459231
931 It should be clear that objects of class @code{numeric} should be used
932 for constructing numbers or for doing arithmetic with them. The objects
933 one deals with most of the time are the polymorphic expressions @code{ex}.
935 @subsection Tests on numbers
937 Once you have declared some numbers, assigned them to expressions and
938 done some arithmetic with them it is frequently desired to retrieve some
939 kind of information from them like asking whether that number is
940 integer, rational, real or complex. For those cases GiNaC provides
941 several useful methods. (Internally, they fall back to invocations of
942 certain CLN functions.)
944 As an example, let's construct some rational number, multiply it with
945 some multiple of its denominator and test what comes out:
948 #include <ginac/ginac.h>
950 using namespace GiNaC;
952 // some very important constants:
953 const numeric twentyone(21);
954 const numeric ten(10);
955 const numeric five(5);
959 numeric answer = twentyone;
962 cout << answer.is_integer() << endl; // false, it's 21/5
964 cout << answer.is_integer() << endl; // true, it's 42 now!
968 Note that the variable @code{answer} is constructed here as an integer
969 by @code{numeric}'s copy constructor but in an intermediate step it
970 holds a rational number represented as integer numerator and integer
971 denominator. When multiplied by 10, the denominator becomes unity and
972 the result is automatically converted to a pure integer again.
973 Internally, the underlying @acronym{CLN} is responsible for this
974 behaviour and we refer the reader to @acronym{CLN}'s documentation.
975 Suffice to say that the same behaviour applies to complex numbers as
976 well as return values of certain functions. Complex numbers are
977 automatically converted to real numbers if the imaginary part becomes
978 zero. The full set of tests that can be applied is listed in the
982 @multitable @columnfractions .30 .70
983 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
984 @item @code{.is_zero()}
985 @tab @dots{}equal to zero
986 @item @code{.is_positive()}
987 @tab @dots{}not complex and greater than 0
988 @item @code{.is_integer()}
989 @tab @dots{}a (non-complex) integer
990 @item @code{.is_pos_integer()}
991 @tab @dots{}an integer and greater than 0
992 @item @code{.is_nonneg_integer()}
993 @tab @dots{}an integer and greater equal 0
994 @item @code{.is_even()}
995 @tab @dots{}an even integer
996 @item @code{.is_odd()}
997 @tab @dots{}an odd integer
998 @item @code{.is_prime()}
999 @tab @dots{}a prime integer (probabilistic primality test)
1000 @item @code{.is_rational()}
1001 @tab @dots{}an exact rational number (integers are rational, too)
1002 @item @code{.is_real()}
1003 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1004 @item @code{.is_cinteger()}
1005 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1006 @item @code{.is_crational()}
1007 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1012 @node Constants, Fundamental containers, Numbers, Basic Concepts
1013 @c node-name, next, previous, up
1015 @cindex @code{constant} (class)
1018 @cindex @code{Catalan}
1019 @cindex @code{Euler}
1020 @cindex @code{evalf()}
1021 Constants behave pretty much like symbols except that they return some
1022 specific number when the method @code{.evalf()} is called.
1024 The predefined known constants are:
1027 @multitable @columnfractions .14 .30 .56
1028 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1030 @tab Archimedes' constant
1031 @tab 3.14159265358979323846264338327950288
1032 @item @code{Catalan}
1033 @tab Catalan's constant
1034 @tab 0.91596559417721901505460351493238411
1036 @tab Euler's (or Euler-Mascheroni) constant
1037 @tab 0.57721566490153286060651209008240243
1042 @node Fundamental containers, Lists, Constants, Basic Concepts
1043 @c node-name, next, previous, up
1044 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1048 @cindex @code{power}
1050 Simple polynomial expressions are written down in GiNaC pretty much like
1051 in other CAS or like expressions involving numerical variables in C.
1052 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1053 been overloaded to achieve this goal. When you run the following
1054 code snippet, the constructor for an object of type @code{mul} is
1055 automatically called to hold the product of @code{a} and @code{b} and
1056 then the constructor for an object of type @code{add} is called to hold
1057 the sum of that @code{mul} object and the number one:
1061 symbol a("a"), b("b");
1066 @cindex @code{pow()}
1067 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1068 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1069 construction is necessary since we cannot safely overload the constructor
1070 @code{^} in C++ to construct a @code{power} object. If we did, it would
1071 have several counterintuitive and undesired effects:
1075 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1077 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1078 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1079 interpret this as @code{x^(a^b)}.
1081 Also, expressions involving integer exponents are very frequently used,
1082 which makes it even more dangerous to overload @code{^} since it is then
1083 hard to distinguish between the semantics as exponentiation and the one
1084 for exclusive or. (It would be embarassing to return @code{1} where one
1085 has requested @code{2^3}.)
1088 @cindex @command{ginsh}
1089 All effects are contrary to mathematical notation and differ from the
1090 way most other CAS handle exponentiation, therefore overloading @code{^}
1091 is ruled out for GiNaC's C++ part. The situation is different in
1092 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1093 that the other frequently used exponentiation operator @code{**} does
1094 not exist at all in C++).
1096 To be somewhat more precise, objects of the three classes described
1097 here, are all containers for other expressions. An object of class
1098 @code{power} is best viewed as a container with two slots, one for the
1099 basis, one for the exponent. All valid GiNaC expressions can be
1100 inserted. However, basic transformations like simplifying
1101 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1102 when this is mathematically possible. If we replace the outer exponent
1103 three in the example by some symbols @code{a}, the simplification is not
1104 safe and will not be performed, since @code{a} might be @code{1/2} and
1107 Objects of type @code{add} and @code{mul} are containers with an
1108 arbitrary number of slots for expressions to be inserted. Again, simple
1109 and safe simplifications are carried out like transforming
1110 @code{3*x+4-x} to @code{2*x+4}.
1112 The general rule is that when you construct such objects, GiNaC
1113 automatically creates them in canonical form, which might differ from
1114 the form you typed in your program. This allows for rapid comparison of
1115 expressions, since after all @code{a-a} is simply zero. Note, that the
1116 canonical form is not necessarily lexicographical ordering or in any way
1117 easily guessable. It is only guaranteed that constructing the same
1118 expression twice, either implicitly or explicitly, results in the same
1122 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1123 @c node-name, next, previous, up
1124 @section Lists of expressions
1125 @cindex @code{lst} (class)
1127 @cindex @code{nops()}
1129 @cindex @code{append()}
1130 @cindex @code{prepend()}
1132 The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
1133 These are sometimes used to supply a variable number of arguments of the same
1134 type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
1135 should have a basic understanding about them.
1137 Lists of up to 15 expressions can be directly constructed from single
1142 symbol x("x"), y("y");
1143 lst l(x, 2, y, x+y);
1144 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1148 Use the @code{nops()} method to determine the size (number of expressions) of
1149 a list and the @code{op()} method to access individual elements:
1153 cout << l.nops() << endl; // prints '4'
1154 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1158 Finally you can append or prepend an expression to a list with the
1159 @code{append()} and @code{prepend()} methods:
1163 l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
1164 l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
1169 @node Mathematical functions, Relations, Lists, Basic Concepts
1170 @c node-name, next, previous, up
1171 @section Mathematical functions
1172 @cindex @code{function} (class)
1173 @cindex trigonometric function
1174 @cindex hyperbolic function
1176 There are quite a number of useful functions hard-wired into GiNaC. For
1177 instance, all trigonometric and hyperbolic functions are implemented
1178 (@xref{Built-in Functions}, for a complete list).
1180 These functions are all objects of class @code{function}. They accept
1181 one or more expressions as arguments and return one expression. If the
1182 arguments are not numerical, the evaluation of the function may be
1183 halted, as it does in the next example, showing how a function returns
1184 itself twice and finally an expression that may be really useful:
1186 @cindex Gamma function
1187 @cindex @code{subs()}
1190 symbol x("x"), y("y");
1192 cout << tgamma(foo) << endl;
1193 // -> tgamma(x+(1/2)*y)
1194 ex bar = foo.subs(y==1);
1195 cout << tgamma(bar) << endl;
1197 ex foobar = bar.subs(x==7);
1198 cout << tgamma(foobar) << endl;
1199 // -> (135135/128)*Pi^(1/2)
1203 Besides evaluation most of these functions allow differentiation, series
1204 expansion and so on. Read the next chapter in order to learn more about
1208 @node Relations, Indexed objects, Mathematical functions, Basic Concepts
1209 @c node-name, next, previous, up
1211 @cindex @code{relational} (class)
1213 Sometimes, a relation holding between two expressions must be stored
1214 somehow. The class @code{relational} is a convenient container for such
1215 purposes. A relation is by definition a container for two @code{ex} and
1216 a relation between them that signals equality, inequality and so on.
1217 They are created by simply using the C++ operators @code{==}, @code{!=},
1218 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1220 @xref{Mathematical functions}, for examples where various applications
1221 of the @code{.subs()} method show how objects of class relational are
1222 used as arguments. There they provide an intuitive syntax for
1223 substitutions. They are also used as arguments to the @code{ex::series}
1224 method, where the left hand side of the relation specifies the variable
1225 to expand in and the right hand side the expansion point. They can also
1226 be used for creating systems of equations that are to be solved for
1227 unknown variables. But the most common usage of objects of this class
1228 is rather inconspicuous in statements of the form @code{if
1229 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1230 conversion from @code{relational} to @code{bool} takes place. Note,
1231 however, that @code{==} here does not perform any simplifications, hence
1232 @code{expand()} must be called explicitly.
1235 @node Indexed objects, Non-commutative objects, Relations, Basic Concepts
1236 @c node-name, next, previous, up
1237 @section Indexed objects
1239 GiNaC allows you to handle expressions containing general indexed objects in
1240 arbitrary spaces. It is also able to canonicalize and simplify such
1241 expressions and perform symbolic dummy index summations. There are a number
1242 of predefined indexed objects provided, like delta and metric tensors.
1244 There are few restrictions placed on indexed objects and their indices and
1245 it is easy to construct nonsense expressions, but our intention is to
1246 provide a general framework that allows you to implement algorithms with
1247 indexed quantities, getting in the way as little as possible.
1249 @cindex @code{idx} (class)
1250 @cindex @code{indexed} (class)
1251 @subsection Indexed quantities and their indices
1253 Indexed expressions in GiNaC are constructed of two special types of objects,
1254 @dfn{index objects} and @dfn{indexed objects}.
1258 @cindex contravariant
1261 @item Index objects are of class @code{idx} or a subclass. Every index has
1262 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1263 the index lives in) which can both be arbitrary expressions but are usually
1264 a number or a simple symbol. In addition, indices of class @code{varidx} have
1265 a @dfn{variance} (they can be co- or contravariant), and indices of class
1266 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1268 @item Indexed objects are of class @code{indexed} or a subclass. They
1269 contain a @dfn{base expression} (which is the expression being indexed), and
1270 one or more indices.
1274 @strong{Note:} when printing expressions, covariant indices and indices
1275 without variance are denoted @samp{.i} while contravariant indices are
1276 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1277 value. In the following, we are going to use that notation in the text so
1278 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1279 not visible in the output.
1281 A simple example shall illustrate the concepts:
1284 #include <ginac/ginac.h>
1285 using namespace std;
1286 using namespace GiNaC;
1290 symbol i_sym("i"), j_sym("j");
1291 idx i(i_sym, 3), j(j_sym, 3);
1294 cout << indexed(A, i, j) << endl;
1299 The @code{idx} constructor takes two arguments, the index value and the
1300 index dimension. First we define two index objects, @code{i} and @code{j},
1301 both with the numeric dimension 3. The value of the index @code{i} is the
1302 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1303 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1304 construct an expression containing one indexed object, @samp{A.i.j}. It has
1305 the symbol @code{A} as its base expression and the two indices @code{i} and
1308 Note the difference between the indices @code{i} and @code{j} which are of
1309 class @code{idx}, and the index values which are the sybols @code{i_sym}
1310 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1311 or numbers but must be index objects. For example, the following is not
1312 correct and will raise an exception:
1315 symbol i("i"), j("j");
1316 e = indexed(A, i, j); // ERROR: indices must be of type idx
1319 You can have multiple indexed objects in an expression, index values can
1320 be numeric, and index dimensions symbolic:
1324 symbol B("B"), dim("dim");
1325 cout << 4 * indexed(A, i)
1326 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1331 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1332 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1333 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1334 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1335 @code{simplify_indexed()} for that, see below).
1337 In fact, base expressions, index values and index dimensions can be
1338 arbitrary expressions:
1342 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1347 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1348 get an error message from this but you will probably not be able to do
1349 anything useful with it.
1351 @cindex @code{get_value()}
1352 @cindex @code{get_dimension()}
1356 ex idx::get_value(void);
1357 ex idx::get_dimension(void);
1360 return the value and dimension of an @code{idx} object. If you have an index
1361 in an expression, such as returned by calling @code{.op()} on an indexed
1362 object, you can get a reference to the @code{idx} object with the function
1363 @code{ex_to_idx()} on the expression.
