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 tries to give dumy indices that appear in different terms of a sum
1609 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
1610 @item it (symbolically) calculates all possible dummy index summations/contractions
1611 with the predefined tensors (this will be explained in more detail in the
1613 @item as a special case of dummy index summation, it can replace scalar products
1614 of two tensors with a user-defined value
1617 The last point is done with the help of the @code{scalar_products} class
1618 which is used to store scalar products with known values (this is not an
1619 arithmetic class, you just pass it to @code{simplify_indexed()}):
1623 symbol A("A"), B("B"), C("C"), i_sym("i");
1627 sp.add(A, B, 0); // A and B are orthogonal
1628 sp.add(A, C, 0); // A and C are orthogonal
1629 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1631 e = indexed(A + B, i) * indexed(A + C, i);
1633 // -> (B+A).i*(A+C).i
1635 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1641 The @code{scalar_products} object @code{sp} acts as a storage for the
1642 scalar products added to it with the @code{.add()} method. This method
1643 takes three arguments: the two expressions of which the scalar product is
1644 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1645 @code{simplify_indexed()} will replace all scalar products of indexed
1646 objects that have the symbols @code{A} and @code{B} as base expressions
1647 with the single value 0. The number, type and dimension of the indices
1648 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1650 @cindex @code{expand()}
1651 The example above also illustrates a feature of the @code{expand()} method:
1652 if passed the @code{expand_indexed} option it will distribute indices
1653 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1655 @cindex @code{tensor} (class)
1656 @subsection Predefined tensors
1658 Some frequently used special tensors such as the delta, epsilon and metric
1659 tensors are predefined in GiNaC. They have special properties when
1660 contracted with other tensor expressions and some of them have constant
1661 matrix representations (they will evaluate to a number when numeric
1662 indices are specified).
1664 @cindex @code{delta_tensor()}
1665 @subsubsection Delta tensor
1667 The delta tensor takes two indices, is symmetric and has the matrix
1668 representation @code{diag(1,1,1,...)}. It is constructed by the function
1669 @code{delta_tensor()}:
1673 symbol A("A"), B("B");
1675 idx i(symbol("i"), 3), j(symbol("j"), 3),
1676 k(symbol("k"), 3), l(symbol("l"), 3);
1678 ex e = indexed(A, i, j) * indexed(B, k, l)
1679 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1680 cout << e.simplify_indexed() << endl;
1683 cout << delta_tensor(i, i) << endl;
1688 @cindex @code{metric_tensor()}
1689 @subsubsection General metric tensor
1691 The function @code{metric_tensor()} creates a general symmetric metric
1692 tensor with two indices that can be used to raise/lower tensor indices. The
1693 metric tensor is denoted as @samp{g} in the output and if its indices are of
1694 mixed variance it is automatically replaced by a delta tensor:
1700 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1702 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1703 cout << e.simplify_indexed() << endl;
1706 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1707 cout << e.simplify_indexed() << endl;
1710 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1711 * metric_tensor(nu, rho);
1712 cout << e.simplify_indexed() << endl;
1715 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1716 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1717 + indexed(A, mu.toggle_variance(), rho));
1718 cout << e.simplify_indexed() << endl;
1723 @cindex @code{lorentz_g()}
1724 @subsubsection Minkowski metric tensor
1726 The Minkowski metric tensor is a special metric tensor with a constant
1727 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1728 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1729 It is created with the function @code{lorentz_g()} (although it is output as
1734 varidx mu(symbol("mu"), 4);
1736 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1737 * lorentz_g(mu, varidx(0, 4)); // negative signature
1738 cout << e.simplify_indexed() << endl;
1741 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1742 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1743 cout << e.simplify_indexed() << endl;
1748 @cindex @code{spinor_metric()}
1749 @subsubsection Spinor metric tensor
1751 The function @code{spinor_metric()} creates an antisymmetric tensor with
1752 two indices that is used to raise/lower indices of 2-component spinors.
1753 It is output as @samp{eps}:
1759 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
1760 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
1762 e = spinor_metric(A, B) * indexed(psi, B_co);
1763 cout << e.simplify_indexed() << endl;
1766 e = spinor_metric(A, B) * indexed(psi, A_co);
1767 cout << e.simplify_indexed() << endl;
1770 e = spinor_metric(A_co, B_co) * indexed(psi, B);
1771 cout << e.simplify_indexed() << endl;
1774 e = spinor_metric(A_co, B_co) * indexed(psi, A);
1775 cout << e.simplify_indexed() << endl;
1778 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
1779 cout << e.simplify_indexed() << endl;
1782 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
1783 cout << e.simplify_indexed() << endl;
1788 The matrix representation of the spinor metric is @code{[[ [[ 0, 1 ]], [[ -1, 0 ]] ]]}.
1790 @cindex @code{epsilon_tensor()}
1791 @cindex @code{lorentz_eps()}
1792 @subsubsection Epsilon tensor
1794 The epsilon tensor is totally antisymmetric, its number of indices is equal
1795 to the dimension of the index space (the indices must all be of the same
1796 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1797 defined to be 1. Its behaviour with indices that have a variance also
1798 depends on the signature of the metric. Epsilon tensors are output as
1801 There are three functions defined to create epsilon tensors in 2, 3 and 4
1805 ex epsilon_tensor(const ex & i1, const ex & i2);
1806 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1807 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1810 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1811 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1812 Minkowski space (the last @code{bool} argument specifies whether the metric
1813 has negative or positive signature, as in the case of the Minkowski metric
1816 @subsection Linear algebra
1818 The @code{matrix} class can be used with indices to do some simple linear
1819 algebra (linear combinations and products of vectors and matrices, traces
1820 and scalar products):
1824 idx i(symbol("i"), 2), j(symbol("j"), 2);
1825 symbol x("x"), y("y");
1827 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
1829 cout << indexed(A, i, i) << endl;
1832 ex e = indexed(A, i, j) * indexed(X, j);
1833 cout << e.simplify_indexed() << endl;
1834 // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
1836 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
1837 cout << e.simplify_indexed() << endl;
1838 // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
1842 You can of course obtain the same results with the @code{matrix::add()},
1843 @code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
1844 don't have to worry about transposing matrices.
1846 Matrix indices always start at 0 and their dimension must match the number
1847 of rows/columns of the matrix. Matrices with one row or one column are
1848 vectors and can have one or two indices (it doesn't matter whether it's a
1849 row or a column vector). Other matrices must have two indices.
1851 You should be careful when using indices with variance on matrices. GiNaC
1852 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
1853 @samp{F.mu.nu} are different matrices. In this case you should use only
1854 one form for @samp{F} and explicitly multiply it with a matrix representation
1855 of the metric tensor.
1858 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
1859 @c node-name, next, previous, up
1860 @section Non-commutative objects
1862 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
1863 non-commutative objects are built-in which are mostly of use in high energy
1867 @item Clifford (Dirac) algebra (class @code{clifford})
1868 @item su(3) Lie algebra (class @code{color})
1869 @item Matrices (unindexed) (class @code{matrix})
1872 The @code{clifford} and @code{color} classes are subclasses of
1873 @code{indexed} because the elements of these algebras ususally carry
1876 Unlike most computer algebra systems, GiNaC does not primarily provide an
1877 operator (often denoted @samp{&*}) for representing inert products of
1878 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
1879 classes of objects involved, and non-commutative products are formed with
1880 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
1881 figuring out by itself which objects commute and will group the factors
1882 by their class. Consider this example:
1886 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
1887 idx a(symbol("a"), 8), b(symbol("b"), 8);
1888 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
1890 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
1894 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
1895 groups the non-commutative factors (the gammas and the su(3) generators)
1896 together while preserving the order of factors within each class (because
1897 Clifford objects commute with color objects). The resulting expression is a
1898 @emph{commutative} product with two factors that are themselves non-commutative
1899 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
1900 parentheses are placed around the non-commutative products in the output.
1902 @cindex @code{ncmul} (class)
1903 Non-commutative products are internally represented by objects of the class
1904 @code{ncmul}, as opposed to commutative products which are handled by the
1905 @code{mul} class. You will normally not have to worry about this distinction,
1908 The advantage of this approach is that you never have to worry about using
1909 (or forgetting to use) a special operator when constructing non-commutative
1910 expressions. Also, non-commutative products in GiNaC are more intelligent
1911 than in other computer algebra systems; they can, for example, automatically
1912 canonicalize themselves according to rules specified in the implementation
1913 of the non-commutative classes. The drawback is that to work with other than
1914 the built-in algebras you have to implement new classes yourself. Symbols
1915 always commute and it's not possible to construct non-commutative products
1916 using symbols to represent the algebra elements or generators. User-defined
1917 functions can, however, be specified as being non-commutative.
1919 @cindex @code{return_type()}
1920 @cindex @code{return_type_tinfo()}
1921 Information about the commutativity of an object or expression can be
1922 obtained with the two member functions
1925 unsigned ex::return_type(void) const;
1926 unsigned ex::return_type_tinfo(void) const;
1929 The @code{return_type()} function returns one of three values (defined in
1930 the header file @file{flags.h}), corresponding to three categories of
1931 expressions in GiNaC:
1934 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
1935 classes are of this kind.
1936 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
1937 certain class of non-commutative objects which can be determined with the
1938 @code{return_type_tinfo()} method. Expressions of this category commute
1939 with everything except @code{noncommutative} expressions of the same
1941 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
1942 of non-commutative objects of different classes. Expressions of this
1943 category don't commute with any other @code{noncommutative} or
1944 @code{noncommutative_composite} expressions.
1947 The value returned by the @code{return_type_tinfo()} method is valid only
1948 when the return type of the expression is @code{noncommutative}. It is a
1949 value that is unique to the class of the object and usually one of the
1950 constants in @file{tinfos.h}, or derived therefrom.
1952 Here are a couple of examples:
1955 @multitable @columnfractions 0.33 0.33 0.34
1956 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
1957 @item @code{42} @tab @code{commutative} @tab -
1958 @item @code{2*x-y} @tab @code{commutative} @tab -
1959 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1960 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
1961 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
1962 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
1966 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
1967 @code{TINFO_clifford} for objects with a representation label of zero.
1968 Other representation labels yield a different @code{return_type_tinfo()},
1969 but it's the same for any two objects with the same label. This is also true
1973 @cindex @code{clifford} (class)
1974 @subsection Clifford algebra
1976 @cindex @code{dirac_gamma()}
1977 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
1978 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
1979 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
1980 is the Minkowski metric tensor. Dirac gammas are constructed by the function
1983 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
1986 which takes two arguments: the index and a @dfn{representation label} in the
1987 range 0 to 255 which is used to distinguish elements of different Clifford
1988 algebras (this is also called a @dfn{spin line index}). Gammas with different
1989 labels commute with each other. The dimension of the index can be 4 or (in
1990 the framework of dimensional regularization) any symbolic value. Spinor
1991 indices on Dirac gammas are not supported in GiNaC.
