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 -3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
370 > b = x^2 + 4*x*y - y^2;
373 3*y^6+x^6-24*x*y^5+43*x^2*y^4+16*x^3*y^3+17*x^4*y^2+8*x^5*y
375 3*y^6+48*x*y^4+2*x^2*y^2+x^4*(-y^2+x^2+4*x*y)+4*x^3*y*(-y^2+x^2+4*x*y)
380 You can differentiate functions and expand them as Taylor or Laurent
381 series in a very natural syntax (the second argument of @code{series} is
382 a relation defining the evaluation point, the third specifies the
385 @cindex Zeta function
389 > series(sin(x),x==0,4);
391 > series(1/tan(x),x==0,4);
392 x^(-1)-1/3*x+Order(x^2)
393 > series(tgamma(x),x==0,3);
394 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
395 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
397 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
398 -(0.90747907608088628905)*x^2+Order(x^3)
399 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
400 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
401 -Euler-1/12+Order((x-1/2*Pi)^3)
404 Here we have made use of the @command{ginsh}-command @code{"} to pop the
405 previously evaluated element from @command{ginsh}'s internal stack.
407 If you ever wanted to convert units in C or C++ and found this is
408 cumbersome, here is the solution. Symbolic types can always be used as
409 tags for different types of objects. Converting from wrong units to the
410 metric system is now easy:
418 140613.91592783185568*kg*m^(-2)
422 @node Installation, Prerequisites, What it can do for you, Top
423 @c node-name, next, previous, up
424 @chapter Installation
427 GiNaC's installation follows the spirit of most GNU software. It is
428 easily installed on your system by three steps: configuration, build,
432 * Prerequisites:: Packages upon which GiNaC depends.
433 * Configuration:: How to configure GiNaC.
434 * Building GiNaC:: How to compile GiNaC.
435 * Installing GiNaC:: How to install GiNaC on your system.
439 @node Prerequisites, Configuration, Installation, Installation
440 @c node-name, next, previous, up
441 @section Prerequisites
443 In order to install GiNaC on your system, some prerequisites need to be
444 met. First of all, you need to have a C++-compiler adhering to the
445 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
446 development so if you have a different compiler you are on your own.
447 For the configuration to succeed you need a Posix compliant shell
448 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
449 by the built process as well, since some of the source files are
450 automatically generated by Perl scripts. Last but not least, Bruno
451 Haible's library @acronym{CLN} is extensively used and needs to be
452 installed on your system. Please get it either from
453 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
454 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
455 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
456 site} (it is covered by GPL) and install it prior to trying to install
457 GiNaC. The configure script checks if it can find it and if it cannot
458 it will refuse to continue.
461 @node Configuration, Building GiNaC, Prerequisites, Installation
462 @c node-name, next, previous, up
463 @section Configuration
464 @cindex configuration
467 To configure GiNaC means to prepare the source distribution for
468 building. It is done via a shell script called @command{configure} that
469 is shipped with the sources and was originally generated by GNU
470 Autoconf. Since a configure script generated by GNU Autoconf never
471 prompts, all customization must be done either via command line
472 parameters or environment variables. It accepts a list of parameters,
473 the complete set of which can be listed by calling it with the
474 @option{--help} option. The most important ones will be shortly
475 described in what follows:
480 @option{--disable-shared}: When given, this option switches off the
481 build of a shared library, i.e. a @file{.so} file. This may be convenient
482 when developing because it considerably speeds up compilation.
485 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
486 and headers are installed. It defaults to @file{/usr/local} which means
487 that the library is installed in the directory @file{/usr/local/lib},
488 the header files in @file{/usr/local/include/ginac} and the documentation
489 (like this one) into @file{/usr/local/share/doc/GiNaC}.
492 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
493 the library installed in some other directory than
494 @file{@var{PREFIX}/lib/}.
497 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
498 to have the header files installed in some other directory than
499 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
500 @option{--includedir=/usr/include} you will end up with the header files
501 sitting in the directory @file{/usr/include/ginac/}. Note that the
502 subdirectory @file{ginac} is enforced by this process in order to
503 keep the header files separated from others. This avoids some
504 clashes and allows for an easier deinstallation of GiNaC. This ought
505 to be considered A Good Thing (tm).
508 @option{--datadir=@var{DATADIR}}: This option may be given in case you
509 want to have the documentation installed in some other directory than
510 @file{@var{PREFIX}/share/doc/GiNaC/}.
514 In addition, you may specify some environment variables.
515 @env{CXX} holds the path and the name of the C++ compiler
516 in case you want to override the default in your path. (The
517 @command{configure} script searches your path for @command{c++},
518 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
519 and @command{cc++} in that order.) It may be very useful to
520 define some compiler flags with the @env{CXXFLAGS} environment
521 variable, like optimization, debugging information and warning
522 levels. If omitted, it defaults to @option{-g -O2}.
524 The whole process is illustrated in the following two
525 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
526 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
529 Here is a simple configuration for a site-wide GiNaC library assuming
530 everything is in default paths:
533 $ export CXXFLAGS="-Wall -O2"
537 And here is a configuration for a private static GiNaC library with
538 several components sitting in custom places (site-wide @acronym{GCC} and
539 private @acronym{CLN}). The compiler is pursuaded to be picky and full
540 assertions and debugging information are switched on:
543 $ export CXX=/usr/local/gnu/bin/c++
544 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
545 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
546 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
547 $ ./configure --disable-shared --prefix=$(HOME)
551 @node Building GiNaC, Installing GiNaC, Configuration, Installation
552 @c node-name, next, previous, up
553 @section Building GiNaC
554 @cindex building GiNaC
556 After proper configuration you should just build the whole
561 at the command prompt and go for a cup of coffee. The exact time it
562 takes to compile GiNaC depends not only on the speed of your machines
563 but also on other parameters, for instance what value for @env{CXXFLAGS}
564 you entered. Optimization may be very time-consuming.
566 Just to make sure GiNaC works properly you may run a collection of
567 regression tests by typing
573 This will compile some sample programs, run them and check the output
574 for correctness. The regression tests fall in three categories. First,
575 the so called @emph{exams} are performed, simple tests where some
576 predefined input is evaluated (like a pupils' exam). Second, the
577 @emph{checks} test the coherence of results among each other with
578 possible random input. Third, some @emph{timings} are performed, which
579 benchmark some predefined problems with different sizes and display the
580 CPU time used in seconds. Each individual test should return a message
581 @samp{passed}. This is mostly intended to be a QA-check if something
582 was broken during development, not a sanity check of your system. Some
583 of the tests in sections @emph{checks} and @emph{timings} may require
584 insane amounts of memory and CPU time. Feel free to kill them if your
585 machine catches fire. Another quite important intent is to allow people
586 to fiddle around with optimization.
588 Generally, the top-level Makefile runs recursively to the
589 subdirectories. It is therfore safe to go into any subdirectory
590 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
591 @var{target} there in case something went wrong.
594 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
595 @c node-name, next, previous, up
596 @section Installing GiNaC
599 To install GiNaC on your system, simply type
605 As described in the section about configuration the files will be
606 installed in the following directories (the directories will be created
607 if they don't already exist):
612 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
613 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
614 So will @file{libginac.so} unless the configure script was
615 given the option @option{--disable-shared}. The proper symlinks
616 will be established as well.
619 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
620 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
623 All documentation (HTML and Postscript) will be stuffed into
624 @file{@var{PREFIX}/share/doc/GiNaC/} (or
625 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
629 For the sake of completeness we will list some other useful make
630 targets: @command{make clean} deletes all files generated by
631 @command{make}, i.e. all the object files. In addition @command{make
632 distclean} removes all files generated by the configuration and
633 @command{make maintainer-clean} goes one step further and deletes files
634 that may require special tools to rebuild (like the @command{libtool}
635 for instance). Finally @command{make uninstall} removes the installed
636 library, header files and documentation@footnote{Uninstallation does not
637 work after you have called @command{make distclean} since the
638 @file{Makefile} is itself generated by the configuration from
639 @file{Makefile.in} and hence deleted by @command{make distclean}. There
640 are two obvious ways out of this dilemma. First, you can run the
641 configuration again with the same @var{PREFIX} thus creating a
642 @file{Makefile} with a working @samp{uninstall} target. Second, you can
643 do it by hand since you now know where all the files went during
647 @node Basic Concepts, Expressions, Installing GiNaC, Top
648 @c node-name, next, previous, up
649 @chapter Basic Concepts
651 This chapter will describe the different fundamental objects that can be
652 handled by GiNaC. But before doing so, it is worthwhile introducing you
653 to the more commonly used class of expressions, representing a flexible
654 meta-class for storing all mathematical objects.
657 * Expressions:: The fundamental GiNaC class.
658 * The Class Hierarchy:: Overview of GiNaC's classes.
659 * Symbols:: Symbolic objects.
660 * Numbers:: Numerical objects.
661 * Constants:: Pre-defined constants.
662 * Fundamental containers:: The power, add and mul classes.
663 * Lists:: Lists of expressions.
664 * Mathematical functions:: Mathematical functions.
665 * Relations:: Equality, Inequality and all that.
666 * Indexed objects:: Handling indexed quantities.
670 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
671 @c node-name, next, previous, up
673 @cindex expression (class @code{ex})
676 The most common class of objects a user deals with is the expression
677 @code{ex}, representing a mathematical object like a variable, number,
678 function, sum, product, etc... Expressions may be put together to form
679 new expressions, passed as arguments to functions, and so on. Here is a
680 little collection of valid expressions:
683 ex MyEx1 = 5; // simple number
684 ex MyEx2 = x + 2*y; // polynomial in x and y
685 ex MyEx3 = (x + 1)/(x - 1); // rational expression
686 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
687 ex MyEx5 = MyEx4 + 1; // similar to above
690 Expressions are handles to other more fundamental objects, that often
691 contain other expressions thus creating a tree of expressions
692 (@xref{Internal Structures}, for particular examples). Most methods on
693 @code{ex} therefore run top-down through such an expression tree. For
694 example, the method @code{has()} scans recursively for occurrences of
695 something inside an expression. Thus, if you have declared @code{MyEx4}
696 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
697 the argument of @code{sin} and hence return @code{true}.
699 The next sections will outline the general picture of GiNaC's class
700 hierarchy and describe the classes of objects that are handled by
704 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
705 @c node-name, next, previous, up
706 @section The Class Hierarchy
708 GiNaC's class hierarchy consists of several classes representing
709 mathematical objects, all of which (except for @code{ex} and some
710 helpers) are internally derived from one abstract base class called
711 @code{basic}. You do not have to deal with objects of class
712 @code{basic}, instead you'll be dealing with symbols, numbers,
713 containers of expressions and so on.
717 To get an idea about what kinds of symbolic composits may be built we
718 have a look at the most important classes in the class hierarchy and
719 some of the relations among the classes:
721 @image{classhierarchy}
723 The abstract classes shown here (the ones without drop-shadow) are of no
724 interest for the user. They are used internally in order to avoid code
725 duplication if two or more classes derived from them share certain
726 features. An example is @code{expairseq}, a container for a sequence of
727 pairs each consisting of one expression and a number (@code{numeric}).
728 What @emph{is} visible to the user are the derived classes @code{add}
729 and @code{mul}, representing sums and products. @xref{Internal
730 Structures}, where these two classes are described in more detail. The
731 following table shortly summarizes what kinds of mathematical objects
732 are stored in the different classes:
735 @multitable @columnfractions .22 .78
736 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
737 @item @code{constant} @tab Constants like
744 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
745 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
746 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
747 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
752 @code{sqrt(}@math{2}@code{)}
755 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
756 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
757 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
758 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
759 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
760 @item @code{indexed} @tab Indexed object like @math{A_ij}
761 @item @code{tensor} @tab Special tensor like the delta and metric tensors
762 @item @code{idx} @tab Index of an indexed object
763 @item @code{varidx} @tab Index with variance
767 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
768 @c node-name, next, previous, up
770 @cindex @code{symbol} (class)
771 @cindex hierarchy of classes
774 Symbols are for symbolic manipulation what atoms are for chemistry. You
775 can declare objects of class @code{symbol} as any other object simply by
776 saying @code{symbol x,y;}. There is, however, a catch in here having to
777 do with the fact that C++ is a compiled language. The information about
778 the symbol's name is thrown away by the compiler but at a later stage
779 you may want to print expressions holding your symbols. In order to
780 avoid confusion GiNaC's symbols are able to know their own name. This
781 is accomplished by declaring its name for output at construction time in
782 the fashion @code{symbol x("x");}. If you declare a symbol using the
783 default constructor (i.e. without string argument) the system will deal
784 out a unique name. That name may not be suitable for printing but for
785 internal routines when no output is desired it is often enough. We'll
786 come across examples of such symbols later in this tutorial.
788 This implies that the strings passed to symbols at construction time may
789 not be used for comparing two of them. It is perfectly legitimate to
790 write @code{symbol x("x"),y("x");} but it is likely to lead into
791 trouble. Here, @code{x} and @code{y} are different symbols and
792 statements like @code{x-y} will not be simplified to zero although the
793 output @code{x-x} looks funny. Such output may also occur when there
794 are two different symbols in two scopes, for instance when you call a
795 function that declares a symbol with a name already existent in a symbol
796 in the calling function. Again, comparing them (using @code{operator==}
797 for instance) will always reveal their difference. Watch out, please.