1365 There are also the methods
1368 bool idx::is_numeric(void);
1369 bool idx::is_symbolic(void);
1370 bool idx::is_dim_numeric(void);
1371 bool idx::is_dim_symbolic(void);
1374 for checking whether the value and dimension are numeric or symbolic
1375 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1376 About Expressions}) returns information about the index value.
1378 @cindex @code{varidx} (class)
1379 If you need co- and contravariant indices, use the @code{varidx} class:
1383 symbol mu_sym("mu"), nu_sym("nu");
1384 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1385 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1387 cout << indexed(A, mu, nu) << endl;
1389 cout << indexed(A, mu_co, nu) << endl;
1391 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1396 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1397 co- or contravariant. The default is a contravariant (upper) index, but
1398 this can be overridden by supplying a third argument to the @code{varidx}
1399 constructor. The two methods
1402 bool varidx::is_covariant(void);
1403 bool varidx::is_contravariant(void);
1406 allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
1407 to get the object reference from an expression). There's also the very useful
1411 ex varidx::toggle_variance(void);
1414 which makes a new index with the same value and dimension but the opposite
1415 variance. By using it you only have to define the index once.
1417 @cindex @code{spinidx} (class)
1418 The @code{spinidx} class provides dotted and undotted variant indices, as
1419 used in the Weyl-van-der-Waerden spinor formalism:
1423 symbol K("K"), C_sym("C"), D_sym("D");
1424 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1425 // contravariant, undotted
1426 spinidx C_co(C_sym, 2, true); // covariant index
1427 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1428 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1430 cout << indexed(K, C, D) << endl;
1432 cout << indexed(K, C_co, D_dot) << endl;
1434 cout << indexed(K, D_co_dot, D) << endl;
1439 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
1440 dotted or undotted. The default is undotted but this can be overridden by
1441 supplying a fourth argument to the @code{spinidx} constructor. The two
1445 bool spinidx::is_dotted(void);
1446 bool spinidx::is_undotted(void);
1449 allow you to check whether or not a @code{spinidx} object is dotted (use
1450 @code{ex_to_spinidx()} to get the object reference from an expression).
1451 Finally, the two methods
1454 ex spinidx::toggle_dot(void);
1455 ex spinidx::toggle_variance_dot(void);
1458 create a new index with the same value and dimension but opposite dottedness
1459 and the same or opposite variance.
1461 @subsection Substituting indices
1463 @cindex @code{subs()}
1464 Sometimes you will want to substitute one symbolic index with another
1465 symbolic or numeric index, for example when calculating one specific element
1466 of a tensor expression. This is done with the @code{.subs()} method, as it
1467 is done for symbols (see @ref{Substituting Expressions}).
1469 You have two possibilities here. You can either substitute the whole index
1470 by another index or expression:
1474 ex e = indexed(A, mu_co);
1475 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1476 // -> A.mu becomes A~nu
1477 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1478 // -> A.mu becomes A~0
1479 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1480 // -> A.mu becomes A.0
1484 The third example shows that trying to replace an index with something that
1485 is not an index will substitute the index value instead.
1487 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1492 ex e = indexed(A, mu_co);
1493 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1494 // -> A.mu becomes A.nu
1495 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1496 // -> A.mu becomes A.0
1500 As you see, with the second method only the value of the index will get
1501 substituted. Its other properties, including its dimension, remain unchanged.
1502 If you want to change the dimension of an index you have to substitute the
1503 whole index by another one with the new dimension.
1505 Finally, substituting the base expression of an indexed object works as
1510 ex e = indexed(A, mu_co);
1511 cout << e << " becomes " << e.subs(A == A+B) << endl;
1512 // -> A.mu becomes (B+A).mu
1516 @subsection Symmetries
1518 Indexed objects can be declared as being totally symmetric or antisymmetric
1519 with respect to their indices. In this case, GiNaC will automatically bring
1520 the indices into a canonical order which allows for some immediate
1525 cout << indexed(A, indexed::symmetric, i, j)
1526 + indexed(A, indexed::symmetric, j, i) << endl;
1528 cout << indexed(B, indexed::antisymmetric, i, j)
1529 + indexed(B, indexed::antisymmetric, j, j) << endl;
1531 cout << indexed(B, indexed::antisymmetric, i, j)
1532 + indexed(B, indexed::antisymmetric, j, i) << endl;
1537 @cindex @code{get_free_indices()}
1539 @subsection Dummy indices
1541 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1542 that a summation over the index range is implied. Symbolic indices which are
1543 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1544 dummy nor free indices.
1546 To be recognized as a dummy index pair, the two indices must be of the same
1547 class and dimension and their value must be the same single symbol (an index
1548 like @samp{2*n+1} is never a dummy index). If the indices are of class
1549 @code{varidx} they must also be of opposite variance; if they are of class
1550 @code{spinidx} they must be both dotted or both undotted.
1552 The method @code{.get_free_indices()} returns a vector containing the free
1553 indices of an expression. It also checks that the free indices of the terms
1554 of a sum are consistent:
1558 symbol A("A"), B("B"), C("C");
1560 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1561 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1563 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1564 cout << exprseq(e.get_free_indices()) << endl;
1566 // 'j' and 'l' are dummy indices
1568 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1569 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1571 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1572 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1573 cout << exprseq(e.get_free_indices()) << endl;
1575 // 'nu' is a dummy index, but 'sigma' is not
1577 e = indexed(A, mu, mu);
1578 cout << exprseq(e.get_free_indices()) << endl;
1580 // 'mu' is not a dummy index because it appears twice with the same
1583 e = indexed(A, mu, nu) + 42;
1584 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1585 // this will throw an exception:
1586 // "add::get_free_indices: inconsistent indices in sum"
1590 @cindex @code{simplify_indexed()}
1591 @subsection Simplifying indexed expressions
1593 In addition to the few automatic simplifications that GiNaC performs on
1594 indexed expressions (such as re-ordering the indices of symmetric tensors
1595 and calculating traces and convolutions of matrices and predefined tensors)
1599 ex ex::simplify_indexed(void);
1600 ex ex::simplify_indexed(const scalar_products & sp);
1603 that performs some more expensive operations:
1606 @item it checks the consistency of free indices in sums in the same way
1607 @code{get_free_indices()} does
1608 @item it (symbolically) calculates all possible dummy index summations/contractions
1609 with the predefined tensors (this will be explained in more detail in the
1611 @item as a special case of dummy index summation, it can replace scalar products
1612 of two tensors with a user-defined value
1615 The last point is done with the help of the @code{scalar_products} class
1616 which is used to store scalar products with known values (this is not an
1617 arithmetic class, you just pass it to @code{simplify_indexed()}):
1621 symbol A("A"), B("B"), C("C"), i_sym("i");
1625 sp.add(A, B, 0); // A and B are orthogonal
1626 sp.add(A, C, 0); // A and C are orthogonal
1627 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1629 e = indexed(A + B, i) * indexed(A + C, i);
1631 // -> (B+A).i*(A+C).i
1633 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1639 The @code{scalar_products} object @code{sp} acts as a storage for the
1640 scalar products added to it with the @code{.add()} method. This method
1641 takes three arguments: the two expressions of which the scalar product is
1642 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1643 @code{simplify_indexed()} will replace all scalar products of indexed
1644 objects that have the symbols @code{A} and @code{B} as base expressions
1645 with the single value 0. The number, type and dimension of the indices
1646 doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1648 @cindex @code{expand()}
1649 The example above also illustrates a feature of the @code{expand()} method:
1650 if passed the @code{expand_indexed} option it will distribute indices
1651 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1653 @cindex @code{tensor} (class)
1654 @subsection Predefined tensors
1656 Some frequently used special tensors such as the delta, epsilon and metric
1657 tensors are predefined in GiNaC. They have special properties when
1658 contracted with other tensor expressions and some of them have constant
1659 matrix representations (they will evaluate to a number when numeric
1660 indices are specified).
1662 @cindex @code{delta_tensor()}
1663 @subsubsection Delta tensor
1665 The delta tensor takes two indices, is symmetric and has the matrix
1666 representation @code{diag(1,1,1,...)}. It is constructed by the function
1667 @code{delta_tensor()}:
1671 symbol A("A"), B("B");
1673 idx i(symbol("i"), 3), j(symbol("j"), 3),
1674 k(symbol("k"), 3), l(symbol("l"), 3);
1676 ex e = indexed(A, i, j) * indexed(B, k, l)
1677 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1678 cout << e.simplify_indexed() << endl;
1681 cout << delta_tensor(i, i) << endl;
1686 @cindex @code{metric_tensor()}
1687 @subsubsection General metric tensor
1689 The function @code{metric_tensor()} creates a general symmetric metric
1690 tensor with two indices that can be used to raise/lower tensor indices. The
1691 metric tensor is denoted as @samp{g} in the output and if its indices are of
1692 mixed variance it is automatically replaced by a delta tensor:
1698 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1700 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1701 cout << e.simplify_indexed() << endl;
1704 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1705 cout << e.simplify_indexed() << endl;
1708 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1709 * metric_tensor(nu, rho);
1710 cout << e.simplify_indexed() << endl;
1713 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1714 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1715 + indexed(A, mu.toggle_variance(), rho));
1716 cout << e.simplify_indexed() << endl;
1721 @cindex @code{lorentz_g()}
1722 @subsubsection Minkowski metric tensor
1724 The Minkowski metric tensor is a special metric tensor with a constant
1725 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1726 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1727 It is created with the function @code{lorentz_g()} (although it is output as
1732 varidx mu(symbol("mu"), 4);
1734 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1735 * lorentz_g(mu, varidx(0, 4)); // negative signature
1736 cout << e.simplify_indexed() << endl;
1739 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1740 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1741 cout << e.simplify_indexed() << endl;
1746 @cindex @code{spinor_metric()}
1747 @subsubsection Spinor metric tensor
1749 The function @code{spinor_metric()} creates an antisymmetric tensor with
1750 two indices that is used to raise/lower indices of 2-component spinors.
1751 It is output as @samp{eps}:
1757 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
1758 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
1760 e = spinor_metric(A, B) * indexed(psi, B_co);
1761 cout << e.simplify_indexed() << endl;
1764 e = spinor_metric(A, B) * indexed(psi, A_co);
1765 cout << e.simplify_indexed() << endl;
1768 e = spinor_metric(A_co, B_co) * indexed(psi, B);
1769 cout << e.simplify_indexed() << endl;
1772 e = spinor_metric(A_co, B_co) * indexed(psi, A);
1773 cout << e.simplify_indexed() << endl;
1776 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
1777 cout << e.simplify_indexed() << endl;
1780 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
1781 cout << e.simplify_indexed() << endl;
1786 The matrix representation of the spinor metric is @code{[[ [[ 0, 1 ]], [[ -1, 0 ]]}.
1788 @cindex @code{epsilon_tensor()}
1789 @cindex @code{lorentz_eps()}
1790 @subsubsection Epsilon tensor
1792 The epsilon tensor is totally antisymmetric, its number of indices is equal
1793 to the dimension of the index space (the indices must all be of the same
1794 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1795 defined to be 1. Its behaviour with indices that have a variance also
1796 depends on the signature of the metric. Epsilon tensors are output as
1799 There are three functions defined to create epsilon tensors in 2, 3 and 4
1803 ex epsilon_tensor(const ex & i1, const ex & i2);
1804 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1805 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1808 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1809 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1810 Minkowski space (the last @code{bool} argument specifies whether the metric
1811 has negative or positive signature, as in the case of the Minkowski metric
1814 @subsection Linear algebra
1816 The @code{matrix} class can be used with indices to do some simple linear
1817 algebra (linear combinations and products of vectors and matrices, traces
1818 and scalar products):
1822 idx i(symbol("i"), 2), j(symbol("j"), 2);
1823 symbol x("x"), y("y");
1825 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
1827 cout << indexed(A, i, i) << endl;
1830 ex e = indexed(A, i, j) * indexed(X, j);
1831 cout << e.simplify_indexed() << endl;
1832 // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
1834 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
1835 cout << e.simplify_indexed() << endl;
1836 // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
1840 You can of course obtain the same results with the @code{matrix::add()},
1841 @code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
1842 don't have to worry about transposing matrices.
1844 Matrix indices always start at 0 and their dimension must match the number
1845 of rows/columns of the matrix. Matrices with one row or one column are
1846 vectors and can have one or two indices (it doesn't matter whether it's a
1847 row or a column vector). Other matrices must have two indices.
1849 You should be careful when using indices with variance on matrices. GiNaC
1850 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
1851 @samp{F.mu.nu} are different matrices. In this case you should use only
1852 one form for @samp{F} and explicitly multiply it with a matrix representation
1853 of the metric tensor.