1993 @cindex @code{dirac_ONE()}
1994 The unity element of a Clifford algebra is constructed by
1997 ex dirac_ONE(unsigned char rl = 0);
2000 @cindex @code{dirac_gamma5()}
2001 and there's a special element @samp{gamma5} that commutes with all other
2002 gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
2006 ex dirac_gamma5(unsigned char rl = 0);
2009 @cindex @code{dirac_gamma6()}
2010 @cindex @code{dirac_gamma7()}
2011 The two additional functions
2014 ex dirac_gamma6(unsigned char rl = 0);
2015 ex dirac_gamma7(unsigned char rl = 0);
2018 return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
2021 @cindex @code{dirac_slash()}
2022 Finally, the function
2025 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2028 creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
2029 whose dimension is given by the @code{dim} argument.
2031 In products of dirac gammas, superfluous unity elements are automatically
2032 removed, squares are replaced by their values and @samp{gamma5} is
2033 anticommuted to the front. The @code{simplify_indexed()} function performs
2034 contractions in gamma strings, for example
2039 symbol a("a"), b("b"), D("D");
2040 varidx mu(symbol("mu"), D);
2041 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2042 * dirac_gamma(mu.toggle_variance());
2044 // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
2045 e = e.simplify_indexed();
2047 // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
2048 cout << e.subs(D == 4) << endl;
2049 // -> -2*gamma~symbol10*a.symbol10
2050 // [ == -2 * dirac_slash(a, D) ]
2055 @cindex @code{dirac_trace()}
2056 To calculate the trace of an expression containing strings of Dirac gammas
2057 you use the function
2060 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2063 This function takes the trace of all gammas with the specified representation
2064 label; gammas with other labels are left standing. The last argument to
2065 @code{dirac_trace()} is the value to be returned for the trace of the unity
2066 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2067 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2068 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2069 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2070 This @samp{gamma5} scheme is described in greater detail in
2071 @cite{The Role of gamma5 in Dimensional Regularization}.
2073 The value of the trace itself is also usually different in 4 and in
2074 @math{D != 4} dimensions:
2079 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2080 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2081 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2082 cout << dirac_trace(e).simplify_indexed() << endl;
2089 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2090 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2091 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2092 cout << dirac_trace(e).simplify_indexed() << endl;
2093 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2097 Here is an example for using @code{dirac_trace()} to compute a value that
2098 appears in the calculation of the one-loop vacuum polarization amplitude in
2103 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2104 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2107 sp.add(l, l, pow(l, 2));
2108 sp.add(l, q, ldotq);
2110 ex e = dirac_gamma(mu) *
2111 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2112 dirac_gamma(mu.toggle_variance()) *
2113 (dirac_slash(l, D) + m * dirac_ONE());
2114 e = dirac_trace(e).simplify_indexed(sp);
2115 e = e.collect(lst(l, ldotq, m), true);
2117 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2121 The @code{canonicalize_clifford()} function reorders all gamma products that
2122 appear in an expression to a canonical (but not necessarily simple) form.
2123 You can use this to compare two expressions or for further simplifications:
2127 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2128 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2130 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2132 e = canonicalize_clifford(e);
2139 @cindex @code{color} (class)
2140 @subsection Color algebra
2142 @cindex @code{color_T()}
2143 For computations in quantum chromodynamics, GiNaC implements the base elements
2144 and structure constants of the su(3) Lie algebra (color algebra). The base
2145 elements @math{T_a} are constructed by the function
2148 ex color_T(const ex & a, unsigned char rl = 0);
2151 which takes two arguments: the index and a @dfn{representation label} in the
2152 range 0 to 255 which is used to distinguish elements of different color
2153 algebras. Objects with different labels commute with each other. The
2154 dimension of the index must be exactly 8 and it should be of class @code{idx},
2157 @cindex @code{color_ONE()}
2158 The unity element of a color algebra is constructed by
2161 ex color_ONE(unsigned char rl = 0);
2164 @cindex @code{color_d()}
2165 @cindex @code{color_f()}
2169 ex color_d(const ex & a, const ex & b, const ex & c);
2170 ex color_f(const ex & a, const ex & b, const ex & c);
2173 create the symmetric and antisymmetric structure constants @math{d_abc} and
2174 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2175 and @math{[T_a, T_b] = i f_abc T_c}.
2177 @cindex @code{color_h()}
2178 There's an additional function
2181 ex color_h(const ex & a, const ex & b, const ex & c);
2184 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2186 The function @code{simplify_indexed()} performs some simplifications on
2187 expressions containing color objects:
2192 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2193 k(symbol("k"), 8), l(symbol("l"), 8);
2195 e = color_d(a, b, l) * color_f(a, b, k);
2196 cout << e.simplify_indexed() << endl;
2199 e = color_d(a, b, l) * color_d(a, b, k);
2200 cout << e.simplify_indexed() << endl;
2203 e = color_f(l, a, b) * color_f(a, b, k);
2204 cout << e.simplify_indexed() << endl;
2207 e = color_h(a, b, c) * color_h(a, b, c);
2208 cout << e.simplify_indexed() << endl;
2211 e = color_h(a, b, c) * color_T(b) * color_T(c);
2212 cout << e.simplify_indexed() << endl;
2215 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2216 cout << e.simplify_indexed() << endl;
2219 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2220 cout << e.simplify_indexed() << endl;
2221 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2225 @cindex @code{color_trace()}
2226 To calculate the trace of an expression containing color objects you use the
2230 ex color_trace(const ex & e, unsigned char rl = 0);
2233 This function takes the trace of all color @samp{T} objects with the
2234 specified representation label; @samp{T}s with other labels are left
2235 standing. For example:
2239 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2241 // -> -I*f.a.c.b+d.a.c.b
2246 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2247 @c node-name, next, previous, up
2248 @chapter Methods and Functions
2251 In this chapter the most important algorithms provided by GiNaC will be
2252 described. Some of them are implemented as functions on expressions,
2253 others are implemented as methods provided by expression objects. If
2254 they are methods, there exists a wrapper function around it, so you can
2255 alternatively call it in a functional way as shown in the simple
2260 cout << "As method: " << sin(1).evalf() << endl;
2261 cout << "As function: " << evalf(sin(1)) << endl;
2265 @cindex @code{subs()}
2266 The general rule is that wherever methods accept one or more parameters
2267 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2268 wrapper accepts is the same but preceded by the object to act on
2269 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2270 most natural one in an OO model but it may lead to confusion for MapleV
2271 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2272 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2273 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2274 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2275 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2276 here. Also, users of MuPAD will in most cases feel more comfortable
2277 with GiNaC's convention. All function wrappers are implemented
2278 as simple inline functions which just call the corresponding method and
2279 are only provided for users uncomfortable with OO who are dead set to
2280 avoid method invocations. Generally, nested function wrappers are much
2281 harder to read than a sequence of methods and should therefore be
2282 avoided if possible. On the other hand, not everything in GiNaC is a
2283 method on class @code{ex} and sometimes calling a function cannot be
2287 * Information About Expressions::
2288 * Substituting Expressions::
2289 * Polynomial Arithmetic:: Working with polynomials.
2290 * Rational Expressions:: Working with rational functions.
2291 * Symbolic Differentiation::
2292 * Series Expansion:: Taylor and Laurent expansion.
2293 * Built-in Functions:: List of predefined mathematical functions.
2294 * Input/Output:: Input and output of expressions.
2298 @node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
2299 @c node-name, next, previous, up
2300 @section Getting information about expressions
2302 @subsection Checking expression types
2303 @cindex @code{is_ex_of_type()}
2304 @cindex @code{ex_to_numeric()}
2305 @cindex @code{ex_to_@dots{}}
2306 @cindex @code{Converting ex to other classes}
2307 @cindex @code{info()}
2308 @cindex @code{return_type()}
2309 @cindex @code{return_type_tinfo()}
2311 Sometimes it's useful to check whether a given expression is a plain number,
2312 a sum, a polynomial with integer coefficients, or of some other specific type.
2313 GiNaC provides a couple of functions for this (the first one is actually a macro):
2316 bool is_ex_of_type(const ex & e, TYPENAME t);
2317 bool ex::info(unsigned flag);
2318 unsigned ex::return_type(void) const;
2319 unsigned ex::return_type_tinfo(void) const;
2322 When the test made by @code{is_ex_of_type()} returns true, it is safe to
2323 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
2324 one of the class names (@xref{The Class Hierarchy}, for a list of all
2325 classes). For example, assuming @code{e} is an @code{ex}:
2330 if (is_ex_of_type(e, numeric))
2331 numeric n = ex_to_numeric(e);
2336 @code{is_ex_of_type()} allows you to check whether the top-level object of
2337 an expression @samp{e} is an instance of the GiNaC class @samp{t}
2338 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
2339 e.g., for checking whether an expression is a number, a sum, or a product:
2346 is_ex_of_type(e1, numeric); // true
2347 is_ex_of_type(e2, numeric); // false
2348 is_ex_of_type(e1, add); // false
2349 is_ex_of_type(e2, add); // true
2350 is_ex_of_type(e1, mul); // false
2351 is_ex_of_type(e2, mul); // false
2355 The @code{info()} method is used for checking certain attributes of
2356 expressions. The possible values for the @code{flag} argument are defined
2357 in @file{ginac/flags.h}, the most important being explained in the following
2361 @multitable @columnfractions .30 .70
2362 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
2363 @item @code{numeric}
2364 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
2366 @tab @dots{}a real integer, rational or float (i.e. is not complex)
2367 @item @code{rational}
2368 @tab @dots{}an exact rational number (integers are rational, too)
2369 @item @code{integer}
2370 @tab @dots{}a (non-complex) integer
2371 @item @code{crational}
2372 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
2373 @item @code{cinteger}
2374 @tab @dots{}a (complex) integer (such as @math{2-3*I})
2375 @item @code{positive}
2376 @tab @dots{}not complex and greater than 0
2377 @item @code{negative}
2378 @tab @dots{}not complex and less than 0
2379 @item @code{nonnegative}
2380 @tab @dots{}not complex and greater than or equal to 0
2382 @tab @dots{}an integer greater than 0
2384 @tab @dots{}an integer less than 0
2385 @item @code{nonnegint}
2386 @tab @dots{}an integer greater than or equal to 0
2388 @tab @dots{}an even integer
2390 @tab @dots{}an odd integer
2392 @tab @dots{}a prime integer (probabilistic primality test)
2393 @item @code{relation}
2394 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
2395 @item @code{relation_equal}
2396 @tab @dots{}a @code{==} relation
2397 @item @code{relation_not_equal}
2398 @tab @dots{}a @code{!=} relation
2399 @item @code{relation_less}
2400 @tab @dots{}a @code{<} relation
2401 @item @code{relation_less_or_equal}
2402 @tab @dots{}a @code{<=} relation
2403 @item @code{relation_greater}
2404 @tab @dots{}a @code{>} relation
2405 @item @code{relation_greater_or_equal}
2406 @tab @dots{}a @code{>=} relation
2408 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
2410 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
2411 @item @code{polynomial}
2412 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
2413 @item @code{integer_polynomial}
2414 @tab @dots{}a polynomial with (non-complex) integer coefficients
2415 @item @code{cinteger_polynomial}
2416 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
2417 @item @code{rational_polynomial}
2418 @tab @dots{}a polynomial with (non-complex) rational coefficients
2419 @item @code{crational_polynomial}
2420 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
2421 @item @code{rational_function}
2422 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
2423 @item @code{algebraic}
2424 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
2428 To determine whether an expression is commutative or non-commutative and if
2429 so, with which other expressions it would commute, you use the methods
2430 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
2431 for an explanation of these.