799 @cindex @code{subs()}
800 Although symbols can be assigned expressions for internal reasons, you
801 should not do it (and we are not going to tell you how it is done). If
802 you want to replace a symbol with something else in an expression, you
803 can use the expression's @code{.subs()} method (@xref{Substituting Symbols},
804 for more information).
807 @node Numbers, Constants, Symbols, Basic Concepts
808 @c node-name, next, previous, up
810 @cindex @code{numeric} (class)
816 For storing numerical things, GiNaC uses Bruno Haible's library
817 @acronym{CLN}. The classes therein serve as foundation classes for
818 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
819 alternatively for Common Lisp Numbers. In order to find out more about
820 @acronym{CLN}'s internals the reader is refered to the documentation of
821 that library. @inforef{Introduction, , cln}, for more
822 information. Suffice to say that it is by itself build on top of another
823 library, the GNU Multiple Precision library @acronym{GMP}, which is an
824 extremely fast library for arbitrary long integers and rationals as well
825 as arbitrary precision floating point numbers. It is very commonly used
826 by several popular cryptographic applications. @acronym{CLN} extends
827 @acronym{GMP} by several useful things: First, it introduces the complex
828 number field over either reals (i.e. floating point numbers with
829 arbitrary precision) or rationals. Second, it automatically converts
830 rationals to integers if the denominator is unity and complex numbers to
831 real numbers if the imaginary part vanishes and also correctly treats
832 algebraic functions. Third it provides good implementations of
833 state-of-the-art algorithms for all trigonometric and hyperbolic
834 functions as well as for calculation of some useful constants.
836 The user can construct an object of class @code{numeric} in several
837 ways. The following example shows the four most important constructors.
838 It uses construction from C-integer, construction of fractions from two
839 integers, construction from C-float and construction from a string:
842 #include <ginac/ginac.h>
843 using namespace GiNaC;
847 numeric two(2); // exact integer 2
848 numeric r(2,3); // exact fraction 2/3
849 numeric e(2.71828); // floating point number
850 numeric p("3.1415926535897932385"); // floating point number
851 // Trott's constant in scientific notation:
852 numeric trott("1.0841015122311136151E-2");
854 std::cout << two*p << std::endl; // floating point 6.283...
858 Note that all those constructors are @emph{explicit} which means you are
859 not allowed to write @code{numeric two=2;}. This is because the basic
860 objects to be handled by GiNaC are the expressions @code{ex} and we want
861 to keep things simple and wish objects like @code{pow(x,2)} to be
862 handled the same way as @code{pow(x,a)}, which means that we need to
863 allow a general @code{ex} as base and exponent. Therefore there is an
864 implicit constructor from C-integers directly to expressions handling
865 numerics at work in most of our examples. This design really becomes
866 convenient when one declares own functions having more than one
867 parameter but it forbids using implicit constructors because that would
868 lead to compile-time ambiguities.
870 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
871 This would, however, call C's built-in operator @code{/} for integers
872 first and result in a numeric holding a plain integer 1. @strong{Never
873 use the operator @code{/} on integers} unless you know exactly what you
874 are doing! Use the constructor from two integers instead, as shown in
875 the example above. Writing @code{numeric(1)/2} may look funny but works
878 @cindex @code{Digits}
880 We have seen now the distinction between exact numbers and floating
881 point numbers. Clearly, the user should never have to worry about
882 dynamically created exact numbers, since their `exactness' always
883 determines how they ought to be handled, i.e. how `long' they are. The
884 situation is different for floating point numbers. Their accuracy is
885 controlled by one @emph{global} variable, called @code{Digits}. (For
886 those readers who know about Maple: it behaves very much like Maple's
887 @code{Digits}). All objects of class numeric that are constructed from
888 then on will be stored with a precision matching that number of decimal
892 #include <ginac/ginac.h>
894 using namespace GiNaC;
898 numeric three(3.0), one(1.0);
899 numeric x = one/three;
901 cout << "in " << Digits << " digits:" << endl;
903 cout << Pi.evalf() << endl;
915 The above example prints the following output to screen:
922 0.333333333333333333333333333333333333333333333333333333333333333333
923 3.14159265358979323846264338327950288419716939937510582097494459231
926 It should be clear that objects of class @code{numeric} should be used
927 for constructing numbers or for doing arithmetic with them. The objects
928 one deals with most of the time are the polymorphic expressions @code{ex}.
930 @subsection Tests on numbers
932 Once you have declared some numbers, assigned them to expressions and
933 done some arithmetic with them it is frequently desired to retrieve some
934 kind of information from them like asking whether that number is
935 integer, rational, real or complex. For those cases GiNaC provides
936 several useful methods. (Internally, they fall back to invocations of
937 certain CLN functions.)
939 As an example, let's construct some rational number, multiply it with
940 some multiple of its denominator and test what comes out:
943 #include <ginac/ginac.h>
945 using namespace GiNaC;
947 // some very important constants:
948 const numeric twentyone(21);
949 const numeric ten(10);
950 const numeric five(5);
954 numeric answer = twentyone;
957 cout << answer.is_integer() << endl; // false, it's 21/5
959 cout << answer.is_integer() << endl; // true, it's 42 now!
963 Note that the variable @code{answer} is constructed here as an integer
964 by @code{numeric}'s copy constructor but in an intermediate step it
965 holds a rational number represented as integer numerator and integer
966 denominator. When multiplied by 10, the denominator becomes unity and
967 the result is automatically converted to a pure integer again.
968 Internally, the underlying @acronym{CLN} is responsible for this
969 behaviour and we refer the reader to @acronym{CLN}'s documentation.
970 Suffice to say that the same behaviour applies to complex numbers as
971 well as return values of certain functions. Complex numbers are
972 automatically converted to real numbers if the imaginary part becomes
973 zero. The full set of tests that can be applied is listed in the
977 @multitable @columnfractions .30 .70
978 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
979 @item @code{.is_zero()}
980 @tab @dots{}equal to zero
981 @item @code{.is_positive()}
982 @tab @dots{}not complex and greater than 0
983 @item @code{.is_integer()}
984 @tab @dots{}a (non-complex) integer
985 @item @code{.is_pos_integer()}
986 @tab @dots{}an integer and greater than 0
987 @item @code{.is_nonneg_integer()}
988 @tab @dots{}an integer and greater equal 0
989 @item @code{.is_even()}
990 @tab @dots{}an even integer
991 @item @code{.is_odd()}
992 @tab @dots{}an odd integer
993 @item @code{.is_prime()}
994 @tab @dots{}a prime integer (probabilistic primality test)
995 @item @code{.is_rational()}
996 @tab @dots{}an exact rational number (integers are rational, too)
997 @item @code{.is_real()}
998 @tab @dots{}a real integer, rational or float (i.e. is not complex)
999 @item @code{.is_cinteger()}
1000 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1001 @item @code{.is_crational()}
1002 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1007 @node Constants, Fundamental containers, Numbers, Basic Concepts
1008 @c node-name, next, previous, up
1010 @cindex @code{constant} (class)
1013 @cindex @code{Catalan}
1014 @cindex @code{Euler}
1015 @cindex @code{evalf()}
1016 Constants behave pretty much like symbols except that they return some
1017 specific number when the method @code{.evalf()} is called.
1019 The predefined known constants are:
1022 @multitable @columnfractions .14 .30 .56
1023 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1025 @tab Archimedes' constant
1026 @tab 3.14159265358979323846264338327950288
1027 @item @code{Catalan}
1028 @tab Catalan's constant
1029 @tab 0.91596559417721901505460351493238411
1031 @tab Euler's (or Euler-Mascheroni) constant
1032 @tab 0.57721566490153286060651209008240243
1037 @node Fundamental containers, Lists, Constants, Basic Concepts
1038 @c node-name, next, previous, up
1039 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1043 @cindex @code{power}
1045 Simple polynomial expressions are written down in GiNaC pretty much like
1046 in other CAS or like expressions involving numerical variables in C.
1047 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1048 been overloaded to achieve this goal. When you run the following
1049 code snippet, the constructor for an object of type @code{mul} is
1050 automatically called to hold the product of @code{a} and @code{b} and
1051 then the constructor for an object of type @code{add} is called to hold
1052 the sum of that @code{mul} object and the number one:
1056 symbol a("a"), b("b");
1061 @cindex @code{pow()}
1062 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1063 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1064 construction is necessary since we cannot safely overload the constructor
1065 @code{^} in C++ to construct a @code{power} object. If we did, it would
1066 have several counterintuitive and undesired effects:
1070 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1072 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1073 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1074 interpret this as @code{x^(a^b)}.
1076 Also, expressions involving integer exponents are very frequently used,
1077 which makes it even more dangerous to overload @code{^} since it is then
1078 hard to distinguish between the semantics as exponentiation and the one
1079 for exclusive or. (It would be embarassing to return @code{1} where one
1080 has requested @code{2^3}.)
1083 @cindex @command{ginsh}
1084 All effects are contrary to mathematical notation and differ from the
1085 way most other CAS handle exponentiation, therefore overloading @code{^}
1086 is ruled out for GiNaC's C++ part. The situation is different in
1087 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1088 that the other frequently used exponentiation operator @code{**} does
1089 not exist at all in C++).
1091 To be somewhat more precise, objects of the three classes described
1092 here, are all containers for other expressions. An object of class
1093 @code{power} is best viewed as a container with two slots, one for the
1094 basis, one for the exponent. All valid GiNaC expressions can be
1095 inserted. However, basic transformations like simplifying
1096 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1097 when this is mathematically possible. If we replace the outer exponent
1098 three in the example by some symbols @code{a}, the simplification is not
1099 safe and will not be performed, since @code{a} might be @code{1/2} and
1102 Objects of type @code{add} and @code{mul} are containers with an
1103 arbitrary number of slots for expressions to be inserted. Again, simple
1104 and safe simplifications are carried out like transforming
1105 @code{3*x+4-x} to @code{2*x+4}.
1107 The general rule is that when you construct such objects, GiNaC
1108 automatically creates them in canonical form, which might differ from
1109 the form you typed in your program. This allows for rapid comparison of
1110 expressions, since after all @code{a-a} is simply zero. Note, that the
1111 canonical form is not necessarily lexicographical ordering or in any way
1112 easily guessable. It is only guaranteed that constructing the same
1113 expression twice, either implicitly or explicitly, results in the same
1117 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1118 @c node-name, next, previous, up
1119 @section Lists of expressions
1120 @cindex @code{lst} (class)
1122 @cindex @code{nops()}
1124 @cindex @code{append()}
1125 @cindex @code{prepend()}
1127 The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
1128 These are sometimes used to supply a variable number of arguments of the same
1129 type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
1130 should have a basic understanding about them.
1132 Lists of up to 15 expressions can be directly constructed from single
1137 symbol x("x"), y("y");
1138 lst l(x, 2, y, x+y);
1139 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
1143 Use the @code{nops()} method to determine the size (number of expressions) of
1144 a list and the @code{op()} method to access individual elements:
1148 cout << l.nops() << endl; // prints '4'
1149 cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
1153 Finally you can append or prepend an expression to a list with the
1154 @code{append()} and @code{prepend()} methods:
1158 l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
1159 l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
1164 @node Mathematical functions, Relations, Lists, Basic Concepts
1165 @c node-name, next, previous, up
1166 @section Mathematical functions
1167 @cindex @code{function} (class)
1168 @cindex trigonometric function
1169 @cindex hyperbolic function
1171 There are quite a number of useful functions hard-wired into GiNaC. For
1172 instance, all trigonometric and hyperbolic functions are implemented
1173 (@xref{Built-in Functions}, for a complete list).
1175 These functions are all objects of class @code{function}. They accept
1176 one or more expressions as arguments and return one expression. If the
1177 arguments are not numerical, the evaluation of the function may be
1178 halted, as it does in the next example, showing how a function returns
1179 itself twice and finally an expression that may be really useful:
1181 @cindex Gamma function
1182 @cindex @code{subs()}
1185 symbol x("x"), y("y");
1187 cout << tgamma(foo) << endl;
1188 // -> tgamma(x+(1/2)*y)
1189 ex bar = foo.subs(y==1);
1190 cout << tgamma(bar) << endl;
1192 ex foobar = bar.subs(x==7);
1193 cout << tgamma(foobar) << endl;
1194 // -> (135135/128)*Pi^(1/2)
1198 Besides evaluation most of these functions allow differentiation, series
1199 expansion and so on. Read the next chapter in order to learn more about
1203 @node Relations, Indexed objects, Mathematical functions, Basic Concepts
1204 @c node-name, next, previous, up
1206 @cindex @code{relational} (class)
1208 Sometimes, a relation holding between two expressions must be stored
1209 somehow. The class @code{relational} is a convenient container for such
1210 purposes. A relation is by definition a container for two @code{ex} and
1211 a relation between them that signals equality, inequality and so on.