1856 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
1857 @c node-name, next, previous, up
1858 @section Non-commutative objects
1860 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
1861 non-commutative objects are built-in which are mostly of use in high energy
1865 @item Clifford (Dirac) algebra (class @code{clifford})
1866 @item su(3) Lie algebra (class @code{color})
1867 @item Matrices (unindexed) (class @code{matrix})
1870 The @code{clifford} and @code{color} classes are subclasses of
1871 @code{indexed} because the elements of these algebras ususally carry
1874 Unlike most computer algebra systems, GiNaC does not primarily provide an
1875 operator (often denoted @samp{&*}) for representing inert products of
1876 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
1877 classes of objects involved, and non-commutative products are formed with
1878 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
1879 figuring out by itself which objects commute and will group the factors
1880 by their class. Consider this example:
1884 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
1885 idx a(symbol("a"), 8), b(symbol("b"), 8);
1886 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
1888 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
1892 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
1893 groups the non-commutative factors (the gammas and the su(3) generators)
1894 together while preserving the order of factors within each class (because
1895 Clifford objects commute with color objects). The resulting expression is a
1896 @emph{commutative} product with two factors that are themselves non-commutative
1897 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
1898 parentheses are placed around the non-commutative products in the output.
1900 @cindex @code{ncmul} (class)
1901 Non-commutative products are internally represented by objects of the class
1902 @code{ncmul}, as opposed to commutative products which are handled by the
1903 @code{mul} class. You will normally not have to worry about this distinction,
1906 The advantage of this approach is that you never have to worry about using
1907 (or forgetting to use) a special operator when constructing non-commutative
1908 expressions. Also, non-commutative products in GiNaC are more intelligent
1909 than in other computer algebra systems; they can, for example, automatically
1910 canonicalize themselves according to rules specified in the implementation
1911 of the non-commutative classes. The drawback is that to work with other than
1912 the built-in algebras you have to implement new classes yourself. Symbols
1913 always commute and it's not possible to construct non-commutative products
1914 using symbols to represent the algebra elements or generators. User-defined
1915 functions can, however, be specified as being non-commutative.
1917 @cindex @code{return_type()}
1918 @cindex @code{return_type_tinfo()}
1919 Information about the commutativity of an object or expression can be
1920 obtained with the two member functions
1923 unsigned ex::return_type(void) const;
1924 unsigned ex::return_type_tinfo(void) const;
1927 The @code{return_type()} function returns one of three values (defined in
1928 the header file @file{flags.h}), corresponding to three categories of
1929 expressions in GiNaC:
1932 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
1933 classes are of this kind.
1934 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
1935 certain class of non-commutative objects which can be determined with the
1936 @code{return_type_tinfo()} method. Expressions of this category commute
1937 with everything except @code{noncommutative} expressions of the same
1939 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
1940 of non-commutative objects of different classes. Expressions of this
1941 category don't commute with any other @code{noncommutative} or
1942 @code{noncommutative_composite} expressions.
1945 The value returned by the @code{return_type_tinfo()} method is valid only
1946 when the return type of the expression is @code{noncommutative}. It is a
1947 value that is unique to the class of the object and usually one of the
1948 constants in @file{tinfos.h}, or derived therefrom.
1950 Here are a couple of examples:
1953 @multitable @columnfractions 0.33 0.33 0.34
1954 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
1955 @item @code{42} @tab @code{commutative} @tab -
1956 @item @code{2*x-y} @tab @code{commutative} @tab -
1957 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1958 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1959 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
1960 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
1964 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
1965 @code{TINFO_clifford} for objects with a representation label of zero.
1966 Other representation labels yield a different @code{return_type_tinfo()},
1967 but it's the same for any two objects with the same label. This is also true
1971 @cindex @code{clifford} (class)
1972 @subsection Clifford algebra
1974 @cindex @code{dirac_gamma()}
1975 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
1976 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
1977 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
1978 is the Minkowski metric tensor. Dirac gammas are constructed by the function
1981 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
1984 which takes two arguments: the index and a @dfn{representation label} in the
1985 range 0 to 255 which is used to distinguish elements of different Clifford
1986 algebras (this is also called a @dfn{spin line index}). Gammas with different
1987 labels commute with each other. The dimension of the index can be 4 or (in
1988 the framework of dimensional regularization) any symbolic value. Spinor
1989 indices on Dirac gammas are not supported in GiNaC.
1991 @cindex @code{dirac_ONE()}
1992 The unity element of a Clifford algebra is constructed by
1995 ex dirac_ONE(unsigned char rl = 0);
1998 @cindex @code{dirac_gamma5()}
1999 and there's a special element @samp{gamma5} that commutes with all other
2000 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2004 ex dirac_gamma5(unsigned char rl = 0);
2007 @cindex @code{dirac_gamma6()}
2008 @cindex @code{dirac_gamma7()}
2009 The two additional functions
2012 ex dirac_gamma6(unsigned char rl = 0);
2013 ex dirac_gamma7(unsigned char rl = 0);
2016 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2019 @cindex @code{dirac_slash()}
2020 Finally, the function
2023 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2026 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2027 whose dimension is given by the @code{dim} argument.
2029 The @code{simplify_indexed()} function performs contractions in gamma strings
2030 if possible, for example
2035 symbol a("a"), b("b"), D("D");
2036 varidx mu(symbol("mu"), D);
2037 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2038 * dirac_gamma(mu.toggle_variance());
2040 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2041 e = e.simplify_indexed();
2043 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2044 cout << e.subs(D == 4) << endl;
2045 // -> -2*gamma~symbol10*a.symbol10
2046 // [ == -2 * dirac_slash(a, D) ]
2051 @cindex @code{dirac_trace()}
2052 To calculate the trace of an expression containing strings of Dirac gammas
2053 you use the function
2056 ex dirac_trace(const ex & e, unsigned char rl = 0);
2059 This function takes the trace of all gammas with the specified representation
2060 label; gammas with other labels are left standing. The @code{dirac_trace()}
2061 function is a linear functional that is equal to the usual trace only in
2062 @math{D = 4} dimensions. In particular, the functional is not cyclic in
2063 @math{D != 4} dimensions when acting on expressions containing @samp{gamma5},
2064 so it's not a proper trace. This @samp{gamma5} scheme is described in greater
2065 detail in @cite{The Role of gamma5 in Dimensional Regularization}.
2067 The value of the trace itself is also usually different in 4 and in
2068 @math{D != 4} dimensions:
2073 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2074 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2075 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2076 cout << dirac_trace(e).simplify_indexed() << endl;
2083 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2084 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2085 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2086 cout << dirac_trace(e).simplify_indexed() << endl;
2087 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2091 Here is an example for using @code{dirac_trace()} to compute a value that
2092 appears in the calculation of the one-loop vacuum polarization amplitude in
2097 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2098 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2101 sp.add(l, l, pow(l, 2));
2102 sp.add(l, q, ldotq);
2104 ex e = dirac_gamma(mu) *
2105 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2106 dirac_gamma(mu.toggle_variance()) *
2107 (dirac_slash(l, D) + m * dirac_ONE());
2108 e = dirac_trace(e).simplify_indexed(sp);
2109 e = e.collect(lst(l, ldotq, m), true);
2111 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2116 @cindex @code{color} (class)
2117 @subsection Color algebra
2119 @cindex @code{color_T()}
2120 For computations in quantum chromodynamics, GiNaC implements the base elements
2121 and structure constants of the su(3) Lie algebra (color algebra). The base
2122 elements @math{T_a} are constructed by the function
2125 ex color_T(const ex & a, unsigned char rl = 0);
2128 which takes two arguments: the index and a @dfn{representation label} in the
2129 range 0 to 255 which is used to distinguish elements of different color
2130 algebras. Objects with different labels commute with each other. The
2131 dimension of the index must be exactly 8 and it should be of class @code{idx},
2134 @cindex @code{color_ONE()}
2135 The unity element of a color algebra is constructed by
2138 ex color_ONE(unsigned char rl = 0);
2141 @cindex @code{color_d()}
2142 @cindex @code{color_f()}
2146 ex color_d(const ex & a, const ex & b, const ex & c);
2147 ex color_f(const ex & a, const ex & b, const ex & c);
2150 create the symmetric and antisymmetric structure constants @math{d_abc} and
2151 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2152 and @math{[T_a, T_b] = i f_abc T_c}.
2154 @cindex @code{color_h()}
2155 There's an additional function
2158 ex color_h(const ex & a, const ex & b, const ex & c);
2161 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2163 The function @code{simplify_indexed()} performs some simplifications on
2164 expressions containing color objects:
2169 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2170 k(symbol("k"), 8), l(symbol("l"), 8);
2172 e = color_d(a, b, l) * color_f(a, b, k);
2173 cout << e.simplify_indexed() << endl;
2176 e = color_d(a, b, l) * color_d(a, b, k);
2177 cout << e.simplify_indexed() << endl;
2180 e = color_f(l, a, b) * color_f(a, b, k);
2181 cout << e.simplify_indexed() << endl;
2184 e = color_h(a, b, c) * color_h(a, b, c);
2185 cout << e.simplify_indexed() << endl;
2188 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2189 cout << e.simplify_indexed() << endl;
2190 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2194 @cindex @code{color_trace()}
2195 To calculate the trace of an expression containing color objects you use the
2199 ex color_trace(const ex & e, unsigned char rl = 0);
2202 This function takes the trace of all color @samp{T} objects with the
2203 specified representation label; @samp{T}s with other labels are left
2204 standing. For example:
2208 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2210 // -> -I*f.a.c.b+d.a.c.b
2215 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2216 @c node-name, next, previous, up
2217 @chapter Methods and Functions
2220 In this chapter the most important algorithms provided by GiNaC will be
2221 described. Some of them are implemented as functions on expressions,
2222 others are implemented as methods provided by expression objects. If
2223 they are methods, there exists a wrapper function around it, so you can
2224 alternatively call it in a functional way as shown in the simple
2229 cout << "As method: " << sin(1).evalf() << endl;
2230 cout << "As function: " << evalf(sin(1)) << endl;
2234 @cindex @code{subs()}
2235 The general rule is that wherever methods accept one or more parameters
2236 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2237 wrapper accepts is the same but preceded by the object to act on
2238 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2239 most natural one in an OO model but it may lead to confusion for MapleV
2240 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2241 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2242 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2243 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2244 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2245 here. Also, users of MuPAD will in most cases feel more comfortable
2246 with GiNaC's convention. All function wrappers are implemented
2247 as simple inline functions which just call the corresponding method and
2248 are only provided for users uncomfortable with OO who are dead set to
2249 avoid method invocations. Generally, nested function wrappers are much
2250 harder to read than a sequence of methods and should therefore be
2251 avoided if possible. On the other hand, not everything in GiNaC is a
2252 method on class @code{ex} and sometimes calling a function cannot be
2256 * Information About Expressions::
2257 * Substituting Expressions::
2258 * Polynomial Arithmetic:: Working with polynomials.
2259 * Rational Expressions:: Working with rational functions.
2260 * Symbolic Differentiation::
2261 * Series Expansion:: Taylor and Laurent expansion.
2262 * Built-in Functions:: List of predefined mathematical functions.
2263 * Input/Output:: Input and output of expressions.
2267 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2268 @c node-name, next, previous, up
2269 @section Getting information about expressions
2271 @subsection Checking expression types
2272 @cindex @code{is_ex_of_type()}
2273 @cindex @code{ex_to_numeric()}
2274 @cindex @code{ex_to_@dots{}}
2275 @cindex @code{Converting ex to other classes}
2276 @cindex @code{info()}
2277 @cindex @code{return_type()}
2278 @cindex @code{return_type_tinfo()}
2280 Sometimes it's useful to check whether a given expression is a plain number,
2281 a sum, a polynomial with integer coefficients, or of some other specific type.