2434 @subsection Accessing subexpressions
2435 @cindex @code{nops()}
2437 @cindex @code{has()}
2439 @cindex @code{relational} (class)
2441 GiNaC provides the two methods
2444 unsigned ex::nops();
2445 ex ex::op(unsigned i);
2448 for accessing the subexpressions in the container-like GiNaC classes like
2449 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
2450 determines the number of subexpressions (@samp{operands}) contained, while
2451 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
2452 In the case of a @code{power} object, @code{op(0)} will return the basis
2453 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
2454 is the base expression and @code{op(i)}, @math{i>0} are the indices.
2456 The left-hand and right-hand side expressions of objects of class
2457 @code{relational} (and only of these) can also be accessed with the methods
2467 bool ex::has(const ex & other);
2470 checks whether an expression contains the given subexpression @code{other}.
2471 This only works reliably if @code{other} is of an atomic class such as a
2472 @code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
2473 @code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
2476 @subsection Comparing expressions
2477 @cindex @code{is_equal()}
2478 @cindex @code{is_zero()}
2480 Expressions can be compared with the usual C++ relational operators like
2481 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
2482 the result is usually not determinable and the result will be @code{false},
2483 except in the case of the @code{!=} operator. You should also be aware that
2484 GiNaC will only do the most trivial test for equality (subtracting both
2485 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
2488 Actually, if you construct an expression like @code{a == b}, this will be
2489 represented by an object of the @code{relational} class (@xref{Relations}.)
2490 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
2492 There are also two methods
2495 bool ex::is_equal(const ex & other);
2499 for checking whether one expression is equal to another, or equal to zero,
2502 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2503 GiNaC header files. This method is however only to be used internally by
2504 GiNaC to establish a canonical sort order for terms, and using it to compare
2505 expressions will give very surprising results.
2508 @node Substituting Expressions, Polynomial Arithmetic, Information About Expressions, Methods and Functions
2509 @c node-name, next, previous, up
2510 @section Substituting expressions
2511 @cindex @code{subs()}
2513 Algebraic objects inside expressions can be replaced with arbitrary
2514 expressions via the @code{.subs()} method:
2517 ex ex::subs(const ex & e);
2518 ex ex::subs(const lst & syms, const lst & repls);
2521 In the first form, @code{subs()} accepts a relational of the form
2522 @samp{object == expression} or a @code{lst} of such relationals:
2526 symbol x("x"), y("y");
2528 ex e1 = 2*x^2-4*x+3;
2529 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2533 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2538 @code{subs()} performs syntactic substitution of any complete algebraic
2539 object; it does not try to match sub-expressions as is demonstrated by the
2544 symbol x("x"), y("y"), z("z");
2546 ex e1 = pow(x+y, 2);
2547 cout << e1.subs(x+y == 4) << endl;
2550 ex e2 = sin(x)*cos(x);
2551 cout << e2.subs(sin(x) == cos(x)) << endl;
2555 cout << e3.subs(x+y == 4) << endl;
2557 // (and not 4+z as one might expect)
2561 If you specify multiple substitutions, they are performed in parallel, so e.g.
2562 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2564 The second form of @code{subs()} takes two lists, one for the objects to be
2565 replaced and one for the expressions to be substituted (both lists must
2566 contain the same number of elements). Using this form, you would write
2567 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
2570 @node Polynomial Arithmetic, Rational Expressions, Substituting Expressions, Methods and Functions
2571 @c node-name, next, previous, up
2572 @section Polynomial arithmetic
2574 @subsection Expanding and collecting
2575 @cindex @code{expand()}
2576 @cindex @code{collect()}
2578 A polynomial in one or more variables has many equivalent
2579 representations. Some useful ones serve a specific purpose. Consider
2580 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2581 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2582 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
2583 representations are the recursive ones where one collects for exponents
2584 in one of the three variable. Since the factors are themselves
2585 polynomials in the remaining two variables the procedure can be
2586 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
2587 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
2590 To bring an expression into expanded form, its method
2596 may be called. In our example above, this corresponds to @math{4*x*y +
2597 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
2598 GiNaC is not easily guessable you should be prepared to see different
2599 orderings of terms in such sums!
2601 Another useful representation of multivariate polynomials is as a
2602 univariate polynomial in one of the variables with the coefficients
2603 being polynomials in the remaining variables. The method
2604 @code{collect()} accomplishes this task:
2607 ex ex::collect(const ex & s, bool distributed = false);
2610 The first argument to @code{collect()} can also be a list of objects in which
2611 case the result is either a recursively collected polynomial, or a polynomial
2612 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
2613 by the @code{distributed} flag.
2615 Note that the original polynomial needs to be in expanded form in order
2616 for @code{collect()} to be able to find the coefficients properly.
2618 @subsection Degree and coefficients
2619 @cindex @code{degree()}
2620 @cindex @code{ldegree()}
2621 @cindex @code{coeff()}
2623 The degree and low degree of a polynomial can be obtained using the two
2627 int ex::degree(const ex & s);
2628 int ex::ldegree(const ex & s);
2631 which also work reliably on non-expanded input polynomials (they even work
2632 on rational functions, returning the asymptotic degree). To extract
2633 a coefficient with a certain power from an expanded polynomial you use
2636 ex ex::coeff(const ex & s, int n);
2639 You can also obtain the leading and trailing coefficients with the methods
2642 ex ex::lcoeff(const ex & s);
2643 ex ex::tcoeff(const ex & s);
2646 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
2649 An application is illustrated in the next example, where a multivariate
2650 polynomial is analyzed:
2653 #include <ginac/ginac.h>
2654 using namespace std;
2655 using namespace GiNaC;
2659 symbol x("x"), y("y");
2660 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
2661 - pow(x+y,2) + 2*pow(y+2,2) - 8;
2662 ex Poly = PolyInp.expand();
2664 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
2665 cout << "The x^" << i << "-coefficient is "
2666 << Poly.coeff(x,i) << endl;
2668 cout << "As polynomial in y: "
2669 << Poly.collect(y) << endl;
2673 When run, it returns an output in the following fashion:
2676 The x^0-coefficient is y^2+11*y
2677 The x^1-coefficient is 5*y^2-2*y
2678 The x^2-coefficient is -1
2679 The x^3-coefficient is 4*y
2680 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
2683 As always, the exact output may vary between different versions of GiNaC
2684 or even from run to run since the internal canonical ordering is not
2685 within the user's sphere of influence.
2687 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
2688 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
2689 with non-polynomial expressions as they not only work with symbols but with
2690 constants, functions and indexed objects as well:
2694 symbol a("a"), b("b"), c("c");
2695 idx i(symbol("i"), 3);
2697 ex e = pow(sin(x) - cos(x), 4);
2698 cout << e.degree(cos(x)) << endl;
2700 cout << e.expand().coeff(sin(x), 3) << endl;
2703 e = indexed(a+b, i) * indexed(b+c, i);
2704 e = e.expand(expand_options::expand_indexed);
2705 cout << e.collect(indexed(b, i)) << endl;
2706 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
2711 @subsection Polynomial division
2712 @cindex polynomial division
2715 @cindex pseudo-remainder
2716 @cindex @code{quo()}
2717 @cindex @code{rem()}
2718 @cindex @code{prem()}
2719 @cindex @code{divide()}
2724 ex quo(const ex & a, const ex & b, const symbol & x);
2725 ex rem(const ex & a, const ex & b, const symbol & x);
2728 compute the quotient and remainder of univariate polynomials in the variable
2729 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
2731 The additional function
2734 ex prem(const ex & a, const ex & b, const symbol & x);
2737 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
2738 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
2740 Exact division of multivariate polynomials is performed by the function
2743 bool divide(const ex & a, const ex & b, ex & q);
2746 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
2747 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
2748 in which case the value of @code{q} is undefined.
2751 @subsection Unit, content and primitive part
2752 @cindex @code{unit()}
2753 @cindex @code{content()}
2754 @cindex @code{primpart()}
2759 ex ex::unit(const symbol & x);
2760 ex ex::content(const symbol & x);
2761 ex ex::primpart(const symbol & x);
2764 return the unit part, content part, and primitive polynomial of a multivariate
2765 polynomial with respect to the variable @samp{x} (the unit part being the sign
2766 of the leading coefficient, the content part being the GCD of the coefficients,
2767 and the primitive polynomial being the input polynomial divided by the unit and
2768 content parts). The product of unit, content, and primitive part is the
2769 original polynomial.
2772 @subsection GCD and LCM
2775 @cindex @code{gcd()}
2776 @cindex @code{lcm()}
2778 The functions for polynomial greatest common divisor and least common
2779 multiple have the synopsis
2782 ex gcd(const ex & a, const ex & b);
2783 ex lcm(const ex & a, const ex & b);
2786 The functions @code{gcd()} and @code{lcm()} accept two expressions
2787 @code{a} and @code{b} as arguments and return a new expression, their
2788 greatest common divisor or least common multiple, respectively. If the
2789 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
2790 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
2793 #include <ginac/ginac.h>
2794 using namespace GiNaC;
2798 symbol x("x"), y("y"), z("z");
2799 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
2800 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
2802 ex P_gcd = gcd(P_a, P_b);
2804 ex P_lcm = lcm(P_a, P_b);
2805 // 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
2810 @subsection Square-free decomposition
2811 @cindex square-free decomposition
2812 @cindex factorization
2813 @cindex @code{sqrfree()}
2815 GiNaC still lacks proper factorization support. Some form of
2816 factorization is, however, easily implemented by noting that factors
2817 appearing in a polynomial with power two or more also appear in the
2818 derivative and hence can easily be found by computing the GCD of the
2819 original polynomial and its derivatives. Any system has an interface
2820 for this so called square-free factorization. So we provide one, too:
2822 ex sqrfree(const ex & a, const lst & l = lst());
2824 Here is an example that by the way illustrates how the result may depend
2825 on the order of differentiation:
2828 symbol x("x"), y("y");
2829 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
2831 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
2832 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
2834 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
2835 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
2837 cout << sqrfree(BiVarPol) << endl;
2838 // -> depending on luck, any of the above
2843 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
2844 @c node-name, next, previous, up
2845 @section Rational expressions
2847 @subsection The @code{normal} method
2848 @cindex @code{normal()}
2849 @cindex simplification
2850 @cindex temporary replacement
2852 Some basic form of simplification of expressions is called for frequently.
2853 GiNaC provides the method @code{.normal()}, which converts a rational function
2854 into an equivalent rational function of the form @samp{numerator/denominator}
2855 where numerator and denominator are coprime. If the input expression is already
2856 a fraction, it just finds the GCD of numerator and denominator and cancels it,
2857 otherwise it performs fraction addition and multiplication.