1212 They are created by simply using the C++ operators @code{==}, @code{!=},
1213 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1215 @xref{Mathematical functions}, for examples where various applications
1216 of the @code{.subs()} method show how objects of class relational are
1217 used as arguments. There they provide an intuitive syntax for
1218 substitutions. They are also used as arguments to the @code{ex::series}
1219 method, where the left hand side of the relation specifies the variable
1220 to expand in and the right hand side the expansion point. They can also
1221 be used for creating systems of equations that are to be solved for
1222 unknown variables. But the most common usage of objects of this class
1223 is rather inconspicuous in statements of the form @code{if
1224 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1225 conversion from @code{relational} to @code{bool} takes place. Note,
1226 however, that @code{==} here does not perform any simplifications, hence
1227 @code{expand()} must be called explicitly.
1230 @node Indexed objects, Methods and Functions, Relations, Basic Concepts
1231 @c node-name, next, previous, up
1232 @section Indexed objects
1234 GiNaC allows you to handle expressions containing general indexed objects in
1235 arbitrary spaces. It is also able to canonicalize and simplify such
1236 expressions and perform symbolic dummy index summations. There are a number
1237 of predefined indexed objects provided, like delta and metric tensors.
1239 There are few restrictions placed on indexed objects and their indices and
1240 it is easy to construct nonsense expressions, but our intention is to
1241 provide a general framework that allows you to implement algorithms with
1242 indexed quantities, getting in the way as little as possible.
1244 @cindex @code{idx} (class)
1245 @cindex @code{indexed} (class)
1246 @subsection Indexed quantities and their indices
1248 Indexed expressions in GiNaC are constructed of two special types of objects,
1249 @dfn{index objects} and @dfn{indexed objects}.
1253 @cindex contravariant
1256 @item Index objects are of class @code{idx} or a subclass. Every index has
1257 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1258 the index lives in) which can both be arbitrary expressions but are usually
1259 a number or a simple symbol. In addition, indices of class @code{varidx} have
1260 a @dfn{variance} (they can be co- or contravariant).
1262 @item Indexed objects are of class @code{indexed} or a subclass. They
1263 contain a @dfn{base expression} (which is the expression being indexed), and
1264 one or more indices.
1268 @strong{Note:} when printing expressions, covariant indices and indices
1269 without variance are denoted @samp{.i} while contravariant indices are denoted
1270 @samp{~i}. In the following, we are going to use that notation in the text
1271 so instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions
1272 are not visible in the output.
1274 A simple example shall illustrate the concepts:
1277 #include <ginac/ginac.h>
1278 using namespace std;
1279 using namespace GiNaC;
1283 symbol i_sym("i"), j_sym("j");
1284 idx i(i_sym, 3), j(j_sym, 3);
1287 cout << indexed(A, i, j) << endl;
1292 The @code{idx} constructor takes two arguments, the index value and the
1293 index dimension. First we define two index objects, @code{i} and @code{j},
1294 both with the numeric dimension 3. The value of the index @code{i} is the
1295 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1296 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1297 construct an expression containing one indexed object, @samp{A.i.j}. It has
1298 the symbol @code{A} as its base expression and the two indices @code{i} and
1301 Note the difference between the indices @code{i} and @code{j} which are of
1302 class @code{idx}, and the index values which are the sybols @code{i_sym}
1303 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1304 or numbers but must be index objects. For example, the following is not
1305 correct and will raise an exception:
1308 symbol i("i"), j("j");
1309 e = indexed(A, i, j); // ERROR: indices must be of type idx
1312 You can have multiple indexed objects in an expression, index values can
1313 be numeric, and index dimensions symbolic:
1317 symbol B("B"), dim("dim");
1318 cout << 4 * indexed(A, i)
1319 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1324 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1325 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1326 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1327 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1328 @code{simplify_indexed()} for that, see below).
1330 In fact, base expressions, index values and index dimensions can be
1331 arbitrary expressions:
1335 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1340 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1341 get an error message from this but you will probably not be able to do
1342 anything useful with it.
1344 @cindex @code{get_value()}
1345 @cindex @code{get_dimension()}
1349 ex idx::get_value(void);
1350 ex idx::get_dimension(void);
1353 return the value and dimension of an @code{idx} object. If you have an index
1354 in an expression, such as returned by calling @code{.op()} on an indexed
1355 object, you can get a reference to the @code{idx} object with the function
1356 @code{ex_to_idx()} on the expression.
1358 There are also the methods
1361 bool idx::is_numeric(void);
1362 bool idx::is_symbolic(void);
1363 bool idx::is_dim_numeric(void);
1364 bool idx::is_dim_symbolic(void);
1367 for checking whether the value and dimension are numeric or symbolic
1368 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1369 About Expressions}) returns information about the index value.
1371 @cindex @code{varidx} (class)
1372 If you need co- and contravariant indices, use the @code{varidx} class:
1376 symbol mu_sym("mu"), nu_sym("nu");
1377 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1378 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1380 cout << indexed(A, mu, nu) << endl;
1382 cout << indexed(A, mu_co, nu) << endl;
1384 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1389 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1390 co- or contravariant. The default is a contravariant (upper) index, but
1391 this can be overridden by supplying a third argument to the @code{varidx}
1392 constructor. The two methods
1395 bool varidx::is_covariant(void);
1396 bool varidx::is_contravariant(void);
1399 allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
1400 to get the object reference from an expression). There's also the very useful
1404 ex varidx::toggle_variance(void);
1407 which makes a new index with the same value and dimension but the opposite
1408 variance. By using it you only have to define the index once.
1410 @subsection Substituting indices
1412 @cindex @code{subs()}
1413 Sometimes you will want to substitute one symbolic index with another
1414 symbolic or numeric index, for example when calculating one specific element
1415 of a tensor expression. This is done with the @code{.subs()} method, as it
1416 is done for symbols (see @ref{Substituting Symbols}).
1418 You have two possibilities here. You can either substitute the whole index
1419 by another index or expression:
1423 ex e = indexed(A, mu_co);
1424 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1425 // -> A.mu becomes A~nu
1426 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1427 // -> A.mu becomes A~0
1428 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1429 // -> A.mu becomes A.0
1433 The third example shows that trying to replace an index with something that
1434 is not an index will substitute the index value instead.
1436 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1441 ex e = indexed(A, mu_co);
1442 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1443 // -> A.mu becomes A.nu
1444 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1445 // -> A.mu becomes A.0
1449 As you see, with the second method only the value of the index will get
1450 substituted. Its other properties, including its dimension, remain unchanged.
1451 If you want to change the dimension of an index you have to substitute the
1452 whole index by another one with the new dimension.
1454 Finally, substituting the base expression of an indexed object works as
1459 ex e = indexed(A, mu_co);
1460 cout << e << " becomes " << e.subs(A == A+B) << endl;
1461 // -> A.mu becomes (B+A).mu
1465 @subsection Symmetries
1467 Indexed objects can be declared as being totally symmetric or antisymmetric
1468 with respect to their indices. In this case, GiNaC will automatically bring
1469 the indices into a canonical order which allows for some immediate
1474 cout << indexed(A, indexed::symmetric, i, j)
1475 + indexed(A, indexed::symmetric, j, i) << endl;
1477 cout << indexed(B, indexed::antisymmetric, i, j)
1478 + indexed(B, indexed::antisymmetric, j, j) << endl;
1480 cout << indexed(B, indexed::antisymmetric, i, j)
1481 + indexed(B, indexed::antisymmetric, j, i) << endl;
1486 @cindex @code{get_free_indices()}
1488 @subsection Dummy indices
1490 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1491 that a summation over the index range is implied. Symbolic indices which are
1492 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1493 dummy nor free indices.
1495 To be recognized as a dummy index pair, the two indices must be of the same
1496 class and dimension and their value must be the same single symbol (an index
1497 like @samp{2*n+1} is never a dummy index). If the indices are of class
1498 @code{varidx}, they must also be of opposite variance.
1500 The method @code{.get_free_indices()} returns a vector containing the free
1501 indices of an expression. It also checks that the free indices of the terms
1502 of a sum are consistent:
1506 symbol A("A"), B("B"), C("C");
1508 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1509 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1511 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1512 cout << exprseq(e.get_free_indices()) << endl;
1514 // 'j' and 'l' are dummy indices
1516 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1517 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1519 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1520 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1521 cout << exprseq(e.get_free_indices()) << endl;
1523 // 'nu' is a dummy index, but 'sigma' is not
1525 e = indexed(A, mu, mu);
1526 cout << exprseq(e.get_free_indices()) << endl;
1528 // 'mu' is not a dummy index because it appears twice with the same
1531 e = indexed(A, mu, nu) + 42;
1532 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1533 // this will throw an exception:
1534 // "add::get_free_indices: inconsistent indices in sum"
1538 @cindex @code{simplify_indexed()}
1539 @subsection Simplifying indexed expressions
1541 In addition to the few automatic simplifications that GiNaC performs on
1542 indexed expressions (such as re-ordering the indices of symmetric tensors
1543 and calculating traces and convolutions of matrices and predefined tensors)
1547 ex ex::simplify_indexed(void);
1548 ex ex::simplify_indexed(const scalar_products & sp);
1551 that performs some more expensive operations:
1554 @item it checks the consistency of free indices in sums in the same way
1555 @code{get_free_indices()} does
1556 @item it (symbolically) calculates all possible dummy index summations/contractions
1557 with the predefined tensors (this will be explained in more detail in the
1559 @item as a special case of dummy index summation, it can replace scalar products
1560 of two tensors with a user-defined value
1563 The last point is done with the help of the @code{scalar_products} class
1564 which is used to store scalar products with known values (this is not an
1565 arithmetic class, you just pass it to @code{simplify_indexed()}):
1569 symbol A("A"), B("B"), C("C"), i_sym("i");
1573 sp.add(A, B, 0); // A and B are orthogonal
1574 sp.add(A, C, 0); // A and C are orthogonal
1575 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1577 e = indexed(A + B, i) * indexed(A + C, i);
1579 // -> (B+A).i*(A+C).i
1581 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1587 The @code{scalar_products} object @code{sp} acts as a storage for the
1588 scalar products added to it with the @code{.add()} method. This method
1589 takes three arguments: the two expressions of which the scalar product is
1590 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1591 @code{simplify_indexed()} will replace all scalar products of indexed
1592 objects that have the symbols @code{A} and @code{B} as base expressions
1593 with the single value 0. The number, type and dimension of the indices
1594 doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1596 @cindex @code{expand()}
1597 The example above also illustrates a feature of the @code{expand()} method:
1598 if passed the @code{expand_indexed} option it will distribute indices
1599 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1601 @cindex @code{tensor} (class)
1602 @subsection Predefined tensors
1604 Some frequently used special tensors such as the delta, epsilon and metric
1605 tensors are predefined in GiNaC. They have special properties when
1606 contracted with other tensor expressions and some of them have constant
1607 matrix representations (they will evaluate to a number when numeric
1608 indices are specified).
1610 @cindex @code{delta_tensor()}
1611 @subsubsection Delta tensor
1613 The delta tensor takes two indices, is symmetric and has the matrix
1614 representation @code{diag(1,1,1,...)}. It is constructed by the function
1615 @code{delta_tensor()}:
1619 symbol A("A"), B("B");
1621 idx i(symbol("i"), 3), j(symbol("j"), 3),
1622 k(symbol("k"), 3), l(symbol("l"), 3);
1624 ex e = indexed(A, i, j) * indexed(B, k, l)
1625 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1626 cout << e.simplify_indexed() << endl;
1629 cout << delta_tensor(i, i) << endl;
1634 @cindex @code{metric_tensor()}
1635 @subsubsection General metric tensor
1637 The function @code{metric_tensor()} creates a general symmetric metric
1638 tensor with two indices that can be used to raise/lower tensor indices. The
1639 metric tensor is denoted as @samp{g} in the output and if its indices are of
1640 mixed variance it is automatically replaced by a delta tensor:
1646 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1648 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1649 cout << e.simplify_indexed() << endl;
1652 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1653 cout << e.simplify_indexed() << endl;
1656 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1657 * metric_tensor(nu, rho);
1658 cout << e.simplify_indexed() << endl;
1661 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1662 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1663 + indexed(A, mu.toggle_variance(), rho));
1664 cout << e.simplify_indexed() << endl;
1669 @cindex @code{lorentz_g()}
1670 @subsubsection Minkowski metric tensor
1672 The Minkowski metric tensor is a special metric tensor with a constant
1673 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1674 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1675 It is created with the function @code{lorentz_g()} (although it is output as
1680 varidx mu(symbol("mu"), 4);
1682 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1683 * lorentz_g(mu, varidx(0, 4)); // negative signature
1684 cout << e.simplify_indexed() << endl;
1687 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1688 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1689 cout << e.simplify_indexed() << endl;
1694 @subsubsection Epsilon tensor
1696 The epsilon tensor is totally antisymmetric, its number of indices is equal
1697 to the dimension of the index space (the indices must all be of the same
1698 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1699 defined to be 1. Its behaviour with indices that have a variance also
1700 depends on the signature of the metric. Epsilon tensors are output as
1703 There are three functions defined to create epsilon tensors in 2, 3 and 4
1707 ex epsilon_tensor(const ex & i1, const ex & i2);
1708 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1709 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1712 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1713 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1714 Minkowski space (the last @code{bool} argument specifies whether the metric
1715 has negative or positive signature, as in the case of the Minkowski metric
1718 @subsection Linear algebra
1720 The @code{matrix} class can be used with indices to do some simple linear
1721 algebra (linear combinations and products of vectors and matrices, traces
1722 and scalar products):
1726 idx i(symbol("i"), 2), j(symbol("j"), 2);
1727 symbol x("x"), y("y");
1729 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
1731 cout << indexed(A, i, i) << endl;
1734 ex e = indexed(A, i, j) * indexed(X, j);
1735 cout << e.simplify_indexed() << endl;
1736 // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
1738 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
1739 cout << e.simplify_indexed() << endl;
1740 // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
1744 You can of course obtain the same results with the @code{matrix::add()},
1745 @code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
1746 don't have to worry about transposing matrices.