2282 GiNaC provides a couple of functions for this (the first one is actually a macro):
2285 bool is_ex_of_type(const ex & e, TYPENAME t);
2286 bool ex::info(unsigned flag);
2287 unsigned ex::return_type(void) const;
2288 unsigned ex::return_type_tinfo(void) const;
2291 When the test made by @code{is_ex_of_type()} returns true, it is safe to
2292 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
2293 one of the class names (@xref{The Class Hierarchy}, for a list of all
2294 classes). For example, assuming @code{e} is an @code{ex}:
2299 if (is_ex_of_type(e, numeric))
2300 numeric n = ex_to_numeric(e);
2305 @code{is_ex_of_type()} allows you to check whether the top-level object of
2306 an expression @samp{e} is an instance of the GiNaC class @samp{t}
2307 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2308 e.g., for checking whether an expression is a number, a sum, or a product:
2315 is_ex_of_type(e1, numeric); // true
2316 is_ex_of_type(e2, numeric); // false
2317 is_ex_of_type(e1, add); // false
2318 is_ex_of_type(e2, add); // true
2319 is_ex_of_type(e1, mul); // false
2320 is_ex_of_type(e2, mul); // false
2324 The @code{info()} method is used for checking certain attributes of
2325 expressions. The possible values for the @code{flag} argument are defined
2326 in @file{ginac/flags.h}, the most important being explained in the following
2330 @multitable @columnfractions .30 .70
2331 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2332 @item @code{numeric}
2333 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
2335 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2336 @item @code{rational}
2337 @tab @dots{}an exact rational number (integers are rational, too)
2338 @item @code{integer}
2339 @tab @dots{}a (non-complex) integer
2340 @item @code{crational}
2341 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2342 @item @code{cinteger}
2343 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2344 @item @code{positive}
2345 @tab @dots{}not complex and greater than 0
2346 @item @code{negative}
2347 @tab @dots{}not complex and less than 0
2348 @item @code{nonnegative}
2349 @tab @dots{}not complex and greater than or equal to 0
2351 @tab @dots{}an integer greater than 0
2353 @tab @dots{}an integer less than 0
2354 @item @code{nonnegint}
2355 @tab @dots{}an integer greater than or equal to 0
2357 @tab @dots{}an even integer
2359 @tab @dots{}an odd integer
2361 @tab @dots{}a prime integer (probabilistic primality test)
2362 @item @code{relation}
2363 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
2364 @item @code{relation_equal}
2365 @tab @dots{}a @code{==} relation
2366 @item @code{relation_not_equal}
2367 @tab @dots{}a @code{!=} relation
2368 @item @code{relation_less}
2369 @tab @dots{}a @code{<} relation
2370 @item @code{relation_less_or_equal}
2371 @tab @dots{}a @code{<=} relation
2372 @item @code{relation_greater}
2373 @tab @dots{}a @code{>} relation
2374 @item @code{relation_greater_or_equal}
2375 @tab @dots{}a @code{>=} relation
2377 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
2379 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
2380 @item @code{polynomial}
2381 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2382 @item @code{integer_polynomial}
2383 @tab @dots{}a polynomial with (non-complex) integer coefficients
2384 @item @code{cinteger_polynomial}
2385 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2386 @item @code{rational_polynomial}
2387 @tab @dots{}a polynomial with (non-complex) rational coefficients
2388 @item @code{crational_polynomial}
2389 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2390 @item @code{rational_function}
2391 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2392 @item @code{algebraic}
2393 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2397 To determine whether an expression is commutative or non-commutative and if
2398 so, with which other expressions it would commute, you use the methods
2399 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2400 for an explanation of these.
2403 @subsection Accessing subexpressions
2404 @cindex @code{nops()}
2406 @cindex @code{has()}
2408 @cindex @code{relational} (class)
2410 GiNaC provides the two methods
2413 unsigned ex::nops();
2414 ex ex::op(unsigned i);
2417 for accessing the subexpressions in the container-like GiNaC classes like
2418 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2419 determines the number of subexpressions (@samp{operands}) contained, while
2420 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2421 In the case of a @code{power} object, @code{op(0)} will return the basis
2422 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2423 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2425 The left-hand and right-hand side expressions of objects of class
2426 @code{relational} (and only of these) can also be accessed with the methods
2436 bool ex::has(const ex & other);
2439 checks whether an expression contains the given subexpression @code{other}.
2440 This only works reliably if @code{other} is of an atomic class such as a
2441 @code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
2442 @code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
2445 @subsection Comparing expressions
2446 @cindex @code{is_equal()}
2447 @cindex @code{is_zero()}
2449 Expressions can be compared with the usual C++ relational operators like
2450 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2451 the result is usually not determinable and the result will be @code{false},
2452 except in the case of the @code{!=} operator. You should also be aware that
2453 GiNaC will only do the most trivial test for equality (subtracting both
2454 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2457 Actually, if you construct an expression like @code{a == b}, this will be
2458 represented by an object of the @code{relational} class (@xref{Relations}.)
2459 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2461 There are also two methods
2464 bool ex::is_equal(const ex & other);
2468 for checking whether one expression is equal to another, or equal to zero,
2471 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2472 GiNaC header files. This method is however only to be used internally by
2473 GiNaC to establish a canonical sort order for terms, and using it to compare
2474 expressions will give very surprising results.
2477 @node Substituting Expressions, Polynomial Arithmetic, Information About Expressions, Methods and Functions
2478 @c node-name, next, previous, up
2479 @section Substituting expressions
2480 @cindex @code{subs()}
2482 Algebraic objects inside expressions can be replaced with arbitrary
2483 expressions via the @code{.subs()} method:
2486 ex ex::subs(const ex & e);
2487 ex ex::subs(const lst & syms, const lst & repls);
2490 In the first form, @code{subs()} accepts a relational of the form
2491 @samp{object == expression} or a @code{lst} of such relationals:
2495 symbol x("x"), y("y");
2497 ex e1 = 2*x^2-4*x+3;
2498 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2502 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2507 @code{subs()} performs syntactic substitution of any complete algebraic
2508 object; it does not try to match sub-expressions as is demonstrated by the
2513 symbol x("x"), y("y"), z("z");
2515 ex e1 = pow(x+y, 2);
2516 cout << e1.subs(x+y == 4) << endl;
2519 ex e2 = sin(x)*cos(x);
2520 cout << e2.subs(sin(x) == cos(x)) << endl;
2524 cout << e3.subs(x+y == 4) << endl;
2526 // (and not 4+z as one might expect)
2530 If you specify multiple substitutions, they are performed in parallel, so e.g.
2531 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2533 The second form of @code{subs()} takes two lists, one for the objects to be
2534 replaced and one for the expressions to be substituted (both lists must
2535 contain the same number of elements). Using this form, you would write
2536 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2539 @node Polynomial Arithmetic, Rational Expressions, Substituting Expressions, Methods and Functions
2540 @c node-name, next, previous, up
2541 @section Polynomial arithmetic
2543 @subsection Expanding and collecting
2544 @cindex @code{expand()}
2545 @cindex @code{collect()}
2547 A polynomial in one or more variables has many equivalent
2548 representations. Some useful ones serve a specific purpose. Consider
2549 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2550 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2551 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
2552 representations are the recursive ones where one collects for exponents
2553 in one of the three variable. Since the factors are themselves
2554 polynomials in the remaining two variables the procedure can be
2555 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
2556 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
2559 To bring an expression into expanded form, its method
2565 may be called. In our example above, this corresponds to @math{4*x*y +
2566 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
2567 GiNaC is not easily guessable you should be prepared to see different
2568 orderings of terms in such sums!
2570 Another useful representation of multivariate polynomials is as a
2571 univariate polynomial in one of the variables with the coefficients
2572 being polynomials in the remaining variables. The method
2573 @code{collect()} accomplishes this task:
2576 ex ex::collect(const ex & s, bool distributed = false);
2579 The first argument to @code{collect()} can also be a list of objects in which
2580 case the result is either a recursively collected polynomial, or a polynomial
2581 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
2582 by the @code{distributed} flag.
2584 Note that the original polynomial needs to be in expanded form in order
2585 for @code{collect()} to be able to find the coefficients properly.
2587 @subsection Degree and coefficients
2588 @cindex @code{degree()}
2589 @cindex @code{ldegree()}
2590 @cindex @code{coeff()}
2592 The degree and low degree of a polynomial can be obtained using the two
2596 int ex::degree(const ex & s);
2597 int ex::ldegree(const ex & s);
2600 which also work reliably on non-expanded input polynomials (they even work
2601 on rational functions, returning the asymptotic degree). To extract
2602 a coefficient with a certain power from an expanded polynomial you use
2605 ex ex::coeff(const ex & s, int n);
2608 You can also obtain the leading and trailing coefficients with the methods
2611 ex ex::lcoeff(const ex & s);
2612 ex ex::tcoeff(const ex & s);
2615 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
2618 An application is illustrated in the next example, where a multivariate
2619 polynomial is analyzed:
2622 #include <ginac/ginac.h>
2623 using namespace std;
2624 using namespace GiNaC;
2628 symbol x("x"), y("y");
2629 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
2630 - pow(x+y,2) + 2*pow(y+2,2) - 8;
2631 ex Poly = PolyInp.expand();
2633 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
2634 cout << "The x^" << i << "-coefficient is "
2635 << Poly.coeff(x,i) << endl;
2637 cout << "As polynomial in y: "
2638 << Poly.collect(y) << endl;
2642 When run, it returns an output in the following fashion:
2645 The x^0-coefficient is y^2+11*y
2646 The x^1-coefficient is 5*y^2-2*y
2647 The x^2-coefficient is -1
2648 The x^3-coefficient is 4*y
2649 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
2652 As always, the exact output may vary between different versions of GiNaC
2653 or even from run to run since the internal canonical ordering is not
2654 within the user's sphere of influence.
2656 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
2657 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
2658 with non-polynomial expressions as they not only work with symbols but with
2659 constants, functions and indexed objects as well:
2663 symbol a("a"), b("b"), c("c");
2664 idx i(symbol("i"), 3);
2666 ex e = pow(sin(x) - cos(x), 4);
2667 cout << e.degree(cos(x)) << endl;
2669 cout << e.expand().coeff(sin(x), 3) << endl;
2672 e = indexed(a+b, i) * indexed(b+c, i);
2673 e = e.expand(expand_options::expand_indexed);
2674 cout << e.collect(indexed(b, i)) << endl;
2675 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
2680 @subsection Polynomial division
2681 @cindex polynomial division
2684 @cindex pseudo-remainder
2685 @cindex @code{quo()}
2686 @cindex @code{rem()}
2687 @cindex @code{prem()}
2688 @cindex @code{divide()}
2693 ex quo(const ex & a, const ex & b, const symbol & x);
2694 ex rem(const ex & a, const ex & b, const symbol & x);
2697 compute the quotient and remainder of univariate polynomials in the variable
2698 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
2700 The additional function
2703 ex prem(const ex & a, const ex & b, const symbol & x);
2706 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
2707 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
2709 Exact division of multivariate polynomials is performed by the function
2712 bool divide(const ex & a, const ex & b, ex & q);
2715 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
2716 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
2717 in which case the value of @code{q} is undefined.
2720 @subsection Unit, content and primitive part
2721 @cindex @code{unit()}
2722 @cindex @code{content()}
2723 @cindex @code{primpart()}
2728 ex ex::unit(const symbol & x);
2729 ex ex::content(const symbol & x);
2730 ex ex::primpart(const symbol & x);
2733 return the unit part, content part, and primitive polynomial of a multivariate
2734 polynomial with respect to the variable @samp{x} (the unit part being the sign
2735 of the leading coefficient, the content part being the GCD of the coefficients,
2736 and the primitive polynomial being the input polynomial divided by the unit and
2737 content parts). The product of unit, content, and primitive part is the
2738 original polynomial.
2741 @subsection GCD and LCM
2744 @cindex @code{gcd()}
2745 @cindex @code{lcm()}
2747 The functions for polynomial greatest common divisor and least common
2748 multiple have the synopsis
2751 ex gcd(const ex & a, const ex & b);
2752 ex lcm(const ex & a, const ex & b);
2755 The functions @code{gcd()} and @code{lcm()} accept two expressions
2756 @code{a} and @code{b} as arguments and return a new expression, their
2757 greatest common divisor or least common multiple, respectively. If the
2758 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
2759 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
2762 #include <ginac/ginac.h>
2763 using namespace GiNaC;
2767 symbol x("x"), y("y"), z("z");
2768 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
2769 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
2771 ex P_gcd = gcd(P_a, P_b);
2773 ex P_lcm = lcm(P_a, P_b);
2774 // 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
2779 @subsection Square-free decomposition
2780 @cindex square-free decomposition
2781 @cindex factorization
2782 @cindex @code{sqrfree()}
2784 GiNaC still lacks proper factorization support. Some form of
2785 factorization is, however, easily implemented by noting that factors
2786 appearing in a polynomial with power two or more also appear in the
2787 derivative and hence can easily be found by computing the GCD of the
2788 original polynomial and its derivatives. Any system has an interface
2789 for this so called square-free factorization. So we provide one, too:
2791 ex sqrfree(const ex & a, const lst & l = lst());
2793 Here is an example that by the way illustrates how the result may depend
2794 on the order of differentiation:
2797 symbol x("x"), y("y");
2798 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
2800 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
2801 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
2803 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
2804 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
2806 cout << sqrfree(BiVarPol) << endl;
2807 // -> depending on luck, any of the above
2812 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
2813 @c node-name, next, previous, up
2814 @section Rational expressions
2816 @subsection The @code{normal} method
2817 @cindex @code{normal()}
2818 @cindex simplification
2819 @cindex temporary replacement
2821 Some basic form of simplification of expressions is called for frequently.
2822 GiNaC provides the method @code{.normal()}, which converts a rational function
2823 into an equivalent rational function of the form @samp{numerator/denominator}
2824 where numerator and denominator are coprime. If the input expression is already
2825 a fraction, it just finds the GCD of numerator and denominator and cancels it,
2826 otherwise it performs fraction addition and multiplication.
2828 @code{.normal()} can also be used on expressions which are not rational functions
2829 as it will replace all non-rational objects (like functions or non-integer
2830 powers) by temporary symbols to bring the expression to the domain of rational
2831 functions before performing the normalization, and re-substituting these
2832 symbols afterwards. This algorithm is also available as a separate method
2833 @code{.to_rational()}, described below.