2859 @code{.normal()} can also be used on expressions which are not rational functions
2860 as it will replace all non-rational objects (like functions or non-integer
2861 powers) by temporary symbols to bring the expression to the domain of rational
2862 functions before performing the normalization, and re-substituting these
2863 symbols afterwards. This algorithm is also available as a separate method
2864 @code{.to_rational()}, described below.
2866 This means that both expressions @code{t1} and @code{t2} are indeed
2867 simplified in this little program:
2870 #include <ginac/ginac.h>
2871 using namespace GiNaC;
2876 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
2877 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
2878 std::cout << "t1 is " << t1.normal() << std::endl;
2879 std::cout << "t2 is " << t2.normal() << std::endl;
2883 Of course this works for multivariate polynomials too, so the ratio of
2884 the sample-polynomials from the section about GCD and LCM above would be
2885 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
2888 @subsection Numerator and denominator
2891 @cindex @code{numer()}
2892 @cindex @code{denom()}
2894 The numerator and denominator of an expression can be obtained with
2901 These functions will first normalize the expression as described above and
2902 then return the numerator or denominator, respectively.
2905 @subsection Converting to a rational expression
2906 @cindex @code{to_rational()}
2908 Some of the methods described so far only work on polynomials or rational
2909 functions. GiNaC provides a way to extend the domain of these functions to
2910 general expressions by using the temporary replacement algorithm described
2911 above. You do this by calling
2914 ex ex::to_rational(lst &l);
2917 on the expression to be converted. The supplied @code{lst} will be filled
2918 with the generated temporary symbols and their replacement expressions in
2919 a format that can be used directly for the @code{subs()} method. It can also
2920 already contain a list of replacements from an earlier application of
2921 @code{.to_rational()}, so it's possible to use it on multiple expressions
2922 and get consistent results.
2929 ex a = pow(sin(x), 2) - pow(cos(x), 2);
2930 ex b = sin(x) + cos(x);
2933 divide(a.to_rational(l), b.to_rational(l), q);
2934 cout << q.subs(l) << endl;
2938 will print @samp{sin(x)-cos(x)}.
2941 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
2942 @c node-name, next, previous, up
2943 @section Symbolic differentiation
2944 @cindex differentiation
2945 @cindex @code{diff()}
2947 @cindex product rule
2949 GiNaC's objects know how to differentiate themselves. Thus, a
2950 polynomial (class @code{add}) knows that its derivative is the sum of
2951 the derivatives of all the monomials:
2954 #include <ginac/ginac.h>
2955 using namespace GiNaC;
2959 symbol x("x"), y("y"), z("z");
2960 ex P = pow(x, 5) + pow(x, 2) + y;
2962 cout << P.diff(x,2) << endl; // 20*x^3 + 2
2963 cout << P.diff(y) << endl; // 1
2964 cout << P.diff(z) << endl; // 0
2968 If a second integer parameter @var{n} is given, the @code{diff} method
2969 returns the @var{n}th derivative.
2971 If @emph{every} object and every function is told what its derivative
2972 is, all derivatives of composed objects can be calculated using the
2973 chain rule and the product rule. Consider, for instance the expression
2974 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
2975 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
2976 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
2977 out that the composition is the generating function for Euler Numbers,
2978 i.e. the so called @var{n}th Euler number is the coefficient of
2979 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
2980 identity to code a function that generates Euler numbers in just three
2983 @cindex Euler numbers
2985 #include <ginac/ginac.h>
2986 using namespace GiNaC;
2988 ex EulerNumber(unsigned n)
2991 const ex generator = pow(cosh(x),-1);
2992 return generator.diff(x,n).subs(x==0);
2997 for (unsigned i=0; i<11; i+=2)
2998 std::cout << EulerNumber(i) << std::endl;
3003 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
3004 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
3005 @code{i} by two since all odd Euler numbers vanish anyways.
3008 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
3009 @c node-name, next, previous, up
3010 @section Series expansion
3011 @cindex @code{series()}
3012 @cindex Taylor expansion
3013 @cindex Laurent expansion
3014 @cindex @code{pseries} (class)
3016 Expressions know how to expand themselves as a Taylor series or (more
3017 generally) a Laurent series. As in most conventional Computer Algebra
3018 Systems, no distinction is made between those two. There is a class of
3019 its own for storing such series (@code{class pseries}) and a built-in
3020 function (called @code{Order}) for storing the order term of the series.
3021 As a consequence, if you want to work with series, i.e. multiply two
3022 series, you need to call the method @code{ex::series} again to convert
3023 it to a series object with the usual structure (expansion plus order
3024 term). A sample application from special relativity could read:
3027 #include <ginac/ginac.h>
3028 using namespace std;
3029 using namespace GiNaC;
3033 symbol v("v"), c("c");
3035 ex gamma = 1/sqrt(1 - pow(v/c,2));
3036 ex mass_nonrel = gamma.series(v==0, 10);
3038 cout << "the relativistic mass increase with v is " << endl
3039 << mass_nonrel << endl;
3041 cout << "the inverse square of this series is " << endl
3042 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
3046 Only calling the series method makes the last output simplify to
3047 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
3048 series raised to the power @math{-2}.
3050 @cindex M@'echain's formula
3051 As another instructive application, let us calculate the numerical
3052 value of Archimedes' constant
3056 (for which there already exists the built-in constant @code{Pi})
3057 using M@'echain's amazing formula
3059 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
3062 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
3064 We may expand the arcus tangent around @code{0} and insert the fractions
3065 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
3066 carries an order term with it and the question arises what the system is
3067 supposed to do when the fractions are plugged into that order term. The
3068 solution is to use the function @code{series_to_poly()} to simply strip
3072 #include <ginac/ginac.h>
3073 using namespace GiNaC;
3075 ex mechain_pi(int degr)
3078 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
3079 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
3080 -4*pi_expansion.subs(x==numeric(1,239));
3086 using std::cout; // just for fun, another way of...
3087 using std::endl; // ...dealing with this namespace std.
3089 for (int i=2; i<12; i+=2) @{
3090 pi_frac = mechain_pi(i);
3091 cout << i << ":\t" << pi_frac << endl
3092 << "\t" << pi_frac.evalf() << endl;
3098 Note how we just called @code{.series(x,degr)} instead of
3099 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
3100 method @code{series()}: if the first argument is a symbol the expression
3101 is expanded in that symbol around point @code{0}. When you run this
3102 program, it will type out:
3106 3.1832635983263598326
3107 4: 5359397032/1706489875
3108 3.1405970293260603143
3109 6: 38279241713339684/12184551018734375
3110 3.141621029325034425
3111 8: 76528487109180192540976/24359780855939418203125
3112 3.141591772182177295
3113 10: 327853873402258685803048818236/104359128170408663038552734375
3114 3.1415926824043995174
3118 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
3119 @c node-name, next, previous, up
3120 @section Predefined mathematical functions
3122 GiNaC contains the following predefined mathematical functions:
3125 @multitable @columnfractions .30 .70
3126 @item @strong{Name} @tab @strong{Function}
3129 @item @code{csgn(x)}
3131 @item @code{sqrt(x)}
3132 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
3139 @item @code{asin(x)}
3141 @item @code{acos(x)}
3143 @item @code{atan(x)}
3144 @tab inverse tangent
3145 @item @code{atan2(y, x)}
3146 @tab inverse tangent with two arguments
3147 @item @code{sinh(x)}
3148 @tab hyperbolic sine
3149 @item @code{cosh(x)}
3150 @tab hyperbolic cosine
3151 @item @code{tanh(x)}
3152 @tab hyperbolic tangent
3153 @item @code{asinh(x)}
3154 @tab inverse hyperbolic sine
3155 @item @code{acosh(x)}
3156 @tab inverse hyperbolic cosine
3157 @item @code{atanh(x)}
3158 @tab inverse hyperbolic tangent
3160 @tab exponential function
3162 @tab natural logarithm
3165 @item @code{zeta(x)}
3166 @tab Riemann's zeta function
3167 @item @code{zeta(n, x)}
3168 @tab derivatives of Riemann's zeta function
3169 @item @code{tgamma(x)}
3171 @item @code{lgamma(x)}
3172 @tab logarithm of Gamma function
3173 @item @code{beta(x, y)}
3174 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
3176 @tab psi (digamma) function
3177 @item @code{psi(n, x)}
3178 @tab derivatives of psi function (polygamma functions)
3179 @item @code{factorial(n)}
3180 @tab factorial function
3181 @item @code{binomial(n, m)}
3182 @tab binomial coefficients
3183 @item @code{Order(x)}
3184 @tab order term function in truncated power series
3185 @item @code{Derivative(x, l)}
3186 @tab inert partial differentiation operator (used internally)
3191 For functions that have a branch cut in the complex plane GiNaC follows
3192 the conventions for C++ as defined in the ANSI standard as far as
3193 possible. In particular: the natural logarithm (@code{log}) and the
3194 square root (@code{sqrt}) both have their branch cuts running along the
3195 negative real axis where the points on the axis itself belong to the
3196 upper part (i.e. continuous with quadrant II). The inverse
3197 trigonometric and hyperbolic functions are not defined for complex
3198 arguments by the C++ standard, however. In GiNaC we follow the
3199 conventions used by CLN, which in turn follow the carefully designed
3200 definitions in the Common Lisp standard. It should be noted that this
3201 convention is identical to the one used by the C99 standard and by most
3202 serious CAS. It is to be expected that future revisions of the C++
3203 standard incorporate these functions in the complex domain in a manner
3204 compatible with C99.