1748 Matrix indices always start at 0 and their dimension must match the number
1749 of rows/columns of the matrix. Matrices with one row or one column are
1750 vectors and can have one or two indices (it doesn't matter whether it's a
1751 row or a column vector). Other matrices must have two indices.
1753 You should be careful when using indices with variance on matrices. GiNaC
1754 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
1755 @samp{F.mu.nu} are different matrices. In this case you should use only
1756 one form for @samp{F} and explicitly multiply it with a matrix representation
1757 of the metric tensor.
1760 @node Methods and Functions, Information About Expressions, Indexed objects, Top
1761 @c node-name, next, previous, up
1762 @chapter Methods and Functions
1765 In this chapter the most important algorithms provided by GiNaC will be
1766 described. Some of them are implemented as functions on expressions,
1767 others are implemented as methods provided by expression objects. If
1768 they are methods, there exists a wrapper function around it, so you can
1769 alternatively call it in a functional way as shown in the simple
1774 cout << "As method: " << sin(1).evalf() << endl;
1775 cout << "As function: " << evalf(sin(1)) << endl;
1779 @cindex @code{subs()}
1780 The general rule is that wherever methods accept one or more parameters
1781 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1782 wrapper accepts is the same but preceded by the object to act on
1783 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1784 most natural one in an OO model but it may lead to confusion for MapleV
1785 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1786 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1787 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1788 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1789 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1790 here. Also, users of MuPAD will in most cases feel more comfortable
1791 with GiNaC's convention. All function wrappers are implemented
1792 as simple inline functions which just call the corresponding method and
1793 are only provided for users uncomfortable with OO who are dead set to
1794 avoid method invocations. Generally, nested function wrappers are much
1795 harder to read than a sequence of methods and should therefore be
1796 avoided if possible. On the other hand, not everything in GiNaC is a
1797 method on class @code{ex} and sometimes calling a function cannot be
1801 * Information About Expressions::
1802 * Substituting Symbols::
1803 * Polynomial Arithmetic:: Working with polynomials.
1804 * Rational Expressions:: Working with rational functions.
1805 * Symbolic Differentiation::
1806 * Series Expansion:: Taylor and Laurent expansion.
1807 * Built-in Functions:: List of predefined mathematical functions.
1808 * Input/Output:: Input and output of expressions.
1812 @node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions
1813 @c node-name, next, previous, up
1814 @section Getting information about expressions
1816 @subsection Checking expression types
1817 @cindex @code{is_ex_of_type()}
1818 @cindex @code{ex_to_numeric()}
1819 @cindex @code{ex_to_@dots{}}
1820 @cindex @code{Converting ex to other classes}
1821 @cindex @code{info()}
1823 Sometimes it's useful to check whether a given expression is a plain number,
1824 a sum, a polynomial with integer coefficients, or of some other specific type.
1825 GiNaC provides two functions for this (the first one is actually a macro):
1828 bool is_ex_of_type(const ex & e, TYPENAME t);
1829 bool ex::info(unsigned flag);
1832 When the test made by @code{is_ex_of_type()} returns true, it is safe to
1833 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
1834 one of the class names (@xref{The Class Hierarchy}, for a list of all
1835 classes). For example, assuming @code{e} is an @code{ex}:
1840 if (is_ex_of_type(e, numeric))
1841 numeric n = ex_to_numeric(e);
1846 @code{is_ex_of_type()} allows you to check whether the top-level object of
1847 an expression @samp{e} is an instance of the GiNaC class @samp{t}
1848 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
1849 e.g., for checking whether an expression is a number, a sum, or a product:
1856 is_ex_of_type(e1, numeric); // true
1857 is_ex_of_type(e2, numeric); // false
1858 is_ex_of_type(e1, add); // false
1859 is_ex_of_type(e2, add); // true
1860 is_ex_of_type(e1, mul); // false
1861 is_ex_of_type(e2, mul); // false
1865 The @code{info()} method is used for checking certain attributes of
1866 expressions. The possible values for the @code{flag} argument are defined
1867 in @file{ginac/flags.h}, the most important being explained in the following
1871 @multitable @columnfractions .30 .70
1872 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
1873 @item @code{numeric}
1874 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
1876 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1877 @item @code{rational}
1878 @tab @dots{}an exact rational number (integers are rational, too)
1879 @item @code{integer}
1880 @tab @dots{}a (non-complex) integer
1881 @item @code{crational}
1882 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1883 @item @code{cinteger}
1884 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1885 @item @code{positive}
1886 @tab @dots{}not complex and greater than 0
1887 @item @code{negative}
1888 @tab @dots{}not complex and less than 0
1889 @item @code{nonnegative}
1890 @tab @dots{}not complex and greater than or equal to 0
1892 @tab @dots{}an integer greater than 0
1894 @tab @dots{}an integer less than 0
1895 @item @code{nonnegint}
1896 @tab @dots{}an integer greater than or equal to 0
1898 @tab @dots{}an even integer
1900 @tab @dots{}an odd integer
1902 @tab @dots{}a prime integer (probabilistic primality test)
1903 @item @code{relation}
1904 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
1905 @item @code{relation_equal}
1906 @tab @dots{}a @code{==} relation
1907 @item @code{relation_not_equal}
1908 @tab @dots{}a @code{!=} relation
1909 @item @code{relation_less}
1910 @tab @dots{}a @code{<} relation
1911 @item @code{relation_less_or_equal}
1912 @tab @dots{}a @code{<=} relation
1913 @item @code{relation_greater}
1914 @tab @dots{}a @code{>} relation
1915 @item @code{relation_greater_or_equal}
1916 @tab @dots{}a @code{>=} relation
1918 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
1920 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
1921 @item @code{polynomial}
1922 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
1923 @item @code{integer_polynomial}
1924 @tab @dots{}a polynomial with (non-complex) integer coefficients
1925 @item @code{cinteger_polynomial}
1926 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
1927 @item @code{rational_polynomial}
1928 @tab @dots{}a polynomial with (non-complex) rational coefficients
1929 @item @code{crational_polynomial}
1930 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
1931 @item @code{rational_function}
1932 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
1933 @item @code{algebraic}
1934 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
1939 @subsection Accessing subexpressions
1940 @cindex @code{nops()}
1942 @cindex @code{has()}
1944 @cindex @code{relational} (class)
1946 GiNaC provides the two methods
1949 unsigned ex::nops();
1950 ex ex::op(unsigned i);
1953 for accessing the subexpressions in the container-like GiNaC classes like
1954 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
1955 determines the number of subexpressions (@samp{operands}) contained, while
1956 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
1957 In the case of a @code{power} object, @code{op(0)} will return the basis
1958 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
1959 is the base expression and @code{op(i)}, @math{i>0} are the indices.
1961 The left-hand and right-hand side expressions of objects of class
1962 @code{relational} (and only of these) can also be accessed with the methods
1972 bool ex::has(const ex & other);
1975 checks whether an expression contains the given subexpression @code{other}.
1976 This only works reliably if @code{other} is of an atomic class such as a
1977 @code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
1978 @code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
1981 @subsection Comparing expressions
1982 @cindex @code{is_equal()}
1983 @cindex @code{is_zero()}
1985 Expressions can be compared with the usual C++ relational operators like
1986 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
1987 the result is usually not determinable and the result will be @code{false},
1988 except in the case of the @code{!=} operator. You should also be aware that
1989 GiNaC will only do the most trivial test for equality (subtracting both
1990 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
1993 Actually, if you construct an expression like @code{a == b}, this will be
1994 represented by an object of the @code{relational} class (@xref{Relations}.)
1995 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
1997 There are also two methods
2000 bool ex::is_equal(const ex & other);
2004 for checking whether one expression is equal to another, or equal to zero,
2007 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2008 GiNaC header files. This method is however only to be used internally by
2009 GiNaC to establish a canonical sort order for terms, and using it to compare
2010 expressions will give very surprising results.
2013 @node Substituting Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions
2014 @c node-name, next, previous, up
2015 @section Substituting symbols
2016 @cindex @code{subs()}
2018 Symbols can be replaced with expressions via the @code{.subs()} method:
2021 ex ex::subs(const ex & e);
2022 ex ex::subs(const lst & syms, const lst & repls);
2025 In the first form, @code{subs()} accepts a relational of the form
2026 @samp{symbol == expression} or a @code{lst} of such relationals. E.g.
2030 symbol x("x"), y("y");
2031 ex e1 = 2*x^2-4*x+3;
2032 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2034 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2038 will print @samp{73} and @samp{-10}, respectively.
2040 If you specify multiple substitutions, they are performed in parallel, so e.g.
2041 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2043 The second form of @code{subs()} takes two lists, one for the symbols and
2044 one for the expressions to be substituted (both lists must contain the same
2045 number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))}
2046 to exchange @samp{x} and @samp{y}.
2049 @node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, Methods and Functions
2050 @c node-name, next, previous, up
2051 @section Polynomial arithmetic
2053 @subsection Expanding and collecting
2054 @cindex @code{expand()}
2055 @cindex @code{collect()}
2057 A polynomial in one or more variables has many equivalent
2058 representations. Some useful ones serve a specific purpose. Consider
2059 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2060 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2061 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
2062 representations are the recursive ones where one collects for exponents
2063 in one of the three variable. Since the factors are themselves
2064 polynomials in the remaining two variables the procedure can be
2065 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
2066 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
2069 To bring an expression into expanded form, its method
2075 may be called. In our example above, this corresponds to @math{4*x*y +
2076 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
2077 GiNaC is not easily guessable you should be prepared to see different
2078 orderings of terms in such sums!
2080 Another useful representation of multivariate polynomials is as a
2081 univariate polynomial in one of the variables with the coefficients
2082 being polynomials in the remaining variables. The method
2083 @code{collect()} accomplishes this task:
2086 ex ex::collect(const ex & s);
2089 Note that the original polynomial needs to be in expanded form in order
2090 to be able to find the coefficients properly.
2092 @subsection Degree and coefficients
2093 @cindex @code{degree()}
2094 @cindex @code{ldegree()}
2095 @cindex @code{coeff()}
2097 The degree and low degree of a polynomial can be obtained using the two
2101 int ex::degree(const ex & s);
2102 int ex::ldegree(const ex & s);
2105 which also work reliably on non-expanded input polynomials (they even work
2106 on rational functions, returning the asymptotic degree). To extract
2107 a coefficient with a certain power from an expanded polynomial you use
2110 ex ex::coeff(const ex & s, int n);
2113 You can also obtain the leading and trailing coefficients with the methods
2116 ex ex::lcoeff(const ex & s);
2117 ex ex::tcoeff(const ex & s);
2120 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
2123 An application is illustrated in the next example, where a multivariate
2124 polynomial is analyzed:
2127 #include <ginac/ginac.h>
2128 using namespace std;
2129 using namespace GiNaC;
2133 symbol x("x"), y("y");
2134 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
2135 - pow(x+y,2) + 2*pow(y+2,2) - 8;
2136 ex Poly = PolyInp.expand();
2138 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
2139 cout << "The x^" << i << "-coefficient is "
2140 << Poly.coeff(x,i) << endl;
2142 cout << "As polynomial in y: "
2143 << Poly.collect(y) << endl;
2147 When run, it returns an output in the following fashion:
2150 The x^0-coefficient is y^2+11*y
2151 The x^1-coefficient is 5*y^2-2*y
2152 The x^2-coefficient is -1
2153 The x^3-coefficient is 4*y
2154 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
2157 As always, the exact output may vary between different versions of GiNaC
2158 or even from run to run since the internal canonical ordering is not
2159 within the user's sphere of influence.
2162 @subsection Polynomial division
2163 @cindex polynomial division
2166 @cindex pseudo-remainder
2167 @cindex @code{quo()}
2168 @cindex @code{rem()}
2169 @cindex @code{prem()}
2170 @cindex @code{divide()}
2175 ex quo(const ex & a, const ex & b, const symbol & x);
2176 ex rem(const ex & a, const ex & b, const symbol & x);
2179 compute the quotient and remainder of univariate polynomials in the variable
2180 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
2182 The additional function
2185 ex prem(const ex & a, const ex & b, const symbol & x);
2188 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
2189 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
2191 Exact division of multivariate polynomials is performed by the function
2194 bool divide(const ex & a, const ex & b, ex & q);
2197 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
2198 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
2199 in which case the value of @code{q} is undefined.