2835 This means that both expressions @code{t1} and @code{t2} are indeed
2836 simplified in this little program:
2839 #include <ginac/ginac.h>
2840 using namespace GiNaC;
2845 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
2846 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
2847 std::cout << "t1 is " << t1.normal() << std::endl;
2848 std::cout << "t2 is " << t2.normal() << std::endl;
2852 Of course this works for multivariate polynomials too, so the ratio of
2853 the sample-polynomials from the section about GCD and LCM above would be
2854 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
2857 @subsection Numerator and denominator
2860 @cindex @code{numer()}
2861 @cindex @code{denom()}
2863 The numerator and denominator of an expression can be obtained with
2870 These functions will first normalize the expression as described above and
2871 then return the numerator or denominator, respectively.
2874 @subsection Converting to a rational expression
2875 @cindex @code{to_rational()}
2877 Some of the methods described so far only work on polynomials or rational
2878 functions. GiNaC provides a way to extend the domain of these functions to
2879 general expressions by using the temporary replacement algorithm described
2880 above. You do this by calling
2883 ex ex::to_rational(lst &l);
2886 on the expression to be converted. The supplied @code{lst} will be filled
2887 with the generated temporary symbols and their replacement expressions in
2888 a format that can be used directly for the @code{subs()} method. It can also
2889 already contain a list of replacements from an earlier application of
2890 @code{.to_rational()}, so it's possible to use it on multiple expressions
2891 and get consistent results.
2898 ex a = pow(sin(x), 2) - pow(cos(x), 2);
2899 ex b = sin(x) + cos(x);
2902 divide(a.to_rational(l), b.to_rational(l), q);
2903 cout << q.subs(l) << endl;
2907 will print @samp{sin(x)-cos(x)}.
2910 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
2911 @c node-name, next, previous, up
2912 @section Symbolic differentiation
2913 @cindex differentiation
2914 @cindex @code{diff()}
2916 @cindex product rule
2918 GiNaC's objects know how to differentiate themselves. Thus, a
2919 polynomial (class @code{add}) knows that its derivative is the sum of
2920 the derivatives of all the monomials:
2923 #include <ginac/ginac.h>
2924 using namespace GiNaC;
2928 symbol x("x"), y("y"), z("z");
2929 ex P = pow(x, 5) + pow(x, 2) + y;
2931 cout << P.diff(x,2) << endl; // 20*x^3 + 2
2932 cout << P.diff(y) << endl; // 1
2933 cout << P.diff(z) << endl; // 0
2937 If a second integer parameter @var{n} is given, the @code{diff} method
2938 returns the @var{n}th derivative.
2940 If @emph{every} object and every function is told what its derivative
2941 is, all derivatives of composed objects can be calculated using the
2942 chain rule and the product rule. Consider, for instance the expression
2943 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
2944 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
2945 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
2946 out that the composition is the generating function for Euler Numbers,
2947 i.e. the so called @var{n}th Euler number is the coefficient of
2948 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
2949 identity to code a function that generates Euler numbers in just three
2952 @cindex Euler numbers
2954 #include <ginac/ginac.h>
2955 using namespace GiNaC;
2957 ex EulerNumber(unsigned n)
2960 const ex generator = pow(cosh(x),-1);
2961 return generator.diff(x,n).subs(x==0);
2966 for (unsigned i=0; i<11; i+=2)
2967 std::cout << EulerNumber(i) << std::endl;
2972 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
2973 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
2974 @code{i} by two since all odd Euler numbers vanish anyways.
2977 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
2978 @c node-name, next, previous, up
2979 @section Series expansion
2980 @cindex @code{series()}
2981 @cindex Taylor expansion
2982 @cindex Laurent expansion
2983 @cindex @code{pseries} (class)
2985 Expressions know how to expand themselves as a Taylor series or (more
2986 generally) a Laurent series. As in most conventional Computer Algebra
2987 Systems, no distinction is made between those two. There is a class of
2988 its own for storing such series (@code{class pseries}) and a built-in
2989 function (called @code{Order}) for storing the order term of the series.
2990 As a consequence, if you want to work with series, i.e. multiply two
2991 series, you need to call the method @code{ex::series} again to convert
2992 it to a series object with the usual structure (expansion plus order
2993 term). A sample application from special relativity could read:
2996 #include <ginac/ginac.h>
2997 using namespace std;
2998 using namespace GiNaC;
3002 symbol v("v"), c("c");
3004 ex gamma = 1/sqrt(1 - pow(v/c,2));
3005 ex mass_nonrel = gamma.series(v==0, 10);
3007 cout << "the relativistic mass increase with v is " << endl
3008 << mass_nonrel << endl;
3010 cout << "the inverse square of this series is " << endl
3011 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3015 Only calling the series method makes the last output simplify to
3016 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3017 series raised to the power @math{-2}.
3019 @cindex M@'echain's formula
3020 As another instructive application, let us calculate the numerical
3021 value of Archimedes' constant
3025 (for which there already exists the built-in constant @code{Pi})
3026 using M@'echain's amazing formula
3028 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3031 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3033 We may expand the arcus tangent around @code{0} and insert the fractions
3034 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3035 carries an order term with it and the question arises what the system is
3036 supposed to do when the fractions are plugged into that order term. The
3037 solution is to use the function @code{series_to_poly()} to simply strip
3041 #include <ginac/ginac.h>
3042 using namespace GiNaC;
3044 ex mechain_pi(int degr)
3047 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3048 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3049 -4*pi_expansion.subs(x==numeric(1,239));
3055 using std::cout; // just for fun, another way of...
3056 using std::endl; // ...dealing with this namespace std.
3058 for (int i=2; i<12; i+=2) @{
3059 pi_frac = mechain_pi(i);
3060 cout << i << ":\t" << pi_frac << endl
3061 << "\t" << pi_frac.evalf() << endl;
3067 Note how we just called @code{.series(x,degr)} instead of
3068 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3069 method @code{series()}: if the first argument is a symbol the expression
3070 is expanded in that symbol around point @code{0}. When you run this
3071 program, it will type out:
3075 3.1832635983263598326
3076 4: 5359397032/1706489875
3077 3.1405970293260603143
3078 6: 38279241713339684/12184551018734375
3079 3.141621029325034425
3080 8: 76528487109180192540976/24359780855939418203125
3081 3.141591772182177295
3082 10: 327853873402258685803048818236/104359128170408663038552734375
3083 3.1415926824043995174
3087 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
3088 @c node-name, next, previous, up
3089 @section Predefined mathematical functions
3091 GiNaC contains the following predefined mathematical functions:
3094 @multitable @columnfractions .30 .70
3095 @item @strong{Name} @tab @strong{Function}
3098 @item @code{csgn(x)}
3100 @item @code{sqrt(x)}
3101 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3108 @item @code{asin(x)}
3110 @item @code{acos(x)}
3112 @item @code{atan(x)}
3113 @tab inverse tangent
3114 @item @code{atan2(y, x)}
3115 @tab inverse tangent with two arguments
3116 @item @code{sinh(x)}
3117 @tab hyperbolic sine
3118 @item @code{cosh(x)}
3119 @tab hyperbolic cosine
3120 @item @code{tanh(x)}
3121 @tab hyperbolic tangent
3122 @item @code{asinh(x)}
3123 @tab inverse hyperbolic sine
3124 @item @code{acosh(x)}
3125 @tab inverse hyperbolic cosine
3126 @item @code{atanh(x)}
3127 @tab inverse hyperbolic tangent
3129 @tab exponential function
3131 @tab natural logarithm
3134 @item @code{zeta(x)}
3135 @tab Riemann's zeta function
3136 @item @code{zeta(n, x)}
3137 @tab derivatives of Riemann's zeta function
3138 @item @code{tgamma(x)}
3140 @item @code{lgamma(x)}
3141 @tab logarithm of Gamma function
3142 @item @code{beta(x, y)}
3143 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3145 @tab psi (digamma) function
3146 @item @code{psi(n, x)}
3147 @tab derivatives of psi function (polygamma functions)
3148 @item @code{factorial(n)}
3149 @tab factorial function
3150 @item @code{binomial(n, m)}
3151 @tab binomial coefficients
3152 @item @code{Order(x)}
3153 @tab order term function in truncated power series
3154 @item @code{Derivative(x, l)}
3155 @tab inert partial differentiation operator (used internally)
3160 For functions that have a branch cut in the complex plane GiNaC follows
3161 the conventions for C++ as defined in the ANSI standard as far as
3162 possible. In particular: the natural logarithm (@code{log}) and the
3163 square root (@code{sqrt}) both have their branch cuts running along the
3164 negative real axis where the points on the axis itself belong to the
3165 upper part (i.e. continuous with quadrant II). The inverse
3166 trigonometric and hyperbolic functions are not defined for complex
3167 arguments by the C++ standard, however. In GiNaC we follow the
3168 conventions used by CLN, which in turn follow the carefully designed
3169 definitions in the Common Lisp standard. It should be noted that this
3170 convention is identical to the one used by the C99 standard and by most
3171 serious CAS. It is to be expected that future revisions of the C++
3172 standard incorporate these functions in the complex domain in a manner
3173 compatible with C99.
3176 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3177 @c node-name, next, previous, up
3178 @section Input and output of expressions
3181 @subsection Expression output
3183 @cindex output of expressions
3185 The easiest way to print an expression is to write it to a stream:
3190 ex e = 4.5+pow(x,2)*3/2;
3191 cout << e << endl; // prints '(4.5)+3/2*x^2'
3195 The output format is identical to the @command{ginsh} input syntax and
3196 to that used by most computer algebra systems, but not directly pastable
3197 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3198 is printed as @samp{x^2}).
3200 It is possible to print expressions in a number of different formats with
3204 void ex::print(const print_context & c, unsigned level = 0);
3207 @cindex @code{print_context} (class)
3208 The type of @code{print_context} object passed in determines the format
3209 of the output. The possible types are defined in @file{ginac/print.h}.
3210 All constructors of @code{print_context} and derived classes take an
3211 @code{ostream &} as their first argument.
3213 To print an expression in a way that can be directly used in a C or C++
3214 program, you pass a @code{print_csrc} object like this:
3218 cout << "float f = ";
3219 e.print(print_csrc_float(cout));
3222 cout << "double d = ";
3223 e.print(print_csrc_double(cout));
3226 cout << "cl_N n = ";
3227 e.print(print_csrc_cl_N(cout));
3232 The three possible types mostly affect the way in which floating point
3233 numbers are written.
3235 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3238 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3239 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3240 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3243 The @code{print_context} type @code{print_tree} provides a dump of the
3244 internal structure of an expression for debugging purposes:
3248 e.print(print_tree(cout));
3255 add, hash=0x0, flags=0x3, nops=2
3256 power, hash=0x9, flags=0x3, nops=2
3257 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
3258 2 (numeric), hash=0x80000042, flags=0xf
3259 3/2 (numeric), hash=0x80000061, flags=0xf
3262 4.5L0 (numeric), hash=0x8000004b, flags=0xf
3266 This kind of output is also available in @command{ginsh} as the @code{print()}
3269 Another useful output format is for LaTeX parsing in mathematical mode.
3270 It is rather similar to the default @code{print_context} but provides
3271 some braces needed by LaTeX for delimiting boxes and also converts some
3272 common objects to conventional LaTeX names. It is possible to give symbols
3273 a special name for LaTeX output by supplying it as a second argument to
3274 the @code{symbol} constructor.
3276 For example, the code snippet
3281 ex foo = lgamma(x).series(x==0,3);
3282 foo.print(print_latex(std::cout));
3288 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
3291 If you need any fancy special output format, e.g. for interfacing GiNaC
3292 with other algebra systems or for producing code for different
3293 programming languages, you can always traverse the expression tree yourself:
3296 static void my_print(const ex & e)
3298 if (is_ex_of_type(e, function))
3299 cout << ex_to_function(e).get_name();
3301 cout << e.bp->class_name();
3303 unsigned n = e.nops();
3305 for (unsigned i=0; i<n; i++) @{
3317 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
3325 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
3326 symbol(y))),numeric(-2)))
3329 If you need an output format that makes it possible to accurately
3330 reconstruct an expression by feeding the output to a suitable parser or
3331 object factory, you should consider storing the expression in an
3332 @code{archive} object and reading the object properties from there.
3333 See the section on archiving for more information.
3336 @subsection Expression input
3337 @cindex input of expressions
3339 GiNaC provides no way to directly read an expression from a stream because
3340 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
3341 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
3342 @code{y} you defined in your program and there is no way to specify the
3343 desired symbols to the @code{>>} stream input operator.
3345 Instead, GiNaC lets you construct an expression from a string, specifying the
3346 list of symbols to be used:
3350 symbol x("x"), y("y");
3351 ex e("2*x+sin(y)", lst(x, y));
3355 The input syntax is the same as that used by @command{ginsh} and the stream
3356 output operator @code{<<}. The symbols in the string are matched by name to
3357 the symbols in the list and if GiNaC encounters a symbol not specified in
3358 the list it will throw an exception.