3207 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
3208 @c node-name, next, previous, up
3209 @section Input and output of expressions
3212 @subsection Expression output
3214 @cindex output of expressions
3216 The easiest way to print an expression is to write it to a stream:
3221 ex e = 4.5+pow(x,2)*3/2;
3222 cout << e << endl; // prints '(4.5)+3/2*x^2'
3226 The output format is identical to the @command{ginsh} input syntax and
3227 to that used by most computer algebra systems, but not directly pastable
3228 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
3229 is printed as @samp{x^2}).
3231 It is possible to print expressions in a number of different formats with
3235 void ex::print(const print_context & c, unsigned level = 0);
3238 @cindex @code{print_context} (class)
3239 The type of @code{print_context} object passed in determines the format
3240 of the output. The possible types are defined in @file{ginac/print.h}.
3241 All constructors of @code{print_context} and derived classes take an
3242 @code{ostream &} as their first argument.
3244 To print an expression in a way that can be directly used in a C or C++
3245 program, you pass a @code{print_csrc} object like this:
3249 cout << "float f = ";
3250 e.print(print_csrc_float(cout));
3253 cout << "double d = ";
3254 e.print(print_csrc_double(cout));
3257 cout << "cl_N n = ";
3258 e.print(print_csrc_cl_N(cout));
3263 The three possible types mostly affect the way in which floating point
3264 numbers are written.
3266 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
3269 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3270 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
3271 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
3274 The @code{print_context} type @code{print_tree} provides a dump of the
3275 internal structure of an expression for debugging purposes:
3279 e.print(print_tree(cout));
3286 add, hash=0x0, flags=0x3, nops=2
3287 power, hash=0x9, flags=0x3, nops=2
3288 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
3289 2 (numeric), hash=0x80000042, flags=0xf
3290 3/2 (numeric), hash=0x80000061, flags=0xf
3293 4.5L0 (numeric), hash=0x8000004b, flags=0xf
3297 This kind of output is also available in @command{ginsh} as the @code{print()}
3300 Another useful output format is for LaTeX parsing in mathematical mode.
3301 It is rather similar to the default @code{print_context} but provides
3302 some braces needed by LaTeX for delimiting boxes and also converts some
3303 common objects to conventional LaTeX names. It is possible to give symbols
3304 a special name for LaTeX output by supplying it as a second argument to
3305 the @code{symbol} constructor.
3307 For example, the code snippet
3312 ex foo = lgamma(x).series(x==0,3);
3313 foo.print(print_latex(std::cout));
3319 @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
3322 If you need any fancy special output format, e.g. for interfacing GiNaC
3323 with other algebra systems or for producing code for different
3324 programming languages, you can always traverse the expression tree yourself:
3327 static void my_print(const ex & e)
3329 if (is_ex_of_type(e, function))
3330 cout << ex_to_function(e).get_name();
3332 cout << e.bp->class_name();
3334 unsigned n = e.nops();
3336 for (unsigned i=0; i<n; i++) @{
3348 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
3356 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
3357 symbol(y))),numeric(-2)))
3360 If you need an output format that makes it possible to accurately
3361 reconstruct an expression by feeding the output to a suitable parser or
3362 object factory, you should consider storing the expression in an
3363 @code{archive} object and reading the object properties from there.
3364 See the section on archiving for more information.
3367 @subsection Expression input
3368 @cindex input of expressions
3370 GiNaC provides no way to directly read an expression from a stream because
3371 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
3372 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
3373 @code{y} you defined in your program and there is no way to specify the
3374 desired symbols to the @code{>>} stream input operator.
3376 Instead, GiNaC lets you construct an expression from a string, specifying the
3377 list of symbols to be used:
3381 symbol x("x"), y("y");
3382 ex e("2*x+sin(y)", lst(x, y));
3386 The input syntax is the same as that used by @command{ginsh} and the stream
3387 output operator @code{<<}. The symbols in the string are matched by name to
3388 the symbols in the list and if GiNaC encounters a symbol not specified in
3389 the list it will throw an exception.
3391 With this constructor, it's also easy to implement interactive GiNaC programs:
3396 #include <stdexcept>
3397 #include <ginac/ginac.h>
3398 using namespace std;
3399 using namespace GiNaC;
3406 cout << "Enter an expression containing 'x': ";
3411 cout << "The derivative of " << e << " with respect to x is ";
3412 cout << e.diff(x) << ".\n";
3413 @} catch (exception &p) @{
3414 cerr << p.what() << endl;
3420 @subsection Archiving
3421 @cindex @code{archive} (class)
3424 GiNaC allows creating @dfn{archives} of expressions which can be stored
3425 to or retrieved from files. To create an archive, you declare an object
3426 of class @code{archive} and archive expressions in it, giving each
3427 expression a unique name:
3431 using namespace std;
3432 #include <ginac/ginac.h>
3433 using namespace GiNaC;
3437 symbol x("x"), y("y"), z("z");
3439 ex foo = sin(x + 2*y) + 3*z + 41;
3443 a.archive_ex(foo, "foo");
3444 a.archive_ex(bar, "the second one");
3448 The archive can then be written to a file:
3452 ofstream out("foobar.gar");
3458 The file @file{foobar.gar} contains all information that is needed to
3459 reconstruct the expressions @code{foo} and @code{bar}.
3461 @cindex @command{viewgar}
3462 The tool @command{viewgar} that comes with GiNaC can be used to view
3463 the contents of GiNaC archive files:
3466 $ viewgar foobar.gar
3467 foo = 41+sin(x+2*y)+3*z
3468 the second one = 42+sin(x+2*y)+3*z
3471 The point of writing archive files is of course that they can later be
3477 ifstream in("foobar.gar");
3482 And the stored expressions can be retrieved by their name:
3488 ex ex1 = a2.unarchive_ex(syms, "foo");
3489 ex ex2 = a2.unarchive_ex(syms, "the second one");
3491 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
3492 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
3493 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
3497 Note that you have to supply a list of the symbols which are to be inserted
3498 in the expressions. Symbols in archives are stored by their name only and
3499 if you don't specify which symbols you have, unarchiving the expression will
3500 create new symbols with that name. E.g. if you hadn't included @code{x} in
3501 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
3502 have had no effect because the @code{x} in @code{ex1} would have been a
3503 different symbol than the @code{x} which was defined at the beginning of
3504 the program, altough both would appear as @samp{x} when printed.
3506 You can also use the information stored in an @code{archive} object to
3507 output expressions in a format suitable for exact reconstruction. The
3508 @code{archive} and @code{archive_node} classes have a couple of member
3509 functions that let you access the stored properties:
3512 static void my_print2(const archive_node & n)
3515 n.find_string("class", class_name);
3516 cout << class_name << "(";
3518 archive_node::propinfovector p;
3519 n.get_properties(p);
3521 unsigned num = p.size();
3522 for (unsigned i=0; i<num; i++) @{
3523 const string &name = p[i].name;
3524 if (name == "class")
3526 cout << name << "=";
3528 unsigned count = p[i].count;
3532 for (unsigned j=0; j<count; j++) @{
3533 switch (p[i].type) @{
3534 case archive_node::PTYPE_BOOL: @{
3536 n.find_bool(name, x);
3537 cout << (x ? "true" : "false");
3540 case archive_node::PTYPE_UNSIGNED: @{
3542 n.find_unsigned(name, x);
3546 case archive_node::PTYPE_STRING: @{
3548 n.find_string(name, x);
3549 cout << '\"' << x << '\"';
3552 case archive_node::PTYPE_NODE: @{
3553 const archive_node &x = n.find_ex_node(name, j);
3575 ex e = pow(2, x) - y;
3577 my_print2(ar.get_top_node(0)); cout << endl;
3585 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
3586 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
3587 overall_coeff=numeric(number="0"))
3590 Be warned, however, that the set of properties and their meaning for each
3591 class may change between GiNaC versions.
3594 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
3595 @c node-name, next, previous, up
3596 @chapter Extending GiNaC
3598 By reading so far you should have gotten a fairly good understanding of
3599 GiNaC's design-patterns. From here on you should start reading the
3600 sources. All we can do now is issue some recommendations how to tackle
3601 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
3602 develop some useful extension please don't hesitate to contact the GiNaC
3603 authors---they will happily incorporate them into future versions.
3606 * What does not belong into GiNaC:: What to avoid.
3607 * Symbolic functions:: Implementing symbolic functions.
3608 * Adding classes:: Defining new algebraic classes.
3612 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
3613 @c node-name, next, previous, up
3614 @section What doesn't belong into GiNaC
3616 @cindex @command{ginsh}
3617 First of all, GiNaC's name must be read literally. It is designed to be
3618 a library for use within C++. The tiny @command{ginsh} accompanying
3619 GiNaC makes this even more clear: it doesn't even attempt to provide a
3620 language. There are no loops or conditional expressions in
3621 @command{ginsh}, it is merely a window into the library for the
3622 programmer to test stuff (or to show off). Still, the design of a
3623 complete CAS with a language of its own, graphical capabilites and all
3624 this on top of GiNaC is possible and is without doubt a nice project for
3627 There are many built-in functions in GiNaC that do not know how to
3628 evaluate themselves numerically to a precision declared at runtime
3629 (using @code{Digits}). Some may be evaluated at certain points, but not
3630 generally. This ought to be fixed. However, doing numerical
3631 computations with GiNaC's quite abstract classes is doomed to be
3632 inefficient. For this purpose, the underlying foundation classes
3633 provided by @acronym{CLN} are much better suited.
3636 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
3637 @c node-name, next, previous, up
3638 @section Symbolic functions
3640 The easiest and most instructive way to start with is probably to
3641 implement your own function. GiNaC's functions are objects of class
3642 @code{function}. The preprocessor is then used to convert the function
3643 names to objects with a corresponding serial number that is used
3644 internally to identify them. You usually need not worry about this
3645 number. New functions may be inserted into the system via a kind of
3646 `registry'. It is your responsibility to care for some functions that
3647 are called when the user invokes certain methods. These are usual
3648 C++-functions accepting a number of @code{ex} as arguments and returning
3649 one @code{ex}. As an example, if we have a look at a simplified
3650 implementation of the cosine trigonometric function, we first need a
3651 function that is called when one wishes to @code{eval} it. It could
3652 look something like this:
3655 static ex cos_eval_method(const ex & x)
3657 // if (!x%(2*Pi)) return 1
3658 // if (!x%Pi) return -1
3659 // if (!x%Pi/2) return 0
3660 // care for other cases...