2202 @subsection Unit, content and primitive part
2203 @cindex @code{unit()}
2204 @cindex @code{content()}
2205 @cindex @code{primpart()}
2210 ex ex::unit(const symbol & x);
2211 ex ex::content(const symbol & x);
2212 ex ex::primpart(const symbol & x);
2215 return the unit part, content part, and primitive polynomial of a multivariate
2216 polynomial with respect to the variable @samp{x} (the unit part being the sign
2217 of the leading coefficient, the content part being the GCD of the coefficients,
2218 and the primitive polynomial being the input polynomial divided by the unit and
2219 content parts). The product of unit, content, and primitive part is the
2220 original polynomial.
2223 @subsection GCD and LCM
2226 @cindex @code{gcd()}
2227 @cindex @code{lcm()}
2229 The functions for polynomial greatest common divisor and least common
2230 multiple have the synopsis
2233 ex gcd(const ex & a, const ex & b);
2234 ex lcm(const ex & a, const ex & b);
2237 The functions @code{gcd()} and @code{lcm()} accept two expressions
2238 @code{a} and @code{b} as arguments and return a new expression, their
2239 greatest common divisor or least common multiple, respectively. If the
2240 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
2241 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
2244 #include <ginac/ginac.h>
2245 using namespace GiNaC;
2249 symbol x("x"), y("y"), z("z");
2250 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
2251 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
2253 ex P_gcd = gcd(P_a, P_b);
2255 ex P_lcm = lcm(P_a, P_b);
2256 // 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
2261 @subsection Square-free decomposition
2262 @cindex square-free decomposition
2263 @cindex factorization
2264 @cindex @code{sqrfree()}
2266 GiNaC still lacks proper factorization support. Some form of
2267 factorization is, however, easily implemented by noting that factors
2268 appearing in a polynomial with power two or more also appear in the
2269 derivative and hence can easily be found by computing the GCD of the
2270 original polynomial and its derivatives. Any system has an interface
2271 for this so called square-free factorization. So we provide one, too:
2273 ex sqrfree(const ex & a, const lst & l = lst());
2275 Here is an example that by the way illustrates how the result may depend
2276 on the order of differentiation:
2279 symbol x("x"), y("y");
2280 ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
2282 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
2283 // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
2285 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
2286 // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
2288 cout << sqrfree(BiVarPol) << endl;
2289 // -> depending on luck, any of the above
2294 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
2295 @c node-name, next, previous, up
2296 @section Rational expressions
2298 @subsection The @code{normal} method
2299 @cindex @code{normal()}
2300 @cindex simplification
2301 @cindex temporary replacement
2303 Some basic form of simplification of expressions is called for frequently.
2304 GiNaC provides the method @code{.normal()}, which converts a rational function
2305 into an equivalent rational function of the form @samp{numerator/denominator}
2306 where numerator and denominator are coprime. If the input expression is already
2307 a fraction, it just finds the GCD of numerator and denominator and cancels it,
2308 otherwise it performs fraction addition and multiplication.
2310 @code{.normal()} can also be used on expressions which are not rational functions
2311 as it will replace all non-rational objects (like functions or non-integer
2312 powers) by temporary symbols to bring the expression to the domain of rational
2313 functions before performing the normalization, and re-substituting these
2314 symbols afterwards. This algorithm is also available as a separate method
2315 @code{.to_rational()}, described below.
2317 This means that both expressions @code{t1} and @code{t2} are indeed
2318 simplified in this little program:
2321 #include <ginac/ginac.h>
2322 using namespace GiNaC;
2327 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
2328 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
2329 std::cout << "t1 is " << t1.normal() << std::endl;
2330 std::cout << "t2 is " << t2.normal() << std::endl;
2334 Of course this works for multivariate polynomials too, so the ratio of
2335 the sample-polynomials from the section about GCD and LCM above would be
2336 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
2339 @subsection Numerator and denominator
2342 @cindex @code{numer()}
2343 @cindex @code{denom()}
2345 The numerator and denominator of an expression can be obtained with
2352 These functions will first normalize the expression as described above and
2353 then return the numerator or denominator, respectively.
2356 @subsection Converting to a rational expression
2357 @cindex @code{to_rational()}
2359 Some of the methods described so far only work on polynomials or rational
2360 functions. GiNaC provides a way to extend the domain of these functions to
2361 general expressions by using the temporary replacement algorithm described
2362 above. You do this by calling
2365 ex ex::to_rational(lst &l);
2368 on the expression to be converted. The supplied @code{lst} will be filled
2369 with the generated temporary symbols and their replacement expressions in
2370 a format that can be used directly for the @code{subs()} method. It can also
2371 already contain a list of replacements from an earlier application of
2372 @code{.to_rational()}, so it's possible to use it on multiple expressions
2373 and get consistent results.
2380 ex a = pow(sin(x), 2) - pow(cos(x), 2);
2381 ex b = sin(x) + cos(x);
2384 divide(a.to_rational(l), b.to_rational(l), q);
2385 cout << q.subs(l) << endl;
2389 will print @samp{sin(x)-cos(x)}.
2392 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
2393 @c node-name, next, previous, up
2394 @section Symbolic differentiation
2395 @cindex differentiation
2396 @cindex @code{diff()}
2398 @cindex product rule
2400 GiNaC's objects know how to differentiate themselves. Thus, a
2401 polynomial (class @code{add}) knows that its derivative is the sum of
2402 the derivatives of all the monomials:
2405 #include <ginac/ginac.h>
2406 using namespace GiNaC;
2410 symbol x("x"), y("y"), z("z");
2411 ex P = pow(x, 5) + pow(x, 2) + y;
2413 cout << P.diff(x,2) << endl; // 20*x^3 + 2
2414 cout << P.diff(y) << endl; // 1
2415 cout << P.diff(z) << endl; // 0
2419 If a second integer parameter @var{n} is given, the @code{diff} method
2420 returns the @var{n}th derivative.
2422 If @emph{every} object and every function is told what its derivative
2423 is, all derivatives of composed objects can be calculated using the
2424 chain rule and the product rule. Consider, for instance the expression
2425 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
2426 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
2427 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
2428 out that the composition is the generating function for Euler Numbers,
2429 i.e. the so called @var{n}th Euler number is the coefficient of
2430 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
2431 identity to code a function that generates Euler numbers in just three
2434 @cindex Euler numbers
2436 #include <ginac/ginac.h>
2437 using namespace GiNaC;
2439 ex EulerNumber(unsigned n)
2442 const ex generator = pow(cosh(x),-1);
2443 return generator.diff(x,n).subs(x==0);
2448 for (unsigned i=0; i<11; i+=2)
2449 std::cout << EulerNumber(i) << std::endl;
2454 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
2455 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
2456 @code{i} by two since all odd Euler numbers vanish anyways.
2459 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
2460 @c node-name, next, previous, up
2461 @section Series expansion
2462 @cindex @code{series()}
2463 @cindex Taylor expansion
2464 @cindex Laurent expansion
2465 @cindex @code{pseries} (class)
2467 Expressions know how to expand themselves as a Taylor series or (more
2468 generally) a Laurent series. As in most conventional Computer Algebra
2469 Systems, no distinction is made between those two. There is a class of
2470 its own for storing such series (@code{class pseries}) and a built-in
2471 function (called @code{Order}) for storing the order term of the series.
2472 As a consequence, if you want to work with series, i.e. multiply two
2473 series, you need to call the method @code{ex::series} again to convert
2474 it to a series object with the usual structure (expansion plus order
2475 term). A sample application from special relativity could read:
2478 #include <ginac/ginac.h>
2479 using namespace std;
2480 using namespace GiNaC;
2484 symbol v("v"), c("c");
2486 ex gamma = 1/sqrt(1 - pow(v/c,2));
2487 ex mass_nonrel = gamma.series(v==0, 10);
2489 cout << "the relativistic mass increase with v is " << endl
2490 << mass_nonrel << endl;
2492 cout << "the inverse square of this series is " << endl
2493 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
2497 Only calling the series method makes the last output simplify to
2498 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
2499 series raised to the power @math{-2}.
2501 @cindex M@'echain's formula
2502 As another instructive application, let us calculate the numerical
2503 value of Archimedes' constant
2507 (for which there already exists the built-in constant @code{Pi})
2508 using M@'echain's amazing formula
2510 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
2513 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
2515 We may expand the arcus tangent around @code{0} and insert the fractions
2516 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
2517 carries an order term with it and the question arises what the system is
2518 supposed to do when the fractions are plugged into that order term. The
2519 solution is to use the function @code{series_to_poly()} to simply strip
2523 #include <ginac/ginac.h>
2524 using namespace GiNaC;
2526 ex mechain_pi(int degr)
2529 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
2530 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
2531 -4*pi_expansion.subs(x==numeric(1,239));
2537 using std::cout; // just for fun, another way of...
2538 using std::endl; // ...dealing with this namespace std.
2540 for (int i=2; i<12; i+=2) @{
2541 pi_frac = mechain_pi(i);
2542 cout << i << ":\t" << pi_frac << endl
2543 << "\t" << pi_frac.evalf() << endl;
2549 Note how we just called @code{.series(x,degr)} instead of
2550 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
2551 method @code{series()}: if the first argument is a symbol the expression
2552 is expanded in that symbol around point @code{0}. When you run this
2553 program, it will type out:
2557 3.1832635983263598326
2558 4: 5359397032/1706489875
2559 3.1405970293260603143
2560 6: 38279241713339684/12184551018734375
2561 3.141621029325034425
2562 8: 76528487109180192540976/24359780855939418203125
2563 3.141591772182177295
2564 10: 327853873402258685803048818236/104359128170408663038552734375
2565 3.1415926824043995174
2569 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
2570 @c node-name, next, previous, up
2571 @section Predefined mathematical functions
2573 GiNaC contains the following predefined mathematical functions:
2576 @multitable @columnfractions .30 .70
2577 @item @strong{Name} @tab @strong{Function}
2580 @item @code{csgn(x)}
2582 @item @code{sqrt(x)}
2583 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
2590 @item @code{asin(x)}
2592 @item @code{acos(x)}
2594 @item @code{atan(x)}
2595 @tab inverse tangent
2596 @item @code{atan2(y, x)}
2597 @tab inverse tangent with two arguments
2598 @item @code{sinh(x)}
2599 @tab hyperbolic sine
2600 @item @code{cosh(x)}
2601 @tab hyperbolic cosine
2602 @item @code{tanh(x)}
2603 @tab hyperbolic tangent
2604 @item @code{asinh(x)}
2605 @tab inverse hyperbolic sine
2606 @item @code{acosh(x)}
2607 @tab inverse hyperbolic cosine
2608 @item @code{atanh(x)}
2609 @tab inverse hyperbolic tangent
2611 @tab exponential function
2613 @tab natural logarithm
2616 @item @code{zeta(x)}
2617 @tab Riemann's zeta function
2618 @item @code{zeta(n, x)}
2619 @tab derivatives of Riemann's zeta function
2620 @item @code{tgamma(x)}
2622 @item @code{lgamma(x)}
2623 @tab logarithm of Gamma function
2624 @item @code{beta(x, y)}
2625 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
2627 @tab psi (digamma) function
2628 @item @code{psi(n, x)}
2629 @tab derivatives of psi function (polygamma functions)
2630 @item @code{factorial(n)}
2631 @tab factorial function
2632 @item @code{binomial(n, m)}
2633 @tab binomial coefficients
2634 @item @code{Order(x)}
2635 @tab order term function in truncated power series
2636 @item @code{Derivative(x, l)}
2637 @tab inert partial differentiation operator (used internally)
2642 For functions that have a branch cut in the complex plane GiNaC follows
2643 the conventions for C++ as defined in the ANSI standard as far as
2644 possible. In particular: the natural logarithm (@code{log}) and the
2645 square root (@code{sqrt}) both have their branch cuts running along the
2646 negative real axis where the points on the axis itself belong to the
2647 upper part (i.e. continuous with quadrant II). The inverse
2648 trigonometric and hyperbolic functions are not defined for complex
2649 arguments by the C++ standard, however. In GiNaC we follow the
2650 conventions used by CLN, which in turn follow the carefully designed
2651 definitions in the Common Lisp standard. It should be noted that this
2652 convention is identical to the one used by the C99 standard and by most
2653 serious CAS. It is to be expected that future revisions of the C++
2654 standard incorporate these functions in the complex domain in a manner
2655 compatible with C99.
2658 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
2659 @c node-name, next, previous, up
2660 @section Input and output of expressions
2663 @subsection Expression output
2665 @cindex output of expressions
2667 The easiest way to print an expression is to write it to a stream:
2672 ex e = 4.5+pow(x,2)*3/2;
2673 cout << e << endl; // prints '(4.5)+3/2*x^2'
2677 The output format is identical to the @command{ginsh} input syntax and
2678 to that used by most computer algebra systems, but not directly pastable
2679 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
2680 is printed as @samp{x^2}).
2682 It is possible to print expressions in a number of different formats with
2686 void ex::print(const print_context & c, unsigned level = 0);
2689 The type of @code{print_context} object passed in determines the format
2690 of the output. The possible types are defined in @file{ginac/print.h}.