3360 With this constructor, it's also easy to implement interactive GiNaC programs:
3365 #include <stdexcept>
3366 #include <ginac/ginac.h>
3367 using namespace std;
3368 using namespace GiNaC;
3375 cout << "Enter an expression containing 'x': ";
3380 cout << "The derivative of " << e << " with respect to x is ";
3381 cout << e.diff(x) << ".\n";
3382 @} catch (exception &p) @{
3383 cerr << p.what() << endl;
3389 @subsection Archiving
3390 @cindex @code{archive} (class)
3393 GiNaC allows creating @dfn{archives} of expressions which can be stored
3394 to or retrieved from files. To create an archive, you declare an object
3395 of class @code{archive} and archive expressions in it, giving each
3396 expression a unique name:
3400 using namespace std;
3401 #include <ginac/ginac.h>
3402 using namespace GiNaC;
3406 symbol x("x"), y("y"), z("z");
3408 ex foo = sin(x + 2*y) + 3*z + 41;
3412 a.archive_ex(foo, "foo");
3413 a.archive_ex(bar, "the second one");
3417 The archive can then be written to a file:
3421 ofstream out("foobar.gar");
3427 The file @file{foobar.gar} contains all information that is needed to
3428 reconstruct the expressions @code{foo} and @code{bar}.
3430 @cindex @command{viewgar}
3431 The tool @command{viewgar} that comes with GiNaC can be used to view
3432 the contents of GiNaC archive files:
3435 $ viewgar foobar.gar
3436 foo = 41+sin(x+2*y)+3*z
3437 the second one = 42+sin(x+2*y)+3*z
3440 The point of writing archive files is of course that they can later be
3446 ifstream in("foobar.gar");
3451 And the stored expressions can be retrieved by their name:
3457 ex ex1 = a2.unarchive_ex(syms, "foo");
3458 ex ex2 = a2.unarchive_ex(syms, "the second one");
3460 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
3461 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
3462 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
3466 Note that you have to supply a list of the symbols which are to be inserted
3467 in the expressions. Symbols in archives are stored by their name only and
3468 if you don't specify which symbols you have, unarchiving the expression will
3469 create new symbols with that name. E.g. if you hadn't included @code{x} in
3470 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
3471 have had no effect because the @code{x} in @code{ex1} would have been a
3472 different symbol than the @code{x} which was defined at the beginning of
3473 the program, altough both would appear as @samp{x} when printed.
3475 You can also use the information stored in an @code{archive} object to
3476 output expressions in a format suitable for exact reconstruction. The
3477 @code{archive} and @code{archive_node} classes have a couple of member
3478 functions that let you access the stored properties:
3481 static void my_print2(const archive_node & n)
3484 n.find_string("class", class_name);
3485 cout << class_name << "(";
3487 archive_node::propinfovector p;
3488 n.get_properties(p);
3490 unsigned num = p.size();
3491 for (unsigned i=0; i<num; i++) @{
3492 const string &name = p[i].name;
3493 if (name == "class")
3495 cout << name << "=";
3497 unsigned count = p[i].count;
3501 for (unsigned j=0; j<count; j++) @{
3502 switch (p[i].type) @{
3503 case archive_node::PTYPE_BOOL: @{
3505 n.find_bool(name, x);
3506 cout << (x ? "true" : "false");
3509 case archive_node::PTYPE_UNSIGNED: @{
3511 n.find_unsigned(name, x);
3515 case archive_node::PTYPE_STRING: @{
3517 n.find_string(name, x);
3518 cout << '\"' << x << '\"';
3521 case archive_node::PTYPE_NODE: @{
3522 const archive_node &x = n.find_ex_node(name, j);
3544 ex e = pow(2, x) - y;
3546 my_print2(ar.get_top_node(0)); cout << endl;
3554 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
3555 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
3556 overall_coeff=numeric(number="0"))
3559 Be warned, however, that the set of properties and their meaning for each
3560 class may change between GiNaC versions.
3563 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
3564 @c node-name, next, previous, up
3565 @chapter Extending GiNaC
3567 By reading so far you should have gotten a fairly good understanding of
3568 GiNaC's design-patterns. From here on you should start reading the
3569 sources. All we can do now is issue some recommendations how to tackle
3570 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
3571 develop some useful extension please don't hesitate to contact the GiNaC
3572 authors---they will happily incorporate them into future versions.
3575 * What does not belong into GiNaC:: What to avoid.
3576 * Symbolic functions:: Implementing symbolic functions.
3577 * Adding classes:: Defining new algebraic classes.
3581 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
3582 @c node-name, next, previous, up
3583 @section What doesn't belong into GiNaC
3585 @cindex @command{ginsh}
3586 First of all, GiNaC's name must be read literally. It is designed to be
3587 a library for use within C++. The tiny @command{ginsh} accompanying
3588 GiNaC makes this even more clear: it doesn't even attempt to provide a
3589 language. There are no loops or conditional expressions in
3590 @command{ginsh}, it is merely a window into the library for the
3591 programmer to test stuff (or to show off). Still, the design of a
3592 complete CAS with a language of its own, graphical capabilites and all
3593 this on top of GiNaC is possible and is without doubt a nice project for
3596 There are many built-in functions in GiNaC that do not know how to
3597 evaluate themselves numerically to a precision declared at runtime
3598 (using @code{Digits}). Some may be evaluated at certain points, but not
3599 generally. This ought to be fixed. However, doing numerical
3600 computations with GiNaC's quite abstract classes is doomed to be
3601 inefficient. For this purpose, the underlying foundation classes
3602 provided by @acronym{CLN} are much better suited.
3605 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
3606 @c node-name, next, previous, up
3607 @section Symbolic functions
3609 The easiest and most instructive way to start with is probably to
3610 implement your own function. GiNaC's functions are objects of class
3611 @code{function}. The preprocessor is then used to convert the function
3612 names to objects with a corresponding serial number that is used
3613 internally to identify them. You usually need not worry about this
3614 number. New functions may be inserted into the system via a kind of
3615 `registry'. It is your responsibility to care for some functions that
3616 are called when the user invokes certain methods. These are usual
3617 C++-functions accepting a number of @code{ex} as arguments and returning
3618 one @code{ex}. As an example, if we have a look at a simplified
3619 implementation of the cosine trigonometric function, we first need a
3620 function that is called when one wishes to @code{eval} it. It could
3621 look something like this:
3624 static ex cos_eval_method(const ex & x)
3626 // if (!x%(2*Pi)) return 1
3627 // if (!x%Pi) return -1
3628 // if (!x%Pi/2) return 0
3629 // care for other cases...
3630 return cos(x).hold();
3634 @cindex @code{hold()}
3636 The last line returns @code{cos(x)} if we don't know what else to do and
3637 stops a potential recursive evaluation by saying @code{.hold()}, which
3638 sets a flag to the expression signaling that it has been evaluated. We
3639 should also implement a method for numerical evaluation and since we are
3640 lazy we sweep the problem under the rug by calling someone else's
3641 function that does so, in this case the one in class @code{numeric}:
3644 static ex cos_evalf(const ex & x)
3646 return cos(ex_to_numeric(x));
3650 Differentiation will surely turn up and so we need to tell @code{cos}
3651 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
3652 instance are then handled automatically by @code{basic::diff} and
3656 static ex cos_deriv(const ex & x, unsigned diff_param)
3662 @cindex product rule
3663 The second parameter is obligatory but uninteresting at this point. It
3664 specifies which parameter to differentiate in a partial derivative in
3665 case the function has more than one parameter and its main application
3666 is for correct handling of the chain rule. For Taylor expansion, it is
3667 enough to know how to differentiate. But if the function you want to
3668 implement does have a pole somewhere in the complex plane, you need to
3669 write another method for Laurent expansion around that point.
3671 Now that all the ingredients for @code{cos} have been set up, we need
3672 to tell the system about it. This is done by a macro and we are not
3673 going to descibe how it expands, please consult your preprocessor if you
3677 REGISTER_FUNCTION(cos, eval_func(cos_eval).
3678 evalf_func(cos_evalf).
3679 derivative_func(cos_deriv));
3682 The first argument is the function's name used for calling it and for
3683 output. The second binds the corresponding methods as options to this
3684 object. Options are separated by a dot and can be given in an arbitrary
3685 order. GiNaC functions understand several more options which are always
3686 specified as @code{.option(params)}, for example a method for series
3687 expansion @code{.series_func(cos_series)}. Again, if no series
3688 expansion method is given, GiNaC defaults to simple Taylor expansion,
3689 which is correct if there are no poles involved as is the case for the
3690 @code{cos} function. The way GiNaC handles poles in case there are any
3691 is best understood by studying one of the examples, like the Gamma
3692 (@code{tgamma}) function for instance. (In essence the function first
3693 checks if there is a pole at the evaluation point and falls back to
3694 Taylor expansion if there isn't. Then, the pole is regularized by some
3695 suitable transformation.) Also, the new function needs to be declared
3696 somewhere. This may also be done by a convenient preprocessor macro:
3699 DECLARE_FUNCTION_1P(cos)
3702 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
3703 implementation of @code{cos} is very incomplete and lacks several safety
3704 mechanisms. Please, have a look at the real implementation in GiNaC.
3705 (By the way: in case you are worrying about all the macros above we can
3706 assure you that functions are GiNaC's most macro-intense classes. We
3707 have done our best to avoid macros where we can.)
3710 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
3711 @c node-name, next, previous, up
3712 @section Adding classes
3714 If you are doing some very specialized things with GiNaC you may find that
3715 you have to implement your own algebraic classes to fit your needs. This
3716 section will explain how to do this by giving the example of a simple
3717 'string' class. After reading this section you will know how to properly
3718 declare a GiNaC class and what the minimum required member functions are
3719 that you have to implement. We only cover the implementation of a 'leaf'
3720 class here (i.e. one that doesn't contain subexpressions). Creating a
3721 container class like, for example, a class representing tensor products is
3722 more involved but this section should give you enough information so you can
3723 consult the source to GiNaC's predefined classes if you want to implement
3724 something more complicated.
3726 @subsection GiNaC's run-time type information system
3728 @cindex hierarchy of classes
3730 All algebraic classes (that is, all classes that can appear in expressions)
3731 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
3732 @code{basic *} (which is essentially what an @code{ex} is) represents a
3733 generic pointer to an algebraic class. Occasionally it is necessary to find
3734 out what the class of an object pointed to by a @code{basic *} really is.
3735 Also, for the unarchiving of expressions it must be possible to find the
3736 @code{unarchive()} function of a class given the class name (as a string). A
3737 system that provides this kind of information is called a run-time type
3738 information (RTTI) system. The C++ language provides such a thing (see the
3739 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
3740 implements its own, simpler RTTI.
3742 The RTTI in GiNaC is based on two mechanisms:
3747 The @code{basic} class declares a member variable @code{tinfo_key} which
3748 holds an unsigned integer that identifies the object's class. These numbers
3749 are defined in the @file{tinfos.h} header file for the built-in GiNaC
3750 classes. They all start with @code{TINFO_}.
3753 By means of some clever tricks with static members, GiNaC maintains a list
3754 of information for all classes derived from @code{basic}. The information
3755 available includes the class names, the @code{tinfo_key}s, and pointers
3756 to the unarchiving functions. This class registry is defined in the
3757 @file{registrar.h} header file.
3761 The disadvantage of this proprietary RTTI implementation is that there's
3762 a little more to do when implementing new classes (C++'s RTTI works more
3763 or less automatic) but don't worry, most of the work is simplified by
3766 @subsection A minimalistic example
3768 Now we will start implementing a new class @code{mystring} that allows
3769 placing character strings in algebraic expressions (this is not very useful,
3770 but it's just an example). This class will be a direct subclass of
3771 @code{basic}. You can use this sample implementation as a starting point
3772 for your own classes.
3774 The code snippets given here assume that you have included some header files
3780 #include <stdexcept>
3781 using namespace std;
3783 #include <ginac/ginac.h>
3784 using namespace GiNaC;
3787 The first thing we have to do is to define a @code{tinfo_key} for our new
3788 class. This can be any arbitrary unsigned number that is not already taken
3789 by one of the existing classes but it's better to come up with something
3790 that is unlikely to clash with keys that might be added in the future. The
3791 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
3792 which is not a requirement but we are going to stick with this scheme:
3795 const unsigned TINFO_mystring = 0x42420001U;
3798 Now we can write down the class declaration. The class stores a C++
3799 @code{string} and the user shall be able to construct a @code{mystring}
3800 object from a C or C++ string:
3803 class mystring : public basic
3805 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
3808 mystring(const string &s);
3809 mystring(const char *s);
3815 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
3818 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
3819 macros are defined in @file{registrar.h}. They take the name of the class
3820 and its direct superclass as arguments and insert all required declarations
3821 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
3822 the first line after the opening brace of the class definition. The
3823 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
3824 source (at global scope, of course, not inside a function).