3661 return cos(x).hold();
3665 @cindex @code{hold()}
3667 The last line returns @code{cos(x)} if we don't know what else to do and
3668 stops a potential recursive evaluation by saying @code{.hold()}, which
3669 sets a flag to the expression signaling that it has been evaluated. We
3670 should also implement a method for numerical evaluation and since we are
3671 lazy we sweep the problem under the rug by calling someone else's
3672 function that does so, in this case the one in class @code{numeric}:
3675 static ex cos_evalf(const ex & x)
3677 return cos(ex_to_numeric(x));
3681 Differentiation will surely turn up and so we need to tell @code{cos}
3682 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
3683 instance are then handled automatically by @code{basic::diff} and
3687 static ex cos_deriv(const ex & x, unsigned diff_param)
3693 @cindex product rule
3694 The second parameter is obligatory but uninteresting at this point. It
3695 specifies which parameter to differentiate in a partial derivative in
3696 case the function has more than one parameter and its main application
3697 is for correct handling of the chain rule. For Taylor expansion, it is
3698 enough to know how to differentiate. But if the function you want to
3699 implement does have a pole somewhere in the complex plane, you need to
3700 write another method for Laurent expansion around that point.
3702 Now that all the ingredients for @code{cos} have been set up, we need
3703 to tell the system about it. This is done by a macro and we are not
3704 going to descibe how it expands, please consult your preprocessor if you
3708 REGISTER_FUNCTION(cos, eval_func(cos_eval).
3709 evalf_func(cos_evalf).
3710 derivative_func(cos_deriv));
3713 The first argument is the function's name used for calling it and for
3714 output. The second binds the corresponding methods as options to this
3715 object. Options are separated by a dot and can be given in an arbitrary
3716 order. GiNaC functions understand several more options which are always
3717 specified as @code{.option(params)}, for example a method for series
3718 expansion @code{.series_func(cos_series)}. Again, if no series
3719 expansion method is given, GiNaC defaults to simple Taylor expansion,
3720 which is correct if there are no poles involved as is the case for the
3721 @code{cos} function. The way GiNaC handles poles in case there are any
3722 is best understood by studying one of the examples, like the Gamma
3723 (@code{tgamma}) function for instance. (In essence the function first
3724 checks if there is a pole at the evaluation point and falls back to
3725 Taylor expansion if there isn't. Then, the pole is regularized by some
3726 suitable transformation.) Also, the new function needs to be declared
3727 somewhere. This may also be done by a convenient preprocessor macro:
3730 DECLARE_FUNCTION_1P(cos)
3733 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
3734 implementation of @code{cos} is very incomplete and lacks several safety
3735 mechanisms. Please, have a look at the real implementation in GiNaC.
3736 (By the way: in case you are worrying about all the macros above we can
3737 assure you that functions are GiNaC's most macro-intense classes. We
3738 have done our best to avoid macros where we can.)
3741 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
3742 @c node-name, next, previous, up
3743 @section Adding classes
3745 If you are doing some very specialized things with GiNaC you may find that
3746 you have to implement your own algebraic classes to fit your needs. This
3747 section will explain how to do this by giving the example of a simple
3748 'string' class. After reading this section you will know how to properly
3749 declare a GiNaC class and what the minimum required member functions are
3750 that you have to implement. We only cover the implementation of a 'leaf'
3751 class here (i.e. one that doesn't contain subexpressions). Creating a
3752 container class like, for example, a class representing tensor products is
3753 more involved but this section should give you enough information so you can
3754 consult the source to GiNaC's predefined classes if you want to implement
3755 something more complicated.
3757 @subsection GiNaC's run-time type information system
3759 @cindex hierarchy of classes
3761 All algebraic classes (that is, all classes that can appear in expressions)
3762 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
3763 @code{basic *} (which is essentially what an @code{ex} is) represents a
3764 generic pointer to an algebraic class. Occasionally it is necessary to find
3765 out what the class of an object pointed to by a @code{basic *} really is.
3766 Also, for the unarchiving of expressions it must be possible to find the
3767 @code{unarchive()} function of a class given the class name (as a string). A
3768 system that provides this kind of information is called a run-time type
3769 information (RTTI) system. The C++ language provides such a thing (see the
3770 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
3771 implements its own, simpler RTTI.
3773 The RTTI in GiNaC is based on two mechanisms:
3778 The @code{basic} class declares a member variable @code{tinfo_key} which
3779 holds an unsigned integer that identifies the object's class. These numbers
3780 are defined in the @file{tinfos.h} header file for the built-in GiNaC
3781 classes. They all start with @code{TINFO_}.
3784 By means of some clever tricks with static members, GiNaC maintains a list
3785 of information for all classes derived from @code{basic}. The information
3786 available includes the class names, the @code{tinfo_key}s, and pointers
3787 to the unarchiving functions. This class registry is defined in the
3788 @file{registrar.h} header file.
3792 The disadvantage of this proprietary RTTI implementation is that there's
3793 a little more to do when implementing new classes (C++'s RTTI works more
3794 or less automatic) but don't worry, most of the work is simplified by
3797 @subsection A minimalistic example
3799 Now we will start implementing a new class @code{mystring} that allows
3800 placing character strings in algebraic expressions (this is not very useful,
3801 but it's just an example). This class will be a direct subclass of
3802 @code{basic}. You can use this sample implementation as a starting point
3803 for your own classes.
3805 The code snippets given here assume that you have included some header files
3811 #include <stdexcept>
3812 using namespace std;
3814 #include <ginac/ginac.h>
3815 using namespace GiNaC;
3818 The first thing we have to do is to define a @code{tinfo_key} for our new
3819 class. This can be any arbitrary unsigned number that is not already taken
3820 by one of the existing classes but it's better to come up with something
3821 that is unlikely to clash with keys that might be added in the future. The
3822 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
3823 which is not a requirement but we are going to stick with this scheme:
3826 const unsigned TINFO_mystring = 0x42420001U;
3829 Now we can write down the class declaration. The class stores a C++
3830 @code{string} and the user shall be able to construct a @code{mystring}
3831 object from a C or C++ string:
3834 class mystring : public basic
3836 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
3839 mystring(const string &s);
3840 mystring(const char *s);
3846 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
3849 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
3850 macros are defined in @file{registrar.h}. They take the name of the class
3851 and its direct superclass as arguments and insert all required declarations
3852 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
3853 the first line after the opening brace of the class definition. The
3854 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
3855 source (at global scope, of course, not inside a function).
3857 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
3858 declarations of the default and copy constructor, the destructor, the
3859 assignment operator and a couple of other functions that are required. It
3860 also defines a type @code{inherited} which refers to the superclass so you
3861 don't have to modify your code every time you shuffle around the class
3862 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
3863 constructor, the destructor and the assignment operator.
3865 Now there are nine member functions we have to implement to get a working
3871 @code{mystring()}, the default constructor.
3874 @code{void destroy(bool call_parent)}, which is used in the destructor and the
3875 assignment operator to free dynamically allocated members. The @code{call_parent}
3876 specifies whether the @code{destroy()} function of the superclass is to be
3880 @code{void copy(const mystring &other)}, which is used in the copy constructor
3881 and assignment operator to copy the member variables over from another
3882 object of the same class.
3885 @code{void archive(archive_node &n)}, the archiving function. This stores all
3886 information needed to reconstruct an object of this class inside an
3887 @code{archive_node}.
3890 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
3891 constructor. This constructs an instance of the class from the information
3892 found in an @code{archive_node}.
3895 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
3896 unarchiving function. It constructs a new instance by calling the unarchiving
3900 @code{int compare_same_type(const basic &other)}, which is used internally
3901 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
3902 -1, depending on the relative order of this object and the @code{other}
3903 object. If it returns 0, the objects are considered equal.
3904 @strong{Note:} This has nothing to do with the (numeric) ordering
3905 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
3906 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
3907 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
3908 must provide a @code{compare_same_type()} function, even those representing
3909 objects for which no reasonable algebraic ordering relationship can be
3913 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
3914 which are the two constructors we declared.
3918 Let's proceed step-by-step. The default constructor looks like this:
3921 mystring::mystring() : inherited(TINFO_mystring)
3923 // dynamically allocate resources here if required
3927 The golden rule is that in all constructors you have to set the
3928 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
3929 it will be set by the constructor of the superclass and all hell will break
3930 loose in the RTTI. For your convenience, the @code{basic} class provides
3931 a constructor that takes a @code{tinfo_key} value, which we are using here
3932 (remember that in our case @code{inherited = basic}). If the superclass
3933 didn't have such a constructor, we would have to set the @code{tinfo_key}
3934 to the right value manually.
3936 In the default constructor you should set all other member variables to
3937 reasonable default values (we don't need that here since our @code{str}
3938 member gets set to an empty string automatically). The constructor(s) are of
3939 course also the right place to allocate any dynamic resources you require.
3941 Next, the @code{destroy()} function:
3944 void mystring::destroy(bool call_parent)
3946 // free dynamically allocated resources here if required
3948 inherited::destroy(call_parent);
3952 This function is where we free all dynamically allocated resources. We don't
3953 have any so we're not doing anything here, but if we had, for example, used
3954 a C-style @code{char *} to store our string, this would be the place to
3955 @code{delete[]} the string storage. If @code{call_parent} is true, we have
3956 to call the @code{destroy()} function of the superclass after we're done
3957 (to mimic C++'s automatic invocation of superclass destructors where
3958 @code{destroy()} is called from outside a destructor).
3960 The @code{copy()} function just copies over the member variables from
3964 void mystring::copy(const mystring &other)
3966 inherited::copy(other);
3971 We can simply overwrite the member variables here. There's no need to worry
3972 about dynamically allocated storage. The assignment operator (which is
3973 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
3974 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
3975 explicitly call the @code{copy()} function of the superclass here so
3976 all the member variables will get copied.
3978 Next are the three functions for archiving. You have to implement them even
3979 if you don't plan to use archives, but the minimum required implementation
3980 is really simple. First, the archiving function:
3983 void mystring::archive(archive_node &n) const
3985 inherited::archive(n);
3986 n.add_string("string", str);
3990 The only thing that is really required is calling the @code{archive()}
3991 function of the superclass. Optionally, you can store all information you
3992 deem necessary for representing the object into the passed
3993 @code{archive_node}. We are just storing our string here. For more
3994 information on how the archiving works, consult the @file{archive.h} header
3997 The unarchiving constructor is basically the inverse of the archiving
4001 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
4003 n.find_string("string", str);
4007 If you don't need archiving, just leave this function empty (but you must
4008 invoke the unarchiving constructor of the superclass). Note that we don't
4009 have to set the @code{tinfo_key} here because it is done automatically
4010 by the unarchiving constructor of the @code{basic} class.
4012 Finally, the unarchiving function:
4015 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
4017 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
4021 You don't have to understand how exactly this works. Just copy these four
4022 lines into your code literally (replacing the class name, of course). It
4023 calls the unarchiving constructor of the class and unless you are doing
4024 something very special (like matching @code{archive_node}s to global
4025 objects) you don't need a different implementation. For those who are
4026 interested: setting the @code{dynallocated} flag puts the object under
4027 the control of GiNaC's garbage collection. It will get deleted automatically
4028 once it is no longer referenced.