2691 All constructors of @code{print_context} and derived classes take an
2692 @code{ostream &} as their first argument.
2694 To print an expression in a way that can be directly used in a C or C++
2695 program, you pass a @code{print_csrc} object like this:
2699 cout << "float f = ";
2700 e.print(print_csrc_float(cout));
2703 cout << "double d = ";
2704 e.print(print_csrc_double(cout));
2707 cout << "cl_N n = ";
2708 e.print(print_csrc_cl_N(cout));
2713 The three possible types mostly affect the way in which floating point
2714 numbers are written.
2716 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
2719 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2720 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2721 cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
2724 The @code{print_context} type @code{print_tree} provdes a dump of the
2725 internal structure of an expression for debugging purposes:
2729 e.print(print_tree(cout));
2736 add, hash=0x0, flags=0x3, nops=2
2737 power, hash=0x9, flags=0x3, nops=2
2738 x (symbol), serial=3, hash=0x44a113a6, flags=0xf
2739 2 (numeric), hash=0x80000042, flags=0xf
2740 3/2 (numeric), hash=0x80000061, flags=0xf
2743 4.5L0 (numeric), hash=0x8000004b, flags=0xf
2747 This kind of output is also available in @command{ginsh} as the @code{print()}
2751 @subsection Expression input
2752 @cindex input of expressions
2754 GiNaC provides no way to directly read an expression from a stream because
2755 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
2756 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
2757 @code{y} you defined in your program and there is no way to specify the
2758 desired symbols to the @code{>>} stream input operator.
2760 Instead, GiNaC lets you construct an expression from a string, specifying the
2761 list of symbols to be used:
2765 symbol x("x"), y("y");
2766 ex e("2*x+sin(y)", lst(x, y));
2770 The input syntax is the same as that used by @command{ginsh} and the stream
2771 output operator @code{<<}. The symbols in the string are matched by name to
2772 the symbols in the list and if GiNaC encounters a symbol not specified in
2773 the list it will throw an exception.
2775 With this constructor, it's also easy to implement interactive GiNaC programs:
2780 #include <stdexcept>
2781 #include <ginac/ginac.h>
2782 using namespace std;
2783 using namespace GiNaC;
2790 cout << "Enter an expression containing 'x': ";
2795 cout << "The derivative of " << e << " with respect to x is ";
2796 cout << e.diff(x) << ".\n";
2797 @} catch (exception &p) @{
2798 cerr << p.what() << endl;
2804 @subsection Archiving
2805 @cindex @code{archive} (class)
2808 GiNaC allows creating @dfn{archives} of expressions which can be stored
2809 to or retrieved from files. To create an archive, you declare an object
2810 of class @code{archive} and archive expressions in it, giving each
2811 expression a unique name:
2815 using namespace std;
2816 #include <ginac/ginac.h>
2817 using namespace GiNaC;
2821 symbol x("x"), y("y"), z("z");
2823 ex foo = sin(x + 2*y) + 3*z + 41;
2827 a.archive_ex(foo, "foo");
2828 a.archive_ex(bar, "the second one");
2832 The archive can then be written to a file:
2836 ofstream out("foobar.gar");
2842 The file @file{foobar.gar} contains all information that is needed to
2843 reconstruct the expressions @code{foo} and @code{bar}.
2845 @cindex @command{viewgar}
2846 The tool @command{viewgar} that comes with GiNaC can be used to view
2847 the contents of GiNaC archive files:
2850 $ viewgar foobar.gar
2851 foo = 41+sin(x+2*y)+3*z
2852 the second one = 42+sin(x+2*y)+3*z
2855 The point of writing archive files is of course that they can later be
2861 ifstream in("foobar.gar");
2866 And the stored expressions can be retrieved by their name:
2872 ex ex1 = a2.unarchive_ex(syms, "foo");
2873 ex ex2 = a2.unarchive_ex(syms, "the second one");
2875 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
2876 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
2877 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
2881 Note that you have to supply a list of the symbols which are to be inserted
2882 in the expressions. Symbols in archives are stored by their name only and
2883 if you don't specify which symbols you have, unarchiving the expression will
2884 create new symbols with that name. E.g. if you hadn't included @code{x} in
2885 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
2886 have had no effect because the @code{x} in @code{ex1} would have been a
2887 different symbol than the @code{x} which was defined at the beginning of
2888 the program, altough both would appear as @samp{x} when printed.
2892 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
2893 @c node-name, next, previous, up
2894 @chapter Extending GiNaC
2896 By reading so far you should have gotten a fairly good understanding of
2897 GiNaC's design-patterns. From here on you should start reading the
2898 sources. All we can do now is issue some recommendations how to tackle
2899 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
2900 develop some useful extension please don't hesitate to contact the GiNaC
2901 authors---they will happily incorporate them into future versions.
2904 * What does not belong into GiNaC:: What to avoid.
2905 * Symbolic functions:: Implementing symbolic functions.
2906 * Adding classes:: Defining new algebraic classes.
2910 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
2911 @c node-name, next, previous, up
2912 @section What doesn't belong into GiNaC
2914 @cindex @command{ginsh}
2915 First of all, GiNaC's name must be read literally. It is designed to be
2916 a library for use within C++. The tiny @command{ginsh} accompanying
2917 GiNaC makes this even more clear: it doesn't even attempt to provide a
2918 language. There are no loops or conditional expressions in
2919 @command{ginsh}, it is merely a window into the library for the
2920 programmer to test stuff (or to show off). Still, the design of a
2921 complete CAS with a language of its own, graphical capabilites and all
2922 this on top of GiNaC is possible and is without doubt a nice project for
2925 There are many built-in functions in GiNaC that do not know how to
2926 evaluate themselves numerically to a precision declared at runtime
2927 (using @code{Digits}). Some may be evaluated at certain points, but not
2928 generally. This ought to be fixed. However, doing numerical
2929 computations with GiNaC's quite abstract classes is doomed to be
2930 inefficient. For this purpose, the underlying foundation classes
2931 provided by @acronym{CLN} are much better suited.
2934 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
2935 @c node-name, next, previous, up
2936 @section Symbolic functions
2938 The easiest and most instructive way to start with is probably to
2939 implement your own function. GiNaC's functions are objects of class
2940 @code{function}. The preprocessor is then used to convert the function
2941 names to objects with a corresponding serial number that is used
2942 internally to identify them. You usually need not worry about this
2943 number. New functions may be inserted into the system via a kind of
2944 `registry'. It is your responsibility to care for some functions that
2945 are called when the user invokes certain methods. These are usual
2946 C++-functions accepting a number of @code{ex} as arguments and returning
2947 one @code{ex}. As an example, if we have a look at a simplified
2948 implementation of the cosine trigonometric function, we first need a
2949 function that is called when one wishes to @code{eval} it. It could
2950 look something like this:
2953 static ex cos_eval_method(const ex & x)
2955 // if (!x%(2*Pi)) return 1
2956 // if (!x%Pi) return -1
2957 // if (!x%Pi/2) return 0
2958 // care for other cases...
2959 return cos(x).hold();
2963 @cindex @code{hold()}
2965 The last line returns @code{cos(x)} if we don't know what else to do and
2966 stops a potential recursive evaluation by saying @code{.hold()}, which
2967 sets a flag to the expression signaling that it has been evaluated. We
2968 should also implement a method for numerical evaluation and since we are
2969 lazy we sweep the problem under the rug by calling someone else's
2970 function that does so, in this case the one in class @code{numeric}:
2973 static ex cos_evalf(const ex & x)
2975 return cos(ex_to_numeric(x));
2979 Differentiation will surely turn up and so we need to tell @code{cos}
2980 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
2981 instance are then handled automatically by @code{basic::diff} and
2985 static ex cos_deriv(const ex & x, unsigned diff_param)
2991 @cindex product rule
2992 The second parameter is obligatory but uninteresting at this point. It
2993 specifies which parameter to differentiate in a partial derivative in
2994 case the function has more than one parameter and its main application
2995 is for correct handling of the chain rule. For Taylor expansion, it is
2996 enough to know how to differentiate. But if the function you want to
2997 implement does have a pole somewhere in the complex plane, you need to
2998 write another method for Laurent expansion around that point.
3000 Now that all the ingredients for @code{cos} have been set up, we need
3001 to tell the system about it. This is done by a macro and we are not
3002 going to descibe how it expands, please consult your preprocessor if you
3006 REGISTER_FUNCTION(cos, eval_func(cos_eval).
3007 evalf_func(cos_evalf).
3008 derivative_func(cos_deriv));
3011 The first argument is the function's name used for calling it and for
3012 output. The second binds the corresponding methods as options to this
3013 object. Options are separated by a dot and can be given in an arbitrary
3014 order. GiNaC functions understand several more options which are always
3015 specified as @code{.option(params)}, for example a method for series
3016 expansion @code{.series_func(cos_series)}. Again, if no series
3017 expansion method is given, GiNaC defaults to simple Taylor expansion,
3018 which is correct if there are no poles involved as is the case for the
3019 @code{cos} function. The way GiNaC handles poles in case there are any
3020 is best understood by studying one of the examples, like the Gamma
3021 (@code{tgamma}) function for instance. (In essence the function first
3022 checks if there is a pole at the evaluation point and falls back to
3023 Taylor expansion if there isn't. Then, the pole is regularized by some
3024 suitable transformation.) Also, the new function needs to be declared
3025 somewhere. This may also be done by a convenient preprocessor macro:
3028 DECLARE_FUNCTION_1P(cos)
3031 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
3032 implementation of @code{cos} is very incomplete and lacks several safety
3033 mechanisms. Please, have a look at the real implementation in GiNaC.
3034 (By the way: in case you are worrying about all the macros above we can
3035 assure you that functions are GiNaC's most macro-intense classes. We
3036 have done our best to avoid macros where we can.)
3039 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
3040 @c node-name, next, previous, up
3041 @section Adding classes
3043 If you are doing some very specialized things with GiNaC you may find that
3044 you have to implement your own algebraic classes to fit your needs. This
3045 section will explain how to do this by giving the example of a simple
3046 'string' class. After reading this section you will know how to properly
3047 declare a GiNaC class and what the minimum required member functions are
3048 that you have to implement. We only cover the implementation of a 'leaf'
3049 class here (i.e. one that doesn't contain subexpressions). Creating a
3050 container class like, for example, a class representing tensor products is
3051 more involved but this section should give you enough information so you can
3052 consult the source to GiNaC's predefined classes if you want to implement
3053 something more complicated.
3055 @subsection GiNaC's run-time type information system
3057 @cindex hierarchy of classes
3059 All algebraic classes (that is, all classes that can appear in expressions)
3060 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
3061 @code{basic *} (which is essentially what an @code{ex} is) represents a
3062 generic pointer to an algebraic class. Occasionally it is necessary to find
3063 out what the class of an object pointed to by a @code{basic *} really is.
3064 Also, for the unarchiving of expressions it must be possible to find the
3065 @code{unarchive()} function of a class given the class name (as a string). A
3066 system that provides this kind of information is called a run-time type
3067 information (RTTI) system. The C++ language provides such a thing (see the
3068 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
3069 implements its own, simpler RTTI.
3071 The RTTI in GiNaC is based on two mechanisms:
3076 The @code{basic} class declares a member variable @code{tinfo_key} which
3077 holds an unsigned integer that identifies the object's class. These numbers
3078 are defined in the @file{tinfos.h} header file for the built-in GiNaC
3079 classes. They all start with @code{TINFO_}.
3082 By means of some clever tricks with static members, GiNaC maintains a list
3083 of information for all classes derived from @code{basic}. The information
3084 available includes the class names, the @code{tinfo_key}s, and pointers
3085 to the unarchiving functions. This class registry is defined in the
3086 @file{registrar.h} header file.
3090 The disadvantage of this proprietary RTTI implementation is that there's
3091 a little more to do when implementing new classes (C++'s RTTI works more
3092 or less automatic) but don't worry, most of the work is simplified by
3095 @subsection A minimalistic example
3097 Now we will start implementing a new class @code{mystring} that allows
3098 placing character strings in algebraic expressions (this is not very useful,
3099 but it's just an example). This class will be a direct subclass of
3100 @code{basic}. You can use this sample implementation as a starting point
3101 for your own classes.
3103 The code snippets given here assume that you have included some header files
3109 #include <stdexcept>
3110 using namespace std;
3112 #include <ginac/ginac.h>
3113 using namespace GiNaC;
3116 The first thing we have to do is to define a @code{tinfo_key} for our new
3117 class. This can be any arbitrary unsigned number that is not already taken
3118 by one of the existing classes but it's better to come up with something
3119 that is unlikely to clash with keys that might be added in the future. The
3120 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
3121 which is not a requirement but we are going to stick with this scheme:
3124 const unsigned TINFO_mystring = 0x42420001U;
3127 Now we can write down the class declaration. The class stores a C++
3128 @code{string} and the user shall be able to construct a @code{mystring}
3129 object from a C or C++ string:
3132 class mystring : public basic
3134 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
3137 mystring(const string &s);
3138 mystring(const char *s);
3144 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
3147 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
3148 macros are defined in @file{registrar.h}. They take the name of the class
3149 and its direct superclass as arguments and insert all required declarations
3150 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
3151 the first line after the opening brace of the class definition. The
3152 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
3153 source (at global scope, of course, not inside a function).