3826 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
3827 declarations of the default and copy constructor, the destructor, the
3828 assignment operator and a couple of other functions that are required. It
3829 also defines a type @code{inherited} which refers to the superclass so you
3830 don't have to modify your code every time you shuffle around the class
3831 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
3832 constructor, the destructor and the assignment operator.
3834 Now there are nine member functions we have to implement to get a working
3840 @code{mystring()}, the default constructor.
3843 @code{void destroy(bool call_parent)}, which is used in the destructor and the
3844 assignment operator to free dynamically allocated members. The @code{call_parent}
3845 specifies whether the @code{destroy()} function of the superclass is to be
3849 @code{void copy(const mystring &other)}, which is used in the copy constructor
3850 and assignment operator to copy the member variables over from another
3851 object of the same class.
3854 @code{void archive(archive_node &n)}, the archiving function. This stores all
3855 information needed to reconstruct an object of this class inside an
3856 @code{archive_node}.
3859 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
3860 constructor. This constructs an instance of the class from the information
3861 found in an @code{archive_node}.
3864 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
3865 unarchiving function. It constructs a new instance by calling the unarchiving
3869 @code{int compare_same_type(const basic &other)}, which is used internally
3870 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
3871 -1, depending on the relative order of this object and the @code{other}
3872 object. If it returns 0, the objects are considered equal.
3873 @strong{Note:} This has nothing to do with the (numeric) ordering
3874 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
3875 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
3876 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
3877 must provide a @code{compare_same_type()} function, even those representing
3878 objects for which no reasonable algebraic ordering relationship can be
3882 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
3883 which are the two constructors we declared.
3887 Let's proceed step-by-step. The default constructor looks like this:
3890 mystring::mystring() : inherited(TINFO_mystring)
3892 // dynamically allocate resources here if required
3896 The golden rule is that in all constructors you have to set the
3897 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
3898 it will be set by the constructor of the superclass and all hell will break
3899 loose in the RTTI. For your convenience, the @code{basic} class provides
3900 a constructor that takes a @code{tinfo_key} value, which we are using here
3901 (remember that in our case @code{inherited = basic}). If the superclass
3902 didn't have such a constructor, we would have to set the @code{tinfo_key}
3903 to the right value manually.
3905 In the default constructor you should set all other member variables to
3906 reasonable default values (we don't need that here since our @code{str}
3907 member gets set to an empty string automatically). The constructor(s) are of
3908 course also the right place to allocate any dynamic resources you require.
3910 Next, the @code{destroy()} function:
3913 void mystring::destroy(bool call_parent)
3915 // free dynamically allocated resources here if required
3917 inherited::destroy(call_parent);
3921 This function is where we free all dynamically allocated resources. We don't
3922 have any so we're not doing anything here, but if we had, for example, used
3923 a C-style @code{char *} to store our string, this would be the place to
3924 @code{delete[]} the string storage. If @code{call_parent} is true, we have
3925 to call the @code{destroy()} function of the superclass after we're done
3926 (to mimic C++'s automatic invocation of superclass destructors where
3927 @code{destroy()} is called from outside a destructor).
3929 The @code{copy()} function just copies over the member variables from
3933 void mystring::copy(const mystring &other)
3935 inherited::copy(other);
3940 We can simply overwrite the member variables here. There's no need to worry
3941 about dynamically allocated storage. The assignment operator (which is
3942 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
3943 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
3944 explicitly call the @code{copy()} function of the superclass here so
3945 all the member variables will get copied.
3947 Next are the three functions for archiving. You have to implement them even
3948 if you don't plan to use archives, but the minimum required implementation
3949 is really simple. First, the archiving function:
3952 void mystring::archive(archive_node &n) const
3954 inherited::archive(n);
3955 n.add_string("string", str);
3959 The only thing that is really required is calling the @code{archive()}
3960 function of the superclass. Optionally, you can store all information you
3961 deem necessary for representing the object into the passed
3962 @code{archive_node}. We are just storing our string here. For more
3963 information on how the archiving works, consult the @file{archive.h} header
3966 The unarchiving constructor is basically the inverse of the archiving
3970 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
3972 n.find_string("string", str);
3976 If you don't need archiving, just leave this function empty (but you must
3977 invoke the unarchiving constructor of the superclass). Note that we don't
3978 have to set the @code{tinfo_key} here because it is done automatically
3979 by the unarchiving constructor of the @code{basic} class.
3981 Finally, the unarchiving function:
3984 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
3986 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
3990 You don't have to understand how exactly this works. Just copy these four
3991 lines into your code literally (replacing the class name, of course). It
3992 calls the unarchiving constructor of the class and unless you are doing
3993 something very special (like matching @code{archive_node}s to global
3994 objects) you don't need a different implementation. For those who are
3995 interested: setting the @code{dynallocated} flag puts the object under
3996 the control of GiNaC's garbage collection. It will get deleted automatically
3997 once it is no longer referenced.
3999 Our @code{compare_same_type()} function uses a provided function to compare
4003 int mystring::compare_same_type(const basic &other) const
4005 const mystring &o = static_cast<const mystring &>(other);
4006 int cmpval = str.compare(o.str);
4009 else if (cmpval < 0)
4016 Although this function takes a @code{basic &}, it will always be a reference
4017 to an object of exactly the same class (objects of different classes are not
4018 comparable), so the cast is safe. If this function returns 0, the two objects
4019 are considered equal (in the sense that @math{A-B=0}), so you should compare
4020 all relevant member variables.
4022 Now the only thing missing is our two new constructors:
4025 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4027 // dynamically allocate resources here if required
4030 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4032 // dynamically allocate resources here if required
4036 No surprises here. We set the @code{str} member from the argument and
4037 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4039 That's it! We now have a minimal working GiNaC class that can store
4040 strings in algebraic expressions. Let's confirm that the RTTI works:
4043 ex e = mystring("Hello, world!");
4044 cout << is_ex_of_type(e, mystring) << endl;
4047 cout << e.bp->class_name() << endl;
4051 Obviously it does. Let's see what the expression @code{e} looks like:
4055 // -> [mystring object]
4058 Hm, not exactly what we expect, but of course the @code{mystring} class
4059 doesn't yet know how to print itself. This is done in the @code{print()}
4060 member function. Let's say that we wanted to print the string surrounded
4064 class mystring : public basic
4068 void print(const print_context &c, unsigned level = 0) const;
4072 void mystring::print(const print_context &c, unsigned level) const
4074 // print_context::s is a reference to an ostream
4075 c.s << '\"' << str << '\"';
4079 The @code{level} argument is only required for container classes to
4080 correctly parenthesize the output. Let's try again to print the expression:
4084 // -> "Hello, world!"
4087 Much better. The @code{mystring} class can be used in arbitrary expressions:
4090 e += mystring("GiNaC rulez");
4092 // -> "GiNaC rulez"+"Hello, world!"
4095 (note that GiNaC's automatic term reordering is in effect here), or even
4098 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4100 // -> "One string"^(2*sin(-"Another string"+Pi))
4103 Whether this makes sense is debatable but remember that this is only an
4104 example. At least it allows you to implement your own symbolic algorithms
4107 Note that GiNaC's algebraic rules remain unchanged:
4110 e = mystring("Wow") * mystring("Wow");
4114 e = pow(mystring("First")-mystring("Second"), 2);
4115 cout << e.expand() << endl;
4116 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4119 There's no way to, for example, make GiNaC's @code{add} class perform string
4120 concatenation. You would have to implement this yourself.
4122 @subsection Automatic evaluation
4124 @cindex @code{hold()}
4126 When dealing with objects that are just a little more complicated than the
4127 simple string objects we have implemented, chances are that you will want to
4128 have some automatic simplifications or canonicalizations performed on them.
4129 This is done in the evaluation member function @code{eval()}. Let's say that
4130 we wanted all strings automatically converted to lowercase with
4131 non-alphabetic characters stripped, and empty strings removed:
4134 class mystring : public basic
4138 ex eval(int level = 0) const;
4142 ex mystring::eval(int level) const
4145 for (int i=0; i<str.length(); i++) @{
4147 if (c >= 'A' && c <= 'Z')
4148 new_str += tolower(c);
4149 else if (c >= 'a' && c <= 'z')
4153 if (new_str.length() == 0)
4156 return mystring(new_str).hold();
4160 The @code{level} argument is used to limit the recursion depth of the
4161 evaluation. We don't have any subexpressions in the @code{mystring} class
4162 so we are not concerned with this. If we had, we would call the @code{eval()}
4163 functions of the subexpressions with @code{level - 1} as the argument if
4164 @code{level != 1}. The @code{hold()} member function sets a flag in the
4165 object that prevents further evaluation. Otherwise we might end up in an
4166 endless loop. When you want to return the object unmodified, use
4167 @code{return this->hold();}.
4169 Let's confirm that it works:
4172 ex e = mystring("Hello, world!") + mystring("!?#");
4176 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4181 @subsection Other member functions
4183 We have implemented only a small set of member functions to make the class
4184 work in the GiNaC framework. For a real algebraic class, there are probably
4185 some more functions that you will want to re-implement, such as
4186 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4187 or the header file of the class you want to make a subclass of to see
4188 what's there. You can, of course, also add your own new member functions.
4189 In this case you will probably want to define a little helper function like
4192 inline const mystring &ex_to_mystring(const ex &e)
4194 return static_cast<const mystring &>(*e.bp);
4198 that let's you get at the object inside an expression (after you have verified
4199 that the type is correct) so you can call member functions that are specific
4202 That's it. May the source be with you!
4205 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4206 @c node-name, next, previous, up
4207 @chapter A Comparison With Other CAS
4210 This chapter will give you some information on how GiNaC compares to
4211 other, traditional Computer Algebra Systems, like @emph{Maple},
4212 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4213 disadvantages over these systems.
4216 * Advantages:: Stengths of the GiNaC approach.
4217 * Disadvantages:: Weaknesses of the GiNaC approach.
4218 * Why C++?:: Attractiveness of C++.
4221 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4222 @c node-name, next, previous, up
4225 GiNaC has several advantages over traditional Computer
4226 Algebra Systems, like
4231 familiar language: all common CAS implement their own proprietary
4232 grammar which you have to learn first (and maybe learn again when your
4233 vendor decides to `enhance' it). With GiNaC you can write your program
4234 in common C++, which is standardized.
4238 structured data types: you can build up structured data types using
4239 @code{struct}s or @code{class}es together with STL features instead of
4240 using unnamed lists of lists of lists.
4243 strongly typed: in CAS, you usually have only one kind of variables
4244 which can hold contents of an arbitrary type. This 4GL like feature is
4245 nice for novice programmers, but dangerous.
4248 development tools: powerful development tools exist for C++, like fancy
4249 editors (e.g. with automatic indentation and syntax highlighting),
4250 debuggers, visualization tools, documentation generators...
4253 modularization: C++ programs can easily be split into modules by
4254 separating interface and implementation.
4257 price: GiNaC is distributed under the GNU Public License which means
4258 that it is free and available with source code. And there are excellent
4259 C++-compilers for free, too.
4262 extendable: you can add your own classes to GiNaC, thus extending it on
4263 a very low level. Compare this to a traditional CAS that you can
4264 usually only extend on a high level by writing in the language defined
4265 by the parser. In particular, it turns out to be almost impossible to
4266 fix bugs in a traditional system.
4269 multiple interfaces: Though real GiNaC programs have to be written in
4270 some editor, then be compiled, linked and executed, there are more ways
4271 to work with the GiNaC engine. Many people want to play with
4272 expressions interactively, as in traditional CASs. Currently, two such
4273 windows into GiNaC have been implemented and many more are possible: the
4274 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
4275 types to a command line and second, as a more consistent approach, an
4276 interactive interface to the @acronym{Cint} C++ interpreter has been put
4277 together (called @acronym{GiNaC-cint}) that allows an interactive
4278 scripting interface consistent with the C++ language.
4281 seemless integration: it is somewhere between difficult and impossible
4282 to call CAS functions from within a program written in C++ or any other
4283 programming language and vice versa. With GiNaC, your symbolic routines
4284 are part of your program. You can easily call third party libraries,
4285 e.g. for numerical evaluation or graphical interaction. All other
4286 approaches are much more cumbersome: they range from simply ignoring the
4287 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
4288 system (i.e. @emph{Yacas}).
4291 efficiency: often large parts of a program do not need symbolic
4292 calculations at all. Why use large integers for loop variables or
4293 arbitrary precision arithmetics where @code{int} and @code{double} are
4294 sufficient? For pure symbolic applications, GiNaC is comparable in
4295 speed with other CAS.
4300 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
4301 @c node-name, next, previous, up
4302 @section Disadvantages
4304 Of course it also has some disadvantages:
4309 advanced features: GiNaC cannot compete with a program like
4310 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
4311 which grows since 1981 by the work of dozens of programmers, with
4312 respect to mathematical features. Integration, factorization,
4313 non-trivial simplifications, limits etc. are missing in GiNaC (and are
4314 not planned for the near future).