4030 Our @code{compare_same_type()} function uses a provided function to compare
4034 int mystring::compare_same_type(const basic &other) const
4036 const mystring &o = static_cast<const mystring &>(other);
4037 int cmpval = str.compare(o.str);
4040 else if (cmpval < 0)
4047 Although this function takes a @code{basic &}, it will always be a reference
4048 to an object of exactly the same class (objects of different classes are not
4049 comparable), so the cast is safe. If this function returns 0, the two objects
4050 are considered equal (in the sense that @math{A-B=0}), so you should compare
4051 all relevant member variables.
4053 Now the only thing missing is our two new constructors:
4056 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
4058 // dynamically allocate resources here if required
4061 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
4063 // dynamically allocate resources here if required
4067 No surprises here. We set the @code{str} member from the argument and
4068 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
4070 That's it! We now have a minimal working GiNaC class that can store
4071 strings in algebraic expressions. Let's confirm that the RTTI works:
4074 ex e = mystring("Hello, world!");
4075 cout << is_ex_of_type(e, mystring) << endl;
4078 cout << e.bp->class_name() << endl;
4082 Obviously it does. Let's see what the expression @code{e} looks like:
4086 // -> [mystring object]
4089 Hm, not exactly what we expect, but of course the @code{mystring} class
4090 doesn't yet know how to print itself. This is done in the @code{print()}
4091 member function. Let's say that we wanted to print the string surrounded
4095 class mystring : public basic
4099 void print(const print_context &c, unsigned level = 0) const;
4103 void mystring::print(const print_context &c, unsigned level) const
4105 // print_context::s is a reference to an ostream
4106 c.s << '\"' << str << '\"';
4110 The @code{level} argument is only required for container classes to
4111 correctly parenthesize the output. Let's try again to print the expression:
4115 // -> "Hello, world!"
4118 Much better. The @code{mystring} class can be used in arbitrary expressions:
4121 e += mystring("GiNaC rulez");
4123 // -> "GiNaC rulez"+"Hello, world!"
4126 (note that GiNaC's automatic term reordering is in effect here), or even
4129 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
4131 // -> "One string"^(2*sin(-"Another string"+Pi))
4134 Whether this makes sense is debatable but remember that this is only an
4135 example. At least it allows you to implement your own symbolic algorithms
4138 Note that GiNaC's algebraic rules remain unchanged:
4141 e = mystring("Wow") * mystring("Wow");
4145 e = pow(mystring("First")-mystring("Second"), 2);
4146 cout << e.expand() << endl;
4147 // -> -2*"First"*"Second"+"First"^2+"Second"^2
4150 There's no way to, for example, make GiNaC's @code{add} class perform string
4151 concatenation. You would have to implement this yourself.
4153 @subsection Automatic evaluation
4155 @cindex @code{hold()}
4157 When dealing with objects that are just a little more complicated than the
4158 simple string objects we have implemented, chances are that you will want to
4159 have some automatic simplifications or canonicalizations performed on them.
4160 This is done in the evaluation member function @code{eval()}. Let's say that
4161 we wanted all strings automatically converted to lowercase with
4162 non-alphabetic characters stripped, and empty strings removed:
4165 class mystring : public basic
4169 ex eval(int level = 0) const;
4173 ex mystring::eval(int level) const
4176 for (int i=0; i<str.length(); i++) @{
4178 if (c >= 'A' && c <= 'Z')
4179 new_str += tolower(c);
4180 else if (c >= 'a' && c <= 'z')
4184 if (new_str.length() == 0)
4187 return mystring(new_str).hold();
4191 The @code{level} argument is used to limit the recursion depth of the
4192 evaluation. We don't have any subexpressions in the @code{mystring} class
4193 so we are not concerned with this. If we had, we would call the @code{eval()}
4194 functions of the subexpressions with @code{level - 1} as the argument if
4195 @code{level != 1}. The @code{hold()} member function sets a flag in the
4196 object that prevents further evaluation. Otherwise we might end up in an
4197 endless loop. When you want to return the object unmodified, use
4198 @code{return this->hold();}.
4200 Let's confirm that it works:
4203 ex e = mystring("Hello, world!") + mystring("!?#");
4207 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
4212 @subsection Other member functions
4214 We have implemented only a small set of member functions to make the class
4215 work in the GiNaC framework. For a real algebraic class, there are probably
4216 some more functions that you will want to re-implement, such as
4217 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
4218 or the header file of the class you want to make a subclass of to see
4219 what's there. You can, of course, also add your own new member functions.
4220 In this case you will probably want to define a little helper function like
4223 inline const mystring &ex_to_mystring(const ex &e)
4225 return static_cast<const mystring &>(*e.bp);
4229 that let's you get at the object inside an expression (after you have verified
4230 that the type is correct) so you can call member functions that are specific
4233 That's it. May the source be with you!
4236 @node A Comparison With Other CAS, Advantages, Adding classes, Top
4237 @c node-name, next, previous, up
4238 @chapter A Comparison With Other CAS
4241 This chapter will give you some information on how GiNaC compares to
4242 other, traditional Computer Algebra Systems, like @emph{Maple},
4243 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
4244 disadvantages over these systems.
4247 * Advantages:: Stengths of the GiNaC approach.
4248 * Disadvantages:: Weaknesses of the GiNaC approach.
4249 * Why C++?:: Attractiveness of C++.
4252 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
4253 @c node-name, next, previous, up
4256 GiNaC has several advantages over traditional Computer
4257 Algebra Systems, like
4262 familiar language: all common CAS implement their own proprietary
4263 grammar which you have to learn first (and maybe learn again when your
4264 vendor decides to `enhance' it). With GiNaC you can write your program
4265 in common C++, which is standardized.
4269 structured data types: you can build up structured data types using
4270 @code{struct}s or @code{class}es together with STL features instead of
4271 using unnamed lists of lists of lists.
4274 strongly typed: in CAS, you usually have only one kind of variables
4275 which can hold contents of an arbitrary type. This 4GL like feature is
4276 nice for novice programmers, but dangerous.
4279 development tools: powerful development tools exist for C++, like fancy
4280 editors (e.g. with automatic indentation and syntax highlighting),
4281 debuggers, visualization tools, documentation generators...
4284 modularization: C++ programs can easily be split into modules by
4285 separating interface and implementation.
4288 price: GiNaC is distributed under the GNU Public License which means
4289 that it is free and available with source code. And there are excellent
4290 C++-compilers for free, too.
4293 extendable: you can add your own classes to GiNaC, thus extending it on
4294 a very low level. Compare this to a traditional CAS that you can
4295 usually only extend on a high level by writing in the language defined
4296 by the parser. In particular, it turns out to be almost impossible to
4297 fix bugs in a traditional system.
4300 multiple interfaces: Though real GiNaC programs have to be written in
4301 some editor, then be compiled, linked and executed, there are more ways
4302 to work with the GiNaC engine. Many people want to play with
4303 expressions interactively, as in traditional CASs. Currently, two such
4304 windows into GiNaC have been implemented and many more are possible: the
4305 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
4306 types to a command line and second, as a more consistent approach, an
4307 interactive interface to the @acronym{Cint} C++ interpreter has been put
4308 together (called @acronym{GiNaC-cint}) that allows an interactive
4309 scripting interface consistent with the C++ language.
4312 seemless integration: it is somewhere between difficult and impossible
4313 to call CAS functions from within a program written in C++ or any other
4314 programming language and vice versa. With GiNaC, your symbolic routines
4315 are part of your program. You can easily call third party libraries,
4316 e.g. for numerical evaluation or graphical interaction. All other
4317 approaches are much more cumbersome: they range from simply ignoring the
4318 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
4319 system (i.e. @emph{Yacas}).
4322 efficiency: often large parts of a program do not need symbolic
4323 calculations at all. Why use large integers for loop variables or
4324 arbitrary precision arithmetics where @code{int} and @code{double} are
4325 sufficient? For pure symbolic applications, GiNaC is comparable in
4326 speed with other CAS.
4331 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
4332 @c node-name, next, previous, up
4333 @section Disadvantages
4335 Of course it also has some disadvantages:
4340 advanced features: GiNaC cannot compete with a program like
4341 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
4342 which grows since 1981 by the work of dozens of programmers, with
4343 respect to mathematical features. Integration, factorization,
4344 non-trivial simplifications, limits etc. are missing in GiNaC (and are
4345 not planned for the near future).
4348 portability: While the GiNaC library itself is designed to avoid any
4349 platform dependent features (it should compile on any ANSI compliant C++
4350 compiler), the currently used version of the CLN library (fast large
4351 integer and arbitrary precision arithmetics) can be compiled only on
4352 systems with a recently new C++ compiler from the GNU Compiler
4353 Collection (@acronym{GCC}).@footnote{This is because CLN uses
4354 PROVIDE/REQUIRE like macros to let the compiler gather all static
4355 initializations, which works for GNU C++ only.} GiNaC uses recent
4356 language features like explicit constructors, mutable members, RTTI,
4357 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
4358 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
4359 ANSI compliant, support all needed features.
4364 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
4365 @c node-name, next, previous, up
4368 Why did we choose to implement GiNaC in C++ instead of Java or any other
4369 language? C++ is not perfect: type checking is not strict (casting is
4370 possible), separation between interface and implementation is not
4371 complete, object oriented design is not enforced. The main reason is
4372 the often scolded feature of operator overloading in C++. While it may
4373 be true that operating on classes with a @code{+} operator is rarely
4374 meaningful, it is perfectly suited for algebraic expressions. Writing
4375 @math{3x+5y} as @code{3*x+5*y} instead of
4376 @code{x.times(3).plus(y.times(5))} looks much more natural.
4377 Furthermore, the main developers are more familiar with C++ than with
4378 any other programming language.
4381 @node Internal Structures, Expressions are reference counted, Why C++? , Top
4382 @c node-name, next, previous, up
4383 @appendix Internal Structures
4386 * Expressions are reference counted::
4387 * Internal representation of products and sums::
4390 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
4391 @c node-name, next, previous, up
4392 @appendixsection Expressions are reference counted
4394 @cindex reference counting
4395 @cindex copy-on-write
4396 @cindex garbage collection
4397 An expression is extremely light-weight since internally it works like a
4398 handle to the actual representation and really holds nothing more than a
4399 pointer to some other object. What this means in practice is that
4400 whenever you create two @code{ex} and set the second equal to the first
4401 no copying process is involved. Instead, the copying takes place as soon
4402 as you try to change the second. Consider the simple sequence of code:
4405 #include <ginac/ginac.h>
4406 using namespace std;
4407 using namespace GiNaC;
4411 symbol x("x"), y("y"), z("z");
4414 e1 = sin(x + 2*y) + 3*z + 41;
4415 e2 = e1; // e2 points to same object as e1
4416 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
4417 e2 += 1; // e2 is copied into a new object
4418 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
4422 The line @code{e2 = e1;} creates a second expression pointing to the
4423 object held already by @code{e1}. The time involved for this operation
4424 is therefore constant, no matter how large @code{e1} was. Actual
4425 copying, however, must take place in the line @code{e2 += 1;} because
4426 @code{e1} and @code{e2} are not handles for the same object any more.