3155 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
3156 declarations of the default and copy constructor, the destructor, the
3157 assignment operator and a couple of other functions that are required. It
3158 also defines a type @code{inherited} which refers to the superclass so you
3159 don't have to modify your code every time you shuffle around the class
3160 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
3161 constructor, the destructor and the assignment operator.
3163 Now there are nine member functions we have to implement to get a working
3169 @code{mystring()}, the default constructor.
3172 @code{void destroy(bool call_parent)}, which is used in the destructor and the
3173 assignment operator to free dynamically allocated members. The @code{call_parent}
3174 specifies whether the @code{destroy()} function of the superclass is to be
3178 @code{void copy(const mystring &other)}, which is used in the copy constructor
3179 and assignment operator to copy the member variables over from another
3180 object of the same class.
3183 @code{void archive(archive_node &n)}, the archiving function. This stores all
3184 information needed to reconstruct an object of this class inside an
3185 @code{archive_node}.
3188 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
3189 constructor. This constructs an instance of the class from the information
3190 found in an @code{archive_node}.
3193 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
3194 unarchiving function. It constructs a new instance by calling the unarchiving
3198 @code{int compare_same_type(const basic &other)}, which is used internally
3199 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
3200 -1, depending on the relative order of this object and the @code{other}
3201 object. If it returns 0, the objects are considered equal.
3202 @strong{Note:} This has nothing to do with the (numeric) ordering
3203 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
3204 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
3205 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
3206 must provide a @code{compare_same_type()} function, even those representing
3207 objects for which no reasonable algebraic ordering relationship can be
3211 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
3212 which are the two constructors we declared.
3216 Let's proceed step-by-step. The default constructor looks like this:
3219 mystring::mystring() : inherited(TINFO_mystring)
3221 // dynamically allocate resources here if required
3225 The golden rule is that in all constructors you have to set the
3226 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
3227 it will be set by the constructor of the superclass and all hell will break
3228 loose in the RTTI. For your convenience, the @code{basic} class provides
3229 a constructor that takes a @code{tinfo_key} value, which we are using here
3230 (remember that in our case @code{inherited = basic}). If the superclass
3231 didn't have such a constructor, we would have to set the @code{tinfo_key}
3232 to the right value manually.
3234 In the default constructor you should set all other member variables to
3235 reasonable default values (we don't need that here since our @code{str}
3236 member gets set to an empty string automatically). The constructor(s) are of
3237 course also the right place to allocate any dynamic resources you require.
3239 Next, the @code{destroy()} function:
3242 void mystring::destroy(bool call_parent)
3244 // free dynamically allocated resources here if required
3246 inherited::destroy(call_parent);
3250 This function is where we free all dynamically allocated resources. We don't
3251 have any so we're not doing anything here, but if we had, for example, used
3252 a C-style @code{char *} to store our string, this would be the place to
3253 @code{delete[]} the string storage. If @code{call_parent} is true, we have
3254 to call the @code{destroy()} function of the superclass after we're done
3255 (to mimic C++'s automatic invocation of superclass destructors where
3256 @code{destroy()} is called from outside a destructor).
3258 The @code{copy()} function just copies over the member variables from
3262 void mystring::copy(const mystring &other)
3264 inherited::copy(other);
3269 We can simply overwrite the member variables here. There's no need to worry
3270 about dynamically allocated storage. The assignment operator (which is
3271 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
3272 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
3273 explicitly call the @code{copy()} function of the superclass here so
3274 all the member variables will get copied.
3276 Next are the three functions for archiving. You have to implement them even
3277 if you don't plan to use archives, but the minimum required implementation
3278 is really simple. First, the archiving function:
3281 void mystring::archive(archive_node &n) const
3283 inherited::archive(n);
3284 n.add_string("string", str);
3288 The only thing that is really required is calling the @code{archive()}
3289 function of the superclass. Optionally, you can store all information you
3290 deem necessary for representing the object into the passed
3291 @code{archive_node}. We are just storing our string here. For more
3292 information on how the archiving works, consult the @file{archive.h} header
3295 The unarchiving constructor is basically the inverse of the archiving
3299 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
3301 n.find_string("string", str);
3305 If you don't need archiving, just leave this function empty (but you must
3306 invoke the unarchiving constructor of the superclass). Note that we don't
3307 have to set the @code{tinfo_key} here because it is done automatically
3308 by the unarchiving constructor of the @code{basic} class.
3310 Finally, the unarchiving function:
3313 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
3315 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
3319 You don't have to understand how exactly this works. Just copy these four
3320 lines into your code literally (replacing the class name, of course). It
3321 calls the unarchiving constructor of the class and unless you are doing
3322 something very special (like matching @code{archive_node}s to global
3323 objects) you don't need a different implementation. For those who are
3324 interested: setting the @code{dynallocated} flag puts the object under
3325 the control of GiNaC's garbage collection. It will get deleted automatically
3326 once it is no longer referenced.
3328 Our @code{compare_same_type()} function uses a provided function to compare
3332 int mystring::compare_same_type(const basic &other) const
3334 const mystring &o = static_cast<const mystring &>(other);
3335 int cmpval = str.compare(o.str);
3338 else if (cmpval < 0)
3345 Although this function takes a @code{basic &}, it will always be a reference
3346 to an object of exactly the same class (objects of different classes are not
3347 comparable), so the cast is safe. If this function returns 0, the two objects
3348 are considered equal (in the sense that @math{A-B=0}), so you should compare
3349 all relevant member variables.
3351 Now the only thing missing is our two new constructors:
3354 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
3356 // dynamically allocate resources here if required
3359 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
3361 // dynamically allocate resources here if required
3365 No surprises here. We set the @code{str} member from the argument and
3366 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
3368 That's it! We now have a minimal working GiNaC class that can store
3369 strings in algebraic expressions. Let's confirm that the RTTI works:
3372 ex e = mystring("Hello, world!");
3373 cout << is_ex_of_type(e, mystring) << endl;
3376 cout << e.bp->class_name() << endl;
3380 Obviously it does. Let's see what the expression @code{e} looks like:
3384 // -> [mystring object]
3387 Hm, not exactly what we expect, but of course the @code{mystring} class
3388 doesn't yet know how to print itself. This is done in the @code{print()}
3389 member function. Let's say that we wanted to print the string surrounded
3393 class mystring : public basic
3397 void print(const print_context &c, unsigned level = 0) const;
3401 void mystring::print(const print_context &c, unsigned level) const
3403 // print_context::s is a reference to an ostream
3404 c.s << '\"' << str << '\"';
3408 The @code{level} argument is only required for container classes to
3409 correctly parenthesize the output. Let's try again to print the expression:
3413 // -> "Hello, world!"
3416 Much better. The @code{mystring} class can be used in arbitrary expressions:
3419 e += mystring("GiNaC rulez");
3421 // -> "GiNaC rulez"+"Hello, world!"
3424 (note that GiNaC's automatic term reordering is in effect here), or even
3427 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
3429 // -> "One string"^(2*sin(-"Another string"+Pi))
3432 Whether this makes sense is debatable but remember that this is only an
3433 example. At least it allows you to implement your own symbolic algorithms
3436 Note that GiNaC's algebraic rules remain unchanged:
3439 e = mystring("Wow") * mystring("Wow");
3443 e = pow(mystring("First")-mystring("Second"), 2);
3444 cout << e.expand() << endl;
3445 // -> -2*"First"*"Second"+"First"^2+"Second"^2
3448 There's no way to, for example, make GiNaC's @code{add} class perform string
3449 concatenation. You would have to implement this yourself.
3451 @subsection Automatic evaluation
3453 @cindex @code{hold()}
3455 When dealing with objects that are just a little more complicated than the
3456 simple string objects we have implemented, chances are that you will want to
3457 have some automatic simplifications or canonicalizations performed on them.
3458 This is done in the evaluation member function @code{eval()}. Let's say that
3459 we wanted all strings automatically converted to lowercase with
3460 non-alphabetic characters stripped, and empty strings removed:
3463 class mystring : public basic
3467 ex eval(int level = 0) const;
3471 ex mystring::eval(int level) const
3474 for (int i=0; i<str.length(); i++) @{
3476 if (c >= 'A' && c <= 'Z')
3477 new_str += tolower(c);
3478 else if (c >= 'a' && c <= 'z')
3482 if (new_str.length() == 0)
3485 return mystring(new_str).hold();
3489 The @code{level} argument is used to limit the recursion depth of the
3490 evaluation. We don't have any subexpressions in the @code{mystring} class
3491 so we are not concerned with this. If we had, we would call the @code{eval()}
3492 functions of the subexpressions with @code{level - 1} as the argument if
3493 @code{level != 1}. The @code{hold()} member function sets a flag in the
3494 object that prevents further evaluation. Otherwise we might end up in an
3495 endless loop. When you want to return the object unmodified, use
3496 @code{return this->hold();}.
3498 Let's confirm that it works:
3501 ex e = mystring("Hello, world!") + mystring("!?#");
3505 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
3510 @subsection Other member functions
3512 We have implemented only a small set of member functions to make the class
3513 work in the GiNaC framework. For a real algebraic class, there are probably
3514 some more functions that you will want to re-implement, such as
3515 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
3516 or the header file of the class you want to make a subclass of to see
3517 what's there. You can, of course, also add your own new member functions.
3518 In this case you will probably want to define a little helper function like
3521 inline const mystring &ex_to_mystring(const ex &e)
3523 return static_cast<const mystring &>(*e.bp);
3527 that let's you get at the object inside an expression (after you have verified
3528 that the type is correct) so you can call member functions that are specific
3531 That's it. May the source be with you!
3534 @node A Comparison With Other CAS, Advantages, Adding classes, Top
3535 @c node-name, next, previous, up
3536 @chapter A Comparison With Other CAS
3539 This chapter will give you some information on how GiNaC compares to
3540 other, traditional Computer Algebra Systems, like @emph{Maple},
3541 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
3542 disadvantages over these systems.
3545 * Advantages:: Stengths of the GiNaC approach.
3546 * Disadvantages:: Weaknesses of the GiNaC approach.
3547 * Why C++?:: Attractiveness of C++.
3550 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
3551 @c node-name, next, previous, up
3554 GiNaC has several advantages over traditional Computer
3555 Algebra Systems, like
3560 familiar language: all common CAS implement their own proprietary
3561 grammar which you have to learn first (and maybe learn again when your
3562 vendor decides to `enhance' it). With GiNaC you can write your program
3563 in common C++, which is standardized.
3567 structured data types: you can build up structured data types using
3568 @code{struct}s or @code{class}es together with STL features instead of
3569 using unnamed lists of lists of lists.
3572 strongly typed: in CAS, you usually have only one kind of variables
3573 which can hold contents of an arbitrary type. This 4GL like feature is
3574 nice for novice programmers, but dangerous.
3577 development tools: powerful development tools exist for C++, like fancy
3578 editors (e.g. with automatic indentation and syntax highlighting),
3579 debuggers, visualization tools, documentation generators...
3582 modularization: C++ programs can easily be split into modules by
3583 separating interface and implementation.
3586 price: GiNaC is distributed under the GNU Public License which means
3587 that it is free and available with source code. And there are excellent
3588 C++-compilers for free, too.
3591 extendable: you can add your own classes to GiNaC, thus extending it on
3592 a very low level. Compare this to a traditional CAS that you can
3593 usually only extend on a high level by writing in the language defined
3594 by the parser. In particular, it turns out to be almost impossible to
3595 fix bugs in a traditional system.
3598 multiple interfaces: Though real GiNaC programs have to be written in
3599 some editor, then be compiled, linked and executed, there are more ways
3600 to work with the GiNaC engine. Many people want to play with
3601 expressions interactively, as in traditional CASs. Currently, two such
3602 windows into GiNaC have been implemented and many more are possible: the
3603 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
3604 types to a command line and second, as a more consistent approach, an
3605 interactive interface to the @acronym{Cint} C++ interpreter has been put
3606 together (called @acronym{GiNaC-cint}) that allows an interactive
3607 scripting interface consistent with the C++ language.
3610 seemless integration: it is somewhere between difficult and impossible
3611 to call CAS functions from within a program written in C++ or any other
3612 programming language and vice versa. With GiNaC, your symbolic routines
3613 are part of your program. You can easily call third party libraries,
3614 e.g. for numerical evaluation or graphical interaction. All other
3615 approaches are much more cumbersome: they range from simply ignoring the
3616 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
3617 system (i.e. @emph{Yacas}).
3620 efficiency: often large parts of a program do not need symbolic
3621 calculations at all. Why use large integers for loop variables or
3622 arbitrary precision arithmetics where @code{int} and @code{double} are
3623 sufficient? For pure symbolic applications, GiNaC is comparable in
3624 speed with other CAS.
3629 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
3630 @c node-name, next, previous, up
3631 @section Disadvantages
3633 Of course it also has some disadvantages:
3638 advanced features: GiNaC cannot compete with a program like
3639 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
3640 which grows since 1981 by the work of dozens of programmers, with
3641 respect to mathematical features. Integration, factorization,
3642 non-trivial simplifications, limits etc. are missing in GiNaC (and are
3643 not planned for the near future).