4317 portability: While the GiNaC library itself is designed to avoid any
4318 platform dependent features (it should compile on any ANSI compliant C++
4319 compiler), the currently used version of the CLN library (fast large
4320 integer and arbitrary precision arithmetics) can be compiled only on
4321 systems with a recently new C++ compiler from the GNU Compiler
4322 Collection (@acronym{GCC}).@footnote{This is because CLN uses
4323 PROVIDE/REQUIRE like macros to let the compiler gather all static
4324 initializations, which works for GNU C++ only.} GiNaC uses recent
4325 language features like explicit constructors, mutable members, RTTI,
4326 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
4327 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
4328 ANSI compliant, support all needed features.
4333 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
4334 @c node-name, next, previous, up
4337 Why did we choose to implement GiNaC in C++ instead of Java or any other
4338 language? C++ is not perfect: type checking is not strict (casting is
4339 possible), separation between interface and implementation is not
4340 complete, object oriented design is not enforced. The main reason is
4341 the often scolded feature of operator overloading in C++. While it may
4342 be true that operating on classes with a @code{+} operator is rarely
4343 meaningful, it is perfectly suited for algebraic expressions. Writing
4344 @math{3x+5y} as @code{3*x+5*y} instead of
4345 @code{x.times(3).plus(y.times(5))} looks much more natural.
4346 Furthermore, the main developers are more familiar with C++ than with
4347 any other programming language.
4350 @node Internal Structures, Expressions are reference counted, Why C++? , Top
4351 @c node-name, next, previous, up
4352 @appendix Internal Structures
4355 * Expressions are reference counted::
4356 * Internal representation of products and sums::
4359 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
4360 @c node-name, next, previous, up
4361 @appendixsection Expressions are reference counted
4363 @cindex reference counting
4364 @cindex copy-on-write
4365 @cindex garbage collection
4366 An expression is extremely light-weight since internally it works like a
4367 handle to the actual representation and really holds nothing more than a
4368 pointer to some other object. What this means in practice is that
4369 whenever you create two @code{ex} and set the second equal to the first
4370 no copying process is involved. Instead, the copying takes place as soon
4371 as you try to change the second. Consider the simple sequence of code:
4374 #include <ginac/ginac.h>
4375 using namespace std;
4376 using namespace GiNaC;
4380 symbol x("x"), y("y"), z("z");
4383 e1 = sin(x + 2*y) + 3*z + 41;
4384 e2 = e1; // e2 points to same object as e1
4385 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
4386 e2 += 1; // e2 is copied into a new object
4387 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
4391 The line @code{e2 = e1;} creates a second expression pointing to the
4392 object held already by @code{e1}. The time involved for this operation
4393 is therefore constant, no matter how large @code{e1} was. Actual
4394 copying, however, must take place in the line @code{e2 += 1;} because
4395 @code{e1} and @code{e2} are not handles for the same object any more.
4396 This concept is called @dfn{copy-on-write semantics}. It increases
4397 performance considerably whenever one object occurs multiple times and
4398 represents a simple garbage collection scheme because when an @code{ex}
4399 runs out of scope its destructor checks whether other expressions handle
4400 the object it points to too and deletes the object from memory if that
4401 turns out not to be the case. A slightly less trivial example of
4402 differentiation using the chain-rule should make clear how powerful this
4406 #include <ginac/ginac.h>
4407 using namespace std;
4408 using namespace GiNaC;
4412 symbol x("x"), y("y");
4416 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
4417 cout << e1 << endl // prints x+3*y
4418 << e2 << endl // prints (x+3*y)^3
4419 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
4423 Here, @code{e1} will actually be referenced three times while @code{e2}
4424 will be referenced two times. When the power of an expression is built,
4425 that expression needs not be copied. Likewise, since the derivative of
4426 a power of an expression can be easily expressed in terms of that
4427 expression, no copying of @code{e1} is involved when @code{e3} is
4428 constructed. So, when @code{e3} is constructed it will print as
4429 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
4430 holds a reference to @code{e2} and the factor in front is just
4433 As a user of GiNaC, you cannot see this mechanism of copy-on-write
4434 semantics. When you insert an expression into a second expression, the
4435 result behaves exactly as if the contents of the first expression were
4436 inserted. But it may be useful to remember that this is not what
4437 happens. Knowing this will enable you to write much more efficient
4438 code. If you still have an uncertain feeling with copy-on-write
4439 semantics, we recommend you have a look at the
4440 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
4441 Marshall Cline. Chapter 16 covers this issue and presents an
4442 implementation which is pretty close to the one in GiNaC.
4445 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
4446 @c node-name, next, previous, up
4447 @appendixsection Internal representation of products and sums
4449 @cindex representation
4452 @cindex @code{power}
4453 Although it should be completely transparent for the user of
4454 GiNaC a short discussion of this topic helps to understand the sources
4455 and also explain performance to a large degree. Consider the
4456 unexpanded symbolic expression
4458 $2d^3 \left( 4a + 5b - 3 \right)$
4461 @math{2*d^3*(4*a+5*b-3)}
4463 which could naively be represented by a tree of linear containers for
4464 addition and multiplication, one container for exponentiation with base
4465 and exponent and some atomic leaves of symbols and numbers in this
4470 @cindex pair-wise representation
4471 However, doing so results in a rather deeply nested tree which will
4472 quickly become inefficient to manipulate. We can improve on this by
4473 representing the sum as a sequence of terms, each one being a pair of a
4474 purely numeric multiplicative coefficient and its rest. In the same
4475 spirit we can store the multiplication as a sequence of terms, each
4476 having a numeric exponent and a possibly complicated base, the tree
4477 becomes much more flat:
4481 The number @code{3} above the symbol @code{d} shows that @code{mul}
4482 objects are treated similarly where the coefficients are interpreted as
4483 @emph{exponents} now. Addition of sums of terms or multiplication of
4484 products with numerical exponents can be coded to be very efficient with
4485 such a pair-wise representation. Internally, this handling is performed
4486 by most CAS in this way. It typically speeds up manipulations by an
4487 order of magnitude. The overall multiplicative factor @code{2} and the
4488 additive term @code{-3} look somewhat out of place in this
4489 representation, however, since they are still carrying a trivial
4490 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
4491 this is avoided by adding a field that carries an overall numeric
4492 coefficient. This results in the realistic picture of internal
4495 $2d^3 \left( 4a + 5b - 3 \right)$:
4498 @math{2*d^3*(4*a+5*b-3)}:
4504 This also allows for a better handling of numeric radicals, since
4505 @code{sqrt(2)} can now be carried along calculations. Now it should be
4506 clear, why both classes @code{add} and @code{mul} are derived from the
4507 same abstract class: the data representation is the same, only the
4508 semantics differs. In the class hierarchy, methods for polynomial
4509 expansion and the like are reimplemented for @code{add} and @code{mul},
4510 but the data structure is inherited from @code{expairseq}.
4513 @node Package Tools, ginac-config, Internal representation of products and sums, Top
4514 @c node-name, next, previous, up
4515 @appendix Package Tools
4517 If you are creating a software package that uses the GiNaC library,
4518 setting the correct command line options for the compiler and linker
4519 can be difficult. GiNaC includes two tools to make this process easier.
4522 * ginac-config:: A shell script to detect compiler and linker flags.
4523 * AM_PATH_GINAC:: Macro for GNU automake.
4527 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
4528 @c node-name, next, previous, up
4529 @section @command{ginac-config}
4530 @cindex ginac-config
4532 @command{ginac-config} is a shell script that you can use to determine
4533 the compiler and linker command line options required to compile and
4534 link a program with the GiNaC library.
4536 @command{ginac-config} takes the following flags:
4540 Prints out the version of GiNaC installed.
4542 Prints '-I' flags pointing to the installed header files.
4544 Prints out the linker flags necessary to link a program against GiNaC.
4545 @item --prefix[=@var{PREFIX}]
4546 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
4547 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
4548 Otherwise, prints out the configured value of @env{$prefix}.
4549 @item --exec-prefix[=@var{PREFIX}]
4550 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
4551 Otherwise, prints out the configured value of @env{$exec_prefix}.
4554 Typically, @command{ginac-config} will be used within a configure
4555 script, as described below. It, however, can also be used directly from
4556 the command line using backquotes to compile a simple program. For
4560 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
4563 This command line might expand to (for example):
4566 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
4567 -lginac -lcln -lstdc++
4570 Not only is the form using @command{ginac-config} easier to type, it will
4571 work on any system, no matter how GiNaC was configured.
4574 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
4575 @c node-name, next, previous, up
4576 @section @samp{AM_PATH_GINAC}
4577 @cindex AM_PATH_GINAC
4579 For packages configured using GNU automake, GiNaC also provides
4580 a macro to automate the process of checking for GiNaC.
4583 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
4591 Determines the location of GiNaC using @command{ginac-config}, which is
4592 either found in the user's path, or from the environment variable
4593 @env{GINACLIB_CONFIG}.
4596 Tests the installed libraries to make sure that their version
4597 is later than @var{MINIMUM-VERSION}. (A default version will be used
4601 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
4602 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
4603 variable to the output of @command{ginac-config --libs}, and calls
4604 @samp{AC_SUBST()} for these variables so they can be used in generated
4605 makefiles, and then executes @var{ACTION-IF-FOUND}.
4608 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
4609 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
4613 This macro is in file @file{ginac.m4} which is installed in
4614 @file{$datadir/aclocal}. Note that if automake was installed with a
4615 different @samp{--prefix} than GiNaC, you will either have to manually
4616 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
4617 aclocal the @samp{-I} option when running it.
4620 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
4621 * Example package:: Example of a package using AM_PATH_GINAC.
4625 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
4626 @c node-name, next, previous, up
4627 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
4629 Simply make sure that @command{ginac-config} is in your path, and run
4630 the configure script.
4637 The directory where the GiNaC libraries are installed needs
4638 to be found by your system's dynamic linker.
4640 This is generally done by
4643 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
4649 setting the environment variable @env{LD_LIBRARY_PATH},
4652 or, as a last resort,
4655 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
4656 running configure, for instance:
4659 LDFLAGS=-R/home/cbauer/lib ./configure
4664 You can also specify a @command{ginac-config} not in your path by
4665 setting the @env{GINACLIB_CONFIG} environment variable to the
4666 name of the executable
4669 If you move the GiNaC package from its installed location,
4670 you will either need to modify @command{ginac-config} script
4671 manually to point to the new location or rebuild GiNaC.
4682 --with-ginac-prefix=@var{PREFIX}
4683 --with-ginac-exec-prefix=@var{PREFIX}
4686 are provided to override the prefix and exec-prefix that were stored
4687 in the @command{ginac-config} shell script by GiNaC's configure. You are
4688 generally better off configuring GiNaC with the right path to begin with.
4692 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
4693 @c node-name, next, previous, up
4694 @subsection Example of a package using @samp{AM_PATH_GINAC}
4696 The following shows how to build a simple package using automake
4697 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
4700 #include <ginac/ginac.h>
4704 GiNaC::symbol x("x");
4705 GiNaC::ex a = GiNaC::sin(x);
4706 std::cout << "Derivative of " << a
4707 << " is " << a.diff(x) << std::endl;
4712 You should first read the introductory portions of the automake
4713 Manual, if you are not already familiar with it.
4715 Two files are needed, @file{configure.in}, which is used to build the
4719 dnl Process this file with autoconf to produce a configure script.
4721 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
4727 AM_PATH_GINAC(0.7.0, [
4728 LIBS="$LIBS $GINACLIB_LIBS"
4729 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
4730 ], AC_MSG_ERROR([need to have GiNaC installed]))
4735 The only command in this which is not standard for automake
4736 is the @samp{AM_PATH_GINAC} macro.
4738 That command does the following: If a GiNaC version greater or equal
4739 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
4740 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
4741 the error message `need to have GiNaC installed'
4743 And the @file{Makefile.am}, which will be used to build the Makefile.
4746 ## Process this file with automake to produce Makefile.in
4747 bin_PROGRAMS = simple
4748 simple_SOURCES = simple.cpp
4751 This @file{Makefile.am}, says that we are building a single executable,
4752 from a single sourcefile @file{simple.cpp}. Since every program
4753 we are building uses GiNaC we simply added the GiNaC options
4754 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
4755 want to specify them on a per-program basis: for instance by
4759 simple_LDADD = $(GINACLIB_LIBS)
4760 INCLUDES = $(GINACLIB_CPPFLAGS)
4763 to the @file{Makefile.am}.
4765 To try this example out, create a new directory and add the three
4768 Now execute the following commands:
4771 $ automake --add-missing
4776 You now have a package that can be built in the normal fashion
4785 @node Bibliography, Concept Index, Example package, Top
4786 @c node-name, next, previous, up
4787 @appendix Bibliography
4792 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
4795 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
4798 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
4801 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
4804 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
4805 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
4808 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
4809 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
4810 Academic Press, London
4813 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
4818 @node Concept Index, , Bibliography, Top
4819 @c node-name, next, previous, up
4820 @unnumbered Concept Index