4427 This concept is called @dfn{copy-on-write semantics}. It increases
4428 performance considerably whenever one object occurs multiple times and
4429 represents a simple garbage collection scheme because when an @code{ex}
4430 runs out of scope its destructor checks whether other expressions handle
4431 the object it points to too and deletes the object from memory if that
4432 turns out not to be the case. A slightly less trivial example of
4433 differentiation using the chain-rule should make clear how powerful this
4437 #include <ginac/ginac.h>
4438 using namespace std;
4439 using namespace GiNaC;
4443 symbol x("x"), y("y");
4447 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
4448 cout << e1 << endl // prints x+3*y
4449 << e2 << endl // prints (x+3*y)^3
4450 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
4454 Here, @code{e1} will actually be referenced three times while @code{e2}
4455 will be referenced two times. When the power of an expression is built,
4456 that expression needs not be copied. Likewise, since the derivative of
4457 a power of an expression can be easily expressed in terms of that
4458 expression, no copying of @code{e1} is involved when @code{e3} is
4459 constructed. So, when @code{e3} is constructed it will print as
4460 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
4461 holds a reference to @code{e2} and the factor in front is just
4464 As a user of GiNaC, you cannot see this mechanism of copy-on-write
4465 semantics. When you insert an expression into a second expression, the
4466 result behaves exactly as if the contents of the first expression were
4467 inserted. But it may be useful to remember that this is not what
4468 happens. Knowing this will enable you to write much more efficient
4469 code. If you still have an uncertain feeling with copy-on-write
4470 semantics, we recommend you have a look at the
4471 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
4472 Marshall Cline. Chapter 16 covers this issue and presents an
4473 implementation which is pretty close to the one in GiNaC.
4476 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
4477 @c node-name, next, previous, up
4478 @appendixsection Internal representation of products and sums
4480 @cindex representation
4483 @cindex @code{power}
4484 Although it should be completely transparent for the user of
4485 GiNaC a short discussion of this topic helps to understand the sources
4486 and also explain performance to a large degree. Consider the
4487 unexpanded symbolic expression
4489 $2d^3 \left( 4a + 5b - 3 \right)$
4492 @math{2*d^3*(4*a+5*b-3)}
4494 which could naively be represented by a tree of linear containers for
4495 addition and multiplication, one container for exponentiation with base
4496 and exponent and some atomic leaves of symbols and numbers in this
4501 @cindex pair-wise representation
4502 However, doing so results in a rather deeply nested tree which will
4503 quickly become inefficient to manipulate. We can improve on this by
4504 representing the sum as a sequence of terms, each one being a pair of a
4505 purely numeric multiplicative coefficient and its rest. In the same
4506 spirit we can store the multiplication as a sequence of terms, each
4507 having a numeric exponent and a possibly complicated base, the tree
4508 becomes much more flat:
4512 The number @code{3} above the symbol @code{d} shows that @code{mul}
4513 objects are treated similarly where the coefficients are interpreted as
4514 @emph{exponents} now. Addition of sums of terms or multiplication of
4515 products with numerical exponents can be coded to be very efficient with
4516 such a pair-wise representation. Internally, this handling is performed
4517 by most CAS in this way. It typically speeds up manipulations by an
4518 order of magnitude. The overall multiplicative factor @code{2} and the
4519 additive term @code{-3} look somewhat out of place in this
4520 representation, however, since they are still carrying a trivial
4521 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
4522 this is avoided by adding a field that carries an overall numeric
4523 coefficient. This results in the realistic picture of internal
4526 $2d^3 \left( 4a + 5b - 3 \right)$:
4529 @math{2*d^3*(4*a+5*b-3)}:
4535 This also allows for a better handling of numeric radicals, since
4536 @code{sqrt(2)} can now be carried along calculations. Now it should be
4537 clear, why both classes @code{add} and @code{mul} are derived from the
4538 same abstract class: the data representation is the same, only the
4539 semantics differs. In the class hierarchy, methods for polynomial
4540 expansion and the like are reimplemented for @code{add} and @code{mul},
4541 but the data structure is inherited from @code{expairseq}.
4544 @node Package Tools, ginac-config, Internal representation of products and sums, Top
4545 @c node-name, next, previous, up
4546 @appendix Package Tools
4548 If you are creating a software package that uses the GiNaC library,
4549 setting the correct command line options for the compiler and linker
4550 can be difficult. GiNaC includes two tools to make this process easier.
4553 * ginac-config:: A shell script to detect compiler and linker flags.
4554 * AM_PATH_GINAC:: Macro for GNU automake.
4558 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
4559 @c node-name, next, previous, up
4560 @section @command{ginac-config}
4561 @cindex ginac-config
4563 @command{ginac-config} is a shell script that you can use to determine
4564 the compiler and linker command line options required to compile and
4565 link a program with the GiNaC library.
4567 @command{ginac-config} takes the following flags:
4571 Prints out the version of GiNaC installed.
4573 Prints '-I' flags pointing to the installed header files.
4575 Prints out the linker flags necessary to link a program against GiNaC.
4576 @item --prefix[=@var{PREFIX}]
4577 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
4578 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
4579 Otherwise, prints out the configured value of @env{$prefix}.
4580 @item --exec-prefix[=@var{PREFIX}]
4581 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
4582 Otherwise, prints out the configured value of @env{$exec_prefix}.
4585 Typically, @command{ginac-config} will be used within a configure
4586 script, as described below. It, however, can also be used directly from
4587 the command line using backquotes to compile a simple program. For
4591 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
4594 This command line might expand to (for example):
4597 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
4598 -lginac -lcln -lstdc++
4601 Not only is the form using @command{ginac-config} easier to type, it will
4602 work on any system, no matter how GiNaC was configured.
4605 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
4606 @c node-name, next, previous, up
4607 @section @samp{AM_PATH_GINAC}
4608 @cindex AM_PATH_GINAC
4610 For packages configured using GNU automake, GiNaC also provides
4611 a macro to automate the process of checking for GiNaC.
4614 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
4622 Determines the location of GiNaC using @command{ginac-config}, which is
4623 either found in the user's path, or from the environment variable
4624 @env{GINACLIB_CONFIG}.
4627 Tests the installed libraries to make sure that their version
4628 is later than @var{MINIMUM-VERSION}. (A default version will be used
4632 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
4633 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
4634 variable to the output of @command{ginac-config --libs}, and calls
4635 @samp{AC_SUBST()} for these variables so they can be used in generated
4636 makefiles, and then executes @var{ACTION-IF-FOUND}.
4639 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
4640 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
4644 This macro is in file @file{ginac.m4} which is installed in
4645 @file{$datadir/aclocal}. Note that if automake was installed with a
4646 different @samp{--prefix} than GiNaC, you will either have to manually
4647 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
4648 aclocal the @samp{-I} option when running it.
4651 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
4652 * Example package:: Example of a package using AM_PATH_GINAC.
4656 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
4657 @c node-name, next, previous, up
4658 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
4660 Simply make sure that @command{ginac-config} is in your path, and run
4661 the configure script.
4668 The directory where the GiNaC libraries are installed needs
4669 to be found by your system's dynamic linker.
4671 This is generally done by
4674 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
4680 setting the environment variable @env{LD_LIBRARY_PATH},
4683 or, as a last resort,
4686 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
4687 running configure, for instance:
4690 LDFLAGS=-R/home/cbauer/lib ./configure
4695 You can also specify a @command{ginac-config} not in your path by
4696 setting the @env{GINACLIB_CONFIG} environment variable to the
4697 name of the executable
4700 If you move the GiNaC package from its installed location,
4701 you will either need to modify @command{ginac-config} script
4702 manually to point to the new location or rebuild GiNaC.
4713 --with-ginac-prefix=@var{PREFIX}
4714 --with-ginac-exec-prefix=@var{PREFIX}
4717 are provided to override the prefix and exec-prefix that were stored
4718 in the @command{ginac-config} shell script by GiNaC's configure. You are
4719 generally better off configuring GiNaC with the right path to begin with.
4723 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
4724 @c node-name, next, previous, up
4725 @subsection Example of a package using @samp{AM_PATH_GINAC}
4727 The following shows how to build a simple package using automake
4728 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
4731 #include <ginac/ginac.h>
4735 GiNaC::symbol x("x");
4736 GiNaC::ex a = GiNaC::sin(x);
4737 std::cout << "Derivative of " << a
4738 << " is " << a.diff(x) << std::endl;
4743 You should first read the introductory portions of the automake
4744 Manual, if you are not already familiar with it.
4746 Two files are needed, @file{configure.in}, which is used to build the
4750 dnl Process this file with autoconf to produce a configure script.
4752 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
4758 AM_PATH_GINAC(0.7.0, [
4759 LIBS="$LIBS $GINACLIB_LIBS"
4760 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
4761 ], AC_MSG_ERROR([need to have GiNaC installed]))
4766 The only command in this which is not standard for automake
4767 is the @samp{AM_PATH_GINAC} macro.
4769 That command does the following: If a GiNaC version greater or equal
4770 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
4771 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
4772 the error message `need to have GiNaC installed'
4774 And the @file{Makefile.am}, which will be used to build the Makefile.
4777 ## Process this file with automake to produce Makefile.in
4778 bin_PROGRAMS = simple
4779 simple_SOURCES = simple.cpp
4782 This @file{Makefile.am}, says that we are building a single executable,
4783 from a single sourcefile @file{simple.cpp}. Since every program
4784 we are building uses GiNaC we simply added the GiNaC options
4785 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
4786 want to specify them on a per-program basis: for instance by
4790 simple_LDADD = $(GINACLIB_LIBS)
4791 INCLUDES = $(GINACLIB_CPPFLAGS)
4794 to the @file{Makefile.am}.
4796 To try this example out, create a new directory and add the three
4799 Now execute the following commands:
4802 $ automake --add-missing
4807 You now have a package that can be built in the normal fashion
4816 @node Bibliography, Concept Index, Example package, Top
4817 @c node-name, next, previous, up
4818 @appendix Bibliography
4823 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
4826 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
4829 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
4832 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
4835 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
4836 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
4839 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
4840 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
4841 Academic Press, London
4844 @cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
4849 @node Concept Index, , Bibliography, Top
4850 @c node-name, next, previous, up
4851 @unnumbered Concept Index