3646 portability: While the GiNaC library itself is designed to avoid any
3647 platform dependent features (it should compile on any ANSI compliant C++
3648 compiler), the currently used version of the CLN library (fast large
3649 integer and arbitrary precision arithmetics) can be compiled only on
3650 systems with a recently new C++ compiler from the GNU Compiler
3651 Collection (@acronym{GCC}).@footnote{This is because CLN uses
3652 PROVIDE/REQUIRE like macros to let the compiler gather all static
3653 initializations, which works for GNU C++ only.} GiNaC uses recent
3654 language features like explicit constructors, mutable members, RTTI,
3655 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
3656 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
3657 ANSI compliant, support all needed features.
3662 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
3663 @c node-name, next, previous, up
3666 Why did we choose to implement GiNaC in C++ instead of Java or any other
3667 language? C++ is not perfect: type checking is not strict (casting is
3668 possible), separation between interface and implementation is not
3669 complete, object oriented design is not enforced. The main reason is
3670 the often scolded feature of operator overloading in C++. While it may
3671 be true that operating on classes with a @code{+} operator is rarely
3672 meaningful, it is perfectly suited for algebraic expressions. Writing
3673 @math{3x+5y} as @code{3*x+5*y} instead of
3674 @code{x.times(3).plus(y.times(5))} looks much more natural.
3675 Furthermore, the main developers are more familiar with C++ than with
3676 any other programming language.
3679 @node Internal Structures, Expressions are reference counted, Why C++? , Top
3680 @c node-name, next, previous, up
3681 @appendix Internal Structures
3684 * Expressions are reference counted::
3685 * Internal representation of products and sums::
3688 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
3689 @c node-name, next, previous, up
3690 @appendixsection Expressions are reference counted
3692 @cindex reference counting
3693 @cindex copy-on-write
3694 @cindex garbage collection
3695 An expression is extremely light-weight since internally it works like a
3696 handle to the actual representation and really holds nothing more than a
3697 pointer to some other object. What this means in practice is that
3698 whenever you create two @code{ex} and set the second equal to the first
3699 no copying process is involved. Instead, the copying takes place as soon
3700 as you try to change the second. Consider the simple sequence of code:
3703 #include <ginac/ginac.h>
3704 using namespace std;
3705 using namespace GiNaC;
3709 symbol x("x"), y("y"), z("z");
3712 e1 = sin(x + 2*y) + 3*z + 41;
3713 e2 = e1; // e2 points to same object as e1
3714 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
3715 e2 += 1; // e2 is copied into a new object
3716 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
3720 The line @code{e2 = e1;} creates a second expression pointing to the
3721 object held already by @code{e1}. The time involved for this operation
3722 is therefore constant, no matter how large @code{e1} was. Actual
3723 copying, however, must take place in the line @code{e2 += 1;} because
3724 @code{e1} and @code{e2} are not handles for the same object any more.
3725 This concept is called @dfn{copy-on-write semantics}. It increases
3726 performance considerably whenever one object occurs multiple times and
3727 represents a simple garbage collection scheme because when an @code{ex}
3728 runs out of scope its destructor checks whether other expressions handle
3729 the object it points to too and deletes the object from memory if that
3730 turns out not to be the case. A slightly less trivial example of
3731 differentiation using the chain-rule should make clear how powerful this
3735 #include <ginac/ginac.h>
3736 using namespace std;
3737 using namespace GiNaC;
3741 symbol x("x"), y("y");
3745 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
3746 cout << e1 << endl // prints x+3*y
3747 << e2 << endl // prints (x+3*y)^3
3748 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
3752 Here, @code{e1} will actually be referenced three times while @code{e2}
3753 will be referenced two times. When the power of an expression is built,
3754 that expression needs not be copied. Likewise, since the derivative of
3755 a power of an expression can be easily expressed in terms of that
3756 expression, no copying of @code{e1} is involved when @code{e3} is
3757 constructed. So, when @code{e3} is constructed it will print as
3758 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
3759 holds a reference to @code{e2} and the factor in front is just
3762 As a user of GiNaC, you cannot see this mechanism of copy-on-write
3763 semantics. When you insert an expression into a second expression, the
3764 result behaves exactly as if the contents of the first expression were
3765 inserted. But it may be useful to remember that this is not what
3766 happens. Knowing this will enable you to write much more efficient
3767 code. If you still have an uncertain feeling with copy-on-write
3768 semantics, we recommend you have a look at the
3769 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
3770 Marshall Cline. Chapter 16 covers this issue and presents an
3771 implementation which is pretty close to the one in GiNaC.
3774 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
3775 @c node-name, next, previous, up
3776 @appendixsection Internal representation of products and sums
3778 @cindex representation
3781 @cindex @code{power}
3782 Although it should be completely transparent for the user of
3783 GiNaC a short discussion of this topic helps to understand the sources
3784 and also explain performance to a large degree. Consider the
3785 unexpanded symbolic expression
3787 $2d^3 \left( 4a + 5b - 3 \right)$
3790 @math{2*d^3*(4*a+5*b-3)}
3792 which could naively be represented by a tree of linear containers for
3793 addition and multiplication, one container for exponentiation with base
3794 and exponent and some atomic leaves of symbols and numbers in this
3799 @cindex pair-wise representation
3800 However, doing so results in a rather deeply nested tree which will
3801 quickly become inefficient to manipulate. We can improve on this by
3802 representing the sum as a sequence of terms, each one being a pair of a
3803 purely numeric multiplicative coefficient and its rest. In the same
3804 spirit we can store the multiplication as a sequence of terms, each
3805 having a numeric exponent and a possibly complicated base, the tree
3806 becomes much more flat:
3810 The number @code{3} above the symbol @code{d} shows that @code{mul}
3811 objects are treated similarly where the coefficients are interpreted as
3812 @emph{exponents} now. Addition of sums of terms or multiplication of
3813 products with numerical exponents can be coded to be very efficient with
3814 such a pair-wise representation. Internally, this handling is performed
3815 by most CAS in this way. It typically speeds up manipulations by an
3816 order of magnitude. The overall multiplicative factor @code{2} and the
3817 additive term @code{-3} look somewhat out of place in this
3818 representation, however, since they are still carrying a trivial
3819 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
3820 this is avoided by adding a field that carries an overall numeric
3821 coefficient. This results in the realistic picture of internal
3824 $2d^3 \left( 4a + 5b - 3 \right)$:
3827 @math{2*d^3*(4*a+5*b-3)}:
3833 This also allows for a better handling of numeric radicals, since
3834 @code{sqrt(2)} can now be carried along calculations. Now it should be
3835 clear, why both classes @code{add} and @code{mul} are derived from the
3836 same abstract class: the data representation is the same, only the
3837 semantics differs. In the class hierarchy, methods for polynomial
3838 expansion and the like are reimplemented for @code{add} and @code{mul},
3839 but the data structure is inherited from @code{expairseq}.
3842 @node Package Tools, ginac-config, Internal representation of products and sums, Top
3843 @c node-name, next, previous, up
3844 @appendix Package Tools
3846 If you are creating a software package that uses the GiNaC library,
3847 setting the correct command line options for the compiler and linker
3848 can be difficult. GiNaC includes two tools to make this process easier.
3851 * ginac-config:: A shell script to detect compiler and linker flags.
3852 * AM_PATH_GINAC:: Macro for GNU automake.
3856 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
3857 @c node-name, next, previous, up
3858 @section @command{ginac-config}
3859 @cindex ginac-config
3861 @command{ginac-config} is a shell script that you can use to determine
3862 the compiler and linker command line options required to compile and
3863 link a program with the GiNaC library.
3865 @command{ginac-config} takes the following flags:
3869 Prints out the version of GiNaC installed.
3871 Prints '-I' flags pointing to the installed header files.
3873 Prints out the linker flags necessary to link a program against GiNaC.
3874 @item --prefix[=@var{PREFIX}]
3875 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
3876 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
3877 Otherwise, prints out the configured value of @env{$prefix}.
3878 @item --exec-prefix[=@var{PREFIX}]
3879 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
3880 Otherwise, prints out the configured value of @env{$exec_prefix}.
3883 Typically, @command{ginac-config} will be used within a configure
3884 script, as described below. It, however, can also be used directly from
3885 the command line using backquotes to compile a simple program. For
3889 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
3892 This command line might expand to (for example):
3895 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
3896 -lginac -lcln -lstdc++
3899 Not only is the form using @command{ginac-config} easier to type, it will
3900 work on any system, no matter how GiNaC was configured.
3903 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
3904 @c node-name, next, previous, up
3905 @section @samp{AM_PATH_GINAC}
3906 @cindex AM_PATH_GINAC
3908 For packages configured using GNU automake, GiNaC also provides
3909 a macro to automate the process of checking for GiNaC.
3912 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
3920 Determines the location of GiNaC using @command{ginac-config}, which is
3921 either found in the user's path, or from the environment variable
3922 @env{GINACLIB_CONFIG}.
3925 Tests the installed libraries to make sure that their version
3926 is later than @var{MINIMUM-VERSION}. (A default version will be used
3930 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
3931 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
3932 variable to the output of @command{ginac-config --libs}, and calls
3933 @samp{AC_SUBST()} for these variables so they can be used in generated
3934 makefiles, and then executes @var{ACTION-IF-FOUND}.
3937 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
3938 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
3942 This macro is in file @file{ginac.m4} which is installed in
3943 @file{$datadir/aclocal}. Note that if automake was installed with a
3944 different @samp{--prefix} than GiNaC, you will either have to manually
3945 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
3946 aclocal the @samp{-I} option when running it.
3949 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
3950 * Example package:: Example of a package using AM_PATH_GINAC.
3954 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
3955 @c node-name, next, previous, up
3956 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
3958 Simply make sure that @command{ginac-config} is in your path, and run
3959 the configure script.
3966 The directory where the GiNaC libraries are installed needs
3967 to be found by your system's dynamic linker.
3969 This is generally done by
3972 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
3978 setting the environment variable @env{LD_LIBRARY_PATH},
3981 or, as a last resort,
3984 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
3985 running configure, for instance:
3988 LDFLAGS=-R/home/cbauer/lib ./configure
3993 You can also specify a @command{ginac-config} not in your path by
3994 setting the @env{GINACLIB_CONFIG} environment variable to the
3995 name of the executable
3998 If you move the GiNaC package from its installed location,
3999 you will either need to modify @command{ginac-config} script
4000 manually to point to the new location or rebuild GiNaC.
4011 --with-ginac-prefix=@var{PREFIX}
4012 --with-ginac-exec-prefix=@var{PREFIX}
4015 are provided to override the prefix and exec-prefix that were stored
4016 in the @command{ginac-config} shell script by GiNaC's configure. You are
4017 generally better off configuring GiNaC with the right path to begin with.
4021 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
4022 @c node-name, next, previous, up
4023 @subsection Example of a package using @samp{AM_PATH_GINAC}
4025 The following shows how to build a simple package using automake
4026 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
4029 #include <ginac/ginac.h>
4033 GiNaC::symbol x("x");
4034 GiNaC::ex a = GiNaC::sin(x);
4035 std::cout << "Derivative of " << a
4036 << " is " << a.diff(x) << std::endl;
4041 You should first read the introductory portions of the automake
4042 Manual, if you are not already familiar with it.
4044 Two files are needed, @file{configure.in}, which is used to build the
4048 dnl Process this file with autoconf to produce a configure script.
4050 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
4056 AM_PATH_GINAC(0.7.0, [
4057 LIBS="$LIBS $GINACLIB_LIBS"
4058 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
4059 ], AC_MSG_ERROR([need to have GiNaC installed]))
4064 The only command in this which is not standard for automake
4065 is the @samp{AM_PATH_GINAC} macro.
4067 That command does the following: If a GiNaC version greater or equal
4068 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
4069 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
4070 the error message `need to have GiNaC installed'
4072 And the @file{Makefile.am}, which will be used to build the Makefile.
4075 ## Process this file with automake to produce Makefile.in
4076 bin_PROGRAMS = simple
4077 simple_SOURCES = simple.cpp
4080 This @file{Makefile.am}, says that we are building a single executable,
4081 from a single sourcefile @file{simple.cpp}. Since every program
4082 we are building uses GiNaC we simply added the GiNaC options
4083 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
4084 want to specify them on a per-program basis: for instance by
4088 simple_LDADD = $(GINACLIB_LIBS)
4089 INCLUDES = $(GINACLIB_CPPFLAGS)
4092 to the @file{Makefile.am}.
4094 To try this example out, create a new directory and add the three
4097 Now execute the following commands:
4100 $ automake --add-missing
4105 You now have a package that can be built in the normal fashion
4114 @node Bibliography, Concept Index, Example package, Top
4115 @c node-name, next, previous, up
4116 @appendix Bibliography
4121 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
4124 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
4127 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
4130 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
4133 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
4134 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
4137 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
4138 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
4139 Academic Press, London
4144 @node Concept Index, , Bibliography, Top
4145 @c node-name, next, previous, up
4146 @unnumbered Concept Index